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arXiv:1111.5077 · Physical Review E

Efficiency at maximum power output of quantum heat engines under finite-time operation

Jianhui Wang, Jizhou He, Zhaoqi Wu

Published Mar 29, 2012• 66 citations
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Abstract

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 ${T}_{h}$ and ${T}_{c}$, 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.

DOI
10.1103/physreve.85.031145

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We study the efficiency at maximum power, \ensuremathηm{\ensuremath{\eta}}_{m}\ensuremathηm​, 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), \ensuremathηm{\ensuremath{\eta}}_{m}\ensuremathηm​ becomes identical to the Carnot efficiency \ensuremathηC=1\ensuremath−Tc/Th{\ensuremath{\eta}}_{{}_{C}}=1\ensuremath{-}{T}_{c}/{T}_{h}\ensuremathηC​​=1\ensuremath−Tc​/Th​. For QCE cycles in which nonadiabatic dissipation and the time spent on two adiabats are included, the efficiency \ensuremathηm{\ensuremath{\eta}}_{m}\ensuremathηm​ at maximum power output is bounded from above by \ensuremathηC/(2\ensuremath−\ensuremathηC){\ensuremath{\eta}}_{{}_{C}}/(2\ensuremath{-}{\ensuremath{\eta}}_{{}_{C}})\ensuremathηC​​/(2\ensuremath−\ensuremathηC​​) and from below by \ensuremathηC/2{\ensuremath{\eta}}_{{}_{C}}/2\ensuremathηC​​/2. In the case of symmetric dissipation, the Curzon-Ahlborn efficiency \ensuremathηCA=1\ensuremath−Tc/Th{\ensuremath{\eta}}_{{}_{\mathit{CA}}}=1\ensuremath{-}\sqrt{{T}_{c}/{T}_{h}}\ensuremathηCA​​=1\ensuremath−Tc​/Th​​ is recovered under the condition that the time allocation between the adiabats and the contact time with the reservoir satisfy a certain relation.
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 consider a quantum endoreversible Carnot engine cycle and its inverse operation–Carnot refrigeration cycle, working between a hot bath of inverse temperature βh and a cold bath at inverse temperature βc. For the engine model, the hot bath is constructed to be squeezed, whereas for the refrigeration cycle, the cold bath is set to be squeezed. In the high-temperature limit, we analyze efficiency at maximum power and coefficient of performance at maximum figure of merit, revealing the effects of the times allocated to two thermal-contact and two adiabatic processes on the machine performance. We show that, when the total time spent along the two adiabatic processes is negligible, the efficiency at maximum power reaches its upper bound, which can be analytically expressed in terms of squeezing parameter r: ηana∗=1−sech[2r]1−ηC, with the Carnot efficiency ηC=1−βh/βc and the coefficient of performance at maximum figure of merit is bounded from the upper side by the analytical function: εana∗=1+εCsech[2r](1+εC)−εC−1, where εC=βh/(βc−βh).
We investigate the finite-power performance of quantum heat engines working in the linear response regime where the temperature gradient is small. The engine cycles with working substances of ideal harmonic systems consist of two heat transfer and two adiabatic processes, such as the Carnot cycle, Otto cycle, and Brayton cycle. By analyzing the optimal protocol under maximum power we derive the explicitly analytic expression for the irreversible entropy production, which becomes the low dissipation form in the long duration limit. Assuming the engine to be endoreversible, we derive the universal expression for the efficiency at maximum power, which agrees well with that obtained from the phenomenological heat transfer laws holding in the classical thermodynamics. Through appropriate identification of the thermodynamic fluxes and forces that a linear relation connects, we find that the quantum engines under consideration are tightly coupled, and the universality of efficiency at maximum power is confirmed at the linear order in the temperature gradient.
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.
Heat engines constitute the major building blocks of modern technologies. However, conventional heat engines with higher power yield lesser efficiency and vice versa and respect various power-efficiency trade-off relations. This is also assumed to be true for the engines operating in the quantum regime. Here we show that these relations are not fundamental. We introduce quantum heat engines that deliver maximum power with Carnot efficiency in the one-shot finite-size regime. These engines are composed of working systems with a finite number of quantum particles and are restricted to one-shot measurements. The engines operate in a one-step cycle by letting the working system simultaneously interact with hot and cold baths via semilocal thermal operations. By allowing quantum entanglement between its constituents and, thereby, a coherent transfer of heat from hot to cold baths, the engine implements the fastest possible reversible state transformation in each cycle, resulting in maximum power and Carnot efficiency. Finally, we propose a physically realizable engine using quantum optical systems.
We consider the finite-time performance of a quantum Otto engine working between a hot squeezed and a cold thermal bath at inverse temperatures ${\ensuremath{\beta}}_{h}$ and ${\ensuremath{\beta}}_{c}(>{\ensuremath{\beta}}_{h})$ with $({k}_{B}\ensuremath{\equiv}1)\ensuremath{\beta}=1/T$. We derive the analytical expressions for work, efficiency, power, and power fluctuations, in which the squeezing parameter is involved. By optimizing the power output with respect to two frequencies, we derive the efficiency at maximum power as ${\ensuremath{\eta}}_{mp}={({\ensuremath{\eta}}_{C}^{\mathrm{gen}})}^{2}/[{\ensuremath{\eta}}_{C}^{\mathrm{gen}}\ensuremath{-}(1\ensuremath{-}{\ensuremath{\eta}}_{C}^{\mathrm{gen}})ln(1\ensuremath{-}{\ensuremath{\eta}}_{C}^{\mathrm{gen}})]$, where the generalized Carnot efficiency ${\ensuremath{\eta}}_{C}^{\mathrm{gen}}$ in the high-temperature or small squeezing limit simplifies to an analytic function of squeezing parameter $\ensuremath{\gamma}$: ${\ensuremath{\eta}}_{C}^{\mathrm{gen}}=1\ensuremath{-}{\ensuremath{\beta}}_{h}/[{\ensuremath{\beta}}_{c}cosh(2\ensuremath{\gamma})]$. Within the context of irreversible thermodynamics, we demonstrate that the expression of efficiency at maximum power satisfies a general form derived from nonlinear steady state heat engines. We show that, the power fluctuations are considerably increased, although the engine efficiency is enhanced by squeezing.