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arXiv:2203.02436 · Quantum

Bending the rules of low-temperature thermometry with periodic driving

Jonas Glatthard, Luis A. Correa

Published May 3, 2022• 14 citations
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Abstract

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.

DOI
10.22331/q-2022-05-03-705

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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.
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.
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.
The precision of a quantum thermometer is fundamentally constrained by the spectral structure of the probe itself, and a systematic mapping between the configurations of energy levels and thermometric performance provides relevant information to design optimized devices. In this work, we establish such a mapping by analyzing a broad class of quantum systems, ranging from finite spin ensembles and degenerate atoms to confining potentials, quantum walks, and continuous-spectrum models. We derive exact scaling laws for the quantum Fisher information, revealing two distinct high-temperature universality classes: finite-spectrum probes exhibit a T−4T^{-4}T−4 decay, while unbounded or continuous spectra yield a slower T−2T^{-2}T−2 decay. At low temperatures, we show that sensitivity, though universally exponentially suppressed, can be enhanced arbitrarily by engineering degenerate excited states or a quantum walk on a fully connected topology. By contrast, specific quantum walk topologies provide a distinct enhancement mechanism based on gap engineering, whereby an optimal network size yields an optimized T−2T^{-2}T−2 low-temperature scaling. Furthermore, power-law spectra enable tunable scaling of thermometric performance with system size, offering a design principle for optimal probes in specific temperature windows. Our results contribute to transform spectral information into a resource for quantum thermometry, providing both fundamental bounds and practical guidelines to tailored temperature sensing.
Precise low-temperature thermometry is a key requirement for virtually any quantum technological application. Unfortunately, as the temperature TTT decreases, the errors in its estimation diverge very quickly. In this paper, we determine exactly how quickly this may be. We rigorously prove that the ``conventional wisdom'' of low-TTT thermometry being exponentially inefficient is limited to local thermometry on translationally invariant systems with short-range interactions, featuring a nonzero gap above the ground state. This result applies very generally to spin and harmonic lattices. On the other hand, we show that a power-law-like scaling is the hallmark of local thermometry on gapless systems. Focusing on thermometry on one node of a harmonic lattice, we obtain valuable physical insight into the switching between the two types of scaling. In particular, we map the problem to an equivalent setup, consisting of a Brownian thermometer coupled to an equilibrium reservoir. This mapping allows us to prove that, surprisingly, the relative error of local thermometry on gapless harmonic lattices does not diverge as $T\ensuremath{\rightarrow}0$; rather, it saturates to a constant. As a useful by-product, we prove that the low-TTT sensitivity of a harmonic probe arbitrarily strongly coupled to a bosonic reservoir by means of a generic Ohmic interaction, always scales as T2{T}^{2}T2 for $T\ensuremath{\rightarrow}0$. Our results thus identify the energy gap between the ground and first excited states of the global system as the key parameter in local thermometry, and ultimately provide clues to devising practical thermometric strategies deep in the ultracold regime.
Abstract High-precision low-temperature thermometry is a challenge for experimental quantum physics and quantum sensing. Here we consider a thermometer modeled by a dynamically-controlled multilevel quantum probe in contact with a bath. Dynamical control in the form of periodic modulation of the energy-level spacings of the quantum probe can dramatically increase the maximum accuracy bound of low-temperatures estimation, by maximizing the relevant quantum Fisher information. As opposed to the diverging relative error bound at low temperatures in conventional quantum thermometry, periodic modulation of the probe allows for low-temperature thermometry with temperature-independent relative error bound. The proposed approach may find diverse applications related to precise probing of the temperature of many-body quantum systems in condensed matter and ultracold gases, as well as in different branches of quantum metrology beyond thermometry, for example in precise probing of different Hamiltonian parameters in many-body quantum critical systems.
Quantum thermometry aims at determining temperature with ultimate precision in the quantum regime. Standard equilibrium approaches, limited by the Quantum Fisher Information given by static energy fluctuations, lose sensitivity outside a fixed temperature window. Non-equilibrium strategies have therefore been recently proposed to overcome these limits, but their advantages are typically model-dependent or tailored for a specific purpose. This Letter establishes a general, model-independent result showing that any temperature-dependent unitary driving applied to a thermalized probe enhances its quantum Fisher information with respect to its equilibrium value. Such information gain is expressed analytically through a positive semi-definite kernel of information currents that quantify the flow of statistical distinguishability. Our results, together with an analysis of the relation between information gain and control cost, are benchmarked on a driven spin-1/21/21/2 thermometer, furthermore showing that resonant modulations remarkably restore the quadratic-in-time scaling of the Fisher information and allow to shift the sensitivity peak across arbitrary temperature ranges.