Improved description of Bose-Einstein Correlation function

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二维吸引Bose—Einstein凝聚的临界值和稳定性

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两分量Bose-Einstein凝聚态系统的混沌与分支

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文章编号：０１０１２０）ｌ０１７１０－５（０８０－０－５００
两分量ｏｅＥｎｔｉ聚态系统的混沌与分支Ｂｓ—ｉｓｎ凝ｅ
’ 李继彬，陈凤娟
（．江师范大学数理与信息工程学院，江金华３１０２昆明理工大学非线性研究中心，南１浙浙２０４；．云
维普资讯
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浙江师范大学学报（自然科学版）
２００８正
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bose-einstein 凝聚态

“Bose-Einstein凝聚态”是凝聚态物理学中的一个重要概念，它描述了当大量粒子（如原子、分子、离子等）处于相同的量子态时，由于相互作用力的影响，这些粒子会发生集体行为的相变，形成一个整体的量子态。

这个概念最初是由Bose和Einstein在20世纪初提出，以解释统计物理中的一些基本问题。

Bose-Einstein凝聚态在许多领域都有应用，包括量子计算、量子密码学、量子模拟等。

此外，它还可以用于描述物质中的一些特殊现象，如超导电性、自旋冰效应等。

在实验上，Bose-Einstein 凝聚态可以通过利用激光冷却技术和强磁场等特殊实验条件来实现。

值得注意的是，除了直接观测到Bose-Einstein凝聚态之外，还有其他一些方法可以在理论上证明这种凝聚态的存在。

在祖国纪念爱因斯坦

物理质中的电磁现象,提出电子论.他引入带有电子运动的分子和微观电磁场的概念,后者的局部多分子的统计平均即是麦克斯韦的宏观电磁场;这样解释了物质对光的折射率随光的波长变化的色散现象.但对电子及其运动规律的较清楚认识,则尚待更多的近代物理实验和与其伴随的20世纪才发现的相对论和量子论.在这两个理论中,对时间和空间,粒子和波,概念上比以前有所深入,有些人称之为革命.但这与现代科学史前的革命,即推翻错误旧概念,建立正确新概念,大不相同.它实际上不过是,随着认识到更深一层次,原来认为割裂的时间和空间却是统一的时空,原来认为对立的粒子和波却是同一的量子.而回过头来看,原来的认识,在原来所概括的范围内仍是正确可靠到一定的近似程度而已.今后随着认识范围的扩大和认识能力的提高,统一更广,近似更精,类似地还会出现某些概念的变革和规律的发展———如某些人所称的“物理学中即将来临的革命”,我想仍将如此.周培源先生谈论物理学中理论的发展时,特别指出新理论与旧理论的如下关系,说新理论不但要说明旧理论已经说明的现象,还要说明旧理论不能说明的现象,而且还要预言现在还没有观察到的现象,我想也是根据如上的判断.我认为理论物理是有用的.作为工程设计原理的早已成熟的那些部分虽已分属各门工科,不是当今理论物理的主流,但对其发展仍应关怀.在开展理论、实验及与工程技术相结合的工作时,理论工作先行一步,可以减少实验和工程的工作量.我还认为理论物理方法是有生命力的,尚可尝试扩大其应用面.很难确切描述理论物理方法,但如下两点必不可少:联系实际获得启发和验证,反复辩证达到系统化统一.请参考毛主席的两篇文章,实践论和矛盾论中有关感性认识和理性认识以及相对真理和绝对真理的论述.下面转引爱因斯坦的两段话,对我们摸索正确的理论物理方法是会有帮助的.“Pure l ogical thinking cannot yield us any knowledge of the e mp irical world .A ll knowledge of re 2ality starts fr om experience and ends in it .”其汉译为“纯粹的逻辑思维不能给我们关于经验世界的任何知识;一切关于实在的知识,都是从经验开始,又终结于经验.”和“W e now realize,with s pecial clarity,how much in err or are those theorists who believe theo 2ry comes inductively fr o m experience .Even the great Ne wt on could not free hi m self fr om this err or (Hypoth 2eses non fingo ).”其汉译为“我们现在特别清楚地领会到,那些相信理论是从经验中归纳出来的理论家是多么错误呀.甚至伟大的牛顿也不能摆脱这种错误(我不作假设).”虽然由于某些原因,或者纯属偶然,几百年前物理学特别是理论物理的发祥地不在中国而在欧洲.但我想借陶渊明的归去来辞中的两句话“悟已往之不谏,知来者之可追”表示一个祝愿:祝愿物理学和理论物理能在中国得到很好的发展.并能在联合国决议所指出的三方面充分发挥它们的基础作用!在祖国纪念爱因斯坦李　政　道(中国高等科学技术中心　北京　100080) 爱因斯坦生于1879年3月14日,为此中国人民邮政在1979年发行了阿尔伯特・爱因斯坦诞辰100周年纪念邮票.1905年爱因斯坦发表了四篇文章:第一篇是A ne w method f or deter m ining the molecular size (测量分子大小的新方法),是他的博士论文;第二篇是L ight quantum (光量子)(Annalen der Physick 17,132);第三篇是B r own moti on (布朗运动)(Annalen der Phys 2ick 17,549);第四篇是Special theory of relativity (狭义相对论)(Annalen der Physick 17,891).联合国教科文组织,为了纪念爱因斯坦,将今年确定为“世界物理年”.为了纪念上面列出的这几篇里程碑式的文章,并响应联合国的决定,中国科协和中国物理学会在今年举办“世界物理年在中国”系列纪念活动.今天,能和大家一起在中国纪念爱因斯坦,我感到很荣幸.爱因斯坦一生对中国关怀很深.爱因斯坦1922・493・“世界物理年”专稿　34卷(2005年)6期年两次来上海.因此,中国学者和青年对相对论有高度的兴趣和了解.中国人民对爱因斯坦也有崇高的尊敬.民国日报(1922年11月15日)载爱因斯坦到沪后获瑞典正式通告得诺贝尔奖的消息.戴念祖先生在《物理》杂志最近发表的文章《上海,爱因斯坦及其诺贝尔奖》对爱因斯坦两次来上海有详细的记录(戴念祖.物理,2005,34:9).1931年爱因斯坦在美国遇见卓别林(Charles Chap lin ).卓别林在一般的相片中都是有小胡子的,可是这一次相会时有小胡子的一位不是卓别林(图1).卓别林对爱因斯坦说:我们两个都是名人,可是我们出名的原因不一样.我成名,因为随便哪一个人都知道我在做什么;可是你成名,是因为没有人知道你在做什么.当然,这是卓别林的幽默.爱因斯坦的成功是因为他了解自然界的规律,他的理论也符合于整个自然界的演变.爱因斯坦对20世纪的科学有极大的影响;很可能他对21世纪的科学有同样,或更大的影响.图1　爱因斯坦和卓别林(1931年1月30日)图2是2004年中国澳门出的邮票:怎样的宇宙?从我们的天文观察已经知道,我们人类能感知到的常规物质的能量(也就是已了解部分的宇宙),只占整个宇宙能量5%或者更小些,其他95%的能量都不是由我们现在所知的物质构成的.地球、太阳和所有我们看得见的星云都是由电子、质子、中子构成的,其中有一些极少数的反物质:正电子、反质子……可是,像我们知道的这类物质在我们宇宙中仅仅占了不到5%.大多数我们宇宙中的能量是暗物质和暗能量.看不见,也不知道是什么图2　2004年中国澳门出的邮票:怎样的宇宙?东西.暗物质对所有我们能测量的光、电场、磁场、强作用(核的能力场)都不起任何作用,可是暗物质有引力场(地心吸力就是引力场).通过引力场我们知道有暗物质存在,而且,暗物质的总能量比我们这类物质的总能量要大了五倍,或五倍以上.可是对暗物质的其他性质,我们完全不知道!暗能量的性质更是奇怪,它能产生一种负的压力.爱因斯坦在20世纪早期就曾假设过负压力这种性质的存在.后来,因为没有实验的支持,爱因斯坦就放弃了这一个方向.在裂变和聚变反应中,反应前后物质的质量有少量的差异.按照爱因斯坦的著名质能公式E =m c 2,这些少量的质量差异能够转化为巨大的能量.而暗能量可以将物质质量完全转化为能量!最近几年,通过哈勃太空望远镜(Hubble s pace telescope ),我们发现,我们的宇宙不仅是在膨胀而且是在“加速地”膨胀.从它膨胀的加速度可以推算出,它是由于一种负压力,也就是暗能量的存在才膨胀的.而这暗能量的总量占据全宇宙能量的百分之七十.关于这一个方向,我最近也在做一些新的理论探讨(李政道.中国物理快报,2004,21(6):1187;Lee T D.Nuclear Physics A,2005,750:1).暗能量在我们的宇宙中占据了如此重要的位置,所以爱因斯坦对21世纪的科学发展的影响,很可能比对20世纪的更大!1952年我和杨振宁合作写了两篇统计力学的文章.爱因斯坦看过之后,请他的助手考夫曼(B ru 2ria Kauf man )来找我们询问,是否可以和他讨论.我们立刻说,当然可以.我们走到他的办公室.他的桌子上就放着我们的文章,爱因斯坦说,他看了这两篇・593・“世界物理年”专稿物理文章觉得很有意思.同时,我看到他面前的纸上写得有很密的算式,他原来在重复我们的一些计算.爱因斯坦先问关于文章中所用巨正则系统(grand canon 2ical ense mble )的基础.显然,他并不熟悉这一观念.这很出我的意外,因为我以为巨正则系统是为了他1925年玻色-爱因斯坦凝聚(Bose -Einstein con 2densati on )的工作而创造的.爱因斯坦又问了我们文章中的格气(lattice gas )的细则.他的问题都着重于物理的基本概念.我的回答使他很满意.他说的英语带有相当重的德国口音,他讲得很慢.我们的讨论范围十分广泛,也谈了很长时间,约一个多小时.最后,他站起来,和我握手并且说:“W ish you future success in physics ”(祝你未来在物理学中获得成功).我记得他的手大,厚而温暖.对我来说,这实在是一次最难忘的经历.他的祝福使我深深感动.我们的讨论就在这张椅子前(图3),三年后爱因斯坦过世了,这张照片是在他过世之后一、二天时照的.爱因斯坦死于1955年4月18日,今天是4月15日,再过3天就是爱因斯坦逝世的50周年日.我们纪念100年前1905年爱因斯坦对物理的贡献,我们也纪念爱因斯坦一生为人类的贡献,为科学献身.我们的地球在太阳系是一个不大的行星.而我们的太阳在整个银河星云系四千亿颗恒星中也好像不是怎么出奇的星,而我们整个银河星云系在整个宇宙中也是非常渺小的.可是,因为爱因斯坦在我们小小的地球上生活过,我们这颗蓝色的地球就比其他宇宙的部分有特色,有智慧,有人的道德.图3　爱因斯坦的办公桌前进中的我国物理学研究张　杰(中国科学院物理研究所　北京　100080) 非常荣幸同我最尊敬的前辈在人民大会堂一起纪念爱因斯坦、一起纪念世界物理年.刚才,前辈们的报告介绍了爱因斯坦对物理学的贡献、探讨了爱因斯坦的人生观和思想体系、表达了对爱因斯坦的怀念和敬仰、阐述了对物理学的精辟认识、论述了物理学发展对人类社会的重要影响,使我深受教益.今天我想在这个纪念大会上主要汇报的是我国物理学工作者特别是青年一代物理学工作者在我国所做的一些工作.1　引言让我们暂时回到一百多年以前.19世纪,建立在热力学和统计力学、经典力学以及电磁场理论这3大支柱之上的经典物理学晴空万里,当是有些乐观的物理学家甚至认为,物理学的发展已经达到顶峰.但是,随着物理学的进一步发展,20世纪初,经典物理学的晴空出现了两朵乌云:解释黑体辐射能谱带来的“紫外灾难”和麦克尔逊-莫雷关于以太的实验带来的“难题”,给经典物理带来极大的冲击.1905年,是物理学的奇迹年.在这一年里,29岁的爱因斯坦发表了具有划时代意义的5篇科学论文,孕育了量子力学和相对论,带动了整个20世纪物理学的蓬勃发展并产生了20世纪里4项最重要的发明:原子能、半导体、计算机和激光器.・693・“世界物理年”专稿。

19 Bose-Einstein Condensation

Chapter19Bose-Einstein CondensationAbstract Bose-Einstein condensation(BEC)refers to a prediction of quantum sta-tistical mechanics(Bose[1],Einstein[2])where an ideal gas of identical bosons undergoes a phase transition when the thermal de Broglie wavelength exceeds the mean spacing between the particles.Under these conditions,bosons are stimulated by the presence of other bosons in the lowest energy state to occupy that state as well,resulting in a macroscopic occupation of a single quantum state.The con-densate that forms constitutes a macroscopic quantum-mechanical object.BEC was ﬁrst observed in1995,seventy years after the initial predictions,and resulted in the award of2001Nobel Prize in Physics to Cornell,Ketterle and Weiman.The exper-imental observation of BEC was achieved in a dilute gas of alkali atoms in a mag-netic trap.Theﬁrst experiments used87Rb atoms[3],23Na[4],7Li[5],and H[6] more recently metastable He has been condensed[7].The list of BEC atoms now includes molecular systems such as Rb2[8],Li2[9]and Cs2[10].In order to cool the atoms to the required temperature(∼200nK)and densities(1013–1014cm−3) for the observation of BEC a combination of optical cooling and evaporative cooling were employed.Early experiments used magnetic traps but now optical dipole traps are also common.Condensates containing up to5×109atoms have been achieved for atoms with a positive scattering length(repulsive interaction),but small con-densates have also been achieved with only a few hundred atoms.In recent years Fermi degenerate gases have been produced[11],but we will not discuss these in this chapter.BECs are now routinely produced in dozens of laboratories around the world. They have provided a wonderful test bed for condensed matter physics with stunning experimental demonstrations of,among other things,interference between conden-sates,superﬂuidity and vortices.More recently they have been used to create opti-cally nonlinear media to demonstrate electromagnetically induced transparency and neutral atom arrays in an optical lattice via a Mott insulator transition.Many experiments on BECs are well described by a semiclassical theory dis-cussed below.Typically these involve condensates with a large number of atoms, and in some ways are analogous to describing a laser in terms of a semiclassi-cal meanﬁeld.More recent experiments however have begun to probe quantum39739819Bose-Einstein Condensation properties of the condensate,and are related to the fundamental discreteness of the ﬁeld and nonlinear quantum dynamics.In this chapter,we discuss some of these quantum properties of the condensate.We shall make use of“few mode”approxi-mations which treat only essential condensate modes and ignore all noncondensate modes.This enables us to use techniques developed for treating quantum optical systems described in earlier chapters of this book.19.1Hamiltonian:Binary Collision ModelThe effects of interparticle interactions are of fundamental importance in the study of dilute–gas Bose–Einstein condensates.Although the actual interaction potential between atoms is typically very complex,the regime of operation of current exper-iments is such that interactions can in fact be treated very accurately with a much–simpliﬁed model.In particular,at very low temperature the de Broglie wavelengths of the atoms are very large compared to the range of the interatomic potential.This, together with the fact that the density and energy of the atoms are so low that they rarely approach each other very closely,means that atom–atom interactions are ef-fectively weak and dominated by(elastic)s–wave scattering.It follows also that to a good approximation one need only consider binary collisions(i.e.,three–body processes can be neglected)in the theoretical model.The s–wave scattering is characterised by the s–wave scattering length,a,the sign of which depends sensitively on the precise details of the interatomic potential [a>0(a<0)for repulsive(attractive)interactions].Given the conditions described above,the interaction potential can be approximated byU(r−r )=U0δ(r−r ),(19.1) (i.e.,a hard sphere potential)with U0the interaction“strength,”given byU0=4π¯h2am,(19.2)and the Hamiltonian for the system of weakly interacting bosons in an external potential,V trap(r),can be written in the second quantised form asˆH=d3rˆΨ†(r)−¯h22m∇2+V trap(r)ˆΨ(r)+12d3rd3r ˆΨ†(r)ˆΨ†(r )U(r−r )ˆΨ(r )ˆΨ(r)(19.3)whereˆΨ(r)andˆΨ†(r)are the bosonﬁeld operators that annihilate or create a par-ticle at the position r,respectively.19.2Mean–Field Theory —Gross-Pitaevskii Equation 399To put a quantitative estimate on the applicability of the model,if ρis the density of bosons,then a necessary condition is that a 3ρ 1(for a >0).This condition is indeed satisﬁed in the alkali gas BEC experiments [3,4],where achieved densities of the order of 1012−1013cm −3correspond to a 3ρ 10−5−10−6.19.2Mean–Field Theory —Gross-Pitaevskii EquationThe Heisenberg equation of motion for ˆΨ(r )is derived as i¯h ∂ˆΨ(r ,t )∂t = −¯h 22m ∇2+V trap (r ) ˆΨ(r ,t )+U 0ˆΨ†(r ,t )ˆΨ(r ,t )ˆΨ(r ,t ),(19.4)which cannot in general be solved.In the mean–ﬁeld approach,however,the expec-tation value of (19.4)is taken and the ﬁeld operator decomposed asˆΨ(r ,t )=Ψ(r ,t )+˜Ψ(r ,t ),(19.5)where Ψ(r ,t )= ˆΨ(r ,t ) is the “condensate wave function”and ˜Ψ(r )describes quantum and thermal ﬂuctuations around this mean value.The quantity Ψ(r ,t )is in fact a classical ﬁeld possessing a well–deﬁned phase,reﬂecting a broken gauge sym-metry associated with the condensation process.The expectation value of ˜Ψ(r ,t )is zero and,in the mean–ﬁeld theory,its effects are assumed to be small,amounting to the assumption of the thermodynamic limit,where the number of particles tends to inﬁnity while the density is held ﬁxed.For the effects of ˜Ψ(r )to be negligibly small in the equation for Ψ(r )also amounts to an assumption of zero temperature (i.e.,pure condensate).Given that this is so,and using the normalisationd 3r |Ψ(r ,t )|2=1,(19.6)one is lead to the nonlinear Schr¨o dinger equation,or “Gross–Pitaevskii equation”(GP equation),for the condensate wave function Ψ(r ,t )[13],i¯h ∂Ψ(r ,t )∂t = −¯h 22m ∇2+V trap (r )+NU 0|Ψ(r ,t )|2 Ψ(r ,t ),(19.7)where N is the mean number of particles in the condensate.The nonlinear interaction term (or mean–ﬁeld pseudo–potential)is proportional to the number of atoms in the condensate and to the s –wave scattering length through the parameter U 0.A stationary solution forthe condensate wavefunction may be found by substi-tuting ψ(r ,t )=exp −i μt ¯h ψ(r )into (19.7)(where μis the chemical potential of the condensate).This yields the time independent equation,40019Bose-Einstein Condensation−¯h2 2m ∇2+V trap(r)+NU0|ψ(r)|2ψ(r)=μψ(r).(19.8)The GP equation has proved most successful in describing many of the meanﬁeld properties of the condensate.The reader is referred to the review articles listed in further reading for a comprehensive list of references.In this chapter we shall focus on the quantum properties of the condensate and to facilitate our investigations we shall go to a single mode model.19.3Single Mode ApproximationThe study of the quantum statistical properties of the condensate(at T=0)can be reduced to a relatively simple model by using a mode expansion and subsequent truncation to just a single mode(the“condensate mode”).In particular,one writes the Heisenberg atomicﬁeld annihilation operator as a mode expansion over single–particle states,ˆΨ(r,t)=∑αaα(t)ψα(r)exp−iμαt/¯h=a0(t)ψ0(r)exp−iμ0t/¯h+˜Ψ(r,t),(19.9) where[aα(t),a†β(t)]=δαβand{ψα(r)}are a complete orthonormal basis set and {μα}the corresponding eigenvalues.Theﬁrst term in the second line of(19.9)acts only on the condensate state vector,withψ0(r)chosen as a solution of the station-ary GP equation(19.8)(with chemical potentialμ0and mean number of condensate atoms N).The second term,˜Ψ(r,t),accounts for non–condensate atoms.Substitut-ing this mode expansion into the HamiltonianˆH=d3rˆΨ†(r)−¯h22m∇2+V trap(r)ˆΨ(r)+(U0/2)d3rˆΨ†(r)ˆΨ†(r)ˆΨ(r)ˆΨ(r),(19.10)and retaining only condensate terms,one arrives at the single–mode effective Hamil-tonianˆH=¯h˜ω0a †a0+¯hκa†0a†0a0a0,(19.11)where¯h˜ω0=d3rψ∗0(r)−¯h22m∇2+V trap(r)ψ0(r),(19.12)and¯hκ=U02d3r|ψ0(r)|4.(19.13)19.5Quantum Phase Diffusion:Collapses and Revivals of the Condensate Phase401 We have assumed that the state is prepared slowly,with damping and pumping rates vanishingly small compared to the trap frequencies and collision rates.This means that the condensate remains in thermodynamic equilibrium throughout its prepara-tion.Finally,the atom number distribution is assumed to be sufﬁciently narrow that the parameters˜ω0andκ,which of course depend on the atom number,can be re-garded as constants(evaluated at the mean atom number).In practice,this proves to be a very good approximation.19.4Quantum State of the CondensateA Bose-Einstein condensate(BEC)is often viewed as a coherent state of the atomic ﬁeld with a deﬁnite phase.The Hamiltonian for the atomicﬁeld is independent of the condensate phase(see Exercise19.1)so it is often convenient to invoke a symmetry breaking Bogoliubovﬁeld to select a particular phase.In addition,a coherent state implies a superposition of number states,whereas in a single trap experiment there is aﬁxed number of atoms in the trap(even if we are ignorant of that number)and the state of a simple trapped condensate must be a number state(or,more precisely, a mixture of number states as we do not know the number in the trap from one preparation to the next).These problems may be bypassed by considering a system of two condensates for which the total number of atoms N isﬁxed.Then,a general state of the system is a superposition of number difference states of the form,|ψ =N∑k=0c k|k,N−k (19.14)As we have a well deﬁned superposition state,we can legitimately consider the relative phase of the two condensates which is a Hermitian observable.We describe in Sect.19.6how a particular relative phase is established due to the measurement process.The identiﬁcation of the condensate state as a coherent state must be modiﬁed in the presence of collisions except in the case of very strong damping.19.5Quantum Phase Diffusion:Collapsesand Revivals of the Condensate PhaseThe macroscopic wavefunction for the condensate for a relatively strong number of atoms will exhibit collapses and revivals arising from the quantum evolution of an initial state with a spread in atom number[21].The initial collapse has been described as quantum phase diffusion[20].The origins of the collapses and revivals may be seen straightforwardly from the single–mode model.From the Hamiltonian40219Bose-Einstein CondensationˆH =¯h ˜ω0a †0a 0+¯h κa †0a †0a 0a 0,(19.15)the Heisenberg equation of motion for the condensate mode operator follows as˙a 0(t )=−i ¯h [a 0,H ]=−i ˜ω0a 0+2κa †0a 0a 0 ,(19.16)for which a solution can be written in the form a 0(t )=exp −i ˜ω0+2κa †0a 0 t a 0(0).(19.17)Writing the initial state of the condensate,|i ,as a superposition of number states,|i =∑n c n |n ,(19.18)the expectation value i |a 0(t )|i is given byi |a 0(t )|i =∑n c ∗n −1c n √n exp {−i [˜ω0+2κ(n −1)]t }=∑nc ∗n −1c n √n exp −i μt ¯h exp {−2i κ(n −N )t },(19.19)where the relationship μ=¯h ˜ω0+2¯h κ(N −1),(19.20)has been used [this expression for μuses the approximation n 2 =N 2+(Δn )2≈N 2].The factor exp (−i μt /¯h )describes the deterministic motion of the condensate mode in phase space and can be removed by transforming to a rotating frame of reference,allowing one to writei |a 0(t )|i =∑nc ∗n −1c n √n {cos [2κ(n −N )t ]−isin [2κ(n −N )t ]}.(19.21)This expression consists of a weighted sum of trigonometric functions with different frequencies.With time,these functions alternately “dephase”and “rephase,”giving rise to collapses and revivals,respectively,in analogy with the behaviour of the Jaynes–Cummings Model of the interaction of a two–level atom with a single elec-tromagnetic ﬁeld mode described in Sect.10.2.The period of the revivals follows di-rectly from (19.21)as T =π/κ.The collapse time can be derived by considering the spread of frequencies for particle numbers between n =N +(Δn )and n =N −(Δn ),which yields (ΔΩ)=2κ(Δn );from this one estimates t coll 2π/(ΔΩ)=T /(Δn ),as before.From the expression t coll T /(Δn ),it follows that the time taken for collapse depends on the statistics of the condensate;in particular,on the “width”of the initial distribution.This dependence is illustrated in Fig.19.1,where the real part of a 0(t )19.5Quantum Phase Diffusion:Collapses and Revivals of the Condensate Phase403Fig.19.1The real part ofthe condensate amplitudeversus time,Re { a 0(t ) }for an amplitude–squeezed state,(a )and a coherent state (b )with the same mean numberof atoms,N =250.20.40.60.81-11234560b a is plotted as a function of time for two different initial states:(a)an amplitude–squeezed state,(b)a coherent state.The mean number of atoms is chosen in each case to be N =25.The timescales of the collapses show clear differences;the more strongly number–squeezed the state is,the longer its collapse time.The revival times,how-ever,are independent of the degree of number squeezing and depend only on the interaction parameter,κ.For example,a condensate of Rb 2,000atoms with the ω/2π=60Hz,has revival time of approximately 8s,which lies within the typical lifetime of the experimental condensate (10–20s).One can examine this phenomenon in the context of the interference between a pair of condensates and indeed one ﬁnds that the visibility of the interference pat-tern also exhibits collapses and revivals,offering an alternative means of detecting this effect.To see this,consider,as above,that atoms are released from two conden-sates with momenta k 1and k 2respectively.Collisions within each condensate are described by the Hamiltonian (neglecting cross–collisions)ˆH =¯h κ a †1a 1 2+ a †2a 22 ,(19.22)from which the intensity at the detector follows asI (x ,t )=I 0 [a †1(t )exp i k 1x +a †2(t )expi k 2x ][a 1(t )exp −i k 1x +a 2(t )exp −i k 2x ] =I 0 a †1a 1 + a †2a 2+ a †1exp 2i a †1a 1−a †2a 2 κt a 2 exp −i φ(x )+h .c . ,(19.23)where φ(x )=(k 2−k 1)x .If one assumes that each condensate is initially in a coherent state of amplitude |α|,with a relative phase φbetween the two condensates,i.e.,assuming that|ϕ(t =0) =|α |αe −i φ ,(19.24)40419Bose-Einstein Condensation then one obtains for the intensityI(x,t)=I0|α|221+exp2|α|2(cos(2κt)−1)cos[φ(x)−φ].(19.25)From this expression,it is clear that the visibility of the interference pattern under-goes collapses and revivals with a period equal toπ/κ.For short times t 1/2κ, this can be written asI(x,t)=I0|α|221+exp−|α|2κ2t2,(19.26)from which the collapse time can be identiﬁed as t coll=1/κ|α|.An experimental demonstration of the collapse and revival of a condensate was done by the group of Bloch in2002[12].In the experiment coherent states of87Rb atoms were prepared in a three dimensional optical lattice where the tunneling is larger than the on-site repulsion.The condensates in each well were phase coherent with constant relative phases between the sites,and the number distribution in each well is close to Poisonnian.As the optical dipole potential is increased the depth of the potential wells increases and the inter-well tunneling decreases producing a sub-Poisson number distribution in each well due to the repulsive interaction between the atoms.After preparing the states in each well,the well depth is rapidly increased to create isolated potential wells.The nonlinear interaction of(19.15)then determines the dynamics in each well.After some time interval,the hold time,the condensate is released from the trap and the resulting interference pattern is imaged.As the meanﬁeld amplitude in each well undergoes a collapse the resulting interference pattern visibility decreases.However as the meanﬁeld revives,the visibility of the interference pattern also revives.The experimental results are shown in Fig.19.2.Fig.19.2The interference pattern imaged from the released condensate after different hold times. In(d)the interference fringes have entirely vanished indicating a complete collapse of the am-plitude of the condensate.In(g),the wait time is now close to the complete revival time for the coherent amplitude and the fringe pattern is restored.From Fig.2of[12]19.6Interference of Two Bose–Einstein Condensates and Measurement–Induced Phase405 19.6Interference of Two Bose–Einstein Condensatesand Measurement–Induced PhaseThe standard approach to a Bose–Einstein condensate assumes that it exhibits a well–deﬁned amplitude,which unavoidably introduces the condensate phase.Is this phase just a formal construct,not relevant to any real measurement,or can one ac-tually observe something in an experiment?Since one needs a phase reference to observe a phase,two options are available for investigation of the above question. One could compare the condensate phase to itself at a different time,thereby ex-amining the condensate phase dynamics,or one could compare the phases of two distinct condensates.This second option has been studied by a number of groups, pioneered by the work of Javanainen and Yoo[23]who consider a pair of statisti-cally independent,physically–separated condensates allowed to drop and,by virtue of their horizontal motion,overlap as they reach the surface of an atomic detec-tor.The essential result of the analysis is that,even though no phase information is initially present(the initial condensates may,for example,be in number states),an interference pattern may be formed and a relative phase established as a result of the measurement.This result may be regarded as a constructive example of sponta-neous symmetry breaking.Every particular measurement produces a certain relative phase between the condensates;however,this phase is random,so that the symme-try of the system,being broken in a single measurement,is restored if an ensemble of measurements is considered.The physical conﬁguration we have just described and the predicted interference between two overlapping condensates was realised in a beautiful experiment per-formed by Andrews et al.[18]at MIT.The observed fringe pattern is shown in Fig.19.8.19.6.1Interference of Two Condensates Initially in Number States To outline this effect,we follow the working of Javanainen and Yoo[23]and consider two condensates made to overlap at the surface of an atom detector.The condensates each contain N/2(noninteracting)atoms of momenta k1and k2,respec-tively,and in the detection region the appropriateﬁeld operator isˆψ(x)=1√2a1+a2exp iφ(x),(19.27)whereφ(x)=(k2−k1)x and a1and a2are the atom annihilation operators for the ﬁrst and second condensate,respectively.For simplicity,the momenta are set to±π, so thatφ(x)=2πx.The initial state vector is represented simply by|ϕ(0) =|N/2,N/2 .(19.28)40619Bose-Einstein Condensation Assuming destructive measurement of atomic position,whereby none of the atoms interacts with the detector twice,a direct analogy can be drawn with the theory of absorptive photodetection and the joint counting rate R m for m atomic detections at positions {x 1,···,x m }and times {t 1,···,t m }can be deﬁned as the normally–ordered averageR m (x 1,t 1,...,x m ,t m )=K m ˆψ†(x 1,t 1)···ˆψ†(x m ,t m )ˆψ(x m ,t m )···ˆψ(x 1,t 1) .(19.29)Here,K m is a constant that incorporates the sensitivity of the detectors,and R m =0if m >N ,i.e.,no more than N detections can occur.Further assuming that all atoms are in fact detected,the joint probability density for detecting m atoms at positions {x 1,···,x m }follows asp m (x 1,···,x m )=(N −m )!N ! ˆψ†(x 1)···ˆψ†(x m )ˆψ(x m )···ˆψ(x 1) (19.30)The conditional probability density ,which gives the probability of detecting an atom at the position x m given m −1previous detections at positions {x 1,···,x m −1},is deﬁned as p (x m |x 1,···,x m −1)=p m (x 1,···,x m )p m −1(x 1,···,x m −1),(19.31)and offers a straightforward means of directly simulating a sequence of atom detections [23,24].This follows from the fact that,by virtue of the form for p m (x 1,···,x m ),the conditional probabilities can all be expressed in the simple formp (x m |x 1,···,x m −1)=1+βcos (2πx m +ϕ),(19.32)where βand ϕare parameters that depend on {x 1,···,x m −1}.The origin of this form can be seen from the action of each measurement on the previous result,ϕm |ˆψ†(x )ˆψ(x )|ϕm =(N −m )+2A cos [θ−φ(x )],(19.33)with A exp −i θ= ϕm |a †1a 2|ϕm .So,to simulate an experiment,one begins with the distribution p 1(x )=1,i.e.,one chooses the ﬁrst random number (the position of the ﬁrst atom detection),x 1,from a uniform distribution in the interval [0,1](obviously,before any measurements are made,there is no information about the phase or visibility of the interference).After this “measurement,”the state of the system is|ϕ1 =ˆψ(x 1)|ϕ0 = N /2 |(N /2)−1,N /2 +|N /2,(N /2)−1 expi φ(x 1) .(19.34)That is,one now has an entangled state containing phase information due to the fact that one does not know from which condensate the detected atom came.The corre-sponding conditional probability density for the second detection can be derived as19.6Interference of Two Bose–Einstein Condensates and Measurement–Induced Phase 407n u m b e r o f a t o m s n u m b e r o f a t o m s 8position Fig.19.3(a )Numerical simulation of 5,000atomic detections for N =10,000(circles).The solid curve is a least-squares ﬁt using the function 1+βcos (2πx +ϕ).The free parameters are the visibility βand the phase ϕ.The detection positions are sorted into 50equally spaced bins.(b )Collisions included (κ=2γgiving a visibility of about one-half of the no collision case.From Wong et al.[24]40819Bose-Einstein Condensationp (x |x 1)=p 2(x 1,x )p 1(x 1)=1N −1 ˆψ†(x 1)ˆψ†(x )ˆψ(x )ˆψ(x 1) ˆψ†(x 1)ˆψ(x 1) (19.35)=12 1+N 2(N −1)cos [φ(x )−φ(x 1)] .(19.36)Hence,after just one measurement the visibility (for large N )is already close to 1/2,with the phase of the interference pattern dependent on the ﬁrst measurement x 1.The second position,x 2,is chosen from the distribution (19.36).The conditional proba-bility p (x |x 1)has,of course,the form (19.32),with βand ϕtaking simple analytic forms.However,expressions for βand ϕbecome more complicated with increasing m ,and in practice the approach one takes is to simply calculate p (x |x 1,···,x m −1)numerically for two values of x [using the form (19.30)for p m (x 1,...,x m −1,x ),and noting that p m −1(x 1,...,x m −1)is simply a number already determined by the simu-lation]and then,using these values,solve for βand ϕ.This then deﬁnes exactly the distribution from which to choose x m .The results of simulations making use of the above procedure are shown in Figs 19.3–19.4.Figure 19.3shows a histogram of 5,000atom detections from condensates initially containing N /2=5,000atoms each with and without colli-sions.From a ﬁt of the data to a function of the form 1+βcos (2πx +ϕ),the visibil-ity of the interference pattern,β,is calculated to be 1.The conditional probability distributions calculated before each detection contain what one can deﬁne as a con-000.10.20.30.40.50.60.70.80.91102030405060number of atoms decided 708090100x=0x=1x=2x=4x=6Fig.19.4Averaged conditional visibility as a function of the number of detected atoms.From Wong et al.[13]19.7Quantum Tunneling of a Two Component Condensate40900.51 1.520.500.5Θz ο00.51 1.520.500.5Θx ο(b)1,234elliptic saddle Fig.19.5Fixed point bifurcation diagram of the two mode semiclassical BEC dynamics.(a )z ∗,(b )x ∗.Solid line is stable while dashed line is unstable.ditional visibility .Following the value of this conditional visibility gives a quantita-tive measure of the buildup of the interference pattern as a function of the number of detections.The conditional visibility,averaged over many simulations,is shown as a function of the number of detections in Fig.19.4for N =200.One clearly sees the sudden increase to a value of approximately 0.5after the ﬁrst detection,followed by a steady rise towards the value 1.0(in the absence of collisions)as each further detection provides more information about the phase of the interference pattern.One can also follow the evolution of the conditional phase contained within the conditional probability distribution.The ﬁnal phase produced by each individual simulation is,of course,random but the trajectories are seen to stabilise about a particular value after approximately 50detections (for N =200).19.7Quantum Tunneling of a Two Component CondensateA two component condensate in a double well potential is a non trivial nonlinear dynamical model.Suppose the trapping potential in (19.3)is given byV (r )=b (x 2−q 20)2+12m ω2t (y 2+z 2)(19.37)where ωt is the trap frequency in the y –z plane.The potential has elliptic ﬁxed points at r 1=+q 0x ,r 2=−q 0x near which the linearised motion is harmonic withfrequency ω0=q o (8b /m )1/2.For simplicity we set ωt =ω0and scale the length in units of r 0= ¯h /2m ω0,which is the position uncertainty in the harmonic oscillatorground state.The barrier height is B =(¯h ω/8)(q 0/r 0)2.We can justify a two mode expansion of the condensate ﬁeld by assuming the potential parameters are chosen so that the two lowest single particle energy eigenstates are below the barrier,with41019Bose-Einstein Condensation the next highest energy eigenstate separated from the ground state doublet by a large gap.We will further assume that the interaction term is sufﬁciently weak that, near zero temperature,the condensate wave functions are well approximated by the single particle wave functions.The potential may be expanded around the two stableﬁxed points to quadratic orderV(r)=˜V(2)(r−r j)+...(19.38) where j=1,2and˜V(2)(r)=4bq2|r|2(19.39) We can now use as the local mode functions the single particle wave functions for harmonic oscillators ground states,with energy E0,localised in each well,u j(r)=−(−1)j(2πr20)3/4exp−14((x−q0)2+y2+z2)/r20(19.40)These states are almost orthogonal,with the deviation from orthogonality given by the overlap under the barrier,d3r u∗j(r)u k(r)=δj,k+(1−δj,k)ε(19.41) withε=e−12q20/r20.The localised states in(19.40)may be used to approximate the single particle energy(and parity)eigenstates asu±≈1√2[u1(r)±u2(r)](19.42)corresponding to the energy eigenvalues E±=E0±R withR=d3r u∗1(r)[V(r)−˜V(r−r1)]u2(r)(19.43)A localised state is thus an even or odd superposition of the two lowest energy eigenstates.Under time evolution the relative phase of the superposition can change sign after a time T=2π/Ω,the tunneling time,where the tunneling frequency is given byΩ=2R¯h=38ω0q20r2e−q20/2r20(19.44)We now make the two-mode approximation by expanding theﬁeld operator asˆψ(r,t)=c1(t)u1(r)+c2(t)u2(r)(19.45) where。

Spectrum of light scattered from a deformed Bose-Einstein condensate

a r X i v :q u a n t -p h /9907084v 1 26 J u l 1999Spectrum of light scattered from a “deformed”Bose–Einstein condensateStefano Mancini †and Vladimir I.Man’ko ‡†Dipartimento di Fisica and Unit`a INFM,Universit`a di Milano,Via Celoria 16,I-20133Milano,Italy ‡P.N.Lebedev Physical Institute,Leninskii Prospekt 53,Moscow 117924,Russia (Date:May 25,1999)Abstract The spectrum of light scattered from a Bose–Einstein condensate is studied in the limit of particle-number conservation.To this end,a description in terms of deformed bosons is invoked and this leads to a deviation from the usual predict spectrum’s shape as soon as the number of particles decreases.PACS number(s):03.65.Fd (Algebraic methods),03.75.Fi (Bose condensation),42.50.Ct (Quantum statistical description of interaction of light and matter)The recent achievements of Bose–Einstein condensate(BEC)with a gas of atoms conﬁned by a magnetic trap[1]has stimulated renewed interest in the question as to what signatures Bose–Einstein condensation imprints in the spectrum of light scattered from atoms in such a condensate[2,3].As well known,to deal with the dynamics of BEC gas the Bogolubov approxima-tion in quantum many-body theory[4]is an eﬃcient approach,in which the creation and annihiliation operators for condensated atoms are substituted by c-numbers.One shortcoming of this method is that the total atomic particle-number may not be con-served after the approximation.Or a symmetry may be broken.To remedy this default,Gardiner[5]suggested a modiﬁed Bogolubov approximation by introducing phonon operators which conserve the total atomic particle number N and obey the bosonic commutation relation in the case of N→∞.In this sense,this phonon oper-ator approach gives an elegant inﬁnite atomic particle-number approximation theory for BEC taking into account the conservation of the total atomic number.Along this line,the case ofﬁnite number of particle has been recentely investi-gated[6],and the algebraic method of treating the eﬀects ofﬁnite particle number in the atomic BEC has been developed.It results a physical and natural realization of the quantum group theory[7]in the BEC systems,whose possibility was already suggested in[8],thought in a diﬀerent manner.Here,we shall use the deformed algebra to study the response of a condensate withﬁnite number of atoms to the laser light and focus our attention on steady-state excitation.We consider a system of weakly interacting Bose gas in a trap and a classical radiationﬁeld interacting with these two-level atoms,where b†,b denote the creation and annihiliation operators for the atoms in the excited state;a†,a,the creation and annihiliation operators for the atoms in the ground state.These operators satisfy theusual bosonic commutation relations.The Hamiltonian of the model readsH=¯h̟b†b+¯h g(t)b†a+g∗(t)ba† ,(1) where g(t)is a time-dependent coupling coeﬃcient for the(classical)laserﬁeld coupled to those two states with level diﬀerence¯h̟.Usually,the time dependence of g(t)is given by g exp(−iΩt),withΩbeing the frequency of the laser beam.Note that,with the above Hamiltonian,the total atomic particle number N= b†b+a†a is conserved.In the thermodynamic limit N→∞,the Bogolubov approxi-mation[4]is usually applied,in which the ladder operators a†,a of the ground state√are replaced by a c-numberN c g(t)b†+h.c.+ kΩk c†k c k+¯hN c−Γb(t)+√N c [3],whereγis the one-atom linewidth[9].Finally,b in(t)is the vacuum noise operatorb†in(t)b in(t′) = b in(t)b in(t′) =0,b in(t)b†in(t′) =δ(t−t′).(4)The solution of Eq.(3)is well known [9],and in the steady-state regime it becomesb (t ) ≡β=−ig √Γ+i ∆,(5)δb (ω)=√Γ+i ∆b in (ω),(6)where the semiclassical approximation b (t )=β+δb (t )has been used.In (6),δb (ω)is the Fourier component of the operator δb (t ).The spectrum of the light scattered from the atoms is given by the correlation of the operators b †(t )and b (t )[3].Hence,in the steady state,the spectrum of ﬂuctuations δb †(ω)δb (ω′) results zero everywhere,by virtue of (6)and (4).This means that in the long time limit,only the equal time correlations survive.Let us now come back to the Bogolubov approximation [4].It destroyes the sym-metry of Hamiltonian (1),i.e.,the conservation of the total particle number is violated because [N,H ]=0.Then,to preserve the property of the initial model,it is possible to determine the following phonon operators [5]B =1N a †b ,B †=1N ab †.(7)These operators obey a deformed algebra [6].In fact,a straightforward calculation leads to the following commutation relationB,B † =1−2ηb †b ,(8)where we have introduced a small operator parameter η=1/N ,which for suﬃcientely large number of atoms is considered as c -number.The algebra deﬁned by Eq.(8)belongs to the f -deformed algebra [10],where in general the deformed operator is related to the undeformed one through an operator valued function f asB =bf (b †b ).(9)In our particular case,we havef (b †b )=and for small deformation we getB≈b 1−ηN g(t)B†+h.c.+ kΩk c†k c k+¯h√Ng b†+b −¯h √2 b†b2+b†2b+ kΩk c†k c k+¯h√NgηN−Γb(t)+√NgηN−Γβ.(15)Of course,the solution of the above equation will be diﬀerent from that of Eq.(5)(we refer to the latter asβ∞),but they approach each other as soon as N increases,as can be seen in Fig.1.The dynamics of the smallﬂuctuations is given by∂tδb(t)=Aδb(t)+Bδb†(t)+√Ngη(β+β∗),(17)B=i√Ξ(ω) [iω−A∗]b in(ω)+B b†in(ω) ,(19)whereΞ(ω)=|A|2−|B|2−ω2−iω(A+A∗).(20) Finally,the spectrum,by means of Eqs.(19),(20)and(4),readsS(ω)= dω′ δb†(ω)δb(ω′) =|B|2may lead(in the limiting case of small number of particles)to observable eﬀects on a probe lightﬁeld.Beyond the oversimpliﬁed model used,we retain the measurement of the light spectrum in presence of few condensed atoms a promising experimental challenge.On the other hand,the use of a BEC with small number of atoms would be the subject of next generation experiments[13].The same aim could be pursued in elementary particleﬁeld as well.In fact,the BEC may also describe theﬁnal state of pions in high-energy-heavy-ion collisions[14,15].AcknowledgementsV.I.Man’ko is grateful to Russsian Foundation for Basic Research under the Project No.99-2-17753.References[1]M.H.Anderson J.R.Ensher,M.R.Matthews,C.E.Wieneman,E.A.Cornell,Science269,198(1995);K.B.Davies,M.-O.Mewes,M.R.Andrews.N.J.van Druten,D.S.Durfee,D.M.Kurn,W.Ketterle,Phys.Rev.Lett.75,3969(1995);C.C.Bradley,C.A.Sackett,J.J.Tollett and R.G.Hulet,Phys.Rev.Lett.75,1687(1995).[2]M.Levenstein and L.You,Phys.Rev.Lett.71,1339(1993);L.You,M.Lewensteinand J.Cooper,Phys.Rev.A50,R3565(1994);R.Graham and D.F.Walls,Phys.Rev.Lett.76,1774(1996).[3]J.Javanainen,Phys.Rev.Lett.72,2375(1994);75,1927(1995).[4]N.N.Bogolubov,J.Phys.(Moscow)2,23(1947).[5]C.W.Gardiner,Phys.Rev.A56,1414(1997).[6]C.P.Sun,S.Yu and Y.B.Gao,eprint qunt-ph/9809079.[7]L.C.Biedenharn,J.Phys.A22,L873(1989);A.J.Macfarlane,J.Phys.A22,4581(1989).[8]S.Mancini,V.I.Man’ko and P.Tombesi,Physica Scripta57,486(1998).[9]C.W.Gardiner,Quantum Noise(Springer,Heidelberg,1991).[10]V.I.Man’ko,G.Marmo,F.Zaccaria and E.C.G.Sudarshan,Physica Scripta55,528(1997).[11]S.Mancini,Physica Scripta59,195(1999).[12]B.R.Mollow,Phys.Rev.188,1969(1969).[13]G.M.Tino,private communication.[14]Proceedings of the Quark Matter’96Conference,(P.Braun-Munzinger et al.,eds.),Nucl.Phys.A610,1c-565c(1996);Proceedings of the Strangeness in Hadronic Matter’96Conference,(T.Csorgo et al.,eds.),Heavy Ion Physics4,1–440(1996).[15]T.Cs¨o rgo and J.Zimanyi,eprint hep-ph/9705432,eprint hep-ph/9705433.FIGURE CAPTIONSFig.1.A plot of the quantity||β|−|β∞||as a function of N.Values of the parameters are:∆=0and g=2.5γ.Furthermore,arg[β]=arg[β∞]=π/2∀N.Fig.2.The spectrum S as a function ofωand N.Values of parameters as in Fig.1.Nβοβο||||| − |501502503500.10.20.3240NFig.2 S. Mancini and V. I. Man'ko Spectrum of light scattered..........。

AlbertEinstein英文简介

Albert Einstein英文简介阿尔伯特·爱因斯坦，犹太裔物理学家，为核能开发奠定了理论基础，开创了现代科学技术新纪元，下面是店铺为你整理的Albert Einstein英文简介，希望对你有用!阿尔伯特·爱因斯坦简介Albert Einstein (Albert Einstein, on March 14, 1879 - April 18, 1955), the jewsphysicists。

Albert Einstein was born in Germany in 1879Ulm,The city of aThe jewsFamilies (parents are jewish), in 1900 graduated from the schoolThe federal institute of technology in Zurich, into theThe SwissNationality.In 1905,The university of ZurichPh.D.Degree, Einstein was put forwardThe photonAssumptions, explains the successThe photoelectric effectSo in 1921The Nobel Prize for physics, the creation ofSpecial theory of relativity.Founded in 1915General theory of relativity。

Einstein asNuclear energyThus laid a foundation for the development, ushered in a new era of modern science and technology, is acknowledged as the followingGalileo、NewtonSince one of the greatestphysicists.On December 26, 1999, Einstein is the United States"Time magazineFor the"The great man”。

Bose-Einstein condensation

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>。

2023-2024学年四川省内江市第六中学高三下期入学考试英语试题

2023-2024学年四川省内江市第六中学高三下期入学考试英语试题Not everyone knows that Mogao Caves in China has a “little sister” — Yulin Caves, which is smaller but better preserved than Mogao Caves. This “little sister” has the most brilliant murals (壁画) which are large in scale (规模) and diverse in forms and skills, with much art value. Here is some information to refer to when planning your tour here.Daily Itinerary (行程)DAY 1: Visit Mingshashan in the middle afternoon when it is not very hot and sunburn is low.DAY 2: Visit Yulin Caves. You can visit up to 6 caves. After that, drive about 30 minutes to Suoyangcheng.DAY 3: Visit the newly opened Mogao Caves Digital Exhibition Center. You will watch two short but well-produced educational documentary movies about Mogao Caves. After the movies, take the interzonal bus to Mogao Caves.Highlights of This Tour●In-depth tour of Mogao Caves and Yulin Caves●Visiting Suoyangcheng — an ancient ruined city 2,000 years ago●Private tour package covering airport pick-up, accommodation, sightseeing and private tour guideBasic Information about Price●$380 per person●This price is based on a group of 2 adults sharing one standard double-bed room in 4-star hotels and traveling with our private tour in low seasons.●This price is subject to change according to your traveling season, group size, hotel, class, etc. If you want a lower price, you can get more people to join you, or use economy class hotels.1. When can tourists see the documentary movies?A．On Day 1. B．On Day 2. C．On Day 3. D．Any day.2. What is a most interesting part of the tour?A．An old living city. B．In-depth travel in private.C．Free accommodations. D．Two brilliant tour guides.3. Where can you find the text probably?A．In a history textbook. B．On a travel agency website.C．In an academic journal. D．On an exhibition of murals.Jeremiah Thoronka was born in the fighting of the Sierra Leone civil war and grew up in the gutter (贫民窟) on the suburbs of the capital Freetown, having to burn wood for lighting and heating. Jeremiah saw with his own eyes how, in addition to the smog making breathing problems common, his young contemporaries fell behind in their schoolwork because of a lack of decent lighting.Energy poverty is a major issue in Sierra Leone—with just 26% of the population having access to electricity. In rural parts of the country, only 6% of people have electricity access, most of whom turn to solar lanterns and dry-cell batteries. As a result, it’s led to the destruction of forests as people cut down trees for firewood, which leaves Sierra Leone highly sensitive to extreme events like flooding and landslides. Families’ reliance on firewood also leads to frequent house fires.These life-threatening disadvantages and hardships fuelled Jeremiah’s passion for renewable energy and climate change advocacy. At 17, when studying at the African Leadership University in Rwanda, he launched a start-up called Optim Energy that transforms vibrations (震动) from vehicles and footfall on roads into an electric current. It is different from established renewable energy sources including wind or solar because it generates power without relying on changeable weather.Optim Energy ran a successful pilot program in Jeremiah’s neighbourhoods, Makawo in the northern part of Sierra Leone and Kuntoluh east of Freetown. With just two devices, the start-up provided free electricity to 150 households comprising around 1,500 citizens, as well as 15 schools where more than 9,000 students attend.Jeremiah is currently developing plans to expand into the healthcare field, which needs power to cool medicines and create enough light for treating patients after dark.4. What affected the young fellows’ academic performance?A．The war. B．The smog. C．The poorlighting. D．The breathing problem.5. What can we learn about “energy poverty” from paragraph 2?A．It is caused by solar lanterns and dry-cellbatteries.B．It has a worse impact on the city area.C．It brings about the destruction of forests. D．It increases the risk of forest fires.6. What is special about Optim Energy?A．It draws on vibrations to makeelectricity.B．It upgrades the use of solar energy.C．It runs a pilot program throughout thecity.D．It helps 9,000 students return to school.7. What message does the text want to convey?A．Electricity is the lifeblood of the city.B．Whoever is happy will make others happy too.C．Life is either a daring adventure or nothing at all.D．We are all in the gutter, but some are making it better.Carl Wieman, a Nobel Prize-winning physicist at Stanford University, excelled in the lab, where he created the Bose-Einstein condensate (玻色-爱因斯坦凝聚态). However, his mastery in the lab did not extend to the classroom. For years, he wrestled with what seemed to be a straightforward task: making undergraduates comprehend physics as he did. Laying it out for them—explaining, even demonstrating the core concepts of the discipline—was not working. Despite his clear explanations, his students’ cap acity to solve the problems he posed to them remained inadequate.It was in an unexpected place that he found the key to the problem: not in his classrooms but among the graduate students (研究生) who came to work in his lab. When his PH. D. candidates entered the lab, Wieman noticed, their habits of thought were no less narrow and rigid than the undergraduates. Within a year or two, however, these same graduate students transformed into the flexible thinkers he was trying so earnestly, and unsuccessfully, to cultivate. “Some kind of intellectual process must have been missing from the traditional education,” Wieman recounts.A major factor in the graduate students’ transformation. Wieman concluded, was their experience of intense social engagement around a body of knowledge — the hours they spent advising, debating with, and recounting anecdotes to one another. In 2019, a study published in the Proceedings of the National Academy of Sciences backed this idea. Tracking the intellectual advancement of several hundred graduate students in the sciences over the course of four years, its authors found that the development of crucial skills such as generating hypotheses (假设), designing experiments, and analyzing data was closely related to the students’ engagement w ith their peers in the lab, rather than the guidance they received from their faculty mentors (导师).Wieman is one of a growing number of Stanford professors who are bringing this “active learning” approach to their courses. His aspiration is to move science education away from the lecture format, toward a model that is more active and more engaged.8. What problem did Carl Wieman have with his undergraduates?A．Making them excel in the lab. B．Demonstrating lab experiments.C．Facilitating their all-round development. D．Enhancing their physics problem-solving.9. Which of the following best describes the graduate students who first joined Wieman’s lab?A．Limited in thinking. B．Resistant to new ideas.C．Flexible and earnest. D．Experienced and cooperative.10. What is crucial for developing students’ intelligent thought according to the 2019 study?A．Intense lab work. B．Peer pressure and evaluation.C．Academic interaction with fellows. D．Engagement with external society.11. Which of the following can be a suitable title for the text?A．Transforming Graduates’ Habits B．Carl Wieman’s Nobel Prize JourneyC．The Nobel-Prize Winner’s Struggles D．Carl Wieman’s Education Innovation Since the 1950s, some 9.2 billion tonnes of plastic have been produced globally, of which only about10% has ever been recycled. Yet environmentally conscious companies and consumers continue to look to recycling as a way to ease the plastic problem. Manufacturing giants claim to be committedto making more of their products and packaging from recycled materials. However, this confidence masks (掩饰) a complex web of issues around plastic recycling. Recycling rates remain extremely low and critics argue that we should look at alternative ways to tackle plastic pollution.While many plastics have the potential to be recycled, most are not because the process is costly, complicated and the resulting product of a lower quality than the original. Despite rising demand for recycled plastic, few waste companies turn a profit. Part of this is because virgin plastic-linked to oil prices - is often cheaper than recycled plastic, meaning there is little economic incentive to use it. Worse yet, much of our plastic waste is difficult to recycle. Lightweight food packaging, like a mozzarella packet, contains different plastics, dyes and toxic additives (添加剂). This dirty mix means plastic recycled through mechanical methods- the most common form- can only be melted down and moulded (浇铸,塑造) again a couple of times before it becomes too fragile to be reused. And the nature of the process means plastic recycling has a carbon footprint of its own.Given all of these difficulties, environmental critics say recycling is not the solution-and argue that creating more products from recycled material to attract environmental consciousness merely worsens the problem. “The solution is to use less plastic and to stop misleading the public about the recyclability,” says Enck, president of Beyond Plastics, a US campaign group with a mission to end single-use plastic. “They should stop making false claims about the recyclability of plastics since they know most will either be littered (乱扔) or burned or landfilled (填埋). Using less plastics means shifting to reusable products and relying more on paper, cardboard, glass and metal- -all of which should be made from recycled content.”12. What is an environmentally conscious customer’s attitude towards recycling plastics?A．Suspicious. B．Favorable. C．Indifferent. D．Disapproving. 13. What does the underlined word “incentive” in Paragraph 2 probably mean?A．motive. B．issue. C．crisis. D．policy.14. What is Paragraph 2 mainly about?A．The recycling process of plastics. B．Pollutants contained in recycled plastics.C．Reasons why users dislike recycled plastics. D．Contributing factors to low plastic recycling rates.15. What will the environmental critics be happy to see according to the text?A．Using metal or glass food containers. B．Littering recycled plastics in a landfill.C．Processing plastics in a mechanical way. D．Launching campaigns to promoterecyclability.Science fiction television has done a lot to shape how we view the meals of the future, from an evening dinner in pill form to machines giving us any meal we desire, on demand. 16 However, while these ideas for food creation are more than fifty years old, the way we produce and consume food has not changed very much. Even though there is enough food available today to feed the world, more than 870 million people do not have enough to eat. 17With the global population expected to grow to more than nine billion by 2050, demand for food will only increase. No pills or machines have solved our food problems yet.However, agricultural science has been responsible for saving huge numbers of lives. Science and technology helped us out of starvation during the 1960s and 70s when the world’s population exploded. 18 The result of this panic was the “green revolution”, which saw the intr oduction to farming of high production grains, improved irrigation systems and hybrid seeds, saving over one billion people from starvation19 Having enough food is only the first step. There are complications with natural disasters, conflict, poverty and environmental problems. All these can mean that the food supply is put at risk 20 Many people are concerned about “playing around with nature”, adjusting how food looks and tastes and smells, and other human interferences(干预), all of which are widespread.Anyhow, the world has not been completely saved from starvation, but we are on the way to get there.Have you ever found yourself in a situation where something you enjoy on a daily basis is suddenly not available to you? For several weeks I volunteered on a ________ on the big island of Hawaii. My ________ in doing so: to learn how to grow my own food in a sustainable way, along with experiencing a simpler ________.Being an average twenty-something citizen of the ________ world, I spend a lot of free time using mobile devices. Before coming to Hawaii, I’d n ever made an effort to ________ that. When I arrived at the farm and discovered that the WiFi, ________ mainly by solar panels on the roof of my small room, wasn’t working normally, I knew a ________ to learn about myself had come knocking. Over the first couple days, I frequently thought about ________ my social media before realizing that I couldn’t. I ________ this source of entertainment and felt ________ from the world. I was________ that someone would message me and think I was ________ them. After it fully sank in that I couldn’t connect to my social networks, I felt more ________ in my immediate environment.My attention was less ________ while working on tasks. Though I still periodically felt ________, my anxiety faded and suddenly, I realized how much more time I had for the things that gave me a deeper, more real sense of achievement and ________: for me, this included yoga and mindful movement, reading, and being out in ________.My digital detox was not something I chose to do ________, but I’m thankful that it happened this way. ________ I value my devices just as much as I did before the digital detox, I realize that being ________ about how I use them is key to keeping my phone a positive addition to my life.21.A．farm B．beach C．playground D．river22.A．method B．purpose C．advantage D．suggestion23.A．solution B．custom C．principle D．lifestyle24.A．unique B．modern C．perfect D．complex25.A．change B．blame C．explore D．destroy26.A．challenged B．repaired C．powered D．closed27.A．vacation B．difficulty C．routine D．chance28.A．checking B．inventing C．building D．quitting29.A．respected B．missed C．praised D．accepted30.A．broken up B．fed up C．cut off D．paid off31.A．anxious B．surprised C．embarrassed D．cautious32.A．repeating B．answering C．ignoring D．criticizing33.A．experienced B．absorbed C．concerned D．disappointed34.A．received B．escaped C．caught D．distracted35.A．delighted B．greedy C．lonely D．convinced36.A．peace B．urgency C．ambition D．loss37.A．return B．place C．danger D．nature38.A．nervously B．reasonably C．carelessly D．willingly39.A．If B．After C．While D．Because40.A．bored B．selective C．familiar D．patient阅读下面材料，在空白处填入适当的内容(1个单词)或括号内单词的正确形式。

论文列表 - 河北工业大学

论文题目（英文题目要译成中文）期刊名倾斜极化磁多层膜的铁磁共振理论Phys. Rev. B Rapid response mechanism of pi cell（Pi 盒的快速响应机制）Appl. Phys. Lett.Response times in Pi-cell liquidcrystal displaysLiq. Cryst.弱锚定垂面排列液晶显示器的响应时间液晶与显示栅状表面液晶盒的光学属性计算物理中文图形12864点阵液晶显示模块与51单片机的并行接口电路及C51程序设计现代显示Monte Carlo 模拟空间各向异性势向列相液晶微滴计算物理基于修正Gruhn-Hess 两体势模型的内禀锚定研究Physics letters. A基于分级模型的钾离子通道选择性通透机制的研究IEEE 会议论文考虑视轴方向的个性化眼模型的构建光学学报基于个性化模型的人眼色差对视功能影响的研究光子学报The study of wavelength-dependentwavefront aberrations based onindividual eye model （基于个性化眼模型的人眼波像差随波长变化的研究）Optik 基于个体眼光学结构的角膜与晶状体的像差补偿研究光学学报The lower-valence coexistingferrimagnetic Cr2VX (X=Ga, Si, Ge,Sb) Heusler compounds: A first-principles study低价共存的赫斯勒合金Cr2VX(X=Ga, Si, Ge, Sb)的第一性原理研究Ab initio investigation of half-metalstate in 钾通道的结构：钾离子传导和选择性的分子基础《科学》杂志精选论文基于离子通道选择性的水合碱金属阳离子结构的研究中国物理快报Physica BScripta Materialiazinc-blende MnSn and MnC闪锌矿结构的MnSn和MnC的半金属态的从头计算研究用临界点附近的涨落讨论Landau相变理论的适用范围河北工业大学学报“建设国家级实验教学示范中心的探索与实践”《全国高等学校物理基础课程教育学术研讨会论文集》坚持创新教育实验教学理念建设物理实验教学示范中心《第五届全国高等学校物理实验教学研讨会论文集》创建国家级实验教学示范中心及其辐射示范作用的研究与实践《实验技术与管理》大学生自主创新教育应适应社会需求性与尊重学生选择性《第五届全国高等学校物理实验教学研讨会论文集》坚持创新教育实验教学理念建设物理实验教学示范中心《物理实验》（特刊）《关于如何提高基础物理实验课质量的探讨》《科学时代》带电粒子致细胞失活的理论模型中国原子能科学研究院年报2007Two-photon spectroscopic behaviorsand photodynamic effect on the BEL-7402 cancer cells of the newchlorophyll photosensitizer《中国科学》（一种新型叶绿素光敏剂的双光子光谱特性B 辑及其对BEL-7402肝癌细胞的双光子光动力效应）（英文版）Tunneling effect of two horizons froma Gibbons-Maeda black holeChinese Physics LettersTunneling Effect from a Non-static Black Hole with the Internal GlobalMonopole International Journal of TheoreticalPhysicsTunneling effect of two horizons from a Reissner-Nordstrom black hole International Journal of TheoreticalPhysics用新乌龟坐标计算任意直线加速带电黑洞的熵北京师范大学学报自然科学版液晶与显示Physica BTheoretical Models of Cell Inactivation by Ionizing Particles（带电粒子致细胞失活的理论模型）Annual Report of China Institute of Atomic Energy 2007片剂硬度测试仪的液晶显示界面设计克尔黑洞的内视界隧道效应北京师范大学学报自然科学版Order Parameter of theAntiferroelectric Phase Transition ofLead Zirconate,Ferroelectric LettersSymmetry of the AntiferroelectricLead ZirconateFerroelectric LettersInvestigation of Ferroelectric PhaseTransition of Rochelle SaltFerroelectrics 锆酸铅的反铁电相对称性研究人工晶体学报方硼盐本征铁电相变的研究河北工业大学学报The synchronization ofFitzHugh--Nagumo neuronnetwork coupled by gapjunctionChinese Physics BStudy of intrinsic anchoring in nematicliquid crystals based on modifiedGruhn-hess pair potential（基于修正的Gruhn-hess两体势研究向列相液晶的内禀锚定）Phys. Lett. AThe Electrod Effect in LCD Cell（LCD盒中的电极效应）电子器件聚合物稳定向列相液晶显示的上升时间常数现代显示全Heusler 合金Cr2MnAl的第一性原理研究Appl. Phys. Lett.大范围Mn掺杂Heusler-型 Cr3Al的半金属化合物的理论设计J. Appl. Phys.局域键近似下的固体材料热驱动下的弹性软化J. Phys. D: Appl. Phys. Heusler 相 Co2YBi 和半-Heusler 相CoYBi (Y=Mn, Cr)的带结构计算J. Magn. Magn. Mater含时外势作用下玻色-爱因斯坦凝聚的非自治孤子自旋极化电流导致的铁磁金属多层膜中的铁磁共振J. Appl. Phys.“液晶显示器件物理专业方向”的实验设置实验技术与管理混合排列向列相边缘场效应电光特性的模拟计算现代显示Opt. Commun.个性化视觉矫正中人眼波前像差测量数据的修正研究Photonics and Optoelectronics MeetingsDynamical Model of P53-Mdm2-P14/19ARF Network to Radiation inPopulation of Cells 电离辐射作用下多细胞P53-Mdm2-P14/19ARF 网络动力学模型机械波的半波损失条件问题探析河北工业大学成人教育学院学报地方工科高校大学物理课程建设的探索与实践2009年全国高等学校物理基础课程教育学术研讨会论文集A death-survival switch in cell: cross talk between Akt and p53International Conference on Bioinformatics and Biomedical EngineeringModeling of ATM accumulation and ATM-mediated oscillation of p53International Conference on Bioinformatics and Biomedical EngineeringFunctions of oligochitosan inducedprotein kinase in tobacco mosaic virusresistance and pathogenesis related proteins in tobacco （壳寡糖诱导蛋白激酶在烟草对烟草花叶病毒抗性和病理相关蛋白中的作用）Plant Physiology and Biochemistry Neutrino Oscillation in the Space-Time with a Global Monopole 有拓扑缺陷时空的中微子振荡Thermodynamics Properties of theInner Horizon of a Kerr-Newman Black Hole K-N 黑洞的内视界的热性质Tunneling Effect of Two Horizonsfrom a Reissner-Nordstrom Black Hole R-N 黑洞的双视界的隧穿效应太赫兹波在空芯椭圆波导中的传播特性Photonics and Optoelectronics Meetings (POEM) 2008:TerahertzScience and Technology自旋波背景下单轴各向异性铁磁中磁振子密度的新特性Ann. Phys. (New York)BIC-TA2009International Journal of TheoreticalPhysicsInternational Journal of Theoretical Physics International Journal of TheoreticalPhysicsNeutrino oscillationinterference phase in Kerrspace--time克尔时空的中微子振荡干涉相位指形场驱动垂面排列模式的响应机制,Liq. Cryst.,两块栅状表面基板构成的向列相液晶盒的阈值属性Chinese Physics B 挠曲电效应对向列相液晶盒电光效应的影响计算物理双轴向列相液晶在沟槽表面的弹性形变和稳定性Liquid Crystals用光学导波方法区分液晶盒中的预倾角与弱锚定现代显示栅状表面液晶盒的光学特性现代显示平面和高斯光束的非线性吸收系数β的微分分析Photonics and OptoelectronicsMeetingsModel for influences of magneticfields on intracellular calcium oscillations(磁场对细胞内钙振荡影响的理论研究)Commun. Theor. Phys.液晶与显示Chinese Physics B太赫兹波在空芯椭圆波导中的传播特性Photonics and Optoelectronics Meetings (POEM) 2008:Terahertz Science and Technology平行排列向列相液晶的导波研究太赫兹波在空芯镀不同介质膜圆波导中的传播特性Photonics and Optoelectronics Meetings (POEM) 2008:Terahertz Science and TechnologyStability Control System for Four-In-Wheel-Motor Drive Electric Vehicle.The Sixth International Conference onFuzzy Systems and Knowledge Discovery, Tianjin, China, 14–16August ,2009.Propagation characteristics of THzradiation in hollow ellipticalwaveguide Pro. of SPIE光电混合机器人视觉系统中光寻址空间光调制器的噪声处理光子学报A First Principles study on the fullHeusler Compond Compond利用慢度曲面研究晶体的纯模轴人工晶体学报演化网络的拓扑结构河北工业大学学报帐篷映射归宿的完整性探析天津师范大学学报用二进制讨论面包师变换的动力学特性河北工业大学学报克尔-纽曼时空的中微子振荡Class. Quantum Grav.史瓦西-德西特是空的中微子振荡干涉相位Commun. Theor. Phys.Quantitative deviation of the two-photon absorption coefficient based onthree pulse model Chinese Optics LettersTwo-photon absorption coefficientdeviation from the application of laserultra-short pulse models Chinese Physics lettersTwo-photon absorption coefficient inrelation to the typical pulse models oflaser Optics CommunicationsAppl. Phys. Lett.MnAlCr 2MnAl Cr 2罗伯特-沃克度规的中微子振荡和振荡长度的宇宙学蓝移Int J Theor Phys Domain-wall solutions of spinor Bose-Einstein condensates in an opticallattice,Phys. Rev. ANonautonomous bright and darksolitons of Bose-Einstein condensateswith Feshbach- managed time-dependent scattering lengthOpt. Commun.Formation of combined solitons intwo-component Bose-Einsteincondensates Chin. Phys. BBright and Dark Soliton Solutions inGrowing Bose-Einstein CondensatesChin. Phys. B 基于三种激光脉冲模型的双光子吸收系数偏离量研究CHINESE OPTICS LETTERS 扭曲向列相液晶显示器中的响应时间现代显示基于128_64点阵液晶显示的智能温度控制器的设计与实现实验室科学基于TRIZ 原理的航空用毛巾卷圈机折卷部件创新设计机械设计液晶全漏导模的实验研究物理实验温度对向列相液晶阈值电压的影响现代显示FlexoelectricEffectinaHAN-IPSCell Grey Solitons and Soliton interactionof Higher Nonlinear SchrödingerEquation Can. J. Phys.Screw-pitch effect and velocityoscillationof a domain wall in aferromagnetic nanowire driven byspin- polarized current J.Phys.: Condens. MatterTwo-photon absorption coefficient inrelation to the typical pulse models oflaser与若干激光典型超短脉冲模型相关的双光子吸收系数Quantitative deviation of two-photonabsorption coefficient based on threelaser pulse models基于三种激光脉冲模型的双光子吸收系数之间的量化差别Approach dealing with the pulseprofile of pump laser in Z-scanJournal of Optics Z-扫描中处理泵浦激光脉冲时域结构的一种方法曾用名：Journal of Optics A ：Pure and Applied Optics双面高斯形刻槽金属纳米光栅表面等离子体传感芯片设计IEEE，Photonics and OptoelectronicA first-principles study onthelower-valencecoexisting Cr2TiX (X ¼ Al,Ga,Si, Ge, Sn,Sb) HeusleralloysJournal of Magnetism and Magnetic Materials 基于修正的Gruhn-Hess 两体势模型研究弯曲形变向列相液晶盒计算物理表面沟槽诱导双轴向列相液晶的弹性畸变液晶与显示LCD 点状缺陷分析与研究现代显示温度对向列相液晶阈值电压的影响现代显示离子与通道相互作用对NaK 通道通透特性影响的研究物理学报Extracelluar Potassium ions Play Important Roles in the Selectivity of Mutant KcsA Channel 2011 4th International Conference on Biomedical Enginerrint andInformaticsFlexibility Between the Linker of theCD and G-Loops Determines the Gating Dynamics of Hte Kir2.1ChannelBiophysical Journal Direct or Indirect Regulation ofCalcium-Activated Chloride Channel by CalciumJ Membrane Biol.离子与通道相互作用对NaK 通道通透特性影响的研究物理学报Characteristics and molecular basis ofcelecoxib modulation on Kv7potassium channelsBr. J. Pharmacol.Direct or Indirect Regulation ofCalcium-Activated Chloride Chinese Optics Letters Optics CommunicationsThe Journal of Membrane BiologyChannel by CalciumFabrication of subwavelength metallic structures using laserinterference lithographyPhotonics and Optoelectronics,IEEEENonautonomous helical motion ofmagnetization in ferromagneticnanowire driven by spin-polarizedcurrent and magnetic fieldEur. Phys. J. BMatter rogue wave in Bose-Einsteincondensates with attractive atomicinteractionEur. Phys. J. DNeutrino Oscillations in the de Sitterand the Anti-de Sitter Space-Timethe Anti-de Sitter Space-TimeSpin-Rotation Coupling in theTeleparallelism Description in HighSpeed Rotation SystemInt J Theor PhysA TETRAD DESCRIPTION ONTHE DIRACSPIN-ROTATION EFFECTOCB模式液晶器件的优化液晶与显示A high-transmittance verticalalignment liquid crystal display usinga fringe and in-plane electrical fieldLiquid CrystalsFast-response vertical alignmentliquid crystal display driven by in-plane switching, Liquid CrystalLiquid Crystals Low voltage and high transmittanceblue-phase LCDs with double-side in-plane switching electrodesLiquid CrystalsThe Journal of Membrane BiologyPhase diagram of magneticmultilayers with tilted dual spintorques J. Appl. Phys.Combined periodic wave and solitarywave solutions in two-componentBose-Einstein condensatesChin. Phys. BInt J Theor PhysInternational Journal of ModernPhysics D混和排列向列相液晶盒中挠曲电效应引起的电压漂移Commun. Theor. Phys.指导教师负责制的资助选课教学模式河北工业大学学报-社会科学版栅状基板表面几何参数对向列相液晶盒指向矢的影响Commun. Theor. Phys.液晶全漏导模透射率的实验研究大学物理实验离子与通道相互作用对NaK 通道通透特性影响的研究物理学报Extracelluar Potassium ions Play Important Roles in the Selectivity of Mutant KcsA Channel 2011 4th International Conference on Biomedical Enginerrint andInformaticsThe Difference Analysis of Nonlinear Absorption Coefficient β in the Beam Sections of Plain and Gaussian Distribution Nonlinear AbsorptionCoefficient βin the Beam Sections of Plain andGaussian DistributionEffects of Magnetic Fields on the Synchronization of Calcium Oscillations in Coupled Cells Journal of Computational and Theoretical NanoscienceOptimal design of hollow elliptical waveguide for THz radiationJournal of Physics: Conference Series，2011,Volume:276，No.1012229Approach dealing with the pulse profile of pump laser in Z-scantechniqueJournal of Physics: Conference Series，2011,Volume:276，No.1012229Propagation characteristics of THz radiation in hollow rectangle metalwaveguideJournal of Physics: Conference Series，2011,Volume:276，No.1012229ProcSPIE,作者（名次）时间李再东2008孙玉宝，马红梅，张志东3月孙玉宝（2）7月马红梅，王娜红，孙玉宝2月叶文江2008.9李志广2008.7第1张艳君2008.52张艳君2008安海龙(2)May-08刘铭1Feb-08刘铭1Aug-08刘铭1Jun-08刘铭1Aug-08安海龙(1)李佳120082008-3-1安海龙(1)Nov-08李佳12008王双进（第一作者）2008.1魏怀鹏（排名1）、李再东、安莉、张志东、展永，2008.7魏怀鹏（排名1）、张志东、展永2008.1魏怀鹏（排名1）、张志东、展永2008.11，温春东，魏怀鹏（排名2），段雪松，孔祥明，丁军锋，甄芳芳2008.1魏怀鹏（排名1）、张志东、展永2008.11魏怀鹏第一作者2008.4曹天光1Jun-08ZHAO PeiDe et al第一作者（论文通讯联系人）任军12008.5任军12008.3任军1Jul-08任军12008年2月淮俊霞（第一作者）李佳120082008.6曹天光1Jun-08Jun-08任军12008年10月周国香12008周国香12008周国香12008周国香12008周国香12008展永12008张志东，张艳君2008.1张志东，赵金良2008.1刘丽媛（学生），刘艳玲（学生），张志东2008.9盖翠丽甄晓玲王纪刚张志东李佳（1）2009李佳（1）2009李佳（1）2009李佳（1）2009李再东第二(通信作者)李再东第二2009张志东，范志新20092008.72008.6李再东第三(通信作者)刘铭（1）2009.03柳辉（1）2009.3柳辉（2）2009.7.Liu Hui (4)2009.7Liu Hui (4)2009.6Liu Hui (8)2009.8刘铭（5）2009任军（第一作者）2009年8月2009.03Liu Hui (1)2009.1任军（第一作者）2009年2月任军（第一作者）2009年7月马红梅，孙玉宝9月叶文江12009.1叶文江12009.1叶文江22009.8叶文江12009.3叶文江22009.11张勇（2）2009.03张玉红（第一）2009叶文江2任军（第一作者）Dec-092009.12张勇（1）2009.032009.03张勇（2）张玉红（第二）2009赵培德第一作者2010通讯联系人第一期赵培德第七2009赵培德第一作者通讯联系人赵培德第二作者通讯联系人周国香第二2009，1周国香第二2009，10周国香第二2009，6周国香第三2009，7周国香第三2009，8任军(1)2010年3月任军(1)2010年4月20092009周国香第三2009，5任军(1)2010年9月李再东,李秋艳，贺鹏斌，梁九卿，刘伍明，傅广生2010李秋艳，李再东，李录，傅广生2010李秋艳，李再东(*)，姚淑芳，李录，傅广生2010李秋艳，李再东，贺鹏斌，宋伟为，傅广生。

+全国2009年7月高等教育自学考试英语科技文选

英语科技文选试题课程代码：00836PART A: VOCABULARYI. Directions: Add the affix to each word according to the given Chinese, makingchanges when necessary. (8 %)1.accelerate 加速装置 1._________________2.contrast 对比的 2._________________3.alternance 可选择的 3._________________pass 包含 4._________________pare 可比的 5._________________6.bewilder 迷惑不解(名词) 6._________________7.attractive 诱引剂7._________________8.different 区分8._________________II. Directions：Fill in the blanks，each using one of the given words or phrases below in its prope form.(12％)stem from in addition at randompile of in contrast bump intoin the event of in all probability bear outwithin reach of／one’s reach be associated with take one’s place9.I_______________ an old friend of mine at the gas station.10.The new work of his will_______________ among the most important paintings of this century.11.Dependence on alcohol often________________ unhappiness in the home.12.He asked his sister to look after his children________________ his death.13.I’ve no idea where last Saturday’s newspaper is；_______________，it might have been thrown away.14.The facts don’t______________your fears.15.Isabelle placed a wine cup on the table__________________.16.The first thing the secretary does is to sort out the_______________ documents and letters on his desk.17.The lottery numbers are chosen_______________.18.His bad behavior__________________ his difficult childhood.19.It is hot in the day time，but________________it’s very cold at night.20.I need your help.________________ ，I need her support.Ⅲ.Directions：Fill in each blank with a suitable word given below.(10％)massive likely whether Galaxy long hits from off was MilkyThe cloud, called Smith’s Cloud，after the astronomer who discovered it in 1963，contains enough hydrogen to make a million stars like the Sun. Eleven thousand light-years _21_ and 2,500light-years wide，it is only 8,000 light-years_22_our Galaxy’s disk. It is rushing toward our _23_ at more than 150 miles per second，aimed to strike the Milky Way’s disk at an angle of about 45degrees."This is most _24_ a gas cloud left over from the formation of the Milky Way or gas stripped froma neighbor galaxy. When it _25_，it could set off a tremendous burst of star formation. Many of those stars will be very _26_. Over a few million years，it'll look like a celestial New Year’s celebration，with huge firecrackers going _27_in that region of the Galaxy," Lockman said. When Smith’s Cloud _28_ first discovered，and for decades after, the available images did not have enough detail to show _29_ the cloud was part of the Milky Way，somethingbeing blown out of the _30_ Way，or something falling in.PART B：TRANSLATIONⅣ. Directions：Translate the following sentences into English，each using one of the given words or phrases below. (10％)attribute customary subject reminiscent of come into its own31.你的讲述让我想起十年前的一次经历。

玻色-爱因斯坦凝聚态的分子间作用力

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综述BEC(Bose-Einstein Condensation) 【英文版】

Theory of Bose-Einstein condensation in trapped gasesFranco Dalfovo and Stefano GiorginiDipartimento di Fisica,Universita`di Trento and Istituto Nazionale per la Fisica dellaMateria,I-38050Povo,ItalyLev P.PitaevskiiDipartimento di Fisica,Universita`di Trento and Istituto Nazionale per la Fisica dellaMateria,I-38050Povo,Italy;Department of Physics,TECHNION,Haifa32000,Israel;and Kapitza Institute for Physical Problems,ul.Kosygina2,117334MoscowSandro StringariDipartimento di Fisica,Universita`di Trento and Istituto Nazionale per la Fisica dellaMateria,I-38050Povo,ItalyThe phenomenon of Bose-Einstein condensation of dilute gases in traps is reviewed from a theoretical perspective.Mean-ﬁeld theory provides a framework to understand the main features of the condensation and the role of interactions between particles.Various properties of these systems are discussed,including the density proﬁles and the energy of the ground-state conﬁgurations,the collective oscillations and the dynamics of the expansion,the condensate fraction and the thermodynamic functions.The thermodynamic limit exhibits a scaling behavior in the relevant length and energy scales.Despite the dilute nature of the gases,interactions profoundly modify the static as well as the dynamic properties of the system;the predictions of mean-ﬁeld theory are in excellent agreement with available experimental results.Effects of superﬂuidity including the existence of quantized vortices and the reduction of the moment of inertia are discussed,as well as the consequences of coherence such as the Josephson effect and interference phenomena.The review also assesses the accuracy and limitations of the mean-ﬁeld approach.[S0034-6861(99)00103-8]CONTENTSI.Introduction463II.The Ideal Bose Gas in a Harmonic Trap466A.The condensate of noninteracting bosons466B.Trapped bosons atﬁnite temperature:thermodynamic limit468C.Finite-size effects470D.Role of dimensionality471E.Nonharmonic traps and adiabatic transformations472 III.Effects of Interactions:Ground State472A.Order parameter and mean-ﬁeld theory472B.Ground state474C.Collapse for attractive forces477rge-N limit for repulsive forces478E.Beyond mean-ﬁeld theory479 IV.Effects of Interactions:Dynamics480A.Excitations of the condensate and time-dependent Gross-Pitaevskii equation480rge Na/a ho limit and collisionlesshydrodynamics482C.Sum rules and collective excitations485D.Expansion and large-amplitude oscillations486E.Density of states:collective vs single-particleexcitations489 V.Effects of Interactions:Thermodynamics491A.Relevant energy scales491B.Critical temperature492C.Below T c493D.Thermodynamic limit and scaling495E.Results for the thermodynamic functions496F.Collective modes atﬁnite temperature498 VI.Superﬂuidity and Coherence Phenomena500A.Rotational properties:vortices and moment ofinertia501B.Interference and Josephson effect504C.Collapse and revival of collective oscillations506 VII.Conclusions and Outlook507 Acknowledgments509 References509 I.INTRODUCTIONBose-Einstein condensation(BEC)(Bose,1924;Ein-stein,1924,1925)was observed in1995in a remarkable series of experiments on vapors of rubidium(Anderson et al.,1995)and sodium(Davis et al.,1995)in which the atoms were conﬁned in magnetic traps and cooled down to extremely low temperatures,of the order of fractions of microkelvins.Theﬁrst evidence for condensation emerged from time-of-ﬂight measurements.The atoms were left to expand by switching off the conﬁning trap and then imaged with optical methods.A sharp peak in the velocity distribution was then observed below a cer-tain critical temperature,providing a clear signature for BEC.In Fig.1,we show one of theﬁrst pictures of the atomic clouds of rubidium.In the same year,ﬁrst signa-tures of the occurrence of BEC in vapors of lithium were also reported(Bradley et al.,1995).Though the experiments of1995on the alkalis should be considered a milestone in the history of BEC,the experimental and theoretical research on this unique phenomenon predicted by quantum statistical mechanics is much older and has involved different areas of physics (for an interdisciplinary review of BEC see Grifﬁn, Snoke,and Stringari,1995).In particular,from the very beginning,superﬂuidity in helium was considered by463Reviews of Modern Physics,Vol.71,No.3,April19990034-6861/99/71(3)/463(50)/$25.00©1999The American Physical SocietyLondon (1938)as a possible manifestation of BEC.Evi-dence for BEC in helium later emerged from the analy-sis of the momentum distribution of the atoms measured in neutron-scattering experiments (Sokol,1995).In re-cent years,BEC has been also investigated in the gas of paraexcitons in semiconductors (see Wolfe,Lin,and Snoke,1995,and references therein),but an unambigu-ous signature for BEC in this system has proven difﬁcult to ﬁnd.Efforts to Bose condense atomic gases began with hy-drogen more than 15years ago.In a series of experi-ments hydrogen atoms were ﬁrst cooled in adilutionFIG.2.(Color)Collective excitations of a Bose-Einstein condensate.Shown are in situ repeated phase-contrast images taken of a ‘‘pure’’condensate.The excitations were produced by modulating the magnetic ﬁelds which conﬁne the condensate,and then letting the condensate evolve freely.Both the center-of-mass and the shape oscillations are visible,and the ratio of their oscillation frequencies can be accurately measured.The ﬁeld of view in the vertical direction is about 620␮m,corresponding to a condensate width of the order of 200–300␮m.The time step is 5ms per frame.From Stamper-Kurn and Ketterle(1998).FIG.1.(Color)Images of the velocity distribution of rubidium atoms in the experiment by Anderson et al.(1995),taken by means of the expansion method.The left frame corresponds to a gas at a temperature just above condensation;the center frame,just after the appearance of the condensate;the right frame,after further evaporation leaves a sample of nearly pure condensate.The ﬁeld of view is 200␮m ϫ270␮m,and corresponds to the distance the atoms have moved in about 1/20s.The color corresponds to the number of atoms at each velocity,with red being the fewest and white being the most.From Cornell (1996).464Dalfovo et al.:Bose-Einstein condensation in trapped gasesRev.Mod.Phys.,Vol.71,No.3,April 1999refrigerator,then trapped by a magneticﬁeld and fur-ther cooled by evaporation.This approach has come very close to observing BEC,but is still limited by re-combination of individual atoms to form molecules(Sil-vera and Walraven,1980and1986;Greytak and Klepp-ner,1984;Greytak,1995;Silvera,1995).At the time of this review,ﬁrst observations of BEC in spin-polarized hydrogen have been reported(Fried et al.,1998).In the 1980s laser-based techniques,such as laser cooling and magneto-optical trapping,were developed to cool and trap neutral atoms[for recent reviews,see Chu(1998), Cohen-Tannoudji(1998),and Phillips(1998)].Alkali at-oms are well suited to laser-based methods because their optical transitions can be excited by available lasers and because they have a favorable internal energy-level structure for cooling to very low temperatures.Once they are trapped,their temperature can be lowered fur-ther by evaporative cooling[this technique has been re-cently reviewed by Ketterle and van Druten(1996a)and by Walraven(1996)].By combining laser and evapora-tive cooling for alkali atoms,experimentalists eventually succeeded in reaching the temperatures and densities re-quired to observe BEC.It is worth noticing that,in these conditions,the equilibrium conﬁguration of the system would be the solid phase.Thus,in order to observe BEC,one has to preserve the system in a metastable gas phase for a sufﬁciently long time.This is possible be-cause three-body collisions are rare events in dilute and cold gases,whose lifetime is hence long enough to carryout experiments.So far BEC has been realized in87Rb(Anderson et al.,1995;Han et al.,1998;Kasevich,1997;Ernst,Marte et al.,1998;Esslinger et al.,1998;So¨dinget al.,1999),in23Na(Davis et al.,1995;Hau,1997and 1998;Lutwak et al.,1998),and in7Li(Bradley et al.,1995and1997).The number of experiments on BEC invapors of rubidium and sodium is now growing fast.Inthe meanwhile,intense experimental research is cur-rently carried out also on vapors of caesium,potassium,and metastable helium.One of the most relevant features of these trappedBose gases is that they are inhomogeneous andﬁnite-sized systems,the number of atoms ranging typicallyfrom a few thousands to several millions.In most cases,the conﬁning traps are well approximated by harmonicpotentials.The trapping frequency␻ho also provides a characteristic length scale for the system,a ho ϭ͓ប/(m␻ho)͔1/2,of the order of a few microns in the available samples.Density variations occur on this scale.This is a major difference with respect to other systems,like,for instance,superﬂuid helium,where the effects ofinhomogeneity take place on a microscopic scaleﬁxedby the interatomic distance.In the case of87Rb and 23Na,the size of the system is enlarged as an effect of repulsive two-body forces and the trapped gases can be-come almost macroscopic objects,directly measurable with optical methods.As an example,we show in Fig.2 a sequence of in situ images of an oscillating condensate of sodium atoms taken at the Massachusetts Institute of Technology(MIT),where the mean axial extent is of the order of0.3mm.The fact that these gases are highly inhomogeneous has several important consequences.First BEC shows up not only in momentum space,as happens in super-ﬂuid helium,but also in coordinate space.This double possibility of investigating the effects of condensation is very interesting from both the theoretical and experi-mental viewpoints and provides novel methods of inves-tigation for relevant quantities,like the temperature de-pendence of the condensate,energy and density distributions,interference phenomena,frequencies of collective excitations,and so on.Another important consequence of the inhomogene-ity of these systems is the role played by two-body inter-actions.This aspect will be extensively discussed in the present review.The main point is that,despite the very dilute nature of these gases(typically the average dis-tance between atoms is more than ten times the range of interatomic forces),the combination of BEC and har-monic trapping greatly enhances the effects of the atom-atom interactions on important measurable quantities. For instance,the central density of the interacting gas at very low temperature can be easily one or two orders of magnitude smaller than the density predicted for an ideal gas in the same trap,as shown in Fig.3.Despite the inhomogeneity of these systems,which makes the solution of the many-body problem nontrivial,the dilute nature of the gas allows one to describe the effects of the interaction in a rather fundamental way.In practice a single physical parameter,the s-wave scattering length, is sufﬁcient to obtain an accuratedescription.FIG.3.Density distribution of80000sodium atoms in the trap of Hau et al.(1998)as a function of the axial coordinate.The experimental points correspond to the measured optical den-sity,which is proportional to the column density of the atom cloud along the path of the light beam.The data agree well with the prediction of mean-ﬁeld theory for interacting atoms (solid line)discussed in Sec.III.Conversely,a noninteracting gas in the same trap would have a much sharper Gaussian distribution(dashed line).The same normalization is used for the three density proﬁles.The central peak of the Gaussian is found at about5500␮mϪ2.Theﬁgure points out the role of atom-atom interaction in reducing the central density and en-larging the size of the cloud.465Dalfovo et al.:Bose-Einstein condensation in trapped gases Rev.Mod.Phys.,Vol.71,No.3,April1999The recent experimental achievements of BEC in al-kali vapors have renewed a great interest in the theoret-ical studies of Bose gases.A rather massive amount of work has been done in the last couple of years,both to interpret the initial observations and to predict new phe-nomena.In the presence of harmonic conﬁnement,the many-body theory of interacting Bose gases gives rise to several unexpected features.This opens new theoretical perspectives in this interdisciplinaryﬁeld,where useful concepts coming from different areas of physics(atomic physics,quantum optics,statistical mechanics,and condensed-matter physics)are now merging together. The natural starting point for studying the behavior of these systems is the theory of weakly interacting bosons which,for inhomogeneous systems,takes the form of the Gross-Pitaevskii theory.This is a mean-ﬁeld ap-proach for the order parameter associated with the con-densate.It provides closed and relatively simple equa-tions for describing the relevant phenomena associated with BEC.In particular,it reproduces typical properties exhibited by superﬂuid systems,like the propagation of collective excitations and the interference effects origi-nating from the phase of the order parameter.The theory is well suited to describing most of the effects of two-body interactions in these dilute gases at zero tem-perature and can be naturally generalized to also ex-plore thermal effects.An extensive discussion of the application of mean-ﬁeld theory to these systems is the main basis of the present review article.We also give,whenever possible, simple arguments based on scales of length,energy,and density,in order to point out the relevant parameters for the description of the various phenomena.There are several topics which are only marginally discussed in our paper.These include,among others, collisional and thermalization processes,phase diffusion phenomena,light scattering from the condensate,and analogies with systems of coherent photons.In this sense our work is complementary to other recent review ar-ticles(Burnett,1996;Parkins and Walls,1998).Further-more,in our paper we do not discuss the physics of ul-tracold collisions and the determination of the scattering length which have been recently the object of important experimental and theoretical studies in the alkalis (Heinzen,1997;Weiner et al.,1999).The plan of the paper is the following:In Sec.II we summarize the basic features of the non-interacting Bose gas in harmonic traps and we introduce theﬁrst relevant length and energy scales,like the oscil-lator length and the critical temperature.We also com-ment onﬁnite-size effects,on the role of dimensionality and on the possible relevance of anharmonic traps.In Sec.III we discuss the effects of the interaction on the ground state.We develop the formalism of mean-ﬁeld theory,based on the Gross-Pitaevskii equation.We consider the case of gases interacting with both repulsive and attractive forces.We then discuss in detail the large-N limit for systems interacting with repulsive forces,leading to the so-called Thomas-Fermi approxi-mation,where the ground-state properties can be calcu-lated in analytic form.In the last part,we discuss the validity of the mean-ﬁeld approach and give explicit re-sults for theﬁrst corrections,beyond meanﬁeld,to the ground-state properties,including the quantum deple-tion of the condensate,i.e.,the decrease in the conden-sate fraction produced by the interaction.In Sec.IV we investigate the dynamic behavior of the condensate using the time-dependent Gross-Pitaevskii equation.The equations of motion for the density and the velocityﬁeld of the condensate in the large-N limit, where the Thomas-Fermi approximation is valid,are shown to have the form of the hydrodynamic equations of superﬂuids.We also discuss the dynamic behavior in the nonlinear regime(large amplitude oscillations and free expansion),the collective modes in the case of at-tractive forces,and the transition from collective to single-particle states in the spectrum of excitations.In Sec.V we discuss thermal effects.We show how one can deﬁne the thermodynamic limit in these inho-mogeneous systems and how interactions modify the be-havior compared to the noninteracting case.We exten-sively discuss the occurrence of scaling properties in the thermodynamic limit.We review several results for the shift of the critical temperature and for the temperature dependence of thermodynamic functions,like the con-densate fraction,the chemical potential,and the release energy.We also discuss the behavior of the excitations atﬁnite temperature.In Sec.VI we illustrate some features of these trapped Bose gases in connection with superﬂuidity and phase coherence.We discuss,in particular,the structure of quantized vortices and the behavior of the moment of inertia,as well as interference phenomena and quantum effects beyond mean-ﬁeld theory,like the collapse re-vival of collective oscillations.In Sec.VII we draw our conclusions and we discuss some further future perspectives in theﬁeld.The overlap between current theoretical and experi-mental investigations of BEC in trapped alkalis is al-ready wide and rich.Various theoretical predictions, concerning the ground state,dynamics,and thermody-namics are found to agree very well with observations; others are stimulating new experiments.The comparison between theory and experiments then represents an ex-citing feature of these novel systems,which will be fre-quently emphasized in the present review.II.THE IDEAL BOSE GAS IN A HARMONIC TRAPA.The condensate of noninteracting bosonsAn important feature characterizing the available magnetic traps for alkali atoms is that the conﬁning po-tential can be safely approximated with the quadratic formV ext͑r͒ϭm2͑␻x2x2ϩ␻y2y2ϩ␻z2z2͒.(1) Thus the investigation of these systems starts as a text-466Dalfovo et al.:Bose-Einstein condensation in trapped gases Rev.Mod.Phys.,Vol.71,No.3,April1999book application of nonrelativistic quantum mechanics for identical pointlike particles in a harmonic potential. Theﬁrst step consists in neglecting the atom-atom in-teraction.In this case,almost all predictions are analyti-cal and relatively simple.The many-body Hamiltonian is the sum of single-particle Hamiltonians whose eigenval-ues have the form␧nx n y n zϭͩn xϩ12ͪប␻xϩͩn yϩ12ͪប␻yϩͩn zϩ12ͪប␻z,(2)where͕n x,n y,n z͖are non-negative integers.The ground state␾(r1,...,r N)of N noninteracting bosons conﬁned by the potential(1)is obtained by putting all the particles in the lowest single-particle state(n xϭn y ϭn zϭ0),namely␾(r1,...,r N)ϭ⌸i␸0(r i),where␸0(r) is given by␸0͑r͒ϭͩm␻ho␲បͪ3/4expͫϪm2ប͑␻x x2ϩ␻y y2ϩ␻z z2͒ͬ,(3)and we have introduced the geometric average of the oscillator frequencies:␻hoϭ͑␻x␻y␻z͒1/3.(4) The density distribution then becomes n(r)ϭN͉␸0(r)͉2 and its value grows with N.The size of the cloud is in-stead independent of N and isﬁxed by the harmonic oscillator length:a hoϭͩបm hoͪ1/2(5)which corresponds to the average width of the Gaussian in Eq.(3).This is theﬁrst important length scale of the system.In the available experiments,it is typically of the order of a hoϷ1␮m.Atﬁnite temperature only part of the atoms occupy the lowest state,the others being ther-mally distributed in the excited states at higher energy. The radius of the thermal cloud is larger than a ho.A rough estimate can be obtained by assuming k B T ӷប␻ho and approximating the density of the thermal cloud with a classical Boltzmann distribution n cl(r)ϰexp͓ϪV ext(r)/k B T͔.If V ext(r)ϭ(1/2)m␻ho2r2,the width of the Gaussian is R Tϭa ho(k B T/ប␻ho)1/2,and hence larger than a ho.The use of a Bose distribution function does not change signiﬁcantly this estimate. The above discussion reveals that Bose-Einstein con-densation in harmonic traps shows up with the appear-ance of a sharp peak in the central region of the density distribution.An example is shown in Fig.4,where we plot the prediction for the condensate and thermal den-sities of5000noninteracting particles in a spherical trap at a temperature Tϭ0.9T c0,where T c0is the temperature at which condensation occurs(see discussion in the next section).The curves correspond to the column density, namely the particle density integrated along one direc-tion,n(z)ϭ͐dx n(x,0,z);this is a typical measured quantity,the x direction being the direction of the light beam used to image the atomic cloud.By plotting di-rectly the density n(r),the ratio of the condensed and noncondensed densities at the center would be even larger.By taking the Fourier transform of the ground-state wave function,one can also calculate the momentum distribution of the atoms in the condensate.For the ideal gas,it is given by a Gaussian centered at zero mo-mentum and having a width proportional to a hoϪ1.The distribution of the thermal cloud is,also in momentum space,ing a classical distribution function oneﬁnds that the width is proportional to(k B T)1/2.Ac-tually,the momentum distributions of the condensed and noncondensed particles of an ideal gas in harmonic traps have exactly the same form as the density distribu-tions n0and n T shown in Fig.4.The appearance of the condensate as a narrow peak in both coordinate and momentum space is a peculiar fea-ture of trapped Bose gases having important conse-quences in both the experimental and theoretical analy-sis.This is different from the case of a uniform gas where the particles condense into a state of zero mo-mentum,but BEC cannot be revealed in coordinate space,since the condensed and noncondensed particles ﬁll the same volume.Indeed,the condensate has been detected experimen-tally as the occurrence of a sharp peak over a broader distribution,in both the velocity and spatial distribu-tions.In theﬁrst case,one lets the condensate expand freely,by switching off the trap,and measures the den-sity of the expanded cloud with light absorption(Ander-son et al.,1995).If the particles do not interact,the ex-pansion is ballistic and the imaged spatial distribution of the expanding cloud can be directly related to the initial momentum distribution.In the second case,one mea-sures directly the density of the atoms in the trapby FIG.4.Column density for5000noninteracting bosons in a spherical trap at temperature Tϭ0.9T c0.The central peak is the condensate,superimposed on the broader thermal distri-bution.Distance and density are in units of a ho and a hoϪ2,re-spectively.The density is normalized to the number of atoms. The same curves can be identiﬁed with the momentum distri-bution of the condensed and noncondensed particles,provided the abscissa and the ordinate are replaced with p z,in units of a hoϪ1,and the momentum distribution,in units of a ho2,respec-tively.467Dalfovo et al.:Bose-Einstein condensation in trapped gases Rev.Mod.Phys.,Vol.71,No.3,April1999means of dispersive light scattering(Andrews et al., 1996).In both cases,the appearance of a sharp peak is the main signature of Bose-Einstein condensation.An important theoretical task consists of predicting how the shape of these peaks is modiﬁed by the inclusion of two-body interactions.As anticipated in Fig.3,the interac-tions can change the picture drastically.This effect will be deeply discussed in Sec.III.The shape of the conﬁningﬁeld alsoﬁxes the symme-try of the problem.One can use spherical or axially sym-metric traps,for instance.Theﬁrst experiments on ru-bidium and sodium were carried out with axial symmetry.In this case,one can deﬁne an axial coordi-nate z and a radial coordinate rЌϭ(x2ϩy2)1/2and the corresponding frequencies,␻z and␻Ќϭ␻xϭ␻y.The ra-tio between the axial and radial frequencies,␭ϭ␻z/␻Ќ,ﬁxes the asymmetry of the trap.For␭Ͻ1the trap is cigar shaped while for␭Ͼ1is disk shaped.In terms of␭the ground state Eq.(3)for noninteracting bosons can be rewritten as␸0͑r͒ϭ␭1/43/4aЌ3/2expͫϪ12aЌ2͑rЌ2ϩ␭z2͒ͬ.(6)Here aЌϭ(ប/m␻Ќ)1/2is the harmonic-oscillator length in the xy plane and,since␻Ќϭ␭Ϫ1/3␻ho,one has also aЌϭ␭1/6a ho.The choice of an axially symmetric trap has proven useful for providing further evidence of Bose-Einstein condensation from the analysis of the momentum distri-bution.To understand this point,let us take the Four-ier transform of the wave function Eq.(6):␸˜0(p)ϰexp͓ϪaЌ2(pЌ2ϩ␭Ϫ1p z2)/2ប2͔.From this one can calculate the average axial and radial widths.Their ratio,ͱ͗z2͗͘Ќ2͘ϭͱ␭,(7) isﬁxed by the asymmetry parameter of the trap.Thus the shape of the expanded cloud in the xz plane is an ellipse,the ratio between the two axis(aspect ratio)be-ing equal toͱ␭.If the particles,instead of being in the lowest state(condensate),were thermally distributed among many eigenstates at higher energy,their distribu-tion function would be isotropic in momentum space, according to the equipartition principle,and the aspect ratio would be equal to1.Indeed,the occurrence of anisotropy in the condensate peak has been interpreted from the very beginning as an important signature of BEC(Anderson et al.,1995;Davis et al.,1995;Mewes et al.,1996a).In the case of the experiment at the Joint Institute for Laboratory Astrophysics(JILA)in Boul-der,the trap is disk-shaped with␭ϭͱ8.Theﬁrst mea-sured value of the aspect ratio was about50%larger than the prediction,ͱ␭,of the noninteracting model (Anderson et al.,1995).Of course,a quantitative com-parison can be obtained only including the atom-atom interaction,which affects the dynamics of the expansion (Holland and Cooper,1996;Dalfovo and Stringari,1996; Holland et al.,1997;Dalfovo,Minniti,Stringari,and Pi-taevskii,1997).However,the noninteracting model al-ready points out this interesting effect due to anisotropy.B.Trapped bosons atﬁnite temperature:thermodynamic limitAt temperature T the total number of particles is given,in the grand-canonical ensemble,by the sumNϭ͚n x,n y,n z͕exp͓␤͑␧n x n y n zϪ␮͔͒Ϫ1͖Ϫ1,(8) while the total energy is given byEϭ͚n x,n y,n z␧n x n y n z͕exp͓␤͑␧n x n y n zϪ␮͔͒Ϫ1͖Ϫ1,(9) where␮is the chemical potential and␤ϭ(k B T)Ϫ1.Be-low a given temperature the population of the lowest state becomes macroscopic and this corresponds to the onset of Bose-Einstein condensation.The calculation of the critical temperature,the fraction of particles in the lowest state(condensate fraction),and the other ther-modynamic quantities,starts from Eqs.(8)and(9)with the appropriate spectrum␧nxn y n z(de Groot,Hooman, and Ten Seldam,1950;Bagnato,Pritchard,and Klepp-ner,1987).Indeed the statistical mechanics of these trapped gases is less trivial than expected atﬁrst sight. Several interesting problems arise from the fact that these systems have aﬁnite size and are inhomogeneous. For example,the usual deﬁnition of thermodynamic limit(increasing N and volume with the average density kept constant)is not appropriate for trapped gases. Moreover,the traps can be made very anisotropic, reaching the limit of quasi-two-dimensional and quasi-one-dimensional systems,so that interesting effects of reduced dimensionality can be also investigated.As in the case of a uniform Bose gas,it is convenient to separate out the lowest eigenvalue␧000from the sum (8)and call N0the number of particles in this state.This number can be macroscopic,i.e.,of the order of N,when the chemical potential becomes equal to the energy of the lowest state,␮→␮cϭ32ប␻¯,(10)where␻¯ϭ(␻xϩ␻yϩ␻z)/3is the arithmetic average of the trapping frequencies.Inserting this value in the rest of the sum,one can writeNϪN0ϭ͚n x,n y,n z 01exp͓␤ប͑␻x n xϩ␻y n yϩ␻z n z͔͒Ϫ1.(11) In order to evaluate this sum explicitly,one usually as-sumes that the level spacing becomes smaller and smaller when N→ϱ,so that the sum can be replaced by an integral:NϪN0ϭ͵0ϱdn x dn y dn zexp͓␤ប͑␻x n xϩ␻y n yϩ␻z n z͔͒Ϫ1.(12) This assumption corresponds to a semiclassical descrip-tion of the excited states.Its validity implies that the relevant excitation energies,contributing to the sum (11),are much larger than the level spacingﬁxed by the468Dalfovo et al.:Bose-Einstein condensation in trapped gases Rev.Mod.Phys.,Vol.71,No.3,April1999。

Bose-Einstein condensates in the presence of a magnetic trap and optical lattice two-mode a

University of New Mexico Albuquerque, NM 87131
P. G. Kevrekidis † Department of Mathematics and Statistics
University of Massachusetts Amherst, MA 01003-4515
4. Numerical results
11
4.1. Veriﬁcation of the two-mode analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2. Connection to the large nonlinearity solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Bose-Einstein condensates in the presence of a magnetic trap and optical lattice: two-mode approximation
Todd Kapitula ∗ Department of Mathematics and Statistics
form
iqt
+
1 2 qxx
+
ωq
+
δ|q|2q
=
( 1 Ω2x2 2
+
p(x + ξ))q
(1.1)
[18]. Here δ ∈ {−1, +1}, ξ ∈ R, and p : R → R is L-periodic. The term Ω2x2/2 represents the eﬀect of the magnetic trap, with the parameter Ω being the reduced frequency of the trap [18], while the term p(x + ξ) represents the eﬀect of the optical lattice. The parameter ξ corresponds to a shift of the optical lattice relative to the magnetic trap. As it will be seen in the following sections, this shift plays an important role in the stability of the steady-states.

控制Bose-Einstein凝聚混沌

ＢＣ的实现为人们研究诸如在原子物理、凝Ｅ
聚态物理、量子光学等领域中的许多物较大时会改变原来的动力学行为而把系统的混沌控制成某种周期行为。当调制
了一个完美的平台。由于多体作用而将非线性效应引入到这样的量子系统，其中的混沌特征已经广为人们所关注。混沌与凝聚体的塌缩 … ，宏观量子
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bose-einstein_statistic_博斯-爱因斯坦统计量

bose-einstein statistic 博斯-爱因斯坦统计量1. 引言1.1 概述博斯-爱因斯坦统计量是一种描述粒子在量子力学体系中分布情况的统计方法。

它是由印度物理学家博斯和奥地利物理学家爱因斯坦在20世纪早期提出的，用于研究玻色子（Bosons）这类具有整数自旋的基本粒子。

1.2 文章结构本文将依次介绍博斯-爱因斯坦统计量的相关概念和原理，探讨其在凝聚态物理、超冷原子系统以及光子和声子系统中的应用，并深入讨论实验验证与实现方法。

最后，对整篇文章进行总结和结论。

1.3 目的本文旨在全面介绍博斯-爱因斯坦统计量这一重要的物理概念，揭示其在不同领域中的应用与意义。

通过对博斯-爱因斯坦分布函数及其相关实验观测和验证方法的详细阐述，读者将能够更加全面地了解并深入探索该统计方法对现代物理学领域的重要性。

2. 博斯-爱因斯坦统计量2.1 统计力学基础：博斯-爱因斯坦统计量是统计力学中的一个重要概念。

在研究粒子或量子系统的行为时，统计力学提供了一种描述粒子分布和性质的数学工具。

而博斯-爱因斯坦统计量则针对玻色子（Bose particle）这类具有整数自旋的粒子提供了适用的统计分布函数。

2.2 玻色子与费米子：在粒子物理学中，玻色子和费米子是两类最常见的基本粒子。

相比之下，玻色子具有整数自旋，例如光子就是一种典型的玻色子；而费米子则具有半整数自旋，例如电子就是一种典型的费米子。

这两类粒子服从截然不同的统计分布规律，即波尔兹曼（Boltzmann）分布和博斯-爱因斯坦分布。

2.3 博斯-爱因斯坦分布函数：博斯-爱因斯坦分布函数描述了玻色子在不同能级上分布概率与温度之间的关系。

根据该函数，当温度趋近于绝对零度时，玻色子有极大的几率集聚在能级的基态上（基态是系统具有的最低能量状态）。

这一现象被称为玻色-爱因斯坦凝聚（Bose-Einstein condensation），是博斯-爱因斯坦统计量的一个重要特征。

高阶Bose-Einstein关联的动力学奇异性

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Ａｂｓｔｒａｃｔ：Ｉｎｑｕａｎｔｕｍｉｆｅｌｄｔｈｅｏｒｙ，ｔｈｅｃｌａｓｓｉｃｌａｍａｔｈｅｍａｔｉｃｓｃａｎｎｏｔｄｅｓｃｉｒｂｅｔｈｅｃｏｕｐｌｉｎｇｃｏｎｓｔａｎｔｇ２ — ｄｅｐｅｎｄｅｎｃｅｏｆｔｈｅｓｉｎｇｕｌａｉｔｒｉｅｓ．ＩｎｔｈｅｍｏｄｅｒｎｍａｔｈｅｍａｔｉｃｓｔｈｅｓｉｎｇｕｌａｉｒｔｉｅｓｏｆＮ－ｐｏｉｎｔｔｒａｎｃａｔｅｄＧｒｅｅｎ’ ｓｇｅｎｅｒａｌｉｚｅｄｆｎｃｔｉｏｎｓ

运用拉格朗日待定乘子法推导bose-einstein分布。

运用拉格朗日待定乘子法推导Bose-Einstein分布Bose-Einstein分布是描述玻色子（Bosons）在热力学平衡下的分布情况的统计分布。

在物理学中，玻色子是一类具有整数自旋的粒子，如光子、声子等。

为了推导Bose-Einstein分布，我们将运用拉格朗日待定乘子法，这是一种常用于求解约束条件下的极值问题的数学方法。

首先，我们考虑一个系统中有N个玻色子，每个玻色子可以占据的能级数目为g。

假设系统的总能量为E，我们希望求解出每个能级的玻色子数目n_i。

根据玻色子的特性，每个能级上的玻色子数目可以是0、1、2、3...直到无穷大。

为了求解这个问题，我们引入一个约束条件，即总玻色子数目为常数N。

这个约束条件可以表示为：n_1 + n_2 + n_3 + ... + n_i + ... = N为了推导出满足这个约束条件下的极值，我们引入拉格朗日乘子λ。

考虑到每个能级上的能量为ε_i，我们可以写出总能量E与每个能级上的玻色子数目n_i之间的关系：E = ε_1 * n_1 + ε_2 * n_2 + ε_3 * n_3 + ... +ε_i * n_i + ...现在，我们的目标是求解出每个能级上的玻色子数目n_i，使得总能量E满足约束条件。

为了达到这个目标，我们可以构建一个拉格朗日函数L：L = E - λ(n_1 + n_2 + n_3 + ... + n_i + ...)其中，λ是拉格朗日乘子。

我们的目标是最小化L，即求解出满足约束条件下的能量最小值。

为了求解L的极值，我们需要对L分别对n_i和λ求偏导数，并令偏导数为零。

首先对n_i求偏导数，我们得到：∂L/∂n_i = ε_i - λ = 0解上述方程可得到：ε_i = λ这个结果告诉我们，在满足约束条件下，每个能级上的能量ε_i与拉格朗日乘子λ相等。

接下来，我们对λ求偏导数，我们有：∂L/∂λ = n_1 + n_2 + n_3 + ... + n_i + ... - N = 0这个方程告诉我们，在满足约束条件下，总玻色子数目与拉格朗日乘子λ之间存在一个关系。

爱因斯坦英文介绍

I. 广义相对论
II. 狭义相对论
III. 布朗运动
IV. 光电效应
V. Energy is equal to mass multiplied by the square of the speed of light
VI. 爱因斯坦场方程
VII. 玻色-爱因斯坦统计
Einstein's famous sayings
A wise teacher
At sixteen Einstein applied for ETH（瑞士苏黎世的联邦工业大学）, but the entrance examination ended with failure. Mr. Weber read his papers of mathematics and physics and praised him: “You are a very intelligent child, Einstein, a very smart child, but you have a big drawback （缺点） that you do not want to express yourself."
The most incomprehensible thing about the world is that it
is comprehensible
Albert Einstein was a world famous physicist and he was considered as the greatest scientist of the 20th century. He won a Nobel Prize of physic in 1921.
1.真理就是在实践面前站得住脚的东西。 Truth is what stands the test of experience.

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a rXiv:h ep-e x /05615v17Jun25Improved description of Bose-Einstein Correlation function ∗T.Nov ´a k Representing L3Collaboration Institute for Mathematics,Astrophysics and Particle Physics,Radboud University Nijmegen 6525ED Nijmegen,The Netherlands The L3data on Bose-Einstein correlations of equally charged pion pairs produced in hadronic Z decays are analyzed in terms of various parametriza-tions.Preliminary results are presented here.PACS numbers:13.38.Dg,25.75.Gz 1.Introduction In particle and nuclear physics intensity interferometry provides a direct experimental method for the determination of sizes,shapes and lifetimes of particle emitting sources (for recent reviews see [1,2]).In particular,boson interferometry provides a powerful tool for the investigation of the space-time structure of particle production processes,since Bose-Einstein correlations (BEC)of two identical bosons reﬂect both geometrical anddynamical properties of the particle radiating source.Originally,the method of Bose-Einstein correlations was invented by the radio astronomers R.Hanbury Brown and R.Q.Twiss (HBT),who applied it to determine the angular diameters of main sequence stars [3,4].The ﬁrst experimental evidence for Bose-Einstein correlations in high energy physics dates back to 1959[5].The results were interpreted as BEC by G.and S.Goldhaber,Lee and Pais (GGLP)[6].Angular distributions of pions could be described more exactly by applying Bose-Einstein statistics instead of a classical statistical model.This eﬀect is frequently referred as either the HBT,or GGLP eﬀect,or simply Bose-Einstein correlations.The L3experiment provides a good opportunity to perform detailed investigation of Bose-Einstein correlations in e +e −annihilation at a center2arxiv-rhic04printed on February5,2008√of mass energy of,(1)ρ1(p1)ρ1(p2)whereρ2(p1,p2)is the two-particle invariant momentum distribution,ρ1(p i) is the single-particle invariant momentum distribution and p i stands for the four-momentum of particle i.Since it is diﬃcult to create the product of the single-particle distribution it is replaced by a“reference sample”,the two-particle density that would occur in the absence of BE interference.If long-range correlations can be neglected or corrected for,and if the short-range correlations are dominated by Bose-Einstein correlations,this two-particle BEC function is related to the Fourier-transformed source dis-tribution.If we assume that f(x)is the density distribution of the source of the pions then the correlation function is found to beC2(p1,p2)=1+|˜f(Q)|2,(2)where Q is the invariant four-momentum diﬀerence,Q=−(p1−p2)2and ˜f(Q)is the Fourier transform of f(x)˜f(Q)= d x exp(iQx)f(x).(3)2.1.Gaussian distributed sourceThe simplest assumption is that the source has a Gaussian distribu-tion,in which case the Fourier transformed source function is determined as˜f(Q)=exp −Q2arxiv-rhic04printed on February 5,200830.811.21.41.61.82Q (GeV)C 2(Q )Fig.1.The Bose-Einstein correlation function C 2.The full line corresponds to the ﬁt by Eq.4,while the dashed line stands for the long-range momentum correlations.A ﬁt of this correlation function results in an unacceptably low conﬁ-dence level from which one concludes that the shape of the source deviates from a Gaussian.The ﬁt is particularly bad at low Q values,as shown in Fig.1.2.2.L´e vy distributed sourceThe Central limit theorem states that,under certain conditions,the sum of a large number of random variables behaves as a Gaussian distribution.The generalization of the central limit theorem gives the classiﬁcation of the stable distributions.The study of these stable distributions was begun by Paul L´e vy in the 1920’s.For recent results see [8].According to the generalized Central Limit Theorem,if particle produc-tion is a result of a multifold probabilistic process such as the branching of gluons into gluons into gluons etc.,then the class of possible limiting probability distributions coincides with the class of L´e vy distributions [8].The characteristic function of symmetric stable distributions is [9]˜f (Q )=exp iQδ−|RQ |α4arxiv-rhic04printed on February 5,2008Here we utilize the notation of Nolan[8].The index of stability,α,satisﬁes the inequality 0<α≤2.The case α=2corresponds to a Gaussian source distribution.For more details see [8].Thus the Bose-Einstein correlation function for L´e vy stable distributions has the following,relatively simple form:C 2(Q )=γ[1+λexp (−(RQ )α)](1+δQ )(6)After ﬁtting the data with Eq.6it is clear that the correlation function is far from Gaussian:α≈1.3.However,the conﬁdence level is still unacceptably low.The curve of this ﬁt is shown in Fig.2.0.811.21.41.61.82Q (GeV)C 2 (Q )Fig.2.The Bose-Einstein correlation function C 2.The full line corresponds to the ﬁt by Eq.4,while the dashed line stands for the long-range momentum correlations.Since there is no particle production before the onset of the collision,a more appropriate form of the source distribution for the time component is the asymmetric stable distribution.In this case,one obtains the following result for the correlation function:C 2(Q )=γ[1+λcos [(R a Q )α]exp (−(RQ )α)](1+δQ ),(7)where R a is an additional parameter.The ﬁt of Eq.7to the data is statistically acceptable.The data points are well described by the ﬁt curve which is shown in Fig.3.Note that forarxiv-rhic04printed on February 5,200850.811.21.41.61.82Q (GeV)C 2 (Q )Fig.3.The Bose-Einstein correlation function C 2.The full line corresponds to the ﬁt by Eq.4,while the dashed line stands for the long-range momentum correlations.Q between 0.5GeV and 1.5GeV the ﬁtted curve goes below one.This is caused by the cosine term,which comes from the asymmetric L´e vy assump-tion,in Eq.7.The ﬁtted value of the index of stability,αis found to be approximately 0.8.3.ConclusionsThe assumption that the source has a Gaussian shape is too simple.A good description of the Bose-Einstein correlation function is achieved using L´e vy stable distributions as the source function.AcknowledgmentsThe author would like to thank Tam´a s Cs¨o rg˝o ,Wolfram Kittel and Wes Metzger for inspiration,support and careful attention.He is also grateful to the members of the L3collaboration.REFERENCES[1]W.Kittel,Acta Phys.Pol.B32(2001)3927.6arxiv-rhic04printed on February5,2008[2]T.Cs¨o rg˝o,Heavy Ion Physics15(2002)1.[3]R.Hanbury Brown and R.Q.Twiss,Phil.Mag.45(1954)663.[4]R.Hanbury Brown and R.Q.Twiss,Nature178(1956)1046.[5]G.Goldhaber,W.B.Fowler,S.Goldhaber,T.F.Hoang,Phys.Rev.Lett.3(1959)181.[6]G.Goldhaber,S.Goldhaber,W.Lee and A.Pais,Phys.Rev.120(1960)300.[7]S.Bethke,Nucl.Phys.B370(1992)310.[8]J.P.Nolan,Stable Distributions:Models for heavy Tailed Data∼jpnolan/stable/CHAP1.PDF[9]T.Cs¨o rg˝o,S.Hegyi and W.A.Zajc,nucl-th/0310042,Eur.Phys.J.C36,(2004)67.。