Erwin Rudolf

Josef Alexander Schrodinger

Gaoren Borsalino

Hui Wang

Eliane Li

Maxence Luc

Goulven Lepers

Maykel L Quéméner

Anne Gonzalez-Martinez

Jean-François Crubellier

Nadia Wyart

Romain Boulo- Ufa

Andrea Vexiau

most recently, Adrien Devolder Orbàn

Etienne As

Frédéric Brion

Grosshans

It

Dr Franco A Gianturco

Amir Mahdian

Alexander D D Örfler

Ibrokhim Iskandarov

Steffen Meyer

Kaveh Najafian

Johannes Heinrich

Lorenzo Petralia

Ivan Kortunov

Milán J Negyedi

Ilia Sergachev

Karin Fisher

Chang J Kwong

Ashwin Boddeti

Johannes Kombe

Still

Dra M Aymar

Dr P S Zuchowski

Maciej Kosicki

Dr T Mukaiyama

Dr R Wester

Dr M Hernández

Fabio Carelli

Dr J Hecker

Juan Pablo

S Castillo

Gemma G Pi

Aline Farias

Keywords: changing rates under ion-trap conditions

Such a life that would not be possible

I N T R O D U C T I O N

from cold/ultracold atoms to cold/ultracold molecules and beyond: ultracold molecular ions At 10:54 on June 5, 1995, in a laboratory at JILA, the world's first Bose-Einstein condensate (BEC) of Rb atoms was observed, in a joint work led by Eric A. Cornell (NIST) and Carl E. Wieman (University of Colorado) [START_REF] Anderson | Observations of Bose-Einstein condensation in a dilute atomic vapor[END_REF][START_REF] Bradley | Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions[END_REF]. In the same year, a study carried out by Wolfgang Ketterle (MIT) and co-workers came out to shed light on the properties of a BEC of Na atoms [START_REF] Davis | Bose-Einstein Condensation in a Gas of Sodium Atoms[END_REF]. For such an achievement the three researchers were awarded with the Nobel prize of Physics in 2001. Besides the BEC observation, as a climax of significant advances in cooling and trapping atoms over the last decades, it is worthwhile to mention as well: (i) the atom interferometry development [START_REF] Adams | Atom interferometery[END_REF][START_REF] Adams | Atom optics[END_REF]; (ii) the optical lattice realization [START_REF] Deutsch | Optical lattices[END_REF]; (iii) matter-wave holography manipulation [START_REF] Morinaga | Holographic manipulation of a cold atomic beam[END_REF]; (iv) hydrodynamic nature of the collective oscillations [START_REF] Jin | Collective Excitations of a Bose-Einstein Condensate in a Dilute Gas[END_REF][START_REF] Mewes | Output Coupler for Bose-Einstein Condensed Atoms[END_REF]; (v) propagation of solitons [START_REF] Burger | Dark Solitons in Bose-Einstein Condensates[END_REF][START_REF] Jochim | Bose-Einstein Condensation of Molecules[END_REF][START_REF] Khaykovich | Formation of a Matter-Wave Bright Soliton[END_REF]; (vi) observation of spinor condensates [START_REF] Stenger | Spin domains in ground-state Bose-Einstein condensates[END_REF]; (vii) observation of interaction effects in the Bloch oscillations [START_REF] Morsch | Bloch Oscillations and Mean-Field Effects of Bose-Einstein Condensates in 1D Optical Lattices[END_REF]; and, (ix) the remarkable achievement of quantum degeneracy in trapped Fermi gases [START_REF] Demarco | Onset of Fermi Degeneracy in a Trapped Atomic Gas[END_REF], which on itself opened an entire field, seeking to observe the BEC-like counterpart phenomena of fermionic species: the Fermi superfluidity. The starting point been in the earlier 80's when it was demonstrated, for the first time, the usage of light to spatially trap and cool down atoms up to a few micro to millikelvin regime [START_REF] Phillips | Laser cooling and electromagnetic trapping of neutral atoms[END_REF][START_REF] Dalibard | Dressed-atom approach to atomic motion in laser light: The dipole force revisted[END_REF][START_REF] Phillips | Nobel Lecture: Laser cooling and trapping of neutral atoms[END_REF][START_REF] Chu | Nobel Lecture: The manipulation of neutral particles[END_REF][START_REF] Cohen-Tannoudji | Nobel Lecture: Manipulating atoms with photons[END_REF]. The examples given above shall evince the interest of the scientific community in the cold/ultracold domain. As earlier observed by Kevin E. Strecker and David W. Chandler [START_REF] Smith | Low temperatures and cold molecules[END_REF], the thermal de Broglie wavelength, λ db , for an ensemble of ultracold atoms with mass m, is such that

λ db = 2πh 2 mk b T (1)
where, k b is the Boltzmann constant. In a regime in which the temperature is T -→ 0 and recalling the canonical interpretation of λ db as the quantum "size" of the particles, it should become evident that in the cold/ultracold domain the de Broglie wavelength is much larger than the particle's classical size, exhibiting thus a strong quantum character. It is worthwhile to notice, in particular in the ultracold regime, that the thermal 1 2 introduction de Broglie wavelength is of the same magnitude as the average interparticle separation. Or, in other words, the quantum "size" of the particles begins to spatially overlap. That is, the particle interactions are no longer classical hard spheres interactions and are driven by their respective quantum statistics, opening therefore, the pathway for new physical phenomena, such as those cited above [START_REF] Smith | Low temperatures and cold molecules[END_REF].

In the following, the "cold" and "ultracold" terminology stands to translational temperatures between 1 mK to 1 K and below a mK, respectively. The "true ultracold" regime, however, corresponding to translational temperatures at which only a single partial wave contributes, i.e. s-wave for bosons and distinguishable particles and pwave for identical fermions. For some heavier molecules a single partial wave regime may even be well below 1 µK though. Nevertheless, we shall adopt a soft notation, dropping out the usage of the "cold" term and referring to as "ultracold" systems, since most of the temperatures used here are below the coldest temperatures found in nature (a few K in the interstellar medium).

Trapping ultracold molecules could offer a significant advantage over atoms since there are many additional tunable experimental parameters. For instance, an electric dipole moment could be induced by an external electric field, and transitions between internal rotational states could be driven by means of resonant microwave fields. Or, the presence of rotationally excited states would open the possibility to dynamically tune electric-field-induced dipole-dipole interactions to an effective short or long range, as shown in Ref. [START_REF] Büchler | Strongly Correlated 2D Quantum Phases with Cold Polar Molecules: Controlling the Shape of the Interaction Potential[END_REF] for bosonic polar molecules in quantum gases. Not to mention that the choice of molecules could also lead to fine or hyperfine structures, which can be addressed together with the rotational states. Moreover, these additional structures combined with the vibrational states would lead to a large internal Hilbert space, i.e. a huge workspace for an experimental exploration. And, indeed, soon enough molecules have been proposed for a number of applications in the low-temperature research field as a case study to provide a fundamental insight on molecular interactions and reaction dynamics, essentially dominated by quantum mechanics 1 . Thus, similarly to how ultracold atoms have revolutionized AMO physics, cold and ultracold molecules have the potential to dramatically influence fundamental physics, physical chemistry and few-body physics. Some of the most prominent ideas [START_REF] Dulieu | Formation and interactions of cold and ultracold molecules: new challenges for interdisciplinary physics[END_REF] go from (i) high-resolution molecular spectroscopy [START_REF] Stwalley | Photoassociation of ultracold atoms: a new spectroscopic technique[END_REF][START_REF] Van De Meerakker | Direct Measurement of the Radiative Lifetime of Vibrationally Excited OH Radicals[END_REF][START_REF] Jones | Ultracold photoassociation spectroscopy: Long-range molecules and atomic scattering[END_REF][START_REF] Gilijamse | The radiative lifetime of metastable CO (a 3Pi, v=0)[END_REF]; (ii) qubits for the implementation of quantum gates [START_REF] Demille | Quantum computation with trapped polar molecules[END_REF][START_REF] Tesch | Quantum computation with vibrationally excited molecules[END_REF][START_REF] Rabl | Hybrid Quantum Processors: Molecular Ensembles as Quantum Memory for Solid State Circuits[END_REF][START_REF] Rabl | Molecular dipolar crystals as high-fidelity quantum memory for hybrid quantum computing[END_REF]; (iii) molecular optics devices [START_REF] Kallush | Evanescent-wave mirror for ultracold diatomic polar molecules[END_REF]; (iv) to probe the existence of an electron permanent dipole moment [START_REF] Demille | Investigation of PbO as a system for measuring the electric dipole moment of the electron[END_REF][START_REF] Hudson | Measurement of the Electron Electric Dipole Moment Using YbF Molecules[END_REF][START_REF] Kozlov | Enhancement of the Electric Dipole Moment of the Electron in PbO[END_REF][START_REF] Kawall | Precision Zeeman-Stark Spectroscopy of the Metastable a(1) 3 Σ+ State of PbO[END_REF]; (v) to check the charge-parity violation [START_REF] Hinds | Experiment to search for P-and T -violating interactions in the hyperfine structure of thallium fluoride[END_REF][START_REF] Cho | Search for time-reversal-symmetry violation in thallium fluoride using a jet source[END_REF][START_REF] Ravaine | Atomic CP-violating polarizability[END_REF]; (vi) to study the time-dependence, if any, on the 1.1 from cold/ultracold atoms to cold/ultracold molecules and beyond 3 electron-to-nuclear mass ratios [START_REF] Van Veldhoven | Decelerated molecular beams for high-resolution spectroscopy[END_REF][START_REF] Ph | Energy levels and two-photon transition probabilities in the HD + ion[END_REF][START_REF] Schiller | Tests of time independence of the electron and nuclear masses with ultracold molecules[END_REF][START_REF] Chin | Enhanced Sensitivity to Fundamental Constants In Ultracold Atomic and Molecular Systems near Feshbach Resonances[END_REF][START_REF] Demille | Enhanced Sensitivity to Variation of me/mp in Molecular Spectra[END_REF][START_REF] Zelevinsky | Precision Test of Mass-Ratio Variations with Lattice-Confined Ultracold Molecules[END_REF] and (vii) the fine-structure constant [START_REF] Hudson | Cold Molecule Spectroscopy for Constraining the Evolution of the Fine Structure Constant[END_REF]; (viii) quantum magnetism [START_REF] Barnett | Quantum Magnetism with Multicomponent Dipolar Molecules in an Optical Lattice[END_REF]; (ix) quantum phase transitions [START_REF] Büchler | Strongly Correlated 2D Quantum Phases with Cold Polar Molecules: Controlling the Shape of the Interaction Potential[END_REF][START_REF] Yi | Trapped atomic condensates with anisotropic interactions[END_REF][START_REF] Baranov | Superfluid pairing of a polarized dipolar Fermi gas[END_REF][START_REF] Gorál | Quantum Phases of Dipolar Bosons in Optical Lattices[END_REF][START_REF] Santos | Roton-Maxon Spectrum and Stability of Trapped Dipolar Bose-Einstein Condensates[END_REF][START_REF] Micheli | Cold polar molecules in two-dimensional traps: Tailoring interactions with external fields for novel quantum phases[END_REF]; and (x) lowenergy collisions [START_REF] Avdeenkov | Ultracold collisions of oxygen molecules[END_REF][START_REF] Avdeenkov | Collisional dynamics of ultracold OH molecules in an electrostatic field[END_REF][START_REF] Avdeenkov | Linking Ultracold Polar Molecules[END_REF][START_REF] Balakrishnan | The He-CaH( 2 Σ + ) interaction. II. Collisions at cold and ultracold temperatures[END_REF][START_REF] Avdeenkov | Ultracold collisions of fermionic OD radicals[END_REF][START_REF] Avdeenkov | Suppression of inelastic collisions of polar 1 Σ state molecules in an electrostatic field[END_REF][START_REF] Lee | State-to-state rotational transitions in H 2 +H 2 collisions at low temperatures[END_REF][START_REF] Quéméner | Vibrational energy transfer in ultracold molecule-molecule collisions[END_REF][START_REF] Quéméner | Quantum calculations of H 2 -H 2 collisions: from ultracold to thermal energies[END_REF][START_REF] Micheli | Universal Rates for Reactive Ultracold Polar Molecules in Reduced Dimensions[END_REF][START_REF] Quéméner | Universalities in ultracold reactions of alkali-metal polar molecules[END_REF]. Among all the prospects mentioned above, one deserves, on itself, special attention: the so-called ultracold chemistry, often refereed to as cold controlled chemistry [START_REF] Balakrishnan | Perspective: Ultracold molecules and the dawn of cold controlled chemistry[END_REF], and its applications to astrochemistry, [START_REF] Balakrishnan | Perspective: Ultracold molecules and the dawn of cold controlled chemistry[END_REF][START_REF] Carr | Focus on Cold and Ultracold Molecules[END_REF]. While the lowest temperatures of interest in astrochemistry are only of a few kelvins, ultracold molecules shall give insights into the chemistry and molecule formation at the low temperature regime of astrophysical environments. That is, in order to understand cold chemistry needed for modeling dense interstellar clouds or deposition of cold molecules on surfaces, one should attack the quantum mechanical details of the cold chemical systems [START_REF] Schlemmer | Non-destructive, high resolution and absolute mass determination of single, charged particles in a 3D quadrupole trap[END_REF].

The field of cold/ultracold molecules grew over the tradition of molecular beams experiments, in both physics and chemistry communities, where researchers have made steady advances in manipulation and cooling of molecular beams [START_REF] Smith | Low temperatures and cold molecules[END_REF]. One should mention, for instance, the development of buffer gas cooling [START_REF] Weinstein | Magnetic trapping of calcium monohydride molecules at millikelvin temperatures[END_REF][START_REF] Carvalho | Buffer-gas loaded magnetic trap for atoms and molecules: a primer[END_REF] and the supersonic expansion-based cooling [START_REF] Friedrich | Slowing of supersonically cooled atoms and molecules by timevarying nonresonant induced dipole forces[END_REF], as well the control of neutral molecules motion through an inhomogeneous electric or magnetic field [START_REF] Friedrich | Net polarization of a molecular beam by strong electrostatic or radiative fields[END_REF]. In order to cool down molecules, however, there is an evident challenge: not only the translational motion needs to slow down, but also the internal ones, namely the rotational and vibrational degrees of freedom. The usual laser-cooling techniques applied to atoms rely on a cyclic absorption-emission pattern that requires a sort of simple two-level system, e.g. electronic structure of alkali or alkaline earth metals. The many molecular degrees of freedom, however, give rise to a multi-level structure which results in the traditional atomic laser-cooling techniques being very inefficient when applied to molecules.

In fact, only following the advent of devices to store ultracold atoms, such as optical dipole traps and optical lattices, was possible to form ultracold molecules by light-assisted association of two atoms nearly at rest, e.g. the photoassociation process [START_REF] Jones | Ultracold photoassociation spectroscopy: Long-range molecules and atomic scattering[END_REF][START_REF] Thorsheim | Laser-induced photoassociation of ultracold sodium atoms[END_REF][START_REF] Napolitano | Line shapes of high resolution photoassociation spectra ofoptically cooled atoms[END_REF][START_REF] Marcassa | Optical suppression of photoassociative ionization in a magneto-optical trap[END_REF][START_REF] Williams | Molecular hyperfine structure in the photoassociation spectroscopyof laser cooled atoms[END_REF][START_REF] Lett | Photoassociative Spectroscopy of laser-cooled atoms[END_REF][START_REF] Tiesinga | Photoassociative Spectroscopy of Highly Excited Vibrational Levels of Alkali Dimers: Greens Function Approach for Eigenvalue Solvers[END_REF][START_REF] Jones | Fitting line shapes in photoassociation spectroscopy of ultracold atoms: A useful approximation[END_REF][START_REF] Simoni | Intensity effects in ultracold photoassociation line shapes[END_REF][START_REF] Mckenzie | Photoassociation of sodium in a Bose-Einstein condensate[END_REF][START_REF] Machholm | Subthermal linewidths in photoassociation spectra of cold alkaline-earth-metal atoms[END_REF][START_REF] Zelevinsky | Narrow Line Photoassociation in an Optical Lattice[END_REF][START_REF] Kemmann | Near-threshold photoassociation of 87 Rb 2[END_REF][START_REF] Kitagawa | Two-color photoassociation spectroscopy of ytterbium atoms and the precise determinations of s-wave scattering lengths[END_REF]. Or, magneto-association, when Feshbach resonances are tuned during the collision [START_REF] Chin | Feshbach resonances in ultracold gases[END_REF]. Although these ultracold molecules were indeed slow -up to a few microkelvins for the homonuclear alkaline diatomic case, in a translational sense -they were not necessarily internally cold, since there was still some degree of internal excitation. Not to mention that optical dipole traps and optical lattices are very much dependent on how efficient the internal electric properties of each atomic species couple with light, being therefore, on the whole not a general method of production of ultracold molecules. Whereas, on the other hand, magnetic trapping with buffer gas cooling seems a general recipe for the trapping and cooling of any system, for actual purpose of application, however, it is limited to those systems with an unpaired electron, besides further technical limitations concerning the efficiency of the buffer gas, introduction namely, the ratio between elastic and inelastic/reactive collisions, between the buffer gas and the target, leading to a trap loss. In general, elastic collisions lead to thermalization and cooling while inelastic collisions may lead to energy release and heating [START_REF] Smith | Low temperatures and cold molecules[END_REF].

Charged cold/ultracold molecules, however, are much simpler to trap due to their strong interactions with electric and magnetic fields. Notice that, only the presence of a previously trapped and cooled charged species is enough to cool down a neighboring hot ensemble, due to their strong Coulombic interactions. Thus, cooling the translational motion, and trapping of molecular ions, is quite a general process, with respect the method of producing them. That is, any type of molecular ion may be produced by either REMPI2 schemes, photoionization or electron-impact ionization of neutral species, and trapped afterwards, where the specific cooling scheme for the internal motion would become a subsequent step. Charged species became, therefore, fundamental for long time storage in a well-controlled environment.

As a typical example, several of the fundamental aspects of light-atom interactions are also extended to molecules, thanks to one of the most important example of molecular ions, HD + [START_REF] Koelemeij | Vibrational spectroscopy of HD + with 2-ppb accuracy[END_REF]. It is worthwhile to notice that one of the outstanding challenges in the field of few-body quantum physics and metrology is the achievement of ultrahigh-precision laser vibrational spectroscopy of the molecular hydrogen ion (MHI) H + 2 , a member of the family of one-electron molecules. These ab initio calculable systems at the intersection of atomic and molecular physics have enormous potential for ultra-precise spectroscopy at the 10 -17 fractional accuracy level [START_REF] Schiller | Simplest Molecules as Candidates for Precise Optical Clocks[END_REF][START_REF] Karr | H + 2 and HD + : Candidates for a molecular clock[END_REF]. So far, experiments have reached the 10 -9 level [START_REF] Koelemeij | Vibrational spectroscopy of HD + with 2-ppb accuracy[END_REF][START_REF] Bressel | Manipulation of Individual Hyperfine States in Cold Trapped Molecular Ions and Application to HD + Frequency Metrology[END_REF][START_REF] Biesheuvel | Probing QED and fundamental constants through laser spectroscopy of vibrational transitions in HD +[END_REF]. If a number of appropriately chosen rovibrational transitions of at least two of the three ions H + 2 , HD + , D + 2 are determined, it is possible to test the accuracy of the ab initio calculations and to determine the values of constants such as the proton-to-electron mass ratio, m e /m p . Such determinations would be complementary to those performed on other systems and thus would provide an important consistency check, and could possibly lead to higher accuracy of the constants, and improved tests of their time-independence. Everything relying on the single-state preparation of such MHIs and, ultimately, the capability of trapping and cooling these molecular ions.

In the following we restrict our attention to the many aspects of molecular ions in a variety of experimental setups for ultracold studies, to which the reasons are properly presented by the end of this introductory part. Before, however, it is worthwhile making a few remarks regarding some specific definitions. From the above discussion, it is already clear that there is a conceptual difference between cold and slow. In the classical thermodynamical point of view, the simple relation between macroscopic units of temperature, T, and the absolute mean kinetic energy, E kin , of each particle, E kin ≈ k b T, still holds here. Moreover, during this work we may use temperature and kinetic energy interchangeably in units, often, of millikelvin. But ultracold molecules might not represent, however, a thermodynamic state in equilibrium, since the system still carries a relatively huge amount of energy on its internal degrees of freedom. As a typical example, suppose a collision between an ultracold H atom and a HeH + molecular ion, in its internal ground state, at a collision energy of a few mK. One possible outcome of such a process is the following chemical reaction, H + HeH + -→ He + H + 2 [START_REF] Bradley | Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions[END_REF] where, the ground state of H + 2 is about 0.8 eV below the energy threshold of the reactants. That is, a collision of a few mK could release about 0.8 eV (≈ 9000 K), with ground state products being ejected from the collision zone at a relative kinetic energy of thousands of kelvins with respect to the slow reactants. Some authors used to define, strictly, different kinds of temperature, namely, translational, rotational and vibrational temperatures. This is not the case, in this particular work, but we shall often refer to ultracold systems as being simply slow on its translational motion but not necessarily rotationally and/or vibrationally cold.

more on ultracold molecular ions

With charged particles and various cooling schemes, it is possible to overcome the many extra difficulties faced when using neutral systems, which are still restricted to a small number of specific molecules. We shall now focus entirely on its current trends, technical difficulties, open questions and to propose the general motivations of this work, concerning the use of molecular ions in the ultracold field of research.

Notice, however, that the respective field of ultracold atomic ions is a well established research field, as traditional as the one of ultracold atoms. Laser cooling of trapped ions has made it possible to lower the temperature, down to a range of a few mK, such that spatially ordered structures are observed: the so-called Coulomb crystals of ions [START_REF] Bowe | Sympathetic Crystallization of Trapped Ions[END_REF][START_REF] Drewsen | Ion Coulomb crystals and some applications[END_REF][START_REF] Drewsen | Ion Coulomb crystals: A tool for studying ion processes[END_REF]. These structures have been observed in Penning and Paul traps [START_REF] Wineland | Atomic-ion coulomb clusters in an ion trap[END_REF]. As an example, see Fig. [START_REF] Anderson | Observations of Bose-Einstein condensation in a dilute atomic vapor[END_REF], where sympathetically cooled N + 2 molecular ions into a Coulomb crystal of ultracold Ca + is presented. The Coulomb crystal gives a remarkable control over a single specific ion, which can be addressed and manipulated. For instance, strings of well-localized ions have become very attractive physical systems for studying multi particle entanglement as well as for quantum computing.

And thanks to such steady advances brought by the knowledge on Coulomb crystals, it has become possible to set up one of the most powerful and general techniques to obtain ultracold molecular ions, i.e. by sympathetically cooling them down, in the presence of laser-cooled atomic ions, just as depicted in the example given by Fig. [START_REF] Anderson | Observations of Bose-Einstein condensation in a dilute atomic vapor[END_REF]. It relies on the advantage that it is nowadays possible to replace any single laser cooled ion in a Coulomb crystal by another ionic species, which efficiently freezes the newcomer translational motion. Although some mass/charge ratios are known to rule the efficiency of the method, it became broad and general for many applications, due the variety of laser cooled ions available, e.g. ranging from light 9 Be + to heavy 198 Hg + ions, as pointed out by D. Gerlich in his Chapter The Production and Study of Ultra-Cold Molecular Ions -see Ref. [START_REF] Smith | Low temperatures and cold molecules[END_REF] for more. 2 molecular ions inside (dark core regions). The molecular ions are produced by leaking a N 2 neutral source into the chamber and raising the background gas pressure to 1×10 -8 mbar, followed by a 2 + 1 REMPI scheme for ionization, with a pulsed 202 nm laser running at 10 Hz for 30 seconds at a power of 3 mW per pulse. The rovibrational distribution is kept the same as at room temperature. The picture was taken with a CMOS camera system, at ×20 averaging to reduce the background noise, and was kindly provided by Alexander D örfler and Stefan Willitsch (University of Basel).

As typical examples of such a multi-species crystals, the earliest observation was reported by T. Baba and I. Waki, who trapped small singly-charged organic fragments, e.g. C 2 H + 5 , into crystals of laser-cooled Mg + , at temperatures of a few dozens of K [START_REF] Baba | In: Japan[END_REF]; K. Mølhave and co-workers produced MgH + and MgD + by means of photochemical reactions, followed by sympathetic cooling, into Mg + crystals, at temperatures below 100 mK [START_REF] Mølhave | Formation of translationally cold MgH + and MgD + molecules in an ion trap[END_REF]; and a few years later, B. Roth et al. were able to cool down, now, a variety of molecular ions, such as diatomic H + 2 , D + 2 , N + 2 , HD + , ArH + , ArD + , and triatomic H + 3 , D + 3 , H 2 D + , N 2 H + , N 2 D + within Be + crystals at temperatures between 20 to 30 mK [START_REF] Roth | Ion-neutral chemical reactions between ultracold localized ions and neutral molecules with singleparticle resolution[END_REF][START_REF] Roth | Ion-neutral chemical reactions between ultracold localized ions and neutral molecules with singleparticle resolution[END_REF][START_REF] Roth | Production of ultracold diatomic and triatomic molecular ions of spectroscopic and astrophysical interest[END_REF][START_REF] Roth | Motional resonance coupling in cold multispecies Coulomb crystals[END_REF]. The detection is often performed either by (i) monitoring a fluorescence signal, from the original laser cooled ions, where the new species embedded in the crystals will behave as dark spots. Due to different mass/charge ratios, the different species may be characterized by the changes in the spatial structure of the crystal, often relying on the help of accurate numerical simulations [START_REF] Roth | Motional resonance coupling in cold multispecies Coulomb crystals[END_REF][START_REF] Schiffer | Temperature, ordering, and equilibrium with time-dependent confining forces[END_REF][START_REF] Schiller | Molecular dynamics simulation of sympathetic crystallization of molecular ions[END_REF][START_REF] Zhang | Moleculardynamics simulations of cold single-species and multispecies ion ensembles in a linear Paul trap[END_REF]. Or, (ii) a nondestructive identification of species via resonant excitation with an oscillating electric field resulting in high mass-to-charge resolution, widely used by F. Hall et al. [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF][START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF]. In such a case the secular frequencies of ions in the effective harmonic potential of the quadrupole trap are shifted and broadened by the Coulomb interaction. In addition to these relatively simple and reliable methods of detection, the background gas that would eventually induce chemical processes and, therefore, losses of sympathetically cooled molecular ions may be kept at very low densities, approaching ultrahigh vacuum conditions, and thus, multi species Coulomb crystals are known to last for several hours, being the key point which brought sympathetic cooling to its current know-how.

As earlier observed, for instance by A. Bertelsen and co-workers [START_REF] Bertelsen | The rotational temperature of polar molecular ions in Coulomb crystals[END_REF][START_REF] Bertelsen | The rotational temperature of polar molecular ions in Coulomb crystals[END_REF], whereas the Coulomb interaction couples efficiently with the translational motion, the internal degrees of freedom are left almost unaltered, subject to the production mechanisms of the charged molecules and the following respective decay processes, such as radiative transitions. Even when molecular ions with relatively high permanent electric dipole moments (PEDM) are used, such as MgH + [START_REF] Bertelsen | The rotational temperature of polar molecular ions in Coulomb crystals[END_REF][START_REF] Bertelsen | The rotational temperature of polar molecular ions in Coulomb crystals[END_REF], no efficient internal cooling has been observed. We shall also demonstrate, later on, the inefficiency of the PEDM for this purpose. Thus, a specific cooling scheme, for the internal degrees of freedom, should be chosen and applied afterwards. One of the pioneer methods, for a general cooling purpose, is the buffer gas cooling mentioned earlier [START_REF] Weinstein | Magnetic trapping of calcium monohydride molecules at millikelvin temperatures[END_REF][START_REF] Gerlich | Ion-neutral collision in a 22-pole trap at very low energies[END_REF][START_REF] Best | Absolute Photodetachment Cross-section Measurements for Hydrocarbon Chain Anions[END_REF][START_REF] Hansen | Efficient rotational cooling of Coulomb-crystallized molecular ions by a helium buffer gas[END_REF], and which we shall discuss in more details later. Briefly however, the method relies entirely on the gas friction between a buffer gas -often, cryogenic He atoms -and a target, where the only requirement is that the target can survive successive collisions. That is, both thermalization driven by elastic collisions and inelastic internal de-excitation mechanisms are dominant over inelastic internal excitation, rearrangements and collision induced dissociations (or the inverse process, i.e. clustering induced by collisions [START_REF] Paul | On the dynamics of the reaction of positive hydrogen cluster ions (H + 5 to H + 23 ) with para and normal hydrogen at 10 K[END_REF]). From a quantum mechanical point of view, the perturbation caused by the approaching atom, from the buffer gas, induces a non-negligible radial coupling between the internal states of the target -either completely orthogonal with respect to each other or only weakly coupled, before -promoting, therefore, a flux of population between them. As a remarkable example, Wang et al. were able to reduce the internal energy of a hot ensemble of C - 60 molecular ions in collisions with He atoms up to around 15 K [START_REF] Wang | Vibrational cooling in a cold ion trap: Vibrationally resolved photoelectron spectroscopy of cold C -60 anions[END_REF]. Among many methods for production of molecular ions, to which we shall add the recently proposed atom-ion photoassociation [START_REF] Gacesa | Production of NaCa + molecular ions in the ground state from cold atom-ion mixtures by photoassociation via an intermediate state[END_REF], similar to what is routinely done in order to achieve neutral ultracold molecules, molecular ions have been also introduction observed in one of the novel developments of ultracold matter research, aiming to merge a trap of ultracold atoms and a trap of laser-cooled ions. Such a hybrid setup offers the opportunity to study collisional dynamics in the quantum s-wave scattering regime associated to the long-range interaction between the ion and the atom, just like it has been extensively studied for ultracold neutral atom-atom collisions. And several chemical phenomena have been observed, opening the way to a rich chemistry at temperatures of a few millikelvin, or less [START_REF] Willitsch | Cold Reactive Collisions between Laser-Cooled Ions and Velocity-Selected Neutral Molecules[END_REF]. Following the theoretical prospect of Ref. [START_REF] Makarov | Radiative charge-transfer lifetime of the excited state of (NaCa) +[END_REF], with ultracold Na trapped in a magneto-optical trap (MOT) and lasercooled Ca + ions in a Paul trap, other groups have carried out pioneering experiments with various combinations of atoms and atomic ions, which we shall briefly discuss in the next section. It is worthwhile to notice, however, that among all these experiments, the direct observation of molecular ions resulting from the association of an ultracold atom and an ultracold ion has been reported only in two cases (described in the next section), being therefore, one of the central problems we seek to shed light on herein.

hybrid ultracold atom trap with a trap of laser-cooled ions

In 2003 a hybrid setup composed by a trap of ultracold atoms and a trap of lasercooled ions, in the same confined space, was proposed by O. P. Makarov and coworkers [START_REF] Makarov | Radiative charge-transfer lifetime of the excited state of (NaCa) +[END_REF], finally successfully implemented later in 2005 [START_REF] Smith | Cold ion-neutral collisions in a hybrid trap[END_REF], in order to investigate the collision of laser cooled Ca + ions by ultracold Na atoms. Becoming, therefore, the starting point of a previously experimentally unexplored regime of ultracold atomion collisions. Such collision between atoms and ions [START_REF] Makarov | Radiative charge-transfer lifetime of the excited state of (NaCa) +[END_REF][START_REF] Côté | Ultracold atom-ion collisions[END_REF][START_REF] Idziaszek | Quantum theory of ultracold atom-ion collisions[END_REF][START_REF] Smith | Experiments with an ion-neutral hybrid trap: cold charge-exchange collisions[END_REF], pertain into an intermediate regime, between atom-atom with long-range interaction potentials varying as ∝ 1/R 6 (i.e. van der Waals forces) and ion-ion repulsive interactions, as ∝ 1/R, where the actual long-range interaction is an ion-induced-dipole attractive potential, varying as ∝ 1/R 4 , with R the internuclear distance. Namely, the leading term in the long-range atom-ion interaction, V = V (R), being of the form of lim

R-→∞ V (R) = D e - C 4 2R 4 (3)
where, D e is the dissociation energy in the equilibrium position of an atom-ion molecular complex, and

C 4 = αq 2 (4π 0 ) 2 (4)
with α the static ion-induced-dipole polarizability of the atom and q the charge. Such an atom-ion regime of collisions leads to a semiclassical behavior for a wide range of collision energies, E coll , to which the dependence on a collisional crosssection, σ = σ (E coll ), scales as

σ (E coll ) ∝ 1 √ E coll (5)
A strong quantum phenomenon is known to dominate at very low energies, however, where contributions from individual partial waves can be resolved. Typical atomatom interactions lead to collision cross sections of a few atomic units over a wide range of energies, whereas atom-ion collisions can have much larger total cross sections, especially in the temperature range below 1 mK.

And, thus, hybrid setups have been earlier proposed for many applications, e.g. quantum computation [START_REF] Doerk | Atom-ion quantum gate[END_REF][START_REF] Joger | Quantum dynamics of an atomic double-well system interacting with a trapped ion[END_REF] and many-body physics [START_REF] Bissbort | Emulating Solid-State Physics with a Hybrid System of Ultracold Ions and Atoms[END_REF]. As we shall see below, the standard implementation of hybrid traps went through the usage of MOT for atoms and laser-cooling/Paul traps for ions up to more sophisticated approaches relying on a single ion in a Bose-Einstein condensate of atoms [START_REF] Schmid | Dynamics of a Cold Trapped Ion in a Bose-Einstein Condensate[END_REF][START_REF] Zipkes | Cold Heteronuclear Atom-Ion Collisions[END_REF][START_REF] Zipkes | Cold Heteronuclear Atom-Ion Collisions[END_REF] and the use of optical traps rather than a MOT [START_REF] Haze | Observation of elastic collisions between lithium atoms and calcium ions[END_REF]. After a gap of about six years since the original proposal of O. P. Makarov et al., many hybrid setups were developed for similar purposes. In the following we list the most relevant of these experiments:

• 2009: A. T. Grier et al. investigated the charge-exchange collision between Yb atoms and Yb + [START_REF] Grier | Observation of Cold Collisions between Trapped Ions and Trapped Atoms[END_REF]. The experiment introduced the very ingenious detection method, mentioned earlier, in which the monitored signal is the decay of the fluorescence of Yb + crystals, at those conditions, by means of real-time images, due to the average density of atoms experienced by the ions. The charge-exchange rate found is about 6 × 10 -10 cm 3 /s, in overall agreement with theoretical calculations [START_REF] Zhang | Scattering of Yb and Yb +[END_REF].

• 2010: C. Zipkes et al. investigated Rb atoms and Yb + [START_REF] Zipkes | Cold Heteronuclear Atom-Ion Collisions[END_REF][START_REF] Zipkes | Cold Heteronuclear Atom-Ion Collisions[END_REF], followed by S. Schmid et al. who performed collisions between Rb atoms and Rb + /Ba + [START_REF] Schmid | Dynamics of a Cold Trapped Ion in a Bose-Einstein Condensate[END_REF]. Both experiments, however, observed a charge-exchange rate much smaller than the one of Grier's work, about 10 -14 cm 3 /s.

• 2011: W. G. Rellergert et al. introduced the collision between Ca atoms and Yb + , observing a higher charge-exchange rate, of 2 × 10 -10 cm 3 /s [START_REF] Rellergert | Measurement of a Large Chemical Reaction Rate between Ultracold Closed-Shell 40 Ca Atoms and Open-Shell 174 Yb + Ions Held in a Hybrid Atom-Ion Trap[END_REF]. The authors were the first to point out the striking fact of charge-exchange being the dominant outcome of such an experiment, even though an eventual formation of molecular ion is the most accepted mechanism. Independently and almost at the same time, however, F. Hall et al. in a collaboration with O. Dulieu and coworkers, from Laboratoire Aimé Cotton (LAC), were the first to observe a direct signal of RbCa + molecular ions in collisions between ultracold Rb atoms and introduction Ca + , in a combined experimental/theoretical study. The work of F. Hall et al. used high-level ab initio calculations of the molecular ion electronic structure, to figure out the main mechanisms of formation as being light-assisted collisions, i.e. radiative association (RA) and radiative charge-transfer (RCT). The role of non-radiative processes are also discussed [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF].

• 2012: Many experiments were performed. Briefly, (i) charge-exchange is used as argument to prove the cooling, instead of heating, of Rb + in the presence of ultracold Rb atoms [START_REF] Ravi | Combined ion and atom trap for low-temperature ion-atom physics[END_REF][START_REF] Ravi | Cooling and stabilization by collisions in a mixed ion-atom system[END_REF]; (ii) sympathetic cooling of trapped Na + ions in the presence of ultracold Na atoms [START_REF] Sivarajah | Evidence of sympathetic cooling of Na + ions by a Na magneto-optical trap in a hybrid trap[END_REF]; (iii) exothermic three-body recombination between a single trapped Rb + ion and two neutral Rb atoms in an ultracold atom cloud [START_REF] Härter | Single ion as a three-body reaction center in an ultracold atomic gas[END_REF]; (iv) single trapped Rb + /Ba + in a Bose-Einstein condensation of Rb atoms [START_REF] Schmid | An apparatus for immersing trapped ions into an ultracold gas of neutral atoms[END_REF]; (v) observation of the influence of hyperfine interaction on the charge-exchange process between a single Yb + and ultracold Rb atoms [START_REF] Ratschbacher | Controlling chemical reactions of a single particle[END_REF]; and (vi) collisions between ultracold Ca atoms and Ba + ions where the role of different electronic states is investigated [START_REF] Sullivan | Role of Electronic Excitations in Ground-State-Forbidden Inelastic Collisions Between Ultracold Atoms and Ions[END_REF]. No further observations of molecules is reported.

• 2013: F. Hall et al. using the same experimental setup as in their earlier work, back in 2011, now performing collisions between ultracold Rb atoms and Ca + /Ba + [START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF][START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF], observed clear signals of both RbCa + and RbBa + molecules being, therefore, the only cases of direct observation of molecular ions in such a sort of experiment. In the same year, S. Haze et al. investigated elastic collisions between laser-cooled fermionic Li atoms and Ca + at the energy range of 100 mK to 3 K [START_REF] Haze | Observation of elastic collisions between lithium atoms and calcium ions[END_REF]; and S. Lee et al. were able to resolve the rate coefficients of Rb/Rb + ultracold collisions, in which case the ions are optically dark [START_REF] Lee | Measurement of collisions between rubidium atoms and optically dark rubidium ions in trapped mixtures[END_REF].

• 2014: Using the original experimental setup proposed by O. P. Makarov, W. W. Smith and co-workers measured charge exchange rates of the order of 2×10 -11 cm 3 /s between ultracold Na atoms and Ca + [START_REF] Smith | Experiments with an ion-neutral hybrid trap: cold charge-exchange collisions[END_REF].

• 2015-today: Z. Meir and co-workers performed, for the first time, collisions between 87 Rb atoms and 88 Sr + ions, in which the similar masses, in contrast to previous experiments where light atoms and heavy ions were used, are pointed out as the main reason behind the observed deviation from a usual Maxwell-Boltzmann to a power-law energy distribution, not seen in the experiments before [START_REF] Meir | Dynamics of a Ground-State Cooled Ion Colliding with Ultracold Atoms[END_REF]. And the original experiment of S. Haze et al. is now able to resolve the charge-exchange collisions between ultracold 6 Li atoms and 40 Ca + at different electronic states [START_REF] Haze | Charge-exchange collisions between ultracold fermionic lithium atoms and calcium ions[END_REF]. Finally, A. Kr ükow and co-workers using an experimental setup similar to Ref. [START_REF] Härter | Single ion as a three-body reaction center in an ultracold atomic gas[END_REF], observed two-body and three-body collisions of a single Ba + inside relatively dense clouds of ultracold Rb atoms [START_REF] Krükow | Reactive two-body and three-body collisions of Ba + in an ultracold Rb gas[END_REF].

From this brief overview given above, it becomes striking that formation of molecular ions, in atom-ion collisions, is as a matter of fact not the dominant process in the ultracold domain. Remarkably, the light-assisted collision observed by F. Hall et al. [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF][START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF], leading to formation of molecular ions, i.e.

Rb + Ca + -→ RbCa + + hν [START_REF] Deutsch | Optical lattices[END_REF] or Rb + Ba + -→ RbBa + + hν [START_REF] Morinaga | Holographic manipulation of a cold atomic beam[END_REF] where, hν is a quanta of energy of a photon released during the collision, competes with the respective radiative charge transfer process Rb + Ca + -→ Rb + + Ca + hν [START_REF] Jin | Collective Excitations of a Bose-Einstein Condensate in a Dilute Gas[END_REF] or Rb + Ba + -→ Rb + + Ba + hν [START_REF] Mewes | Output Coupler for Bose-Einstein Condensed Atoms[END_REF] and is dominated by its non-radiative counterpart, in the cases of Rb/Ca + experiments [START_REF] Tacconi | Computing charge-exchange cross sections for Ca + collisions with Rb at low and ultralow energies[END_REF]. It is worthwhile to notice, further, that these are not the only molecular ions observed, since a clear signal of the presence of Rb + 2 is presented. It may indicate that a somewhat more complicated collision-induced rearrangement takes place. For instance, after the production of RbCa + ions, the molecule could undergo subsequent collisions with the remaining neighboring ultracold Rb atoms leading to chemical processes, such as RbCa + + Rb -→ Rb + 2 + Ca [START_REF] Burger | Dark Solitons in Bose-Einstein Condensates[END_REF] being therefore an extra source of trap losses of RbCa + and/or RbBa + . As we shall see later, by means of electronic structure calculations, Rb + 2 is relatively more stable than either RbCa + or RbBa + . In both experimental cases, however, cooling lasers were alternately blocked using a mechanical chopper at a frequency of 1 kHz in order to prevent photoionization of Rb out from the (5p) 2 P 3/2 level by 397 nm photons. Such a characteristic, with respect the many other experiments available, and briefly described above, may become an evidence of destruction of molecular ions by photodissociation processes if the exposure of collisional products to cooling lights is not properly understood. We shall explore such a mechanism of photodissociation, within the context of A. Kr ükow's work [START_REF] Krükow | Reactive two-body and three-body collisions of Ba + in an ultracold Rb gas[END_REF][START_REF] Krükow | Energy Scaling of Cold Atom-Atom-Ion Three-Body Recombination[END_REF], i.e. a single Ba + in a cloud of ultracold Rb atoms, which undergo three-body recombination where no RbBa + molecule is observed.

The aim of this work is to carry out a theoretical study of formation/destruction mechanisms of molecular ions, and their efficiency, in such a hybrid setup. As a case study we shall focus on those observations coming from Rb/Ba + [START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF][START_REF] Krükow | Reactive two-body and three-body collisions of Ba + in an ultracold Rb gas[END_REF] and Rb/Ca + [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF] experiments. Nevertheless, a systematic and consistent analysis of a few other relevant inelastic collisions, e.g. Rb/Sr + , Rb/Yb + , Li/Yb + and Li/Ca + pairs, in their electronic ground state, considering the two competitive channels of RA and RCT, is also presented.

Once the general ideas of forming molecular ions are properly established, we shall then present an investigation on the role of inelastic and reactive collisions in the issue of internal cooling by buffer gas cooling, now using as a case study the most popular molecular ions, H + 2 , D + 2 and HD + in collisions with He atoms intended for high precision measurements and metrology. Following the trend of H + 2 -He collisions, we finally present an investigation on the eventual quenching of Rb + 2 internal states in ultracold collisions with residual Rb atoms, due to its relevance in many aspects to atom-ion collisions in hybrid traps, as briefly shown above and exemplified by Eq. [START_REF] Burger | Dark Solitons in Bose-Einstein Condensates[END_REF] and Ref. [START_REF] Jyothi | Photodissociation of Trapped Rb + 2 : Implications for Simultaneous Trapping of Atoms and Molecular Ions[END_REF].

This work was realized and funded within the context of a Marie-Curie Initial Training Network (ITN) named Cold Molecular Ions at the Quantum Limit (COMIQ), supported by the European Union's Seventh Framework Programme (grant agreement no. 607491). It was intended to investigate how cooling, trapping, and control techniques of molecular ions can contribute in the domain of quantum technology, enhance precision measurements on molecular systems and lead to chemistry in the ultracold regime. A total of 13 early-stage researchers (ESR) were recruited and trained through undertaking PhD projects at the respective host institutions. The formation was achieved by means of three fundamental components: (i) host-based training through individual research; (ii) "secondment" (collaborations with other network members) and scientific visits; and, (iii) network-based training comprising schools, practical courses in the laboratory, scientific meetings etc.

In the present case the main project was carried out in the Laboratoire Aimé Cotton (CNRS-LAC), under the supervision of Dr. Olivier Dulieu and Dr. Maurice Raoult, devoted to developments of theoretical methods for electronic structure and dynamics of ultracold neutral and charged molecules; seeking to probe the actual mechanisms and therefore their efficiency on producing molecular ions within the atom-ion hybrid setups briefly introduced earlier. We shall focus on the context of those observations coming from Rb/Ba + and Rb/Ca + seminal experiments performed by the network members, namely the groups headed by Prof. Stefan Willitsch [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF][START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF] and Prof. Johannes Hecker Denschlag [START_REF] Härter | Single ion as a three-body reaction center in an ultracold atomic gas[END_REF][START_REF] Krükow | Reactive two-body and three-body collisions of Ba + in an ultracold Rb gas[END_REF]. Nevertheless, a systematic and consistent analysis of several other relevant atomic colliding partners, such as Rb/Sr + , Rb/Yb + , Li/Yb + , Li/Ca + and Li/Ba + pairs, were also carried out, in their electronic ground state, considering the two competitive channels of RA and RCT. Thus, on Chapter (4) we describe the electronic structure (potential energy curves, transition and permanent electric dipole moments) of all molecular ions involved, using accurate quantum chemistry calculations which are compared to other determinations, when available. We present our results for both RA and RCT cross sections and rate constants, emphasizing then the presence of narrow shape resonances induced by the centrifugal barrier in the entrance channel, and the vibrational distribution of the molecular ions, which are not expected to be created in their ground level. A general discussion and prospects about possible explanation for the absence of molecular ions in several of the experiments quoted above are also carried out. In the Chapter [START_REF] Adams | Atom optics[END_REF] we present an ex-perimental energy dependence study of the charge transfer collision cross sections in a mixture of 6 Li atoms and 40 Ca + ions, performed in the group of Prof. T. Mukaiyama (Institute for Laser Science, University of Electro-Communications in Tokyo), where our electronic structure calculations are used to interpret and understand their observations.

Two collaborations were carried out. The first one, under the supervision of Prof. Franco A. Gianturco, seeking to understand the eventual internal cooling of H + 2 , D + 2 and HD + molecular ions in collisions with He atoms, intended for subsequent high precision measurements in a new experimental setup at the network member lead by Prof. Stephan Schiller. Our main contribution concerns the preparation (described later) of the required potential energy surface, originally computed elsewhere [START_REF] Ramachandran | Revisiting the potential energy surface for the reaction at the full configuration interaction level[END_REF], and the respective computation of the coupling matrix, as the principal ingredient in order to understand the underlying mechanisms during the atom-diatom collisions, and to obtain actual cross sections. Thus, on Chapter [START_REF] Deutsch | Optical lattices[END_REF] we present an accurate representation of the atom-diatom interaction, from an ab initio description of the relevant potential energy surface, with the molecular ion in the lowest vibrational level of its ground electronic state, and the respective rotationally state-to-state cross sections by solving the quantum scattering equations. In particular, it is shown that the sum of the individual state-to-state cross sections accounting for the fine-structure components is dynamically equivalent to directly treating the collision problem of a molecular ion as a structureless spherical rotor interacting with the He atom. Chapter [START_REF] Morinaga | Holographic manipulation of a cold atomic beam[END_REF] presents a detailed discussion on the couplings between the various vibrational states of the systems and provide a brief summary of the formula employed for the relevant vibrational couplings. Some illustrative computational results on the excitation and de-excitation cross sections for the rotationally inelastic collisions when vibrational effects are incorporated are also presented. And finally, a further illustrative example of the size and energy behavior of the rovibrational cooling cross sections between the lowest two vibrational states of the H + 2 molecular ion. Inspired by H + 2 /He prospects and given the relevance of Rb + 2 species in atom-ion hybrid setups based on sources of Rb atoms (see the introductory part), on Chapter [START_REF] Jin | Collective Excitations of a Bose-Einstein Condensate in a Dilute Gas[END_REF] we present the ab initio computation of the first 20 electronic states of a Rb + 3 complex; and discuss the theoretical aspects of an eventual internal cooling of sympathetically cold Rb + 2 ions in collisions with Rb atoms, as an experimental proposal for the scientific community. In particular, the extra difficulty imposed by the homonuclear nature of the problem, inducing losses from reactive collisions, e.g. ortho-Rb + 2 + Rb -→ para-Rb + 2 + Rb (or vice-versa) and ortho/para-Rb + 2 + Rb -→ ortho/para-Rb 2 + Rb + , is also discussed.

Finally, in the context of the second collaboration, now under the supervision of Prof. Johannes Hecker Denschlag, Chapter [START_REF] Mewes | Output Coupler for Bose-Einstein Condensed Atoms[END_REF] presents a theoretical study of the radiative lifetime of a hypothetical weakly-bound RbBa + molecular ion produced in an experimental setup similar to A. Kr ükow's work [START_REF] Krükow | Reactive two-body and three-body collisions of Ba + in an ultracold Rb gas[END_REF][START_REF] Krükow | Energy Scaling of Cold Atom-Atom-Ion Three-Body Recombination[END_REF], i.e. a single Ba + into a relative dense cloud of ultracold Rb atoms, which undergoes a three-body recombination process, with no RbBa + molecule being observed. We shall discuss the influence of many continuous, and relatively intense, sources of light in the experimental setup, in particular the ones used for dipole traps, as one of the main drawbacks in order to observe long lived molecular ions. The hypothesis of weakly-bound species by means of three-body recombination is also analyzed.

Chapters (4), ( 5), ( 6) and ( 7) are adapted ipsis litteris from the respective publications produced to date.

Before, however, Chapter (3) is devoted to a short description of the main theoretical methods used. • The keywords "atom-atom" and "atom-diatom", a standard notation in the literature, are also used all along this work, in order to refer to each respective type of collisional problem. Notice, however, that both keywords are used herein for collisions between neutral and charged colliding partners, implying therefore either ion-molecule or atom-molecular ion possibilities, as well as atom-ion type of collisions. For instance, in the Chapter (8), the same notation "atom-diatom" shall be used to refer either to Rb + 2 + Rb or Rb 2 + Rb + collisions. Any occurrence of ambiguities shall be resolved at the given context.

• The usual assumption of the absence of external forces acting on the colliding system is always implied. Therefore, the system center-of-mass (COM) moves uniformly. In such a body-fixed reference frame (BF), i.e. the center-of-mass is at rest, the position vector, x, for each of the atoms involved are x 1 , x 2 , x 3 , . . ., relative to a given origin in the space-fixed (SF) reference frame. Thus, most of the non-trivial physics of an atom-atom and atom-diatom collision will arise from the interaction along a relative motion between the colliding partners. As a typical example, using the atom-atom case, R being the vector connecting the two atomic cores, we shall use as relative coordinate the interatomic distance,

R = |R| = |x 2 -x 1 |, with R ∈ (0, ∞).
• The interaction potentials between the colliding partners of interest, e.g. V = V (R), are radially symmetric, R ∈ (0, ∞), and vanishing asymptotically, V (∞) = 0. We shall use the symbol R always for the relative motion between the projectile and the target, i.e. a scattering coordinate. In those cases of atom-diatom collisions, the symbols r and θ are used for the diatomic internuclear distance and its orientation with respect to R, respectively. In contrast with some old studies found in the literature, where R always represents the diatomic internuclear distance even within an atom-diatom problem, the notation chosen here agrees with the most recent publications on the subject.

• The adiabatic separation between the nuclear and electronic motions is also always implied, i.e. the nuclear masses are considered infinite with respect the electronic ones and therefore electrons are allowed to move in a static field produced by fixed nuclei. We shall take advantage of this ansatz, also known as the Born-Oppenheimer approximation (BO), to uncouple a somewhat complicated electronic-nuclear motion into two independent problems, which rely first on the electronic coordinates and parametrically to each nuclear position. Once the electronic problem is solved, where the typical outcome are the interaction potentials themselves, the nuclear one is resolved, giving access to nuclear wavefunctions and, thus, all the properties of the dynamical problem of interest.

Thus, the next sections are devoted to recall the current widely used time-independent formalisms for the collisional part. The atom-diatom problem is treated on a fully ab initio sense, that is, both electronic and nuclear problems (standard quantum chemistry and close coupling methods, respectively) treated entirely in a quantum mechanical basis. Whereas the atom-atom case, as we shall see below, is enough described by a semiclassical matter-radiation interaction formalism, with the electronic part based on first principles and the radiation described classically. Atomic units (a.u.) are used in the formalism, i.e.

m e = e = h = 1 4π 0 = 1 (11)
and, therefore, the Planck's constant is h = 2π. But results may be expressed in different sets of units: often, either wavenumber for energy in a spectroscopic context (1 a.u. = 219474.63137054 cm -1 ), or millikelvin in an ultracold collisional context (0.695 cm -1 = 1 K = 10 3 mK); for units of length we shall use either Angstroms (1 a.u. = 0.5291772083 Å) or Bohr radii (1 a.u. = 1 a 0 ); units of Debye for electric dipole moments (1 a.u. = 2.54158059 D); and, finally, units of cm 2 or m 2 for cross sections (1 a.u. 2 = 28.0028520055×10 -18 cm 2 ).

the light-assisted atom-ion problem Some of the ultracold atom-ion collisions of interest in this work, briefly described in the introductory part, are those in which the colliding partners may exit the interaction range (i) bound into a molecular ion, i.e. A + B + -→ AB + + hν (radiative association, RA); or, (ii) with the charge exchanged between the nuclei, i.e. A + B + -→ A + + B + hν (radiative charge transfer, RCT); or even, (iii) once a bound molecular state is formed, absorbing a photon and undergoing a photodissociation process, i.e. AB + + hν -→ A + B + . That is, all the processes are assisted either by emitting or absorbing a quanta of hν, in order to ensure the conservation of the total momentum, promoting transitions between rovibronic states. As a typical example, in the Fig. [START_REF] Bradley | Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions[END_REF] is presented the ab initio interaction potential of Li(2s) atoms interacting with Ca + (3s) ions, which happens to correlate with the first excited electronic state, (2) 1 Σ + , of the complex [LiCa] + . The collision could exit in the radiative formation of LiCa + 1 1 Σ + molecular ions in the electronic ground state or the radiative charge transfer leading to Li + and Ca 1 S atoms.

The atom-ion problem is treated in an easy-to-implement semiclassical formalism, which is known by its overall good agreement with the full quantum mechanical treatment, i.e. quantum electrodynamics. The model relies on a matter-radiation interaction scheme of absorption and emission, where, the wave-matter counterpart is described by its quantal form whereas the radiation is given by the classical theory. Such a methodology has been successfully applied for the description of RA [START_REF] Zygelman | The radiative association of He(+) and H[END_REF][START_REF] Gianturco | Radiative association of LiH(X 1 Σ + ) from electronically excited lithium atoms[END_REF][START_REF] Gianturco | Radiative Association Rates and Structure of Resonances for Li and Li + Colliding with H and H +[END_REF][START_REF] Zygelman | Stimulated Radiative Association of He and H +[END_REF][START_REF] Zygelman | Cold ion-atom chemistry driven by spontaneous radiative relaxation: a case study for the formation of the YbCa + molecular ion[END_REF] and the RCT of elementary systems such as Li(2s) + H + [START_REF] Stancil | Radiative Charge Transfer in Collisions of Li with H +[END_REF].

Franck-Condon theory

It is worthwhile, however, to start revisiting the somewhat phenomenological treatment of atomic transitions developed by Einstein [START_REF] Einstein | The quantum theory of radiation[END_REF], which is generalized in this section for any transition. Thus, recalling the seminal works of James Franck and Edward U. Condon, according to Einstein, the probability of a spontaneous emission from an initial state |i with energy E i and population N i , in the instant t i , towards a fi- nal state |f with energy E f at an instant t > t i is A f←i . Therefore, the initial population as a function of time is simply

dN i dt = -∑ f A f←i N i ( 12
)
Where, according to Bohr's rule, E f -E i = hν f←i is the emitted photon of frequency ν. Moreover, the lifetime τ i of |i is

τ i = 1 ∑ f A f←i (13)
i.e. the mean time of the population which remains in the initial state |i . Likewise, Einstein ascribes the probability of absorbing a photon, B i→f , and promoting an upward transition, due to a stimulation of the radiation field with an energy density ρ (ν i→f ) dν. Such a rate of transition is N i B i→f ρ (ν i→f ). In the following, the notation f←i is used for an emission (spontaneous or stimulated) process and, thus, E f < E i ; or, i→f for a stimulated absorption process, i.e. E f > E i .

theoretical approach

Using a detailed balance argument, i.e. at equilibrium the rates of each elementary process and its inverse are equal,

N i B i→f ρ (ν i→f ) = N f [A i←f + B i←f ρ (ν i←f )] (14)
and therefore,

N i N f = A i←f + B i←f ρ (ν i←f ) B i→f ρ (ν i→f ) (15)
Such a ratio of populations is also obtained from the Boltzmann distribution law,

N i N f = exp E f -E i k b T (16)
where, k b is the Boltzmann constant. And thus, using these two definitions and solving for ρ (ν),

ρ (ν i→f ) = A i←f B i→f exp E f -E i k b T -B i←f (17)
assuming ρ (ν i→f ) ≡ ρ (ν i←f ). Likewise, from the Planck's law of a blackbody radia- tion at a given temperature T, the spectral density is known as

ρ (ν) = 8πhν 3 c 3 1 exp hν k b T -1 (18)
c being the speed of light. Finally, comparing Eq. ( 17) and Eq. ( 18),

B i→f = B i←f (19)
and

A i←f = 8πhν 3 i→f c 3 B i→f ( 20
)
By inspection of Eq. ( 20), an intrinsic relation between a spontaneous emission and a stimulated absorption is found. Despite only classical theory based arguments been unable to demonstrate such a result, it is known, from the quantum electrodynamics formalism that a spontaneous emission is, for practical purposes, another sort of stimulated process. In such a case, the vacuum of a quantized radiation field is the perturbation which induces atomic and molecular systems to emit radiation. Which, at a first glance, may be seen macroscopically as a spontaneous process [START_REF] Milonni | Why spontaneous emission?[END_REF]. (ii) an outgoing continuum level in the exit channel, Li + + Ca 1 S ; and, (iii) a single bound rovibrational level in the exit channel, LiCa + 1 Σ + .

theoretical approach

Still remains the problem of computing A and B Einstein coefficients. The usual procedure is the use of a first-order perturbation theory picture: given a system in the initial state |i with energy E i at the instant t i , that evolves towards a final state |f with energy E f at an instant t > t i , by means of a perturbation H , the Hamiltonian which describes the system is H = H 0 (t i ) + H (t > t i ), where H 0 (t i ) describes the unperturbed one, with eigenstates {|i } and energies {E i }. From the time-dependent Schr ödinger equation,

ih ∂ ∂t |ψ = H 0 + H |ψ (21)
and using the usual solutions |ψ being spanned in a basis of those internal states from the known unperturbed Hamiltonian, {|i },

|ψ ≡ ∑ i g i (t) exp - iE i t h |i (22)
where, the time-dependent perturbation is set into the expansion coefficients, g i = g i (t), and the exponential part carries out the time-dependence of the unperturbed Hamiltonian. In order to retrieve only perturbation-dependent terms, plugging Eq. ( 22) into Eq. ( 21) and ignoring terms such H 0 |ψ , we obtain,

H |ψ = ih ∑ i d dt g i (t) exp - iE i t h |i (23)
Projecting this result on a final state |f , belonging to a subspace {|f },

f| H |ψ = f| ih ∑ i d dt g i (t) exp - iE i t h |i (24)
where, we seek to solve for dg i (t)/dt, since it accounts for the time-evolution of the wavefunctions g i (t), components of the state vector H |ψ in the space vector of {|i }.

Using the orthonormalization relation

f | i = δ fi , ih ∑ i d dt g i (t) exp - iE i t h δ fi = ∑ i g i (t) exp - iE i t h f| H |i (25)
Where, the matrix elements f| H |i are transition amplitudes between the initial and final states, i→f, due to the perturbation described by H . Thus,

ih d dt g f (t) = ∑ i g i (t) exp -iE i t h exp -iE f t h f| H |i = ∑ i g i (t) exp i h (E f -E i ) t f| H |i (26)
Equation [START_REF] Jones | Ultracold photoassociation spectroscopy: Long-range molecules and atomic scattering[END_REF] gives the first-order transition rate to which the system evolves from |i to |f due the perturbation.

Setting up the nature of H explicitly as a classically described electromagnetic radiation of frequency ν -for sake of simplicity polarized over the x-axis and generalized later on -, i.e. when the electric vector of the radiation induces a dipole moment along its direction:

H = 1 2 ξ 0 x ∑ η q η x η [exp (i2πνt) + exp (-i2πνt)] ( 27
) =ξ 0 x ∑ η q η x η cos (2πνt) (28)
With q being the charge in the position x; the quantity ∑ η q η x η redefined as the x-axis component of the electric dipole moment, ∑ η q η x η = dx [START_REF] Tesch | Quantum computation with vibrationally excited molecules[END_REF] and ξ 0 x the electric field amplitude. Notice that, (i) only the electric component of the electromagnetic radiation is taken into account due the negligible magnetic field, if compared with the electric one, at least for the purposes and molecular species of interest in this work. And, (ii) only the time-dependence of the radiation is expressed, since the spatial dependence is expected to be negligible, as the actual transitions aimed are those between molecular rovibronic levels and/or continuum scattering states, with a wavelength of many orders of magnitude higher than the size of molecular systems themselves. From these considerations, using the exponential form for the perturbation, given by Eq. ( 27), and plugging into Eq. ( 26), yields

ih d dt g f (t) = g i (0) δ fi + 1 2 ξ 0 x f| dx |i exp 2πi h (E f -E i + hν) t + exp 2πi h (E f -E i -hν) t (30)
theoretical approach

Notice that δ fi = 0 in the initial instant, when t = 0, since i = f. Thus, solving 1 for g f (t),

ihg f (t) = 1 2 ξ 0 x f| dx |i t 0 exp 2πi h (E f -E i + hν) t dt + t 0 exp 2πi h (E f -E i -hν) t dt (31) = 1 2 ξ 0 x f| dx |i exp 2πi h (E f -E i + hν) t -1 2πi h (E f -E i + hν) + exp 2πi h (E f -E i -hν) t -1 2πi h (E f -E i -hν) (32)
It is worthwhile to notice that g f (t) is non-negligible only in the surroundings of the resonance E f -E i hν, where the second denominator tends to zero, i.e. either E f -E i = +hν (light absorption) or E f -E i = -hν (stimulated emission). Thus, in agreement with the Bohr's rule. The first denominator, however, tends to increase as the transition E f -E i increases and, therefore, the first exponential function governed by E f -E i + hν becomes essentially negligible. From these considerations and neglecting terms driven by E f -E i + hν, the transition probability per unit of time,

g * f (t)g f (t) = |g f (t)| 2 , is given by |g f (t)| 2 = ξ 0 x 2 f| dx |i 2 sin 2 π h (E f -E i -hν) t (E f -E i -hν) 2 (33)
Where, the following relation is used:

exp 2πi h (E f -E i -hν) t -1 = 2i exp πi h (E f -E i -hν) t sin π h (E f -E i -hν) t (34)
Thus, integrating over a full range of frequencies 2 , ν ∈ (-∞, +∞), with negative frequencies being of no special meaning for any physical purposes described here, since all non-negligible transitions that contributes to the integral are only those in 1 The following relation is used:

t 0 {exp [i (a + b) t] + exp [i (a -b) t]} dt = -i exp [i (a + b) t] -1 (a + b) + exp [i (a -b) t] -1 (a -b) 2 Given the constant a ∈ R, +∞ -∞ sin 2 πx a x 2 dx = π 2 |a| the domain of frequencies ν i→f ∼ ν = (E f -E i ) /h due the resonance argument given earlier, yields |g f (t)| 2 = ξ 0 x 2 f| D |i 2 +∞ -∞ sin 2 π h (E f -E i -hν) t (E f -E i -hν) 2 dν = π 2 h 2 ξ 0 x 2 f| D |i 2 t (35)
Equation [START_REF] Kozlov | Enhancement of the Electric Dipole Moment of the Electron in PbO[END_REF] gives the probability of transition proportional to an amount of time, t, in which the system is exposed to a x-polarized radiation. For a general non-polarized radiation, we shall take into account the energy flow per unit of time and area, i.e. the intensity, given by the Poynting vector, S, in the direction of propagation over the x-axis, with magnitude

S = 1 4π ξ 2 x ( 36
)
where an isotropic radiation is implied,

ξ 2 x = ξ 2 y = ξ 2 z = ξ 2 3 (37)
Therefore,

S = 3 4π ξ 2 x ( 38
)
and the average value of ξ 2 x over one cycle is ξ 0 x cos (2πνt) 2 = 1/2, which yields

S = 3 8π ξ 0 x 2 ≡ ρ (ν) (39)
i.e. an average isotropic radiation density. Moreover, defining the dipole moment operator as

D = dx i + dy j + dz k (40)
where, i, j and k are the unit vectors forming the basis of a Cartesian space and the transition dipole moment (TDM) integrals being f| D |i with mean value,

d dt |g f (t)| 2 = 8π 3 3h 2 ρ (ν i→f ) f| D |i 2 (41)
and the Einstein B coefficient is now explicitly known:

B i→f = 8π 3 3h 2 f| D |i 2 (42)
likewise, the A coefficient is

A i←f = 8πhν 3 i→f c 3 B i→f = 64π 4 ν 3 i→f 3hc 3 f| D |i 2 ( 43
)
Line strengths

In the case of RA and RCT problems, the initial state vector, |i , is a continuum complex-valued scattering wavefunction which describes an incoming wave representing the colliding partners, e.g. Li(2s) + Ca + (3s), in a given electronic state Λ, total energy E total , relative kinetic energy h2 k 2 /2µ, and diatomic angular momentum Ĵ2 with projection Ĵz over the z-axis. That is, |i = |Λ, J, k . Such a state vector is a solution of the radial time-independent Schr ödinger equation,

- h2 2µ
d 2 dR 2 + h2 2µ Ĵ2 R 2 + V(Λ; R) |Λ, J, k = E total |Λ, J, k (44)
Where, µ is the system reduced mass and V(Λ; R) is the radially symmetric interaction potential between the colliding partners as a function of the internuclear distance, R ∈ (0, ∞). Also notice that

E total -V (Λ; ∞) = h2 k 2µ 2 ( 45
)
with k being the associated wavevector of the relative atomic motion, i.e. its initial linear momentum.

When the two atoms are far apart, as R -→ ∞, the asymptotic condition is such that V (Λ; ∞) = 0. Thus, using the relation E total = h2 k 2 /2µ and multiplying Eq. ( 44) by 2µ/h 2 , yields the equation which represents the angular momentum components of a free-particle wave equation, the so-called spherical Bessel equation:

- d 2 dR 2 + Ĵ2 R 2 |Λ, J, k = k 2 |Λ, J, k (46)
which, admits two linearly independent spherical solutions on kR, asymptotically, behaving either as

sin kR -J π 2 + O 1 kR (47)
a physical regular solution which is zero at the origin, or,

cos kR -J π 2 + O 1 kR (48)
not square integrable for J > 0. Therefore, their superposition is taken as the full asymptotic form: lim

R-→∞ |Λ, J, k ∼ sin kR -J π 2 + δ J (49)
which is proportional to a fully free-particle wave, e.g. sin kR -J π 2 , except that the phase of its oscillations is shifted by an amount δ J induced by the interaction potential 3 .

Solutions of an effective scattering process, with positive energies, E total > V (Λ; ∞), are orthogonal state vectors for each k = k, Λ, J, k | Λ, J, k = 0, and thus Λ, J, k | Λ, J, k ∝ δ (kk). It can be shown for pure sine waves of unit amplitude, such as sin (kR

) for k > 0, that ∞ 0 sin k R sin (kR) dR = π 2 δ k -k (50)
and a normalized asymptotic form would become lim

R-→∞ |Λ, J, k ∼ 2 π sin kR -J π 2 + δ J ( 51
)
3 The characterization of the phase shift is often understood from the fact that the kinetic energy in the short range zone of the potental is higher, that is V (R) 0, and pulls the wavefunction inward (R -→ 0). Thus, giving a positive phase shift when it emerges from the potential well. Likewise, a repulsive potential pushes the wavefunction outward (R -→ ∞), i.e. gives a negative phase shift. In the total absence of interactions, no shift would be considered.

theoretical approach

Moreover, notice that from the following identity

δ k -k = dE total dk δ E total -E total = h2 k µ δ E total -E total (52)
an energy normalized asymptotic form would be better expressed as lim

R-→∞ |Λ, J, k ∼ 2µ πh 2 k sin kR -J π 2 + δ J (53) to obey Λ, J, k | Λ, J, k = δ E total -E total .
Therefore, in oder to obtain a continuum scattering wavefunction which describes an atom-ion collision, one needs to integrate the radial equation, given by Eq. ( 44) -either analytically or numericallyfrom R = 0 outward, as R -→ ∞, respecting the boundary condition of such a function vanishing at R = 0. The wavefunction integrated well beyond the influence of the potential shall behave as Eq. ( 53) and once compared to a free partial wave, i.e. sin kR -J π 2 in the absence of any interaction, gives the last unknown term which is the phase shift, δ J , induced by the potential V (Λ; R). For practical purposes of com- putation, solutions of the Schr ödinger equation are taken as the product of a radial wavefunction, g J (Λ, E total ; R), and an angular wavefunction, Y J,m J (θ), eigenstate of Ĵ2 with J (J + 1) eigenvalues and projections m J , eigenvalue of Ĵz . In coordinate representation:

|Λ, J, k ≡ 2µ πh 2 k g J (Λ, E total ; R) Y J,m J (θ) (54)
Where, θ is the angle between the diatomic internuclear axis and the SF z-axis. In such a case spherical harmonics, Y J,m J , are known analytically and the radial wavefunctions are calculated numerically by standard methods, e.g. the renormalized Numerov method of B. R. Johnson [START_REF] Johnson | New numerical methods applied to solving the one-dimensional eigenvalue problem[END_REF][START_REF] Johnson | The renormalized Numerov method applied to calculating bound states of the coupledchannel Schr öedinger equation[END_REF].

Therefore, in the case of a RCT process, both initial and final levels belonging to two different electronic states Λ and Λ with hν difference in energy, are represented by a continuum incoming wave, |i = |Λ, J, k , and a continuum outgoing wave, |f = |Λ , J , k , i.e. continuum-to-continuum transitions. Thus, the dipole matrix elements used in Eq. ( 43) would become,

E total 0 | Λ , J , k D (R) |Λ, J, k | 2 dE total ( 55
)
with the radial dependence of the transition electric dipole moment explicitly written, D = D (R).

The final state of a RA process is an internal bound rovibrational state of a molecular ion AB + in the electronic state Λ , i.e. |f = |Λ , J , υ , with vibrational quantum number υ and binding energy ε Λ ,J ,υ , solution of the bound diatomic eigenvalue problem,

- h2 2µ
d 2 dR 2 + h2 2µ Ĵ2 R 2 + V(Λ ; R) Λ , J , υ = ε Λ ,J ,υ Λ , J , υ (56)
Or, on its secular form,

- d 2 dR 2 + Ĵ2 R 2 + 2µ h2 V(Λ ; R) -ε Λ ,J ,υ Λ , J , υ = 0 ( 57
)
which may be computed numerically nowadays subject to a relatively high level of accuracy, using for instance the discrete variable representation (DVR) method [START_REF] Yurchenko | Duo: A general program for calculating spectra of diatomic molecules[END_REF]. And, therefore, the respective RA dipole matrix element, driven by continuum-tobound transitions, is

υ max ∑ υ | Λ , J , υ D (R) |Λ, J, k | 2 (58)
Likewise, the photodissociation process, described by Eq. ( 42), would lead to matrix elements of the same sort,

υ max ∑ υ | Λ , J , k D (R) |Λ, J, υ | 2 (59)
where, a photon is absorbed and promotes a bound-to-continuum transition instead.

Cross sections

From the dipole moment matrix elements given by Eq. ( 55), Eq. ( 58), Eq. ( 59) and the Einstein coefficients, Eq. ( 43) and Eq. ( 42), describing different kinds of processes, one can determine the respective state-to-state and total transition cross sections.

As we shall see in the Chapter ( 4) and [START_REF] Mewes | Output Coupler for Bose-Einstein Condensed Atoms[END_REF], in this work, the atom-ion collisions and/or photodissociation processes of interest are only those at electronic manifolds Λ = Σ + and Λ = Π with electronic spin multiplicities singlet or triplet. Since the state vectors used in Eq. ( 55), Eq. ( 58) and Eq. ( 59) are products of an angular component and a radial one, the probability for a transition can be factored into a part which theoretical approach depends on the quantum numbers for the angular momentum, S (J , J), and a part which is independent of the nuclear angular momentum, with S (J , J) the so-called H önl-London factors [START_REF] Hansson | A comment on H önl-London factors[END_REF]. Such a factorization yields, for Eq. ( 55),

E total 0 S J , J | Λ , k D (R) |Λ, k | 2 dE total (60)
Likewise, for Eq. ( 58),

υ max ∑ υ S J , J | Λ , υ D (R) |Λ, k | 2 (61)
and for Eq. ( 59),

υ max ∑ υ S J , J | Λ , k D (R) |Λ, υ | 2 (62)
Explicit forms for the H önl-London factors can be derived analytically by expressing the dipole moment operator, D (R), in terms of spherical tensor components in the BF reference frame, applied between the respective angular momentum components of the total state vectors, Y J,m J . Since such a derivation is beyond the scope of this section and H önl-London factors are nowadays available as extensive tables in the literature -see for instance Ref. [START_REF] Hansson | A comment on H önl-London factors[END_REF] -, we shall use explicitly the S (J , J) val- ues required by those transitions of interest here, as follow: at a given initial angular momentum J, and its three possible final values J , namely the respective R-lines (J = J -1), Q-lines (J = J) and P-lines (J = J + 1), from Σ + -→ Σ + and Σ + -→ Π transition bands, we have

S J , J =                                  J ; Σ + -→ Σ + , (R -line) 0 ; Σ + -→ Σ + , (Q -line) J + 1 ; Σ + -→ Σ + , (P -line) J -1 ; Σ + -→ Π, (R -line) 2J + 1 ; Σ + -→ Π, (Q -line) J + 2 ; Σ + -→ Π, (P -line) (63)
Equation ( 63) is adapted from the Table (2) of Ref. [START_REF] Hansson | A comment on H önl-London factors[END_REF], currently implemented in the LEVEL code written by Prof. Robert J. LeRoy (University of Waterloo) and widely used during this work -for a brief comment on the LEVEL code, see Appendix (C.2).

In the Chapter (4), the RA and RCT processes are discussed only for 1 Σ + electronic states where both processes are spontaneous emissions. Thus, in the absence of Qlines for Σ + -→ Σ + transition band, plugging Eq. ( 60) and Eq. ( 61) into Eq. ( 43), and using S (J , J) explicitly, yields

σ RCT = g 64π 4 3hc 3 k 2 ∞ ∑ J=0 E total 0 ν 3 R-line J | Λ , k D (R) |Λ, k | 2 dE total + E total 0 ν 3 P-line (J + 1) | Λ , k D (R) |Λ, k | 2 dE total ( 64
)
σ RA = g 64π 4 3hc 3 k 2 ∞ ∑ J=0 υ max ∑ υ=0 ν 3 R-line J | Λ , υ D (R) |Λ, k | 2 + ν 3 P-line (J + 1) | Λ , υ D (R) |Λ, k | 2 (65)
Where, σ RCT and σ RA are the total RCT and RA cross sections, respectively; g = 1/4 is the statistical weight of an initial singlet electronic state; and the whole process is weighted by the initial velocity of the colliding partners, k 2 .

In the photodissociation problems treated on Chapter (9) both Σ + -→ Σ + and Σ + -→ Π transition bands are studied. Therefore, plugging Eq. ( 62) into the Einstein absorption coefficient definition, given by Eq. ( 42), yields to total photodissociation cross sections σ PD ,

σ PD Σ + →Σ + = g 8π 3 3h 2 υ max ∑ υ=0 ∞ ∑ J=0 ν R-line J Λ = Σ + , k D (R) Λ = Σ + , υ 2 + ν P-line (J + 1) Λ = Σ + , k D (R) Λ = Σ + , υ 2 (66)
and

σ PD Σ + →Π = g 8π 3 3h 2 υ max ∑ υ=0 ∞ ∑ J=0 ν R-line (J -1) Λ = Π, k D (R) Λ = Σ + , υ 2 + ν Q-line (2J + 1) Λ = Π, k D (R) Λ = Σ + , υ 2 +ν P-line (J + 2) Λ = Π, k D (R) Λ = Σ + , υ 2 (67)
where, g = 1/4 and g = 3/4 are properly chosen if either an singlet or a triplet electronic state is chosen as the initial state.

Due the relation between k and the total energy E total , becomes convenient to express the RCT and RA total cross sections as functions of E total , i.e. σ RCT (E total ) and σ RA (E total ). A detailed discussion about the features observed in those cross sections is carried out at the Chapter (4).

theoretical approach a brief review of stationary multichannel scattering theory Often during this work, scattering theory is invoked in order to describe atom-ion and atom-diatom collisions. Therefore, it turns out worthwhile a brief review of the key points about its formulation, in a quantum mechanical sense. It does not recall, however, the details of atomic and molecular collisions which are left to the respective chapters. Notice a usual assumption, mentioned earlier, and always implied during the discussion below, of no external forces acting on the colliding system. Therefore, the system center-of-mass moves uniformly. In such a body-frame (BF) of reference, i.e. the center-of-mass is at rest, the position vector, x, for each of the particles involved, e.g. x 1 and x 2 , is defined relative to a given origin in the space-fixed (SF) reference frame. Thus, most of the non-trivial physics in a collision and scattering processes will arise from the interaction along its relative motion, e.g. the interparticle distances,

R = R 12 = |x 2 -x 1 |.
Assume we wish to perform a target-projectile collision at an energy E total , in which the target carries an internal structure, described by a previously known internal Hamiltonian, Ĥ0 , with eigenstates {|η } and energies ε η . Thus, the internal Schr ödinger equation is,

Ĥ0 |η = ε η |η (68)
Where, the symbol η is a composite index for all the quantum numbers and labels associated to the respective internal degrees of freedom, from now on referred to as channels. It is worthwhile to notice that {|η } forms a convenient basis set to work with, in order to express the changes in the internal degrees of freedom induced by the collision. Moreover, if the eigenstates {|η } are chosen asymptotically, as R -→ ∞, i.e. target and projectiles are far apart, those are known as diabatic representation. Nevertheless one may also describe the problem by means of an adiabatic basis, changing as the projectile approach the target. In any case, however, both representations shall be identical at R = ∞.

The scattering state vector, |E total , is a linear expansion on the basis of all internal degrees of freedom,

|E total = ∑ η g η (R) |η (69)
Where, the expansion coefficients g η (R) are the actual scattering wavefunctions, i.e. the coordinates of the state vector |E total spanned in the vector-space of {|η }. Notice that, if E total < ε η , implies such channel wavefunctions to obey bound-state boundary conditions at large distances, g η (0) = 0 and g η (∞) = 0. While in the case E total > ε η , g η (R) behaves as an unbound continuum state. Thus, ε η defines a relative channel threshold between the respective accessible E total > ε η and forbidden E total < ε η channels η, at a collision energy given by E total . Concerning the actual notation used in scattering theory we shall refer to accessible and forbidden as "open" and "closed".

One of the fundamental key points of scattering theory is to assume that in an instant, t, before the collision, as t -→ -∞, when the colliding partners are far apart, the scattering state vector is well represented by a wave packet of free incoming planewaves (herein assumed to travel over the SF z-axis),

lim t-→-∞ |E total ∼ exp ik η z |η (70)
at a given internal state η with initial associated linear momentum hk η . When the collision happens, at t = 0 say, the actual form of |E total is unknown due the inter- action potential, V = V (R), and a somewhat complicated scattering process that can take place. Far beyond the interaction range, after the collision, as t -→ +∞, the final state is now a set of scattered spherical waves, modulated by a scattering amplitude f,

lim t-→+∞ |E total ∼ ∑ ς f ς,η exp (ik ς R) R |ς (71)
where, {|ς } is a subspace of all final, and open, channels available with kinetic energy

E total -ε ς = h2 k 2 ς 2µ (72)
With µ the target-projectile reduced mass. Notice that R in the denominator of Eq. ( 71) is required to preserve the probability within a given solid angle. In order to introduce some anisotropy we shall further define the scattering amplitude matrix elements depending explicitly on the position of a detector in the surroundings of the collisional event, f ς,η = f ς,η (θ, φ) -using spherical coordinates. And, therefore, it shall carry all the information concerning the scattering process.

A second important point is, therefore, the motivation under the asymptotic forms given in the Eq. ( 70) and Eq. ( 71): the target-projectile interaction potential, V (R), falls off faster than 1/R 2 along the scattering coordinate R, that is V (±∞) = 0. Notice that it may have the property of being radially symmetric, R ∈ (0, ∞) with V (∞) = 0, e.g. scattering by central forces, and if that is the case the system is now rotationally invariant. That is, the collision process can be resolved parametrically to each value of the incoming orbital angular momentum, , of the projectile with respect the target. Thus, the scattering wavefunctions, g η (R), are further spanned in a basis of eigenstates of angular momentum, {| , m }, with projection m along the scattering coordinate R:

g η (R) = ∞ ∑ =0 ∑ m =- g η, ,m (R) R | , m (73)
As we shall see below, during this work, radially symetric potentials are the sort of interactions driving the processes of our interest.

The flux of scattered particles passing by the detector at (θ, φ), with solid angle dΩ = sin (θ) dθdφ, is simply the outgoing flux of particles f ς,η (θ, φ)

2 dΩ normalized to its initial flux. The differential scattered cross section is

dσ ς,η = f ς,η (θ, φ) 2 dΩ (74)
or,

dσ ς,η dΩ = f ς,η (θ, φ) 2 ( 75
)
where, integrating over θ and φ,

2π 0 π 0 f ς,η (θ, φ) 2 sin (θ) dθ dφ = σ ς,η (76)
gives the cross section, σ, at a given total energy, σ = σ (E total ), in units of area and represents a plane perpendicular to the direction of incidence through which the incoming flux passes and is scattered in the direction dΩ. The state-to-state integral cross sections, σ ς,η (E total ), gives then the ratio of population which evolves from an initial state |η to a final state |ς , accessible at E total , due to the collision process. Strictly speaking, elastic collisions if η = ς and inelastic (reactive) collisions if η = ς.

Therefore, given a collision process driven by an interaction potential V (R) which asymptotically is V (∞) = 0, as R -→ ∞, the main task in scattering theory is the computation of the actual scattering state |E total up to some spatial range in which the asymptotic assumption becomes valid, i.e. R -→ ∞. Thus, once the solution is projected on the superposition of its incoming and outgoing asymptotic forms, Eq. ( 70) and Eq. ( 71), lim

R-→∞ |E total ∼ exp ik η z |η + ∑ ς f ς,η (θ, φ) exp (ik ς R) R |ς (77)
the respective scattering amplitude f ς,η (θ, φ) can be retrieved, in order to resolve cross sections and eventual rate constants.

The time-independent Schr ödinger equation, to which |E total is solution of, is

- h2 2µ d 2 dR 2 + h2 2µ ( + 1) R 2 + Ĥ0 + V (R) |E total = E total |E total ( 78
)
Plugging Eq. ( 69) in the Schr ödinger equation and projecting the result into the outcome channel |ς , yields

- h2 2µ
d 2 dR 2 + h2 2µ ( + 1) R 2 g η, ,m (R) + ∑ ς, ,m V ς,η, ,m , ,m g ς, ,m (R) = E total -ε η g η, ,m (R) (79)
Where, the definition

ς; , m V 0 + V (R) |η; , m = V ς,η, ,m , ,m (R) (80)
with V 0 representing the interaction potential from the internal Hamiltonian Ĥ0 , within the target; the orthonormalization ς | η = δ ς,η , and Eq. ( 68) are used. Or,

- d 2 dR 2 + ( + 1) R 2 g η, ,m (R) + 2µ h2 ∑ ς, ,m V ς,η, ,m , ,m g ς, ,m (R) = k 2 η g η, ,m (R) (81)
With E totalε η = h2 k 2 η /2µ and multiplying the whole expression by 2µ/h 2 . We shall adopt, however, a more compact form:

d 2 dR 2 + ( + 1) R 2 + k 2 η g η, ,m (R) = 2µ h2 ∑ ς, ,m V ς,η, ,m , ,m g ς, ,m (R) (82)
Equation ( 82) is, strictly speaking, an infinite set of coupled-by-potential inhomogeneous differential equations. But for practical purposes of computation, the basis {|η } is finite and, nevertheless, taken large enough in order to describe properly the physical process of interest. Thus, solving Eq. ( 82), reduced to a n-coupled problem, is often referred to as the close-coupling approximation (CC). The CC formalism still remains remarkably complicated sometimes and more approximations are invoked whenever possible, e.g. decoupling schemes for the states |η etc.

A comment on the rotational diatomic basis set

Along this thesis atom-diatom collision, and the subsequent scattering process, is primarily used to study the internal excitation and/or de-excitation of such a molecular target under collision energies of a few mK. Therefore, it is also worthwhile a brief comment on the internal structure of such molecules, which are used to define the basis set {|η } introduced earlier. In the following we shall consider a structureless atom with polar coordinates, (R, θ, φ) in the SF reference frame, in the surroundings of a molecule, such that the polar angles of the diatom, may be described either at the SF frame or the BF frame, (θ, φ) or (θ , φ ), respectively. The orientation of such a BF frame with respect the SF frame is denoted by (Θ, Φ). For sake of simplicity we shall consider the molecule as a rigid rotor only, in a similar fashion of the original SF formulation of molecular scattering by Arthurs and Dalgarno [START_REF] Arthurs | The theory of scattering by a rigid rotator[END_REF], however accounting to its vibrational motion is straightforward. We shall also keep track of the BF frame counterpart formulation, alike the seminal works of C. F. Curtiss and co-workers [START_REF] Curtiss | [END_REF][159][160].

Thus, we wish to describe the internal rotational (de)excitation of a diatom due to collisions with an atom, which implies, therefore, to express the total atom-diatom (scattering) wavefunction in a complete basis set of orthonormal diatomic rotational eigenstates, with rotational quantum number j and projection m j over the SF frame z-axis, j, m j , where -j ≤ m j ≤ j. The normalized spherical harmonics, Y j,m j = Y j,m j (θ, φ), form such a set, i.e. j, m j ≡ Y j,m j (θ, φ), in which

Y j,-m j (θ, φ) = (-1) m Y * j,m j (θ, φ) (83)
In the BF frame, Y j,Ω j = Y j,Ω j (θ , φ ), with Ω j being the j-projection, now, on the BF frame z-axis. The two representations are connected by the rotation matrix, D = D (α, β, γ), i.e. a unitary transformation that brings the coordinate representation at the SF frame to those at the BF frame, by means of successive rotations over the three Euler angles: α = φ about the z-axis and β = θ about the y-axis, with γ, often, arbitrary taken to zero. The D matrix elements are coefficients of the irreducible representations of the rotation group, given by D j m ,m (α, β, γ) = j, m j exp (-iαj z ) exp -iβj y exp (-iγj x ) j, m j [START_REF] Chin | Feshbach resonances in ultracold gases[END_REF] with j x , j y , j z being the components of the angular momentum, j 2 x + j 2 y + j 2 z = ĵ2 . And therefore,

j, Ω j = j ∑ m j =-j D j Ω,m (α, β, γ) j, m j (85)
or,

j, m j = j ∑ Ω j =-j D j m,Ω (α, β, γ) j, Ω j (86)
Equations [START_REF] Koelemeij | Vibrational spectroscopy of HD + with 2-ppb accuracy[END_REF] or Eq. ( 86), are eigenfunctions of the angular momentum operators, ĵ2 and ĵz , with eigenvalues j (j + 1) and m j (or Ω j ), respectively, at the proper reference frame.

Moreover, the angular momentum associated with the motion of the incoming atom around the diatom, introduced in the previous section, is described such that a total angular momentum, J, is

J = j + (87)
and

J z = j z + z ( 88
)
with a z -projection over the SF frame z-axis being m , where similar to Eq. ( 86), the eigenstates of ˆ 2 and ˆ z are | , m ≡ Y ,m (Θ, Φ), with eigenvalues ( + 1) and m , respectively. For sake of simplicity, a usual definition is to let the radial distance, R, between the incoming atom and the diatom coincide with the z-axis in the BF frame and, therefore, the orbital angular momentum , perpendicular to z, has no projection on this axis. That is, in the BF frame, Ω j represents the projections of either j or J.

The uncoupled representation given by the direct product, j, m j × | , m = j, m j ; , m , is not an eigenvector of the total angular momentum, Ĵ2 and Ĵz , i.e. j, m j ; , m = |J, m J . But such a function is readily known [157] as

|J, m J = j ∑ m j =-j ∑ m =-   j J m j m m J   j, m j ; , m (89)
or, in coordinate representation,

|J, m J ≡ j ∑ m j =-j ∑ m =-   j J m j m m J   Y j,m j (θ, φ) Y ,m (Θ, Φ) (90)
Where, the total projection is m J = m j + m and the symbol

  j J m j m m J   rep-
resents a Clebsch-Gordan coefficient, i.e. a unitary transformation to couple the two representations of angular momenta, which herein span a vector space of dimension (2j + 1) (2 + 1), into a single space with (2J + 1) components. It is worthwhile to recall the actual dependence of such a vector coupling coefficient on the Wigner 3-j symbol:

  j J m j m m J   ≡ (-1) j--m J 2J + 1   j J m j m -m J   (91)
From Eq. ( 90), clearly

j, m j ; , m | J, m J =   j J m j m m J   (92)
where becomes evident a few properties of a Clebsch-Gordan coefficient, being the most important for our purposes the orthonormality relations

j ∑ m j =-j ∑ m =- J , m J | j, m j ; , m j, m j ; , m | J, m J = δ J ,J δ m J ,m J (93)
and

∑ J j, m j ; , m | J, m J J, m J | j, m j ; , m = δ m j ,m j δ m ,m (94)
Since the number of linearly independent basis function are constant under unitary transformation,

∑ J (2J + 1) = (2j + 1) (2 + 1) (95)
Part I

F O R M AT I O N O F M O L E C U L A R I O N S I N AT O M -I O N H Y B R I D T R A P S
In the following, Chapter (4) is adapted from our published work Formation of molecular ions by radiative association of cold trapped atoms and ions available in the special issue Focus on New Frontiers of Cold Molecules Research from the New Journal of Physics (NJP) -see Ref. [START_REF] Da Silva Jr | Formation of molecular ions by radiative association of cold trapped atoms and ions[END_REF] -; and, Chapter (5), adapted from Ref. [START_REF] Saito | Characterization of charge-exchange collisions between ultracold 6 Li atoms and 40 Ca + ions[END_REF], published in a collaboration with Prof. T. Mukaiyama (Institute for Laser Science, University of Electro-Communications in Tokyo).

F O R M AT I O N O F M O L E C U L A R I O N S I N AT O M -I O N H Y B R I D T R A P S
As introduced earlier, in this first part we shall focus on probing the efficiency of RA and RCT processes on producing molecular ions, mainly in the context of those observations coming from Rb/Ba + and Rb/Ca + seminal experiments performed in the groups headed by Prof. Stefan Willitsch [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF][START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF] and Prof. J. Denschlag [START_REF] Härter | Single ion as a three-body reaction center in an ultracold atomic gas[END_REF][START_REF] Krükow | Reactive two-body and three-body collisions of Ba + in an ultracold Rb gas[END_REF]. Nevertheless, we achieved a systematic and consistent analysis of a few other relevant inelastic collisions of atom-ion pairs, namely Rb/Sr + , Rb/Yb + , Li/Yb + and Li/Ca + , in their ground state, considering the two competitive channels.

For such systems we have described the electronic structure, i.e. potential energy curves, transition and permanent electric dipole moments, of all molecular ions involved, using accurate effective core-potential-based quantum chemistry calculations which have been already used in our group to describe a number of diatomic systems [START_REF] Aymar | Structure of the Alkali-metal-atom-Strontium molecular ions: towards photoassociation and formation of cold molecular ions[END_REF]. The atom-ion systems are modeled as molecules with two valence electrons moving in the field of ionic cores represented with effective core potentials including relativistic scalar effects. These potentials are further complemented with core polarization potentials depending on the orbital angular momentum of the valence electron of each one-electron system and parametrized to reproduce their experimental atomic energies. See Section (4.3) for details. The two valence electrons are used to calculate the Hartree-Fock and the excitation determinants, in an atom-centered Gaussian basis set, through the usual self-consistent field methodology, followed by a full configuration interaction treatment. To this end, all the calculations are accomplished using the Configuration Interaction by Perturbation of a Multiconfiguration Wave Function Selected Iteratively (CIPSI) [START_REF] Huron | Iterative perturbation calculations of ground and excited state energies from multiconfigurational zeroth-order wavefunctions[END_REF] package developed at "Laboratoire de Chimie et Physique Quantiques, Université Toulouse III -Paul Sabatier" (France). Over the years, this methodology has been proved accurate and efficient allowing us to compute many single-point energies (within the Born-Oppenheimer approximation), as well as many excited states, in a reasonable amount of computing time. The same electronic structure calculations are used for the study of the photodissociation of molecular ions, using the RbBa + species as a case study -see Chapter (9) -, and it is pushed forward in the Chapter [START_REF] Jin | Collective Excitations of a Bose-Einstein Condensate in a Dilute Gas[END_REF], where we used it to describe a triatomic system, namely Rb + 3 . In the following we present our results for both RA and RCT cross sections and rate constants, emphasizing then the presence of narrow shape resonances induced by the centrifugal barrier in the entrance channel, and on the vibrational distribution of the molecular ions, which are not expected to be created in their internal ground level. A general discussion and prospects about possible explanations for the absence of molecular ions in several of the experiments is also carried out. We performed about 40000 calculations per system in the collisional energy range between 100 µK to 80 mK, currently accessible in most experiments. This requires to consider an upper limit of total angular momentum values, J, of about 50 partial waves, in order to obtain converged cross sections. The integrations over the internuclear distance R are performed with the Milne phase-amplitude method [START_REF] Milne | The Numerical Determination of Characteristic Numbers[END_REF]166]. Such high J values and low initial temperature impose to propagate out the continuum wave function in the entrance channel up to large R values, namely 20000 a.u. This ensures that the outward propagation is stopped well within the classically allowed region situated at long range beyond the centrifugal barrier. The energy scan used a fixed 2 µK step size in order to properly locate all the shape resonances. In particular, all the atomion collisional cross sections present the same dependence with the collision energy corresponding to a Langevin-type regime. Moreover, the RA process is predicted to dominate for all systems in comparison with RCT, by a factor ≈ 2.4 for RbCa + , ≈ 3.8 for RbSr + , ≈ 13.6 for RbBa + , ≈ 2.1 for RbYb + , and ≈ 22.4 for LiYb + .

We have further used our electronic structure calculations to interpret and understand the experimental energy dependence of the charge transfer collision cross sections in a mixture of 6 Li atoms and 40 Ca + ions, performed by Prof. T. Mukaiyama and co-workers, also the most recent atom-ion experiment using hybrid setups. Whereas no molecular ions are detected, lighter species such as Li/Ca + are the most suited systems for an eventual observation of shape resonances due to relatively large resonances width, in units of energy -as we shall see in the Chapter [START_REF] Adams | Atom interferometery[END_REF]. The full study is reproduced in the Chapter [START_REF] Adams | Atom optics[END_REF].

F O R M AT I O N O F M O L E C U L A R I O N S B Y R A D I AT I V E A S S O C I AT I O N O F C O L D T R A P P E D AT O M S A N D I O N S introduction
One of the novel developments of ultracold matter research is exemplified by the experiments aiming at merging a cold atom trap and a trap of laser-cooled ions. Such hybrid setups could offer the opportunity to study collisional dynamics in the quantum s-wave scattering regime associated to the long-range interaction between the ion and the atom varying as R -4 (where R is the ion-atom distance) [START_REF] Idziaszek | Quantum theory of ultracold atom-ion collisions[END_REF][START_REF] Côté | Enhanced Cooling of Hydrogen Atoms by Lithium Atoms[END_REF][START_REF] Gao | Quantum Langevin model for exoergic ion-molecule reactions and inelastic processes[END_REF][START_REF] Krych | Sympathetic cooling of the Ba + ion by collisions with ultracold Rb atoms: Theoretical prospects[END_REF], just like it has been extensively studied for ultracold neutral atom-atom collisionssee for instance Ref. [START_REF] Quéméner | Ultracold molecules under control[END_REF]. These hybrid devices have also been proposed to emulate solid-state physics where a band structure in the fermionic atoms -thus simulating electrons -could be induced by periodic structure generated by the laser-cooled ions [START_REF] Bissbort | Emulating Solid-State Physics with a Hybrid System of Ultracold Ions and Atoms[END_REF]. Other prospects concern the implementation of a Josephson junction through a pair of cold atoms controlled by a single ion [START_REF] Joger | Quantum dynamics of an atomic double-well system interacting with a trapped ion[END_REF][START_REF] Gerritsma | Bosonic Josephson Junction Controlled by a Single Trapped Ion[END_REF], the immersion of a single ion into a Tonk-Girardeau gas to generate an ionic density "bubble" [START_REF] Goold | Ion-induced density bubble in a strongly correlated one-dimensional gas[END_REF], or the elaboration of an atom-ion quantum gate [START_REF] Doerk-Bendig | Atom-ion quantum gate[END_REF]. The variety of combinations of laser-cooled alkali-metal atoms and alkaline-earth (or rare earth) atomic ions allows to explore the limit of sympathetic cooling between them depending on their mass ratio [START_REF] Krych | Sympathetic cooling of the Ba + ion by collisions with ultracold Rb atoms: Theoretical prospects[END_REF][START_REF] Cetina | Micromotion-Induced Limit to Atom-Ion Sympathetic Cooling in Paul Traps[END_REF][START_REF] Huy Nguyên | Micromotion in trapped atom-ion systems[END_REF][START_REF] Tomza | Cold interactions between an Yb + ion and a Li atom: Prospects for sympathetic cooling, radiative association, and Feshbach resonances[END_REF].

Also, a wealth of cold molecular ions species could be created, opening the way to a rich chemistry at temperatures of a few millikelvin, or less [START_REF] Willitsch | Cold Reactive Collisions between Laser-Cooled Ions and Velocity-Selected Neutral Molecules[END_REF]. Following the theoretical prospect of Ref. [START_REF] Makarov | Radiative charge-transfer lifetime of the excited state of (NaCa) +[END_REF] with cold Na trapped in a magneto-optical trap (MOT) and laser-cooled Ca + ions in a Paul trap (then followed by further developments [START_REF] Smith | Experiments with an ion-neutral hybrid trap: cold charge-exchange collisions[END_REF]), other groups have carried out pioneering experiments with various combinations of atoms and atomic ions: Yb atoms with Yb + ions [START_REF] Grier | Observation of Cold Collisions between Trapped Ions and Trapped Atoms[END_REF], Rb atoms with Ca + [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF] and Ba + [START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF] ions, and Ca atoms with Yb + ions [START_REF] Rellergert | Measurement of a Large Chemical Reaction Rate between Ultracold Closed-Shell 40 Ca Atoms and Open-Shell 174 Yb + Ions Held in a Hybrid Atom-Ion Trap[END_REF] or Ba + ions [START_REF] Sullivan | Role of Electronic Excitations in Ground-State-Forbidden Inelastic Collisions Between Ultracold Atoms and Ions[END_REF]. Rubidium atoms can also be confined in a magnetic trap in (or close to) the Bose-Einstein condensation (BEC) regime thus ensuring a density larger than in a MOT, and then interacting with a few Yb + ions [START_REF] Zipkes | Cold Heteronuclear Atom-Ion Collisions[END_REF][START_REF] Zipkes | Cold Heteronuclear Atom-Ion Collisions[END_REF][START_REF] Ratschbacher | Controlling chemical reactions of a single particle[END_REF][START_REF] Zipkes | Hybrid quantum systems of atoms and ions[END_REF]. Similarly, an optical dipole trap of Rb atoms has been merged in a Paul trap containing a few Ba + atoms [START_REF] Schmid | Dynamics of a Cold Trapped Ion in a Bose-Einstein Condensate[END_REF][START_REF] Schmid | An apparatus for immersing trapped ions into an ultracold gas of neutral atoms[END_REF] (or with the combination Li/Ca + [START_REF] Haze | Observation of elastic collisions between lithium atoms and calcium ions[END_REF]). When atomic ions like Rb + or Na + cannot be laser-cooled, they can be sympathetically cooled by another species in the Paul trap in 48 formation of molecular ions by ra of cold trapped atoms and ions the presence of trapped Rb atoms [START_REF] Härter | Single ion as a three-body reaction center in an ultracold atomic gas[END_REF][START_REF] Schmid | An apparatus for immersing trapped ions into an ultracold gas of neutral atoms[END_REF] or Na atoms [START_REF] Sivarajah | Evidence of sympathetic cooling of Na + ions by a Na magneto-optical trap in a hybrid trap[END_REF], or they can be created in situ inside the Paul trap by photoionization of trapped Rb atoms [START_REF] Ravi | Combined ion and atom trap for low-temperature ion-atom physics[END_REF][START_REF] Ravi | Cooling and stabilization by collisions in a mixed ion-atom system[END_REF][START_REF] Lee | Measurement of collisions between rubidium atoms and optically dark rubidium ions in trapped mixtures[END_REF]. To be complete, sympathetically-cooled molecular ions created from an external source have been utilized in two remarkable experiments: one has demonstrated the vibrational quenching of BaCl + ions by laser-cooled Ca atoms [START_REF] Rellergert | Evidence for sympathetic vibrational cooling of translationally cold molecules[END_REF], and another one has recorded a particularly large charge-exchange rate between N + 2 ions and laser-cooled Rb atoms [START_REF] Hall | Millikelvin Reactive Collisions between Sympathetically Cooled Molecular Ions and Laser-Cooled Atoms in an Ion-Atom Hybrid Trap[END_REF]. A comprehensive review of most of these experiments can be found in Ref. [START_REF] Härter | Cold atom-ion experiments in hybrid traps[END_REF].

Elastic and inelastic collisions as well as elementary reactions like charge transfer between the trapped ions and the trapped atoms have been studied in these experiments. Due to the residual micromotion of the trapped ions, these studies have been limited to energies larger than a few millikelvin, well beyond the s-wave quantum regime. It has been recognized in several of these experiments (see for instance Ref. [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF]) that the cooling and/or trapping light most often assist the reaction through the excitation of one of the colliding partners, inducing large rates described by the Langevin regime close to the unitary limit.

Among all these experiments, it is striking that so far, the direct observation of molecular ions resulting from the association of a cold atom and a cold ion has been reported only in two cases, namely RbCa + [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF] and RbBa + [START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF]. These observations have been assigned to the formation of ions by radiative association (RA), i.e. Rb + Alke + -→ RbAlke + + hν (where Alke = Ca, Ba), and hν is the energy of the photon emitted during the collision in order to stabilize the molecular ion [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF][START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF]. The RA process competes with radiative charge transfer (RCT) Rb + Alke + -→ Rb + + Alke + hν when Alke = Ba, and is dominated by non-radiative charge transfer (NRCT) in the Rb/Ca + case [START_REF] Tacconi | Computing charge-exchange cross sections for Ca + collisions with Rb at low and ultralow energies[END_REF]. The RA of CaYb + has been invoked -but molecular ions have not been directly detected -in Ref. [START_REF] Rellergert | Measurement of a Large Chemical Reaction Rate between Ultracold Closed-Shell 40 Ca Atoms and Open-Shell 174 Yb + Ions Held in a Hybrid Atom-Ion Trap[END_REF] to explain the measured fast reaction rate, but not confirmed by a subsequent theoretical investigation [START_REF] Zygelman | Cold ion-atom chemistry driven by spontaneous radiative relaxation: a case study for the formation of the YbCa + molecular ion[END_REF].

In order to clarify the possible molecular ion formation processes and their efficiency in such merged ion and atom traps, we achieved in this paper a systematic and consistent analysis of the inelastic collisions of cold Rb/Alke + pairs in their ground state considering the two competitive channels of RA and RCT. Here the notation Alke refers to the alkaline-earth atoms Ca, Sr, Ba. For convenience the lanthanide atom Yb, with its closed 4 f shell and its external 6s 2 shell, will be included in this Alke notation. The Li-Yb + complex is also studied as its significantly different mass ratio motivated the analogy for using this combination to emulate solid-state physics [START_REF] Bissbort | Emulating Solid-State Physics with a Hybrid System of Ultracold Ions and Atoms[END_REF]. Occasionally, the Rb and Li alkali-metal atoms will be referred to as the Alk species.

theoretical approach for ra and rct

The spontaneous emission of a photon during a collisional process is modeled by a well-established formalism. It is a special case of a well-known semiclassical matterradiation interaction scheme of absorption and emission. A 1 st -order time-dependent perturbation approach expresses the atomic or molecular transition between an initial state |i , with energy E i at the instant t i and a final state |f , with energy E f at t > t i , induced by the perturbation of the electromagnetic field H = H (t > t i ). Such a methodology has been successfully applied to the description of the radiative association [START_REF] Zygelman | The radiative association of He(+) and H[END_REF][START_REF] Gianturco | Radiative association of LiH(X 1 Σ + ) from electronically excited lithium atoms[END_REF][START_REF] Gianturco | Radiative Association Rates and Structure of Resonances for Li and Li + Colliding with H and H +[END_REF][START_REF] Zygelman | Stimulated Radiative Association of He and H +[END_REF][START_REF] Zygelman | Cold ion-atom chemistry driven by spontaneous radiative relaxation: a case study for the formation of the YbCa + molecular ion[END_REF] and of the radiative charge transfer in atom-ion collisions, for elementary systems such as Li(2s) + H + [START_REF] Stancil | Radiative Charge Transfer in Collisions of Li with H +[END_REF]. In the following we only recall the main steps, and a full description can be found in the given references and the references therein.

In the collisions between ground-state Alke + (ns) ions (n = 4, 5, 6, 6 for Ca, Sr, Ba, Yb, respectively) and Rb(5s), or between Yb + (6s) and Li(2s) atoms, the entrance channel is associated with a single adiabatic Born-Oppenheimer (BO) potential energy curve (PEC) V i (R) of 1 Σ + symmetry which actually corresponds to the first excited electronic state A 1 Σ + of the [Alk-Alke] + molecular ion Fig. (4a). Note that this may not be the case for this class of mixed species, as for instance in LiCa + and LiBa + where the Alke + (ns) + Li(2s) dissociation limit is the lowest one. The spontaneous emission during the collision thus leads either to the dissociation continuum (for RCT) or to the bound rovibrational level manifold (for RA) of the PEC V f (R) of the [Alk-Alke] + ground electronic state X 1 Σ + correlated to the lowest Alke ns 2 , 1 S + Alk + asymptote. Note that below the entrance channel there is an additional asymptotic limit only in RbCa + , namely Rb + + Ca 4s4p, 3 P , to which the PEC lowest 3 Π state is correlated and which is coupled to the A state by spin-orbit interaction, thus inducing non-adiabatic transition, i.e. NRCT [START_REF] Tacconi | Computing charge-exchange cross sections for Ca + collisions with Rb at low and ultralow energies[END_REF]. This feature is not treated here. Also, as illustrated for instance in our previous work on alkali-metal atom and Sr + compounds [START_REF] Aymar | Structure of the Alkali-metal-atom-Strontium molecular ions: towards photoassociation and formation of cold molecular ions[END_REF], the lowest 3 Σ + PEC is correlated to the entrance channel Alke + (ns) + Alk(n s), but cannot give rise to spontaneous emission to the X state, as long as no second-order spin-orbit coupling is introduced. The hyperfine structure of the colliding partnerswhich has been found to be an important feature in Ref. [START_REF] Zipkes | Cold Heteronuclear Atom-Ion Collisions[END_REF][START_REF] Zipkes | Cold Heteronuclear Atom-Ion Collisions[END_REF] -is neglected, so that the entrance channel A 1 Σ + is assumed to be populated with a statistical weight of g = 1/4.

Following our previous investigation [START_REF] Ayouz | Formation of H -3 by radiative association of H 2 and Hin the interstellar medium[END_REF] and similar ones (see for instance Ref. [START_REF] Gianturco | Radiative Association Rates and Structure of Resonances for Li and Li + Colliding with H and H +[END_REF][START_REF] Zygelman | Radiative quenching of He(2 1 S) induced by collisions with ground-state helium atoms[END_REF]), the total spontaneous emission cross section during the collision at energy E coll is formulated as the sum of two contributions from the RCT and RA processes formation of molecular ions by ra of cold trapped atoms and ions

σ RCT = g 8π 2 3C 3 1 k 2 i ∞ ∑ J=0 max f 0 ω 3 i, f J J -1, f D (R) | i , J 2 + ω 3 i, f (J + 1) J -1, f D (R) | i , J 2 d f (96) σ RA = g 8π 2 3C 3 1 k 2 i ∞ ∑ J=0 υ max ∑ υ=0 ω 3 i,J,υ,(J-1) J | J -1, υ| D (R) | i , J | 2 + ω 3 i,J,υ,(J-1) (J + 1) | J -1, υ| D (R) | i , J | 2 (97)
All the quantities in these equations are expressed in atomic units. In Eq. ( 96) the integral is limited to the largest possible energy max f in the exit channel, while in Eq. ( 97) the summation is limited to the uppermost vibrational level υ max of the X state. The initial ket | i , J with energy i = E coll -V i (R = ∞) and the associated wavenumber k i = 2µ i (µ is the reduced mass of the Alk-Alke + system) represents a partial-wave component J of an energy-normalized continuum wavefunction of the two nuclei. As only 1 Σ + states are involved, J is the total angular momentum in the entrance channel, and J = J ± 1 are the two possible allowed total angular momentum quantum numbers in the exit channel. In both equations above, the summation on J is actually limited for each collisional energy to the maximal value for which the induced rotational barrier in the entrance channel prevents the collision to occur. The transition electric dipole moment (TEDM) function D(R) is represented in Fig. (4c) for each system.

For RCT, the final state is represented by

| f , J ± 1 with energy f = E coll -V f (R = ∞) -hω if , where ω if (or, 2πhν if /2π
) is the energy of the emitted photon. Eq. ( 96) contains the transition dipole moment between | i , J and | f ,

J J , f |D(R)| i , J = ∞ 0 F f J ( f , R)D(R)F i J ( i , R)dR (98)
For RA, the final state |v, J is a bound rovibrational level of the X 1 Σ + ground electronic state of the [Alk-Alke] + molecular ion. The corresponding transition dipole moment is

J , υ|D(R)| i , J = ∞ 0 χ f υ,J (R)D(R)F i J ( i , R)dR (99)
where, χ υ is a radial rovibrational wave function of the molecule normalized to unity.

Our calculations cover an energy range in the entrance channel from 1 mK up to 80 mK, currently accessible in most experiments. This requires to consider J values in the range [15;40], [20;50], [20;50], [20;55], [5;15] [5;20] for RbCa + , RbSr + , RbBa + , RbYb + , LiYb + and LiCa + respectively, in order to obtain converged cross sections.

The upper limit max f = 1000 cm -1 of the relative energy in the integral of Eq. ( 96) and the integration step d f = 1 cm -1 are taken to ensure the integral convergence.

Anticipating on the results presented below, Fig. (3a) illustrates this well-known feature in the case of RbSr + , where examples of radial wave functions are drawn for three collision energies: (i) at 0.08 cm -1 , above the centrifugal barrier; (ii) at 0.015 cm -1 , showing a major tunneling through the barrier, (iii) at 0.01 cm -1 , with no more tunneling. The last two cases correspond to the Wigner threshold law energy range, and the related partial cross sections reported in Fig. (3b) present an energy dependence proportional to k 2 as expected for high partial wave value associated with a R -4 long-range interaction. The wavefunction of type (ii) above gives rise to sharp shape resonances with widths from a fraction of µK up to about 4 mK, requiring a very small energy step (2 µK) to locate them. The first case depicts a regime similar to the Langevin one, with the same k -1 energy dependence of the cross-section at high energy in every partial-wave, but with a magnitude smaller than the one predicted by the Langevin model, as discussed below. Note also the alternation of the occurrence of the shape resonances: if a shape resonance shows up for the partial wave , another also occurs for the + 2 partial-wave as demonstrated by B. Gao in the case of a R -4 long range interaction [START_REF] Gao | Quantum-defect theory for -1/r 4 -type interactions[END_REF].

electronic structure calculations for [alk-alke] + molecular ions

The potential energy curves, permanent electric dipole moments (PEDMs) and transition electric dipole moments are computed following the same method described in details in Ref. [START_REF] Aymar | Structure of the Alkali-metal-atom-Strontium molecular ions: towards photoassociation and formation of cold molecular ions[END_REF][START_REF] Aymar | Calculation of accurate permanent dipole moments of the lowest 1,3 Σ + states of heteronuclear alkali dimers using extended basis sets[END_REF][START_REF] Aymar | Comment on Calculation of accurate permanent dipole moments of the lowest 1,3 Σ + states of heteronuclear alkali dimers using extended basis sets[END_REF]. We briefly recall here the main steps of the calculations, carried out using the Configuration Interaction by Perturbation of a Multiconfiguration Wave Function Selected Iteratively (CIPSI) [START_REF] Huron | Iterative perturbation calculations of ground and excited state energies from multiconfigurational zeroth-order wavefunctions[END_REF] package developed at "Laboratoire de Chimie et Physique Quantiques, Université Toulouse III -Paul Sabatier" (France). The Alk + Alke + systems are modeled as molecules with two valence electrons moving in the field of the Alk + and Alke 2+ ions represented with ECPs including relativistic scalar effects. These ECPs are taken from Refs. [START_REF] Durand | New atomic pseudopotentials for electronic structure calculations of molecules and solids[END_REF][START_REF] Durand | A theoretical method to determine atomic pseudopotentials for electronic structure calculations of molecules and solids[END_REF] for all species but Sr 2+ and Ba 2+ [START_REF] Fuentealba | Pseudopotential calculations for alkaline-earth atoms[END_REF][START_REF] Fuentealba | Pseudopotential calculations on the ground state of the alkaline-earth monohydride ions[END_REF] and Yb 2+ [START_REF] Wang | Fine-structure predissociation of ultracold photoassociated 39 K 2 molecules observed by fragmentation spectroscopy[END_REF]. The ECPs are complemented with core polarization potentials (CPPs) depending on the orbital angular momentum of the valence electron of Rb (or Li) and Alke + [START_REF] Müller | Treatment of intershell correlation effects in ab initio calculations by use of core polarization potentials. Method and application to alkali and alkaline earth atoms[END_REF][START_REF] Foucrault | Nonpertubative method for corevalence correlation in pseudopotential[END_REF], and parametrized with the Rb + (or Li + ) and Alke 2+ static polarizabilities α and two sets of three cut-off radii ρ s , ρ p , and ρ d . All these parameters (except for Yb + ) were determined in our previous works on alkali-atom-strontium neutral molecules [START_REF] Guérout | Ground state of the polar alkali-metalatom-strontium molecules: Potential energy curve and permanent dipole moment[END_REF] and on alkaline-earth hydride ions [START_REF] Aymar | The electronic structure of the alkaline-earth-atom (Ca, Sr, Ba) hydride molecular ions[END_REF]. The value for the Yb 2+ polarizability is α = 6.388 a.u. [START_REF] Dzuba | Dynamic polarizabilities and related properties of clock states of the ytterbium atom[END_REF], and the cut-off radii ρ s = 1.8869 a.u., ρ p = 0.89235 a.u., and ρ d = 2.051150 a.u. have been adjusted to reproduce the experimental energies of the f 14 6s, 4 f 14 6 p and 4 f 14 5d states of the Yb + ion [START_REF] Yu | NIST Atomic Spectra Database (version 4.1) [Online[END_REF]. Only the remaining two valence electrons are used to calculate the Hartree-Fock and the excitation determinants, in an atom-centered Gaussian basis set, through the usual self-consistent field methodology. The basis set used for the Rb and Li atoms are from Ref. [START_REF] Aymar | Calculation of accurate permanent dipole moments of the lowest 1,3 Σ + states of heteronuclear alkali dimers using extended basis sets[END_REF][START_REF] Aymar | Comment on Calculation of accurate permanent dipole moments of the lowest 1,3 Σ + states of heteronuclear alkali dimers using extended basis sets[END_REF] and from Ref. [START_REF] Aymar | The electronic structure of the alkaline-earth-atom (Ca, Sr, Ba) hydride molecular ions[END_REF][START_REF] Bouissou | A comprehensive theoretical investigation of the electronic states of Ca2 up to the Ca(4s(2) 1S) + Ca(4s5p 1P) dissociation limit[END_REF] for the Alke + ions (except for Yb + ). We used for Yb + a large uncontracted (5s5 p6d ) Gaussian basis set with the series of exponents (0.785942, 0.370489, 0.068843, 0.031984, 0.015622), (0.685598, 0.338376, 0.073598, 0.034232, 0.015836) and (1.377363, 0.785942, 0.492827, 0.1680780, 0.058334, 0.02) for s, p, and d orbitals, respectively. It is worthwhile to note that to our knowledge this is the first time that the Yb + ion is represented in a so-called "large core" model, apart from Ref. [START_REF] Wang | Fine-structure predissociation of ultracold photoassociated 39 K 2 molecules observed by fragmentation spectroscopy[END_REF].

A full configuration interaction (FCI) is finally achieved to obtain the relevant PECs and TEDMs displayed in Fig. ( 4), limited to the X and A states and their transition. The origin of energies is taken at the Alke + (ns) + Rb(5s) -or Yb + (6s) + Li(2s) -disso- ciation limit corresponding to the entrance channel relevant for the present study. By construction of our approach, the energies of the Rb(5s), Li(2s), and Yb + (6s) ground levels are adjusted to the experimental values. Thus the dissociation energy matches the experimental one in the former case, while for the latter case it results from the CI calculation performed on the neutral Alke atom. These calculated Alke ground-state ionization energies are reported in Ref. [START_REF] Aymar | The electronic structure of the alkaline-earth-atom (Ca, Sr, Ba) hydride molecular ions[END_REF] for all atoms except Yb. In this latter case, we obtained an ionization energy of 50330 cm -1 , smaller by 113 cm -1 compared to the experimental one (50443 cm -1 [START_REF] Yu | NIST Atomic Spectra Database (version 4.1) [Online[END_REF]). This discrepancy of 113 cm -1 also holds for the spacing between the Yb + (6s) + Alk(ns) and the Yb(6s 2 , 1 S) + Alk + asymptotes (equal to 16864.8 cm -1 in our method, in good agreement with the experimental value of 16752.2 cm -1 [START_REF] Yu | NIST Atomic Spectra Database (version 4.1) [Online[END_REF]). It favorably compares to the discrepancy of 322 cm -1 reported in Ref. [START_REF] Lamb | Structure and interactions of ultracold Yb ions and Rb atoms[END_REF], and to Ref. [START_REF] Sayfutyarova | Charge transfer in cold Yb + +Rb collisions[END_REF] which reported an energy difference of 16279 cm -1 between these two asymptotes. Finally, in their work on LiYb + [START_REF] Tomza | Cold interactions between an Yb + ion and a Li atom: Prospects for sympathetic cooling, radiative association, and Feshbach resonances[END_REF] the authors report an ionization energy for the Yb ground state of 50267 cm -1 .

All PECs for the A state consistently behave in the same way at large distances (Fig. 4a), varying as -C 4 /(2R 4 ) determined by the Rb (or Li) static polarizability α ≡ 2C 4 . A rough fit with a single parameter of the long-range part of the PECs yields 2C 4 (Rb) ≈ 360 ± 20 a.u. (depending on the RbAlke + species) and 2C 4 (Li) ≈ 192 atomic units. Note that a more relevant determination using our quantum chemistry approach above, in the context of the finite field method, yields 2C 4 (Rb) = 318 a.u. and 2C 4 (Li) = 164 a.u. [200] in good agreement with the various determinations of the Rb static polarizability [START_REF] Derevianko | Electric dipole polarizabilities at imaginary frequencies for the alkali-metal, alkaline-earth, and inert gas atoms[END_REF]. The potential well of this state results from a strongly avoided crossing with the lowest X 1 Σ + PEC, which is located at about the same distance range for all species.

The well-depth decreases with increasing reduced mass along the [RbAlke] + series.

As also observed in previous works [START_REF] Krych | Sympathetic cooling of the Ba + ion by collisions with ultracold Rb atoms: Theoretical prospects[END_REF][START_REF] Knecht | Large-scale parallel configuration interaction. I. Nonrelativistic and scalar-relativistic general active space implementation with application to RbBa+[END_REF][START_REF] Knecht | Accurate calculations of the ground state and low-lying excited states of the (RbBa) + molecular ion: a proposed system for ultracold reactive collisions[END_REF], the RbBa + PEC exhibits a double well induced by an avoided crossing with an upper 1 Σ + curve not displayed here. This feature is imprinted in the variation of the corresponding A-X TEDM Fig. (4c). Surprisingly, the PEC of RbCa + and RbYb + are very similar, which could be related to the well-known f -shell contraction in the lanthanide atoms, yielding to the Yb the character of a much smaller atom than expected from its mass [START_REF] Cowan | The theory of atomic structure and spectra[END_REF]. In contrast, the A potential well in LiYb + is significantly less deep than for the [RbAlke] + series due to the low value of the C 4 long-range coefficient.

The PECs for the X state all look very similar along the [RbAlke] + series Fig. (4b). Due to the smaller size of the Li + ion compared to the Rb + one, the minimum of the well for LiYb + is located at quite short distances, while its depth is larger than for the Rb-Alke + compounds. As above, the long-range part of the PEC varies as R -4 , with a coefficient depending on the static polarizability of the (Alke) neutral species. Again, a simple fit of this long-range part with a single parameter gives 2C 4 ≈ 140 a.u., 220 a.u., 300 a.u., 154 a.u., and 154 a.u., for RbCa + , RbSr + , RbCa + , RbYb + and LiYb + respectively, in reasonable agreement with the static polarizabilities of Ca, Sr, Ba, and Yb from our previous calculations [START_REF] Aymar | The electronic structure of the alkaline-earth-atom (Ca, Sr, Ba) hydride molecular ions[END_REF] or from [START_REF] Derevianko | Electric dipole polarizabilities at imaginary frequencies for the alkali-metal, alkaline-earth, and inert gas atoms[END_REF]. Note again that to determine more rigorously these quantities within our quantum chemistry approach, a method like the finite-field approach should be used, as we have done for instance in the Sr case [START_REF] Aymar | Structure of the Alkali-metal-atom-Strontium molecular ions: towards photoassociation and formation of cold molecular ions[END_REF].

The main spectroscopic constants of these PECs are deduced from a parabolic fit of the bottom of the wells around their minimum, and are reported in Tables [START_REF] Bradley | Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions[END_REF] and (3), allowing for comparison with the few other theoretical determinations available in the literature. The choice of isotopologues is somewhat arbitrary, as it has no significant influence on the present scattering calculations, as it will be discussed later.

There is only one other work reporting about the molecular structure of RbCa + [START_REF] Tacconi | Computing charge-exchange cross sections for Ca + collisions with Rb at low and ultralow energies[END_REF][START_REF] Belyaev | Resonances in Ca + + Rb nonadiabatic collisions at ultralow energies[END_REF] for the study of NRCT. The authors used a so-called "small core" ECPs where electrons from the 4s and 4p shells in Rb, and from the 3s and 3p shells in Ca + are explicitly treated besides the valence electrons. As for the 1 Σ + states, the authors only provided figures for the PECs from which we estimated equilibrium distances and potential well depths in good agreement with our results. This is a strong indication in favor of the ability of our approach employing large-core ECPs to represent corevalence electronic correlation.

The RbBa + molecule already attracted several detailed non-relativistic studies which are in quite good agreement among each other. We observe the same trend than for ). The origin of energies is taken at the dissociation of the A 1 Σ + PEC for all systems. The lowest 3 Π PEC of Rb-Ca + is also displayed for completeness. (c) Transition electric dipole moments between the X and A states. Open circles:

Ref. [START_REF] Mclaughlin | Ultracold, radiative charge transfer in hybrid Yb ion and Rb atom traps[END_REF]. The vertical dotted lines tag the inner turning points of the incoming continuum wave function -in the entrance channel -with the respective points in the exit channel and the transition dipole moment magnitude at that position.

RbCa + , i.e. for both X and A states we find slightly smaller equilibrium distances, and slightly deeper potential wells than in Ref. [START_REF] Krych | Sympathetic cooling of the Ba + ion by collisions with ultracold Rb atoms: Theoretical prospects[END_REF][START_REF] Knecht | Large-scale parallel configuration interaction. I. Nonrelativistic and scalar-relativistic general active space implementation with application to RbBa+[END_REF][START_REF] Knecht | Accurate calculations of the ground state and low-lying excited states of the (RbBa) + molecular ion: a proposed system for ultracold reactive collisions[END_REF]. In Ref. [START_REF] Krych | Sympathetic cooling of the Ba + ion by collisions with ultracold Rb atoms: Theoretical prospects[END_REF] the authors investigated the role of the triple excitations in their approach, yielding a double-well potential for the A state with a barrier located below the dissociation limit at about the same position (estimated from their figure) than in our calculation. The TEDM between the X and A states display a very similar shape with two maxima in both our results and in those of Ref. [START_REF] Krych | Sympathetic cooling of the Ba + ion by collisions with ultracold Rb atoms: Theoretical prospects[END_REF], reflecting the double-well structure of the A PEC. The largest value of d A m =2.84 a.u. is reached at a distance of R A m =13 a. u. in our work, compared to d A m ≈2.65 a.u. at R A m =13.5 a.u. in Ref. [START_REF] Krych | Sympathetic cooling of the Ba + ion by collisions with ultracold Rb atoms: Theoretical prospects[END_REF]. Again, these are strong arguments in favor of the present approach based on ECP and CPP to represent core-valence correlation effects with the same quality than in other methods.

For RbSr + , there is no other calculation available than [START_REF] Aymar | Structure of the Alkali-metal-atom-Strontium molecular ions: towards photoassociation and formation of cold molecular ions[END_REF], but given the favorable comparisons above, and as this ion is treated consistently with the same method than the three other species, the present results are probably of satisfactory quality.

As already quoted above, the cases of RbYb + and LiYb + are peculiar, as the Yb + ion and Yb atom have not been treated with large ECP before. Tables [START_REF] Bradley | Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions[END_REF] and ( 3) reveal a contrasted situation for RbYb + in comparison with the results of Refs. [START_REF] Lamb | Structure and interactions of ultracold Yb ions and Rb atoms[END_REF][START_REF] Sayfutyarova | Charge transfer in cold Yb + +Rb collisions[END_REF][START_REF] Mclaughlin | Ultracold, radiative charge transfer in hybrid Yb ion and Rb atom traps[END_REF] that are all obtained with similar approaches based on small core ECPs. Our values for R e and D e of the X PEC are in good agreement with those of Ref. [START_REF] Sayfutyarova | Charge transfer in cold Yb + +Rb collisions[END_REF], while the potential well depth from Refs. [START_REF] Lamb | Structure and interactions of ultracold Yb ions and Rb atoms[END_REF][START_REF] Mclaughlin | Ultracold, radiative charge transfer in hybrid Yb ion and Rb atom traps[END_REF] is twice smaller than ours, with a minimum located at a significantly larger distance. This has been quoted by the authors of Ref. [START_REF] Mclaughlin | Ultracold, radiative charge transfer in hybrid Yb ion and Rb atom traps[END_REF] but no explanation has been provided. On the other hand for the A state, the three previous papers have obtained results in good agreement among each other, while we find an equilibrium distance shorter by about 1 a.u. and the well depth deeper by about 400 cm -1 . Surprisingly the magnitude and the R variation of the TEDM of the X-A transition look very similar among all determinations including ours, with a maximal value of about 3 a.u. Fig. [START_REF] Adams | Atom interferometery[END_REF], suggesting that electronic wave functions behave in the same way in all methods.

The only other work about the LiYb + structure that we are aware of is the very recent article by Tomza et al. [START_REF] Tomza | Cold interactions between an Yb + ion and a Li atom: Prospects for sympathetic cooling, radiative association, and Feshbach resonances[END_REF], using the same approach than in Ref. [START_REF] Krych | Sympathetic cooling of the Ba + ion by collisions with ultracold Rb atoms: Theoretical prospects[END_REF] for RbBa + . The conclusions looks similar to the case of RbYb + and RbBa + for both the X and A PECs. The spectroscopic constants of the X PEC are in good agreement, with a smaller harmonic constant and a smaller potential well by about 5% in our results compared to those of Ref. [START_REF] Tomza | Cold interactions between an Yb + ion and a Li atom: Prospects for sympathetic cooling, radiative association, and Feshbach resonances[END_REF]. We found the depth of the A PEC about 30% larger, with a minimum located at a distance by about 1.15 a.u., and an harmonic constant smaller by about 3.5%, than the values obtained in Ref. [START_REF] Tomza | Cold interactions between an Yb + ion and a Li atom: Prospects for sympathetic cooling, radiative association, and Feshbach resonances[END_REF]. In contrast with these large differences, the present TEDM between the X and A states are in very good agreement with this latter work, as already observed for the other ions above.

For completeness we report in Fig. [START_REF] Adams | Atom optics[END_REF] the PEDMs for both the A and X states of these molecular ions. They have been computed with respect to the origin of coordinates placed at the center-of-mass of the molecule, with the axis oriented from the Rb or Li atom towards the Alke + ion. We note the total mass M = M (Alk + ) + M(Alke) and the mass difference δM = M (Alk + ) -M (Alke). We checked that at large dis- tances -beyond 20 a.u., typically the PEDMs linearly diverge as R/2(1 + δM/M) and -R/2(1 -δM/M) for the A and X states respectively, as the charge is carried either by the Alke species or by the Alk one. The PEDM of the A state in RbBa + has an abrupt change of slope around 12 a.u. due to the occurrence of the barrier between the two potential wells. Tomza et al. [START_REF] Tomza | Cold interactions between an Yb + ion and a Li atom: Prospects for sympathetic cooling, radiative association, and Feshbach resonances[END_REF] have displayed the absolute value of the X and A PEDMs in LiYb + , showing very similar variation to ours but with a slightly smaller amplitude.

cross sections and rates for ra and rct

The main objective of this study is to evaluate, through the comparison of five molecular ions treated consistently by the same approaches for both their electronic structure and their dynamics, the efficiency of the formation of molecular ions, and the possibility to observe shape resonances with the current experimental resolution. We performed about 40000 calculations per system in the collisional energy range between 100 µK to 80 mK, for all the total rotational quantum numbers between 0 and 80. The energy step was fixed to 2 µK in order to properly locate all the shape resonances.

Figure [START_REF] Deutsch | Optical lattices[END_REF] displays the total cross sections (RA + RCT) for all five systems, which present several similar features. As anticipated above, they all present the same -1/2 i dependence with the entrance channel energy i which corresponds to the Langevintype regime. However the magnitude of the cross sections is much smaller than the one predicted by the original Langevin model [207]

σ lang = gP lang π 2C 4 i ( 100
)
with the statistical factor g = 1/4 and a reaction probability P lang following the capture equal to 1. As the baseline of our calculated cross sections (i.e. ignoring shape resonances) varies as Table 2: Main spectroscopic constants for the X 1 Σ + electronic state of RbAlke + and LiYb + : the bond length, r e (a. u.), the harmonic constant, ω e (cm -1 ), the potential well depths, D e (cm -1 ), the rotational constant, B e (cm -1 ). Other published values are also provided when available. Two sets of results were published in Ref. [START_REF] Knecht | Accurate calculations of the ground state and low-lying excited states of the (RbBa) + molecular ion: a proposed system for ultracold reactive collisions[END_REF], (a) using a multireference configuration interaction (MRCI), and (b) using the coupled-cluster method. The * symbol for Ref. [START_REF] Tomza | Cold interactions between an Yb + ion and a Li atom: Prospects for sympathetic cooling, radiative association, and Feshbach resonances[END_REF] indicates that the authors reported spectroscopic constants for the 7 Li 172 Yb + isotope. Table 3: Same as Table (2) for the A 1 Σ + electronic states of RbAlke + and LiYb + . In addition the transition energy T e (cm -1 ) between the bottom of the A and X potential wells is reported. For RbBa + , values for the inner and outer wells are displayed on the same line. The position of the top of the barrier, and its energy below the dissociation limit, are also indicated under the r e and D e columns. The * symbol for Ref. [START_REF] Tomza | Cold interactions between an Yb + ion and a Li atom: Prospects for sympathetic cooling, radiative association, and Feshbach resonances[END_REF] indicates that the authors reported spectroscopic constants for the 7 Li 172 Yb + isotope.

2.2 × 10 -5 , 3.0 × 10 -5 , 3.9 × 10 -5 , 5.9 × 10 -5 and 2.0 × 10 -6 for LiYb + , RbBa + , RbSr + , RbCa + , RbYb + and LiCa + , respectively. Numerous narrow shape resonances arising from the entrance channel can be observed in each system, with a density increasing with the reduced mass. In this respect, only a few resonances are predicted in LiYb + due to its comparatively small reduced mass. The resonances have obviously energy positions which dramatically depend on the calculated potential. By comparing the baseline above to a -1/2 i fit of the total cross-sections, we estimate the contribution of the shape resonances to about 20±2 % for all these species over the displayed energy range. Apart from the resonances, the hierarchy of the cross sections reflected by their baselines mostly reflects the increasing magnitude of the ω 3 if factor in Eq. ( 96) and ( 97) along the series LiYb + , RbBa + , RbCa + , RbSr + , and RbYb + which can be roughly appreciated from the relative position in energy of the relevant turning points in the X and A PECs Fig. [START_REF] Adams | Atom interferometery[END_REF].

The RA process is predicted to dominate for all systems in comparison with RCT, by a factor ≈ 2.4 for RbCa + , ≈ 3.8 for RbSr + , ≈ 13.6 for RbBa + , ≈ 2.1 for RbYb + , and ≈ 22.4 for LiYb + Fig. [START_REF] Morinaga | Holographic manipulation of a cold atomic beam[END_REF]. This is due to the favorable relative position of the X and A potential wells which are shifted against each other in R, so that the classical inner turning point of the A potential is aligned with the one of quite deeply bound levels of the X potential -see dotted lines in Fig. [START_REF] Adams | Atom interferometery[END_REF]. The very similar behavior between RbCa + and RbYb + reflects the strong similarities of the PECs and TEDMs functions. The RA process is even more favored in RbBa + due to the double well in the A potential curve, which enhances the spontaneous emission towards X bound levels around the top of the barrier where the TEDM is large. In all cases, the shape resonances in the RA and RCT cross sections are the same, as they are due to the centrifugal barrier in the entrance channel (see also Refs. [START_REF] Sayfutyarova | Charge transfer in cold Yb + +Rb collisions[END_REF][START_REF] Belyaev | Resonances in Ca + + Rb nonadiabatic collisions at ultralow energies[END_REF]). In the framework of the present model, the choice of the isotopologues for each ionic species does not yield significantly different cross sections in magnitude, as the reduced mass (which differs by less than one percent among a series of isotopologues of a given ion) is involved only through the kinetics described by the radial wave functions. Sayfutyarova et al. reached the same conclusion for isotopologues of RbYb + [START_REF] Sayfutyarova | Charge transfer in cold Yb + +Rb collisions[END_REF], as well as Tomza et al. for LiYb + [START_REF] Tomza | Cold interactions between an Yb + ion and a Li atom: Prospects for sympathetic cooling, radiative association, and Feshbach resonances[END_REF]. But of course, the energy positions of the shape resonances will vary with the chosen isotopologue.

In Fig. [START_REF] Jin | Collective Excitations of a Bose-Einstein Condensate in a Dilute Gas[END_REF] are presented the vibrational distributions of the ground state vibrational levels υ X produced by RA for all species, in the J = 1 case and i = 0.1 mK as a representative example. The fraction of population of a given vibrational level υ X of the X state is defined as the ratio of the matrix element in Eq. ( 97) for J = 1 and υ = υ X , divided by the sum over all vibrational levels of the X state, for J = 1, of the matrix elements. As expected, these distributions are similar for the three ions RbCa + , RbSr + , and RbYb + , due to their similar electronic structure. The peak in the distributions fulfills the Franck-Condon principle as it is located at a transition energy close to the energy difference between the inner classical turning point of the A potential (located in the 10-12 a.u. range) and the outer turning point of the X potential -see vertical dotted lines in Fig. [START_REF] Adams | Atom interferometery[END_REF]. In contrast, the double-well pattern in RbBa + is manifested by a distribution extended to much lower vibrational levels than for the other species, due to the inner turning point of the A potential located around 8 atomic units. The vibrational distribution is double-peaked as well, the peak at low υ X (resp. high υ X ) corresponding to the energy difference between the inner classical turning point of the A potential and the inner (resp. outer) turning point of the X potential. The peak at low υ X is small however, due to the small magnitude of the TEDM in the corresponding distance range -see Fig. (4c). Finally, the low reduced mass of LiYb + is reflected in the low density of vibrational levels in the X state compared to the species above. But the trend is similar, i.e. only the uppermost levels are expected to be populated by RA. Note that the overall shape of the vibrational distribution looks the same than in Ref. [START_REF] Tomza | Cold interactions between an Yb + ion and a Li atom: Prospects for sympathetic cooling, radiative association, and Feshbach resonances[END_REF], with slightly shifted extrema due to the obtained differences in the potential well depths. The molecular ions are all expected vibrationally hot, as well as rotationally hot given the number of partial waves involved in the present energy range, even around 1 mK where many partial waves are still involved. Due to their non-vanishing PEDM Fig. [START_REF] Adams | Atom optics[END_REF] they can radiatively decay in principle down two lower vibrational levels. We computed the radiative lifetime of a given level υ for decaying down to the level υ -1 for all systems, and found that it always exceeds 10 s because of the slow R variation of the PEDM.

In order to mimic the measured rate constant behavior as a function of the collision energy, the total cross sections (RA + RCT) above must be convoluted with a reliable energy distribution. Such distributions have been modeled using molecular dynamics simulations -see Ref. [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF][START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF], reflecting the dominant role of the micromotion of the trapped ions which is much faster than the thermal motion of the ultracold atoms. These energy distributions reflect both the fact that ions are located at well-defined places in the ion trap, and that their micromotion strongly depends on the ion number and thus on their places. This results into quite broad distributions which completely smooth out the shape resonances. Nevertheless, new experimental developments are in progress in various experimental groups aiming at better controlling the relative velocity of the ion and the atom. In order to illustrate the required width which would allow to observe the predicted shape resonances, we convolved the RbSr + computed rates with Gaussian distributions of half-width varying from 1 mK to 5 mK Fig. [START_REF] Burger | Dark Solitons in Bose-Einstein Condensates[END_REF]. We see that a width of about 2 mK would be acceptable. We extended this result to all studied species Fig. [START_REF] Mewes | Output Coupler for Bose-Einstein Condensed Atoms[END_REF], leading to the same statement. As expected from the -1/2 i variation of the cross sections, the obtained rates are almost constant over the 1-80 mK collision energy range -see the dashed lines in Fig. [START_REF] Mewes | Output Coupler for Bose-Einstein Condensed Atoms[END_REF]. Their relative magnitude still reflects the change in the ω 3 if factor invoked earlier for the cross section. For instance, this factor is larger by a magnitude of almost 8 in RbYb + compared to LiYb + , which can be retrieved in the average level of the rates. It is worthwhile to note that Tomza et al. [START_REF] Tomza | Cold interactions between an Yb + ion and a Li atom: Prospects for sympathetic cooling, radiative association, and Feshbach resonances[END_REF] reported a thermally averaged RA rate almost 100 times larger than the RCT one around 10 mK, and a total rate of about 1.5 × 10 -14 cm 3 /s, namely about 4 times larger than ours. This latter issue may be due to the statistical factor of 1/4 for populating the singlet state in the entrance channel, which is not mentioned in Ref. [START_REF] Tomza | Cold interactions between an Yb + ion and a Li atom: Prospects for sympathetic cooling, radiative association, and Feshbach resonances[END_REF].

discussion and prospects

We have computed in a consistent way the electronic structure of a series of molecular ions involved in ongoing hybrid experiment merging cold ion and cold atom traps, as well as their radiative emission rate leading either to the formation of cold molecular ions, or to charge exchange. For the first time, molecular ions involving Yb + or Yb have radiative association, as a function of the binding energies and the corresponding vibrational quantum numbers, v X , for the rotational quantum number J = 1, at i = 0.1 mK. Note that the vibrational distribution is the same over the displayed energy range. For comparison purpose, the energy scale is common for all species, and thus vibrational indexes are shifted from one species to the other.

formation of molecular ions by ra of cold trapped atoms and ions been modeled using effective large-core potentials completed with a core polarization potential, in the framework of a full configuration interaction approach. Such a simple description provides results for the electronic structure which are generally in good agreement with other determinations, when available. Results for radiative emission rate are computed for the first time for the ionic species RbSr + mixing an ion and an atom of close mass. The consistent calculation of rates for a series of molecular ions allows us to infer a uniform quality and accuracy over the entire series, which is of great help for comparison with experimental results, as discussed below.

In the experiments quoted in the introduction, the rate measurements are based on the observation of the decreasing of the number of trapped laser-cooled ions with time, due to all possible processes. In some cases, a careful analysis of the data allows for extracting a rate for a particular process. Before discussing the comparison with the published experimental results, it is first worthwhile to check the consistency among all theoretical studies focusing on the radiative processes (RA and RCT) occurring during the collision between two ground state particles.

The dashed lines in Fig. [START_REF] Mewes | Output Coupler for Bose-Einstein Condensed Atoms[END_REF] indicates the average level of the calculated total radiative rates (RA + RCT), yielding 3.1 × 10 -14 cm 3 /s, 1.9 × 10 -14 cm 3 /s, 1.2 × 10 -14 cm 3 /s, 3.1 × 10 -14 cm 3 /s, and 0.31 × 10 -14 cm 3 /s, for RbCa + , RbSr + , RbBa + , RbYb + , and LiYb + , respectively. The remarkably close values of these rates along the species is well understood from the similarities of their electronic structure invoked earlier, while their differences play only a minor role. This is exemplified by the results from Ref. [START_REF] Sayfutyarova | Charge transfer in cold Yb + +Rb collisions[END_REF][START_REF] Mclaughlin | Ultracold, radiative charge transfer in hybrid Yb ion and Rb atom traps[END_REF] of 2.9 × 10 -14 cm 3 /s for the RA + RCT rate in RbYb + , where the large difference of the A potential well compared to the present one only slightly influences the final rate, as the main contribution is controlled by the inner classical turning point of the PEC. The magnitude of the present rates is also consistent with the smaller theoretical rate of 1.5 × 10 -15 cm 3 /s computed for the same processes in Ca-Yb + mixture [START_REF] Zygelman | Cold ion-atom chemistry driven by spontaneous radiative relaxation: a case study for the formation of the YbCa + molecular ion[END_REF], as the energy difference δR t X-A between the inner classical turning point of the A 2 Σ + potential curve in the entrance channel, and the outer classical turning point of the X 2 Σ + potential curve in the exit channel is smaller (about 9000 cm -1 ) than in the present series of species. Similarly, Makarov et al. modeled the same processes in Na-Ca + , and obtained a total rate of 2.3 × 10 -16 cm 3 /s which reflects the relatively small difference δR t X-A ≈ 10700 cm -1 (similar to the situation of RbBa + and LiYb + ), combined with a significantly smaller TEDM of about 0.6 a.u. compared to the present cases. 6) with a Gaussian energy distribution of 2 mK half-width. A linear fit of this curves yields an estimate of the average magnitude of the rates (dashed lines). For RbYb + , the calculated rate obtained in Ref. [START_REF] Sayfutyarova | Charge transfer in cold Yb + +Rb collisions[END_REF][START_REF] Mclaughlin | Ultracold, radiative charge transfer in hybrid Yb ion and Rb atom traps[END_REF] is indicated with a dot-dashed line.

Rate constant (×10 Collision energy (mK) @1.0 mK @2.0 mK @3.0 mK @4.0 mK @5.0 mK As visible from Fig. (4a) the case of RbCa + is peculiar, due to the presence of the b 3 Π state dissociating to the Rb + + Ca 4s4p, 3 P limit which is energetically open to the entrance channel. Tacconi et al. [START_REF] Tacconi | Computing charge-exchange cross sections for Ca + collisions with Rb at low and ultralow energies[END_REF] modeled the non-adiabatic coupling related to the spin-orbit interaction between the b 3 Π state and the A 1 Σ + state, thus inducing non-radiative charge transfer (NRCT) which is predicted to dominate the dynamics of the charge exchange process. We note that this process cannot lead to subsequent radiative formation of molecular ions due to spin selection rule. In a further work [START_REF] Belyaev | Resonances in Ca + + Rb nonadiabatic collisions at ultralow energies[END_REF], the NRCT cross sections are found to approximately vary as expected by the Langevin model as -1/2 i , and to exhibit shape resonances of the same kind than those expected in the RA and RCT cross sections, due to the centrifugal barrier in the entrance channel. Over the present energy range, the NRCT cross section, is evaluated to be about 200 times larger than the total radiative cross section [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF]. Around 2 K the NRCT rate is found of the same order of magnitude (3.5 × 10 -12 cm 3 /s) than the experimental rate, conservatively estimated of the order of 2 to 3 × 10 -12 cm 3 /s [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF]. The hypothesis for NRCT has been also invoked in the Ca-Yb + system, with a rate evaluated to a few 10 -14 cm 3 /s [START_REF] Rellergert | Measurement of a Large Chemical Reaction Rate between Ultracold Closed-Shell 40 Ca Atoms and Open-Shell 174 Yb + Ions Held in a Hybrid Atom-Ion Trap[END_REF].

As for the Rb-Yb + system, the experimental rates measured in Ref. [START_REF] Zipkes | Cold Heteronuclear Atom-Ion Collisions[END_REF][START_REF] Zipkes | Cold Heteronuclear Atom-Ion Collisions[END_REF] are 3 ± 1 × 10 -14 cm 3 /s and 4.5 ± 1.5 × 10 -14 cm 3 /s, corresponding to the 172 Yb + and the 174 Yb + isotope, respectively, and when Rb is prepared in the (F = 2, M F = 2) hyperfine level (F being the total angular momentum including nuclear spin and M F its projection on the magnetic field axis used for the preparation). The present computed rate as well as those of Refs. [START_REF] Sayfutyarova | Charge transfer in cold Yb + +Rb collisions[END_REF][START_REF] Mclaughlin | Ultracold, radiative charge transfer in hybrid Yb ion and Rb atom traps[END_REF] are thus in remarkable agreement within the error bars with the experimental results. It is noteworthy to quote that the same experimental group later discovered a huge effect of the initial preparation of the Rb atoms in a given hyperfine level, with a rate 35 times larger than the above one when Rb is in the (F = 1, M F = 1) hyperfine level [START_REF] Ratschbacher | Controlling chemical reactions of a single particle[END_REF]. This effect cannot be retrieved at the present level of the theoretical model which does not include the still unexplored dynamics of the numerous hyperfine channels. Moreover the present agreement between experiment and theory suggests that our computed values for LiYb + rates are correct as well.

The situation for the Rb-Ba + system is somewhat intermediate in comparison with the two previous cases above. Indeed, there is no possibility for NRCT, and only an upper bound of 5 × 10 -13 cm 3 /s is given for the experimental rate in Ref. [START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF], compatible with the present calculation but suggesting that it is at most 40 times larger than the calculated one. In another experiment with Rb atoms prepared in the (F = 1, M F = -1) state in an almost Bose-condensed sample, Schmid et al. [START_REF] Schmid | Dynamics of a Cold Trapped Ion in a Bose-Einstein Condensate[END_REF] estimated an inelastic cross section presumably due to radiative processes between ground state particles in the range of 10 -15 cm 2 to 10 -14 cm 2 at about 30 mK, corresponding to rate 3 to 30 times larger than the calculated one.

To be complete with such comparisons with experimental results, it is worthwhile to mention the exceptional case of Ca-Yb + -not interpreted yet -where a considerable rate for inelastic collision between ground state particles of ≈ 2 × 10 -10 cm 3 /s, i.e. of the same order of magnitude than quasi-resonant charge transfer reaction between an atom and an ion of the same species but different isotopes (Grier et al. [START_REF] Grier | Observation of Cold Collisions between Trapped Ions and Trapped Atoms[END_REF] measured a rate of ≈ 6 × 10 -10 cm 3 /s in Yb-Yb + ; see also Ref. [START_REF] Côté | Ultracold atom-ion collisions[END_REF][START_REF] Lee | Measurement of collisions between rubidium atoms and optically dark rubidium ions in trapped mixtures[END_REF]). Table 4: Energies (in cm -1 ) of the quoted atom-molecule complexes in their ground state with respect to an origin of energy taken when all three atomic fragments (one Alk atom, one Alk + ion, one Alke atom) are at infinity. Therefore the reported quantities corresponds to the well-depths of the relevant molecular ions (from the present work) or neutrals in their electronic ground state. The Rb + 2 well depth is taken from Ref. [START_REF] Aymar | Model-potential calculations for ground and excited Sigma states of Rb + 2 , Cs + 2 and RbCs + ions[END_REF], and the Li + 2 (experimental) one from Ref. [START_REF] Bernheim | Rydberg states of 7 Li 2 by pulsed optical-optical double resonance spectroscopy: Molecular constants of 7 Li + 2[END_REF].

Last but not least, beside this quite contrasted situation for the comparison between experimental rate for inelastic collisions and calculated rates for radiative processes, there is another puzzling and not yet understood feature in this kind of experiments: while the theoretical models all consistently predict that RA should dominate RCT, molecular ions are directly observed by mass spectrometry only in the experiments of Hall et al. on Rb-Ca + [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF] and Rb-Ba + [START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF]. An indirect detection of CaYb + has been also reported in Ref. [START_REF] Rellergert | Measurement of a Large Chemical Reaction Rate between Ultracold Closed-Shell 40 Ca Atoms and Open-Shell 174 Yb + Ions Held in a Hybrid Atom-Ion Trap[END_REF]. This statement strongly suggests that the dynamic in such merged cold atom-cold ion traps is much more complicated than anticipated. Several elements may have a significant impact on the interpretation of the experimental results:

• The laser light used for cooling and trapping atoms and ions is recognized to have a strong influence on the measured rates, as inelastic processes with excited species may be very large (see for instance Ref. [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF][START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF][START_REF] Sullivan | Role of Electronic Excitations in Ground-State-Forbidden Inelastic Collisions Between Ultracold Atoms and Ions[END_REF]). Furthermore, such -quite intense -light could prevent the produced molecular ions to be detected before being photodissociated. On the theoretical side, such light-assisted processes may be difficult to describe accurately, as much more collision channels are open due to the presence of numerous excited electronic states (see for instance Ref. [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF]) which are more sensitive to the quality of quantum chemistry calculations.

• The quoted experiment are achieved with various conditions of atomic densities: Rb atoms are either trapped in MOT with a quite low density of a few 10 9 atoms/cm 3 , or in almost Bose-condensed gases with density up to 10 12 atoms/cm 3 . Therefore collisions between surrounding Rb atoms and the formed molecular ions could well occur, and they are strongly exoergic for all systems studied here -see Table ( 4) -, assuming that there is no activation potential barrier in the triatomic collisional complex which would prevent the reaction to occur. One product would be Rb + 2 ions which are indeed observed in the Rb-Ca + experiment of Ref. [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF], and presumably present in the Rb-Ba + experiment of Ref. [START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF]. Note that the energy of a Rb + -RbAlke complex at zero collisional energy is higher than the one of a Rb-RbAlke + complex (with respect to their common dissociation limit, Rb + + Rb + Ca) due to the weak binding energy D e of the RbAlke molecule compared to the one of the related RbAlke + molecule. Therefore charge exchange reactions between a RbAlke + ion and a Rb atom is energetically forbidden.

• At high atomic density, three-body collisions could well occur, as it has been recently probed [START_REF] Härter | Single ion as a three-body reaction center in an ultracold atomic gas[END_REF][START_REF] Härter | Population distribution of product states following three-body recombination in an ultracold atomic gas[END_REF]. Such processes may contribute to the formation of molecular ions, as well as to their destruction if they are radiatively created. Experiments of Ref. [START_REF] Schmid | Dynamics of a Cold Trapped Ion in a Bose-Einstein Condensate[END_REF] on Rb-Ba + and of Ref. [START_REF] Zipkes | Cold Heteronuclear Atom-Ion Collisions[END_REF][START_REF] Zipkes | Cold Heteronuclear Atom-Ion Collisions[END_REF] on Rb-Yb + both use an almost Bose-condensed atomic sample and have not observed yet any molecular ions.

To summarize, the present work focused on the cold inelastic collisions inducing radiative decay between ground state Rb or Li atoms and a series of ground state ionic species (Ca + , Sr + , Ba + , Yb + ). The consistent treatment of these series of pairs with the same quantum chemistry approach and the same dynamical model allows for general conclusions about the magnitude of the rate constants, and the efficiency of the formation of cold molecular ions. Dynamical results for RbSr + are published here for the first time, as well as the modeling of a heavy diatomic compound containing Yb with a large core ECP. These results could stimulate future joint experimental and theoretical investigations. For instance, the detection of a couple of shape resonances in one of these systems would put strong constraints on its electronic structure. Photodissociation experiments of the produced molecular ions such like those performed on MgH + [START_REF] Bertelsen | Photodissociation of Cold MgH + ions: Towards rotational temperature measurements and controlled dissociation[END_REF] could also provide more insight on their spectroscopy, and on their vibrational distribution (see for instance Ref. [START_REF] Blangé | Vibrational state distribution of Na + ions created in ultracoldcollisions[END_REF]). The exploration of a family of such systems like in the present work where a Rb atom interacts with an ion of a given class allows for varying -discontinuously -a characteristic parameter of the system, its reduced mass, revealing then the changes in the dynamics while the nature of the interactions are similar over the entire series. The same study is already achievable on the series of Alk-Sr + (with Alk = Li, Na, K, Rb, Cs) species that we previously studied [START_REF] Aymar | Structure of the Alkali-metal-atom-Strontium molecular ions: towards photoassociation and formation of cold molecular ions[END_REF]. Based on the arguments developed in the present work, we can already draw some conclusions for this family:

• the LiSr + will not be favorable for the formation of deeply-bound molecular ion.

Indeed, despite a noticeable TEDM of about 2 a.u., the A PEC of the entrance channel -which is almost repulsive -has an inner turning point which coincides in position with the outer turning point of the uppermost vibrational levels of the X PEC. Moreover, the energy difference δR t X-A is quite small, lower than 3000 cm -1 .

• the RbSr + has been explicitly treated here, and the NaSr + and KSr + species smoothly progress toward the RbSr + with increasing δR t X-A concerning deeper vibrational levels.

• the CaSr + case is specific: due to the large spin-orbit in Sr 5s5p, 3 P , the entrance channel Cs(6s) + Sr + (5s) is located at an energy of about 20 cm -1 and 200 cm -1 above the Cs + + Sr 5s5p, 3 P 1 and the Cs + + Sr 5s5p, 3 P 0 dissociation limits, respectively. Therefore, just like in the Rb-Ca + combination, NRCT should dominate the dynamics of this system.

Note finally that the electronic ground state of species like LiBa + and LiCa + dissociates into Li + Ba + and Li + Ca + respectively, so that in the context of hybrid trap experiments invoked here, the entrance channel is the lowest one and no loss is expected due to radiative processes. A central goal remains to progress towards lower temperature in order to approach the quantum s-wave collisional regime, which will be of crucial importance for the understanding and the control of such collisions which clearly departs from the semiclassical Langevin model, and for improving theoretical models.

C A S E S T U D Y: E X P E R I M E N TA L C H A R A C T E R I Z AT I O N O F C H A R G E -E X C H A N G E C O L L I S I O N S B E T W E E N U LT R A C O L D 6 L I AT O M S A N D 4 0 C A + I O N S introduction
A system of laser-cooled ions immersed in an ultracold atomic gas introduces a new experimental degree of freedom, a charge, into cold atom experiments [103, 105, 116, 123-127, 129, 131, 135-137, 139, 218]. A single ultracold ion would be useful as a local probe of an ultracold atomic cloud with high spatial resolution, since ions can be controlled using an electric field without exerting much influence on the atoms [START_REF] Kollath | Scanning tunneling microscopy for ultracold atoms[END_REF][START_REF] Sherkunov | Probing ultracold Fermi atoms with a single ion[END_REF]. A system of several crystallized ions mixed with neutral atoms would nicely model atomic cores comprising lattice mixed with an electron gas in a solid-state material [START_REF] Bissbort | Emulating Solid-State Physics with a Hybrid System of Ultracold Ions and Atoms[END_REF]. The charge of the ions would activate the atoms and ions to chemical reactions, and therefore the atom-ion hybrid system would make an ideal platform to study chemical reactions at a single atom level.

When atoms and ions are trapped together, the ions attract the atoms by their static electric field and sometimes lead to inelastic collisions. So far, charge-exchange collisions either with or without emission of radiation [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Schmid | Dynamics of a Cold Trapped Ion in a Bose-Einstein Condensate[END_REF][START_REF] Zipkes | Cold Heteronuclear Atom-Ion Collisions[END_REF][START_REF] Rellergert | Measurement of a Large Chemical Reaction Rate between Ultracold Closed-Shell 40 Ca Atoms and Open-Shell 174 Yb + Ions Held in a Hybrid Atom-Ion Trap[END_REF][START_REF] Sullivan | Role of Electronic Excitations in Ground-State-Forbidden Inelastic Collisions Between Ultracold Atoms and Ions[END_REF][START_REF] Haze | Charge-exchange collisions between ultracold fermionic lithium atoms and calcium ions[END_REF] and molecular association due to radiative processes [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF][START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF][START_REF] Da Silva Jr | Formation of molecular ions by radiative association of cold trapped atoms and ions[END_REF] or three-body collisions [START_REF] Härter | Single ion as a three-body reaction center in an ultracold atomic gas[END_REF][START_REF] Krükow | Reactive two-body and three-body collisions of Ba + in an ultracold Rb gas[END_REF][START_REF] Krükow | Energy Scaling of Cold Atom-Atom-Ion Three-Body Recombination[END_REF] have been studied in ultracold atom-ion hybrid systems. A systematic study of such inelastic collisions, which are elementary chemical reaction processes, is essential to realize the full quantum control of the chemical reactions in an ultra low temperature regime. The knowledge of the molecular interaction potentials and the potential couplings is important for understanding such chemical reactions. The experimental characterization of the ultracold inelastic collisions under the precise control of the collision energies and the internal states of the reacting atoms and ions allows for testing the interaction potentials which are obtained from the elaborated quantum chemistry calculations.

In this article, we systematically study the energy dependence and the internal-state dependence of the charge-exchange cross sections in an ultracold 6 Li-4 0 Ca + mixture. Our choice of the atom-ion combination is unique and promising for the observation of quantum collisions since the 6 Li-4 0 Ca + combination has a high s-wave threshold energy of 10 µK and a quite low limit of micromotion-induced heating rate of a few microkelvin per second [START_REF] Cetina | Micromotion-Induced Limit to Atom-Ion Sympathetic Cooling in Paul Traps[END_REF]. To control the collision energies precisely, we deliberately excite an excess ion micromotion in a linear Paul trap. We determine the energy of the ions from the micromotion-modulated fluorescence spectra [START_REF] Devoe | Role of laser damping in trapped ion crystals[END_REF][START_REF] Blümel | Chaos and order of lasercooled ions in a Paul trap[END_REF][START_REF] Berkeland | Minimization of ion micromotion in a Paul trap[END_REF]. The energy dependence of the charge-exchange cross section σ CE ( E ) is scaled with the energy-dependent Langevin cross section σ lang ( E ) and the cross section can be expressed as σ CE ( E ) = A × σ lang ( E ). The factor A describes the probability of undergoing the charge-exchange collision reflecting the details of the potential mixing of the incoming and outgoing channels. We measure A for the 4 0 Ca + ions in the ground S 1 / 2 state and the excited P 1 / 2 , D 3 / 2 and D 5 / 2 states. We also performed the calculation of the LiCa + potential energy curves, and from the comparison between the experimental results with theory calculations, we have identified the route of the charge-exchange collisions. A profound understanding of the interaction potentials is an important step to predict the route for the molecular ion formation, which may allow us to realize the creation of translationally cold molecules composed of large number of atoms, and to reveal the physics of the growth of mesoscopic molecular ions [START_REF] Côté | Mesoscopic Molecular Ions in Bose-Einstein Condensates[END_REF].

experimental setup

In our experiment, 6 Li atoms and 40 Ca + ions were trapped and cooled separately in a different part of a vacuum chamber as shown in Fig. [START_REF] Jochim | Bose-Einstein Condensation of Molecules[END_REF]. After the preparation of the atoms and the ions, the atoms were transferred to the center between the ion trap electrodes using an optical tweezer, and mixed with the ions. The ions were trapped by a conventional linear Paul trap with an RF voltage with amplitude 44 V and frequency Ω = 2π × 4.8 Mhz, realizing the radial and the axial trap frequencies of (ω r , ω z ) = 2π × (252, 95) kHz, respectively. Neutral 40 Ca atoms from the oven were photoionized using 423 nm and 375 nm lasers. Then the ions were cooled with a 397 nm laser, whose frequency was red-detuned from the S 1/2 -P 1/2 transition and with a 866 nm laser, as a repump light, tune to the resonance of the D 3/2 -P 1/2 transition. Fluorescence from the ions was detected by a photomultiplier tube (PMT) and an electron multiplying CCD (EMCCD) camera. The ion trap system included compensation electrodes to minimize the excess micromotion of the trapped ions. Minimization of the micromotion was accomplished by the RF photon correlation method, and the residual micromotion energy at the optimum condition was estimated to be k b × 0.2 mK. 6 Li atoms were first trapped by a conventional magneto-optical trap (MOT) and then transferred into a cavity-enhanced optical dipole trap (ODT) [START_REF] Mosk | Resonator-enhanced optical dipole trap for fermionic lithium atoms[END_REF]. Eventually, they were transferred into a single optical dipole trap created by a 1064 nm laser [START_REF] Inada | Collisional Properties of p-Wave Feshbach Molecules[END_REF].

Then, the atoms were transported to the position of the ion trap to create the atomion mixture by moving one lens placed on the translation stage. The trap frequencies for the atoms were 670 Hz in the radial direction and 1.7 Hz in the axial direction. The peak density of the atoms was (4.8 ± 0.3) × 10 8 cm -3 . The temperature of the 6 Li atoms was (5.8 ± 0.1) µK. 6 Li was prepared in the ground hyperfine state F = 1/2 of 2 S 1/2 when they were mixed with the 40 Ca + ions in the current experiment.

calibration of the collision energy

Under the condition that the kinetic energy of the ion is much larger than that of the atoms, the atom-ion collision energy E coll is described by the kinetic energy of the ion E ion as

E coll = µ m ion E ion ( 101
)
where µ is the reduced mass of 6 Li-40 Ca + combination. Since µ/m ion = 0.13 in our case, the collision energy E coll is considerably smaller but proportional to E ion . In this work, we tune E coll through the tuning of E ion due to deliberate excitation of the excess micromotion in a controlled way explained below.

We start from the condition in which the micromotion of the ions is well compensated, and then we apply an offset electric field to displace the ions in the radial direction (x and y directions) to intentionally add an excess micromotion energy to the ions. The added kinetic energy of the ions is determined from the shape of the ions' fluorescence spectra. When the micromotion is excited, the spectral shape gets broadened and deformed, as shown in Fig. (12a). The spectral intensity P can be described as

P = χ 2 4 ∞ ∑ n=-∞ J 2 n (β) (∆ω + nΩ) 2 + (γ/2) 2 (102)
where ∆ω, γ, and χ are the laser detuning, the natural line width of the S 1/2 -P 1/2 transition, and the electric dipole moment of the transition, respectively, and J n is the n-th order Bessel function for β [START_REF] Devoe | Role of laser damping in trapped ion crystals[END_REF][START_REF] Blümel | Chaos and order of lasercooled ions in a Paul trap[END_REF][START_REF] Berkeland | Minimization of ion micromotion in a Paul trap[END_REF]. The value β is the modulation index defined by β = | kυ r cos(θ) Ω |, where k, is the wavenumber of the cooling laser, v r is the velocity of the ion along the direction of the micromotion (namely x and y directions in our case), and θ is the angle of the propagation direction of the cooling laser to the x-y plane, respectively. When β = 0, the Bessel function can only be n = 0, and the spectral shape becomes a simple Lorenz function. When β has a finite value, the case study: collisions between ultracold 6 li atoms and 40 ca + ions spectral shape is deformed and shows a peak around |∆ω| ≈ βΩ. We determine β by fitting the spectra with Eq. [START_REF] Zhang | Moleculardynamics simulations of cold single-species and multispecies ion ensembles in a linear Paul trap[END_REF]. Figure (12a) shows the fluorescence intensity of the Ca + ions measured by the PMT plotted against cooling laser detuning. The red, blue and green markers show the fluorescence spectra obtained at the applied electric field strength of 0, 5.2, and 19.2 V/m, respectively. The background PMT signal intensities due to stray light are subtracted from the spectra. After the background subtraction, the fluorescence spectra are normalized to be from zero to unity. We fit the spectra with Eq. ( 102) taking β as a free parameter. The resonance position is chosen to be at the frequency where the fluorescence from the ion drastically decreases. The solid red, green and blue curves show the results of the fitting with Eq. ( 102), and β = 0.73, 5.0, 10 for the red, blue and green data, respectively.

The excess micromotion energy of the ion E emm is determined by β from the expression [START_REF] Berkeland | Minimization of ion micromotion in a Paul trap[END_REF],

E emm = m ion Ω 2 2k 2 cos 2 θ β 2 (103)
where, m ion is the mass of the Ca + ion. The total kinetic energy of the Ca + ion is

E ion = E min + E emm ( 104
)
with the first term of the right hand side being the minimum ion energy we can reach in the experiment.

Figure (12b) illustrates the kinetic energy of the ion determined from the fluorescence spectra plotted against the applied offset electric field. The electric field is enforced by applying the voltage to the compensation electrode, and the electric field strength is calibrated through the displacement of the ions measured using the CCD camera. The dashed curve shows the quadratic dependence of E emm to the applied electric field which can be described as [START_REF] Brunner | Minimization of ion micromotion using ultracold atomic probes[END_REF]

E emm = 4 m ion eE r qΩ 2 ( 105
)
where e, E r and q are an elementary charge, the applied electric field strength, and the stability parameter of the ion trap determined from the secular frequency measurements (q = 0.15), respectively [START_REF] Meir | Dynamics of a Ground-State Cooled Ion Colliding with Ultracold Atoms[END_REF][START_REF] Berkeland | Minimization of ion micromotion in a Paul trap[END_REF][START_REF] Brunner | Minimization of ion micromotion using ultracold atomic probes[END_REF]. The dashed curve shows the relation described by Eq. ( 105) taking E min as a fitting parameter. E min /k b ∼10 mK is limited by the imperfection of the laser cooling condition. The experimental data points and the expected quadratic curve shows a great consistency and therefore we have been able to experimentally confirm the validity of Eq. ( 105) for the ion energy calibration. The dashed curve is used to interpolate the data for the determination of the energy of the ion at the intermediate electric field strength.

energy dependence of the charge-exchange collision cross sections

The inelastic collisions that we observe in our system have been identified as chargeexchange collisions through the mass spectrometry of the reaction product ions in a previous work [START_REF] Haze | Charge-exchange collisions between ultracold fermionic lithium atoms and calcium ions[END_REF] of Prof. T. Mukaiyama's team. In the current experiment, three to seven Ca + ions are loaded in the ion trap to form a one-dimensional crystal. Then ions are mixed with Li atoms and held for one second, and we count the number of ions lost from the trap. Within one second of the holding, the number of ions decays in time. We repeat the measurement 50 times and the ion-loss probability P CE as a function of holding time is measured. We derive the charge-exchange collision rate Γ CE from the solution of the rate equation which takes into account the radiative decay of the ion from the metastable D 3/2 and D 5/2 states,P CE = Γ γ+Γ e -(γ+Γ)t + γ γ+Γ , where γ is the spontaneous decay rate of the D 3/2 and D 5/2 states and t is the mixing time of the atoms and ions [START_REF] Haze | Charge-exchange collisions between ultracold fermionic lithium atoms and calcium ions[END_REF]. The energy-dependent charge-exchange collision cross section σ CE (E) can be derived from the relation σ CE (E) = Γ CE /nv, where n is the atomic density and v is the mean relative velocity of the colliding atom and ion. The atomic density n at the ion position is determined from the information on the number of atoms, the trap frequencies for the atoms, and the displacement of the ion position from the center of the atomic cloud. Although an elastic collision between a Li atom and a Ca + ion causes the loss of the atoms, the reduction of the number of atoms in the current experiment is negligible. The relative velocity of the Li atom and the Ca + ion is determined solely by the velocity of the ion, since the velocity of the ion is much greater than the velocity of the atom.

σ lang (E coll ) = π 2C 4 E coll ≈ 3.3 × 10 -28 × E -1/2 coll [m 2 ] ( 106
)
for the 6 Li- 40 Ca + combination, where C 4 = αQ 2 /4π 0 , α, Q and 0 are the static polarizability of the 6 Li atoms, the charge of the Ca + ion and the vacuum permittivity, respectively. Since the charge-exchange collision happens when the atom-ion distance is very small, the charge-exchange collision is a part of the Langevin collision and therefore σ CE (E coll ) is proportional to but smaller than σ lang (E coll ). The energy dependence of the data shown in Fig. (13a) is consistent with that of the Langevin model in the collision energy range of our measurements.

In order to reveal the internal-state dependence of the charge-exchange collision cross sections, we prepare 40 Ca + ions in the various internal states using an optical pumping. To prepare the ion in ground S 1/2 state, we turn off the cooling laser 20 ms earlier than the repump laser. To prepare the ion in the D 3/2 state, the repump laser is turned off 20 ms earlier than the cooling laser. To prepare the ion in the D 5/2 state, we turn off the repump laser and then the 850 nm laser that drives the ions from D 3/2 to P 3/2 is irradiated together with the cooling laser for 20 ms. The pumping of the ions is done just before the atoms are transported to the ion trap position. We have confirmed that the fluctuation of the transportation timing does not affect our measurement of the charge-exchange rate.

Figures (13b) and (13c) show the charge-exchange collision cross sections vs E coll for the Ca + ions prepared in the metastable D 3/2 and D 5/2 states. In addition, in contrast to the case of ions under cooling lasers, ions prepared in the metastable D states are not irradiated by the cooling lasers and therefore ions may be exposed to heating by an RF noise [START_REF] Wineland | Experimental Issues in Coherent Quantum-State Manipulation of Trapped Atomic Ions[END_REF][START_REF] Brownnutt | Ion-trap measurements of electric-field noise near surfaces[END_REF] and the coupling between the micromotion and the secular motion in the presence of multiple ions [START_REF] Berkeland | Minimization of ion micromotion in a Paul trap[END_REF]. The rate of the heating due to the RF noise depends on the spectral density of the electric field noise, which obeys the empirically obtained power law to the ion-electrode distance [START_REF] Wineland | Experimental Issues in Coherent Quantum-State Manipulation of Trapped Atomic Ions[END_REF][START_REF] Brownnutt | Ion-trap measurements of electric-field noise near surfaces[END_REF]. We estimate the rate of the heating due to RF noise using the typical value of the spectral density from the power law, and it is estimated to be 70 mK/s for the geometric layout of our ion trap. Since our measurement of the charge-exchange cross section may suffer from the heating due to the RF noise, we exclude the data of the collision energy less than 10 mK (shown with the open markers in Fig. (13b-c) from the analysis of energy dependence of the cross sections provided below. On the other hand, a precise determination of the heating rate due to the micromotion-secular-motion coupling is rather an open issue.

However, this heating effect does not seriously affect our conclusion1 . Therefore this coupling effect is not taken into account in our analysis.

We fit the data shown in Fig. (13a-c) to the function σ CE (E coll ) = A × σ lang (E coll ) with the pre-factor A as a free parameter. The dashed lines in Fig. (13a-c) are the fitting result to the data. The obtained values of A are summarized in Table [START_REF] Adams | Atom optics[END_REF]. To derive A for the P 1/2 state, we estimate the population distribution of the ion in the S 1/2 , P 1/2 and D 3/2 states. From the information on the intensity and the detuning of the cooling and the repump lasers, the population distribution is determined to be (p S , p P , p D ) = (0.42, 0.29, 0.29). Since the collision cross section for the mixed state is described as σ mixed = p S σ S + p P σ P + p D σ D , A coefficient for the P 1/2 state is determined from A for the S 1/2 and D 3/2 states. As for the ions in the S 1/2 state, the charge-exchange collision rate is too small to be measured accurately in the current experiment. We have determined the upper limit of A for the S 1/2 state to be 1 × 10 -3 which corresponds to our measurement limit of the smallest charge-exchange rate of 10 -3 s -1 . 40

calculation of the potential energy curves

The knowledge of the molecular properties of the LiCa + system yields useful insight to interpret the results above. Surprisingly enough, no information was available in the literature for this system up to now. Following the approach developed in Ref. [START_REF] Aymar | Structure of the Alkali-metal-atom-Strontium molecular ions: towards photoassociation and formation of cold molecular ions[END_REF][START_REF] Aymar | The electronic structure of the alkaline-earth-atom (Ca, Sr, Ba) hydride molecular ions[END_REF], we calculated the LiCa + potential energy curves (PECs) for all symmetries up to the Li(2s) + Ca + (4p) dissociation limit, namely the 9 th asymptote. Computing such highly-excited states is indeed easily tractable with our approach. Generally speaking, a diatomic molecules composed of an alkali-metal atom and an alkaline-earth ion can be modeled as a system where the two valence electrons are moving in the field of the two ionic cores Li + and Ca 2+ . The latter are represented by two effective core potentials which include core polarization effects. The Schr ödinger equation for the two valence electrons can then be solved in a configuration space spanned by a large Gaussian basis set. With only two active electrons, a full configuration interaction can thus be achieved [START_REF] Aymar | Structure of the Alkali-metal-atom-Strontium molecular ions: towards photoassociation and formation of cold molecular ions[END_REF][START_REF] Aymar | The electronic structure of the alkaline-earth-atom (Ca, Sr, Ba) hydride molecular ions[END_REF] yielding ground state PEC and numerous excited state PECs, displayed on Fig. [START_REF] Morsch | Bloch Oscillations and Mean-Field Effects of Bose-Einstein Condensates in 1D Optical Lattices[END_REF].

In contrast with heavier systems like RbCa + or RbBa + [103-105], the LiCa + species is light enough to allow accessing to the PECs up to Li(2s) + Ca + (4p), thus representing an appealing species to sort out the complicated dynamics taking place in such a hybrid trap. Assuming that only two-body reactions are expected in the present experimental conditions, one can immediately infer several statements allowing to understand the present experimental results. First there is a strong avoided crossing -circled in Fig. [START_REF] Morsch | Bloch Oscillations and Mean-Field Effects of Bose-Einstein Condensates in 1D Optical Lattices[END_REF] -between the (2) 3 Σ + and the (3) 3 Σ + PECs which is probably responsible for the large observed charge exchange rate in the Li(2s) + Ca + (3d) entrance channel. As it is well isolated from other curves of the same symmetry, this avoided crossing can be linearized following the standard Landau-Zener approach, so that a single path transition probability can be estimated according to

P LZ = exp(-2πW 2 c /(v(R c )∆F c ).
In this expression the interaction strength parameter W c = 0.0015025 a. u. is half of the potential energy spacing between the two PECs at the crossing point located at R c = 9.69a 0 (a 0 is the Bohr radius). The classical relative velocity v c = 2U c /µ = 0.002082 a.u. at R c of the particles with reduced mass µ is obtained through the value U c = 0.023533 a. u. of the potential energy at R c with respect to the Li(2s) + Ca + (3d), assuming a negligible incoming velocity at large distance. Finally the ∆F c = 0.011612 a.u. is the difference of slopes between the two branches of the linearized crossing. All together we obtain P LZ = 0.556 which is large enough to explain the large observed charge exchange rate. Note however that several further ) issues should be taken in account for a more quantitative estimate. First, assuming a statistical population of the initial molecular states, the (3) 3 Σ + state would be populated with a weight of 3/20. On the other hand, spin-orbit interaction probably plays a significant role in the dynamics; in particular, the (2) 3 Π, (1) 1 Π and (1) 3 ∆ state are expected to be efficiently coupled to the (3) 3 Σ + state, so that the effective incoming flux toward the avoided crossing is probably higher than the one inferred by the statistical weight. Moreover, the inclusion of the spin-orbit interaction in the model would allow to interpret the observed difference of rates for the D 3/2 and the D 5/2 states. The reported almost unmeasurable rate for the charge exchange reaction starting from the Li(2s) + Ca + (4s) entrance channel is also readily interpreted from the behavior of the PECs. Just like in other similar species -see Chapter (4) -, two molecular states are correlated to this asymptote, namely the (1) 3 Σ + and the (2) 1 Σ + . The former state is the lowest one in this symmetry so that only elastic collisions are expected in this channel. The latter state lies far above the LiCa + electronic ground state so that no clear efficient avoided crossing is present. Therefore the only possible two-body reaction proceeds through spontaneous emission, leading to either radiative charge exchange leading to two ground state species (a Li + ion and a neutral Ca atom), or to the formation of a ground state molecular ion LiCa + as already reported in Refs. [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF]. Following the method exposed in the Chapter (4), we estimated that the cross section for these radiative processes is about 50 times smaller than the one calculated for RbCa + , namely smaller than 10 -17 cm 2 . It is likely that this process could be observed provided that an improved detection using mass spectrometry is implemented.

1 Σ + 3 Σ + 1 Π 3 Π 1 ∆ 3 ∆ Li + +Ca( 1 S) Li + +Ca( 3 P) Li(2s)+Ca + (4s) Li(2s)+Ca + (3d) Li + +Ca( 1 P) Li + +Ca( 1 D) Li + +Ca( 3 D) Li(2p)+Ca + (4s) Li(2s)+Ca + (4p) Li + +Ca( 3 S) (1) 1 Σ + (2) 1 Σ
Finally, the situation is more involved for the Li(2s) + Ca + (4p) entrance channel. This asymptote is quite well isolated from neighboring ones so that there is no obvious avoided crossing in the relevant PECs. However we see that at almost vanishing initial energy (see the horizontal dotted line in the figure) the (5) 3 Π PEC crosses several curves correlated to lower asymptotes to which it can be coupled through spin-orbit interaction, thus allowing for efficient charge exchange. An estimate of the transition probability would require a model for the molecular spin-orbit for these states, and thus further quantum chemistry calculations which are out of the scope of the present paper.

conclusion

In conclusion, we have systematically studied the energy dependence and the internalstate dependence of the charge-exchange collision cross sections for the 6 Li -40 Ca + combination. The energy dependence is confirmed to be consistent with the Langevin model in the collision energy range realized in this work. The inelastic collision rate depends on which internal state the ions are prepared, and the direct comparison between experiment and theory enable us to pin down the route of the charge-exchange process and also to confirm the calculation result of the potential curves with the experiment. Considering that the 6 Li- 40 Ca + combination has a quite low heating rate arising from a micromotion [START_REF] Cetina | Micromotion-Induced Limit to Atom-Ion Sympathetic Cooling in Paul Traps[END_REF], the 6 Li- 40 Ca + combination is one of the most promising candidates to investigate the quantum collisions in an atom-ion hybrid system. The detailed understanding of the inelastic collisions and interaction potentials will be an important step toward the realization of the full quantum control of the elementary processes of chemical reactions in an ultracold temperature regime.

Part II

T O WA R D I N T E R N A L C O O L I N G O F M O L E C U L A R I O N S I N C O L L I S I O N S W I T H AT O M S
In the following, Chapters ( 6) and ( 7) are adapted from the published works in our collaboration with the team headed by Prof. Franco A. Gianturco (from a total of three joint publications) -see Ref. [START_REF] Hernandez-Vera | Rotationally inelastic collisions of H + 2 ions with He buffer gas: Computing cross sections and rates[END_REF] and [START_REF] Iskandarov | Shape and strength of dynamical couplings between vibrational levels of the H + 2 , HD + and D + 2 molecular ions in collision with He as a buffer gas[END_REF]; and Chapter ( 8) is currently intended for a publication elsewhere.

T O WA R D I N T E R N A L C O O L I N G O F M O L E C U L A R I O N S I N C O L L I S I O N S W I T H AT O M S
Chapter [START_REF] Deutsch | Optical lattices[END_REF], adapted from the published work in the Ref. [START_REF] Hernandez-Vera | Rotationally inelastic collisions of H + 2 ions with He buffer gas: Computing cross sections and rates[END_REF], introduces an extensive and complete discussion on the internal quenching of H

+ 2 X 2 Σ + g , D + 2 X 2 Σ + g
and HD + X 2 Σ + g molecular ions, in collisions with He atoms from a buffer gas, as an alternative solution to the fact that such homonuclear diatomic systems lack electricdipole-allowed rovibrational transitions, and are unable to promote their own internal relaxation (except HD + ). This work has also led to another publication, not reproduced here and found in the Ref. [232], where the collisional H + 2 -He cross sections (and the respective D + 2 -He case) computed here are further used to interpret the temperature-dependent and time-dependent rate coefficients for several rotational excitation and de-excitation processes in an experimental context. To this end, Ref.

[232] also proposes a complete quantum state preparation scheme for both H + 2 and D + 2 cases, produced by electron impact ionization, with these molecular ions being trapped in an ion trap, of linear quadrupole kind, and sympathetically cooled by cotrapped laser-cooled Be + [START_REF] Blythe | Production of Ultracold Trapped Molecular Hydrogen Ions[END_REF]. It is worthwhile to notice that in such implementation the trapped molecular ions are lighter than or of equal mass as the buffer gas (H + 2 and D + 2 vs. He), and no experimental result on the effect of a buffer gas on the translational degrees of freedom of such trapped ions is available, to our knowledge. The technical details concerning the experimental setup implementation and the subsequent precision measurements aimed are probably beyond the scope of this work and we shall, therefore, focus on the theoretical description of the atom-diatom collision formalism needed to model the H + 2 -He problem (with the respective isotopologues). The H + 2 -He ground state potential energy surface used in this work was calculated, and kindly provided, by C. N. Ramachandran et al. [START_REF] Ramachandran | Revisiting the potential energy surface for the reaction at the full configuration interaction level[END_REF]. They have used a full configuration interaction method with an extended basis set, where the selection of geometries is based on previous multireference configuration interaction (MRCI) calculations available in the literature. In particular, the interconnection between the longrange two-body interactions with the short-range three-body contributions, specifically described in a hyperspherical reference frame, has required special attention of the authors. For the actual purpose of use of such a surface in scattering calculations, a continuous global potential surface was built by fitting the many-body expansion form of A. Aguado and co-workers [START_REF] Paniagua | Global fit of ab initio potential energy surfaces I. Triatomic systems[END_REF] to the ab initio energies -see Appendix (C.5) for a brief comment on Aguado's method.

Ramachandran's global surface is the full reactive PES accounting for all possible atomic arrangements, for instance H + HeH + and H + 2 + He, as a function of the three interatomic distances, V = V (R H-H , R H-He , R H -He ), preserving thus all the permuta- tion symmetries. In the particular experimental implementation and physical conditions intended in [232], the possible exit reactive channels available for a H + 2 + He collision are either energetically closed or negligible (more details below and in Ref.

[232]), and therefore the internal cooling of H + 2 , by collisions with He atoms, can be studied by fixing the H + 2 + He arrangement of the interaction potential, which yields to the use of a single set of Jacobi coordinates fixed at the H + 2 center-of-mass (diatomic internuclear distance given by r) and H + 2 -He scattering coordinate represented by R, with the orientation angle θ between r and R. Thus, we shall set up a proper transformation between the internuclear distance coordinates and Jacobi coordinates, and use the arrangement-fixed PES, V = V (r, R, θ), for close-coupling calculations, in order to model excitation and de-excitation of rotational states. For the sake of simplicity, in Chapter ( 6), the ground vibrational manifold of H + 2 , D + 2 and HD + is kept fixed but vibrational effects are included and discussed in Chapter [START_REF] Morinaga | Holographic manipulation of a cold atomic beam[END_REF]. Since the triatomic H + 2 /He system is heteronuclear and the He atom has a zero nuclear spin, the total nuclear spin is conserved during the collisions and ortho-to-para conversions (or viceversa) are of no concern. Therefore, in the following, only rotational quantum number changes ∆N = N -N = 0, ±2, ±4, . . . are allowed. It is worthwhile to briefly recall the particular rotational structure of homonuclear diatomic molecules, in which allowed rotational states are those restricted to odd (ortho-molecules) or even (para-molecules) quantum numbers, i.e. anti symmetric or symmetric rotational wavefunctions, such that properly combined with symmetric/anti symmetric nuclear spin wavefunctions, respectively, would give rise to the correct symmetry of a total molecular wavefunction, under the operation of exchange of the two nuclei.

To this end, one particular aspect discussed below is the influence of the coupling between the electronic spin and the molecular rotation on collisional cross sections, to which is verified numerically that the sum of the individual state-to-state cross sections accounting for the fine-structure components is dynamically equivalent to directly treating the collision problem of a molecular ion as a structureless spherical rotor, i.e. a pseudo-H + 2 X 1 Σ + g , interacting with the He atom. Chapter [START_REF] Morinaga | Holographic manipulation of a cold atomic beam[END_REF], adapted from the published work in the Ref. [START_REF] Iskandarov | Shape and strength of dynamical couplings between vibrational levels of the H + 2 , HD + and D + 2 molecular ions in collision with He as a buffer gas[END_REF], extends the discussion started above, where the H + 2 -He arrangement-fixed PES, in Jacobi coordinates, prepared previously and assuming the diatomic molecular ion as a pseudo-H + 2 X 1 Σ + g , is used to discuss the influence of higher diatomic vibrational manifolds in the H + 2 -He state-to-state cross sections (as well as for its isotopologues). It is worthwhile to notice that due the absence of the doublet spin multiplicity, the splitted molec-ular rotational states represented by the N j symbol, used in Chapter [START_REF] Deutsch | Optical lattices[END_REF], becomes now represented by the symbol j only. And besides some cross sections calculations provided as examples, for a few rovibrational transitions, the full discussion is carried out mainly by studying the shape and strength of the respective dynamical coupling between the vibrational levels. For such, we express the atom-diatom interaction potential in terms of a set of multipolar radial coefficients as a function of the distance R between the incoming He atom and the respective molecular ion, to which, in turn, shall evince the overall anisotropy of the interaction between the colliding partners.

Therefore, we present the computation of the vibrational levels of each diatomic system, using the LEVEL code [START_REF] Le Roy | A Computer Program for Solving the Radial Schr ödinger Equation for Bound and Quasibound Levels[END_REF], written by Prof. Robert J. LeRoy (University of Waterloo), comparing the level energies with other theoretical values available in the literature, some of them subject to relative high accuracy, accounting for relativistic and radiative corrections. Then, a detailed discussion is carried out on how to compute the multipolar coefficients and how to interpret the state-to-state cross sections hierarchy based on the strength of dynamical coupling. In particular, the dominant multipolar coefficients are those driving rotational transitions of ∆j = 2 followed by ∆j = 4 and elastic transitions, ∆j = 0, which are directly translated in the respective numerical cross sections hierarchy, in terms of magnitude. Moreover, a single collision may lead to vibrational de-excitation in exchange of producing rotationally excited molecular ions, i.e. the products being cold in a vibrational sense but excited in rotation. As a typical example, we shall see below the dominance of transitions such (υ = 1, j = 0) -→ (υ = 0, j = 2) over the whole range of collision energies studied (with υ the vibrational quantum number).

R O TAT I O N A L LY I N E L A S T I C C O L L I S I O N S O F H

+ 2 I O N S W I T H H E B U F F E R G A S : C O M P U T I N G C R O S S S E C T I O N S A N D R AT E S introduction
The physical problem of collisionally cooling simple molecular ions with an additional atomic gas is of intrinsic interest as a benchmark problem in scattering theory and in the manipulations of molecules at low temperatures. More specifically, H + 2 being the simplest of all molecules, there is an additional, profound motivation for its study coming from fundamental physics.

In this field, increasing the accuracy of the values of fundamental constants and testing whether they are truly constant in time and space [START_REF] Uzan | Varying Constants, Gravitation and Cosmology[END_REF] are a key endeavor that has been ongoing for nearly a century. Because of the central role played by the constants for the laws of physics and for the science of measurement, it is of fundamental importance to develop independent, as well as more accurate and sensitive, measurements for the same constants, since this allows a strong consistency check and, in the case of discrepancy, possibly the discovery of new effects. One such approach is based on the precision optical spectroscopy of rovibrational transitions in the simplest molecules, i.e., the molecular hydrogen ions (MHIs) H + 2 , D + 2 , and HD + , confined in an ion trap. Since the preparation of such species in specific initial states is an important issue in trap confinements, the present work shall chiefly analyze the collisional physics that will lead to such a state-selected preparation of ions in traps.

These one-electron MHIs have recently emerged as a family of systems at the intersection of atomic and molecular physics, with enormous potential for ultra-high accuracy in the experiments involving them [START_REF] Schiller | Simplest Molecules as Candidates for Precise Optical Clocks[END_REF]. This potential is due, on the theoretical side, to the possibility to compute ab initio all their properties, in particular the levels of their internal state energies [START_REF] Korobov | m7-Order Corrections in the Hydrogen Molecular Ions and Antiprotonic Helium[END_REF], including hyperfine structure [START_REF] Bakalov | High-Precision Calculation of the Hyperfine Structure of the HD + Ion[END_REF][START_REF] Karr | Vibrational spectroscopy of H 2 + : Precise evaluation of the Zeeman effect[END_REF] as well as the interaction with external fields [START_REF] Karr | H + 2 and HD + : Candidates for a molecular clock[END_REF][START_REF] Bakalov | High-Precision Calculation of the Hyperfine Structure of the HD + Ion[END_REF][START_REF] Bakalov | The electric quadrupole moment of molecular hydrogen ions and their potential for a molecular ion clock[END_REF][START_REF] Schiller | Static and dynamic polarizability and the Stark and blackbody-radiation frequency shifts of the molecular hydrogen ions H + 2 , HD + , and D + 2[END_REF]. Since the theoretical predictions depend on the values of certain fundamental constants, a comparison between ab initio results and data allows for the determination of these constants.

On the experimental side, MHIs can be robustly confined in an ion trap, where sympathetic cooling by laser-cooled atomic ions permits the reduction of their kinetic energies down to a few mK. The variety of physical methods for internal and externalstate manipulation known from the field of atomic manipulation, highly successful in 93 94 rotationally inelastic collisions of h + 2 ions with he atomic clocks and in quantum information studies, should in principle be applicable to MHIs as well. However, the MHIs offer no strongly bound excited electronic states, so that fast spontaneous electronic emission to their ground state is not available. Furthermore, the homonuclear MHIs do not have electric dipole-allowed rovibrational transitions. Hence, the standard methods from atomic physics are not transferable in a simple way. Instead, specifically adapted techniques must be developed. In particular, it is desirable to prepare MHIs in a single quantum state, as the most convenient initial condition for spectroscopy. Suitable states of interest are those having zero or small rotational angular momentum N, in which case the hyperfine structure is simpler and thus simpler spectra result. For example, in a N = 2 level of H + 2 , there are only two hyperfine states (not counting Zeeman sublevels), and only one in N = 0 [START_REF] Karr | Vibrational spectroscopy of H 2 + : Precise evaluation of the Zeeman effect[END_REF]. Preparation in the vibrational ground state υ = 0 is also desirable as it cannot be further vibrationally deactivated by collisions. In addition, these states are highly suitable for achieving high spectroscopic accuracy [START_REF] Karr | H + 2 and HD + : Candidates for a molecular clock[END_REF][START_REF] Schiller | Simplest Molecules as Candidates for Precise Optical Clocks[END_REF].

Here we consider a preparation approach that starts with the MHIs produced by standard electron impact ionization, a procedure resulting in populating a large number of rovibrational levels of the produced ion. This can also occur within an ion trap, so that the MHIs are immediately trapped. A co-trapping with continuously lasercooled beryllium atomic ions can be implemented in order to reduce the (secular, kinetic) temperature of the MHIs to a level of 10 mK by sympathetic cooling. This preparation of the external degrees of freedom can be followed by interaction with a neutral gas at an appropriately chosen temperature, which would then act as a buffer gas for rotational and vibrational cooling: in particular, cryogenic He buffer gas is a practical option from an experimental point of view. The laser cooling of the atomic ions and the resulting sympathetic cooling on the MHIs may be important to counteract any translational heating effect that the buffer gas may exert on the trapped ions. We are not aware of any experimental result on the translational buffer gas cooling of trapped ions which are lighter than the present gas molecules: for simulations see Ref. [START_REF] Ralph | Power-Law Distributions for a Trapped Ion Interacting with a Classical Buffer Gas[END_REF].

In explicit terms, the goal of He buffer gas cooling is to prepare para-H + 2 (total nuclear spin I = 0, even N) in the ground rovibrational level (υ = 0, N = 0), which has only a single hyperfine state (consisting of two Zeeman substates), or in (0,2) level, which has two hyperfine sublevels. Ortho-H + 2 (total nuclear spin I = 1, odd N) shall be prepared in its lowest rotational level (0, 1), which exhibits five hyperfine sublevels.

If the rotationally and vibrationally excited H + 2 is cooled by He, especially at very low He densities of the order of n He ≈ 10 9 cm -3 , reactive collisions should not be relevant and the vibrational deactivation (i.e., cooling to υ = 0) will occur with a time constant of several seconds [START_REF] Bovino | Ion chemistry in the early universe -Revisiting the role of HeH + with new quantum calculations[END_REF][START_REF] Bovino | Cold Chemistry with Ionic Partners: Quantum Features of HeH + ( 1 Σ) with H( 1 S) at Ultralow Energies[END_REF][START_REF] Bovino | Chemical destruction of rotationally hot HeH + : Quantum cross sections and mechanisms of its reaction with H[END_REF]. In other words, since the reaction forming HeH + + H only opens when the molecular ion H + 2 is highly vibrationally excited, we expect that the density of such species formed during the ionization process will not significantly cause losses of the ion of interest here.

Therefore, one of the important questions is whether this deactivation can also be accompanied by rotational cooling within the vibrational ground state and what is its characteristic relaxation time. In this work we shall address these questions within an accurate computational analysis of the relevant quantum processes.

More specifically, we shall first explore the detail of the anisotropic atom-ion interaction in the simpler initial case of a vibrationally cooled ortho/para-H + 2 , molecular partner with a He atom. Following the analysis of the spatial characteristics of the ab initio derivation of the forces at play, we shall generate an orientationally anisotropic potential energy surface (PES) that will be employed to investigate rotational statechanging collisional cross sections (both for internal rotational "heating" and internal "cooling" processes). We will initially treat the molecular partner as a rigid rotor (RR) within the harmonic approximation at its equilibrium geometry, r eq , given by r eq = 1.057 Å. We shall further show that to include the anharmonic effects on the vibrational ground state, the target will slightly modify the shape and energy behavior of the final, state-changing rates.

We shall additionally show that, from the present computational study, the inclusion of either hyperfine structure effect or fine structure coupling during the inelastic dynamics has negligible consequences in obtaining the averaged inelastic cross sections which are produced without fine structure contributions, thereby markedly simplifying the calculation of the final inelastic rates.

Finally, we evaluate a broad variety of the state-changing rates, over a range of temperatures which are representative of those encountered in cryogenic ion traps, and analyze their overall behavior in terms of propensity rules involving different transitions and their relative magnitudes as a function of the initial rotational level of the molecular partner.

interaction forces and anisotropic potential

One of the earlier ab initio calculations of the interaction potential between H + 2 and He was carried out by Falcetta and Siska [247]. They conducted an extensive, high level ab initio study of the ground electronic state of HeH + 2 , accounting for the longrange interaction, the short-range anisotropy, and the dependence of the potential on H + 2 bond length r. They found an energy minimum for the collision complex of 2728 cm -1 . The complex had a collinear structure with the distance R = 1.568 Å between the He atom and the center-of-mass of H + 2 and a slightly elongated geometry of H + 2 with r = 1.098 Å. Subsequently, more extensive calculations of the same-non-reactive PES were carried out by Meuwly and Hudson [START_REF] Meuwly | The potential energy surface and near-dissociation states of He-H + 2[END_REF] and later on by Mrugala et al. [START_REF] Mrugala | Radiative association of HeH + 2[END_REF] who employed coupled-cluster treatments using very large basis sets (aug-cc-PVQZ) and obtained very similar results. A more extensive comparison of these findings will be carried out on Chapter [START_REF] Morinaga | Holographic manipulation of a cold atomic beam[END_REF] in relation to our present ab initio potential which we shall discuss below.

As a matter of fact, an extensive series of ab initio calculations for the reactive threeatom system had been carried out by various groups over the years. An example of that type of work also computed the isotopic variants HD + and D + 2 [START_REF] Palmieri | Ab initio dynamics of the He + H + 2 → HeH + + H reaction: a new potential energy surface and quantum mechanical cross-sections[END_REF][START_REF] Kumar | Effect of Reagent Rotation on Isotopic Branching in (He, HD + ) Collisions[END_REF]. Additional, and more recent, work has also been directed to investigate one of the reactive channels [START_REF] Ramachandran | Revisiting the potential energy surface for the reaction at the full configuration interaction level[END_REF],

H + 2 + He -→ HeH + + H ( 107
)
and the outcomes of this reaction have also been studied along the exothermic path (the right-to-left path of the reaction above) that leads to the formation of H + 2 and thus to the destruction of HeH + molecules. The study followed the low-temperature environment of the interstellar medium, and the most recent reactive potential energy surface (RPES) of Ramachandran et al. [START_REF] Ramachandran | Revisiting the potential energy surface for the reaction at the full configuration interaction level[END_REF] has been employed for the computational analysis of the reaction Eq. ( 107) [START_REF] Bovino | Ion chemistry in the early universe -Revisiting the role of HeH + with new quantum calculations[END_REF][START_REF] Bovino | Cold Chemistry with Ionic Partners: Quantum Features of HeH + ( 1 Σ) with H( 1 S) at Ultralow Energies[END_REF][START_REF] Bovino | Chemical destruction of rotationally hot HeH + : Quantum cross sections and mechanisms of its reaction with H[END_REF] at the expected conditions of the early universe chemical environment.

The PES employed by the above studies, and described in detail by Ramachandran et al. [START_REF] Ramachandran | Revisiting the potential energy surface for the reaction at the full configuration interaction level[END_REF] was obtained at a very high level of ab initio calculations, using the multireference configuration interaction (MRCI) approach that included all single and double excitation from the initial complete active space self consistent field (CASSCF). Since it turns out to be of a very similar level of accuracy as those cited earlier in this section, we decided to follow in our present work this latest calculation to extract the non-reactive potential from that surface, as outlined below.

That analysis [START_REF] Ramachandran | Revisiting the potential energy surface for the reaction at the full configuration interaction level[END_REF] provided several choices for the final, full PES and also accurate numerical fittings of their extended production of ab initio raw points. They reported in fact what they called the M = 8 fitting choice using FCI/cc-pVQZ level for the basis set expansion. It is the latter choice which we have employed in the present work, by accurately generating from the initial interacting potential in interparticle coordinates, V M8 (R 1 , R 2 , R 3 ) of the (HeHH) + complex, the corresponding non-reactive potential in Jacobi coordinates, V M8 r eq ; R, θ for the initial system of H + 2 -He,

V M8 (R 1 , R 2 , R 3 ) -→ V M8 r eq ; R, θ (108)
where the [R, θ] two dimensional (2D) space represents the distance R from the He atom to the mid-bond in H + 2 and the angle θ is formed between the distance vector and the molecular axis, kept fixed at the ground vibrational state equilibrium value r eq . After the transformation of Eq. ( 108), we generate four thousand points on the above 2D grid and further produced the corresponding multipolar coefficients V λ r eq ; R ,

V λ r eq ; R = λ max ∑ λ=0 V λ r eq ; R P λ (cos θ) (109)
where,

V λ r eq ; R = 2λ + 1 2 +π -π V M8 r eq ; R, θ P λ (cos θ) d cos θ (110)
The summation in Eq. ( 109) was extended up to λ max = 20 to ensure a representation of the potential surface's anisotropy that would reproduce the points from Eq. ( 108) within a few cm -1 . We have also extended the calculations to the larger dimensionality of the 3D PES that includes the vibrational effects in the target -see Chapter [START_REF] Morinaga | Holographic manipulation of a cold atomic beam[END_REF].

In order to show a more immediate, pictorial view of the potential energy surface for the present situation, we report in Fig. [START_REF] Demarco | Onset of Fermi Degeneracy in a Trapped Atomic Gas[END_REF] the overall behavior of the present interaction V M8 r eq ; R, θ . The data in the figure show strong similarities with the same figures reported by the earlier calculations of Ref. [247] and of Ref. [START_REF] Meuwly | The potential energy surface and near-dissociation states of He-H + 2[END_REF] that directly generated the non-reactive interaction. The locations of the minima, the relative well depths, and the comparative strengths of the multipolar coefficients we have found here are very close to those given in the earlier work of Ref. [START_REF] Meuwly | The potential energy surface and near-dissociation states of He-H + 2[END_REF]. The data confirm the agreement between the present findings from the improved calculations of Ramachandran et al. [START_REF] Ramachandran | Revisiting the potential energy surface for the reaction at the full configuration interaction level[END_REF] and the earlier ab initio results discussed above in this section.

One clearly sees in the figure the steep repulsive region at short distances, which remains so for all orientational approaches, as well as the appearance of the attractive wells symmetrically located along the two collinear approaches for the He partner: the more positively charged, external regions of the H atoms therefore show the most attractive interactions with the polarizable He atom.

That the anisotropy of the interaction is dominated by forces (V λ=0 for odd λ values) which are symmetric with respect to the homonuclear partner is also shown by the relative strength of the multipolar coefficients from Eq. ( 109), which are depicted by the curves of Fig. [START_REF] Phillips | Laser cooling and electromagnetic trapping of neutral atoms[END_REF]. The most striking result from the shapes of the coefficients reported by Fig. ( 16) is the fact that the λ = 2 anisotropic term is the dominant term in the region of the well of the three-atom complex. The spherical interaction, λ = 0, extends further out since it corresponds to the term controlled by the spherical (i.e. scalar), α 0 , polarizability of He [START_REF] Bishop | Static electric properties of H and He[END_REF]. We can see that the asymptotic coefficient given by the product α 0 with the H + 2 permanent quadrupole moment [247] goes more rapidly to zero but describes also the strongest anisotropic torque, controlled by the P 2 (cos θ) Legendre polynomial, which is applied during collisions to the molecular rotor by the impinging atom. We shall see below more specific consequences of this feature in our discussion of the relative magnitudes of the inelastic cross sections.

the quantum treatment of the collisions

The rotational structure of a 2 Σ + target molecule A rotating diatomic molecule with non-zero electron spin and nuclear spin can be approximately described by a Hamiltonian that contains spin-rotation coupling (H sr ) and hyperfine coupling (H hfs ) terms in addition to the nuclear rotation term (H rot ),

H total = H rot + H sr + H hfs = BN 2 + γS e • N + bI • S e + cI z S z e + f N • I (111)
Here, I is the total nuclear spin, S e is the electron spin, and N is the molecular rotational angular momentum. The largest coefficient is the rotational constant B (29.65 cm -1 , 890 GHz), followed by b (880.162 MHz), by c (128.490 MHz), by γ (42.416 MHz), and by f (-0.0417 MHz) [START_REF] Korobov | Hyperfine structure in the hydrogen molecular ion[END_REF].

As one can see from the relative values of the coupling terms, the effects of including explicitly the dynamical coupling of spin angular momentum to the rotational angular momentum are likely to influence only marginally the cross section behavior. We shall, however, carry out the final rate evaluations using only the pseudo-closed shell (i.e. spin-zero) formulation of the scattering problem outlined below.

The coupled channel (CC) equations: a short outline

In the space-fixed (SF) reference frame, due to the invariance of the potential under rotations of the system, it is convenient to expand the scattering wavefunctions into eigenfunctions |j, , J, m J of the total angular momentum J = l + j to be defined later,

ψ J (r, R, θ) = 1 R ∑ j, g J j, (R) |j, , J, m J ( 112
)
100 rotationally inelastic collisions of h + 2 ions with he where designates the relative orbital angular momentum of the collision partners. The conservation of the parity also constrains the quantum numbers and j, but we do not include the parity index explicitly in the following in order to simplify the notation. The substitution of Eq. ( 112) in the usual time-independent Schr ödinger equation which describes the collisional process leads to the CC equations given by Ref. [START_REF] Martinazzo | A modified Variable-Phase algorithm for multichannel scattering with long-range potentials[END_REF]

d 2 dR 2 + k 2 -V - l 2 R 2 g J = 0 (113)
and, as usual, each element of the energy matrix, k 2 ς,η , is given as δ ς,η 2µ E collη , a diagonal matrix for the asymptotic (squared) wavevectors. Here µ is the reduced mass of the system.

The potential coupling matrix V = 2µU contains the full interaction potential representation: U, of which the individual terms are typically given by Eq. ( 109) and ( 110). l 2 is the matrix representation of the square of the relative orbital angular momentum l 2 ς,η = δ ς,η η η + 1 . The solution matrix g J yields the radial solutions for each selected value of the total angular momentum J. In the asymptotic region, the wavefunction solution matrix can be rewritten in order to obtain from it the K matrix and the scattering matrix, S: S = (1

+ iK) • (1 -iK) -1 .
For the special case of purely rotational state-changing collisions, the corresponding final, state-to-state, inelastic cross sections are obtained as

σ j-→j = π (2j + 1) k 2 ∑ J (2J + 1) ∑ , δ ,j ; ,j -S J
,j ; ,j 2 [START_REF] Willitsch | Cold Reactive Collisions between Laser-Cooled Ions and Velocity-Selected Neutral Molecules[END_REF] from the S-matrix elements. Here j denotes the rotational quantum number of the initial rotational state and j corresponds to the final state. The cross sections describe either a "rotational internal cooling" (RIC) process or a "rotational internal heating" (RIH) process, which for the systems of the present study means j < j or j > j, respectively. Both situations will be analyzed below for the case of the target molecule H + 2 .

The role of the hyperfine structure effects on energy-transfer dynamics

The exact scattering calculations including hyperfine structure terms are obviously able to clarify the role of the electronic and nuclear spins present in H + 2 during the energy-transfer dynamics of the collision. However, given the smallness of such effects, as mentioned before, we will not report the formalism for the quantum dynamics employed to compute the hyperfine-state-resolved cross sections for the title systems. The details of the scattering formulation for the full coupling situation of the open-shell case will be collected in our Appendix (A), while in the following we will present, as an example of it, the results for the rotational excitation of para-H + 2 system. We shall only report the averaged results for the ortho-H + 2 case, following the simplified scheme that we shall explain below.

The vibration-averaged potential approach

In this work we are chiefly interested in the rotational (de-)excitation of H + 2 due collisions with He. If we restrict our analysis to the ground vibrational state of H + 2 , the simplest approximation is to consider the H + 2 as a rigid rotor (RR) for which the ground-state vibrational wavefunction is further described as a harmonic oscillator (HO). In such a case, then, the convolution of the full interaction over the internal vibrational coordinate yields a delta function around the r eq value discussed earlier.

In other words, the final 2D potential energy surface described by Eq. ( 109) and Eq. ( 110) is valid whenever the HO approximation describes well the target vibrational state in question. Thus, the above formulation will be called the HO-RR scheme in the following discussion.

Another possible choice is to explicitly consider the vibrational motion of the molecular target by averaging the PES over the specifically computed ground-state vibrational wave function as given by the potential of the isolated molecular target. Namely, one can then write

V 00 (R, θ) = ∞ 0 χ * υ =0,N =0 (r) V (r, R, θ) χ υ=0,N=0 (r) dr (115)
where χ υ,N are the eigenfunctions computed from the Hamiltonian of the isolated vibrating rotor. This scheme shall be called the Vibration-Averaged (VA) computational scheme. We may fix the rotational quantum number N to zero in the above integration because this number does not affect significantly the χ υ=0,N wavefunctions over the range of rotational energies considered in this work. The V 00 (R, θ) function has to be also expanded in terms of Legendre polynomials, as described above, in order to implement the new PES in our scattering code in the same way that we did for the calculations within the HO-RR scheme. As we shall see below, the possible deviations of V 00 (R, θ) of the VA approach from V r = r eq ; R, θ of the HO-RR approach, albeit small, can cause changes in the scattering calculations and therefore in the final state-changing rates. The changes would in effect provide a measure of the anharmonic nature of the ground vibrational state of our target molecules and shall be discussed below.

the computed state-changing cross sections

In all calculations we have extended the effects of the anisotropic coupling potential out to the external radial region up to around R max = 50 Å. The multipolar coefficients of Eq. ( 109) were included up to λ max = 20 to represent the full 2D potential of Eq. ( 108). The integral cross sections were obtained by summing the partial cross sections over a sufficiently large number of values of the total angular momentum J until convergence reached around 1%. The number of N values included in the asymptotic expansion of target states was checked for convergence by having at least two closed channels above each energy threshold as part of the full expansion of the coupledchannel wavefunction of [START_REF] Wang | Vibrational cooling in a cold ion trap: Vibrationally resolved photoelectron spectroscopy of cold C -60 anions[END_REF].

We also carried out numerically converged calculations for the elastic processes, i.e. for the collisions where the molecular rotational quantum numbers did not change. Some examples for the quantities which we have obtained for a very broad range on initial rotational states of the para-target are shown by Fig. (17-top). The range of collision energies presented runs, for the sake of the example, over a more limited span that which was actually computed for the later studies of the internal cooling in ion traps.

One clearly sees in all cases that the ion-neutral potential of the present systems is yielding very large elastic cross sections: they are all much larger than the molecular "size", as measured by the square of the target bond length. This is also to be expected for scattering processes guided by long-range interactions. Furthermore, we notice also that the larger cross sections occur for elastic collisions with the target in its lower initial rotational states, with the N = 2 case producing the largest cross sections, especially at the lower collision energies.

We have also investigated the behavior of the cross sections for two different sets of state-changing collisional events: (i) the RIC processes, which de-excite the target molecules down to lower rotational internal states, and (ii) the RIH processes, which excite the molecular partners to higher rotational internal states. The results for both sets of processes involving the para-H + 2 molecular ion are reported by the two panels of Fig. (17-bottom). Referring to the panels in that figure, we see that the RIC cross sections decrease as ∆N increases, in accordance with the fact that increasing the energy differences between the involved levels decreases the collisional efficiency of the energy transfer dynamics. Thus, the RIC cross sections at, for example, 200 cm -1 of collision energy, drop by a factor of about 4 in the case of para-H + 2 when ∆N increases from ∆N = 2 to ∆N = 8. Furthermore we see that the strong anisotropy of the Legendre polynomial linked to the λ = 2 multipolar coefficient of Fig. [START_REF] Phillips | Laser cooling and electromagnetic trapping of neutral atoms[END_REF] is mirrored in the dynamics by making the ∆N = 2 transitions to be the largest, since they are dominated by the direct torque caused by that term of the anisotropic potential during collisional encounters.

Another clear feature of the cross sections shown in Fig. (17-bottom) is the dominance of the ∆N = ∆j propensity rule: when this happens for a RIC process, the corresponding probability of the process is always larger.

The rotational excitation cross sections, i.e. the RIH collisions, are shown in the right panel of the same Fig. (17-bottom). The effect of the decrease of the cross sections as ∆N increases is here very prominent for the processes involving para-H + 2 which are shown in that panel. The same processes for ortho-H + 2 (not shown in this thesis) exhibit smaller cross sections because the corresponding amounts of energy transferred are now larger than for para-H + 2 , as shown by the energy level spacings [START_REF] Brown | Rotational Spectroscopy of Diatomic Molecules[END_REF]. Additionally, the RIH collisions show in general a weaker presence of resonances than the RIC processes. In any event, the presence of resonances is not expected to play a significant role on the behavior of the corresponding rates at the temperatures of interest, as we shall further discuss below. In the case of the RIC processes, we also verify here the same propensity rule of ∆N = ∆j discussed before.

As mentioned earlier, the coupling between molecular rotations and electronic spin is fairly weak and it therefore affects the rotational state-changing process only indirectly, via the F • N recoupling after the collisional partners recede from each other. Thus, we expect that both the fine and hyperfine structure contributions to overall rotational inelasticity should be fairly limited and easily modeled by the averaging procedures that we shall mention below. An example of such options is shown by the calculations in the four panels of Fig. [START_REF] Phillips | Nobel Lecture: Laser cooling and trapping of neutral atoms[END_REF], where the computed cross sections are obtained by treating the two target molecules as HO-RR systems without either fine or hyperfine structure effects. In the case of the para-H + 2 system, the present couplingaveraged results are also compared with those more correctly obtained by using the complete angular momentum coupling scheme of the Hund's case mentioned before.

In the top-left panel of Fig. [START_REF] Phillips | Nobel Lecture: Laser cooling and trapping of neutral atoms[END_REF] we clearly see that the size and behavior of the rotational de-excitation cross sections computed by including hyperfine structure effects, and summed over all their components, are essentially coincident with the cross sections for the same processes, obtained without including fine structure effects, i.e., considering the collision dynamics of a pseudo-1 Σ target molecule.

When we now turn to the RIH processes shown by the bottom left panel of the same figure, we can see again there strong similarities between the results coming Figure 18: Left panels: Comparison between pseudo-1 Σ + rotational cross sections (continuous lines) and summed fine-structure cross sections (dashed lines) for para-H + 2 . Right panels: All cross sections are calculated among pseudo-1 Σ + rotational state representations for both orthoand para-MHIs.

from the two angular momenta coupling procedures: they indeed still remain, both in size, energy dependence, and relative behaviors, essentially the same in the two sets of calculations: the difference between the two cross sections in the bottom-left panel is never larger than 10%.

We can conclude from these examples that to obtain the relative strength and energy dependence of the state-changing collisional rates at the typical temperatures in the ion traps, it is quantitatively correct to treat the collisional processes as if occurring with a simpler, pseudo-1 Σ-type molecular partner. The corresponding results for the ortho-H + 2 are directly shown in the two right panels only for the case of the pseudo-1 Σ target molecule. We can notice, like before, that the RIH cross sections for this system are uniformly smaller than those of the corresponding right panel since larger energy transfers are involved here.

In order to extend the simplified coupling procedure to the analysis of the elastic collisions, we further report in the two panel of Fig. ( 19) some examples for both varieties of molecular targets which we have examined in this work. Because of the strength of the interaction potential, both sets of cross sections exhibit very interesting resonant structures all across the examined energy range, with the features being particularly marked in the low-energy range for both systems. One interesting feature of these resonances is that they are more prominent in the cross sections where their initial molecular ion is either non-rotating or in low rotational states. This is due to the fact that, when the molecular partner in the complex is rotationally "hot" the interaction potential supports fewer bound states. Hence, the trapping of He is less efficient, thereby making the resonances more short lived and less noticeable on the general cross section energy profiles.

To analyze the origin of such resonances is outside the scope of the present study, although we know already that in such ionic interactions the presence of rotational Feshbach resonances and of shape resonances is known to be very likely to occur. We must however note that the large size of the elastic cross sections, in contrast with the sizes of the state-changing processes discussed in the previous figures, is again a clear indication of the strong interaction existing between these colliding partners in the traps.

computed state-changing rates under ion-trap conditions

To establish a sort of benchmark evaluation of the network of state-changing rates between rotational levels, we have first focused on a range of temperatures from a few K to about 100 K in the modelling of the conditions expected in a cryogenically cooled ion trap.

We have carried out the required convolution over a Boltzmann's distribution of relative collision energies E to obtain the rates at a common temperature T of both the ions and the He atoms,

k j→j (T) = 8 πµk 3 b T 3 2 ∞ 0 Eσ j→j (E) exp - E k b T dE (116)
where k b is the Boltzmann's constant. To simplify the notation, the label j now collectively represents all the relevant quantum numbers already discussed before and used there to classify the collisional processes.

We have evaluated the above rates from a few K up to 100 K employing σ j→j (E) values up to 4000 cm -1 . For the integration we started with E values above threshold by about 10 -3 cm -1 and employed an unevenly distributed set of energy values which included a total of more than 1000 points. Numerical convergences were checked to be better than 10 -2 in fractional terms. In the calculations for the rates we have employed the cross sections which were computed using, for comparison, the HO-RR but also the more accurate VA scheme described above.

We report in the two panels of Fig. (20-top) the temperature behavior of the rates for the collisionally elastic processes: those for the para-system on the left and those for the ortho-system on the right. As expected, and as further confirmed by the comparison with the state-changing rates reported in the additional figures given later, we see that the elastic processes are much larger than the inelastic ones and also have a much more rapid dependence on the temperature: in both panels the rates change by more than one order of magnitude when going from a few kelvin to 100 K. The role above reduction effect is present on all transitions shown in the figure for the para-H + 2 molecular partner. The only exception is the relative behavior of the 6 6.5 -→ 4 4.5 vs 6 5.5 -→ 4 3.5 transitions, involving the states already defined, in the lowest T range from 0 to above 50 K, where the expected reduction is inverted: no numerical reason was found for this effect. (iii) All the computed rates were found to become smaller when the initial rotational angular momentum N increases. These decreases are not very large and become less important than those associated with the propensity rule effects discussed in (i).

On the whole, however, the state-changing rates around 50 K were found to be of the order of 10 -10 to 10 -11 cm 3 /s, indicating the cooling/heating processes for the present system to be fairly efficient.

As already discussed earlier, the case ∆N = 2 yields the largest cross sections because the anisotropic interaction between the He atom and the molecular rotor directly associates with the strongest multipolar coefficient with λ = 2, which means that the corresponding Legendre polynomial efficiently couples initial and final rotational states. It is also interesting to see what happens to these most efficient processes when the target initial N changes. This is shown in Fig. [START_REF] Smith | Low temperatures and cold molecules[END_REF] where we also report, for the case of para-H + 2 , another comparison between calculations of the rates using the full treatment of the hyperfine structure effects and those obtained using the simpler pseudo-1 Σ target molecule dynamics.

• The most dramatic difference between the RIC and RIH rates is seen when increasing the initial rotational state of the ions: for the case of para-H + 2 the RIH rates at 80 K change approximately by eight orders of magnitude when the rotational state of the molecular partner increases from |N = 0 to |N = 8 or from |N = 1 state to |N = 9 . This increase and also the strong increase of these rates with temperature are both due to a simple threshold effect: the collision energy must exceed the rotational state energy difference. Thus, at low temperatures, it is only the tail of the Boltzmann velocity distribution that contributes to the reaction rate coefficients.

• In the case of the RIC rates (top two panels), their values vary only moderately, albeit non-uniformly, with the increasing initial rotational state quantum number N. This is again a truly interaction-related effect, where we see that the particular initial-value of N plays a minor role in the RIC processes.

• The plots in Fig. [START_REF] Smith | Low temperatures and cold molecules[END_REF] show once more that the results obtained with the simplest approximation, i.e. the VA model (without spin effects included), are very similar to, and at times practically coincident with, the quantum results obtained from the more accurate treatment of angular momenta's couplings. This is an important result which suggests that one can carry out computational simplifications without substantial losses of numerical accuracy in evaluation the statechanging rates.

In Fig. (22-top) we further examine the dependence of the RIC rate coefficients on the magnitude of the N-changing value during the inelastic processes. Thus, we report there the dependence of the rates on the initial N and also on the ∆N values for a temperature of 10 K, which, from a practical point of view, is similar to the lowest temperature that will occur in an ion trap cooled by a standard closed-cycle cooler. We clearly see there that the rate coefficients markedly decrease as ∆N increases. This is a consequence of at least two factors: (i) the angular momentum reorientation factor, K, in Eq. ( 108) the energy factor associated with the amount of energy being transferred away from the internal energy content of the partner molecules. The former factor, the change of K, correctly reflects into yielding smaller rates as K increases and also |∆N| increases. The latter factor, the energy exchange, also reduces the size of the cross sections as |∆N| increases.

We also see, however, that the dependence on the initial N of the molecular partner is not uniform: all the |∆N| cases shown in both panels vary rather uniformly with the mutual rotational state of the targets, although no uniform decrease with increasing mutual ∆N is observed.

We further report in (22-bottom) the RIC rates for N = 10 and N = 11 as a function of N and for three different values of relevant temperatures. We see now that the rate coefficients decrease with the increase of ∆N at all the examined temperatures. This decrease is by nearly two orders of magnitude when ∆N = -10. We can therefore infer from the above findings that the most efficient processes for bringing the rotationally excited initial states of both para-H + 2 and ortho-H + 2 down to their |N = 0, 1 levels would chiefly occur via a multistep chain of processes which come down the population ladder by |∆N| = 2 jumps.

We report additional results, which further illustrate the behavior of the present systems during state-changing collisions, in the two panels of Fig. [START_REF] Dulieu | Formation and interactions of cold and ultracold molecules: new challenges for interdisciplinary physics[END_REF]. We show there the RIC rates as a function of temperature and for transitions involving ∆N = -2 transitions that start from different N values. We further compare the rates, for both para-H + 2 and ortho-H + 2 systems, for two different numerical schemes that treat the ground vibrational level occupied by the target molecules: the HO approximation of the RR scheme, given by the solid lines, and the VA treatment of the ground vibrational state given by the dashed lines.

There are several observations which can be made by examining the results of Fig. (23): • Over the range of examined temperatures we clearly see that, at least for N up to N max = 10-11, the variations of the rates are all rather modest. The rates on the left panel, in fact, vary at most by a factor of 3 to 4, while those in the right panel vary by even a smaller factor. In other words, for the ∆N = -2, an inelastic process which associates with the most efficient state-changing collisions, the actual initial rotational state of the molecular partner plays a rather minor role.

• We also see from the data in the figure that the scaling of the rates as the |N state changes is not uniform since the increase of the energy gap when going from initial |N = 2 to initial |N = 10 does not cause a uniform reduction of the rates, the same being true also for ortho-H + 2 .

If we now observe the different rates computed either within the HO-RR scheme (continuous lines) or by using the VA scheme (dashed lines), then the following comments can be made:

• The consequence of including anharmonic effects when building the rotationcoupling potential, i.e. the VA scheme, is mainly that of making the corresponding rates smaller over the whole range of temperatures examined here.

• The size of the changes, however, is not the same for all RIC transitions. For both molecular partners, in fact, we see that the largest changes occur when the lowest initial N states are considered, while for cooling processes from the higher N states the changes are fairly small and practically of not much importance.

• It therefore seems, from the present analysis, that the actual inclusion of the correct physical wavefunction to describe the ground vibrational state of the present systems within the rotationally inelastic dynamics does play a noticeable role on the final sizes of the corresponding rates. Thus, in our further study of modeling rotational relaxation during actual trap physical conditions we shall employ for both systems the corresponding VA-computed rates within the pseudo-1 Σ treatment of the CC dynamics.

conclusion

The results of the present calculations provide the fundamental input for evaluating the efficiency of collisional cooling in the case of rotationally excited states of both para-H + 2 and ortho-H + 2 by a cold gas of He atoms. We have focused our study on the exit channels associated with the rotation-only internal state-changing collisions, with the molecular ion in its ground vibrational level. We have further tested the numerical consequences of two different computational schemes for producing an interaction potential that includes the presence of the ground vibrational state in each system.

To this end, we have first obtained the full anisotropic 2D and 3D potential energy surfaces using high-level ab initio calculations [START_REF] Ramachandran | Revisiting the potential energy surface for the reaction at the full configuration interaction level[END_REF] and then cast the latter in the multipolar form indicated by Eq. ( 109) and [START_REF] Hansen | Efficient rotational cooling of Coulomb-crystallized molecular ions by a helium buffer gas[END_REF]. This shows the dominant strength of the coupling potential component associated with the P 2 (cos θ) Legendre polynomial. This characteristic of the interaction potential is reflected into the dominance of direct |∆N = 2| processes in the state-changing dynamics. This feature affects all the cross sections and rates reported in our work.

The target molecule, even in its ground vibrational state, is not exactly a harmonic oscillator. This means, for the present dynamical study, that small anharmonic effects need to be included and indeed modify the size of the final rates associated with pure rotational state-changing processes.

Another important issue of the quantum dynamics for the title system is that regarding the most efficient choice of the total angular momentum representation for the scattering wavefunction during collisions. The correct physical choice turns out to be the one which explicitly includes spin-rotation effects during the dynamical evolution, a property due to the physics stemming from the 2 Σ electronic state of the molecular ion. However, our calculations show that summing over fine-structure recoupling terms produces very similar summed cross section values and rate coefficients as those obtained by directly treating the target molecule as a simpler rigid rotor in a pseudo-1 Σ electronic state. This is an important result which strongly suggests that all numerical modeling for the collisionally inelastic dynamics can be safely, and accurately, carried out by using the simpler angular momentum representation of the pure rigid rotor target interacting with a closed-shell atom in its zero total angular momentum. Given the nature of intermolecular force between diatomic ions and He atoms, we feel that such a reduction of computational efforts could also yield similar results for many other systems.

We have also investigated the validity of simple propensity rules to scale the cross sections (and hence the ensuing rates) within a manifold of different processes among diverse initial and final levels. The tests we have carried out indicate propensity rules with respect to angular momentum reorientations during collisions are indeed valid for the present case. Our calculations indicate that such rules tend to favour the ∆N = ∆j processes and increase their efficiency when the rotational angular momentum transfer values during inelastic transitions become increasingly smaller and approach zero: the marked increase in the size of the elastic cross sections is an indicator of the extreme tendency of this effect.

Finally, the changes of the rate magnitudes when changing the initial and/or final N value of the target state indicate that the |∆N| = 2 RIC and RIH processes are the dominant ones within the manifold of the many transitions involved. This therefore means that any cooling process to the ground rotational level would evolve via a cascade of transitions from the highest initial levels occupied by the molecular ions in the trap.

S H A P E A N D S T R E N G T H O F D Y N A M I C

A L C O U P L I N G S B E T W E E N V I B R AT I O N A L L E V E L S O F T H E H + 2 , H D + A N D D + 2 M O L E C U L A R I O N S I N C O L L I S I O N W I T H H E A S A B U F F E R G A S introduction
Over the past 15 years or so, sympathetic cooling [START_REF] Ravi | Cooling and stabilization by collisions in a mixed ion-atom system[END_REF][START_REF] Wester | Radiofrequency multipole traps: tools for spectroscopy and dynamics of cold molecular ions[END_REF] has emerged as a versatile method for bringing a broad range of molecular ions down to millikelvin secular temperatures. Sympathetic cooling relies on the simultaneous trapping of molecular ions with laser-cooled atomic ions. The ions exchange kinetic energy by elastic collisions which is removed by laser cooling of the atomic species. Due to the long range Coulomb interaction, the sympathetic cooling technique turns out to be efficient in the translational cooling while leaving the molecular ions internally hot [START_REF] Bertelsen | The rotational temperature of polar molecular ions in Coulomb crystals[END_REF][START_REF] Okada | Characterization of ion Coulomb crystals in a linear Paul trap[END_REF]. Inelastic collisions with neutral buffer gas, however, have been repeatedly shown to represent an effective way to also cool the internal molecular degrees of freedom and of going beyond the translational cooling step [START_REF] Hansen | Efficient rotational cooling of Coulomb-crystallized molecular ions by a helium buffer gas[END_REF]. This approach rests on the fact that in the framework of the neutral buffer gas cooling scheme, the rovibrational molecular temperature can be tuned to a value lower than the translational one, therefore useful for preparing a cloud of trapped molecular ions in a state of thermodynamic equilibrium. A typical reaction scenario of a molecular ion AB + with He buffer gas, in which a broad variety of initial and final rovibrational levels can be involved, is given by: A B + (j, υ) + He -→ AB + j , υ + He [START_REF] Côté | Ultracold atom-ion collisions[END_REF] In order to estimate the relative efficiency of all the possible final channels for this reaction, one has to evaluate first the strength of the coupling between the rovibrational levels. Therefore, our preliminary analysis in this chapter will focus on computing the vibrational coupling potentials between the vibrational levels of H + 2 , HD + , D + 2 , in their electronic ground state, as induced by their interactions with He atoms in the trap. To better illustrate the effect of our final potential coupling terms, we shall also briefly provide some quantitative results on state-to-state inelastic cross sections. It is also important to be reminded that the orientational anisotropy of such potential terms will also drive the rotational state-changing collisions and therefore will ultimately control the relative efficiency of the overall rovibrationally collisional cooling kinetics. While the quantum physics of buffer gas cooling is of intrinsic interest as a benchmark problem in scattering theory, the species in our studies are the simplest of all one-electron molecules and can thus provide interesting data when employed as molecular targets in cold traps. Experiments involving these systems can also increase the accuracy of our evaluation of several existing fundamental constants, such as the Rydberg constant, nuclear radii and nucleus-to-electron mass ratios [START_REF] Mohr | Advances in Determination of Fundamental Constants[END_REF]. A relevant issue in the production of H + 2 ion from neutral H 2 molecule is that H + 2 ions formed from the impact of energetic electrons on H + 2 will have several of its vibrational states populated according to a Franck-Condon distribution from the ionization processes of the ground-states of the initial neutral molecules. In illustrating this problem, Theard and Huntress [START_REF] Theard | DVR3D: a program suite for the calculation of rotation-vibration spectra of triatomic molecules[END_REF] have shown that H + 2 ions can be vibrationally de-excited in nonreactive encounters with helium and hydrogen gases, thereby lowering the population of the excited vibrational states for the ion in the trap. It is therefore at this stage of the molecular manipulation process that we need to have specific information on the structural features of the interaction potentials which would drive the internal cooling dynamics in the ion traps.

interaction potentials between h + 2 , hd + , d + 2 and the he buffer gas

The interaction of H + 2 , HD + and D + 2 with He atoms have been the subject of various investigations both experimentally [260] and theoretically [261], mainly involving studies where the aim was to look at the proton transfer channel in the following reaction:

H + 2 + He -→ HeH + + H ( 118
)
It had been shown earlier that this reaction is strongly endothermic [START_REF] Bovino | Cold Chemistry with Ionic Partners: Quantum Features of HeH + ( 1 Σ) with H( 1 S) at Ultralow Energies[END_REF] and can occur at low temperatures only when starting from vibrationally excited states of H + 2 . However, under the low-temperature conditions of the traps we intend to model, the molecular ions will be largely produced in their ground vibrational level thereby excluding any significant formation of the final products on the right-hand side of Eq. [START_REF] Idziaszek | Quantum theory of ultracold atom-ion collisions[END_REF].

The processes we intend to study will then simply follow the scenario of Eq. ( 117), i.e. the collision physics of the trap partners will be controlled by the interaction potential for the internal energy-transfer collisions without any chemical change. Such collisional channels will therefore be evolving under a non-reactive anisotropic potential energy surface (PES) which shall be the main object of the present study.

interaction potentials between h +

2 , hd + , d + 2 and the he buffer gas 121

One of the earliest one-dimensional (1D) potential energy curve (PEC) calculations for the collinear geometry of H + 2 with He had been performed by Edminston et al. [START_REF] Edmiston | Hydrogen Bonding of H + 2 to the He Atom: Energy Surface for Linear HeH + 2[END_REF]. A more advanced study of the 2D surface including nonlinear geometries was constructed by Kuntz, where a semi-empirical diatomic-in-molecule (DIM) formalism was used to obtain the interaction potential [START_REF] Kuntz | Use of the method of diatomics-in-molecules in fitting ab initio potential surfaces: the system HeH + 2[END_REF]. Further calculations have been devoted to improve the level of accuracy of the surface and McLaughlin and Thompson were the first to perform truncated configuration interaction (CI) [264] studies. A mainly two-dimensional (2D) PES, with only a few angular "cuts" at different geometries of the partner molecular ion, was computed by Falcetta and Siska (F-S) who used an MCSCF MRCI level of quantum-chemical calculations with a 5s3p1d basis set for H and a 6s3p1d basis set for He [START_REF] Falcetta | The long-range interaction between He and H + 2 : an ab initio and dynamical study[END_REF]. More extensive calculations of the full 3D PES were carried out later on by Meuwly and Hutson who employed the QCISD(T) level of quantum-chemical calculation and a correlation-consistent basis set expansion, the Dunning's aug-cc-pVQZ basis set [START_REF] Meuwly | The potential energy surface and near-dissociation states of He-H + 2[END_REF]. Calculations of similar qualities were also carried out by Kraemer et al. [START_REF] Mrugala | Radiative association of HeH + 2[END_REF][START_REF] Kraemer | Bound and low-lying quasi-bound rotation-vibration energy levels of the ground and first excited electronic states of HeH + 2[END_REF] who employed a CASSCF approach with an active space consisting of four a and two a orbitals in order to obtain at the same time the ground and first excited electronic states of the system. The chosen basis set was 9s4p3d for He and 8s4p3d for H, contracted to 4s3p2d in each case. All the involved acronyms reported above are standard terminology within quantum chemical calculations so we refer the interested readers to the current, extensive literature on the subject.

A comparison of the well depth values obtained by the latest three independent calculations for the collinear geometry indicates that the values and locations of the well which are very similar to each other, with relative differences of no more than a few percent [START_REF] Meuwly | The potential energy surface and near-dissociation states of He-H + 2[END_REF]. A more detailed comparison of the most recent PESs will be discussed further below in relation with the present work.

The most recent PES employed for the system, and described in detail by [START_REF] Ramachandran | Revisiting the potential energy surface for the reaction at the full configuration interaction level[END_REF], was obtained at very high level of quantum chemical calculations, using the MRCI approach that included all single and double excitations from the initial CASSCF space. The calculated ab initio points were fitted using Aguado-Paniagua many-body expansion [START_REF] Paniagua | Global fit of ab initio potential energy surfaces I. Triatomic systems[END_REF] in which the PES of the triatomic system ABC can be expressed as a sum of three monoatomic terms (V

(1) A , V (2) B , V (3) C ), three 2-body terms (V (1) AB , V (2) BC , V (3)
CA ) and one 3-body term V

(3) ABC . The latter term defined as:

V (3) ABC (R AB , R AC , R BC ) = M ∑ ijk d ijk ρ i AB ρ j AC ρ k BC ( 119
)
where, the optimal fitting was found by setting the parameter M = 8 for those geometries at FCI/cc-pVQZ level of theory, as further discussed in [START_REF] Paniagua | Global fit of ab initio potential energy surfaces I. Triatomic systems[END_REF]. Here M is 122 vibrational couplings of h + 2 , hd + and d + 2 in collision with he an order index of the expansion of a product function that decays exponentially with the distance and is defined in reference [START_REF] Ramachandran | Revisiting the potential energy surface for the reaction at the full configuration interaction level[END_REF]. The detailed meaning of all terms has been given extensively in [START_REF] Ramachandran | Revisiting the potential energy surface for the reaction at the full configuration interaction level[END_REF] and therefore will not be repeated here. It is the latter choice for representing the relevant PES which we have employed in the present work. The data of [START_REF] Ramachandran | Revisiting the potential energy surface for the reaction at the full configuration interaction level[END_REF] were provided by the authors as an interpolated potential function V M8 (R H-H , R H-He , R H -He ) which can be used for calculations related to the reaction reported by Eq. [START_REF] Idziaszek | Quantum theory of ultracold atom-ion collisions[END_REF]. Since the present work requires the use of Jacobi coordinates for the potential V M8 (r, R, θ), which are defined for the inelastic processes of interest in the Eq. ( 117), we carried out the corresponding transformation of the PES of Ref. [START_REF] Ramachandran | Revisiting the potential energy surface for the reaction at the full configuration interaction level[END_REF].

Thus, the vector, R, connecting the He atom and the diatom's center-of-mass is simply

R = x He - m H x H + m H x H m H + m H ( 120
)
where, the H

+ 2 interatomic distance is R H-H = |r| = |x H -x H |,
with θ being the angle between r and R. In the case when m H = m H , the H + 2 center-of-mass reduces to its midpoint,

R = x He - 1 2 x H + x H (121)
The remaining two interatomic distances, R H-He and R H -He , are readily given by geometric considerations,

R 2 H-He = [x H -R cos (θ)] 2 + [R sin(θ)] 2 (122) R 2 H -He = x H -R cos (θ) 2 + [R sin(θ)] 2 (123)
Here, x are the position vectors of each center with respect the origin of a spacefixed (SF) reference frame.

One can easily extend the above coordinate transformation to the systems HD + and D + 2 although the relation given by Eq. ( 120) will change for the case of HD + , since H and D have different masses. In Fig. [START_REF] Stwalley | Photoassociation of ultracold atoms: a new spectroscopic technique[END_REF] the contour plot of H + 2 interacting with He is given for the equilibrium geometry of the target molecular ion, where r = r eq . The calculations have been performed for θ values from 0 • up to 180 • degrees with 10 • degrees step size. As it can be clearly seen, there is still a markedly repulsive interaction when He comes along the 90 • degrees approach, while the two deepest wells of the potential correspond to the collinear geometries. It is also important to 7.2 interaction potentials between h + 2 , hd + , d + 2 and the he buffer gas 123

note here that the calculated potential for the case of a rigid rotor (RR), V RR r eq , R, θ is obtained at fixed equilibrium internuclear distance r eq of H + 2 , just for the sake of providing a simpler example through the 2D cut of the fuller PES. In the general case, which is of interest to us in this work, of a vibrating molecular ion we shall of course obtain the full 3D PES as we shall be discussing further below.

In order to better assess the overall quality of the general shape and strength of the presently chosen PES, we also compare some of its features with the earlier calculations of M. Meuwly [START_REF] Meuwly | The potential energy surface and near-dissociation states of He-H + 2[END_REF] over three different values of the molecular bond distance which are within the range of the data presented by those authors. The comparison is given by the two panels of Fig. [START_REF] Van De Meerakker | Direct Measurement of the Radiative Lifetime of Vibrationally Excited OH Radicals[END_REF]: in the upper panel we show the radial cuts for the different values of the r-coordinate at the collinear approach, while the lower panel shows the same set of curves but for the case of the T-shape approach between partners.

There is a clear overall good agreement between the two sets of computed PESs: the two interaction potentials are essentially the same in terms of well depths for bond distances close to the equilibrium value. The fitting procedure of Ref. [START_REF] Meuwly | The potential energy surface and near-dissociation states of He-H + 2[END_REF], in fact, produces small deviations for the stretched values, thereby enhancing the graphical differences.

In order to employ the presently obtained PES for non-reactive quantum scattering calculations, one has to further transform the PES in terms of the usual multipolar expansion. In other words, one has to first obtain the radial coefficients for the associated Legendre polynomial expansion of the original PES

V total (r, R, θ) = λ max ∑ λ V total λ (r, R)P λ (cos θ) (124)
Here V total λ (r, R) are the radial terms of the multipolar expansion and P λ (cos θ) are the corresponding orthogonal polynomials. The above relation is written in a more general form which is considering the ionic target as a vibrating molecule. In the case where r is not any more a variable but it is frozen at its equilibrium value, then we can simply obtain the 2D cuts of the full 3D PES and therefore write

V(r eq , R, θ) = ∑ λ V λ (r eq , R)P λ (cos θ) (125)
The last equation will be employed for the simpler problem of pure rotationally inelastic collisions which do not require generating vibrational coupling matrix ele-124 vibrational couplings of h + 2 , hd + and d + 2 in collision with he

-2 -2 -2 -2 -2 -2 -2 -2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -0.

interaction potentials between h +

2 , hd + , d + 2 and the he buffer gas 125

ments. In the following subsection we shall further discuss the 3D interaction potential required for the more general rovibrational state-changing processes.

- vibrational couplings of h X 2 Σ + g potential energy curves for their ground electronic state. Such diatomic potentials are needed to obtain the corresponding wavefunctions and energies for their vibrational bound states of total angular momentum J = 0, i.e. for the non-rotating Rigid-Rotor (RR) model of the isolated molecule. We shall not consider at this stage the splitting effects of both hyperfine structure and spin-rotation couplings that need to be considered in the full angular momentum coupling scheme for the open-shell target ions, as we have discussed in detail in the Chapter [START_REF] Deutsch | Optical lattices[END_REF]. To generate the potential energy curves of the molecular ions, we have used exactly the same level of theory as it was done for H + 2 /He and we therefore refer the interested reader to the original work for further details [START_REF] Ramachandran | Revisiting the potential energy surface for the reaction at the full configuration interaction level[END_REF].

To obtain the vibrational wavefunctions a total of about 300 ab initio points were fitted and extrapolated for H + 2 and HD , while for D + 2 about 500 ab initio computed points were involved.

In Table [START_REF] Deutsch | Optical lattices[END_REF] a comparison between the first few vibrational levels computed by us for the three systems and other computed values existing in the literature. We can see from the data in the table that the agreement between our new calculations and the existing energy levels is rather good and confirms the overall quality of the present potential calculations. The rovibrational wavefunctions will be obtained as products of each of the presently computed vibrational wavefunctions, times the corresponding spherical harmonics describing the involved rotational state. Albeit a simplified picture, such partionioning has been used many times in the past and it is not expected to vary substantially the relevant physical processes. It is therefore this representation of the molecular rovibrational wavefunctions that will be employed in the calculations described in the following subsections.

vibrational couplings between h + 2 , hd + and d + 2 interacting with he atoms

We already saw from Fig. ( 24) that a clear angular anisotropy exists for the PES of our system. We can further extend this analysis by varying the intramolecular bond distance r which will then show how strongly r and R are coupled via the initial and final target's asymptotic vibrational states and through the presence of the impinging He atom. It will also show how their couplings vary with the orientation coordinate θ. For this purpose we have fixed the R at the equilibrium geometry of the complex and HD + , at J = 0, and those calculated in this work. Values using different approaches (some also accounting for relativistic and radiative corrections) are provided. Comparisons with experimental data are available in the respective references and the references therein. All in units of wavenumber with respect to a chosen threshold at 0 cm -1 .

vibrational couplings of h + 2 , hd + and d + 2 in collision with he which occurs, following Fig. [START_REF] Stwalley | Photoassociation of ultracold atoms: a new spectroscopic technique[END_REF], for R = 1.02 Å. Figure [START_REF] Jones | Ultracold photoassociation spectroscopy: Long-range molecules and atomic scattering[END_REF] therefore illustrates the dependence of the non-reactive interaction potential obtained in the present work on r for various angles and for a broad range of stretching values of the internal vibrational coordinate. As it could be expected, the overall potential appears to be less strongly influenced by the H + 2 geometry when θ = 90 • while the variation is far more noticeable along the collinear approach on either of the hydrogen sides and as the projectile atom forms a stable complex with the molecular ion at the shown value of R. One should note here, in fact, that at the geometry of the complex's minimum energy configuration the couplings within the vibrating target states are expected to be significant at all angles, as we shall further illustrate below for different vibrational levels. The crucial quantity of interest for carrying out the quantum dynamics is the coupling potential matrix between the various vibrational levels. The latter has its matrix elements defined as follows:

V υ,υ (R, θ) = φ υ (r)V(r, R, θ)φ υ (r)dr ( 126
)
7.3 vibrational couplings between h + 2 , hd + and d + 2 interacting with he atoms 129

Each of the elements of the coupling matrix therefore describes the distortions of the vibrational states of the target ion by the interaction with the impinging He atom. If we consider now the full 3D multipolar expansion of the non-reactive PES the last relation can be rewritten within a similar expansion as the one of Eq. ( 125) before,

V υ,υ (R, θ) = ∑ λ V λ υ,υ (R)P λ (cos θ) ( 127
)
The spatial dependence of the coupling potentials for H + 2 and HD + are respectively given in Fig. ( 27) and [START_REF] Demille | Quantum computation with trapped polar molecules[END_REF]. We further show in Fig. [START_REF] Gilijamse | The radiative lifetime of metastable CO (a 3Pi, v=0)[END_REF] the coupling potential matrix elements for D + 2 (which are specified by the dashed lines). For H + 2 and D + 2 interacting with He, we are presenting examples only for two cases, collinear and T-shape geometries, although our actual calculations were done for the full range of angles reported by Fig. [START_REF] Jones | Ultracold photoassociation spectroscopy: Long-range molecules and atomic scattering[END_REF]. On the other hand, for the HD + target ion, given the reduced symmetry of this system, we are explicitly showing in Fig. [START_REF] Gilijamse | The radiative lifetime of metastable CO (a 3Pi, v=0)[END_REF] three different angles of approach of the He partner.

As can be seen from the figures, we have considered up to 8 vibrational levels when calculating the spatial features of the coupling potential matrix elements: they have been computed over the relevant range of atom-molecule radial distances within which the coupling potential is non-negligible. One sees from Fig. [START_REF] Gilijamse | The radiative lifetime of metastable CO (a 3Pi, v=0)[END_REF] that the strongest vibrational coupling involves the approach on the H side where the lighter part of the ion is more easily affected by the perturbing He atom. Additionally the comparisons of the coupling potentials involving various vibrational states between H + 2 and D + 2 , presented in detail by Fig. [START_REF] Gilijamse | The radiative lifetime of metastable CO (a 3Pi, v=0)[END_REF], show that the strength of the coupling for H + 2 case is uniformly stronger. This fact could be explained as due to the mass differences between H and D since the latter's heavier isotope generates a higher density of vibrational levels that are more closely spaced than is the case for the lighter molecular ion. The corresponding coupling matrix elements will therefore be smaller in value.

As already discussed earlier, the anisotropy of the present interaction potentials affects the magnitude of the vibrational coupling matrix elements as one varies the angle of approach between partners. More specifically, the radial coefficients generated by the multipolar expansions of Eq. ( 125) will show different strength and different radial range of their coupling values as the anisotropy index λ is changed.

Since our present task is to analyze the collisional dynamics of the complexes, it is instructive to see the behavior of those multipolar coefficients, V λ υ,υ (R), as a function of R and for various vibrational states present in the off-diagonal elements of that coupling matrix.

In Fig. [START_REF] Tesch | Quantum computation with vibrationally excited molecules[END_REF] and [START_REF] Rabl | Hybrid Quantum Processors: Molecular Ensembles as Quantum Memory for Solid State Circuits[END_REF] we present the computed V λ υ,υ (R) respectively. In Fig. (29), we present several panels corresponding to the lower values of λ = 0, 2, 4, 6, showing the vibrational couplings of h + 2 , hd + and d + 2 in collision with he V λ υ,υ (R) couplings for several off-diagonal indeces involving the lower vibrational states as labeled λ in that figure. Additionally, in the second top panel (for λ = 2) of Fig. [START_REF] Tesch | Quantum computation with vibrationally excited molecules[END_REF], the V λ υ,υ (R) for the heavier isotopic variant D + 2 are given.

θ = 90°θ = 0°H 2 + (υ, υ ' ) D 2 + (υ, υ ' )
Figure 27: Computed off-diagonal matrix elements from Eq. ( 126) as a function of R and for two values of the angle θ for H + 2 and D + 2 . The plot on the top shows the T-shape while the bottom one is for collinear geometries.

vibrational couplings between h +

2 , hd + and d + 2 interacting with he atoms 131

For the case of the polar ion HD + we present in Fig. [START_REF] Rabl | Hybrid Quantum Processors: Molecular Ensembles as Quantum Memory for Solid State Circuits[END_REF] the V λ υ,υ (R) coupling coefficients for a series of values involving λ = 0, 1, 2, 3: we are covering the same range of vibrational states as in Fig. [START_REF] Tesch | Quantum computation with vibrationally excited molecules[END_REF]. One sees from a comparison with similar data in Fig. [START_REF] Tesch | Quantum computation with vibrationally excited molecules[END_REF] that the λ = 2 involves the strongest coupling, which maintains a marked strength over the whole radial range presented. θ = 0°θ = 180° (dashed lines)

Figure 28: Computed off-diagonal matrix elements from Eq. ( 126) for the case of the HD + molecular ion, as a function of R and for three values of the angle θ. The plot on the top shows the T-shape while the bottom one is for collinear geometries. The latter curves are given by bold characters.

vibrational couplings of h + 2 , hd + and d + 2 in collision with he Furthermore it is the odd values of λ which now are as strong as those which are even in Fig. [START_REF] Tesch | Quantum computation with vibrationally excited molecules[END_REF]. Additionally, the couplings between the higher vibrational states show reduced strength in comparison with the even values of the λ multipolar coefficients. )

H 2 + (υ, υ ' ) λ = 0 λ = 2 λ = 4 λ = 6 D 2 + (υ, υ ' )
Figure 29: Some of the computed multipolar expansion coefficients for the matrix elements V λ υ,υ (R) as a function of R for H + 2 (solid lines) and D + 2 (dashed lines) with λ = 0, 2, 4, 6. vibrational couplings of h + 2 , hd + and d + 2 in collision with he effects from vibrational averaging: the υ = 0 case One of the common approximations for the calculations of rotationally state-changing cross sections within the ground vibrational levels of a diatomic molecule, neutral or ionic, is to employ the rigid rotor (RR) approximation where the internuclear distance is kept fixed during the process at its r = r eq value. The more general case is of course, the one where r is no longer kept fixed and therefore the rotationally inelastic cross sections are computed via a more realistic description of the vibrational motion of the target molecule, especially when excited vibrational states are involved as in the present study. This also holds when the target ion is kept in its ground vibrational state during the rotationally inelastic processes.

vibrational couplings between h

In this Section we shall therefore analyze the implications from using a more extended coupling potential for the exemplary case of the H + 2 molecular ion. The results can be extended to the other molecular ions following the same approach.

The potential of a rigid rotor can be written as,

V RR (R, θ) = δ r -r eq V (r, R, θ)}dr (128)
Here δ rr eq is a delta function which basically plays the role of a symmetric vibrational wavefunction centered around the r = r eq value of the target molecule: for this specific case the wavefunction is obviously only a sharp peak around that value. It physically describes schematically a highly harmonic oscillator for the υ = 0 state. We shall call this approximation the HO-RR scheme for including vibrational effects. On the other hand, the PES of the full 3D potential where r is a variable should be written as in Eq. ( 125), which we repeat below for the special case of the ground vibrational wavefunction of the molecular partner:

V VA (R, θ) = ψ υ=0 (r)V(r, R, θ)ψ υ=0 (r)dr (129)
In this case the PES is averaged over the physical ground state vibrational wavefunction and the resultant interaction scheme is here called the vibration-averaged (VA) potential scheme. To provide specific examples of its behavior, we present in Fig. (31top) several angular cuts of the computed PES for the H + 2 -He following Eq. ( 128). For completeness we have also shown in Fig. (31-bottom) the dependence of the various multipolar coefficients V λ on the radial atom-molecule variable R.

In the figures we have compared the (HO-RR) and (VA) potential schemes, the latter obtained for the υ = 0 level of the lightest homonuclear ion. They also give an overview of the angular anisotropy and the strength of the potential coupling for the ground vibrational state of the targets.

By looking at the locations of the well depth as a function of the chosen angles, as it was already discussed in an earlier section, we can see that the collinear cases exhibit the deepest wells while the bent approaches show weaker interactions: the well depths decrease up to more than 90% from the collinear value. It is also evident that the (VA) potential shows a coupling strength around the potential well which is slightly stronger than (HO-RR). This means that even in the ground vibrational level the target ions present a small, but visible anharmonicity to affect the interaction potential, which will in turn generate the rotationally inelastic cross sections as we shall further discuss later.

HO-RR and VA rovibrational state-changing cross sections

To calculate the relative inelastic cross sections we used the quantum Coupled-Channel (CC) method for solving the scattering equations, employing an in-house developed scattering code: the ASPIN code [START_REF] López-Durán | ASPIN: An all spin scattering code for atom-molecule rovibrationally inelastic cross sections[END_REF]. Since that code has been presented and discussed many times before, we shall omit the details of its numerical implementation of the well-known CC approach. Suffices it to say that the number of closed rotational channels needed to be included in the multichannel expansion in order to achieve a 2% accuracy in the numerical convergence of the individual cross sections turned out to be four.

The inelastic cross sections that we are interested in correspond to the state-changing processes for the target ion H + 2 in an initially excited rotational state and we shall initially keep the vibrational state fixed at υ = 0 during the collisional events.

In Fig. (32-top) and (32-bottom), the excitation and de-excitation cross sections are given as a function of collisional energy for various pure rotational transitions. The dashed and solid lines correspond to the same transitions but using either the (HO-RR) or the (VA) potentials respectively. From Fig. (32-top) it can be seen that the largest excitation cross section corresponds to the transition from j = 0 to j = 2 rotational states which is nearly a factor of four larger than all the others. Thus, this transition is markedly favored since it is directly linked to the corresponding strength of the anisotropic coefficient for λ = 2, as shown in Fig. (31-bottom).

The calculations of the de-excitation cross sections reported by Fig. that the de-excitation cross sections strength markedly depends on the energy gaps involved. Thus, the quenching of the ions turns out to be most efficient when the deexcitation dynamics is chiefly controlled by the λ = 2 anisotropic coefficient of the interaction potential. We can also see that the treating of the potential either as a rigid rotor or by taking its vibrational average (VA scheme) affects the final cross sections mainly in the lowest range of collision energies: the anharmonicity becomes therefore more significant when the collision energies approach their relative thresholds. The excitation process for the 0 -→ 2 transition shows the largest difference between the two different schemes for generating the coupling potentials discussed above at energies very close to threshold -see Fig. (32-top).

As mentioned in the introduction, the present analysis of the shape and strength of the vibrational coupling potential matrix elements is ultimately directed to the calculation of inelastic processes involving state-changing rovibrational collisions.

To briefly illustrate the behavior of some specific examples involving rovibrational heating and cooling cross sections, we report some of the relevant cross sections (identified by two-number labels) in Fig. (33-top) and (33-bottom). The first label indicates the initial vibrational state and the second specifies the initial rotational state within the chosen vibrational level, respectively. The numerical convergence tests have been carried out by including up to four closed rotational levels and up to two closed vibrational levels, as described for our in-house computer coding in a separate publication [START_REF] López-Durán | ASPIN: An all spin scattering code for atom-molecule rovibrationally inelastic cross sections[END_REF].

We can clearly see that the order of magnitude of the cross sections, for cases where the collision dynamics vibrationally cools but rotationally excites the target ion -Fig.

(33-bottom) -, follows a similar strength-ordering as that shown by the pure rotational de-excitation processes. On the other hand, the opposite happens when one looks at the excitation rovibrational channels reported by Fig. (33-bottom): starting with target molecular ions which are already rovibrationally excited produces cross sections which are around one order of magnitude smaller than those for the channel where only vibrational excitation occurs. This indicates that the pure vibrational excitation probability, albeit smaller than the rotational excitation probability seen before, is to be the most efficient excitation process within the rovibrational manifold. A more detailed and extensive analysis of a larger range of rovibrational processes will be presented elsewhere and is being prepared for a future publication. We provide here a brief analysis of their features to demonstrate that the present full interaction potential can indeed yield meaningful results for the state-to-state quantum energy transfer probabilities: the latter will be more extensively discussed elsewhere. ), of para-H + 2 , in their ground vibrational level (υ = 0) to a vibrationally excited (υ = 1) molecular products in their j = 0 rotational levels. Bottom: Computed cross sections as a function of collision energy for various transitions to lower vibrational states, of para-H + 2 , and producing rotationally excited molecular ions (j = 0, 2, 4, 6).

vibrational couplings of h + 2 , hd + and d + 2 in collision with he conclusions

In this Chapter we have analyzed in some detail the full three-dimensional interaction of H + 2 , HD + and D + 2 with He atoms by computationally exploring the strength and radial range of the coupling potential matrix elements between the various vibrational levels in their ground electronic states. In order to realize this, first the potential energy surfaces of the ions have been obtained earlier [START_REF] Ramachandran | Revisiting the potential energy surface for the reaction at the full configuration interaction level[END_REF], by using accurate ab initio methods and covering the full range of the 3D space configuration. In the next step we have fitted the obtained ab initio points with an accurate high-level many-body expansion in Jacobi coordinates and have then computed the corresponding asymptotic, isolated wavefunctions of all the vibrational states of H + 2 , HD + and D + 2 by using the same level of theory used in [START_REF] Ramachandran | Revisiting the potential energy surface for the reaction at the full configuration interaction level[END_REF] to obtain the required PEC of each diatom. We have also provided some illustrative examples for the state-to-state rovibrationally inelastic cross sections obtained from our own multichannel quantum computing code [START_REF] López-Durán | ASPIN: An all spin scattering code for atom-molecule rovibrationally inelastic cross sections[END_REF] and have briefly discussed their relative size and energy dependence.

Since the previous studies on the production of these molecular ions in traps containing He atoms indicated the first 6 to 8 vibrational levels to be most abundantly present, we have focussed on the first eight vibrational levels in each molecular ion and have additionally computed the full 3D PES of the H + 2 -He complex. When examining the 2D cuts of the PES where the target ions were kept at their equilibrium geometries, we have further found that the collinear approaches provide the stronger interaction with respect to the bent approaches of the impinging He atom. This behavior turned out to be true when we have analyzed the V λ couplings between the various vibrational states for both H + 2 , D + 2 and HD + at different orientations. We have also generated the corresponding multipolar coefficients that measure the overall anisotropy of the interaction between partners.

In order to further provide specific examples of the effects from the potential features on different scattering observables, we have initially looked at the role of the averaged potential over the ground vibrational states and compared it with the case of a rigid rotor scheme that provides a simpler coupling potential. We then found that the vibrationally averaged (VA) potential scheme provided small but significant differences in the potential shape with respect to the HO-RR potential.

The effects on the final cross sections from using the VA potential scheme have been analyzed via the calculations of the collisional excitation and de-excitation between pure rotational levels for H + 2 . We found that this effect is fairly small, although it gets more marked when the collisional energies are close to the corresponding thresholds.

As an additional example of the effects on scattering observables from the vibrational coupling matrix elements, we have shown preliminary results involving the use of the (2×2) vibrational matrix for the lowest two vibrational levels and computed with it the rovibrational heating and cooling cross sections at the same energies of the purely rotationally inelastic processes. We found that pure vibrational excitation cross sections are the largest within the manifold, although in general the opening of vibrational levels strongly reduces the efficiency of the energy transfer dynamics in comparison with the pure rotationally inelastic channels.

In conclusion our present work has provided detailed information on the structural features of a specific set of interaction potentials involving He atoms and Hydrogen molecular ions. Such information will be used in a later work for a more extensive analysis of the relative efficiency of the rovibrational excitation and de-excitation of a broad range of collisional channels. The latter will be those involved in the cooling dynamics of the present molecular ions by He as the buffer gas under trap conditions.

A F U L L C O N F I G U R AT I O N I N T E R A C T I O N S T U D Y O F T H E F I R S T T W E N T Y P O T E N T I A L E N E R G Y S U R FA C E S O F [ R B 2 -R B ] + S Y S T E M introduction
Rubidium atoms and their molecular counter parts, such as Rb 2 and Rb + 2 , have been playing an important role in many theoretical and experimental studies on atomic and molecular physics. As typical examples, one shall recall the remarkable first Bose-Einstein condensate (BEC) [START_REF] Anderson | Observations of Bose-Einstein condensation in a dilute atomic vapor[END_REF][START_REF] Bradley | Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions[END_REF]; sympathetic cooling of different isotopologues [START_REF] Bloch | Sympathetic cooling of 85 Rb and 87 Rb[END_REF]; theoretical studies of Rb 2 as a model heavy two-electron system [START_REF] Spiegelmann | Theoretical study of the excited states of the heavier alkali dimers. II. The Rb 2 molecule[END_REF]; spectroscopy of homonuclear dimers [START_REF] Spiegelmann | Theoretical study of the excited states of the heavier alkali dimers. II. The Rb 2 molecule[END_REF] (see the references therein); and becoming, more recently, the most relevant atomic source used in the implementation of the novels hybrid traps of ultracold atoms and a trap of laser-cooled ions [103, 124, 125, 130, 131, 133-135, 137, 138, 140, 142], due to its favorable scattering and electronic structure properties, for instance the 8 7 Rb isotope [START_REF] Bloch | Sympathetic cooling of 85 Rb and 87 Rb[END_REF]. In particular, Rb + 2 -Rb interactions at ultracold collisional energies [START_REF] Jyothi | Photodissociation of Trapped Rb + 2 : Implications for Simultaneous Trapping of Atoms and Molecular Ions[END_REF] and immersion of a single Rb + ions in a relatively dense cloud of ultracold Rb atoms [START_REF] Schmid | An apparatus for immersing trapped ions into an ultracold gas of neutral atoms[END_REF], where three-body recombination has been recently observed for the neutral case [START_REF] Härter | Population distribution of product states following three-body recombination in an ultracold atomic gas[END_REF], are currently two experimental ongoing works fully dependent on the knowledge of triatomic interactions of Rb atoms and ions.

It is worthwhile to notice, among the many ion-rubidium ultra cold collisions performed in hybrid atom-ion traps, the detection of molecular ion formation. Namely, formation of either RbCa + and Rb + 2 observed in the Ref. [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF] or RbBa + and Rb + 2 in the Ref. [START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF], where a cloud of ultracold Rb atoms overlaps a Coulomb crystal of laser cooled Ca + /Ba + respectively. The detection of two molecular species indicates further a somewhat more complicated triatomic collision-induced rearrangement revealing a rich ultracold chemistry [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF].

Following the advances of hybrid traps of ultracold atoms and a trap of laser-cooled ions, well suited for single quantum state preparation and precision measurements, fundamental chemistry at ultra-low ranges of temperatures, molecule-ion or molecular ion-atom collisions, it has been necessary to shed light on the electronic structure of many diatomic and triatomic species involved, mainly alkali or alkaline earth atomsbased compounds. From a theoretical point of view, however, the description of such species has been always challenging due the relatively large number of electrons to a fci study of the first 20 pes of [rb 2rb] + system represent by standard quantum chemistry methods. In this work, we describe the complex of Rb + 3 by pushing forward a hybrid technique based on configuration interaction (CI) with relativistic effective core potential (ECP) corrected with core polarization potential (CPP), which has been successful in the description of many heavy diatomic systems used for cold/ultracold purposes [START_REF] Aymar | Structure of the Alkali-metal-atom-Strontium molecular ions: towards photoassociation and formation of cold molecular ions[END_REF][START_REF] Aymar | Calculation of accurate permanent dipole moments of the lowest 1,3 Σ + states of heteronuclear alkali dimers using extended basis sets[END_REF][START_REF] Aymar | Comment on Calculation of accurate permanent dipole moments of the lowest 1,3 Σ + states of heteronuclear alkali dimers using extended basis sets[END_REF][START_REF] Guérout | Ground state of the polar alkali-metalatom-strontium molecules: Potential energy curve and permanent dipole moment[END_REF]. The methodology reduces the electronic structure problem to an effective two-electron problem, which turns out to simplify systematic studies and to accurately compute either ground or excited electronic states. The method is also benchmarked by a somewhat more expensive spin-restricted variant of a coupled cluster description. Besides that Rb + 3 represents a perfect heavy two-electron model for our formalism, the system is relevant for experimental purposes [START_REF] Schmid | An apparatus for immersing trapped ions into an ultracold gas of neutral atoms[END_REF][START_REF] Jyothi | Photodissociation of Trapped Rb + 2 : Implications for Simultaneous Trapping of Atoms and Molecular Ions[END_REF].

In the following, we briefly describe the formalism and discuss the many aspects of the first lowest electronic states of Rb + 3 , to which we have constructed global potential energy surfaces (PES) using numerical interpolation in the framework of a manybody expansion [START_REF] Paniagua | Global fit of ab initio potential energy surfaces I. Triatomic systems[END_REF][START_REF] Sorbie | Analytical potentials for triatomic molecules from spectroscopic data[END_REF][START_REF] Murrell | Analytical potentials for triatomic molecules from spectroscopic data[END_REF][START_REF] Farantos | Analytical potentials for triatomic molecules from spectroscopic data[END_REF][START_REF] Murrell | Analytical potentials for triatomic molecules from spectroscopic data[END_REF][START_REF] Murrell | Laser photoelectron spectroscopy of the formyl anion[END_REF][START_REF] Carter | ANALYTICAL POTENTIAL ENERGY FUNCTIONS AND TETRA-ATOMIC MOLECULES OF TRIATOMIC[END_REF][START_REF] Paniagua | A new functional form to obtain analytical potentials of triatomic molecules[END_REF]. Since one of the main purposes of such global PESs is scattering calculations, a brief discussion concerning such prospects is also carried out.

spatial description

The body-fixed (BF) frame of Rb + 3 , herein, is described by means of Jacobi coordinates, upon one of the three Rb-Rb centers-of-mass. Thus, the vector R connecting the incoming Rb atom and the diatom is simply

R τ = x τ - m τ+1 x τ+1 + m τ+2 x τ+2 m τ+1 + m τ+2 = x τ - 1 2 (x τ+1 + x τ+2 ) (130)
with a diatom interatomic distance as r τ = |r τ | = |x τ+2x τ+1 |, and the orientation angle between R τ and r τ being θ τ . The cyclic indexes for the three possible arrangements (τ = 1, 2 and 3), m stands for the atomic masses and x are the position vectors of each center with respect the origin of the space-fixed (SF) reference frame. The remaining two interatomic distances are readily known by geometric considerations. Thus, if r 12 is the distance between atoms 1 and 2,

r 12 =r τ r 2 23 = r 12 2 2 + R 2 τ + r τ R τ cos (θ τ ) r 2 31 = r 12 2 2 + R 2 τ + r τ R τ cos (π -θ τ ) (131)
8.3 full ci and rccsd(t) levels of theory

In the collinear geometries the singularity at interatomic distances r τ = 0, i.e. two Rb atoms completely overlapped, happens at each r τ = 2R τ , for a fixed R τ value.

In order to understand the magnitude of the Cartesian components of the electric dipole moments, a few remarks are worthwhile on how the calculations are performed. Building up all the possible geometries within Jacobi coordinates, one finds out the three usual permutation point groups of three centers: C 2v , C s and C ∞v (or D ∞h ). Calculations of each point group are performed with a different orientation unambiguously to each BF frame, (r, R, θ), coordinate. Thus, in standard Cartesian coordinates, (i) the collinear geometries are built with the first Rb atom siting at the systems origin, for sake of simplicity, and the next two centers confined to move on the z-axis only.

(ii) The bent geometries lies in the xy-plane, with two atoms over the y-axis and the third one orbiting the diatom center-of-mass. And (iii) T-shaped geometries are built with two atoms over the y-axis, while the third one lies on z.

full ci and rccsd(t) levels of theory

The potential energy surfaces, potential energy curves and all the electric dipole moments of Rb + 3 -and the respective isolated diatomic potential, Rb + 2 and Rb 2 -are calculated in the frame of Born-Oppenheimer approximation using two different approaches. The first one is a direct generalization, for three dimensions, of the semiempirical methodology which has been already used to describe a number of diatomic systems in our group, and detailed in Ref. [START_REF] Aymar | Structure of the Alkali-metal-atom-Strontium molecular ions: towards photoassociation and formation of cold molecular ions[END_REF][START_REF] Aymar | Calculation of accurate permanent dipole moments of the lowest 1,3 Σ + states of heteronuclear alkali dimers using extended basis sets[END_REF][START_REF] Aymar | Comment on Calculation of accurate permanent dipole moments of the lowest 1,3 Σ + states of heteronuclear alkali dimers using extended basis sets[END_REF], also used during this work in the Chapter (4) and Chapter [START_REF] Mewes | Output Coupler for Bose-Einstein Condensed Atoms[END_REF]. We shall, therefore, only recall the main steps herein.

The Rb atoms are modeled as an one-electron system, where the valence electron is moving in the field of a Rb + core represented with a relativistic effective core potential (ECP) [START_REF] Durand | New atomic pseudopotentials for electronic structure calculations of molecules and solids[END_REF][START_REF] Durand | A theoretical method to determine atomic pseudopotentials for electronic structure calculations of molecules and solids[END_REF]. The ECP is complemented with a core polarization potential (CPP) depending on the orbital angular momentum, , of each one-electron system [START_REF] Müller | Treatment of intershell correlation effects in ab initio calculations by use of core polarization potentials. Method and application to alkali and alkaline earth atoms[END_REF][START_REF] Foucrault | Nonpertubative method for corevalence correlation in pseudopotential[END_REF], and parametrized with the Rb + static polarizability, α = 9.245, and a set of three cutoff radii ρ s = 2.5365, ρ p = 2.279, and ρ d = 2.5217 (all in atomic units). All parameters were determined in our previous work on alkali-atom-strontium neutral molecules [START_REF] Guérout | Ground state of the polar alkali-metalatom-strontium molecules: Potential energy curve and permanent dipole moment[END_REF]. Thus, the Rb + 3 ion reduces to an effective two-electron system, such that only the remaining two valence electrons are used to calculate the Hartree-Fock and the excitation determinants, in an atom-centered Gaussian basis set, through the usual self-consistent field methodology. The Rb basis set used are from [START_REF] Aymar | Calculation of accurate permanent dipole moments of the lowest 1,3 Σ + states of heteronuclear alkali dimers using extended basis sets[END_REF][START_REF] Aymar | Comment on Calculation of accurate permanent dipole moments of the lowest 1,3 Σ + states of heteronuclear alkali dimers using extended basis sets[END_REF]. Finally, a full configuration interaction (FCI) is achieved. The method is implemented in the Configuration Interaction by Perturbation of a Multiconfiguration Wave Func-tion Selected Iteratively (CIPSI) [START_REF] Huron | Iterative perturbation calculations of ground and excited state energies from multiconfigurational zeroth-order wavefunctions[END_REF] package, developed at "Laboratoire de Chimie et Physique Quantiques, Université Toulouse III -Paul Sabatier" (France).

The second method is the spin-restricted variant of coupled clusters theory with single-, double-and non-iterative triple-excitations, RCCSD(T), using the spin-restricted Hartree-Fock (RHF) reference wave functions [START_REF] Knowles | Coupled cluster theory for high spin, open shell reference wave functions[END_REF]. We employed the spin-restricted variant to avoid the problem of spin contamination in a description of an open-shell Rb + 2 ion as well as to have agreement with the description of Rb + 3 and Rb + ions which are closed-shell systems. At this level of theory, electrons from outer-core orbitals (4s4p and 5s) of rubidium were included in calculations. A large basis set containing spdfgh orbitals with added diffused even tempered functions was used to describe the active electrons on a small core represented by relativistic pseudopotentials (ECP28MDF) [282]. According to FCI calculations described above, we treat the interaction in Rb + 3 as a two-electron problem by a subtracting of energy components in the following way:

E = E Rb + 3 -3E Rb +
, where E Rb + 3 and E Rb + terms correspond to energies for 26 and 8 electrons, respectively. All energies for monomers we computed in the presence of dummy atoms to carry calculations in the basis sets of the full system and likewise to reduce basis set superposition error in the interaction energy. In the same way the interaction in Rb + 2 molecular ion was simplified to one-electron system by subtracting energies of the dimer and two rubidium ions.

Both, FCI and RCCSD(T) methods were also employed to study electric properties in the frame of the finite-field approach. The electric permanent dipole moment (PDM) and static polarizability tensor components from first and second order of the numerical energy derivatives with respect to an applied weak field were obtained. For illustrative purposes, we also computed a static polarizability in the time-independent variant of a polarization propagator at the single and double excitations within a coupled clusters linear response functions theory framework (TI-CCSD) [START_REF] Korona | Time-independent coupled cluster theory of the polarization propagator. Implementation and application of the singles and doubles model to dynamic polarizabilities and Van der Waals constants[END_REF]. All RCCSD(T) calculations were performed using the Molpro package [START_REF] Werner | MOLPRO, a package of ab initio programs[END_REF].

results

We have computed the first twenty electronic states of Rb + 3 using the FCI formalism described above for each A 1 , E 1 and E 2 symmetries of collinear geometries; A and A symmetries for bent geometries; and A 1 , A 2 , B 1 and B 2 symmetries for T-shaped geometries, upon 70000 geometries in total. The electronic states span the first four triatomic asymptotic energies, as r -→ ∞ and R -→ ∞, at 0 cm -1 (by definition), 8.4 results 147 12729.5 cm -1 , 19313.8 cm -1 and 23703.3 cm -1 , which agrees (for instance, the NIST Atomic Spectra Database), with the first four following asymptotes: (i) Rb(5s) + Rb(5s) + Rb + , (ii) Rb(5s) + Rb(5p) + Rb + , (iii) Rb(5s) + Rb(5d) + Rb + and (iv) Rb(5s) + Rb(6p) + Rb + , respectively. In the Table [START_REF] Morinaga | Holographic manipulation of a cold atomic beam[END_REF] we summarize the calculations for the first six electronic states, in the first asymptote, and the respective global PES described below.

The electronic ground state belongs to the 1 A 1 symmetry (i.e. Σ + state in diatomic notation), directly correlated to 1 A of bent geometries. The equilibrium geometry is predicted to be a T-shaped structure with intermolecular distances of 8.5 a. u. and dissociation energy of 12632.7 cm -1 , with respect the triatomic asymptote, Rb(5s) + Rb(5s) + Rb + . As R -→ ∞, asymptotically, the ground PES now as a quasi-diatomic potential, as a function of r, presents spectroscopic characteristics (r e = 9.09 a. u., D e = 6139.4 cm -1 , B e = 0.01714 cm -1 , ω e = 47.33 cm -1 ) in a very good agreement with those, predicted and observed, for Rb + 2 2 Σ + g , as shown in the Table [START_REF] Jin | Collective Excitations of a Bose-Einstein Condensate in a Dilute Gas[END_REF]. Thus, the first PES, 1 1 A 1 , correlates asymptotically with Rb + 2 2 Σ + g + Rb(5s). The first exited surface, 1 3 A 1 , happens to behave asymptotically in a similar fashion, but presenting a collinear equilibrium geometry instead. Therefore, each doublet Rb + 2 subsystem are found embedded into two consecutive Rb + 3 electronics states, singlet and triplet. Figure [START_REF] Hudson | Measurement of the Electron Electric Dipole Moment Using YbF Molecules[END_REF] and ( 35) presents 2D cutoffs, as functions of r and R, for both 1 1 A 1 and 1 3 A 1 electronic states of Rb + 3 at FCI level of theory, where the yellow zones represents the atom-diatom limit correlated to Rb + 2 2 Σ + g + Rb(5s), the red zones are the minimum of the complex Rb + 3 and the zero energies (green zones) are the atom-atom-atom limit, Rb(5s) + Rb(5s) + Rb + . In particular, it is worthwhile to notice that at collinear

Asymptote C ∞v C s C 2v Relative dissociation Global PES order Rb(5s) + Rb(5s) + Rb + 1 1 A 1 1 1 A 1 1 A 1 Rb + 2 (1 2 Σ + g ) + Rb(5s) 1
1 3 A 1 1 3 A 1 3 A 1 Rb + 2 (1 2 Σ + g ) + Rb(5s) 2 2 1 A 1 2 1 A 2 1 A 1 Rb 2 (1 1 Σ + g ) + Rb + 3 2 3 A 1 2 3 A 1 3 B 1 Rb 2 (1 3 Σ + u ) + Rb + 4 3 1 A 1 3 1 A 1 1 B 1 Rb + 2 (1 2 Σ + u ) + Rb(5s) 5
3 3 A 1 3 3 A 2 3 B 1 Rb + 2 (1 2 Σ + u ) + Rb(5s) 6
Table 7: Correlation chart of the first six lowest Rb + 3 electronic states calculated in this work, belonging to the first triatomic asymptote, as R -→ ∞. The remaining quasi-diatomic potentials as a function of r (reported in the relative dissociation column) are used to characterize the respective electronic states.

geometries, where the two atom-diatom arrangements becomes evident, the two yellow zones are interconnected by an attractive potential and the deep minima of the Rb + 3 complex -red zones at short-range. Therefore, an eventual rearrangement process in the electronic ground state of Rb + 3 is expected as barrierless.

Comparisons between the FCI and RCCSD(T) calculations, for both 1 1 A 1 and 1 3 A 1 states, are presented in the Fig. [START_REF] Hinds | Experiment to search for P-and T -violating interactions in the hyperfine structure of thallium fluoride[END_REF] for the T-shape orientation as function of R parametric to a set of r values. In both cases we observed a poor agreement in the most repulsive zones -for instance the black curves at r = 6 a. u. -which improves when r increases. In the 1 3 A 1 case, despite the repulsive region, the two methods converge into an overall good agreement very quickly. The different results at small r values may be attributed to a less-appropriate core-valence and core-core correlation description in the FCI approach. In the RCCSD(T) method a small pseudo potential, leaving a few active core electrons, might describe better such small geometries.

The first occurrence quasi-diatomic potentials of Rb 2 , are found from the third excited surface. The spectroscopic characteristics presented in the 2 1 A 1 PES, asymptotically (r e = 7.936 a. u., D e = 3880.96 cm -1 , B e = 56.66 cm -1 , ω e = 0.02250 cm -1 ), agrees with Rb 2 1 Σ + g + Rb(5s), shown in the Table [START_REF] Mewes | Output Coupler for Bose-Einstein Condensed Atoms[END_REF]. The respective ungerade Rb 2 3 Σ + u and Rb + 2 2 Σ + u , essentially repulsive, are found in the 1 3 B 1 and 1 1 B 1 /2 3 B 1 PES, respectively.

These six surfaces, briefly described above, sum up the main features observed in the first triatomic asymptote. In the second asymptote, the molecular states are built from combinations of two atomic s and one p orbitals, which give rise to a more complicated electronic structure, now adding not only spherical symmetric molecular states but also out of plane states. Thus, we shall leave a detailed description, of each highly excited PES, for the moment when these states eventually become relevant for a given dynamical problem. Nevertheless, in the sake of example, the diatomic spectroscopic characteristics of Rb 2 1 Π g found correlated to a highly excited PES, belonging to the second triatomic asymptote, is already presented in the Table [START_REF] Jochim | Bose-Einstein Condensation of Molecules[END_REF] and compared with the literature. An overview of the first 10 surfaces as 1D cuts are presented in the Fig. [START_REF] Cho | Search for time-reversal-symmetry violation in thallium fluoride using a jet source[END_REF]. ) at θ = 90F igure 34: Heat-maps of the first electronic state of Rb + 3 1 1 A 1 , at FCI level of theory and two orientations (collinear on top and T-shape in the bottom). The PES is, asymptotically, correlated to Rb +

Electric properties

We have also calculated some electric properties since these are more sensitive tests for the quality of an electronic wave function, namely the permanent electric dipole moment and the orientation-averaged static dipole polarizability for Rb + 3 ion in the electronic ground state. Obtained values of the permanent dipole moment are gathered in the Table [START_REF] Stenger | Spin domains in ground-state Bose-Einstein condensates[END_REF] and reveal a noticeable negative PDM at the equilibrium geometry. For both FF-CIPSI and FF-RCCSD(T) methods we find an accordance of about 99 % between their results: -1.116 and -1.103 Debye, respectively. A change of the sign in the PDM was observed above R = 13 and below R = 15 a.u. in the finite-field approach as well. Although, for these and larger distances, the limitation of using the RCCSD(T) method is visible: monomers are far enough away with respect to each other and nomixed spin functions should be included in coupled cluster calculations to achieve more quantitative results. An excellent agreement near the minimum was also observed for diagonal components of the static polarizability tensor. These results computed in the finite-field method are collected in the Table [START_REF] Khaykovich | Formation of a Matter-Wave Bright Soliton[END_REF] as well as components obtained in TI-CCSD approach. TI-CCSD method was used as additional validation of our result and additional test. Its accuracy is lower in comparison to the finite-field calculations mainly due to the smaller level of excitations in the cluster operator. This method also gives worse results in the description of the stretched geometry, although works well for short interatomic distances. Values of average static polarizability for all employed methods are collected in Table [START_REF] Morsch | Bloch Oscillations and Mean-Field Effects of Bose-Einstein Condensates in 1D Optical Lattices[END_REF]. The difference between TI-CCSD results in comparison to finite-field results is about 5%. From a qualitative point of view, however, such a result is sufficient to confirm the quality of both FCI and RCCSD(T) methods. One of the main advantages of the semi-empirical FCI approach described above is its computational efficiency. In order to build a continuous global PES from those individual potential surfaces of each point group, which can be further used for dynamical calculations, and due to such a high number of raw ab initio points, we are able to avoid the, always cumbersome, fitting procedures, using functional forms, applying a numerical interpolation instead. In the following, we shall use the notation "fit" for the typical procedure of approximating a given functional form to a set of raw data, where an intrinsic error is always implied at a given data point. In contrast, the word "interpolation" herein stands to finding a functional form which reproduces exactly the raw data and any intrinsic error is implied between each data point. Our choice for an interpolation procedure is an attempt to minimize any error overhead and to maximize the fidelity, i.e. topology-dependence, of the global PES on the original ab initio points.

Therefore, we have checked the efficiency of modified quadratic Shepard interpolation (bivariate and trivariate) [START_REF] Renka | Multivariate Interpolation of Large Sets of Scattered Data[END_REF], trivariate b-splines, univariate l-splines and Akima's variant of splines for univariate curves [START_REF] Akima | A New Method of Interpolation and Smooth Curve Fitting Based on Local Procedures[END_REF][START_REF] Akima | A Method of Univariate Interpolation Has the Accuracy of a Third-Degree Polynomial[END_REF]. Although all the methods were able to interpolate specific zones precisely, they were completely unable to reproduce a whole surface, handling eventual changes in the grid step or gaps of points. The ab initio points are not originally scattered over an equally spaced rectangular grid, which may give rise to misleading topological features.

In order to overcome these difficulties, we have applied a numerical interpolation within a minimal functional form, in the framework of a many-body expansion, similar to the method developed by A. Aguado and co-workers [START_REF] Paniagua | Global fit of ab initio potential energy surfaces I. Triatomic systems[END_REF][START_REF] Paniagua | A new functional form to obtain analytical potentials of triatomic molecules[END_REF], also widely discussed by J. N. Murrell et al. [START_REF] Sorbie | Analytical potentials for triatomic molecules from spectroscopic data[END_REF][START_REF] Murrell | Analytical potentials for triatomic molecules from spectroscopic data[END_REF][START_REF] Farantos | Analytical potentials for triatomic molecules from spectroscopic data[END_REF][START_REF] Murrell | Analytical potentials for triatomic molecules from spectroscopic data[END_REF][START_REF] Murrell | Laser photoelectron spectroscopy of the formyl anion[END_REF] and S. Carter [START_REF] Carter | ANALYTICAL POTENTIAL ENERGY FUNCTIONS AND TETRA-ATOMIC MOLECULES OF TRIATOMIC[END_REF]. Thus, the global PES for a given electronic state Λ, V = V (Λ; r, R, θ), is defined as

V (Λ; r, R, θ) = V 1 (Λ 1 ) + V 2 (Λ 2 ) + V 3 (Λ 3 ) + V 12 (Λ 12 ; r 12 ) + V 23 (Λ 23 ; r 23 ) + V 31 (Λ 31 ; r 31 ) + V 123 (Λ 123 ; r, R, θ) (132)
i.e. the sum of three one-body terms (V 1 , V 2 , V 3 ) which are the energies of the separated atoms in their respective electronic states; three diatomic-terms (V 12 , V 23 , V 31 ) which are the Rb-Rb diatomic potentials as a function of their internuclear distances; and one non-additive triatomic term (V 123 ), where the symbol Λ is used as a composite index and stands for the many symmetries of electronic states, e.g. s-states, Σ-states or A 1 -states, respectively. No approximations are made in Eq. ( 132), and it is worthwhile to notice the simultaneous use of internal atomic distances and Jacobi coordinates. Expressing the PES in such a form gives clearly advantages, namely: (i) the diatomic terms which are known to be a source of most of the energy fluctuations [START_REF] Vance | Representation of ab initio energy surfaces by analytic functions[END_REF], are now reduced to a simple one variable problem; moreover (ii) due to the homonuclear symmetry of Rb + 3 , the three diatomic terms are identical, i.e. for practical reasons only a single diatomic potential is needed; (iii) the triatomic-term is a smooth monotonic decreasing function, as r -→ ∞ and R -→ ∞, which is non-zero only in the zones of significant chemical interaction; and, (iv) due to the well-behaved shape of V 123 , spurious energies, if any, usually close to r = 0 and/or R = 0, can be easily detected and eliminated. Each many-body term is extracted from the raw ab initio energies by subtracting the respective terms appropriately.

After preliminary tests we have chosen, as interpolation method, the one-variable Akima's variant of splines, since it accounts for a number of features: (i) it carries a fundamental property, of spline-based interpolation methods, of passing through the original points; (ii) its interpolant function is defined locally, preserving the shape of the curve and causing no spurious oscillations, whenever possible; (iii) the method has the same accuracy of third-degree polynomials unless higher degrees are invoked; (iv) it does not require equally spaced grids and (v) there is no concerns about computational stability or convergence features as usually found in fitting procedures. Thus, a single diatomic-term is interpolated straightforwardly. The triatomic term, besides being a well behaved function, still remain as a 3D function and it is first splitted into one-variable curves parametrically to (r, θ), as a function of R, V 123 = V (r,θ) 123 (R), and interpolated. Two successive 1D interpolations are then carried out over r and θ, respectively. Therefore, the final short range PES relies entirely on an one-variable Akima spline interpolation, which is known to be more accurate than its multi-dimension versions [START_REF] Sathyamurthy | COMPUTATIONAL FITTING OF AB INITIO POTENTIAL ENERGY SURFACES[END_REF]. Spline theory has become popular, over the years, in order to construct analytical PES; for more details see Ref. [START_REF] Sathyamurthy | COMPUTATIONAL FITTING OF AB INITIO POTENTIAL ENERGY SURFACES[END_REF] and the references therein.

The method described above is used to interpolate the ab initio energies up to r = R = 90 a. u. Beyond this interval a long range behavior is imposed by means of extrapolation. Thus, the diatomic terms are defined, asymptotically, as

V ς,η Λ ς,η ; r ς,η -→ ∞ = D e Λ ς,η - C 4 r 4 ς,η (133)
for those triatomic electronic states correlated to Rb + 2 potentials, and

V ς,η Λ ς,η ; r ς,η -→ ∞ = D e Λ ς,η - C 6 r 6 ς,η (134)
to Rb 2 potentials. The dissociation energy, D e , and the long range leading terms, C 4 and C 6 , are estimated, on demand, from the last two points of each respective ab initio diatomic energies. Notice that the usage of only two points are in an excellent agreement with a more elaborated non-linear fitting, since the ab initio diatomic potentials are computed up to fairly large internuclear distances, e.g. 50 a. u. to 70 a. u., where, contributions from the electronic wavefunction overlapping are expected to be negligible. The repulsive parts of the diatomic potentials beyond the ab initio range are taken as a growing exponential form, as r ς,η -→ 0.

The non-additive term, V 123 , is assumed to decay exponentially,

V 123 (Λ 123 ; r, R -→ ∞, θ) ≡ V 123 Λ 123 ; r, R , θ exp R -R κ (r, θ) (135)
or

V 123 (Λ 123 ; r -→ ∞, R, θ) ≡ V 123 Λ 123 ; r , R, θ exp r -r κ (R, θ) (136)
where, r and R are the last ab initio grid point available and the decay rates, κ (r, θ) or κ (R, θ), are estimated using the two last points. Such a methodology gives a smooth transition between the atom-diatom and the atom-atom-atom regimes. For sake of comparisons, we present in the Fig. [START_REF] Van Veldhoven | Decelerated molecular beams for high-resolution spectroscopy[END_REF] several 1D cuts of the Rb + 3 ground state surface, for the original raw ab initio points (circles) and the interpolation method described above (solid lines).

prospects for [rb 2rb] + dynamics The choice of constructing the global PES with a hybrid implementation of many-body expansion plus numerical interpolation, and therefore avoiding any error overhead induced by numerical fitting schemes, is motivated by the need of interaction potentials well suited to describe collisional processes, due to the well known sensitivity of some scattering properties on topological features of potential energy surfaces, in particular, cold and ultracold collisions.

Thus, the knowledge of Rb + 3 PES opens an avenue for theoretical investigations describing [Rb 2 -Rb] + dynamical problems, relevant for a number of experimental situations. As mentioned in the introductory part, typical examples are (i) a single Rb + ion immersed in a relatively dense cloud of ultracold Rb atoms [START_REF] Schmid | An apparatus for immersing trapped ions into an ultracold gas of neutral atoms[END_REF], where three-body recombination has been recently observed for the neutral case [START_REF] Härter | Population distribution of product states following three-body recombination in an ultracold atomic gas[END_REF]; (ii) the interaction between trapped Rb + 2 ions in the presence of neighboring Rb atoms from a magneto optical trap (MOT) [START_REF] Jyothi | Photodissociation of Trapped Rb + 2 : Implications for Simultaneous Trapping of Atoms and Molecular Ions[END_REF]; or, (iii) the use of collisions with Rb atoms in order to internally cool down Rb + 2 ions, since such homonuclear diatomic systems are unable of rovibrational relaxation, within reasonable time scales, due the absence of permanent dipole moments [START_REF] Rellergert | Evidence for sympathetic vibrational cooling of translationally cold molecules[END_REF][START_REF] Thierry Stoecklin | Explanation of efficient quenching of molecular ion vibrational motion by ultracold atoms[END_REF].

A preliminary study of the PES topology, however, helps to shed light on the many challenges which are needed to be proper understood. To this end we have plugged our ground state global PES into the ABC quantum reactive scattering code [START_REF] Skouteris | ABC: a quantum reactive scattering program[END_REF], attempting to describe a typical collision between Rb + 2 2 Σ + g and Rb atoms at ultracold regime of energies. The choice of using quantum reactive formalism instead of standard inelastic scattering will be clarified below.

One shall face a first challenge concerning the propagation of total wavefunctions up to ranges where the Rb + 2 -Rb interaction potential is essentially dominated by the relative angular orbital momentum of the incoming Rb atom around Rb + 2 , ( + 1) R -2 V (R), as R -→ ∞ and V (∞) = 0. Due to the actual long-range ion-induced-dipole attractive interaction regime, varying as ∝ 1/R 4 , the R-range needed to obtain converged total wavefunctions can easily go beyond many dozens of thousands, in units of length. Therefore, it becomes highly recommended to implement a hybrid propagation scheme using, for instance, a standard log-derivative propagator for the short range zone of interactions followed by a more long-range suited algorithm, such as either the Airy-propagator of M. Alexander [START_REF] Alexander | Hybrid quantum scattering algorithms for long-range potentials[END_REF] -see also Ref. [START_REF] Alexander | A stable linear reference potential algorithm for solution of the quantum close-coupled equations in molecular scattering theory[END_REF] -, the modified variable-phase method of F. Gianturco and co-workers [START_REF] Martinazzo | A modified Variable-Phase algorithm for multichannel scattering with long-range potentials[END_REF] or multichannel quantum defect theory [START_REF] Hazra | Quantum defect theory for cold chemistry with product-quantum-state resolution[END_REF]. For the time being, the available ABC implementation uses a constant step size log-derivative formalism. An extra challenge becomes evident from the internal structure of Rb + 2 2 Σ + g -see for instance Table [START_REF] Jin | Collective Excitations of a Bose-Einstein Condensate in a Dilute Gas[END_REF] -where, the harmonic frequency ω e ≈ 50 cm -1 , is relatively small compared to most of the atom-diatom collisions studied to date. Thus, a relatively high number of vibrational components are expected to play a significant role as closed channels in the asymptotic basis chosen to describe the diatom. Our preliminary ABC calculations have delivered not converged elastic S-matrix elements for many different basis sizes, for instance υ max = 7, 8, 9, 10 and j max = 60. It is worthwhile to notice that converged state-to-state cross sections, for H + 2 -He collisions, calculated in the Chapter ( 6) have required a much smaller basis, relying on υ max = 4 solely. By neglecting the anharmonicity of a Rb + 2 2 Σ + g potential energy curve, which supports about υ max ≈ 298 vibrational levels, one can show that more than 100 vibrational components should be taken into account in order to have all relevant closed vibrational states in the coupling matrix. Such an upper limit imposes, therefore, many technical difficulties due to the huge sizes of matrices needed. Moreover, in contrast with the small contribution from the electronic spin-rotation coupling, invoked on Chapter [START_REF] Deutsch | Optical lattices[END_REF], for H + 2 -He collision cross sections, one needs to describe Rb + 2 2 Σ + g basis in terms of N j rotational manifolds, due to its heaviness, which would imply a substantial increasing of the rotational subspace on the total diatomic basis set.

The homonuclear character of the system imposes extra difficulties, due the underlying rearrangement during the collision, inducing conversions between rotational states driven by the Fermi-Dirac statistics, i.e. even-j -→ odd-j (or, para-Rb + 2 + Rb -→ ortho-Rb + 2 + Rb and vice-versa), due to nuclear spin-rotation coupling. By this point, only a full quantum reactive scattering formalism is suited to properly describe the problem. However one should notice that the system becomes nearly impossible to treat -at least by standard numerical methods -since it requires now two basis sets to describe the two possible diatomic species involved (para-Rb + 2 and ortho-Rb +

2 ). Our preliminary reactive scattering calculations, taking into account 10 vibrational levels in the basis set (υ max = 10, j max = 60), for each diatom, as a pseudo-1 Σ diatomic electronic state, does not reach the accuracy needed, in order to have converged state-to-state cross sections and, therefore, it still remains an open question if the rearrangement-induced ortho-to-para conversion is non-negligible. For experiments aiming at internal cooling of trapped molecular ions by means of collisions with ultracold Rb atoms, such a reactive branch would lead to unfortunate trap losses. We have performed tests for total angular momentum J + = 0 and both diatomic parities, ortho and para, leading to initial basis sets already well above the average size used for lighter species, typically 1000 × 1000 channels. It is important to note that comparisons with experimental data would require many partial waves for both triatomic parities, J + and J -.

From the electronic manifold point of view, there are extra problems to overcome. As shown in the Fig. [START_REF] Hudson | Measurement of the Electron Electric Dipole Moment Using YbF Molecules[END_REF] and Fig. [START_REF] Kozlov | Enhancement of the Electric Dipole Moment of the Electron in PbO[END_REF], two surfaces (the respective singlet and triplet ground states) are correlated with the same Rb + 2 2 Σ + g + Rb(5s) colliding partners, where both potential wells are relatively superimposed mainly at collinear geometries. For some scattering conditions, such as hot-collisions or highly-excited entrance channels on the singlet ground state, one may need, therefore, to involve a more general basis set, spanned over the eigenstates of both PES. If, on the other hand, one wants to describe collisions between Rb 2 1 Σ + g and Rb + , driven by the third electronic state of Rb + 3 , namely the 2 1 A 1 PES, all the problematics discussed above are enhanced by the evident fact that two ground states (1 1 A 1 and 1 3 A 1 ) are now open, adding an extra reactive branch such as charge exchange by collision, ortho/para-Rb 2 + Rb + -→ ortho/para-Rb + 2 + Rb. The energy splitting between the two electronic manifolds supporting Rb 2 and Rb + 2 states is, however, of the order of 4000 cm -1 and no strong non-adiabatic interactions are expected -or, topologically observed between the two PES sheets. Finally, isotopic substitutions processes can be also studied by properly setting the diatomic center-of-mass, given by Eq. ( 130) in the PES coordinate definition, opening a way to explore reactive collisions such as 85 Rb 87 Rb + + 85 Rb -→ 85 Rb + 2 + 87 Rb. In such a particular case, the limitations imposed by the Fermi-Dirac statistics are of no concern, since rotational states are now allowed to have any value. It is worthwhile to notice, however, that the many drawbacks imposed either by neutral-charged long range interactions or the heaviness of the system are still valid. As shown in the Fig. [START_REF] Hudson | Measurement of the Electron Electric Dipole Moment Using YbF Molecules[END_REF], and briefly commented above, in particular at collinear geometries where the two arrangements becomes evident, the rearrangement process is barrierless, with the two yellow zones interconnected by the deep minima of the Rb + 3 complex -see the red zones at short-ranges of Fig. [START_REF] Hudson | Measurement of the Electron Electric Dipole Moment Using YbF Molecules[END_REF]. For Rb 2 + Rb + case, driven by the 2 1 A 1 electronic state, the rearrangement process is, however, limited by the presence of a significant large barrier between reactants and products (the two soft-yellow zones) -see for instance Fig. [START_REF] Kawall | Precision Zeeman-Stark Spectroscopy of the Metastable a(1) 3 Σ+ State of PbO[END_REF] at the short-range zone of collinear geometries.

conclusions

We have reported an ab initio study of the first twenty electronic states of Rb + 3 molecular ion. The lowest two electronic states 1 A 1 and 3 A 1 correlated, asymptotically, with Rb + 2 2 Σ + g + Rb(5s). A promising result for the possibility of theoretically studying reactive and inelastic Rb + 2 -Rb collisions, shedding light into many experimental context; for instance, the internal cooling of Rb + 2 2 Σ + g molecular ions, formed into hybrid atom-ion traps, by collisions with ultracold gases of Rb atoms, including sympathetic cooling. To this end, we have provided highly-accurate interaction energy surfaces expecting that these results will be beneficial for performing scattering calculations to study collisions in the ultra-low temperature. Note that collisional properties are strongly sensitive for the quality of the interaction potential and, therefore we have studied the low-and high-spin electronic ground states using two high-level ab initio methods: FCI and RCCSD(T). Both, the lowest electronic states, were also briefly studied in the point of view spectroscopy and interactions in an external fields: rotational constants, harmonic frequencies as well as electric properties were estimated. A comparison of the obtained results reveals the high efficiency for the FCI approach confirmed by relatively expensive RCCSD(T) method.

A brief qualitative discussion concerning the main challenges of further scattering studies is also carried out, where the atom-diatom long range interaction, the heavi-ness of the system and its homonuclear symmetry imposes many computational limitations in order to proper describe the underlying collisional physics. The presence of many sources of light currently used on the implementation of a hybrid trap of ultracold atoms and a trap of laser-cooled ions is known to have a strong influence on the measured rates, as inelastic processes with excited species may be very large -see for instance Ref. [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF][START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF][START_REF] Sullivan | Role of Electronic Excitations in Ground-State-Forbidden Inelastic Collisions Between Ultracold Atoms and Ions[END_REF]. Furthermore, such -quite intense -light could prevent the produced molecular ions to be detected before being photodissociated. Most recently, S. Jyothi and co-workers have probed the possibility of simultaneous trapping of 8 5 Rb + 2 molecular ions with ultracold 8 5 Rb atoms in a magneto-optical trap (MOT) [START_REF] Jyothi | Photodissociation of Trapped Rb + 2 : Implications for Simultaneous Trapping of Atoms and Molecular Ions[END_REF], where a rapid decay of the measured molecular ion signal has been observed, with an expected lifetime of about 495±80 ms in the presence of lasers with 780.2 nm wavelength (the cooling light for the Rb MOT).

Among many experimental setups developed, in order to implement atom-ion hybrid setups, the direct observation of molecular ion formation (namely RbCa + , RbBa + and Rb +

2 ), in Rb-Alke + collisions (where, Alke = Ca and Ba), is currently achieved only in the seminal works of F. Hall et al. [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF][START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF], in the team headed by Prof. Stefan Willitsch (University of Basel). One of the main open questions nowadays relies, therefore, on the lifetime of such molecular species in the presence of many sources of light, e.g. cooling lasers, repumping lasers etc. It is worthwhile to notice unique features of the experimental implementation of Basel group, which may eventually prevent an effective photodissociation process, for instance the usage of a mechanical chopper at a frequency of 1 kHz, blocking the Ca + and Rb cooling lasers, in order to prevent excitations and to ensure, thus, atom-ion collisions with colliding partners in their respective ground state [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF].

The typical density of ultracold Rb atoms, trapped in a magneto-optical trap (MOT), used in Basel experiment, is of the order of 1 × 1 0 9 cm -3 at temperatures ranging between 150 and 200 µK, where only two-body (2B) collisions are observed [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF]. In contrast, A. Kr ükow and co-workers from the group of Prof. J. H. Denschlag (University of Ulm), have recently observed two-and three-body collisions (3B) of a single Ba + ion into a relative dense cloud (≈ 1 0 1 2 cm -3 ) of ultracold Rb atoms [START_REF] Krükow | Reactive two-body and three-body collisions of Ba + in an ultracold Rb gas[END_REF], in a combination of a Bose-Einstein condensation apparatus and a linear Paul trap [START_REF] Schmid | An apparatus for immersing trapped ions into an ultracold gas of neutral atoms[END_REF].

a case study: photodissociation dynamics of weakly-bound rbba + By mapping out the Ba + loss rate dependence on the Rb atom density A. Kr ükow et al. can resolve two-body reactive collisions from three-body ones and determine both rate coefficients, where ternary recombination dominates over binary reactions even at moderate atom densities: Rb(5s) + Rb(5s) + Ba + (6s). Assuming a non-thermal ion energy distribution, the three-body recombination rate agrees with theoretical predictions, ∝ E -3/4 coll (with E coll the collision energy), determined by at least two different micromotion-induced energy scales. By means of molecular dynamics classical trajectory calculations, A. Kr ükow and co-workers have found evidences that three-body recombination leads to formation of molecular ions scaling with the collision energy (E coll , of a few millikelvin), being therefore a weakly-bound molecule [START_REF] Krükow | Energy Scaling of Cold Atom-Atom-Ion Three-Body Recombination[END_REF] (see also Ref. [START_REF] Perez-Rios | Communication: Classical threshold law for ion-neutral-neutral three-body recombination[END_REF]), i.e. binding energies of the same order of magnitude of E coll . In Ulm experiment, however, no direct signal of molecular ions is observed and its absence may be attributed to a shortened molecular lifetime in the presence of light, in particular, photons with 1064 nm wavelength for a continuous and relatively intense crossed dipole trap beam.

In this work, we shall use accurate quantum chemistry calculations -described previously in the Section (4.3) -to compute the electronic structure and all relevant transition and permanent electric dipole moments of RbBa + , in order to model a weakly-bound molecular ion produced within the context of the work of Ulm group, that is Rb(5s) + Rb(5s) + Ba + (6s) -→ RbBa + 2 1 Σ + ; υ 0, J + Rb(5s) [START_REF] Lee | Measurement of collisions between rubidium atoms and optically dark rubidium ions in trapped mixtures[END_REF] in the presence of photons with 1024, 650, 614 and 493 nm wavelength, from the many different sources of light. We shall probe the lifetime of a weakly-bound molecular ion in the presence of light for a few different experimental conditions and see that, besides an overall good agreement between experimental and theoretical total absorbing cross sections, agreement between state-to-state cross sections may be found only assuming deeply-bound species, in contrast with Ulm's weakly-bound assumption mentioned earlier. Such a strong evidence may indicate that the internal structure of the initial weakly-bound molecular ion may be strongly perturbed either by subsequent collisions with ultracold Rb atoms, i.e. internal cooling; or its own permanent electric dipole moment; or, spontaneous emission towards the electronic ground state manifold, (1 ←-2) 1 Σ + .

new experimental evidences

New experimental evidences are available [A. Mohammadi et al., private communication], using the original setup of A. Kr ükow's work [START_REF] Krükow | Reactive two-body and three-body collisions of Ba + in an ultracold Rb gas[END_REF], where a single laser cooled 138 Ba + (confined in a linear Paul trap) is immersed into three different regimes of densities (≈ 10 11 , ≈ 10 12 and ≈ 10 13 cm -3 ) of ultracold 87 Rb atoms (about 750 nK). In these new experiments the ultracold atoms are held, now, in the presence of either on or off crossed dipole trap beams. When no light is present, however, the atomic cloud is allowed to expand and naturally heats up, which implies a time-dependent atomic density description. We shall, therefore, investigate the influence of light at time scales such that atomic heating is hopefully of no concern. All the experimental details concerning the setup calibration, cooling/trapping schemes, detection methods, measurement and results from A. Mohammadi's work is currently under preparation to be published elsewhere soon and we shall, herein, focus on its main findings, which are as follow:

(i) In the absence of 1064 nm wavelength lasers (switched off a few hundred µs before the atomic cloud interacts with the ion), a clear signal assigned to different charged species, as a function of interaction time, is observed. Not surprising, besides a RbBa + signal (of 30%) being directly measured, for the second time since the seminal work of F. Hall et al. [START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF], Rb + ions (45%) and Rb + 2 (23%) are also observed. (ii) In the presence of light, in all atomic densities probed, the signal of initial cold Ba + ions presents the same exponential-like decay, as a function of the interaction time. This might be taken as an evidence that reactive/inelastic events happens independently of the presence of light. However, only at the presence of light, a 3:2 ratio of hot Ba + /Rb + ions is observed instead (with residual 2% of initial cold Ba + ions), indicating an effective photodissociation process driven by absorbing photons with 1064 nm wavelength. In the following, the use of "hot" or "cold" notation stands for the needed (or not) of re-cooling the respective atomic products, e.g. hot Ba + meaning that a re-colling scheme was applied after the atom-ion interaction step.

(iii) By monitoring the signal decay of Ba + ions, parametrically for different light intensities, and fitting the experimental data, A. Mohammadi can estimate an experimental total photodissociation cross section of σ exp = (1 ± 0.2) × 10 -19 cm 2 , which is now subject to theoretical comparisons. electronic structure calculations for [rb-ba] + complex The potential energy curves (PEC), permanent electric dipole moments (PEDM) and transition electric dipole moments (TEDM) are computed following the same method described in details in Ref. [START_REF] Aymar | Structure of the Alkali-metal-atom-Strontium molecular ions: towards photoassociation and formation of cold molecular ions[END_REF][START_REF] Aymar | Calculation of accurate permanent dipole moments of the lowest 1,3 Σ + states of heteronuclear alkali dimers using extended basis sets[END_REF][START_REF] Aymar | Comment on Calculation of accurate permanent dipole moments of the lowest 1,3 Σ + states of heteronuclear alkali dimers using extended basis sets[END_REF] and in the Section (4.3). Briefly, however, we recall its main features. All the calculations are carried out using the Configuration Interaction by Perturbation of a Multiconfiguration Wave Function Selected Iteratively (CIPSI) [START_REF] Huron | Iterative perturbation calculations of ground and excited state energies from multiconfigurational zeroth-order wavefunctions[END_REF] package developed at "Laboratoire de Chimie et Physique Quantiques, Université Toulouse III -Paul Sabatier" (France). The [Rb-Ba] + complex is modeled as a molecule with two valence electrons moving in the field of the Rb + and Ba 2+ ions represented with effective core potentials (ECP) including relativistic scalar effects, taken from Ref. [START_REF] Durand | New atomic pseudopotentials for electronic structure calculations of molecules and solids[END_REF][START_REF] Durand | A theoretical method to determine atomic pseudopotentials for electronic structure calculations of molecules and solids[END_REF] and Ba 2+ [START_REF] Fuentealba | Pseudopotential calculations for alkaline-earth atoms[END_REF][START_REF] Fuentealba | Pseudopotential calculations on the ground state of the alkaline-earth monohydride ions[END_REF]. The ECPs are complemented with core polarization potentials (CPPs) depending on the orbital angular momentum of the valence electron [START_REF] Müller | Treatment of intershell correlation effects in ab initio calculations by use of core polarization potentials. Method and application to alkali and alkaline earth atoms[END_REF][START_REF] Foucrault | Nonpertubative method for corevalence correlation in pseudopotential[END_REF], and parametrized with the Rb + and Ba 2+ static polarizabilities α and two sets of three cut-off radii ρ s , ρ p , and ρ d . All parameters determined in our previous works on alkali-atom-strontium neutral molecules [START_REF] Guérout | Ground state of the polar alkali-metalatom-strontium molecules: Potential energy curve and permanent dipole moment[END_REF] and on alkaline-earth hydride ions [START_REF] Aymar | The electronic structure of the alkaline-earth-atom (Ca, Sr, Ba) hydride molecular ions[END_REF]. Only the remaining two valence electrons are used to calculate the Hartree-Fock and the excitation determinants, in an atomcentered Gaussian basis set, through the usual self-consistent field methodology. The basis set used for the Rb atoms are from Ref. [START_REF] Aymar | Calculation of accurate permanent dipole moments of the lowest 1,3 Σ + states of heteronuclear alkali dimers using extended basis sets[END_REF][START_REF] Aymar | Comment on Calculation of accurate permanent dipole moments of the lowest 1,3 Σ + states of heteronuclear alkali dimers using extended basis sets[END_REF] and from Ref. [START_REF] Aymar | The electronic structure of the alkaline-earth-atom (Ca, Sr, Ba) hydride molecular ions[END_REF][START_REF] Bouissou | A comprehensive theoretical investigation of the electronic states of Ca2 up to the Ca(4s(2) 1S) + Ca(4s5p 1P) dissociation limit[END_REF] for the Ba + ion. A full configuration interaction (FCI) is finally achieved to obtain the relevant PECs, PEDMs and TEDMs. See Chapter (4) for a somewhat extensive discussion concerning the comparison between these calculations and the ones available in the literature.

Figure [START_REF] Ph | Energy levels and two-photon transition probabilities in the HD + ion[END_REF] shows the first few potential energy curves of the complex [Rb-Ba] + , as a function of the internuclear distance R, for different molecular symmetries, where the zero energy is taken coinciding with the (2) 1 Σ + asymptote, correlated with Rb(5s) + Ba + (6s), as R -→ ∞, which are the respective atomic species in those quantum states prepared by A. Kr ükow et al. [START_REF] Krükow | Reactive two-body and three-body collisions of Ba + in an ultracold Rb gas[END_REF]. Thus, after a three-body recombination we shall assume a weakly-bound molecular ion created either on (2) 1 Σ + or (1) 3 Σ + diatomic electronic states, with a statistical weight of g = 1/4 and g = 3/4, respectively. In particular, the 2 1 Σ + electronic state presents a strong avoided crossing with upper states leading to a pronounced double-well feature, which supports about 174 vibrational levels (at J = 0), and dissociation energies of 964 and 679 cm -1 (in the respective equilibrium distances of each potential well), being therefore, well shallow if compared to its triplet counter part, (1) 3 Σ + , or the electronic ground state, (1) 1 Σ + . The (1) 3 Σ + state, however, with a potential well of 7165 cm -1 depth, supports about 322 vibrational levels, and plays a fundamental role in the problem since it behaves as a local ground state, whereas the molecular ions described by the (2) 1 Σ + electronic state can undergo spontaneous emission towards the singlet ground state, (1 ←-2) 1 Σ + . Notice that contributions from the magnetic transition dipole moment inducing forbidden singlet-to-triplet transitions, if any, are neglected.

In experimental conditions, a crossed dipole trap beam with 1064 nm wavelength, at relatively high power, is always shinning the eventual products of a three-body recombination process [START_REF] Schmid | An apparatus for immersing trapped ions into an ultracold gas of neutral atoms[END_REF], being therefore, the first source of light to be investigated. We shall consider two scenarios:

• In the presence of lasers with 1064 nm wavelength, weakly-bound molecular ions, either at the (2) 1 Σ + or the (1) 3 Σ + entrance channels, are photodissociated straight away, leading to formation of a number of different atom-ion species within the upper limit imposed by the total transition energy of E 1064nm = 9396 cm -1 . Figure [START_REF] Schiller | Tests of time independence of the electron and nuclear masses with ultracold molecules[END_REF] summarizes PECs and TEDMs used to describe such a scenario.

• In the absence of light, as shown in the scheme given in the Fig. [START_REF] Chin | Enhanced Sensitivity to Fundamental Constants In Ultracold Atomic and Molecular Systems near Feshbach Resonances[END_REF], we shall first evaluate the radiative lifetime of an initial weakly-bound molecular state, either at the (2) 1 Σ + or the (1) 3 Σ + entrance channels, against internal relaxation towards lower levels in the same electronic manifold, due to the presence of a PEDM; and, in those particular cases of (2) 1 Σ + molecules, their spontaneous emission towards the singlet ground state, (1) 1 Σ + , coupled by the respective TEDM, (1 ←-2) 1 Σ + -see Chapter (4) for more details. If that is the case, a sub- sequent photodissociation process would be modeled from each rovibrational level, in the singlet ground state, properly populated after the spontaneous emission, as well as from those levels belonging to the triplet ground state.

state-to-state absorbing cross sections

At collision energies assumed to be of the order of E coll = 1 mK, chosen in agreement with the overall temperatures during the collisions performed in the experiment of Ulm [START_REF] Krükow | Reactive two-body and three-body collisions of Ba + in an ultracold Rb gas[END_REF][START_REF] Krükow | Energy Scaling of Cold Atom-Atom-Ion Three-Body Recombination[END_REF], the 3B process is not described here and, due the lack of information about how such a process happens, it is considered as a disconnected event well before the photodissociation process itself, resulting in a single weakly-bound molecular state, (2) 1 Σ + ; υ max , J or (1) 3 Σ + ; υ max , J , with binding energy of the same order of magnitude of E coll [START_REF] Krükow | Energy Scaling of Cold Atom-Atom-Ion Three-Body Recombination[END_REF]. Where, υ max is the highest vibrational quantum number at a given rotational manifold J within the initial electronic state, (2) 1 Σ + or (1)

+ 3 Σ + 1 Π 3 Π 1 ∆ 3 ∆ Figure 41: (6s) 1 Σ
The potential energy curves of the complex [Rb-Ba] + , as a function of the internuclear distance R, for different molecular symmetries. The zero energy is taken coinciding with the 2 1 Σ + asymptote, correlated with Rb(5s) + Ba + (6s), as R -→ ∞, which are the respective atomic states prepared experimentally [START_REF] Krükow | Reactive two-body and three-body collisions of Ba + in an ultracold Rb gas[END_REF].

- Transition electric dipole moment (a. u.) It is assumed, previously, a spontaneous emission (green curve) from the original entrance channel, 2 1 Σ + , correlated with the respective initial atomic states prepared experimentally [START_REF] Krükow | Reactive two-body and three-body collisions of Ba + in an ultracold Rb gas[END_REF]. Middle panel: all the transition electric dipole moments between the respective electronic states for Σ -→ Σ and Σ -→ Π transitions. Only singlet electronic states are displayed due the forbidden singlet-to-triplet transitions. Lower panel: vibrational distribution created by spontaneous emission, from the 2 1 Σ + entrance channel toward the 1 1 Σ + manifold (υ max ∼ 248) at J = 0, as a function of the vibrational quantum number.

+ 3 Σ + 1 Π 3 Π -2 0 2 4 Σ + Σ + (b) (2-3) 1 Σ + (2-4) 1 Σ + (2-5) 1 Σ + (a) 1 Σ
Σ + Π (c) (2) 1 Σ + -(2) 1 Π (2) 1
Σ + -(3) 1 Π (1) 3
Σ + -(2) 3 Π (1)
+ (1) 1 Σ + -(1) 1 Π (1-3) 1 Σ + (1) 1 Σ -(2) 1 Π (1) 1 Σ + -(3) 1 Π (1-4) 1 Σ + ( (1-2) 1 Σ
The probability of a (2) 1 Σ + ; υ max , J level spontaneously decaying towards the lev- els on the electronic ground state manifold, (1 ←-2) 1 Σ + , is the Einstein probability of emission, A 1←2 ,

A 1←2 = 8πhν 3 1→2 c 3 B 1→2 (138)
where, ν 1→2 is the frequency of an emitted photon, h is the Planck's constant, c is the speed of light and B 1→2 is the Einstein probability of absorbing a photon,

B 1→2 = B 1←2 = 8π 3 3h 2 η 1 | D (R) |η 2 2 (139)
with, |η 1 and |η 2 chosen properly, e.g. |η 2 = (2) 1 Σ + ; υ max , J and |η 1 represent- ing all bound plus continuum levels in the (1) 1 Σ + manifold; and D (R) being the transition electric dipole moment which couples the two electronic states. Therefore, the population, N η 2 , of the initial level |η 2 , in an instant t, is simply

dN η 2 dt = -∑ η 1 A η 1 ←η 2 N η 2 (140)
And its radiative lifetime, τ 2 ,

τ 2 = 1 ∑ η 1 A η 1 ←η 2 (141)
Thus, in the first scenario of hypothesis assuming a photodissociation process well beyond the upper limit given by τ 2 , the final population of singlet molecular ions shall be considered in the (1) 1 Σ + electronic manifold, with individual levels populated accordingly. That is, taking N η 1 as the population in a given η 1 bound level,

N η 1 = A η 1 ←η 2 ∑ η 1 A η 1 ←η 2 (142)
where, bound-to-continuum transition probabilities are not taken into the normalization of Eq. ( 142) and, therefore, are taken as a loss of molecular ions during the spontaneous emission, estimated on the order of 10%, producing Rb + + Ba(6s 2 ).

In the case of our fist scenario described in the last section, the state-to-state absorbing cross section summed over all rotational levels, from a unity-normalized single vibrational level, |Λ; υ max , J with Λ = (2) 1 Σ + and Λ = (1) 3 Σ + , towards an energy- normalized continuum level in the electronic manifold Λ Λ = 1 Σ + , 3 Σ + , 1 Π, 3 Π , at a given transition energy E trans , where the photodissociated species has a final associated wavevector k, such that h2 k (143) with S (J , J) being the usual H önl-London coefficients for Σ -→ Σ and Σ -→ Π transitions [START_REF] Hansson | A comment on H önl-London factors[END_REF], µ the RbBa + reduced mass and g a statistical weight of g = 1/4 and g = 3/4, for initial singlet states and triplet states, respectively. Transitions given by Eq. ( 143) are defined only for those h2 k 2 > 0, otherwise σ Λ ←Λ (E trans ) = 0, i.e. closed transitions. Besides considering an initial collisional energy of 1 mK, as we shall see below, new experimental evidences demonstrate qualitatively that collisional products are observed rotationally excited and, therefore, the initial number of total J values included in Eq. ( 143), J max , is unknown.

2 = 2µ [E trans -V (Λ , R = ∞)], is given by σ Λ ←Λ (E trans ) = g 8π 3 3h 2 J max ∑ J ∑ J ν Λ ,J ,k;Λ,υ,J S J , J Λ ; k D (R) |Λ; υ max | 2
In the case of our second scenario, the photodissociation calculations are performed from the electronic ground state manifold, Λ = (1) 1 Σ + , as discussed in the previous section, where all rovibrational levels are taken into account instead (υ max = 248), thus Eq. ( 143) becomes

σ Λ ←Λ (E trans ) = g 8π 3 3h 2 υ max ∑ υ J max ∑ J ∑ J ν Λ ,k;Λ,υ,J N υ,J S J , J Λ ; J , k D (R) (1) 1 Σ + ; υ 2 ( 144
)
All the unity-normalized rovibrational wavefunctions and energy-normalized radial continuum wavefunctions are computed with the Milne phase-amplitude method [START_REF] Milne | The Numerical Determination of Characteristic Numbers[END_REF]166], for R ∈ (0, 2000) atomic units. All the diatomic potentials are extrapo- lated beyond the ab initio domain using the familiar functional form based on the ion-induced-dipole potential for atom-ion interactions, varying as ∝ 1/R 4 , with leading C 4 terms, lim

R-→∞ V (R) ≈ D e - C 4 R 4 (145)
where, D e are the respective dissociation energies. Table [START_REF] Demarco | Onset of Fermi Degeneracy in a Trapped Atomic Gas[END_REF] summarize all the D e and C 4 parameters estimated, by fitting Eq. [START_REF] Zygelman | The radiative association of He(+) and H[END_REF] to the ab initio tail potential, for each electronic state.

theoretical results

Besides being still a work in progress, some preliminary calculations in the presence of light with 1064 nm wavelength, indicates that cross sections of the same order of magnitude as those measured by A. Mohammadi are possible only by considering a relatively deeply-bound molecular ion. Thus, the current ongoing work tends to demonstrate that after a 3B process, the molecular ion may undergo an efficient internal cooling and, the most reasonable explanation for such an internal quenching, rely on successive collisions with the remaining ultracold Rb Atoms.

In the absence of light during the collisions, those weakly-bound molecular ions described in the (2) 1 Σ + entrance channel are predicted to decay towards the singlet ground state, (1) 1 Σ + . Using the same spontaneous emission methodology applied previously, in the Chapter (4), we describe the (1 ←-2) 1 Σ + transition for a single partial wave, and from Eq. ( 141), a lifetime estimation within at least 30 ms for the highest rovibrational levels is retrieved. Figure [START_REF] Demille | Enhanced Sensitivity to Variation of me/mp in Molecular Spectra[END_REF] presents the radiative lifetime of the highest four vibrational levels, in the (2) 1 Σ + entrance channel, with binding ener- gies smaller than 1 mK. Thus, considering the (1) 1 Σ + levels to be populated, as shown in the lower panel of Fig. [START_REF] Chin | Enhanced Sensitivity to Fundamental Constants In Ultracold Atomic and Molecular Systems near Feshbach Resonances[END_REF], in the following, we shall probe the photodissociation cross section at total transition energies of E 650nm = 15380.6 cm -1 , E 614nm = 16282.2 cm -1 and E 493nm = 20278.6 cm -1 , in the presence of photons with 650 nm, 614 nm and 493 nm wavelength. In contrast with those lasers, the transition energy available by absorbing photons with 1064 nm wavelength is not enough to reach higher electronic manifolds, having therefore, one single photodissociation exit channel: boundto-continuum transitions in the same electronic manifold with relatively high momentum hk, i.e. a small probability of producing only Rb + + Ba 6s 2 .

We shall focus on a description of an initial rotationless molecule, J = 0 (and, therefore, J = 1), accounting only to vibrational levels populated by spontaneous emission, in order to check qualitatively the hierarchy of the products if compared with the A. Mohammadi's observations. The total cross section is therefore summed over all the vibrational contributions available. Typical examples of such vibrationally summed photodissociation cross sections in a wide range of transition energies are shown in the Fig. [START_REF] Zelevinsky | Precision Test of Mass-Ratio Variations with Lattice-Confined Ultracold Molecules[END_REF].

As we can see in the Table [START_REF] Phillips | Laser cooling and electromagnetic trapping of neutral atoms[END_REF], at total transition energies of E 650nm = 15380.6 cm -1 and E 614nm = 16282.2 cm -1 , deeper vibrational levels are unable to contribute for tran-

Σ + → (3) 1 Σ + (1) 1 Σ + → (4) 1 Σ + (1) 1 Σ + → (2) 1 Π
Figure 45: Vibrationally summed total cross sections at J = 0, as a function of the transition energy for three given transitions from the (1) 1 Σ + electronic manifold upward. The energies of two given lasers, with 650 nm and 614 nm wavelength respectively, are tagged by the dashed lines.

sitions towards higher electronic states -i.e. binding energy plus transition energy < V (Λ ; R = ∞) -except for the case of (1 -→ 2) 1 Σ + , producing Rb(5s) + Ba + (6s).

The majority of the levels, however, contributes to higher transitions dominated by (1 -→ 4) 1 Σ + and (1) 1 Σ + -→ (2) 1 Π, producing essentially Ba + (5d). In those cases of transition energies of E 493nm = 20278.6 cm -1 -shown in the Table [START_REF] Phillips | Nobel Lecture: Laser cooling and trapping of neutral atoms[END_REF] -, all the products described up to now are possible, dominated once more by (1 -→ 5) 1 Σ + and (1) 1 Σ + -→ (3) 1 Π, transitions leading to Rb + + Ba 6s6p, 1 P . In this particular case, at the presence of photons with 493 nm wavelength, the formation of Rb + is expected to be dominant, as well as Ba + formation dominant at the presence 650 nm photons. Preliminary experimental data (not shown, to be published elsewhere soon) shows the exact same discrepancy as before, with more hot Ba + formation in the presence of 493 nm light and more hot Rb + for 650 nm. It is worthwhile to notice, however, that the above description clearly ignores the contributions of those molecules described by the (1) 3 Σ + electronic states. Transition energies such as E 493nm = 20278.6 cm -1 , for instance from the (1) 3 Σ + electronic manifold, would have required even higher PECs and TEDMs, which becomes laborious and less accurate to obtain at highly excited asymptotes, for the current quantum chemistry standards. Nevertheless, by inspection of the scheme presented in the Fig. [START_REF] Ph | Energy levels and two-photon transition probabilities in the HD + ion[END_REF], we expect transitions either from the (2) 1 Σ + or the (1) 3 Σ + manifold towards the highest triplet molecular states, lead- ing to Rb(5p) + Ba + (6s) formation to be non-negligible and even dominant (with a smallest hk compared to lower asymptotes), for instance at total transition energies of E 650nm = 15380.6 cm -1 , which agrees experimentally.

Conclusions

We have presented a theoretical investigation about the possible destruction of a weakly-bound RbBa + molecular ion produced by three-body recombination in a hybrid atom-ion trap, by a photodissociation mechanism, due to the many lasers involved in a usual experimental setup. The potential energy curves and all the relevant permanent and transition electric dipole moments are calculated in a semiempirical methodology, where both atoms are represented by a single valence electron moving in the field of a relativistic effective core potential with a core polarization potential correction, followed by a full configuration interaction. With these data we have com-

(5d) (1) 1 Σ + →(5) 1 Σ + - - 0.0 0.0 Rb + + Ba(6s6p, 1 P) (1) 1 Σ + →(3) 1 Π - - 0.0 0.0 Rb + + Ba(6s6p, 1 P) Total - -1.17×10 -17 - -
Table 17: Vibrationally summed photodissociation cross sections (in cm 2 ) for transitions between the (υ, J = 0)-levels, in the singlet electronic ground state, (1) 1 Σ + , of RbBa + at total transition energy of E 614nm = 16282.2 cm -1 . A population for the set of vibrational states is assumed after a spontaneous emission from the initial weakly-bound level (υ max = 174, J = 0) belonging to the first excited electronic state, (2) 1 Σ + (the collisional entrance channel).

found evidences that after a 3B process, such molecular ion may undergo an efficient internal cooling. In particular, when exposed to photons with 1064 nm wavelength, we have probed an effective shortened lifetime, if molecular ions are considered relatively deeply-bound. To this end, the most reasonable explanation rely on the fact that a weakly-bound molecular ion is not the correct assumption, in order to explain the experimental conditions. Moreover, despite an extreme slow internal relaxation due to the presence of a permanent electric dipole moment, at a reasonably time scales, the internal structure of those molecules may be still strongly influenced by extra collisions with neighboring ultracold Rb atoms, favored by higher atomic densities and relatively long atom-ion interaction time.

a case study: photodissociation dynamics of weakly-bound rbba + Experiment Theory

λ = 493 nm Vib. states involved σ (cm 2 ) σ (cm 2 ) σσ total (%) Products (1) 1 Σ + →(2) 1 Σ + - -Negligible 0 Rb(5s) + Ba + (6s) (1) 1 Σ + →(3) 1 Σ + 0 ≤ υ ≤ 247 - 4.8×10 -19 0.4 Rb + + Ba(6s5s, 1 D) (1) 1 Σ + →(1) 1 Π 0 ≤ υ ≤ 247 - 1.7×10 -24 1.5×10 -6 Rb + + Ba(6s5s, 1 D) (1) 1 Σ + →(4) 1 Σ + 9 < υ ≤ 247 - 5.6×10 -18 4.9 Rb(5s) + Ba + (5d) (1) 1 Σ + →(2) 1 Π 0 ≤ υ ≤ 247 - 1.4×10 -18 1.2 Rb(5s) + Ba + (5d) (1) 1 Σ + →(5) 1 Σ + 56 < υ ≤ 247 - 3.3×10 -17 28.6 Rb + + Ba(6s6p, 1 P) (1) 1 Σ + →(3) 1 Π 56 < υ ≤ 247 - 7.4×10 -17 65.1 Rb + + Ba(6s6p, 1 P) Total - 5.2×10 -18 1.14×10 -16 - -
Table 18: Vibrationally summed photodissociation cross sections (in cm 2 ) for transitions between the (υ, J = 0)-levels, in the singlet electronic ground state, (1) 1 Σ + , of RbBa + at total transition energy of E 493nm = 20278.6 cm -1 . A population for the set of vibrational states is assumed after a spontaneous emission from the initial weakly-bound level (υ max = 174, J = 0) belonging to the first excited electronic state, (2) 1 Σ + (the collisional entrance channel).

Part IV

O U T L O O K 10 C O N C L U S I O N
During this work we have explored three aspects of molecular ions in hybrid traps merging an ultracold atom trap and a laser-cooled ion trap: (i) formation, (ii) destruction and (iii) internal cooling in collisions with ultracold atoms. Radiative emission during binary ultracold collisions between Rb atoms and alkaline-earth ions (Ca + , Sr + , Ba + ) and Yb + ions, as well as between Li atoms and Ca + /Yb + ions, are studied theoretically, using accurate effective core-potential-based quantum chemistry calculations for potential energy curves and transition dipole moments of the related species.

Radiative association of molecular ions is predicted to occur for all systems with a cross section two to ten times larger than the radiative charge transfer one. Partial and total rate constants are also calculated and compared to available experiments. Narrow shape resonances are expected, which could be detectable at low temperature with an experimental resolution at the limit of the present standards. Vibrational distributions are also calculated, showing that the final molecular ions are not created in their ground state level, as well as dozens of partial waves are required in order to reach converged cross sections. Agreement between theory and experiments, however, has been found limited by the sensitivity, of such processes of formation, on the actual underlying physical conditions of each experimental setup and, therefore, the choice of the most suited theoretical model invoked to describe them. In particular, it becomes crucial to distinguish the regime of atomic densities used and the actual interaction time between atoms and ions. As a typical example given all along this work, those experiments carried out in the team of Prof. Stefan Willitsch (University of Basel), where an ultracold atom cloud with densities around ≈ 10 9 cm -3 is trapped in a magnetic optical trap (whereas, the laser-cooled ions are kept in a Coulomb crystal), have induced only binary collisions; in contrast with those experiments carried out in the team of Prof. Johannes Hecker Denschlag (University of Ulm), with larger densities of atoms (≈ 10 12 cm -3 , sometimes ≈ 10 13 cm -3 ) trapped in a crossed dipole trap beam in contact with a single ion, showing also three-body recombination processes. To this end, we shall recall some prospects concerning the trap of ultracold atoms and a trap of laser-cooled ions discussed previously in the Chapter (4): (i) High atomic densities and three-body collisions, as it has been recently probed [START_REF] Härter | Single ion as a three-body reaction center in an ultracold atomic gas[END_REF][START_REF] Krükow | Reactive two-body and three-body collisions of Ba + in an ultracold Rb gas[END_REF][START_REF] Krükow | Energy Scaling of Cold Atom-Atom-Ion Three-Body Recombination[END_REF][START_REF] Härter | Population distribution of product states following three-body recombination in an ultracold atomic gas[END_REF], may contribute to formation of molecular ions, as well as to their destruction if they are radiatively created. Experiments of Ref. [START_REF] Schmid | Dynamics of a Cold Trapped Ion in a Bose-Einstein Condensate[END_REF] on Rb-Ba + and of Ref. [START_REF] Zipkes | Cold Heteronuclear Atom-Ion Collisions[END_REF][START_REF] Zipkes | Cold Heteronuclear Atom-Ion Collisions[END_REF] on Rb-Yb + both use an almost Bose-condensed atomic sample, with similar regimes of densities, and have not observed yet any molecular ions.

(ii) During the overlap between the atom cloud and the ions, collisions between surrounding ultracold Rb atoms and the formed molecular ions could well occur, and they are strongly exoergic for all systems studied here -see Table (4) -, assuming that there is no activation potential barrier in the triatomic collisional complex which would prevent the reaction to occur. One product would be Rb + 2 ions which are indeed observed in the Rb-Ca + experiment of Ref. [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF], and presumably present in the Rb-Ba + experiment of Ref. [START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF].

(iii) The laser light used for cooling and trapping atoms and ions is recognized to have a strong influence on the measured rates, as inelastic processes with excited species may be very large (see for instance Ref. [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF][START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF][START_REF] Sullivan | Role of Electronic Excitations in Ground-State-Forbidden Inelastic Collisions Between Ultracold Atoms and Ions[END_REF]). Furthermore, such quite intense lasers could prevent the produced molecular ions to be detected before being photodissociated. On the theoretical side, such light-assisted processes may be tedious to describe accurately, as much more collision channels are open due to the presence of numerous excited electronic states (see for instance Ref. [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF]) which are more sensitive to the quality of quantum chemistry calculations.

Nevertheless, addressing the problematic of point (iii), once light-assisted formation of molecular ions is probed, we have checked their effective lifetime due to the presence of several lasers in the experiments, which turns out to efficiently photodissociate the created ions. Moreover, besides an extremely slow internal relaxation due to the presence of a permanent electric dipole moment, at reasonably time scales, we have found evidences for the population of molecular levels being indeed strongly influenced by collisions with surrounding ultracold atoms, in agreement with the earlier hypothesis (i) and (ii).

Due to the relevance of ultracold collisions between atoms and molecular ions, depicted above, we have then investigated such processes for two sorts of systems: (a) a series of lighter atom-diatom pairs, H + 2 , HD + and D + 2 colliding with He atoms employed as buffer gas within ion trap experiments; and, (b) heavier sympathetically cold homonuclear Rb + 2 ions in ultracold collisions with Rb atoms as it would occur in the experiments cited above -for instance Ref. [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF].

In the case (a) we have computed ab initio rotationally-and in part spin-stateresolved elastic and inelastic (non-reactive) cross sections, using the close-coupling magnitude, where the rotational cooling rates are found to be of similar size as those for vibrational cooling. In a complete picture, a sample of initially rotationally and vibrationally excited H + 2 made to interact with cold He buffer gas will undergo the competing processes of vibrationally-excitation, vibrational cooling and rotational cooling. We probed an effective complete rotational cooling into the rotational ground state (fixed at the vibrational ground level), at already relatively hot collisional temperatures, for instance 20 K, within a time window of approximately 40 s for a He density of ≈ 10 9 cm -3 . For initially vibrationally excited molecules, more relaxation channels exist, but the cooling time is expected to be of the same order of magnitude. These results also helped to propose experimental implementation for achieving single-quantum state preparation of H + 2 and, on the whole, we find that because of its relative simplicity, cryogenic buffer gas cooling of homonuclear molecular hydrogen ions by helium atoms appears as an attractive approach for enabling next-generation spectroscopy experiments on these fundamental one-electron molecular systems.

In the case (b) we have computed and characterized the first twenty potential energy surfaces of a Rb + 3 complex, using the same effective core potential based quantum chemistry method applied above in the description of the diatomic systems. The additional difficulties of a close-coupling treatment for an effective internal cooling of sympathetically cold Rb + 2 ions in collisions with Rb atoms are discussed along the following lines: (i) the homonuclear nature of the problem, inducing losses from reactive collisions, e.g. ortho-Rb + 2 + Rb -→ para-Rb + 2 + Rb (or vice-versa) and ortho/para-Rb + 2 + Rb -→ ortho/para-Rb 2 + Rb + ; (ii) the higher density of internal states due to the heaviness of the system; and, (iii) its long-range neutral-charged interaction. Nevertheless, strong evidences of an efficient internal cooling by inelastic collisions with Rb atoms is found, if the collisional temperature is kept cold enough, and may indicate a general trend for all similar heavy species currently found in hybrid trap experiments.

S Y N T H È S E E N F R A N Ç A I S

Au cours de ce travail, nous avons exploré trois aspects de la dynamique d'ions moléculaires dans des pièges hybrides superposant un piège d'atomes ultra-froids et un piège d'ions refroidis par laser : (i) formation, (ii) destruction et (iii) refroidissement interne par collisions avec des atomes ultra-froids. L'émission radiative au cours des collisions ultra-froides binaires entre les atomes de Rb et les ions alcalino-terreux (Ca + , Sr + , Ba + ) et Yb + , ainsi qu'entre les atomes de Li et les ions Ca + /Yb + , est étudiée théoriquement, en utilisant des calculs de chimie quantique employant des potentiels effectifs de coeur avec polarisation, donnant les courbes d'énergie potentielle et les moments dipolaires de transition des espèces concernées avec une bonne précision. L'association radiative d'ions moléculaires est prévue pour tous les systèmes, avec une section efficace deux à dix fois plus grande que celle du transfert de charge radiatif. Les constantes de vitesse partielle et totale sont également calculées et comparées aux expériences disponibles. Des résonances de forme étroite sont attendues, qui pourraient être détectables à basse température avec une résolution expérimentale à la limite de celle des expériences actuelles. Les distributions vibrationnelles sont également calculées, montrant que les ions moléculaires finaux ne sont pas créés dans leur niveau d'état fondamental. Cependant, l'accord entre la théorie et les expériences s'est trouvé limité par la sensibilité de ces processus de formation aux conditions physiques sous-jacentes réelles de chaque configuration expérimentale et, par conséquent, au choix du modèle théorique le plus approprié pour les décrire. En particulier, il devient crucial de distinguer le régime des densités atomiques utilisées et le temps d'interaction réel entre les atomes et les ions. L' exemple typique donné tout au long de ce travail, concerne les expériences menées dans l'équipe du Professeur Willitsch (Université de Bâle), où un nuage d'atomes ultra-froids avec une densité autour de ≈ 10 9 cm -3 est piégé dans un piège optique magnétique, et les ions refroidis par laser sont piégés dans un cristal de Coulomb. Seules des collisions binaires sont observées. Au contraire, les expériences menées dans l'équipe du Professeur Denschlag (Université d'Ulm), avec un piège atomique de plus grande densité (≈ 10 12 cm -3 , parfois ≈ 10 13 cm -3 ) et un ion piégé unique, détectent des processus de recombinaison à trois corps conduisant à la formation d'ions moléculaires. Nous rappellerons quelques faits majeurs concernant le piègeage simultané d'atomes ultra-froids et d'ions refroidis au laser discuté précédemment dans le chapitre (4) :

(i) Les densités atomiques élevées et les collisions à trois corps, [START_REF] Härter | Single ion as a three-body reaction center in an ultracold atomic gas[END_REF][START_REF] Krükow | Reactive two-body and three-body collisions of Ba + in an ultracold Rb gas[END_REF][START_REF] Krükow | Energy Scaling of Cold Atom-Atom-Ion Three-Body Recombination[END_REF][START_REF] Härter | Population distribution of product states following three-body recombination in an ultracold atomic gas[END_REF], peuvent contribuer à la formation d'ions moléculaires, ainsi qu'à leur destruction s'ils sont créés par rayonnement. Les expériences de Réf. [START_REF] Schmid | Dynamics of a Cold Trapped Ion in a Bose-Einstein Condensate[END_REF] sur Rb-Ba + et de Réf. [START_REF] Zipkes | Cold Heteronuclear Atom-Ion Collisions[END_REF][START_REF] Zipkes | Cold Heteronuclear Atom-Ion Collisions[END_REF] sur Rb-Yb + utilisent toutes deux un échantillon atomique presque dégénéré, avec des régimes similaires de densités, et n'ont pas encore observé d'ions moléculaires.

(ii) Lors du recouvrement entre le nuage d'atomes et les ions, des collisions entre les atomes de Rb ultra-froids environnants et les ions moléculaires formés peuvent se produire, fortement exoergiques pour tous les systèmes étudiés ici -voir le tableau (4) -, en supposant qu'il n'y a pas de barrière potentielle d'activation dans le complexe collisionnel triatomique qui empêcherait la réaction de se produire. Un des produits serait l'ion Rb + 2 , effectivement observés dans l'expérience sur Rb-Ca + [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF], et probablement aussi dans l'expérience sur Rb-Ba + [START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF].

(iii) La lumière laser utilisée pour refroidir et piéger les atomes et les ions est reconnue comme ayant une forte influence sur les taux de réaction mesurées, car les processus inélastiques avec des espèces excitées par ces lasers peuvent être très importants (voir par exemple [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF][START_REF] Hall | Light-assisted cold chemical reactions of barium ions with rubidium atoms[END_REF][START_REF] Sullivan | Role of Electronic Excitations in Ground-State-Forbidden Inelastic Collisions Between Ultracold Atoms and Ions[END_REF]). De plus, de tels lasers assez intenses pourraient empêcher la détection des ions moléculaires produits avant leur photodissociation. Du point de vue théorique, de tels processus assistés par la lumière peuvent être fastidieux à décrire avec précision, car beaucoup de canaux de collision sont ouverts en raison de la présence de nombreux états électroniques excités (voir par exemple [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF]) qui sont plus sensibles à la qualité des calculs de chimie quantique.

Néanmoins, en abordant la problématique du point (iii), une fois que la formation assistée par la lumière des ions moléculaires est sondée, nous avons vérifié leur durée de vie effective en présence de plusieurs lasers dans les expériences, qui photodissocient les ions créés. De plus, outre une relaxation interne extrêmement lente malgré la présence d'un moment dipolaire électrique permanent, à des échelles de temps raisonnables, nous avons trouvé des preuves que la population des niveaux moléculaires était fortement influencée par les collisions avec les atomes ultra-froids environnants, en accord avec les hypothèses antérieures (i) et (ii).

Nous avons ensuite étudié des processus similaires pour deux sortes de systèmes qui présentent une analogie formelle mais dans des régimes très différents : (a) une série de paires atome-diatomées plus légères, H + 2 , HD + et D + 2 en collision avec les atomes de He utilisés comme gaz tampon dans le cadre d'expériences consacrées à la spectroscopie de très haute précision d'ions moléculaires ; (b) un système homonucléaire plus lourd, Rb + 2 , dans des collisions ultra-froides avec des atomes de Rb comme cela se produirait dans les expériences citées ci-dessus -par exemple Réf. [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: radiative molecule formation vs. charge exchange[END_REF][START_REF] Hall | Ion-neutral chemistry at ultralow energies: dynamics of reactive collisions between laser-cooled Ca+ ions and Rb atoms in an ion-atom hybrid trap[END_REF].

Dans le cas (a), nous avons calculé dès le départ des sections efficaces élastiques et inélastiques (non réactives) résolues par rotation par une approche « close-coupling », où les vitesses de refroidissement de la rotation sont trouvées du même ordre de grandeur que celles du refroidissement de la vibration. Dans une image complète, un échantillon de H + 2 initialement excité rotationnellement et vibrationnellement interagit avec les atomes He froid, le gaz tampon induisant ainsi les processus compétitifs d'excitation et de refroidissement des degrés de liberté interne des ions. Nous avons étudié le refroidissement rotationnel vers l'état fondamental de rotation, à des températures de collision déjà relativement hautes (≈ 20 K), dans une fenêtre temporelle d'environ 40 s avec une densité atomique de ≈ 10 9 cm -3 . Pour les molécules initialement excitées par vibration, il existe plus de canaux de relaxation, mais on s'attend à ce que le temps de refroidissement soit du même ordre de grandeur. Ces résultats ont également contribué à proposer une mise en oeuvre expérimentale pour atteindre la préparation de l'ion H + 2 dans un état quantique unique. En raison de sa relative simplicité, le refroidissement cryogénique des gaz moléculaires homonucléaires par des atomes d'hélium apparaît comme une approche attrayante pour permettre des expériences de spectroscopie de nouvelle génération sur ces systèmes moléculaires à un électron.

Dans le cas (b), nous avons calculé et caractérisé les vingt premières surfaces d'énergie potentielle d'un complexe Rb + 3 , en utilisant la même méthode de chimie quantique basée sur le potentiel effectif de coeur avec polarisaton appliquée dans la description des systèmes diatomiques. Les difficultés supplémentaires pour un refroidissement interne efficace des ions Rb + 2 en collision avec les atomes de Rb sont discutées selon les lignes suivantes : (i) la nature homonucléaire du problème, induisant des pertes de collisions réactives, par exemple ortho-Rb + 2 + Rb -→ para-Rb + 2 + Rb (ou vice versa) et ortho/para-Rb + 2 + Rb -→ ortho/para-Rb 2 + Rb + ; (ii) la densité plus élevée des états internes due à la masse du système ; et (iii) les interactions à longue distance. Néanmoins, de bonnes présomptions d'un refroidissement interne efficace par des collisions inélastiques avec des atomes de Rb sont mises en évidence, si la température de collision est maintenue assez froide, pouvant indiquer une tendance générale pour toutes les espèces massives similaires actuellement employées dans des expériences en pièges hybrides. In the following a brief discussion about the spin-rotation and hyperfine coupling for H + 2 molecular ions is carried out, as it is central to understand the results presented in the Chapter [START_REF] Deutsch | Optical lattices[END_REF].

In the Hund's case (b), the electronic spin S and the total nuclear spin I couple together to yield a total spin vector F = S + I. Since, in the general case, the vector F is further coupled to the molecular rotation vector N, we additionally have that each hyperfine level is designated by a quantum number j, j = F + N, varying between | N -F | and | N + F |. In a homonuclear case, I takes on integer values, including zero. The even values of I are called para, while the odd values are called ortho. In the case of H + 2 , the para state has I = 0. The symmetrization of the total molecular wavefunction, under the operation of exchange of the two nuclei, requires that parastates have rotational angular momenta N = 0 , 2 , 4 , . . . while ortho-states have odd N values.

In para-H + 2 ( I = 0 ), we have the direct coupling only between N and S. This simplification leads to the fine structure levels with j = N + 1 / 2 and to those with j = N -1 / 2. In this specific case, therefore, we can directly evaluate the expectation value of the spin-rotation interaction,

H sr = γ S • N = γ 1 2 j 2 -S 2 -N 2 = γ 1 2 j ( j + 1 ) - 3 4 -N ( N + 1 ) =      1 2 γ N , j = N + 1 2 -1 2 γ ( N + 1 ) , j = N -1 2 . ( 146
)
It therefore follows that the expressions for the rotational energies are

E N j = N + 1 / 2 = B N ( N + 1 ) + 1 2 γ N , (147)
Since the above coupling sequence is satisfied in our systems, in the N = 2 case, for example, we should have shifts of γ and -3γ/2. As a matter of fact, the earlier calculations of Karr et al. [START_REF] Karr | Vibrational spectroscopy of H 2 + : Precise evaluation of the Zeeman effect[END_REF] find the shifts +42.16 MHz for the j = 3/2 level and -63.24 MHz for the j = 1/2 level, as it is to be expected.

In the different situation of the ortho-H + 2 (I = 1) system, we now have two initial sublevels corresponding to F = 1/2 and F = 3/2. These sublevels are further split into five (for N = 1) or six (for N ≥ 3) j sublevels due to the spin-rotation interaction. It is actually known [START_REF] Karr | Vibrational spectroscopy of H 2 + : Precise evaluation of the Zeeman effect[END_REF] that in the ortho variety the spin-spin coupling is one order of magnitude larger than the spin-rotation coupling. One can therefore approximate that, for a given N, each of the states N, F = 1/2, j, m j and N, F = 3/2, j, m j can be considered to be degenerate, thereby simplifying the hyperfine structure effects.

We can therefore evaluate the expectation values of the hyperfine interaction by neglecting the terms I z S z and S • N, which are found to be smaller when compared to S • I,

H h f s = b S • I = b 1 2 F 2 -S 2 -I 2 = b 1 2 F (F + 1) - 3 4 -2 =      1 2 b, F = 3 2 -b, F = 1 2 . ( 149
)
From the above formulation, the total energies are now given as

E N F=3/2 = BN (N + 1) + 1 2 b, (150)
and

E N F=1/2 = BN (N + 1) -b, (151)
which is in agreement with what has been found by Karr et al. [START_REF] Karr | Vibrational spectroscopy of H 2 + : Precise evaluation of the Zeeman effect[END_REF], who, for the state N = 1, gave a shift of 461 MHz for the F = 3/2 level and -923 MHz for the spin-rotation and hyperfine coupling for h + 2 209 F = 1/2 level. These two levels are further split by the spin-rotation interaction, but this splitting is only 90 MHz and 20 MHz, respectively. It therefore becomes acceptable to neglect such splitting in the following discussion and calculations for the ortho-H + 2 system.

One should also be reminded at this point that the ortho-to-para conversion between molecular states is forbidden during pure inelastic collisional processes studied in the Chapters ( 6) and [START_REF] Morinaga | Holographic manipulation of a cold atomic beam[END_REF]. For such one should consider a rearrangement-induced collision and a correct post symmetrization of the total wavefunction. Therefore, the two types of molecular states are treated as entirely independent systems during the scattering calculations.

B D Y N A M I C S W I T H F I N E -S T R U C T U R E E F F E C T S

In the following a brief discussion is carried out about the modifications required by the close-coupling formalism, in order to treat diatomic targets with doublet spin multiplicity, such as H + 2 , presented in Chapter [START_REF] Deutsch | Optical lattices[END_REF]. The standard space-fixed (SF) formulation of the quantum scattering within the close-coupling (CC) formalism outlined earlier deals only with a 1 Σ + molecule interacting with a 1 S atom [START_REF] Arthurs | The theory of scattering by a rigid rotator[END_REF]. The changes necessary for the present case of a 2 Σ + molecule have been worked out in detail by many authors over the years -see for instance [START_REF] Corey | Inelastic differential and integral cross sections for 2S+1 Σ linear molecule-1 S atom scattering: the use of Hund's case b representation[END_REF] -and will not be extensively reviewed again here. In analogy with the standard CC formalism [START_REF] Arthurs | The theory of scattering by a rigid rotator[END_REF], one defines a total angular momentum representation by expanding the atom-molecule wavefunction in eigenfunctions of the total angular momentum J, thereby producing |(N, F) , j, ; J, m J = ∑ m j ,m j, m j , , m |J, m J × N, F, j, m j | , m , (

where N, F, j, m j is the molecular wave function and | , m is the basis set for the relative orbital motion of the colliding pair, with (. . . |..) being Clebsch-Gordan coefficient. The above representation may also be given by the following vector coupling scheme:

N + F = j; j + = J. [START_REF] Johnson | New numerical methods applied to solving the one-dimensional eigenvalue problem[END_REF] The ensuing development of the coupled equations that now modify the CC scheme -presented in Chapter (6) -is carried out in the new total-J representation [START_REF] Corey | Inelastic differential and integral cross sections for 2S+1 Σ linear molecule-1 S atom scattering: the use of Hund's case b representation[END_REF] that produces the scattering matrix elements. They will be in turn employed to provide the corresponding state-to-state scattering amplitudes between the relevant channels. From the latter quantities we shall then produce the state-to-state partial integral cross sections summed over final and averaged over initial degeneracies, i.e. the σ N,F,j-→N ,F,j cross sections already defined in earlier work [START_REF] Corey | Inelastic differential and integral cross sections for 2S+1 Σ linear molecule-1 S atom scattering: the use of Hund's case b representation[END_REF].

The CC formulation outlined above for the scattering from 2 Σ target molecules were introduced earlier in the computational code [START_REF] López-Durán | ASPIN: An all spin scattering code for atom-molecule rovibrationally inelastic cross sections[END_REF] used during this work, and we have therefore generated with it the relevant network of cross sections for the case of para-H + 2 .

dynamics with fine-structure effects

One should also keep in mind that in the Hund's case (b) coupling scheme, the total spin and rotational angular momentum are only weakly coupled. In fact, since the internuclear forces are basically electrostatic in origin, there is no direct effect coming from the spin state of the target, so that binary collisions in the trap do not directly influence either the magnitude or the orientation of the spin angular momentum F. There is, however, an indirect effect of the collision dynamics on the rotational angular momentum j since both N and F recouple again after the particles recede from each other. One can therefore take advantage of this mechanism to decouple the total spin F from the rotational-centrifugal motion by utilizing a total-J representation defined by the coupling scheme,

N + = ; + F = J. ( 154
)
As a matter of fact, it was already shown by different authors [START_REF] Corey | Inelastic differential and integral cross sections for 2S+1 Σ linear molecule-1 S atom scattering: the use of Hund's case b representation[END_REF] that using the above coupling one can express the S-or T-matrix elements already mentioned before in terms of simpler spin-independent matrix elements, and still be able to get information on the hyperfine-structure transitions.

This problem was further considered in detail in an earlier work of M. H. Alexander et al., who analyzed the physical effects of such decoupling on the hyperfine structure transitions [START_REF] Alexander | Rotationally inelastic collisions between a diatomic molecule in a 2 Σ + electronic state and a structureless target[END_REF][START_REF] Alexander | On the physical origin of propensity rules in collisions involving molecules in 2 Σ electronic states[END_REF]. They derived physically based propensity rules that govern the relative probabilities of collision-induced transitions between hyperfine-structure levels of 2 Σ molecular targets. Thus, one obtains the following relationship for the cross section of a particular (N, F, j -→ N , F, j ) transition as a sum of irreducible tensorial components of order K: σ N,F,j-→N ,F,j = π

k 2 2j + 1 ∑ K    N N K j j F    2 P K N , N , (155)
where {• • • } is a 6j-symbol and each term P K (N , N) contains the transition proba- bility (i.e. opacity) value which are summed together to yield the relevant cross section. The angular momentum transfer index K can be interpreted as a measure of the degree of reorientation of N. It is also useful to remember that the (F = 3/2 -→ F = 1/2) collision-induced transitions are forbidden during the excitation of the ortho-H + 2 because the total (nuclear + electronic) spin state is not affected by the collision.

Equation [START_REF] Yurchenko | Duo: A general program for calculating spectra of diatomic molecules[END_REF] indicates that the small collision transfers of angular momentum correlate with small nuclear rotational angular momentum reorientation angles. The contribution of each term decreases as K increases, which is in fact a manifestation of the propensity toward the conservation of the orientation of N. Furthermore, for N = j transitions, the angular momentum coupling described by the 6j-symbol in Eq. ( 155), preferentially weights collisions that cause a small reorientation of N. This is the main reason for the observed propensity rules in favor of N = j, a feature that was also observed in earlier calculations [START_REF] Alexander | On the physical origin of propensity rules in collisions involving molecules in 2 Σ electronic states[END_REF].

C C O M P U TAT I O N A L T O O L S

In the following, a brief comment on each computational tool used in this work is provided, mainly focusing on how to obtain the respective program, to provide its reference (and/or documentation) and to highlight technical issues found during its use, if any. For the sake of completeness, it is also provided those computational tools intended to use, which eventually were either not used or not used for production runs.

abc quantum reactive scattering program

The ABC source code [START_REF] Skouteris | ABC: a quantum reactive scattering program[END_REF] is available for download, under the catalogue identifier ADMX, in the Computer Physics Communications Program Library1 . In order to build its binary, one shall link the usual BLAS and LAPACK linear algebra libraries and, in this particular work, it is accomplished by using the BLAS/LAPACK versions already available in the Intel MKL library. The ABC code becomes orders of magnitude more efficient if compiled with a parallel BLAS/LAPACK library if, and only if, the size of matrices used are relatively high, e.g. 1000 × 1000. If small matrices are used, however, the parallel implementation may become a bottleneck and a sequential compilation is highly recommended.

For an optimum efficiency ABC requires several modifications in order to implement a hybrid sequential-parallel propagation scheme of the total wavefunction [START_REF] Baraglia | A Parallel Code for Time Independent Quantum Reactive Scattering on CPU-GPU Platforms[END_REF]. And it is worthwhile to notice that scattering solutions are integrated by means of a constant step size log-derivative method, which may become an extra overhead for a proper description of neutral-charged colliding partners, where the long-range potential impose propagating solutions up to relatively large distances over the scattering coordinate. Thus, a further implementation of a hybrid propagation scheme is recommended, i.e. log-derivative for a short-range zone followed by a more long-range suited algorithm, such as either the Airy-propagator of M. Alexander [START_REF] Alexander | Hybrid quantum scattering algorithms for long-range potentials[END_REF] -see also Ref. [START_REF] Alexander | A stable linear reference potential algorithm for solution of the quantum close-coupled equations in molecular scattering theory[END_REF] -, the modified variable-phase method of F. Gianturco and co-workers [START_REF] Martinazzo | A modified Variable-Phase algorithm for multichannel scattering with long-range potentials[END_REF] or multichannel quantum defect theory [START_REF] Hazra | Quantum defect theory for cold chemistry with product-quantum-state resolution[END_REF].

The ABC code comes with three potential energy surfaces ready to use -for details see Ref. [START_REF] Skouteris | ABC: a quantum reactive scattering program[END_REF]. In order to use third-party potentials one shall implement the following modifications (using standard Fortran77):

• Main program: The three atomic masses used are implemented in-code and only a few atomic species are available. Thus, one shall include the respective masses, if still not available, in the first if-else block at the main part of the code. The variable imass, an integer value provided in the input namelist (see Ref. [START_REF] Skouteris | ABC: a quantum reactive scattering program[END_REF] for details), is a label to identify a given atom and the array mass shall carry its real value.

• Subroutine potsub: an external subroutine (or function) interface to retrieve the potential energy as a function of the three internuclear distances should be designed and further called by potsub. The if-else statements, at the end of potsub, selects the proper potential energy surface depending on the atomic masses. For safety, one can replace the whole if-else block by a single call of its own potential interface though. The order of the respective internuclear distances are going to control the respective order of arrangements internally used by ABC and, therefore, selecting the respective reactant-arrangement and product-arrangements.

• Subroutine parset: in order to compute the asymptotic basis, i.e. internal vibrational/rotational states of each arrangement, one can use the variables m and ds to increase the size of each respective diatomic potentials. Where, m is the number of quadrature points used and ds represents the step size on the internuclear distance.

• Subroutine driver: the parameter nmax represents the total number of channels and defines the size of all arrays used in the code. Should be increased accordingly.

• Subroutine setrec: The ABC code uses scratch files, in order to store the temporary energy-dependent log-derivative matrix, during the propagation. The routine setrec is designed to set up and open such files properly. However, ABC avoids calculations for those files bigger than an arbitrary chosen limit of 1000 Mb. Thus, due to the relatively big amount of storage available nowadays, it is highly recommended to remove such a limitation. It may be done by simply suppressing the if-else block, in the sertec subroutine, leaving only the open statement.

ABC is used for preliminary reactive scattering calculations of Rb + 2 + Rb, presented in Chapter [START_REF] Jin | Collective Excitations of a Bose-Einstein Condensate in a Dilute Gas[END_REF]. The LEVEL code is a computer program for solving the radial Schr ödinger of diatomic systems, written by R. J. Le Roy [START_REF] Le Roy | A Computer Program for Solving the Radial Schr ödinger Equation for Bound and Quasibound Levels[END_REF], and available for download2 under request. It is suited for bound and/or quasi-bound levels of any smooth single or double-minimum potential, and calculate inertial rotation and centrifugal distortion constants, expectation values, and/or Franck-Condon factors and other off-diagonal matrix elements, either for levels of a single potential or between levels of two different potentials. It is worthwhile to notice the use of revised H önl-London factors for both Π -→ Σ and Σ -→ Σ transition bands [START_REF] Hansson | A comment on H önl-London factors[END_REF].

An extensive documentation including examples is also available. In order to build a binary file, the source code may be compiled by standard methods at any modern compiler and no special requirements are needed. LEVEL 8.2 is used to compute the H + 2 , D + 2 and HD + rovibrational spectra presented in Chapter [START_REF] Morinaga | Holographic manipulation of a cold atomic beam[END_REF].

aspin

The ASPIN source code [START_REF] López-Durán | ASPIN: An all spin scattering code for atom-molecule rovibrationally inelastic cross sections[END_REF] is available for download, under the catalogue identifier AEBO _ v1 _ 0, in the Computer Physics Communications Program Library. The code is provided with a prototypical makefile which may be modified in order to build a binary file. It is designed to compute atom-diatom close-coupling problems in which the atom is taken to be structureless while the molecule can be in its singlet, doublet, or triplet spin states and can be treated as either a rigid rotor or a rovibrational target. All the relevant state-to-state integral cross sections, and their sums over final states, can be calculated with the present code. In particular, two schemes of propagations are available: the usual log-derivative method for the short-range part of the potential and the modified variable-phase [START_REF] Martinazzo | A modified Variable-Phase algorithm for multichannel scattering with long-range potentials[END_REF], more suited for long-range potentials. The ASPIN code becomes, therefore, a complete solution for the most sophisticated atomdiatom problems, such as heavy neutral-charged systems. One shall notice, however, that ASPIN is designed for the description of elastic and inelastic processes, during an atom-diatom collision, not accounting to the possibility of rearrangements. For such a reactive system one should use, for instance, the ABC quantum reactive scattering program described above (singlet spin states only).

It is worthwhile to notice, however, one technical issue that may be found:

• Main program: the code uses a MPI scheme of parallelization in which some of the MPI broadcast instructions in the #ifdef blocks, i.e. the data exchange from the master processor towards the remaining processors, are apparently misplaced. Depending on the use it may induce the code to crash. In particular, during this work, the problem was verified when dealing with doublet cases.

ASPIN is used to produce the collisional cross sections presented in Chapters ( 6) and [START_REF] Morinaga | Holographic manipulation of a cold atomic beam[END_REF].

dvr3d suite

The DVR3D suite is a library of codes, developed and maintained by Jonathan Tennyson and co-workers [START_REF] Tennyson | DVR3D: a program suite for the calculation of rotation-vibration spectra of triatomic molecules[END_REF], designed to calculate energy levels, wavefunctions, and where appropriate transition dipole moments, for rotating and vibrating triatomic molecules. Expectation values of geometrically defined functions can be calculated, a feature which is particularly useful for fitting potential energy surfaces. The programs use an exact (within the Born-Oppenheimer approximation) Hamiltonian and offer a choice of Jacobi or Radau internal coordinates and several body-fixed axes. The source code is available for download, under the catalogue identifier ADTI, in the Computer Physics Communications Program Library. In order to build its binary, one shall link the usual BLAS and LAPACK linear algebra libraries and, in this particular work, it is accomplished by using the BLAS/LAPACK versions already available in the Intel MKL library. It was earlier intended for the computation of bound states of Rb + 3 -Chapter (8) -, but finally not used for production runs.

gfit3c

The GFIT3C code, developed by Alfredo Aguado and co-workers [START_REF] Paniagua | Global fit of ab initio potential energy surfaces I. Triatomic systems[END_REF], is a computational program intended to fit ab initio electronic structure calculations of triatomic systems, to a given many-body functional form, detailed in Ref. [START_REF] Paniagua | Global fit of ab initio potential energy surfaces I. Triatomic systems[END_REF] -see also [START_REF] Paniagua | A new functional form to obtain analytical potentials of triatomic molecules[END_REF]. The global fit obtained maintains all the symmetry properties of the system including the permutation symmetry and fulfills the strict criteria needed for molecular dynamical calculations. The source code is available for download, under the catalogue identifier ADHK, in the Computer Physics Communications Program Library, and can be compiled by standard methods at any modern compiler with no special requirements. The method is used in Chapter [START_REF] Deutsch | Optical lattices[END_REF] and Chapter [START_REF] Morinaga | Holographic manipulation of a cold atomic beam[END_REF] to build the global potential energy surface (PES) of the He/H + 2 system. Attempts to use it for the Rb/Rb + 2 system -Chapter (8) -were also carried out, facing, however, several limitations. Mainly due to the relatively high number of ab initio points to fit, in a reasonably amount of time, in which, in turn, a reduced number of points produces a very inaccurate global PES. In particular, at highly curved zones of the PES.

C.5 gfit3c

Figure 1 :

 1 Figure 1: A Coulomb crystal of Ca + with sympathetically cooled N +2 molecular ions inside (dark core regions). The molecular ions are produced by leaking a N 2 neutral source into the chamber and raising the background gas pressure to 1×10 -8 mbar, followed by a 2 + 1 REMPI scheme for ionization, with a pulsed 202 nm laser running at 10 Hz for 30 seconds at a power of 3 mW per pulse. The rovibrational distribution is kept the same as at room temperature. The picture was taken with a CMOS camera system, at ×20 averaging to reduce the background noise, and was kindly provided by Alexander D örfler and Stefan Willitsch (University of Basel).

Figure 2 :

 2 Figure 2: Potential energy curves of the two first electronic states of LiCa + molecular ion and the respective transition electric dipole moment. The horizontal dotted lines representing (i) an incoming continuum level in the entrance channel, Li(2s) + Ca + (3s);(ii) an outgoing continuum level in the exit channel, Li + + Ca 1 S ; and, (iii) a single bound rovibrational level in the exit channel, LiCa + 1 Σ + .

Figure 3 :

 3 Figure 3: (a) Centrifugal barrier for J = 35 in the A 1 Σ + potential energy curve of RbSr + (dashed line), with radial wave functions vertically placed at the appropriate collision energies, namely 0.08, 0.015, and 0.1 cm -1 to illustrate the various tunneling regimes. (b) Partial cross sections as a function of the collision energy (in mK) at given values of rotational quantum numbers (J = 29 to 39). Shape resonances gradually evolve with increasing J.

Figure 4 :

 4 Figure 4: Potential energy curves of (a) the A 1 Σ + state, and (b) of the X 1 Σ + state, for the [Alk-Alke] + molecules, with Alk = Li or Rb (n = 2, 5), and Alke = Ca, Sr, Ba, and Yb (n = 4, 5, 6, 6). The origin of energies is taken at the dissociation of the A 1 Σ + PEC for all systems. The lowest 3 Π PEC of Rb-Ca + is also displayed for completeness. (c) Transition electric dipole moments between the X and A states. Open circles: Ref.[START_REF] Mclaughlin | Ultracold, radiative charge transfer in hybrid Yb ion and Rb atom traps[END_REF]. The vertical dotted lines tag the inner turning points of the incoming continuum wave function -in the entrance channel -with the respective points in the exit channel and the transition dipole moment magnitude at that position.

Figure 5 :

 5 Figure 5: Permanent electric dipole moments of (a) the A 1 Σ + state, and (b) of the X 1 Σ + state, for the RbAlke + molecules, with Alke = Ca, Sr, Ba, and Yb, and for Li-Yb + .

Figure 6 :

 6 Figure 6: Calculated total cross sections (RA + RCT), as functions of the collision energy (with both axes scaling logarithmically). The dashed lines correspond to the classical cross section of Eq. (100).

Figure 7 :

 7 Figure 7: Calculated total cross sections as functions of the collision energy (with both axes scaling logarithmically), for radiative association (RA) and radiative charge transfer (RCT).

Figure 8 :

 8 Figure8: Calculated vibrational distributions of the ground state molecular ions produced by radiative association, as a function of the binding energies and the corresponding vibrational quantum numbers, v X , for the rotational quantum number J = 1, at i =

Figure 9 :

 9 Figure 9: Calculated total (RA + RCT) rate constants convoluted from the cross sections of Fig. (6) with a Gaussian energy distribution of 2 mK half-width. A linear fit of this curves yields an estimate of the average magnitude of the rates (dashed lines). For RbYb + , the calculated rate obtained in Ref.[START_REF] Sayfutyarova | Charge transfer in cold Yb + +Rb collisions[END_REF][START_REF] Mclaughlin | Ultracold, radiative charge transfer in hybrid Yb ion and Rb atom traps[END_REF] is indicated with a dot-dashed line.

Figure 10 :

 10 Figure 10: Calculated total rate constants for RbSr + convoluted with a Gaussian distribution half-width from 1 mK to 5 mK.

Figure 11 :

 11 Figure 11: Schematic drawing of the experimental setup. The upper part shows the atom chamber whereas the lower part presents the ion chamber. 6 Li atoms are supplied from right side of the atom chamber and are decelerated by a Zeeman slower. First, 6 Li atoms are trapped by a conventional MOT, and, then, they are trapped by the cavityenhanced trap and transferred into an ODT at the center of the atom chamber. Ca + ions are trapped at the center of the ion trap electrode placed inside the ion chamber. The 397 nm cooling laser for the ions is incident along both the radial and the axial directions of the ion trap, and the 866 nm repump laser is incident along the axial direction. Fluorescence of the ions is collected by an objective lens and detected by a PMT and EMCCD camera. Enegy level diagram of the 40 Ca + ion is shown in the inset.

FigureFigure 12 :

 12 Figure (13a) shows the measured charge-exchange collision cross sections as a function of the collision energy E coll for the Ca + ions under the irradiation of the cooling and repump lasers. In this condition, ions are considered to be in the mixture state

Figure 13 :

 13 Figure 13: Charge-exchange collision cross sections as a function of the collision energy for the Ca + ions (a) in the mixture of the S 1/2 , P 1/2 and D 3/2 states, (b) D 3/2 state, and (c) D 5/2 state, respectively. The data shown with the open circles are the points which may suffer from the ion heating due to RF noise, and also excluded from the further analysis. The black solid lines indicate Langevin cross section given by Eq. (106). The dashed lines show the fitting result of the data with the E -1/2 dependence predicted from the Langevin model.

Figure 14 :

 14 Figure14: LiCa + potential energy curves computed in the present work relevant for the experiment. The circle locates the avoided crossing which is expected to be active for the charge-exchange process starting from Li(2s) + Ca + (3d) for the 3 Σ + symmetry.

Figure 15 :

 15 Figure 15: 3D view and contour plot of the anisotropic potential given by Eq. (108). The potential isolines are separated by 50 cm -1 . Angles are given in degrees.

Figure 16 :

 16 Figure 16: The computed radial coefficients from the multipolar expansion of the present PES,following Eq. (108) and (109). Only the first four coefficients are shown.

Figure 17 :

 17 Figure 17: Top: Kinetic energy dependence of fine-structure elastic cross sections for the para-H + 2 .The labeling of the transitions is the following: N j -→ N j . Bottom: Kinetic energy dependence of the fine-structure de-excitation and excitation cross sections for the para-H + 2 . The transitions are labeled by the main N levels involved and by their fine-structure sub levels as subscripts.

Figure 19 :

 19 Figure 19: Kinetic energy dependence of the elastic cross sections for the pseudo-1 Σ, para-H + 2 , and ortho-H + 2 . The transitions are labeled by the main N rotational levels involved.

Figure 21 :

 21 Figure 21: Left panels: Comparison between the simpler pseudo-1 Σ RR rate coefficients (continuous lines) and the summed hyperfine-structure rate coefficients (dashed lines) for para-H + 2 . Right panels: Behavior of the simpler pseudo-1 Σ RR rate coefficients for ortho-H + 2 . The bottom panels are in a logarithmic scale. The individual curves are labeled by the indexes of the corresponding N rotational levels.

Figure 22 :

 22 Figure 22: Top: De-excitation rate coefficients at 10 K for different initial values of N for para-H + 2 -He (left panel) and ortho-H + 2 -He (right panel) collisional systems. Bottom: Deexcitation inelastic rate coefficients from N = 10 and N = 11 at different temperatures for para-H + 2 -He (left panel) and ortho-H + 2 -He (right panel) collisional systems.

Figure 23 :

 23 Figure 23: Comparison between rates coefficients for ∆N = -2 transitions computed with the HO-RR approach (continuous lines) and the VA potential (dashed lines). Left panels: results for the para-H + 2 target molecule. Right panel: for the ortho-H + 2 target. The labeling refers to the initial and final N rotational levels involved in the transitions.

Figure 24 :

 24 Figure 24: Contour plot of the PES H + 2 /He at r = 1.0574 Å. The attractive wells are located collinearly on both sides of H + 2 target. Distances are in Å and energies in ×10 3 cm -1 .

igure 25 :

 25 9 Å (Ramachandran et al) r = 1.1 Å (Ramachandran et al) r = 1.2 Å (Ramachandran et al) r = 0.9 Å (Meuwly et al) r = 1.1 Å (Meuwly et al) r = 1.2 Å (Meuwly et al) Comparisons of the PECs for three different values of r from the two outlined references.

Figure 26 :

 26 Figure 26: Interaction potential as a function of the H + 2 internuclear distance r for various angles θ. The value of R is kept fixed at the one of the equilibrium geometry of the triatomic complex.

  D + (υ, υ ' )

+ 2 ,

 2 hd + and d + 2 interacting with he atoms 133

Figure 30 :

 30 Figure 30: Computed multipolar expansion coefficients for the coupling matrix elementsV λ υ,υ ( R ) as a function of R for HD + with λ = 0, 1, 2, 3.

Figure 31 :

 31 Figure 31: Top: Different "cuts" of the PES for the H + 2 -He system computed at different orientation angles. Solid lines: HO-RR potential; dashed lines: VA potential for the υ = 0 case. Bottom: Radial dependence of the corresponding multipolar expansion V λ coefficients of H + 2 -He complex computed for different values of the index λ. The solid and the dashed lines have the same meaning as in the top figure.

vibrational couplings of h + 2 , 2 Figure 32 :Figure 33 :

 223233 Figure 32: Top: Computed cross sections as a function of collision energy for transitions fromthe lowest to various rotational excited rotational states of para-H + 2 . The solid lines report the OH-RR scheme calculations, while the dashes describe the results obtained by the VA method for generating the coupling potential in the ground vibrational levels of the molecular targets. Bottom: Computed cross sections as a function of collision energy for various transitions to lower rotational states and starting from different initial rotational levels of para-H + 2 .

2 2 Σigure 35 :

 235 + g + Rb(5s), yellow zones. The zero energy (green zone) is correlated to the first triatomic asymptote, Rb(5s) + Rb(5s) + Rb + , and the blue zones correspond to repulsive energies typically when two or more Rb atoms are closer than 5 a 0 . a fci study of the first 20 pes of [rb 2rb] + system Heat-maps of the second electronic state of Rb + 3 1 3 A 1 , at FCI level of theory and two orientations (collinear on top and T-shape in the bottom). The PES is, asymptotically, correlated to Rb + 2 2 Σ + g + Rb(5s), yellow zones. The zero energy (green zone) is correlated to the first triatomic asymptote, Rb(5s) + Rb(5s) + Rb + , and the blue zones correspond to repulsive energies typically when two or more Rb atoms are closer than 5 a 0 .

Figure 36 : 152 aFigure 37 : 3 BFigure 38 :

 3615237338 Figure 36: Heat-maps of the third electronic state of Rb + 3 2 1 A 1 , at FCI level of theory and two orientations (collinear on top and T-shape in the bottom). The PES is, asymptotically, correlated to Rb 2 1 Σ + g + Rb + , soft-yellow zones. The zero energy (green zone) is correlated to the first triatomic asymptote, Rb(5s) + Rb(5s) + Rb + , and the blue zones correspond to repulsive energies typically when two or more Rb atoms are closer than 5 a 0 .

Figure 39 :

 39 Figure 39: The (a) permanent electric dipole moment of the ground state, 1 1 A 1 , and (b) the transition electric dipole moment between the first two 1 A 1 electronic states of Rb + 3 , as a function of the intermolecular distance, R.

8. 6 Figure 40 :

 640 Figure 40: Comparison between the ab initio points (circles) and the PES interpolation (solid lines) for Rb + 3 1 1 A 1 . The vertical dashed line represents the limit (4 a. u.) to which, at smaller values of R, the energy is represented as a short range extrapolation.

8. 7

 7 conclusions Part III S TA B I L I T Y O F C O L D M O L E C U L A R I O N S I N H Y B R I D T R A P S A C A S E S T U D Y: P H O T O D I S S O C I AT I O N D Y N A M I C S O F W E A K LY-B O U N D R B B A + M O L E C U L A R I O N S introduction

3 ΠFigure 42 :

 342 Figure 42: (a) Potential energy curves of the complex [Rb-Ba] + , as a function of the internuclear distance R, for different molecular symmetries, addressed by photons with1064 nm wavelength, from the Rb(5s) + Ba + (6s) asymptote (entrance channel). (b, c) The electric dipole moments between the respective electronic states for Σ -→ Σ (red curves) and Σ -→ Π (blue curves) transitions. For actual purposes of computation the Delta electronic states are not taken into account. For comparison purposes, the black vertical arrow represents a transition (in the singlet entrance channel) from an initial weakly-bound level close to threshold and the brown vertical arrow represents a transition from a deeply-bound level (about 300 cm -1 below the same threshold).

Figure 43 :

 43 Figure43: Top panel: potential energy curves of the complex [Rb-Ba] + , as a function of the internuclear distance R, for different molecular symmetries, addressed by photons with 650 nm (red arrow), 614 nm (yellow arrow) and 493 nm (blue arrow) wavelength, from the electronic ground state 1 1 Σ + . It is assumed, previously, a spontaneous emission (green curve) from the original entrance channel, 2 1 Σ + , correlated with the respective initial atomic states prepared experimentally[START_REF] Krükow | Reactive two-body and three-body collisions of Ba + in an ultracold Rb gas[END_REF]. Middle panel: all the transition electric dipole moments between the respective electronic states for Σ -→ Σ and Σ -→ Π transitions. Only singlet electronic states are displayed due the forbidden singlet-to-triplet transitions. Lower panel: vibrational distribution created by spontaneous emission, from the 2 1 Σ + entrance channel toward the 1 1 Σ + manifold (υ max ∼ 248) at J = 0, as a function of the vibrational quantum number.

Figure 44 :

 44 Figure 44: Radiative lifetime for the highest vibrational levels at (J = 0), in the (2) 1 Σ + electronic state, for spontaneous emission towards the singlet ground state, (1←-2) 1 Σ + , J = 1.Such an estimation should be taken as a lower limit.

  N -R O TAT I O N A N D H Y P E R F I N E C O U P L I N G F O R H + 2

207 208 spin-rotation and hyperfine coupling for h + 2 andE

 2072082 N j = N -1 / 2 = B N ( N + 1 ) -1 2 γ ( N + 1 ) .

Table 1 :

 1 Chart of publications used in this work.

	Chapter	References	Supervision COMIQ member
	(4)
	(8)	(in preparation)	O. Dulieu	CNRS-LAC
	(9)	(in preparation) Johannes H. Denschlag	UULM

H. da Silva Jr et al. New J. Phys. 17 045015 (2015) O. Dulieu CNRS-LAC (5) R. Saito et al. Phys. Rev. A 95, 032709 (2017) (6) M. H. Vera et al. J. Chem. Phys. 146, 124310 (2017) F. Gianturco UIBK S. Schiller et al. Phys. Rev. A 95, 043411(2017) (7) I. Iskandarov et al. Eur. Phys. J. D 71, 141 (2017)

Table 5 :

 5 Ca + internal state A A for various internal states of 40 Ca + ions.

	S 1/2	1 × 10 -3
	D 3/2	0.15±0.01
	D 5/2	0.53±0.06
	Mixed states of S 1/2 , P 1/2 and D 3/2 0.23±0.03
	P 1/2	0.64±0.04

  + 2 , hd + and d + 2 in collision with he Structure of the vibrational levels of H + 2 , HD + and D +

	2
	Initially, we analyze the isolated H +	2

2

X 2 Σ + g , HD + X 2 Σ + g and D +

Table 6 :

 6 A comparison between different calculated values for the first few vibrational levels of H + 2 , D + 2

Table 10 :

 10 a fci study of the first 20 pes of [rb 2rb] + system Comparison between the published spectroscopic characteristics of Rb 2 3 Σ g/u and the ones estimated from the isolated quasi-diatomic potential of Rb + 3 , as R → ∞ and θ = 90 • , where, the rotational constant, B e , and the harmonic frequency, ω e , are estimated for a85 Rb atom with mass 84.9117893×1822.8839 atomic units. (a. u.) D e (cm -1 ) ω e (cm -1 ) B e (cm -1 )

	Rb 2	3 Σ + g	r e (a. u.) D e (cm -1 ) ω e (cm -1 ) B e (cm -1 )
	This work, FCI	9.781	3341.03	36.83	0.01482
	[273]	9.637	3408	36	0.01517
	[294]	9.732	3345.2	36.55	0.01487
	[295]	9.883	3209	35	0.01442
	Rb 2	3 Σ + u ; inner well only
	This work, FCI	11.466	243.59	13.35	0.01078
	[273]	10.091	1096	26	0.01384
	[294]	11.452	237	13.34	0.01074
	[295]	11.584	240	14	0.01050
	[291]	11.849	202	12	0.01004
	[192]	11.432	325	14	0.01078
	[296]	10.960	391	17.02	0.01173
	[298]	11.515	241.50	-	0.01062

Table 11 :

 11 Comparison between the published spectroscopic characteristics of Rb 2 1 Π g and the ones estimated from the isolated quasi-diatomic potential of Rb + 3 , as R → ∞ and θ = 90 • , where, the rotational constant, B e , and the harmonic frequency, ω e , are estimated for a85 Rb atom with mass 84.9117893×1822.8839 atomic units.

	R (a. u.)		FF-RCCSD(T) (a. u.)		TIP-CCSD (a. u.)		FF-FCI (a. u.)
		θ xx	θ yy	θ zz	θ xx	θ yy	θ zz	θ xx	θ yy	θ zz
	9	-308.92 -563.15	-648.58	312.6 599.43	695.05	301.90 567.74	655.72
	11	-352.1 -624.03	-1006.68	355.52 673.24 1052.13	345.61 626.02 1038.45
	13	-393.86 -662.49	-2043.98	395.36 730.82 1916.49	387.44 660.57 2033.54
	15	-430.82 -661.51	-4907.89	429.27 766.25	-	424.56 664.65 3766.04
	17	-458.09 -570.91 -11426.15	-	-	-	449.19 644.37 3846.51

Table 12 :

 12 Static polarizability components using the finite field (FF) and the time-independent propagator (TIP) methods, for RCCSD(T), CCSD and FCI level of theories, at r = 9 a. u. and θ = 90 • .

	R (a. u.) RCCSD(T) (a. u.) FCI (a. u.)
	9	-0.434	-0.439
	11	-1.080	-0.988
	13	-1.112	-0.796
	15	0.227	0.609
	17	4.708	3.007

Table 13 :

 13 The z-component of the permanent dipole moment using the finite field (FF) method, for RCCSD(T) and FCI levels of theories, at r = 9 a. u. and θ = 90 • .

	R (a. u.) TI-CCSD (a. u.) FF-RCCSD(T) (a. u.) FF-FCI (a. u.)
	9	535.69	506.88	508.45
	11	693.63	660.94	670.03
	13	1014.2	1033.4	1027.2

Table 14 :

 14 The average static polarizability, θ (in atomic units), computed for r = 9 a. u. and θ = 90 • .

  3 Σ + .

		16000
		14000
		12000				Rb(5p) + Ba +
		10000
						Rb + + Ba(6s6p,	1 P)
		8000
	Potential energy (cm -1 )	-2000 0 2000 4000 6000				Rb(5s) + Ba + (6s) Rb + + Ba(6s5d, 3 D) Rb + + Ba(6s5d, 1 D) Rb + + Ba(6s6p, 3 P) Rb(5s) + Ba + (5d)
		-4000
		-6000
		-8000				Rb + + Ba(6s 2 )
		-10000
		-12000
		-14000	10	15	20	25
				Interatomic distance (a. u.)

  puted the state-to-state absorption cross sections, at the presence of those most relevant lasers in an experimental context. Besides being still a work in progress, we have a case study: photodissociation dynamics of weakly-bound rbba +

			Experiment	Theory
	λ = 650 nm Vib. states involved	σ (cm 2 )	σ (cm 2 ) σσ total (%)	Products
	(1) 1 Σ + →(2) 1 Σ +	0 ≤ υ ≤ 247	-	2.5×10 -24 4.0×10 -5	Rb(5s) + Ba + (6s)
	(1) 1 Σ + →(3) 1 Σ +	43 < υ ≤ 247	-	1.3×10 -18	21.8 Rb + + Ba(6s5s, 1 D)
	(1) 1 Σ + →(1) 1 Π	43 < υ ≤ 247	-	8.8×10 -24 1.4×10 -4 Rb + + Ba(6s5s, 1 D)
	(1) 1 Σ + →(4) 1 Σ +	89 < υ ≤ 247	-	2.2×10 -18	36.4	Rb(5s) + Ba + (5d)
	(1) 1 Σ + →(2) 1 Π	89 < υ ≤ 247	-	2.5×10 -18	41.8	Rb(5s) + Ba + (5d)
	(1) 1 Σ + →(5) 1 Σ +	-	-	0.0	0.0 Rb + + Ba(6s6p, 1 P)
	(1) 1 Σ + →(3) 1 Π	-	-	0.0	0.0 Rb + + Ba(6s6p, 1 P)
	Total	-	3.7×10 -18 6.14×10 -18	-	-

Table 16 :

 16 Vibrationally summed photodissociation cross sections (in cm 2 ) for transitions between the (υ, J = 0)-levels, in the singlet electronic ground state, (1) 1 Σ + , of RbBa + at total transition energy of E 650nm = 15380.6 cm -1 . A population for the set of vibrational states is assumed after a spontaneous emission from the initial weakly-bound level (υ max = 174, J = 0) belonging to the first excited electronic state, (2) 1 Σ + (the collisional entrance channel).

			Experiment	Theory
	λ = 614 nm Vib. states involved	σ (cm 2 )	σ (cm 2 ) σσ total (%)	Products
	(1) 1 Σ + →(2) 1 Σ +	0 ≤ υ ≤ 247	-	5.0×10 -27 4.3×10 -8	Rb(5s) + Ba + (6s)
	(1) 1 Σ + →(3) 1 Σ +	23 < υ ≤ 247	-	1.2×10 -18	10.2 Rb + + Ba(6s5s, 1 D)
	(1) 1 Σ + →(1) 1 Π	23 < υ ≤ 247	-	4.9×10 -24 4.2×10 -5 Rb + + Ba(6s5s, 1 D)
	(1) 1 Σ + →(4) 1 Σ +	63 < υ ≤ 247	-	5.4×10 -18	46.0	Rb(5s) + Ba + (5d)
	(1) 1 Σ + →(2) 1 Π	63 < υ ≤ 247	-	5.1×10 -18	43.7	Rb(5s) + Ba +

Resonance-enhanced multiphoton ionization.1.2 more on ultracold molecular ions

If we assume a factor of five larger heating for the data in Fig.(13b) and (13c), we still have data points enough to determine the energy dependence of the charge-exchange collisions, and we obtain the same results in the pre-factor A. Therefore, we conclude that the extra heating due to the micromotion-secularmotion coupling does not change the conclusion of our results.

case study: collisions between ultracold 6 li atoms and 40 ca + ions

6.6 conclusion

, hd + and d + 2 in collision with he

http://www.cpc.cs.qub.ac.uk/

computational tools

http://scienide2.uwaterloo.ca/~rleroy/LEVEL16/

of the elastic processes is therefore expected to be important in modeling the overall relaxation under trap conditions. The results given in the two panels of Fig. (20-bottom) show the behavior of the computed rates involving two exemplary situations: (i) the internal cooling rates for transitions where ∆N = ∆j (solid line) and (ii) the collisional cooling rates for transitions in which ∆N = ∆j (dashed lines). For simplicity, only the results for the para-H + 2 system are shown there.

The following considerations follow from the inspection of all the results in that figure: (i) The strength of the state-changing rates are strongly reduced when the ∆N = ∆j propensity rule is not satisfied, this reduction being increasingly larger when the internal molecular rotational state becomes more excited (larger N). (ii) The Rb + 2 2 Σ + g r e (a. u.) D e (cm -1 ) ω e (cm -1 ) B e (cm , where, the rotational constant, B e , and the harmonic frequency, ω e , are estimated for a 85 Rb atom with mass 84.9117893×1822.8839 atomic units. Estimations using RCCSD(T) level of theory are for an isolated diatomic system. , where, the rotational constant, B e , and the harmonic frequency, ω e , are estimated for a 85 Rb atom with mass 84.9117893×1822.8839 atomic units.

Elec. state

Long range form, Eq. ( 145) (a. u.) Asymptote

(1) 1 Σ + -0.559964 -151.57/R 4 Rb + + Ba 6s 2

(2) 1 Σ + -0.521128 -187.99/R 4 Rb(5s) + Ba + (6s)

Table 15: The functional form used to describe the [Rb-Ba] + long range potential. The parameters are estimated by fitting the form given by Eq. ( 145) to the ab initio tail potential of each electronic state of interest. Notice that positive C 4 coefficients are obtained to those potentials behaving slightly repulsive at large distances. All the parameters refers to an origin, in units of Hartree, which is the electronic ground state(1) 1 Σ + , asymptotically leading to -0.559964 a.u. as R -→ ∞.

Title : Structure and dynamics of cold molecular ions: formation and destruction processes Keywords : photoassociation, cold molecular ions, ultracold gases, inelastic collisions

Abstract :

The work sheds light on the mechanisms, and their efficiency, for (i) formation, (ii) destruction and (iii) internal cooling of cold molecular ions by inelastic ultracold collisions, such as those studied in hybrid setups merging an ultracold atom trap and a laser-cooled ion trap. We have carried out a systematic and consistent analysis of light assisted binary collisions of many relevant atom/ion pairs using accurate effective core potential based quantum chemistry calculations. Radiative association is predicted to occur for all systems with a cross section two to ten times larger than the competitive channel of radiative charge transfer. Partial and total rate constants are also calculated and compared to available experiments. Narrow shape resonances are expected, which could be detectable at low temperature with an experimental resolution at the limit of the present standards. Vibrational distributions are also calculated, showing that the final molecular ions are not created in their internal ground state level. Once light-assisted formation of molecular ions is probed, we have checked their effective radiative lifetime due to the presence of several lasers in the experiments, which turns out to efficiently photodissociate the created ions. Moreover, besides an extremely slow internal relaxation due to the presence of a permanent electric dipole moment, at reasonably time scales, we have found evidences for the population of molecular levels being strongly influenced by collisions with surrounding ultracold atoms. We have further investigated the internal cooling of both H + 2 and Rb + 2 (with the respective isotopologues) in collisions with ultracold He and Rb atoms, respectively, due to their relevance for experimental implementations. We have described the calculation of the interaction potentials needed to obtain the coupling potential matrix elements used in a multichannel close coupling formalism. In particular, it is shown that the sum of the He-H + 2 state-tostate cross sections (with and without vibrational effects) accounting for the coupling between electronic spin and molecular rotation is dynamically equivalent to directly treating the collision problem of a molecular ion as a structureless spherical rotor interacting with the He atom. The additional difficulties of a close-coupling treatment for an effective internal cooling of sympathetically cold Rb + 2 ions in collisions with Rb atoms are discussed along the following lines: (i) the homonuclear nature of the problem, inducing trap losses from reactive collisions; (ii) the higher density of internal states due to the heaviness of the system; and, (iii) its long-range neutral-charged interaction. Nevertheless, strong evidence of an efficient internal cooling by inelastic collisions with Rb atoms is found, and may indicate a general trend for all similar heavy species currently found in hybrid trap experiments.

Université Paris-Saclay

Espace Technologique / Immeuble Discovery Route de l'Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France Titre : Structure et dynamique des ions moléculaires froids : processus de formation et de destruction Mots-clés : photoassociation, ions moléculaires froids, gaz ultra-froids, collisions inélastiques Résumé : Ce travail concerne les mécanismes de formation, de destruction et de relaxation interne des ions moléculaires formes au cours de collisions inélastiques ultra-froides. Ces collisions sont étudiées expérimentalement dans des pièges hybrides mêlant un piège d'atomes alcalins ultra-froids et un piège d'ions alcalino-terreux refroidis par laser. Nous avons effectué une analyse théorique systématique des collisions binaires, assistées par la lumière, impliquant plusieurs sortes de paires atome/ion. Leur interaction mutuelle est décrite par des calculs de chimie quantique basés sur l'utilisation de potentiels de coeur effectifs très précis calculés dans l'équipe. La formation d'ions moléculaires par 'association radiative est prédite efficace pour tous les systèmes étudiés : RbCa + , RbSr + , RbBa + , RbYb + , LiCa + , LiYb + . La section efficace a une valeur calculée deux à dix fois plus grande que celle du processus concurrent, le transfert de charge radiatif. Les constantes de vitesse partielles et totales sont aussi calculées et comparées aux valeurs expérimentales disponibles, révélant un accord satisfaisant. Des résonances de forme sont attendues avec une largeur très faible, qui pourraient être observées avec la meilleure résolution expérimentale atteinte aujourd'hui. Les populations des niveaux vibrationnels des ions moléculaires formés dans leur état électronique fondamental par association radiativees ont aussi été calculées. Elles montrent que les ions moléculaires formés ne sont pas créés dans leur état vibrationnel fondamental, mais dans un grand nombre de niveaux situés à des énergies de liaison intermédiaires. Nous avons ensuite montré que ces ions moléculaires pouvaient être photodissociés par les lasers utilisés dans l'expérience pour le refroidissement et le piégeage, ce qui leur confère un caractère transitoire. En d'autres termes, dans les montages expérimentaux actuels, ces ions ne sont pas protégés de la dissociation. Par ailleurs nous avons étudié la relaxation vibrationnelle des ions formés. Dans les échelles de temps des différentes expériences, nous avons montré que celle-ci ne résulte pas d'une relaxation radiative associée au moment dipolaire permanent de l'ion, a priori trop longue, mais plutôt à des collisions avec les atomes alcalins ultra-froids environnants. Nous avons ainsi étudié cette relaxation interne pour deux systèmes différents afin de proposer une vision comparative : les ions H + 2 (resp. Rb + 2 ) lors d'une collision avec des atomes ultra-froids de He (resp. Rb). Ces deux cas sont importants pour des expériences en cours. Nous avons décrit le calcul du potentiel d'interaction nécessaire pour l'obtention de la matrice de couplage qui intervient dans la résolution des équations couplées multi-voies. En particulier il a été montré, dans le cas He-H + 2 , que les interactions spin-rotation, du fait de la structure doublet de l'ion, n'ont qu'une faible influence sur la somme des sections efficaces d'état à état (avec ou sans effet vibrationnel). Ces calculs peuvent être donc simplifiés en traitant l'ion à un électron actif comme étant dans un état de spin nul. Dans le cas Rb-Rb + 2 , l'étude de la relaxation vibrationnelle de l'ion Rb + 2 suite à des collisions froides avec des atomes de Rb se heurte à des difficultés supplémentaires: (i) l'échange des atomes identiques conduisant à des collisions réactives qui induisent des pertes dans le piège.