Optimal limits of continuously monitored thermometers and their Hamiltonian structure

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.

Figures (5)

• Figure 1: (a) An $N$-level probe is in contact with a thermal bath at temperature $T$. The transition rates between different energy levels, $\Gamma_{ij}$, are temperature dependent. (b) The trajectory that the $N$-level system takes contains information about the temperature. The precision with which the trajectory can estimate the temperature highly depends on the energy structure. (c) The optimal energy structure---as we find in this work---is effectively 2-level, with ground state and exited state degeneracy that both increase linearly with $N$, and at some optimal gap $\epsilon^*$. In the figure we have color coded the degenerate gaps to tell them apart visually.
• Figure 2: Numerical results for the optimal dimensionless FI rate \ref{['eq:FIrate']}. We compare the exact solution obtained via global optimisation with the optimal $2$-level ansatz \ref{['eq:2level']} (optimised over $N_0$ and $x$). The analytic value \ref{['eq:fermion_asympt']} obtained in the asymptotic $N\gg 1$ limit is also plotted and shows good agreement with the exact solution for $n\geq 3$. Inset: This plot depicts a histogram of the energy levels found from a global optimisation for $n=6$. Our numerics suggest that an effective $2$-level structure is optimal. We have thus also considered an effective $2$-level ansatz which can be optimised more efficiently due to fewer parameters. The FI rate of the $2$-level ansatz is shown by the red curve in the primary figure which also as one can see agrees very well with the global optimum for $n\geq 3$.
• Figure 3: Left: Plot of the optimal free parameters for the $2$-level ansatz \ref{['eq:2level']} for a fermionic probe as a function of $n=\log_2 N$, with optimal ground state degeneracy $N_0^*$ (black) and optimal gap $x^*$ (red). The dashed lines give the values for the asymptotic optimal solution \ref{['eq:fermion_asympt']}. Right: Plot of FI rate normalised by its maximum value, for different values of $n$. Here the gap is tuned to $\epsilon_0 = x^* T_0$, such that the thermometer is optimal for some assumed temperature $T_0$.
• Figure 4: Left: the optimal FI rate \ref{['eq:FIrateboson']} for a bosonic case with ohmicity $s=2$, using a global optimisation (solid) and the 2-level ansatz \ref{['eq:2levelboson']} (red-dashed). The asymptotic values are taken from Table \ref{['table:bos_opt']}. Right: The optimal energy gap $x^*$ (red) for the 2-level ansatz and its asymptotic prediction from Table \ref{['table:bos_opt']} and the ground state degeneracy of the optimal structure for the 2-level ansatz (black) and its asymptotic behaviour from Table \ref{['table:bos_opt']}.
• Figure 5: Left: The energy structure chosen as two Gaussian distributions with standard deviation $\sigma$ and mean values that are the two optimal energies (i.e., $0$ and $x^*\approx 2.9682$). Here, we set $N=2^{10}$. Right: The Fisher information of such Hamiltonian structures are quite close to the optimal two level setting, proving the robustness of our results.