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Acta Physica Sinica

Performance analysis of a quantum heat engine working with a particle in a one-dimensional harmonic trap

Wang Jian-Hui, Xiong Shuang-Quan, He Ji-Zhou, Jiangtao Liu

Published Jan 1, 2012• 6 citations
Publisher

Abstract

A heat engine model of a sing particle confined in a one-dimensional harmonic trap is established. The model, as a quantum-mechanical analog of the Carnot engine, does work to outside when both the width of the potential and the quantum state change very slowly. On the assumption that the width of the potential moves at a very slow speed and the heat leakage is included, we derive the general expressions for several performance parameters, such as work, power output, efficiency, etc. We find the optimally operating regions and the optimal values of performance parameters of the heat engine cycle.

DOI
10.7498/aps.61.080509

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A heat engine model of a sing particle confined in a one-dimensional harmonic trap is established. The model, as a quantum-mechanical analog of the Carnot engine, does work to outside when both the width of the potential and the quantum state change very slowly. On the assumption that the width of the potential moves at a very slow speed and the heat leakage is included, we derive the general expressions for several performance parameters, such as work, power output, efficiency, etc. We find the optimally operating regions and the optimal values of performance parameters of the heat engine cycle.
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).
The cycle model of a regenerative quantum heat engine working with many noninteracting harmonic oscillators is established. The cycle consists of two isothermal and two constant-frequency processes. The performance of the cycle is investigated, based on the quantum master equation and semigroup approach. The inherent regenerative losses in the two constant-frequency processes are calculated. The expressions of several important performance parameters such as the efficiency, power output, and rate of the entropy production are derived for several interesting cases. Especially, the optimal performance of the cycle in high-temperature limit is discussed in detail. The maximum power output and the corresponding parameters are calculated. The optimal region of the efficiency and the optimal ranges of the temperatures of the working substance in the two isothermal processes are determined.
A new model of a quantum heat engine cycle is established in which the working substance consists of many non-interacting spin-1/2 systems.Performance characteristics of the cycle and time evolution of the spin angular momentum are investigated,based on the quantum master equation and a semi-group approach.The general expressions of efficiency and power output are derived.Further,at high temperatures the optimal relations of efficiency and power output are studied in detail.Optimal performance of heat engine cycles is analyzed from a microscopic model.
The unavoidable irreversible loss of power in a heat engine is found to be of quantum origin. Following thermodynamic tradition, a model quantum heat engine operating in an Otto cycle is analysed, where the working medium is composed of an ensemble of harmonic oscillators and changes in volume correspond to changes in the curvature of the potential well. Equations of motion for quantum observables are derived for the complete cycle of operation. These observables are sufficient to determine the state of the system and with it all thermodynamical variables. Once the external controls are set, the engine settles to a limit cycle. Conditions for optimal work, power and entropy production are derived. At high temperatures and quasistatic operating conditions, the efficiency at maximum power coincides with the endoreversible result . The optimal compression ratio varies from in the quasistatic limit where the irreversibility is dominated by heat conductance to in the sudden limit when the irreversibility is dominated by friction. When the engine deviates from adiabatic conditions, the performance is subject to friction. The origin of this friction can be traced to the noncommutability of the kinetic and potential energy of the working medium.
It was reported that, if and only if the specific heat, correlation length, and dynamical exponents α,να, να,ν and zzz, fulfill the condition α−zν>0α-zν>0α−zν>0, the phase transitions can enable a quantum heat engine to approach Carnot efficiency at finite power. We start our analysis via a different approach in which the effects of interaction and fluctuations on the Hamiltonian of a trapped dilute Bose gas belonging to the same universality as XYXYXY model. Based on models of quantum Otto heat engines, we find the general expression of the efficiency which includes the correction due to interaction and fluctuations at the critical point, and show that, near the Bose-Einstein-condensation point with α−zν<0α-zν<0α−zν<0, energy fluctuations could enable attaintment of the Carnot efficiency with nonvanishing power. Such quantum heat engines can also be realized by changing the shape of the trap confining the ideal and weakly interacting Bose gas during the adiabatic processes of the cycle. These quantum heat engines working with the trapped Bose gases, which are based on techniques of cooling Bose condensates and could be realizable at present technology.
In their work [J. Phys. A 33, 4427 (2000)], Bender, Brody, and Meister have shown by employing a two-state model of a particle confined in the one-dimensional infinite potential well that it is possible to construct a quantum-mechanical analog of the Carnot engine through changes of both the width of the well and the quantum state in a specific manner. Here, a discussion is developed about realizing the maximum power of such an engine, where the width of the well moves at low but finite speed. The efficiency of the engine at the maximum power output is found to be universal independently of any of the parameters contained in the model.