Improved mirror position estimation using resonant quantum smoothing

The field of quantum metrology can be described as using quantum resources to enhance measurement precision beyond what is achievable with classical resources alone. A related discipline is quantum parameter estimation that is focused on precisely estimating the classical parameters (e.g. optical phase, mirror position) of a quantum system. Precise measurement of these parameters is important to fields like quantum control, gravitational wave detection, opto-mechanical force sensing and quantum metrology. Many applications of quantum parameter estimation rely on an optical probe, e.g. a coherent state or a squeezed state, to make precision measurements ultimately limited by quantum mechanics. In cases where delays are tolerable estimation precision can be further enhanced by the acausal technique known as quantum smoothing, which takes advantage of future information when forming the estimate. Quantum smoothing has been shown to give a precision enhancement by up to (but not exceeding) a factor two over optimal filtering on non-resonant systems. In this work, we use a coherent state probe to show that implementing quantum smoothing on a mechanically resonant structure driven by a resonant forcing function can achieve a greater improvement in precision than on non-resonant systems. We show that it is possible to achieve a smoothing enhancement factor of in excess of three over optimal filtering. Additionally, by taking advantage of an intra-cavity light probe, we demonstrate this technique to estimate the position of a cavity mirror to a finer precision than has been achieve with equivalent quantum resource in free-space. These results are published in [T. A. Wheatley, M. Tsang, I. R. Petersen and E. H.  Huntington, EPJ Quantum Technology, 2:13 (2015)].