Second-order adiabatic expansion of heat and charge currents within the nonequilibrium Green's function approach

Due to technological needs, nanoscale heat management, energy conversion, and quantum thermodynamics have become key areas of research, putting heat pumps and nanomotors center stage. The treatment of these particular systems often requires the use of adiabatic expansions in terms of the frequency of the external driving or the velocity of some classical degree of freedom. However, due to the difficulty of getting the expressions, most works have only explored first-order terms. Despite this, adiabatic expansions have allowed the study of intriguing phenomena such as adiabatic quantum pumps and motors, or electronic friction. Here, we use nonequilibrium Green's functions, within a Schwinger-Keldysh approach, to develop second-order expressions for the energy, heat, and charge currents. We illustrate, through two simple models, how the obtained formulas produce physically consistent results, and allow for the thermodynamic study of unexplored phenomena, such as second-order monoparametric pumping.