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arXiv:2003.07608 · Physical Review Research

Optimal operation of a three-level quantum heat engine and universal nature of efficiency

Varinder Singh

Published Nov 5, 2020• 44 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevresearch.2.043187

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arXiv:2106.01193•Physical Review Research•Feb 25, 2022•29 citations
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.
Abstract We work on a comprehensive theoretical investigation of a three-level quantum heat engine operating with engineered squeezed thermal reservoirs. Unlike conventional quantum heat engines that use thermal reservoirs, we demonstrate that squeezed reservoirs can significantly enhance the engine's performance by increasing both efficiency and output power. Using a rigorous open-quantum-system approach, we derive the full Lindblad master equation for a three-level system interacting with hot and cold squeezed thermal baths. Through a detailed analysis of the steady-state thermodynamics, we obtain explicit expressions for heat currents, work output, and efficiency. Our results reveal that moderate squeezing parameters (r≈0.5r \approx 0.5r≈0.5) combined with optimal squeezing phase (ϕ=π/2\phi = \pi/2ϕ=π/2) enhance the engine efficiency by up to 40\% compared to the classical Carnot limit for the same temperature ratio. The power output shows a non-monotonic behavior with squeezing, exhibiting an optimal squeezing parameter that maximizes performance. We identify three distinct operational regimes: (i) heat engine regime, (ii) refrigerator regime, and (iii) heat pump regime, which can be tuned by adjusting the squeezing phase. The efficiency at maximum power follows a universal scaling ηMP≈ηCarnot/2+Δη(r)\eta_{MP} \approx \eta_{Carnot}/2 + \Delta\eta(r)ηMP​≈ηCarnot​/2+Δη(r), where Δη(r)\Delta\eta(r)Δη(r) represents the squeezing-induced enhancement. Our findings provide fundamental insights into the role of non-thermal reservoirs in quantum thermodynamics and offer practical guidelines for designing high-performance quantum heat engines using squeezed light and engineered baths.
Abstract We present the quantum Otto machine under different optimization criterion when function either as a heat engine or a refrigerator. We examine the optimal performance of the heat engine and refrigerator depending on their efficiency, output power and maximum entropy production. For heat engine case, we obtain the expression for the upper and lower bounds efficiencies at maximum power and maximum ecological function. The optimal ecological performance is laying between the maximum efficiency and the maximum power output. On the other hand, the quantum Otto refrigerator coefficient of performance is optimized for three different criterion – cooling power, χ criterion(product of performance and power) and ecological function. We further study the dimen-sionless power loss to the cold reservoir when the machine is operating as a heat engine as well as its counterpart for the refrigerator case. We find that the maximum operation of the heat engine (refrigerator) cycle is when optimized with respect to hot (cold) reservoir frequency.
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 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 study the optimal performance of a three-level quantum refrigerator using two different objective functions: cooling power and χ function. For both cases, we obtain general expressions for the coefficient of performance (COP) and derive its well-known lower and upper bounds for the limiting cases when the ratio of system-bath coupling constants at the hot and cold contacts approaches infinity and zero, respectively. We also show that the cooling power can be maximized with respect to one control frequency, while χ function can be maximized globally with respect to two control frequencies. Additionally, we show that in the low-temperature regime, our model of refrigerator can be mapped to Feynman's ratchet and pawl model, a classical mesoscopic heat engine. In the parameter regime where both cooling power and χ function can be maximized, we compare the cooling power of the quantum refrigerator at maximum χ function with the maximum cooling power.
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.