Using Polarons for sub-nK Quantum Nondemolition Thermometry in a Bose-Einstein Condensate

We introduce a novel minimally disturbing method for sub-nK thermometry in a Bose-Einstein condensate (BEC). Our technique is based on the Bose polaron model; namely, an impurity embedded in the BEC acts as the thermometer. We propose to detect temperature fluctuations from measurements of the position and momentum of the impurity. Crucially, these cause minimal backaction on the BEC and hence, realize a nondemolition temperature measurement. Following the paradigm of the emerging field of quantum thermometry, we combine tools from quantum parameter estimation and the theory of open quantum systems to solve the problem in full generality. We thus avoid any simplification, such as demanding thermalization of the impurity atoms, or imposing weak dissipative interactions with the BEC. Our method is illustrated with realistic experimental parameters common in many labs, thus showing that it can compete with state-of-the-art destructive techniques, even when the estimates are built from the outcomes of accessible (suboptimal) quadrature measurements.

Figure 1

Optimal relative error $(\delta T{)}_{\min }/T$ ($\nu =1$) as a function of the temperature of the BEC in a logarithmic scale (black curves). Specifically, we work with impurities of Yb in a sea of ultracold K. The temperature range for the BEC is $200\text{pK}\le T\le 2\mathrm{nK}$. The trapping frequency of the gas (with $N=5000$ atoms) was set to ${\omega }_B=2\pi \times 100\mathrm{Hz}$, while $\mathrm{\Omega }=2\pi \times 10\mathrm{Hz}$ ($g_{rmB}=3\times {10}^{-39}\mathrm{J}\mathrm{m}$). Different probe-sample coupling ratios $\eta =g_{IB}/g_B$ were considered, namely $\eta =1$ (solid), $\eta =3$ (dashed), and $\eta =6$ (dotted). For comparison, we also depicted the relative error of a fully thermalized impurity (i.e., $\eta \to 0$) (dot-dashed red). Note that, for $\eta =1$, the relative error can be kept below $\sim 14\%$ from only $\nu \sim 100$ measurements. This is quantitatively close to state-of-the-art destructive experimental techniques. See text for discussion.

Figure 2

Relative error for the position quadrature $\delta T({\widehat{x}}^2)/T$ (dashed blue curve) and the momentum quadrature $\delta T({\widehat{p}}^2)/T$ (dotted red curve) as a function of $T$ for $\eta =1$ (left panel) and $\eta =6$ (right panel). All parameters are the same as in Fig. 1. The minimum relative error (solid black curve) is superimposed for reference. Even though both measurement schemes are suboptimal, they still might allow us to draw estimates with relative errors as low as 18% for $\nu \sim 400$. Note that for $T\ge 0.5\mathrm{nK}$ and by using the same data size $\nu =400$, one can achieve a relative error below 10%.