Optimal limits of continuously monitored thermometers and their Hamiltonian structure

We investigate the fundamental and practical precision limits of thermometry in bosonic and fermionic environments by coupling an <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>N</a:mi></a:math>-level probe to them and continuously monitoring it. Our findings show that the ultimate precision limit, quantified by the Fisher information, scales linearly with <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>N</b:mi></b:math>, offering an exponential improvement over equilibrium thermometry, where the scaling is only <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:msup><c:mo form="prefix">log</c:mo><c:mn>2</c:mn></c:msup><c:mi>N</c:mi></c:mrow></c:math>. 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>N</e:mi></e:math>—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.