Enhancement of sensitivity in low-temperature quantum thermometry via reinforcement learning

We present a quantum thermometry method utilizing a probe-meter system and leveraging reinforcement learning (RL) to enhance sensitivity at low temperatures. Temperature information is exclusively obtained by coupling the probe qubit with the bath, facilitating efficient information extraction by measuring the meter system. Our evaluations, considering both dephasing and dissipative probe-bath couplings, demonstrate a significant enhancement in temperature sensitivity through meter-assisted thermometry, excluding the ultralow-temperature range. To further optimize performance, RL is employed to adjust both the interaction strength between the probe-meter system and the frequency of the meter system. Results showcase that RL effectively enhances temperature sensitivity across a broad low-temperature spectrum by improving average sensitivity within the specified temperature range. This optimization successfully expands the operational range of quantum thermometers. These findings highlight the promising potential of the RL approach for high-resolution, low-temperature quantum thermometry applications.

Figure 1

(a) A schematic representation of a probe-meter quantum thermometer optimized through reinforcement learning, featuring interactions between the agent (a neural network) and the system involving iterative actions $a_t=\{{\omega }_m\left(t\right),f\left(t\right)\}$ and corresponding rewards $r_t$. (b), (c) Diagrams illustrating the energy levels of bare states and eigenstates within the Hamiltonian $H_S$ for the probe-meter coupled system: (b) ${\mathcal{S}}_p={\sigma }_p^z$ (dephasing channel) and (c) ${\mathcal{S}}_p={\sigma }_p^x$ (dissipative channel). The bath-induced dissipative processes occur among the four eigenstates. The effective transition rates from states $|{\lambda }_i\rangle$ to $|{\lambda }_j\rangle$ are denoted as ${\mathrm{\Gamma }}_{ij}$.

Figure 2

QSNR ${\mathcal{Q}}_T$ in the case of ${\mathcal{S}}_p={\sigma }_p^z$. (a) Dynamic behavior of ${\mathcal{Q}}_T$ when $T/{\omega }_p=0.1$. (b) Optimal QSNR ${\mathcal{Q}}_T^{*}$ vs temperature $T/{\omega }_a$. Here, the red solid line corresponds to the situation with meter assistance, while the dashed line corresponds to the scenario without meter assistance. Other parameters are $f/{\omega }_p=0.02, {\omega }_m/{\omega }_p=0.1, {\omega }_c/{\omega }_p=10$, and $\eta /{\omega }_p=0.05$. It is worth noting that ${\mathcal{Q}}_T^{*}$ for the case without meter assistance is determined at the optical time ${\omega }_p{t}^{*}$, whereas for the case with meter assistance, it is obtained at steady time.

Figure 3

QSNR ${\mathcal{Q}}_T$ in the case of ${\mathcal{S}}_p={\sigma }_p^x$. (a) Dynamic behavior of ${\mathcal{Q}}_T$ when $T/{\omega }_p=0.2$. (b) Optimal QSNR ${\mathcal{Q}}_T^{*}$ as a function of temperature $T/{\omega }_p$. Here, the red solid line corresponds to the situation with meter assistance, while the dashed line corresponds to the scenario without meter assistance. Other parameters are $f/{\omega }_p=0.4, {\omega }_m/{\omega }_p=0.5, {\omega }_c/{\omega }_p=10$, and $\eta /{\omega }_p=0.05$.

Figure 4

Reinforcement learning assisted probe-meter quantum thermometry in the case of ${\mathcal{S}}_p={\sigma }_p^z$. (a) Optimal action generated by the RL agent. (b) Dynamics of the mean QSNR ${\overline{\mathcal{Q}}}_T$ due to optimal action. Inset: Temperature-dependent behavior of ${\mathcal{Q}}_T$ in the steady-state regime, featuring both RL (red solid line) and non-RL (black dashed line) methods. (c) Dynamics of the mean QFI ${\overline{\mathcal{F}}}_T$ due to optimal action. Inset: Temperature-dependent behavior of ${\mathcal{F}}_T$ in the steady-state regime, featuring both RL (red solid line) and non-RL (black dashed line) methods. Here, the probe-meter system is initially prepared as $\mathrm{\Psi }\left(0\right)=|\downarrow \uparrow \rangle =sin(\theta /2)|{\lambda }_2\rangle +cos(\theta /2)|{\lambda }_3\rangle$. Other parameters are given as ${\omega }_c/{\omega }_p=10$ and $\eta /{\omega }_p=0.05$.

Figure 5

Reinforcement learning assisted probe-meter quantum thermometry in the case of ${\mathcal{S}}_p={\sigma }_p^x$ with the parameters being the same as those in Fig. 4.