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

From the thermodynamic equilibrium properties of a two-level system with variable energy-level gap ∆, and a careful distinction between the Gibbs relation dE = T dS + (E/∆) d∆ and the energy balance equation dE = δQ← − δW→, we infer some important aspects of the second law of thermodynamics and, contrary to a recent suggestion based on the analysis of an Otto-like thermodynamic cycle between two values of ∆ of a spin-1/2 system, we show that a quantum thermodynamic Carnot cycle, with the celebrated optimal efficiency 1− (Tlow/Thigh), is possible in principle with no need of an infinite number of infinitesimal processes, provided we cycle smoothly over at least three (in general four) values of ∆, and we change ∆ not only along the isoentropics, but also along the isotherms, e.g., by use of the recently suggested maser-laser tandem technique. We derive general bounds to the net-work to high-temperature-heat ratio for a Carnot cycle and for the ’inscribed’ Otto-like cycle, and represent these cycles on useful thermodynamic diagrams.