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Optimal nonequilibrium thermometry in finite time

Pavel Sekatski, Martí Perarnau-Llobet

Published Jul 9, 2021• 2 citations
OpenAlex

Abstract

What is the minimum time required to take the temperature? In this paper, we solve this question for any process where temperature is inferred by measuring a probe (the thermometer) weakly coupled to the sample of interest, so that the probe's evolution is well described by a quantum Markovian master equation. Considering the most general control strategy on the probe (adaptive measurements, arbitrary control on the probe's state and Hamiltonian), we provide bounds on the achievable measurement precision in a finite amount of time, and show that in many scenarios these fundamental limits can be saturated with a relatively simple experiment. We find that for a general class of sample-probe interactions the scaling of the measurement uncertainty is inversely proportional to the time of the process, a shot-noise like behaviour that arises due to the dissipative nature of thermometry. As a side result, we show that the Lamb shift induced by the probe-sample interaction can play a relevant role in thermometry, allowing for finite measurement resolution in the low-temperature regime (more precisely, the measurement uncertainty decays polynomially with the temperature as $T\rightarrow 0$, in contrast to the usual exponential decay with $T^{-1}$). We illustrate these general results for (i) a qubit probe interacting with a bosonic sample, where the role of the Lamb shit is highlighted, and (ii) a collective superradiant coupling between a $N$-qubit probe and a sample, which enables a quadratic decay with $N^2$ of the measurement uncertainty.

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What is the minimum time required to take the temperature? In this paper, we solve this question for any process where temperature is inferred by measuring a probe (the thermometer) weakly coupled to the sample of interest, so that the probe's evolution is well described by a quantum Markovian master equation. Considering the most general control strategy on the probe (adaptive measurements, arbitrary control on the probe's state and Hamiltonian), we provide bounds on the achievable measurement precision in a finite amount of time, and show that in many scenarios these fundamental limits can be saturated with a relatively simple experiment. We find that for a general class of sample-probe interactions the scaling of the measurement uncertainty is inversely proportional to the time of the process, a shot-noise like behaviour that arises due to the dissipative nature of thermometry. As a side result, we show that the Lamb shift induced by the probe-sample interaction can play a relevant role in thermometry, allowing for finite measurement resolution in the low-temperature regime (more precisely, the measurement uncertainty decays polynomially with the temperature as T→0T\rightarrow 0T→0, in contrast to the usual exponential decay with T−1T^{-1}T−1). We illustrate these general results for (i) a qubit probe interacting with a bosonic sample, where the role of the Lamb shit is highlighted, and (ii) a collective superradiant coupling between a NNN-qubit probe and a sample, which enables a quadratic decay with N2N^2N2 of the measurement uncertainty.
What is the minimum time required to take a temperature? In this paper, we solve this question for a large class of processes where temperature is inferred by measuring a probe (the thermometer) weakly coupled to the sample of interest, so that the probe's evolution is well described by a quantum Markovian master equation. Considering the most general control strategy on the probe (adaptive measurements, arbitrary control on the probe's state and Hamiltonian), we provide bounds on the achievable measurement precision in a finite amount of time, and show that in many scenarios these fundamental limits can be saturated with a relatively simple experiment. We find that for a general class of sample-probe interactions the scaling of the measurement uncertainty is inversely proportional to the time of the process, a shot-noise like behaviour that arises due to the dissipative nature of thermometry. As a side result, we show that the Lamb shift induced by the probe-sample interaction can play a relevant role in thermometry, allowing for finite measurement resolution in the low-temperature regime. More precisely, the measurement uncertainty decays polynomially with the temperature as T→0 , in contrast to the usual exponential decay with T−1 . We illustrate these general results for (i) a qubit probe interacting with a bosonic sample, where the role of the Lamb shift is highlighted, and (ii) a collective superradiant coupling between a N -qubit probe and a sample, which enables a quadratic decay with N of the measurement uncertainty.
We consider the problem of estimating the temperature TTT of a very cold equilibrium sample. The temperature estimates are drawn from measurements performed on a quantum Brownian probe strongly coupled to it. We model this scenario by resorting to the canonical Caldeira-Leggett Hamiltonian and find analytically the exact stationary state of the probe for arbitrary coupling strength. In general, the probe does not reach thermal equilibrium with the sample, due to their nonperturbative interaction. We argue that this is advantageous for low-temperature thermometry, as we show in our model that (i) the thermometric precision at low TTT can be significantly enhanced by strengthening the probe-sampling coupling, (ii) the variance of a suitable quadrature of our Brownian thermometer can yield temperature estimates with nearly minimal statistical uncertainty, and (iii) the spectral density of the probe-sample coupling may be engineered to further improve thermometric performance. These observations may find applications in practical nanoscale thermometry at low temperatures---a regime which is particularly relevant to quantum technologies.
While the ability to measure low temperatures accurately in quantum systems is important in a wide range of experiments, the possibilities and the fundamental limits of quantum thermometry are not yet fully understood theoretically. Here we develop a general approach to low-temperature quantum thermometry, taking into account restrictions arising not only from the sample but also from the measurement process. {We derive a fundamental bound on the minimal uncertainty for any temperature measurement that has a finite resolution. A similar bound can be obtained from the third law of thermodynamics. Moreover, we identify a mechanism enabling sub-exponential scaling, even in the regime of finite resolution. We illustrate this effect in the case of thermometry on a fermionic tight-binding chain with access to only two lattice sites, where we find a quadratic divergence of the uncertainty}. We also give illustrative examples of ideal quantum gases and a square-lattice Ising model, highlighting the role of phase transitions.
We study low-temperature non-equilibrium quantum thermometry with a bosonic probe: a quantum harmonic oscillator strongly coupled to a bosonic bath at temperature TTT through a Drude--Ohmic spectral density. We treat the probe--bath dynamics both exactly, using the quadratic solution of Boyanovsky and Jasnow, and within a renormalized Gorini--Kossakowski--Lindblad--Sudarshan (GKLS) master equation. From the time-dependent covariance matrix we extract the quantum Fisher information (QFI) for general single-mode Gaussian probe states, including squeezed ones. In the strong-coupling, non-Markovian regime the QFI is non-monotonic in time, displaying bath-memory revivals that make a finite interrogation time t∗>0t^*>0t∗>0 strictly optimal. By contrast, we prove that the Markovian QFI rises monotonically to its stationary value and develops no interior optimum, so that its optimum is always pinned to the boundary t∗→∞t^*\to\inftyt∗→∞; this complements existing Markovian precision-rate bounds, which concern (F(t)/t)(\mathcal F(t)/t)(F(t)/t) rather than the single-shot QFI (F(t))(\mathcal F(t))(F(t)). Squeezed initial states yield a large transient advantage that thermalisation eventually erases, establishing squeezing and interrogation time as complementary thermometric resources. At equilibrium, strong coupling replaces the exponential Boltzmann suppression of the low-temperature relative error by a far milder polynomial divergence. As the model maps directly onto circuit quantum electrodynamics, these protocols appear within current experimental reach.
We develop a general perturbative theory of finite-coupling quantum thermometry up to second order in probe-sample interaction. By assumption, the probe and sample are in thermal equilibrium, so the probe is described by the mean-force Gibbs state. We prove that the ultimate thermometric precision can be achieved – to second order in the coupling – solely by means of local energy measurements on the probe. Hence, seeking to extract temperature information from coherences or devising adaptive schemes confers no practical advantage in this regime. Additionally, we provide a closed-form expression for the quantum Fisher information, which captures the probe's sensitivity to temperature variations. Finally, we benchmark and illustrate the ease of use of our formulas with two simple examples. Our formalism makes no assumptions about separation of dynamical timescales or the nature of either the probe or the sample. Therefore, by providing analytical insight into both the thermal sensitivity and the optimal measurement for achieving it, our results pave the way for quantum thermometry in setups where finite-coupling effects cannot be ignored.
Accurate temperature estimation in the quantum and cryogenic regimes remains a fundamental challenge. Here, we investigate nonequilibrium quantum thermometry using a single-qubit probe coupled to a bosonic bath through noncommuting interaction operators, which unify pure dephasing and dissipative dynamics within a spin-boson model. We show that the interference between these two coupling channels induces strong non-Markovian feedback between populations and coherences, leading to coherence trapping and enhanced thermal sensitivity. Remarkably, by tuning the coupling structure, the probe's temperature sensitivity exhibits a quadratic low-temperature scaling, even under weak coupling. Moreover, while coherence-based measurements are formally suboptimal, they become the most informative in the early nonequilibrium regime, where memory effects dominate. Our findings identify noncommutative system-bath couplings as a practical and tunable resource for achieving high-precision quantum thermometry in realistic open-system architectures.