Direct temperature readout in nonequilibrium quantum thermometry

TL;DR

This work tackles the lack of a direct temperature readout in nonequilibrium quantum thermometry by introducing a thermodynamic-inference framework that assigns a time-dependent reference temperature $T_r(t)$ via the maximum-entropy principle using the thermometer’s mean energy. It then provides positive semi-definite error bounds $\mathcal{E}_1(t)$ and $\mathcal{E}_2(t)$ to construct a corrected dynamical temperature $T_{\rm corr}(t)$ (and $\beta_{\rm corr}(t)$) that converge to the true temperature $T$ as thermalization proceeds. The approach is validated on a qubit thermometer modeled by a Lindblad equation, showing that initial quantum coherence can boost readout precision while dephasing can reduce the metrological advantage, and that the corrected readout outperforms the raw reference temperature in finite time. The results offer a practically feasible route to real-time thermal monitoring in quantum technologies, enabling more reliable temperature control in nanoscale and cryogenic devices.

Abstract

Quantum thermometry aims to measure temperature in nanoscale quantum systems, paralleling classical thermometry. However, temperature is not a quantum observable, and most theoretical studies have therefore concentrated on analyzing fundamental precision limits set by the quantum Fisher information through the quantum Cramer-Rao bound. In contrast, whether a direct temperature readout can be achieved in quantum thermometry remains largely unexplored, particularly under the nonequilibrium conditions prevalent in real-world applications. To address this, we develop a direct temperature readout scheme based on a thermodynamic inference strategy. The scheme integrates two conceptual developments: (i) By applying the maximum entropy principle with the thermometer's mean energy as a constraint, we assign a reference temperature to the nonequilibrium thermometer. We demonstrate that this reference temperature outperforms a commonly used effective temperature defined through equilibrium analogy. (ii) We obtain positive semi-definite error functions that lower-bound the deviation of the reference temperature from the true temperature-in analogy to the quantum Cramer-Rao bound for the mean squared error-and vanish upon thermalization with the sample. Combining the reference temperature with these error functions, we introduce a notion of corrected dynamical temperature which furnishes a postprocessed temperature readout under nonequilibrium conditions. We validate the corrected dynamical temperature in a qubit-based thermometer under a range of nonequilibrium initial states, confirming its capability to estimate the true temperature. Importantly, we find that increasing quantum coherence can enhance the precision of this readout.

Figures (4)

• Figure 1: Dynamics of $\mathcal{F}_T(t)$ with dephasing strength $\gamma_0=0$ (green line), $\gamma_0=-.2$ (orange line) and $\gamma_0=0.5$ (blue line) starting from a coherent initial state $\rho_p(0)=0.5\mathrm{I}+0.4\sigma_x-0.2\sigma_z$ with $\mathrm{I}$ the identity matrix. For comparison, the red curve shows $\mathcal{F}^{\rm{in}}_T(t)$ for an incoherent initial state $\rho_p(0)=0.5\mathrm{I}-0.2\sigma_z$. The black dashed line marks the value of thermal QFI given by Eq. (\ref{['eq:F_T_s']}) which equals the stationary value of both $\mathcal{F}_T(t)$ and $\mathcal{F}^{\rm{in}}_T(t)$ as $t\to \infty$. Other parameters are $\omega=1$, $T=0.5$ and $\gamma=1$.
• Figure 2: Dynamics of the reference temperature $\beta_r(t)$ under a coherent initial state $\rho_0(0) = 0.5 \text{I} + 0.2 \sigma_x - 0.2 \sigma_z$ for different dephasing strengths $\gamma_0$. The blue solid line represents the effective temperature $\beta_e(t)$ for $\gamma_0=0.5$. The blacked dashed line marks the value of actual inverse temperature $\beta$. Other parameters are $\omega=1$, $T=0.5$ and $\gamma=1$.
• Figure 3: Performance of the corrected dynamical temperatures $T_{\mathrm{corr}}^1(t)$ (upper panel, green solid line) and $\beta_{\mathrm{corr}}^2(t)$ (lower panel, green solid line) defined in Eqs. (\ref{['eq:corr_T']}) and (\ref{['eq:corr_beta']}), respectively. All plots correspond to initial states with fixed coherence $\rho_{p,12}(0)=\rho_{p,21}(0)=0.2$ but varying populations: Left column (a,d) $\rho_{p,11}(0)=0.3,\rho_{p,22}(0)=0.7$; middle column (b,e) $\rho_{p,11}(0)=0.2,\rho_{p,22}(0)=0.8$; right column (c,f) $\rho_{p,11}(0)=0.1,\rho_{p,22}(0)=0.9$. For comparison, the reference temperature $T_{\mathrm{r}}(t)$ (upper panel, orange dashed line) and its inverse $\beta_{\mathrm{r}}(t)$ (lower panel, orange dashed line) are also shown. The horizontal black dashed line marks the actual temperature $T$ (upper panel) or its inverse $\beta$ (lower panel). Insets in the upper panel display the temperature deviation $\Delta T(t)=|T_r(t)-T|$ (green solid line) and its lower bound $\epsilon_1(t)$ from Eq. (\ref{['eq:epson1']}) (orange solid line). Insets in the lower panel show the inverse-temperature deviation $\Delta\beta(t)=|\beta_r(t)-\beta|$ (green solid line) and its lower bound $\epsilon_{2}(t)$ from Eq. (\ref{['eq:epson2']}) (orange solid line). Parameters are $\gamma_0=0$, $\omega=1$, $T=0.5$ and $\gamma=1$.
• Figure 4: Performance of the corrected dynamical temperatures $T_{\mathrm{corr}}^1(t)$ (upper panel, green solid line) and $\beta_{\mathrm{corr}}^2(t)$ (lower panel, green solid line) defined in Eqs. (\ref{['eq:corr_T']}) and (\ref{['eq:corr_beta']}), respectively. All plots correspond to initial states with fixed populations $\rho_{p,11}(0)=0.3, \rho_{p,22}(0)=0.7$ but varying coherences: Left column (a, c) $\rho_{p,12}(0)=\rho_{p,21}(0)=0.3$; right column (b, d) $\rho_{p,12}(0)=\rho_{p,21}(0)=0.4$. For comparison, the reference temperature $T_{\mathrm{r}}(t)$ (upper panel, orange dashed line) and its inverse $\beta_{\mathrm{r}}(t)$ (lower panel, orange dashed line) are also shown. The horizontal black dashed line marks the actual temperature $T$ (upper panel) or its inverse $\beta$ (lower panel). Insets in the upper panel display the temperature deviation $\Delta T(t)=|T_r(t)-T|$ (green solid line) and its lower bound $\epsilon_1(t)$ from Eq. (\ref{['eq:epson1']}) (orange solid line). Insets in the lower panel show the inverse-temperature deviation $\Delta\beta(t)=|\beta_r(t)-\beta|$ (green solid line) and its lower bound $\epsilon_{2}(t)$ from Eq. (\ref{['eq:epson2']}) (orange solid line).Parameters are $\gamma_0=0$, $\omega=1$, $T=0.5$ and $\gamma=1$.