Non-Markovian temperature sensing

We investigate the sensing performance of a single-qubit quantum thermometer within a non-Markovian dynamical framework. By employing an exactly numerical hierarchical equations of the motion method, we go beyond traditional paradigms of the Born-Markov theory, the pure dephasing mechanism, and the weak-coupling approximation, which were commonly used in many previous studies of quantum thermometry. We find (i) the non-Markovian characteristics may boost the estimation efficiency, (ii) the sensitivity of quantum thermometry can be effectively optimized by engineering the proportions of different coupling operators in the whole sensor-reservoir interaction Hamiltonian, and (iii) a threshold, above which the strong sensor-reservoir coupling can significantly enhance the sensing precision in the long-encoding-time regime. Our results may have certain applications for high-resolution quantum thermometry.

Figure 1

Schematic diagram of our quantum thermometry scheme in which a two-level system is used to as the sensor estimate the temperature of a bosonic reservoir.

Figure 2

The dynamics QFI with different cutoff frequencies: ${\omega }_{\mathrm{c}}=0.5\mathrm{\Delta }$ (blue rectangles), ${\omega }_{\mathrm{c}}=0.8\mathrm{\Delta }$ (red circles), ${\omega }_{\mathrm{c}}=4\mathrm{\Delta }$ (purple rhombuses), and ${\omega }_{\mathrm{c}}=10\mathrm{\Delta }$ (magenta triangles). The inset figure depicts the steady-state QFI in the long-time regime. Parameters are chosen as $\alpha =\pi /4$, $\phi =\pi /2$, $\epsilon =0.5\mathrm{\Delta }$, $\beta \mathrm{\Delta }=0.06$, $\theta =0$, and $\chi =0.06\mathrm{\Delta }$ with $\mathrm{\Delta }=1{\mathrm{cm}}^{-1}$.

Figure 3

The maximum QFI with respect to the optimal encoding time versus the coupling angle for different cutoff frequencies: ${\omega }_{\mathrm{c}}=0.5\mathrm{\Delta }$ (blue rectangles), ${\omega }_{\mathrm{c}}=0.8\mathrm{\Delta }$ (red circles), ${\omega }_{\mathrm{c}}=4\mathrm{\Delta }$ (purple rhombuses), and ${\omega }_{\mathrm{c}}=10\mathrm{\Delta }$ (magenta triangles). Other parameters are the same as those of Fig. 2.

Figure 4

Schematic diagram of the influence of frequency renormalization $\mathrm{\Omega }\to \tilde{\mathrm{\Omega }}$ on (a) the steady-state population ratio and (b) the QFI. (c) The steady-state population ratio ${\varrho }_{\mathrm{gg}}\left(\infty \right)/{\varrho }_{\mathrm{ee}}\left(\infty \right)$ is plotted as a function of the sensor-reservoir coupling strength $\chi$. (d) The steady-state QFI versus $\chi$. The red five-point stars are the numerical results obtained by the HEOM method, the blue rectangles are the steady-state solutions from the Born-Markov master equation approach, while the blue solid lines represent the results from the canonical Gibbs state. Parameters are chosen as $\alpha =\pi /4$, $\phi =\pi /2$, $\epsilon =0.5\mathrm{\Delta }$, $\beta \mathrm{\Delta }=5$, and ${\omega }_{\mathrm{c}}=0.5\mathrm{\Delta }$ with $\mathrm{\Delta }=50{\mathrm{cm}}^{-1}$. Here, ${\mathrm{\Omega }}^{*}\simeq 24{\mathrm{cm}}^{-1}$ and $\mathrm{\Omega }\simeq 56{\mathrm{cm}}^{-1}$.

Figure 5

The same with Fig. 4, but $\beta \mathrm{\Delta }=1$ and $\mathrm{\Delta }=10{\mathrm{cm}}^{-1}$. Here, ${\mathrm{\Omega }}^{*}\simeq 24{\mathrm{cm}}^{-1}$ and $\mathrm{\Omega }\simeq 11{\mathrm{cm}}^{-1}$.

Figure 6

The dynamics of population difference $\langle {\widehat{\sigma }}_z\left(t\right)\rangle$ with different cutoff frequencies: (a) ${\omega }_{\mathrm{c}}=30\mathrm{\Delta }$ (the deep fast-reservoir regime), (b) ${\omega }_{\mathrm{c}}=20\mathrm{\Delta }$, (c) ${\omega }_{\mathrm{c}}=10\mathrm{\Delta }$, (d) ${\omega }_{\mathrm{c}}=0.07\mathrm{\Delta }$, and (e) ${\omega }_{\mathrm{c}}=0.05\mathrm{\Delta }$ (the deep slow-reservoir regime). The red circles are numerical results predicted by the HEOM method, while the blue solid lines are from the Born-Markovian master equation approach. Other parameters are chosen as $\alpha =\pi /4$, $\phi =\pi /2$, $\epsilon =0.5\mathrm{\Delta }$, $\chi =0.06\mathrm{\Delta }$, $\beta \mathrm{\Delta }=0.25$, $\theta =3\pi /8$, and $\mathrm{\Delta }=1{\mathrm{cm}}^{-1}$.