We formulate computationally efficient classical stochastic measurement
trajectories for a multimode quantum system under continuous observation.
Specifically, we consider the nonlinear dynamics of an atomic Bose-Einstein
condensate contained within an optical cavity subject to continuous monitoring
of the light leaking out of the cavity. The classical trajectories encode
within a classical phase-space representation a continuous quantum measurement
process conditioned on a given detection record. We derive a Fokker-Planck
equation for the quasi-probability distribution of the combined
condensate-cavity system. We unravel the dynamics into stochastic classical
trajectories that are conditioned on the quantum measurement process of the
continuously monitored system, and that these trajectories faithfully represent
measurement records of individual experimental runs. Since the dynamics of a
continuously measured observable in a many-atom system can be closely
approximated by classical dynamics, the method provides a numerically efficient
and accurate approach to calculate the measurement record of a large multimode
quantum system. Numerical simulations of the continuously monitored dynamics of
a large atom cloud reveal considerably fluctuating phase profiles between
different measurement trajectories, while ensemble averages exhibit local
spatially varying phase decoherence. Individual measurement trajectories lead
to spatial pattern formation and optomechanical motion that solely result from
the measurement backaction. The backaction of the continuous quantum
measurement process, conditioned on the detection record of the photons,
spontaneously breaks the symmetry of the spatial profile of the condensate and
can be tailored to selectively excite collective modes.Comment: 22 pages, 11 figure
