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  • Letter
  • Open Access

Optimal limits of continuously monitored thermometers and their Hamiltonian structure

Mohammad Mehboudi1,*, Florian Meier1,†, Marcus Huber1,2,‡, and Harry J. D. Miller3,§

  • *Contact author: mohammad.mehboudi@tuwien.ac.at
  • †Contact author: florian.meier@tuwien.ac.at
  • ‡Contact author: marcus.huber@tuwien.ac.at
  • §Contact author: harry.miller@manchester.ac.uk
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Phys. Rev. A 111, L020401 – Published 5 February, 2025

DOI: https://doi.org/10.1103/PhysRevA.111.L020401

Abstract

We investigate the fundamental and practical precision limits of thermometry in bosonic and fermionic environments by coupling an N-level probe to them and continuously monitoring it. Our findings show that the ultimate precision limit, quantified by the Fisher information, scales linearly with N, offering an exponential improvement over equilibrium thermometry, where the scaling is only log2N. For a fixed Hamiltonian structure, we develop a maximum-likelihood estimation strategy that maps the observed continuously monitored trajectories of the probe into temperature estimates with minimal error. By optimizing over all possible Hamiltonian structures, we discover that the optimal configuration is an effective two-level system, with both levels exhibiting degeneracy that increases with N—a stark contrast to equilibrium thermometry, where the ground state remains nondegenerate. Our results have practical implications. First, continuous monitoring is experimentally feasible on several platforms and accounts for the preparation time of the probe, which is often overlooked in other approaches such as prepare and reset. Second, the linear scaling is robust against deviations from the effective two-level structure of the optimal Hamiltonian. Additionally, this robustness extends to cases of initial ignorance about the temperature. Thus, in global estimation problems, the linear scaling remains intact even without adaptive strategies.

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Phys. Rev. A 111, L020401– Published 5 February, 2025
  • Received 3 September 2024
  • Accepted 15 January 2025
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