Quantum thermometry based on a cavity-QED setup

We present a quantum thermometry scheme based on a cavity-QED setup, which attains a sensitivity with Heisenberg scaling. A stream of identical two-level systems passes through a thermal bath to be tested. Each system partially thermalizes, carrying information on the temperature of the thermal bath, and it then interacts with a dissipative single-mode cavity. The Heisenberg scaling is attained from the initial coherence of each system. The systems cannot be fully thermalized by the thermal bath. The optimal interaction time between the system and the thermal bath is derived. Direct photon detection is proved to be an optimal measurement when there is no extra thermal bath in the cavity. In the case where there is an extra thermal bath in the cavity independent of the thermal bath to be tested, homodyne detection is the optimal measurement. Moreover, we show that direct photon detection obtains less information in the case where the extra thermal bath in the cavity is in thermal equilibrium with the thermal bath. However, the optimal measurement can utilize the information from the extra thermal bath to improve the estimation precision of the temperature.

Figure 1

Schematic diagram of the proposed quantum thermometry sequence. The TLSs pass through the thermal bath with the temperature $T$ to be tested. Then the TLSs traverse a cavity with the decay rate $\kappa$. There is an additional thermal bath with temperature $T_c$ in the cavity. By measuring the steady state of the cavity, the temperature $T$ of the thermal bath can be obtained.

Figure 2

Diagram of estimation temperature $T$ with direct photon detection. In this diagram, “ind” denotes the case where ETBC is independent of the thermal bath and “equ” denotes the case where ETBC is thermalized with the thermal bath. In the case of “ind,” the high temperature of ETBC will reduce the measurement precision of $T$. For the same value of ETBC, the uncertainty of $T$ in the case of “equ” is larger than that in the case of “ind.” Here, the parameters are given by $\omega =1$, $g{\tau }_2=0.3$, $N_c=10$, $\lambda =0.5$, in arbitrary units.

Figure 3

Diagram of estimation temperature $T$ with optimal detection. In this diagram, “ind” denotes the case of that ETBC is independent with the thermal bath and “equ” denotes the case where ETBC is thermalized with the thermal bath. For the same value of $T_c$, the uncertainty in $T$ in the case of “equ” is less than that in the case of “ind.” Here, the parameters are given by $\omega =1$, $g{\tau }_2=0.3$, $N_c=10$, $\lambda =0.5$ in arbitrary units.

Figure 4

Diagram of estimation temperature $T$ with three ways of measurements in the case where ETBC is in thermal equilibrium with the thermal sample. This figure shows that homodyne detection performs better than direct detection. However, homodyne detection does not perform as well as the optimal measurement. Here, the parameters are the same as in the previous figures.