A convenient Keldysh contour for thermodynamically consistent perturbative and semiclassical expansions

The work fluctuation theorem (FT) is a symmetry connecting the moment
generating functions (MGFs) of the work extracted in a given process and in its
time-reversed counterpart. We show that, equivalently, the FT for work in
isolated quantum systems can be expressed as an invariance property of a
modified Keldysh contour. Modified contours can be used as starting points of
perturbative and path integral approaches to quantum thermodynamics, as
recently pointed out in the literature. After reviewing the derivation of the
contour-based perturbation theory, we use the symmetry of the modified contour
to show that the theory satisfies the FT at every order. Furthermore, we extend
textbook diagrammatic techniques to the computation of work MGFs, showing that
the contributions of the different Feynman diagrams can be added to obtain a
general expression of the work statistics in terms of a sum of independent
rescaled Poisson processes. In this context, the FT takes the form of a
detailed balance condition linking every Feynman diagram with its time-reversed
variant. In the second part, we study path integral approaches to the
calculation of the MGF, and discuss how the arbitrariness in the choice of the
contour impacts the final form of the path integral action. In particular, we
show how using a symmetrized contour makes it possible to easily generalize the
Keldysh rotation in the context of work statistics, a procedure paving the way
to a semiclassical expansion of the work MGF. Furthermore, we use our results
to discuss a generalization of the detailed balance conditions at the level of
the quantum trajectories.