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A fundamental prediction of quantum theory that is derived from the"projection postulate" is that under continuous measurement, the state of asystem traces out a "quantum trajectory" in time that depends upon itsmeasurement record, and that in certain situations, the quantum state of asystem stochastically collapses at a finite rate toward a random eigenstate. Onthe other hand, quantum theory also predicts that environmental couplingproduces the opposite effect of spreading the wave function into a mixed state.Studying the effect of these two competing processes, as the ratio of theirstrengths is varied, on the distribution of a quantum state is of fundamentalinterest in any setting, but especially so in the mechanical realm. A keyissue, however, is that a fundamental study of such physics can not entail an apriori assumption of the projection postulate itself, and Bayesian inferencemay therefore not be employed to observe the quantum state. In this theoreticalwork we propose a feasible way to perform such a fundamental test with amechanical wave function in the energy eigenbasis. The scheme is based on anoptomechanical setup wherein optical fields provide a direct probe of theenergy variance of the mechanical wave function. Such an energy variance probeis unprecedented in any type of system. Numerical simulations that incorporatethe projection postulate predict a monotonic decrease in the time-averagedenergy variance as the ratio of continuous measurement strength to dissipationis increased. The measurement strength in the proposed scheme is tunable insitu, and the behavior predicted by the simulations therefore implies that thescheme amounts to a way to verifiably control in situ the time-averagedvariance of a mechanical wave function over the course of a single quantumtrajectory.

Observing controlled quantum state collapse in a single quantum trajectory of a mechanical oscillator