Heat transport in overdamped quantum systems

In classical and statistical physics, the overdamped limit of systems interacting with their environments is a very useful approximation allowing for the simplification of the Fokker-Plank equation in phase space to the Smoluchowski equation for the position variable alone. For
quantum systems, the same limit can still be studied using path integral techniques. This leads to the quantum version of the Smoluchowski equation for systems in thermal equilibrium with only one thermal bath.
However, to study the stochastic and quantum thermodynamics, one needs to deal with systems in a nonequilibrium situation where the quantum Smoluchowski equation is not valid anymore.
In this work we obtain an analytical expression for the heat current between two overdamped quantum oscillators interacting with local thermal baths at different temperatures. The total heat current is split into classical and quantum contributions. We show how to evaluate both contributions by taking advantage of the time scale separation associated with the overdamped regime, and without assuming the usual weak coupling and Markovian approximations. We find that non-trivial quantum corrections survive even when the temperatures are high compared to the frequency scale relevant for the overdamped dynamics of the system.