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arXiv:quant-ph/0611275 · Physical Review E

Quantum thermodynamic cycles and quantum heat engines

H. T. Quan, Yu-xi Liu, C. P. Sun, Franco Nori

Published Sep 7, 2007• 787 citations
arXivHTMLPDF

Abstract

In order to describe quantum heat engines, here we systematically study isothermal and isochoric processes for quantum thermodynamic cycles. Based on these results the quantum versions of both the Carnot heat engine and the Otto heat engine are defined without ambiguities. We also study the properties of quantum Carnot and Otto heat engines in comparison with their classical counterparts. Relations and mappings between these two quantum heat engines are also investigated by considering their respective quantum thermodynamic processes. In addition, we discuss the role of Maxwell's demon in quantum thermodynamic cycles. We find that there is no violation of the second law, even in the existence of such a demon, when the demon is included correctly as part of the working substance of the heat engine.

DOI
10.1103/physreve.76.031105

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In order to describe quantum heat engines, here we systematically study isothermal and isochoric processes for quantum thermodynamic cycles. Based on these results the quantum versions of both the Carnot heat engine and the Otto heat engine are defined without ambiguities. We also study the properties of quantum Carnot and Otto heat engines in comparison with their classical counterparts. Relations and mappings between these two quantum heat engines are also investigated by considering their respective quantum thermodynamic processes. In addition, we discuss the role of Maxwell's demon in quantum thermodynamic cycles. We find that there is no violation of the second law, even in the existence of such a demon, when the demon is included correctly as part of the working substance of the heat engine.
Quantum Otto and Carnot engines have recently been receiving attention due to their ability to achieve high efficiencies and powers based on the laws of quantum mechanics. This paper discusses the theory, progress, and possible applications of quantum Otto and Carnot engines, such as energy production, cooling, and nanoscale technologies. In particular, we investigate a two-spin Heisenberg system that works as a substance in quantum Otto and Carnot cycles while exposed to an external magnetic field with both Dzyaloshinsky-Moriya and dipole-dipole interactions. The four stages of engine cycles are subject to analysis with respect to the heat exchanges that occur between the hot and cold reservoirs, alongside the work done during each stage. The operating conditions of the heat engine, refrigerator, thermal accelerator, and heater are all achieved. Moreover, our results demonstrate that the laws of thermodynamics are strictly upheld and the Carnot cycle produces more useful work than that of the Otto cycle.
From an entropy-based formulation of the first law of thermodynamics in the quantum regime, we investigate the performance of Otto-like and Carnot-like engines for a single-qubit working medium. Within this framework, the first law includes an additional contribution-coherence work-that quantifies the energetic cost of deviating the quantum trajectory from its natural unitary evolution. We focus on the efficiency of the heat-to-coherence work conversion and show that the Carnot cycle achieves the classical Carnot efficiency, while the performance of the Otto cycle is upper-bounded by the Carnot efficiency corresponding to the extreme temperatures of the cycle. We identify entropy generation during the isochoric stages as the key source of irreversibility limiting the Otto cycle's efficiency.
With a class of quantum heat engines which consists of two-energy-eigenstate systems undergoing, respectively, quantum adiabatic processes and energy exchanges with heat baths at different stages of a cycle, we are able to clarify some important aspects of the second law of thermodynamics. The quantum heat engines also offer a practical way, as an alternative to Szilard's engine, to physically realize Maxwell's demon. While respecting the second law on the average, they are also capable of extracting more work from the heat baths than is otherwise possible in thermal equilibrium.
We study the physical mechanism of Maxwell's demon (MD), which helps do extra work in thermodynamic cycles with the heat engine. This is exemplified with one molecule confined in an infinitely deep square potential with a movable solid wall. The MD is modeled as a two-level system (TLS) for measuring and controlling the motion of the molecule. The processes in the cycle are described in a quantum fashion. It is discovered that a MD with quantum coherence or one at a temperature lower than the molecule's heat bath can enhance the ability of the whole working substance, formed by the heat engine plus the MD, to do work outside. This observation reveals that the essential role of the MD is to drive the whole working substance off equilibrium, or equivalently, to work between two heat baths with different effective temperatures. The elaborate studies with this model explicitly reveal the effect of finite size off the classical limit or thermodynamic limit, which contradicts common sense on a Szilard heat engine (SHE). The quantum SHE's efficiency is evaluated in detail to prove the validity of the second law of thermodynamics.
After a brief historical perspective, we introduce the key notions of work and heat for quantum systems, to then apply them to quantum engines operating on quantum Otto and Carnot cycles. The irreversible and dissipative character of the quantum Otto cycle is briefly analyzed, contrasting with the energetic optimality of the quantum Carnot cycle. The central question of quantum effects is also addressed and illustrated with several examples. Finally, the last part strives to explain the role that quantum thermodynamics plays for quantum applications and quantum technologies, particularly in relation to energy optimization and the trade-off between performances and energy costs.
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 − ( T low / T high ) 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 d E = T d S + ( E /Δ) dΔ and the energy balance equation . 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.