Quantum Simulation of Single-Qubit Thermometry Using Linear Optics

Standard thermometry employs the thermalization of a probe with the system of interest. This approach can be extended by incorporating the possibility of using the nonequilibrium states of the probe and the presence of coherence. Here, we illustrate how these concepts apply to the single-qubit thermometer introduced by Jevtic et al. [Phys. Rev. A 91, 012331 (2015)] by performing a simulation of the qubit-environment interaction in a linear-optical device. We discuss the role of the coherence and how this affects the usefulness of nonequilibrium conditions. The origin of the observed behavior is traced back to how the coherence affects the propensity to thermalization. We discuss this aspect by considering the availability function.

Figure 1

Conceptual scheme of the protocol. First, the qubit is initialized in a suitable probe state, and then it is put in contact with a thermal bath of unknown temperature, either $T_1$ or $T_2$. Finally, the qubit is removed from the interaction after a time $\tau$ and measured to infer the working temperature.

Figure 2

Experimental linear-optical simulation. Light is provided by a diode laser emitting $680\mu \mathrm{W}$ at 810 nm. Its polarization is controlled by means of the H0 wave plate. The SLM, embedded in a displaced Sagnac interferometer, realizes the coupling between the polarization and path, as detailed in the text. At the two outputs, two polarization analyzers, consisting of a quarter-wave plate, a half-wave plate, and a polarizing beam splitter, are used to characterize the state after the simulated interaction. The two analysis channels 1 and 2 are kept distinct for practicality, but the results are combined for the analysis. Intensities are detected by a linear diode. Inset: Detail of the loops in the Sagnac interferometer. The presence of H3 and H4, both set at an angle of 22.5°, makes the polarization sensitive to the birefringent phase $\varphi$ imparted by the SLM. A phase mask is applied, presenting two phase settings: In order to implement ($E_0$, $E_1$), one half on the clockwise loop is kept fixed at $\varphi =0$, while the other half is varied to simulated different interaction times. The mask is then inverted to implement ($E_2$, $E_3$).

Figure 3

Simulated temperature discrimination. The expectation values of $\widehat{\mathbb{G}}(\tau )$ have been inferred from the experimentally reconstructed density matrices, corresponding to three different input states. In the three panels, red dots are for the hot bath $N_2=9.5$ and blue dots for the cold bath $N_1=5.5$; the solid lines show the predicted behavior. The vertical dashed lines indicate the optimal discrimination time, i.e., the time for which the difference of the expectation values is maximal. Vertical error bars are obtained through a Monte Carlo routine that takes into account the uncertainties on the measured intensities, while horizontal error bars reflect the uncertainties in the calibration of the SLM birefringent phase $\varphi$, which simulates the interaction time $\tau$ [27].

Figure 4

Variation of the availability during the evolution of the system. The points are the free energies of the output states extracted from the experimental density matrices, using different input states: $|H⟩$ (dark red and blue), $|D⟩$ (red and purple), and $|V⟩$ (orange and cyan). The solid curves are the predicted behaviors. The evolution in the presence of the hot (cold) bath results in a larger (smaller) variation of the availability. Inset: Predicted variation of the availability, normalized to its limit value $\mathrm{\Delta }{F}_{\infty }$ at large times.