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Stochastic collapse and decoherence of a non-dissipative forced harmonic oscillator

Careful monitoring of harmonically bound (or as a limiting case, free) masses is the basis of current and future gravitational wave detectors, and of nanomechanical devices designed to access the quantum regime. We analyse the effects of stochastic localization models for state vector reduction, and of related models for environmental decoherence, on such systems, focusing our analysis on the non-dissipative forced harmonic oscillator and its free mass limit. We derive an explicit formula for the time evolution of the expectation of a general operator in the presence of stochastic reduction or environmentally induced decoherence, for both the non-dissipative harmonic oscillator and the free mass. In the case of the oscillator, we also give a formula for the time evolution of the matrix element of the stochastic expectation density matrix between general coherent states. We show that the stochastic expectation of the variance of a Hermitian operator in any unraveling of the stochastic process is bounded by the variance computed from the stochastic expectation of the density matrix, and we develop a formal perturbation theory for calculating expectation values of operators within any unraveling. Applying our results to current gravitational wave interferometer detectors and nanomechanical systems, we conclude that the deviations from quantum mechanics predicted by the continuous spontaneous localization (CSL) model of state vector reduction are at least five orders of magnitude below the relevant standard quantum limits for these experiments. The proposed LISA gravitational wave detector will be two orders of magnitude away from the capability of observing an effect.