Density Matrix of a Bose--Einstein Condensate Steady--State versus Mean--Field Approach

1/4波片quarter-wave plateCG矢量耦合系数Clebsch-Gordan vector coupling coefficient; 简称“CG[矢耦]系数”。

X射线摄谱仪X-ray spectrographX射线衍射X-ray diffractionX射线衍射仪X-ray diffractometer[玻耳兹曼]H定理[Boltzmann] H-theorem[玻耳兹曼]H函数[Boltzmann] H-function[彻]体力body force[冲]击波shock wave[冲]击波前shock front[狄拉克]δ函数[Dirac] δ-function[第二类]拉格朗日方程Lagrange equation[电]极化强度[electric] polarization[反射]镜mirror[光]谱线spectral line[光]谱仪spectrometer[光]照度illuminance[光学]测角计[optical] goniometer[核]同质异能素[nuclear] isomer[化学]平衡常量[chemical] equilibrium constant[基]元电荷elementary charge[激光]散斑speckle[吉布斯]相律[Gibbs] phase rule[可]变形体deformable body[克劳修斯-]克拉珀龙方程[Clausius-] Clapeyron equation[量子]态[quantum] state[麦克斯韦-]玻耳兹曼分布[Maxwell-]Boltzmann distribution[麦克斯韦-]玻耳兹曼统计法[Maxwell-]Boltzmann statistics[普适]气体常量[universal] gas constant[气]泡室bubble chamber[热]对流[heat] convection[热力学]过程[thermodynamic] process[热力学]力[thermodynamic] force[热力学]流[thermodynamic] flux[热力学]循环[thermodynamic] cycle[事件]间隔interval of events[微观粒子]全同性原理identity principle [of microparticles][物]态参量state parameter, state property[相]互作用interaction[相]互作用绘景interaction picture[相]互作用能interaction energy[旋光]糖量计saccharimeter[指]北极north pole, N pole[指]南极south pole, S pole[主]光轴[principal] optical axis[转动]瞬心instantaneous centre [of rotation][转动]瞬轴instantaneous axis [of rotation]t 分布student's t distributiont 检验student's t testＫ俘获K-captureＳ矩阵S-matrixＷＫＢ近似WKB approximationＸ射线X-rayΓ空间Γ-spaceα粒子α-particleα射线α-rayα衰变α-decayβ射线β-rayβ衰变β-decayγ矩阵γ-matrixγ射线γ-rayγ衰变γ-decayλ相变λ-transitionμ空间μ-spaceχ 分布chi square distributionχ 检验chi square test阿贝不变量Abbe invariant阿贝成象原理Abbe principle of image formation阿贝折射计Abbe refractometer阿贝正弦条件Abbe sine condition阿伏伽德罗常量Avogadro constant阿伏伽德罗定律Avogadro law阿基米德原理Archimedes principle阿特伍德机Atwood machine艾里斑Airy disk爱因斯坦-斯莫卢霍夫斯基理论Einstein-Smoluchowski theory 爱因斯坦场方程Einstein field equation爱因斯坦等效原理Einstein equivalence principle爱因斯坦关系Einstein relation爱因斯坦求和约定Einstein summation convention爱因斯坦同步Einstein synchronization爱因斯坦系数Einstein coefficient安[培]匝数ampere-turns安培[分子电流]假说Ampere hypothesis安培定律Ampere law安培环路定理Ampere circuital theorem安培计ammeter安培力Ampere force安培天平Ampere balance昂萨格倒易关系Onsager reciprocal relation凹面光栅concave grating凹面镜concave mirror凹透镜concave lens奥温电桥Owen bridge巴比涅补偿器Babinet compensator巴耳末系Balmer series白光white light摆pendulum板极plate伴线satellite line半波片halfwave plate半波损失half-wave loss半波天线half-wave antenna半导体semiconductor半导体激光器semiconductor laser半衰期half life period半透[明]膜semi-transparent film半影penumbra半周期带half-period zone傍轴近似paraxial approximation傍轴区paraxial region傍轴条件paraxial condition薄膜干涉film interference薄膜光学film optics薄透镜thin lens保守力conservative force保守系conservative system饱和saturation饱和磁化强度saturation magnetization本底background本体瞬心迹polhode本影umbra本征函数eigenfunction本征频率eigenfrequency本征矢[量] eigenvector本征振荡eigen oscillation本征振动eigenvibration本征值eigenvalue本征值方程eigenvalue equation比长仪comparator比荷specific charge; 又称“荷质比(charge-mass ratio)”。

2022年自考专业(英语)英语科技文选考试真题及答案一、阅读理解题Directions: Read through the following passages. Choose the best answer and put the letter in the bracket. (20%)1、 (A) With the recent award of the Nobel Prize in physics, the spectacular work on Bose-Einstein condensation in a dilute gas of atoms has been honored. In such a Bose-Einstein condensate, close to temperatures of absolute zero, the atoms lose their individuality and a wave-like state of matter is created that can be compared in many ways to laser light. Based on such a Bose-Einstein condensate researchers in Munich together with a colleague from the ETH Zurich have now been able to reach a new state of matter in atomic physics. In order to reach this new phase for ultracold atoms, the scientists store a Bose-Einstein condensate in a three-dimensional lattice of microscopic light traps. By increasing the strength of the lattice, the researchers are able to dramatically alter the properties of the gas of atoms and can induce a quantum phase transition from the superfluid phase of a Bose-Einsteincondensate to a Mott insulator phase. In this new state of matter it should now be possible to investigate fundamental problems of solid-state physics, quantum optics and atomic physics. For a weak optical lattice the atoms form a superfluid phase of a Bose-Einstein condensate. In this phase, each atom is spread out over the entire lattice in a wave-like manner as predicted by quantum mechanics. The gas of atoms may then move freely through the lattice. For a strong optical lattice the researchers observe a transition to an insulating phase, with an exact number of atoms at each lattice site. Now the movement of the atoms through the lattice is blocked due to therepulsive interactions between them. Some physicists have been able to show that it is possible to reversibly cross the phase transition between these two states of matter. The transition is called a quantum phase transition because it is driven by quantum fluctuations and can take place even at temperatures of absolute zero. These quantum fluctuations are a direct consequence of Heisenberg’s uncertainty relation. Normally phase transitions are driven by thermal fluctuations, which are absent at zero temperature. With their experiment, the researchers in Munich have been able to enter a new phase in the physics of ultracold atoms. In the Mott insulator state theatoms can no longer be described by the highly successful theories for Bose-Einstein condensates. Now theories are required that take into account the dominating interactions between the atoms and which are far less understood. Here the Mott insulator state may help in solving fundamental questions of strongly correlated systems, which are the basis for our understanding of superconductivity. Furthermore, the Mott insulator state opens many exciting perspectives for precision matter-wave interferometry and quantum computing.What does the passage mainly discuss?A.Bose-Einstein condensation.B.Quantum phase transitions.C.The Mott insulator state.D.Optical lattices.2、What will the scientists possibly do by reaching the new state of matter in atomic physics?A.Store a Bose-Einstein condensate in three-dimensional lattice of microscopic light traps.B.Increase the strength of the lattice.C.Alter the properties of the gas of atoms.D.Examine fundamental problems of atomic physics.3、Which of the following is NOT mentioned in relation to aweak optical lattice?A.The atoms form a superfluid phase of a Bose-Einstein condensate.B.Each atom is spread out over the entire lattice.C.The gas of atoms may move freely through the lattice.D.The superfluid phase changes into an insulating phase.4、What can be said about the quantum phase transition?A.It can take place at temperatures of absolute zero.B.It cannot take place above the temperatures of absolute zero.C.It is driven by thermal fluctuations.D.It is driven by the repulsive interactions between atoms.5、The author implies all the following about the Mott insulator state EXCEPT that______.A.the theory of Bose-Einstein condensation can’t possibly account for the atoms in the Mott insulator stateB.not much is known about the dominating interactions between the atoms in the Mott insulator stateC.it offers new approaches to exact quantum computingD.it forms a superfluid phase of a Bose-Einstein condensate6、 (B) Gene therapy and gene-based drugs are two ways we would benefit from our growing mastery of genetic science. But therewill be others as well. Here is one of the remarkable therapies on the cutting edge of genetic research that could make their way into mainstream medicine in the c oming years. While it’s true that just about every cell in the body has the instructions to make a complete human, most of those instructions are inactivated, and with good reason: the last thing you want for your brain cells is to start churning out stomach acid or your nose to turn into a kidney. The only time cells truly have the potential to turn into any and all body parts is very early in a pregnancy, when so-called stem cells haven’t begun to specialize. Most diseases involve the death of healthy cells—brain cells in Alzheimer’s, cardiac cells in heart disease, pancreatic cells in diabetes, to name a few; if doctors could isolate stem cells, then direct their growth, they might be able to furnish patients with healthy replacement tissue. It was incredibly difficult, but last fall scientists at the University of Wisconsin managed to isolate stem cells and get them to grow into neural, gut, muscle and bone cells. The process still can’t be controlled, and may have unforeseen limitations; but if efforts to understand and master stem-cell development prove successful, doctors will have a therapeutic tool of incredible power. The same applies to cloning, whichis really just the other side of the coin; true cloning, as first shown, with the sheep Dolly two years ago, involves taking a developed cell and reactivating the genome within, resenting its developmental instructions to a pristine state. Once that happens, the rejuvenated cell can develop into a full-fledged animal, genetically identical to its parent. For agriculture, in which purely physical characteristics like milk production in a cow or low fat in a hog have real market value, biological carbon copies could become routine within a few years. This past year scientists have done for mice and cows what Ian Wilmut did for Dolly, and other creatures are bound to join the cloned menagerie in the coming year. Human cloning, on the other hand, may be technically feasible but legally and emotionally more difficult. Still, one day it will happen. The ability to reset body cells to a pristine, undeveloped state could give doctors exactly the same advantages they would get from stem cells: the potential to make healthy body tissues of all sorts. And thus to cure disease.That could prove to be a true “miracle cu re”.What is the passage mainly about?A.Tomorrow’s tissue factory.B.A terrific boon to medicine.C.Human cloning.D.Genetic research.7、 According to the passage, it can be inferred that which of the following reflects the author’s opinion?A.There will inevitably be human cloning in the coming year.B.The potential to make healthy body tissues is undoubtedly a boon to human beings.C.It is illegal to clone any kind of creatures in the world.D.It is legal to clone any kind of creatures in the world except human.8、Which of the following is NOT true according to the passage?A.Nearly every cell in the human brain has the instructions to make a complete human.B.It is impossible for a cell in your nose to turn into a kidney.C.It is possible to turn out healthy replacement tissues with isolated stem cells.D.There will certainly appear some new kind of cloned animal in the near future.9、All of the following are steps involved in true cloning EXCEPT_______.A.selecting a stem cellB.taking a developed cellC.reactivating the genome within the developed cellD.resetting the developmental instructions in the cell to its original state10、The word “rejuvenated” in para. 5 is closest in meaning to_______.A.rescuedB.reactivatedC.recalledD.regulated参考答案：【一、阅读理解题】1~5CDDAD6~10DBBA。

Bose-Einstein condensationShihao LiBJTU ID#:13276013;UW ID#:20548261School of Science,Beijing Jiaotong University,Beijing,100044,ChinaJune1,20151What is BEC?To answer this question,it has to begin with the fermions and bosons.As is known,matters consist of atoms,atoms are made of protons,neutrons and electrons, and protons and neutrons are made of quarks.Also,there are photons and gluons that works for transferring interaction.All of these particles are microscopic particles and can be classified to two families,the fermion and the boson.A fermion is any particle characterized by Fermi–Dirac statistics.Particles with half-integer spin are fermions,including all quarks,leptons and electrons,as well as any composite particle made of an odd number of these,such as all baryons and many atoms and nuclei.As a consequence of the Pauli exclusion principle,two or more identical fermions cannot occupy the same quantum state at any given time.Differing from fermions,bosons obey Bose-Einstein statistics.Particles with integer spin are bosons,such as photons,gluons,W and Z bosons,the Higgs boson, and the still-theoretical graviton of quantum gravity.It also includes the composite particle made of even number of fermions,such as the nuclei with even number ofnucleons.An important characteristic of bosons is that their statistics do not restrict the number of them that occupy the same quantum state.For a single particle,when the temperature is at the absolute zero,0K,the particle is in the state of lowest energy,the ground state.Supposing that there are many particle,if they are fermions,there will be exactly one of them in the ground state;if they are bosons,most of them will be in the ground state,where these bosons share the same quantum states,and this state is called Bose-Einstein condensate (BEC).Bose–Einstein condensation(BEC)—the macroscopic groundstate accumulation of particles of a dilute gas with integer spin(bosons)at high density and low temperature very close to absolute zero.According to the knowledge of quantum mechanics,all microscopic particles have the wave-particle duality.For an atom in space,it can be expressed as well as a wave function.As is shown in the figure1.1,it tells the distribution but never exact position of atoms.Each distribution corresponds to the de Broglie wavelength of each atom.Lower the temperature is,lower the kinetic energy is,and longer the de Broglie wavelength is.p=mv=h/λ(Eq.1.1)When the distance of atoms is less than the de Broglie wavelength,the distribution of atoms are overlapped,like figure1.2.For the atoms of the same category,the overlapped distribution leads to a integral quantum state.If those atoms are bosons,each member will tend to a particular quantum state,and the whole atomsystem will become the BEC.In BEC,the physical property of all atoms is totally identical,and they are indistinguishable and like one independent atom.Figure1.1Wave functionsFigure1.2Overlapped wave functionWhat should be stressed is that the Bose–Einstein condensate is based on bosons, and BEC is a macroscopic quantum state.The first time people obtained BEC of gaseous rubidium atoms at170nK in lab was1995.Up to now,physicists have found the BEC of eight elements,most of which are alkali metals,calcium,and helium-4 atom.Always,the BEC of atom has some amazing properties which plays a vital role in the application of chip technology,precision measurement,and nano technology. What’s more,as a macroscopic quantum state,Bose–Einstein condensate gives a brand new research approach and field.2Bose and Einstein's papers were published in1924.Why does it take so long before it can be observed experimentally in atoms in1995?The condition of obtaining the BEC is daunting in1924.On the one hand,the temperature has to approach the absolute zero indefinitely;on the other hand,the aimed sample atoms should have relatively high density with few interactions but still keep in gaseous state.However,most categories of atom will easily tend to combine with others and form gaseous molecules or liquid.At first,people focused on the BEC of hydrogen atom,but failed to in the end. Fortunately,after the research,the alkali metal atoms with one electron in the outer shell and odd number of nuclei spin,which can be seen as bosons,were found suitable to obtain BEC in1980s.This is the first reason why it takes so long before BEC can be observed.Then,here’s a problem of cooling atom.Cooling atom make the kinetic energy of atom less.The breakthrough appeared in1960s when the laser was invented.In1975, the idea of laser cooling was advanced by Hänsch and Shallow.Here’s a chart of the development of laser cooling:Year Technique Limit Temperature Contributors 1980～Laser cooling of the atomic beam～mK Phillips,etc. 19853-D Laser cooling～240μK S.Chu,etc. 1989Sisyphus cooling～0.1~1μK Dalibard,etc. 1995Evaporative cooling～100nK S.Chu,etc. 1995The first realization of BEC～20nK JILA group Until1995,people didn’t have the cooling technique which was not perfect enough,so that’s the other answer.By the way,the Nobel Prize in Physics1997wasawarded to Stephen Chu,Claude Cohen－Tannoudji,and William D.Phillips for the contribution on laser cooling and trapping of atoms.3Anything you can add to the BEC phenomena(recent developments,etc.)from your own Reading.Bose–Einstein condensation of photons in an optical microcavity BEC is the state of bosons at extremely low temperature.According to the traditional view,photon does not have static mass,which means lower the temperature is,less the number of photons will be.It's very difficult for scientists to get Bose Einstein condensation of photons.Several German scientists said they obtained the BEC of photon successfully in the journal Nature published on November24th,2011.Their experiment confines photons in a curved-mirror optical microresonator filled with a dye solution,in which photons are repeatedly absorbed and re-emitted by the dye molecules.Those photons could‘heat’the dye molecules and be gradually cooled.The small distance of3.5 optical wavelengths between the mirrors causes a large frequency spacing between adjacent longitudinal modes.By pumping the dye with an external laser we add to a reservoir of electronic excitations that exchanges particles with the photon gas,in the sense of a grand-canonical ensemble.The pumping is maintained throughout the measurement to compensate for losses due to coupling into unconfined optical modes, finite quantum efficiency and mirror losses until they reach a steady state and become a super photons.(Klaers,J.,Schmitt,J.,Vewinger, F.,&Weitz,M.(2010).Bose-einstein condensation of photons in an optical microcavity.Nature,468(7323), 545-548.)With the BEC of photons,a brand new light source is created,which gives a possible to generate laser with extremely short wavelength,such as UV laser and X-ray laser.What’s more,it shows the future of powerful computer chip.Figure3.1Scheme of the experimental setup.4ConclusionA Bose-Einstein condensation(BEC)is a state of matter of a dilute gas of bosons cooled to temperatures very close to absolute zero.Under such conditions,a large fraction of bosons occupy the lowest quantum state,at which point macroscopic quantum phenomena become apparent.This state was first predicted,generally,in1924-25by Satyendra Nath Bose and Albert Einstein.And after70years,the Nobel Prize in Physics2001was awarded jointly to Eric A.Cornell,Wolfgang Ketterle and Carl E.Wieman"for theachievement of Bose-Einstein condensation in dilute gases of alkali atoms,and for early fundamental studies of the properties of the condensates".This achievement is not only related to the BEC theory but also the revolution of atom-cooling technique.5References[1]Pethick,C.,&Smith,H.(2001).Bose-einstein condensation in dilute gases.Bose-Einstein Condensation in Dilute Gases,56(6),414.[2]Klaers J,Schmitt J,Vewinger F,et al.Bose-Einstein condensation of photons in anoptical microcavity[J].Nature,2010,468(7323):545-548.[3]陈徐宗,&陈帅.(2002).物质的新状态——玻色-爱因斯坦凝聚——2001年诺贝尔物理奖介绍.物理,31(3),141-145.[4]Boson(n.d.)In Wikipedia.Retrieved from:</wiki/Boson>[5]Fermion(n.d.)In Wikipedia.Retrieved from:</wiki/Fermion>[6]Bose-einstein condensate(n.d.)In Wikipedia.Retrieved from:</wiki/Bose%E2%80%93Einstein_condensate>[7]玻色-爱因斯坦凝聚态(n.d.)In Baidubaike.Retrieved from:</link?url=5NzWN5riyBWC-qgPhvZ1QBcD2rdd4Tenkcw EyoEcOBhjh7-ofFra6uydj2ChtL-JvkPK78twjkfIC2gG2m_ZdK>。

偶极玻色-爱因斯坦凝聚体中孤子碰撞的理论研究SHI Yu-ren;YANG Xue-ying;TANG Na;LI Xiao-lin;SONG Lin【摘要】对准一维情形下具有偶极相互作用的玻色-爱因斯坦凝聚体(Bose-Einstein condensate,BEC)中孤子的碰撞进行了理论研究.运用虚时演化法数值求解了Gross-Pitaevskii方程的孤子态解,然后构造了实验中可实现的双孤子态以研究其碰撞规律.发现既存在完全弹性碰撞,也存在完全非弹性碰撞.通过调节偶极作用强度,可实现从弹性碰撞到非弹性碰撞的转变.初始时刻孤子的相位差不仅会影响系统的对称性,也会改变孤子的碰撞类型.【期刊名称】《西北师范大学学报（自然科学版）》【年(卷),期】2019(055)004【总页数】8页(P44-51)【关键词】偶极BEC;孤子;碰撞【作者】SHI Yu-ren;YANG Xue-ying;TANG Na;LI Xiao-lin;SONG Lin【作者单位】;;;;【正文语种】中文【中图分类】O145玻色-爱因斯坦凝聚(Bose-Einstein condensate, BEC)是物质的一种新型状态.自从实验上发现BEC中的亮孤子后[1]，冷原子中孤子的行为便受到广泛关注[2-8].在不同的囚禁外势下，孤子的周期、能量变化都有很大不同.研究表明，通过减小轴向频率和径向频率的比值和原子的损耗，可以延长孤子的寿命，产生非常丰富的新奇现象[9-11].混合冷原子中孤子的特性也令研究者产生了极大兴趣[12-13].由于短距离自旋极化费米子之间强烈的泡利阻塞排斥作用，在反射玻色子-费米子相互作用中不可能存在费米亮孤子.Sadhan[14]等证明稳定的费米亮孤子可以在玻色-费米混合气体中形成.此外，当孤子间相互作用不同时，孤子的性质也会发生改变15-19].例如分子类型的相互作用，使得许多孤子可以存在，其中包括串形、环状或规则格子型孤子分子，其动力学行为会发生很大变化[20].近年来，在实验和理论的研究中量子简并气体的远程相互作用受到很多关注.偶极相互作用是长程力，且各向异性，这些特性会影响凝聚体的基态、稳定性及动力学性质.这些性质提供了一种研究多体量子效应的方式，例如,超流晶体的量子相变、超固体，甚至是拓扑序列等.许多学者针对偶极BEC中孤子间相互作用下的动力学性质也做了大量研究[21-24].Pedri[25]数值研究了准二维情形下具有偶极相互作用的亮孤子碰撞，发现亮孤子在碰撞后合并，这是一种典型的完全非弹性碰撞.文中主要研究准一维偶极BEC中孤子的碰撞.采用虚时演化法得到凝聚体的孤子态，在谐振子势下探究初始孤子的相位差对碰撞的影响.1 理论模型考虑束缚在谐振子势阱中的偶极BEC，在平均场理论框架内，体系的动力学行为可以用Gross-Pitaevskii(GP)方程描述[26]其中,m为粒子质量;Ψ为波函数;满足为总粒子数;原子间相互作用强度g=4πћ2as/m,as为s-波散射长度;外势分别为径向和轴向频率;ρ和z为径向和轴向坐标;偶极相互作用项为极化方向单位矢量;对于磁偶极情形，Cdd=μ0μ2,μ0为真空磁导率，对于电偶极情形，为真空介电常数，为玻尔半径.引入可对方程(1)进行无量纲化.当ωρ≫ωz时，方程(1)可化为准一维GP方程[27]其中,变量上面的“～”已略去;“*”表示傅氏卷积;表征变换后的接触相互作用强度;表征变换后偶极作用强度，为偶极作用与接触作用强度的比值，为z方向的特征长度;为余误差函数;波函数满足2 数值结果2.1 准一维单分量BEC的孤子态单分量BEC在实验上容易操控，对其进行深入探究将有利于对BEC的特性更深入的了解.Thierry[28]等在实验上实现了准二维的具有强偶极相互作用的52Cr原子BEC，通过调节外加磁场减弱s-波散射长度，从而使52Cr原子的偶极相互作用强度变得与接触相互作用可比拟，这将导致原子云的长宽比发生变化.下面研究准一维情形下偶极相互作用与接触相互作用对52Cr原子BEC的影响，采用虚时演化法[29]可得到GP方程(2)的孤子态.文献[27]中给出了动能项与偶极项均忽略时孤子态的解析结果.下面用变分法求解当动能项保留而偶极项忽略(即εdd=0)时的孤子解，即GP方程(2).其拉格朗日密度为[30]采用高斯波包作为拟设(假设波包中心位于z=0处)则有效拉格朗日量为变分参数wz的欧拉—拉格朗日方程为(6)一定条件下，方程(6)反映了在给定初始条件下孤立波的振幅及波宽随时间变化的规律.考虑将N=104个52Cr原子束缚在谐振子势阱中. 52Cr原子磁偶极矩为6μB(μB 为玻尔磁子)，原子质量m=8.63×10-26 kg，极化方向取为n=(0,0,1)，谐振子频率(ωρ，ωz)=2π(350,35)Hz,则利用Feshbach共振技术[31]可调节接触相互作用系数β1D，调节外磁场可以改变偶极相互作用强度εdd，这样可保证与Thierry 等实验所用参数[28]一致.图1给出了不同参数情况下的粒子数分布图，其中，NS(Numerical solution)表示用虚时演化法得到的数值结果;AS(Analytical solution)表示用变分法得到的解析结果.可以看出，粒子数分布均呈现出钟状孤立子态.图1(a～b)中所用参数分别为β1D=10,εdd=0.4和β1D=100，εdd=0.6，由此计算得到η≈9.55,λ≈-21.07和η≈63.66,λ≈-316.12.在该参数情形下，均有η>0且λ<0，表明近程作用表现为排斥而长程作用为吸引.从图1(a～b)可看出，解析结果与数值结果在较大范围内基本吻合.但图1a中数值结果的振幅比解析结果的要大，这是因为在变分法计算时忽略了偶极相互作用；而在此参数条件下，偶极作用表现为吸引，这会使得孤子振幅增大.图1b中亦是如此.图1c给出了εdd=0.4时粒子数密度随β1D的变化.可以看出，粒子数密度的峰值(可视为孤子的振幅)随着β1D的增加而增大.图1d给出了β1D=100时粒子数密度随εdd的变化.可以看出，孤子振幅随εdd增加而变大，同时孤子的宽度减小.这是由于偶极作用(此时为吸引)增强的缘故.这些结论与文献[28]中实验结果一致.图1 不同参数情形下粒子数密度分布Fig 1 Particle number density distribution under different parameters为进一步研究粒子数密度与原子间接触相互作用及偶极相互作用的关系，图2给出了孤立波振幅max(|Ψ|2)随β1D及εdd的变化.从图2a可以看出，当εdd较小时，孤立波振幅随β1D的增大而逐渐减小；而当εdd较大时，孤立波振幅则随β1D的增加而增大.图2b给出了不同β1D时孤立波振幅随εdd的变化.可以看出，β1D一定时，孤立波振幅随εdd的增加而单调增加(这一点在图2(a)中也有所体现).这是偶极作用与接触作用相互竞争的结果.可解释如下：在图2所示参数条件下，可见始终有λ<0，表明偶极作用始终表现为吸引.吸引作用将会导致孤立波振幅增加,而排斥作用则相反.当εdd>1时，η<0；而当εdd<1时，η>0.说明通过调节偶极作用强度，也可改变接触相互作用的性质.另外，此时当εdd较小时(接近0)η>0而相对较大，说明接触相互作用表现为排斥且排斥作用强于吸引作用，故孤立波振幅较小.而当εdd较大时，相对较小，说明排斥作用减弱，故会导致孤立波振幅变大.图2 不同参数情形下孤立波振幅随β1D及εdd的变化Fig 2 T he variation of solitary wave amplitude with β1D and εdd under different parameters2.2 准一维单分量BEC孤子态的稳定性孤子的动力学稳定性是一个非常重要的问题.不稳定的孤波结构不能长时间存在，而稳定的孤波具有强的抗干扰能力，可以长时间存在，便于实验上观察和进一步研究.Ueda考虑该系统的平衡态(粒子均匀分布的情形)，通过计算能量得出,无外势情形下，当-0.5≤εdd≤1时,该态呈稳定性,在外势作用下系统将更加稳定[32].我们用数值方法对系统孤子态的稳定性进行研究，发现在所计算的参数范围0<εdd<2内，孤子态均呈现非常强的稳定性.此结果与系统平衡态下的稳定性有很大不同，也符合孤子的特性.数值研究时采用以下做法,用虚时演化法得到GP方程的孤子态Ψ=φ0(z,t)后，当t=0时刻，在该态上加一微小扰动做为初态Ψ(z,0)=φ0(z,0)然后用时间劈裂傅里叶谱方法[33]对GP方程进行长时间动力学演化，便可研究该孤子态的动力学稳定性. 计算时，取A=0.001，W根据孤子的宽度做适当调整.在不同参数情况下的时间步长也需要调整以保证数值稳定性.图3 不同参数下粒子数密度随时间t的变化Fig 3 Variation of particle number density with time t under different parameters图3给出了不同接触作用和偶极作用强度时，粒子数密度随时间t的变化.可以看出，粒子数密度呈钟状孤子态分布，在扰动下并不随时间发生明显的变化，表明该态是动力学稳定的.为保证数值计算精度，图3中空间网格数取为2 048；为保证数值稳定性，图3a计算时时间步长(无量纲化的)取为10-7；图3b中则需取为10-8.这使得计算量急剧增大.比较图3a,b可以看出，随着偶极作用系数的增加，波包明显变得窄而“瘦高”，这是因为偶极作用表现为很强的吸引作用.从图3b可看出，即使在εdd=1.8的情形下，孤子态仍保持稳定.数值计算时，在尝试的参数范围0<β1D≤200,0<ε<εdd内，均没发现不稳定的孤子态.这种较强的稳定性对于孤子在量子信息、非线性光学、原子输运和原子干涉仪等领域内的应用有着重要的理论指导意义.2.3 双孤子碰撞碰撞是孤子重要的动力学性质之一，影响碰撞的因素也有很多.实验中，可以在系统中放置两份制备好的孤子态BEC以观察孤子之间的碰撞现象.理论研究中可采取如下方式来模拟此碰撞过程,首先用虚时演化法得到GP方程(2)的孤立波解，记此态为Ψ=Ψs(z,0).然后通过空间坐标平移，从而得到两份(甚至更多份)孤子态BEC，分别记为Ψ1=Ψs(z-z0,0)和Ψ2=Ψs(z+z0,0)，其中z0为空间平移量.这样制备的双孤子在初始时刻空间位置沿z=0点呈对称分布(理论和实验中均可研究非对称情形，但由于篇幅原因，本文仅研究对称情形).接着取Ψ(z,0)=Ψ1+Ψ2eiΔθ作为初始条件对GP方程(2)进行动力学演化，便可研究双孤子之间的碰撞，这里为初始时刻两孤子态的相位差.进行动力学演化时，采用时间劈裂傅里叶谱方法[33].这种方法精度高，有保持粒子数守恒的优点，被广泛应用于BEC系统的理论模拟[34-37].文献[25]中，Pedri等数值研究了准二维情形下具有偶极相互作用的亮孤子碰撞，发现亮孤子在碰撞后发生合并.这是一种典型的完全非弹性碰撞.但在准一维情形下，我们发现存在两孤立波的完全弹性碰撞.图4a给出了Δθ=0,β1D=50,β1D=50,εdd=0.3时两孤立波的完全弹性碰撞过程.刚开始时两孤立波静止，但在外势和吸引作用下会逐渐加速，相向而行.一段时间后发生碰撞，且在碰撞过程中伴随物质波的干涉现象.碰撞结束后，两孤立波穿过彼此后变为背向而行，并逐渐减速，速度减为0后又重复前述碰撞过程.图4b～e放大给出了孤立波的碰撞过程，从中可以清晰地观察到物质波的干涉现象.Δθ的变化会对干涉条纹产生影响；当Δθ=0时(图4b)，两孤子碰撞时出现五个峰值，且中心位置处也为极大值.当Δθ=π时(图4d)，虽然干涉条纹仍以z=0为中心呈左右对称分布，但中心位置处变为极小值；和时(图4c,e)，碰撞过程中干涉条纹的对称性也不复存在；Δθ取其它值时，也有类似的干涉现象.图4 两孤立波的碰撞(β1D=50,εdd=0.3)Fig 4 Collision of two solitarywaves(β1D=50,εdd=0.3)准一维情形下，也存在类似Pedri等发现的非弹性碰撞.图5给出了β1D=200,εdd=0.6时不同初始相位差情形下双孤子的碰撞.图5a中Δθ=0，初始时刻两孤子均静止.在外势和偶极作用下，它们逐渐加速，相向而行，过段时间后发生碰撞.碰撞后合二为一，然后在z=0附近左右振荡，且振荡幅度随时间逐渐变小.这是一种完全非弹性碰撞,图5b中两孤子在第一次碰撞后穿过彼此，且发生能量转移，对称性也被破坏.这是较典型的非完全弹性碰撞.碰撞后向右运动的孤子振幅变大，且运动相对较短的距离后便向左返回；而向左运动的孤子振幅变小，运动相对较长的距离后便向右运动,然后两孤子再次发生碰撞.二次碰撞后两孤子合并，为完全非弹性碰撞,合并后的孤子在z=0附近左右振荡，但振荡幅度较Δθ=0的情形(参看图5a)大得多.类似的情形在图5c中也存在，和图5b相比仅是在空间上发生了翻转.图5d给出了Δθ=π时双孤子的碰撞.可以看出，两孤子在经过几次非完全弹性碰撞后合二为一，然后在z=0附近作周期性振荡.图5 势场中不同初始相位差时双孤子的碰撞Fig 5 Collisions of double solitons under different initial phase differences in the potential field4 结束语数值研究了准一维偶极玻色-爱因斯坦凝聚体中双孤子的碰撞.运用虚时演化法数值求出GP方程的孤子态解，然后构造了实验中可实现的双孤子态以研究其碰撞规律.发现不仅存在完全弹性碰撞和完全非弹性碰撞，还存在从弹性碰撞到非弹性碰撞的转变.碰撞过程中存在明显的物质波干涉现象.初始时刻孤子的相位差不仅会影响系统的对称性，还会改变孤子的碰撞类型.当初始相位差为0或π时，粒子密度分布具有很好地空间对称性；当初始相位差为其它值时，这种对称性被破坏.这些性质表明BEC在原子运输和量子信息方面具有潜在的应用价值.研究结果可为实验上偶极BEC在实验上的研究提供可能的理论指导.参考文献:【相关文献】[1] STRECKER K E,PARTRIDGE G B,TRUSCOTT A G,et al.Formation and propagation of matter-wave soliton 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玻色-爱因斯坦凝聚论文：玻色-爱因斯坦凝聚孤子非谐势阱散射长度【中文摘要】自从实验观察到二元玻色-爱因斯坦凝聚体(Bose-Einstein Condensates, BEC)现象以来,有关多组分BEC中的非线性研究已成为目前物质波研究领域中广泛关注的热点之一。

实验上,可调控的宏观物理量有:囚禁BEC的外部势阱和可利用Feshbach 共振技术来控制的原子间相互作用强度。

对于多组分BEC,原子间的相互作用不仅存在种内相互作用,还存在种间相互作用。

理论上,二元BEC的相关物理性质均可采用平均场近似理论下的耦合Gross-Pitaevskii(GP)方程来描述。

本文从GP方程出发,利用多重尺度方法,研究了非谐外部势阱中的二元BEC中的孤子动力学行为和随时间变化的种间相互作用强度对二元BEC中孤子碰撞行为的影响。

全文共分为四章,主要结构如下:第一章,介绍了BEC的相关基础知识、基本理论,简要回顾了二元BEC的相关实验及当前的理论研究现状。

同时,基于平均场理论,简扼推导出描述BEC动力学的GP方程。

最后,对我们所采用的研究方法—多重尺度方法和论文的研究内容进行了简明扼要的介绍。

第二章,利用多重尺度方法,解析地研究了四次非谐势调制下的二元BEC中的孤子融合现象。

结果表明,凝聚体中两个不同组分中的孤子会发生融合现象。

且随着四次非谐外部势阱强度的增加,融合现象变得更加迅速。

从而证实,二元BEC中两孤子的融合行为可通过外部非谐势阱调控。

第三章,解析地研究了随时间变化的种间散射长度对二元BEC中孤子动力学行为的影响。

结果表明,两个孤子间发生碰撞的位置、时间和频率均与种间散射长度密切相关。

也就是说,二元BEC中的孤子碰撞行为可以通过种间散射长度来调控。

与此同时,我们发现孤子的幅度也可以利用种间散射长度来调控。

最后一章,对本文做了一个简单的总结,且对下一步研究工作进行了展望。

【英文摘要】Since the observation of two-componentBose-Einstein condensates (BECs) in the experiments, there are plenty of researches concentrating on the nonlinear phenomena of multi-component BEC. Experimentally, the controllable two macroscopical parameters are the external trapping potential and the strength of interatomic interactions. And the interatomic interactions could be modulated by means of Feshbach Resonance. In multi-component BECs, the interatomic interactions include both the intraspecies interactions and the interspecies interactions. Theoretically, the ultra-cold two-component BECs system in the mean field approximation can be well described by coupled time-dependent Gross-Pitaevskii (GP) equations. In this thesis, beginning with GP equations and applying multiple-scale method, we analytically study the dynamical properties of the two-component Bose-Einstein condensates trapped in a harmonic plus quartic anharmonic potential, and the dynamical properties and collision properties of the solitons in two-component BECs withtime-dependent interspecies interactions. The thesis is organized as follows:In chapter one, we introduce the elementary knowledge, the basic concept and theory of BEC. Then, we state the related experimental implementation and current theoretical research of two-component BECs. Based on the mean-field theory, we briefly deduce the GP equations, which govern the dynamics of the condensates. Finally, we present the multiple-scale method, which will be used in the following chapters for theoretical analysis, and give a summary of our work in this thesis at the end of this chapter.In chapter two, by using the multiple-scale method, we analytically study dynamical properties of two-component BECs trapped in a harmonic plus quartic anharmonic potential. It is shown that the anharmonic potential has an important effect on the dark solitons of the condensates. Especially, when the strength of the anharmonic external potential increases, the fusion of the two solitons becomes faster. This implies that the fusion of the two solitons can be controlled by an anharmonic potential.In chapter three, we analytically study the soliton dynamical properties of two-component BECs with time-dependent interspecies scattering length by using the multiple-scale method. It is shown that the interspecies scattering length hasan important effect on the solitons collision property of the condensates. The position, the time, and the frequency of the collision between two solitons are relative to thetime-dependent interspecies scattering length of the condensates. That is to say, the collision property of the two solitons in two-component Bose-Einstein condensates can be controlled by the time-dependent interspecies scattering length. Additionally, the amplitude of the solitons is also close related to the time-dependent interspecies scattering length.In Final chapter, we make a summary of our work and give some prospects in future works.【关键词】玻色-爱因斯坦凝聚孤子非谐势阱散射长度【英文关键词】Bose-Einstein condensates Soliton Anharmonic potential Scattering length【目录】二元玻色—爱因斯坦凝聚中的孤子动力学摘要4-5Abstract5-6第1章绪论8-22 1.1 玻色-爱因斯坦凝聚简介8-11 1.2 二元玻色-爱因斯坦凝聚11-13 1.3 二元玻色-爱因斯坦凝聚的研究现状13-18 1.3.1 二元玻色-爱因斯坦凝聚的实验观察14-16 1.3.2 二元玻色-爱因斯坦凝聚中的非线性物理研究16-18 1.4 本文主要研究方法、内容及意义18-22 1.4.1本文的主要研究方法—多重尺度方法18-20 1.4.2 本文的主要研究内容及意义20-22第2章非谐外部势对二元BEC 中孤子动力学的影响22-31 2.1 引言22 2.2 理论模型22-24 2.3 多重尺度展开及孤子解析解24-28 2.4 非谐外部势强度对二元BEC 孤子融合的影响28-30 2.5 本章小结30-31第3章含时种间相互作用下二元BEC 中孤子的碰撞行为31-39 3.1 引言31 3.2含时情况下的理论模型31-33 3.3 变系数KdV 方程及其孤子解析解33-35 3.4 二元BEC 中孤子的碰撞行为35-38 3.5 小结38-39第4章总结与展望39-41 4.1 论文工作总结39 4.2下一步工作的展望39-41参考文献41-47致谢47-48个人简历及攻读硕士学位期间完成的学术论文及研究成果48【采买全文】1.3.9.9.38.8.4.8 1.3.8.1.13.7.2.1 同时提供论文写作一对一辅导和论文发表服务.保过包发【说明】本文仅为中国学术文献总库合作提供，无涉版权。

云南大学学报(自然科学版),2004,26(2):132~133CN53-1045/N ISSN0258-7971 Journal of Yunnan University*一类非线性薛定谔方程的孤子解刘良桂,李云德(云南大学物理系,云南昆明650091)摘要:研究了具有V(x,t)=f1(t)x+f2(t)x2形式的外部势的非线性薛定谔方程的单一孤立子解.结果表明:当孤立子的中心满足带有势V(x,t)的牛顿方程,孤立子的内部结构由/体固定0坐标系决定.孤立子的结构与f1(t)无关.若f2(t)与t无关,孤立子是固定的.原则上,若f2(t)剧烈变化,则孤立子将扩散.但数值计算表明,在一定条件下,孤立子还是经得起f2(t)的剧烈变化.关键词:玻色-爱因斯坦凝聚;非线性薛定谔方程;孤立子解;牛顿方程中图分类号:O413;O414文献标识码:A文章编号:0258-7971(2004)02-0132-02在诺贝尔奖设立百年之际,2001年诺贝尔奖授予了3位科学家,瑞典皇家科学院称颂他们得奖的原因是:/由于在碱性原子的稀薄气体中获得了玻色-爱因斯坦凝聚(BEC)和对这类凝聚体特性的早期基础研究0.一时间,这项世界上/最冷0的研究领域成了最热的话题.BEC的实验上的进展极大地推动了理论上努力预测这一宏观量子系统的性质.预测的起点往往是带有谐和捕获势的Gross-Pitaevskii方程.这是个非线性薛定谔方程,假设它对于零温度的稀薄气体(na3n1,这里n为平均密度,a为S波散射长度)也成立,这里量子和热涨落可以忽略.很多文献[1~4]已经讨论了该方程的解.但对于带有和时间有关的线性谐和势的非线性薛定谔方程的孤立子的情况却鲜有文章报道.本文讨论1+1维外部势具有二次形式V(x,t)=V1(x,t)+V2(x,t),V n(x,t)S f n(t)x n的非线性薛定谔方程的孤立子行为和结构,并导出相关结论.1方法和举例考虑非线性薛定谔方程i 9W9t=-12m92W9x2-g|W|2W+V(x,t)W,(1)其中W=W(x,t),m为/质量0,g>0,为一常数,V(x,t)为外部势.将W归一化,Q]-]|W(x,t)|2d x=1.外部势具有二次形式V(x,t)=V1(x,t)+V2(x,t),V n(x,t)S f n(t)x n,(2)其中f1(t),f2(t)均为t的函数,只是f2(t)还要满足下面将要讨论的条件.在缺少外部势的情况下,V(x,t)=0,方程(1)的单孤立子解为[5]W(x,t)=A0(x-M t)e i[m M x-(E0+m M2/2)t],(3)A0(x)=(J/2)1/2sech(J x),(4)E0=-J22m,J=12mg,(5)其中A0(x)决定了/自由0孤立子的形状,是与时间无关的方程-12m A0x x-gA30=E0A0(6)的束缚态解.运用Husimi变换[6],x c=x-N(t),(7)这里,x c为关于运动原点N(t)的坐标,后面,把N(t)看成孤立子的质心,将W(x,t)改写为W(x,t)=U(x c,t)e i m N#x c,(8)*收稿日期:2003-04-08基金项目:国家自然科学基金资助项目(10347011).作者简介:刘良桂(1976-),男,江西人,硕士,主要从事理论物理研究.其中N#=d N(t)/d t.可得W(x,t)=V(x c,t)ex p i m N#x c+i Q L(t c)d t c,(9)L(t)=12m N#2-V(N,t),(10)i V t=-12mV x c x c-g|V|2V+V2(x c,t)V.(11)(11)式与(1)式有2点不同:¹(1)式是对/实验室0系而言,而(11)式是对运动系而言,其原点定在x=N(t);º(11)式少了V1项,且关于x c的宇称是好量子数.由º可知(11)式具有束缚态解,且有Q]-]|V(x c,t c)|2x c d x c=0.束缚态结构由(11)式决定,且与N(t)无关.束缚态的质心位于x c=0,亦即x=N(t).x c坐标系是束缚态的/体固定0系.当g=0,f2(t)>0时,(11)式显然有束缚态解.当添加引力非线性项时,这些束缚态仍存在.非线性相互作用强烈影响着最低态,孤立子由此形成.若f2(t)<0,则V2(x,t)为一反转谐振子势;若g=0,则(11)式不允许束缚态存在.若g>0,即使f2(t)<0,仍存在束缚孤立子态.(Ñ)线性势V(x,t)=V1(x,t)=m A(t)x,(12)由V2=0,可知:A(x)=A0(x),E=E0.N(t)由m N=-A(t)决定.由N(t)可求出W(x,t).(Ò)谐振子模式V(x)=V2(x)=12m X2x2,(13)其中X为常数,式(18)变为-12mA x x-gA3+12m X2x2A=E A.(14)对非线性自相互作用,外部势相对强度可用(m X)1/2/J度量,这里J=12mg.系统的能量由下式给出E=Q]-]-12m(A x)2-12gA4+12m X2x2A2d x.(15)(Ó)反转谐振子模型V(x)=V2(x)=-12m X2x2,(16)W(x,t)可仿(Ò)求出.若非线性项-gA3不存在,则没有束缚态.设想一个定位在原点周围的函数A,该函数产生一有效的引力势-gA2.若对A 退定域,则系统能量增加.若进一步对A在原点之外退定域,则能量开始下降.若采用参量K来量度A的定域化度,则能量作为K的函数将有局域最小值,亦即意味着束缚态的存在.(Ô)受迫的谐振子模型V(x,t)=12m X2x2-m X2F(t)x,(17)这里m X2F(t)x是附加的微扰.由牛顿方程,可知N(t)=X Q]-]F(t c)sin[X(t-t c)]d t c.(18)可在式(18)右边加上类似N0cos X t的项.振幅函数A(x-N)与(Ò)中的相同.由(18)可求出W(x,t).无论F(t)变化多剧烈,该解仍保持有效. 2结语总而言之,对带有式(2)所示的与时间有关的二次势的非线性薛定谔方程所描写的孤立子而言,它的内部结构可从孤立子的质心运动分离出来.对质心而言,方程(1)可化简为牛顿方程m N##= -V N(N,t),对于孤立子的结构而言,方程(1)又可化简为式(11).孤立子结构与线性项V1(x,t)无关,因此孤立子经得起f1(t)的剧烈变化.原则上,若f2(t)剧烈变化,则孤立子将扩散.但数值计算表明,孤立子经得起f2(t)的剧烈变化.详细地讨论了V(x,t)为反转谐振子势(即f2(t)<0)时的孤立子行为.参考文献:[1]EDWAR DS M,DODD R J,CLA RK C W,et al.Proper-ties of a Bose-Einstein condensate in an anisotropic har-monic pot ential[J].Phys Rev A,1996,53(4):1950)1953.[2]BAYM G,PETHICK C.Ground-State properties of mag-netically trapped Bose-Condensed rubidium gas[J].PhysR ev L ett,1996,76(1):6)9.[3]HOL LAN D M,COOPER J.Ex pansion o f a Bose-Ein-stein condensate in a harmonic potential[J].Phys RevA,1996,53(4):1954)1957.[4]RU PRECHT P A,HOL LA ND M J,BU RNET T K,etal.Rea-l T ime Bose-Einstein condensationin a finite vo-lume w ith a discrete spectrum[J].Ibid,1995,51(3):4704)4708.(下转第138页)133第2期刘良桂等:一类非线性薛定谔方程的孤子解Preparation of matrix of quantum dots using AAO template by PLDCH EN Da-peng1,YANG Ru-i ming1,ZHANG Peng-x iang1,2,FANG Yan2,YANG Zhi2(1.Institute of Advanced M ater ials for Photoelectronics,K unming Universityof Science and T echnology,Kunming650051,China;2.L aboratory o f N ano-P hotoelectro nics of Beijing,Capital Nor mal University,Beijing10053,China;3.Department of Chemistry,Yunnan N ormal U niversity,Kunming650092,China)Abstract:M atrix of quantum dots is fabricated using anodic aluminum ox ide(AAO)tem plate and pulsed laser deposition(PLD).The cylindrical pore array structure of AAO serves as a template for the preparation of the quantum dots.T he morphology of the matrix of dots and AAO template are characterized by scanning elec-tron microscope,the luminescence spectra of the dots and target materials are recorded by a m icro-Ram an spec-trometer.The structure and photoluminescence of the dots are discussed,w hich demonstrates that one can make m atrix of quantum dots by this method.It is proofed that this method can fabricate the m atrix of a quan-tum dots of the other materials in the future.Key words:anodic alum inum ox ide template;nanostucture system;fluorescence materials;pulsed laser de-position;quantum dots********************************* (上接第133页)[5]刘良桂,李云德.一维Bose-Einstein凝聚中的孤波[J].云南大学学报(自然科学版),2003,25(4):332)334.[6]CH EN H H,L IU C S.No nlinear w ave and soliton prop-agation in media w ith arbitrary inhomog eneities[J].P hys Fluids,1978,21(3):377)380.Soliton solution of certain nonlinear SchrÊdinger equationLIU Liang-gui,LI Yun-de(Depar tment of Physics,Y unnan U niv ersity,Kunming650091,China)Abstract:The one-soliton solution of the nonlinear SchrÊdinger equation w ith an external potential of the form of V(x,t)=f1(t)x+f2(t)x2is ex amined.It is show n that,w hile the center of the soliton obeys Newton.s equation w ith the potential V(x,t),the internal structure of the soliton is determined by the NLSE of the/body-fix ed0coordinate system.The soliton structure is found to be independent of f1(t).In principle,the soliton can be diffused if f2(t)varies rapidly.Through num erical method,how ever,that the sol-i ton is ex tremely tenacious against rapid variations of f2(t).Key words:Bose-Einstein condensate;nonlinear SchrÊdinger equation;soliton solution;New ton.s equation 138云南大学学报(自然科学版)第26卷。

玻色-爱因斯坦凝聚：量子宏观现象玻色-爱因斯坦凝聚（Bose-Einstein condensation，简称BEC）是一种量子宏观现象，最早由印度物理学家萨蒂扬德拉·纳特·玻色和德国物理学家阿尔伯特·爱因斯坦在1924年独立提出。

BEC是指在极低温下，一群玻色子（具有整数自旋的粒子）会聚集在能量最低的量子态，形成一个宏观量子态。

这种凝聚态具有许多奇特的性质，对于研究量子力学和凝聚态物理有着重要的意义。

玻色-爱因斯坦凝聚的基本原理玻色-爱因斯坦凝聚的基本原理可以通过统计力学和量子力学的理论来解释。

根据波尔兹曼分布和玻色-爱因斯坦统计，当温度趋近绝对零度时，粒子会趋向于占据能量最低的状态。

对于费米子（具有半整数自旋的粒子），由于泡利不相容原理的限制，不同粒子不能占据相同的量子态。

而对于玻色子，由于它们可以占据相同的量子态，当温度趋近绝对零度时，大量玻色子会聚集在能量最低的量子态，形成一个凝聚态。

玻色-爱因斯坦凝聚的实验观测玻色-爱因斯坦凝聚最早是在1995年由美国科学家埃里克·科尔曼和卡尔·韦曼等人在铷原子气体中实现的。

他们通过使用激光冷却和磁场操控技术，将铷原子冷却到极低温度，并将其限制在一个磁性陷阱中。

当温度足够低时，铷原子会进入玻色-爱因斯坦凝聚态，形成一个超流体。

这一实验观测为玻色-爱因斯坦凝聚的研究奠定了基础。

随后的实验中，科学家们还在其他物质中观测到了玻色-爱因斯坦凝聚现象，包括钠、锂、氢等原子气体，以及凝聚态固体中的激子和极化子等。

这些实验观测进一步验证了玻色-爱因斯坦凝聚的普适性和重要性。

玻色-爱因斯坦凝聚的应用玻色-爱因斯坦凝聚不仅在基础物理研究中具有重要意义，还在其他领域有着广泛的应用。

量子计算与量子通信玻色-爱因斯坦凝聚可以作为实现量子计算和量子通信的基础。

由于玻色-爱因斯坦凝聚具有宏观量子态的特性，可以用来存储和处理大量的量子信息。

銣原子之玻色-愛因斯坦凝聚文/韓殿君摘要利用雷射冷卻，磁阱囚禁與蒸發冷卻等方式，可將銣原子氣體冷卻至達成玻色-愛因斯坦凝聚所需之數百nK之低溫。

本文將簡介達成此一量子簡併態之實驗原理、方式與過程。

一、前言玻色-愛因斯坦凝聚(Bose-Einstein condensation，以下簡稱玻愛凝聚)之物理現象由愛因斯坦於1924年，以印度物理學家玻色(Bose)之光子統計原理為基礎所提出[1, 2]。

愛因斯坦與玻色之統計原理可推廣至所有玻色子(bosons)，此即所謂玻色-愛因斯坦統計(Bose-Einstein statistics)。

一群由相同(identical)[3]玻色子構成之系統(ensemble)，即使該群玻色子間並無任何作用，隨著溫度降低，並達一臨界值(critical temperature)時，該群粒子將大量且巨觀群聚於該系統之能量最基態，此即所謂玻色-愛因斯坦凝聚，為另一物質態(new state of matter)。

玻愛凝聚與一般所熟知於空間之凝聚現象，如水蒸氣凝結成水等不同。

玻愛凝聚乃系統之組成粒子凝聚於動量空間(momentum space)，雖於特殊情況下亦同時伴隨空間之上之凝聚。

氣態中性原子玻愛凝聚體，因粒子間之距離遠較其為液態及固態時為長，因而粒子間之作用力極弱，且極為接近一理想氣體(ideal gas)之系統。

雖玻愛凝聚現象早於其他系統中被觀測，如液態氦中的超流性(superfluidity)與液態氦庫柏對(Cooper pairs)之形成等[4, 5]。

然而，氣態玻愛凝聚體則提供一極單純、理論上極易分析與處理、且實驗上可操控之絕佳系統。

氣態中性原子玻愛凝聚於1995年由美國科羅拉多大學的康乃爾(E. Cornell)、魏曼(C. Wieman)[6]與麻省理工學院的凱特利(W. Ketterle)[7]等首度於實驗室中達成。

至今全球已超過30個實驗群有能力進行該類實驗。