Quantum smoothing of mechanical motion

Measurements provide a powerful tool to prepare quantum states. A key question is how to optimally combine the measurement outcomes to best estimate the state, that is: what filter should be chosen? An important result from quantum control theory is that for linear systems, such as most optically read-out mechanical resonators, classical estimation theory can be directly applied to determine this filter so long as only past measurements are used (i.e., the state is “predicted”). It is not widely appreciated that this result breaks down when future measurements are used (i.e., the state is “retrodicted”), or when past and future measurements are combined (i.e., the state is “smoothed”). This talk will present experimental results that demonstrate this divergence of quantum and classical estimation theory for the first time for a mechanical system, in the particular case of smoothing of a continuously measured mechanical resonator that is close to its ground state. The talk will, further, present theory showing that, for sufficiently high detection efficiency, classical estimation theory predicts an apparent violation of the Heisenberg uncertainty principle. Taken together, our results highlight the need to take proper account of quantum effects when performing retrodiction. This is important, for instance, for optimal quantum sensing since this uses all available measurement data and when using retrodiction to validate states prepared using past measurement.