Geometric optimization of non-equilibrium adiabatic thermal machines and implementation in a qubit system

We adopt a geometric approach to describe the performance of adiabatic quantum machines, operating under slow time-dependent driving and in contact to two or more reservoirs with a temperature bias during all the cycle. We show that the problem of optimizing the power generation of a heat engine and the efficiency of both the heat engine and refrigerator operational modes is reduced to an isoperimetric problem with non-trivial underlying metrics and curvature. This corresponds to the maximization of the ratio between the area enclosed by a closed curve and its corresponding length. We illustrate this procedure in a qubit coupled to two reservoirs operating as a thermal machine by means of an adiabatic protocol.