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Losses in interacting quantum gases: Ultraviolet divergence and its regularization

We investigate the effect of losses on an interacting quantum gas. We show that, for gases in dimension higher than one, assuming together a vanishing correlation time of the reservoir where dissipation occurs, and contact interactions leads to a divergence of the energy increase rate. This divergence is a combined effect of the contact interactions, which impart arbitrary large momenta to the atoms, and the infinite energy width of the reservoir associated to its vanishing correlation time. We show how the divergence is regularized when taking into account the finite energy width of the reservoir, and, for large energy width, we give an expression for the energy increase rate that involves the contact parameter. We then consider the specific case of a weakly interacting Bose Einstein condensate, that we describe using the Bogoliubov theory. Assuming slow losses so that the gas is at any time described by a thermal equilibrium, we compute the time evolution of the temperature of the gas. Using a Bogoliubov analysis, we also consider the case where the regularization of the divergence is due to the finite range of the interaction between atoms.