Quantum limits for stationary force sensing

State-of-the-art linear sensors of force, motion, and magnetic fields have reached the sensitivity where the quantum noise of the meter is significant or even dominant. In particular, the sensitivity of the best optomechanical devices has reached the standard quantum limit (SQL), which directly follows from the Heisenberg uncertainty relation and corresponds to balancing the measurement imprecision and the perturbation of the probe by the quantum back-action of the meter. The SQL is not truly fundamental and several methods for its overcoming have been proposed and demonstrated. At the same time, two quantum sensitivity constraints which are more fundamental are known. The first limit arises from the finiteness of the probing strength (in the case of optical interferometers—of the circulating optical power) and is known as the energetic quantum limit or, in a more general context, as the quantum Cramér-Rao bound (QCRB). The second limit arises from the dissipative dynamics of the probe, which prevents full efficacy of the quantum back-action evasion techniques developed for overcoming the SQL. No particular name has been assigned to this limit; we propose the term dissipative quantum limit (DQL) for it. Here we develop a unified theory of these two fundamental limits by deriving the general sensitivity constraint for stationary, linear systems from which they follow as particular cases. Our analysis reveals a phase transition occurring at the boundary between the QCRB-dominated and the DQL regimes, manifested by the discontinuous derivatives of the optimal spectral densities of the meter quantum noise. This leads to the counterintuitive (but favorable) finding that quantum-limited sensitivity can be achieved with meter noise cross-correlations having a nonzero imaginary component, a feature arising in lossy meter systems. Finally, we show that the DQL originates from the nonautocommutativity of the internal thermal noise of the probe and that it can be overcome in nonstationary measurements.

Figure 1

Generic linear force sensor consisting of probe and meter subsystems. The probe is subjected to an unknown classical signal force $F_{\text{sig}}$, a thermal force ${\widehat{F}}_T$ due to dissipation, and a back-action force ${\widehat{F}}_{\text{ba}}$ from the meter. $\chi$ is the “bare” probe susceptibility. $\tilde{x}$ is the meter output referenced to the probe position.

Figure 2

The linear force sensing scheme of Fig. 1 with an additional feedback loop based on the measurement record.

Figure 3

Plots of the sum quantum noise spectral density $S_{\mathrm{sum}}$ as a function of the back-action noise spectral density $S_{\mathcal{F}\mathcal{F}}$. Shown cases are the full optimization (49) (thin solid line), the optimization under the constraint $\mathrm{Im}{S}_{x\mathcal{F}}=0$ (50) (dashed line), and the negative-mass-reference-frame measurement scheme (discussed in Sec. 4) with suboptimally matched susceptibilities of the mechanical object and the atomic ensemble subsystem, Eq. (64) (dotted line).

Figure 4

Plots of $S_{\mathrm{UB}}$, Eq. (57), as a function of $S_{FF}$ for (top) fixed $\hslash \left|\mathrm{Im}K\right|/S_{\mathrm{thr}}$ and (bottom) fixed $\hslash \left|\mathrm{Im}K\right|/S_{FF}$. Note that the plot domains are constrained by $\hslash \left|\mathrm{Im}K\right|\le S_{FF},S_{\mathrm{thr}}$.