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

TL;DR

This paper develops a mean-field theoretical framework to analyze atomic self-organization in optical cavities, capturing equilibrium and dynamical behaviors, and compares it with detailed N-body simulations to understand its accuracy and limitations.

Contribution

It introduces a systematic mean-field model for atomic self-organization in optical cavities and evaluates its validity against N-body simulations and existing theoretical approaches.

Findings

Mean-field theory reproduces short-time dynamics of N-body equations.

Predicted thermalization times are significantly shorter than N-body results.

Discrepancies are due to neglecting long-range incoherent forces in the mean-field model.

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.