Olivier Brodier

A semiclassical limit for work in the Jarzynski equality

The Jarzynski equality is a simple identity that generalizes an old thermodynamical result, found in every textbook, relating the "maximal useful work" with the free energy variation. The identity appeared in the 90s and has, since then, stimulated many studies, discussions and experimental verifications, bringing a new point of view in the growing field of non-equilibrium thermodynamics. Quite naturally, attempts have been made to find a quantum analogue of the Jarzynski equality, even though there is a central difficulty : whereas the Jarzynski equality is about the work received by a system along trajectories, the latter don't exist in quantum mechanics. This problem can be circumvented when the system has no environment, because work is then just energy difference. In Markovian systems one can also access work indirectly through quantum trajectories, in the picture of a system continuously monitored by the environment. However, in both approaches, no natural quantity appears as a "quantum work". Our approach has been to try to write a Jarzynski equality in the semiclassical limit, by using the "double trajectories" that appear naturally in the Weyl-Wigner representation of a quantum propagation. An expression then emerges of a work in the sense of Jarzynski, which turns out to be similar to the Hamiltonians that are used to "rapidely drive" quantum systems, in the field of "Shortcut To Adiabaticity".