激光控制周期势中玻色—爱因斯坦凝聚体的宏观量子态
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摘要
玻色-爱因斯坦凝聚是一个非常奇异的量子现象,近几十年来被广泛关注.它涉及物理学很多领域,包括原子分子物理、量子光学、统计物理、凝聚态物理等等.1995年这个现象在稀薄原子气体中第一次被观察到,随后掀起了理论和实验的研究热潮.自从实现碱金属原子在光学晶格中的玻色-爱因斯坦凝聚以来,人们对周期势中的玻色-爱因斯坦凝聚体实验和理论上的研究与日俱增.本文在平均场理论框架下,以Gross-Pitaevskii方程为主要模型,采用理论分析和数值模拟相结合的方法,研究了周期势中玻色-爱因斯坦凝聚体规则和混沌的宏观量子态,提出利用激光来控制所得量子态的方法,得到一些有意义的结论。全文一共分为六章,主要内容如下:
第一章主要介绍研究背景,简单介绍玻色-爱因斯坦凝聚的研究历史,稀薄玻色原子气体的平均场理论以及与玻色-爱因斯坦凝聚体的规则态及混沌现象相关的理论和研究进展.
第二章主要研究在一种特殊的空间周期势即Kronig-Penney势中的准一维静态玻色-爱因斯坦凝聚体系统。这个系统由一维定态Gross-Pitaevskii方程来描述。我们研究了排斥相互作用情况的边值问题,并得到在零边界值和周期边界条件下,对应于不同化学势{μ_n}的Gross-Pitaevskii方程一组精确的离散本征态{R_n}。而第n个本征态R_n为一个Jacobian椭圆函数,它的周期是Kronig-Penney势周期的2/n(n为正整数)倍,且零点包含了组成Kronig-Penney势的所有势垒的位置坐标.因此,R_n在空间任意点都是可微的,而粒子数密度R_n~2在Kronig-Penney势每个周期中会出现n个完整的波包。在此基础上,我们研究了通过附加单个激光脉冲来控制玻色-爱因斯坦凝聚体在不同的宏观量子态间的跃迁。所增的激光脉冲由单个δ势来描述,它的位置为x_i。激光脉冲的增加能使系统从零点不包含x_i的{R_n}态跃迁到零点包含x=x_i的{R_(n′)}态.这些结论为在实验上利用玻色-爱因斯坦凝聚体研究边值问题以及实现不同宏观量子态间的跃迁提供了新的方法。
在第三章中我们主要讨论在弱激光驻波势和弱δ激光脉冲微扰下的玻色-爱因斯坦凝聚体系统的空间混沌.利用微扰的混沌解,我们得到一种新型的Melnikov混沌区域,它与由边界条件决定的积分常数c_0密切相关.当|c_0|较小,混沌区域对应较小的激光波矢k值;对应|c_0|较大的情况,混沌区域中的k值也较大。我们从两个不同的参数区域中分别取参数并保持其它参数不变作Poincar(?)截面图,得到对应混沌区域的混沌轨道和对应规则区域的规则轨道,从而证实了理论分析的结果.因此,对应一个确定的c_0,将波矢k从一个较小的值调至较大的值,可以将混沌区域转化为规则区域,或者将规则区域转化成混沌区域。这为消除或产生Melnikov混沌提供了可行的办法.
在第四章中我们研究了弱运动超光格中具有吸引相互作用的玻色-爱因斯坦凝聚体系统.我们发现,在一组随机初始条件下和给定的参数区域内,从孤子到混沌的跃迁会以一定几率发生.这个几率与Melnikov函数的零点个数密切相关.而Melnikov函数的零点个数则依赖于可控的外场参数.我们从分析和数值两方面来研究光格强度和波矢对混沌几率的影响,并找到对应不同混沌几率的混沌区域.这为我们在实验上通过调节超光格来消除或加强混沌提供了非常重要的思路.
在第五章中,对于存在不能忽略的阻尼效应时,我们讨论了在一个运动光格中的玻色-爱因斯坦凝聚体系统的混沌行为,得到Melnikov混沌解和参数空间的混沌区域。由于阻尼效应,系统存在不会出现混沌的规则参数区域.在混沌区域混沌解的时空演化是有界的,但在其上下极限之间是不可预测的。而增大光格的速度,对应的混沌区域将减小;若加入第二个光格,混沌区域将随着第二个光格强度的增加而增大,在这种双光格情况下,混沌区域也随着光格速度增加而减小。这对我们利用或者抑制Melnikov混沌非常有帮助。
最后一章,我们对上述工作进行了简单的总结和讨论,并对玻色-爱因斯坦凝聚体系统的应用前景作了展望。我们的主要工作集中在第二至第五章.
Bose-Einstein condensaton is a very fantastic quantum phenomenon,which has been an attractive subject in recent decades.Actually,it is a physical phenomenon referring to many aspects,such as atomic and molecular physics,quantum optics,statistical physics,and condensed matter physics.Shortly after the first experimental realization of Bose-Einstein condensates in 1995 in dilute gases of alkali metals,intense efforts have been devoted to the study of the new properties of Bose-Einstein condensates.Since the realization of Bose-Einstein condensates of alkali-metal atoms in an optical lattice,the studies of Bose-Einstein condensates in periodic potentials have been the subject of an explosion of research,both theoretically and experimentally.In the framework of mean-field theory,we study the Bose-Einstein condensates held in periodic potentials,which are governed by Gross-Pitaevskii equations.We investigate the regular or chaotic macroscopic quantum states of the systems by theoretical analysis and numerical method,and how to manipulate the macroscopic quantum states by using laser potentials.Some meaningful results are obtained after the investigations.This paper is organized as the foolowing six chapters:
In the first chapter,we give a simple introduction of the background of Bose-Einstein condensates,and review the research history of it.We also introduce the mean-field theory of dilute gases of alkali metals,the research status and improvements about the regular states of Bose-Einstein condensate and chaos.
In chapter 2,We investigate the boundary value problem(BVP)of a one- dimensional Gross-Pitaevskii equation with the spatially periodic Kronig-Penney potential(KPP)of period d,which governs a quasi-one-dimensional repulsive Bose-Einstein condensate.Under the zero and periodic boundary conditions,we show how to determine n exact stationary eigenstates{R_n}corresponding to different chemical potentials{μ_n}from the known solutions of the system.The n-th eigenstate R_n is the Jacobian elliptic function with period 2d/n for n=1,2,…,and with zero points containing the potential barrier positions.So R_n is differentiable at any spatial point and R_n~2 describes n complete wave-packets in each period of the KPP.It is revealed that one can use a laser pulse modeled by aδpotential at site x_i to manipulate the transitions from the states of{R_n}with zero point x≠x_i to the states of{R_(n')}with zero point x=x_i.The results suggest an experimental scheme for applying Bose-Einstein condensate to test the BVR and to observe the macroscopic quantum transitions.
In chapter 3,the spatial chaos of the Bose-Einstein condensate perturbed by a weak laser standing wave and a weak laserδpulse is studied.By using the perturbed chaotic solution we investigate the new type of Melnikov chaotic regions,which depend on an integration constant co determined by the boundary conditions.It is shown that when the|c_0|values are small,the chaotic region is corresponded to small values of laser wave vector k,and the chaotic region for the larger k values is related to the large|c_0|values.The result is confirmed numerically by finding the chaotic and regular orbits on the Poincar(?)section for the two different parameter regions.So for a fixed c_0 the adjustment of k from a small value to large value can transform the chaotic region into the regular one or on the contrary,that suggests a feasible method for eliminating or generating Melnikov chaos.
In chapter 4,we investigate an attractive Bose-Einstein condensate perturbed by a weak traveling optical superlattice.It is demonstrated that under a stochastic initial set and in a given parameter region the solitonic chaos appears with a certain probability which is tightly related to the zero-point number of Melnikov function,and the latter depends on the controllable potential parameters.Effects of the lattice depths and wave vectors on the chaos probability are studied analytically and numerically,and different chaotic regions of parameter space are found. The results suggest a feasible method for strengthening or weakening chaos by modulating the potential parameters experimentally.
In chapter 5,we study the spatiotemporal chaotic evolution a Bose-Einstein condensate in a moving optical lattice considering damping effects.Melnikov chaotic solution and chaotic region of parameter space are found by using the direct perturbation method.Due to the damping effects,there is a regular region with zero chaos probability for the system.In the chaotic region,the spatiotemporal evolution of the chaotic solution is analytically bounded but unpredictable between the superior and inferior limits.We demonstrate theoretically and numerically that the chaotic region reduces as the propagating velocity of the optical lattice increases.On the other hand,adding a second lattice,we find the chaotic region would widen as the intensity of it increases.In the double lattice case,the chaotic region is also strongly influenced by the velocity of the lattices.This is helpful for eliminating or generating Melnikov chaos.
In the last chapter,we give a simlpe summary and discussion to the abovementioned works,and also discuss the prospects of applicions of the Bose-Einstein condensate systems.Our main works are involded in the sencod,third,fourth and fifth chapters.
第一章主要介绍研究背景,简单介绍玻色-爱因斯坦凝聚的研究历史,稀薄玻色原子气体的平均场理论以及与玻色-爱因斯坦凝聚体的规则态及混沌现象相关的理论和研究进展.
第二章主要研究在一种特殊的空间周期势即Kronig-Penney势中的准一维静态玻色-爱因斯坦凝聚体系统。这个系统由一维定态Gross-Pitaevskii方程来描述。我们研究了排斥相互作用情况的边值问题,并得到在零边界值和周期边界条件下,对应于不同化学势{μ_n}的Gross-Pitaevskii方程一组精确的离散本征态{R_n}。而第n个本征态R_n为一个Jacobian椭圆函数,它的周期是Kronig-Penney势周期的2/n(n为正整数)倍,且零点包含了组成Kronig-Penney势的所有势垒的位置坐标.因此,R_n在空间任意点都是可微的,而粒子数密度R_n~2在Kronig-Penney势每个周期中会出现n个完整的波包。在此基础上,我们研究了通过附加单个激光脉冲来控制玻色-爱因斯坦凝聚体在不同的宏观量子态间的跃迁。所增的激光脉冲由单个δ势来描述,它的位置为x_i。激光脉冲的增加能使系统从零点不包含x_i的{R_n}态跃迁到零点包含x=x_i的{R_(n′)}态.这些结论为在实验上利用玻色-爱因斯坦凝聚体研究边值问题以及实现不同宏观量子态间的跃迁提供了新的方法。
在第三章中我们主要讨论在弱激光驻波势和弱δ激光脉冲微扰下的玻色-爱因斯坦凝聚体系统的空间混沌.利用微扰的混沌解,我们得到一种新型的Melnikov混沌区域,它与由边界条件决定的积分常数c_0密切相关.当|c_0|较小,混沌区域对应较小的激光波矢k值;对应|c_0|较大的情况,混沌区域中的k值也较大。我们从两个不同的参数区域中分别取参数并保持其它参数不变作Poincar(?)截面图,得到对应混沌区域的混沌轨道和对应规则区域的规则轨道,从而证实了理论分析的结果.因此,对应一个确定的c_0,将波矢k从一个较小的值调至较大的值,可以将混沌区域转化为规则区域,或者将规则区域转化成混沌区域。这为消除或产生Melnikov混沌提供了可行的办法.
在第四章中我们研究了弱运动超光格中具有吸引相互作用的玻色-爱因斯坦凝聚体系统.我们发现,在一组随机初始条件下和给定的参数区域内,从孤子到混沌的跃迁会以一定几率发生.这个几率与Melnikov函数的零点个数密切相关.而Melnikov函数的零点个数则依赖于可控的外场参数.我们从分析和数值两方面来研究光格强度和波矢对混沌几率的影响,并找到对应不同混沌几率的混沌区域.这为我们在实验上通过调节超光格来消除或加强混沌提供了非常重要的思路.
在第五章中,对于存在不能忽略的阻尼效应时,我们讨论了在一个运动光格中的玻色-爱因斯坦凝聚体系统的混沌行为,得到Melnikov混沌解和参数空间的混沌区域。由于阻尼效应,系统存在不会出现混沌的规则参数区域.在混沌区域混沌解的时空演化是有界的,但在其上下极限之间是不可预测的。而增大光格的速度,对应的混沌区域将减小;若加入第二个光格,混沌区域将随着第二个光格强度的增加而增大,在这种双光格情况下,混沌区域也随着光格速度增加而减小。这对我们利用或者抑制Melnikov混沌非常有帮助。
最后一章,我们对上述工作进行了简单的总结和讨论,并对玻色-爱因斯坦凝聚体系统的应用前景作了展望。我们的主要工作集中在第二至第五章.
Bose-Einstein condensaton is a very fantastic quantum phenomenon,which has been an attractive subject in recent decades.Actually,it is a physical phenomenon referring to many aspects,such as atomic and molecular physics,quantum optics,statistical physics,and condensed matter physics.Shortly after the first experimental realization of Bose-Einstein condensates in 1995 in dilute gases of alkali metals,intense efforts have been devoted to the study of the new properties of Bose-Einstein condensates.Since the realization of Bose-Einstein condensates of alkali-metal atoms in an optical lattice,the studies of Bose-Einstein condensates in periodic potentials have been the subject of an explosion of research,both theoretically and experimentally.In the framework of mean-field theory,we study the Bose-Einstein condensates held in periodic potentials,which are governed by Gross-Pitaevskii equations.We investigate the regular or chaotic macroscopic quantum states of the systems by theoretical analysis and numerical method,and how to manipulate the macroscopic quantum states by using laser potentials.Some meaningful results are obtained after the investigations.This paper is organized as the foolowing six chapters:
In the first chapter,we give a simple introduction of the background of Bose-Einstein condensates,and review the research history of it.We also introduce the mean-field theory of dilute gases of alkali metals,the research status and improvements about the regular states of Bose-Einstein condensate and chaos.
In chapter 2,We investigate the boundary value problem(BVP)of a one- dimensional Gross-Pitaevskii equation with the spatially periodic Kronig-Penney potential(KPP)of period d,which governs a quasi-one-dimensional repulsive Bose-Einstein condensate.Under the zero and periodic boundary conditions,we show how to determine n exact stationary eigenstates{R_n}corresponding to different chemical potentials{μ_n}from the known solutions of the system.The n-th eigenstate R_n is the Jacobian elliptic function with period 2d/n for n=1,2,…,and with zero points containing the potential barrier positions.So R_n is differentiable at any spatial point and R_n~2 describes n complete wave-packets in each period of the KPP.It is revealed that one can use a laser pulse modeled by aδpotential at site x_i to manipulate the transitions from the states of{R_n}with zero point x≠x_i to the states of{R_(n')}with zero point x=x_i.The results suggest an experimental scheme for applying Bose-Einstein condensate to test the BVR and to observe the macroscopic quantum transitions.
In chapter 3,the spatial chaos of the Bose-Einstein condensate perturbed by a weak laser standing wave and a weak laserδpulse is studied.By using the perturbed chaotic solution we investigate the new type of Melnikov chaotic regions,which depend on an integration constant co determined by the boundary conditions.It is shown that when the|c_0|values are small,the chaotic region is corresponded to small values of laser wave vector k,and the chaotic region for the larger k values is related to the large|c_0|values.The result is confirmed numerically by finding the chaotic and regular orbits on the Poincar(?)section for the two different parameter regions.So for a fixed c_0 the adjustment of k from a small value to large value can transform the chaotic region into the regular one or on the contrary,that suggests a feasible method for eliminating or generating Melnikov chaos.
In chapter 4,we investigate an attractive Bose-Einstein condensate perturbed by a weak traveling optical superlattice.It is demonstrated that under a stochastic initial set and in a given parameter region the solitonic chaos appears with a certain probability which is tightly related to the zero-point number of Melnikov function,and the latter depends on the controllable potential parameters.Effects of the lattice depths and wave vectors on the chaos probability are studied analytically and numerically,and different chaotic regions of parameter space are found. The results suggest a feasible method for strengthening or weakening chaos by modulating the potential parameters experimentally.
In chapter 5,we study the spatiotemporal chaotic evolution a Bose-Einstein condensate in a moving optical lattice considering damping effects.Melnikov chaotic solution and chaotic region of parameter space are found by using the direct perturbation method.Due to the damping effects,there is a regular region with zero chaos probability for the system.In the chaotic region,the spatiotemporal evolution of the chaotic solution is analytically bounded but unpredictable between the superior and inferior limits.We demonstrate theoretically and numerically that the chaotic region reduces as the propagating velocity of the optical lattice increases.On the other hand,adding a second lattice,we find the chaotic region would widen as the intensity of it increases.In the double lattice case,the chaotic region is also strongly influenced by the velocity of the lattices.This is helpful for eliminating or generating Melnikov chaos.
In the last chapter,we give a simlpe summary and discussion to the abovementioned works,and also discuss the prospects of applicions of the Bose-Einstein condensate systems.Our main works are involded in the sencod,third,fourth and fifth chapters.
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