Efficiency at the maximum power output for simple two-level heat engine

We introduce a simple two-level heat engine to study the efficiency in the condition of the maximum power output, depending on the energy levels from which the net work is extracted. In contrast to the quasi-statically operated Carnot engine whose efficiency reaches the theoretical maximum, recent research on more realistic engines operated in finite time has revealed other classes of efficiency such as the Curzon-Ahlborn efficiency maximizing the power output. We investigate yet another side with our heat engine model, which consists of pure relaxation and net work extraction processes from the population difference caused by different transition rates. Due to the nature of our model, the time-dependent part is completely decoupled from the other terms in the generated work. We derive analytically the optimal condition for transition rates maximizing the generated power output and discuss its implication on general premise of realistic heat engines. In particular, the optimal engine efficiency of our model is different from the Curzon-Ahlborn efficiency, although they share the universal linear and quadratic coefficients at the near-equilibrium limit. We further confirm our results by taking an alternative approach in terms of the entropy production at hot and cold reservoirs.