Master Equation for Quantum Self-Organization of Atoms and Molecules in Cavities

Quantum gases of atoms and molecules in optical cavities offer a formidable laboratory for studying the out-of-equilibrium dynamics of long-range interacting systems. Multiple scattering of cavity photons mediate interactions, and the emerging phases of matter are determined by the interplay of photon-mediated forces, dissipation, and quantum and thermal fluctuations. Control of these dynamics requires a detailed understanding of the mechanisms at play. Due to the number of degrees of freedom, however, effective theoretical models often work in specific limits, where either the cavity field is treated as a semiclassical variable or the cavity state is a slightly perturbed vacuum state. In this work, we present the derivation of an effective Lindblad master equation for the dynamics of the sole motional variables of polarizable particles, such as atoms or molecules, that dispersively couple to the cavity field. The master equation is valid even for relatively large intracavity photon numbers, and is apt to study both the steady-state regime and the out-of-equilibrium dynamics where quantum fluctuations of the field seed the onset of macroscopic coherences. We validate the theoretical description by showing that it captures the dynamics across a wide temperature interval, from Doppler cooling down to the ultra-cold regime, and from weak to strong cavity-mediated interactions. Our theory provides a powerful framework for the description of the dynamics of quantum gases in cavities and permits, amongst others, to connect models and hypotheses of statistical mechanics with a versatile experimental platform.