Limitations of strong coupling in non-Markovian quantum thermometry

Abstract

We investigate quantum thermometry using a single-qubit probe embedded in a non-Markovian environment, employing the numerically exact hierarchical equations of motion (HEOM) to overcome the limitations of Born-Markov approximations. Through a systematic analysis of the dynamical and steady-state behavior of the quantum signal-to-noise ratio (QSNR) for temperature estimation, we identify several key findings that challenge the conventional expectation that strong coupling necessarily enhances thermometric performance. In non-equilibrium dynamical thermometry, weak system-environment coupling generally yields the optimal QSNR, whereas in the steady-state regime, strong coupling enhances sensitivity only in the ultra-low-temperature limit, while weak coupling significantly improves precision at moderately low temperatures. To optimize performance across coupling regimes, we develop a hybrid computational framework that integrates HEOM with quantum-enhanced particle swarm optimization, enabling precise quantum dynamical control under varying coupling strengths. Our results reveal fundamental constraints and opportunities in quantum thermometry, offering practical strategies for the design of high-performance quantum thermometers operating in realistic open quantum systems.

Figures (7)

• Figure 1: (a)–(c) QSNR $\mathcal{Q}_T$ dynamics for varying coupling strengths (quantified by the dimensionless parameter $\lambda/\omega_0$) under different environmental parameters, computed using the HEOM method. Results correspond to $T/\omega_0 = 0.2$ with $\omega_c/\omega_0 = 0.05$, $0.1$, and $0.5$, respectively. Insets show the short-timescale $\mathcal{Q}_T$ dynamics. (d)–(f) QSNR $\mathcal{Q}_T$ dynamics for varying $\lambda$ under identical environmental parameters, calculated via the Bloch-Redfield master equation (BRME). The insets depict how increasing the cutoff frequency $\omega_c$ induces spectral smoothing in $J(\omega)$, thereby suppressing environmental memory effects. Here, the system is initialized in the equal superposition state $|\psi(0)\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$ of the two eigenstates of $\hat{H}_S$.
• Figure 2: The variation of optimal QSNR $\mathcal{Q}^{*}_T$ with respect to $\lambda$ under different $\omega_c$ values, with $T/\omega_0 = 0.2$.
• Figure 3: Optimal non-Markovianity measure $\mathcal{N}$, evaluated using the orthogonal states $|\pm\rangle = (|0\rangle \pm |1\rangle)/\sqrt{2}$, as a function of $\omega_c$ for varying $\lambda$. The gray-shaded region corresponds to the low cutoff frequency regime $\omega_c/\omega_0 < 0.06$, while the pink-shaded region denotes high cutoff frequency regime $\omega_c/\omega_0 >0.38$. All other parameters match those in Fig. \ref{['fig1']}.
• Figure 4: (a) Steady-state QSNR $\mathcal{Q}_T(\infty)$ as a function of temperature $T/\omega_0$ for varying system-bath coupling strengths $\lambda/\omega_0$, computed using the HEOM method. Dashed curves indicate thermal equilibrium values for reference, which reveals that complete thermalization occurs only at $\lambda/\omega_0 \lesssim 0.001$, with stronger couplings exhibiting pronounced deviations from equilibrium. (b) Low-temperature regime ($T/\omega_0 < 0.12$), highlighting the emergence of enhanced $\mathcal{Q}_T(\infty)$ under strong coupling conditions.
• Figure 5: (a-c) Optimization of QSNR using the QPSO algorithm for $\lambda = 0.03\omega_0$: (a) Evolution of the fitness value over optimization epochs; (b) Control parameter sequence corresponding to the optimal fitness value; (c) Comparison of the QSNR dynamics with and without optimal control. (d-f) Optimization of QSNR using the QPSO algorithm for $\lambda = 0.04\omega_0$: (d) Evolution of the fitness value over optimization epochs; (e) Control parameter sequence corresponding to the optimal fitness value; (f) Comparison of the QSNR dynamics with and without optimal control. Parameters: $\omega_c=0.05\omega_0$ and $T/\omega_0=0.2$.