Thermodynamics of two-stroke engine based on periodically driven two-level system

We investigate a microscopic motor based on an externally driven two-level system. One cycle of the motor operation consists of two strokes. Within each stroke, the two energy levels are driven with a constant rate. The occupation probabilities of the two states evolve according to the Pauli rate equation and represent the delayed system's response to the external driving. We give the exact solution of the Pauli rate equation and discuss its thermodynamical consequences. In particular, we calculate the motor's efficiency, the power output, and the performance dependence on the control parameters. Secondly, we introduce an augmented stochastic process which reflects, at a given time, both the occupation probabilities for the two states and the time spent in the individual states during the previous evolution. Our exact calculation of the evolution operator for the augmented process allows one to discuss in detail the probability density for the work during the limit cycle. In the strongly irreversible regime, the density shows strong deviations from a Gaussian shape.