Multimode mean-field model for the quantum phase transition of a Bose-Einstein condensate in an optical resonator

We develop a mean-field model describing the Hamiltonian interaction of ultracold atoms and the optical field in a cavity. The Bose-Einstein condensate is properly defined by means of a grand-canonical approach. The model is efficient because only the relevant excitation modes are taken into account. However, the model goes beyond the two-mode subspace necessary to describe the self-organization quantum phase transition observed recently. We calculate all the second-order correlations of the coupled atom field and radiation field hybrid bosonic system, including the entanglement between the two types of fields.Comment: 10 page

A Vlasov approach to bunching and selfordering of particles in optical resonators

We develop a Vlasov type continuum density description for the coupled nonlinear dynamics of polarizable particles moving in the light field of a high Q optical resonator. The intracavity light field, which exerts optical forces on the particles, depends itself on the dynamics of the particle density, which constitutes a time dependent refractive index. This induces mode frequency shifts, losses and coupling. For typical geometries we find solid analytic criteria for the stability of an initial homogeneous particle density for a wide class of initial velocity distributions including thermal distributions. These agree with previously found bunching and self-ordering instabilities but are extended to a wider range of parameters and initial conditions. Using a linear perturbation expansion we calculate the growth exponents of small density perturbations in the parameter region beyond this instability threshold. Numerical solutions of the full equations as well as simulations of the underlying many particle trajectories confirm these results. In addition the equations allow to extract analytical scaling laws to extrapolate cavity cooling and selfordering dynamics to higher particle numbers