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

Abstract

From the thermodynamic equilibrium properties of a two-level system with variable energy-level gap $\Delta$, and a careful distinction between the Gibbs relation $dE=TdS+(E/\Delta )d\Delta$ and the energy balance equation $dE=\delta Q^{\leftarrow }-\delta W^{\to }$, 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 $\Delta$ of a spin-1/2 system, we show that a quantum thermodynamic Carnot cycle, with the celebrated optimal efficiency $1-(T_{low}/T_{high})$, 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 $\Delta$, and we change $\Delta$ 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.

RevTex4, 4 pages, 1 figure

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