Master equation for a quantum gas of polarizable particles in cavities

Quantum gases of atoms and molecules in optical cavities offer a formidable laboratory for studying the out-of-equilibrium dynamics of open quantum systems with long-range interactions. Long-range interactions are here mediated by multiple scattering of cavity photons and can induce the formation of quantum structures in space and time. Control of these dynamics requires a detailed understanding of all relevant mechanisms at play. Due to the strong correlations induced by light, however, perturbative theoretical models, which reduce the number of degrees of freedom, do not correctly capture the regime where the interplay of photon-mediated long-range forces and quantum fluctuations of light and matter become significant, such as across the transition to self-organization. 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 cavity fields. 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 ultracold regime, and from weak to strong cavity-mediated interactions. Our theory provides a powerful framework for the description of cavity-induced dynamics of quantum matter. In doing so, it permits to connect models of statistical mechanics with cavity-QED experimental platforms, thus enabling quantum simulation of long-range-interacting matter.