玻色—爱因斯坦凝聚的混沌动力学研究
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摘要
玻色-爱因斯坦凝聚(BEC)是近几十年来被广泛关注的课题。它不仅提供了一个研究量子力学基本问题的宏观系统,而且在原子光学,量子计算等领域有着光明的应用前景。本文在平均场理论的框架下以Gross-Pitaevskii(G-P)方程为主要模型,讨论了双阱中的混沌、不稳二维环面、宏观量子自囚禁和运动光学晶格中混沌控制及同步。
全文共分六章,各章研究内容如下。第一章简单介绍了光学混沌的研究进展,BEC中的混沌,光学混沌控制及反控制,原子光学,BEC及其性质,原子激光等的研究现状。
第二章主要研究了在运动光学晶格中BEC的线性稳定性和时空演化性质,在这一章中重点讨论的是类Bloch解的时空混沌性质。当考虑到阻尼效应时,利用李雅普诺夫指数(Lyapunov exponent),给出了系统的混沌参数区域。同时通过数值分析我们研究了系统的暂态混沌特征,并模拟了由暂态混沌向定态混沌的转变过程。我们发现在这一过程中暂态混沌的最终吸引子经历了一系列的倍周期分岔。我们还数值模拟了粒子数密度的时间系列和功率谱,进一步阐明了BEC的混沌特性。
第三章主要研究了对光学晶格中BEC空间混沌分布的控制和同步。提出了四种控制方法。第二节中的周期参数调制法控制BEC混沌,用正弦信号调制外加的激光光强,将运动光学晶格的BEC混沌行为控制到周期运动。数值模拟结果表明,选择不同的调制强度和调制频率,只要满足系统的最大Lyapunov指数小于零,即可实现混沌控制。在同样的调制频率下,不同的调制强度会出现不同的周期运动,或者在相同的调制强度下,不同的调制频率也会出现不同的周期运动。第三节中的周期信号驱动法控制BEC混沌。数值模拟结果表明,用小的周期信号控制系统混沌,选择各种共振条件,采用恰当的信号相位和强度,使系统从混沌运动转变成周期运动。第四节中的线性反馈法控制BEC混沌。利用原子反射镜,将运动光学晶格中BEC混沌行为控制到周期运动。数值模拟结果表明,选择恰当的系数来反馈单一系统变量,只要满足系统的最大Lyapunov指数小于零,即可实现混沌控制。第五节中的调s-波散射长度控制BEC混沌。通过feshbach共振调节s-波散射长度,能够将BEC混沌控制到周期运动。数值模拟结果表明,只要最大Lyapunov指数小于零,不同的s-波散射长度对应不同的周期轨道。第六节是研究BEC混沌同步。对光学晶格中BEC空间混沌分布的控制和同步的研究都是利用李雅普诺夫指数来讨论的。
第四章主要研究了对光学晶格中BEC空间混沌分布的反控制。提出了两种反控制方法。第一节中混沌反控制的方法-周期参数调制法,通过调制外加激光的光强,使BEC从周期运动转化为混沌态。数值模拟结果表明,选择不同的调制强度和调制角频率,只要满足系统的最大Lyapunov指数大于零,即可实现不同的混沌轨道重构。第二节中提出了另一种实现BEC混沌反控制的方法-外部周期信号驱动法,数值模拟结果表明,用小的周期信号控制系统,采用恰当的调制相位和强度,只要满足系统的最大Lyapunov指数大于零,即可使周期运动进入混沌状态。调制相位在混沌轨道重构中起了很重要的作用。
第五章主要研究了在三体相互作用下BEC的混沌、不稳二维环面特征和自囚禁现象。利用Lyapunov指数研究了三体相互作用下的BEC相对粒子数布居的时间演化。数值分析了混沌运动的各种吸引子及相应的时间变化图和功率谱,本章从理论上和数值模拟上来研究BEC的混沌和不稳二维环面性质。当BEC气体原子是相互吸引的相互作用时,系统受到三体复合损失和外部非凝聚热云对凝聚体的补充。相应的在G-P方程中加入虚的相互作用项-五次方项。第二节研究了三体相互作用下BEC中的混沌特性。第三节讨论了三体相互作用下BEC中不稳二维环面的行为。双势阱模型是一个非常基本的简单物理模型,人们可以用它来研究BEC原子的量子隧穿和量子相干等相关特性。最近人们在实验上观测到双势阱中许多有趣的物理现象,其中之一是自囚禁(self-trapping)现象,第四节讨论了三体相互作用下BEC中的自囚禁现象。
最后,在第六章对本文做了简单的总结,并对该领域前景做了一点展望。本文中,作者的研究主要集中在第三、四、五章。
Bose-Einstein condensates (BEC) have been attractive subjects in recent decades. They not only offer the perfect macroscopic quantum systems to investigate many fundamental problems in quantum mechanics but also have extensively application foregrounds such as in atom laser and quantum computation. In the framework of mean-field theory, the BEC is governed by the Gross-Pitaevskii equation. Based on the Gross-Pitaevskii equation we shall study the chaos, unstable cycle in double-well and chaos control, synchronism in an optical lattice.
Our paper is organized as the following six parts. In the first part we shall give a simple introduction to development of optical chaos, chaos in BEC, controlling chaos, anti-control of chaos ,atomic optical, BEC and properties, atomic laser and so on.
In part two, we studied the space-time evolution properties of the BEC held in a traveling optical lattice. In this chapter, the dynamic equation of the BEC is deeply investigated and the stability of its stable solution is analyzed. We focus on the features of spatiotemporal chaos of the Bloch-like solution of the system. When the damping is our considerations, we use Lyapunov exponent to present the chaotic region in parametrical space. Features of the transient chaos are studied through numerical method. The transition procession from transient chaos to stationary one has been numerically simulated. In this procession, we find that the final attractors of the transient chaos undergo a series of period-doubling bifurcations. We also simulate the time-series and power spectra.
In part three, chaos control and synchronism in BEC are investigated by four methods. Firstly, we suggest a method for eliminating chaos by modulating periodic signals to convert the chaotic state into regular state. As a function of modulation intensity and modulation frequency the maximal Lyapunov exponent is calculated respectively, and the periodic orbits associated with the negative Lyapunov exponent. Secondly, BEC chaos is controlled with the outer period signal parameter modulation. Numerical simulation shows that the chaotic behavior can be well controlled to enter into periodicity by choosing condition of resonance and the best phase matching or intensity. Thirdly, a method of chaos control with linear feedback is presented. By using the method, we propose a scheme of controlling chaotic behavior in a BEC with atomic mirror. The results of the computer simulation show that the chaos into the stables could be realized by adjusting the coefficient of feedback only if the maximal Lyapunov exponent of the system is negative. Fourth, through changing s-wave scattering length by using feshbach resonance, the chaotic behavior can be well controlled to enter into periodicity. Numerical simulation shows that there are different periodic orbits according to different s-wave scattering length only if the maximal Lyapunov exponent of the system is negative. Finally, chaotic synchronization in BEC is investigated.
The fourth chapter anti-control of chaos in BEC is put forward by two methods. We suggest a method for generating chaos in BEC by modulating periodic signals to convert the regular states. As a function of modulation intensity and modulation, frequency the maximal Lyapunov exponent is calculated respectively and the chaotic orbits associated with the positive Lyapunov exponent. Finally, we present a method anti-control of chaos in BEC by applying outer periodic signals to convert the periodic state into chaotic state. The periodicity can be well-controlled to enter into chaotic behavior by choosing condition of resonance and best phase matching or intensity phase. Numerical simulation shows that there are chaotic orbits corresponding to different phase matching or intensity only if the maximal Lyapunov exponent is positive.
In part five, by using Lyapunov exponent, chaotic time evolutions and the macroscopic quantum self-trapping in BEC are investigated for the particle number density of BEC with three-body interaction. Chaotic attractors, the time series and power spectra are simulated numerically. Properties of chaos are revealed theoretically and numerically in this part. Firstly, we study the chaotic properties in BEC. In the second part, unstable cycle in BEC with three-body interaction is investigated by using Lyapunov exponent. Finally, we study the macroscopic quantum self-trapping in BEC. Double-well trapping in BEC is a simple model. People use it to study many phenomenons.
Finally, we briefly summarize our contributions and discuss future work in the last chapter. Here, our main works are involved in chapter's three, four and five.
全文共分六章,各章研究内容如下。第一章简单介绍了光学混沌的研究进展,BEC中的混沌,光学混沌控制及反控制,原子光学,BEC及其性质,原子激光等的研究现状。
第二章主要研究了在运动光学晶格中BEC的线性稳定性和时空演化性质,在这一章中重点讨论的是类Bloch解的时空混沌性质。当考虑到阻尼效应时,利用李雅普诺夫指数(Lyapunov exponent),给出了系统的混沌参数区域。同时通过数值分析我们研究了系统的暂态混沌特征,并模拟了由暂态混沌向定态混沌的转变过程。我们发现在这一过程中暂态混沌的最终吸引子经历了一系列的倍周期分岔。我们还数值模拟了粒子数密度的时间系列和功率谱,进一步阐明了BEC的混沌特性。
第三章主要研究了对光学晶格中BEC空间混沌分布的控制和同步。提出了四种控制方法。第二节中的周期参数调制法控制BEC混沌,用正弦信号调制外加的激光光强,将运动光学晶格的BEC混沌行为控制到周期运动。数值模拟结果表明,选择不同的调制强度和调制频率,只要满足系统的最大Lyapunov指数小于零,即可实现混沌控制。在同样的调制频率下,不同的调制强度会出现不同的周期运动,或者在相同的调制强度下,不同的调制频率也会出现不同的周期运动。第三节中的周期信号驱动法控制BEC混沌。数值模拟结果表明,用小的周期信号控制系统混沌,选择各种共振条件,采用恰当的信号相位和强度,使系统从混沌运动转变成周期运动。第四节中的线性反馈法控制BEC混沌。利用原子反射镜,将运动光学晶格中BEC混沌行为控制到周期运动。数值模拟结果表明,选择恰当的系数来反馈单一系统变量,只要满足系统的最大Lyapunov指数小于零,即可实现混沌控制。第五节中的调s-波散射长度控制BEC混沌。通过feshbach共振调节s-波散射长度,能够将BEC混沌控制到周期运动。数值模拟结果表明,只要最大Lyapunov指数小于零,不同的s-波散射长度对应不同的周期轨道。第六节是研究BEC混沌同步。对光学晶格中BEC空间混沌分布的控制和同步的研究都是利用李雅普诺夫指数来讨论的。
第四章主要研究了对光学晶格中BEC空间混沌分布的反控制。提出了两种反控制方法。第一节中混沌反控制的方法-周期参数调制法,通过调制外加激光的光强,使BEC从周期运动转化为混沌态。数值模拟结果表明,选择不同的调制强度和调制角频率,只要满足系统的最大Lyapunov指数大于零,即可实现不同的混沌轨道重构。第二节中提出了另一种实现BEC混沌反控制的方法-外部周期信号驱动法,数值模拟结果表明,用小的周期信号控制系统,采用恰当的调制相位和强度,只要满足系统的最大Lyapunov指数大于零,即可使周期运动进入混沌状态。调制相位在混沌轨道重构中起了很重要的作用。
第五章主要研究了在三体相互作用下BEC的混沌、不稳二维环面特征和自囚禁现象。利用Lyapunov指数研究了三体相互作用下的BEC相对粒子数布居的时间演化。数值分析了混沌运动的各种吸引子及相应的时间变化图和功率谱,本章从理论上和数值模拟上来研究BEC的混沌和不稳二维环面性质。当BEC气体原子是相互吸引的相互作用时,系统受到三体复合损失和外部非凝聚热云对凝聚体的补充。相应的在G-P方程中加入虚的相互作用项-五次方项。第二节研究了三体相互作用下BEC中的混沌特性。第三节讨论了三体相互作用下BEC中不稳二维环面的行为。双势阱模型是一个非常基本的简单物理模型,人们可以用它来研究BEC原子的量子隧穿和量子相干等相关特性。最近人们在实验上观测到双势阱中许多有趣的物理现象,其中之一是自囚禁(self-trapping)现象,第四节讨论了三体相互作用下BEC中的自囚禁现象。
最后,在第六章对本文做了简单的总结,并对该领域前景做了一点展望。本文中,作者的研究主要集中在第三、四、五章。
Bose-Einstein condensates (BEC) have been attractive subjects in recent decades. They not only offer the perfect macroscopic quantum systems to investigate many fundamental problems in quantum mechanics but also have extensively application foregrounds such as in atom laser and quantum computation. In the framework of mean-field theory, the BEC is governed by the Gross-Pitaevskii equation. Based on the Gross-Pitaevskii equation we shall study the chaos, unstable cycle in double-well and chaos control, synchronism in an optical lattice.
Our paper is organized as the following six parts. In the first part we shall give a simple introduction to development of optical chaos, chaos in BEC, controlling chaos, anti-control of chaos ,atomic optical, BEC and properties, atomic laser and so on.
In part two, we studied the space-time evolution properties of the BEC held in a traveling optical lattice. In this chapter, the dynamic equation of the BEC is deeply investigated and the stability of its stable solution is analyzed. We focus on the features of spatiotemporal chaos of the Bloch-like solution of the system. When the damping is our considerations, we use Lyapunov exponent to present the chaotic region in parametrical space. Features of the transient chaos are studied through numerical method. The transition procession from transient chaos to stationary one has been numerically simulated. In this procession, we find that the final attractors of the transient chaos undergo a series of period-doubling bifurcations. We also simulate the time-series and power spectra.
In part three, chaos control and synchronism in BEC are investigated by four methods. Firstly, we suggest a method for eliminating chaos by modulating periodic signals to convert the chaotic state into regular state. As a function of modulation intensity and modulation frequency the maximal Lyapunov exponent is calculated respectively, and the periodic orbits associated with the negative Lyapunov exponent. Secondly, BEC chaos is controlled with the outer period signal parameter modulation. Numerical simulation shows that the chaotic behavior can be well controlled to enter into periodicity by choosing condition of resonance and the best phase matching or intensity. Thirdly, a method of chaos control with linear feedback is presented. By using the method, we propose a scheme of controlling chaotic behavior in a BEC with atomic mirror. The results of the computer simulation show that the chaos into the stables could be realized by adjusting the coefficient of feedback only if the maximal Lyapunov exponent of the system is negative. Fourth, through changing s-wave scattering length by using feshbach resonance, the chaotic behavior can be well controlled to enter into periodicity. Numerical simulation shows that there are different periodic orbits according to different s-wave scattering length only if the maximal Lyapunov exponent of the system is negative. Finally, chaotic synchronization in BEC is investigated.
The fourth chapter anti-control of chaos in BEC is put forward by two methods. We suggest a method for generating chaos in BEC by modulating periodic signals to convert the regular states. As a function of modulation intensity and modulation, frequency the maximal Lyapunov exponent is calculated respectively and the chaotic orbits associated with the positive Lyapunov exponent. Finally, we present a method anti-control of chaos in BEC by applying outer periodic signals to convert the periodic state into chaotic state. The periodicity can be well-controlled to enter into chaotic behavior by choosing condition of resonance and best phase matching or intensity phase. Numerical simulation shows that there are chaotic orbits corresponding to different phase matching or intensity only if the maximal Lyapunov exponent is positive.
In part five, by using Lyapunov exponent, chaotic time evolutions and the macroscopic quantum self-trapping in BEC are investigated for the particle number density of BEC with three-body interaction. Chaotic attractors, the time series and power spectra are simulated numerically. Properties of chaos are revealed theoretically and numerically in this part. Firstly, we study the chaotic properties in BEC. In the second part, unstable cycle in BEC with three-body interaction is investigated by using Lyapunov exponent. Finally, we study the macroscopic quantum self-trapping in BEC. Double-well trapping in BEC is a simple model. People use it to study many phenomenons.
Finally, we briefly summarize our contributions and discuss future work in the last chapter. Here, our main works are involved in chapter's three, four and five.
引文
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