Self-organization of a laser-driven cold gas in a ring cavity

– A gas of linearly polarizable particles transversely pumped by an off-resonant laser and interacting with the counterpropagating radiation modes of a ring cavity is studied. Depending on the pump intensity and the detunings the gas can form a self-organized density grating that enhances the feeding of the cavity. We investigate the system via a mean-field approach and find the thermodynamic phases of i) uniform distribution, ii) self-organized Bragg lattice, iii) lattice with defects, and iv) instability. The occurrence of these phases as a function of the pump intensity and particle density is fully mapped. The physics of ultracold atoms has been a proliferating field these years. There is a renewed interest in fundamental many-body phenomena which can be investigated in well-controlled, weakly interacting atomic ensembles. Phase transition into a Bose-Einstein condensate is being routinely realized with alkali atoms, and various other manifestations of quantum statistics, superfluidity, etc., are observed. The system of cold atoms is particularly attractive because the collisional properties are partially tunable. The nature of atom-atom interaction is a central issue in the understanding of many-body effects. This interaction is dominated by the dipole-dipole coupling in a dense cloud of cold atoms illuminated by quasi-resonant laser fields. On the one hand, this amounts to a measurable modification of the optical properties of the cloud, e.g., a nonlinear density dependence of the refractive index [1], or slow diffusion of light [2]. On the other hand, it induces interatomic forces, which eventually cause an instability of the homogeneous distribution of the cloud [3]. In this letter we analyze a system presenting a strong interplay of the electromagnetic field generating a collective behaviour of an ensemble of polarizable particles, and the backreaction of those particles on the dynamics of the field. The system is shown to produce a phase transition between the homogeneous spatial distribution and a regular pattern bound by the electromagnetic field. This effect can be described in terms of analytical expressions accounting for the density of the particles and laser pumping strength by virtue of the simple geometry. c © EDP Sciences Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx.doi.org/10.1209/epl/i2005-10521-4 D. Nagy et al.: Self-organization in a ring cavity 255 Instead of the three-dimensional dipole-dipole interaction in free space, we envisage a system in which the interparticle communication is predominantly mediated by a few radiation modes selected by a high-finesse optical resonator. Within the cavity, particles experience the field change induced by any remote particle regardless of the distance. This is due to the fast photon round-trips in the resonator on the time scale given by the other interaction strengths [4,5], i.e., the longitudinal mode spacing is much larger than the atomic and cavity linewidth as well as the relevant detunings. To be specific, we assume a one-dimensional ring resonator with two counterpropagating modes, which corresponds to the experimental setups in [6–8]. As a difference to the correlated atomic recoil laser (CARL), we consider a transverse pump scheme, i.e., the particles are laser driven from a direction orthogonal to the resonator axis. This geometry has the virtue that the polarizing field is separated from the cavity modes mediating the multiple scattering between the particles. In addition, the system is completely translationally invariant along the cavity axis. Apart from the straightforward geometry, the atom-light interaction considered in this paper is also simply described in terms of scattering processes. Due to the large detuning, effects of the polarization dynamics can be neglected [4]. Hence the predicted phase transitions rely on basic properties of light-matter interaction and should occur in quite general setups, with molecules or other scatterers. We developed a mean-field method which can be applied to study phase transitions in the geometry of a standing-wave cavity filled with transversely pumped atoms [9–11] and in that of the CARL [12–16]. The method yields a complete mapping of various phases as a function of the system parameters. The dynamics of the field. – The system is composed of a gas of N polarizable particles coupled to two degenerate optical modes of a ring resonator, described by the plane-wave mode functions f1(x) = e and f2(x) = e−ikx, with coherent amplitudes α1 and α2. The particles are driven by a pump laser oriented perpendicular to the cavity axis. The electric field in the cavity is then given by E(x) = f1(x)α1+f2(x)α2+Epump, where the last pumping field term is assumed to be constant along the resonator axis “x”. For simplicity, we consider the system only along this spatial dimension: the atoms are supposed to be confined near the resonator axis by, e.g., a strong dipole trap. The interaction is in the dispersive regime, i.e., the pump laser is very far detuned with respect to the resonance frequencies of the gas particles: no real excitations need take place. The gas can then in principle even be composed of molecules, but we refer to the gas particles as “atoms”. The atoms redistribute photons by coherent scattering between the two modes and the pump field. This process feeds the cavity modes with an effective amplitude η. The dynamics of the cavity modes with the scatterers at xj , j = 1, . . . , N is given by the following differential equations [4]: d dt ( α1 α2 ) = A ( α1 α2 ) − iη N ∑ j=1 ( f∗ 1 (xj) f∗ 2 (xj) )