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We introduce a novel minimally-disturbing method for sub-nK thermometry in aBose-Einstein condensate (BEC). Our technique is based on the Bose-polaronmodel; namely, an impurity embedded in the BEC acts as the thermometer. Wepropose to detect temperature fluctuations from measurements of the positionand momentum of the impurity. Crucially, these cause minimal back-action on theBEC and hence, realize a non-demolition temperature measurement. Following theparadigm of the emerging field of \textit{quantum thermometry}, we combinetools from quantum parameter estimation and the theory of open quantum systemsto solve the problem in full generality. We thus avoid \textit{any}simplification, such as demanding thermalization of the impurity atoms, orimposing weak dissipative interactions with the BEC. Our method is illustratedwith realistic experimental parameters common in many labs, thus showing thatit can compete with state-of-the-art \textit{destructive} techniques, even whenthe estimates are built from the outcomes of accessible (sub-optimal)quadrature measurements.

Using polarons for sub-nK quantum non-demolition thermometry in a Bose-Einstein condensate