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arXiv:2607.06215

Coherence Estimation Beyond the Liouvillian Gap in a Finite Nonequilibrium System

Sonali Brahma, Trishna Kalita, Himangshu Prabal Goswami

quant-phcond-mat.mes-hallSubmitted Jul 7, 2026• Announced Jul 8, 2026• 0 citations
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Abstract

We investigate the estimation of bath-induced coherence in a finite quantum system interacting with thermal reservoirs. Enhancement of coherence estimation is transient and the estimation precision totally disappears at the steady state despite the system retaining finite coherence. By analyzing the full Liouvillian eigenspectrum, we demonstrate that the optimal sensing window emerges from the competition between identifiable contributory modes' temporal relaxation and statistical importance. Neither is the linear inverse scaling of Liouvillian gap with transient optimal time a signature of unimodal contribution to optimal sensing, nor is the existence of multimodal dynamics a signature of nonlinear scaling. The inverse Liouvillian gap does not obey any general scaling with the optimal sensing time of coherence and we prove our numerical results analytically using a general Markovian framework. We further show that coupling the finite system to a quantum cavity and maintaining a thermal bias, transforms the transient metrological optimization into a sustained steady-state resource.

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We investigate the estimation of bath-induced coherence in a finite quantum system interacting with thermal reservoirs. Enhancement of coherence estimation is transient and the estimation precision totally disappears at the steady state despite the system retaining finite coherence. By analyzing the full Liouvillian eigenspectrum, we demonstrate that the optimal sensing window emerges from the competition between identifiable contributory modes' temporal relaxation and statistical importance. Neither is the linear inverse scaling of Liouvillian gap with transient optimal time a signature of unimodal contribution to optimal sensing, nor is the existence of multimodal dynamics a signature of nonlinear scaling. The inverse Liouvillian gap does not obey any general scaling with the optimal sensing time of coherence and we prove our numerical results analytically using a general Markovian framework. We further show that coupling the finite system to a quantum cavity and maintaining a thermal bias, transforms the transient metrological optimization into a sustained steady-state resource.
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We unveil a fundamental temperature bias in transient quantum thermometry under Markovian dynamics. For qubit probes evolving in a thermal Markovian environment, we prove that transient precision beyond the steady-state benchmark can be achieved if and only if the probe is initially colder than the bath temperature to be estimated. Cold probes are therefore both necessary and sufficient for enhanced transient precision in the Markovian regime. We then investigate the fate of this bias in the presence of environmental memory. In particular, in a non-Markovian scenario generated by an auxiliary-mediated system-bath coupling, we find that the cold-probe requirement for enhanced transient precision persists, indicating that the temperature bias survives certain forms of memory effects. In contrast, for a non-Markovian collisional model with perfect swap interactions between bath ancillas, transient enhancement is entirely absent regardless of the probe's initial temperature. This indicates that strong non-Markovianity can lead to the complete disappearance of the enhancement effect, placing hot and cold probes on equal footing, with neither capable of achieving enhanced precision in this regime.
The metrological limits of thermometry operated in nonequilibrium dynamical regimes are analyzed. We consider a finite-dimensional quantum system, employed as a quantum thermometer, in contact with a thermal bath inducing Markovian thermalization dynamics. The quantum thermometer is initialized in a generic quantum state, possibly including quantum coherence with respect to the Hamiltonian basis. We prove that the precision of the thermometer, quantified by the Quantum Fisher Information, is enhanced by the quantum coherence in its initial state. We analytically show this in the specific case of qubit thermometers for which the maximization of the Quantum Fisher Information occurs at a finite time during the transient thermalization dynamics. Such a finite-time precision enhancement can be better than the precision that is achieved asymptotically.
We investigate quantum thermometry in a driven--dissipative Kerr cavity coupled to a thermal reservoir. The system exhibits a finite-size precursor of a dissipative phase transition, characterized by a pronounced minimum of the Liouvillian gap and a sharp jump in the steady-state physical observables such as average photon number at the critical driving strength. We show that this regime leads to strong enhancement of the quantum Fisher information (QFI) for temperature estimation. Using an effective two-branch description, we show that the enhancement originates from temperature-induced redistribution of the weight factors in photon number distribution between low- and high-photon-number branches, which is described by an effective binary Fisher information. By optimizing the coherent drive, the enhanced response persists over an extended low-temperature, low-thermal-occupation regime and yields a favorable relative temperature-uncertainty bound. These results identify finite-size precursors of dissipative phase transitions in Kerr-cavity platforms as useful resources for tunable nonequilibrium quantum thermometry. We further show that the predicted thermometric enhancement is accessible in a parameter regime compatible with circuit quantum electrodynamics (circuit-QED) platforms.
Quantum thermometry provides a key capability for nanoscale devices and quantum technologies, but most existing strategies rely on probes initialized near equilibrium. This equilibrium paradigm imposes intrinsic limitations: sensitivity is tied to long-time thermalization and often cannot be improved in fast, noisy, or nonstationary settings. In contrast, the \textit{Mpemba effect}, the counterintuitive phenomenon where hotter states relax faster than colder ones, has mostly been viewed as a thermodynamic anomaly. Here, we bridge this gap by proving that Mpemba-type inversions generically yield a finite-time enhancement of the quantum Fisher information (QFI) for temperature estimation, thereby converting an anomalous relaxation effect into a concrete metrological resource. Through explicit analyses of two-level and ΛΛΛ-level probes coupled to bosonic baths, we show that nonequilibrium initializations can transiently outperform both equilibrium strategies and colder states, realizing a \emph{metrological Mpemba effect}. Our results establish anomalous relaxation as a general design principle for nonequilibrium quantum thermometry, enabling ultrafast and nanoscale sensing protocols that exploit, rather than avoid, transient dynamics.
We investigate the non-monotonic temperature sensitivity of a coherently driven two-level quantum system coupled to an Ohmic phonon environment. By employing a unitary polaron transformation, we account for phonon-induced renormalization effects that go beyond the standard weak-coupling approximations. Our analysis reveals that the Quantum Fisher Information (QFI) exhibits a prominent peak at an intermediate system-environment coupling strength, identifying an optimal regime for thermal sensing. This behavior emerges from a fundamental competition between environment-induced dissipation enhancement and the exponential suppression of system parameters due to phonon dressing. We demonstrate that while thermometric precision vanishes in both the ultra-weak and strong coupling limits, a properly tuned nonequilibrium steady state can significantly enhance sensitivity. These results suggest that environmental interactions, often viewed as detrimental decoherence sources, can be engineered as a resource to optimize the performance of solid-state quantum thermometers.