Continuous quantum measurement of a Bose-Einstein condensate: A stochastic Gross-Pitaevskii equation

We analyze the dynamics of a Bose-Einstein condensate undergoing a continuous dispersive imaging by using a Lindblad operator formalism. Continuous strong measurements drive the condensate out of the coherent-state description assumed within the Gross-Pitaevskii mean-field approach. Continuous weak measurements allow us instead to replace, for time scales short enough, the exact problem with its mean-field approximation through a stochastic analog of the Gross-Pitaevskii equation. The latter is used to show the unwinding of a dark soliton undergoing a continuous imaging.