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

Beretta, G P

Published Mar 28, 2007• 18 citations
OpenAlex

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 $dE = T dS + (E/\\Delta) d\\Delta$ 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.

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From the thermodynamic equilibrium properties of a two-level system with variable energy-level gap Delta\\DeltaDelta, and a careful distinction between the Gibbs relation dE=TdS+(E/Delta)dDeltadE = T dS + (E/\\Delta) d\\DeltadE=TdS+(E/Delta)dDelta 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\\DeltaDelta of a spin-1/2 system, we show that a quantum thermodynamic Carnot cycle, with the celebrated optimal efficiency 1−(Tlow/Thigh)1 - (T_{low}/T_{high})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 Delta\\DeltaDelta, and we change Delta\\DeltaDelta 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.
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.
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.
In (1) a thermodynamical-statistical mechanical approach is presented for a two energy level quantum system for a particle in a box with infinite potential walls. It is shown that classical statistical mechanical efficiency of a Carnot cycle 1- Tc/Th holds for this quantum system. In this note we argue that the Carnot cycle efficiency holds in general if one uses delta E = delta Q - delta Work with delta Q/T = delta S if E and S are state variables as they are. One does not need to perform the two energy level specific integrals presented in (1) because the Carnot efficiency result is guaranteed for any number of energy levels.
The quantum analog of Carnot cycles in few-particle systems consists of two quantum adiabatic steps and two isothermal steps. This construction is formally justified by use of a minimum work principle. It is then shown, using minimal assumptions of work or heat in nanoscale systems, that the heat-to-work efficiency of such quantum heat engine cycles can be further optimized via two conditions regarding the expectation value of some generalized force operators evaluated at equilibrium states. In general the optimized efficiency is system specific, lower than the Carnot efficiency, and dependent upon both temperatures of the cold and hot reservoirs. Simple computational examples are used to illustrate our theory. The results should be an important guide towards the design of favorable working conditions of a realistic quantum heat engine.