Unified trade-off optimization of quantum Otto heat engines with squeezed thermal reservoirs
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Unified trade‑off optimization of quantum Otto heat engines with squeezed thermal reservoirs Yanchao Zhang1 · Juncheng Guo2 · Jincan Chen3 Received: 16 February 2020 / Accepted: 17 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We consider a quantum Otto heat engine cycle operating between two squeezed thermal reservoirs that are characterized by a temperature as well as additional parameters that quantify the degree of squeezing. The optimal efficiency at the unified trade-off optimization criterion representing a compromise between energy benefits and losses for a quantum Otto heat engine is systematically investigated. The analytical expressions for the optimal efficiency are determined in the limit adiabatic and nonadiabatic processes as well as in the high- and low-temperature regimes, respectively. The general unified trade-off efficiency is given as a nonequilibrium efficiency that extends the standard unified trade-off efficiency to a more general nonequilibrium condition. Keywords Quantum thermodynamics · Quantum otto cycle · Quantum heat engine · Unified trade-off efficiency · Squeezed thermal reservoir
1 Introduction Heat engines as important energy conversion devices that convert heat into useful mechanical work are receiving special attention in thermodynamics, engineering, and nanotechnologies. Standard heat engines are usually assumed to be in contact with two equilibrium thermal reservoirs with temperatures Th and Tc (Tc 𝜔c ) while the system is isolated. During the isochoric processes, the system is alternatingly coupled to two squeezed thermal reservoirs, which
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Fig.1 Energy-frequency diagram of a quantum Otto cycle
/ are characterized by an inverse temperature 𝛽i = 1 (kB Ti )(i = h, c) and squeezed parameter ri , where kB is the Boltzmann constant and Ti is the corresponding temperature. We first consider that the cycle starts from a stationary squeezed thermal state A(𝜔c , 𝛽c , rc ) that is prepared by coupling the system to a squeezed thermal reservoir at inverse temperature 𝛽c and squeezing parameter rc . As a result, the mean energy at the initial state A(𝜔c , 𝛽c , rc ) is given by [26] � � � � ℏ𝛽c 𝜔c ℏ𝜔c ΔH rc , coth ⟨H⟩A = (3) 2 2 � � �−1 � � � � where ΔH rc = 1 + 2 + 1 ⟨nc ⟩ sinh2 rc , and ⟨nc ⟩ = exp(𝛽c ℏ𝜔c ) − 1 is the thermal occupation number of the squeezed thermal reservoir. Thus, the model of the quantum Otto heat engine with squeezed thermal reservoirs can be described by the following four processes: (1) Isentropic compression process: In this process, the system is isolated and the system states from A(𝜔c , 𝛽c , rc ) to B(𝜔h , 𝛽c , rc , Q∗c ) and the frequency increases from 𝜔c to 𝜔h . This transformation is unitary and the von Neumann entropy is constant. The mean energy at the state B(𝜔h , 𝛽c , rc , Q∗c ) is given by � � � � ℏ𝛽c 𝜔c ℏ𝜔h ΔH rc Q∗c , coth ⟨H⟩B = (4) 2 2 where Q∗c (and Q∗h later) is a dimensionless adiabaticity parameter that characterizes the degree of adiabaticity of the i
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