Microscopic expression for the heat in the adiabatic basis

TL;DR

This paper derives a microscopic expression for the heat generated in a quantum system during a time-dependent process, linking it to transition probabilities and initial state properties, with implications for thermodynamics and observable expansions.

Contribution

It introduces a microscopic formula for heat in adiabatic basis systems, applicable to arbitrary processes and initial states, enhancing understanding of quantum thermodynamics.

Findings

Heat is expressed via transition probabilities between energy levels.

For passive initial states, heat is shown to be non-negative.

The results facilitate systematic expansions around the adiabatic limit.

Abstract

We derive a microscopic expression for the instantaneous diagonal elements of the density matrix $\rho_{nn}(t)$ in the adiabatic basis for an arbitrary time dependent process in a closed Hamiltonian system. If the initial density matrix is stationary (diagonal) then this expression contains only squares of absolute values of matrix elements of the evolution operator, which can be interpreted as transition probabilities. We then derive the microscopic expression for the heat defined as the energy generated due to transitions between instantaneous energy levels. If the initial density matrix is passive (diagonal with $\rho_{nn}(0)$ monotonically decreasing with energy) then the heat is non-negative in agreement with basic expectations of thermodynamics. Our findings also can be used for systematic expansion of various observables around the adiabatic limit.