Non-Markovian quantum thermodynamics: laws and fluctuation theorems

Abstract

This work brings together Keldysh non-equilibrium quantum theory and thermodynamics, by showing that a real-time diagrammatic technique is an equivalent of stochastic thermodynamics for non-Markovian quantum machines (heat engines, refrigerators, etc). Symmetries are found between quantum trajectories and their time-reverses on the Keldysh contour, for any interacting quantum system coupled to ideal reservoirs of electrons, phonons or photons. These lead to quantum fluctuation theorems the same as the well-known classical ones (Jarzynski and Crooks equalities, integral fluctuation theorem, etc), whether the system's dynamics are Markovian or not. Some of these are also shown to hold for non-factorizable initial states. The sequential tunnelling approximation and the cotunnelling approximation are both shown to respect the symmetries that ensure the fluctuation theorems. For all initial states, energy conservation ensures that the first law of thermodynamics holds on average, while the above symmetries ensures that the second law of thermodynamics holds on average, even if fluctuations violate it. [ERRATUM added: March 2021]

This version corrects an error in Section VIII D of the version published in Phys. Rev. B (an error in the condition for the validity of the Crooks equaton)

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