Illustration of thermometry using a single spin

Optimal quantum thermometry with coarse-grained measurements

This work is about taking the temperature of a quantum system. Ordinary thermometers, like the ones used to check if you have a fewer, work well for large systems with a temperature which is not too cold and not too hot. When measuring temperatures at the quantum scale, however, we need to design thermometers which can operate at very low temperatures and which won’t disturb the system too much. An intriguing possibility is to use individual quantum probes as thermometers. For example we could take a single atom, allow the atom to interact with the thermal system for some time, and then measure the energy of the atom. The outcome of the measurement reveals something about the system temperature.

In our paper we investigate how to design optimal thermometers, that is, thermometers with maximal precision, given that the available measurements themselves have limited precision. It is, in fact, well known that the measurements from which give the most information about temperature are precise measurements of the total energy of the full system. Energy in quantum systems is quantised (hence the name) and the best measurement should distinguish all the distinct energy levels of the system. However, this becomes extremely difficult in even moderately sized systems, as the number of levels grows rapidly and they become closely spaced. Realistic energy measurements in such systems always involve some coarse graining over the individual energy levels. Using tools from signal processing theory, we derive an equation describing the structure of optimal coarse-grained measurements. Surprisingly, we find that good temperature estimates can generally be attained even the number of distinct measurement outcomes is small. That is, for very coarse-grained measurements.

We apply our results to many-body quantum systems and nonequilibrium thermometry. For the latter, in particular, we consider a probe of given dimension interacting with the sample, followed by a measurement of the probe. We derive an upper bound on arbitrary, nonequilibrium strategies for such probe-based thermometry and illustrate it for thermometry on an ultra cold gas (specifically, a Bose-Einstein condensate) using a single atom as a probe. We find that even for a probe with just two energy levels, the coarse-graining constraint still allows approximately 64% of the best possible thermometric precision.