Optimal limits of continuously monitored thermometers and their Hamiltonian structure

TL;DR

This paper demonstrates that continuously monitored quantum probes can achieve a linear scaling of thermometry precision with the number of levels, surpassing equilibrium methods, and identifies the optimal Hamiltonian structure for such measurements.

Contribution

It introduces a maximum likelihood estimation method for continuous quantum thermometry and finds that an effective two-level Hamiltonian with degeneracy optimizes precision.

Findings

Fisher information scales linearly with N, improving over equilibrium thermometry.

Optimal Hamiltonian is an effective two-level system with degeneracy increasing with N.

Linear scaling is robust against deviations from the ideal Hamiltonian and initial ignorance.

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 $\log^2 N$. 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 non-degenerate. 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.