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Physical Review E

Efficiency at maximum power of a heat engine working with a two-level atomic system

Rui Wang, Jianhui Wang, Jizhou He, Yongli Ma

Published Apr 23, 2013• 78 citations
Publisher

Abstract

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 ${T}_{h}$ and ${T}_{c}$$(<$${T}_{h})$, 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.

DOI
10.1103/physreve.87.042119

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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 {T}_{c}$$(<$${T}_{h}), 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 \ensuremathηC{\ensuremath{\eta}}_{C}\ensuremathηC​, \ensuremathη+=\ensuremathηC2/[\ensuremathηC\ensuremath−(1\ensuremath−\ensuremathηC)ln(1\ensuremath−\ensuremathηC)]{\ensuremath{\eta}}_{+}={\ensuremath{\eta}}_{C}^{2}/[{\ensuremath{\eta}}_{C}\ensuremath{-}(1\ensuremath{-}{\ensuremath{\eta}}_{C})\mathrm{ln}(1\ensuremath{-}{\ensuremath{\eta}}_{C})]\ensuremathη+​=\ensuremathηC2​/[\ensuremathηC​\ensuremath−(1\ensuremath−\ensuremathηC​)ln(1\ensuremath−\ensuremathηC​)], with \ensuremathηC=1\ensuremath−Tc/Th{\ensuremath{\eta}}_{C}=1\ensuremath{-}{T}_{c}/{T}_{h}\ensuremathηC​=1\ensuremath−Tc​/Th​. 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.
We consider the efficiency at maximum power of a quantum Otto engine, which uses a spin or a harmonic system as its working substance and works between two heat reservoirs at constant temperatures Th{T}_{h}Th​ and Tc{T}_{c}Tc​ (<Th)(<{T}_{h})(<Th​). Although the behavior of spin-1/21/21/2 system differs substantially from that of the harmonic system in that they obey two typical quantum statistics, the efficiencies at maximum power based on these two different kinds of quantum systems are bounded from the upper side by the same expression ${\ensuremath{\eta}}_{\mathrm{mp}}\ensuremath{\le}{\ensuremath{\eta}}_{+}\ensuremath{\equiv}{\ensuremath{\eta}}_{C}^{2}/[{\ensuremath{\eta}}_{C}\ensuremath{-}(1\ensuremath{-}{\ensuremath{\eta}}_{C})ln(1\ensuremath{-}{\ensuremath{\eta}}_{C})]$ with ${\ensuremath{\eta}}_{C}=1\ensuremath{-}{T}_{c}/{T}_{h}$ as the Carnot efficiency. This expression ${\ensuremath{\eta}}_{\mathrm{mp}}$ possesses the same universality of the CA efficiency ${\ensuremath{\eta}}_{\mathrm{CA}}=1\ensuremath{-}\sqrt{1\ensuremath{-}{\ensuremath{\eta}}_{C}}$ at small relative temperature difference. Within the context of irreversible thermodynamics, we calculate the Onsager coefficients and show that the value of ${\ensuremath{\eta}}_{\mathrm{CA}}$ is indeed the upper bound of EMP for an Otto engine working in the linear-response regime.
In this paper, we consider a model of two-level quantum heat engine to investigate the explicit analytic expression for the thermodynamics quantities in different condition under the finite-time operation. In this engine, the working substance is composed of a spin-half particles immersed in a magnetic field. The finite-time thermodynamic processes consisting of two quantum adiabatic and two quantum isothermal processes. This processes working between two heat reservoirs with an inverse temperatures β1β_{1}β1​ and β2β_{2}β2​ (<β1<β_{1}<β1​). In this processes, we obtain the work, heat, power and efficiency at maximum power output of the model. Our result of the efficiency at maximum power agree with the universal value in the first order of Carnot efficiency.
We study the efficiency at maximum power, ${\ensuremath{\eta}}_{m}$, of irreversible quantum Carnot engines (QCEs) that perform finite-time cycles between a hot and a cold reservoir at temperatures Th{T}_{h}Th​ and Tc{T}_{c}Tc​, respectively. For QCEs in the reversible limit (long cycle period, zero dissipation), ${\ensuremath{\eta}}_{m}$ becomes identical to the Carnot efficiency ${\ensuremath{\eta}}_{{}_{C}}=1\ensuremath{-}{T}_{c}/{T}_{h}$. For QCE cycles in which nonadiabatic dissipation and the time spent on two adiabats are included, the efficiency ${\ensuremath{\eta}}_{m}$ at maximum power output is bounded from above by ${\ensuremath{\eta}}_{{}_{C}}/(2\ensuremath{-}{\ensuremath{\eta}}_{{}_{C}})$ and from below by ${\ensuremath{\eta}}_{{}_{C}}/2$. In the case of symmetric dissipation, the Curzon-Ahlborn efficiency ${\ensuremath{\eta}}_{{}_{\mathit{CA}}}=1\ensuremath{-}\sqrt{{T}_{c}/{T}_{h}}$ is recovered under the condition that the time allocation between the adiabats and the contact time with the reservoir satisfy a certain relation.
Thermodynamic constraints impose a trade-off between power and efficiency in heat engines, preventing the simultaneous achievement of high power and high efficiency. For classical microscopic engines, explicit inequalities have been discovered, demonstrating the inherent inevitability of this power-efficiency trade-off. However, extensions of these results to quantum engines have so far been limited to cases of slow operation. In this study, we derive a power-efficiency trade-off relation for a paradigmatic quantum engine operating within a finite time, specifically the Otto cycle of a quantum harmonic oscillator. By utilizing a phase-space approach based on quasiprobability representations, we establish a universal trade-off relation applicable to arbitrary time-dependent protocols during the adiabatic processes. Our results reveal that the power of the quantum engine vanishes as the efficiency approaches the quantum-mechanical efficiency bound, which is stricter than the Carnot bound. Furthermore, we identify the conditions under which the upper bound is attained, which indicate maximum power is achieved when the generation of quantum coherence is reduced, and the difference in time durations of the isochoric processes increases. These findings are validated through numerical calculations, which confirm their applicability across various types of protocols for heat engine cycles.
We investigate the efficiency at maximum power (EMP) of irreversible quantum Carnot engines that perform finite-time cycles between two temperature tunable baths. The temperature form we adopt can be experimentally realized in squeezed baths in the high temperature limit, which makes our proposal of practical relevance. Focusing on low dissipation engines, we first generalize the pervious upper as well as lower bounds for the EMP to temperature tunable cases in which they are solely determined by a generalized Carnot limit. As an illustrative example, we then consider a minimal heat engine model with a two-level spin as the working medium. It mimics a low dissipation engine as confirmed by finite time thermodynamic optimization results. The so-obtained EMP, being constrained by the generalized bounds, is well described by a generalized Curzon- Ahlborn efficiency as consequences of a left/right symmetry for a rate constant and low dissipations. Intriguing features of this minimal heat engine under optimal power output are also demonstrated.
We generalize the quantum heat engine (QHE) model which was first proposed by Bender et al. [J. Phys. A 33, 4427 (2000)] to the case in which an ideal Fermi gas with an arbitrary number N of particles in a box trap is used as the working substance. Besides two quantum adiabatic processes, the engine model contains two isoenergetic processes, during which the particles are coupled to energy baths at a high constant energy E(h) and a low constant energy E(c), respectively. Directly employing the finite-time thermodynamics, we find that the power output is enhanced by increasing particle number N (or decreasing minimum trap size L(A)) for given L(A) (or N), without reduction in the efficiency. By use of global optimization, the efficiency at possible maximum power output (EPMP) is found to be universal and independent of any parameter contained in the engine model. For an engine model with any particle-number N, the efficiency at maximum power output (EMP) can be determined under the condition that it should be closest to the EPMP. Moreover, we extend the heat engine to a more general multilevel engine model with an arbitrary 1D power-law potential. Comparison between our engine model and the Carnot cycle shows that, under the same conditions, the efficiency η = 1 - E(c)/E(h) of the engine cycle is bounded from above the Carnot value η(c) =1 - T(c)/T(h).