Equilibrium, Relaxation and Fluctuations in homogeneous Bose-Einstein Condensates: Linearized Classical Field Analysis

We present a thorough analysis of the linearized stochastic projected Gross-Pitaevskii equation (SPGPE) describing finite temperature in Bose-Einstein condensates (BECs). Our study reveals an optimal choice for the cut-off that divides the Bose gas into a low energy coherent region forming a classical wave and a high energy thermal cloud acting as a reservoir. Moreover, it highlights the relevance of energy damping, the number conserving scattering between thermal and coherent atoms. We analyze the equilibrium properties and near equilibrium relaxation of a homogeneous BEC in one, two and three dimensions at high phase space density, and calculate the autocorrelation function and power spectrum of the density and phase fluctuations. Simulations of the full non-linear SPGPE are in close agreement, and extend our arguments beyond the linear regime. Our work suggests the need for a re-examination of decay processes in BECs studied under the neglect of energy damping.