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arXiv:0709.4125 · Physical Review E

Work extremum principle: Structure and function of quantum heat engines

ARMEN E. ALLAHVERDYAN, Ramandeep S. Johal, G. Mahler

Published Apr 18, 2008• 149 citations
arXivHTMLPDF

Abstract

We consider a class of quantum heat engines consisting of two subsystems interacting with a work-source and coupled to two separate baths at different temperatures Th>Tc. The purpose of the engine is to extract work due to the temperature difference. Its dynamics is not restricted to the near equilibrium regime. The engine structure is determined by maximizing the extracted work under various constraints. When this maximization is carried out at finite power, the engine dynamics is described by well-defined temperatures and satisfies the local version of the second law. In addition, its efficiency is bounded from below by the Curzon-Ahlborn value 1-radical Tc/Th and from above by the Carnot value 1-(Tc/Th). The latter is reached-at finite power--for a macroscopic engine, while the former is achieved in the equilibrium limit Th-->Tc . The efficiency that maximizes the power is strictly larger than the Curzon-Ahloborn value. When the work is maximized at a zero power, even a small (few-level) engine extracts work right at the Carnot efficiency.

DOI
10.1103/physreve.77.041118

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We consider a class of quantum heat engines consisting of two subsystems interacting with a work-source and coupled to two separate baths at different temperatures Th>Tc. The purpose of the engine is to extract work due to the temperature difference. Its dynamics is not restricted to the near equilibrium regime. The engine structure is determined by maximizing the extracted work under various constraints. When this maximization is carried out at finite power, the engine dynamics is described by well-defined temperatures and satisfies the local version of the second law. In addition, its efficiency is bounded from below by the Curzon-Ahlborn value 1-radical Tc/Th and from above by the Carnot value 1-(Tc/Th). The latter is reached-at finite power--for a macroscopic engine, while the former is achieved in the equilibrium limit Th-->Tc . The efficiency that maximizes the power is strictly larger than the Curzon-Ahloborn value. When the work is maximized at a zero power, even a small (few-level) engine extracts work right at the 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.
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.
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.
Sadi Carnot's theorem regarding the maximum efficiency of heat engines is considered to be of fundamental importance in thermodynamics. This theorem famously states that the maximum efficiency depends only on the temperature of the heat baths used by the engine, but not on the specific structure of baths. Here, we show that when the heat baths are finite in size, and when the engine operates in the quantum nanoregime, a revision to this statement is required. We show that one may still achieve the Carnot efficiency, when certain conditions on the bath structure are satisfied; however if that is not the case, then the maximum achievable efficiency can reduce to a value which is strictly less than Carnot. We derive the maximum efficiency for the case when one of the baths is composed of qubits. Furthermore, we show that the maximum efficiency is determined by either the standard second law of thermodynamics, analogously to the macroscopic case, or by the non increase of the max relative entropy, which is a quantity previously associated with the single shot regime in many quantum protocols. This relative entropic quantity emerges as a consequence of additional constraints, called generalized free energies, that govern thermodynamical transitions in the nanoregime. Our findings imply that in order to maximize efficiency, further considerations in choosing bath Hamiltonians should be made, when explicitly constructing quantum heat engines in the future. This understanding of thermodynamics has implications for nanoscale engineering aiming to construct small thermal machines.
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.
We present a detailed study of a three-level quantum heat engine operating at maximum efficient power function, a trade-off objective function defined by the product of the efficiency and power output of the engine.The efficient power function represents a trade-off between the power output and efficiency of a heat engine.Engines working in the maximum efficient power regime operate at finite power, with finite efficiency lying in between the regimes of maximum power and maximum efficiency (Carnot efficiency).First, for near-equilibrium conditions, we find a general expression for the efficiency and establish the universal nature of efficiency at maximum power and maximum efficient power.Then in the high-temperature limit, optimizing with respect to one parameter while constraining the other one, we obtain the lower and upper bounds on the efficiency for both strong as well as weak matter-field coupling conditions.Except for the weak matter-field coupling condition, the obtained bounds on the efficiency exactly match with the bounds already known for some models of classical heat engines.Further for weak matter-field coupling, we derive some new bounds on the efficiency of the engine which lie beyond the range covered by bounds obtained for strong matter-field coupling.We conclude by comparing the performance of our three-level quantum heat engine in maximum power and maximum efficient power regimes and show that the engine operating at maximum efficient power produces at least 88.89% of the maximum power output while considerably reducing the power loss due to entropy production.Finally, to complete our analysis, we review the experimental setup of a recently realized effective three-level quantum engine.