Enhancement of low-temperature thermometry by strong coupling

We consider the problem of estimating the temperature $T$ of a very cold equilibrium sample. The temperature estimates are drawn from measurements performed on a quantum Brownian probe strongly coupled to it. We model this scenario by resorting to the canonical Caldeira-Leggett Hamiltonian and find analytically the exact stationary state of the probe for arbitrary coupling strength. In general, the probe does not reach thermal equilibrium with the sample, due to their nonperturbative interaction. We argue that this is advantageous for low-temperature thermometry, as we show in our model that (i) the thermometric precision at low $T$ can be significantly enhanced by strengthening the probe-sampling coupling, (ii) the variance of a suitable quadrature of our Brownian thermometer can yield temperature estimates with nearly minimal statistical uncertainty, and (iii) the spectral density of the probe-sample coupling may be engineered to further improve thermometric performance. These observations may find applications in practical nanoscale thermometry at low temperatures—a regime which is particularly relevant to quantum technologies.

Figure 1

(a) Log-log plot of the best-case relative error $\delta T/T=1/\left(T\sqrt{{\mathcal{F}}_T}\right)$ vs the sample temperature $T$ for different dissipation strengths $\gamma$; namely, $\gamma /{\omega }_0=0.1$ (solid black), $\gamma /{\omega }_0=1$ (dashed black), and $\gamma /{\omega }_0=5$ (dotted black). The relative error of a single-mode probe at thermal equilibrium (dot-dashed red) has been super-imposed for comparison. $\delta T/T$ diverges as $T\to 0$; while for the thermal mode it would diverge exponentially, our exact solution yields $\delta T/T\sim T^{-2}$ at low $T$. Whenever $T/{\omega }_0\ll 1$, increasing the dissipation strength results in a significant reduction of the minimum $\delta T/T$. On the contrary, at larger temperatures, the best-case relative error need not be monotonically decreasing with $\gamma$. This is shown in the inset, which zooms into the bottom-right corner of the plot. (b) Log-log plot of ${\mathcal{F}}_T$ as a function of $\gamma$ for $T=1$ (solid), $T=0.1$ (dashed), and $T=0.01$ (dotted). It becomes again clear that, while not strictly monotonic in $\gamma$, the QFI always grows with the dissipation strength for $\gamma /{\omega }_0\gtrsim 1$ at $T/{\omega }_0\ll 1$. Furthermore, as $T/{\omega }_0\to 0$, we observe such a sensitivity enhancement at arbitrarily weak probe-sample coupling. In both cases ${\omega }_c=100{\omega }_0$ and $\hslash =k_B={\omega }_0=1$.

Figure 2

Log-log plot of the QFI ${\mathcal{F}}_T$ (solid black on all panels), thermal sensitivity of the energy of the probe $F_T\left(H_{\text{p}}\right)$ (dashed black on the left-hand panels), and $F_T\left(x^2\right)$ (dashed black on the right-hand panels), for different values of the dissipation strength: $\gamma /{\omega }_0=5\times {10}^{-3}$ (top), $\gamma /{\omega }_0=5\times {10}^{-2}$ (middle), and $\gamma /{\omega }_0=0.5$ (bottom). Note that the thermal sensitivity $H_{\text{p}}$ is deterred as the dissipation strength grows, while $x^2$ becomes a quasioptimal temperature estimator. As in Fig. 1, ${\omega }_c=100$ and $\hslash =k_B={\omega }_0=1$.

Figure 3

Log-log plot of the QFI ${\mathcal{F}}_T$ as a function of temperature for Ohmic (solid) and super-Ohmic (dashed) spectral density $J_s\left(\omega \right)$ with exponential high-frequency cutoff ($s=1$ and $s=2$, respectively). In the inset, both spectral densities are compared. Note that the Ohmic form largely outperforms the super-Ohmic one at low temperatures ($\gamma /{\omega }_0=0.1$, ${\omega }_c=100{\omega }_0$, and $\hslash =k_B={\omega }_0=1$).