Energy measurements remain thermometrically optimal beyond weak coupling

Jonas Glatthard1, Karen V. Hovhannisyan2, Martí Perarnau-Llobet3, Luis A. Correa4,1, and Harry J. D. Miller5

1Department of Physics and Astronomy, University of Exeter, Exeter EX4 4QL, United Kingdom
2University of Potsdam, Institute of Physics and Astronomy, Karl-Liebknecht-Str. 24–25, 14476 Potsdam, Germany
3Département de Physique Appliquée, Université de Genève, 1211 Genève, Switzerland
4Departamento de Física, Universidad de La Laguna, La Laguna 38203, Spain
5Department of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom

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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 common notion of thermometry is putting a probe (the “thermometer”) in contact with the sample, waiting for them to reach a joint thermal equilibrium, and then measuring the probe. When the probe–sample interaction is weak, the probe is itself thermal and optimal thermometry is achieved by simply measuring the probe in its local energy eigenbasis. This picture, while convenient, becomes fundamentally flawed at low temperatures: No nonzero interaction can be considered weak near absolute zero. And pushing interactions to zero is no solution, as doing so hinders probe thermalisation.
When the probe–sample coupling is strong, the probe is not in a thermal state when at equilibrium with the sample. It is instead described by the so-called mean-force Gibbs state, which in general has complicated dependence on the coupling parameters and even temperature itself. As a result, the optimal thermometric measurement loses its simplicity, and it remains an open challenge to find general prescriptions for optimal thermometric measurements beyond the weak coupling regime.
Nonetheless, here we prove under minimal assumptions that that—surprisingly—energy measurements of the probe remain nearly optimal even at moderate coupling, beyond the weak coupling regime. This means that sophisticated measurement schemes exploiting coherences or using adaptive strategies do not confer any practical advantage as long as the coupling is not too strong.
Our take-home message? The experimental ability to measure a probe in its local basis will often be sufficient for precise thermometry.

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