Qubit-assisted thermometry of a quantum harmonic oscillator

We use the theory of quantum estimation in two different qubit-boson coupling models to demonstrate that the temperature of a quantum harmonic oscillator can be estimated with high precision by quantum-limited measurements on the qubit. The two models that we address embody situations of current physical interest due to their connection with ongoing experimental efforts on the control of mesoscopic dynamics. We show that population measurements performed over the qubit probe are near optimal for a broad range of temperatures of the harmonic oscillator.

Figure 1

A physical model for the realization of the coupling Hamiltonian ${\widehat{H}}_I=g{\widehat{X}}_o\otimes {\widehat{\sigma }}_x$ between a harmonic oscillator and a qubit. An electrically driven nanomechanical oscillator (bias voltage $V_x$) is coupled to a CPB through the capacitance $C_x$. The state of the CPB is controlled by the gate voltage $V_g$ (coupled to the box through the capacitance $C_g$) and the Josephson energy $E_J$. We work at the charge degeneracy point.

Figure 2

(Left) The functional form of ${\tau }_{\mathrm{opt}}\left(\beta \right)$, which is the optimal time at which the Fisher information is maximum for the coupling model ${\widehat{H}}_1=g\widehat{X}\otimes {\widehat{\sigma }}_x$ at set temperature $\beta$ and for an optimal preparation of the probe qubit. (Right) For the same interaction model, we show a typical behavior of the Fisher information $F(\beta =3)$ against the dimensionless interaction time $\tau$ between probe and oscillator and the angle $\theta$ that parametrizes the initial preparation of the probe. Clearly, the Fisher information is optimized at $\theta =0,\pi$ and quickly decays as $\tau$ grows.

Figure 3

Temporal evolution of the quantum Fisher information for the interaction model ${\widehat{H}}_1=g\widehat{X}\otimes {\widehat{\sigma }}_x$ for inverse temperature $\beta =10$. The probe qubit is prepared in states having $\varphi =0$ and $\theta =0$ (blue solid), $\theta =\frac{\pi }{4}$ (magenta dashed), and $\theta =\frac{\pi }{2}+0.1$ (green dot-dashed). For $\theta =0$, the quantum Fisher information is maximum (independently on $\varphi$) and equals the Fisher information associated with population measurements, while for $\theta =\frac{\pi }{2}$ $H\left(\beta \right)$ identically vanishes. (Inset) The quantum Fisher information evaluated at the optimal time, plotted against $\theta$ and for $\varphi =0$ (blue solid), $\varphi =\frac{\pi }{4}$ (red dashed), and $\varphi =\frac{\pi }{2}$ (orange dot-dashed). The last choice leads to the same maximum value of the quantum Fisher information and does not depend on $\theta$.

Figure 4

(Left) We consider the model ${\widehat{H}}_2=g{\widehat{a}}^{†}\widehat{a}\otimes {\widehat{\sigma }}_x$ and plot the Fisher information against the dimensionless interaction time $\tau$ for inverse temperature $\beta =10$ and the probe qubit prepared in states having $\varphi =\frac{\pi }{2}$ and $\theta =0$ (blue solid), $\theta =0.01$ (magenta dashed), and $\theta =0.1$ (green dot-dashed). For $\theta =0$ the Fisher information does not depend on $\varphi$. (Right) Fisher information for $\beta =10$ and $\theta =\frac{\pi }{4}$ as a function of time for different $\varphi$: $\varphi =\frac{\pi }{2}$ (blue solid), $\varphi =\frac{\pi }{3}$ (magenta dashed), and $\varphi =\frac{\pi }{6}$ (green dot-dashed). Although the choice $\varphi =\frac{\pi }{2}$ maximizes the Fisher information, it is evident that the relevant parameter in setting the qubit is $\theta$.

Figure 5

We consider the model ${\widehat{H}}_2=g{\widehat{a}}^{†}\widehat{a}\otimes {\widehat{\sigma }}_x$ and show the logarithmic plot of the quantity $[H_{\mathrm{opt}}\left(\beta \right)-F_{\mathrm{opt}}\left(\beta \right)]/H_{\mathrm{opt}}\left(\beta \right)$ for $\beta =10$ (blue solid), $\beta =5$ (magenta dashed), and $\beta =1$ (green dot-dashed).