Output of a pulsed atom laser

The experimental breakthrough to Bose-Einstein condensation with small numbers of atoms in magnetic traps [1] has raised much interest in the properties of mesoscopic quantum gases. Bose-Einstein condensates with atoms in a single magnetic sublevel have been studied experimentally and theoretically. The recent experimental and theoretical investigations of interference between two independent Bose-Einstein condensates convincingly proved their macroscopic coherence [2,3]. Moreover, the laser-like coherence of the atoms is preserved in the presence of a matter-wave splitter based on rf-transitions pumping the atoms into untrapped magnetic sublevels [4]. These states are either strong-field seeking or have no magnetic moment at all, and leave the trap. Alternatively, optical Raman transitions can be used for the transfer [5,6]. Such schemes provide controllable output couplers for coherent atom lasers. In analogy to a laser one can distinguish between a cw laser, based on continuous refilling of the condensate, and a pulsed atom laser where the condensate is periodically refilled and slowly released, similar to [4]. Whereas a continuous wave atom laser has been studied only theoretically [6–8], current Bose-Einstein condensation experiments are limited to the pulsed mode of operation. The closest approximation of a cw atom laser by a pulsed one can be reached in the limit of a weak coupling rf field. In this case we are able to describe the decay of the trapped condensate and its energy width analytically. Previous calculations addressed the opposite limit of strong coupling by numerical calculations [9] or neglected the important influence of atom-atom interactions [10]. The output coupler consists of a monochromatic resonant rf field of frequency ωrf transferring Na atoms in the F = 1 hyperfine state from the trapped m = −1 into the untrapped m = 0 and the repelled m = 1 magnetic sublevels. For simplicity an isotropic harmonic trap potential V−1(r) = Voff + Mω 2 Tr /2, V+1(r) = −V−1(r) and V0(r) ≡ 0 are assumed while effects of gravity are neglected. The three coupled coherent matter waves are described by a three-component Gross-Pitaevskii equation (GPE) with resonant excitation in rotating wave approximation first studied for a generic two-level system in Ref. [9]. In the following we adopt the point of view of spontaneously broken gauge symmetry for a Bose gas initially at zero temperature. The system of equations for the macroscopic wave funtion ψ̃m(t) = e −imωrf 〈ψ̂m(t)〉 in rotating wave approximation for m,m′ ∈ {−1, 0,+1} now reads