Qubit thermometry for micromechanical resonators

We address estimation of temperature for a micromechanical oscillator lying arbitrarily close to its quantum ground state. Motivated by recent experiments, we assume that the oscillator is coupled to a probe qubit via Jaynes-Cummings interaction and that the estimation of its effective temperature is achieved via quantum-limited measurements on the qubit. We first consider the ideal unitary evolution in a noiseless environment and then take into account the noise due to nondissipative decoherence. We exploit local quantum estimation theory to assess and optimize the precision of estimation procedures based on the measurement of qubit population and to compare their performances with the ultimate limit posed by quantum mechanics. In particular, we evaluate the Fisher information (FI) for population measurement, maximize its value over the possible qubit preparations and interaction times, and compare its behavior with that of the quantum Fisher information (QFI). We found that the FI for population measurement is equal to the QFI, i.e., population measurement is optimal, for a suitable initial preparation of the qubit and a predictable interaction time. The same configuration also corresponds to the maximum of the QFI itself. Our results indicate that the achievement of the ultimate bound to precision allowed by quantum mechanics is in the capabilities of the current technology.

Figure 1

(top) FI for $\beta =10$ as a function of the effective time $\tau$ for different $\vartheta$ values: $\vartheta =\pi$ (dashed blue line), $\vartheta =0.95\pi$ (dot-dashed magenta line) and $\vartheta =0$ (solid green line). FI takes a pronounced global maximum at $(\vartheta ,\tau )=\left(\pi ,\frac{\pi }{2}\right)$, while it is possible to see a secondary extremely peaked maximum, which occurs for $\tau =\pi$ and by initially preparing the qubit in $\left|0\right.〉$. (bottom) Log-linear plot of the FI for $\beta =10$ as a function of $\vartheta$ for $\tau =\frac{\pi }{2}$ (dashed blue line), $\tau =\frac{\pi }{2}+\varepsilon$ (dot-dashed magenta line), $\tau =\pi$ (solid green gand), $\tau =\pi +\varepsilon$ (dotted red line), with $\varepsilon =0.01$.

Figure 2

Log-linear plot of the FI as a function of effective time $\tau$ for different values of $\beta$. The qubit is prepared in the ground state $\left|1\right.〉$ ($\vartheta =\pi$). From bottom to top, $\beta =15$ (solid blue line), $\beta =10$ (dashed magenta line), $\beta =5$ (dot-dashed green line), and $\beta =1$ (dotted red line). Upon raising the temperature the FI no longer keeps a scale-free shape: thermal excitations modify its profile, making it irregular. In particular, the global maximum comes earlier in time.

Figure 3

(top) Log-log plot of the FI maximized over $\tau$ as a function of $\beta$, with $\theta =\pi$ for different values of detuning: $\gamma =0$ (solid blue line), $\gamma =1$ (dashed magenta line), and $\gamma =1.5$ (dot-dashed green line). (bottom) The times ${\tau }_{\max }$ that maximize the FI as a function of $\beta$, with $\theta =\pi$ for different values of $\gamma$ (the same values and color scheme as in the top panel).

Figure 4

(top) QFI for $\beta =10$ as a function of $\tau$ for $\vartheta =\pi$ (dashed blue line), $\vartheta =0.95\pi$ (dot-dashed magenta line), and $\vartheta =0$ (solid green line). QFI behaves like FI for $\vartheta =\pi$, leading to the same maximum, while for smaller angles it shows a smoother profile. For angles $0<\vartheta <\pi$ one may find measurements that improve the precision of temperature estimation. (bottom) QFI for $\beta =10$ as a function of $\vartheta$ for $\tau =\frac{\pi }{2}$ (dashed blue line), $\tau =\frac{\pi }{2}+\varepsilon$ (dot-dashed magenta line), $\tau =\pi$ (solid green line), and $\tau =\pi +\varepsilon$ (dotted red line), with $\varepsilon =0.01$.

Figure 5

(top) Fisher information $F\left(\beta \right)$ for $\beta =10$ as a function of $\tau$ in the presence of decoherence and for different qubit preparations. The decoherence parameters are chosen as $a=0.1$ and $b={10}^{-5}$. Dashed blue line stands for $\vartheta =\pi$, dot-dashed magenta line stands for $\vartheta =0.95\pi$, and solid green line stands for $\vartheta =0$. Having included the decoherence treatment enables us not to restrict the evolution to the first Rabi half period. (bottom) Log-log plot of the Fisher information $F_M\left(\beta \right)$ maximized over the interaction time and in the presence of decoherence as a function of $\beta$ and for fixed $\vartheta =\pi$ for $b=0$ (solid green line), $b={10}^{-5}$ (dashed magenta line), and $b={10}^{-4}$ (dot-dashed blue line).