What Temperature is Schrödinger's Cat?

If thermodynamical quantities can be associated with quantum systems, can temperature have quantum features, similar to how time can exhibit them in association with quantum clocks?

In this work we explore two scenarios in which the notion of a ‘temperature superposition’ may arise. In the first, a probe interacts with different baths depending on another quantum system’s state. In the second, the probe interacts with only one bath, but the bath, with its purification, is now correlated with another quantum system. The bath temperature depends on the state of this added system. We derive final system states, discuss conditions for probe thermalisation and temperature coherence (coherence in the system on which temperature depends) and show the two cases are surprisingly different. In case 1 the probe does not thermalise and temperature coherence is reduced even for baths of same temperature. In case 2, the probe can thermalise and reach maximal coherence. We also find the probe states are sensitive to particular Kraus representations of the channels, which may explain results in quantum interference of relativistic particle detectors thermalising with Unruh/Hawking radiation. Our results extend to partial/pre-thermalisation, where we introduce a collisional model of thermalising interactions between system and bath(s).