  The quantum analog of Carnot cycles in few-particle systems consists of two
quantum adiabatic steps and two isothermal steps. This construction is formally
justified by use of a minimum work principle. It is then shown, without relying
on any microscopic interpretations of work or heat, that the heat-to-work
efficiency of the quantum Carnot cycle thus constructed may be further
optimized, provided that two conditions regarding the expectation value of some
generalized force operators evaluated at equilibrium states are satisfied. In
general the optimized efficiency is system-specific, lower than the Carnot
efficiency, and dependent upon both temperatures of the cold and hot
reservoirs. Simple computational examples are used to illustrate our theory.
The results should be an important guide towards the design of favorable
working conditions of a realistic quantum heat engine.
