Invasiveness of nonequilibrium pure-dephasing quantum thermometry

One of the main advantages expected from using quantum probes as thermometers is noninvasiveness, i.e., a negligible perturbation to the thermal sample. However, invasiveness is rarely investigated explicitly. Here, focusing on a spin probe undergoing pure dephasing due to the interaction with a bosonic sample, we show that there is a nontrivial relation between the information on the temperature gained by a quantum probe and the heat absorbed by the sample due to the interaction. We show that time-optimal probing schemes obtained by considering the total experiment time as a resource also have the benefit of limiting the heat absorbed by the sample in each shot of the experiment. For such time-optimal protocols, we show that it is advantageous to have strong probe-sample coupling, since in this regime the accuracy increases linearly with the coupling strength, while the amount of heat per shot saturates to a finite value. Since in pure-dephasing models the absorbed heat corresponds to the external work needed to couple and decouple the probe and the sample, our results also represent a first step towards the analysis of the thermodynamic and energetic cost of quantum thermometry.