Bridging thermodynamics and metrology in nonequilibrium quantum thermometry

Single-qubit thermometry presents the simplest tool to measure the temperature of thermal baths with reduced invasivity. At thermal equilibrium, the temperature uncertainty is linked to the heat capacity of the qubit, however, the best precision is achieved outside the equilibrium condition. Here, we discuss a way to generalize this relation in a nonequilibrium regime, taking into account purely quantum effects such as coherence. We support our findings with an experimental photonic simulation.

Figure 1

Plots of the nonequilibrium QFI $Q_T\left(\tau \right)$ (solid purple curves) and of the quantity $C_T^2\left(\tau \right)/{\mathrm{\Delta }}^2{E}_T\left(\tau \right)$ (dashed red curves) appearing on the right-hand side of Eq. (3) and which determines the accuracy of the estimation procedure based on direct energy measurement of the probe $\mathcal{S}$. In all plots we assume the input state $\rho \left(0\right)$ of $\mathcal{S}$ to be pure with zero azimuthal angle and with the polar angle given by $\theta \left(0\right)=0$ (excited state) for the first panel; $\theta \left(0\right)=\pi /4$ for the second; $\theta \left(0\right)=\pi /2$ for the third; and $\theta \left(0\right)=0$ (excited state) for the last panel. The temperature $T$ is set equal to 2 in units of $\hslash \omega /k_B$ while the time is measured in units of ${\gamma }^{-1}$. Notice that when the system is initially prepared in a diagonal state, i.e., for $\theta \left(0\right)=0$ and $\theta \left(0\right)=\pi$, the bound (3) is saturated and the two curves coincide.

Figure 2

Quantum simulation via quantum photonics. Upper panel: Quantum circuits for the implementation of the AD ($k=0$), and of the IAD ($k=1$) channels [41]. The circuital elements are as follows: X and Z, that implement the Pauli rotations ${\sigma }_x$ and ${\sigma }_z$; CZ, representing a controlled-${\sigma }_z$ gate; $R\left(\varphi \right)$ is a rotation by an angle $\varphi$ around the $y$ axis. The measurements are performed in the computational basis. Lower panel: Experimental scheme. One photon is employed to simulate the single-qubit thermometer, while a second one is used as an ancilla to simulate the system-bath interaction. The computational basis is encoded in the vertical and the horizontal polarizations of the single photons. The single-qubit gates in the upper panel are implemented by means of half-wave plates (HWPs), while the CZ is realized by a partially polarizing beam splitter. More details are given in the Supplemental Material [36].

Figure 3

Comparison between the experimental errors $\mathrm{\Delta }T$, and metrological figures of merit related to the temperature parameter. In the three panels, the solid purple curves represent the theoretical QCRB, the dashed orange curve represents the theoretical CRB, and the purple points represent the experimental uncertainties on the temperature. In the left panel, we confirm that the ground state allows one to reach the greatest sensitivity of the single-qubit probe as it permits to reach the lowest value of $\mathrm{\Delta }T$; in the right panel, we show the behavior of the probe prepared in the excited state, and we observe a divergence in the QCRB due to the presence of a zero in the QFI—see the first panel of Fig. 1; in the middle panel, we show the coherent strategy. Here, the experimental uncertainties on the temperature do not reach the QCRB but they are well captured by its classical counterpart.