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arXiv:2108.10469

Thermometric machine for ultraprecise thermometry of low temperatures

Ivan Henao, Karen V. Hovhannisyan, Raam Uzdin

Published Aug 24, 2021• 3 citations
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Abstract

Thermal equilibrium states are exponentially hard to distinguish at very low temperatures, making equilibrium quantum thermometry in this regime a formidable task. We present a thermometric scheme that circumvents this limitation, by using a two-level probe that does not thermalize with the sample whose temperature is measured. This is made possible thanks to a suitable interaction that couples the probe to the sample and to an auxiliary thermal bath known to be at a higher temperature. Provided a reasonable upper bound on the temperature of the sample, the resulting 'thermometric machine' drives the probe towards a steady state whose signal-to-noise ratio can achieve values as high as $\mathcal{O}(1/T)$. We also characterize the transient state of the probe and numerically illustrate an extreme reduction in the number of measurements to attain a given precision, as compared to optimal measurements on a thermalized probe.

DOI
10.48550/arxiv.2108.10469

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Thermal equilibrium states are exponentially hard to distinguish at very low temperatures, making equilibrium quantum thermometry in this regime a formidable task. We present a thermometric scheme that circumvents this limitation, by using a two-level probe that does not thermalize with the sample whose temperature is measured. This is made possible thanks to a suitable interaction that couples the probe to the sample and to an auxiliary thermal bath known to be at a higher temperature. Provided a reasonable upper bound on the temperature of the sample, the resulting 'thermometric machine' drives the probe towards a steady state whose signal-to-noise ratio can achieve values as high as O(1/T)\mathcal{O}(1/T)O(1/T). We also characterize the transient state of the probe and numerically illustrate an extreme reduction in the number of measurements to attain a given precision, as compared to optimal measurements on a thermalized probe.
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.
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.
The precise measurement of low temperatures is significant for both the fundamental understanding of physical processes and technological applications. In this work, we present a method for low-temperature measurement that improves thermal range and sensitivity by generating quantum coherence in a thermometer probe. Typically, in temperature measurements, the probes thermalize with the sample being measured. However, we use a two-level quantum system, or qubit, as our probe and prevent direct probe access to the sample by introducing a set of ancilla qubits as an interface. We describe the open system dynamics of the probe using a global master equation and demonstrate that while the ancilla-probe system thermalizes with the sample, the probe per se evolves into a nonthermal steady state due to nonlocal dissipation channels. The populations and coherences of this steady state depend on the sample temperature, allowing for precise and wide-range low-temperature estimation. We characterize the thermometric performance of the method using quantum Fisher information and show that the quantum Fisher information can exhibit multiple and higher peaks at different low temperatures with increasing quantum coherence and the number of ancilla qubits. Our analysis reveals that the proposed approach, using a nonthermal qubit thermometer probe with temperature-dependent quantum coherence generated by a multiple qubit interface between a thermal sample and the probe qubit, can enhance the sensitivity of temperature estimation and broaden the measurable low-temperature range.
There exist severe limitations on the accuracy of low-temperature thermometry, which poses a major challenge for future quantum-technological applications. Low-temperature sensitivity might be manipulated by tailoring the interactions between probe and sample. Unfortunately, the tunability of these interactions is usually very restricted. Here, we focus on a more practical solution to boost thermometric precision – driving the probe. Specifically, we solve for the limit cycle of a periodically modulated linear probe in an equilibrium sample. We treat the probe-sample interactions exactly and hence, our results are valid for arbitrarily low temperatures T and any spectral density. We find that weak near-resonant modulation strongly enhances the signal-to-noise ratio of low-temperature measurements, while causing minimal back action on the sample. Furthermore, we show that near-resonant driving changes the power law that governs thermal sensitivity over a broad range of temperatures, thus `bending' the fundamental precision limits and enabling more sensitive low-temperature thermometry. We then focus on a concrete example – impurity thermometry in an atomic condensate. We demonstrate that periodic driving allows for a sensitivity improvement of several orders of magnitude in sub-nanokelvin temperature estimates drawn from the density profile of the impurity atoms. We thus provide a feasible upgrade that can be easily integrated into low- T thermometry experiments.
Quantum thermometry has long promised sensitivities beyond classical limits, with recent paradigms highlighting quantum coherence as a primary driver of precision. While most studies focus on single-point temperature estimation, we extend this framework to the spatial domain. We consider a probe two-level system (TLS) coupled asymmetrically to an auxiliary TLS that equilibrates with a sample of profile T(x). The reduced probe state carries coherence in its local basis that remains temperature-sensitive after the probe transition has frozen out, yielding a Fisher information exponentially larger at low temperature than that of a population-only readout. An exact reparametrization converts the thermometric precision into a local spatial distinguishability bound; its validity requires only a sufficiently small relative temperature uncertainty and yields an explicit measurement budget. Adding auxiliary TLSs shifts the array's optimal working temperature upward and broadens the useful sensing window, with the achievable gain ultimately limited by the total available coupling rather than by an indefinite scaling law. Finally, we propose a microwave Mach-Zehnder interferometer that maps the temperature-dependent probe coherence onto an experimentally accessible quadrature observable.
Equilibrium quantum thermometry seeks to enhance temperature estimation for reservoirs at thermal equilibrium using quantum resources, yet achieving the Heisenberg scaling in a physical protocol remains a significant challenge. Moreover, quantum measurements face an intrinsic limitation arising from an exponential suppression of thermometric sensitivity as the temperature approaches zero. This limitation severely hinders access to high precision in the low-temperature regime. Here, we demonstrate a simple but powerful quantum engineering protocol to overcome these limitations and reach high precision for temperature estimation. By engineering a small subset of central probes and their linear coupling to a larger array of adjacent probes, we show that the composite partition function can be tailored to yield Heisenberg-limited precision for temperature estimation. Critically, we show that this protocol can simultaneously saturate the Landau bound at low temperatures, an inverse quadratic scaling with the temperature, thereby fully exploiting the potential of equilibrium thermometry and circumventing the exponential suppression of thermometric sensitivity in the low-temperature regime. This scheme relies solely on linear interactions, avoiding complicated many-body couplings and ensuring scalability with the number of probes. Moreover, we find that the comparable precision can be achieved using only local measurements on the central probes, substantially reducing the implementation complexity compared to global measurement schemes and offering a promising approach toward high-precision equilibrium thermometry in physical systems.