Bath-Induced Correlations Enhance Thermometry Precision at Low Temperatures

We study the role of bath-induced correlations in temperature estimation of cold bosonic baths. Our protocol includes multiple probes, that are not interacting, nor are they initially correlated to each other. They interact with a bosonic sample and reach a nonthermal steady state, which is measured to estimate the temperature of the sample. It is well known that in the steady state such noninteracting probes may get correlated to each other and even entangled. Nonetheless, the impact of these correlations in metrology has not been deeply investigated yet. Here, we examine their role for thermometry of cold bosonic gases and show that, although being classical, bath-induced correlations can lead to significant enhancement of precision for thermometry. The improvement is especially important at low temperatures, where attaining high precision thermometry is particularly demanding. The proposed thermometry scheme does not require any precise dynamical control of the probes and tuning the parameters and is robust to noise in initial preparation, as it is built upon the steady state generated by the natural dissipative dynamics of the system. Therefore, our results put forward new possibilities in thermometry at low temperatures, of relevance, for instance, in cold gases and Bose-Einstein condensates.

Figure 1

Schematic of the thermometry protocols. (a) Independent baths: each thermometer is in contact with a separate bath, thus no correlations are built among them. Hence, the precision is additive. (b) Common bath—focus of this work: all thermometers are in contact with the same bath. The $i$th and the $j$th thermometers have a distance $r_{ij}$, they do not interact with each other nor do they share initial correlations, yet they get correlated thanks to their interactions with the common bath. Our results show that such correlations might lead to superadditive precision at low temperatures. The two scenarios are equivalent when the distance between the probes is very large or at high temperatures.

Figure 2

Minimum relative error [$\delta T_{\min }/T$ from r.h.s of Eq. (14)] against temperature for different system-bath couplings $\gamma$ (quantitative description in the Supplemental Material [60]) for a fixed number of oscillators. The solid curves represent the common bath scenario (b), whereas the dashed lines are obtained considering independent baths scenario (a). We see that at low temperatures by increasing $\gamma$ the error reduces, whereas at higher temperatures the opposite holds (not shown). Moreover, embedding the oscillators at the same bath can significantly decrease the relative error. Here, we set the parameters to ${\omega }_0=1$ in arbitrary units, ${\omega }_i={\omega }_0\forall i$, $r_{21}/c{\omega }_0=0.1$, $\mathrm{\Omega }=100{\omega }_0$, $g_{ij}=0$, and $N=10$.

Figure 3

Minimum relative error normalized by the number of oscillators (that is $\sqrt{N}\delta T_{\min }/T$), as a function of temperature, in the independent baths scenario (a) for arbitrary $N$ (dashed black) and in the common bath scenario (b): for different number of oscillators $N=1$ (also dashed black), $N=5$ (solid black), $N=10$ (solid blue), and $N=15$ (solid red). Unlike the scenario (a) which is additive, the scenario (b) is superadditive, as the normalized error reduces by increasing $N$. The parameters are set to ${\omega }_0=1$ in arbitrary units, and ${\omega }_i={\omega }_0\forall i$, $r_{21}/c{\omega }_0=0.1$, $\mathrm{\Omega }=100{\omega }_0$, $g_{ij}=0$, and $\gamma =\sqrt{{\omega }_0}$.

Figure 4

Log-log plot of the quantum Fisher information vs $N$ in solid blue for scenario (b) and solid black for scenario (a). Here, we tune the setting to the low temperature limit with $T_0=0.01{\omega }_0$. Scenario (b) behaves superlinearly; our data fitting shows that $\mathcal{F}(T_0)\propto N^{2.5}$ in the range of $1<N<30$, although the slope is decreasing as we see for $N\approx 80$ we have $\mathcal{F}(T_0)\propto N^{1.1}$ (Inset). Nonetheless, for $N=30$ we already see that the QFI in scenario (b) is 2 orders of magnitude bigger than (a)—which shows a linear behavior as suggested by the additivity of the QFI for a tensor product state. The CFI for measuring $X={\otimes }_{i=1}^N{x}_i$ is also depicted in dashed black for (a) and dashed blue for (b). Albeit suboptimal, these local measurements show a similar behavior to the optimal ones in both scenarios. Specifically, for scenario (b) they are superlinear with $N$. The rest of the parameters are set to ${\omega }_0=1$, ${\omega }_i={\omega }_0\forall i$, $\gamma =\sqrt{{\omega }_0}$, $\mathrm{\Omega }=100{\omega }_0$, $g_{ij}=0$, and $r_{21}/c{\omega }_0=0.1$.