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arXiv:quant-ph/0703261 · Europhysics Letters (EPL)

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

Gian Paolo Beretta

Published Jul 1, 2012• 32 citations
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Abstract

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.

DOI
10.1209/0295-5075/99/20005

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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.
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.
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.
Abstract Recent theoretical and experimental studies in quantum heat engines show that, in the quasi-static regime, it is possible to have higher efficiency than the limit imposed by Carnot, provided that engineered reservoirs are used. The quasi-static regime, however, is a strong limitation to the operation of heat engines, since an infinitely long time is required to complete a cycle. In this paper we propose a two-level model as the working substance to perform a quantum Otto heat engine surrounded by a cold thermal reservoir and a squeezed hot thermal reservoir. Taking advantage of this model we show a striking achievement, that is to attain unity efficiency even at non-null power.
Enhancement of thermal efficiency of quantum heat engines has been discussed mainly in terms of quantumness of working substances. In the present study, we propose as another possibility to utilize a quantum-mechanical (reversible) feature of system-reservoir interaction, which reveals itself in the short-time regime as a non-Markovian effect on the reduced dynamics of the substance, before asymptotically approaching irreversible processes. For this purpose, we study a limit cycle of the quantum Otto engine comprising a working two-level system and two reservoirs, including the non-Markovian effect for the finite-time isochoric processes while the work-extracting processes are kept quantum adiabatic. Assuming that the system interacts weakly with the reservoirs consisting of infinite bosons, we find a parameter regime beyond the Carnot efficiency in which a positive work is extracted by including a non-Markovian effect called energy backflow. We also find that the interaction energy is finite and contributes to the positive work even for the weak interaction. The work diminishes if we include into the work the energy for detaching the working substance from reservoirs.
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.
We consider the finite-time operation of a quantum heat engine whose working substance is composed of a two-level atomic system. The engine cycle, consisting of two quantum adiabatic and two quantum isochoric (constant-frequency) processes and working between two heat reservoirs at temperatures Th{T}_{h}Th​ and Tc{T}_{c}Tc​(<(<(<Th){T}_{h})Th​), is a quantum version of the classical Otto cycle. By optimizing the power output with respect to two frequencies, we obtain the efficiency at maximum power output (EMP) and analyze numerically the effects of the times taken for two adiabatic and two isochoric processes on the EMP. In the absence of internally dissipative friction, we find that the EMP is bounded from the upper side by a function of the Carnot efficiency ${\ensuremath{\eta}}_{C}$, ${\ensuremath{\eta}}_{+}={\ensuremath{\eta}}_{C}^{2}/[{\ensuremath{\eta}}_{C}\ensuremath{-}(1\ensuremath{-}{\ensuremath{\eta}}_{C})\mathrm{ln}(1\ensuremath{-}{\ensuremath{\eta}}_{C})]$, with ${\ensuremath{\eta}}_{C}=1\ensuremath{-}{T}_{c}/{T}_{h}$. This analytic expression is confirmed by our exact numerical result and is identical to the one derived in an engine model based on a mesoscopic or macroscopic system. If the internal friction is included, we find that the EMP decreases as the friction coefficient increases.