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We formulate computationally efficient classical stochastic measurementtrajectories for a multimode quantum system under continuous observation.Specifically, we consider the nonlinear dynamics of an atomic Bose-Einsteincondensate contained within an optical cavity subject to continuous monitoringof the light leaking out of the cavity. The classical trajectories encodewithin a classical phase-space representation a continuous quantum measurementprocess conditioned on a given detection record. We derive a Fokker-Planckequation for the quasi-probability distribution of the combinedcondensate-cavity system. We unravel the dynamics into stochastic classicaltrajectories that are conditioned on the quantum measurement process of thecontinuously monitored system, and that these trajectories faithfully representmeasurement records of individual experimental runs. Since the dynamics of acontinuously measured observable in a many-atom system can be closelyapproximated by classical dynamics, the method provides a numerically efficientand accurate approach to calculate the measurement record of a large multimodequantum system. Numerical simulations of the continuously monitored dynamics ofa large atom cloud reveal considerably fluctuating phase profiles betweendifferent measurement trajectories, while ensemble averages exhibit localspatially varying phase decoherence. Individual measurement trajectories leadto spatial pattern formation and optomechanical motion that solely result fromthe measurement backaction. The backaction of the continuous quantummeasurement process, conditioned on the detection record of the photons,spontaneously breaks the symmetry of the spatial profile of the condensate andcan be tailored to selectively excite collective modes.

Classical stochastic measurement trajectories: Bosonic atomic gases in an optical cavity and quantum measurement backaction