Local Quantum Thermometry using Unruh-De Witt detectors

We propose an operational definition for the local temperature of a quantum field employing Unruh-DeWitt detectors, as used in the study of the Unruh and Hawking effects. With this definition, an inhomogeneous quantum system in equilibrium can have different local temperatures, in analogy with the Tolman-Ehrenfest theorem from general relativity. We study the local temperature distribution on the ground state of hopping fermionic systems on a curved background. The observed temperature tends to zero as the thermometer-system coupling $g$ vanishes. Yet, for small but finite values of $g$, we show that the product of the observed local temperature and the logarithm of the local speed of light is approximately constant. Our predictions should be testable on ultracold atomic systems.