Nonequilibrium Quantum Thermometry

• Nonequilibrium quantum thermometry is defined as temperature estimation using quantum probes interacting for finite times, where quantum features enhance sensitivity beyond equilibrium limits.
• Methodologies optimize probe coherences, entanglement, and non-Markovian memory to boost the quantum Fisher information, enabling rapid and minimally invasive temperature measurements.
• Practical strategies span single-qubit, Gaussian, and critical probes, employing time-resolved control and adaptive measurements to achieve high precision in nanoscale quantum technologies.

Nonequilibrium quantum thermometry is the theory and practice of temperature estimation using quantum probes that interact with a thermal reservoir for finite times, typically without waiting for the probe to reach equilibrium. In this regime, quantum features such as coherence, nonclassical correlations, and non-Markovian memory fundamentally alter both the achievable sensitivity and the optimal estimation protocols compared to their equilibrium counterparts. The precision of temperature inference is characterized by the quantum Fisher information (QFI) of the nonequilibrium probe state, which need not coincide with traditional thermodynamic quantities such as heat capacity. Nonequilibrium thermometric strategies are crucial for rapid, minimally invasive sensing at nanoscale, in ultrafast, strongly correlated, or dissipative environments, and in quantum technologies where full equilibration is either unattainable or suboptimal.

1. Conceptual Foundation and Nonequilibrium Thermometric Paradigms

In classical and equilibrium quantum thermometry, the uncertainty in temperature estimation is fundamentally constrained by the probe's equilibrium energy fluctuations, encapsulated in the heat capacity. For a probe with Hamiltonian $H$ at temperature $T$, the equilibrium precision bound is given by

$$\Delta^2 T \ge \frac{k_B^2 T^4}{\Delta^2 E_T^{(\mathrm{eq})}}, \qquad C_T^{(\mathrm{eq})} = \frac{\Delta^2 E_T^{(\mathrm{eq})}}{k_B T^2}$$

where $\Delta^2 E_T^{(\mathrm{eq})}$ is the energy variance in the equilibrium Gibbs state. In nonequilibrium quantum thermometry, as formalized in (Cavina et al., 2018), the protocol consists of preparing an initial probe state $\rho(0)$, allowing it to interact with the bath for time $\tau$, and then subjecting it to a measurement while $\rho(\tau) \neq \rho_T^{(\mathrm{eq})}$. The resulting metrological framework requires full characterization via the quantum Fisher information $Q_T(\tau)$ of $\rho(\tau)$. This generalizes thermodynamic uncertainty to a quantum estimation theory context, establishing a regime where quantum coherences, correlations, and time-resolved control can enable sensitivity enhancements beyond equilibrium thermodynamics.

2. Thermo-Metrological Bounds and the Role of Quantum Coherence

The core nonequilibrium sensitivity bound is a generalized Cramér–Rao inequality,

$$\Delta^2 T \ge \frac{1}{M Q_T(\tau)}\,,\quad Q_T(\tau)\ge \frac{C_T^2(\tau)}{\Delta^2 E_T(\tau)}$$

where $M$ is the number of repeated independent interrogations, $Q_T(\tau)$ the QFI for temperature at time $\tau$, $C_T(\tau)$ a generalized (nonequilibrium) heat capacity, and $\Delta^2 E_T(\tau)$ the probe's energy variance at $\tau$ (Cavina et al., 2018). For a qubit probe, when the state maintains quantum coherences, $Q_T(\tau)$ receives additional contributions beyond those accessible to energy measurements alone. Explicitly, in the Bloch vector parametrization $\vec r(\tau)$,

$$Q_T(\tau)=\frac{[\partial_T r(\tau)]^2}{1-r^2(\tau)+r^2(\tau) [\partial_T \theta(\tau)]^2}$$

where $r(\tau)=|\vec r|$ and $\theta(\tau)$ encodes quantum phase. The denominator's coherence-dependent term $r^2(\tau)[\partial_T\theta(\tau)]^2$ is a direct witness of quantum enhancement: in the absence of coherence, the denominator reduces and $Q_T$ matches the classical (energy-based) sensitivity. The maximization of $Q_T(\tau)$ over initial state and interrogation time identifies regimes where the best precision is obtained before equilibration, and generally requires measurements in the SLD (symmetric logarithmic derivative) basis (Cavina et al., 2018).

3. Uncertainty Relations Linking Heat, Correlations, and Estimation Precision

Beyond probe energy fluctuations, the nonequilibrium temperature–heat uncertainty relation formalizes that both the stochastic trajectory heat and measurement-induced correlation heat are intrinsic quantum resources for temperature estimation (Zhang et al., 2023). The relation

$$\Delta\beta\,\Delta Q \geq 1,\qquad Q \equiv \delta H_{tra} + H_{cor}$$

clarifies that the total information on $T$ comes from the joint fluctuations of heat exchanged along the probe's open-system trajectory ($H_{tra}$) and the heat change due to probe-bath measurement-induced correlations ($H_{cor}$). In strong-coupling or transient regimes, the latter can dominate and directly link quantum correlations (including entanglement) to metrological advantage. Explicit model calculations demonstrate regimes where correlation heat constitutes the dominant contribution to Fisher information, particularly for probes subject to dephasing dynamics (Zhang et al., 2023).

4. Quantum Resources in Nonequilibrium Thermometry

Different quantum resources provide operational advantages in nonequilibrium thermometry depending on probe architecture and system-bath interaction:

• Coherence: Initial quantum superpositions aligned transverse to the probe Hamiltonian amplify sensitivity in the transient regime, producing peak QFI prior to equilibration. This enhancement is strictly monotonic with the degree of coherence and can exceed equilibrium sensitivity, but typically vanishes asymptotically (Frazao et al., 2024, Trombetti et al., 4 Jul 2025).
• Entanglement and Correlations: In multipartite probe ensembles, preexisting local coherences and global quantum correlations (e.g., GHZ-type entanglement) act synergistically to amplify early-time QFI peaks, though the maximal achievable scaling with probe number remains linear under local Markovian evolution (Trombetti et al., 4 Jul 2025).
• Nonclassical Gaussian Probes: For continuous-variable (infinite-dimensional) thermometers, single- and two-mode squeezed states offer an unbounded QFI rate enhancement over all classical Gaussian probes for short time and strong squeezing, operationally accessible via joint Gaussian measurement schemes (Mirkhalaf et al., 2022). However, ultimate sensitivity is realized only at steady state for weakly coupled Markovian probes (Mancino et al., 2020).

These findings establish that the effective engineering and utilization of quantum resources depends on finite interaction time, dissipative and memory channel structure, and available measurement bases.

5. Nonequilibrium Strategies and Practical Protocols

Practical nonequilibrium thermometric protocols span a broad range of implementations and inference strategies:

• Single-Qubit Probes: Photonic (Mancino et al., 2016, Tham et al., 2016), superconducting, and trapped-ion based probes implement generalized amplitude-damping dynamics, allowing for time-resolved discrimination of bath temperatures by exploiting finite-time transient dynamics and optimizing over initial states and measurement basis.
• Interferometric and Berry Phase Schemes: Protocols leveraging quantum phases acquired under nonequilibrium evolution, including geometric (Berry) phases, achieve temperature sensing without any need for probe thermalization, demonstrated in both theoretical (Mann et al., 2014) and photonic implementations.
• Work Fluctuation Thermometry: Qubit interferometry protocols using Tasaki–Crooks fluctuation relations can extract temperature from universal non-equilibrium work distributions, requiring minimal prior system knowledge and non-destructive interaction with, e.g., cold atomic gases (Johnson et al., 2015).
• Critical and Strongly-Coupled Probes: Open quantum Rabi models and strongly-coupled fermionic probes present non-Markovian dynamics with optimal QFI at short times leveraging critical-enhancement and non-Markovian revivals for rapid and ultrasensitive thermometry (Xie et al., 2021, Rodríguez et al., 2023).
• Sequential and Adaptive Measurements: Non-equilibrium sequential measurement schemes enable precision beyond the energy-measurement equilibrium QFI by utilizing repeated, rapid local sampling before full system relaxation. Such strategies are robust to partial system access and experimentally feasible in many-body probes (Yang et al., 2024).

A representative taxonomy of some protocols and their key operational features is summarized below.

Probe Type	Quantum Resource	Optimality Regime
Qubit (finite-d)	Coherence, correlations	Transient/pre-equilibrium
Gaussian (CV)	Squeezing, entanglement	Short time; steady state
Critical probe	Susceptibility divergence	At or near phase transition
Many-body/sequential	Sequential correlations	Sub-thermalization

6. Fundamental and Operational Limits

The ultimate precision of nonequilibrium quantum thermometry is limited by several fundamental considerations:

• Probe Dimension and Energy Measurement: For a $d$-level probe, no protocol can surpass the coarse-grained energy-measurement bound, $F_P(T) \le B(d)$, which is always less than or equal to the full quantum Fisher information obtainable by projective energy measurement on the bath itself (Hovhannisyan et al., 2020).
• Scaling Laws: Under Markovian, weak-coupling, and secular approximations, temperature precision scales at best as shot-noise with total interrogation time, i.e., $\Delta T \sim 1/\sqrt{\tau}$, and, for multi-probe strategies, linear in the number of probes $N$ (no Heisenberg scaling) (Sekatski et al., 2021, Trombetti et al., 4 Jul 2025).
• Lamb Shift and Low-$T$ Regimes: Inclusion of temperature-dependent Lamb shifts in weak-coupling theory leads to polynomial scaling $\Delta T \sim T^{-\alpha}/\sqrt{\tau}$ at low $T$, replacing the exponential suppression typical of standard dissipative thermometry (Sekatski et al., 2021).
• Heat–Temperature Uncertainty: The central nonequilibrium temperature–heat uncertainty relation $\Delta\beta \cdot \Delta Q \ge 1$ unifies thermodynamic and estimation-theoretic constraints, explicitly identifying the operational quantum resources for finite-time, quantum-limited thermometric protocols (Zhang et al., 2023).

7. Emerging Themes and Design Perspectives

Recent developments reveal several key principles for the design of high-precision nonequilibrium quantum thermometers:

• Harnessing Mpemba and Anomalous Relaxation: Nonequilibrium probes initialized “hotter” than equilibrium can, via the Mpemba effect, achieve transiently superior QFI at finite times over both colder and equilibrium-initialized probes. This inversion is generic in open quantum dynamics with a nontrivial Liouvillian spectrum and constitutes a new paradigm for ultrafast quantum thermometry (Chattopadhyay et al., 8 Jan 2026).
• Exploiting Noncommutative Interactions and Memory: Probes coupled via noncommuting system-bath channels (simultaneous dissipative and dephasing couplings) exhibit coherence trapping, enhanced back-action, and quadratic low-$T$ scaling of QFI, providing tunable resources and extended sensitivity windows even at weak coupling (Aiache et al., 22 Dec 2025).
• Criticality and Non-Markovianity: Proximity to dissipative phase transitions or non-Markovian memory-induced revivals provides a robust operational resource for precision enhancement, with the optimal measurement time strictly finite and protocol-dependent (Xie et al., 2021, Rodríguez et al., 2023).
• Measurement and Control Considerations: Optimal nonequilibrium thermometry often requires precise time-resolved control, initialization, and measurement basis adaptation—in particular, projection in the SLD basis rather than energy or populations. Practical schemes in photonics, circuit QED, trapped ions, and cold atoms increasingly realize these protocols with growing sophistication (Mancino et al., 2016, Johnson et al., 2015, Rodríguez et al., 2023).

Taken together, nonequilibrium quantum thermometry constitutes a unifying framework for fast, precise, and quantum-enhanced temperature sensing, linking quantum metrology, thermodynamics, and open system dynamics across diverse quantum platforms.