光学晶格中玻色—爱因斯坦凝聚体系的Melnikov混沌
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摘要
原子气体玻色-爱因斯坦凝聚(BEC)在实验上的实现，具有十分重大的理论研究意义和应用价值。玻色-爱因斯坦凝聚体系不仅为量子理论的研究提供了一个可靠的宏观量子系统，而且在原子激光、芯片技术、精密测量等方面有着非常美好的应用前景。本文在平均场理论的框架下，以Gross-Pitaevskii(GP)方程为主要研究模型，采用理论解析和数值模拟相结合的方法，讨论和展示了光学晶格中玻色-爱因斯坦凝聚体系的Melnikov混沌行为，并提出了控制凝聚原子混沌运动的方法，得到了一些有意义的结论。全文一共分为六章，各章研究的主要内容如下：
第一章简要介绍了玻色-爱因斯坦凝聚与混沌的有关概念和定律，回顾了在该领域的研究现状和展望，揭示了玻色-爱因斯坦凝聚体系的Melnikov混沌特征，并阐述了几种控制混沌的方法。
第二章讨论了两个弱耦合玻色-爱因斯坦凝聚体中的混沌Josephson效应。利用与时间关联的自洽场方法和宏观量子单体波函数，从GP方程出发导出了对称囚禁和非对称囚禁势两种情形下描述相差φ随时间变化的玻色子Josephson结方程。运用直接微扰法，求出了小幅振荡情形下的玻色子Josephson结方程的混沌解。研究结果表明，在与时间无关联的驱动作用下，凝聚原子的运动是有规律的运动。但是，对于处在非对称囚禁势情形下的玻色-爱因斯坦凝聚体系，当加上与时间相关联的周期性驱动后，则引起凝聚原子发生混沌运动。此外，我们还利用所求得的混沌解证明了系统混沌轨道的解析有界与数值无界特性，从而揭示出了混沌系统的不可计算性。
第三章主要分析和研究了一维光学晶格中玻色-爱因斯坦凝聚的Wannier-Stark混沌。在Melnikov混沌准则下，利用直接微扰技术，我们构建了具有平凡相和非平凡相两种情形下的基于Groos-Pitaevskii方程的经典混沌解，并对这两种不同相情形下在参数空间的混沌区域进行了图解和分析说明。混沌解描述了原子的数密度和能量密度的混沌空间演化，而混沌区域则提供了产生或者消除混沌的一种方法，这就是通过调节可控参数。此外，混沌解的不稳定性在这章也被讨论。研究表明，混沌出现在玻色-爱因斯坦凝聚坍塌的进程中，它对玻色-爱因斯坦凝聚体系统起着毁灭性作用。因此，对于一个玻色-爱因斯坦凝聚的形成和应用而言，预测混沌与控制混沌是相当重要的。
第四章主要研究和讨论了两种囚禁势(Gaussian势与Ratchet势)情形下玻色-爱因斯坦凝聚体系中凝聚原子的非线性输运问题。从GP方程出发，将凝聚波函数的时、空变量分离，得到了该系统所满足的定态非线性Schr(?)dinger方程，然后运用解析方法和数值计算方法对该方程进行了求解。解析结果表明，在Gaussian势情况下，凝聚系统具有稳定的微扰解，此表明凝聚原子为稳定的非线性输运。但在Ratchet势作用下，系统却具有混沌的解析解，因而表明凝聚原子的输运过程为混沌输运，相应的数值模拟结果印证了理论解析的结论。并且，我们的研究结果说明了这种混沌的输运过程可以通过改变系统参数或边界条件对其加以有效的控制。
第五章探索性地研究了倾斜光学品格中两元玻色-爱因斯坦体系(TBECs)的Melnikov混沌。解析结果指出，在Melnikov混沌准则下，该体系所满足的两个耦合非线性Schr(?)dinger方程的定态微扰解一般为不稳定的混沌解，因而两种凝聚体都具有Smale马蹄意义下的混沌特征。相应的数值结果表明，在相同的初、边界条件下，当两种凝聚体的系统参数取值相同时，它们在相空间的轨道完全相同，即两种凝聚体的混沌运动是完全同步的。而当其中一凝聚体的某一参数哪怕有一微小的改变，则它们的相图变得完全不一样，两者的混沌运动不再同步。此一方面说明了混沌系统对系统参数的敏感依赖性，另一方面表明了系统参数在混沌控制和混沌同步中扮演了一个重要的角色。
第六章对全文所得到的结果进行了简要地回顾和总结，并对玻色-爱因斯坦凝聚体的混沌运动和稳定性问题以及凝聚体在实际应用方面的价值作出了相应的研究展望。本文中，作者的研究工作主要集中在第二章、第三章，第四章和第五章。
The realization of atomic gases Bose-Einstein condensation in the experimentation has very importent purports of the theory and application. Bose-Einstein condensates not only offer the perfect macrosciopic quantum systems to investigate many foundmental problems in quantum mechanics but also have extensively application foregrounds such as in atom laser, CMOS and the exactitude measure. In the framework of the mean-field theory, the Bose-Einstein condensates is govonered by the Gross-Pitaeviskii equation. Based on the Gross-Pitaeviskii equation we had been study of the Melnikov chaos in Bose-Einstein condensates by theoretical analysis and numerical method. Meanwhile, we have also discussed the problem of controlling the chaotic motion of the condensated atoms, and a series of significant results are obtained in this paper. Our paper is organized as the following six parts:
In chapter 1, we have introduced the conceptions and the laws of Bose-Einstein condensation and chaos.Their actuality and expectation are retrospected in this chaper. We also studied the chaotic behavior of Bose-Einstein condensates, and discussed the motheds of controlling chaos.
In chapter 2, we have studied the chaotic Josephson effects in two weakly coupled Bose-Einstein condensates. Using the time-dependent self-consistent field method and the macroscopic one-body wave function, we investigated the BJJ equations that de- scribe time evolutions of the phase difference in symmetric and asymmetric trapped case. Wehave obtained the chaotic solution of the BJJ equation for a small-amplitude oscillation by using the direct perturbation method. The results reveal that the motion of the condensate atoms is regular without the time-dependent drive and the periodical drive causes the chaotic motion of the Bose-Einstein condensates in the asymmetric trap. Using the chaotic solution we have demonstrated the analytical boundedness and numerical unboundedness of the chaotic orbits that implies the incomputabilityof the system.
In chapter 3, we have considered a trapped BEC in a tilted potential of the
Wannier-Stark framework. Under the Melnikov's chaos criterion, we investigated the chaotic solutions based on the GP equation with trivial and non-trivial phases are constructed by using the direct perturbation technique. The chaotic regions on the parameter space are illustrated for the two different phases. The chaotic solutions describe the chaotic spatial evolutions of the atomic number density and energy density, and the chaotic regions supply a method for producing or eliminating chaos, through the adjustments of the controllable parameters. The instability of chaotic solution is discussed and the qualitative agreements with the work reported recently is found. It is well known that the chaos emerges in the processions of the BEC collapsing which plays a destructive role for the BEC system. Therefore, predicting and controlling chaos are quite important for the formation and application of a BEC.
In chapter 4, we investigated the nonlinear transport of BEC atoms in Gaussian potential and ratchet potential. We have obtained a stationary nonlinear Schro from GP equation, and sloved this equation theoretically and numerically. The analytical results showed that the condensed system has a steady perturbation solution, and reveal that the nonlinear transport of the condensate atoms in Gaussian potential case. But in ratchet potential case, there is chaotic analytical solution in this system, and show that the chaotic transport of BEC atoms. The corresponding numerical results agree well with the theoretical analytical results. Besides, we can control the chaotic transport of condensate atoms by change the system parameters or boundary conditions.
In chapter 5, we studied the Melnikov chaos of two-component Bose-Einstein condensates in tilted optical lattices. The analytical results show that the stationary perturbation solutions of the two coupled nonlinear Schrodinger equations are the instable chaotic solutions in Melnikov criterion for chaos, and there are Smale horseshoes chaos in the two condensates. The corresponding numerical results show that the phase orbits of the two condensates are identical when the system parameters and initial conditions or boundary conditions, which means that the two condensates are chaotic synchronized. However, they are different and asynchronous when there is any very small difference between the system parameters, which show that the system parameters play a very important role in controlling chaos and chaos synchronization.
In chapter 6, we give a simple summary and discussion to the above-mentioned works. We also give a expectaton in the study for the chaotic motion of BEC,the stability of BEC system and the applications of the condensates. Here, our main works are involved in chapters two, three, four and five.
第一章简要介绍了玻色-爱因斯坦凝聚与混沌的有关概念和定律，回顾了在该领域的研究现状和展望，揭示了玻色-爱因斯坦凝聚体系的Melnikov混沌特征，并阐述了几种控制混沌的方法。
第二章讨论了两个弱耦合玻色-爱因斯坦凝聚体中的混沌Josephson效应。利用与时间关联的自洽场方法和宏观量子单体波函数，从GP方程出发导出了对称囚禁和非对称囚禁势两种情形下描述相差φ随时间变化的玻色子Josephson结方程。运用直接微扰法，求出了小幅振荡情形下的玻色子Josephson结方程的混沌解。研究结果表明，在与时间无关联的驱动作用下，凝聚原子的运动是有规律的运动。但是，对于处在非对称囚禁势情形下的玻色-爱因斯坦凝聚体系，当加上与时间相关联的周期性驱动后，则引起凝聚原子发生混沌运动。此外，我们还利用所求得的混沌解证明了系统混沌轨道的解析有界与数值无界特性，从而揭示出了混沌系统的不可计算性。
第三章主要分析和研究了一维光学晶格中玻色-爱因斯坦凝聚的Wannier-Stark混沌。在Melnikov混沌准则下，利用直接微扰技术，我们构建了具有平凡相和非平凡相两种情形下的基于Groos-Pitaevskii方程的经典混沌解，并对这两种不同相情形下在参数空间的混沌区域进行了图解和分析说明。混沌解描述了原子的数密度和能量密度的混沌空间演化，而混沌区域则提供了产生或者消除混沌的一种方法，这就是通过调节可控参数。此外，混沌解的不稳定性在这章也被讨论。研究表明，混沌出现在玻色-爱因斯坦凝聚坍塌的进程中，它对玻色-爱因斯坦凝聚体系统起着毁灭性作用。因此，对于一个玻色-爱因斯坦凝聚的形成和应用而言，预测混沌与控制混沌是相当重要的。
第四章主要研究和讨论了两种囚禁势(Gaussian势与Ratchet势)情形下玻色-爱因斯坦凝聚体系中凝聚原子的非线性输运问题。从GP方程出发，将凝聚波函数的时、空变量分离，得到了该系统所满足的定态非线性Schr(?)dinger方程，然后运用解析方法和数值计算方法对该方程进行了求解。解析结果表明，在Gaussian势情况下，凝聚系统具有稳定的微扰解，此表明凝聚原子为稳定的非线性输运。但在Ratchet势作用下，系统却具有混沌的解析解，因而表明凝聚原子的输运过程为混沌输运，相应的数值模拟结果印证了理论解析的结论。并且，我们的研究结果说明了这种混沌的输运过程可以通过改变系统参数或边界条件对其加以有效的控制。
第五章探索性地研究了倾斜光学品格中两元玻色-爱因斯坦体系(TBECs)的Melnikov混沌。解析结果指出，在Melnikov混沌准则下，该体系所满足的两个耦合非线性Schr(?)dinger方程的定态微扰解一般为不稳定的混沌解，因而两种凝聚体都具有Smale马蹄意义下的混沌特征。相应的数值结果表明，在相同的初、边界条件下，当两种凝聚体的系统参数取值相同时，它们在相空间的轨道完全相同，即两种凝聚体的混沌运动是完全同步的。而当其中一凝聚体的某一参数哪怕有一微小的改变，则它们的相图变得完全不一样，两者的混沌运动不再同步。此一方面说明了混沌系统对系统参数的敏感依赖性，另一方面表明了系统参数在混沌控制和混沌同步中扮演了一个重要的角色。
第六章对全文所得到的结果进行了简要地回顾和总结，并对玻色-爱因斯坦凝聚体的混沌运动和稳定性问题以及凝聚体在实际应用方面的价值作出了相应的研究展望。本文中，作者的研究工作主要集中在第二章、第三章，第四章和第五章。
The realization of atomic gases Bose-Einstein condensation in the experimentation has very importent purports of the theory and application. Bose-Einstein condensates not only offer the perfect macrosciopic quantum systems to investigate many foundmental problems in quantum mechanics but also have extensively application foregrounds such as in atom laser, CMOS and the exactitude measure. In the framework of the mean-field theory, the Bose-Einstein condensates is govonered by the Gross-Pitaeviskii equation. Based on the Gross-Pitaeviskii equation we had been study of the Melnikov chaos in Bose-Einstein condensates by theoretical analysis and numerical method. Meanwhile, we have also discussed the problem of controlling the chaotic motion of the condensated atoms, and a series of significant results are obtained in this paper. Our paper is organized as the following six parts:
In chapter 1, we have introduced the conceptions and the laws of Bose-Einstein condensation and chaos.Their actuality and expectation are retrospected in this chaper. We also studied the chaotic behavior of Bose-Einstein condensates, and discussed the motheds of controlling chaos.
In chapter 2, we have studied the chaotic Josephson effects in two weakly coupled Bose-Einstein condensates. Using the time-dependent self-consistent field method and the macroscopic one-body wave function, we investigated the BJJ equations that de- scribe time evolutions of the phase difference in symmetric and asymmetric trapped case. Wehave obtained the chaotic solution of the BJJ equation for a small-amplitude oscillation by using the direct perturbation method. The results reveal that the motion of the condensate atoms is regular without the time-dependent drive and the periodical drive causes the chaotic motion of the Bose-Einstein condensates in the asymmetric trap. Using the chaotic solution we have demonstrated the analytical boundedness and numerical unboundedness of the chaotic orbits that implies the incomputabilityof the system.
In chapter 3, we have considered a trapped BEC in a tilted potential of the
Wannier-Stark framework. Under the Melnikov's chaos criterion, we investigated the chaotic solutions based on the GP equation with trivial and non-trivial phases are constructed by using the direct perturbation technique. The chaotic regions on the parameter space are illustrated for the two different phases. The chaotic solutions describe the chaotic spatial evolutions of the atomic number density and energy density, and the chaotic regions supply a method for producing or eliminating chaos, through the adjustments of the controllable parameters. The instability of chaotic solution is discussed and the qualitative agreements with the work reported recently is found. It is well known that the chaos emerges in the processions of the BEC collapsing which plays a destructive role for the BEC system. Therefore, predicting and controlling chaos are quite important for the formation and application of a BEC.
In chapter 4, we investigated the nonlinear transport of BEC atoms in Gaussian potential and ratchet potential. We have obtained a stationary nonlinear Schro from GP equation, and sloved this equation theoretically and numerically. The analytical results showed that the condensed system has a steady perturbation solution, and reveal that the nonlinear transport of the condensate atoms in Gaussian potential case. But in ratchet potential case, there is chaotic analytical solution in this system, and show that the chaotic transport of BEC atoms. The corresponding numerical results agree well with the theoretical analytical results. Besides, we can control the chaotic transport of condensate atoms by change the system parameters or boundary conditions.
In chapter 5, we studied the Melnikov chaos of two-component Bose-Einstein condensates in tilted optical lattices. The analytical results show that the stationary perturbation solutions of the two coupled nonlinear Schrodinger equations are the instable chaotic solutions in Melnikov criterion for chaos, and there are Smale horseshoes chaos in the two condensates. The corresponding numerical results show that the phase orbits of the two condensates are identical when the system parameters and initial conditions or boundary conditions, which means that the two condensates are chaotic synchronized. However, they are different and asynchronous when there is any very small difference between the system parameters, which show that the system parameters play a very important role in controlling chaos and chaos synchronization.
In chapter 6, we give a simple summary and discussion to the above-mentioned works. We also give a expectaton in the study for the chaotic motion of BEC,the stability of BEC system and the applications of the condensates. Here, our main works are involved in chapters two, three, four and five.
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