Nonstationary force sensing under dissipative mechanical quantum squeezing

We study the stationary and nonstationary measurement of a classical force driving a mechanical oscillator coupled to an electromagnetic cavity under two-tone driving. For this purpose, we develop a theoretical framework based on the signal-to-noise ratio to quantify the sensitivity of linear spectral measurements. Then we consider stationary force sensing and study the necessary conditions to minimize the added force noise. We find that imprecision noise and back-action noise can be arbitrarily suppressed by manipulating the amplitudes of the input coherent fields; however, the force noise power spectral density cannot be reduced below the level of thermal fluctuations. Therefore, we consider a nonstationary protocol that involves nonthermal dissipative state preparation followed by a finite time measurement, which allows one to perform measurements with a signal-to-noise ratio much greater than the maximum possible in a stationary measurement scenario. We analyze two different measurement schemes in the nonstationary transient regime: a back-action-evading measurement, which implies modifying the drive asymmetry configuration upon arrival of the force, and a nonstationary measurement that leaves the drive asymmetry configuration unchanged. Conditions for optimal force noise sensitivity are determined, and the corresponding force noise power spectral densities are calculated.