Mean-field theory of atomic self-organization in optical cavities

Abstract

Photons mediate long-range optomechanical forces between atoms in high finesse resonators, which can induce the formation of ordered spatial patterns. When a transverse laser drives the atoms, the system undergoes a second order phase transition, that separates a uniform spatial density from a Bragg grating maximizing scattering into the cavity and is controlled by the laser intensity. Starting from a Fokker-Planck equation describing the semiclassical dynamics of the $N$-atom distribution function, we systematically develop a mean-field model and analyse its predictions for the equilibrium and out-of-equilibrium dynamics. The validity of the mean-field model is tested by comparison with the numerical simulations of the $N$-body Fokker-Planck equation and by means of a BBGKY hierarchy. The mean-field theory predictions well reproduce several results of the $N$-body Fokker-Planck equation for sufficiently short times, and are in good agreement with existing theoretical approaches based on field-theoretical models. Mean-field, on the other hand, predicts thermalization time scales which are at least one order of magnitude shorter than the ones predicted by the $N$-body dynamics. We attribute this discrepancy to the fact that the mean-field ansatz discards the effects of the long-range incoherent forces due to cavity losses.

17 pages, 8 figures

References in corpus (9)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Cold atoms in cavity-generated dynamical optical potentials
• Cavity Nonlinear Optics at Low Photon Numbers from Collective Atomic Motion
• Keldysh approach for non-equilibrium phase transitions in quantum optics: beyond the Dicke model in optical cavities
• Mott insulator states of ultracold atoms in optical resonators
• Self-organization of atoms in a cavity field: threshold, bistability and scaling laws
• Dynamical Coupling between a Bose-Einstein Condensate and a Cavity Optical Lattice
• Prethermalization of atoms due to photon-mediated long-range interactions
• Quantum kinetics of ultracold fermions coupled to an optical resonator

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• Superfluid flow of polaron polaritons above Landau's critical velocity
• Dynamical phases of a BEC in a bad optical cavity at optomechanical resonance
• Spin self-organization in an optical cavity facilitated by inhomogeneous broadening
• Cooperative Cooling in a 1D Chain of Optically Bound Cold Atoms
• Master Equation for a Quantum Gas of Polarizable Particles in Cavities
• Canonical equilibrium of mean-field $O(n)$~models in the presence of random fields