Riccardo Spezia

Cédric Nicolas

François-Xavier Coudert

Pierre Archirel

Rodolphe Vuilleumier

Anne Boutin
email: anne.boutin@lcp.u-psud.fr

Reactivity of an excess electron with monovalent cations in bulk water by mixed quantum classical molecular dynamics simulations

A mixed quantum classical molecular dynamics simulation of the silver and sodium cations in presence of an excess electron is reported. The silver cation is shown to be reduced by the hydrated electron and to form a stable, highly polarized, neutral atom. On the contrary, the sodium cation is not reduced and a metastable contact ion pair is observed. The resulting absorption spectra of both species are compared with experiments and shown to be in good agreement. Furthermore, the free energy curve for the charge separation was calculated and rationalized in terms of a thermodynamic cycle. Finally, a direct, reactive, molecular dynamics trajectory provides some useful informations on the reduction mechanism.

I. INTRODUCTION

Understanding the elementary chemical events in terms of basic physical properties is a subject of considerable interest today. Among those, the electron transfer process has been widely studied, due to the importance of redox reactions in everyday physical chemistry and biophysics. The reduction of a metal ion was thoroughly investigated using pulse radiolysis techniques [START_REF] Buxton | Critical review of rate constants for reactions hydrated electrons, hydrogen atoms and hydroxyl radicals in aqueous solution[END_REF]. Several theoretical investigations have also been reported [START_REF] Sprik | Electron-ion interactions and ionization in a polar solvent[END_REF][START_REF] Borgis | Excited states of hydrated electron and aqueous chloride by computer simulations[END_REF][START_REF] Staib | Reaction pathways in the photodetachment of an electron from a queous chloride: A quantum molecular dynamics study[END_REF]. Even in the simplest case of metal atoms in polar solvents, the optical and electrical properties of the solvated atom, which strongly depend on the metal-solvent system, are not fully understood.

In some cases, such as sodium in water, the ionization is complete and leads to two well separated solvated species, i.e. the electron and the ion. In contrast, the reduction of silver cations in a radiolysis experiment or when a solvated silver salt is exposed to light leads to the formation of silver atoms, followed by a multistep coalescence into charged metallic clusters [START_REF] Mostafavi | Pulse radiolysis study of absorption spectra of Ag 0 and Ag 2 + in water from room temperature up to 380 • C[END_REF]. The role of the solvent is of critical importance as illustrated by the effect of the polarity of the solvent on the optical properties of solvated metal atoms [START_REF] Belloni | Ultrafast reaction Dynamics and Solvent Effects[END_REF]. The solvent indeed mediates effective interactions between the ion and the electron. These interactions depend on both the solvent and the ion, or atom properties.

The chemical nature of the cation is crucial for the equilibrium of the elementary redox processes in solution :

e - aq + M + M 0 (1)
Silver cation solutions have been widely studied, owing to the existence of a monoelectronic reduction of the ion to the zerovalent state, to the stability of the bare cation in solution and to the easy complexation of the cation with various ligands. The silver aqueous solution was the first system studied by pulse radiolysis [START_REF] Mulvaney | Formation of unstabilized oligomeric silver clusters during the reduction of Ag + ions in aqueous solution[END_REF] and was recently revisited [START_REF] Mostafavi | Pulse radiolysis study of absorption spectra of Ag 0 and Ag 2 + in water from room temperature up to 380 • C[END_REF]. The optical absorption spectrum of transient single silver atoms has been observed in various solvents. This is followed by dimerization and multistep silver cluster formation.

In contrast, the alkaline and earth-alkaline cations are not spontaneously reduced by an excess electron in solution, and only contact pairs are believed to be formed [START_REF] Gelabert | Short-time electron transfer processes in ionic aqueous solution: Counterion and H/D isotope effects on electron-atom pairs relaxation[END_REF][START_REF] Renou | Solvated electron pairing with earth alkaline metals in THF. 1. Formation and structure of the pair with divalent Magnesium[END_REF].

Computer simulations are expected to contribute to the understanding of the difference in behavior between the two opposite cases: silver (Ag + ) and sodium cation (Na + ) in presence of an excess electron in bulk water. Here, we report a simulation study on these systems using mixed quantum classical molecular dynamics. In the following section, the simulation technique is presented. The results of this study are reported in section III and we finally conclude in section IV.

II. MIXED QUANTUM CLASSICAL MOLECULAR DYNAMICS

We have performed a mixed quantum classical molecular dynamics (QCMD) of an excess electron in bulk water and also in presence of a silver or sodium cation. The excess electron is treated quantum mechanically, using the Born-Oppenheimer approximation: a certain solvent/cation configuration S generates an effective electron-solvent/cation interaction potential V int ( r, S) where r denotes the electronic coordinates. The stationary Schrödinger equation for the excess charge in this effective potential is then solved at fixed solvent/cation coordinates:

(T e + V int ( r, S))ψ n ( r, S) = E n (S)ψ n ( r, S) (2)
where T e denotes the electronic kinetic energy operator. The solution of the Schrödinger equation gives the electronic energy levels E n (S) and the corresponding wave function ψ n ( r, S), by diagonalization of Eq. 2 at each time step, i.e. for each solvent/cation configuration. Once the wave function is obtained, the quantum contribution to the forces acting on each classical particle is determined via the Hellmann-Feynman theorem [START_REF] Feynman | Forces in molecules[END_REF][START_REF] Hellmann | Einführung in die quantenchemie[END_REF] for each specific quantum state of the electron.

In the present work, the forces are always computed for the ground state of the excess electron (n = 0). The single-electron wave functions ψ n ( r, S) are expanded into a basis of 7 × 7 × 7 spherical gaussian functions centered on the nodes of a cubic lattice. The basis set parameters are identical to those used in Ref. [START_REF] Nicolas | Molecular simulation of a hydrated electron at different thermodynamic state points[END_REF] for simulation of a hydrated electron: the gaussian exponent α is 0.357 Å-2 and the basis grid extension is 10 Å, which corresponds roughly to half of the simulation box length. For the excess electron-water interaction we used the pseudo-potential developed by Turi and Borgis [START_REF] Turi | Analytical investigations of an electron-water molecule pseudo potential. II. Development of a new pair potential and molecular dynamics simulations[END_REF]. This pseudo-potential is based on a quantum ab initio calculation for one water molecule plus an additional electron confined in a box in the static exchange theory limit. Details on the use of this pseudo-potential with a gaussian grid basis set can be found in Ref. [START_REF] Nicolas | Molecular simulation of a hydrated electron at different thermodynamic state points[END_REF].

The electron/cation interaction was modeled using the one-electron semi-local pseudopotential proposed by Durand and Barthelat [START_REF] Durand | A theoretical method to determine atomic pseudopotentials for electronic structure calculations of molecules and solids[END_REF]. The parameters were re-optimized in order to reproduce the properties of the isolated atoms using the gaussian grid basis set used in the MD simulations. More details on the pseudo-potential parameters can be found in ref [START_REF] Spezia | Molecular dynamics simulations of the Ag + or Na + cation with an excess electron in bulk water[END_REF].

Water and cations interact via classical interaction potentials made of a coulombic and a 6-12 Lennard Jones (LJ) term. Water-water interactions are described by the SPC model [START_REF] Berendsen | Interaction models for water in relation to protein hydratation[END_REF], while Na + -water interaction by the Dang model [START_REF] Dang | Mechanism and thermodynamics of ion selectivity in aqueous solutions of 18-Crown-Ether: a molecular dynamics study[END_REF]. For Ag + -water interaction there is no 6-12 LJ parameters in the literature, to our knowledge. We have thus adjusted silver-water LJ parameters in order to better reproduce structural and energetic properties of silver cation in water, performing purely classical MD runs of Ag + in 300 SPC water molecules, with the same simulation conditions used for the QCMD simulation. We obtained Ag + /O = 0.40 kJ/mol and σ Ag + /O = 2.78 Å, with a point charge on silver for coulombic interactions, q Ag + = +1e [START_REF] Spezia | Molecular dynamics simulations of the Ag + or Na + cation with an excess electron in bulk water[END_REF].

The simulations were carried out in a box of 300 and 800 SPC water molecules, a cation, Ag + or Na + , and an excess electron at ambient condition (T = 298 K and ρ = 1 g/cm 3 ).

Standard periodic boundary conditions were used with the Ewald summation technique to calculate the electrostatic interactions. The simulations were performed in the NVT ensemble using the Nosé-Hoover thermostat [START_REF] Nosé | A unified formulation of the constant temperature molecular dynamics methods[END_REF][START_REF] Hoover | Canonical dynamics: Equilibrium phase-space distributions[END_REF] to ensure isothermal conditions. The integration of translation and rotation equation of motion was performed using the Gear predictor-corrector algorithm. The time step was fixed at 0.5 fs and simulations were run typically during a few tens of picoseconds after a stabilization time. The Lanczos algorithm (ARPACK library) was used to solve the electron eigenvalue problem.

III. RESULTS AND DISCUSSIONS A. Absorption spectra

To calculate the electron absorption spectrum we used the cumulant expansion and the vibration-rotation decorrelation hypothesis [START_REF] Tarjus | Theory of infrared and raman spectra of diluted van der Waals solutions: Simple product approximation and its extension[END_REF], leading to the formula

I(E) = E(1 -e -βE ) M n=1 |µ 0n | 0 exp - (E -E 0n 0 ) 2 δE 2 0n (3)
where E 0n ≡ E 0n (S) is the solvent-dependent energy gap between the ground and the n th excited state, whereas µ 0n ≡ µ 0n (S) is the corresponding transition dipole matrix. . . . 0 indicates an average over the solvent configurations with an electron equilibrated in its ground state. The vibration-rotation decorrelation approximation used in equation 3 assumes decorrelation between transition dipoles and transition energies: this approximation which may seem to be a drastic one, represents an excellent approximation to the absorption spectrum in the "slow modulation" (or "inhomogeneous broadening") limit [START_REF] Spezia | Molecular dynamics simulations of the Ag + or Na + cation with an excess electron in bulk water[END_REF][START_REF] Staib | Molecular dynamics simulation of an excess charge in water using mobile Gaussian orbitals[END_REF] and reduces the computational burden for a given Signal/Noise ratio.

The calculated and experimental spectra for the hydrated electron and in presence of a sodium or a silver cation are shown in figure 1. In all cases, the spectrum results from three sub-spectra corresponding to transitions from the s-type ground state to the three p-type excited states [START_REF] Nicolas | Molecular simulation of a hydrated electron at different thermodynamic state points[END_REF][START_REF] Spezia | Molecular dynamics simulations of the Ag + or Na + cation with an excess electron in bulk water[END_REF]. Within the model used, with no adjustment of the pseudopotential on the condensed phases properties, the overall agreement between experiments and simulations is fairly good for the three studied systems. However the calculated spectrum for the hydrated electron does not reproduce the asymmetry of the experimental spectrum.

Recent Car-Parrinello molecular simulation of the hydrated electron have been reported [START_REF] Boero | First-principles molecular-dynamics simulations of a hydrated electron in normal and supercritical water[END_REF]. The optical absorption spectra calculated is more asymmetric in this case. The spatial distribution of the electronic charge of the excess electron often departs from an isotropic s-like shape and both localized lobes and solvation shells dynamically break and reform.

This effect was not observed in our one electron mixed quantum classical simulation and we only observed a s-type electron in a water cavity. In the case of sodium and silver cation however, our simulation model reproduces both the location and the width of the experimental spectra. A contact ion-electron pair is observed in the case of sodium [START_REF] Spezia | Molecular dynamics simulations of the Ag + or Na + cation with an excess electron in bulk water[END_REF] with an average electron-cation distance of 2 Å. Such a contact ion pair has also been observed recently by MD simulation of an electron in a reverse micelle [START_REF] Laria | Electron solvation in aqueous reverse micelles: Equilibrium properties[END_REF]. In the case of silver, we observed the reduction of the cation. We found an average electron/silver distance of ∼ 0.5 Å.

Although noticeably non-zero this value is well within the electron radius estimated from the width of the electron wavefunctions, i.e. 1.98 Å. The characterization of the neutral atom, i.e. an hydrophobic species in a polar solvent, was previously achieved by comparing our QCMD simulation with a Car-Parrinello study of the same system [START_REF] Spezia | Molecular dynamics simulations of a silver atom in water: dipolar excitonic state evidence[END_REF]. A large fluctuating induced atomic dipole of ∼ 2 D on the neutral silver atom arose from both simulations, in very good agreement with the atomic dipolar excitonic state theory [START_REF] Logan | Comment on "Electron-Ion interactions and ionization in a polar solvent[END_REF] that predicts for this system an atomic dipole of ∼ 1.96 D.

B. Free energy curves

Apart from the difference in nature of the reduced species, the two cations also differ by their ability to be reduced. In order to better characterize this property, we studied the free energy curves as a function of the electron/cation distance r n (S),

r n (S) = | Ψ 0 | r|Ψ 0 -R cat | = ξ (4)
For ξ = 0 the electron is located on the atom. A contact electron/cation pair is formed as ξ increases and, at large distances there are two distinct solvated species, the electron and the cation. We use the method proposed by Borgis and Staib to adapt the Umbrella Sampling technique in a system where a quantum particle is present [START_REF] Spezia | Molecular dynamics simulations of the Ag + or Na + cation with an excess electron in bulk water[END_REF][START_REF] Borgis | Quantum adiabatic umbrella sampling: the excited state free energy surface of an electron-atom pair in solution[END_REF].

The free energy curves in the case of sodium and silver are shown in figure 2. The sodium curve reveals a minimum at ∼ 2.1 Å (or ∼ 2.3 Å), and the silver one a minimum at 0.5 Å (or 0.6 Å), for systems containing 300 or 800 water molecules respectively. These values are similar to the average values found when performing standard (free) simulations. In the case of sodium the free energy strongly increases at short distances leading to difficulty in constraining the cation-electron distance to values smaller than 1.5 Å. The energy barrier to escape from the minimum is ∼ 0.08 eV for sodium and ∼ 0.7 eV in the case of silver.

The sodium-electron pair appears to be metastable and a small perturbation is sufficient to separate the two species. The evolution of the energy levels of the excess electron are shown in figure 3 as a function of the cation-electron distance (it is found to be identical for systems with 300 and 800 water molecules). The general trend is a stabilization of the ground state when getting closer to the cation. The first three excited states are also stabilized but in a smaller quantity. In the case of sodium, the overall effect is small and the electron never gets closer than 2 Å to the cation. The stabilization is much larger in the case of silver when the electron-cation distance gets lower than 2 Å. The three excited states are however much less stabilized than the ground state leading to the strong shift observed in the absorption spectrum compared to the hydrated electron case. The energy difference between the excited states and the ground state are plotted in figure 4 and compared to the energy differences in the simple case of the hydrated electron. The excess electron in presence of a cation tends to behave like the hydrated electron at large electron-cation distance as expected.

The influence of the cation on the electron absorption spectrum is almost unnoticeable for distances greater than 9 Å.

C. Thermodynamic cycle

This difference in behavior between sodium and silver can be rationalized in terms of a thermodynamic cycle. The free enthalpy, ∆G * , of the reduction reaction, eq. ( 1), of the cation by the hydrated electron can be estimated by considering the solvation free enthalpies of the different species and the ionization potential in the gas phase. The thermodynamic cycle is presented in figure 5. For solvation free enthalpy of the hydrated electron, ∆G solv (e -),

we have taken the value of -1.7 eV [START_REF] Zhan | The nature and absolute hydration free energy of the solvated electron in water[END_REF]. The solvation free enthalpy of both neutral atoms are probably very small and were fixed to zero [START_REF] Dubois | Monte Carlo simulations of Ag + and Ag in aqueous solution. Redox potential of the Ag + /Ag couple[END_REF]. The solvation free enthalpy of the cation in water were estimated to -4.40 eV and -5.12 eV for sodium and silver respectively [START_REF] Burgess | Metal ions in solution[END_REF][START_REF] Tissandier | The proton's absolute aqueous enthalpy and Gibbs free energy of solvation from cluster ion solvation data[END_REF].

The gas phase ionization potential of sodium and silver are 5.14 and 7.56 eV respectively [START_REF] Parr | Density-Functional Theory of Atoms and Molecules[END_REF]. Using the following equation :

∆G * = -IP g + ∆G solv (M 0 ) -∆G solv (M + ) -∆G solv (e -) ( 5
)
∆G * is thus estimated to be +0.96 and -0.74 eV for sodium and silver respectively. Thermodynamics considerations show that the silver cation can be reduced by the hydrated electron in bulk water. On the other hand, as observed in our simulations, sodium can not be reduced to form the neutral metal atom.

D. Direct simulation of the reduction reaction

From the above results we can conclude that the silver atom should form spontaneously from the silver cation and the solvated electron. In order to check this point, we have performed direct simulations of the silver cation reduction reaction. Unconstrained simulations have been carried out, starting from 10 different configurations, for both 300 and 800 water boxes. The starting configurations correspond to a solvated electron in a solvent cavity separated from the cation by at least one water layer. At this stage, the electron properties are very similar to the free solvated electron ones (see figure 4). The evolution in time of the electron-cation distance along a selection of 7 different trajectories is presented in figure 6.

All but one trajectories spontaneously yield to the reduction of the silver cation in less than 10 ps, for both the 300 and 800 water molecules boxes. These trajectories reveal a general trend: during the first period the electron-cation distance is oscillating around an average distance equal or larger than 4 Å, then it suddenly drops to 1 Å in ∼ 0.15 ps.

Then the electron-cation distance undergoes a rather slowly decrease (from 1 Å to 0.5 Å in about 0.8 ps) to the final excitonic state, oscillating around 0.5 Å. This second part of the reduction reaction is governed by the solvent reorganization. There is an important reorientation of water molecules in the first solvation shell between the contact ion-electron pair and the solvated silver atom. In the former, the water molecules are "hydrogen-bonded" to the electron [START_REF] Nicolas | Molecular simulation of a hydrated electron at different thermodynamic state points[END_REF], whereas the latter case is typical of a neutral atom solvation [START_REF] Spezia | Molecular dynamics simulations of a silver atom in water: dipolar excitonic state evidence[END_REF].

The activation energy of the reduction reaction can be estimated to 0.09 eV (∼ 4 kT ) from the free energy curve for 800 water molecules (figure 2). This value is twice as small as the apparent activation energy of 0.18 eV obtained experimentally by fitting the global rate constants on a wide range of temperature [START_REF] Mostafavi | Pulse radiolysis study of absorption spectra of Ag 0 and Ag 2 + in water from room temperature up to 380 • C[END_REF]. In the case of a 300 water molecules simulation, there is no apparent activation energy. We observe a surprising dependence of the free energy curve on the box size, which is presumably due to statistical uncertainties. Our results support the conclusion that the reaction between hydrated electrons and the silver ions is diffusion controlled, consistently with the evolution observed for the free trajectories.

IV. CONCLUSIONS

The reactivity of an hydrated electron towards two types of cations have been studied in bulk water by mixed quantum classical simulation. We have found that the sodium cation is not reduced by an hydrated electron but forms an ion-electron contact pair. Free energy analysis reveals that this pair is metastable with respect to the fully separated state. On the other hand, the silver cation is reduced by an hydrated electron. The resulting neutral atom is very stable and is akin to a dipolar excitonic state. The absorption spectra have been calculated in all cases and a good agreement with experiments is observed. Constrained molecular dynamics have been performed in order to investigate the evolution of the electron energy levels and the total free energy of the system with mean electron-cation distances.

The observed shift of absorption spectra in presence of a cation is due to a stabilization of the electron ground state with decreasing electron-cation distance. For distances larger than 10 Å, no influence of the cation on the hydrated electron properties can be noticed.

The spontaneous reduction reaction of the silver cation has also been observed by direct molecular dynamics simulation. It shows that a complete description of the reaction should include the solvent reorganization as the electron-cation distance is decreased. Such simple mixed quantum-classical molecular dynamics simulations are shown to reproduce the details of solvated species properties in bulk fluid and help in better understanding the chemistry of solvated atoms and cations. A study of the reduction process over a wide range of temperature is now in progress in order to increase our insights in the charge-transfer-tosolvent processes.

FIG. 1 :FIG. 2 :FIG. 3 :FIG. 4 :FIG. 5 :FIG. 6 :

 123456 FIG.1: Absorption spectra of an hydrated electron (solid line), in presence of a sodium cation (dashed line) and in presence of a silver cation (dashed-dotted line). Simulation results (top panel) and experimental data (lower panel).

Acknowledgments

We wish to acknowledge B. Lévy, M. Mostafavi and A. H. Fuchs for fruitful discussions.

We thank the Institut du Développement et des Ressources en Informatique Scientifique (IDRIS) of the CNRS for generous allocations of computer time and the European Union Marie Curie training site UDYCHEM for foundings and a grant for one of us (RS).


J Billy

V Josse

Z Zuo

W Guérin

A Aspect

P Bouyer

Guided atom laser : a new tool for guided atom optics

We present a guided atom laser. A Bose-Einstein condensate (BEC) is created in a crossed hybrid magnetic and an elongated optical trap, which acts as a matterwave guide. Atoms are extracted from the BEC by radio frequency (rf) outcoupling and then guided in the horizontal optical matterwave guide. This method allows to control the acceleration of the beam and to achieve large de Broglie wavelength. We also measure the longitudinal energy of the guided atom laser using atom optical elements based on a blue light barrier.

I. INTRODUCTION

The field of Bose-Einstein condensation of atomic vapors is undergoing a rapid experimental development, providing a rich new phenomenology. Along these lines, the possibility of building atom interferometers [1,2] and guiding condensed particles [3,4] through various geometries opens up the prospects of a rich variety of interference [5,6], transport [7,8], and/or coherence phenomena [9]. The occupation of a single quantum state by a large number of identical bosons is a matter-wave analog to the storage of photons in a single mode of a laser cavity. Just as one extracts a coherent, directed beam of photons from a laser cavity by using a partially transmitting mirror as an output coupler, one can analogously extract directed matter waves from a condensate [10,11,12] which are now used as a coherent sources atoms. Such atom laser is important in the field of atom optics, the manipulation of atoms being analogous to the manipulation of light. Its development is providing atom sources that are as different from ordinary atomic beams as lasers are from light bulbs. In parallel to the development of atom lasers, there are numerous efforts for miniaturizing the structures used to confine the atoms [13,14]. Following the successful trapping and guiding of thermal atoms using self-supported miniature wires [15,16,17,18,19] and substrate-supported microfabricated wire arrays [20,21,22,23], recent experiments merged wire traps on the millimeter scale and microfabricated electronic devices [24,25] with Bose-Einstein condensation.

As for classical optical sources, the transfer of cold atoms from magneto-optic traps into this small atom guides represents a problem because the source volume of the order 1 mm 3 cannot be accommodated inside small atom guides. Hence, the coupling of cold atoms into a microscopic atomic waveguide has been accomplished by injecting fast moving atoms (10 m/s) from a low velocity source such as LVIS or 2D-MOT [21,27]. While this approach allows for a high flux of atoms coupled into the atom guide, the large temperatures of the injected atoms pose a disadvantage for future atom interferometry experiments. Alternatively, slower atoms can be coupled into an atom guide by placing the entrance of the guide vertically below a MOT and dropping the atoms into the guide using gravity [17,22]. In this approach, the expansion of the atom cloud during the free fall leads to a low coupling efficiency.

Although many cold atoms sources have already been injected in various guides [16,17,22,28] better efficiency can be achieved with coherent sources of atoms which have recently been designed using various output coupling schemes. Nevertheless, even though continuous guided beams of condensed atoms will be accessible in the near future [29], no continuous or quasi continuous coupling of atoms coherently into a guide was yet acheived until our realization of a guided quasicontinuous atom laser [36]. In our realization of a guided atom laser [36], the coherent source, i.e. the trapped BEC, and the guide are merged together in a hybrid combination of a magnetic Ioffe-Pritchard trap and an horizontally elongated far off-resonance optical trap constituting an atomic waveguide (see Fig. 1). The BEC, in a state sensitive to both trapping potentials, is submitted to a rf outcoupler yielding atoms in a state sensitive only to the optical potential, resulting in an atom laser propagating along the weak confining axis of the optical trap. In addition to canceling the effect of gravity, this configuration has several advantages.

• Coupling into a guide from a BEC rather than from a thermal sample allows us to couple a significant flux into a small number of transverse modes of the guide. Hence, in spite of the lensing effect due to the interaction of the atom laser with the trapped BEC [34,35], adiabatic transverse mode matching results into the excitation of only a small number of transverse modes, and we discussed the possibility of achieving single transverse mode operation of the atom laser in [36].

• Using a rf outcoupler rather than releasing a BEC into a guide results into quasicon-tinuous operation.

• It is possible to compensate the weak longitudinal trapping potential of the guide by the antitrapping potential due to the second order Zeeman effect acting onto the outcoupled atoms, resulting in an atom laser with a quasiconstant large de Broglie wavelength (> 1 µm over 0.1 s of propagation in this work).

• Changing the frequency of the outcoupler allows one to tune the value of the de Broglie wavelength of the atom laser. One can also vary the atom-laser density from the interacting regime to the noninteracting regime, offering the prospect to study linear as well as nonlinear atom optics phenomena.

We finally present our first investigations on the development of guided atom optical elements using light microstructures to reflect or transmit the atomic beam.

II. EXPERIMENTAL SET-UP

Our setup [30,36] The guided atom laser is obtained by rf-induced magnetic transition [12] between the |1, -1 state and the |1, 0 state, which is submitted to the same transverse confinement due to the optical guide, but is not sensitive (at first order) to the magnetic trapping. We thus obtain a quasicontinuous guided coherent matter wave freely propagating along the optical guide [Fig. 1(b)]. This configuration, where the optical guide dominates the transverse trapping of both the source BEC and the atom laser, enables to collect the outcoupled atoms into the optical guide with 100% efficiency.

III. A LARGE DE BROGLIE WAVELENGHT ATOM LASER

Taking advantage of having a horizontal guide, we can control the acceleration of the beam, which is not dominated by gravity any more, as for usual atom laser experiment (see [36] for additional explanaitions). This is of great interest to obtain a coherent beam with a large de Broglie wavelength. The combination of the expulsive Zeeman quadratic effect and of the dipole potential, whose center is shifted from z 0 , results in a nearly linear potential whose slope is mω 2 z z 0 . Typical experimental parameters are ω z = 2π × 2.5 Hz and z 0 = 1 mm. This gives an acceleration of 0.25 m.s -2 , which corresponds to a reduction by a factor 40 compared to gravity. Moreover, we can tune this acceleration by changing the power of the guiding laser, thus changing ω z , or the position of the waist z 0 [36]. Ultimately, we can cancel the residual acceleration so that the movement of the atoms is dominated by an initial velocity "kick" induced by mean-field interactions (see fig. 2). Indeed, as can be seen on the diagram of fig. 2, the outcoupled atoms experience first the repulsive meanfield potential, which gives the beam an initial kinetic energy of the order of the chemical potential µ(z E ) where z E is the position from which the atoms are extracted. We have studied this effect by measuring the atom laser wavefront velocity (the method used and explained in [36]) with respect to the RF outcoupler position by changing the outcoupler frequency [34,35]. Changing this position changes the local density which is extracted and thus modulates the atom laser flux as shown in fig. 2. It also changes the initial velocity "kick" induced by the local mean field energy. As shown in the preliminary studies on fig. 2, we can change this velocity between 0.4 mm/s and 1.2 mm/s, corresponding to de Broglie wavelength between 4 and 12 µm, well above the de Broglie wavelength expected from "classical" energy considerations [42].

IV. TOWARDS A FABRY-PEROT CAVITY FOR THE ATOM LASER

Among the many possibilities opened up by the guided atom laser, the atom optical analog of a Fabry Perot cavity comes immediately to mind [31]. In our experiment, two potential barriers will be created by a laser beam at 405 nm in the TEM01 mode (with a node line in the center). The short optical wavelength is chosen to render the barriers sufficiently thin that the tunnelling amplitude cannot be prohibitively small. When the FIG. 2: above : Principle of RF outcoupling. Because atoms can only be outcoupled (see [35]) from where the resonance condition is fulfilled, fixing the RF frequency fixes the initial density and flux (and thus the amount of interaction energy) that is transferred to the atom laser. below : Coupling curve (left) and corresponding velocity (right) with respect to the RF outcoupling frequency.

barriers are kept sufficiently stable in height and position, one expects the build up of a wave in the space between them.

Our laser source is a 405 nm fibered diode laser delivering 30 mW of maximum power.

The outpout of the fiber is collimated and a 2 mm diameter beam is sent to an anamorphic telescope, creating an anisotropic beam (20 × 0.5 mm 2 ) which is focused onto the atom laser with a numerical aperture of 0.15. This allows to create optical structure of width of the order of 1.3 µm FWHM along the propagation direction and 20 µm in the transverse direction. We calibrated our optical structure by sending clouds of atoms above and below BEC threshold onto a single optical barrier. This allows to measure the energy distribution within the cloud since in this "classical" regime, only the atom with kinetic energy higher than the barrier can be transmitted. With this 'spectroscopic' technique, we could record spectra of an atomic sample above and below BEC threshhold and demonstrate the control and stability of our barrier to better than 1 % (100 Hz).

We now switch to our preliminary study of the transmission of the guided atom laser through the barrier. For that, first create a BEC in our hybrid trap as in [36] and turn on the RF outcoupler for 100 ms. By choosing the proper configuration, it is possible to create two atom lasers propagating in opposite directions from two symetric positions in the BEC. We set the optical barrier on one side of the BEC so that only one atom laser will be affected. This allows for a single shot calibration of the transmission factor through the barrier. The result for two different RF outcoupler frequencies, with estimated corresponding de Broglie wavelength of ∼ 1µm and ∼ 1.5µm are shown in figure 3. It is interesting to see that a smooth transition occurs from total transmission (at 0 V) to total reflexion (at maximum barrier height). For an ideal atom laser, we expect nonetheless a step function since the energy spread should be negligible. This difference could be explained by unwanted broadening of the atom laser due to technical noise. We estimate this noise from the resonance curve of figure 2 to be below 1 kHz, to low to explain the transmission curve.

Other explanations, such as more fundamental broadening due to residual excitation in the coupling process, are under investigation. A last explanation could come from the fact that the wavelength of the laser is of the order or bigger than the width of the barrier. In this case, tunneling is also expecting to broaden the transmission curve.

V. CONCLUSION

We present the realization of the first guided atom laser [36]. Unlike other atom lasers, in which the atoms accelerate under gravity after being outcoupled from a BEC in a trap, our laser beam can be made to propagate horizontally and thus free of any acceleration.

The atomic velocity can be chosen by the outcoupling parameters and then the matter wave propagates with a constant de Broglie wavelength (typically of order 1 µm or more) over several mm. One can also control the atomic density in the beam and thus the amount of interaction between atoms. It becomes possible to explore various regimes from noninteracting to strongly interacting beams. This type of laser beam is ideal for the study of transport phenomena, especially those related to tunneling and quantum reflection in which low energies and weak interactions are favorable [37]. Among the many possibilities opened up by the guided atom laser, the atom optical analog of a Fabry Perot cavity comes immediately to mind. Nevertheless, by contrast to the typical optical case, the atoms interact.

Thus any atom laser Fabry-Pérot cavity will resemble an optical cavity containing a Kerr medium. The high confinement in this configuration means that the resonance energies are strongly shifted by the presence of other atoms, rendering the cavity bistable and producing a profound modification of the statistical properties of the atoms. Depending on the interaction strength, which is quantified by the maximum number of atoms N max allowed in the cavity at the bistability threshold, different behaviors are expected for the quantum properties of the transmitted atomic beam. As has been discussed [38,39] and observed in the optical domain [40,41], squeezing appears in the weak interacting regime (N max ≥ 1), where the presence or absence of one atom does not significantly perturb the dynamics. Such an observation will constitute in itself a pioneering experiment in the context of quantum atoms optics.

FIG. 1 :

 1 FIG. 1: (a) Schematic view of the setup. The BEC is produced at the intersection of a magnetic trap and a horizontal elongated optical trap acting as an atomic waveguide for the outcoupled atom laser. An "rf knife" provides outcoupling into the waveguide and an atom laser is emitted on both sides. (b) Absorption image (along x) of a guided atom laser after 100 ms of outcoupling.

  produces magnetically trapped cold clouds of 87 Rb in the |F, m F = |1, -1 state. During the evaporative cooling, an optical guide produced by 120 mW of Nd:YAG laser (λ = 1064 nm) focussed on a waist of 30 µm is superimposed along the z direction and Bose-Einstein condensation is directly obtained in the optomagnetic trap. In this hybrid trap, the optical guide ensures a tight transverse confinement, with oscillation frequencies ω x,y /2π = ω ⊥ /2π = 360 Hz, large compared to the frequencies characterizing the magnetic confinement along the same axes (ω m x /2π = 8 Hz and ω m y /2π = 35 Hz). In contrast, the longitudinal confinement of the BEC along the z axis is due to the shallow magnetic trap with an oscillation frequency ω m z /2π = ω m /2π = 35 Hz. A BEC of 10 5 atoms has then a chemical potential µ BEC /h ≃ 3.2 kHz and Thomas-Fermi radii R z = 25 µm and R ⊥ = 2.4 µm.

FIG. 3 :

 3 FIG. 3: above A dual beam atom laser used to study the transmission through a light barrier. The left side is used to calibrate the flux in a single shot while the barrier is applied on the right. below The transmission curves versus the laser intensity (the height in V is the amplitude control command, and correspond roughly to 2 kHz/V) for two different atom laser energy (with estimated de Broglie wavelength of ∼ 1µm (left) and ∼ 1.5µm (right).