玻色—爱因斯坦凝聚动力学性质的数值研究
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摘要
玻色-爱因斯坦凝聚(BEC)的意义在于为物理学研究打开了一个崭新的领域。它是一种新的物质形态,可以在大尺度上展示量子特性,具有广阔的应用前景。玻色-爱因斯坦凝聚是指大量无相互作用的玻色气体在一定的低温下,一部分粒子将占据最低能量单粒子态。在玻色-爱因斯坦凝聚提出近70年后,于1995年分别实现了近理想碱金属原子铷,纳,锂气体的BEC,此后对BEC的研究如雨后春笋逐年增加。
在BEC的理论研究方面,多数理论研究都是从Gross-Pitaevskii方程(GP方程)出发的。在BEC的静态性质中通过GP方程讨论凝聚体的基态解,本征值问题,转变温度等；在BEC的动态性质中应用GP方程讨论凝聚体的涡旋,呼吸子,振荡等动力学性质；在BEC的相干性质中通过GP方程讨论凝聚体的干涉现象。GP方程应用了平均场近似,具有非线性Schrodinger方程的形式,非线性Schrodinger方程是物理学中的一个重要模型。
鉴于BEC的某些动态性质以及干涉现象还有许多值得深入研究的地方,同时为了更好的求解GP方程使计算结果更有效,本文探讨了玻色-爱因斯坦凝聚动力学性质的数值研究,以描述一维简谐势阱中中性原子的玻色-爱因斯坦凝聚的GP方程为基本方程,研究了辛格式在求解立方五次方非线性Schrodinger方程中的应用,提出了求解定态GP方程的改进的打靶法,给出了求解含时GP方程的辛算法；对比地研究了两个以及三个凝聚体的干涉现象,讨论了处于同一势阱中的两个以及三个凝聚体的动力学演化；提出将隧穿现象与干涉现象联合考虑,研究凝聚体在简谐势阱和高斯能垒中的动力学,以及之后可能发生的干涉现象。
本文研究了GP方程的求解,好的基态波函数是计算物理量以及数值模拟物理过程的基础。GP方程是描述BEC的数学模型,它具有非线性Schrodinger方程的形式,因而要得到一个准确的解析解是十分困难的,数值计算是有效途径。非线性Schrodinger方程是一个无穷维的Hamilton系统,Hamilton系统具有辛结构,Hamilton系统正则方程在辛变换下形式不变,Hamilton系统的时间演化是辛变换的演化。辛算法是Ruth和冯康提出的保持Hamilton系统辛结构的差分方法,它在保持系统整体结构上和长时间、多步数的计算中明显的好于其它算法。本文用辛算法求解立方五次方非线性Schrodinger方程,讨论其动力学性质。数值地给出了立方非线性Schrodinger方程的N孤子解；随着立方非线性参数的增加,立方非线性Schrodinger方程的解存在从拟周期状态、混沌状态到周期状态的交替演化；随着五次方非线性参数的增加,立方五次方非线性Schrodinger方程的呼吸子解退化为单孤子解。用辛算法求解了GP方程,提出了具有两个参数的改进的打靶法,将其应用于求解一维定态GP方程,给出凝聚体的基态本征值和相应的波函数,其稳定性是可以被检验的；给出了求解一维含时GP方程的辛算法,并采用缓慢增加非线性参数的方法逐步计算,这种方法既能求解凝聚体的基态波函数又能做动力学演化；介绍了谱方法和虚时演化法等其他数值方法。数值结果表明,本文的算法与其他文献方法的结果符合得很好,是合理有效的,且计算过程中能自动保持波函数的模方守恒,并在长时间多步数的计算中具有优越性。
研究了凝聚体间的千涉现象。目前实验室中可以在势阱中间建立偶极力势垒排斥原子造成两团原子云,亦可以利用光镊等技术来移动原子云,这些技术为进一步研究凝聚体的动力学性质提供了实验基础。本文数值地讨论凝聚体的干涉现象,用辛算法研究了粒子数相同,相位相同的两个凝聚体间的干涉,进而研究了三个凝聚体间的干涉,并与前者进行了比较,给出了干涉图样。数值结果表明两个凝聚体重叠后产生干涉条纹,条纹间距随着时间的演化变宽,且由于计算中在足够远处选取零边界条件,防止了计算中在边界处出现反射而影响到中心区域的现象,反射对称点处的几率密度平滑的演化。三个凝聚体间的干涉分为两个过程,首先是相邻两个凝聚体间干涉,其行为与前者相同,然后是三个凝聚体共同干涉,此后干涉条纹出现了振荡,表现为反射对称点处的粒子数几率密度出现振荡,振荡随着凝聚体间相位差的改变而改变,非线性相互作用的增大对振荡的出现起到了重要作用。研究了处于同一陷俘势中的两个及三个凝聚体的动力学演化,凝聚体呈现周期性演化,类似于简谐振动,凝聚体间的相互作用类似于孤子间的相互通过,且由于陷俘势的存在,三个凝聚体相互重叠时出现的干涉现象很强烈。两个凝聚体,尤其是三个凝聚体的动力学行为携带了丰富的物理信息,因而讨论它们的动力学现象是十分有意义的。
结合实验上的可能性,提出研究凝聚体在简谐势阱和高斯能垒中的动力学,讨论隧穿以及可能发生的干涉现象。提出先将凝聚体平移,使之偏离势阱最低点,凝聚体由于偏离了平衡位置而开始做类似简谐振动的周期运动。一段时间后将势阱关掉,释放凝聚体,同时建立高斯能垒,讨论凝聚体与势垒间的隧穿现象,结果表明当释放的凝聚体具有较大的动能时,一部分凝聚体隧穿过势垒形成一个小的波包,另一部分被反射回来与入射波形成干涉,且干涉过后演化成一个小的波包。为了控制这两个小波包的运动方向,研究两个运动的小波包可能发生的干涉,提出在建立高斯能垒的同时保持简谐势阱的束缚。研究表明发生了隧穿效应,入射波与反射波之间发生了干涉效应,且干涉随着势垒高度的增加而增强。当势垒高度增到一定程度时,隧穿现象基本被抑制。隧穿过后可以得到两个运动的凝聚体,一定条件下撤掉势阱与势垒将这两个运动的凝聚体释放,两凝聚体重叠之后发生干涉。两个运动的凝聚体干涉的发生一方面可以验证整体相位的存在,另一方面为实验探测提供了方便条件。
本文将打靶法改进并用于求解GP方程,结果是合理而有效的,可以尝试用该方法求解其他类似的非线性Schrodinger方程。将凝聚体的隧穿与干涉联合考虑研究简谐势阱和高斯能垒中凝聚体的动力学是一件十分有意义的工作,并且给出了实验上实现干涉现象的一种新途径。BEC的理论研究逐渐进入了更广阔的领域,对它的数值讨论也十分广泛,辛算法在其中亦有广泛的应用前景。
Bose-Einstein condensation (BEC) opens a new area of research in physics. It is a new form of matter which can show quantum property in large scale, and has wide prospect for application. A large number of Bose gases without interaction at certain low temperatures, a part of the atoms can all reside in the same lowest energy single atom quantum state which is called Bose-Einstein condensation. After 70 years of efforts, BEC was realized in nearly ideal alkali gases Rubidium, Sodium, and Lithium in 1995, and then the research of BEC began prosperous year after year.
In the theoretical research of BEC, many of the numerical research are based on the Gross-Pitaevskii (GP) equation. In the static property of BEC, the ground state solution of the condensate, the eigenvalue problem, and the transition temperature etc can be studied by GP equation. In the dynamic property of BEC, vortex, breather, and oscillation etc can be examined by GP equation. In the coherence study of BEC, the interference effect can be studied by GP equation. GP equation utilizes mean field approximation, and has a form of nonlinear Schrodinger equation, which is an important model in physics.
Since some dynamic property and the interference of BEC deserve more researches in depth, this paper concentrates on the numerical study of the Bose-Einstein condensates dynamics. Based on the GP equation describing the one dimensional neutral BEC in harmonic potential, apply the symplectic method in studying the dynamics of the cubic and quintic Schrodinger equation, propose an improved shooting method for solving the time independent GP equation, and present the symplectic method for solving the time dependent GP equation; Study the interference of two and three condensates and make comparison, discuss the dynamics of two and three condensates in a global potential; Take tunneling effect and interference effect as consequence process, study the dynamics of condensate in the harmonic trap and the Gauss energy barrier, and study the possible interference effect.
This paper does some research on how to solve the GP equation, because a good ground state wavefunction is the basis for solving physics quantities and numerical simulating. GP equation is a mathematic model for describing BEC, and it has a form similar with the nonlinear Schrodinger equation. It is difficult to find the analytical solution of it, and numerical computation is more efficient in solving it. Nonlinear Schrodinger equation is a Hamiltonian system of infinite dimensions, and the time evolution of the system is the evolution of symplectic transformation, that is, the Hamiltonian system has symplectic structure. Based on these, Ruth and Feng Kang derive out the symplectic method for solving the Hamiltonian system. Symplectic method is a difference method which can preserve the symplectic structure of the Hamiltonian system. It is superior to many other methods on long-time, many-step computing, and on preserving the general structure of the system. Solve the cubic and quintic nonlinear Schrodinger equation by symplectic method and studies its dynamic property. Give out the N soliton solution for the cubic nonlinear Schrodinger equation; The solution of the cubic nonlinear Schrodinger equation changes from the quasiperiodic solution, the chaotic solution to the periodic solution with the increasing of the cubic nonlinear parameter; With the increasing of the quintic nonlinear parameter, the breather solution of the cubic and quintic nonlinear Schrodinger equation collapses into the fundamental soliton solution. Solve the GP equation by symplectic method, propose an improved shooting method with two parameters, and apply it to solve the one dimension time independent GP equation, and give out the ground state eigenvalue and the corresponding wavefunction of the condensate which can be tested to be stable; Give out the symplectic method for solving the one dimension time dependent GP equation, and adopt the procedure of gradually increasing the nonlinear parameter in our calculation. This method not only can solve out the ground state wavefunction, but also can be used for dynamical calculation; Introduce other numerical methods such as the spectrum method and the imaginary time evolution method. The results reached are consist with those obtained by other methods and are tested to be correct and efficient, and our methods can preserve the norm of the wavefunction simultaneously during calculation, and are superior to other methods in long-time many-step computation.
Study the interference phenomenon of the condensates. Today in laboratory a dipole barrier can be erected in the middle of the trapping potential which can repulse the atoms, thus produce two atom clouds, and the technique of optical tweezers also can be used to displace the atom cloud, etc, all these lay the experimental ground for further studying the dynamics of the condensate. Mainly discuss the interference phenomenon of the condensates numerically, and study the interference of two condensates of the same particles and the same phases, further study the interference of three condensates, comparison is made between them and the interference patterns are presented. It shows that interference happen when the two condensates overlap, and the spacing between the interference fringes widens with time evolution. In order to prevent the reflection at the boundary from influencing the computation, adopt zero boundary condition at large enough distance, and it shows that the probability density at the reflection point of the interference pattern evolutes smoothly with time. The interference of three condensates has two stages, firstly the two neighbor condensates interfere, and it is the same as the former case, then the three condensates interfere together, in this stage the interference pattern shows oscillation. This phenomenon manifested itself by the oscillation of the probability density at the reflection point of the interference pattern, and the oscillation changes with the phase difference of the condensates, and it is believed that the increasing nonlinear interaction plays an important role in the oscillation phenomenon. This paper also studies the dynamics of two and three condensates in a global trapping potential. The condensates present periodic evolution, just like the harmonic oscillation, and the condensates evolute as the inter-passing of solitons. Due to the existence of the trapping, the interference is much stronger when the condensates overlap. The dynamic behavior of two and especially of three condensates brings much physics information, and the discussion of its dynamic phenomenon is of much interest.
Considering the possibility of realizing in laboratory, propose to study the dynamics of the condensate in a harmonic trapping potential and a Gauss energy barrier, discuss the tunneling effect and the possible interference effect. Firstly propose to displace the condensate in the harmonic trapping potential with a distance from its equilibrium, and the condensate is expected to make periodic evolution like the harmonic oscillation. Secondly shut off the potential after some time of evolution and let free the condensates, at the same time erect a Gauss energy barrier, and the tunneling effect is discussed. It shows that when the released condensate has relatively large kinetic energy, part of the condensate tunnels through the barrier and forms a small wave packet; the other part reflected back to interfere with the incident wave and a small wave packet forms after the interference. In order to control the moving of the two small wave packets, and study the interference of two moving condensates, propose to retain the existing of the trapping potential when the barrier is erected. Study shows that tunneling effect could happen and interference effect happens between the incident and reflected wave. and the interference becomes stronger with the increasing of the height of the barrier.The tunneling effect is inhibited when the height of the barrier is increased to a certain value. Two moving condensates are obtained after the tunneling process, and interference could happen when they overlap after being released under certain conditions. It not only proves the existence of the global phase, but also provides convenience for experimental probing.
This paper improves the shooting method and applies it for solving GP equation, and it shows to be correct and efficient. This method can be expected to apply for solving other similar nonlinear Schrodinger equation. This paper considers the tunneling effect and the interference effect together and studies the dynamics of condensates in a harmonic trapping potential and a Gauss energy barrier, it is a very interesting work and provides a new method for interference study experimentally. Theoretical study of BEC has reached wider area, and the numerical discussion of it is also very widely, and the symplectic method has broad application in the study of BEC.
在BEC的理论研究方面,多数理论研究都是从Gross-Pitaevskii方程(GP方程)出发的。在BEC的静态性质中通过GP方程讨论凝聚体的基态解,本征值问题,转变温度等；在BEC的动态性质中应用GP方程讨论凝聚体的涡旋,呼吸子,振荡等动力学性质；在BEC的相干性质中通过GP方程讨论凝聚体的干涉现象。GP方程应用了平均场近似,具有非线性Schrodinger方程的形式,非线性Schrodinger方程是物理学中的一个重要模型。
鉴于BEC的某些动态性质以及干涉现象还有许多值得深入研究的地方,同时为了更好的求解GP方程使计算结果更有效,本文探讨了玻色-爱因斯坦凝聚动力学性质的数值研究,以描述一维简谐势阱中中性原子的玻色-爱因斯坦凝聚的GP方程为基本方程,研究了辛格式在求解立方五次方非线性Schrodinger方程中的应用,提出了求解定态GP方程的改进的打靶法,给出了求解含时GP方程的辛算法；对比地研究了两个以及三个凝聚体的干涉现象,讨论了处于同一势阱中的两个以及三个凝聚体的动力学演化；提出将隧穿现象与干涉现象联合考虑,研究凝聚体在简谐势阱和高斯能垒中的动力学,以及之后可能发生的干涉现象。
本文研究了GP方程的求解,好的基态波函数是计算物理量以及数值模拟物理过程的基础。GP方程是描述BEC的数学模型,它具有非线性Schrodinger方程的形式,因而要得到一个准确的解析解是十分困难的,数值计算是有效途径。非线性Schrodinger方程是一个无穷维的Hamilton系统,Hamilton系统具有辛结构,Hamilton系统正则方程在辛变换下形式不变,Hamilton系统的时间演化是辛变换的演化。辛算法是Ruth和冯康提出的保持Hamilton系统辛结构的差分方法,它在保持系统整体结构上和长时间、多步数的计算中明显的好于其它算法。本文用辛算法求解立方五次方非线性Schrodinger方程,讨论其动力学性质。数值地给出了立方非线性Schrodinger方程的N孤子解；随着立方非线性参数的增加,立方非线性Schrodinger方程的解存在从拟周期状态、混沌状态到周期状态的交替演化；随着五次方非线性参数的增加,立方五次方非线性Schrodinger方程的呼吸子解退化为单孤子解。用辛算法求解了GP方程,提出了具有两个参数的改进的打靶法,将其应用于求解一维定态GP方程,给出凝聚体的基态本征值和相应的波函数,其稳定性是可以被检验的；给出了求解一维含时GP方程的辛算法,并采用缓慢增加非线性参数的方法逐步计算,这种方法既能求解凝聚体的基态波函数又能做动力学演化；介绍了谱方法和虚时演化法等其他数值方法。数值结果表明,本文的算法与其他文献方法的结果符合得很好,是合理有效的,且计算过程中能自动保持波函数的模方守恒,并在长时间多步数的计算中具有优越性。
研究了凝聚体间的千涉现象。目前实验室中可以在势阱中间建立偶极力势垒排斥原子造成两团原子云,亦可以利用光镊等技术来移动原子云,这些技术为进一步研究凝聚体的动力学性质提供了实验基础。本文数值地讨论凝聚体的干涉现象,用辛算法研究了粒子数相同,相位相同的两个凝聚体间的干涉,进而研究了三个凝聚体间的干涉,并与前者进行了比较,给出了干涉图样。数值结果表明两个凝聚体重叠后产生干涉条纹,条纹间距随着时间的演化变宽,且由于计算中在足够远处选取零边界条件,防止了计算中在边界处出现反射而影响到中心区域的现象,反射对称点处的几率密度平滑的演化。三个凝聚体间的干涉分为两个过程,首先是相邻两个凝聚体间干涉,其行为与前者相同,然后是三个凝聚体共同干涉,此后干涉条纹出现了振荡,表现为反射对称点处的粒子数几率密度出现振荡,振荡随着凝聚体间相位差的改变而改变,非线性相互作用的增大对振荡的出现起到了重要作用。研究了处于同一陷俘势中的两个及三个凝聚体的动力学演化,凝聚体呈现周期性演化,类似于简谐振动,凝聚体间的相互作用类似于孤子间的相互通过,且由于陷俘势的存在,三个凝聚体相互重叠时出现的干涉现象很强烈。两个凝聚体,尤其是三个凝聚体的动力学行为携带了丰富的物理信息,因而讨论它们的动力学现象是十分有意义的。
结合实验上的可能性,提出研究凝聚体在简谐势阱和高斯能垒中的动力学,讨论隧穿以及可能发生的干涉现象。提出先将凝聚体平移,使之偏离势阱最低点,凝聚体由于偏离了平衡位置而开始做类似简谐振动的周期运动。一段时间后将势阱关掉,释放凝聚体,同时建立高斯能垒,讨论凝聚体与势垒间的隧穿现象,结果表明当释放的凝聚体具有较大的动能时,一部分凝聚体隧穿过势垒形成一个小的波包,另一部分被反射回来与入射波形成干涉,且干涉过后演化成一个小的波包。为了控制这两个小波包的运动方向,研究两个运动的小波包可能发生的干涉,提出在建立高斯能垒的同时保持简谐势阱的束缚。研究表明发生了隧穿效应,入射波与反射波之间发生了干涉效应,且干涉随着势垒高度的增加而增强。当势垒高度增到一定程度时,隧穿现象基本被抑制。隧穿过后可以得到两个运动的凝聚体,一定条件下撤掉势阱与势垒将这两个运动的凝聚体释放,两凝聚体重叠之后发生干涉。两个运动的凝聚体干涉的发生一方面可以验证整体相位的存在,另一方面为实验探测提供了方便条件。
本文将打靶法改进并用于求解GP方程,结果是合理而有效的,可以尝试用该方法求解其他类似的非线性Schrodinger方程。将凝聚体的隧穿与干涉联合考虑研究简谐势阱和高斯能垒中凝聚体的动力学是一件十分有意义的工作,并且给出了实验上实现干涉现象的一种新途径。BEC的理论研究逐渐进入了更广阔的领域,对它的数值讨论也十分广泛,辛算法在其中亦有广泛的应用前景。
Bose-Einstein condensation (BEC) opens a new area of research in physics. It is a new form of matter which can show quantum property in large scale, and has wide prospect for application. A large number of Bose gases without interaction at certain low temperatures, a part of the atoms can all reside in the same lowest energy single atom quantum state which is called Bose-Einstein condensation. After 70 years of efforts, BEC was realized in nearly ideal alkali gases Rubidium, Sodium, and Lithium in 1995, and then the research of BEC began prosperous year after year.
In the theoretical research of BEC, many of the numerical research are based on the Gross-Pitaevskii (GP) equation. In the static property of BEC, the ground state solution of the condensate, the eigenvalue problem, and the transition temperature etc can be studied by GP equation. In the dynamic property of BEC, vortex, breather, and oscillation etc can be examined by GP equation. In the coherence study of BEC, the interference effect can be studied by GP equation. GP equation utilizes mean field approximation, and has a form of nonlinear Schrodinger equation, which is an important model in physics.
Since some dynamic property and the interference of BEC deserve more researches in depth, this paper concentrates on the numerical study of the Bose-Einstein condensates dynamics. Based on the GP equation describing the one dimensional neutral BEC in harmonic potential, apply the symplectic method in studying the dynamics of the cubic and quintic Schrodinger equation, propose an improved shooting method for solving the time independent GP equation, and present the symplectic method for solving the time dependent GP equation; Study the interference of two and three condensates and make comparison, discuss the dynamics of two and three condensates in a global potential; Take tunneling effect and interference effect as consequence process, study the dynamics of condensate in the harmonic trap and the Gauss energy barrier, and study the possible interference effect.
This paper does some research on how to solve the GP equation, because a good ground state wavefunction is the basis for solving physics quantities and numerical simulating. GP equation is a mathematic model for describing BEC, and it has a form similar with the nonlinear Schrodinger equation. It is difficult to find the analytical solution of it, and numerical computation is more efficient in solving it. Nonlinear Schrodinger equation is a Hamiltonian system of infinite dimensions, and the time evolution of the system is the evolution of symplectic transformation, that is, the Hamiltonian system has symplectic structure. Based on these, Ruth and Feng Kang derive out the symplectic method for solving the Hamiltonian system. Symplectic method is a difference method which can preserve the symplectic structure of the Hamiltonian system. It is superior to many other methods on long-time, many-step computing, and on preserving the general structure of the system. Solve the cubic and quintic nonlinear Schrodinger equation by symplectic method and studies its dynamic property. Give out the N soliton solution for the cubic nonlinear Schrodinger equation; The solution of the cubic nonlinear Schrodinger equation changes from the quasiperiodic solution, the chaotic solution to the periodic solution with the increasing of the cubic nonlinear parameter; With the increasing of the quintic nonlinear parameter, the breather solution of the cubic and quintic nonlinear Schrodinger equation collapses into the fundamental soliton solution. Solve the GP equation by symplectic method, propose an improved shooting method with two parameters, and apply it to solve the one dimension time independent GP equation, and give out the ground state eigenvalue and the corresponding wavefunction of the condensate which can be tested to be stable; Give out the symplectic method for solving the one dimension time dependent GP equation, and adopt the procedure of gradually increasing the nonlinear parameter in our calculation. This method not only can solve out the ground state wavefunction, but also can be used for dynamical calculation; Introduce other numerical methods such as the spectrum method and the imaginary time evolution method. The results reached are consist with those obtained by other methods and are tested to be correct and efficient, and our methods can preserve the norm of the wavefunction simultaneously during calculation, and are superior to other methods in long-time many-step computation.
Study the interference phenomenon of the condensates. Today in laboratory a dipole barrier can be erected in the middle of the trapping potential which can repulse the atoms, thus produce two atom clouds, and the technique of optical tweezers also can be used to displace the atom cloud, etc, all these lay the experimental ground for further studying the dynamics of the condensate. Mainly discuss the interference phenomenon of the condensates numerically, and study the interference of two condensates of the same particles and the same phases, further study the interference of three condensates, comparison is made between them and the interference patterns are presented. It shows that interference happen when the two condensates overlap, and the spacing between the interference fringes widens with time evolution. In order to prevent the reflection at the boundary from influencing the computation, adopt zero boundary condition at large enough distance, and it shows that the probability density at the reflection point of the interference pattern evolutes smoothly with time. The interference of three condensates has two stages, firstly the two neighbor condensates interfere, and it is the same as the former case, then the three condensates interfere together, in this stage the interference pattern shows oscillation. This phenomenon manifested itself by the oscillation of the probability density at the reflection point of the interference pattern, and the oscillation changes with the phase difference of the condensates, and it is believed that the increasing nonlinear interaction plays an important role in the oscillation phenomenon. This paper also studies the dynamics of two and three condensates in a global trapping potential. The condensates present periodic evolution, just like the harmonic oscillation, and the condensates evolute as the inter-passing of solitons. Due to the existence of the trapping, the interference is much stronger when the condensates overlap. The dynamic behavior of two and especially of three condensates brings much physics information, and the discussion of its dynamic phenomenon is of much interest.
Considering the possibility of realizing in laboratory, propose to study the dynamics of the condensate in a harmonic trapping potential and a Gauss energy barrier, discuss the tunneling effect and the possible interference effect. Firstly propose to displace the condensate in the harmonic trapping potential with a distance from its equilibrium, and the condensate is expected to make periodic evolution like the harmonic oscillation. Secondly shut off the potential after some time of evolution and let free the condensates, at the same time erect a Gauss energy barrier, and the tunneling effect is discussed. It shows that when the released condensate has relatively large kinetic energy, part of the condensate tunnels through the barrier and forms a small wave packet; the other part reflected back to interfere with the incident wave and a small wave packet forms after the interference. In order to control the moving of the two small wave packets, and study the interference of two moving condensates, propose to retain the existing of the trapping potential when the barrier is erected. Study shows that tunneling effect could happen and interference effect happens between the incident and reflected wave. and the interference becomes stronger with the increasing of the height of the barrier.The tunneling effect is inhibited when the height of the barrier is increased to a certain value. Two moving condensates are obtained after the tunneling process, and interference could happen when they overlap after being released under certain conditions. It not only proves the existence of the global phase, but also provides convenience for experimental probing.
This paper improves the shooting method and applies it for solving GP equation, and it shows to be correct and efficient. This method can be expected to apply for solving other similar nonlinear Schrodinger equation. This paper considers the tunneling effect and the interference effect together and studies the dynamics of condensates in a harmonic trapping potential and a Gauss energy barrier, it is a very interesting work and provides a new method for interference study experimentally. Theoretical study of BEC has reached wider area, and the numerical discussion of it is also very widely, and the symplectic method has broad application in the study of BEC.
引文
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