Optimization of asymmetric quantum Otto engine cycles

We consider the optimization of the work output and fluctuations of a finite-time quantum Otto heat engine cycle consisting of compression and expansion work strokes of unequal duration. The asymmetry of the cycle is characterized by a parameter r_{u} giving the ratio of the times for the compression and expansion work strokes. For such an asymmetric quantum Otto engine cycle, with working substance chosen as a harmonic oscillator or a two-level system, we find that the optimal values of r_{u} maximizing the work output and the reliability (defined as the ratio of average work output to its standard deviation) shows discontinuities as a function of the total time taken for the cycle. Moreover we identify cycles of some specific duration where both the work output and the reliability take their largest values for the same value of the asymmetry parameter r_{u}.