Collisional effects on the collective laser cooling of trapped bosonic gases

In the recent years, laser cooling has constituted one of the most active research fields in atomic physics [1]. However, the laser cooling techniques by themselves have not allowed to reach temperatures for which the quantum statistical effects become evident. In particular, only the combination of laser cooling, and evaporative [2] or sympathetic cooling [3] has permited in the last years to observe experimentally, the Bose–Einstein condensation (BEC) in alkali gases, seventy years after its theoretical prediction [4]. The question whether it is or it is not possible to achieve the BEC only with laser cooling techniques remain, at least as an intellectual challenge. The laser–induced BEC is, however, not only an academic problem, but has several advantages with respect to the nowadays widely–employed collisional mechanisms (as evaporative cooling). These advantages are: (i) the number of atoms does not decrease during the cooling process; (ii) it is possible to design a non–destructive BEC detection by fluorescence measurements; (iii) reacher effects can appear, since now the system is open, i.e. it is not in thermal equilibrium; (iv) laser–induced condensation can be used to design techniques to pump atoms into the condensate. The latter is specially important in the contex of future atom–laser devices [5–8]. The main problem which prevents experimentalists from obtaining BEC by optical means is the reabsorption of spontaneously emitted photons. The most effective laser–cooling techniques (such as VSCPT [9], or Raman cooling [10]), are based on the crucial concept of dark states [11], i.e. states which cannot absorb the laser light, but can receive population via incoherent pumping, i.e. via spontaneous emission. Unfortunately, the atoms occupying the dark states are not unaffected by the photons spontaneously emitted by other atoms. This problem turns to be very important at high densities, as those required for the BEC [12]; in such conditions dark–state cooling techniques cease to work adequately. Several remedies to the reabsorption problem have been proposed, as the reduction of the dimensionality of the trap from three to two or one dimensions [12], or the use of traps with frequencies, ω, of the order of the recoil frequency (ωR = h̄k L/2M , where kL is the laser wavevector and M is the atomic mass) [6]. Other, perhaps more promising, idea consists in exploiting the dependence of the reabsorption probability on the fluorescence rate γ. In particular, in the so–called Festina Lente limit [14], when γ < ω with ω the trap frequency, the heating effects of the reabsorption can be neglected. Another proposal consists in working in the so–called Bosonic Accumulation Regime [8], in which the reabsorption can, under certain conditions, even help to build up the condensate. In the following we shall assume that the considered system fulfills the Festina Lente limit. In a series of papers [15,16], we have proposed a cooling mechanism (which we have called Dynamical cooling) which permits the cooling of an atomic sample into an arbitrary single state of an harmonic trap, beyond the Lamb–Dicke limit (i.e. when the Lamb–Dicke parameter η > 1, with η = ωR/ω). The cooling mechanism employs laser pulses of different frequencies (and eventually different directions, phases and intensities), in such a way that a particular state of the trap remains dark during the cooling process, acting as a trapping state. Therefore, the population is finally transferred to this particular state. We have first analysed the particular situation of a single atom in the trap [15], and extended the analysis to a collection of trapped bosons [16]. We have shown that the bosonic statistics helps to achieve more robust and rapid condensation, as well as to produce non–linear effects, such as hysteresis and multistability phenomena. However, all the calculations performed so far in the analysis of the dynamical cooling scheme do not take into account the atom–atom collisions, i.e. are considered in the so–called ideal gas limit. The ideal gas limit imposes important restrictions to the physical system, in particular the atomic density cannot be very large. An interesting possibility in order to achieve quasi–ideal gases consists in the “switching–off” of the s–wave scattering length a (which is the main contribution to the atom– atom collisions for sufficiently low energies), either by employing magnetic fields (tuning the so–called Feshbach resonances [17]) or by using a red–detuned laser tuned between molecular resonances as proposed by Fedichev et al [18]. However, without special precautions, the ef-