Shake before use: universal enhancement of quantum thermometry by unitary driving

TL;DR

The paper proves a universal, model-independent enhancement of quantum thermometry by any temperature-dependent unitary driving applied to a thermally prepared probe, showing the QFI decomposes as $\mathcal{F}^\beta_t = \mathcal{F}^\beta_{\pi_0} + \mathcal{I}^\beta_t$ with $\mathcal{I}^\beta_t \ge 0$. It expresses the dynamical increment via a positive semidefinite kernel of information currents, $K(s,u,\beta)$, capturing two-time correlations of the driving-induced distinguishability flow. In a driven spin-1/2 thermometer, resonance with the Bohr frequency recovers a quadratic-in-time scaling of the QFI at long times and enables shifting the sensitivity window across temperatures by shaping the drive's $\beta$-dependence. The results position external unitary control as a universal resource for precision metrology and have broad implications for driven quantum estimation beyond thermometry.

Abstract

Quantum thermometry aims at determining temperature with ultimate precision in the quantum regime. Standard equilibrium approaches, limited by the Quantum Fisher Information given by static energy fluctuations, lose sensitivity outside a fixed temperature window. Non-equilibrium strategies have therefore been recently proposed to overcome these limits, but their advantages are typically model-dependent or tailored for a specific purpose. This Letter establishes a general, model-independent result showing that any temperature-dependent unitary driving applied to a thermalized probe enhances its quantum Fisher information with respect to its equilibrium value. Such information gain is expressed analytically through a positive semi-definite kernel of information currents that quantify the flow of statistical distinguishability. Our results are benchmarked on a driven spin-$1/2$ thermometer, furthermore showing that resonant modulations remarkably restore the quadratic-in-time scaling of the Fisher information and allow to shift the sensitivity peak across arbitrary temperature ranges. Our findings identify external unitary control as a universal resource for precision metrology and pave the way for future implementations in quantum sensing.

Figures (3)

• Figure 1: Schematic illustration of the protocol. In the initial stage (left), the probe is allowed to thermalize with the sample, encoding the unknown inverse temperature $\beta^*$ into the Gibbs state $\pi_0(\beta^*)$, and yielding an estimation precision bounded by the equilibrium quantum Cramér–Rao bound in \ref{['eq:CramerRaoB']}. At $t=0$ the driving is switched on, perturbing the now isolated probe. The resulting QFI of the non-equilibrium state $\rho(t,\beta^*)$ evolves according to \ref{['eq:main']}, increasing whenever the drive depends non-trivially on temperature and thereby enhancing the attainable precision in the estimation (here illustrated as the acquisition of an additional significant digit in the estimator $\hat{\beta}$).
• Figure 2: Time evolution of the QFI for a single-spin probe under a temperature-dependent transverse driving. Time $t$ is measured in units of the spin characteristic period $2\pi/\Omega$, with $\Omega=1$. Solid and dashed lines denote, respectively, the QFI as given by \ref{['eq:main']} and by the standard expression in terms of the instantaneous eigenvalues of $\rho(t)$art:Liu_QFIM, showing perfect agreement. The dash-dotted line marks the equilibrium QFI benchmark. The three panels illustrate the transition from bounded dynamics to resonant quadratic scaling, identifying the optimal driving frequency. The Gaussian temperature control profile is chosen with a standard deviation saturating the equilibrium quantum Cramér–Rao bound, as if an optimal equilibrium estimation had been performed beforehand, and is randomly centered within the interval $\max\bigl(0,\,\beta^* \pm 1/\sqrt{\mathcal{F}^{\beta^*}_{\pi_0}}\bigr)$. For all panels $\phi = 0$ and $\lambda_0 = 0.1$. (a) Off-resonant regime with $\omega_{\mathrm d}=0.5\,\Omega$, where the QFI exhibits bounded oscillations. (b) Resonant regime with $\omega_{\mathrm d}=\Omega$, where the QFI displays an almost monotonic $t^2$ growth, signalling a build-up of sensitivity. (c) High-frequency regime with $\omega_{\mathrm d}=2\,\Omega$, where one observes once again a bounded, oscillatory response, albeit at a faster characteristic timescale.
• Figure 3: Behaviour of the maximum QFI at a fixed evolution time $t=12\,s$ in the resonant regime, as a function of the inverse temperature $\beta$. The magenta dashed line (with squares) represents the equilibrium baseline $\mathcal{F}^\beta_{\pi_0}$, while the blue circles and yellow triangles show the maxima for the driven case, for $G(\beta)$ with $(\beta_0,s_\beta)=(10,3)$ and $(5,3)$, respectively. By tuning the mean and standard deviation of the Gaussian envelope, the sensitivity window can be shifted to different temperature ranges. The resulting QFI profiles display two symmetric lobes around $\beta_0$, reflecting the underlying Gaussian modulation of the driving amplitude, as exemplified by the yellow curve.