Characteristics of the limit cycle of a reciprocating quantum heat engine.

After starting a reciprocating heat engine it eventually settles to a stable mode of operation. A first principle quantum heat engine also approaches this stable limit cycle. The studied engine is based on a working medium consisting of an ensemble of quantum systems composed of two coupled spins. A four-stroke cycle of operation is studied, with two isochore branches where heat is transferred from the hot/cold baths and two adiabats where work is exchanged. The dynamics is generated by a completely positive map. It has been shown that the performance of this model resembles an engine with intrinsic friction. The quantum conditional entropy is employed to prove the monotonic approach to a limit cycle. Other convex measures such as the quantum distance display the same monotonic approach. The equations of motion of the engine are solved for the different branches and are combined to a global propagator that relates the state of the engine in the beginning of the cycle to the state after one period of operation of the cycle. The eigenvalues of the propagator define the rate of relaxation toward the limit cycle. A longitudinal and transverse mode of approach to the limit cycle is thus identified. The entropy balance is used to explore the necessary conditions which lead to a stable limit cycle. The phenomena of friction can be identified with a zero change in the von Neumann entropy of the working medium.