Xiaodong Xing

Humberto Da Silva

Romain Vexiau

Nadia Bouloufa- Maafa

Stefan Willitsch

Olivier Dulieu

Ion loss caused by rich dynamics in Ca + Coulomb crystals

Keywords: Cold molecular ions, ultracold collisions, photodissociation, nonradiative dynamics

The thesis concerns the theoretical study of ion losses in hybrid systems of ultracold atoms and trapped cold ions, an emerging and active field of research in atomic and molecular physics. Understanding the interactions and processes involved in collisions is essential for both fundamental research and the more advanced applications. In general, we are interested in the quantum dynamical processes of both radiative and non-radiative scattering reactions for a variety of atom-ion combinations which are of interest to the experimentalists: (i) Rb/Ba + , (ii) H 2 /Ba + , (iii) Rb/Ca + , and (iv) Li/Ba + . The calculated properties are directly associated to the corresponding experimental measurements. We first introduce the experimental background of the study, illustrating the essential cooling and trapping technologies employed in the atom-ion hybrid experiment. Then the basics of radiative theory and scattering theory are presented. CONTENTS III A Spin Orbit Coupling for 2-electron atom B Description of the shifted PECs for RbCa + 155 C Selected channels in SF frame for s + s dissociation limit D Selected channels in SF frame for s + p dissociation limit E Selected channels in SF frame for s + d dissociation limit Bibliography 163

 The angular momenta and quantum numbers relevant for a diatomic molecule. The projection quantum numbers are signed quantities. The nuclear spin is not considered. . . . . . . . . . . . 2.2 Symmetries and quantum numbers of the molecular states with a given total angular momentum J, and correlated to dissociation limits of two atomic systems A and B (neutral or ionic) , each possessing a single valence electron (in the present case, Rb+Ca + , Rb+Ba + , or Li+Ba + ). Displayed cases are A(s)+B(s), A(s)+B(p) and A(s)+B(d). The relevant quantum numbers are given for the three Hund's cases a,c,e. The value of the rotational quantum number l varies in order to ensure the correct parity to the total wavefunction. . . . . . . . . . . . . . . . .

Synthèse

Mots clés : Ions moléculaires froids, collisions ultra-froides, photodissociation, dynamique non-radiative.

La thèse porte sur l'étude théorique des pertes ioniques dans les systèmes hybrides d'atomes ultra-froids et d'ions froids piégés, un domaine de recherche émergent et actif en physique atomique et moléculaire. La compréhension des interactions et des processus impliqués dans les collisions est essentielle tant pour la recherche fondamentale que pour les applications les plus avancées. En général, nous nous intéressons à la dynamique quantique des réactions de diffusion radiative et non radiative pour une variété de combinaisons atome-ion qui présentent un intérêt pour les expérimentateurs : (i) Rb/Ba + , (ii) H 2 /Ba + , (iii) Rb/Ca + , et (iv) Li/Ba + . Les propriétés calculées sont directement associées aux mesures expérimentales correspondantes. Nous présentons d'abord le contexte expérimental de l'étude, en illustrant les techniques principales de refroidissement et de piégeage employées dans les expériences de pièges hybrides atome-ion. Ensuite, les bases de la théorie radiative et de la théorie de la diffusion sont présentées.

(i) La thèse étudie la photodissociation d'un ion moléculaire faiblement lié RbBa + créé par des collisions à trois corps. Nous mettons en place un modèle de photodissociation incluant le couplage spin-orbite pour le canal de dissociation afin de prédire la section efficace de formation de l'ion Ba + 'chaud', induite par le laser du piège optique dipolaire à 1064 nm. Il s'agit d'une extension du travail publié dans [Phys. Rev. Research 3, 013196 (2021)].

(ii) Ensuite, nous proposons un scénario de « recyclage » en trois étapes comprenant une réaction d'hydrogénation, une photodissociation et un refroidissement par laser Doppler, pour accompagner une expérience en cours visant à préparer un seul Ba + à longue durée de vie dans un piège de Paul. Le premier processus réactif concerne la formation de l'ion moléculaire HBa + en raison de la présence du gaz résiduel de H 2 dans le dispositif expérimental. Nous proposons d'utiliser le faisceau laser à 271.7 nm avec une intensité de 100 W/cm 2 pour réaliser la photodissociation de HBa + et récupérer un ion Ba + "chaud", ce qui est considéré comme la réaction 4 SYNTHÈSE principale. Enfin, l'ion Ba + chaud peut être recapturé dans le piège de Paul au moyen d'un mécanisme de refroidissement laser de type Doppler.

(iii) Sur la base de nouvelles courbes d'énergie potentielle ab initio et de couplages spin-orbite dépendant de la distance interatomique, nous confirmons que l'échange de charge non-radiatif est le processus de perte d'ions dominant dans l'expérience Rb-Ca + . Nousobtenons des taux de diffusion en bon accord avec les mesures expérimentales. Nous démontrons également que l'observation directe d'ions RbCa + est effectivement robuste face à une série de processus: association radiative, photodissociation et rayonnement du corps noir. Seule une petite fraction de RbCa + associée radiativement peut être directement dissociée par le laser de refroidissement de 397 nm, tandis que la photodissociation induite par le rayonnement du corps noir est négligeable.

(iv) Nous développons une série de modèles de diffusion de complexité croissante allant de la théorie semi-classique à la théorie quantique complète pour étudier les interactions non radiatives lorsqu'un Ba + unique préparé à l'état métastable 2 D 3/2 ou 2 D 5/2 entre en collision avec des atomes ultra-froids de Li à l'état fondamental. L'échange de charge non-radiatif dépendant de la parité, les processus non-radiatifs de changement de structure fine et de changement d'excitation sont distingués expérimentalement et théoriquement. Le couplage rotationnel est identifié comme l'interaction principale qui couple fortement les moments angulaires internes pendant les collisions. Enfin, les taux de réaction calculés théoriquement concordent bien avec les mesures expérimentales.

(i) The thesis investigates the photodissociation of a weakly-bound molecular ion RbBa + created by three-body collisions. We set up a photodissociation model including the spin-orbit coupling for the dissociation channel to predict the cross section for the formation of the 'hot' Ba + ion induced by the optical dipole trap laser 1064 nm as an extension of the published work in [Phys. Rev. Research 3, 013196 (2021)].

(ii) Next, we propose a three-step recycling reaction scenario including hydrogenation reaction, photodissociation and Doppler laser cooling to support an ongoing experiment to prepare a single long lifetime Ba + in a Paul trap. The first reactive process is for the formation of HBa + due to the presence of background gas H 2 in the experimental chamber. Then, we suggest taking use of the 271.7 nm laser beam with 100W/cm 2 intensity to achieve photodissociation of HBa + to retrieve a 'hot' Ba + ion, which is considered as the main reaction. Finally, the hot Ba + ion can be recaptured in the Paul trap by means of a Doppler laser cooling mechanism.

(iii) Based on novel ab initio potential energy curves and distance-dependent spin-orbit couplings, we confirm that the non-radiative charge exchange is a dominant ion-loss process in the Rb-Ca + experiment and obtain the 6 SYNOPSIS scattering rates in a good agreement with experimental measurements. We also demonstrate that the direct observation of RbCa + is indeed robust against a series of processes: radiative association, photodissociation and black body radiation. Only a small fraction of radiative associated RbCa + can be directly dissociated by the 397 nm cooling laser, while the photodissociation induced by black body radiation is negligible.

(iv) We develop a series of scattering models with an increasing complexity from semi-classical to full quantum theory to investigate the non-radiative interactions when a single Ba + prepared at metastable state 2 D 3/2 or 2 D 5/2 collides with ground state Li gas. The parity-dependent non-radiative charge exchange, non-radiative fine structure quenching and non-radiative quenching are distinguished experimentally and theoretically. The rotation couplings are demonstrated as the main interaction which strongly couple the internal angular momenta during collisions. Meanwhile, the theoretically calculated reaction rates agree well with the experimental measurements.

I still remember that the first day I arrived in France, the Jet lag was killing me, a slim but strong man helped me to carry my super heavy luggage through 8 floors of stairs to my accommodation. I would like to thank this man, my supervisor, Olivier Dulieu, for taking good care of me and training me with his extraordinary intelligence and guidance. I would like to thank the other supervisor, Nadia Bouloufa-Maafa, for her kindness and generous help in both my personal life and academic research. I will never forget that she always has the patience to answer my naive questions when we were sharing the same office in IDRIS.

I would like to thank Eliane Luc-Koenig. I am very lucky to have this opportunity to work with her. I respect her rigorous scientific attitude. I appreciate the effort she put into my thesis.

I would like to thank Maurice Raoult for the pleasant coffee break conversation, for the motivating academic discussion, and for the valuable mentoring even after he retired.

I would like to thank Humberto da Silva Jr. who is extremely friendly and has helped me tremendously. During the first year of my Ph.D., this guy spent almost all his afternoons supervising my studies. I am also very much enjoying the Friday movie night with him and Adrien Devolder.

Despite I have been in France for quite a long time, I am ashamed of my French. I would like to thank Adrien Devolder and Lucas Lassablière for helping me with all the language-related matters, namely, everything! Also, I enjoyed the Star Wars discussions with Adrien. Thanks to Lucas, I started to like climbing.

For the past two years, unfortunately, Covid-19 has made our entire planet sick, and seriously disturbed the life, work, and study of almost everyone. I would like to thank Ting Xie for his generous assistance which enabled me to complete my final research in my thesis during this difficult time.

I would like to thank Amrendra Pandey for the enthusiastic discussion, especially about ab initio method. I hope you can secure a permanent research position in India very soon.

I would like to thank again all the THEOMOL members: Olivier Dulieu, Nadia Bouloufa-Maafa, Eliane Luc-Koenig, Maurice Raoult, Laurent Wiesenfeld, Goulven Quéméner, Andrea Orbàn, Humberto da Silva Jr., Ting Xie, Amendra The respective energy level scheme with hyperfine states F of 87 Rb/ 6 Li for the laser cooling. [START_REF] Bransden | Physics of atoms and molecules[END_REF] The Euler angles describing the orientation of the molecular (x, y, z) rotated axis (purple) from the laboratory (X, Y, Z) fixed axis (green). ON (red) defines the node line, which is not only the intersection of the X-Y and the x-y planes but also perpendicular to both the Z and z axes. α is the angle between the X-axis and the ON , β is the angle between the Z-axis and the z-axis, γ is the angle between the ON and the x-axis. . . With respect to the centre of mass O, the position vectors of electrons and nuclei are indicated by r e 1 , r e 2 , r A and r B , respectively. 32 2.9 The Cauchy-Lorentzian distribution (Picture from P.218 of Ref. [START_REF] Bransden | Physics of atoms and molecules[END_REF]). 37 2.10 Schematic diagram of a scattering event between an atom A and a single ion B + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.11 The cross section of elastic collision in a single channel, and the decomposed cross section for the selected partial waves. . . . . . 46 2.12 Structureless particles A (b a > b ma x ), B (b a = b ma x ) and C (b a > b ma x ) collide a particle D with a finite size a along z axis. The maximum impact parameter b ma x = a. . . . . . . . . . . . . . . . . 47 2.13 Long-range effective potentials of a general atom-singly-charged ion system for selected partial waves s, p, d, f , g in atomic units. The location R * ma x (l = 1) and height E * ma x (l = 1) of the centrifugal barrier of the p wave is marked by the blue arrow. . . . . . . . . . 48 2.14 Top panel: The cross section of l = 3 from Figure 2.11 as a function of the collision temperature, the position (3.41 mK) of the peak is marked by a blue bar; Bottom panel: The angle of the phase shift as a function of the collision temperature, the position of the shape resonance marked by blue which is corresponding to the indicated shape resonance of l = 3 at 3.41 mK in the top panel. 50 [START_REF] Weckesser | Observation of Feshbach resonances between a single ion and ultracold atoms[END_REF] The unscaled (see text) cross section of 1064 nm laser induced photodissociation from weakly bound vibrational states of either 2 1 Σ + or 1 3 Σ + entrance channel towards the possible exit channels as a function of the vibrational quantum number of the entrance channels in Hund's case (a) basis. . . . . . . . . . . . . . . . . . . . 3. [START_REF] Bransden | Physics of atoms and molecules[END_REF] The unscaled (see text) cross section of photodissociation from the last several vibrational levels with energies (-0.1 cm -1 → -21.3cm -1 ) of either 2 1 Σ + or 1 3 Σ + entrance channel towards the possible exit channels as a function of the transition energy. The dashed line indicates the transition energy of the laser 1064 nm. 3.6 For the transition 2 1 Σ + -4 1 Σ + , unscaled cross section in panel (a) and the corresponding transition dipole moment in panel (i).

List of Figures

In each of panels (b, c, d) and zoomed-in panel (b , c , d ), the vibrational wavefunction (pink) of binding energy 10.5 cm -1 is amplified by a scaling factor of 300 for the sake of clarity. The continuum wavefunctions (blue) with explicit energies which are corresponding to the red stars in panel (a). For panels ( j, k, l) and zoomed-in panel ( j , k , l ), the integrand < vi br.|D|cont. > (green) and the cummulative sum (blue). . . . . . . . . . . . . . . for Ω = 0 +/-, 1, 2 in Hund's case c basis. The corresponding potential energy curves (thin curves) in Hund's case a basis are presented, respectively. The dashed pink arrow indicates the 1064 nm laser. The exit kinetic energy of the system is ∼ 5350 K or ∼ 6470 K with respect to 2 D 5/2 or 2 D 3/2 , respectively. 68 3.9 From the top to the bottom as a function of the internuclear distance in atomic units: (1-2) the proportion of 4 1 Σ + and 3 3 Π for (a)0 + and (b)0 + states, respectively; (3) TEDMs in Hund's case (a); [START_REF] Weckesser | Observation of Feshbach resonances between a single ion and ultracold atoms[END_REF][START_REF] Bransden | Physics of atoms and molecules[END_REF] TEDMs in Hund's case (c). . . . . . . . . . . . . . . 70 3.10 From the top to the bottom as a function of the internuclear distance in atomic units: (1-2) the proportion of 4 3 Σ + and 3 3 Π for (a)0 -and (b)0 -states, respectively; (3) TEDMs in Hund's case (a); (4) TEDMs in Hund's case (c). . . . . . . . . . . . . . . . 71 3.11 From the top to the bottom as a function of the interatomic distance in atomic units: (1-4) the proportion of 4 3 Σ + , 3 3 Π, 2 1 Π and 2 3 ∆ for (a)1, (b)1, (c)1 and (d)1 states, respectively; [START_REF] Bransden | Physics of atoms and molecules[END_REF] TEDMs in Hund's case (a); [START_REF] Zrafi | Ab initio electronic structure and prospects for the formation of ultracold Calcium-alkali-metal-atom molecular ions[END_REF][START_REF] Hall | Cold ion-neutral reactions[END_REF] TEDMs in Hund's case (c). . . . 72 3.12 From the top to the bottom as a function of the interatomic distance in atomic units: (1-3) the proportion of 3 3 Π, 2 1 ∆ and 2 3 ∆ for (a)2, (b)2 and (c)2 states, respectively; (4) TEDMs in Hund's case (a); [START_REF] Bransden | Physics of atoms and molecules[END_REF] TEDMs in Hund's case (c). . . . . . . . . . . 73 3.13 The unscale photodissociation cross section from several bound levels with energies (-0.1 cm -1 → -21.3 cm -1 ) of either 2 1 Σ + or 1 3 Σ + entrance channel, towards the exit channels of Ω = 0 +/- as a function of the transition energy in Hund's case c basis. The vertical dashed line locates the position of 1064 nm. . . . . . . . 76 3.14 The unscale photodissociation cross section from several bound levels with energies (-0.1 cm -1 → -21.3 cm -1 ) of either 2 1 Σ + or 1 3 Σ + entrance channel, towards the exit channels of Ω = 1 as a function of the transition energy in Hund's case c basis. The vertical dashed line locates the position of 1064 nm. . . . . . . . 77 3.15 The unscale photodissociation cross section from several bound levels with energies (-0.1 cm -1 → -21.3 cm -1 ) of either 2 . At the end of the atom-ion interaction phase, the xODT is switched off. After a short expansion time (∼ 100µs typically) the atomic could is absorption-imaged at high magnetic field, which is then switched off. In the following sequence, the ion state is detected and classified as one of four possible outcomes. First, the Doppler cooling lasers are switched on for 1 s, followed by 300 ms of fluorescence imaging. An ion detected in this stage is classified as cold 6S 1/2 or 5D 3/2 . Secondly, a far-detuned cooling beam is switch on, followed by fluorescence detection. An ion appearing in this stage is classified as hot. Lastly, the 614.9 nm repumping laser is switched on to detect whether the ion is shelved in the 5D (top panel) or 2 D 5/2 (bottom panel) state, interacting with an ultracold gas of atomic Li. An event is categorized as a survival, if the ion can be fluorescence-detected, either after a short Dopplercooling period (top panel) or after switching on the 615 nm repumping laser for a short period (bottom panel). Each data point shows the averaged survival probability of at least 20 interaction events, the error bars representing the 1δ confidence interval. The black curve is a fit to the data with an exponential decay, where the shaded area represents the uncertainty of the fit. The non-unity survival probability at t = 0 ms originates from the finite size of the atomic cloud, meaning the ion starts interacting with the Li atoms while moving to its final position. . . . . . . . . 5.3 Hund's case (a) LiBa + potential energy curves in the BF frame.

The black circle locates the avoided crossing between the 2 1 Σ + and 3 1 Σ + PECs contributing the NRCE (red arrow): Li + +Ba(6s

Chapter 1

Introduction

In recent years, the development of laser cooling technology has enabled the opportunity to study the exotic phenomena of gas-phase atoms/molecules and ion Coulomb crystals at very low temperatures and revolutionized the development of atomic molecular optical (AMO) physics. Under sub-kelvins, cold/ultracold charged atoms/molecules exhibit novel physical and chemical properties for many applications such as quantum logic spectroscopy [START_REF] Kienzler | Quantum logic spectroscopy with ions in thermal motion[END_REF][START_REF] Sackett | Experimental entanglement of four particles[END_REF][START_REF] Monroe | Entangled states of trapped atoms for quantum logic and spectroscopy[END_REF][START_REF] Peik | Logical spectroscopy[END_REF][START_REF] Rosenband | Single-ion optical clocks[END_REF], quantum communication [START_REF] Calderaro | Towards quantum communication from global navigation satellite system[END_REF][START_REF] Chen | An integrated space-to-ground quantum communication network over 4,600 kilometres[END_REF][START_REF] Bhaskar | Experimental demonstration of memory-enhanced quantum communication[END_REF][START_REF] Bose | Quantum communication through an unmodulated spin chain[END_REF] and quantum computing [START_REF] Bruzewicz | Trappedion quantum computing: Progress and challenges[END_REF][START_REF] Milburn | Ion trap quantum computing with warm ions[END_REF][START_REF] García-Ripoll | Speed optimized two-qubit gates with laser coherent control techniques for ion trap quantum computing[END_REF][START_REF] Blinov | Quantum computing with trapped ion hyperfine qubits[END_REF]. Notably, the rapid growth of research integrating quantum physics with computerization, digitalization, and artificial intelligence pioneers the Fourth Industrial Revolution for humankind. Taking a step back, why do cold-charged species (atoms, molecules, or clusters) matter? The motion of charged species [START_REF] Willitsch | Very cool molecular ions[END_REF] can be well localized and isolated from noisy environments by effectively manipulating the electric and magnetic field. And they can be stored for long periods for further operations. These characteristics make cold species suitable candidates to merge with other species for the detection of collision reactions and other fundamental studies.

Ultracold atoms vs Cold ions

Twenty years back, R. Côte and A. Dalgarno [START_REF] Côté | Ultracold atom-ion collisions[END_REF] put forward the idea of hybrid traps, combining two well-separated fields, ultracold (< 1 mK) atom experiment, and cold (< 1 K) ion experiment, to study charge exchange with an example of Na-Na + collisions at ultralow energies. Five years later, W. Smith and co-workers [START_REF] Smith | Cold ion-neutral collisions in a hybrid trap[END_REF] successfully implemented the first hybrid traps of the Na-Ca + system, and demonstrated that the two experimental devices can indeed coexist stably. The charge exchange as the most intuitive and fundamental dynamic in ion-atom collisions is observed over the collision temperature range (35 mK -45K) in different isotopic systems of Yb + + Yb in 2009 [START_REF] Grier | Observation of cold collisions between trapped ions and trapped atoms[END_REF]. So far, a rich diversity of atom-ion combinations have been investigated theoretically and experimentally, such as, (Ca/Ba/Rb/Yb/Sr/Be/H+Na/Li/Na/Rb/Yb) + [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 ion-neutral reactive processes in the cold regime: Radiative molecule formation versus charge exchange[END_REF][START_REF] Hall | Millikelvin reactive collisions between sympathetically cooled molecular ions and laser-cooled atoms in an ionatom hybrid trap[END_REF][START_REF] Saito | Characterization of charge-exchange collisions between ultracold 6 Li atoms and 40 Ca + ions[END_REF][START_REF] Haze | Cooling dynamics of a single trapped ion via elastic collisions with small-mass atoms[END_REF][START_REF] Ben-Shlomi | Direct observation of ultracold atom-ion excitation exchange[END_REF][START_REF] Sikorsky | Phase locking between different partial waves in atom-ion spin-exchange collisions[END_REF][START_REF] Sikorsky | Spincontrolled atom-ion chemistry[END_REF][START_REF] Zipkes | A trapped single ion inside a Bose-Einstein condensate[END_REF][START_REF] Sayfutyarova | Charge transfer in cold Yb + +Rb collisions[END_REF][START_REF] Ratschbacher | Decoherence of a single-ion qubit immersed in a spin-polarized atomic bath[END_REF][START_REF] Fürst | Prospects of reaching the quantum regime in Li-Yb + mixtures[END_REF][START_REF] Joger | Observation of collisions between cold Li atoms and Yb + ions[END_REF][START_REF] Fürst | Dynamics of a single ion-spin impurity in a spin-polarized atomic bath[END_REF][START_REF] Hirzler | Experimental setup for studying an ultracold mixture of trapped Yb + -6 Li[END_REF][START_REF] Weckesser | Trapping, shaping, and isolating of an ion coulomb crystal via state-selective optical potentials[END_REF][START_REF] Gacesa | Charge transfer in ultracold gases via Feshbach resonances[END_REF][START_REF] Li | Photon-mediated charge exchange reactions between 39 K atoms and 40 Ca + ions in a hybrid trap[END_REF]. For atom-ion 1 2

TRAPPING AND COOLING TECHNIQUE

hybrid traps, cold trapped ions usually collide with ultracold trapped atoms in the millikelvin regime. The general processes are the formation of molecular ions by radiative association [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 ion-neutral reactive processes in the cold regime: Radiative molecule formation versus charge exchange[END_REF][START_REF] Hall | Millikelvin reactive collisions between sympathetically cooled molecular ions and laser-cooled atoms in an ionatom hybrid trap[END_REF][START_REF] Da Silva Jr | Formation of molecular ions by radiative association of cold trapped atoms and ions[END_REF], radiative/non-radiative charge/excitation exchange, light-assisted collision, photoassociation, direct/indirect photodissociation, and three-body many-body collisions depending on the density of atomic gas. Recently, the branches of the study are expanded even further to swave atom-ion collisions [START_REF] Feldker | Buffer gas cooling of a trapped ion to the quantum regime[END_REF], precision-controlled chemistry [START_REF] Sikorsky | Spincontrolled atom-ion chemistry[END_REF], atom-ion entanglements [START_REF] Secker | Controlled long-range interactions between Rydberg atoms and ions[END_REF], quantum gates [START_REF] Doerk | Atom-ion quantum gate[END_REF], impurity physics [START_REF] Pérez-Ríos | Cold chemistry: a few-body perspective on impurity physics of a single ion in an ultracold bath[END_REF], mesoscopic molecular ions [START_REF] Cote | Mesoscopic molecular ions in Bose-Einstein condensates[END_REF] and interstellar molecular formations [START_REF] Wakelam | Reaction networks for interstellar chemical modelling: improvements and challenges[END_REF].

Trapping and cooling technique

Although the focus of this thesis is on the theoretical study of ion-neutral collisions, it is worthwhile to briefly introduce the most important techniques for cooling and trapping atoms or charged atoms in experiments.

Doppler laser cooling

Doppler laser cooling method was respectively proposed by Hänsch-Schawlow [START_REF] Hänsch | Cooling of gases by laser radiation[END_REF] and Wineland-Dehmelt [START_REF] Wineland | Proposed 1014 delta upsilon less than upsilon laser fluorescence spectroscopy on t1+ mono-ion oscillator iii[END_REF] in 1975. The Nobel Prize (1997) [START_REF] Phillips | Nobel lecture: Laser cooling and trapping of neutral atoms[END_REF][START_REF] Cohen-Tannoudji | Nobel lecture: Manipulating atoms with photons[END_REF][START_REF] Chu | Nobel lecture: The manipulation of neutral particles[END_REF] was awarded to S. Chu, C. Cohen-Tannoudji, and W.D. Phillips for the development of cooling and trapping atoms with lasers. This approach has accelerated the development of research in ultracold physics and opened the window to the discovery of the quantum world. The mechanism of this technique is to apply a red-detuned cooling laser within the framework of an absorption-emission cycle to slow down the atoms. As the atom moves against the laser beam, the atom will absorb photons coming from this single direction and this process can slow down the movement of the atom. Since the motion of the spontaneously emitted photons can be in any direction in space, the gains in the momentum of spontaneous emission for the atom can be averaged out over a period of time. Usually, 10 3 -10 4 cooling cycles are needed in a few milliseconds [START_REF] Dorfler | Cold molecular ion-neutral collisions in a dynamic ion-atom hybrid trap[END_REF][START_REF] Tomza | D thesis: Quantum dynamics and control of ultracold molecules in external fields[END_REF]. Then atoms can reach ∼ 240 µK with a low density in the order of 10 6 cm -3 [START_REF] Eberle | Increased control over reaction conditions in a hybrid trap[END_REF].

Because the restoring force is missing in this method, the atoms are not trapped. The cooled atoms have approximately 0.1 second lifetime in the cooling center and eventually escape in a few seconds [START_REF] Dorfler | Cold molecular ion-neutral collisions in a dynamic ion-atom hybrid trap[END_REF]. This technology is well known as optical molasses.

Magneto-optical traps (MOT)

The cooling and trapping scheme of MOT is introduced in 1987 by E. L. Raab et al. [START_REF] Raab | Trapping of neutral sodium atoms with radiation pressure[END_REF] by adding quadrupole magnetic field (2 coils) to the optical molasses, as shown in Figure 1.1 (a). Along z-axis, the magnetic field is B(z) = Az with a constant A. Considering that the Zeeman splitting equation is proportional to B, the linear energy shifts of magnetic sublevels start from z = 0, i.e., the energy The respective energy level scheme with hyperfine states F of 87 Rb/ 6 Li for the laser cooling.

TRAPPING AND COOLING TECHNIQUE

levels simply merge at the center and scale linearly with increasing magnetic field strength, moving away from the center along the horizontal coordinate axis. Figure 1.1 (b) is for a prototype atom with the ground state J = 0 and the excited state J = 1, the magnetic sublevel |J, m > of J = 0 is |0, 0 >, J = 1 has three magnetic sublevels |1, +1 >, |1, 0 > and |1, -1 >, where |1, 0 > is immune to the magnetic field. Assuming that the atoms move out of the center along the σ + direction, the magnetic sublevel |1, -1 > is pulled closer to resonance by the σ -polarisation cooling laser, while |1, +1 > moves out of the resonance range. The transition selection rule from the ground state |0, 0 > requires that |1, -1 > should only be tackled by σ -, while |1, +1 > should only be addressed by σ + polarised light. The σ -laser beam travels in the σ -direction, which means that the net momentum of these photons after scattering is in the same direction, thus forcing the atom back towards z = 0. The picture is reversed when the atom is beginning to travel in the σ -direction, in which case the σ + photons are scattered and the atom reverts to the z = 0 direction.

In the experiments concerned in this thesis, like Rb-Ba + , Rb-Ca + , and Li-Ba + system, MOT is widely used to prepare the ultracold gases of 87 Rb and 6 Li, whose energy level schemes are presented in Figure 1.1 (c) and (d).

Radio frequency (rf) traps

In 1989, W. Paul [START_REF] Paul | Ein neues massenspektrometer ohne magnetfeld[END_REF] shared the Nobel Prize for his research on quadrupole ion traps. Since then, this technique has also been widely known as Paul trap or rf-trap. The ions are sensitive to the electric field due to their charge, which allows creating a quadrupole electric potential to trap them. Various types of traps are available, but the traps covered in this thesis are designed with four rods and two endcaps, displayed in Figure 1.2 (a). A trapped ion's motion has been characterized as a combination of the slower secular motion and the fasterdriven motion. The latter is known as the micromotion due to the displacement of the ion from the rf-axis by the frequency of the applied electric field [START_REF] Zhukas | Direct observation of ion micromotion in a linear paul trap[END_REF], which often heat the ions. This is why the temperature of the ions in the rf-trap is hardly below 1 mK. It should be stated that the method of the laser cooling of ions is similar to these of atoms, but the cooling cycle is more complicated, for instance, the cooling scheme of 138 Ba + and 40 Ca + in Figure 1.2 (c) and (d).

Optical dipole traps (ODT)

Compared to MOT, an optical dipole trap (ODT) has the advantage of being able to capture cold atoms/ions at any Zeeman energy level and, most importantly, to cool the atoms by direct evaporation by reducing the power of the laser that constitutes the optical trap, which is why ODT is widely used for the all-optical preparation of atoms for Bose-Einstein condensations. The theory of an ODT is composed of a focused, far red-detuned, trapping laser beam, and a pair of red-detuned, counter-propagating, cooling beams [START_REF] Garraway | Theory of an optical dipole trap for cold atoms[END_REF]. Documented reviews

OUTLINE OF THESIS

for specific theoretical mechanisms and experimental methods are available in the literature [START_REF] Julian | Optical trapping of ion Coulomb crystals[END_REF][START_REF] Weckesser | Trapping, shaping, and isolating of an ion coulomb crystal via state-selective optical potentials[END_REF][START_REF] Garraway | Theory of an optical dipole trap for cold atoms[END_REF]61] . Due to the shallow depth of the ODT, thereby it can be expected that the density of species in the ODT is quite low, and it is possible to trap individual atoms or particles with sub-mK temperatures in ODT, such as Ba + /Ca + [START_REF] Julian | Optical trapping of ion Coulomb crystals[END_REF][START_REF] Weckesser | Trapping, shaping, and isolating of an ion coulomb crystal via state-selective optical potentials[END_REF][START_REF] Mohammadi | Life and death of a cold BaRb + molecule inside an ultracold cloud of Rb atoms[END_REF] in Figure 1.2 (b).

Hybrid atom-ion trap

Hybrid traps, as the name implies, are combinations of diverse traps for the study of controlling and simulating various quantum processes between species. As shown in Figure 1.3, MOT+rf-trap and MOT+ODT+rf-trap are the common hybrid atom-ion experimental apparatus. As a matter of fact, the hybrid MOT+ODT trap is employed in research on ultracold atom-atom systems, for example, 41 K- 87 Rb dual-species BECs [START_REF] Burchianti | Dual-species Bose-Einstein condensate of 41 K and 87 Rb in a hybrid trap[END_REF]. To enter the ultracold domain, two groups achieved the significant breakthroughs with Li-Yb + collision at the swave energy (or 9.9(2.0) µK) [START_REF] Feldker | Buffer gas cooling of a trapped ion to the quantum regime[END_REF], and with Feshbach resonances in Li-Ba + [START_REF] Weckesser | Observation of Feshbach resonances between a single ion and ultracold atoms[END_REF] by the use of the large ion-atom mass ratio to mitigate the heating process from the ion trap's oscillating rf-fields.

Outline of thesis

This thesis consists of the derivation of the theory and the specific application in atom-ion hybrid traps. 1) Chapter 2 introduces the general notations, units, and formulas in the radiative theory and the scattering theory. We have derived various quantities like the spin-orbit matrix, the rotational coupling, and the radiative transition cross sections and rate based on dimensionality analysis.

2) Chapter 3 studies the laser-induced accidental and intentional molecular dissociation of 87 Rb 138 Ba + and 1 H 138 Ba + , respectively. Due to the observations from the Rb/Ba + experiment in the group of Prof. J.H. Denschlag (Ulm, Germany), the formation and destruction of molecule RbBa + is observed. We model a photodissociation scheme with fine structure effects to predict the output ion with high kinetic energies. The photodissociation model without spin-orbit interactions is used to guide the experiment by Prof. M. Drewsen (Aarhus Denmark) picking the optimal laser to completely remove all hydrides (mainly HBa + ) in the ion clock device.

3) Chapter 4 focuses on the radiative association, the photodissociation, and the black-body radiation to in-depth analysis of the ground state 87 Rb 40 Ca + molecular ions, which are created in Ca + Coulomb crystals in the experiment of Prof S. Willitsch (Basel, Switzerland). The non-radiative charge exchange is revisited with an alternative new set of ab initio potential energy curves and 1.3. OUTLINE OF THESIS 7 spin-orbit couplings depending on the internuclear distance. 4) Chapter 5 investigates a complex non-radiative scattering problem at the metastable state 2 D 3/2,5/2 of 138 Ba + in the cold hybrid 6 Li/ 138 Ba + experiment which is performed by Prof. T. Schaetz (Freiburg, Germany). We have developed scattering models from classical via semi-classical to fully quantum in order to study the interaction with Li of the metastable state of Ba + with high angular moments.

5) Chapter 6 presents the summary of the achievements of this thesis and the perspectives for the future work of ultracold/cold diatomic systems.

Chapter 2

Theoretical approaches

In the ultra-low energy range, the quantum effects in the atomic, molecular, and optical (AMO) system cannot be studied without two important physical pillars: Schrödinger equation and Born-Oppenheimer approximation. The Schrödinger equation of a system is Ĥψ = Eψ, where Ĥ is the Hamiltonian operator, E is the energy, and ψ is the wave function of a quantum state. Without any approximation, all of the kinetic energy of the nuclei and the electrons and the potential energy between the nuclei and between the electrons are included in the Hamiltonian. However, solving such an equation is extremely difficult due to the fact that there are too many variables. Luckily, Born and Oppenheimer proposed an approximation to simplify this equation and to make it solvable. Considering that the nuclei are much heavier than the electrons, the nuclei move much slower than the electron. Every time the nuclei move slowly to a new position, the electrons can quickly adjust themselves to best fit the motion of the nuclei. This approximation is called the adiabatic approximation. With this respect, naturally, the energy of the system at every nuclei configuration can be calculated, i.e., the adiabatic potential energy surface. Taking into account the motion of nuclei, herein, the Schrödinger equation can be solved appropriately. A perfectly defined Hamiltonian is the key to solve the Schrödinger equation.

In this chapter, first and foremost, I first define the essential quantum numbers, coordinate systems, and selection rules. Then I introduce the used methodology for the diatomic molecular structure calculations with or without the fine structure in section 2.1. In section 2.2, I concentrate on the resolution of the Schrödinger equation in body-fixed (BF) frame (Figure 2.1 (a)), and in the space-fixed (SF) frame (Figure 2.1 (c)) including the rotation of the molecules. Both sections actually define the relevant interactions in a molecular Hamiltonian. Note that the nuclear spins are not considered in the current theoretical model. Afterward, the theory of molecular dynamics is discussed in section 2.3 by focusing on the problem of the formation or destruction of molecular ions (marked by the pink arrow in

Molecular structure for diatomic molecules

In my project, my duty is to examine ab initio potential energy surfaces, electronic dipole moments, and spin-orbit couplings which are provided by my colleagues, and to focus mainly on the study of modeling various dynamics in hybrid cold atom-ion experiments. Therefore, the methods and results of ab initio calculations will be discussed only in a brief fashion. Despite that, I would like to emphasize that I have derived various spin-orbit matrices to support the development of the new ab initio r-dependent spin-orbit coupling code in our group.

Quantum numbers, Hund's case (a), (c), (e), and Parity

For the sake of the ensuing discussion, we summarize the angular momenta and quantum numbers relevant for the characterisation of the states of a diatomic molecule in Table 2.1, in which the projection quantum number on molecular axis z is considered to be either positive or negative. If the total nuclear spin I is ignored, the total angular momentum operator J is given by

J = L + S + l = j + l (2.1)
= L + Ŝ j Ω = Λ + Σ Nuclear rotation l l 0 Total angular momentum Ĵ = L + Ŝ + l J Ω
In molecules, the magnetic interaction is able to couple the total electronic orbital angular momenta L, the total electronic spin S, where the magnetic field either originates from the electrons or is generated by the rotation l of the molecule. One of the most important interactions is the spin-orbit (SO) coupling between S and L. It can be expressed as ξ(r) L • S, where ξ(r) is called the spin-orbit coupling factor. To analyze the couplings of L, S and l, F. Hund identifies five cases (a, b, c, d, e) based on the relative strengths of the electrostatic interaction, spin-orbit, and rotational energies. The electrostatic interaction [START_REF] Bransden | Physics of atoms and molecules[END_REF] between the electrons and the nuclei constrains the electronic wave function to rotate as the molecule rotates. A measure of the strength of this interaction is the difference in energy |∆E| between two adjacent electronic states with different values of Λ.

We now proceed to discuss the principal interactions and selection rules in each of Hund's cases.

Hund's case (a) in BF Frame

In Figure 2.2 case (a), the electrostatic interaction is stronger than the spinorbit interaction which in turn is larger than the rotational interaction. The projection quantum numbers Λ and Σ are well-defined on the molecular z axis. The two momenta l and j are coupled to form J. The electronic molecular state is characterized by the symbol by |2S+1| Λ Ω , where Ω = Λ + Σ, |2S + 1| is the spin multiplicity, and Λ = 0, ±1, ±2, ±3... correspond to the notations Σ, Λ, ∆, Φ.... At this stage, it is useful to display the selection rules for electric-dipoleallowed transitions, which can be expressed in the |J, S, Λ, Σ, Ω > basis. As indicated in Figure 2.1 (b), the radiative process requires the transition electric dipole moment as a bridge connecting two states. Due to the fact that the electric dipole operator D is independent of the spin, the first selection rule for

Since the angular momentum carried by a single photon is 0, ±1, the transition is also governed by a selection rule for the total angular momentum,

∆J = 0, ±1 ∆Ω = 0, ±1. (2.3)
We call parallel (resp. perpendicular) transitions when the parallel (resp. perpendicular) component of the dipole moment is active, e.g. Σ + -Σ + transition (resp. Σ + -Π transitions). For a given vibrational band (i.e. a series of rotational transitions based on a given vv vibrational transition) in molecular spectroscopy, transitions corresponding to ∆J = -1, ∆J = 0, ∆J = +1 are usually referred to as P, Q, and R branches respectively). Note that there is no Q-branch for a parallel transition.

Hund's case (c) in BF Frame

In Figure 2.2 case (c), the spin-orbit interaction is larger than the electrostatic interaction which in turn is larger than the rotational interaction. As such Λ and Σ are no longer good quantum numbers. L and S couple to form j, l couples with j to form J, the projection of J is Ω, where |Ω| = 0 + , 0 -, 1, 2.... Here, the ± superscript of 0 represents the plane symmetry with respect to the molecular z axis. Then, the basis is |J, Ω >. The selection rules are same as Hund's case (a) for ∆J and ∆Ω.

Hund's case (e) in SF Frame

In Figure 2.2 case (e), both the spin-orbit and rotational interaction are significantly larger than the electrostatic interaction. Again, L and S couple to form j, where j = j A + j B with j A = L A + S A and j B = L B + S B . Then, l couples j to form J. The good quantum numbers are J, l, and j. The selection rules are the same as Hund's case (a) and (c) for ∆J.

Parity

Space inversion is the operation in which the axes of the reference frame are reflected with respect to the origin O [START_REF] Thankappan | Quantum Mechanics[END_REF], as in Figure 2.3. This operation converts a right-handed coordinate system to left-handed:

x → -x, y → -y, z → -z. A function F ( r) is unchanged by two consecutive applications of the inversion operator Î, Î Î F ( r) = Î F (-r) = F ( r) (2.4)
Therefore Î F ( r) = P F ( r), where P = ±1 is the eigenvalue: it is a good quantum number. Note that the inversion operator only acts on the spatial coordinates, thus leaving the spin unchanged.

The wave function of a single electron i in an atom is characterized by the quantum numbers (n i , l i , m l i ):

u n i ,l i ,m l i ( r, θ , φ) = R n i l i (r)P m l i l i (cosθ )e im l i φ , (2.5)
where P m l i l i (cosθ ) is a Legendre polynomial. Given the spatial inversion (θ → 14

MOLECULAR STRUCTURE FOR DIATOMIC MOLECULES

πθ , φ → φ + π) we have e im l i (φ+π) = (-1) m l i e im l i φ P m l i l i (-cosθ ) = (-1) l i -m l i P l i (cosθ ).

(2.6)

Hence, the parity operator Î acting on the wave function, we get

Îu n i l i m l i = (-1) l i u n i l i m l i (2.7)
The parity of the wave function of a single electron in an atom is

P i = (-1) l i (2.8)
If the atom possesses many electrons, then

P = (-1) l i (2.9)
For a diatomic molecule (AB) with one valence electron on each atom, the parity [START_REF] Mies | Molecular theory of atomic collisions: Fine-structure transitions[END_REF][START_REF] Julienne | Nonadiabatic theory of atomic line broadening: Final-state distributions and the polarization of redistributed radiation[END_REF] now acts on both the electrons and nuclei spatial coordinates so that

P = (-1) l i +l = (-1) l P A P B = (-1) l+l A +l B , (2.10)
where, l = |J -j|, ..., J + j. The total wave function of the molecule is characterized by a well-defined parity. Current conventions established by spectroscopists [START_REF] Brown | The labeling of parity doublet levels in linear molecules[END_REF] refer to e parity when P = (-1) J with integer J (even number of electrons), or P = (-1) J+1/2 with half-integer J (odd number of electrons), while f parity corresponds to P = (-1) J+1 or P = (-1) J+3/2 . It needs to be emphasized that the section about the rotational coupling will be discussed based on good quantum numbers and the parity in the BF frame (Hund's case (a/c)) and the SF frame (Hund's case (e)), which are listed in Table 2.2 for specific examples relevant for my work. The corresponding scattering channels which will emerge in the SF frame are summarized in Appendices C, D, and E.

In the following section on radiative transitions, the treatment of wave functions will not involve specific quantum numbers, but rather simply uses the initial and final states to indicate the allowed transitions between two energy states.

Ab initio calculations without spin-orbit couplings

The first ab initio calculation in our group was carried out by M. Aymar and O. Dulieu in 2005 on permanent dipole moments of the lowest 1,3 Σ + states of heteronuclear alkali dimers [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] taking the CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively) algorithm [START_REF] Huron | Iterative perturbation calculations of ground and excited state energies from multiconfigurational zerothorder wavefunctions[END_REF]. Up to now, this method has been successfully used to correctly predict ground/excited state structures without spin-orbit interactions and electronic dipole moments in good agreement with experimental data for various molecules [START_REF] Da Silva Jr | Formation of molecular ions by radiative association of cold trapped atoms and ions[END_REF][START_REF] Aymar | Electronic properties of francium diatomic compounds and prospects for cold molecule formation[END_REF][START_REF] Aymar | Accurate calculations of electronic properties of alkali dimers for ultracold molecule formation[END_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] Azizi | Electronic structure of alkali polar ions[END_REF][START_REF] Deiglmayr | Calculations of static dipole polarizabilities of alkali dimers: Prospects for alignment of ultracold molecules[END_REF][START_REF] Vexiau | Longrange interactions between polar bialkali ground-state molecules in arbitrary vibrational levels[END_REF]. Table 2.2: Symmetries and quantum numbers of the molecular states with a given total angular momentum J, and correlated to dissociation limits of two atomic systems A and B (neutral or ionic) , each possessing a single valence electron (in the present case, Rb+Ca + , Rb+Ba + , or Li+Ba + ). Displayed cases are A(s)+B(s), A(s)+B(p) and A(s)+B(d). The relevant quantum numbers are given for the three Hund's cases a,c,e. The value of the rotational quantum number l varies in order to ensure the correct parity to the total wavefunction.

(a) ( S Λ |Ω| ) (c) (|Ω|) (e) ( j A , j B , j, l) e f e f e f s+s 1 Σ + 0 3 Σ + 0 0 + 0 - (1/2,1/2,0,J) (1/2,1/2,1,J+1) 3 Σ + 1 3 Σ + 1 1 1 (1/2,1/2,1,J) (1/2,1/2,1,J-1) s+p 1 Σ + 0 3 Σ + 0 0 + 0 - (1/2,1/2,1,J+1) (1/2,1/2,0,J) 3 Σ + 1 3 Σ + 1 1 1 (1/2,1/2,1,J-1) (1/2,1/2,1,J) 1 Π 1 1 Π 1 1 1 (1/2,3/2,1,J+1) (1/2,3/2,1,J) 3 Π 0 3 Π 0 0 + 0 - (1/2,3/2,1,J-1) (1/2,3/2,2,J-2) 3 Π 1 3 Π 1 1 1 (1/2,3/2,2,J+1) (1/2,3/2,2,J) 3 Π 2 3 Π 2 2 2
(1/2,3/2,2,J-1) (1/2,3/2,2,J+2)

s+d 1 Σ + 0 3 Σ + 0 0 + 0 - (1/2,3/2,2,J+2) (1/2,3/2,2,J+1) 3 Σ + 1 3 Σ + 1 1 1 (1/2,3/2,2,J) (1/2,3/2,2,J-1) 1 Π 1 1 Π 1 1 1 (1/2,3/2,2,J-2) (1/2,3/2,1,J+1) 3 Π 0 3 Π 0 0 + 0 - (1/2,3/2,1,J) (1/2,3/2,1,J-1) 3 Π 1 3 Π 1 1 1 (1/2,5/2,3,J+2) (1/2,5/2,3,J+3) 3 Π 2 3 Π 2 2 2 (1/2,5/2,3,J) (1/2,5/2,3,J+1) 1 ∆ 2 1 ∆ 2 2 2 (1/2,5/2,3,J-2) (1/2,5/2,3,J-1) 3 ∆ 1 3 ∆ 1 2 2 (1/2,5/2,2,J+2) (1/2,5/2,3,J-3) 3 ∆ 2 3 ∆ 2 2 2 (1/2,5/2,2,J) (1/2,5/2,2,J+1) 3 ∆ 3 3 ∆ 3 3 3 (1/2,5/2,2,J-2) (1/2,5/2,2,J-1)

MOLECULAR STRUCTURE FOR DIATOMIC MOLECULES

In brief, each ionic core A + and B + of the atoms composing an AB molecule is described by a parametric pseudopotential [START_REF] Durand | A theoretical method to determine atomic pseudopotentials for electronic structure calculations of molecules and solids[END_REF][START_REF] Durand | New atomic pseudopotentials for electronic structure calculations of molecules and solids[END_REF] including scalar relativistic effects, which parameters are adjusted to reproduce the energies and valence orbitals of all-electron Hartree-Fock self-consistent calculations. Therefore, alkali-metal atoms and alkaline-earth ions with one positive charge are treated as one-valence-electron species. In addition, a parametric core-polarization potential is added to mimic the correlation between core and valence electrons, which parameters are fixed to reproduce the lowest energy levels of the effective one-electron systems. The electronic structure of the resulting diatomic molecule is determined by an effective two-valence-electron problem, involving a core-centered large Gaussian basis set. This approach allows for performing a full configuration interaction in the Hilbert space of the two-valence-electron determinants. The potential energy surfaces without spin-orbit couplings and permanent/transition dipole moments are extracted from the diagonalization of the resulting Hamiltonian matrix. In this thesis, we applied this method to several systems: RbBa + , HBa + , RbCa + , and LiBa + . More details about the calculations will be presented in the dedicated chapters.

Ab initio calculations with spin-orbit couplings

The inclusion of spin-orbit interaction is required when it becomes noticeable. This is the case for the heavy elements considered in this thesis, namely Rb, Ba (and Ba + ), and Ca (and Ca + ), where the spin-orbit splittings of the lowest atomic energy levels range between a few tens and a few hundreds of cm -1 . For current precisely-controlled experiments such as ultracold collision experiments in hybrid traps, highly-accurate potential energy surfaces with fine structure is crucial to get insight into the measured quantum effects. However, performing fully relativistic quantum calculations is difficult and costly in terms of computing resources. In this section, we present a simple projection method to calculate the r-dependent spin-orbit couplings and Hund's case (c) potential energy surfaces.

General methodology

A new ab initio program has been recently developed by R. Vexiau in our group, to directly calculate Hund's case (c) potential energy surfaces and the corresponding r-dependent spin-orbit couplings. The central idea was to elaborate a simple algorithm involving only the atomic spin-orbit coupling constants.

The atomic spin-orbit effect is well-characterized by the energy spacing between the energy levels of the multiplet arising from the coupling of electronic orbital angular momentum and spin. The spin-orbit couplings in a diatomic molecule are then easily handled in a separated atomic picture, which is valid for large elongations. An effective term H SO = A SO ( A • s A + B • s B ) is added to the electronic Hamiltonian. The resulting matrix elements are easily evaluated in the framework of the linear combination of atomic orbitals (LCAO) approach (see more details below). After diagonalization of this complete operator, the eigenenergies describe the system at large internuclear distances, well beyond the range of the electron exchange interaction characterized by the so-called "Le Roy radius" [START_REF] Brom | Absorption spectrum of BS 2 at 4 k[END_REF].

The objective of the current method is to obtain accurate spin-orbit couplings also at short internuclear distances in a diatomic molecule (say, above 10 a.u. for the present atom-ion combinations), and at the same time relying on the atomic spin-orbit splittings, as invoked above. This is accomplished by firstly obtaining wave function in the molecular frame and then using a basis transformation from the molecular basis to the separated atom basis. Since we need to retain all the information on the initial molecular basis, the basis transformation is achieved by a unitary transformation rather than a projection. The basics of such a quasi-diabatic approach have been reported in Ref. [START_REF] Cimiraglia | Quasidiabatic states and dynamical couplings from ab initio CI calculations: a new proposal[END_REF], and used once for a van der Waals molecule [START_REF] Angeli | Quasi-diabatic and adiabatic states and potential energy curves for Na Cd collisions and excimer formation[END_REF]. A paper describing the detailed methodology is currently in progress in our group, and I recall below the main steps of the approach.

First, we calculate the N lowest adiabatic energies and the associated electronic wavefunctions |Ψ 0 >= {|Ψ 0 1 >, |Ψ 0 2 >, ..., |Ψ 0 N >} depending parametrically of the internuclear distance: these energies are the eigenvalues of the electronic Hamiltonian, and constitute a N × N diagonal matrix H 0 . Next, we build up a basis of N "reference states" |R >= {|R 1 >, |R 2 >, ..., |R N >} to describe the separated atoms. This basis is conveniently assembled with the adiabatic states at a large internuclear distance r r e f . The third step aims at obtaining, at every internuclear distance r, a unitary T-matrix which connects the adiabatic matrix H 0 to a quasi-diabatic matrix H d ia b = TH 0 T T : the objective is to determine T in order to generate a new basis set |Ψ 0 > T which is as similar as possible to the reference basis set |R >. If completed, the adiabatic Hamiltonian is now expressed in an effective basis representing the two separated atoms. At this stage, this quasi-diabatic matrix is determined for every relevant molecular symmetry (say, in our case, 1 Σ + , 3 Σ + , 1 Π, 3 Π, 1 ∆, 3 ∆, for the previously invoked s + s, s + p, s + d asymptotes). The spin-orbit matrix H so is known in the separated atom representation, and involves atomic splitting energies at each dissociation limit. Therefore an effective Hamiltonian matrix including spin-orbit interaction, H e f f = H ad ia b + H so , is built by putting together the quasidiabatic matrices for the coupled molecular symmetries as diagonal blocks, and distributing off-diagonal terms between blocks, according to the expression of H so in the separated atom representation. The diagonalization of H e f f yields the adiabatic Hund's case c potential energy curves. Alternatively, the inverse transformation of H e f f back to the original adiabatic basis |Ψ 0 > leads to diagonal and off-diagonal r-dependent matrix elements for the molecular spin-orbit couplings between the involved molecular symmetries.

It is easy to realize that the precision of this quasi-diabatic approach is conditioned by the closeness of the new basis set |Ψ 0 > T from the reference basis set |R >. This is the reason why it has been applied some time ago the van

MOLECULAR STRUCTURE FOR DIATOMIC MOLECULES

der Waals molecule NaCd [START_REF] Angeli | Quasi-diabatic and adiabatic states and potential energy curves for Na Cd collisions and excimer formation[END_REF], i.e. where the individual atomic wavefunctions do not overlap too much. In the present ion-atom systems, we will see that the dynamics in which we are interested happen at reasonably large distances where the above assumption remains reasonable.

As stated before, we limit our study to a series of two-valence-electron systems. When a single electron is bound to each of the separated partners, the dissociation limits are of the type s + s, s + p and s + d. We define an atomic spin-orbit constant

A = ξ sp 2 = 1 3 ε ( 2 P 3/2 -2 P 1/2 )
for the dissociation limit s + p, where ε is the fine structure splitting energy. The spin-orbit matrices [START_REF] Beuc | Absorption spectroscopy of the rubidium dimer in an overheated vpor: An accurate check of molecular structure and dynamics[END_REF][START_REF] Comparat | Improved LeRoy-Bernstein near-dissociation expansion formula, and prospect for photoassociation spectroscopy[END_REF] are:

H(|Ω| = 2) = V ( 3 Π) + A (2.11
)

H(|Ω| = 1) =   V ( 3 Π) A A A V ( 1 Π) -A A -A V ( 3 Σ + )   (2.
12)

H(Ω = 0 + ) = V ( 3 Π) -A 2A 2A V ( 1 Σ + ) (2.13)
H(Ω = 0 -) = V ( 3 Π) -A 2A 2A V ( 3 Σ + ) (2.14)
For the s + d limit, similarly, we define

A = ξ sd 2 = 1 5 ε ( 2 D 5/2 -2 D 3/2
) , then we obtain [START_REF] Lozeille | Detection by two-photon ionization and magnetic trapping of cold Rb 2 triplet state molecules[END_REF]:

H(|Ω| = 3) = V ( 3 ∆) + 2A (2.15) H(|Ω| = 2) =   V ( 3 ∆) 2A 2A 2A V ( 1 ∆) -2A 2A -2A V ( 3 Π) + A   (2.16
)

H(|Ω| = 1) =     V ( 3 ∆) -2A 2A 2A 0 2A V ( 3 Π) A 3A 2A A V ( 1 Π) -3/2 0 3A -3A V ( 3 Σ + )     (2.17) H(Ω = 0 + ) = V ( 3 Π) -A 6 6 V ( 1 Σ + ) (2.18) H(Ω = 0 -) = V ( 3 Π) -A 6A 6A V ( 3 Σ + ) (2.19)
In the following we describe various asymptotes where a closed-shell alkalimetal ion is facing a neutral alkaline-earth atom, i.e., both valence electrons lie at the same center.

Spin-orbit coupling matrix for two-electron in one atom

The symmetry of these asymptotes is determined by the one of the neutral alkaline-earth atom, namely 1 S, 3 S, 1 P, 3 P, 1 D, 3 D. Here I derived the corresponding spin-orbit matrices which I report below, while more details about the calculations are reported in Appendix A.

It may happen that a 3 P atomic level is fairly isolated from its 1 P counterpart (like in Ca atom), so than the SO matrix can be limited to this single symmetry.

We define an atomic spin-orbit constant

A = ξ p 2 = 1 3 ε ( 3 P 2 -3 P 0 )
, where ε ( 3 P 2 -3 P 0 ) is the energy spacing between the lowest and the highest energy levels of the atomic fine structure multiplet. The matrices are expressed as follows, and I indicated for each of them in parenthesis the fine structure levels which are concerned.

H(|Ω| = 2) = V ( 3 Π) + A , ( 3 P 2 ).
(2.20)

Ĥ(|Ω| = 1) = V ( 3 Π) A A V ( 3 Σ + ) , ( 3 P 1 , 3 P 2 ).
(2.21)

H(Ω = 0 -) = V ( 3 Π + ) -A 2A 2A V ( 3 Σ + ) , ( 3 P 0 , 3 P 2 ). (2.22)
H(Ω = 0 + ) = V ( 3 Π -) -A , ( 3 P 1 ) (2.23)
When the 3 P and 1 P atomic levels are not so far from each other, they have to be considered simultaneously, thus providing the matrices below.

H(|Ω| = 2) = V ( 3 Π) + A , ( 3 P 2 )
(2.24)

H(|Ω| = 1) =   V ( 3 Π) A -A A V ( 3 Σ + ) A -A A V ( 1 Π)   , ( 3 P 1 , 3 P 2 , 1 P 1 ) (2.25
)

H(Ω = 0 -) = V ( 3 Π + ) -A 2A 2A V ( 3 Σ + ) , ( 3 P 0 , 3 P 2 ) (2.26) H(Ω = 0 + ) = V ( 3 Π -) -A 2A 2A V ( 1 Σ + ) , ( 3 P 1 , 1 P 1 ) (2.27)
A very similar analysis can be done for the 3 D and 1 D atomic energy levels. As for the isolated 3 D state, we define

A = ξ d 2 = 1 5 ε ( 3 D 3 -3 D 1 )
, yielding the matrices:

H(|Ω| = 3) = V ( 3 ∆) + 2A ( 3 D 3 ) (2.28) H(|Ω| = 2) = V ( 3 ∆) 2A 2A V ( 3 Π) + A , ( 3 D 2 , 3 D 3 ) (2.29) 20 2.2. ROTATIONAL MOTION H(|Ω| = 1) =   V ( 3 ∆) -2A 2A 0 2A V ( 3 Π) 3A 0 3A V ( 3 Σ + )   , ( 3 D 1 , 3 D 2 , 3 D 3 ) (2.30) H(Ω = 0 -) = V ( 3 Π + ) -A 6A 6A V ( 3 Σ + ) , ( 3 D 1 , 3 D 3 ) (2.31
)

H(Ω = 0 + ) = V ( 3 Π -) -A , ( 3 D 2 ) (2.32)
When the two levels 3 D and 1 D cannot be isolated from each other, we obtain the following matrices:

H(|Ω| = 3) = V ( 3 ∆) + 2A , ( 3 D 3 ) (2.33) H(|Ω| = 2) =   V ( 3 ∆) 2A -2A 2A V ( 3 Π) + A 2A -2A 2A V ( 1 ∆)   , ( 3 D 2 , 3 D 3 , 1 D 2 ) (2.34) H(|Ω| = 1) =     V ( 3 ∆) -2A 2A 0 -2A 2A V ( 3 Π) 3A -A 0 3A V ( 3 Σ + ) 3A -2A -A 3A V ( 1 Π)     , ( 3 D 1 , 3 D 2 , 3 D 3 , 1 D 2 )
(2.35)

H(Ω = 0 + ) = V ( 3 Π -) -A 6A 6A V ( 1 Σ + ) , ( 3 D 2 , 1 D 2 ) (2.36) H(Ω = 0 -) = V ( 3 Π + ) -A 6A 6A V ( 3 Σ + ) , ( 3 D 1 , 3 D 3 ) (2.37)

Rotational motion 2.2.1 Rotational Hamiltonian

There is no doubt that diatomic molecules are rotating in the universe. In Figure 2.4, we use the rotation model of two hard spheres to analyze such motion in a two-nuclei system (a diatomic molecule). The angular momentum of each particle is

L i = I i ω, (i = 1, 2) with the inertia moment I i = m i r 2 i ,
where ω is the angular velocity, and r i is the length from the center of one ball to the center of mass o. Then, the classical rotational Hamiltonian of a diatomic molecule is

H r ot = L 1 2 2I 1 + L 2 2 2I 2 = 1 2 m 1 r 2 1 ω 2 + 1 2 m 2 r 2 2 ω 2 = 1 2 (m 1 r 2 1 + m 2 r 2 2 ) ω 2 = 1 2 µr 2 ω 2 .
(2.38)

m 1 + m 2 . Finally,
we get

H r ot = 1 2 µr 2 ω 2 = µ 2 r 4 ω 2 2µr 2 = l 2 2µr 2 , (2.39)
where l is used to describe the angular motion for this effective system with reduced mass µ. This term is responsible for the so-called centrifugal barrier in the potential energy involved in a scattering process.

Euler angles and Wigner function

To discuss the rotational aspects of diatomic molecules, we start with the rotations defined by three independent angles (α, β, γ) in three dimensions. A coordinate system (X , Y, Z) can be rotated to another configuration (x, y, z).

The operation sequence will be specified below and remain unchanged. 1) We rotate Z axis with an angle α from {X , Y, Z} to {X 1 , Y 1 , Z 1 }, then the new Z 1 axis is colinear with Z axis, and X 1 axis is called the line of nodes (marked by red N in Figure 2.5). 2) Subsequently, we rotate X 1 axis counter-clockwise with an angle

β from {X 1 , Y 1 , Z 1 } to {X 2 , Y 2 , Z 2 },
R(α, β, γ) =   cos γ sin γ 0 -sin γ cos γ 0 0 0 1     1 0 0 0 cos β sin β 0 -sin β cos β     cos α sin α 0 -sin α cos α 0 0 0 1  
(2.40) where the matrix from left to right represents the rotation around the z axis, the rotation around the line of intersection N , and the rotation around the Z axis. After the mathematical computations, we obtain the overall operator 2.2. ROTATIONAL MOTION Figure 2.5: The Euler angles describing the orientation of the molecular (x, y, z) rotated axis (purple) from the laboratory (X, Y, Z) fixed axis (green). ON (red) defines the node line, which is not only the intersection of the X-Y and the x-y planes but also perpendicular to both the Z and z axes. α is the angle between the X-axis and the ON , β is the angle between the Z-axis and the z-axis, γ is the angle between the ON and the x-axis. (2.41) The Wigner D function gives the matrix elements of the rotation operator R in jm-representation [START_REF] Rose | Elementary theory of angular momentum[END_REF]. When the rotation of a molecule described in terms of Euler angles (α, β, γ), and the components m or m = -j, -j + 1, ..., j of an arbitrary angular momentum vector, the D matrix is written as

[85] D j m m (α, β, γ) =< jm |R(α, β, γ)| jm >, =< jm |e -iα ĵz e -iβ ĵy e -iγ ĵz | jm > = e -1(m α+mγ) d j m m (β), (2.42)
where the components ĵx , ĵy and ĵz satisfy ĵ2 = ĵx 2 + ĵy 2 + ĵz 2 . We define

d j m m (β) =< jm |e -iβ ĵy | jm >,
which is the so-called Wigner small d function. Some useful relations are

d j m m (β) = d j -m-m (β) = (-1) m-m d j mm (β) = (-1) m -m d j m m (-β), (2.43
)

d j m m (π + β) = (-1) j+m d j -m ,m (β), (2.44)
d j m m (π -β) = (-1) j+m d j m ,-m (β).
(2.45)

Transformation between BF and SF frames

The strategy is to derive an r-independent transformation of molecular states between the two frames. The total Hamiltonian of a diatomic molecule AB can be written as a sum of the electronic term H el ec (r), the nuclear kinetic energy term

T (r) = -1 2µr
∂ 2
∂ r 2 r, the internal rotation term H r ot , and the spin-orbit interaction term H so (r) in the BF frame,

H t ot al = H el ec (r) + T (r) + H r ot (r) + H so (r).
(2.46)

The molecular basis (BF frame) is essential in the small r region because the Born-Oppenheimer (BO) solutions diagonalize H el ec (r) and the results of ab initio electronic potential energy calculations or experimental data can be utilized [START_REF] Parlant | A theoretical analysis of the state-spectific decomposition of OH(A 2 Σ + , v , N , F 1 /F 2 ) levels, including the effects of spin-orbit and coriolis interactions[END_REF]. The BO states can be represented by |ΛSΣ >. At a large distance r, |ΛSΣ > can be quantized along the sysmmetry axis of the diatomic, not the SF axis. Then, we can expand |ΛSΣ > in a basis of products of atomic states

|L A Λ A > |L B Λ B > |SΣ > |ΛSΣ >= Λ A Λ B |L A Λ A > |L B Λ B > |SΣ >< Λ A Λ B |Λ >, (2.47)
where

Λ = Λ A + Λ B . The coefficients < Λ A , Λ B |Λ > are
determined by consideration of the leading interatomic term of the multipolar expansion of H el ec at large r. If one of the atoms has s electron, then l A = 0, < Λ A Λ B |Λ >= δ ΛΛ B = 1, otherwise, this value is according to the electronic Hamiltonian. The paritydependent Hund's case (a) basis vectors with parity p for both bound and continuum molecular states at a smaller r is

|ΛSΣJ M J p >=(2 -δ Λ,0 δ Σ,0 ) -1/2 {|ΛSΣ > |JΩM J > +(-1) J-S+p+σ (1 -δ Λ,0 δ Σ,0 )| -ΛS -Σ > |J -ΩM J >}, ( 2.48)
with:

|J ± ΩM J >= 2J + 1 8π 2 D J * M J ±Ω , (2.49)
where The Wigner D function [START_REF] Zare | Angular Momentum[END_REF] indicates the orientation of the molecular axis in the SF frame, and σ = 1 for the Σ -state and σ = 0 otherwise. In SF frame, the total Hamiltonian of two separated atoms (A and B) is the sum of the kinetic term describing the relative radial motion T (r), the rotation term H r ot , and the term accounting for the internal energy of the atoms H int ,

H t ot al = T (r) + H r ot (r) + H int (r).
(2.50)

When r → ∞, it is convenient to form states of the total electronic angular momentum in the SF frame,

| j A j B jm j >= m A j ,m B j | j A m A j > | j A m B j >< j A m A j j B m B j | jm j > . (2.51)

ROTATIONAL MOTION

The Hund's case (e) basis vectors [START_REF] Parlant | A theoretical analysis of the state-spectific decomposition of OH(A 2 Σ + , v , N , F 1 /F 2 ) levels, including the effects of spin-orbit and coriolis interactions[END_REF] with parity p = (-1) L A +L B +l are

| j A j B jlJ M J >= m l ,m j | j A j B jm j > |lm l >< jm j l m l |J M J >, ( 2.52)
where M j , m j and m l are the projections of the angular momenta on the SF axis. < jm j l m l |J M J > is a Clebsch-Gordan coefficient [START_REF] Zare | Angular Momentum[END_REF].

In Eqn [2.48] the electronic part of the wave function |ΛSΣ > in the BF frame (Hund's case (a)) is transformed using angular algebra to introduce the angular momentum j A and j B present in Eqn [2.52] in Hund's case (e). This allows us to write the following equations valid at large distances,

|ΛSΣ > = Λ A ,Λ B |L A Λ A > |L B Λ B > |SΣ >< Λ A Λ B |Λ > = Λ A ,Λ B LΛ < L A Λ A L B Λ B |LΛ > |L A L B LΛ > |SΣ >< Λ A Λ B |Λ > = Λ A ,Λ B LΛ jΩ < L A Λ A L B Λ B |LΛ >< LΛSΣ| jΩ > |L A S A L B S B LS jΩ >< Λ A Λ B |Λ > = Λ A ,Λ B j A , j B jΩ < L A Λ A L B Λ B |LΛ >< LΛSΣ| jΩ > × (2L + 1)(2S + 1)(2 j A + 1)(2 j B + 1)    L A S A j A L B S B j B L S j    × | j A j B jΩ > BF < Λ A Λ B |Λ > .
(2.53) In the first line of Eqn [2.53], L A and L B denote the total orbital angular momentum of atoms A and B. For s + s, s + p and s + d electronic dissociation limits (assuming atoms with a single valence electron),

< Λ A , Λ B |Λ >= 1. Re- coupling |L A Λ A > |L B Λ B >
introduces the molecular total angular momentum and its projection on the BF axis |LΛ >, as written in the second line of Eqn [2.53]. Summations on L, Λ appear as well as a Clebsch-Gordon coefficients. The molecular orbital |LΛ > and spin |SΣ > momenta states are projected in the third line of Eqn [2.53] on states with defined total electronic momentum and its projection on the molecular z axis | jΩ >. Changing from the |LS jΩ > basis to the | j A j B jΩ > basis introduces a 9j Racah coefficient (marked with {}), with summations on j A j B (fourth line of Eqn [2.53]). This function | j A j B jΩ > BF in the last line is marked by BF because it is defined in the BF frame. It can be expressed as a sum over | j A j B jΩ > SF functions defined in the SF frame with m j the projection of j on the Z axis in the SF frame according to

D j * m j Ω (α, β, γ = 0), | j A j B jΩ > BF = m j | jm j j A j B > SF D j * m j Ω (αβγ), (2.54)
where α, β, γ are the Euler angles to describe the position of the BF frame in the SF frame.

RADIATIVE TRANSITIONS 25

We have now to add the rotational wave function |JΩM J > occurring in Eqn [2.48] to obtain a transformed expression of the total but not symmetrized wave function in the BF frame. In this expression, accounting for Eqn [2.48], the product of Wigner functions occurs. Using the Clebsch-Gordon series formula [START_REF] Varshalovich | Quantum Theory Of Angular Momemtum[END_REF], this product is changed into a summation over all the wave functions | j A j B jlJ M J > S F defined in the SF frame.

Considering Eqn [2.52] and Eqn [2.53] at large r, finally, the formula for the frame transformation between BF and SF frames is obtained within a subspace of fixed J M J [START_REF] Parlant | A theoretical analysis of the state-spectific decomposition of OH(A 2 Σ + , v , N , F 1 /F 2 ) levels, including the effects of spin-orbit and coriolis interactions[END_REF][START_REF] Singer | Theory of distomic molecule photodissociation: Electronic angular momentum influence on fragment and fluorescence cross sections[END_REF], which is valid at any internuclear separation,

< j A j B jlJ M J |ΛSΣJ M J p >= (-1) l-Ω-J (2 -δ Λ,0 δ Σ,0 ) -1/2 × [1 + (-1) L A +L B +l+p (1 -δ Λ,0 δ Σ,0 )] × (2S + 1)(2 j A + 1)(2 j B + 1) < l0| j -Ω, JΩ > × L,Λ A ,Λ B 2L + 1 < LΛ|L A Λ A , L B Λ B > ×    L A S A j A L B S B j B L S j    < jΩ|LΛ, SΣ >< Λ A Λ B |Λ > .
(2.55)

Note that these elements are independent of M J . The square of these matrix elements determines the weights of Hund's case (a) |LΛSΣ > channels in the Hund's case (e) channels.

Radiative transitions

In 1917 [START_REF] Einstein | The quantum theory of radiation[END_REF], A. Einstein obtained a derivation of the semi-classical quantum theory of radiation. After 100 years, we can still benefit from his efforts and apply the theory to various interactions between molecules and radiation. The essence of this theory is to use the quantum perspective on atoms, molecules, and electrons, but the classical electromagnetic wave perspective on the radiation field. The atoms (and electrons in the atom) are described in quantum mechanics under the forced vibration of classical electromagnetic fields which generate quantum transitions between energy levels, and at the same time release photons or absorb (annihilate) photons. This semi-quantum theory of atom-field interaction can give the correct result of the interaction between light and matter. But, due to its incompleteness, it cannot explain the problem of spontaneous radiation of excited atoms and the problem of multiphoton processes in strong radiation fields. Regarding the problem of spontaneous emission, A. Einstein dealt with the relationship between spontaneous emission and stimulated radiation semi-phenomenologically but universally according to the general concept of thermodynamic equilibrium.

In this chapter, I discuss the essential concepts, such as Black Body Radiation, the intensity of light, wavefunctions, transition dipole moments, Einstein coefficients, and selection rules. Then, I derive the well-known formulas related to absorption and emission of radiations, and pay attention to units and dimensionality. In the following, the dimensionality of the derived quantities will be indicated with brackets, either in the equations or in the text, using the following symbols:

[M], [L], [T],
[Q] for mass, length, time, and charge, respectively.

The scheme represented in Figure 2.6 depicts three major objectives: (1) spontaneous emission: the process in which an excited atom decays from a high energy level (excited state) to a lower energy level (ground state) spontaneously without any external action, and simultaneously emits a photon, (2) stimulated emission, that is, a photon is radiated by a light-emitting atom or molecule in an excited state when it transitions to a low energy state or a ground state under the action of an external radiation field. At the same time, the energy of the external radiation must be exactly equal to the energy difference between the energies of the two states. (3) stimulated absorption, which is the opposite physical process of stimulated emission. The above dynamics may happen through bound-bound or bound-continuum or continuum-continuum transitions.

Classical electromagnetic field

As we mentioned before, the radiation field should be treated classically in semiclassical quantum theory. Electric E( r, t) and magnetic B( r, t) fields satisfy Maxwell's equations [START_REF] Maxwell | A dynamical theory of the electromagnetic field[END_REF]

                 • E( r, t) - 1 0 ρ e = 0, × B( r, t) - 1 c 2 ∂ ∂ t E( r, t) -µ 0 J = 0, × E( r, t) + ∂ ∂ t B( r, t) = 0, • B( r, t) = 0, (2.56)
where light speed c equals to 1/ 0 µ 0 , the permittivity of free space 0 , the permeability of free space µ 0 , the total charge per unit volume ρ e , the total current per unit area J, as well as electric E( r, t) and magnetic B( r, t) can be expressed with scalar φ( r, t) and vector A( r, t) potentials,

E( r, t) = - ∂ ∂ t A( r, t) -φ( r, t), B( r, t) = × A( r, t), (2.57)
with the relation

• A( r, t) + 1 c 2 ∂ ∂ t φ( r, t) = 0. (2.58)
Choosing the Coulomb gauge, namely • A = 0 and when no sources (ρ e = 0, J = 0) are present one may take φ = 0, E( r, t) and B( r, t) are only depending on A. Then wave equation of A is

2 A( r, t) - 1 c 2 ∂ 2 ∂ 2 t 2 A( r, t) = 0. (2.59)
The solutions of the above equation are not unique. A monochromatic plane wave solution with angular pulsation ω 0 , or frequency

ν 0 = ω 0 2π is A( r, t) = A 0 (ω 0 )ˆ cos( k • r -ω 0 t + δ ω 0 ), [M LT -1 Q -1 ] (2.60)
where the potential vector A has an amplitude |A 0 (ω 0 )| in the direction determined by vector ˆ which is the polarization vector, k is the propagation vector and δ ω 0 is a real constant phase. The magnitude of wave vector

k is k = ω 0 c .
The choice of the Coulomb gauge imposes k • ε = 0, i.e. the wave is plane transverse. Therefore, in the Coulomb gauge,

E 0 ( r, t) = - ∂ ∂ t A( r, t) = -ω 0 A 0 (ω 0 )ˆ sin( k • r -ω 0 t + δ ω 0 ), [M LT -2 Q -1 ] B 0 ( r, t) = × A( r, t) = -A 0 (ω 0 )( k × ˆ ) sin( k • r -ω 0 t + δ ω 0 ), [M T -1 Q -1 ],
(2.61) where the vectors E 0 , B 0 and k are mutually perpendicular. For the monochromatic light with the angular frequency ω 0 , the time-averaged energy density ρ(ω 0 ) of the field can be written as

ρ(ω 0 ) = 1 2 ( 0 | E 0 ( r, t)| 2 + | B 0 ( r, t)| 2 µ 0 ) = 1 2 1 2 0 ω 2 0 A 0 (ω 0 ) 2 + 1 2 0 ω 2 0 A 0 (ω 0 ) 2 = 1 2 0 ω 2 0 A 0 (ω 0 ) 2 = 1 2 0 E 0 (ω 0 ) 2 , [M L -1 T -1 ], (2.62)
where the dimensionality of 0 is

[M -1 L -3 T 2 Q 2 ],
and of µ 0 is [M LQ -2 ]. The top line in the equation above stands for a mathematical operation for an averaging time. The amplitude of the monochromatic electric field is

|E 0 (ω 0 )| = | 2ρ(ω 0 ) 0 )| 1/2 = |2ħ hω 0 N (ω 0 )/( o )| 1/2 , [M LT -2 Q -1 ] (2.63)
with N (ω 0 ) the number of photon with angular frequency ω 0 per unit volume, with dimensionality

[M L -1 T -1 ].
The nearly monochromatic light is the superposition of plane waves with the same directions of propagation and polarisation, but different angular frequencies ω characterized by a ω-dependent amplitude A 0 (ω) strongly peaked around ω 0 . In this case one introduces similarly the time-averaged energy density (or spectral energy density) in the angular frequency range dω denoted by ρ(ω 0 ), and similarly the spectral intensity per unit angular frequency range I(ω 0 ), the number of photons per unit volume in the dω spectral range N (ω 0 ) [T L -3 ]. For these spectral quantities, Eqn [2.64]

remains valid but E 0 (ω 0 ) 2 dimensionality [(M LT -2 Q -1 ) 2 ]
denotes the square of the amplitude of the electric field per unit angular frequency range. The time-averaged rate of energy flow through a unit cross-sectional area perpendicular to the direction of propagation defines the intensity I(ω 0 ),

I(ω 0 ) = ρ(ω 0 )c = ħ hω 0 N (ω 0 )c = 1 2 0 cE 0 (ω 0 ) 2 = 1 2 0 cω 2 0 A 0 (ω 0 ) 2 , [M T -2 ]
(2.64) At last, the frequency-averaged rate of the energy density ρ, the frequencyaveraged intensity I and the number of photons N for a radiation with a range of frequencies per unit volume are

ρ = ∞ 0 ρ(ω)dω, [M L -1 T -2 ] I = ∞ 0 I(ω)dω, [M T -3 ] N = ∞ 0 N (ω)dω, [L -3 ] (2.65)

Black body radiation

The Planck radiation law [START_REF] Planck | On the law of distribution of energy in the normal spectrum[END_REF] defines the energy density ρ(ν)dν for frequencies between ν and ν + dν for a black body radiation at thermal equilibrium characterized by the temperature T by Planck (October 1900) [START_REF] Planck | On the law of distribution of energy in the normal spectrum[END_REF],

ρ(ν)dν = 8πhν 3 c 3 1 ex p hν k B T -1 dν, [M L -1 T -2 ], (2.66)

RADIATIVE TRANSITIONS 29

where ρ(ν) is the energy density per unit frequency (

[M L -1 T -1 ]), h the Planck constant ([M L 2 T -1 ]), ν the frequency of light ([T -1 ]), c the velocity of light ([LT -1 ]), k B T Boltzmann's constant k B multiplied with temperature ([M L 2 T -2 ]).
The number of photons per unit frequency per unit volume can be defined by

N (ν) = ρ(ν) hν = 8πν 2 c 3 1 ex p hν k B T -1 , [T L -3 ] (2.67)

Einstein coefficient relation

The Einstein coefficient The Einstein coefficient A f i and B i f [START_REF] Einstein | The quantum theory of radiation[END_REF] describe the probability of spontaneous, and stimulated emission or absorption processes, respectively, where i and f set respectively for an initial and a final non-degenerate states. Assuming a system of two discrete i and f states with energy E i and E f , respectively, N i particles occupy the lower state i and N f the upper state f . When the equilibrium between emission and absorption occurs, we have the equality between the corresponding rates

Rat e o f absor pt ion

= Rat e o f (spont aneous + st imul at ed) emission N i B i f ρ(ν i f ) = N f (A f i + B f i ρ(ν f i )) (2.68) where A (resp. B) has a dimensionality [T -1 ] (resp. [M -1 L]). Considering Maxwell-Boltzmann distribution law, N i = C(T )e - E i k B T and N f = C(T )e - E f k B T
, where C(T ) is a function only depending on the temperature T , and hν = ħ hω = |E f -E i |, we can express the ratio of these two populations,

N i N f = A f i + B f i ρ(ν f i ) B i f ρ(ν i f ) = e -(E i -E f )/k B T = e hν/k B T . (2.69)
Rearranging, we get the energy density per unit frequency

ρ(ν i f ) = A f i B i f e hν/k B T -B f i . (2.70)
Introducing it into 2.68, we get

B i f = B f i .
(2.71)

A f i = 8πhν 3 c 3 B f i . ( 2

Wave function A free particle

Due to the fact that microscopic particles have wave-particle duality, it is necessary to introduce a concept of a complex wave function to represent complete physical information. For a free particle with its energy E and a mass m, the wave function is written as

ψ( r, t) = ψ 0 ex p[ i ħ h ( p • r -E t)] (2.73)
where k = p ħ h is the wave vector , ψ 0 is the amplitude and p = 2µE is the momentum. In the case of a propagation along x axis the one-dimensional wave function writes

ψ(x, t) = ψ 0 ex p[i(kx -ωt)] (2.74)
The plane wave functions Eqn [2.73] and Eqn [2.74] with a constant amplitude describe a free particle with a complete absence of its localisation in space. To normalize the plane wave function, we have to use the closure relation

ψ * (x )ψ(x)dk = δ(x -x), (2.75)
where one of the ways of representing the Dirac delta function

is δ(x, a → ∞) = a π ex p(-ax 2 ), in effect, δ(x) =0 (if x = 0) or +∞ (if x = 0), and -∞ +∞ δ(x)d x = 1.

Two-atom system

Considering our diatomic system composed of the one-electron atom and oneelectron ion, we will handle with discrete bound state and continuum state which both of them are independent on time. In term of bound wave function, it includes radial functions u nl (r) and spherical harmonics Y l m (θ , φ) (see Figure 2.7 for the illustration of spherical coordinates), where n is the principal quantum number, l the orbital quantum number and m its projection on an arbitrary quantization axis,

ψ nl m (r, θ , φ) = u nl (r) r Y l m (θ , φ). (2.76)
This wave function should be normalised to unity in three dimensions,

∞ 0 r 2 d r π 0 dθ sin θ 2π 0 dφ|ψ nl m (r, θ , φ)| 2 = 1 (2.77)
The dimensionality of ψ is [L -3/2 ]. Then it is easy to get the dimensionality of

u nl (r) which is [L -1/2 ].
To discuss the form of continuum wave functions of a free particle, we begin with radial time-independent Schrödinger equation assuming V (r) = 0,

d 2 d r 2 + 2 r d d r - l(l + 1)
r 2 + k 2 R E,l (r) = 0, (2.78)
where k = 2µE/ħ h, l is the partial wave and E is a continuum energy. To solve this equation, we set ρ = kr and R l (ρ) ≡ R E,l (r), so that it turns out to spherical Bessel differential equation

d 2 dρ 2 + 2 ρ d dρ + 1 - l(l + 1) ρ 2 R l (ρ) = 0. (2.79)
The general solution is a linear combination of a spherical Bessel function

j l (ρ) = π 2ρ 1/2 J -l-1/2 (ρ), (2.80)
and a spherical Neumann function

n l (ρ) = (-1) l+1 π 2ρ 1/2 J l+1/2 (ρ). (2.81)
So the asymptotic behavior of an energy-normalized continuum wave function for Eqn [2.78] is

R E,l (r) ∼ 2µ πħ h 2 k sin(kr -lπ/2 + δ l ), [M -1/2 L -3/2 T ] (2.82)

Transition rate

Eqn [2.72] gives only the relation between A and B with no specific form. In this subsection, we start with a one-electron atomic system to discuss the formula of the stimulated and spontaneous transition rates, which can be easily extended to the same problems in the diatomic molecule frame.

One-electron atom interacting with an electromagnetic field

We first need to set up the Hamiltonian to describe the interaction of oneelectron species with the electromagnetic field including the nucleus charge Z e and mass m u , the electron charge -e and mass m e . Due to the fact that m u is way larger than m e , we could make an infinite nucleus mass approximation ignoring recoil effects and reduced mass effects. The time-dependent Hamiltonian of an one-electron atom/ion in an electromagnetic field is [START_REF] Bransden | Physics of atoms and molecules[END_REF] iħ h

∂ ∂ t ψ( r, t) = [H 0 + H int (t)]ψ( r, t), H 0 = - ħ h 2 2m e 2 - Z e 2 4π o r , H int (t) = -iħ h e m e A • - e 2 2m e A 2 , (2.83)
where the time-independent Hamiltonian H 0 is described in the absence of the radiation field, while H int (t) is the time-dependent Hamiltonian of the interaction of the atom/ion with the field. Since we mainly consider one-photon process in this thesis, namely the field is weak, the magnitude of the term in A 2 is way smaller than that of the linear term A and can be neglected. Herein,

H int (t) = -iħ h e m e A • .
(2.84)

RADIATIVE TRANSITIONS 33

Then the complete time-dependent Schrödinger equation can be represented as

iħ h ∂ ∂ t ψ( r, t) = - ħ h 2 2m e 2 - Z e 2 4π o r -iħ h e m e A • ψ( r, t).
(2.85)

The general normalised solution can be expanded as

ψ( r, t) = k c k (t)φ k ( r)ex p(-iE k t/ħ h), (2.86)
where E k is the eigenvalues of Schrödinger eigenvalue equation

H 0 φ k ( r) = E k φ k ( r),
(0 ≤ λ ≤ 1), c k (t) = c 0 k (t) + λc 1 k (t) + .
.., where the first term and second term are zero-order and first-order solutions, and λ is used to distinguish between the various orders of the perturbation calculation. Now we assume that the initial system is well-defined in a state of energy E i , then its wave function is

ψ i ( r, t) = c i (t)φ i ( r)ex p(-i E i t/ħ h) (2.
iħ h ∂ ∂ t c f (t) =< φ f |H int (t)|φ i > c i (t)ex p(iω i f t), ( 2.88)
where

ω i f = (E f -E i )/ħ h.
As the perturbative term is introduced at the moment t = 0, the state of the system remains constant. Thus the Eqn [2.87] can be rewritten as

ψ i ( r, t) = ψ( r, 0) = k c k (0)φ k ( r). (2.89)
In order to complete the equation, it follows that

c k (0) =< φ i ( r)|φ k ( r) >= δ ik , (2.90)
Where we assume that the notation of δ ik is valid for both discrete and continuum states for convenience. With respect to the final state at t = 0, we get the zeroorder solution

ψ i ( r, t) = ψ f ( r, 0) = c 0 f (0)φ f ( r). (2.91) Then, c 0 f (0) =< φ i ( r)|φ f ( r) >= δ i f . (2.92) 34

RADIATIVE TRANSITIONS

Considering the solution of the first-order equation of c 1 f (t),

∂ ∂ t c 1 f (t) = (iħ h) -1 k < φ f |H int (t)|φ k > ex p(iω f k t)c 0 k (t). (2.93) Substituting Eqn [2.90] into Eqn [2.93], c 1 f (t) = (iħ h) -1 t 0 < φ f |H int (t)|φ i > ex p(iω i f t)d t = - e m e t 0 < φ f | A • |φ i > e x p(iω i f t)d t, (2.94)
where the perturbation term is 0, at t = 0. Then,

c f (t) = c 0 f (t) + c 1 f (t) = δ i f + c 1 f (t) = c 1 f (t), i = f . ( 2
|c 1 f (t)| 2 = π 2 e m e 2 A 0 (ω i f ) 2 |M i f (ω i f )| 2 t, (2.96)
where we define the matrix element M i f as

M i f (ω i f ) = m e ω i f ħ he ˆ • D i f . (2.97)
Subsequently, the transition rate can be defined as

W i f = d d t |c 1 f (t)| 2 = 4π 2 m 2 e c e 2 4π 0 I(ω i f ) ω 2 i f |M i f (ω i f )| 2 = 4π 2 m 2 e c e 2 4π 0 I(ω i f ) ω 2 i f |m e ω i f D i f /(ħ he)| 2 cosΘ 2 = 4π 2 ħ h 2 c 1 4π 0 I(ω i f )| D i f | 2 cosΘ 2 = B i f (ν)ρ(ν), (2.98)
where the angle Θ is defined by the vector ˆ . In non-polarized radiation, the average value of cosΘ 2 over solid angles is 1/3. In the electric dipole approximation, the electric dipole moment operator D

[LQ] is given by

D = e i Z i r i - j r e j , i = 1, 2; j = 1, 2, (2.99)
with a nuclei distance r = i r i and nuclei charge eZ i , as shown in Figure 2.8. Then the transition electronic dipole moment D i f between electronic states i and f can be expressed by < ψ i | D|ψ f >, which is also recognized as the off-diagonal matrix element. Meanwhile,

| D(r)| 2 = |D x (r)| 2 + |D y (r)| 2 + |D z (r)| 2 , [L 2 Q 2 ]
(2.100)

The formula for the coefficient B of the stimulated process can be derived regarding one-dimensional wave functions,

B i f (ν) = W i f ρ(ν) = 8π 3 3h 2 1 4π 0 | < ψ i |D(r)|ψ f > | 2 , [M -1 L]. (2.101)
Regarding Einstein relations Eqn [2.72], the coefficient A for the spontaneous emission can be written as

A f i = 8πhν 3 c 3 B f i (ν) = 64π 4 ν i f 3 3hc 3 1 4π 0 | < ψ i |D(r)|ψ f > | 2 , [T -1 ]. (2.102)

Bound/continuum-bound/continuum stimulated process

In this subsection we focus on the one-photon induced processes with respect to one-dimensional bound wave functions [L -1/2 ] and one-dimensional continuum wave functions [ M -1/2 L -3/2 T ] in a diatomic system with an internuclear distance r. The scheme in Figure 2.6 shows that the radiative transitions can happen through bound-bound, continuum-continuum, or bound-continuum states. If one of the involved wave functions belongs to a continuum state, we should integrate over a small energy interval dε to normalize the wave function in energy. On the other hand, the bound state wave function is normalized to unity in space. The goal here is to ensure that the probabilities are all consistent in dimensionality for all cases.

Bound-to-bound stimulated process

In order to describe the stimulated emission and absorption of one molecule in black-body radiation scheme, the bound-bound transition probability per second

P i f (of dimensionality [T -1 ]) can be expressed as Einstein coefficient B i f (ν) (of dimensionality [M -1 L]) multiplied by the energy density of radiation ρ(ν) (of dimensionality [M L -1 T -1 ]), P i f = B i f (ν)ρ(ν)dν = B i f (ν) I(ν) c dν, [T -1 ], (2.103)
where I(ν) the light intensity per frequency

[M T -2 ] considering Eqn[2.65].

RADIATIVE TRANSITIONS

We introduce the absorption bound to bound cross-section σ BB (of dimensionality [L 2 ]), defined as the rate of absorption of energy (per molecule) divided by the density of energy of the light per unit frequency,

P i f = σ BB (ν) I(ν) hν dν, [T -1 ]. (2.104)
For the present, we obtain the cross section for bound-bound stimulated absorption and emission for a sharp absorption line, namely, the frequency is unique,

B i f (ν) I(ν) c = σ BB (ν) I(ν) hν σ BB (ν) = B i f (ν)hν/c. (2.105)
By comparison, the cross section is

σ BB (ν) = 8π 3 3hc 1 4π 0 ν| < ψ i v |D(r)|ψ f v > | 2 , [L 2 ]. (2.106)
In the scientific literature, the expressions of the scattering cross section can be either a function of ν, of ω or of

∆E = |E i -E f |.
Therefore we consider it is necessary to present a detailed comparison of the forms of these equations for the convenience. We have

σ BB (ω) = 2πσ BB (ν) = 16π 4 3hc 1 4π 0 ν| < ψ i v |D(r)|ψ f v > | 2 = 8π 3 3hc 1 4π 0 ω| < ψ i v |D(r)|ψ f v > | 2 = 4π 2 3ħ hc 1 4π 0 ω| < ψ i v |D(r)|ψ f v > | 2 ,
(2.107)

σ BB (∆E) = hσ BB (ν) = 8π 3 3c 1 4π 0 ν| < ψ i v |D(r)|ψ f v > | 2 = 8π 3 3c 1 4π 0 ∆E h | < ψ i v |D(r)|ψ f v > | 2 = 4π 2 3ħ hc 1 4π 0 ∆E| < ψ i v |D(r)|ψ f v > | 2 .
(2.108)

Unit preferences can also lead to different forms of formulas. In atomic units, i.e., ħ h = e = m e = 1, we may write

σ BB (∆E) = 4π 2 3c ∆E| < ψ i v |D(r)|ψ f v > | 2 , [L 2 ; a.u.] (2.109)
In the brackets, L 2 is the dimension and a.u. refers to the atomic units (hereinafter). Or, it can be written

σ BB (∆E) ≈ 2.69 × 10 -18 ∆E| < ψ i v |D(r)|ψ f v > | 2 , [L 2 ; cm 2 ], (2.110)
Figure 2.9: The Cauchy-Lorentzian distribution (Picture from P.218 of Ref. [START_REF] Bransden | Physics of atoms and molecules[END_REF]).

where both the transition energy and the dipole matrix element are in a.u.. The conversion from atomic units to squared centimeters is 1 a.u. ≈ 2.8 × 10 -17 cm 2 . The corresponding reaction rate coefficient is

K BB (ν) = σ BB (ν) × I hν = 4π 2 I 3c | < ψ i v |D(r)|ψ f v > | 2 , [L 3 T -1 ; a.u.]. (2.111)

Bound-to-continuum stimulated process

The expression for the stimulated bound-to-continuum cross section for photodissociation is straightforward for a continuous absorption laser field with angular frequency ω = ∆E/ħ h,

σ BC = σ BC (∆E)dω = ħ hσ BC (ω)dω = σ BC (ω)d(ħ hω) = ∞ 0 4π 2 3ħ hc 1 4π 0 f (ω)dω| < Ψ i v |D(r)|Ψ f ε > | 2 dε f = ∞ 0 4π 2 3ħ hc 1 4π 0 Γ 2 /4ħ h 2 (ω -ω 0 ) 2 + Γ 2 /4ħ h 2 dω| < ψ i v |D(r)|ψ f ε > | 2 dε f ω=ω 0 4π 2 3ħ hc 1 4π 0 ω 0 ∞ 0 | < ψ i v |D(r)|ψ f ε > | 2 dε f [L 2 ]
(2.112) The relevant quantities are illustrated in Figure 2.9: the central frequency ω 0 of the dimensionless Cauchy-Lorentzian distribution of frequencies f (ω), its half-width Γ /2ħ h at half-maximum, the bound-to-continuum absorption cross section σ BC (ν).

In atomic units, the rate coefficient is then

K BC (ν) = σ BC × I hν = 4π 2 I 3c ∞ 0 | < ψ i v |D(r)|ψ f ε > | 2 dε f [L 3 T -1 ; a.u.].
(2.113)

RADIATIVE TRANSITIONS

Continuum-to-bound stimulated process

The initial state can be a continuum state (like ( 4) and (6-9) in Figure 2.6) as in the case in the photoassociation where a pair of atoms with reduced mass µ approach each other along a given molecular state with a relative kinetic energy ε i . The normalized Maxwell-Boltzmann distribution function f p for the relative momentum p = (2µε i ) 1/2 is

f p = (2πµk B T /h 2 ) -3/2 ex p - p 2 2µk B T = Q T ex p - ε i k B T , (2.114)
where both f p and Q T = (2πµk B T /h 2 ) -3/2 are dimensionless quantities. Herein, we can obtain a rate formula for continuum-to-bound stimulated process,

P i f = B i f (ν)ρ(ν) f p dν = 8π 3 h 2 1 4π 0 I(ν) c Q T ex p - ε i k B T | < ψ i ε | D(r)|ψ f v > | 2 dε i dν = σ C B (ν) I(ν) hν dν, [T -1 ], (2.115)
where σ C B (ν) is the continuum-to-bound absorption cross section at frequency ν

σ C B (ν) = 8π 3 3hc ν 4π 0 Q T ex p - ε i k B T | < ψ i ε |D(r)|ψ f v > | 2 dε i , [L 2 ], (2.116
) yielding then the rate coefficient

K C B (ν) = σ C B (ν) × I hν = 8π 3 3h 2 c I 4π 0 Q T ex p - ε i k B T | < ψ i ε |D(r)|ψ f v > | 2 dε i , [L 3 T -1 ].
(2.117)

Continuum-to-continuum stimulated processes

This is the case for photo-assisted collisions like radiative charge exchange

(A + B + → A + + B + γ), radiative excitation exchange (A + B * → A * + B + γ), or (A + B + γ → A * + B)
, where γ is the energy of a photon and both initial and final wave functions are continuum ones (see [START_REF] Hall | Cold ion-neutral reactions[END_REF][START_REF] Tacconi | Computing chargeexchange cross sections for Ca + collisions with Rb at low and ultralow energies[END_REF] in Figure 2.6). Based on 2.4. ELEMENTS OF SCATTERING THEORY 39 the previous subsection, the cross section is

σ C C (ν) = 8π 3 3hc ν 4π 0 Q T ex p - ε i k B T | < Ψ i ε |D(r)|Ψ f ε > | 2 dε i dε f , (2.118)
where ε i and and ε f are the kinetic energies in the entrance and exit channels. The rate coefficient is

K C C (ν) = σ C C (ν) × I hν = 8π 3 3h 2 c I 4π 0 Q T e x p - ε i k B T | < ψ i ε |D(r)|ψ f ε > | 2 dε i dε f , [L 3 T -1 ]. (2.119)

Elements of scattering theory

The collision between a particle and an atom can reveal the fundamental internal configuration of the atom. In 1911, for instance, Ernest Rutherford [START_REF] Rutherford | The scattering of α and β particles by matter and the structure of the atom[END_REF] demonstrated the atomic solar system model (Rutherford atomic model) using α particle scattering experiment. In 1914, Frank and Hertz [START_REF] Franck | On the collisions between electrons and molecules of mercury vapor and the ionization potential of the same[END_REF] visually confirmed the existence of a series of discontinuous energy states within the atom by colliding mercury atoms with electrons. During the period 1934-1938, Hahn, Meitner, and Strassmann [95,96] bombarded uranium targets with neutrons and discovered many transmutation products. In 1967 Friedman, Kendall, and Taylor [97] proved the existence of quarks in an inelastic scattering experiment of high-energy electron-protons. Accordingly, scattering experiments have become an important scientific technique in modern physics for the investigation of microscopic interactions and internal structures of atoms or molecules. In this section, although we mainly focus on ion-atom collisions, the essential concepts are also valid for other systems: atom-atom, atom-molecule or molecule-molecule. The collision processes can be classified as 1. Elastic collision: the atom A and the ion B + are scattered without changing their internal structure but their kinetic energy, A + B + = A + B + . It should be emphasized that I shall then discuss and derive the fundamental scattering theory based on elastic collisions unless otherwise stated.

Inelastic collision: the atom

General description

Assume that a beam of atoms A is sent from infinity along the Z-axis towards an ion B + . Here, the ion B + can be referred to as the scattering center. If the mass of the ion B + is much larger than that the one of A, the kinetic energy of the ion B + remains almost the same, and the angle θ between the direction of the movement of the scattered A and the Z-axis is the scattering angle in (2.120)

The scaling factor dσ(θ , φ) [L 2 ] denotes the probability that an incident atom A is scattered to the solid angle dΩ, also known as the differential scattering cross section. The total scattering cross section is

σ = σ(θ , φ)dΩ = π 0 2π 0 σ(θ , φ) sin θ dθ dφ. (2.121)

Scattering amplitude

Assuming that a free particle A is colliding a target particle B + , their interaction being characterized by a time-independent potential U(r), where r is the relative coordinate. The initial wave function of the effective particle with reduced mass m propagating along the z-axis can be expressed as a plane wave

ψ i = ψ 0 e ikz (2.122)
With only one incident particle per unit volume, the amplitude ψ 0 of the plane wave function is equal to 1. After the collision, the wave function of particles is

ψ = ψ i + ψ f ∼ e ikz + f (θ , φ) e ikr r , r → ∞, (2.123)
which is the superposition of the incident plane wave function ψ i and the scattered spherical wave function ψ f . The asymptotic behavior at infinity of

ψ f is f (θ , φ) e ikr
r with the time-independent scattering amplitude f (θ , φ). This wave function satisfies the Schrödinger equation

- ħ h 2 2m 2 ψ + U(r)ψ = Eψ, E = ħ h 2 k 2 2m . (2.124)
When r → ∞, U(r) → 0, so that the probability flow J f in spherical coordinates is

J f = iħ h 2m ψ f ∂ ψ * f ∂ r -ψ * f ∂ ψ f ∂ r = iħ h 2m | f (θ , φ)| 2 - ik r 2 - ik r 2 = ħ hk mr 2 | f (θ , φ)| 2 .
(2.125) The physical meaning of J f is the probability to find the particle entering the θ , φ direction per unit time per area ds. Therefore, the probability of the outgoing particle passing through the area dΩ per unit time is

J f ds = J f r 2 dΩ = ħ hk m | f (θ , φ)| 2 dΩ.
(2.126)

Concerning the incident particle on the z-axis, we get

J i = iħ h 2m ψ i ∂ ψ * i ∂ z -ψ * i ∂ ψ i ∂ z = iħ h 2m -ikψ i ψ * i -ikψ * i ψ i = ħ hk m . (2.127)
Then, the probability of the incident particle passing through the area dΩ per unit time is

J i ds = σ(θ , φ)J i dΩ = ħ hk m σ(θ , φ)dΩ. (2.128)
Considering Eqn [2.126] and Eqn [2.128], we obtain

σ(θ , φ) = | f (θ , φ)| 2 .
(2.129)

Until now, we realized that to solve the scattering cross section is actually a matter of computing the scattering amplitude.

Partial wave expansion

In the following, we first introduce elementary scattering theory based on the partial wave expansion to derive the general plane wave function and the asymptotic behavior of the wave function of the scattered particle. We consider an incident beam of particles along the z-axis described by a plane wave scattered by a spherically symmetric potential with a boundary condition U(r) → 0 at infinity. Since the beam is switched on for a longer time compared to the interaction time of the particles with the scattering potential, a steady-state is assumed. This means that the stationary Schrödinger equation for the wave function representing the particle beam should be solved.

The incident wave e ikz is a plane wave propagating along the z-axis and is also the common eigenstate of the z-axis projection of the angular momentum Lz and the kinetic momentum ˆ P. The expansion of the plane wave function is

e ikz = e ikr cosθ = ∞ l=0 (2l + 1)i l j l (kr)P l (cosθ ), (2.130)
where l is the partial wave, j l (kr) is the bessel function

j l (kr) = π 2kr J l+1/2 (kr) J l+1/2 (x) = (-1) l 2 π x l+1/2 d l (x d x) l sinx x .
(2.131) J l+1/2 (kr) is the half-integer Bessel function. Its asymptotic expression is

j l (kr) → 1 kr sin kr - 1 2 lπ , (r → ∞) (2.132)

Stationary Schrödinger equation

The dynamics we have to deal with in this thesis can be categorized into two groups: photo-assisted dynamics and non-radiative collisions. For both cases, the initial state of the ion or atom is in the long-life metastable or ground state. The time of applied lasers is long enough as well. So that we only need to solve time-independent stationary Schrödinger equation

- ħ h 2 2m 2 ψ + U(r)φ = Eψ. (2.133)
Here, U(r) is the central force field.

E = 1 2 m v 2 = ħ h 2 k 2 2m , then 2 ψ + k 2 - 2m ħ h 2 U(r) ψ = 0. (2.134)
The general solution is

ψ(r, θ , φ) = l,m R l (r)Y l,m (θ , φ).
(2.135)

Taking the direction of incidence of the particle through the centre of the vertical force field, i.e. the z-axis, then both the wave function ψ and the scattering amplitude f are independent on the angle φ. Since the eigenvalue of Lz is

zero, we have Y l,0 (θ , φ) = 2l + 1 4π P l (cosθ ), 2l + 1 4π
being the normalisation constant. The formula 2.135 becomes

ψ(r, θ ) = l R l (r)P l (cosθ ), (2.136)
where R l (r)P l (cosθ ) represents a certain partial wave l, and l = 0, 1, 2, 3... indicates the s, p, d, f ... wave, respectively. The radial Schrödinger equation, which is satisfied by the radial function R l (r), is 1

r 2 d d r r 2 dR l (r) d r + k 2 - 2m ħ h 2 U(r) - l(l + 1)
r 2 R l (r) = 0. (2.137)
Setting R l (r) = u l (r) r , then

d 2 u l (r) d r 2 + k 2 - 2m ħ h 2 U(r) - l(l + 1)
r 2 u l (r) = 0. (2.138)
when r tends to infinity, U(r) → 0 and 1

r 2 → 0, so d 2 u l (r) d r 2 + k 2 u l (r) = 0, (2.139)
admitting the solution u l (r) = A i sin(kr + δ i ).

(2.140)

Re-setting u l (r) = R l (r)r, and comparing to the half-integer Bessel equation 2.132, we set A l = kA i and

δ l = δ i + 1 2 lπ, R l (r) r→∞ --→ A i r sin(kr + δ i ) ≡ A l sin(kr - 1 2 lπ + δ l ) kr . (2.141)
Thus, the asymptotic solution is (2.144)

ψ(r, θ ) r→∞ --→ ∞ l=0 A l kr sin(kr - 1 2 lπ + δ l )P l (cos(θ )). ( 2
By using the expression sin α = 1 2i (e iαe -iα ), we are able to transform the above formula into an exponential form. To ensure that this equation is valid for any distance r, the prefactor of e ikr and e -ikr must be 0. Considering that i l = e ilπ/2 we get

         2ki f (θ ) + ∞ l=0 (2l + 1) 1
kr P l (cosθ ) - ∞ l=0 A l e i(δ l -lπ/2) P l (cosθ ) = 0 ∞ l=0 (2l + 1)e ikπ P l (cosθ ) - ∞ l=0
A l e -i(δ l -lπ/2) P l (cosθ ) = 0

(2.145)

According to the orthogonality of the Legendre polynomials,

π 0 P l (cosθ )P l (cosθ ) sin θ dθ = 2 2l + 1 δ l l , (2.146)
and multiplying P l (cosθ ) sin θ on both sides of the second line of the Eqn [2.145], and integrating over θ from 0 to π, we obtain

A l = (2l + 1)i l e iδ l = (2l + 1)e i(δ l +lπ/2) . ( 2
f (θ ) = 1 k ∞ l=0 (2l + 1)e iδ l P l (cosθ ) sin δ l . (2.148)
Eventually, the differential scattering cross section can be written as

σ(θ ) = | f (θ )| 2 = 1 k 2 ∞ l=0 (2l + 1)e iδ l P l (cosθ ) sin δ l 2 , ( 2.149)
2.4. ELEMENTS OF SCATTERING THEORY 45 and the total cross section is

σ t ot = 2π π 0 σ(θ ) sin θ dθ = 2π k 2 ∞ l=0 ∞ l =0 (2l + 1)(2l + 1) π 0 P l (cosθ )P l (cosθ ) sin θ dθ e i(δ l -δ l ) sin δ l sinδ l = 2π k 2 ∞ l=0 ∞ l =0 (2l + 1)(2l + 1) 2δ l l 2l + 1 e i(δ l -δ l ) sin δ l sinδ l = 4π k 2 ∞ l=0 (2l + 1) sin 2 δ l = l σ l , (2.150)
where σ l = 4π k 2 (2l + 1) sin 2 δ l is the partial cross section for a given partial wave l.

It is clear from Eqn [2.148] and Eqn k 2 (2l + 1). In contrast, the contribution of each partial wave to the total scattering cross section is zero as long as δ l (k) = nπ. Figure 2.11 shows that at a given temperature or kinetic energy the total scattering cross section is a superposition of the contributions of several partial waves. For each partial wave, it is apparent that the corresponding scattering cross section oscillates as a function of the collision temperature or the kinetic energy.

Since the initial incoming flux and the final outgoing flux of particles should be equal at the scattering region θ = 0, then P l (1) = 1, the imaginary part of f (θ ) can be expressed as

I m f (θ ) = 1 k ∞ l=0 (2l + 1) sin 2 δ l . (2.151)
Then, the formula 2.150 can be re-expressed as

σ t ot = 4π k I m f (0) (2.152)
which is also known as the optical theorem and conservation of the probability flux.

ELEMENTS OF SCATTERING THEORY

Figure 2.11: The cross section of elastic collision in a single channel, and the decomposed cross section for the selected partial waves.

Hard sphere

To analyze the scattering problem from a quantum mechanical point of view using the derived scattering cross section Eqn [2.150], we can postulate a similar central force field (U 0 > 0) of a hard sphere

U(r) = U 0 (r ≤ a) 0 (r > a) (2.153)
The differential scattering cross section and the total scattering cross section are

σ(θ ) =a 2 t an(ik 0 a) ik 0 a -1 = a 2 t anh(k 0 a) k 0 a -1 σ t ot =2π π 0 σ(θ ) sin θ dθ = 4πa 2 t anh(k 0 a) k 0 a -1 , (2.154)
where

k 2 0 = -2mU 0 ħ h 2 . Considering U 0 → ∞, k 0 → ∞, t anh(k 0 a) = e k 0 a -e -k 0 a
e k 0 a + e -k 0 a → 0, the total cross section is

σ t ot =4πa 2 , (2.155)
which is identical to the total surface area of the hard sphere. The definition of the scattering cross section in quantum mechanics is four times bigger than that of classical mechanics due to the diffraction effect of the incident wave. Under the premise of low energy collision energy k → 0, l = 0 and m = 0, the s wave is isotropic, all regions of the surface of a hard sphere contribute equally to the scattering, such that the total scattering cross section is equal to the surface area of the hard sphere.

Impact parameter

For the sake of clarity, we introduce another concept: the impact parameter b, that is, the vertical distance between the direction of the relative velocity of a structureless particle (A, B, or C), and the parallel line passing through the center of the target structureless particle D in Figure 2.12. The maximum distance of the effective potential energy of particle D is a, so that the maximum impact parameter b ma x is a. A collision can only occur when b is less or equal than b max . Therefore, we can define the possible partial wave depending on b ma x . The angular momentum of an incident structureless particle on the radial axis z is

L z = P z × b max . ( 2

.156)

We know the maximum eigenvalue of L z is lħ h for each partial wave, and the eigenvalue of P z is ħ hk. Then we can obtain the practical conditions of the partial wave method l ≤ kb ma x = ka.

(2.157)

In the perspective of this classical collision model, herein, the maximum total scattering cross section is

σ max cl assical = πb 2 ma x = πa 2 . (2.158)
This is exactly the area of the circle with a radius b ma x .

Centrifugal barrier

For the atom-ion system, the leading long-range interaction is During a collision, the effective long-range potential includes the rotational interaction

V (r) = - c 4 r 4 , ( 2
V (r) = - c 4 r 4 + l(l + 1)ħ h 2 2µr 2 (2.160)
where µ is the reduced mass of the ion-atom system. As r increases, there is a competition between the two terms, resulting in a centrifugal barrier. Figure 2.13 presents the corresponding centrifugal barriers for the selected partial waves. It should be noted that there is no centrifugal barrier for the s wave (l = 0). The position R * max (l) and the energy height E * max (l) of the barrier can be evaluated as

R * max (l) = 4µc 4 ħ h 2 l(l + 1) , (2.161) E * max (l) = ħ h 4 l 2 (l + 1) 2 16µ 2 c 4 .
(2.162)

Capture theory

In 1905, Langevin [98] developed the capture theory to model collision which are dominated by the long-range attractive interaction V (r) between the two colliding particles. However, the capture theory only considers the reaction with the maximum partial wave l max for each collision and does not account for the quantum effects raised by the barriers (l < l ma x ), like shape-resonances. The effective long-range potential is rewritten in atomic units as

V e f f (r) = - c 4 r 4 + l 2 2µr 2 .
(2.163)

Then, Eqn [2.162] is rewritten as

E ma x (l) = l 4 16µ 2 c 4 .
(2.164)

Assuming the collision energy E = E ma x (l), we get the maximum value for l,

l ma x = (16µ 2 c 4 E)
σ L = πb 2 ma x = π l 2 ma x k 2 = π (16µ 2 c 4 E) 1/2 2µE = 2π c 4 E . ( 2

.166)

Assuming q = 1, the Langevin rate is in atomic units

K L = σ L v = 2π c 4 E 2E µ = 2π 2c 4 µ = 2π α µ , [L 3 T -1 ], (2.167)
or in the often used CGS units

K L = 2.34 × 10 -9 α µ , [L 3 T -1 , cm 3 s -1 ], (2.168)
where µ is in atomic mass units (amu) and α in Å 3 .

Shape resonance

Shape resonance as an important quantum effect in the molecular field is used to interpret a quasi-bound state in which a molecule is temporarily trapped due to the centrifugal barrier, through which it may eventually tunnel and escape [START_REF] Dehmer | Fundamental Processes of Atomic Dynamics: Shape resonances in molecular field[END_REF]. In other words, the molecule is essentially restrained in the inner side of the molecular potential with a barrier, and the molecule oscillates with a long lifetime before dissociating. The asymptotic behavior of the energy-normalized continuum wave function of the quasi-bound state is given by [100]

Ψ l,E (R) ∼ 2µ πħ h 2 k 1/2 sin kR - lπ 2 + δ l (E) , (2.169)
where k = 2µE/ħ h. The asymptotic phase shift is

δ l (E) = δ bg + ar c t an Γ l /2 E -E r es l ,
(2.170)

ELEMENTS OF SCATTERING THEORY

51

where δ b g indicates the background phase shift, Γ l is the full width at half maximum (FWHM) of the resonance of the resonant energy E r es l . For a broad resonance, δ l (E) is less than π [START_REF] Friedrich | Theoretical Atomic Physics[END_REF]. Since Γ l is related to the energy and its dimensionality is [M L 2 T -2 ], the lifetime of molecules on quasi-bound states is

T l = ħ h/Γ l .
(2.171)

In Figure 2.14, the top panel displays a shape resonance of l = 3 at 3.41 mK. Its phase shift is displayed in the bottom panel. The FWHM Γ l equals 0.4 mK, then the lifetime of this shape resonance is 19.1 ns.

Scattering matrix

In this subchapter, we briefly describe this multi-partial wave scattering approach with regard to the single-channel problem and the coupled-channel problem.

S, K and T matrix elements

In the previous subchapter, we derive the scattering cross section based on a single-channel model following the partial wave methodology. We note that the expressions of cross sections do not depend on the form of the normalisation factor A l of the wave function R l (r) or u l (r).

If A l is chosen to be -2iex p[iδ l (k)],
the obtained radial wave function Eqn [2.140] can be re-defined as

u l (r) r→∞ --→ A l sin(kr -lπ/2 + δ l ) r→∞ --→ ex p[-i(kr -lπ/2)] -S l (k)ex p[i(kr -lπ/2)], (2.172)
where the single value of the S matrix element is defined as

S l (k) = ex p[2iδ l (k)].
(2.173)

Assuming A l = [cosδ l (k)] -1 , then u l (r) r→∞ --→ sin(kr -lπ/2) + K l (k)cos(kr -lπ/2), ( 2.174)
where the K matrix element is defined by

K l (k) = t anδ l (k) (2.175)
The relation of the S scattering matrix and K reaction matrix element is

S l (k) = 1 + iK l (k) 1 -iK l (k) . ( 2

.176)

Due to the fact that the K matrix is symmetric, the S matrix is symmetric as well. The unitarity of the S matrix also implies that the K matrix is real. If K matrix element is real (complex), S matrix element should be real (complex). However, the outgoing channel a is forbidden.

The partial elastic cross section σ el l , and the partial inelastic cross section σ inel l can be determined in terms of the S matrix

σ el = l (2l + 1)σ el l = π k 2 l (2l + 1)|1 -S l | 2 σ inel = l (2l + 1)σ inel l = π k 2 l (2l + 1)(1 -|S l |) 2 .
(2.177)

The transition T matrix is also useful to explain the state-to-state transition probability, which is defined as

T l (k) = S l (k) -1.
(2.178)

Coupled-channel method

Figure 2.15 displays that two incident particles with given relative collision energy E along with channel b can be elastically scattered to channel b, or inelastically scattered to channel c, but can not be scattered to channel a depending on the interactions. As such, we regard these channels are coupled by this interaction. To generalize this multichannel scattering problem, we assume that there is a total of n open channels and m closed channels for a given incident energy E for a general system of colliding particles. The matrix wave function ψ ψ ψ(r) obeys Schrödinger's equation

d 2 d r 2 + 2µ ħ h 2 EI I I -V V V (r) -H int H int H int - , ψ ψ ψ(r) = 0 (2.179)
where µ is the reduced mass, I I I is the unit matrix, V V V (r) is the potential energy matrix plus the centrifugal barrier, H H H int is the internal interaction which is rdependent matrix or not, is the diagonal matrix containing the kinetic energy of various channels. The scattering wave function in a state can be written as

ψ ψ ψ α (r) = r -1 i φ i (τ)F iα (r), (2.180)
where α runs up to the number of channels. The functions φ i (τ) form a complete orthonormal basis for the motion with all coordinates τ in the system, and the radial channel function F iα (r) is the element of the matrix F F F . The differential equation for each channel function F jα (r) is then

d 2 F jα (r) d r 2 = j W i j -Eδ i j F iα (r) (2.181)
where the coupling matrix element W i j can be expressed by

W i j = 2µ ħ h 2 φ * j (τ) H int + V (r) + L2 2µr 2 φ i (τ)dτ. (2.182)
Eqn [2.179] can be reformulated as

d 2 F F F d r 2 = [W W W (r) -EI I I] F F F (r).
(2.183)

At infinity, the centrifugal barrier term tends to zero, and the wave function matrix

F F F is F F F (r) ∼ r→∞ J J J(r) + N N N (r)K K K, (2.184)
which is a combination of Riccati-Bessel functions [START_REF] Abramowitz | Handbook of mathematical functions[END_REF] for open channels

J i j (r) = δ i j k -1/2 j ĵl j (k j r) N i j (r) = δ i j k -1/2 j nl j (k j r), (2.185)
and of modified spherical Bessel functions of the first and third kinds for the closed channels,

J i j (r) = δ i j (k j r) 1/2 I l j +1/2 (k j r) N i j (r) = δ i j (k j r) 1/2 K l j +1/2 (k j r), (2.186)
where the J J J(r) and N N N (r) matrices are diagonal, k j is the channel wave number. Then, we can take K K K reaction matrix to present the interaction between the closed and open channels, Thereby the generalized state-to-state cross section using explicit quantum numbers is given by

K K K = K K K nn oo K K K nm oc K K K mn co K K K mm cc , ( 2
σ i→ f (E) = π k 2 0 J (2J + 1) l i , j i ,m i l f , j f ,m f |S(J, l f , j f , m f , p ← J, l i , j i , m i , p) -I| 2 .
(2.190) Here, k 2 0 = 2µE/ħ h 2 , p is parity, J is the total angular momentum quantum number, m is the magnetic quantum number, j is the total electronic angular momentum quantum number. |J, l i , j i , m i , p > and |J, l f , j f , m f , p > represent the chosen initial channel and the allowed final channel.

Log-derivative algorithm

The formula [2.190] tells us that the solution to the scattering cross section is to gain the S matrix elements. To avoid numerical instability during the direct propagation of the wave function F F F (r), we apply the multichannel log-derivative method originally developed by B. R. Johnson [START_REF] Johnson | The multichannel log derivative method for scattering calculations[END_REF] and other developers [START_REF] Manolopoulos | An improved log derivative method for inelastic scattering[END_REF][START_REF] Van Dijk | On the nonradiative and radiative decay rates and a modified exponential energy gap law for 4f-4f transitions in rare-earth ions[END_REF][START_REF] Manolopoulos | Close-coupled equations: the log derivative approach to inelastic scattering, bound state and photofragmentation problems[END_REF][START_REF] Alexander | A stable linear reference potential algorithm for solution of the quantum close-coupled equations in molecular scattering theory[END_REF], to propagate the log-derivative matrix Y Y Y (r) from the short distance to large distance until K and S matrices are converged. The log-derivative matrix is then considered as

Y Y Y (r) = F F F (r)F F F -1 (r) (2.191)
Substituting Eqn [2.183] and using Eqn [2.191] to eliminate the second derivative term, we obtain the matricial Ricatti equation

Y Y Y (r) = W W W (r) -Y Y Y 2 (r). (2.192)
From Eqn [2.191] we know that F F F = 0 is not defined as the determinant of the wave function vanishes, so we can not directly propagate Y Y Y (r) matrix. Instead, we take the embedding-type propagator which is defined on an interval [r x , r y ],

F F F (r x ) F F F (r y ) = 1 (r x , r y ) 2 (r x , r y ) 3 (r x , r y ) 4 (r x , r y ) -F F F (r x ) F F F (r y ) (2.193)

ELEMENTS OF SCATTERING THEORY 55

We post-multiply the first and second of Eqn [2.193] by F F F -1 (r x ) and F F F -1 (r y ), respectively. Eliminating F F F (r x )F F F -1 (r y ), we get a recursion relation for the logderivative matrix,

Y Y Y (r y ) = 4 (r x , r y ) -3 (r x , r y ) × Y Y Y (r x ) + 1 (r x , r y ) - 1
2 (r x , r y ). (2.194) In the end, the reaction K K K matrix is obtained,

K K K = -[Y Y Y (r)N N N (r) -N N N (r)] -1 × [Y Y Y (r)J J J(r) -J J J (r)]. (2.195)
here, we choose log-derivative propagators consistently, while it is still necessary to mention that an alternative method is the renormalized Numerov propagator [START_REF] Johnson | The multichannel log derivative method for scattering calculations[END_REF], which also yields stable and accurate outcomes.

Chapter 3

Photodissociation in ultracold experiments

In this chapter, we concentrate on the photodissociation and its applications. Photodissociation [START_REF] Heays | Photodissociation and photoionisation of atoms and molecules of astrophysical interest[END_REF][START_REF] Hollenbach | Photodissociation regions in the interstellar medium of galaxies[END_REF] is of astrophysical interest in many respects, like explaining the existence of small molecules by photodissociation generated by the ultraviolet light from the sun [START_REF] Haser | Distribution d'intensité dans la tête d'une comète[END_REF][START_REF] Crovisier | The spectrum of comet Hale-Bopp (C/1995 O1) observed with the infrared space observatory at 2.9 astronomical units from the sun[END_REF], modeling the astrochemical reactions in clouds of gas and dust [START_REF] Glassgold | Circumstellar Photochemistry[END_REF][START_REF] Hollenbach | Photodissociation regions in the interstellar medium of galaxies[END_REF][START_REF] Tielens | The molecular universe[END_REF][START_REF] Glover | Approximations for modelling CO chemistry in giant molecular clouds: a comparison of approaches[END_REF] and so forth. However, it also plays a critical role in laboratory experiments, particularly the hybrid molecular ion experiment which often consists of MOT, rf-trap, or ODT. These experimental devices require multiple lasers to cool down and trap species, meanwhile inevitably induce very often photodissociation. There are two types of photodissociation: direct photodissociation and indirect photodissociation. Unlike direct photodissociation, the indirect photodissociation mechanism first deals with the discrete transition to the bound vibrational level of the excited electronic state. Indirect photodissociation can be further subdivided into coupled states photodissociation, predissociation and spontaneous radiative dissociation [START_REF] Bates | Advances in atomic, molecular and poptical physics[END_REF].

In this chapter, we will only discuss the direct photodissociation in two independent experiments: a single cold Ba + ion in an ultracold Rb gas (Ulm, Germany) [START_REF] Krükow | Energy scaling of cold atom-atom-ion three-body recombination[END_REF][START_REF] Härter | Single ion as a three-body reaction center in an ultracold atomic gas[END_REF][START_REF] Schmid | Dynamics of a cold trapped ion in a bose-einstein condensate[END_REF][START_REF] Krükow | Reactive two-body and three-body collisions of Ba + in an ultracold Rb gas[END_REF], and ionic clock setup including a single Ba + (Aarhus, Denmark).

RbBa + photodissociation in a hybrid trap

Introduction

Nowadays collisions between atoms and ions at ultracold temperatures have attracted considerable attention, especially for their prospects toward novel cold chemistry dominated by quantum effects [START_REF] Dulieu | Cold chemistry: molecular scattering and reactivity near absolute zero[END_REF]. For this reason, hybrid traps are required, merging a cold atom trap and an ion trap, where both species are laser-cooled. At low atomic density (< 10 11 cm -3 ) the spontaneous formation of cold molecular ions by radiative association is expected for many species, but it 57 58 3.1. RBBA + PHOTODISSOCIATION IN A HYBRID TRAP has been experimentally probed only for two cases [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: Radiative molecule formation versus charge exchange[END_REF][START_REF] Hall | Lightassisted cold chemical reactions of barium ions with rubidium atoms[END_REF]. When the atomic density is higher, these molecular ions are preferentially created by three-body collisions, as demonstrated in the Ulm group with the Ba + +Rb combination [START_REF] Krükow | Energy scaling of cold atom-atom-ion three-body recombination[END_REF]. In these experiments, however, the observation of cold molecular ions is hindered by the strong photodissociation induced by the numerous lasers.

Recently, a further investigation of the experiment of single cold Ba + immersed in the ultracold Rb gas has been investigated specifically for the physicochemical reactions of the weakly-bound RbBa + created by Rb-Rb-Ba + collisions at various stages of its whole life from birth to death [START_REF] Mohammadi | Life and death of a cold BaRb + molecule inside an ultracold cloud of Rb atoms[END_REF]. By analyzing the electronic states and the kinetic energy of several ionic products ( Ba + , Rb + 2 , Rb + ), it was demonstrated that this weakly-bound molecule with a short averaged lifetime of 0.5 ms endures a series of elastic and inelastic processes, where the short lifetime is determined by a combination of dynamics, such as radiative relaxation, spin-flips, and collisional relaxation. Among these effects, it was proved that the 1064 nm optical dipole trap laser acts as a dual-edged sword that assists in the synthesis of RbBa + at the beginning, but decomposes this molecule into 'HOT' Ba + with a high kinetic energy (∼ 0.2 eV (2321 K)) in the end.

In this section, we even further investigate this work by modelling the photodissociation of RbBa + molecular ion induced by the relatively intense 1064 nm wavelength laser used for the optical dipole trap, and the most probable dissociation channel Rb(5s)+ Ba + (5d) which involves six excited molecular states (4 1 Σ + , 4 3 Σ + , 2 1 Π, 3 3 Π, 2 1 ∆, 2 3 ∆), to discuss the influence of the spin-orbit interaction on the photodissociation. The dissociation channel splits into two channels, namely Rb(5s) + Ba + (5D 3/2,5/2 ), where the atomic spin-orbit splitting of Ba + (5D 3/2,5/2 ) amounts to 801 cm -1 (1200 K). Taking into account the atomic spin-orbit coupling, this allows us to predict the number of atomic ions which are produced in the Ba + (5D 3/2 ) and Ba + (5D 5/2 ) states, to be compared with upcoming experimental results.

Photodissociation in Hund's case (a) basis

The top panel of Figure 3.1 shows Hund's case (a) potential energy curves up to the seventh excited dissociation limits. The photodissociation is marked by the black dashed arrows from the weakly bound levels in the entrance dissociation limit Rb(5s)+Ba + (6s) to the possible exit channels in Rb(5s)+Ba + (5d) and Rb + +Ba(6s6p, 3 P) dissociation limits. The red arrow indicates the scattered Rb and Ba + with a high temperature 5800 K, which is compatible with the measured HOT Ba + ion (0.2 eV). The relevant transition electric dipole moments (TEDMs) are presented in the bottom panel of Figure 3.1. According to the atomic transition rules and molecular transition rules, the selected TEDMs curves oscillate in the short-range and converge to zero in the long-range between s + s and s + d limits. These ab initio data have already been published in [START_REF] Da Silva Jr | Formation of molecular ions by radiative association of cold trapped atoms and ions[END_REF][START_REF] Mohammadi | Life and death of a cold BaRb + molecule inside an ultracold cloud of Rb atoms[END_REF].

One of the entrance channels is 2 1 Σ + which owns a double-well character due to the presence of an avoided crossing with 3 1 Σ + correlated to the Rb + +Ba(6s5d, 1 D) dissociation limit. The other entrance channel 1 3 Σ + potential has a much deeper well-depth than that of 2 1 Σ + . Due to the high density of the Rb gas (∼ 10 12 cm -3 ) in the experiment, here, we consider that RbBa + is formed by a three-body collision (Rb+Rb+Ba + ). The temperatures of laser-cooled Rb and rf-trapped Ba + are around 750nK and 100 mK, respectively [START_REF] Mohammadi | Life and death of a cold BaRb + molecule inside an ultracold cloud of Rb atoms[END_REF], RbBa + can only be a cold weakly-bound molecular ion which can either populate the 2 1 Σ + state or the 2 3 Σ + state, that we assume to be statistical, i.e. with a 1/4 and 3/4 population, respectively. Moreover, there is another avoided crossing presented between 2 3 Π and 3 3 Π exit channels. If the dissociated RbBa + can reach the continuum area of Rb(5s)+Ba + (5d), this avoided crossing is expected to result in a charge exchange to generate Rb + ions.

To evaluate the impact of the avoided crossing of 2 3 Π and 3 3 Π, we may use the classical Landau-Zener model to make a rough assessment. In Figure 3.2, linear AB and F D lines are chosen as the most promising approximation. The probability P of decay from one adiabatic surface to the other is given by [START_REF] Nakamura | Semiclassical treatment of nonadiabatic transitions: Multilevel curve crossing and nonadiabatic tunneling problems[END_REF]:

P =ex p(-2π W 2 c ħ hv∆F ), [dimensionless] ex p ... (M L 2 T -2 ) 2 (M L 2 T -1 )(LT -1 )(M L 2 T -2 /L) (3.1)
Here, the second formula is to show the dimensionality of each quantity, 2W c is the energy gap between the two adiabatic states at the crossing point C. The smaller this energy gap goes, the larger P will be. ∆F is the difference of the slopes of lines resulting from the linearization of the avoided crossing around the point C (see Figure 3.2). v is the local relative velocity at the crossing point C which is calculated based on the photodissociation conditions,

v = 2E 0 µ , E 0 = E P D ex i t + |W c | (3.2)
where E 0 is the final kinetic energy of the system, the laser energy is E l aser = hv, the weakly-bound energy of 1 1 Σ + is too small to be considered in this classical model, the exit energy of the dissociated RbBa + is E P D ex i t = E l aser -|E ex i t -E ent r ance |, where E ent r ance and E e x i t are the energies of the dissociation limits of Rb(5s) + Ba + (6s) and Rb(5s) + Ba + (5d), respectively. Then we obtain W c = 0.000737635 a.u., ∆F = 0.00136266 a.u., v = 0.00064742 a.u., so that the single-path transition probability P = 0.02 and the total transition probability 2P(1 -P) = 0.0392. Therefore, we can neglect this avoided crossing effect in our following theoretical model for the photodissociation.

Photodissociation in Hund's case (a)

Concerning Eqn [2.112], the state-to-state absorption cross section is calculated using this simplified expression without considering the spin-orbit coupling, where, D i f (R) is the TEDM between the initial Λ i and final Λ f electronic state, v b is the initial vibrational quantum number, c is the speed of light and E l aser = hv is the transition energy, i.e. the photon energy of 1064 nm. The final continuum energy

σ v = 4π 2 3c hv| < Λ f , E cont. |D i f (R)|Λ i , v b > | 2 (3.3)
E cont = E l aser -|E(v b )| -(|E e x i t | -|E ent r ance |).
The spacenormalized vibrational wave function is determined using the mapping Fourier grid Hamiltonian (MFGH) approach [START_REF] Kokoouline | Mapped Fourier methods for long-range molecules: Application to perturbations in the Rb 2 (0 + u ) photoassociation spectrum[END_REF][START_REF] Willner | Mapped mapped grid methods for long-range molecules and cold collisions[END_REF] which diagonalizes a discrete variable representation matrix of the Hamiltonian, while the energy-normalized continuum wave function is calculated by standard Numerov method [127].

Convergence of the cross section

To test the convergence of the calculated cross section, we firstly check the number of grid points of the continuum wave function with an arbitrary case in Figure 3.3. Because the effective nonzero TEDMs and the vibrational wave function determine the integration interval of the distance in the bound-to-continuum transition, in particular, Figure 3.1 displays that the TEDMs tend to 0 when it is close to 35 a.u., the continuum wave function with a maximum distance of 5000 a.u. is fairly sufficient for the calculation. It is obvious that the cross section changes dramatically as long as the points are fewer than 40,000, but start to converge until 80,000. Taking into account the computational time and the desired convergence, we think that 80,000 is sufficient for our computational needs. However, in the subsequent calculations, we realized that about 200,000 points for a continuum wave function with R max = 5000 a.u. would give better results, but cost a much longer computing time. For instance, the profile of cross sections exhibits odd ripples in the top panel of Figure 3.5, while such features vanish in the right of Figure 3.7. This comes from the convergence of the continuum wave function. As we discussed, obtaining a smooth curve of the cross section requires a large number of points (200,000) to express the continuum wave function but to avoid excessive time-consuming calculations, we use fewer points (80,000) within the acceptable range of accuracy to do the simulations, where odd ripples is acceptable.

Figure 3.4 represents the photodissociation cross section as a function of the vibrational level of two initial states 1 3 Σ + or 2 1 Σ + . The statistical weighting is not taken into account in our calculations, i.e. 1/4 of 1 3 Σ + and 3/4 of 2 1 Σ + , so we term it the unscaled cross section. According to the calculated cross sections, we find that when the produced RbBa + is a strongly bound molecule with vibrational level roughly (v < 240) of 1 3 Σ + or (v < 140) of 2 1 Σ + , the photodissociation is very efficient, except for the path 2 1 Σ + -2 1 Π. However, the binding energy of the RbBa + molecule is transferred from the collision energy of Rb and Ba + , so this molecule can only be created on a very weak vibrational state.

To get another perspective, we produce unscaled cross sections as a function of the transition energy for the last several vibrational levels with energies (-0.1 cm -1 → -21.3 cm -1 ) in Figure 3.5. The profile of cross sections is oscillating pretty heavily for the exit channels 2 1 Π, 4 3 Σ + and 3 3 Π. We realize that the dominant cross section is between (2) 1 Σ + and 4 1 Σ + states with a magnitude of 10 -18 cm 2 . To analyze our computational results and to check the precision of our model, we have chosen a level with binding energy 10.5 cm -1 of 2 1 Σ + to analyze the TEDM and the corresponding bound/continuum wave functions in Figure 3.6. Panel (a) shows the unscaled cross section of the main transition for the photodissociation of a weakly vibrational state (10.5 cm -1 ) with three selected transition energies with respect to the maximum and minimum cross sections (red dots on the graph). The corresponding TEDM is presented in panel (b). In the remaining panels, we intend to discuss the influence of the turning points in the integrals between bound wave function and three continuum wavefunctions of (4)Σ + , where the turning point is described by the first antinode (the first peak) of wavefunctions matching to the repulsive wall. According to the Franck-Condon principle, the overlap of wave functions is the key factor to predict the transition probability. For a bound molecular state, the molecule will spend more time around the edge of the potential energy surface, so that the match at the turning point of both wave functions would be significant. We can see that the match of the turning point from panel (b ) to (d ) gradually gets more and more effective, so do the corresponding integrated dipole matrix elements (green) but the cumulative sum of the integrand < vi br.|D|cont. > in panels ( jl ). The cumulative sum (blue) at a large distance shows that the good match provides a nearly zero value (panel (k )) while the less good overlap gives rise to a bigger number of the cumulative sum in panel ( j ). Since the TEDM essentially weight the entire overlap of wave functions, this implies the match of the turning points is not sufficient to determine a promising transition.

The repulsive wall of PEC

A recent joint theoretical and experimental publication [START_REF] Mohammadi | Life and death of a cold BaRb + molecule inside an ultracold cloud of Rb atoms[END_REF] reports that a maximum value (20×10 -20 cm 2 ) of the theoretical photodissociation cross section should be taken to compare the measuted data (540×10 -20 cm 2 ) around 8.7 K due to the fact that the uncertainty of the PECs. Besides, a discrepancy of a factor of 13.5 ( resp. 7) is pointed out for the transition of 2 1 Σ + -4 1 Σ + ( resp. 1 3 Σ + -3 3 Π), where the specific explanation is still unclear. Please go to the paper [START_REF] Mohammadi | Life and death of a cold BaRb + molecule inside an ultracold cloud of Rb atoms[END_REF] for more details.

First of all, we use the exact same data of 2, 4 1 Σ + [62] by simply shifting the repulsive wall of the exit channel 4 1 Σ + to understand why the maximum value is important. The orginal and shifted curves of 4 1 Σ + are displayed in Figure 3.7. The red curve with a new r new distance of the generated PEC (red curve) is obtained by gradually changing the distance r of the left part of the equilibrium position r e of the black curve (ab initio curve), where the shifted distance is δr =0.1 a.u. at a energy position r t ar g et marked by the red dotted line (1064 nm). The shifting formula is [START_REF] Pandey | Interaction potentials and ultracold scattering cross sections for the 7 Li + --7 Li ion-atom system[END_REF]

r new = r + r -r e r t ar g et -r e × δr (3.4)
where the right part of the equilibrium position r e of the black curve remains the same. In the right panel of Figure 3.7, the calculated cross section confirms that the magnitude can be significant changed with respect to such a slight adjustment of the potential energy surface. Our theoretical calculations suggest a plausible explanation of the maximum value of photodissociation cross section,

but not yet interpret the discrepancy factor 13.5.

Photodissociation in Hund's case (c) basis

Due to the fact that the fine structure exists only in the dissociation limit of Rb(5s)+Ba + (5d), it is necessary to first derive the Hamiltonian containing spinorbit interactions for modeling the photodissociation in Hund's case (c). Then, the dissociation transition is from a bound state of 1 3 Σ + or 2 1 Σ + in Hund's case (a) basis to a continuum state of an excited electronic state in Hund's case (c) basis, where the Hund's case (c) electronic state is a superposition of electronic states in Hund's case (a) basis. Rather than calculating the ab initio potential energy curves in Hund's case (c) basis, we treat this interaction in a perturbative fashion, using the eigenvectors of spin-orbit matrices as weighting/proportion factors to gain Hund's case (c) TEDMs. Finally, the model for photodissociation in Hund's case (c) basis is composed of two parts: entrance and exit electronic states in Hund's case (a) basis and TEDMs in Hund's case (c) basis.

Spin-orbit coupling matrix

In order to take into account the effects of spin-orbit interaction responsible for the mixing of the Hund's case (a) states in the s + d limit, we have built a new set of Hund's case (c) curves according to the projection Ω of the total electronic angular momentum on the molecular axis. The Hamiltonian matrices with symmetry Ω = 0 + , 0 -, ±1, ±2 and ±3 are written as [START_REF] Lozeille | Detection by two-photon ionization and magnetic trapping of cold Rb 2 triplet state molecules*[END_REF][START_REF] Beuc | Absorption spectroscopy of the rubidium dimer in an overheated vapor: An accurate check of molecular structure and dynamics[END_REF]:

H(|Ω| = 3) = A + V ( 3 ∆) (3.5) H(|Ω| = 2) =   V ( 3 ∆) A A/ 2 A V ( 1 ∆) -A/ 2 A/ 2 -A/ 2 V ( 3 Π) + A/2   (3.6) H(|Ω| = 1) =     V ( 3 ∆) -A A/ 2 A/ 2 0 A/ 2 V ( 3 Π) A/2 3A/2 A/ 2 A/2 V ( 1 Π) -3A/2 0 3A/2 -3A/2 V ( 3 Σ + )     (3.7)
H(Ω = 0 -) = V ( 3 Π) -A/2 A 3/2 A 3/2 V ( 3 Σ + ) (3.8)
H(Ω = 0 + ) = V ( 3 Π) -A/2 A 3/2 A 3/2 V ( 1 Σ + ) (3.9)
where the interaction parameter A = 2A so /5 is proportional to the atomic spinorbit splitting of Ba + (5d) A so = 801.46 cm -1 . In general, the Hund's case (c) continuum wave function formalism in s + d limit can be expressed as:

ϕ c (r) = (α n (r) × ϕ a n (r)) (3.10)
where, ϕ c (r) and ϕ a n (r) are the continuum wave functions belonging to the exit channels in Hund's case (c) and (a), respectively. The proportion factor α n (r) is the eigenvector achieved by diagonalization of above matrices. Due to Hund's case (a) transition rules (∆Ω = -1, 0, 1; ∆S = 0), the maximum number of components to consider is n = 2 for a ϕ c (r). The bound wave function of the entrance channels in Hund's case (a) basis is defined by φ a (r).

Hund's case (c) TEDMs and PECs

Then the state-to-state absorbing cross section can be described in Hund's case (c) with two situations:

σ = 4π 2 3c hν| < ϕ c (r)| -→ D |φ a (r) > | 2 = 4π 2 3c hν| < α 1 (r)ϕ a 1 (r)| -→ D |φ a (r) > | 2 (3.11)
Situation 2: when two components are involved:

σ ν = 4π 2 3c hν| < ϕ c (r)| -→ D |φ a (r) > | 2 = 4π 2 3c hν | < α 1 (r)ϕ a 1 (r) + α 2 (r)ϕ a 2 (r)| -→ D |φ a (r) > | 2 ∼ 4π 2 3c hν | < α 1 (r)ϕ a 1 (r)| -→ D |φ a (r) > | 2 + | < α 2 (r)ϕ a 2 (r)| -→ D |φ a (r) > | 2 + 2 × 4π 2 3c hν| < α 1 (r)ϕ a 1 (r)| -→ D |φ a (r) > • < α 2 (r)ϕ a 2 (r)| -→ D |φ a (r) > | (3.
12) In this work, we only take the squared term disregarding the interference term (third item in Eqn [3.12]). Within this approximation we can provide the minimum cross section for the photodissociation in Hund's case (c) basis.

Meanwhile, we are able to produce a qualitative description of squared TEDMs between φ a (r) and the mixture of ϕ a n (r), which are simplified as

D c (r) = α 1 (r) × D a 1 (r) n = 1 (3.13) D c (r) 2 = (α 1 (r) × D a 1 (r)) 2 + (α 2 (r) × D a 2 (r)) 2 n = 2 (3.14)
Here, the TEDM functions are corresponding to the two situations of cross sections, respectively. For simplicity and clarity, we name the linear combination of squared Hund's case (a) TEDMs as approximate TEDM 2 . By diagonalizing these Hamiltonian matrices 3.5-3.9, Hund's case (c) PECs can be straightforwardly obtained in Figure 3.8. The exit channel Rb(5s) + Ba + (5d) splits at large distances into two channels Rb(5s) + Ba + (5D 3/2 ) and Rb(5s) + Ba + (5D 5/2 ). The short-range potentials display a mixture character of Hund's case (a) potentials. It's worth noting that we do not directly use Hund's case (c) PECs in our calculations, but use Hund's case (c) TEDMs.

In the top panel of Figure 3.8, 0 + potentials are the mixture of Hund's case (a) 3 3 Π and 4 1 Σ + PECs, and 0 -potentials are composed of Hund's case (a) 3 3 Π and 4 3 Σ + PECs. As already noted above, the state-to-state transition obey Hund's case (a) selection rules. As for 2 1 Σ + to 0 + , the transition between 2 1 Σ + and 3 3 Π is forbidden. The transition 1 3 Σ + -4 1 Σ + is invalid for 1 3 Σ + to 0 -, neither. In this case, therefore, the TEDM satisfies Situation 1 .

The middle panel displays Hund's case (c) potentials for Ω = 1. All of them involve 4 3 Σ + , 2 1 Π, 3 3 Π and 2 3 ∆ Hund's case (a) molecular states. Due to the rule ∆S = 0, the non-allowed transitions are from 2 1 Σ + to 4 3 Σ + , 3 3 Π or 2 3 ∆. These transitions can be classified as Situation 1 to compute the corresponding transition dipole moments. In terms of the entrance 1 3 Σ + , both 3 3 Π and 4 3 Σ + need to be considered, so we should utilize Situation 2 to calculate the relevant Hund's case (c) TEDMs.

Hund's case (c) PECs of Ω = 2 have double-well behaviors in the bottom panel. This is because of the contribution of Hund's case (a) 2 3 ∆ potential which has an avoided crossing with upper molecular state 3 3 ∆. Since the entrances are Σ + states and the rule ∆Ω = ±1, 0, the outgoing ∆ states are not taken into account in our photodissociation model. Only the transition between 1 3 Σ + and 3 3 Π is valid, the TEDM can be described by Situation 1 . Meanwhile, there is no allowed transition from the entrance 2 1 Σ + .

In Figure 3.9-3.12, the sum of squared proportion curves in each panel at each distance equals one. This is due to the eigenvector matrix is normalized. In general, the curves of proportions converge to some constants in the long-range. As long as there is a crossing of proportion curves, these relevant Hund's case (a) PECs intersect as well. The sign of Hund's case (c) TEDMs is arbitrary for Situation 1 , while the approximate TEDM 2 should be positive. For both cases, the absolute value matters. In Figure 3.12, we use symbols instead of solid lines to plot data for clarity. Hund's case (a) potentials 3 3 Π and 2 3 ∆ cross each other at 8.84 a.u. and 10.3 a.u. distance, resulting in such instantaneous behavior of proportions and Hund's case (c) TEDMs. There is no necessity to consider Ω = 3 case because its spin-orbit matrix has only one matrix element 3 ∆ which is uncoupled with other molecular states with spin-orbit interactions and does not fit the selection rules regarding the entrance 1 , 3Σ + states.

Hund's case (c) photodissociation cross section

To obtain the photodissociation cross section in Hund's case (c) basis, we take equations 3.11 and 3.12 with Hund's case (a) potentials for both entrance and exit channel as well as Hund's case (c) TEDMs/TEDMs 2 considering a small range of transition energies around 1064 nm. The results are represented in Figures 3. 13-3.15. The analysis of the cross sections in Hund's case (a) basis indicates that the transitions 2 1 Σ + -4 1 Σ + is the dominant dissociation for RbBa + . Since 4 1 Σ + belongs to the spin-orbit matrix of Ω = 0 + , the top two panels in Figure 3.13 are actually the respective weight/proportion of 2 1 Σ + -4 1 Σ + cross sections in Hund's case (a) basis relying on the fine structure asymptotes Rb(5s) + Ba + (5D 3/2,5/2 ) in Hund's case (c) basis. As expected, the magnitude of the scattering cross section of the fine structure satisfies the ratio 3:5 of the center of gravity of the fine structure energy level (D 3/2 , D 5/2 ), and the sum of Hund's case (c) cross sections of Ω = 0 + is equal to the value of the transitions 2 1 Σ + -4 1 Σ + in Hund's case (a) basis. Hund's case (c) photodissociation cross sections of the remaining Ω fulfill both of the conclusions. And once again, the odd ripples present in the cross section are from the convergence issue.

Conclusion

In summary, to characterize the photodissociation of the weakly-bound molecular ion RbBa + created by three-body collisions in the hybrid trap, we start by calculating the photodissociation cross section in Hund's case (a) frame. The pathway 2 1 Σ + -4 1 Σ + is identified as the primary destructive transition. For this photodissociation cross section, the discrepancy of a factor 13.5 between theory and experiment is qualitatively explained by pushing the repulsive wall of the exit channle 4 1 Σ + . Then, we included the spin-orbit interaction in the photodissociation model. Our simulations reveal that the HOT Ba + ions generated by the photodissociation of RbBa + would be distributed over the fine structure exit channels Ba + (D 5/2 ) and Ba + (D 3/2 ) with a kinetic energy difference ∼1120 K. This huge thermal difference allows experimentalists to distinguish HOT Ba + at the fine structure level to compare to our theoretical predications.

Single Ba + recycling reaction in a Paul trap

Introduction

The state-of-the-art quantum logic spectroscopy technique [START_REF] Schmidt | Spectroscopy using quantum logic[END_REF] has recently been used for tremendous applications [START_REF] Kienzler | Quantum logic spectroscopy with ions in thermal motion[END_REF][START_REF] Brewer | 27 Al + quantum-logic clock with a systematic uncertainty below 10 -18[END_REF][START_REF] Schulte | Quantum algorithmic readout in multi-ion clocks[END_REF][START_REF] Hansen | Single-ion recycling reactions[END_REF][START_REF] Rosenband | Observation of the 1 S 0 → 3 P 0 clock transition in 27 Al +[END_REF] in ion traps, such as optical clock, internal-atomic-state operations, multiqubit entangling gates and so on. The essential principle [START_REF] Schmidt | Spectroscopy using quantum logic[END_REF] is to trap an auxiliary logic ion and a target spectroscopy ion simultaneously. Due to the lack of a suitable deep-ultraviolet laser to implement the classical Doppler cooling in the spectroscopy ion, we can map the coherent transfer of the spectroscopy ion's internal states acting on the logic ion. To achieve this, one of the fundamental requirements is to make sure the trapped ions are immune to environmental perturbations in the ultra-high vacuum chamber. Unfortunately, the presence of background molecular gas in the vacuum cavity is unavoidable, like H 2 , O 2 , H 2 O, etc, then unexpected collision events or chemical reactions may occur. For instance, cold hydride molecular ion products are identified in previous single-ion Mg + [START_REF] Jørgensen | Intensity and wavelength control of a single molecule reaction: Simulation of photodissociation of cold-trapped MgH +[END_REF][START_REF] Mølhave | Formation of translationally cold MgH + and MgD + molecules in an ion trap[END_REF] and Ca + [START_REF] Hansen | Single-ion recycling reactions[END_REF] experiments. In this respect, photodissociation is considered to be the ideal solution to regenerate the ions with a well-defined laser.

In this section, we focus on a recycling reaction scheme proposed to assist the experimental team of Prof. Michael Drewsen (Aarhus, Denmark) to successfully prepare a single long-life Ba + ion, where the preliminary results of this experiment have proven the existence of BaH + . The schematic flow shown in Figure 3.16 consists of three steps: hydrogenation reaction, photodissociation, Doppler laser cooling cycle. Since the hydrogenation reactions happen spontaneously and the Doppler laser cooling is the initial experimental design, this work is to 3.2. SINGLE BA + RECYCLING REACTION IN A PAUL TRAP build a photodissociation model to predict a laser frequency range with a high efficiency to eliminate HBa + . A successful laser can guarantee the purity of Ba + ion, and protect against detrimental excessive micromotion [START_REF] Schmid | Dynamics of a cold trapped ion in a bose-einstein condensate[END_REF] of ions as an additional benefit. As long as the stable Ba + can be captured for a long enough time, the experimentalists would continue the research on the single-ion optical clock.

Characteristics of HBa +

Before discussing the molecular dissociation, it is necessary to determine the initial quantum state of HBa + . At first, my colleagues calculate the potential energy curves with symmetries Σ + and Π for HBa + , which are presented in the top panel of Figure 3.17. For the ground dissociation limit of Ba + (6s)+H, 1 1 Σ + possesses a very deep well-depth with 26 vibrational states (J = 0) while 1 3 Σ + is almost purely repulsive with 4 vibrational states (J = 0). As for the metastable Ba + (5d)+H limit, another nearly purely repulsive molecular state 2 3 Σ + has 3 vibrational states (J = 0). It is worth clarifying that the surface calculations and the permanent electric dipole moment (PEDM) are consistent with those in a previous publication [START_REF] Aymar | The electronic structure of the alkaline-earthatom (Ca, Sr, Ba) hydride molecular ions[END_REF] by our group, additional calculations for the short-range (2-3 a.u.) have been achieved here. The selected TEDMs are the newly displayed results for the photodissociation. Since the atomic transition s-p is allowed, TEDMs between Ba + (6s)+H and Ba + (6p)+H limits are non-zero at a large interatomic distance. As such it is the reverse for TEDMs of Ba + (6s)+H and Ba + (5d)+H limits.

Black body radiation

According to the experimental protocol, the likely molecular state of HBa + is 1 1 Σ + . Then, we can use black body radiation (BBR) theory and Einstein relations to predict the rovibrational levels populated and the lifetime of HBa + . At thermal equilibrium characterized by the temperature T , the Plank black body radiation law [START_REF] Planck | On the law of distribution of energy in the normal spectrum[END_REF] defines the energy density for a given frequency ω [START_REF] Fedorov | Accurate potential energy, dipole moment curves, and lifetimes of vibrational states of heteronuclear alkali dimers[END_REF],

ρ(ω) = 2ω 3 πc 3 1 ex p ω k B T -1 (3.15)
where ρ(ω) is the spectral energy density, a temperature T = 300 K is from the surrounding environment and N (ω) = 1/(ex p ω k B T -1) stands for the number of BBR photons. All quantities are in atomic units.

In terms of Einstein coefficients, the reaction rate Γ and the lifetime τ can be expressed as

τ -1 = ε v f <ε v i Γ spont-emmi + ε v f <ε v i Γ st imu-emmi + ε v i <ε v f Γ st imu-a bsor , ( 3
Γ spont = 4ω 3 3c 3 | < J f , v f |D(r)|J i , v i > | 2 , ( 3
.17)

Γ st imu = Γ spont N (ω) = 4ω 3 3c 3 | < J f , v f |D(r)|J i , v i > | 2 1 ex p ω k B T -1 , (3.18)
where ω = |ε v fε v i | is the transition frequency between vibrational v i and v f levels of the ground state, D(R) is the r-dependent PEDM. These quantities are in the atomic units. In the top panel of Figure 3.18, the reaction rate of each dynamic proves that the spontaneous emission completely dominates the black body radiation over the vibrational states of 1 1 Σ + . The bottom panel displays the lifetime of each vibrational level. We can see that the lifetime of the ground vibrational state (v = 0) is 4.06 seconds which is much longer than the others. In order to obtain pure Ba + , that is, we shall seek a laser with the ability to efficiently and completely remove HBa + with the lowest energy in the real ground state (v = 0, J = 0) of 1 1 Σ + in the setup.

Photodissociation of HBa +

The goal is to utilize photodissociation methodology to efficiently decompose any HBa + molecule in the experimental chamber. Plus, there are two available lasers at 396.8 nm and 271.7 nm in the experimental setup. Therefore, our objective is to support the experimentalists in setting up the experimental apparatus, via determining the appropriate laser, and estimate its most optimal intensity for the photodissociation using Eqn 3.3. Figure 3.19 shows the photodissociation cross section induced by the 396.8 nm and 271.7 nm lasers as a function of the vibrational state of 1 1 Σ + for four possible transitions:

1 1 Σ + → 2 1 Σ + , 1 1 Σ + → 3 1 Σ + , 1 1 Σ + → 1 1 Π, 1 1 Σ + → 2 1 Π.
Overall, the cross sections of 1 1 Σ + → 3 1 Σ + and 1 1 Σ + → 2 1 Π transitions are relatively large, which usually means that a low-powered laser is efficient enough, but molecules with the lower vibrational states (v ≤ 5) are immune to the mechanism of photodissociation no matter which one the laser is. We can also notice that 396.8 nm laser cannot dissociate the molecule with v = 0 for 1 1 Σ + → 2 1 Σ + and 1 1 Σ + → 1 1 Π transitions, while 396.8 nm laser can destory the molecular ion with all vibrational states. Since both 2 1 Σ + and 1 1 Π belong to Ba + (6p)+H limit, hot Ba + with a temperature 13814.8 K (9602.98 cm -1 ) should be generated by photodissociating an HBa + with v = 0 of 1 1 Σ + .

It is important to emphasize that the sudden decrease of the cross section occuring at v = 6 of the panel of 1 1 Σ + → 3 1 Σ + is due to the fact that the phase of the wave function of the vibrational state v=6 coincidentally provides the minimum value of the cross section. For the sake of clarity, we produce the cross section as a function of the transition energy regarding the neighborhood vibrational states v = 5, 6, 7 in Figure 3.20. We can see that the transition energy of 271.7 nm laser right locates at the bottom of cross section curve for v = 6, Figure 3.18: Top: The spontaneous emission rate, the stimulated absorption rate and the stimulated emission rate as a function of the vibrational quantum number of 1 1 Σ + at 300 K room temperature. Bottom: Black body radiation lifetime as a function of the vibrational quantum number of 1 1 Σ + at 300 K room temperature. whereas the behavior of curves for v = 5, 7 is a slow linear variation. This is a similar situation to what we encountered when discussing the cross section of RbBa + previously.

To efficiently dissociate the HBa + molecule, we need to evaluate the intensity of the laser depending on the cross section of 1 1 Σ + → 2 1 Σ + and 1 1 Σ + → 1 1 Π transitions for vibrational state v = 0. The photodissociation rate is proportional to the photodissociation cross section and the intensity of light. The rate is calculated as

Γ pd = σ pd I hν , (3.19)
where σ pd is the state-to-state cross section, I is the intensity of a laser, and ν is the frequency. Figure 3.21 displays the rate as a function of the intensity of 271.7 nm laser. We discover that once the intensity is above W /cm 2 , then an event per second can occur due to 1 1 Σ + → 2 1 Σ + transition.

Conclusion

In this chapter, we provide theoretical support for the experiments to prepare a single Ba + in a Paul trap. We consider a scheme with a three-step recycling reaction including hydrogenation reaction, photodissociation, and Doppler laser cooling, where photodissociation is considered as the key process. Through the black body radiation model, when the molecular ion HBa + generated by the hydrogenation process is distributed in the absolute ground state (1 1 Σ + , v = 0, J = 0), a long lifetime (4.06 s) is predicted. In our model, we consider that the chosen laser requires the ability to effectively remove molecules in the absolute ground state. Given the available 396.8 nm and 271.7 nm lasers in the experimental setup, we predict that if the intensity of 271.7 nm laser can be adjusted to 100 W /cm 2 then a photodissociation event will occur every second via the 1 1 Σ + → 2 1 Σ + transition for HBa + at the absolute ground state. At last, we hope that our theoretical predictions can be helpful for the ongoing experiments.

Introduction

In the research field of ultracold dilute matter (namely where the kinetic energy of the particles is well below k B ×1 mK and density does not exceed 10 15 cm -3 , k B being the Boltzmann constant), the development of so-called hybrid traps, which merge ultracold atoms and atomic ions opened fascinating topics: the premises of cold chemistry with the formation of cold molecular ions, the dynamics of ultracold inelastic collisions between ground-state atoms and metastable excitedstate ions involving huge internal energy, the three-body collisions between an ion and two atoms, the emergence of ultracold ion-atom collisions in the quantum regime with the observation of spin-exchange. The advantage of such atom-ion experiment is that the characteristics of charged particles are reliable fingerprints to distinguish product branching ratios. In all the corresponding experimental situations, the unavoidable presence of lasers to cool and trap the species has been proven to have huge consequences on the hybrid trap dynamics. For instance, the detection of the above-mentioned inelastic collisions originates from the creation of excited ions during the laser-cooling process.

INTRODUCTION

Cold molecular ions are also expected to be mostly destroyed by these lasers. Up to now, there is only a single group which reported the direct observation of cold molecular ions [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 ion-neutral reactive processes in the cold regime: Radiative molecule formation versus charge exchange[END_REF][START_REF] Hall | Millikelvin reactive collisions between sympathetically cooled molecular ions and laser-cooled atoms in an ionatom hybrid trap[END_REF][START_REF] Willitsch | Coulomb-crystallised molecular ions in traps: methods, applications, prospects[END_REF], namely involving a magnetooptical trap (MOT) of Rb atoms and a Paul trap of Ca + or Ba + ions. Three main processes have been identified and investigated when Rb and Ca + collide in their ground state: non-radiative charge exchange (NRCE), radiative charge exchange (RCE), and radiative association (RA) leading to the formation of molecular ions. The former one dominates the dynamics. Accordingly, by running the Paul trap as a mass spectrometer, four ionic species are detected: Ca + , Rb + , RbCa + , and Rb + 2 . The latter species was considered as a possible outcome of RbCa + surrounded by Rb atoms.

In order to clarify to which extent the RbCa + species is peculiar compared to other similar species, we present here an extension of our previous study devoted to radiative association of laser-cooled Rb atoms and alkaline-earth ions [START_REF] Da Silva Jr | Formation of molecular ions by radiative association of cold trapped atoms and ions[END_REF], within a framework similar to the recent joint experimental and theoretical work aiming at modeling the life and death of a RbBa + molecular ion created in an hybrid trap [START_REF] Mohammadi | Life and death of a cold BaRb + molecule inside an ultracold cloud of Rb atoms[END_REF]. From the very beginning of such studies [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: Radiative molecule formation versus charge exchange[END_REF], it has been acknowledged that the observed ion losses can be properly interpreted only by taking into account the lasers present in any of these experiments. Such lasers could indeed photodissociate the created molecular ions through highly-excited molecular states. In other words, the previous study [START_REF] Da Silva Jr | Formation of molecular ions by radiative association of cold trapped atoms and ions[END_REF], as well as the more recent one of Ref. [START_REF] Zrafi | Ab initio electronic structure and prospects for the formation of ultracold Calcium-alkali-metal-atom molecular ions[END_REF], are not sufficient to assess the possible detection of cold molecular ions in hybrid traps. This requires an accurate knowledge of these highly-excited states, namely their potential energy curves (PECs) and the relevant transition dipole moments (TDMs).

In the current RbCa + hybrid system, several fruitful results of combined theoretical and experimental investigations have been reported [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 ion-neutral reactive processes in the cold regime: Radiative molecule formation versus charge exchange[END_REF][START_REF] Willitsch | Coulomb-crystallised molecular ions in traps: methods, applications, prospects[END_REF]. In particular, three main competing processes have been identified and investigated in the excited reaction channels, i.e., radiative association (RA), radiative charge transfer, and non-radiative charge exchange (NRCE). The resonant-excitation spectrum of the ion trap [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: Radiative molecule formation versus charge exchange[END_REF] exhibits the evidence of four distinct resultant ions: Ca + , Rb + , RbCa + and Rb + 2 . Here, Ca + ions can be understood as a legacy of the initial ions, and Rb + ions come from the two-close real crossings between 2 1 Σ + and 1 3 Π states or radiative charge transfer dynamics. Whereas, a likely reactive pathway to synthesize Rb + 2 ions is through a three-body collision: RbCa + + Rb → Rb + 2 + Ca, where RbCa + ions are formed in course of radiative association or an association reaction by pure collisions at the ground Rb + + Ca(4s 2 , 1 S) reaction channel.

However the picture is not yet complete to describe the dynamics of Rb-Ca + hybrid trap. The influence of the lasers on resultant species is not taken into account. This inspires us with the enthusiasm for in-depth research and exploration. In terms of these experimental conditions [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 ion-neutral reactive processes in the cold regime: Radiative molecule formation versus charge exchange[END_REF], three active lasers are implemented simultaneously to cool Ca + and Rb in the hybrid trap. Ca + coulomb crystals are confined by 397 nm and 866 nm laser beams cycling on the (4s) 2 S 1/2 → (4p) 2 P 1/2 and (3d) 2 D 3/2 → (4p) 2 P 1/2 transitions, respectively. Six intersecting laser beams with the wavelength 780 nm in MOT are set up for the cooling transition (5s) 2 S 1/2 → (5p) 2 P 3/2 for Rb atomic gas.

The current work mainly studies the loss of RbCa + ions caused by the PD and the loss of Ca + ions induced by the NRCE to interpret the ion losses. To achieve these goals, we build up a RA model to characterize the vibrational distribution of the trapped RbCa + varying the temperature range from microkelvin to kelvin. Subsequently, the PD dynamic is modeled and discussed for various laser wavelengths: 866nm,780 nm, and 397 nm. The results manifest that only 397 nm can trigger the dissociation reaction. In addition, we perform the Black body radiation (BBR) calculation to explain the competing balance of spontaneous emission, stimulated emission, and absorption related to the redistribution of vibrational states of RbCa + . Finally, the NRCE is discussed within a new model including spin-orbit couplings.

Ab initio calculations

The potential energy curves (PECs), transition electric dipole moments (TEDMs), and permanent electric dipole moments (PEDMs) for numerous electronic states are calculated following the method described in Chapter 2. The parameters used are updated from the previous publication [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: Radiative molecule formation versus charge exchange[END_REF][START_REF] Aymar | The electronic structure of the alkaline-earthatom (Ca, Sr, Ba) hydride molecular ions[END_REF][START_REF] Guérout | Ground state of the polar alkalimetal-atom-strontium molecules: Potential energy curve and permanent dipole moment[END_REF] in our group. The optimized basis set for Ca + is composed a large set of Gaussian orbitals (9s,8p,8d,2 f ). The cut-off parameters of the core-polarization potential are, in atomic units, (ρ C a + s = 1.89095, ρ C a + p = 1.6528, ρ C a + d = 1.827734), and the ionic core polarizability is α C a 2+ = 3.522a 3 0 [START_REF] Coker | Empirical free-ion polarizabilities of the alkali metal, alkaline earth metal, and halide ions[END_REF]. The basis set of Rb is extended with an additional Gaussian f-orbital, compared to Ref. [START_REF] Guérout | Ground state of the polar alkalimetal-atom-strontium molecules: Potential energy curve and permanent dipole moment[END_REF]. Figure 4.1 displays the computed PECs for 1,3 Σ + and 1,3 Π electronic states up to 8 th dissociation limit, and the energies of dissociation limits of RbCa + are listed in Table 4.1. Our results are found very close to the experimental measurements in the NIST database for Rb(5s)+Ca + limit, while the rest always exhibits a discrepancy to some extent. The comparison with the theoretical literature [START_REF] Zrafi | Ab initio electronic structure and prospects for the formation of ultracold Calcium-alkali-metal-atom molecular ions[END_REF] using the same method demonstrates that we have achieved a better agreement with the experimental data.

The extracted spectroscopic constants are summarized in Table 4.2. The harmonic vibrational frequency ω e and the anharmonic correction ω e χ e are computed in the Taylor series [START_REF] Fedorov | Accurate potential energy, dipole moment curves, and lifetimes of vibrational states of heteronuclear alkali dimers[END_REF]

E v = ω e v + 1 2 -ω e χ e v + 1 2 2 , (4.1)
where E v=0,1,2 are the three lowest vibration energies, so that we obtain [START_REF] Fedorov | Accurate potential energy, dipole moment curves, and lifetimes of vibrational states of heteronuclear alkali dimers[END_REF]]

ω e = 3E 1 -2E 0 -E 2 2ω e χ e = 2E 1 -E 0 -E 2 .
(4.2) Figure 4.1: Hund case a PECs of RbCa + for different molecular symmetries, as a function of interatomic distance. The downwards dashed vertical arrow states the RA from initial channel 2 1 Σ + to final 1 1 Σ + . Upwards solid vertical arrows represent the energies of respective lasers, 397 nm, 780 nm, and 866 nm, from the last bound level of the ground state 1 1 Σ + towards five exit channels (2 1 Σ + , 3 1 Σ + , 4 1 Σ + , 1 1 Π, 2 1 Π) for the PD.

RESULTS AND DISCUSSION

91 Table 4.1: Dissociation limits of RbCa + , the energy difference

E d i f f . = E N IST - E T heo. . Asymptote E T heo. (cm -1 ) E N IS T (cm -1 ) E d i f f . (cm -1 )

This work

Ref. [START_REF] Zrafi | Ab initio electronic structure and prospects for the formation of ultracold Calcium-alkali-metal-atom molecular ions[END_REF] This work Ref. [START_REF] Zrafi | Ab initio electronic structure and prospects for the formation of ultracold Calcium-alkali-metal-atom molecular ions[END_REF] Rb + +Ca(4s The rotational constant is

B e = h 8π 2 cµR 2 e (4.3)
where R e is the equilibrium distance, µ is the reduced mass of the diatomic molecule, µR 2 e is the moment of inertia. In Figure 4.2, PEDMs are calculated to the center of mass. We observe that PEDMs vary over short distances, but linearly diverge with the same slope at a large distance. In Figure 4.3, TEDMs at the large distance has two convergence characteristics, either degenerating to zero or approaching some constant, due to the forbidden/allowed atomic transition. The results of both PEDMs and TEDMs are in good agreement with the Ref. [START_REF] Zrafi | Ab initio electronic structure and prospects for the formation of ultracold Calcium-alkali-metal-atom molecular ions[END_REF]. These data will be included in the following models for the RA, the PD, and the BBR.

Figure 4.4 shows the spin-orbit couplings (SOCs) for Ω = 0 +/-, 1, 2 regarding to Rb(5s)+Ca + (4s) and Rb + +Ca(4s4p, 3 P) dissociation limits. The method was presented in Chapter 2. The SOCs curves associated with 2 3 Σ + show an abrupt change due to an avoided crossing between 2 3 Σ + and 3 3 Σ + at around 12 a.u. Moreover, asymptotic values of the SOCs curves at long-range satisfy the splitting of the fine structure of Ca + ( 3 P). These data will be used later in the NRCE model. 4.2: The spectroscopic constants of RbCa + . Equilibrium interatomic distances R e , well depths D e , transition energies T e , harmonic constants ω e , anharmonic vibrational frequencies ω e χ e , and rotational constants B e .

Results and discussion

Asymptote

States D e (cm -1 ) R e (cm -1 ) B e (cm -1 ) ω e (cm -1 ) ω e χ e (cm -1 ) T e (cm -1 ) Ref. Rb + +Ca(4s

Vibrational distribution of the ground state RbCa +

The previous works [START_REF] Zrafi | Ab initio electronic structure and prospects for the formation of ultracold Calcium-alkali-metal-atom molecular ions[END_REF][START_REF] Da Silva Jr | Formation of molecular ions by radiative association of cold trapped atoms and ions[END_REF] reported that the ground state RbCa + is formed by the RA from the entrance channel 2 1 Σ + , which is indicated by the downward dashed arrow in Figure 4.1. Both works yielded the same order of magnitude of 10 -15 cm 2 with the initial collision temperature 1µK for the RA cross section. The vibrational distribution of ground state RbCa + at the collision temperature 1 mK is discussed additionally in [START_REF] Da Silva Jr | Formation of molecular ions by radiative association of cold trapped atoms and ions[END_REF]. To determine the effect of the collision temperature on the vibrational levels, therefore, we calculate the vibrational distribution of the ground state RbCa + generated by RA. The fraction Γ of population of a given vibrational level v X at 1 1 Σ + is calculated by

Γ v X ,ε i ,J = ω J v X ,ε i J | < J , v X |D(R)|ε i , J > | 2 v max v=0 ω J v,ε i J | < J , v|D(R)|ε i , J > | 2 (4.4)
All the quantities above are in atomic units. The initial ket |ε i , J > with energy

ε i = E -V i (R → ∞)
and associated wavenumber k i = 2µε i represents a partial-wave component J of an energy-normalized continuum wavefunction of the two nuclei, where V i is the energy of the dissociation limit of 2 1 Σ + channel. The final ket |v, J > is a bound rovibrational level of 1 1 Σ + channel. D f i is the transition dipole moment between the initial 2 1 Σ + and final 1 1 Σ + channels. The summation is limited to the uppermost vibrational level v ma x of 1 1 Σ + channel. ħ hω J v X ,ε i J is the energy of the emitted photon. J is the total angular momentum quantum number for 2 1 Σ + channel, J = J ± 1 are the two possible allowed total angular momentum quantum numbers for 1 1 Σ + channel.

For cases of J = 1, J = 0 and ε i = 2.4 µK, 5.8 mK, or 2.8 K, the vibrational distribution of the ground state RbCa + created by RA is presented in Figure 4.5. It can be noticed that RbCa + are preferably formed in deeply-bound molecules around v = 21, which is consistent with previous calculations [START_REF] Zrafi | Ab initio electronic structure and prospects for the formation of ultracold Calcium-alkali-metal-atom molecular ions[END_REF][START_REF] Da Silva Jr | Formation of molecular ions by radiative association of cold trapped atoms and ions[END_REF]. As expected, we find that the biggest probability of vibrational distribution at v = 21 is always the favorite in three panels, while the amplitude of the vibrational distribution decreases around v = 21 when the collision energy grows. This is due to the fact that the selected collision temperatures remain very small comparing the energy of the depth of 2 1 Σ + , the continuum wave function does not change much at the turning point. Hence, we propose a hypothesis that the molecular distribution of RbCa + in the ground state is independent of the practical collision temperatures in the experiment. More importantly, we discover that the deeper bound molecular ions with v ≤ 15 are unlikely formed within RA, in other words, 96%, molecular ions are mainly distributed in v = 16 -60 of the ground state 1 1 Σ + .

Photodissociation

In order to evaluate the effects of light fields on RbCa + , three lasers (866 nm, Figure 4.5: Calculated vibrational distributions of RbCa + as a function of binding energies of vibrational levels of the 1 1 Σ + ground state, created by radiative association. RbCa + is generated at 1 1 Σ + state by the RA from the initial 2 1 Σ + channel with selected collision temperatures 2.4 µK, 5.8 mK and 2.8 K. The red rule indicates the corresponding quantum numbers of the vibrational levels of 1 1 Σ + ground state. 100 4.3. RESULTS AND DISCUSSION mechanism without the fine structure. We start with RbCa + in the ground electronic state Λ = 1 1 Σ + to the scattered electronic states Λ , then the absorption state-to-state cross section is obtained by

σ P D (ω) = 4π 2 3c ω| < Λ , J , ε |D Λ ,Λ (R)|Λ, J, v > | 2 . ( 4.5)
Here, all quantities are expressed in atomic units. ω and D Λ ,Λ (R) are the transition frequency and the TEDM, respectively. The energy of the continuum state of the exit channel is ε . c is the speed of light. The bound energy of a vibrational level v is defined by E v . For a given energy E l aser = ħ hω of one photon,

ε = E laser -|E e x i t -E ent r ance | -|E v |
, where E ex i t and E ent r ance are the energies of the dissociation limits of the entrance and exit channels. J and J denote the total angular momentum quantum number of the entrance and exit channels. A recent paper [START_REF] Mohammadi | Life and death of a cold BaRb + molecule inside an ultracold cloud of Rb atoms[END_REF] reports that the total PD cross section is independent of J by summing over the contributions of R, P, Q branches. Hence, we only perform calculations for the transition J = 0 to J = 1.

In Figure 4.1, the up-arrows indicate the maximum upper limit of photodissociation for the most weakly bound molecular ions. As we have observed, 780 nm and 866 nm can not provide enough energy to dissociate the ground state RbCa + in any excited singlet states. As for the 397 nm laser, there are five possibly channels identified by the selection rule (∆Λ = 0, ±1, ∆S = 0): 2 1 Σ + , 3 1 Σ + , 4 1 Σ + , 1 1 Π and 2 1 Π, where the relevant dissociated channels are highlighted in Figure 4.1. By substituting TEDMs into the above formula, we can straightforwardly obtain the cross section as a function of the vibrational binding energy of the initial channel 1 1 Σ + for each exit channel. The calculated results are represented in Figure 4.6. It is worth noting that the dominant exit channels for the dissociation are 4 1 Σ + and 2 1 Π which could provide the cross section which are larger by four orders magnitude than for the 3 1 Σ + channel, and by seven orders of magnitude than for the 1 1 Π channel, and by nine orders of magnitude than for the 2 1 Σ + channel.

Additionally, we are aware that the PD occurs only for the ground state RbCa + within the binding energy range (-1993 cm -1 -0) or the vibrational range (v = 28 -142) in terms of exit channels 4 1 Σ + or 2 1 Π. Herein, we put forward a threshold v = 28 for the PD which has been marked by red vertical short lines in Figure 4.6. According to this threshold, 41% molecular ions would be dissociated by 397 nm, and 59% molecular ions are left in terms of a given amount of ground state molecular ions. The corresponding PD rates are calculated by σ P D I hν and displayed in Figure 4.7, where the intensity I of 397 nm beam is 400 mW/cm 2 [START_REF] Hall | Cold ion-neutral reactions[END_REF]. We can see indeed the PD rates for the exit channels 4 1 Σ + or 2 1 Π are considerable, with about one hundred dissociation events per second.

Black body radiation

As a matter of fact, the vibrational distribution of RbCa + at the ground state can be rearranged by BBR. Once the deeply bound RbCa + is excited by the stimulated absorption over PD threshold, molecule can further be dissociated by 397 nm. We consider this process is BBR-induced PD. Alternatively, RbCa + may be relaxed beneath PD threshold by BBR emissions before the photodissociation. Herein, we discuss the role of BBR of 1 1 Σ + state at the surrounding environment T = 300K. The BBR-induced dynamics include stimulated absorption and emission which are described by Einstein coefficient B. The Einstein coefficient A describes the spontaneous emission. The formulas used here are Eqn [3.16], Eqn [3.17] and Eqn [3.18] which are demonstrated in Chapter 2. The calculated rates for spontaneous emission, stimulated emission, and stimulated absorption for J = 0 → J = 1 are displayed in Figure 4.8. Among these processes, only the stimulated absorption has the feature to excite RbCa + over the PD threshold. The rate of stimulated radiation (blue) is the sum of stimulated processes, which is much faster than the spontaneous emission (red). The rates of stimulated emission (green) and absorption (orange) are nearly equal for the range of vibrational levels less than 92. We can see that the stimulate absorption and emission are basically cancelled out reagrding the vibrational levels below the PD threshold v = 28. As the vibrational level increases (v > 92), the absorption rate decreases rapidly comparing with other rates, and even less than the spontaneous emission rate at v = 118.

Since the associated RbCa + is mainly allocated in the vibrational range of 16-60, we estimated the level-to-level BBR rate in Figure 4.9 for the initial levels v = 20, 30, 40, 50, 60. It is clear that the transition takes place only in the neighbouring vibrational levels. Comparing the order of magnitude of PD rate, BBR-induced PD/relaxation will take more longer time. We boldly assume that the effect of BBR in this experiment is not significant.

Non-radiative charge exchange

In this section, we investigate the NRCE dynamic between the entrance Rb(5s) + Ca + (4s) and the exit Rb + + Ca( 3 P) asymptotes taking into account the spin-orbit couplings (SOCs). As been reported [START_REF] Tacconi | Computing chargeexchange cross sections for Ca + collisions with Rb at low and ultralow energies[END_REF][START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: Radiative molecule formation versus charge exchange[END_REF][START_REF] Belyaev | Resonances in Ca + +Rb nonadiabatic collisions at ultralow energies[END_REF], the non-radiative collision with the two crossings induced by SOCs between 2 1 Σ + and 1 1 Σ + results in a hight rate for NRCE yielding the loss of Ca + and the formation of Rb + . Although theoretical calculations [START_REF] Tacconi | Computing chargeexchange 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] agree well with the experimental values [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: Radiative molecule formation versus charge exchange[END_REF], Table 4.1 tells the difficulty to accurately calculate the dissociation limit of the potential energy curves in agreement with experimental values, especially for the Rb + + Ca( 3 P) limit. Thus, we are motivated to use a new set of PECs and SOCs, as well as a different theoretical model to explain NRCE and to compare with their results.

NRCE model

For the Rb(5s) + Ca + (4s), there is no SOC involved due to both particles are in their ground state. In Rb + + Ca( 3 P) limit, the good quantum number |Ω| could be 0 + , 0 -, 1. Eventually the total potential energy matrices of Rb(5s) + Ca + (4s) and Rb + + Ca( 3 P) can be built as follows:

H(Ω = 0 -) =   V (2 3 Σ + ) A 12 0 A 21 V (1 3 Π) -A 22 A 23 0 A 32 V (1 3 Σ + )   , (4.6)
H(Ω = 0 + ) = V (2 1 Σ + ) A 12 A 21 V (1 3 Π) -A 22 , (4.7)
H(|Ω| = 1) =   V (2 3 Σ + ) A 12 0 A 21 V (1 3 Π) A 23 0 A 32 V (1 3 Σ + )   . ( 4.8)
The r-dependent spin-orbit interactions (A, A , A ) are plotted in Figure 4.4. (The long-range spin-orbit coupling matrix defined by its spin-orbit splitting constants is listed in Appendix A.) We find that only the matrix of Ω = 0 + includes initial 2 1 Σ + and final 1 3 Π of interest, while the rest of Ω is not concerned with 2 1 Σ + . The following discussion will be carried out only on Ω = 0 + . The first panel in Figure 4.4 shows that the SOCs between 2 1 Σ + and 1 3 Π vary at an inner distance and vanish at a larger distance, while the amplitudes of SOC of 1 3 Π itself raises up rapidly and converge to a constant related to the splitting limit of 3 P 1 . It should be noticed that although we introduce the fine structure interactions into our matrix, 1 3 Σ + will not be taken into account in our calculations for the following reasons: 1 3 Σ + has no direct crossings with entrance state 2 1 Σ + , and the SOC of 2 1 Σ + and 1 3 Π is extremely smaller than the energy gap of 2 1 Σ + and 1 3 Π around crossings zone. Furthermore, the Coriolis coupling is not present between 2 1 Σ + and 1 3 Π due to the different spin multiplicity. A similar argument has been reported in References [START_REF] Tacconi | Computing chargeexchange 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]. Herein, we consider l-unchanging collisions, and each partial wave l is treated independently. ( 3 P 2 -3 P 0 ) at the infinity distance. In the inner range, the two real crossings in Hund's case a turn into two avoided crossings in Hund's case c, whose zoomed-in presentations are displayed in Figure 4.10(b) and 4.10(c), respectively. We notice that the positions of crossings and avoided crossings are shifted by the r-dependent SOC of 3 P 0 . In general, the classical Landau-Zener model is applicable to address single crossing cases. In this work, the cumulative effect of these two avoided crossings have to be taken into account. Therefore, we perform a rigorous quantum scattering two-channel calculation within the close-coupling method [START_REF] Johnson | The multichannel log derivative method for scattering calculations[END_REF] (See Chapter 2 for details) to estimate the transition probability and the reaction rate for NRCE dynamics. The inelastic cross section for the scattering NRCE process is computed from the 4.3. RESULTS AND DISCUSSION 107 off-diagonal elements S i f of S matrix as a sum over partial waves l,

σ N RC E (E) = p πħ h 2 2µE ∞ l=0 (2l + 1)|S i f | 2 (4.9)
where p = 1 4 is the statistical weight of the initial state, E is the collision energy for the entrance channel and µ is the reduced mass of the system. The NRCE transition probability is defined by

|S i f | 2 .
The non-thermalized rate coefficient is then

K N RC E (E) = σ N RC E (E) × v = σ N RC E (E) × 2E µ , (4.10)
where v is the entrance velocity. The thermalized rate coefficient involves Maxwell-Boltzmann distribution,

K N RC E (T e f f ) = 2 π(k B T e f f . ) 3/2 ∞ 0 K N RC E (E) Ee -E/k B T e f f . d E, ( 4.11)
where the effective temperature is evaluated as

T e f f = m Rb T C a + + m C a + T Rb m Rb + m C a + . In
this experiment the micromotion of ion cloud and the associated radiofrequency leads to a higher temperature for T C a + than for T Rb in MOT. Then, the effective temperature is simplified as T e f f . m Rb T C a + m Rb + m C a + = 0.685T C a + .

NRCE transition probability

As illustrated in Figure 4.11, we notice that the transition probability of the NRCE approaches a constant value of 0.006 at low collision temperatures and low partial waves, which is compatible with the theoretical prediction 0.007 of [START_REF] Belyaev | Resonances in Ca + +Rb nonadiabatic collisions at ultralow energies[END_REF]. This also exhibits the phenomenon of the phase-locking which will be discussed later. The sudden increase in amplitude due to shape resonance suggests that for given collision energy, a single partial wave contributes significantly to the total cross section of the NRCE.

NRCE cross section

The cross section is exhibited as a function of an effective temperature from microkelvin to kelvin in Figure 4.12(a). To yield converged cross sections, the adequate partial wave numbers are taken for each collision energy, for instance, 5, 6, 10, 15, 25, 41, and 122 partial waves are considered respectively for 10 -6 K, 10 -4 K, 10 -3 K, 10 -2 K, 10 -1 K, 1K and 10K. The Boltzmann thermalized and non-thermalized rates are represented in Figure 4.12(b). The thermalized rate removes the resonances of the non-thermalized rate reflecting the rate is roughly independent of collision energy in the sub-kelvin range. We can see that our results are in good agreement with experimental and theoretical values. Since the SOC of the exit channel 1 3 Π is included in our scattering model, furthermore, the amplitude and profile of the averaged rate of this work are even closer to experiments [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] than in reference [START_REF] Tacconi | Computing chargeexchange cross sections for Ca + collisions with Rb at low and ultralow energies[END_REF].

Comparison with Ref. [145]

To understand the differences between theoretical models, a detailed comparison between the results of this work and Ref. [START_REF] Belyaev | Resonances in Ca + +Rb nonadiabatic collisions at ultralow energies[END_REF] is done in Figure 4. [START_REF] Peik | Logical spectroscopy[END_REF]. We can see the dissociation energies of 1 3 Π and 2 1 Σ + are close, but the potential wells differ. Although the SOCs trends are quite different over the short-range < 10 au, the rest are similar both at the intersections and the long-range. The profile of the cross section demonstrates a high degree of consistency of the shape resonance. We also find that the cross section curve obtained in our calculation is shifted relatively downwards globally, that is due to the fact that the diagonal spin-orbit coupling of 1 3 Π is accurately taken into account. So that we can get closer to the experimental results. Although the discrepancy of the SOC is huge at the short-range, this good agreement also demonstrates that the short-range SOC does not play a role here. In Table 4.1, we notice that the dissociation energy of the exit charge exchange channel Rb + +Ca(4s4p, 3 P) is not being properly predicted theoretically. Therefore, we construct three sets of data containing the entrance channel 2 1 Σ + , the SOCs (Ω = 0 + ), and the exit channel 1 3 Π with three respective limits E Re f . [START_REF] Zrafi | Ab initio electronic structure and prospects for the formation of ultracold Calcium-alkali-metal-atom molecular ions[END_REF] > E N IS T > E T hiswor k to detect likely effects. The 2 1 Σ + and the SOCs (Ω = 0 + ) are the same in three sets of data, while we artificially make a vertical shift of the entire curve of 1 3 Π from the dissociation limit E T hiswor k to dissociation limits E N IS T and E Re f . We first adopt the Landau-Zener (LZ) model to calculate the transition probability. In terms of the shifted 1 3 Π, the positions of the real crossings are changed, but the shape and the spin-orbit couplings have very limited changes around the crossings. So that the classical LZ transition probability remains unchanged to some extent, the detail is recalled in Appendix B.

Then, we apply our developed NRCE scattering model to calculate the cross section with respect to the shifted PECs. Figure 4.14 shows significant differences of the order of magnitude of the cross section. At higher collision energies (over 10 -2 K), the baseline of the shape resonance is largely oscillatory, while the position of the shape resonance is similar. At the lower temperature domain (10 -6 -10 -3 K), the slopes of the baseline of shape resonances of three cases are similar. Whereas, the order of magnitude of the cross section of three cases is different. At higher collision temperatures, we find a rich oscillating behavior of the baseline of shape resonances. In terms of the energy E N IS T , in particular, a huge suppression (jumping) of the cross section (red) is appearing out from 10 -2 K to 1 K. Within such temperature interval, the magnitude of the scattering cross section is increasing with the increased dissociation energy of the outgoing channel. This can be explained by the overlap of wave functions. The incident wave function (the incident kinetic energy) and the spin-orbit couplings are the same for three independent calculations, while the magnitude of the outgoing wave function is increasing as the decreasing outgoing kinetic energy. However, at the lower temperatures, we believe that the quantum effect plays a key role. In the following, we will introduce two concepts to investigate this problem: Phase-locking and s-wave signature.

Phase-locking

In order to introduce the phase δ channel l of each partial wave l for a given channel, we can rewrite Eqn [4.9] as

σ N RC E (E) = p π k 2 ∞ l=0 (2l + 1) sin 2 (δ i l -δ f l ) (4.12)
where p = 1 4 and k = 2µE. i and f refer to the initial 2 1 Σ + and the final 1 3 Π channels, respectively. In the ultracold regime, the partial-dependent phase is varying slowly. When ∆δ l = δ i lδ f l is a l-independent constant, we term this phenomenon as phase-locking [START_REF] Sikorsky | Phase locking between different partial waves in atom-ion spin-exchange collisions[END_REF]. In Figure 4.11, for instance, the transition probability is proportional to sin 2 ∆δ l , and we get a constant up to l = 8 for the collision energies (57 µK -2.3 mK). Therefore, studying solely the phase of the s-wave is sufficient to understand the collision mechanism of charge exchange at ultra-low temperatures.

s-wave signature Combining Langevin theory and the phase of s-wave, Robin Cote [START_REF] Côté | Signature of the s-wave regime high above ultralow temperatures[END_REF] proposed a new theoretical concept of s-wave signature to explain the inelastic collisions at ultracold temperatures. With respect to the s-wave phase, the formula of the cross section is

σ N RC E (E) = p π k 2 + σ L (E) sin 2 ∆δ 0 (E) (4.13)
The first term π k 2 sin 2 ∆δ 0 (E) is the contribution of l = 0, the second term σ L (E) sin 2 ∆δ 0 (E) is for the upper limit of the partial wave l > 0 defined by the impact parameter b ma x , where ∆δ 0 (E) is the s-wave phase shift depending on the collision energy. In Figure 4.14 the semi-classical results calculated by the second term are able to well describe the evolutionary trends of the baseline shape resonances obtained by the full quantum scattering calculation, where sin 2 ∆δ 0 (E) is acting as a scaling factor on Langevin cross section.

Conclusion

We theoretically investigated the collisional dynamics of laser-cooled Rb atoms and Ca + ions in the context of the hybrid trap experiment of Ref. [START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: Radiative molecule formation versus charge exchange[END_REF][START_REF] Hall | Millikelvin reactive collisions between sympathetically cooled molecular ions and laser-cooled atoms in an ionatom hybrid trap[END_REF][START_REF] Hall | Light-assisted ion-neutral reactive processes in the cold regime: Radiative molecule formation versus charge exchange[END_REF], mostly leading to ion losses. Based on novel molecular data, we confirm that the non-radiative charge exchange, induced by spin-orbit interaction, is a dominant ion-loss process and obtain rates in agreement with experimental observations and a previous calculation [START_REF] Tacconi | Computing chargeexchange cross sections for Ca + collisions with Rb at low and ultralow energies[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]. Cold RbCa + ground-state molecular ions are created by radiative association, and we demonstrate that they are protected against photodissociation by black-body radiation and by the Ca + cooling laser at 397 nm. This study yields an interpretation of the direct observation of RbCa + ions in the experiment, in contrast with all other hybrid trap experiments using other species. Based on the pubished potential energy curves [START_REF] Zrafi | Ab initio electronic structure and prospects for the formation of ultracold Calcium-alkali-metal-atom molecular ions[END_REF], besides, we are aware that the photodissociation-induced ion loss of ground state molecular ion would be commonly present in other ultracold Calcium-ion-alkali-metal-atom systems due to the cooling lasers of Ca + . 116 5.2. EXPERIMENTAL OBSERVATIONS state-of-the-art technique allows experimentalists to precisely control the collision temperature and prepare the internal state of initial species as well as to detect the reactants and their internal state. The limitations of the theory, however, come to the fore. The investigation in a Li-Yb + combination [START_REF] Joger | Observation of collisions between cold Li atoms and Yb + ions[END_REF] reports that the impact of a f -shell electron in the trapped Yb + ion adds complexity for ab initio calculations, hindering precise predictions of charge exchange rates. The intricacy of that problem is coming from the difficulty of calculating the potential energy surface with many electrons and the spin-orbit couplings mixing up the fine-structure levels of P-orbit, D-orbit and F -orbit of Yb + . In the Rb-Sr + experiment, [START_REF] Ben-Shlomi | Direct observation of ultracold atom-ion excitation exchange[END_REF], the observation of the excitation exchange depending on the spin orientation of Sr + evidenced a notable rotation and spin-orbit coupling which remains as a legacy issue. At present we can see that an ideal ion-atom candidate with well-described internal couplings is needed to study the most fundamental quantum effects, to further achieve full quantum control in the excited state.

To achieve our goals, we report on a new study involving a single longlived cold 138 Ba + ( 2 D 3/2,5/2 ) in an ultracold 6 Li condensate. The high ion/atom mass ratio [START_REF] Willitsch | Very cool molecular ions[END_REF] enables ultracold researches around s-wave energy. The corresponding molecular data are displayed in Figures 5.3 below. The fine structure level 2 D 3/2,5/2 is well isolated from the adjacent electronic dissociation limits. Furthermore, there is only one single charge exchange channel 2 1 Σ + coupled by an effectively avoided crossing. The corresponding experiment is performed by P. Weckesser in the group of Prof. Dr. T. Schaetz (Freiburg University, Germany), where the trapped and Doppler-cooled Ba + single ion is in a Paul trap, while laser-cooled Li atoms are in an optical dipole trap (ODT). The competing non-radiative and quenching processes are observed simultaneously, i.e., the non-radiative charge exchange (NRCE) Li(2s)+Ba + (5d) → Li + + Ba( 1 S), the fine structure quenching (FSQ) Li(2s)+Ba + ( 2 D 5/2 ) → Li(2s)+Ba + ( 2 D 3/2 ), and the non-radiative quenching (NRQ) Li(2s)+Ba + (5d) → Li(2s)+Ba + (6s).

In this work, we propose a series of theoretical models of increasing complexity to interpret the experimental observations and identify the main interactions at play: Landau-Zener (LZ) model, Few-Channel Landau-Zener/Quantum Scattering (FCLZ/FCQS) model with Spin-Orbit Couplings (SOCs), and Multi-Channel Quantum Scattering (MCQS) model considering SOCs and rotational couplings.

Experimental observations

In the hybrid setup, a linear Paul trap with comparably large ion-electrode distance (∼ 9 mm) is combined to an all-in-one-spot ultracold atom apparatus. The ion setup has been described in more detail in [START_REF] Weckesser | Trapping, shaping, and isolating of an ion coulomb crystal via state-selective optical potentials[END_REF]. In short, a few Dopplercooled ions are loaded into the trap by ablation loading. Ions are then optically removed down to a single remaining ion Ba + , using a 532 nm optical dipole trap (VIS-ODT). Stray electric fields are compensated down to ∼ 3 mV m -1 by Figure 5.1: Scheme of the product state detection sequence (times are not on scale). At the end of the atom-ion interaction phase, the xODT is switched off. After a short expansion time (∼ 100µs typically) the atomic could is absorptionimaged at high magnetic field, which is then switched off. In the following sequence, the ion state is detected and classified as one of four possible outcomes. First, the Doppler cooling lasers are switched on for 1 s, followed by 300 ms of fluorescence imaging. An ion detected in this stage is classified as cold 6S 1/2 or 5D 3/2 . Secondly, a far-detuned cooling beam is switch on, followed by fluorescence detection. An ion appearing in this stage is classified as hot. Lastly, the 614.9 nm repumping laser is switched on to detect whether the ion is shelved in the 5D 5/2 -state. If the ion is not detected during any of the steps, the event is classified as a loss.

EXPERIMENTAL OBSERVATIONS

lowering the confinement of the ion and cancelling any observed displacement, using voltages applied to three pairs of control electrodes. The atomic part of the experiment is a conventional setup for the creation of an ultracold, spinpolarized gas of Li atoms. Atoms are loaded into a magneto-optical trap and, after a short compression stage, transferred to a crossed optical-dipole trap (xODT). The two xODT beams individually aligned to the position of the ion with two piezo-controlled mirrors. The overlap between the atomic cloud and the ion is independently verified by measuring the inelastic ion-loss probability for different ion displacements.

The Doppler-cooled ion is prepared in the 2 D 3/2 state by first switching off the 2 D 5/2 repumping laser for 50 ms while continuing to cool on the D 1 -line. To shelve it to the 2 D 5/2 state, the off-resonant VIS-ODT laser is used. Here, while Doppler cooling, the ion is illuminated with 532 nm light at ∼ 5 W until it is successfully shelved. The lifetime of the 2 D 3/2 ( 2 D 5/2 ) state is 80 s (32 s), orders of magnitude longer than the duration of the experiments. The ion is then shuttled axially and radially out of the trap center to allow for the preparation of the atomic cloud. Li atoms are loaded in a conventional magneto-optical trap (MOT) and, after a short compression phase, transferred to the xODT. Evaporative cooling of the atoms is performed at B ∼ 345 G to temperatures of 1-3 µK. After evaporation, a 15 s pulse of light resonant with the |m S = -1/2, m I = 1 >→ P 3/2 transition polarizes the atomic cloud in the |m S = -1/2, m I = 0 > state. Subsequently, the magnetic field is shifted to B = 293G for later detection of the atomic cloud. After the ion is shuttled back into the trap center, the initial reaction energy in the center-of-mass (COM) frame, including excess micromotion energy, is estimated on the order of 220 µK.

At the end of the interaction period, the xODT is switched off. After a short time of flight, the atomic cloud is absorption-imaged on a closed transition cycle during 15 µs at B = 293 G. The magnetic fields are then ramped down to B ∼4 G for Ba + state detection. The protocol for Ba + state detection is schematically depicted in Figure 5.1 and consists of three phases. In each phase the ion is illuminated by detection lasers for 1 s, followed by a 300 ms fluorescence image (CCD camera). In the first phase, only the Doppler cooling and 2 D 3/2 -repumping lasers are switched on. This will reveal ions that are either in the 2 D 5/2 or S 1/2 state with a temperature below ∼ 50 K, limited in temperature by the spatial overlap of the cooling beam with the ion. Next, a far detuned (δ ∼ 15 Γ nat ) Doppler cooling laser with a larger waist recools hot ions. Finally, the 2 D 5/2 repumping laser is applied to 'de-shelve' ions that are in the 2 D 5/2 state after the interaction. If the ion is not detected during any of the phases, the event is classified as a loss attributed to NRCE. Because ions in the S 1/2 and 2 D 3/2 states cannot be distinguished, due to the necessity of the 2 D 3/2 repumping laser for fluorescence detection, the outcome of the first detection phase is interpreted depending on the initial state of the ion. If the ion is initially prepared in the 2 D 5/2 state, it is assumed that it has undergone FSQ to the 2 D 3/2 state, because quenching to the S 1/2 state would heat the ion by ∼ 280 K, which is too hot for direct fluorescence detection. Similarly, for an ion initially in the 2 D 3/2 state, it is assumed that it has remained in that state. To obtain the survival probability of an ion in the 2 D 3/2 ( 2 D 5/2 )state, the relative frequency of ions detected in the first (third) detection phase is calculated.

When the experiment is conducted with the ion initially in the 2 D 5/2 state, but without the presence of atoms, a cold ion in the first detection phase is observed in 17.5( 14)% of cases. It is attributed to a leakage of 615 nm repumping light into the chamber. As the interaction duration is orders of magnitude shorter than the preparation of the atomic cloud, it can be assumed that the ion is pumped to the ground state before the interaction is efficient. Having shown that the S 1/2 state is stable up to 1 s of interaction time at the given densities, the respective product rate of 2 D 5/2 experiments is rescaled accordingly.

To compare the observed reaction rates to the Langevin rate K L , the number of atoms N , the radial trap frequency ω r ad , the axial size δ a x and the temperature T of the atomic cloud are carefully measured to obtain the number density

n = 1 (2π) 3/2 m Li ω 2 r ad k B T δ a x N .
The densities for interaction with the ion in the 2 D 3/2 or 2 D 5/2 state is adjusted to n = 1.3(2) × 10 11 cm -3 and 1.6(4) × 10 11 cm -3 respectively. The Langevin collision rate for atom-ion interactions is K L = 2π(2C 4 ) 1/2 µ -1/2 , with the reduced mass µ and the induced dipole coefficient C 4 . The corresponding Langevin rates are K L = 610(100) s -1 and 770(180) s -1 . In Figure 5.2, the experimentalists find reactions of the metastable state occurring on the order of K L . The 2 D 3/2 state reacts with a rate of 0.17 [START_REF] Bransden | Physics of atoms and molecules[END_REF] st at (3) s ys K L , while the 2 D 5/2 state reacts with an even faster rate of 0.30(4) st at [START_REF] Zrafi | Ab initio electronic structure and prospects for the formation of ultracold Calcium-alkali-metal-atom molecular ions[END_REF] s ys K L , a sign of the additional open FSQ channel. The initial non-unitary survival probability of the ion is due to the finite time of the shuttling ramp. Calculating the survival probabilities P for the measured decay rates and known ramp parameters yields initial values of P(0 ms) = 0.75(1) (P(0 ms) = 0.54(8))).

Theoretical models and results

The total Hamiltonian of two colliding particles validates independently of the number of limits and Born-Oppenheimer (BO) molecular states

Ĥ(R) = - ħ h 2 2µ d 2 dR 2 + V (R) + Vsoc (R) + Ĥrot (R), (5.1)
where the first term is the nuclear kinetic energy, µ is the reduced mass, R is the internuclear distance. V (R) denotes the electronic potential energy curves (PECs). Vsoc (R) represents the spin-orbit couplings (SOCs). The last term Ĥrot (R) describes the rotational couplings. The methodologies utilized to generate PECs and SOCs are presented in Appendix A. Initially, a cold trapped Ba + is prepared in the 5d state, while the polarized ultracold Li gas is in the ground 2s state. Therefore, our theoretical calculations only need to include the first three dissociation limits to investigate the possible non-radiative processes. In the inner part of PECs in Figure 5.3, an avoided crossing between 2 1 Σ + and 3

LZ model

Figure 5.3 presents the Hund's case (a) PECs in the absence of SOCs. We note that an avoided crossing (marked by a black circle) of 2 1 Σ + and 3 1 Σ + PECs associated with the neighboring dissociation limits Li + +Ba(6s 2 ) and Li(2s)+Ba + (5d), is well isolated from PECs with the same molecular symmetries. Hence, we build up a two-channel model using LZ theory to estimate the NRCE probability only depending on the non-relativistic electrostatic interaction PECs in the BF frame. We obtain the probability P L Z =0.769 for a single path, while the total NRCE probability 2P L Z (1 -P L Z ) is equal to 0.355. It should be emphasized that this probability is based on the assumption that only the 3 1 Σ + state is populated among the electronic molecular states corresponding to the incoming channel Li(2s)+Ba + (5d). Statistically speaking, however, the total NRCE probability should be 0.355/20, where 20 is the total number of channels. Regardless of the fine structure effects of Ba + (5d), the experimental total NRCE probability can be approximated as 0.089 by dividing the total NRCE events by the total collision events. Thus, the semi-classical LZ model underestimates the experimental measurements by about an order of magnitude. Note that a similar pattern was already evidenced in a previous study on Rb-Sr + system [START_REF] Ben-Shlomi | Direct observation of ultracold atom-ion excitation exchange[END_REF].

FCLZ model

To better acknowledge the reaction rates of NRCE, FSQ, and NRQ, we build up a 4x4 symmetric Hamiltonian matrix involving the four molecular states 1 1 Σ + , 2 1 Σ + , 3 1 Σ + and 1 3 Π, coupled by SOCs. Due to the fact that only Ω = 0 + symmetry contains the molecular states 1 Σ + , and that the rotational couplings are not included in our LZ model, we can ignore the matrices of Ω = 0 -, 1, 2, 3. For the case Ω = 0 + we have in the BF frame

   V (1 1 Σ + ) 0 0 A 14 0 V (2 1 Σ + ) G 23 A 24 0 G 32 V (3 1 Σ + ) A 34 A 41 A 42 A 43 V (1 3 Π) + A 44   
The curves of PECs V and SOCs A nm are displayed in Figure 5. , where the atomic spin-orbit splitting of Ba + ( 2 D 5/2,3/2 ) reaches 801 cm -1 . In the following, for the sake of simplicity we use 2 D 5/2 and 2 D 3/2 referring to the entrance channels Li(2s)+Ba + ( 2 D 5/2 ) and Li(2s)+Ba + ( 2 D 3/2 ) respectively. At the short range, we identify two avoided crossings with quite symmetric pattern.

To obtain the transition probability of the NRCE and FSQ regarding 2 D 5/2 , therefore, we should consider that two colliding atoms actually go through two avoided crossings consecutively. Herein, we obtain that the single path transition 5.1). Although this simple theoretical model does not provide accurate theoretical results, the obtained non-adiabatic potential energy surface (V ), the spin-orbit couplings Table 5.1: The experimental measurements and theoretical results concerning the detected events. A total of 510 or 116 events was evaluated for Ba + prepared in 2 D 3/2 or 2 D 5/2 state, respectively. The measured reaction rate per Langevin collision (K e x p. /K L ) per process is given except for the value of the elastic collision (EC). The non-thermalized reaction rate per Langevin collision (K mod el /K L ) is obtained within theoretical FCQS and MCQS models regarding the collision temperature 220 µK, where the reaction rate K L mod el is estimated by the Langevintype baseline method.

Event

Mechanism Counts Ratio(%) (A ), and the Gaussian coupling (G) can be used for further discussions.

K ex p. /K L K F CQS /K L K M CQS /K L K L M CQS /K L Initial Ba + prepared in 2 D 3/

FCQS model

In this subsection, we propose a 4-channel FCQS model by applying the quantum scattering partial wave expansion theory based on the framework of the previous FCLZ model to do a further investigation in the SF frame. Notably, the rotational coupling is currently not taken into account. As such we shall separately solve Hamiltonian for each partial wave. The Hamiltonian considering the electronic interactions (PECs and SOCs) and the uncoupled rotation of the nuclei, can be described as

H (R) = - ħ h 2 2µ d 2 dR 2 + ħ h 2 l(l + 1) 2µR 2 + V (R) + V soc (R) (5.2)
where µ is the reduced mass, R is the internuclear distance, V (R) denotes the PECs, V soc (R) represents the spin-orbit interactions. l is the partial wave, i.e. the rotational angular momentum of the colliding nuclei in the SF frame. In this simplified model we treat collisions with conserved partial wave l. Then we propagate the coupled equations using the log-derivative method [START_REF] Johnson | The multichannel log derivative method for scattering calculations[END_REF] with a constant step-size 0.005 a.u. for a given collision energy E. Since 1/R 4 vanishes more rapidly than 1/R

(E) = π k 2 i l (2l + 1)|S( f ← i)| 2 , ( 5.3)
where k 2 i = 2µE/ħ h 2 , E is the initial collision energy. The labels i and f denote the initial and final states, respectively. Figure 5.7 illustrates the calculated cross sections for various processes with respect to 2 D 5/2 and 2 D 3/2 . In the classical capture theory the Langevin cross section P L = 2πC 1/2 4 E -1/2 is represented as well. It is worth mentioning that the scattering cross sections of these three processes have a similar profile in terms of 2 D 5/2 or 2 D 3/2 . We find that a shape resonance occurs every two partial waves in Figure 5.7. Regarding 2 D 5/2 , the cross section of the NRQ is negligible compared to the NRCE and FSQ, which is consistent with the experimental findings. However, the NRCE is identified as the principal transition mechanism which is contrary to the experimental measurements. A similar contradictory conclusion between theory and experiment is found for 2 D 3/2 .

It is straightforward to acquire the rate coefficients relying on the calculated cross section. We assume that the thermalized reaction rate coefficient is weighted over the Boltzmann distribution

K(T e f f . ) = 2 π(k B T e f f . ) 3/2 ∞ 0 K(E) Ee -E/k B T e f f . d E (5.4)
where the non-thermalized rate is

K(E) = σ(E) × v = σ(E)(2E) 1/2 µ -1/2
, the effective temperature T e f f . = (m Li T Ba + + m Ba + T Li )/(m Li + m Ba + ), a temperature T Ba + and T Li for Ba + and Li, respectively. And the Langevin rate is

K L = 2π(2C 4 ) 1/2 µ -1/2
. The results are displayed in Figure 5.8. Even though the thermalized reaction rate smoothes out the resonance of the non-thermalized reaction rate, the thermalized and non-thermalized reaction rates remain essentially the same. Ergo, our Boltzmann distribution assumption is reasonable and does not profoundly modify the thermalized rate. However, such a simplified scattering model is incapable of interpreting the experimental phenomena. This motivates the need to build a full multi-channel model.

MCQS model

Motivated by the above approaches, we propose a rigorous MCQS model and introduce the rotational coupling, total angular momenta J, its projection M as well as a parity p to explore non-radiative processes (NRCE, FSQ, and NRQ) in Hund case (e) basis (SF frame). When a typical collision occurs, we assume that any effects of spontaneous emission of a molecule can be neglected in our model. Due to the fact that the time is approximately 10 -13 sec of a collision, that is considerably shorter than a spontaneous emission lifetime of a molecule around 10 -9 sec [START_REF] Julienne | Nonadiabatic theory of atomic line broadening: Final-state distributions and the polarization of redistributed radiation[END_REF]. The basis transformation from Hund's case (a) to Hund's case (e) basis is reported in Chapter 2, which can help us to obtain the S matrix elements. Finally, the cross section is calculated from the parity-and J-dependent but M -independent S matrices

σ(E) = π k 2 0 J (2J + 1) l i , j i l f , j f |S(J, l f , j f , p ← J, l i , j i , p)| 2 .
(5.5)

Here, |J, l i , j i , p > and |J, l f , j f , p > represent the chosen initial incoming channels and the allowed outgoing channels. Note that M is omitted because the S matrix element is not dependent on M . The quantum values for s + s and s + d dissociation limits are summarized in Table 2.1. We notice that the number of states is the same not only among Hund's case (a), (c), and (e) but also between parity e and f . (Please note that there is no association between Hund's case (e) and parity e.) Since the charge exchange exit channel is 2 1 Σ + and the molecular state 1 Σ + only belongs to parity e, the NRCE only takes place for parity e. Figure 5.9 shows out PECs in Hund's case (e) representation depending on parities and J. In the panel of J = 0 and parity e, PECs are obtained for the matrix Ω = 0 + and l = 2. The short-range of Hund's case (e) PECs are similar to those of Hund's case (c) PECs in Figure 5.5. As discussed in FCLZ and FCQS models, the two neighboring avoided crossings make a significant contribution to NRCE and FSQ in the MCQS model. As J increases, the relevant Hamiltonian matrix increases with Ω and l, but the two coupled avoided crossings remain unchanged. As long as the matrix Ω = 1 is included, a new avoided crossing generated solely by SOC (1 3 Σ + -1 3 Π) appears at a rather shorter distance 6.2 a.u., where the matrix of

Ω = 1 is      V (1 3 Σ + ) 0 A 8,10 A 8,11 0 0 V (2 3 Σ + ) A 9,
     .
This avoided crossing is depicted in the zoomed-in panel for parity e. Since parity f differs from parity e by containing the matrix Ω = 0 -, not Ω = 0 + , another new avoided crossing shows up depending on the same real crossing of 1 3 Σ + -1 3 Π with different strength of the relevant SOC (Ω = 0 -) comparing to that of Ω = 1, where the Ω = 0 -matrix is

THEORETICAL MODELS AND RESULTS



 V (1 3 Σ + ) 0 A 5,7 0 V (2 3 Σ + ) A 6,7 A 7,5 A 7,6 V (1 3 Π) + A 7,7   .
When J > 0, i.e. both Ω = 0 -, 1 are taken into account, there will be two avoided crossings at the same time around 6.2 a.u. as exhibited in the bottom panel for parity f . Regardless of parities and entrance channels, these avoided crossings characterize the NRQ. Note that when J is bigger than 1, Ω = 2 should be considered all the time,   V (1 3 Π) + A 13,13 A 13,14 A 13,15 A 14,13

V (1 1 ∆) A 14,15 A 15,13 A 15,14 V (1 3 ∆)   .

Figure 5.9 also indicates that the number of coupled channels is increasing with J < 3 and then remains a constant for J ≥ 3, meanwhile the matrix

[V (1 3 ∆) + A 16,16
] should be taken into account for Ω = 3, which leads to a significant effect on the number of incoming and outgoing channels and the S matrix elements. Taking 2 D 5/2 and parity e as an example, the number of incoming channels for J = 0, 1, 2, 3 is 1, 3, 5, 6, the number of outgoing channels of FSQ is 1, 3, 4, 4, the number of outgoing channels of NRCE is 1, 1, 1, 1, the number of outgoing channels of NRQ is 1, 2, 2, 2, respectively. The parity-dependent cross sections are exhibited for 2 D 5/2 and 2 D 3/2 in the left column of Figure 5.10. For 2 D 5/2 and parity e, we calculate the cross section of FSQ, NRCE, and NRQ, in which we recognize the dominant process is FSQ at the limit of Langevin cross section P L level, while the magnitude of the NRQ is approximately 3 orders of magnitude smaller within the energy scale shown. The profile of the cross section of each process is different as a result of summing over different numbers of S matrix elements. Moreover, we find a prominent shape resonance for both FSQ and NRCE around 10 -4 K, which is around the collision temperature 220µK, and both of them are mainly composed of J |l i = 2, j i = 3 > and J |l i = 2, j i = 2 > channels. As expected, NRCE is not prohibited for 2 D 5/2 and parity f . The contribution of NRQ can be ignored once again. As to FSQ, the probability is considerable and approaches or even exceeds P L due to the presence of a shape resonance.

Regarding the entrance 2 D 3/2 , then, the 2 D 5/2 is closed, the FSQ is not presented for each parity. For parity e, the numerical values and trends of cross sections of NRQ and NRCE processes are similar to each other over the energy range. They are about two orders of magnitude less than P L within the given collision energy. As for parity f , not surprisingly NRCE does not appear. And the enhanced magnitude of NRQ is close to P L according to that the Hamiltonian for parity f is able to generate one more avoided crossing induced by SOC (1 3 Σ + -1 3 Π) for Ω = 0 -than parity e.

Of practical interest, we have separately evaluated the thermalized and non-thermalized rates based on the related cross sections in the right panels in Figure 5.10. The curves of reaction rates show only small differences whether thermally averaged or not. Since the number of incoming channels for each parity is the same, we can simply obtain the parity-average rate used to compare the results of the experiments. In Table 5.1, we take T e f f . = 220 µK to estimate the reaction rate per Langevin collision per process per model K mod el /K L to compare experimental measurements, where K mod el = (K e mod el + K f mod el )/2 with model = FCQS or MCQS ( K f M CQS for NRCE does not exist).

Discussion

To better understand our theoretical calculations and to figure out the likely dependency of our MCQS model, it is worth further examining our theoretical method for several aspects:

(1)In this experiment, although the polarized Li atoms are initially prepared on the hyperfine |F = 1/2, mF = -1/2 > state, the collisions between the polarized Li(2s) atoms with an upward or downward spin and an unpolarized sphere-like Ba + (5d) ion in space are the same and indistinguishable. The Mindependent MCQS model is still valid and sufficient. Only if Li and Ba + are both polarized, |ms Li , ms Ba + , ml Li , ml Ba + > should be taken into account in our model.

(2) For the radial coupling G, as the main parameter, W c , is determined by the PECs in Hund's case (a), we examine the width Γ by varying δ = 0.75 ± 0.5 to assess the sensitivity to the MCQS model, shown in Fig. 5.11. Using the modified G, the recalculated results show that the effect of G is minimal.

(3) In Table 5.1, K M CQS /K L falls in the interval of experimental observations for both 2 D 5/2,3/2 , except for the FSQ with respect to 2 D 5/2 . Our model provides a number of 5 times larger, which is due to the enchanced shape resonances. Considering that the position of the shape resonance strongly depends on the PECs, we attribute this discrepancy to a coincidence due to the calculation method for PECs.

(4) Finally, it is worth verifying the baseline of the Langevin-type cross section, in which we can avoid the effect of the shape resonance position, as shown in the left panel of Fig. 5.10. The Langevin-type reaction rates obtained are listed in Table 5.1. We can see that the difference in the reaction rate K L M CQS /K L between theory and experiment for the FSQ is mitigated to only a factor of 2. The rest is in a good agreement with experiment.

Conclusions

To summarize, we propose a series of theoretical models of progressive complexity to investigate the non-radiative interactions via the metastable state Li(2s)+Ba + ( 2 D 3/2,5/2 ) in a hybrid trap involving ultracold Li gas and a single cold Ba + . The theoretical LZ and FCLZ models demonstrate that the classical model is limited because there are several different interactions at play in the Li(2s)+Ba + ( 2 D 3/2,5/2 ) dissociation limits. By comparing the theoretical FCQS and MCQS models, we realize that the respective efficiency of SOCs and rotational couplings can completely reverse the final results of NRCE and FSQ, in particular, the rotational coupling is demonstrated to enable complete mixing of internal angular momenta. Although the parity cannot be distinguished in experimental measurements, our in-depth designed theoretical model proves a particular parity-dependent quantum effect for non-radiative charge exchange. The proposed MCQS model is proven to be a sufficiently robust approach to explain the cold collisions at excited levels.

138

(3) H 2 -Ba + : I described a three-step recycling scenario to bring back into the trap the accidentally disappeared ions Ba + to its ground state. The first step is the spontaneous collision reaction, H 2 +Ba + → HBa + +H, while the third step is to cool down Ba + , which is performed due to the experimental laser cooling devices. We focused on the second step to design a photodissociation using the two lasers at 396.8 nm and 271.7 nm present in the experimental setup. We also set up a black-body radiation model to estimate the initial quantum state of HBa + . The spontaneous emission rate proved that the absolute ground state 1 1 Σ + (J = 0, v = 0) with a long lifetime (4.06 s) is preferably populated. Subsequently, we predict that when 271.7 nm laser with 100 W /cm 2 is switched on, a photodissociation event will occur every second via the 1 1 Σ + → 2 1 Σ + transition.

(4) Rb-Ca + : I presented theoretical calculations related to four processes: radiative association to generate molecular ions, photodissociation of the created molecular ions, the influence of black-body radiation (BBR), and non-radiative charge exchange based on novel molecular data compared to Ref. [START_REF] Belyaev | Resonances in Ca + +Rb nonadiabatic collisions at ultralow energies[END_REF]. The Ca + cooling laser with 397 nm wavelength is demonstrated to be capable of inducing the dissociation of the cold RbCa + ions, beyond a given threshold for their internal states. The analysis of the BBR effect reveals that the surviving RbCa + are not destructed by BBR-induced photodissociation. The obtained specific range of vibration states 16 ≤ v ≤ 28 implies that photodissociation would destroy the molecules with v ≥ 29, and the deeper bound molecular ions with v ≤ 15 are unlikely formed within the radiative association. Finally, the non-radiative charge exchange calculations, despite the significant differences in our new set of molecular data compared to those of [START_REF] Belyaev | Resonances in Ca + +Rb nonadiabatic collisions at ultralow energies[END_REF], yield rates in good agreement with experimental data. By slightly varying the dissociation limit of the potential energy surface, a huge suppression of the charge exchange cross section appears. Such a quantum effect is discussed within two concepts: phase-locking and s-wave signature.

(5) Li-Ba + : I investigated the non-radiative dynamics to understand the nature of quantum interactions, in close relation with an ongoing experiment in which a hybrid trap uses ultracold polarized Li(2s) atoms colliding with a long-lived ultracold Ba + ion in an excited metastable state (5d 2 D 5/2,3/2 ). Based on counting the ion loss events, non-radiative charge exchange (NRCE), fine structure quenching (FSQ), and non-radiative quenching (NRQ) are probed simultaneously. I designed a series of theoretical models of increasing complexity to identify the main interactions at play during collisions: Landau-Zener (LZ) model, and full quantum scattering (QS) model involving a various number of channels, spin-orbit couplings (SOCs), and rotational couplings. The results for branching ratios between the various processes from the QS model are found in good agreement with experimental observations around 0.2 mK collisional temperature. The full QS approach demonstrates the competing contributions of the SOCs and the rotational couplings induced by the initial high angular momentum of the ion. Although the parity of the total wave function of the colliding complex cannot be distinguished in experimental measurements, our model proves a particular parity-dependent quantum effect for non-radiative collisions.

Outlook

In future investigations, some remaining issues from the current work are still worth be further discussed. The theoretical models and programs developed at present can be applied to other systems as well.

(1)The theoretical predictions of the photodissociation cross section of the weakly-bound RbBa + have a factor of 13.5 discrepancies with experimental measurements. The photodissociation model is mainly governed by the Franck-Condon factor. The approaches to calculating the bound wave function and the continuum wave function are well established and accurate (Fourier grid Hamiltonian method and the Numerov method, respectively). So the most probable uncertainty would come from the ab initio potential energy surface, which would affect the weakly-bound wave functions. A more precise ab initio surface is required for an accurate photodissociation calculation. An alternative option would be to consider that the experimental result would serve as a basis for adjusting the global PEC of the dissociative channel in order to match the experimental rates. This would deliver an effective PEC which could be relevant for further studies.

(2) The mechanism for the formation of the detected Rb + 2 in the Rb-Ca + experiment still remains uncertain. Various pathways are possible, like the reactive collision between RbCa + and surrounding Rb atoms, but also radiative association Rb+Rb + → Rb + 2 ,or three-body recombination Rb+Rb+Ba + → Rb + 2 +Ba. In addition, the cooling laser 397 nm of Ca + has been demonstrated to dissociate the ground state RbCa + molecular ion. Therefore, we consider that photodissociation due to 397 nm should be prevalent in other ultracold calciumion-alkali-metal-atom systems. Studying this very topic can provide important guidance for future experiments.

(3)In the Li-Ba + experiment, the quantum numbers m S and m I of the polarized Li currently are considered as spectators during collisions with an unpolarized ion. In the ongoing Li-Ca + experiment, however, both species are initially prepared in the hyperfine states, while in the Rb-Sr + experiment, the Sr + was polarized. Therefore the BF-SF frame transformation including magnetic sublevels is now needed to predict to which extent reaction rates are indeed influenced by the initial polarization of the reactants.

The various matrix elements are as follows for each value of the projection Ω of J on an axis, assuming that the energy of the 3 P level is put to zero. The corresponding wave functions are displayed in Table A The various matrix elements are as follows for each value of the projection Ω of J on an axis, assuming that the energy of the 3 D level is put to zero. The corresponding wave functions are displayed in Table A Table A.6: The SOC wave function for 1 D. The operate notes that the projection of total angular momentum is reduced by one operation which could be spin or orbital angular momentum. We start with 1 ∆ state.

< 3 Σ + |L z S z | 3 Σ + >= 0 < 3 Π|L z S z | 3 Π >= 1 4 (- A( 3 D) 2 + A( 3 D) 2 - A( 3 D) 2 + A( 3 D) 2 ) = 0 < 3 ∆| L + S -+ L -S + 2 | 3 Π >= 1 2 2 ( 2A( 3 D) + 2A( 3 D) 2 ) = 2 2 A( 3 D) < 3 ∆| L + S -+ L -S + 2 | 3 Σ + >= 0 < 3 Σ + | L + S -+ L -S + 2 | 3 Π >= 1

State

Wave function

Ω = 2 1 ∆ 1 2 [(|0 s + > 1 |2 d -> 2 -|0 s -> 1 |2 d + > 2 ) -(|0 s + > 2 |2 d -> 1 -|0 s -> 2 |2 d + > 1 )
]

Ω = 1 1 Π 1 2 [(|0 s + > 1 |1 d -> 2 -|0 s -> 1 |1 d + > 2 ) -(|0 s + > 2 |1 d -> 1 -|0 s -> 2 |1 d + > 1 )
]

Ω = 0 + 1 Σ + 1 2 [(|0 s + > 1 |0 d -> 2 -|0 s -> 1 |0 d + > 2 ) -(|0 s + > 2 |0 d -> 1 -|0 s -> 2 |0 d + > 1 )] Ĥ(|Ω| = 1) =            V ( 3 ∆) -A 2 2 A 0 - 2A 2 2 2 A V ( 3 Π) 3 2 A - A 2 0 3 2 A V ( 3 Σ + ) 3A 2 - 2A 2 - A 2 3A 2 V ( 1 Π)            , ( 3 D 1 , 3 D 2 , 3 D 3 , 1 D 2 ) Ĥ(Ω = 0 + ) =    V ( 3 Π -) - A 2 6A 2 6A 2 V ( 1 Σ + )    , ( 3 D 2 , 1 D 2 )
Appendix B Description of the shifted PECs for RbCa + B.1: these probabilities are almost independent of the energy shifts, so that one may expect that no significant changes will be observed in the dynamics.

However, we calculated the thermalized non-radiative charge exchange rates for each pair of curves (Figure B.2) based on the relevant cross section in Figure 4.14. While the results of the computations using E T hiswor k are found in good agreement with the experimental values, we observe that the charge exchange rate can be significantly changed with the dissociation energy of the 1 3 Π outgoing channel, from few factors above 0.1 K, up to 2 or 3 orders of magnitude in the low temperature range. For J ≥ 3, each parity has ten channels, and so on and so forth. R ésum é : Cette th èse se concentre sur la description th éorique de la dynamique quantique des collisions ion-atome dans le domaine des temp ératures sub-Kelvin. Le d éveloppement de pi èges hybrides pour les ions et les atomes neutres froids a stimul é l' étude de nombreux processus, notamment la formation d'ions mol éculaires froids, le contr ôle quantique de collisions froides, le transfert de charge. Dans ce travail, nous avons mod élis é la dynamique non radiative, et la dynamique assist ée par les photons qui sont pr ésents dans l'exp érience. Nous avons choisi des syst èmes étudi és dans diff érents groupes exp érimentaux. Tout d'abord, on a montr é que la formation d'un ion mol éculaire RbBa + dans un pi ège hybride Rb/Ba + est annul ée par la destruction de cet ion par le laser de pi égeage. J'ai inclus l'interaction spinorbite dans le mod èle de photodissociation, fournissant ainsi des r ésultats avec une meilleure pr écision. J'ai étendu ce mod èle à la photodissociation des ions BaH + cr é és lors de la collision r éactive entre Ba + froid et H 2 . Ensuite, j'ai explor é divers processus impliqu és dans un pi ège Rb/Ca + : association radiative, photodissociation et rayonnement du corps noir. J'ai inclus de nouveaux calculs de l'interaction spin-orbite effectu és dans le groupe. J'ai obtenu des taux de r éaction en bon accord avec les r ésultats exp érimentaux, et je sugg ère une interpr étation directe de l'observation exp érimentale de RbCa + bas ée sur la robustesse de la distribution vibrationnelle par rapport au laser de pi égeage. Enfin, j'ai mis en place un mod èle rigoureux de diffusion quantique incluant les couplages spinorbite calcul és dans le groupe et les couplages rotationnels pour étudier la collision entre les atomes de Li à l' état fondamental et les ions Ba + m étastables. Les coefficients de taux calcul és pour le changement de structure fine, l' échange de charge non radiatif d épendant de la parit é et les collisions in élastiques non radiatives sont qualitativement et quantitativement coh érents avec les donn ées exp érimentales.

Title : Modelling ion losses in cold ion-atom hybrid traps : photodissociation and non-radiative processes

Keywords : Cold molecular ions, ultracold collisions, photodissociation, non-radiative dynamics Abstract : This thesis focuses on the theoretical description of the quantum dynamics of ion-atom collisions in the sub-Kelvin temperature domain. The development of hybrid traps merging a cold ion trap and a cold neutral trap stimulated the investigation of many processes, including the formation of cold molecular ions, cold quantum-controlled reactions, charge transfer. In this work, we modelled non-radiative dynamics, and dynamics assisted by photons which are present in the experiment. We chose systems under investigation in various experimental groups. First, the formation of RbBa + molecular ion in a hybrid Rb/Ba + trap has been shown to be cancelled by the destruction of this ion by the trapping laser. I included the spin-orbit interaction in the photodissociation model, thus providing results with improved accuracy. I extended this model to the photodissociation of BaH + ions created during the reactive collision between cold Ba + and H 2 . Next, I explored various processes involved in a Rb/Ca + trap : radiative association, photodissociation, and black body radiation. I included novel calculations of the spin-orbit interaction performed in the group. I obtained reaction rates in good agreement with experimental findings, and I suggest a direct interpretation of the experimental observation of RbCa + based on the robustness of the vibrational distribution with respect to the trapping laser. Finally, I set up a rigorous quantum scattering model including spin-orbit couplings calculated in the group and rotational couplings to study the collision between ground state Li atoms and metastable Ba + ions. The calculated rate coefficients of the fine structure quenching, the paritydependent non-radiative charge exchange, and the non-radiative quenching are qualitatively and quantitatively consistent with experimental data.

Universit é Paris-Saclay

Espace Technologique / Immeuble Discovery Route de l'Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France

1. 1

 1 (a): Schematic diagram of the MOT. Six counter-propagating laser beams are marked in red with their respective circular polarisation σ +/-. The anti-Helmholtz configured solenoids are shown in blue and labeled by their respective current flow direction. (Figure from Ref. [1, 2]) (b): Schematic of the magnetic sublevels of atom labeled by |total angular moment J, its projection m>. (Figure from Ref. [1, 2]) (c)/(d):

3

 3 1.2 (a): Schematic diagram of a linear rf-trap (Paul trap) with 12 segments and four rods. (Figure from Ref. [1, 2]) (b) The ions Ba + /Ca + are transferred from rf-trap to ODT.(Figure from Ref. [3]) (c)/(d): The energy level scheme of 40 Ca + / 138 Ba + for the laser cooling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Schematic representation of a single ion (blue) immersed in an polarized atoms (red) by the homogeneous magnetic field B in a hybrid trap. The single cold ion is prepared in a linear segmented rf trap, alternatively in ODT when rf fields are off, while the ultracold atoms are stored in a crossed ODT (xODT). (Figure from Ref. [4]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Schematic diagram of the diatomic molecule AB in body-fixed (BF) frame and space-fixed (SF) frame: (a) the molecule AB with internuclear distance r on the molecular axis z in BF frame; (b) the ab initio potential energy surfaces calculated in BF frame, the radiative process (pink arrow) between a bound level α of the low electronic state and a continuum level of an upper electronic state β, and the scattering process (blue arrow); (c) the rotation of the molecule AB in SF frame. . . . . . . . . . . . . . . . . . . . . 2.2 The vector model for angular momenta in a diatomic molecule AB for Hund's cases a, c, and e. Case a: Λ and Σ are well defined on the z axis, and thus Ω = Λ+Σ; Case c: similar to case a except that Λ and Σ are not defined on the z axis; Case e: this represents the situation in the SF frame with undefined Ω, Λ, and Σ. . . . . The effect of parity operation on the angular coordinates (φ, θ ) with a electron P, where θ → πθ , φ → π + φ. . . . . . . . . . 13 2.4 Schematic diagram of the rotation of two hard balls with the mass m 1 and m 2 . o is the center of mass. r 1 and r 2 are the lengths. ω is the angular velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.

IX 2 .

 2 15 Schematic representation of the scattering channels. The potential energy surface a denotes the closed channel, while potentials b and c are open with respect to the incoming energy E. When the outgoing channel is open channel b or c, we consider this is an elastic or inelastic collision, respectively. However, the outgoing channel a is forbidden. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Top panel: the potential energy curves of RbBa + in Hund's case (a) basis, the bold potential energy curves indicate two entrance channels (2 1 Σ + , 13 Σ + ) and four exit channels (4 1 Σ + , 4 3 Σ + , 3 3 Π, 2 1 Π) for the photodissociation of a weakly bound molecule RbBa + , which are indicated by the laser at 1064 nm (black dashed arrows). The temperature of the scattered Ba + (5d) and Rb(5s) is around 5800 K (red arrow); Bottom panel: the relevant transition electric dipole moments (TEDMs) for the photodissociation in Hund's case (a) basis. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The avoided crossing between 2 3 Π and 3 3 Π electronic states. Blue curves are identical to the blue circles. A, B, D, and F points are picked out to estimate the slope of black lines AB and F D. C is the intersect point. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Test the convergence of interpolated numbers for continuum wave functions. The photodissociation cross sections between bound level v = 172 of 2 1 Σ + and 11 random continuum energy levels of 4 1 Σ + state are calculated, where curves of different colors represent arbitrary transition energies in the range of 8896.01 -9896.01 cm -1 with the energy step 100 cm -1 . . . . . . . . . . . . 3.

X LIST OF FIGURES 3 . 7

 37 Left: the exit channel 4 1 Σ + (black) and the shifted potential (red); right: the corresponding cross sections in the case of an initial binding energy of 10.5 cm -1 state of 2 1 Σ + to the exit 4 1 Σ + . The dashed pink arrow indicates the transition energy of 1064 nm laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.8 Potential energy curves (bold curves) are labelled by (a)Ω, (b)Ω, (c)Ω and (d)Ω

5 . 1

 51 (solid lines) labelled by (α) and (β) for Ω = 0 + , ; the zooms around the left crossing (b) and the right crossing (c). . . . . . . 4.11 The calculated NRCE transition probability |S i f | 2 between the entrance Rb(5s) + Ca + (4s) and the exit Rb + + Ca( 3 P) asymptotes, as a function of the partial wave number l. The peak of the selected collision energy indicates a shape resonance. . . . . . . . 4.12 a): The calculated cross section for the Rb(5s) + Ca + (4s) → Rb + + Ca( 3 P) charge exchange as a function of the collision energy (10 -6 K -1K): the quantum scattering results (black) and the Langevin cross section (blue); b): Based on the NRCE cross section (black) in panel (a), we calculate NRCE rates for non-thermalized rate (green) and Boltzmann thermalized rate (red), comparing with theoretical data (blue triangles) of [8] and the experimental data (black circles) of [9], as a function of the collision energy (10 -6 K -1K). . . . . . . . . . . . . . . . . . . . . . 4.13 This work (red solid line) and Ref. [8] (blue dashed line): (a) Hund' case a PECs 1 3 Π and 2 1 Σ + , the vertical dashed lines indicate the two real crossings; (b) the SOC between 1 3 Π and 2 1 Σ + , the vertical dashed lines mark the values of the SOC corresponding to the two real crossings; (c) The calculated NRCE cross sections (cm 2 ) as a function of the collision energy in unit (eV). 4.14 The NRCE cross section (solid lines) regarding three dissociation limits of the exit channel 1 3 Π, E T hiswor k (blue), E N IST (green) and E Re f . (red) [6]. Black dashed line is the Langevin cross section σ L . The colorful dashed lines are calculated by σ L sin 2 (∆δ 0 ). . . LIST OF FIGURES XIII Scheme of the product state detection sequence (times are not on scale)

Figure 1 . 1 :

 11 Figure 1.1: (a): Schematic diagram of the MOT. Six counter-propagating laser beams are marked in red with their respective circular polarisation σ +/-. The anti-Helmholtz configured solenoids are shown in blue and labeled by their respective current flow direction. (Figure from Ref. [1, 2]) (b): Schematic of the magnetic sublevels of atom labeled by |total angular moment J, its projection m>. (Figure from Ref. [1, 2]) (c)/(d):The respective energy level scheme with hyperfine states F of 87 Rb/6 Li for the laser cooling.

Figure 1 . 2 :

 12 Figure 1.2: (a): Schematic diagram of a linear rf-trap (Paul trap) with 12 segments and four rods. (Figure from Ref. [1, 2]) (b) The ions Ba + /Ca + are transferred from rf-trap to ODT.(Figure from Ref. [3]) (c)/(d): The energy level scheme of 40 Ca + / 138 Ba + for the laser cooling.

Figure 1 . 3 :

 13 Figure 1.3: Schematic representation of a single ion (blue) immersed in an polarized atoms (red) by the homogeneous magnetic field B in a hybrid trap. The single cold ion is prepared in a linear segmented rf trap, alternatively in ODT when rf fields are off, while the ultracold atoms are stored in a crossed ODT (xODT). (Figure from Ref. [4])

  Figure 2.1 (b)), and in section 2.4 the atom-ion non-radiative scattering problem (indicated by the blue arrow in Figure 2.1 (b)).

10 2. 1 .

 101 Figure 2.1: Schematic diagram of the diatomic molecule AB in body-fixed (BF) frame and space-fixed (SF) frame: (a) the molecule AB with internuclear distance r on the molecular axis z in BF frame; (b) the ab initio potential energy surfaces calculated in BF frame, the radiative process (pink arrow) between a bound level α of the low electronic state and a continuum level of an upper electronic state β, and the scattering process (blue arrow); (c) the rotation of the molecule AB in SF frame.

Figure 2 . 2 :

 22 Figure 2.2: The vector model for angular momenta in a diatomic molecule AB for Hund's cases a, c, and e. Case a: Λ and Σ are well defined on the z axis, and thus Ω = Λ + Σ; Case c: similar to case a except that Λ and Σ are not defined on the z axis; Case e: this represents the situation in the SF frame with undefined Ω, Λ, and Σ.

Figure 2 . 3 :

 23 Figure 2.3: The effect of parity operation on the angular coordinates (φ, θ ) with a electron P, where θ → πθ , φ → π + φ.

2. 2 . ROTATIONAL MOTION 21 Figure 2 . 4 :

 22124 Figure 2.4: Schematic diagram of the rotation of two hard balls with the mass m 1 and m 2 . o is the center of mass. r 1 and r 2 are the lengths. ω is the angular velocity.

  γ -cos β sin α sin γ sin α cos γ + cos β cos α sin γ sin β sin γ cos α sin γ -cos β sin α cos γ -sin α sin γ + cos β cos α cos γ sin β cos γ sin β sin α -sin β cos α cos β  

Figure 2 . 6 :

 26 Figure 2.6: Schematic representation of doable molecular transitions marked by arrows regarding different dynamics with or without lights. Stimulated emission (red arrows): (1) bound to bound; (4) free to bound; (7) free to free. Stimulated absorption (green arrows): (2) bound to bound; (5) bond to free; (8) free to free. Spontaneous emission (orange arrows): (3) bound to bound; (6) free to bound; (9) free to free.

Figure 2 . 7 :

 27 Figure 2.7: Spherical polar coordinates (r, θ , φ). The point P in the rectangular Cartesian coordinate: x = rsinθ cosφ; y = rsinθ sin φ; z = r cosφ.

Figure 2 . 8 :

 28 Figure 2.8: Coordinate system for a molecule AB with two electrons (e 1 , e 2 ). With respect to the centre of mass O, the position vectors of electrons and nuclei are indicated by r e 1 , r e 2 , r A and r B , respectively.

  A and/or the ion B + undergo an internal energy change during the collision, A + B + = A * + B + , A + B + * , A * + B + * . 3. Reactive collision: a new product AB + molecule is generated during the collision, A + B + = AB + + γ.

Figure 2 .

 2 Figure 2.10: Schematic diagram of a scattering event between an atom A and a single ion B + .

Figure 1 . 6 .

 16 The number dN (with dimensionality [T -1 ]) of scattered atoms A per unit of time to the solid angle dΩ = ds/r 2 = sin θ dθ φ with a small area ds of an arbitrary sphere, and the intensity N [L -2 T -1 ] of the flux of incident atoms A are related as d N = dσ(θ , φ)N dΩ.

[ 2 . 1 2 1 2

 211 150] that to determine the cross section we should firstly determine the phase shift δ l , where the phases of the incident and scattered waves are (kr -lπ) and (kr -lπ + δ l ) in the Eqn [2.132] and [2.142], respectively. When δ l (k) = (n + 1 2 π), the maximum cross section of a given partial wave is σ ma x l = 4π

Figure 2 . 12 :

 212 Figure 2.12: Structureless particles A (b a > b max ), B (b a = b ma x ) and C (b a > b ma x ) collide a particle D with a finite size a along z axis. The maximum impact parameter b max = a.

Figure 2 .

 2 Figure 2.13: Long-range effective potentials of a general atom-singly-charged ion system for selected partial waves s, p, d, f , g in atomic units. The location R * ma x (l = 1) and height E * max (l = 1) of the centrifugal barrier of the p wave is marked by the blue arrow.

Figure 2 . 14 :

 214 Figure 2.14: Top panel: The cross section of l = 3 from Figure 2.11 as a function of the collision temperature, the position (3.41 mK) of the peak is marked by a blue bar; Bottom panel: The angle of the phase shift as a function of the collision temperature, the position of the shape resonance marked by blue which is corresponding to the indicated shape resonance of l = 3 at 3.41 mK in the top panel.

Figure 2 . 15 :

 215 Figure 2.15: Schematic representation of the scattering channels. The potential energy surface a denotes the closed channel, while potentials b and c are open with respect to the incoming energy E. When the outgoing channel is open channel b or c, we consider this is an elastic or inelastic collision, respectively. However, the outgoing channel a is forbidden.

.187) 54 2. 4 .

 544 ELEMENTS OF SCATTERING THEORY where n and m denote the dimensions of K K K matrices, o and c indicate the open and close channels, respectively. S S S matrix can be expressed by the open channel part of K K K matrix S S S = I I I + iK K K nn oo I I I -iK K K nn oo , (2.188) where |S i j | 2 indicates the transition probability from channel i to channel j. Eqn [2.178] in multi-channel representation is then T T T = S S S -I I I. (2.189)

Figure 3 . 1 :

 31 Figure 3.1: Top panel: the potential energy curves of RbBa + in Hund's case (a) basis, the bold potential energy curves indicate two entrance channels (2 1 Σ + , 1 3 Σ + ) and four exit channels (4 1 Σ + , 4 3 Σ + , 3 3 Π, 2 1 Π) for the photodissociation of a weakly bound molecule RbBa + , which are indicated by the laser at 1064 nm (black dashed arrows). The temperature of the scattered Ba + (5d) and Rb(5s) is around 5800 K (red arrow); Bottom panel: the relevant transition electric dipole moments (TEDMs) for the photodissociation in Hund's case (a) basis.

3. 1 .Figure 3 . 2 :

 132 Figure 3.2: The avoided crossing between 2 3 Π and 3 3 Π electronic states. Blue curves are identical to the blue circles. A, B, D, and F points are picked out to estimate the slope of black lines AB and F D. C is the intersect point.

Figure 3 . 3 :

 33 Figure 3.3: Test the convergence of interpolated numbers for continuum wave functions. The photodissociation cross sections between bound level v = 172 of 2 1 Σ + and 11 random continuum energy levels of 4 1 Σ + state are calculated, where curves of different colors represent arbitrary transition energies in the range of 8896.01 -9896.01 cm -1 with the energy step 100 cm -1 .

62 3. 1 .Figure 3 . 4 :

 62134 Figure 3.4: The unscaled (see text) cross section of 1064 nm laser induced photodissociation from weakly bound vibrational states of either 2 1 Σ + or 1 3 Σ + entrance channel towards the possible exit channels as a function of the vibrational quantum number of the entrance channels in Hund's case (a) basis.

Figure 3 . 5 :

 35 Figure 3.5: The unscaled (see text) cross section of photodissociation from the last several vibrational levels with energies (-0.1 cm -1 → -21.3cm -1 ) of either 2 1 Σ + or 1 3 Σ + entrance channel towards the possible exit channels as a function of the transition energy. The dashed line indicates the transition energy of the laser 1064 nm.

Figure 3 . 6 :

 36 Figure 3.6: For the transition 2 1 Σ + -4 1 Σ + , unscaled cross section in panel (a) and the corresponding transition dipole moment in panel (i). In each of panels (b, c, d) and zoomed-in panel (b , c , d ), the vibrational wavefunction (pink) of binding energy 10.5 cm -1 is amplified by a scaling factor of 300 for the sake of clarity. The continuum wavefunctions (blue) with explicit energies which are corresponding to the red stars in panel (a). For panels ( j, k, l) and zoomed-in panel ( j , k , l ), the integrand < vi br.|D|cont. > (green) and the cummulative sum (blue).

Figure 3 . 7 :

 37 Figure 3.7: Left: the exit channel 4 1 Σ + (black) and the shifted potential (red); right: the corresponding cross sections in the case of an initial binding energy of 10.5 cm -1 state of 2 1 Σ + to the exit 4 1 Σ + . The dashed pink arrow indicates the transition energy of 1064 nm laser.

Figure 3 . 8 :

 38 Figure 3.8: Potential energy curves (bold curves) are labelled by (a)Ω, (b)Ω, (c)Ω and (d)Ω for Ω = 0 +/-, 1, 2 in Hund's case c basis. The corresponding potential energy curves (thin curves) in Hund's case a basis are presented, respectively. The dashed pink arrow indicates the 1064 nm laser. The exit kinetic energy of the system is ∼ 5350 K or ∼ 6470 K with respect to 2 D 5/2 or 2 D 3/2 , respectively.

Figure 3 . 9 :

 39 Figure 3.9: From the top to the bottom as a function of the internuclear distance in atomic units: (1-2) the proportion of 4 1 Σ + and 3 3 Π for (a)0 + and (b)0 + states, respectively; (3) TEDMs in Hund's case (a); (4-5) TEDMs in Hund's case (c).

Figure 3 . 10 :

 310 Figure 3.10: From the top to the bottom as a function of the internuclear distance in atomic units: (1-2) the proportion of 4 3 Σ + and 3 3 Π for (a)0 -and (b)0 -states, respectively; (3) TEDMs in Hund's case (a); (4) TEDMs in Hund's case (c).

Figure 3 . 11 :

 311 Figure 3.11: From the top to the bottom as a function of the interatomic distance in atomic units: (1-4) the proportion of 4 3 Σ + , 3 3 Π, 2 1 Π and 2 3 ∆ for (a)1, (b)1, (c)1 and (d)1 states, respectively; (5) TEDMs in Hund's case (a); (6-7) TEDMs in Hund's case (c).

Figure 3 . 12 :

 312 Figure 3.12: From the top to the bottom as a function of the interatomic distance in atomic units: (1-3) the proportion of 3 3 Π, 2 1 ∆ and 2 3 ∆ for (a)2, (b)2 and (c)2 states, respectively; (4) TEDMs in Hund's case (a); (5) TEDMs in Hund's case (c).

Figure 3 . 13 :

 313 Figure 3.13: The unscale photodissociation cross section from several bound levels with energies (-0.1 cm -1 → -21.3 cm -1 ) of either 2 1 Σ + or 1 3 Σ + entrance channel, towards the exit channels of Ω = 0 +/-as a function of the transition energy in Hund's case c basis. The vertical dashed line locates the position of 1064 nm.

Figure 3 . 14 :

 314 Figure 3.14: The unscale photodissociation cross section from several bound levels with energies (-0.1 cm -1 → -21.3 cm -1 ) of either 2 1 Σ + or 1 3 Σ + entrance channel, towards the exit channels of Ω = 1 as a function of the transition energy in Hund's case c basis. The vertical dashed line locates the position of 1064 nm.

Figure 3 . 15 :

 315 Figure 3.15: The unscale photodissociation cross section from several bound levels with energies (-0.1 cm -1 → -21.3 cm -1 ) of either 2 1 Σ + or 1 3 Σ + entrance channel, towards the exit channels of Ω = 2 as a function of the transition energy in Hund's case c basis. The vertical dashed line locates the position of 1064 nm.

Figure 3 . 16 :

 316 Figure 3.16: The schematic of single ion Ba + recycling reactions: (1) Ba + +H 2 → HBa + +H; (2) HBa + +hν → Ba + (Hot)+H; (3) laser cooled Ba + (cold).

. 16 )Figure 3 . 17 :

 16317 Figure 3.17: From the top panel to the bottom panel, the potential energy curves of HBa + molecule in Hund's case (a) basis; the permanent electric dipole moment (PEDM) of 1 1 Σ + ; the selected TEDMs from 1 1 Σ + to 2, 3 1 Σ + and 1, 2 1 Π states.

Figure 3 . 19 :

 319 Figure 3.19: The cross section of 396.8 nm (black) and 271.7 nm (red) induced photodissociation as a function of the vibrational quantum number of 1 1 Σ + .

Figure 3 . 20 :

 320 Figure 3.20: The photodissociation cross section as a function of the transition energy regarding v = 5, 6, 7 of 1 1 Σ + . The blue dashed line marks the energy position of 271.7 laser.

3. 2 .Figure 3 . 21 :

 2321 Figure 3.21: The photodissociation rates for v = 0 as a function of the intensity of 271.7 nm laser.

Figure 4 . 2 :

 42 Figure 4.2: PEDMs of this work (solid lines) and Ref.[6] (dashed lines).

4. 3 . 93 Figure 4 . 3 :

 39343 Figure 4.3: TEDMs of this work (solid lines) and Ref.[6] (dashed lines).

94 4. 3 .Figure 4 . 4 :

 94344 Figure 4.4: SOCs between states associating with Rb(5s)+Ca + (4s) dissociation limit in the case of Omaga=0 + , 0 -,1 and 2.

4. 3 .

 3 RESULTS AND DISCUSSION 95 Table

Figure 4 . 6 :

 46 Figure 4.6: The PD cross sections as a function of binding energies of vibrational levels of the ground state 1 1 Σ + . The PD induced by 397 nm cooling laser from a vibrational level of 1 1 Σ + towards five exit channels. The PD threshold (v = 28) is marked by the red bar.

Figure 4 . 7 :

 47 Figure 4.7: The PD rates as a function of binding energies of vibrational levels of the ground state 1 1 Σ + , based on the PD cross sections displayed in Figure 4.6. The intensity of the focused 397 nm beam is 400 mW cm -2 [7].

Figure 4 . 8 :

 48 Figure 4.8: The reaction rate of spontaneous emission and stimulated absorption/emission as a function of the vibrational quantum number of the ground state 1 1 Σ + . The stimulated radiation rate is the sum of stimulated absorption and emission rates. Red vertical dashed line marks the PD threshold (v = 28). Pink vertical dashed lines indicate the crossing point v = 92 of curves of stimulated emission and absorption rates and the crossing point v = 118 of curves of spontaneous emission and stimulated absorption rates.

Figure 4 . 9 :

 49 Figure 4.9: For the initial vibrational levels v = 20, 30, 40, 50, and 60, the reaction rate of spontaneous emission and stimulated absorption/emission as a function of the vibrational quantum number of the ground state 1 1 Σ + .

106 4. 3 .

 1063 Figure 4.10: (a) Hund' case a PECs (dashed lines), and Hund' case c PECs (solid lines) labelled by (α) and (β) for Ω = 0 + , ; the zooms around the left crossing (b) and the right crossing (c).

Figure 4 . 1 3 A sp = 1 3

 413 10(a) shows the input PECs (dashed lines) and the output PECs (solid lines) obtained by diagonalizing the matrix. The asymptotic energy difference of 2 1 Σ and (β)0 + equals to

Figure 4 . 11 :

 411 Figure 4.11: The calculated NRCE transition probability |S i f | 2 between the entrance Rb(5s) + Ca + (4s) and the exit Rb + + Ca( 3 P) asymptotes, as a function of the partial wave number l. The peak of the selected collision energy indicates a shape resonance.

Figure 4 .

 4 Figure 4.12: a): The calculated cross section for the Rb(5s) + Ca + (4s) → Rb + + Ca( 3 P) charge exchange as a function of the collision energy (10 -6 K -1K): the quantum scattering results (black) and the Langevin cross section (blue); b): Based on the NRCE cross section (black) in panel (a), we calculate NRCE rates for non-thermalized rate (green) and Boltzmann thermalized rate (red), comparing with theoretical data (blue triangles) of [8] and the experimental data (black circles) of [9], as a function of the collision energy (10 -6 K -1K).

110 4. 3 .Figure 4 . 13 :

 1103413 Figure 4.13: This work (red solid line) and Ref. [8] (blue dashed line): (a) Hund' case a PECs 1 3 Π and 2 1 Σ + , the vertical dashed lines indicate the two real crossings; (b) the SOC between 1 3 Π and 2 1 Σ + , the vertical dashed lines mark the values of the SOC corresponding to the two real crossings; (c) The calculated NRCE cross sections (cm 2 ) as a function of the collision energy in unit (eV).

Figure 4 . 14 :

 414 Figure 4.14:The NRCE cross section (solid lines) regarding three dissociation limits of the exit channel 13 Π, E T hiswor k (blue), E N IST (green) and E Re f . (red)[START_REF] Zrafi | Ab initio electronic structure and prospects for the formation of ultracold Calcium-alkali-metal-atom molecular ions[END_REF]. Black dashed line is the Langevin cross section σ L . The colorful dashed lines are calculated by σ L sin 2 (∆δ 0 ).

[ 6 ]

 6 . The shifted PECs (1 3 Π with three dissociation limits) are represented in Figure B.1 in Appendix B.

Figure 5 . 2 :

 52 Figure 5.2: Experimentally measured decay of a Ba + prepared in the 2 D 3/2 (top panel) or 2 D 5/2 (bottom panel) state, interacting with an ultracold gas of atomicLi. An event is categorized as a survival, if the ion can be fluorescence-detected, either after a short Doppler-cooling period (top panel) or after switching on the 615 nm repumping laser for a short period (bottom panel). Each data point shows the averaged survival probability of at least 20 interaction events, the error bars representing the 1δ confidence interval. The black curve is a fit to the data with an exponential decay, where the shaded area represents the uncertainty of the fit. The non-unity survival probability at t = 0 ms originates from the finite size of the atomic cloud, meaning the ion starts interacting with the Li atoms while moving to its final position.

Figure 5 . 3 :

 53 Figure 5.3: Hund's case (a) LiBa + potential energy curves in the BF frame. The black circle locates the avoided crossing between the 2 1 Σ + and 3 1 Σ + PECs contributing the NRCE (red arrow): Li + +Ba(6s 2 , 1 S) ← Li(2s)+Ba + (5d).

Figure 5 . 4 :

 54 Figure 5.4:The R-dependent spin-orbit couplings regarding the first 3 dissociation limits, for molecular states labeled with Ω = 0 +/-, 1, 2, 3, the projection of the total electronic angular momentum on the molecular axis. We use the notation A denoting the matrix elements in the text.

Figure 5 . 5 :

 55 Figure 5.5: LiBa + potential energy curves for Ω = 0 + in Hund's case (c) basis. FSQ (blue arrow): Li(2s)+Ba + (D 3/2 ) ← Li(2s)+Ba + (D 5/2 ); NRCE (red arrow): Li + +Ba( 1 S) ← Li(2s)+Ba + (D 5/2,3/2 ); NRQ (orange arrow): Li(2s)+Ba + (6s) ← Li(2s)+Ba + (D 5/2,3/2 ).

  3 and 5.4, respectively. n and m are indices. Besides those of V are linearized around the avoided crossing, and the corresponding SOCs A are modified as well. The R-dependent Gaussian coupling G is modeled as W c e -(R-R c ) 2 /2δ 2 , where the estimated strength W c = 0.001795 a.u., the expected value of the distance R c = 11.06 a.u., the full width of the Gaussian distribution Γ = 2 2l n(2)δ with the standard deviation δ = 0.75 a.u.. Diagonalizing this matrix, we can extract Hund's case (c) PECs in Figure 5.5. The dissociation limit Li(2s)+Ba + (5d) actually splits into two limits, namely Li(2s)+Ba + ( 2 D 5/2,3/2 )

Figure 5 . 6 : 2 B 2 B 2 F 2 B

 562222 Figure 5.6: The zoomed-in Hund's case (c) potential energy curves. The LZ probabilities are marked by red and yellow arrows. Yellow arrows: assuming that the entrance channel is fully distributed on the Li(2s)+Ba + ( 2 D 5/2 ) limit with a unitary probability. After two adjacent avoided crossings, the probability of the NRCE is 2(P 5/2 T )(1 -P 5/2 T )(1 -P 5/2 B ) and the probability of the FSQ is 2(P 5/2 T )(1 -P 5/2 T )P 5/2 B . Red arrows: The same initial conditions are applied on the Li(2s)+Ba + ( 2 D 3/2 ) limit. The probability of the NRCE is 2(1 -P 3/2 B )P 3/2 B

Figure 5 . 7 :

 57 Figure 5.7: The cross sections of NRQ, NRCE, and FSQ in the entrance 2 D 5/2 (upper panel) and the cross section of NRQ and NRCE in the entrance 2 D 3/2 (lower panel), as a function of collision energy. The dominant partial wave is indicated for the relevant shape resonance.

Figure 5 . 8 : 130 5. 3 .

 581303 Figure 5.8: The non-thermalized (dashed lines) and thermalized (solid lines) rate constants of considered processes as a function of collision energy and T e f f . , respectively.

Figure 5 . 9 :

 59 Figure 5.9: Diagonalized Hund's case (e) potential energy curves for the first three dissociation limits considering the spin-orbit couplings and rotational couplings are displayed depending on parity e and f . parity e: the number of channels is 4, 9, 12, 13 for J=0, 1, 2, 3, respectively. Parity f : the number of channels is 3, 8, 11, 12 for J=0, 1, 2, 3, respectively. The channel number remains constant for each J ≥ 3 case. Two panels at the bottom represent the zoom-in PECs for the avoided crossings marked by black circles.

Figure 5 . 10 :

 510 Figure 5.10: Left panels: the parity-dependent cross sections of FSQ, NRCE, and NRQ for the entrances 2 D 5/2 and 2 D 3/2 as a function of the collision energy. The Langevin cross section and the Langevin-type cross section with a factor are indicated. Right panels: the non-thermalized (dashed lines) and thermalized (solid lines) reaction rate constants based on the corresponding cross sections shown in the right panels, as a function of the collision energy and T e f f . . The Langevin rate is indicated as well.

Figure 5 . 11 :

 511 Figure 5.11: Gaussian coupling G as a function a interatomic distance. Each curve is labeled by (mean value; half-width). (11.06;0.75) is selected in this work.

. 1 Ω = 2 :< 3 2 Ω = 1 :< 3 Π| 2 (< 3 Σ< 3 ΠΩ

 123213233 Π| Lz Ŝz | 3 Π >= 1 Lz Ŝz | 3 Π >= 0 < 3 Σ + | Lz Ŝz | 3 Σ + >= 0 < 3 Π| L+ Ŝ-+ L-Ŝ+ 2 | 3 Σ + >= 1 4 2A( 3 P) + 2A( 3 P)) = A( + | Lz Ŝz | 3 Σ + >= 0 + | Lz Ŝz | 3 Π + >= 1 P) + 2A( 3 P) + 2A( 3 P) + 2A( 3 P) = 0 + :The case of a 3 D atomic termThe following results are obtained along the same lines than in the 3 P case. The spin-orbit constant is defined as A( 3 D) = 2A d /5 = 2ε ( 3 D 3 -3 D 1 ) /5.

. 4 .Ω = 3 :< 3 Ω = 2 :< 3 < 3 2 < 3 2 | 3 Ω = 1 :< 3

 433233232313 ∆| Lz Ŝz | 3 ∆ >= 1 2 (A( 3 D) + A( 3 D)) = A( 3 D) ∆|L z S z | 3 ∆ >= 1 4 (-A( 3 D) + A( 3 D) -A( 3 D) + A( 3 D)) = 0 Π|L z S z | 3 Π >= 1 ∆| L + S -+ L -S + ∆|L z S z | 3 ∆ >= 1 2 (-A( 3 D) -A( 3 D)) = -A( 3 D)

2 2 (< 3 Σ 2 < 3 4 (< 3 Π

 2232343 6A( 3 D) + 6A( 3 D) + |L z S z | 3 Σ + >= 0 < 3 Π + |L z S z | 3 Π + >= 1 Σ + | L + S -+ L -S + 2 | 3 Π + >= 1 6A( 3 D) + 6A( 3 D) + 6A( 3 D) + 6A( 3 D) -|L z S z | 3 Π -the potential energy matrices for the molecular states including soc are displayed below. In each case, the corresponding dissociation limits are indicated by the atomic spectral term3 D J . Ĥ(|Ω| = 3) = V ( 3 ∆) + A( 3 D) ( 3 D 3 ) = 0 + ) = V ( 3 Π -) -A(

Figure B. 1

 1 Figure B.1 displays the potential energy surfaces of the 2 1 Σ + and 13 Π RbCa + states, where the entire curve of 13 Π is vertically shifted from the dissociation energy calculated in this work E T hiswor k , toward the experimental energy from the NIST database E N IS T , and the one from Ref.[START_REF] Zrafi | Ab initio electronic structure and prospects for the formation of ultracold Calcium-alkali-metal-atom molecular ions[END_REF] E Re f . . We estimated the transition probability at each of the crossings (hereafter referred to as left/right crossing) between the 2 1 Σ + and 1 3 Π PECs (Figure B.1) using the Landau-Zener model, for each pairs of curves. The results are reported in TableB.1: these probabilities are almost independent of the energy shifts, so that one may expect that no significant changes will be observed in the dynamics.However, we calculated the thermalized non-radiative charge exchange rates for each pair of curves (FigureB.2) based on the relevant cross section in Figure4.14. While the results of the computations using E T hiswor k are found in good agreement with the experimental values, we observe that the charge exchange rate can be significantly changed with the dissociation energy of the 1 3 Π outgoing channel, from few factors above 0.1 K, up to 2 or 3 orders of magnitude in the low temperature range.

Table B. 1 :

 1 The single-path Landau-Zener probability P l/r L Z based on the left (l) or the right (r) crossings between the 2 1 Σ + and 1 3 Π PECs of RbCa + , visible in FigureB.1, and using the spin-orbit couplings in Figure4

Figure B. 1 :

 1 Figure B.1: Hund's case a PECs 2 1 Σ + and 1 3 Π of RbCa + , where the curve of 13 Π with E T hiswor k is shifted to dissociation limits E Re f . andE N IS T , respectively.

Figure B. 2 :

 2 Figure B.2: The thermalized and non-thermalized NRCE rates as a function of the collision energy calculated for the three pairs of curves in Figure B.1. The non-thermalized rates exhibit richly shape resonances, while the curves of the thermalized rates are smooth. The black dots are experimental values.

Titre:

  Mod élisation des pertes d'ions dans les pi èges hybrides ions-atomes froids : photodissociation et processus non radiatifsIons mol éculaires froids, collisions ultra-froides, photodissociation, dynamique non radiative.

  . . . . . . . . . . . . . 22 2.6 Schematic representation of doable molecular transitions marked by arrows regarding different dynamics with or without lights.

Stimulated emission (red arrows): (1) bound to bound; (4) free to bound; (7) free to free. Stimulated absorption (green arrows): (2) bound to bound; (5) bond to free; (8) free to free. Spontaneous emission (orange arrows): (3) bound to bound; (6) free to bound; (9) free to free. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.7 Spherical polar coordinates (r, θ , φ). The point P in the rectangular Cartesian coordinate: x = rsinθ cosφ; y = rsinθ sin φ; z = r cosφ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.8 Coordinate system for a molecule AB with two electrons (e 1 , e 2 ).

  1 Σ + or 13 Σ + entrance channel, towards the exit channels of Ω = 2 as a function of the transition energy in Hund's case c basis. The vertical dashed line locates the position of 1064 nm. . . . . . . . 78 XI 3.[START_REF] Chen | An integrated space-to-ground quantum communication network over 4,600 kilometres[END_REF] The schematic of single ion Ba + recycling reactions:(1) Ba + +H 2 → HBa + +H; (2) HBa + +hν → Ba + (Hot)+H; (3) laser cooled Ba + (cold). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.17 From the top panel to the bottom panel, the potential energy curves of HBa + molecule in Hund's case (a) basis; the permanent electric dipole moment (PEDM) of 1 1 Σ + ; the selected TEDMs from 1 1 Σ + to 2, 3 1 Σ + and 1, 2 1 Π states. . . . . . . . . . . . . . . . 3.18 Top: The spontaneous emission rate, the stimulated absorption rate and the stimulated emission rate as a function of the vibrational quantum number of 1 1 Σ + at 300 K room temperature. Bottom: Black body radiation lifetime as a function of the vibrational quantum number of 1 1 Σ + at 300 K room temperature. . . 3.19 The cross section of 396.8 nm (black) and 271.7 nm (red) induced photodissociation as a function of the vibrational quantum number of 1 1 Σ + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20 The photodissociation cross section as a function of the transition energy regarding v = 5, 6, 7 of 1 1 Σ PEDMs of this work (solid lines) and Ref.[6] (dashed lines). . . . 4.3 TEDMs of this work (solid lines) and Ref.[6] (dashed lines). . . . 4.4 SOCs between states associating with Rb(5s)+Ca + (4s) dissociation limit in the case of Omaga=0 + , 0 -,1 and 2. . . . . . . . . . . 4.5 Calculated vibrational distributions of RbCa + as a function of binding energies of vibrational levels of the 1 1 Σ + ground state, created by radiative association. RbCa + is generated at 1 1 Σ + state by the RA from the initial 2 1 Σ + channel with selected collision temperatures 2.4 µK, 5.8 mK and 2.8 K. The red rule indicates the corresponding quantum numbers of the vibrational levels of 1 1 Σ + ground state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 The PD cross sections as a function of binding energies of vibrational levels of the ground state 1 1 Σ The PD rates as a function of binding energies of vibrational levels of the ground state 1 1 Σ + , based on the PD cross sections displayed in Figure 4.6. The intensity of the focused 397 nm beam is 400 mW cm -2 [7]. . . . . . . . . . . . . . . . . . . . . . . . 4.8 The reaction rate of spontaneous emission and stimulated absorption/emission as a function of the vibrational quantum number of the ground state 1 1 Σ + . The stimulated radiation rate is the sum of stimulated absorption and emission rates. Red vertical dashed line marks the PD threshold (v = 28). Pink vertical dashed lines indicate the crossing point v = 92 of curves of stimulated emission and absorption rates and the crossing point v = 118 of curves of spontaneous emission and stimulated absorption rates. . . . . 4.9 For the initial vibrational levels v = 20, 30, 40, 50, and 60, the reaction rate of spontaneous emission and stimulated absorption/emission as a function of the vibrational quantum number of the ground state 1 1 Σ + . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 (a) Hund' case a PECs (dashed lines), and Hund' case c PECs

+ . The blue dashed line marks the energy position of 271.7 laser. . . . . . . . . . . . . . . . . . . . 3.21 The photodissociation rates for v = 0 as a function of the intensity of 271.7 nm laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Hund case a PECs of RbCa + for different molecular symmetries, as a function of interatomic distance. The downwards dashed vertical arrow states the RA from initial channel 2 1 Σ + to final 1 1 Σ + . Upwards solid vertical arrows represent the energies of respective lasers, 397 nm, 780 nm, and 866 nm, from the last bound level of the ground state 1 1 Σ + towards five exit channels (2 1 Σ + , 3 1 Σ + , 4 1 Σ + , 1 1 Π, 2 1 Π) for the PD. . . . . . . . . . . . . . . . 4.2 + . The PD induced by 397 nm cooling laser from a vibrational level of 1 1 Σ + towards five exit channels. The PD threshold (v = 28) is marked by the red bar.101 XII LIST OF FIGURES 4.7

  5/2 -state. If the ion is not detected during any of the steps, the event is classified as a loss. . . . . . . . . . . . . . . . . . . . . . . . 5.2 Experimentally measured decay of a Ba + prepared in the 2 D 3/2

  The cross sections of NRQ, NRCE, and FSQ in the entrance 2 D 5/2 (upper panel) and the cross section of NRQ and NRCE in the entrance Hund's case a PECs 2 1 Σ + and 1 3 Π of RbCa + , where the curve of 13 Π with E T hiswor k is shifted to dissociation limits E Re f . andE N IS T , respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	XIV	LIST OF FIGURES
	5.6 The zoomed-in Hund's case (c) potential energy curves. The
		LZ probabilities are marked by red and yellow arrows. Yellow
	5.7	arrows: assuming that the entrance channel is fully distributed on the Li(2s)+Ba + ( 2 D 5/2 ) limit with a unitary probability. Af-ter two adjacent avoided crossings, the probability of the NRCE is 2(P 5/2 T )(1 -P 5/2 T )(1 -P 5/2 B ) and the probability of the FSQ is 2(P 5/2 T )(1 -P 5/2 T )P 5/2 B . Red arrows: The same initial conditions are applied on the Li(2s)+Ba + ( 2 D 3/2 ) limit. The probability of the NRCE is 2(1 -P 3/2 B )P 3/2 undergoes through one avoided crossing. B
		. .
		5.5 LiBa + potential energy curves for Ω = 0 + in Hund's case (c) basis. FSQ (blue arrow): Li(2s)+Ba + (D 3/2 ) ← Li(2s)+Ba + (D 5/2 ); NRCE (red arrow): Li + +Ba( 1 S) ← Li(2s)+Ba + (D 5/2,3/2 ); NRQ (orange arrow): Li(2s)+Ba + (6s) ← Li(2s)+Ba + (D 5/2,3/2 ). . . . . . . . . . .

2 , 1 S) ← Li(2s)+Ba + (5d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The R-dependent spin-orbit couplings regarding the first 3 dissociation limits, for molecular states labeled with Ω = 0 +/-, 1, 2, 3, the projection of the total electronic angular momentum on the molecular axis. We use the notation A denoting the matrix elements in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 D 3/2 (lower panel), as a function of collision energy. The dominant partial wave is indicated for the relevant shape resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 The non-thermalized (dashed lines) and thermalized (solid lines) rate constants of considered processes as a function of collision energy and T e f f . , respectively. . . . . . . . . . . . . . . . . . . . . . 5.9 Diagonalized Hund's case (e) potential energy curves for the first three dissociation limits considering the spin-orbit couplings and rotational couplings are displayed depending on parity e and f . parity e: the number of channels is 4, 9, 12, 13 for J=0, 1, 2, 3, respectively. Parity f : the number of channels is 3, 8, 11, 12 for J=0, 1, 2, 3, respectively. The channel number remains constant for each J ≥ 3 case. Two panels at the bottom represent the zoom-in PECs for the avoided crossings marked by black circles. 5.10 Left panels: the parity-dependent cross sections of FSQ, NRCE, and NRQ for the entrances 2 D 5/2 and 2 D 3/2 as a function of the collision energy. The Langevin cross section and the Langevintype cross section with a factor are indicated. Right panels: the non-thermalized (dashed lines) and thermalized (solid lines) reaction rate constants based on the corresponding cross sections shown in the right panels, as a function of the collision energy and T e f f . . The Langevin rate is indicated as well. . . . . . . . . . . 5.11 Gaussian coupling G as a function a interatomic distance. Each curve is labeled by (mean value; half-width). (11.06;0.75) is selected in this work. . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1

Table 2 .

 2 1: The angular momenta and quantum numbers relevant for a diatomic molecule. The projection quantum numbers are signed quantities. The nuclear spin is not considered.

	Name of angular momentum	Vector operator	Vector	Projection onto axis
	Total electronic orbital angular momentum	L	L	Λ
	Total electronic spin	Ŝ	S	Σ
	Total electronic angular momentum	ĵ

  the set of the orthogonal eigenfunctions φ k ( r) including both the discrete and the continuum states is complete. c

k (t) is a time-dependent coefficient. |c k (t)| 2 indicates the probability of finding species in a state k at a certain time t. When the perturbation term is weak, we can expand the coefficient in powers of the parameter λ

  .142)2.4. ELEMENTS OF SCATTERING THEORYRegarding the condition m = 0, the Eqn [2.123] can be re-written as

					ψ	r→∞ --→ e ikz + f (θ )	e ikr r	.	(2.143)
	Substituting the Eqn [2.130] into Eqn [2.143] and comparing it with the Eqn [2.142], we get
	∞ l=0	(2l + 1)i l 1 kr	sin kr -	1 2	lπ P l (cosθ ) + f (θ )	e ikr r
						=	∞ l=0	A l kr	sin(kr -	1 2	lπ + δ

l )P l (cos(θ )).

  (cm -1 ) R e (cm -1 ) B e (cm -1 ) ω e (cm -1 ) ω e χ e (cm -1 ) T e (cm -1 )

	98	4.3. RESULTS AND DISCUSSION
	2 , 1 S) Rb + +Ca(4s4p, 3 P) Rb(5s)+Ca + (4s) Rb + +Ca(4s3d, 3 D) Rb(5p)+Ca + (3d) Rb(5p)+Ca + (4s) 5 1 Σ + (1st min) 1 1 Σ + 1 3 Σ + 1 3 Π 2 1 Σ + 2 3 Σ + (1st min) 2 3 Σ + (2ed min) 3 3 Σ + 4 3 Σ + 3 1 Π 3 3 Π (1st min) 3 3 Π (2ed min) 6 1 Σ + 5 3 Σ + 5 1 Σ + (2ed min) Asymptote States 2 3 Π 1 3 ∆ Rb + +Ca(4s3d, 1 D) 3 1 Σ + 1 1 Π 1 1 ∆ 2 1 Π Rb + +Ca(4s4p, 1 P) 4 1 Σ + Asymptote States Π Π ∆ ∆	3811 3714 3666 3851 3717 3730 7559 7494 7455 3195 3022 3308 1249 1160 1284 1272 1170 -1727 -1839 -1496 181 137 121 1353 2051 1801 1745 275 131 233 281 202 685 937 1575 -350 1216 D e Ref. 7.98 0.034540 74.38 0.31 0 This work 7.97 0.034735 73.53 0.35 0 [6] 8.06 0.034100 73.90 -0 [8] 7.96 0.034600 73.02 -0 [44] 8.26 0.032371 73.43 0.50 0 [9] 8.00 ---0 [144] 8.88 0.027877 65.16 0.13 This work 8.90 0.027855 64.21 0.16 [6] 9.15 0.026367 77.92 0.20 [9] 7.40 0.039834 76.35 0.38 7.38 0.040511 75.86 0.40 [6] 7.63 ---[9] 12.88 0.013254 30.24 0.15 This work 12.99 0.010376 29.58 0.15 [6] 12.82 0.013388 28.20 0.09 [9] 12.86 0.010376 29.89 -[44] 13.00 ----[144] 8.32 0.031767 62.14 0.23 This work 8.27 0.032261 61.54 0.49 [6] 8.31 0.031976 --[9] 20.02 0.005488 9.85 0.16 This work 20.82 0.005090 8.37 0.13 [6] 17.82 ---11.54 0.016521 106.02 5.23 This work [9] 13.61 0.011912 16.90 0.31 [6] 18.50 0.006404 18.53 0.04 This work 18.57 0.006398 20.67 0.04 [6] 12.73 0.013571 16.24 0.52 This work 17.00 0.007635 8.70 0.07 [6] 13.48 0.012109 --This work 15.60 0.009032 14.82 0.34 This work 16.19 0.008418 12.17 0.16 [6] 15.40 0.009268 55.72 3.13 This work 14.15 0.010978 53.83 2.07 This work 8.47 0.030755 46.14 0.51 9.59 0.023907 50.28 1.53 20.26 0.005343 20.29 0.05 This work This work [6] This work 1481 11.42 0.016918 61.54 0.49 [6] 2894 8.69 0.029145 58.41 0.21 This work 3027 8.67 0.029353 58.30 0.21 [6] 3226 8.38 0.031347 62.58 0.22 This work 3293 8.35 0.031646 61.17 0.24 [6] 2943 8.71 0.028982 60.68 0.22 This work 3175 8.65 0.029489 62.04 0.34 [6] 3499 8.42 0.031054 58.39 0.21 This work 3542 8.43 0.031048 57.48 0.16 [6] 3981 8.32 0.031772 64.62 0.19 This work 4184 8.29 0.032105 63.51 0.18 [6] 1016 14.89 0.009952 22.61 0.31 1923 8.46 0.030752 51.10 0.38 This work [6] 973 15.01 0.009756 19.42 0.19 This work D Ref. 678 14.96 0.009827 25.87 0.33 32402 This work 997 14.00 0.011228 39.73 1.05 32083 This work 1015 13.59 0.011901 27.39 0.16 32065 This work 1162 13.16 0.012701 29.36 0.16 31918 This work	97 4.3. RESULTS AND DISCUSSION 96 4.3. RESULTS AND DISCUSSION

e (cm -1 ) R e (cm -1 ) B e (cm -1 ) ω e (cm -1 ) ω e χ e (cm -1 ) T e (cm -1 )

  1 Σ + is approximately located around 10 -13 a.u. due to the radial coupling. Such avoided crossing is expected to dominate the NRCE in most cases. Along with other interactions involved in Hamiltonian, we can already distinguish the main couplings acting over consecutive distance ranges, such as SOCs (R < 10 a.u.), SOCs, and the radial coupling G (10 a.u. < R < 13 a.u.), SOCs and the rotational coupling H r ot at larger R, and atomic SOCs at infinity. To capture the role of these couplings during a collision, the idea of our theoretical work is to explore the effects step by step: (1) a semi-classical Landau-Zener (LZ) model only dependent on G in Body-Fixed (BF) frame, (2) a Few-Channel Landau-Zener (FCLZ) model including G and SOCs in BF frame, (3) a Few-Channel Quantum Scattering (FCQS) model with G, SOCs and the decoupled rotation of the nuclei H r ot in Space-Fixed (SF) frame, and (4) a Multi-Channel Quantum Scattering (MCQS) model considering explicitly G, SOCs and the coupled rotation of the nuclei H r ot in the SF frame.

  2 , the scattering wave function is defined by Riccati-Bessel functions at infinity. At the end, we can obtain the reaction matrix K holding5.3. THEORETICAL MODELS AND RESULTS127close and open channels and the S matrix containing only open channels. The S matrix with a large distance of 10000 a.u. is verified in this work. Note that the long-range polarization potential is -C 4 /R4 , where C 4 is the half of the static dipole polarizability a 0 of the neutral atom. In this work we take C 4 = a 0 /2 = 82.05 a.u.[START_REF] Tang | Nonrelativistic ab initio calculations for 2 2 s, 2 2 p, and 3 2 d lithium isotopes: Applications to polarizabilities and dispersion interactions[END_REF][START_REF] Zhang | Dispersion coefficients of the excited states of lithium atoms[END_REF] and 137 a.u.[START_REF] Lim | Four-component and scalar relativistic douglas-kroll calculations for static dipole polarizabilities of the alkalineearth-metal elements and their ions from Ca n to Ra n (n = 0,+1,+2)[END_REF] for Li(2s)+Ba + (6s/5d) and Li + +Ba + (6s 2 , 1 S) limits, respectively. Note that the relevant numerical calculation details are consistent with the following MCQS model.The total cross sections are computed by the off-diagonal elements of S matrix as a sum over partial waves l

	σ

2.4. ELEMENTS OF SCATTERING THEORY

Acknowledgements

All of a sudden, I am a Doctor! ACKNOWLEDGEMENTS Pandey, Melek Ben Khalifa, Adrien Devolder, Lucas Lassablière, and most recently, Charbel Karam. I am confident that Charbel will have fruitful and stimulating doctoral training in this team.

I would like to gratefully acknowledge my jury members: Carine Clavaguera, Nadine Halberstadt, Andrea Simoni, Stefan Willitsch, Piotr Zuchowski. Many thanks for all the questions during my defense. Nadine Halberstadt and Andrea Simoni, thanks again for the corrections and reports for my thesis.

Thank you Alexander Dietmar Dörfler (University of Basel, Switzerland), Pascal Weckesser and Leon Karpa (University of Freiburg, Germany), Maciej Kosicki (Nicolaus Copernicus University, Poland) for my visiting time in your group.

I would like to thank Tobias Schätz, Pascal Weckesser, and Fabien Thielemann for the inspiring discussions about our work. Pascal, many thanks for the nice talk and your visit to our collaboration. I wish you all the best in your defense and your future. I also hope our beautiful results will be published as soon as possible.

I would like to thank funding for this project from China Scholarship Council (CSC) with grant number 201706240178 and University Paris-Saclay.

The most important old man in my life, my father Gaicheng Xing, passed away on Sept. 24, 2018. I have so many regrets for him. In terms of Chinese tradition, I do not have the opportunity to do my filial duty. Thanks to my sister Linghua Xing who is taking care of my mother Bai'e Ji in these years. And I thank my sweet little niece Xueqi Wang for cheering up her grandma. Last but not least, I would like to thank all my friends.

To my father

Chapter 5

Competing non-radiative processes of a single excited Ba + in an ultracold Li bath

Please note that this chapter is set up as a draft of a forthcoming paper

Xiaodong Xing 1 , Pascal Weckesser 2 , Fabien Thielemann 2 , Romain Vexiau 1 , Ting Xie 1 , Eliane Luc-Koenig 1 , Nadia Bouloufa-Maafa 1 , Tobias Schätz Albert-Ludwigs-Universität Freiburg, Physikalisches Institut, 79104 Freiburg, Germany

Introduction

Exceptional control of ultracold chemical reactions at a single quantum level is an ambitious and challenging scientific frontier. The experiments of immersing a single trapped ground-state ion in the neutral ultracold gas achieved remarkable advances [START_REF] Saito | Characterization of charge-exchange collisions between ultracold 6 Li atoms and 40 Ca + ions[END_REF][START_REF] Haze | Cooling dynamics of a single trapped ion via elastic collisions with small-mass atoms[END_REF][START_REF] Ben-Shlomi | Direct observation of ultracold atom-ion excitation exchange[END_REF][START_REF] Sikorsky | Phase locking between different partial waves in atom-ion spin-exchange collisions[END_REF][START_REF] Sikorsky | Spincontrolled atom-ion chemistry[END_REF][START_REF] Zipkes | A trapped single ion inside a Bose-Einstein condensate[END_REF][START_REF] Sayfutyarova | Charge transfer in cold Yb + +Rb collisions[END_REF][START_REF] Ratschbacher | Decoherence of a single-ion qubit immersed in a spin-polarized atomic bath[END_REF][START_REF] Fürst | Prospects of reaching the quantum regime in Li-Yb + mixtures[END_REF][START_REF] Joger | Observation of collisions between cold Li atoms and Yb + ions[END_REF][START_REF] Fürst | Dynamics of a single ion-spin impurity in a spin-polarized atomic bath[END_REF][START_REF] Hirzler | Experimental setup for studying an ultracold mixture of trapped Yb + -6 Li[END_REF] in the uncovering of quantum mechanisms, for example, impurity physics, many-body physics, sympathetic cooling of ions, and even Feshbach resonances in ultracold ion-atom systems. Nonetheless, various competing reactions between a single excited ion and ultracold atoms can drive the complexity of the full quantum control to another degree, due to an abundance of emergent reactive paths and mutual interactions induced by the electronic spin, the electronic orbital momentum and the rotational momentum which are strongly depending on the system.

In recent years some experimental groups [START_REF] Saito | Characterization of charge-exchange collisions between ultracold 6 Li atoms and 40 Ca + ions[END_REF][START_REF] Ben-Shlomi | Direct observation of ultracold atom-ion excitation exchange[END_REF][START_REF] Joger | Observation of collisions between cold Li atoms and Yb + ions[END_REF] are motivated by utilizing the hybrid trap to characterize the reactive collisions involving the metastable state of the ion which is involved in the laser cooling process. This Chapter 6

Summary and outlook

Summary

My doctoral project was to theoretically investigate photo-assisted dynamics and non-radiative scattering in ultracold hybrid atom-ion experiments. This work has strong connections with several well-known experimental groups regarding different systems: Rb-Ba + (Ulm, Germany), H 2 -Ba + (Aarhus, Denmark), Rb-Ca + (Basel, Switzerland), Li-Ba + (Freiburg, Germany). The displayed methodologies and theoretical models could well be adapted to other atom-ion experiments, like Rb-Sr + (Rehovot, Israel) previously treated in a simplified way in our group, and Li-Ca + (Tokyo, Japan).

(1) I first exposed the various theoretical tools that I used to accurately interpret the experimental outcomes. The basic equations for radiative processes have been systematically re-derived in a consistent way based on dimensionality considerations, hopefully clarifying the different forms and units which appeared in the literature. The non-radiative problem of interest in this thesis is mainly inelastic scattering collisions on the excited states, where the spin-orbit coupling and the rotational coupling play a crucial role. The spin-orbit matrices are originally derived for a two-electron system when both electrons sit on the same atom at large interatomic distance. These matrices are well integrated into the ab initio code developed by our group, which can generate distance-dependent spin-orbit coupling. Besides, the general formula for rotational couplings was derived based on SF-BF frame transformation.

(2) Rb-Ba + : I explored the influence of a laser with wavelength 1064 nm on a weakly-bound molecule RbBa + created by a three-body collision, Rb+Rb+Ba + , and proposed a photodissociation model based on an atomic spin-orbit coupling scheme to trace the dissociated Ba + (Hot) ion, and to acknowledge the destruction of RbBa + . It is worth noting that a distance-independent spin-orbit coupling was used as the predicted dynamical region concerns quite large interatomic distances. This yielded a good agreement with experimental observations.

Appendix A Spin Orbit Coupling for 2-electron atom

In this appendix, I present the derivation of the spin-orbit coupling (soc) matrices in the case of an atom with two valence electrons, like alkaline-earth atoms. Indeed, when a singly-charged alkaline-earth ion collides with a neutral alkali atom in an hybrid traps, charge exchange can happen, so that the products are a neutral alkaline-earth atom and an alkali-metal ion. The derivation of these matrices, while conventional, yields results which are not easily available in the literature, and thus are recalled in the following. Two situations relevant for my work will be presented: the spin-orbit for a 3 P atom, and for a 3 D atom. An approximate soc matrix can be derived in both cases for the lowest atomic state of such symmetries, as those levels are quite isolated from their singlet counterpart, namely 1 P and for a 1 D. In a second step, we will include the two latter states in our description in order to properly characterize their influence on the triplet states of interest.

Useful formulas

I first recall basic formula for the sake of completeness.

The ladder operators associated to the electronic angular momentum and spin, and their action of the eigenfunctions:

The electronic spin orbit operator:

The effective operator involving the quantum numbers for the total electronic angular momentum L and total electronic spin S:

where

where ξ l i is for one electron i. For a p orbital electron we have:

For a d electron we have:

The two-electron wavefunction can be expressed as a linear combination of atomic orbitals,

where N is a normalization factor. This form allows for an easy determination of the soc matrices, which are then parametrized by the experimental splitting between energy levels of the same multiplet.

In the case of an alkali-metal ion interacting with a neutral alkaline-earth atom, the molecular problem is identical to the atomic one. In the following, we will consider the total angular momentum J of the neutral atom, its projection Ω on the molecular axis z. The projection L z of the electronic angular momentum L on z will be labeled with the molecular notation, namely Σ for L z = 0, Π for L z = 1, etc. Two cases are treated: 3 P and 3 D atomic terms.

The case of a 3 P atomic term

In atomic physics, the Landé interval rule states that the intervals between two adjacent energy levels are proportional to the energy amplitude A( 3 P) of the spin-orbit multiplet. For 3 P, the total angular momentum J = 0, 1, 2, and

To summarize, the potential energy matrices for the molecular states including soc are displayed below, assuming for convenience the coupling constant

. In each case, the corresponding dissociation limits are indicated by the atomic spectral term 3 P J .

The LCAO wave functions for 3 P, for each Ω value.

State

Wave function

]

]

] Using the experimental value for the spin-orbit splittings ε ( 3 P 2 -3 P 0 ) , the diagonalization of these matrices provide the energy of dissociation limits (or of atomic energy levels 3 P J ). These values are collected in Table A.2 for all the alkaline-earth atoms, assuming that the 3 P 0 is positioned at 0 energy. By construction of these matrices, the 3 P 2 is always found at the right position, while the 3 P 1 level is more and more shifted as the soc increases. This is a manifestation that even if the 1 P level is quite far in energy for all these species, it must be taken into account in the description. The soc matrices for Ω = 1, 0 + must be modified as follows, using the wave functions reported in Table A. [START_REF] Julian | Optical trapping of ion Coulomb crystals[END_REF]:

For Ω = 0 + ,

The potential energy matrices for the molecular states are now modified as (with A = A( 3 P)/2):

Table A.3: The LCAO wave functions for 1 P, for each Ω value.

State

Wave function

]

State

Wave function

]

]

]

] Using the experimental value for the spin-orbit splittings ε ( 3 D 3 -3 D 1 ) , the diagonalization of these matrices provide the energy of dissociation limits (or of atomic energy levels 3 D J ). These values are collected in Table A.5 for all the alkaline-earth atoms, assuming that the 3 D 1 is positioned at 0 energy. By construction of these matrices, the 3 D 3 is always found at the right position, while the 3 D 2 level is more and more shifted as the soc increases. This is again a manifestation that even if the 1 D level is quite far in energy for all these species, it must be taken into account in the description. The soc matrices for Ω = 2, 1, 0 + must be modified as follows, using the wave functions reported in Table A.6:

Appendix C

Selected channels in SF frame for s + s dissociation limit

For J = 0 and parity e [(-1) l+l A +l B = (-1) 0 ], i.e., l = 0, we have one channel |J, j A , j B , j = j A + j B , l = |J -j| > for A(s 1/2 ) -B(s 1/2 ) system:

For J = 0 and parity f [(-1) l+l A +l B = (-1) 0+1 ], i.e., l = 1, we have one channels:

For J = 1 and parity e [(-1) l+l A +l B = (-1) 1 ], i.e., l = 1, we have two channels:

For J = 1 and parity f [(-1) l+l A +l B = (-1) 1+1 ], i.e., l = 0, 2, we have two channels:

For J ≥ 2, each pairty has two channels, and so on and so forth.

Appendix D

Selected channels in SF frame for s + p dissociation limit

For J = 0 and parity e [(-1) l+l A +l B = (-1) 0 ], i.e., l = 1, we have two channel

For J = 0 and parity f [(-1) l+l A +l B = (-1) 0+1 ], i.e., l = 0, 2, we have one channels:

For J = 1 and parity e [(-1) l+l A +l B = (-1) 1 ], i.e., l = 0, 2, we have five channels: