Sound Wave in the Backreaction Affected Spacetime in Analogue Gravity Based on Number-Conserving Approach

Adopting a number-conserving approach, we show that the sound wave in the backreaction-affected dynamical spacetime obeys the equations of motion for a massive scalar field in an analogue spacetime. Even with backreaction, the analogue metric has the same form as that without backreaction, except that the sound velocity, fluid density, and fluid velocity incorporate small corrections that account for backreaction. Moreover, the modification of the classical fluid-dynamical equations by backreaction introduces a spacetime-dependent mass. We also present a linear form of the modified Bogoliubov-de Gennes equation with number conservation, which allows us to solve it numerically. Using a finite-size homogeneous quasi-one-dimensional Bose gas as an example, we find that backreaction enhances the canonically normal-ordered correlation function at the same point, by about 10% in the bulk and up to about 20% near the edges. The bulk enhancement persists when the finite imaging resolution of cold-atom experiments is taken into account.