Optimal Performance of a Quantum Einstein Heat Engine with Ecological Function

We study the optimal performance of a quantum heat engine, in which the heat transfer between the working medium and the heat reservoirs takes place according to Einstein’s law of heat transfer, by using a trade-off objective function known as ecological function which represents a compromise between the power output and power lost due to entropy production in a heat engine. We assume that the constituent particles in the two-level system either obey Maxwell-Boltzmann (MB) or Bose-Einstein (BE) distributions. In each case, we obtain analytic expressions for the efficiency at maximum ecological function in two limiting cases of large contact time and short contact time. We also obtain analytic expressions on the lower and upper bounds on the efficiency at maximum ecological function for the above mentioned cases.