Sensitivity limits of non-stationary quantum sensors

Abstract: The concept of the dissipative quantum limit (DQL) was first put forward in 1980s and was analyzed in detail much later in Ref. [Phys. Rev. A 103, 043721 (2021)] for the particular case of stationary (invariant with respect to a shift of time) systems. Here we extend that analysis to the general non-stationary case.

• The paper demonstrates that for single-parameter estimation with fully known time dependence, DQL constraints can be evaded with non-causal filtering.
• Explicit derivations show that in non-stationary settings, meter output commutators vanish, enabling arbitrarily small measurement noise for known signals.
• For multi-parameter estimation, DQL-like uncertainty relations re-emerge, establishing sensitivity bounds directly tied to physical dissipation.

Sensitivity Limits in Non-Stationary Quantum Sensors

Introduction

This paper undertakes a rigorous analysis of quantum sensitivity limits in linear force sensors under general non-stationary (time-dependent) conditions, extending prior results obtained for stationary systems. It examines explicit constraints arising from probe dissipation (the dissipative quantum limit, DQL) within the framework of linear quantum measurement theory, and situates these results in the context of quantum optomechanical sensing and gravitational-wave detection, where quantum backaction and measurement imprecision jointly set the Standard Quantum Limit (SQL).

Theoretical Framework

The starting point is the established linear theory of quantum measurement, where a quantum probe is monitored by a linear meter and subjected to both classical signals and stochastic quantum and thermal forces. The SQL, defined by the balance of measurement imprecision and quantum backaction, is given (in spectral density units) by $S_{\rm SQL}(\Omega) = \hbar|\chi^{-1}(\Omega)|$ where $\chi^{-1}(\Omega)$ is the probe response function. However, the SQL is not a strict lower bound: correlations between the measurement and back-action noises (e.g., via squeezed states) can evade the SQL. Practical demonstrations in interferometric gravitational-wave detectors such as LIGO have validated such SQL-evasion strategies.

More fundamental limits arise from two mechanisms: (1) the quantum Cramér-Rao bound (QCRB) stemming from finite energy resources, and (2) the DQL, which originates from intrinsic probe dissipation (the imaginary part of $\chi^{-1}$). For stationary systems, the DQL is characterized by the minimum achievable noise spectral density $S_{\rm DQL}(\Omega) = \hbar|\Im\chi^{-1}(\Omega)|$, independent of the measurement protocol.

Non-Stationary Measurement Analysis

The generalization to non-stationary sensors, for which all response functions and noises can be explicitly time-dependent, substantially complicates the uncertainty relations. The DQL in stationary settings is constructed from frequency-domain commutator constraints, but in the absence of stationarity, these Fourier-based derivations are not applicable.

The paper proceeds by analyzing the Heisenberg commutators among the constituent meter noise operators for the general time-dependent case. It is shown that the autocorrelator of the meter output—after accounting for response function inversion and thermal noise contributions—commutes at all times: $[\tilde{F}(t), \tilde{F}(t')]=0$. This confirms that, in principle, the DQL does not constrain the sensitivity of non-stationary quantum sensors to forces of known temporal shape.

By explicit construction, the author demonstrates that, for a general non-stationary quantum meter, it is always possible to engineer a meter state such that, for a fixed known signal, the measurement noise can be made arbitrarily small, unconstrained by the DQL, provided classical data processing (e.g. non-causal filtering) is allowed and measurement strength is unbounded. This contrasts with stationary cases, and with situations where the force is not entirely known a priori.

Multiple Signal Components and DQL-like Constraints

The scenario changes drastically when considering the simultaneous estimation of multiple parameters (e.g., the two quadratures of a narrow-band force). The operator-valued measurement errors associated with distinct filter functions generally do not commute if the probe exhibits dissipation. The author derives a set of DQL-like uncertainty relations for measurement noise, demonstrating that for non-commuting observables the product of variances is bounded from below by the squared dissipation:

$$(\Delta F^{(j)}_{\rm sum})^2 (\Delta F^{(k)}_{\rm sum})^2 \geq \hbar^2 \left( \int \Phi_j(t) \chi_a^{-1}(t,t') \Phi_k(t') \, dtdt' \right)^2.$$

The paper provides a concrete example for narrowband force detection at frequency $\Omega_0$ under stationary friction, showing that the DQL-type constraint re-emerges for simultaneous quadrature measurements, effectively generalizing the DQL for non-stationary, multi-component estimation scenarios. The lower bound now depends on the antisymmetric (dissipative) part $\chi_a^{-1}$ of the response kernel, but not on the full kernel, as in the SQL.

Implications

The results in this paper reveal that for single-parameter estimation with fully known time dependence, non-stationary quantum sensors are not fundamentally bound by the DQL. This suggests opportunities for metrology protocols utilizing time-dependent control or non-stationary states to surpass classic quantum noise constraints in targeted force detection tasks. However, for multi-parameter estimation (such as full quantum state tomography of an external field, or simultaneous quadrature measurement), DQL-like uncertainty relations enforce sensitivity limits directly proportional to physical dissipation, independent of the SQL.

This distinction has consequences both for the theory of quantum measurement—where practical metrology must confront tradeoffs between parameter types, prior knowledge, and dissipation—and for the design of quantum-enhanced sensors where temporal control and tailored input states are feasible.

Conclusion

The paper establishes rigorously that the dissipative quantum limit constrains only multi-component estimation in non-stationary quantum sensors, while single-parameter detection with a priori known time dependence can in principle evade DQL restrictions. These findings underscore the crucial role of task-specific prior knowledge in quantum sensing limits, and provide a formal basis for developing advanced non-stationary quantum metrology protocols. The explicit separation of SQL and DQL roles in stationary versus non-stationary, single versus multi-parameter detection offers theoretical guidance and practical benchmarks for future experiments targeting the ultimate limits of force sensing and optomechanical precision measurement.