Laser Induced Condensation of Trapped Bosonic Gases

Laser cooling has led to spectacular results in recent years [1]. So far, however, it has not allowed to reach temperatures for which quantum statistics become important. In particular, evaporative cooling is used to obtain Bose-Einstein condensation of trapped gases [2]. Nevertheless, several groups are pursuing the challenging goal of condensation via all–optical means [3–5]. In traps of size larger than the inverse wavevector, k−1 L , the temperatures required for condensation are below, or of the order of, the photon recoil energy, ER = h̄ωR = h̄k2 L/2m, where m is the atomic mass. There exist several laser cooling schemes to reach such temperatures [6,7]. They exploit single atom “dark states”, i.e. states which cannot be excited by the laser, but can be populated via spontaneous emission. The main difficulty in applying dark state cooling for dense gases is caused by light reabsorbtion. Unfortunately, these states are not dark with respect to the photons spontaneously emitted by other atoms. Thus, at sufficiently high densities, dark state cooling may cease to work, since multiple reabsorptions can increase the system energy by several recoil energies per atom [8–11]. In particular, laser induced condensation is feasible only if the reabsorbtion probability is smaller than the inverse of the number of energy levels accessible via spontaneous emission processes [9]. Several remedies to the reabsorption problem have been proposed. First, the role of reabsorptions increases with the dimensionality. If the reabsorption cross section for trapped atoms is the same as in free space, i.e. ' 1/k L, the reabsorptions should not cause any problem in 1D, have to be carefully considered in 2D, and forbid condensation in 3D. Working with quasi-1D or 2D systems is thus a possible way to reduce the role of reabsorptions [12]. The most promising remedy against this problem employs the dependence of the reabsorption probability for trapped atoms on the fluorescence rate γ, which can be easily adjusted in dark state cooling [13]. In particular, in the so called Festina Lente limit, when γ is much smaller than the trap frequency ω [14], the reabsorption processes in which the atoms change energy and undergo heating are suppressed. However, neither collective cooling schemes in traps of realistic size have been investigated in this limit, nor has it been shown that laser induced condensation is possible. In this Letter we present such investigation. First, we formulate the Master Equation (ME) that describes Raman cooling in the Festina Lente limit using coarse graining in time. Cooling is dynamical, and consists of sequences of pairs of laser pulses inducing stimulated and spontaneous Raman transitions between the two electronic levels of trapped atoms, |g〉 and |e〉. The stimulated absorption pulses induce the energy selective transition |g〉 → |e〉 that depopulates all motional states except the dark ones; the spontaneous emission pulses are non-selective, and repump the atoms from |e〉 to |g〉 populating all accessible motional states. We simulate the dynamics generated by the ME in 3D, and show that: i) laser induced condensation into an arbitrary trap level is possible; an arbitrary trap level may be made dark; ii) condensation is robust with respect to changes of physical parameters; dark states do not have to be completely dark; iii) multistability and hysteresis occur when the parameters undergo large changes. In the limit of large number of atoms analytic solutions of the ME are found. We consider N bosonic atoms with two levels |g〉 and |e〉 in a non-isotropic trap with the frequencies ω x,y,z, ω e x,y,z different for the ground and the excited states, and non-commensurable one with another. This assumption simplifies enormously the dynamics of the spontaneous emission processes in the Festina Lente limit. We use the coarse graining in time and describe variations of the atomic state after one absorption and one spontaneous emission pulse. After such cooling cycle all atoms are in the ground internal state described by the density matrix ρ(t). This matrix is diagonal in the Fock representation corresponding to the bare trap levels. In order to derive the ME, we separate the effects of laser cooling from the ones due to atom–atom collisions. The latter can be described by a quantum kinetic ME, which has been studied in Ref. [15]. In this paper we concentrate on the laser cooling process. Thus, our results are valid in the case when collision processes are slow compared to laser cooling. For the stimulated Raman transitions, we assume weak pulses of duration τabs. Their effects can thus be described by second order perturbation theory (formally that corresponds to one atom excited at most),