Operational models of temperature superpositions

TL;DR

The paper addresses how to meaningfully define and analyze a quantum system thermalising with baths when temperatures are in quantum superposition, motivated by relativistic settings such as Tolman-Ehrenfest and Unruh/Hawking effects. It develops two operational models: a two-bath, control-dependent thermalisation and a one-bath purification superposition, deriving explicit probe states and revealing cross-terms that depend on channel dilations. The key findings show that the final probe state is not generically thermal, even when bath temperatures coincide, and that the interference visibility between temperature branches is dictated by the purifications and local dilations, with maximal visibility tied to thermal-state fidelity. The work connects foundational questions in quantum thermodynamics to relativistic physics, clarifies when temperature superpositions have operational meaning, and suggests experimental probes and further theoretical exploration in pre-thermalisation regimes and curved-spacetime contexts.

Abstract

A quantum system and a thermal bath can reach thermal equilibrium through an interaction, whereupon the system acquires the same temperature as the bath. But how does a delocalised quantum system thermalise with a bath whose local temperature varies, as, for example, in the Tolman effect? Here we formulate two scenarios in which the notion of a ``superposition of temperatures'' may arise. First: a probe interacting with two different baths dependent on the state of another quantum system (control). Second: a probe interacting with a single bath whose purified state is a superposition of states corresponding to different temperatures. We show that the two scenarios are fundamentally different and can be operationally distinguished. Moreover, we show that the probe does not in general thermalise even when the involved temperatures are equal, and that the final probe state is sensitive to the specific realisation of the thermalising channels. Our models may be applied to scenarios involving joint quantum, gravitational, and thermodynamic phenomena, and explain some recent results found in quantum intereference of relativistic probes thermalising with Unruh or Hawking radiation. Finally, we show that our results are reproduced in partial and pre-thermalisation processes, and thus our approach and conclusions hold beyond the idealised scenarios, where thermalisation is incomplete.

Figures (5)

• Figure 1: A Mach-Zehnder interferometer as an example realisation of the two-bath case. An input probe state enters from the left, is placed in a superposition, and travels along the two arms of the interferometer, on each of which is a bath thermalised to some temperature. The paths recombine at the second beamsplitter, and the output state is then detected at $D^+, D^-$.
• Figure 2: A diagram for the one-bath case in analogy to Figure 1. The bath is placed in a superposition and acquires a different temperature depending on the control state $\ket{0}_C$ or $\ket{1}_C$. The probe interacts/thermalises with the bath while the bath is in superposition.
• Figure 3: (a) Schematic diagram showing the collisional model with each step consisting of the GADC unitary ($U_{BS}^{\eta}$) acting on the probe and the bath, each of whose subsystems are in a Gibbs state at a fixed temperature. Equivalently, one could consider the unitary interaction $U_{BS}=U_{BS}^{\eta}\otimes I_A$ between one of the (purified) bath subsystems and the probe at each "collision". (b) Trace distance between the state of the probe and a thermal state as a function of the collision number in the collisional GADC model. The probe is initialized in the state $|0\rangle$, whereas the initial state of each of the bath subsystems is defined as $|\theta^{\beta=1}\rangle$ corresponding to temperature T=1. $\Delta E =1$ for both probe and bath subsystems.
• Figure 4: Heat map of the visibility of the control as a function of the temperatures of the baths in the two-bath case for (a) pre-thermalisation, $\mathcal{M}=3$ and (b) post-thermalisation, $\mathcal{M}=5$. The interaction parameter is set: $\eta = 0.8$ to allow for partial thermalisation.
• Figure 5: Heat map of the visibility of the control as a function of temperatures of the bath in the one-bath case for (a) pre-thermalisation, $\mathcal{M}=3$ and (b) post-thermalisation, $\mathcal{M}=5$. The interaction parameter is set: $\eta = 0.8$ to allow for partial thermalisation.