Number-conserving solution for dynamical quantum backreaction in a Bose-Einstein condensate

We provide a number-conserving approach to the backreaction problem of small quantum fluctuations onto a classical background for the exactly soluble dynamical evolution of a Bose-Einstein condensate, experimentally realizable in the ultracold gas laboratory. A force density exerted on the gas particles which is of quantum origin is uniquely identified as the deviation from the classical Eulerian force density. The backreaction equations are then explored for the specific example of a finite size uniform density condensate initially at rest. By assuming that the condensate starts from a non-interacting regime, and in its ground state, we fix a well-defined initial vacuum condition, which is driven out-of-equilibrium by instantaneously turning on the interactions. The assumption of this initial vacuum accounts for the ambiguity in choosing a vacuum state for interacting condensates, which is due to phase diffusion and the ensuing condensate collapse. As a major finding, we reveal that the time evolution of the condensate cloud leads to condensate density corrections that cannot in general be disentangled from the quantum depletion in measurements probing the power spectrum of the total density. Furthermore, while the condensate is initially at rest, quantum fluctuations give rise to a nontrivial condensate flux, from which we demonstrate that the quantum force density attenuates the classical Eulerian force. Finally, the knowledge of the particle density as a function of time for a condensate at rest determines, to order $N^0$, where $N$ is the total number of particles, the quantum force density, thus offering a viable route for obtaining experimentally accessible quantum backreaction effects.