Optimization of a relativistic quantum mechanical engine
  We present an optimal analysis for a quantum mechanical engine working between two energy baths within the framework of relativistic quantum mechanics, adopting a first-order correction. This quantum mechanical engine, with the direct energy leakage between the energy baths, consists of two adiabatic and two isoenergetic processes and uses a three-level system of two non-interacting fermions as its working substance. Assuming that the potential wall moves at a finite speed, we derive the expression of power output and, in particular, reproduce the expression for the efficiency at maximum power. 
Francisco J. Pe\~na, Michel Ferr\'e, Pedro Orellana Ren\'e G. Rojas   and P. Vargas
cond-mat.stat-mech