Depletion dynamics of a Bose-Einstein condensate in a dissipative optical cavity

We study the depletion dynamics of a driven homogeneous Bose–Einstein condensate (BEC) strongly interacting with an optical cavity. Working in the bad-cavity regime, we eliminate the photonic degrees of freedom to obtain an effective atom-only master equation. Applying Bogoliubov theory, we derive the dynamics of the covariance matrix of the cavity-coupled atomic fluctuations. The resulting Lyapunov equation captures fluctuations arising from coherent and dissipative cavity-mediated interactions, as well as diffusive cavity shot noise. This system exhibits a self-organization (SO) phase transition and we analyze the depletion of the BEC in the normal (below threshold) and self-organized (above threshold) phases. Below threshold, we define an effective temperature for the cavity-coupled Bogoliubov mode and, in the weak-driving limit, derive an analytic expression consistent with previous results. Above threshold, the dominant depletion mechanism crosses over: near the SO transition, diffusion dominates, whereas deeper in the SO phase cavity-mediated atomic fluctuations prevail. We compare these cavity-induced depletion rates with those from short-range contact interactions obtained from Bogoliubov theory. With this we identify regimes in which cavity dissipation can, in principle, stabilize the condensate.