A quantum-mechanical heat engine operating in finite time. A model consisting of spin-1/2 systems as the working fluid

Abstract

The finite-time operation of a quantum-mechanical heat engine with a working fluid consisting of many noninteracting spin-1/2 systems is considered. The engine is driven by an external, time-dependent and nonrotating magnetic field. The cycle of operation consists of two adiabats and two isotherms. The analysis is based on the time derivatives of the first and second laws of thermodynamics. Explicit relations linking quantum observables to thermodynamic quantities are developed. The irreversible operation of this engine is studied in three cases: (1) The sudden limit, where the performance is found to be the same as that of the spin analog of the Otto cycle. This case provides the lower bound of efficiency. (2) The step-cycle operation scheme. Here, the optimization of power is carried out in the high-temperature limit (the ‘‘classical’’ limit). The results obtained are similar to those of Andresen et al. [Phys. Rev. A 15, 2086 (1977)]. (3) The Curzon–Ahlborn operation scheme. The semigroup approach is used to model the dynamics. Then the power production is optimized. All the results obtained for Newtonian engines operating by the same scheme, such as the Curzon–Ahlborn efficiency, apply in the high-temperature limit. These results are obtained without the additional assumption of proximity to thermal equilibrium, implicitly implied by the use of Newtonian heat conduction in the original derivation. It seems that the results of the Curzon–Ahlborn analysis are always obtained in the high-temperature limit, irrespective of the details of the model. The performance beyond the classical limit is optimized numerically. The classical approximation is found to be valid for most of the spin-polarization range. The deviations from the classical limit depend heavily upon the specific nature of both the working fluid and the heat baths and exhibit great diversity and complexity.