相互作用量子系统热力学循环性能研究
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摘要
近年来受量子信息研究的带动以及实验技术进步的影响,量子力学和统计物理学的一些基本问题受到研究者的关注。在众多学者的努力下,基于量子力学的热力学理论体系正在逐步完善,我们称之为量子热力学。回顾热力学发展的历史,可以看到研究热机的性能在经典热力学的建立和发展过程中发挥了重要的作用。同样地,量子热机是人们研究量子热力学问题的平台,不断改进和完善各种量子热机的过程就是量子热力学理论不断完善的过程。量子热机是指以量子系统(如谐振子系统、自旋系统,无限深势阱中的微观粒子,及量子比特等)为工质并能对外做功的热力学循环模型。由于工质的量子性,量子热机与经典热机相比有很多独特的性质,从而量子热机近年成为一个备受关注的研究热点,人们试图发现一些有别于经典热机的新现象。
本文主要基于各种量子热力学循环模型,研究与量子热力学有关的基本问题,分析不同量子系统为工质的热力学循环过程的能量变化及其效率等并进行优化。从现代热力学的发展和研究现状可以知道,量子热力学从最初研究无相互作用的量子系统的热力学循环发展到研究有各种相互作用,及考虑量子摩擦等不可逆因素影响的量子热力学循环,这是一个值得研究的发展方向。
在第二章,我们构建了一个比较特殊的不可逆谐振子量子热机循环模型,由两个等温,一个绝热和一个等频率过程构成。基于量子主方程和半群方法,我们具体分析了谐振子系统的重要的热力学性能参数,在高温极限下推导了诸如吸、放热,输出功,输出功率、循环效率及熵产率等的表达式。通过数值分析,我们绘制了一系列性能特征曲线,如输出功率和效率的优化曲线,生态学函数和效率的优化曲线,输出功率和工质热温比在最大生态学函数时的关系曲线等。另外,还对在最大生态学函数和最大输出功率下的循环效率分别进行了比较。
在第三章,我们研究了有量子摩擦情况下的不可逆量子奥托制冷循环的性能。基于海森堡量子主方程,我们推导了谐振子系统热力学可观测量的运动方程,并且绘制了制冷循环的模拟图。数值分析了平均摩擦功率、制冷率、制冷系数和输入功率与绝热过程时间的关系。进一步分析了热传导和等频率过程时间对循环性能参数的影响。
在第四章,我们研究了两个有XY相互作用的qubit为工质的四能级量子纠缠热机的性能。这是个有趣的量子奥托热机循环,我们假设其绝热过程中保持量子相互作用常数不变而改变外磁场。根据量子热力学的相关理论我们分析了其做功和吸放热情况,并对最大效率和正功条件进行了详细分析,同时通过数值分析绘制了各热力学参量与热纠缠之间的关系曲线。我们发现量子系统间的纠缠作用对热机性能影响很大,量子热机可能在更宽松的条件下对外做功。
在第五章,我们延续了对量子纠缠作用的研究。基于二量子比特一维各向同性的自旋1/2的海森堡Hxx链模型我们构建了两个qubit量子纠缠系统为工质的四能级制冷循环,分析了在循环中系统与外界交换的热量、输入功、制冷率等热力学参数与量子纠缠之间的关系。我们发现了一些与量子纠缠热机不一样的性质,这可以使我们更全面的了解量子纠缠作用对热力学循环的影响。
在第六章,我们构建了工作在单模场中的两个相互作用的原子构成的四能级热机。通过对这个热机模型的研究,进一步阐述量子系统间的相互作用对热机性能的影响,我们分析了热机正功条件,推导得到了比卡诺效率更低的效率上限,并从局域分析的角度深入探讨了相互作用常数对热流方向的影响。
第七章对本论文进行了总结,并列出了量子热力学理论需要进一步深入研究的课题。本论文的研究结果有助于深入了解量子热力学理论,也为开展与此相关的实验工作提供有益的参考。
In recent years, with rapid developments of quantum information science and the experimental technology, some fundamental issues in quantum mechanics and statistical mechanics have attracted great interest. Under many researchers'effort, the thermodynamic theory, which is based on the quantum mechanics and called quantum thermodynamics, has been discussed in a series of papers. In reviewing the history of thermodynamics, one can find that the study of the heat engine played an important role in the development of classical thermodynamics. Similarly, the quantum heat engines serve as a platform for studying fundamental topics of the quantum thermodynamics. Quantum heat engine produce work using quantum system as the working substance, which includes the spin system, harmonic oscillator system, the particles in the infinite potential trap and the qubit system. There are many differences between the quantum heat engine and the classical counterparts because of the nature of the quantum matter. Therefore, the quantum heat engine become the research field of great interest in recent years, aiming to find some novel phenomenon and interesting results.
In this dissertation, the main task is to analyze and optimize the performance of the quantum cycle. Through studying various quantum thermodynamic cycles, we inverstigate the fundamental issue about the quantum thermodynamics.
In the chapter Ⅱ, a new cycle model of an irreversible quantum harmonic heat engine is established, which is composed of two isothermal processes, an adiabatic process and a constant frequency process. Based on the quantum master equation and semi-group approach, the general optimal performance characteristics of the cycle are investigated. Expressions for several important performance parameters such as the efficiency, the power output and the ecological function are derived in high temperature limit. By using numerical calculation, a set of performance characteristic curves is plotted, such as the optimal curves between the power output and efficiency, the ecological function and efficiency, and so on. The optimal regions of the efficiency, power output and temperature ratio of the working substance at the condition of the maximum ecological function are determined. Moreover, a comparison of the corresponding efficiencies at the maximum ecological function and power output is also presented.
In the chapter III, we discussed the performance characteristics of the irreverable quantum Otto refrigeration cycle with intrinsic quantum friction. An irreversible quantum Otto refrigeration cycle working with harmonic systems is established. Base on Heisenberg quantum master equation, the equations of motion for the set of harmonic systems the thermodynamic observables are derived. The simulated diagrams of the quantum Otto refrigeration cycle are plotted. The relationship between average power of friction, cooling rate, power input, and the time of adiabatic process is analyzed by using numerical calculation. Moreover, the influence of the heat conductance and the time of iso-frequency process on the performance of the cycle is discussed.
In chapter Ⅳ, We study the performance characteristics of the four level entangled quantum heat engine, whose work substance is composed of the two-qubit Heisenberg XY model. Based on the theory of the quantum thermodynamics, we inverstigate the several thermodynamic quantities such as the heat transferred, the work, the efficiency and the condition of the positive work. Moreover, relationship between the thermodynamic quantities and the concurrence of the quantum entanglement is analyzed by using graph. We found that the requirements of an entangled quantum heat engine are looser than that of a non-entangled quantum heat engine.
In chapter Ⅴ, we continued to study the effect of the quantum entanglement in the quantum thermodynamics. Based on a two-qubit Heisenberg XX model, a four-level entangled quantum refrigeration cycle is constructed. We analysed the relationship between several thermodynamic quantities and the concurrence of the quantum entanglement. We also found some novelty performance characteristics which is different with the entangled quantum heat engine, and it is very useful for people to understand the entangle quantum thermodynamic cycle.
In chapter Ⅵ, we set up a four-level heat engine model which consist of two atoms resonantly interact with a quantized electromagnetic field inside a cavity, and the dipole-dipole interaction of the atoms cannot be neglected. Here, we investigated the influence of the interactiong working medium on the heat engine performance, and analysed the positive work conditions in different cases. An upper bound for the efficiency is derived, which is smaller than the carnot efficieny. Additionally, the local description of each atom are discussed, this study further shows that the interaction between two atoms have important effects on the heat flow of the heat engine.
The research topics about quantum thermodynamic theory, which deserve a deeper study for the thermodynamic of the interaction quantum system, are listed in the chapter VII.
The results in this dissertation allow to understand well the quantum thermodynamics, and thus provide the theoretical guidelines for the future experiments including the quantum heat engines and quantum entanglement.
本文主要基于各种量子热力学循环模型,研究与量子热力学有关的基本问题,分析不同量子系统为工质的热力学循环过程的能量变化及其效率等并进行优化。从现代热力学的发展和研究现状可以知道,量子热力学从最初研究无相互作用的量子系统的热力学循环发展到研究有各种相互作用,及考虑量子摩擦等不可逆因素影响的量子热力学循环,这是一个值得研究的发展方向。
在第二章,我们构建了一个比较特殊的不可逆谐振子量子热机循环模型,由两个等温,一个绝热和一个等频率过程构成。基于量子主方程和半群方法,我们具体分析了谐振子系统的重要的热力学性能参数,在高温极限下推导了诸如吸、放热,输出功,输出功率、循环效率及熵产率等的表达式。通过数值分析,我们绘制了一系列性能特征曲线,如输出功率和效率的优化曲线,生态学函数和效率的优化曲线,输出功率和工质热温比在最大生态学函数时的关系曲线等。另外,还对在最大生态学函数和最大输出功率下的循环效率分别进行了比较。
在第三章,我们研究了有量子摩擦情况下的不可逆量子奥托制冷循环的性能。基于海森堡量子主方程,我们推导了谐振子系统热力学可观测量的运动方程,并且绘制了制冷循环的模拟图。数值分析了平均摩擦功率、制冷率、制冷系数和输入功率与绝热过程时间的关系。进一步分析了热传导和等频率过程时间对循环性能参数的影响。
在第四章,我们研究了两个有XY相互作用的qubit为工质的四能级量子纠缠热机的性能。这是个有趣的量子奥托热机循环,我们假设其绝热过程中保持量子相互作用常数不变而改变外磁场。根据量子热力学的相关理论我们分析了其做功和吸放热情况,并对最大效率和正功条件进行了详细分析,同时通过数值分析绘制了各热力学参量与热纠缠之间的关系曲线。我们发现量子系统间的纠缠作用对热机性能影响很大,量子热机可能在更宽松的条件下对外做功。
在第五章,我们延续了对量子纠缠作用的研究。基于二量子比特一维各向同性的自旋1/2的海森堡Hxx链模型我们构建了两个qubit量子纠缠系统为工质的四能级制冷循环,分析了在循环中系统与外界交换的热量、输入功、制冷率等热力学参数与量子纠缠之间的关系。我们发现了一些与量子纠缠热机不一样的性质,这可以使我们更全面的了解量子纠缠作用对热力学循环的影响。
在第六章,我们构建了工作在单模场中的两个相互作用的原子构成的四能级热机。通过对这个热机模型的研究,进一步阐述量子系统间的相互作用对热机性能的影响,我们分析了热机正功条件,推导得到了比卡诺效率更低的效率上限,并从局域分析的角度深入探讨了相互作用常数对热流方向的影响。
第七章对本论文进行了总结,并列出了量子热力学理论需要进一步深入研究的课题。本论文的研究结果有助于深入了解量子热力学理论,也为开展与此相关的实验工作提供有益的参考。
In recent years, with rapid developments of quantum information science and the experimental technology, some fundamental issues in quantum mechanics and statistical mechanics have attracted great interest. Under many researchers'effort, the thermodynamic theory, which is based on the quantum mechanics and called quantum thermodynamics, has been discussed in a series of papers. In reviewing the history of thermodynamics, one can find that the study of the heat engine played an important role in the development of classical thermodynamics. Similarly, the quantum heat engines serve as a platform for studying fundamental topics of the quantum thermodynamics. Quantum heat engine produce work using quantum system as the working substance, which includes the spin system, harmonic oscillator system, the particles in the infinite potential trap and the qubit system. There are many differences between the quantum heat engine and the classical counterparts because of the nature of the quantum matter. Therefore, the quantum heat engine become the research field of great interest in recent years, aiming to find some novel phenomenon and interesting results.
In this dissertation, the main task is to analyze and optimize the performance of the quantum cycle. Through studying various quantum thermodynamic cycles, we inverstigate the fundamental issue about the quantum thermodynamics.
In the chapter Ⅱ, a new cycle model of an irreversible quantum harmonic heat engine is established, which is composed of two isothermal processes, an adiabatic process and a constant frequency process. Based on the quantum master equation and semi-group approach, the general optimal performance characteristics of the cycle are investigated. Expressions for several important performance parameters such as the efficiency, the power output and the ecological function are derived in high temperature limit. By using numerical calculation, a set of performance characteristic curves is plotted, such as the optimal curves between the power output and efficiency, the ecological function and efficiency, and so on. The optimal regions of the efficiency, power output and temperature ratio of the working substance at the condition of the maximum ecological function are determined. Moreover, a comparison of the corresponding efficiencies at the maximum ecological function and power output is also presented.
In the chapter III, we discussed the performance characteristics of the irreverable quantum Otto refrigeration cycle with intrinsic quantum friction. An irreversible quantum Otto refrigeration cycle working with harmonic systems is established. Base on Heisenberg quantum master equation, the equations of motion for the set of harmonic systems the thermodynamic observables are derived. The simulated diagrams of the quantum Otto refrigeration cycle are plotted. The relationship between average power of friction, cooling rate, power input, and the time of adiabatic process is analyzed by using numerical calculation. Moreover, the influence of the heat conductance and the time of iso-frequency process on the performance of the cycle is discussed.
In chapter Ⅳ, We study the performance characteristics of the four level entangled quantum heat engine, whose work substance is composed of the two-qubit Heisenberg XY model. Based on the theory of the quantum thermodynamics, we inverstigate the several thermodynamic quantities such as the heat transferred, the work, the efficiency and the condition of the positive work. Moreover, relationship between the thermodynamic quantities and the concurrence of the quantum entanglement is analyzed by using graph. We found that the requirements of an entangled quantum heat engine are looser than that of a non-entangled quantum heat engine.
In chapter Ⅴ, we continued to study the effect of the quantum entanglement in the quantum thermodynamics. Based on a two-qubit Heisenberg XX model, a four-level entangled quantum refrigeration cycle is constructed. We analysed the relationship between several thermodynamic quantities and the concurrence of the quantum entanglement. We also found some novelty performance characteristics which is different with the entangled quantum heat engine, and it is very useful for people to understand the entangle quantum thermodynamic cycle.
In chapter Ⅵ, we set up a four-level heat engine model which consist of two atoms resonantly interact with a quantized electromagnetic field inside a cavity, and the dipole-dipole interaction of the atoms cannot be neglected. Here, we investigated the influence of the interactiong working medium on the heat engine performance, and analysed the positive work conditions in different cases. An upper bound for the efficiency is derived, which is smaller than the carnot efficieny. Additionally, the local description of each atom are discussed, this study further shows that the interaction between two atoms have important effects on the heat flow of the heat engine.
The research topics about quantum thermodynamic theory, which deserve a deeper study for the thermodynamic of the interaction quantum system, are listed in the chapter VII.
The results in this dissertation allow to understand well the quantum thermodynamics, and thus provide the theoretical guidelines for the future experiments including the quantum heat engines and quantum entanglement.
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