Quantum thermodynamic Carnot and Otto-like cycles for a two-level system

Within the recent revival of interest in quantum heat engines between two thermal
reservoirs whereby the working substance is a two-level system, it has been suggested that the
celebrated Carnot heat-to-work conversion efficiency 1−(Tlow/Thigh) cannot be reached. Contrary
to this suggestion, we show that reaching the Carnot bound not only is not impossible and does
not require an infinite number of heat baths and infinitesimal processes, but it is also within reach
of the current experimental techniques. It is sufficient to cycle smoothly (slowly) over at least three
(in general four) values of the tunable energy level gap Δ of the system, by varying Δ not only along
the isoentropics, but also along the isotherms. This is possible by means of the recently suggested
maser-laser tandem technique. We base our proof on the general thermodynamic equilibrium
properties of a two-level system together with a careful distinction between the Gibbs relation
dE = T dS+(E/Δ)dΔ and the energy balance equation dE =dQ←
−dW→. We derive bounds
to the net-work to high-temperature-heat ratio (energy efficiency) for a Carnot cycle and for the
“inscribed” Otto-like cycle. By representing these cycles on useful thermodynamic diagrams, we
infer and confirm important aspects of the second law of thermodynamics.