Measuring the temperature of cold many-body quantum systems

Abstract

Precise low-temperature thermometry is a key requirement for virtually any quantum technological application. Unfortunately, as the temperature T decreases, the errors in its estimation diverge very quickly. In this paper, we determine exactly how quickly this may be. We rigorously prove that the “conventional wisdom” of low- T thermometry being exponentially inefficient is limited to local thermometry on translationally invariant systems with short-range interactions, featuring a nonzero gap above the ground state. This result applies very generally to spin and harmonic lattices. On the other hand, we show that a power-law-like scaling is the hallmark of local thermometry on gapless systems. Focusing on thermometry on one node of a harmonic lattice, we obtain valuable physical insight into the switching between the two types of scaling. In particular, we map the problem to an equivalent setup, consisting of a Brownian thermometer coupled to an equilibrium reservoir. This mapping allows us to prove that, surprisingly, the relative error of local thermometry on gapless harmonic lattices does not diverge as T → 0 ; rather, it saturates to a constant. As a useful by-product, we prove that the low- T sensitivity of a harmonic probe arbitrarily strongly coupled to a bosonic reservoir by means of a generic Ohmic interaction, always scales as T 2 for T → 0 . Our results thus identify the energy gap between the ground and first excited states of the global system as the key parameter in local thermometry, and ultimately provide clues to devising practical thermometric strategies deep in the ultracold regime.