Bose-Einstein condensates near a microfabricated surface

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Bose-Einstein condensation of scalar fields on hyperbolic manifolds

a rXiv:h ep-th/9213v11O ct1992Bose-Einstein condensation of scalar ﬁelds on hyperbolic manifolds Guido Cognola and Luciano Vanzo Dipartimento di Fisica -Universit`a di Trento ∗,Italia and Istituto Nazionale di Fisica Nucleare,Gruppo Collegato di Trento september 1992PACS numbers:03.70Theory of quantized ﬁelds05.90Other topics in statistical physics and thermodynamics1IntroductionBose-Einstein condensation for a non relativistic ideal gas has a long history[1].The physical phenomenon is well described in many text books (see for example ref.[2])and a rigorous mathematical discussion of it was given by many authors[3,4].The generalization to a relativistic idel Bose gas is non trivial and only recently has been discussed in a series of papers[5,6,7].It is well known that in the thermodynamic limit(inﬁnite volume andﬁxed density)there is a phase transition of theﬁrst kind in correspondence of the critical temperature at which the condensation manifests itself.At that temperature,the ﬁrst derivative of some continuous thermodynamic quantities has a jump.If the volume is keepedﬁnite there is no phase transition,nevertheless the phenomenon of condensation still occurs,but the critical temperature in this case is not well deﬁned.For manifolds with compact hyperbolic spatial part of the kind H N/Γ,Γbe-ing a discrete group of isometries for the N-dimensional Lobachevsky space H N, zero temperature eﬀects as well asﬁnite temperature eﬀects induced by non-trivial topology,have been recently studied in some detail[8,9,10,11,12,?,14,15,16]. To our knowledge,a similar analysis has not yet been carried out for non compact hyperbolic manifolds.Hyperbolic spaces have remarkable properties.For example,the continuous spec-trum of the Laplace-Beltrami operator has a gap determined by the curvature ra-dius of H N,implying that masslessﬁelds have correlation functions exponentially decreasing at inﬁnity(such a gap is not present for the Dirac operator).For that reason,H4was recently proposed as an excellent infrared regulator for massless quantumﬁeld theory and QCD[17].Critical behaviour is even more striking.In two ﬂat dimension vortex conﬁgurations of a complex scalarﬁeld,the XY model for He4ﬁlms have energy logarithmically divergent with distance,while on H2it isﬁnite. This implies that the XY model is disordered at anyﬁnite temperature on H2.Even quantum mechanics on H2has been the subject of extensive investigations[18].The manifold H4is also of interest as it is the Euclidean section of anti-de Sitter space which emerges as the ground state of extended supergravity theories.The stress tensor on this manifold has been recently computed for both boson and fermion ﬁelds using zeta-function methods[19].In the present paper we shall discussﬁnite temperature eﬀects and in particular the Bose-Einstein condensation for a relativistic ideal gas in a3+1dimensional ultrastatic space-time M=R×H3.We focus our attention just on H3,because such a manifold could be really relevant for cosmological and astrophysical applications.To this aim we shall derive the thermodynamic potential for a charged scalarﬁeld of mass m on M,using zeta function,which on H3is exactly known.We shall see that the thermodynamic potential has two branch points when the chemical potentialµriches±ωo,ω2o=κ+m2being the lower bound of the spectrum of the operator L m=−△+m2and−κthe negative constant curvature of H3.The values±ωo will be riched byµ=µ(T)of course for T=0,but also for T=T c>0.This is the critical temperature at which the Bose gas condensates.The paper is organized as follows.In section2we study the elementary properties of the Laplace-Beltrami operator on H3;in particular we derive its spectrum and build up from it the related zeta-function.In section3we brieﬂy recall how zeta-function can be used in order to regularize the partition function and we derive the regularized expression for the thermodynamic potential.In section4we discuss the Bose-Einstein condensation and derive the critical temperatures in both the cases of low and high temperatures.In section5we consider in detail the low and high temperature limits and derive the jump of theﬁrst derivative of the speciﬁc heat.The paper end with some considerations on the results obtained and some suggestions for further developments.2The spectrum and the zeta function of Laplace-Beltrami operator on H3For the aims of the present paper,the3-dimensional Lobachevsky space H3can be seen as a Riemannian manifold of constant negative curvature−κ,with hyperbolic metric dl2=d̺2+sinh2̺(dϑ2+sin2ϑdϑ2)and measure dΩ=sinh2̺d̺dΣ,dΣbeing the measure on S2.For convenience,here we normalize the curvature−κto−1.In these coordinates,the Laplace-Beltrami operator△reads△=∂2∂̺+1so,in order to derive it,it is suﬃcient to study radial wave functions of−△,that is solution of equationd2ud̺+λu=0(2) which reduces tod2vνsinh̺(4) Now,the L2(dΩ)scalar product for uν(̺)is(uν,uν′)=4πν2δ(ν−ν′)(5)from which the density of states̺(ν)=Vν2/2π2directly follows.As usual,we have introduced the large,ﬁnite volume V to avoid divergences.When possible,the limit V→∞shall be understood.At this point the computation of zeta function is straightforward.As we shall see in the following,what we are really interested in,is the zeta function related to the operators Q±=L1/2m±µ.The eigenvalues of L m areω2(ν)=ν2+a2=ν2+κ+m2,then we getζ(s;Q±)=V(4π)3/2Γ(s−1/2)(2a)s−3F(s+1,s−3;s−12a)(6)where F(α,β;γ;z)is the hypergeometric function.For its properties and its integral representations see for example ref.[20].It has to be noted that eq.(6)is the very same one has on aﬂat space for a massiveﬁeld with mass equal to a.Here in fact, the curvature plays the role of an eﬀective mass.As we see from eq.(6),the zeta function related to the pseudo-diﬀerential oper-ators Q±has simple poles at the points s n=3,2,1,−1,−2,−3,...with residues b n(±µ)=Res(ζ(s;Q±),s n)given byb3(±µ)=Vπ2;b1(±µ)=V2a);(8)c−n=(−1)n nV(2a)n+3dz F(α,β;γ;z)=αβgdV(11)where Lµ=−(∂τ−µ)2+L m and the Wick rotationτ=ix0has to be understood. In eq.(11)the integration has to be taken over allﬁeldsφ(τ,x a)withβ-periodicity with respect toτ.The eigenvalues of the whole operator Lµ,sayµn,νreadµn,ν= 2πnlog det(ℓ−2Lµ)=−1β[logℓ2ζ(0;Lµ)+ζ′(0;Lµ)](13)βℓbeing an arbitrary normalization parameter coming from the scalar path-integral measure.Note thatℓ,which has the dimensions of a mass,is necessary in order to keep the zeta-function dimensionless for all s.Theﬁnite temperature andµdependent part of the thermodynamic potential does not suﬀer of the presence of such an arbitrary parameter.On the contrary,ℓenters in the regularized expression of vacuum energy and this creates an ambiguity[12],which is proportional to the heat kernel expansion coeﬃcient K N(L m)related to L m(in general,K N(L m)=0). When the theory has a natural scale parameter,like the mass of the particle or the constant curvature of the manifold,the ambiguity can be removed by an”ad hoc”choice ofℓ[23].Here we would like to study the behaviour of thermodynamic quantities,then we are only interested in theµand T dependent part of the thermodynamic potential; that is a well deﬁned quantity,which does not need regularization.To compute it, it is not necessary to use all the analytic properties of zeta function(for a careful derivation of vacuum energy see for example refs.[12,14]).Then we can proceed in a formal way and directly compute log det Lµdisregarding the vacuum energy divergent term.First of all we observe that∞ n=−∞log(ω2+(2πn/β+iµ)2)=∞ n=−∞ dω24ω cothβ2(ω−µ) dω2(14)=−log 1−e−β(ω+µ) −log 1−e−β(ω−µ) −βωUsing eq.(13),recalling thatω2=ν2+a2and by integrating overνwith the state density that we have derived in the previous section,we get the standard result1Ω(β,µ)=−2π2β log 1−e−β(ω(ν)+µ) +log 1−e−β(ω(ν)−µ) ν2dν(15)V+∞ n=1cosh nβµK2(anβ)π2∞ n=0µ2n2;L m)β−s ds(17)πi1E(β,µ)=−where K2is the modiﬁed Bessel function,c is a suﬃciently large real number and ζR(s)is the usual Riemann zeta-function.The integral representations(17)and (18),which are valid for|µ|<a,are useful for high temperature expansion.On the contrary,the representation(16)in terms of modiﬁed Bessel functions is more useful for the low temperature expansion,since the asymptotics of Kνis well known.4Bose-Einstein condensationIn order to discuss Bose-Einstein condensation we have to analyze the behaviour of the charge density∂Ω(V,β,z)ρ=zV(expβωj−z)(20) and the activity z=expβµhas been introduced.Theωj in the sum are meant to be the Dirichlet eigenvalues for any normal domain V⊂H3.That is,V is a smooth connected submanifold of H3with non empty piecewise C∞boundary.By the inﬁnite volume limit we shall mean that a nested sequence of normal domains V k has been choosen together with Dirichlet boundary conditions and such that V k≡H3.The reason for this choice is the following theorem due to Mac Kean (see for example[24]):—ifωok denotes the smallest Dirichlet eigenvalue for any sequence of normal domains V kﬁlling all of H3thenωok≥a and lim k→∞ωok=a.(Although the above inequality is also true for Neumann boundary conditions,the existence of the limit in not assured to the authors knowledge).Now we can show the convergence of theﬁnite volume activity z k to a limit point ¯z as k→∞.Toﬁx ideas,let us supposeρ≥0:then z k∈(1,expβωok).Since ρ(V,β,z)is an increasing function of z such thatρ(V,β,1)=0andρ(V,β,∞)=∞, for eachﬁxed V k there is a unique z k(¯ρ,β)∈(1,βexpωok)such that¯ρ=ρ(V k,β,z k).By compactness,the sequence z k must have at least one ﬁxed point ¯z and as ωok →a 2as k goes to inﬁnity,by Mc Kean theorem,¯z ∈[1,exp βa ].From this point on,the mathematical analysis of the inﬁnite volume limit exactly parallels the one in ﬂat space for non relativistic systems,as it is done in various references [25,26,4].Inparticular,there is a critical temperature T c over which there are no particles in the ground state.T c is the unique solution of the equation̺=sinh βa cosh βaV ∂µ= ∞0 1e β(ω(ν)−µ)−1 ν22π2 ∞0ν2dνκ+m 2.Thevery diﬀerence between ﬂat and hyperbolic spaces occurs for massless particles.We shall return on this important point in a moment.Solutions of eq.(21)can be easily obtained in the two cases βa ≫1and βa ≪1(in the case of massive bosons these correspond to non relativistic and ultrarela-tivistic limits respectively).We have in fact̺≃T 3e x 2/2a −1= aT 2π2 ∞0x 2dx3;βa ≪1(24)from which we get the corresponding critical temperaturesT c =2πζR (3/2)2/3;βa ≫1(25)T c = 3̺∂µ̺(T,µ)(29)and since ∂µ̺diverges for µ=a we obtain µ′(T +c )=0.This is not the case of µ′′.In fact we shall see that µ′′(T +c )is diﬀerent from zero and therefore µ′′(T )isadiscontinuous function of temperature.This implay that theﬁrst derivative of the speciﬁc heat C V has a jump for T=T c given bydC VdT T−c=µ′′(T+c)∂U(T,µ)∂TT=T+c(30)U(T,µ)being the internal energy,which can be derived by means of equation U(β,µ)=−µ̺V+∂πΓ(k+1)Γ(−k+5/2)(2s)−k(32) Then,for small T we haveE(β,µ)≃−a4Vanβ5/2∞ k=0Γ(k+5/2) 2π 3/2∞ n=1e−nβ(a−|µ|)T A2;T∂̺2̺(35)where A=2.363and C=−2.612are two coeﬃcients of the expansion ∞ n=1e−nxNow,using eq.(30),we have the standard resultdC VdT T−c=3̺C2T c(37) The high temperature expansion could be obtained by using eq.(17),like in ref.[15].Here we shall use eq.(18),because for the aim of the present paper it is more ing the properties ofζ(s;Q±),which we have discussed in section2,we see that the integrand function in eq.(18)ζR(s+1)Γ(s)[ζ(s;Q+)+ζ(s;Q−)]β−(s+1)(38) has simple poles at s=3,1,0,−3,−5,−7,...and a double pole at s=−1.In-tegrating this function on a closed path containing all the poles,we get the high temperature expansion,valid for T>T c(hereγis the Euler-Mascheroni constant)E(β,µ)≃−V π212β2(a2−2µ2)+(a2−µ2)3/224π2(3a2−µ2)+a44π+γ−3(−2π)n(2n+1)where we have used the formulaζ′R(−2n)=Γ(2n+1)ζR(2n+1)3+µT(a2−µ2)1/212π2(41)−2(−2π)nForµ=a,the leading term of this expression gives again the result(24).From eq.(41),by a strightforward computation and taking only the leading terms into account,one getsµ′′(T+c)≃−12π2andﬁnally,from eqs.(30)and(31) dC VdT T−c≃−32̺π2pands adiabatically,can represent a manifold of the form we have considered.The problem we have studied then canﬁnd physical applications in the standard model of the universe.References[1]A.Einstein.Berl.Ber.,22,261,(1924).[2]K.Huang.Statistical Mechanics.J.Wiley and Sons,Inc.,New York,(1963).[3]H.Araki and E.J.Woods.J.Math.Phys.,4,637,(1963).[4]ndau and m.Math.Phys.,70,43,(1979).[5]H.E.Haber and H.A.Weldom.Phys.Rev.Lett.,46,1497,(1981).[6]H.E.Haber and H.A.Weldom.J.Math.Phys.,23,1852,(1982).[7]H.E.Haber and H.A.Weldom.Phys.Rev.D,25,502,(1982).[8]A.A.Bytsenko and Yu.P.Goncharov.Mod.Phys.Lett.A,6,669,(1991).[9]Yu.P.Goncharov and A.A.Bytsenko.Class.Quantum Grav.,8,L211,(1991).[10]A.A.Bytsenko and Y.P.Goncharov.Class.Quantum Grav.,8,2269,(1991).[11]A.A.Bytsenko and S.Zerbini.Class.Quant.Gravity,9,1365,(1992).[12]G.Cognola,L.Vanzo and S.Zerbini.J.Math.Phys.,33,222,(1992).[13]A.A.Bytsenko,L.Vanzo and S.Zerbini.Mod.Phys.Lett.A,7,397,(1992).[14]A.A.Bytsenko,G.Cognola and L.Vanzo.Vacuum energy for3+1dimesionalspace-time with compact hyperbolic spatial part.Technical Report,Universit´a di Trento,UTF255,(1992).to appear in JMP.[15]G.Cognola and L.Vanzo.Thermodynamic potential for scalarﬁelds in space-time with hyperbolic spatial part.Technical Report,Universit´a di Trento,UTF 258,(199).to be published.[16]A,A.Bytsenko,L.Vanzo and S.Zerbini.Zeta-function regularization approachtoﬁnite temperature eﬀects in Kaluza-Klein space-time.Technical Report,Uni-versit´a di Trento,UTF259,(1992).to be published.[17]C.G.Callan and F.Wilczek.Nucl.Phys.B,340,366,(1990).[18]N.Balasz and C.Voros.Phys.Rep.,143,109,(1986).[19]R.Camporesi and A.Higuchi.Phys.Rev.D,45,3591,(1992).[20]I.S.Gradshteyn and I.M.Ryzhik.Table of integrals,series and products.Aca-demic press,Inc.,New York,(1980).[21]A.Actor.Phys.Lett.B,157,53,(1985).[22]m.Math.Phys.,55,133,(1977).[23]J.S.Dowker and J.P.Schoﬁeld.Nucl.Phys.B,327,267,(1989).[24]I.Chavel.Eingenvalues in Riemannian Geometry.Accademic Press,(1984).[25]R.Ziﬀ,G.E.Uhlenbeck and M.Kac.Phys.Rep.,32,169,(1977).[26]J.T.Lewis and J.V.Pul´m.Math.Phys.,36,1,(1974).[27]J.M.Blatt and S.T.Butler.Phys.Rev.Lett.,100,476,(1955).。

2022年自考专业(英语)英语科技文选考试真题及答案37

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

关于原子物理认识的英语作文

关于原子物理认识的英语作文The journey of understanding the fundamental building blocks of our universe has been a captivating and ever-evolving pursuit for scientists and thinkers alike. At the heart of this exploration lies the intriguing field of atomic physics, which delves into the intricate workings of the smallest known particles that make up the matter around us. As we delve deeper into the realm of atomic structure and behavior, we uncover a world of incredible complexity and wonder, shedding light on the very essence of our physical reality.One of the most significant milestones in the understanding of atomic physics was the groundbreaking work of Ernest Rutherford, a New Zealand-born physicist who is often referred to as the father of nuclear physics. In the early 20th century, Rutherford and his colleagues conducted a series of experiments that challenged the prevailing understanding of the atom, leading to the development of the Rutherford model of the atom. This model, which depicted the atom as a dense, positively charged nucleus surrounded by orbiting electrons, was a significant departure from the earlier plum pudding model proposed by J.J. Thomson.Rutherford's experiments involved bombarding thin sheets of gold foil with alpha particles, which are positively charged helium nuclei. The vast majority of the alpha particles passed through the foil undeflected, as expected, but a small percentage were unexpectedly deflected at large angles. This observation led Rutherford to conclude that the atom was not a solid, uniform sphere, as previously believed, but rather a dense, concentrated nucleus with a significant amount of empty space surrounding it. This groundbreaking discovery paved the way for a deeper understanding of the structure and behavior of atoms.Building upon Rutherford's work, the Danish physicist Niels Bohr further refined the understanding of atomic structure by proposing a model that incorporated the concept of quantized energy levels. Bohr's model suggested that electrons within an atom could only occupy specific, discrete energy levels, and that they could only transition between these levels by emitting or absorbing a specific amount of energy in the form of a photon. This model, known as the Bohr model of the atom, provided a more accurate description of the behavior of electrons within an atom and laid the foundation for the development of quantum mechanics.The advent of quantum mechanics, pioneered by physicists such as Max Planck, Werner Heisenberg, and Erwin Schrödinger, marked apivotal shift in our understanding of atomic physics. Quantum mechanics introduced the concept of the wave-particle duality, which posits that particles, including electrons, can exhibit both particle-like and wave-like properties. This revelation challenged the classical, deterministic view of the physical world and led to the development of probabilistic interpretations of atomic and subatomic phenomena.One of the most intriguing aspects of quantum mechanics is the principle of uncertainty, as formulated by Heisenberg. This principle states that there is a fundamental limit to the precision with which certain pairs of physical properties, such as position and momentum or energy and time, can be simultaneously measured. This limitation has profound implications for our understanding of the behavior of atoms and subatomic particles, as it suggests that the act of measurement can itself influence the outcome of an experiment.Another key concept in atomic physics is the wave function, which was in troduced by Schrödinger. The wave function is a mathematical representation of the state of a particle, and its square modulus is interpreted as the probability density of finding the particle in a particular location. The wave function evolves over time according to the Schrödinger equation, which describes the dynamics of quantum systems. The wave function's ability to represent the superposition of multiple possible states, known as quantum superposition, is acornerstone of quantum mechanics and has led to the development of groundbreaking technologies, such as quantum computing.As our understanding of atomic physics has progressed, we have also gained insights into the fundamental forces that govern the interactions between particles at the atomic and subatomic scales. These forces, known as the four fundamental forces of nature, include the strong nuclear force, the weak nuclear force, the electromagnetic force, and the gravitational force. The study of these forces and their interplay has led to the development of theories such as quantum electrodynamics (QED) and quantum chromodynamics (QCD), which provide a comprehensive description of the behavior of particles and the interactions between them.One of the most significant developments in atomic physics in recent decades has been the exploration of the behavior of atoms and molecules at extremely low temperatures, known as the field of atomic, molecular, and optical (AMO) physics. In this realm, researchers have been able to observe and manipulate the behavior of individual atoms and molecules, leading to groundbreaking discoveries and the development of technologies such as atomic clocks, Bose-Einstein condensates, and quantum sensors.The ongoing exploration of atomic physics has not only deepened our understanding of the fundamental nature of matter and energybut has also paved the way for numerous technological advancements that have transformed our world. From the development of nuclear power and medical imaging techniques to the emergence of quantum computing and nanotechnology, the insights gained from the study of atomic physics have had a profound impact on our lives and continue to shape the future of scientific and technological progress.As we continue to delve into the mysteries of the atomic world, we are reminded of the enduring power of human curiosity and the relentless pursuit of knowledge. The journey of understanding atomic physics is a testament to the human spirit, as we strive to unravel the intricacies of the universe, one particle at a time. With each new discovery and every breakthrough, we inch closer to a more comprehensive understanding of the fundamental building blocks of our reality, unlocking the potential to transform our world in ways we can scarcely imagine.。

实验在宏观尺度验证全同粒子的全同性

在宏观尺度验证全同粒子的不可分辨性原作：Michael Fleischhauer 翻译：葛韶锋原文网址：/nature/journal/v445/n7128/full/445605a.html 对微观世界进行正确的描述需要使用量子力学，在量子力学中同种微观粒子具有全同性，更确切地说是不可分辨性（Indistinguishability），然而迄今尚未对这一性质在宏观的尺度上验证过。

本文描述的实验用两团在宏观尺度上隔离开的玻色-爱因斯坦凝聚体（Bose-Einstein Condensates）验证了全同粒子的全同性，这是前所未有的。

将相干光脉冲（Coherent Light Pulse）携带的信息传递给一团原子的自旋（Spin）是量子物理实验中采用标准的技术；但是从远处的第二团原子中重新提取出光脉冲的信息看起来像是不可思议的巫术。

但是在下文的介绍中我们将会看到这并不是什么巫术，而是量子力学的效应。

在量子世界中，同种粒子是不可区分的：描述它们的波函数（Wavefunction）是所有占据了全部可能状态的同种粒子波函数的叠加。

这意味着我们在讨论地球上某一个电子的时候，严格说来必须考虑到月球上的每一个电子对它的影响，它们之间是完全平等的。

幸运的是，我们可以忽略这种影响，因为它们离得太远了，而单个电子的波函数主要分布在微观的尺度上，相距宏观尺度电子间波函数的重叠可以忽略不计。

在实际的计算之中，我们常常忽略它们之间的关联。

但是在本期《自然》（Nautre）杂志上，Ginsberg，Garner和Hau的文章中我们可以看到两团原子形成的玻色爱因斯坦凝聚体之间具有全同性，即使它们之间隔开一定的距离。

虽然这两团原子之间的距离不足一毫米，比起地球和月亮之间的距离来显得微不足道，但是比起只有微观尺度的凝聚体本身来已经是相当大的了。

为了演示这种性质，Ginsberg等人使用了一种在几年前才发展出来的一项技术将光脉冲中包含的信息储存在原子凝聚体中。

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。

西南交通大学教师专业技术职务评审个人信息简表

主要学习、工作简历
2003年9月-2007年6月，西北师范大学，物理与电子工程学院，本科。 2007年9月-2010年6月，西北师范大学，物理与电子工程学院，硕士。 2010年9月-2013年12月，中国科学院兰州化学物理研究所，博士。
Hale Waihona Puke 2014年1月至今，中国科学院宁波材料技术与工程研究所，博士后/助研
学术论文与著作
4
5
Phase transition of the ground state for Xin Jiang, Li-Bin Fu, two-component Bose–Einstein condensates in a Wen-shan Duan*, and Jie Liu triple-well trap Xin Jiang, Mai-mai Lin, Sheng-chang Li and Wen-shan Duan* Xin Jiang, LeXi. Shao, Jun Zhang, JianMin. Chen*.
西南交通大学教师专业技术职务评审个人信息简表
申报系列：█教学科研 □科学研究
姓名姜欣出生年月
申报评审程序类别：□实体性条件评审程序 █程序性评审程序（□校内人员
1984.05 申报学科材料科学与工程所在单位
█公开招聘人员）
中科院宁波材料技术与工程研究所
最高学位及毕业学校、专业现专业技术职务及任职时间助理研究员，2014.01 申报原因：公开招聘
博士，中国科学院大学，材料学拟评聘的专业技术职务副教授
业绩亮点：1.通过复合PVD技术结合涂层多尺度结构设计成功实现涂层润滑耐磨防腐功能一体化，该技术已实现产业化，成功应用于宁波中意液压马达公司高端摆线马达零部件的表面防护，2016年助企业新增利润300万元。 2.负责完成了由上海航天149厂委托的JG项目“空间太阳翼伸展机构用二硫化钼薄膜空间环境申报程序性评审性能研究”及工艺鉴定工作，并制定了相关产品标准，产品目前已进入批量供货阶段。原因与业绩亮点 3.通过调控PVD等离子体能量与状态降低了涂层内应力与反溅射损失，成功制备厚度达100μm 的单层超厚CrN涂层，突破了业内PVD耐磨涂层厚度难以超越50μm的极限，该技术成功应用于油田采油杆、阀门的防护。 4.近三年主持国家青年基金1项，中国博士后基金1项，军工课题2项，企业课题3项，累计经费300余万元。作为主要贡献人或骨干参与国家面上基金、科工局专项、中科院先导专项等项目4项，累计经费2000万元。

玻色-爱因斯坦凝聚论文：玻色-爱因斯坦凝聚孤子非谐势阱散射长度

玻色-爱因斯坦凝聚论文：玻色-爱因斯坦凝聚孤子非谐势阱散射长度【中文摘要】自从实验观察到二元玻色-爱因斯坦凝聚体(Bose-Einstein Condensates, BEC)现象以来,有关多组分BEC中的非线性研究已成为目前物质波研究领域中广泛关注的热点之一。

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

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

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

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

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

同时,基于平均场理论,简扼推导出描述BEC动力学的GP方程。

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

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

结果表明,凝聚体中两个不同组分中的孤子会发生融合现象。

且随着四次非谐外部势阱强度的增加,融合现象变得更加迅速。

从而证实,二元BEC中两孤子的融合行为可通过外部非谐势阱调控。

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

结果表明,两个孤子间发生碰撞的位置、时间和频率均与种间散射长度密切相关。

也就是说,二元BEC中的孤子碰撞行为可以通过种间散射长度来调控。

与此同时,我们发现孤子的幅度也可以利用种间散射长度来调控。

最后一章,对本文做了一个简单的总结,且对下一步研究工作进行了展望。

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

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>。

Sodium Bose-Einstein Condensates in the F=2 State in a Large-volume Optical Trap

a r X i v :c o n d -m a t /0208385v 1 [c o n d -m a t .s o f t ] 20 A u g 2002Sodium Bose-Einstein Condensates in the F=2State in a Large-volume Optical TrapA.G¨o rlitz[*],T.L.Gustavson[†],A.E.Leanhardt,R.L¨o w[*],A.P.Chikkatur,S.Gupta,S.Inouye[‡],D.E.Pritchard and W.KetterleDepartment of Physics,MIT-Harvard Center for Ultracold Atoms,and Research Laboratory of Electronics,Massachusetts Institute of Technology,Cambridge,MA 02139(Dated:February 1,2008)We have investigated the properties of Bose-Einstein condensates of sodium atoms in the upper hyperﬁne ground state in a purely optical trap.Condensates in the high-ﬁeld seeking |F=2,m F =-2 state were created from initially prepared |F=1,m F =-1 condensates using a one-photon microwave transition at 1.77GHz.The condensates were stored in a large-volume optical trap created by a single laser beam with an elliptical focus.We found condensates in the stretched state |F=2,m F =-2 to be stable for several seconds at densities in the range of 1014atoms/cm 3.In addition,we studied the clock transition |F=1,m F =0 →|F=2,m F =0 in a sodium Bose-Einstein condensate and determined a density-dependent frequency shift of (2.44±0.25)×10−12Hz cm 3.PACS numbers:03.75.Fi,32.70.JzSo far,Bose-Einstein condensation in dilute atomic gases [1,2,3,4,5]has been achieved in all stable bosonic alkali isotopes except 39K and 133Cs,as well as in atomic hydrogen [6]and metastable helium [7,8].The physics that can be explored with Bose-Einstein condensates (BEC)is to a large extent governed by the details of in-teratomic interactions.At ultra-low temperatures,these interactions not only vary signiﬁcantly from one atomic species to another but can also change signiﬁcantly for diﬀerent internal states of a single species.While in 87Rb,only minor diﬀerences of the collisional properties are observed within the ground state manifolds,in 7Li,the magnitude of the scattering length diﬀers by a factor of ﬁve between the upper and the lower hyperﬁne manifold and even the sign is inverted [9].The behavior of 23Na with a scattering length of 2.80nm in the |F=1,m F =±1 states and 3.31nm in the |F=2,m F =±2 states [10]is intermediate between these two extreme cases.Thus,sodium might provide a system in which the study of BEC mixtures of states with signiﬁcantly diﬀering scat-tering length is possible.Such a mixture would be a natural extension of earlier work on spinor condensates in 87Rb [11,12]and in the F=1manifold of 23Na [13,14].In this Letter,we report the realization of Bose-Einstein condensates of 23Na in the upper F=2hyperﬁne manifold in a large-volume optical trap [15].In 87Rb,condensates in both the F=1and F=2states had been achieved by loading atoms in either state into a magnetic trap and subsequent evaporative cooling.In contrast,sodium BECs have previously only been produced in the F=1state.Early attempts at MIT and NIST to evapo-ratively cool sodium in the F=2state were discontinued since the evaporative cooling scheme proved to be more robust for the F=1state.Instead of developing an opti-mized evaporation strategy for F=2atoms in a magnetic trap,we took advantage of an optical trap which traps atoms in arbitrary spin states [16].After producing F=1condensates and loading them into an optical trap,wetransferred the population into the F=2manifold using a single-photon microwave transition at 1.77GHz.We found that a BEC in the stretched |F=2,m F =-2 state is stable on timescales of seconds at densities of a few 1014atoms/cm 3.Simultaneous trapping of condensates in the |2,-2 and |1,-1 states for several seconds was also achieved.In contrast,at the same density,a condensate in the |2,0 state decays within milliseconds.Neverthe-less,we were able to observe the so-called clock transi-tion |1,0 →|2,0 in a BEC,which is to lowest order insensitive to stray magnetic ﬁelds.By taking spectra of this transition at various condensate densities,we were able to measure a density-dependent frequency shift of (2.44±0.25)×10−12Hz cm 3.The basic setup of our experiment is described in [17,18]and is brieﬂy summarized here.We have pre-pared condensates of more than 4×10723Na atoms in a so-called ‘clover-leaf’magnetic trap with trapping fre-quencies of νx =16Hz and νy =νz =160Hz by ra-diofrequency evaporation for 20s.After preparation of the condensate in the |1,−1 state,the radial trapping frequencies were adiabatically lowered by a factor of 5to decompress the condensate.Subsequently,an optical trapping potential was superimposed on the condensate by slowly ramping up the light intensity.After turning oﬀthe remaining magnetic ﬁelds,nearly all atoms were loaded into the large-volume optical dipole trap.The re-sulting peak density reached 5×1014atoms/cm 3,slightly higher than the density in the magnetic trap.The large-volume optical trap was realized by shaping the output of a Nd:YAG laser (typically 500mW at 1064nm)with cylindrical lenses leading to an elliptical focus with an aspect ratio of approximately 25.At the loca-tion of the condensate,the focal size was ≈20µm along the tight axis resulting in an optical trapping potential with typical trap frequencies of νx =13Hz axially and νy =36Hz and νz =850Hz transversely.The trap axis with the largest trapping frequency was oriented verti-2zx yg r a v i t yFIG.1:Sodium condensates in the |1,-1 and |2,-2 state,30ms after release from the trap.After preparation of the mixture the atoms were held in the optical trap for 1s.The horizontal separation of the spin states is due to application of a magnetic-ﬁeld gradient during expansion.cally to counteract gravity.The pancake shape of the trap,which we had recently used to create (quasi-)2D condensates [18],provided a much larger trapping volume than our previous cigar-shaped optical traps [16,19]and thus signiﬁcantly larger condensates could be stored.Optically trapped condensates were observed by absorption imaging on the closed |F=2,m F =-2 →|F ′=3,m ′F =-3 cycling transition at 589nm after sudden release from the trap,using light propagating parallel to the trap laser.The ballistic expansion time was typically 30ms,after which the vertical size of the condensate had increased by more than a factor of 100while the horizon-tal expansion was less than a factor of two.To make sure that atoms in both the F=1and the F=2manifold could be detected simultaneously,a short laser pulse resonant with the F=1→F ′=2transition was applied to pump all atoms into the F=2manifold.State-selective detection could be achieved by applying a magnetic ﬁeld gradient of several G/cm during the free expansion of the atomic cloud,leading to a spatial separation of spin states which diﬀer in the orientation of the magnetic moment (Fig.1).In order to test the intrinsic stability of the optical trap,we ﬁrst investigated the lifetime of condensates in the |1,-1 state as shown in Fig.2a).Even after 70s of dwell time,more than 106atoms remained in the conden-sate.Generally,the decay of the number of atoms N in the condensate can be modelled by the rate equationdN3ing the solutions of Eq.1we deduce rate coeﬃ-cients for the atom loss,assuming that only one process is responsible for the loss.Thus,we obtain as upper bounds k2=(2.93±0.28±0.29)×10−15cm3s−1and k3=(1.53±0.13±0.32)×10−29cm6s−1.Both values are in reasonable agreement with theoretical predictions [21,22].Though,at typical densities,the decay rate in the F=2state is roughly an order of magnitude larger than in the F=1state,it should still be compatible with direct condensation in the F=2manifold,provided that the loss coeﬃcients for the magnetically trapable|2,+2 state are similar to those for the|2,-2 state.By transferring only part of the atoms into the up-per hyperﬁne manifold we could also observe mixtures of condensates in the|1,-1 and|2,-2 states(see Fig.1). In the presence of small magneticﬁeld gradients,we ob-served a rapid spatial separation of the two components in a time shorter than100ms due to the fact that the|1,-1 state is low-ﬁeld seeking while the|2,-2 state is high-ﬁeld seeking.During the separation,strong density modulations in both components were observed,which could be attributed to tunnelling processes playing a role in the separation process[23].Afterwards,the two com-ponents lived almost independently side by side in the trap and the individual lifetimes were not signiﬁcantly aﬀected.When we tried to compensate all stray mag-neticﬁeld gradients,we still found that in steady state the two components tend to separate,i.e.we observed domains with only one component[14].This indicates that the two states are intrinsically not miscible.While we found23Na BECs in the|2,-2 state as well as mix-tures of|1,-1 and|2,-2 condensates to be stable for sev-eral seconds,non-stretched states in the F=2manifold as well as F=1,F=2mixtures with|m1+m2|=3de-cayed within several ms for typical condensate densities on the order of1014atoms/cm3.This fast decay is prob-ably due to(two-body)spin-relaxation which is strongly suppressed in87Rb but occurs with rate constants on the order of10−11cm3s−1in23Na[21].A particularly interesting transition within the elec-tronic ground state of alkali atoms is the magnetic-ﬁeld insensitive transition|F,0 →|F+1,0 ,often referred to as clock transition since its equivalent in cesium is used as the primary time standard.Shortly after laser cooling had been realized,the beneﬁts of using ultracold atoms for atomic clocks had become apparent[24]and today the most accurate atomic clocks are operated with laser-cooled atoms[25].Therefore,it seems natural to inves-tigate the use of a BEC with its signiﬁcantly reduced kinetic energy for the study of the clock transition.To observe the clock transition,weﬁrst completely transferred an optically trapped|1,-1 condensate into the|1,0 state with a radiofrequency Landau-Zener sweep.Selective driving of the|1,-1 →|1,0 transi-tion was achieved by applying a3G oﬀsetﬁeld which provided a large enough quadratic Zeeman-shift to liftLineshift(Hz)Average density (1014 cm-3)FIG.3:Magnetic-ﬁeld insensitive transition|1,0 →|2,0 in a BEC.(a)Spectrum in the trap at a mean density of1.6×1014atoms/cm3.(b)Spectrum after12.5ms time-of-ﬂight at a mean density of4.3×1011atoms/cm3.The discrepancy between the center of the line andν=0is probably due to an error in the exact determination of the residual magneticﬁeld. The solid lines are Gaussianﬁts.(c)Transition frequency as a function of density yielding a clock shift of(2.44±0.25)×10−12Hz cm3.the degeneracy with the|1,0 →|1,+1 transition.Sub-sequently,the magneticﬁeld was reduced to a value of typically100mG which keeps the spins aligned and gives rise to a quadratic Zeeman shift of the clock transition of≈20Hz.The|1,0 →|2,0 transition was then ex-cited by using a microwave pulse at1.77GHz with a duration between2and5ms.The fraction of atoms transferred into the|2,0 state was kept below20%in order to ensure a practically constant density in the |1,0 state during the pulse.Immediately afterwards, the optical trap was turned oﬀsuddenly and the num-ber of atoms which made the transition was detected by state-selective absorption imaging after15-30ms of bal-listic expansion.A typical spectrum showing the num-ber of transferred atoms as a function of microwave fre-quency(corrected for the calculated quadratic Zeeman shift)for a BEC with an average density of1.6×1014 atoms/cm3is shown in Fig.3a).The density was de-termined by measuring the release energy[18]of|1,-1 condensates without applying a microwave pulse.The release energy E rel is related to the chemical potential4µby E rel=(2/7)µ=(2/7)(h2a|1,−1 |1,−1 /πm)n o[26].Here,a|a |b is the scattering length between two23Naatoms in states|a and|b (a|1,−1 |1,−1 =2.80nm),mis the23Na mass,h is Planck’s constant and n0is thepeak density in the condensate related to the averagedensity by¯n=(4/7)n0.The spectrum in Fig.3a)issigniﬁcantly broadened compared to the one in Fig.3b),which is taken after ballistic expansion,and the transi-tion frequency is shifted with respect to the unperturbedfrequencyν0=1,771,626,129Hz[24].In the limit of weak excitation,the density-dependentshift of the clock-transition frequency is due to the diﬀer-ence in mean-ﬁeld potential that atoms in the|1,0 and|2,0 state experience within a|1,0 condensate.Takinginto account the inhomogeneous density distribution of atrapped BEC,this leads to a line shape given by[27]I(ν)=15h(ν−ν0)1−h(ν−ν0)πm(a|2,0 |1,0 −a|1,0 |1,0 ),(3)where the center of the line is atν0+2n0∆U/3h and the average frequency isν0+4n0∆U/7h.In our experiment, the line is additionally broadened and the asymmetry of Eq.2smeared out due to theﬁnite width of the mi-crowave pulse which was limited by rapid inelastic losses in the|2,0 state.Therefore,we have used a(symmetric) Gaussian toﬁt the resonances where we have identiﬁed theﬁtted center frequency as the average frequency of the line.By taking spectra of the clock-transition at diﬀerent densities we have determined a density shift of (2.44±0.25)×10−12Hz cm3(Fig.3c).Here,the error is the statistical error from a linearﬁt to the data.Addi-tional systematic errors due toﬁtting of the line with a Gaussian and due to an uncertainty in the determination of the density are estimated to be smaller than20%.Us-ing Eq.3and a|1,0 |1,0 =2.71nm[10],we determine the scattering length a|2,0 |1,0 =3.15±0.05nm for collisions between two atoms in states|1,0 and|2,0 .In conclusion,we have prepared condensates in the up-per F=2hyperﬁne manifold of the sodium ground state in a large-volume optical trap and observed a stable con-densate in the high-ﬁeld seeking stretched state|2,−2 . Since only the stretched state exhibits reasonable stabil-ity,experiments with more complex spinor condensates do not seem to be possible.Furthermore,we have for theﬁrst time observed the alkali clock-transition in a Bose-Einstein condensate and determined the value for the density-dependent mean-ﬁeld shift.In present BEC experiments,the magnitude of the shift precludes the use of trapped condensates for precise atomic clocks.How-ever,under circumstances where the condensate density can be drastically reduced as may be feasible in space-based experiments,the extremely low velocity spread of BECs might help improve the accuracy of atomic clocks. This work was supported by NSF,ONR,ARO,NASA, and the David and Lucile Packard Foundation. A.E.L. acknowledges additional support from the NSF.[*]Current address:5th Phys.Inst.,University of Stuttgart, 70550Stuttgart,Germany[†]Current address:Finisar Corp.,Sunnyvale,CA94089 [‡]Current address:JILA,Boulder,CO80309[1]M.H.Anderson et al.,Science269,198(1995).[2]K.B.Davis et al.,Phys.Rev.Lett.75,3969(1995).[3]C.C.Bradley,C.A.Sackett,and R.G.Hulet,Phys.Rev.Lett.78,985(1997).[4]S.L.Cornish et al.,Phys.Rev.Lett.85,1795(2001).[5]G.Modugno et al.,Science294,1320(2001).[6]D.G.Fried et al.,Phys.Rev.Lett.81,3811(1998).[7]A.Robert et al.,Science292,461(2001).[8]F.Pereira Dos Santos et al.,Phys.Rev.Lett.86,3459(2001).[9]F.Schreck et al.,Phys.Rev.Lett.87,080403(2001).[10]C.Samuelis et al.,Phys.Rev.A63,012710(2000).[11]C.Myatt et al.,Phys.Rev.Lett.78,586(1997).[12]D.Hall et al.,Phys.Rev.Lett.81,4531(1998).[13]J.Stenger et al.,Nature396,345(1998).[14]H.-J.Miesner et al.,Phys.Rev.Lett.82,2228(1999).[15]Meanwhile,we have also realized a23Na BEC in the F=2state in a magnetic trap starting from an optical trap.(A.E.Leanhardt et al.,cond-mat0206303(2002)).[16]D.M.Stamper-Kurn et al.,Phys.Rev.Lett.80,2027(1998).[17]W.Ketterle,D.Durfee,and D.M.Stamper-Kurn(IOSPress,Amsterdam,1999),Proceedings of the Interna-tional School of Physics Enrico Fermi,Course CXL,p.67.[18]A.G¨o rlitz et al.,Phys.Rev.Lett.87,130402(2001).[19]T.L.Gustavson et al.,Phys.Rev.Lett.88,020401(2002).[20]H.M.J.M.Boesten,A.J.Moerdijk,and B.J.Verhaar,Phys.Rev.A54,R29(1996).[21]A.J.Moerdijk and B.J.Verhaar,Phys.Rev.A53,R19(1996).[22]A.J.Moerdijk,H.M.J.M.Boesten,and B.J.Verhaar,Phys.Rev.A53,916(1996).[23]D.M.Stamper-Kurn et al.,Phys.Rev.Lett.83,661(1999).[24]M.Kasevich et al.,Phys.Rev.Lett.63,612(1989).[25]G.Santarelli et al.,Phys.Rev.Lett.82,4619(1999).[26]F.Dalfovo et al.,Rev.Mod.Phys.71,463(1999).[27]J.Stenger et al.,Phys.Rev.Lett.82,4569(1999).。

High-temperature Bose-Einstein condensation of polaritons realization under the intracavity

High-temperature Bose-Einstein condensation of polaritons: realization under the intracavity laserpumping of matter conditionV.A. Averchenko1), A.P. Alodjants2*), S.M. Arakelian2), S.N. Bagayev3), E.A. Vinogradov4),V.S. Egorov1), A.I. Stolyarov1), I.A. Chekhonin1)1) St. Petersburg State University, Ul'yanovskaya ul. 1, 198504 St. Petersburg, Staryi Peterhof, Russia2) Vladimir State University, ul. Gor'kogo 87, 600000 Vladimir, Russia3) Insitute of Laser Physics, Russian Academy of Sciences, prosp. akad. Lavrent'eva 13/3, 630090Novosibirsk, Russia4) Institute of Spectroscopy, Russian Academy of Sciences, 142190 Troitsk, Moscow region, RussiaAbstract.A quantum model of Bose-Einstein condensation based on processes involvingpolaritons excited in an intracavity absorbing cell with resonance atoms, which is manifested inthe spectral characteristics of the system, is considered. It is shown that the spectral'condensation' appears which is directly related to the degeneracy of a weakly interacting gas ofpolaritons resulting in quasi-condensation at room temperature. The possibility of obtainingpolariton condensation as a new phase state by using the confinement of polaritons in an atomicoptical harmonic trap is discussed.Keywords: polaritons, quasi-condensation, Bose-Einstein condensation, polariton laser.1. IntroductionExperiments on the Bose-Einstein condensation (BEC) of macroscopic numbers of atoms (N ≥ 106 ) is one of the most spectacular recent advances, which have made a great influence on the development of various directions in modern quantum and laser physics and newest technologies (see, for example, [1]). In the case of BEC, when under conditions of the temperature phase transition a macroscopic number of atoms are in the ground (lower) quantum level, a new coherent state of matter is formed. This is manifested in the fact that, for example, at the limiting temperature T=0 an ensemble of condensate atoms, as each individual atom, is described by the common wave function corresponding to a coherent state. In this aspect, the BEC phenomenon is similar to lasing, for example, when strict phase locking of laser modes occurs in laser cavities [2, 3]. In addition, in the case of BEC, we can say about the realisation of a Bose laser (boser) emitting coherent ensembles of atoms [3, 4]. A remarkable feature of such macroscopic quantum states of matter is the possibility to use them for the development of new physical principles of quantum information processing and communication [5, 6].However, despite spectacular achievements in this direction, there exist a number of practical difficulties imposing principal restrictions on the possibility of real applications of the atomic BEC for these purposes. Thus, one of the basic problems is the necessity of maintaining extremely low temperatures (tens of nK) to realise such devices. In this connection the problem of obtaining macroscopic coherent (quantum) states of matter at high (room) temperatures becomes very important.One of the most attractive approaches to the solution of this problem is the preparation of a quasi-condensate of the two-dimensional Bose gas of weakly interacting polaritons (in atomic physics [7]) and _______________________________________________________*) Email: alodjants@vpti.vladimir.ruexcitons (in solid state physics [8-10] *)). Such collective states of the medium (quasi-particles) represent a superposition of photons and spin waves in the atomic medium and can be obtained, for example, within the framework of the Dicke model used to describe superradiance [12]. Although these states cannot be treated as a condensate in a strict thermodynamic sense due to the nonequilibrium state of the system as a whole, under certain conditions imposed on the type of atomic optical interactions in the system, polaritons do form a condensate, their distribution being described by the Bose-Einstein distribution function for an ideal gas of bosons [13].In this paper, we considered the interaction of a system of two-level atoms with an electromagnetic field in the cavity in the case of the so-called strong coupling, when the inequality1/220212c coh d n πωωτ⎛⎞=⎜⎟⎝⎠, (1) is fulfilled, where c ω is the cooperative frequency determining the collective interaction of atoms with the field; 0ω is the atomic transition frequency; d is the transition dipole moment; coh τ is the characteristic coherence time of the atomic medium; n is the atomic gas density; and is Planck's constant. In this case, the field itself is weak (in the number of photons).The so-called condensation of the spectrum occurs when inequality (1) is fulfilled [14, 15]. This effect consists in the fact that under some threshold conditions imposed on the concentration of absorbing atoms and pump intensity, radiation of a broadband laser with a narrowband absorbing intracavity cell is concentrated ('condensed') near the strongest absorption lines of matter.This phenomenon was observed experimentally and interpreted within the framework of a clear classical model of parametric excitation of two coupled oscillators (electromagnetic field and atoms of matter) upon coherent energy transfer between them. However, this model is one-dimensional and is not quantum one, which obviously restricts the field of its applications.In this paper, we propose a detailed quantum model of spectral condensation realised for polaritons excited in an intracavity absorbing cell [16]. We show that spectral condensation can be directly related to the condensation (quasi-condensation) of polaritons in the cavity if a strong coupling between the electromagnetic field and medium is provided. The latter statement is in itself of interest, and in this paper we substantiate for the first time the possibility of obtaining the true BEC in the polariton system at high (room) temperatures in the case of spectral condensation. In this respect, of interest are the experimental data [8, 9] obtained in semiconductor microcavities, which confirm the above assumption.2. Basic relationsConsider the interaction of two-level atoms (with levels a and b) with a quantum electromagnetic field, which is described by the photon annihilation (creation) operators ()k k f f + for the k-th mode. Within the framework of dipole approximation, such a system can be described by the Hamiltonian [13]12()()()()ph k k atk k k k k k k k k kH k E k f f E b b a a g f a b b a f ++++++=+−+−∑∑∑, (2) level a, and the inequality_______________________________________________________*) Here, we are dealing with the so-called Kosterlitz-Thouless phase transition to the superfluid state of two-dimensional Bose systems in which the true Bose-Einstein condensation (in the absence of confinement of gas particles in a trap) is impossible [11].k k k k b b a a ++ (3)is fulfilled. In this approximation, Hamiltonian (2) can be diagonalised by using the unitary transformation1,,k k k k k k Фf a b µν+=− 2,,k k k k k k Фf a b νµ+=+ (4)where the introduced annihilation operators ,j k Ф (j=1,2) characterise quasi-particles (polaritons) in the atomicmedium, corresponding to two types of elementary perturbations, which in approximation (3) satisfy the boson commutation relations,,;i k j k ij ФФδ+⎡⎤=⎣⎦, ,1,2i j =. (5)The transformation parameters k µand k v in expression (4) are real Hopfield coefficients satisfying the condition 221k k v µ+=, which determine the contributions of the photon and atomic (excited) components to a polariton, respectively:22221/2221/242(4)(4)k k k k g g g µδδδ=⎡⎤+++⎣⎦, 221/22221/2(4)2(4)k k k k g g δδνδ++=+ , (6a,b)where ()k at ph E E k δ=−is the phase mismatch determining the contributions of the photon and atomic components to expression (4) for polaritons. In particular, in the limiting case, when 2k g δ− , we have 21k µ→ (20k v →), which corresponds to the negligible contribution of the photon part to the polariton 2,k Ф. Inopposite limit, when 2k g δ , we have 20k µ→ (21kv →), which means that the photon contribution to the coherence of polaritons of this type increases. Expression (6) shows that the polariton is a half-matter and half-photon (221/2µν==) quasi-particle under the resonance condition 0k δ=.Taking expressions (4) and (6) into account, Hamiltonian (2) takes the form11,1,22,2,()()()k k k k k kH k E k ФФE k ФФ++=+∑∑, (7)where 1,2()E k determine the dispersion dependence of polaritons: {}1/2221,21()()()42at ph at ph E k E E k E E k g ⎧⎫⎡⎤=+±−+⎨⎬⎣⎦⎩⎭. (8) Figure la presents dispersion dependences 1,2()E k (8) of polaritons for the interaction of atoms with the quantum field in free space. One can see that the two allowed energy states, polaritons of the upper 1[()]E k and lower 2[()]E k branches, correspond to each value of the wave vector k .When the medium is placed into the resonator, the wave-vector component k ⊥ orthogonal to the mirrorsurface is quantised. At the same time, a continuum of modes exists in the direction parallel to the mirror surface due to the absence of boundary conditions. This means that in the single-mode (single-frequency for each value of k ⊥) regime, the dispersion of polaritons is determined only by the wave-vector component k parallel to the mirror surface. Then, under the condition k k ⊥ which corresponds physically to the paraxialapproximation in optics (see, for example, [17]), the dispersion relation for photons in the resonator has the form23221/22()()2ph k k E k c k k c k O k k ⊥⊥⊥⊥⎡⎤⎛⎞=+=++⎢⎥⎜⎟⎜⎟⎢⎥⎝⎠⎣⎦. (9)a bFigure 1. Dispersion dependences 1()E k (upper branch) and 2()E k (lower branch) of polaritons on the wave vector k in free space (a) and resonator (b). The wave vector is plotted on the abscissa in the units of the resonance wave vector k ⊥ on the ordinate the energy is plotted inthe units of the coupling coefficient g .Here /cav k m L π⊥=is the quantised component of the wave vector parallel to the resonator axis, which corre-sponds to the periodic boundary conditions in the standard field quantisation procedure; L cav is the effective resonator length; and the number m corresponds to the selected mode (frequency). In the case of strong coupling(1), the dispersion curves of a polariton are pushed apart, resulting in the appearance of the upper and lower polariton branches in the resonator (Fig. lb). The principal feature of these curves is the presence of the 'potential' well (for 0k =). The width of the lower polariton well can be found from the condition 2220E k ∂∂= . This condition determines the angular parameters of a polariton beam in the resonator. It is important to note that these effects, which are related to the transverse component of the wave vector of a polariton (k in our case), will not be suppressed due to light diffraction if the angular dimensions of the polariton beam exceed the diffraction-limited divergence ϕ of the light beam, which can be estimated from the expression cav d L ϕ≈ [d and cav L are the beam diameter and resonator (or absorbing cell) length, respectively].3. Spectral 'condensation' and condensation of polaritonsWithin the framework of our approach, the narrowing ('condensation') of the polariton spectrum, which was observed in experiments [14, 15], can be simply explained by BEC. In this connection, taking into account paraxial approximation (9), we represent Hamiltonian (7) in the formlong tr H H H =+, (10a)where''11,1,22,2,()()long k k k k k k H E k ФФE k ФФ⊥⊥⊥⊥⊥⊥++⊥⊥=+∑∑, (10b) ''1,1,1,2,2,2,()()tr tr k k tr k k k k H E k ФФE k ФФ++=+∑∑ (10c)The expression for long H describes polaritons formed along the resonator axis, ''1,21,20()()k E k E k ⊥=≡ determines their dispersion dependence [see (8)] for 0k = . The expression for tr H characterises polaritons produced in the two-dimensional plane perpendicular to the resonator axis. The dispersion of these polaritons is described by the expression '22(1,2)11,2,2/tr pol E k m = . Here,()(1,2)1/222214phpol m m g =∆∆+∓(11)is the mass of polaritons of the upper and lower branches; 20ph m k c E c ⊥=≈ is the effective photon mass inthe medium and 0E ck ⊥∆=− is the detuning of the resonator mode (frequency) from the atomic transitionfrequency.Thus, the BEC of polaritons in the resonator is related to the second term in the expression for the Hamiltonian H in (10a). This term leads in fact to the renormalisation of the photon mass in the medium [see(11)]. Quasi-particles (polaritons) appearing in this case can be treated as an ideal two-dimensional gas [see also (10b)]. Indeed, the possibility of BEC assumes the presence of a stable state with the minimal energy - a 'potential' well (at the point 0k = ), which, as shown in section 2, takes place for polaritons in the resonator (the2well expressed in energy units is of the order of the coupling coefficient222effk g m ∆≈ . (12)In this case, it is possible to introduce formally the effective temperature eff T of the two-dimensional Bose gas of polaritons, which is also of the order of the coupling coefficient within the polariton well [13], i.e., B eff K T g ≈, where B K is the Boltzmann constant.The approach discussed above determines the condensation (more exactly, quasi-condensation) of the two-dimensional gas by assuming that polaritons with large k efficiently relax to the bottom of the dispersion-curve well. In our case, unlike the case of semiconductor microcavities considered in [8, 9], the two-dimensional property of the polariton gas can be provided by the fact that an optically dense medium is excited, as a rule, by the wave packet of synchronised electromagnetic modes, which corresponds to the quasi-monochromatic interaction of the field with medium.The efficient relaxation of polaritons to the bottom of the 'dispersion' well can be related to the intense polariton-polariton interaction discussed in a number of papers (mainly concerning the problems with semiconductor micro-cavities [10, 18]).Consider now in more detail the quasi-condensation of a two-dimensional Bose gas of polaritons described by the last term in (10c). The chemical potential of such a gas is described by the expression [19]22ln 1exp()ln 1exp B T B d T K T n K T T µλ⎡⎤⎛⎞⎡⎤=−−≡−−⎢⎥⎜⎟⎣⎦⎢⎥⎝⎠⎣⎦, (13)where 222/()d eff B T n m K π= is the gas degeneracy temperature; 2n is the two-dimensional density of polaritonsin the plane perpendicular to the resonator axis; and 1/2/(2)T eff B m K T λ= is the thermal wavelength (de Broglie wavelength). The temperature T d in (13) is determined by the condition when the thermal wavelength T λ is of the order of the average distance 1/3V between particles (V is the system volume). Due to the interaction between polaritons (nonideal gas), the additional parameter a scat appears, which is the scattering length depending on the interaction potential. This parameter affects the energy spectrum, which becomes a phonon spectrum [9].It follows from (13) that, strictly speaking, the condensation of the two-dimensional polariton gas (0µ=) occurs at 0T →. At the same time, it is known (see, for example, [9]) that already at the temperature224sd KT eff B n T T m K π= (14)the Kosterlitz-Thouless phase transition to the superfluid state occurs in the two-dimensional weakly interacting Bose gas, when isolated condensate droplets with uncorrelated phases are formed on the two-dimensional surface [n s in (14) is the superfluid liquid density on the two-dimensional surface].For polaritons with the effective mass 33510eff m g −=×and density in a three-dimensional resonator 1133 3.510n cm −=×, the gas degeneracy can appear already at room temperature (T d = 300 K). Indeed, in this case the minimal two-dimensional density of the polariton gas estimated from (13) for d T T = gives the value 82230.310T n n cm λ−≈× [19] for the de Broglie wavelength 41.8410T cm λ−≈×. It is for this value of the atomic concentration 3n that spectral condensation was observed near the yellow doublet of sodium in experiments [14] (Fig. 2). For the upper spectrum (Fig. 2a), 103310n cm −<, and for the lower spectrum (Fig. 2c), 1133 3.510n cm −=×. The similar results obtained in [14] for the neon spectrum also demonstrated the spectralcondensation for polaritons.a bcFigure 2. Spectral condensation near the yellow doublet of sodium (see text) at the atomic concentration 10310n ≤(a), 1110 (b), and 1133.510cm −× (c).Let us find now the conditions under which the true (in thermodynamic sense) condensation of polaritons excited in the resonator can be obtained. It is known [19, 20] that, to obtain such condensation in a two-dimensional weakly interacting (ideal) gas, gas particles should be confined in a trap. For example, for a trap with the trapping potential described by the expression (harmonic potential)222020()2eff eff m r U r U r r Ω== (15)(eff Ω is the trapping (oscillation) frequency of particles, 0r is the transverse size of the trapping region, and r isthe transverse coordinate), the critical BEC temperature for a two-dimensional gas is (cf. [20])2221.645c eff Bn T m K π== , (16)where N is the total number of particles. We also took into account in the right-hand side of (16) that the number N 2 of particles trapped by the potential U(r) on the surface is described by the expression [20]2222B eff eff N n K T m π=Ω. In the absence of a trap (0eff Ω=), as should be, BEC does not occur: 0c T = in (16).(measurements of the first- and second-order coherence degree) is one of the main tools for diagnostics of polariton condensation (see [8, 9]). Because a polariton is a linear superposition of a photon and atomic excitation [see (4)], its coherent properties are caused by the coherence of the light field itself and of an ensemble of atoms with which the field interacts, as well as by their possible quantum interference caused by the condensation process. Within the framework of these experiments, when the condition of the exact resonance0∆= is fulfilled, we have 0k δ≈ and obtain 221/2k k v µ== from expressions (6) and (11), which means thatoptical and atomic parts make identical contributions to a polariton. In this case, the coherent properties of the polariton state can be simply caused by a high coherence of the optical field at the input to the atomic medium irrespective of BEC. However, the problem of measuring the coherence of atomic exitations caused by the interaction and of the intrinsic coherence of the polariton condensate (if it is produced in the system) remains open. In our opinion, this problem can be solved, in particular, by producing polariton BEC based on three-level atoms under conditions of electromagnetic induced transparency (EIT) (see below).Here we consider another possibility based on a small variation of the detuning ∆ [and, therefore, k δ, see (6)] in experiments as the parameter governing the contributions of photon and atomic parts to the resulting coherence of resonator polaritons. In this case, the effective mass of polaritons [see (11)] and, hence, the critical temperatures of degeneracy, condensation, and quasi-condensation in (13), (14), and (16) change. This specific property of a polariton gas means in fact that the formation of a Bose-Einstein condensate can be controlled in experiments.Note, however, that we do not consider in this paper the questions concerning the BEC of a photonic gas in the resonator or, more exactly, the condensation of polaritons of the upper branch of the dispersion curve (see Fig. lb) characterised by the first term in expression (10c). This problem is undoubtedly very important for the scope of questions considered in our paper although it was discussed only in connection with the quantum properties of light in media with cubic nonlinearity (see [3]).In addition, the formation of a photon condensate (or a condensate of polaritons of the upper branch), which is directly connected with lasing in the resonator (cf. [2]), upon varying the parameter ∆ also has an interesting feature. Indeed, for 0∆=, it follows from (11) that polaritons of both dispersion branches have equalmasses, i.e., (1)(2)pol pol eff m m m ==, which corresponds to the equal temperatures of their quasi-condensation [see(14)1. However, in the case of 0∆≠, we have from (11) that (1)(2)pol pol m m ≠ which means physically that thephase-transition temperatures (14) for polaritons of the upper (1)()KT T and lower (2)()KT T dispersion branches aredifferent. Thus, by introducing asymmetry with the help of a small change in the detuning ∆, it is possible to produce a very narrow temperature (energy) gap within which the coherent properties of polaritons of both branches should substantially change. These properties can be observed, for example, by measuring the function of their cross correlation or by using probe radiation under resonance conditions.Therefore, the study of this effect will give the answer to the principal question about the properties ofthe coherence of light, atomic system, and polaritons themselves in the case of BEC.4. ConclusionsWe have developed in the paper the quantum approach for solving the problems of formation of quasi-condensation and realisation of the true (in the thermodynamic sense) Bose - Einstein condensation of a two-dimensional gas of polaritons at room temperature. This approach has allowed us to explain some features of spectral condensation of broadband lasing near strong absorption lines in the laser resonator, which were observed in experiments (in particular, the so-called spectral condensation upon non-resonance pumping). In this aspect, BEC reduces the threshold pump power of parametric excitation of cooperative effects. Consider briefly some phenomena that are directly related to the problem studied in the paper.First, this is the condensation of polaritons, which is of interest in the presence of the EIT effect when alight pulse propagates in a resonance atomic medium without changing its shape in the absence of absorption (see, for example, [5, 21, 22]. A remarkable feature of this effect is the appearance of atomic coherence both for hot [21] and ultracold atoms [5, 22]. The EIT effect can be also explained in terms of bright and dark polaritons, which in the adiabatic approximation corresponding to condition (3) in our case, represent the coherent superposition of atoms in the two states of the hyperfine Zeeman structure and the external probe field maintained with the help of the external probe field at the optical frequency through the third (auxiliary) level (the so-called Λ- scheme [5, 6, 21]).Therefore, upon placing an atomic medium into the resonator to produce the BEC of polaritons, the EITeffect would become a tool for obtaining such a quantum state. In this case, the condensation process could be controlled more precisely by coupling directly two atomic levels with an external weak field, which would provide the ejection of 'hot' polaritons from a trap, as, for example, occurs for condensation of alkali atoms in a magneto-optical trap [1]. On the other hand, upon spectral condensation in the case of BEC, a 'bleaching' of the atomic medium in the resonator caused by a change in its refractive properties can be expected. In this case, the group velocity of a light pulse directed into an atomic medium after switching on probe radiation with the delay time del coh ττ< can decrease, in particular, due to polariton condensation. Indeed, it follows from expressions (10c) and (11) that the group velocities of such quasi-particles in the plane perpendicular to the resonator axis are determined by the expression1,2(1,2)(1,2)()tr gr polE k k m υ∂==∂ In the case of the exact atomic optical resonance (for 0∆=), we have from this that(1)(2)/2gr gr k c k υυ⊥= Therefore, in the paraxial approximation, when k k ⊥ the group velocity of condensedpolaritons is estimated as (1,2)gr c υ , which means in fact that the 'slow' light regime is observed for polaritons inthe resonator.Second, the high-temperature BEC of polaritons is of interest for quantum information, for example, forthe development of new physical principles of quantum memory and data storage. Indeed, as we have shown in[6], such macroscopic polariton states can be used in problems of cloning and quantum information storage.Acknowledgements. This work was partially supported by the Russian Foundation for Basic Research (Grants Nos 04-02-17359 and 05-02-16576) and the Ministry of Education and Science of the Russian Federation. A.P. Alodjants thanks the non-profit Dynasty Foundation for support.AppendixLet us discuss the problem of confinement of the BEC of intracavity polaritons in a trap. Consider a special atomic optical trap whose operation is based on the fact that polaritons represent a coherent superposition of a photon and atomic perturbation. Photons can be confined in the region of atomic-optical interaction in such a trap, where polaritons are produced, by focusing a light beam with a special gradient (cylindrical) lens (or inhomogeneous waveguide) with the refractive index varying along the transverse coordinate as2220()(1)n r n n r ′=−, (A.1)where 'n is the required gradient addition to the refractive index of the lens. The potential for trapping (focusing) photons of the light beam produced by such an optical system can be written in the form [17]222020()'()22opt n r n n r U r n −==,which exactly corresponds to the harmonic-trap potential (15) with the inhomogeneity parameter 2'eff eff n m =Ω.In addition, to trap atoms in the plane perpendicular to the resonator axis, we can use a two-dimensional magnetic trap with the oscillation frequency at Ω, which is widely applied in experiments with 'usual' atomic condensates [1].Thus, to confine polaritons in a trap, it is necessary to confine atoms by a standard method and focus simulta-neously the light beam into the region of atomic-optical interaction by selecting the appropriate parameters at Ωand 'n . This determines the value of eff Ω required in the experiment.References1. Ketterle V. Usp. Fiz. Nauk, 173, 1339 (2003).2. Oraevsky A.N. Kvantovaya Elektron., 24, 1127 (1997) [Quantum Electron., 27, 1094 (1997)].3. Chiao R., Boyce J. Phys. Rev. A, 60, 4114 (1999).4. Imamoglu A., Ram R.J., Pau S., Yamamoto Y. Phys. Rev. A, 53, 4250 (1996).5. Liu C, Dutton Z., Behroozi C.H., Hau L.N. Nature, 409, 490 (2001).6. Alodjants A.P, Arakelian S.M. Int. J. Mod. Phys. B, 20, 1593 (2006).7. Averchenko V.A., Bagayev S.N., et al. Abstract in Technical Digest o/ICONO'05 Conf. (Sankt-Petersburg, Russia, 2005).8. Deng H., Weihs G., Santori C, Bloch J., Yamamoto Y. Science, 298, 199 (2002).9. Kavokin A., Malpuech G., Laussy F.P. Phys. Lett. A, 306, 187 (2003); Richard M., Kasprzak J., Andre R., et al. Phys. Rev. B, 72, 201301(R) (2005).10. Gippius N.A., Tikhodeev S.G., Keldysh L.V., Kulakovskii V.D., Usp. Fiz. Nauk, 175, 327 (2005); Kulakovskii V.D., KrzhizhanovskiiD.N., et al. Usp. Fiz. Nauk, 175, 334 (2005).11. Kosterlitz J.M., Thouless D.J. J. Phys. B: Sol. State Phys., 6, 1181 (1973).12. Dicke R.H. Phys.Rev., 93, 99 (1954).13. Eastham P.R., Littlewood P.B. Phys. Rev. B, 64, 235101 (2001).14. Vasil'ev V.V., Egorov V.S., Fedorov A.N., Chekhonon LA. Opt. Spektr., 76, 146 (1994).15. Bagayev S.N., Egorov V.S., Moroshkin P.V., Fedorov A.N., Chekhonon LA. Opt. Spektr., 86, 912 (1999).16. Kocharovskii V.V., Kocharovskii Vl.V. Kvantovaya Elektron., 14, 2246 (1987) [Sov. J. Quantum Electron., 17, 1430 (1987)].17. Marte M.A., Stenholm S. Phys. Rev. A, 56, 2940 (1997).18. Savvidis P.G., Baumberg J.J., Stevenson P.M., et al. Phys. Rev. Lett., 84, 1547 (2000).19. Petrov D.S., Gangardt G.M., Shlyapnikov G.V. J. Phys. IV France, 116, 3 (2004).20. Bagnato V., Kleppner D.K. Phys. Rev. A, 44, 7439 (1991).21. Lukin M.D. Rev. Mod. Phys., 75, 457 (2003).22. Prokhorov A.V., Alodjants A.P., Arakelyan S.M. Pis'ma Zh. Eksp. Tear. Fiz., 80, 870 (2004).。

+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。

科学家称已发现暗物质

科学家称已发现暗物质哈勃太空望远镜产生的复合图片显示了星系群Cl 0024+17附近如幽灵般的暗物质“环”。

凤凰科技讯北京时间12月12日消息，美国生活科学网站报道，近日研究人员表示，利用超导回路他们已经检测到组成宇宙大部分物质的神秘暗物质。

暗物质是宇宙最大的谜团之一——它是一种不可见的物质，据称组成了宇宙全部物质的5/6。

目前进行的科学调查表明暗物质是由一种新型粒子组成，后者会与宇宙中所有已知的力，除了重力，发生微弱的相互作用。

由于暗物质不可见且几乎完全不可触摸，所以只能通过它产生的引力拖拽作用监测它们的存在。

利用深埋在地下的大型传感器阵列进行的实验试图鉴别暗物质与其它粒子发生罕见碰撞时发出的微弱信号。

截至目前，没有任何研究监测到任何暗物质的足迹。

现在理论物理学家表明，较小的台式探测器或可能监测轴粒子，后者是暗物质粒子的首要理论候选者。

近期的理论研究表明轴粒子可能会聚集在一起，形成物理家们所谓的波色-爱因斯坦凝聚（Bose-Einstein condensates）的超级粒子。

“我开始思考的并非单个轴粒子的行为，而是很多轴粒子堆积在一起的集体行为。

”他说道。

科学家注意到描述这些轴粒子运动的方程式非常类似于控制名为S/N/S约瑟夫森结(S/N/S Josephson junction)的特殊回路的方程式，这种设备是由两个超导体组成，两者通过一层薄薄的金属层分开。

科学家计算出轴粒子在流经这些设备时可能会留下可监测的电子信号。

“这开启了寻找轴粒子的、前人未设想过的新途径。

”贝克说道。

如果这一想法是正确的，科学家表示证据可能已经出现了——在2004年探索S/N/S约瑟夫森结噪声级的一项实验里发现了一种未知起源的信号。

如果这一信号来自于轴粒子，那么这意味着，经贝克的计算，这些粒子的质量大约是电子的40亿分之一。

为了证实或者否决2004年实验里产生的信号来自轴粒子，科学家们需要进行进一步的实验，他们必须尤为关注屏蔽任何外来辐射。

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.

physical review letters模板 -回复

physical review letters模板-回复"[physical review letters模板]，以中括号内的内容为主题，写一篇1500-2000字文章，一步一步回答"Title: Understanding the Quantum Tunneling Phenomenon: A Physical Review LettersAbstract:In this paper, we delve into the intriguing concept of Quantum Tunneling, a fundamental phenomenon in quantum mechanics. We explore the theoretical background, experimental evidence, and potential applications of quantum tunneling. By utilizing the format of Physical Review Letters, we present a comprehensive analysis of this captivating topic.1. IntroductionQuantum tunneling refers to the remarkable ability of particles to pass through energy barriers that classical physics would deem impassable. This phenomenon is a direct consequence of the wave-particle duality intrinsic to quantum mechanics. The aim of this article is to unravel the underlying principles behind quantum tunneling and shed light on its profound implications for variousfields of science.2. Theoretical BackgroundTo understand quantum tunneling, we need to first comprehend the Schrödinger equation, which describes the behavior of quantum systems. This equation reveals that particles possess wave-like properties, allowing them to exist in a superposition of states. Furthermore, the Heisenberg uncertainty principle states that the more precisely we know a particle's position, the less certain we are about its momentum.3. Barrier PenetrationThe concept of tunneling emerges when particles encounter an energy barrier. According to classical physics, particles with insufficient energy to overcome the barrier should be completely reflected. However, in quantum mechanics, particles possess wave functions that extend beyond the classical boundaries. Consequently, there is a finite probability for them to penetrate the barrier and appear on the other side.4. Experimental EvidenceExperimental verification of quantum tunneling has been achievedin a variety of areas. For instance, the scanning tunneling microscope has allowed scientists to observe the tunneling of electrons between a conducting tip and a surface, enabling atom manipulation with atomic precision. Additionally, experiments involving tunneling of cold atoms through Bose-Einstein condensates have provided direct evidence of quantum tunneling phenomena.5. ApplicationsQuantum tunneling has numerous applications across diverse scientific disciplines. In the field of electronics, the phenomenon is utilized in the creation of tunneling diodes and transistors, enabling faster and more efficient electronic devices. Tunneling is also pivotal in nuclear fusion, where particles need to overcome the Coulomb barrier to initiate fusion reactions. Moreover, quantum tunneling plays a crucial role in the functioning of enzymes in biological systems.6. Quantum Tunneling in AstrophysicsQuantum tunneling also influences astrophysical phenomena. For instance, nuclear reactions within stars rely on tunneling to overcome the barriers inherent in fusion processes. Additionally,tunneling is vital in explaining the phenomenon of stellar nucleosynthesis, where the synthesis of heavier elements occurs through fusion reactions.7. ConclusionQuantum tunneling is a captivating aspect of quantum mechanics, challenging classical notions of energy barriers. Through a thorough examination of its theoretical foundations, experimental observations, and diverse applications, we have explored the fundamental concepts of quantum tunneling. This phenomenon has revolutionized various scientific realms, from electronics to astrophysics, and continues to be an area of active research and exploration.。

玻色-爱因斯坦凝聚态和相变

玻色-爱因斯坦凝聚态和相变
玻色-爱因斯坦凝聚（Bose-Einstein Condensates，简称BEC）是量子物理中的一种现象，其中遵从玻色–爱因斯坦统计且总粒子数守恒的理想气体，在温度低于一个极低但非零的转变温度 T_c 时，占全部粒子数有限百分比的(宏观数量的)部分将聚集到单一的粒子最低能态上。

这种现象是1924年由印度物理学家玻色和德国物理学家爱因斯坦独立提出的。

BEC的形成可以通过增加粒子浓度（压缩体积）或降低温度来实现。

如果是通过压缩体积达到的BEC，那么这是一种1阶相变，具有相变潜热；而如果是通过降低温度达到的BEC，那么这是一种三阶相变，此时在相变点位置比热容连续但比热容对温度导数不连续。

尽管BEC的概念已经存在了近百年，但相关的研究和探索至今仍在进行中，无论是在数学还是物理领域。

(完整)生活大爆炸第一季台词中英文对照1.09

—Sheldon：Look at me。 Look at me. I’ve got goose bumps。
看看我，看看我，我起鸡皮疙瘩了.
—Howard：Are we ready on the stereo
音响接好了么？
—Raj：Go for stereo.
试试音响。
-Penny：Hey, guys.
through our local ISP racing down fiber-optic cable
从我们的本地网络出发经由光纤电缆，
at the speed of light to San Francisco bouncing off a satellite
以光速抵达旧金山，经由地球同步卫星
in geosynchronous orbit to Lisbon, Portugal，
闭嘴Howard。
Sheldon， we have to do this.
Sheldon我们必须这么做。
但用奥坎氏简化论解释的话有人把它给扔了。（Occam’s Razor 如无必要勿增实体)
—Leonard：It’s from the Institute for Experimental Physics。
这是物理实验协会的来信。
They want us to present our paper on the properties of super solids
玻色—爱因斯坦凝聚的主题会议的派对可是充满传奇色彩啊.
—Leonard：Forget the parties.
别管那些派对了。
-Howard：Forget the parties What a nerd.
别管派对真是个书呆子.

物质中的基本粒子包含了玻色因和费米子

物质中的基本粒子包含了玻色因和费米子In the realm of particle physics, the fundamental building blocks of matter are composed of elementary particles. These particles can be divided into two distinct categories: bosons and fermions. Let's delve into each category and explore their unique properties and contributions to our understanding of the universe.在粒子物理学领域，物质的基本构建单位由基本粒子组成。

这些粒子可以分为两个不同的类别：玻色子和费米子。

让我们深入研究每个类别，并探索它们各自的特性和对我们对宇宙理解的贡献。

Bosons are particles that follow Bose-Einstein statistics, named after Indian physicist Satyendra Nath Bose and Albert Einstein. They have integral spins such as 0, 1, or 2, meaning they possess integer multiples of Planck's constant (h) divided by 2π. One well-known example of a boson isthe photon, which carries electromagnetic force and serves as the mediator for interactions between charged particles.玻色子是遵循玻色-爱因斯坦统计的粒子，以印度物理学家Satyendra Nath Bose和阿尔伯特·爱因斯坦命名。

介绍超流体和玻色–爱因斯坦凝聚的物理性质

介绍超流体和玻色–爱因斯坦凝聚的物理性质超流体和玻色–爱因斯坦凝聚是量子力学的非常重要的现象之一。

它们的物理性质不仅有助于我们探索物质的本质，还有着广泛的应用，如制造更高效的元器件和更精确的传感器等。

本文将介绍这两种物质的主要物理性质和应用。

超流体超流体是指在极低温度下阻力变为零的物质，它与普通物质最大的区别在于其流动时不受到任何阻碍。

这种现象发生在液氦和氢中，也发现在一些固体中。

超流体的特殊性质源于其量子性质。

量子力学中，波函数起着重要的作用。

实际上，如果物质的波函数对称，则其内部的“运动”相当于是同步的，这也即波函数的相位存在一种相干性。

当超冷物质中的粒子的相干性从粒子间的散射中倾向于聚合的时候，就会形成一种简单的波函数，这些粒子将几乎不与其它粒子相互作用，就会变成超流体。

各种物理现象都可以以粒子的波函数的物理意义来解释，但超流体的行为特别适合这种描述，从而被广泛地研究。

超流体的最初研究始于液体氦的实验，参与这项工作的有Peter Kapitza和Landeau等人。

当液氦被冷却到极低温度时，它会从液态变成固态，接着它又会突然变成超流体。

这种转变说明了液氦中存在一些非常特殊的量子效应。

这个发现对物理学的发展产生了深远的影响，极低温度下的物理现象和行为成为了新的物理分支。

液氦的超流体性质也被用在医学和天体物理学的研究中。

例如，在核磁共振成像技术中就广泛使用液氦来制冷。

玻色-爱因斯坦凝聚玻色-爱因斯坦凝聚是另一种量子物质。

它是一种由玻色子组成的物质。

玻色子是一种不同于常见物质中的费米子的粒子，物理学界通常将它们归为是代表光子和体外释放的冷原子等类似事物的基本粒子。

玻色-爱因斯坦凝聚现象是在BEC(Bose-Einstein Condensates)得到观测。

当玻色子被冷却到接近绝对零度时，其量子特性会显著增强，粒子会集聚到一个微小的区域中，而不是像传统的气体一样扩散到周围的区域。

这种相干的聚集现象引起了物理学家的广泛关注。

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a r X i v :c o n d -m a t /0211345v 1 [c o n d -m a t .s o f t ] 16 N o v 2002Bose-Einstein condensates near a microfabricated surfaceA.E.Leanhardt,Y.Shin,A.P.Chikkatur,D.Kielpinski,W.Ketterle,and D.E.Pritchard ∗Department of Physics,MIT-Harvard Center for Ultracold Atoms,and Research Laboratory of Electronics,Massachusetts Institute of Technology,Cambridge,Massachusetts,02139(Dated:February 1,2008)Magnetically and optically conﬁned Bose-Einstein condensates were studied near a microfabri-cated surface.Condensate fragmentation observed in microfabricated magnetic traps was not ob-served in optical dipole traps at the same location.The measured condensate lifetime was ≥20s and independent of the atom-surface separation under both magnetic and optical conﬁnement.Radio-frequency spin-ﬂip transitions driven by technical noise were directly observed for optically conﬁned condensates and could limit the condensate lifetime in microfabricated magnetic traps.PACS numbers:03.75.Fi,03.75.Be,39.20.+q,34.50.DyThe manipulation of gaseous Bose-Einstein conden-sates with magnetic ﬁelds produced by wires microfabri-cated on material surfaces has opened a new frontier in the ﬁeld of atom optics [1,2,3,4].Magnetic conﬁnement using microfabricated wires is tighter and has higher spa-tial resolution than is achievable in macroscopic mag-netic traps [5,6,7,8].Generally,an important feature of magnetic traps is the excellent thermal isolation be-tween the nanokelvin temperature clouds they conﬁne and the 300K laboratory environment.However,de-creasing the atom-surface separation into the micrometer range has raised concerns that the surface may perturb the atoms through ﬂuctuating currents.Theoretical pre-dictions suggest that thermally induced atom-surface in-teractions will not pose limitations for distances >∼1µm from the surface [9,10].While early demonstrations of trapping and guiding of laser-cooled thermal atoms with microfabricated devices reported no evidence of deleteri-ous surface eﬀects [11,12,13,14],recent studies using samples cooled by forced radio-frequency (rf)evapora-tion to <∼2µK have found corrugated potentials [3,15],large heating rates [2,15],and short trap lifetimes [2,15]for atom-surface separations in the 100µm regime.The ultimate applicability of microfabricated devices to atom optics depends on the characterization and elimination of such eﬀects.In this work,we experimentally investigate the behav-ior of Bose-Einstein condensates near a microfabricated surface.The condensates were conﬁned at the same po-sition relative to the surface by either a microfabricated magnetic trap or an optical dipole trap.Since the two traps operate on diﬀerent principles and the electromag-netic ﬁelds for each have diﬀerent sources,this study pro-vides a unique examination of the interaction between Bose-Einstein condensates and a microfabricated surface.For example,while condensates conﬁned near the surface in a microfabricated magnetic trap were found to frag-ment longitudinally [3,15],the clouds remained intact under optical conﬁnement.Signiﬁcantly,the measured condensate lifetime in both the microfabricated magnetic trap and the optical dipole trap was ≥20s,an order of magnitude longer than previ-ous results [2,15],and independent of the atom-surface separation.We have directly observed spin-ﬂip transi-tions driven by rf technical noise for condensates held in the optical dipole trap.The transition rate increased rapidly with decreasing atom-surface separation imply-ing that distance-dependent losses can occur in magnetic traps where the products of such transitions cannot be directly identiﬁed.Bose-Einstein condensates containing over 10723Na atoms were created in the |F =1,m F =−1 state in a macroscopic Ioﬀe-Pritchard magnetic trap,loaded into the focus of an optical tweezers beam,and transported ≈32cm in 2s into an auxiliary “science”chamber as described in Ref.[16].The optical tweezers consisted of ≈50mW of 1064nm laser light focused to a 1/e 2ra-dius of 26µm.This resulted in axial and radial trap frequencies ω||=2π×4Hz and ω⊥=2π×425Hz,re-spectively,and a trap depth of 2.5µK.Condensates con-taining 2−3×106atoms arrived 70−500µm below the microfabricated structures mounted in the science cham-ber.The atom-surface separation was varied by angling the optical tweezers axis before translation and was lim-ited to distances ≥70µm due to the laser beam clipping on the microchip support structures.In the science chamber,the condensate either remained conﬁned by the optical tweezers or was loaded into a mi-crofabricated Ioﬀe-Pritchard magnetic trap formed by a Z-shaped wire carrying current I and an external mag-netic bias ﬁeld,B ⊥,as described in Ref.[3].An addi-tional longitudinal bias ﬁeld,B ||,was applied with exter-nal coils to adjust the magnetic trap bottom and radial trap frequency.The microfabricated wires were litho-graphically patterned on a 600µm thick silicon substrate mounted on an aluminum block.They were 50µm wide and electroplated with copper to a thickness of 10µm.As in previous experiments [3,15],condensates con-ﬁned near the surface in the microfabricated magnetic trap were observed to fragment longitudinally [Fig.1(a)].The condensate density depletions appeared in the same longitudinal position relative to the surface on each re-2(a)(b)FIG.1:Fragmentation of Bose-Einstein condensates.Radial absorption images after10ms ballistic expansion of conden-sates containing≈106atoms after holding at a distance of 85µm from the microfabricated surface for15s in the(a) microfabricated magnetic trap and(b)optical dipole trap. Longitudinal fragmentation occurred for condensates held in the microfabricated magnetic trap,but not for those conﬁned optically at the same location with the microfabricated mag-netic trap oﬀ.The microfabricated magnetic trap was oper-ated with I=130mA,B⊥=3.2G,and B||=1.4G yielding a radial trap frequencyω⊥=2π×450Hz.The optical dipole trap had a radial trap frequencyω⊥=2π×425Hz and was operated with B||=1.8G.For both condensates,the chemi-cal potential wasµ=k B×120nK.The absorption imaging light was resonant with the F=1→F′=2transition.The ﬁeld of view is0.5mm×1.0mm.alization of the experiment,and more fragments formed as the atoms came closer to the microchip[3].In con-trast,condensates conﬁned optically at the same location were not observed to fragment[Fig.1(b)].The same lon-gitudinal biasﬁeld was nominally applied to both mag-netically and optically conﬁned condensates so that any surface magnetization eﬀects would perturb the clouds identically.The lack of condensate fragmentation in the optical dipole trap implies that the longitudinal poten-tial corrugations arise due to the presence of currentﬂow in the microfabricated wires,in agreement with conclu-sions reached elsewhere[17].Deviations of the current ﬂow from a straight line would lead to such corrugations and could arise due to imperfect microfabrication[3]or current instabilities at high current densities[17].The magnetic nature of the potential corrugations and other possible origins are discussed in Ref.[17].It is interesting to note that in our earlier work no fragmentation was observed when condensates conﬁned in a macroscopic Z-shaped wiretrap were brought within ≈10µm of the surface of the wire[16].The wire was made of copper and had a circular cross-section with 1.27mm diameter.The condensates were loaded into the wiretrap740µm from the surface of the wire and brought closer by lowering the wire current.The experimental parameters upon closest approach were I=920mA and B⊥=2.9G,yielding estimated axial and radial trap frequenciesω||=2π×7Hz andω⊥=2π×78Hz,respec-tively[18].The macroscopic wiretrap contained5×105 atoms extended longitudinally over200µm at a chemi-cal potentialµ=k B×30nK.Diﬀerences between the macroscopic and microfabricated wiretraps include vastly diﬀerent fabrication techniques as well as lower current densities in the macroscopic wire.Conﬁned atoms are sensitive to noise at their trap fre-quency and Zeeman splitting frequency[9,10,19,20, 21,22].In this work,typical radial trap frequencies were≈500Hz while Zeeman splitting frequencies were ≈1MHz.Noise at the trap frequency leads to heating and subsequent trap loss after the atoms acquire an en-ergy greater than the trap depth.For atoms conﬁned in a Ioﬀe-Pritchard magnetic trap,radial magnetic biasﬁeld ﬂuctuations cause radial trap-centerﬂuctuations.The amplitude of such trap-centerﬂuctuations is independent of the longitudinal biasﬁeld.However,for optically con-ﬁned atoms,onlyﬂuctuating radial magneticﬁeld gradi-ents cause radial trap-centerﬂuctuations.The eﬀects of such gradients can be minimized by applying a longitudi-nal biasﬁeld that adds in quadrature with theﬂuctuating radial gradients since it is the gradient of the magnitude of the biasﬁeld vector that determines the force on an atom.Spin-ﬂip transitions driven by rf noise at the atomic Zeeman splitting frequency distribute the atomic pop-ulation across magnetically conﬁnable and unconﬁnable states.This causes atom loss for clouds held in magnetic traps.However,all spin states are conﬁned in an opti-cal dipole trap so spin-ﬂip transitions do not lead to loss and the products can be directly observed.Since mag-netically and optically conﬁned condensates react diﬀer-ently to noise,whether it is at their trap frequency or the atomic Zeeman splitting frequency,a systematic study of condensate lifetimes in both magnetic and optical traps provides better noise characterization than studies per-formed in either a magnetic or optical trap exclusively. Any atom-surface coupling,regardless of frequency, should manifest itself as a dependence of the conden-sate lifetime on the atom-surface separation.Figure2 shows a measurement of the magnetically and optically conﬁned condensate lifetime as a function of the distance from the microfabricated surface.No distance depen-dence was observed and the measured condensate life-time was≥20s,ten times longer than previous re-sults[2,15].A distance independent condensate lifetime indicates that atom-surface interactions are unimportant over the70−500µm separation range[23].This is in contrast to results presented in Ref.[15],where a distance dependent lifetime was observed for thermal atoms mag-netically conﬁned near a microfabricated surface.These data are included in Fig.2for comparison.Several experimental details altered the measured con-densate lifetime.Excitations created during the micro-fabricated magnetic trap loading were found to shorten the measured lifetime,and care had to be taken to over-lap the optical and magnetic traps during transfer to min-imize such excitations.Translating the condensate either towards or away from the microfabricated surface by adi-abatically varying I and B⊥to shift the trap center while maintaining a constant radial trap frequency was found to decrease the condensate lifetime.This presumably3L i f e t i m e (s )Atom-Surface Separation (µm)FIG.2:Lifetime of Bose-Einstein condensates near a micro-fabricated surface.The 1/e lifetime of condensates conﬁned in the microfabricated magnetic trap (solid squares)and op-tical dipole trap (solid circles)is shown to be independent of distance from the microfabricated surface.I and B ⊥were varied with distance to maintain the radial magnetic trap fre-quency at ω⊥=2π×450Hz with B ||=1.4G.The vertical line indicates the onset of longitudinal condensate fragmen-tation in the magnetic trap.In the optical dipole trap,the condensate was held directly below the microfabricated wire used to form the magnetic trap with B ||=1.8G.No external connections were made to the microchip.The optical dipole trap had axial and radial trap frequencies ωz =2π×4Hz and ω⊥=2π×425Hz,respectively.Only atoms remaining in the |1,−1 state were resonant with the absorption imaging light.For comparison,thedistance dependence of thermal cloud lifetimes measured in Ref.[15]is shown for atoms conﬁned magnetically by a microstructure (open squares)and copper wire (open diamonds).Error bars smaller than the symbol size are not included.Also,magnetically conﬁned condensate lifetimes reported in Ref.[2](open triangle)and Ref.[15](open circle)are shown for comparison.resulted from excitations induced by irregular current changes due to technical limitations in controlling the power supplies connected to the microchip.As a result,microfabricated magnetic trap lifetime data is only pre-sented for atom-surface separations ≥70µm,where the atoms were loaded into their ﬁnal position directly from the optical tweezers.Occasionally,heating was observed for atoms in both the microfabricated magnetic trap and optical dipole trap due to technical noise at the trap fre-quency,even with care taken to eliminate ground loops and minimize cable lengths [24].Connecting a 10mF capacitor in parallel with the 2Ωmicrofabricated wire (1/RC =2π×8Hz)eliminated such eﬀects.Thereafter,applying rf power the microchip at a frequency chosen to limit the trap depth for magnetically conﬁned atoms did not consistently alter the condensate lifetime.The distance independent condensate lifetime pre-T r a n s i t i o n R a t e (s -1)Atom-Surface Separation (µm)(a)(b)(c)m F = -1m F = +1m F = 0m F = -1m F = +1m F = 0FIG.3:Spin-ﬂip transitions driven by radio-frequency tech-nical noise.Axial absorption images after 22ms ballistic ex-pansion of condensates held in the optical dipole trap at a distance of 185µm from the microfabricated surface for (a)0s and (b)40s.A magnetic ﬁeld gradient was applied during ballistic expansion to separate the diﬀerent spin states.(c)Spin-ﬂip transition rate,Γ,vs distance,d ,from the microfab-ricated surface on a log-log scale.A line Γ∝1/d 2is provided as a guide.The inset shows the fraction of the total atoms in the m F =0state with the transition rate being deﬁned as the initial slope of the data (before any atoms in the m F =+1state were detected).The absorption imaging light was res-onant with the F =2→F ′=3transition.The atoms were optically pumped into the F =2hyperﬁne level with a pulse resonant with the F =1→F ′=2transition.This provided equal imaging sensitivity to each magnetic sublevel.The ﬁeld of view in (a)and (b)is 1.0mm ×2.0mm.sented in Fig.2indicates that our experiment is not cur-rently limited by the proximity of the microfabricated surface.However,we have observed spin-ﬂip transitions driven by rf noise in the microfabricated wires.Figure 3shows the behavior of condensates conﬁned optically di-rectly beneath the microfabricated wire used for mag-netic trapping.Condensate atoms initially in the |1,−1 state [Fig.3(a)]were found to make transitions to other magnetic sublevels [Fig.3(b)].Such transitions would act as a loss mechanism for magnetically conﬁned clouds.The transition rate was found to decrease as the square of the atom-surface separation distance,d .Since the mag-netic ﬁeld of a straight wire decays as 1/d ,and the power scales as the square of the ﬁeld,the 1/d 2dependence of the spin-ﬂip transition rate is expected for atoms in the near ﬁeld (d ≪λ)of the wire,where λ≈300m is the wavelength of ≈1MHz radiation.The transition rate vs distance data presented in Fig.3(c)was taken with all connections necessary to4run the microfabricated magnetic trap made to the mi-crochip,but with no currentﬂowing in the microfabri-cated wires.The atoms were exposed to a longitudinal biasﬁeld B||=1.8G to nearly duplicate theﬁeld conﬁg-uration in the microfabricated magnetic trap.This also maximized their sensitivity toﬂuctuatingﬁelds generated by wire currents since rf transitions are more favorable for magneticﬁelds oscillating orthogonal to a static bias ﬁeld.Spin-ﬂip transitions were suppressed by exposing the optically conﬁned atoms to an orthogonal biasﬁeld, B⊥.The rf driven spin-ﬂip transition rate depended strongly on experimental details,suggesting that antenna eﬀects coupled rf noise into the system.The rate was measured to be of order100times higher if care was not taken to carefully eliminate ground loops and use mini-mal cable lengths[24].Also,with no connections to the microchip,spin-ﬂip transitions were not detectable for condensates held up to60s.The spin-ﬂip rate presented in Fig.3(c)became comparable to the measured con-densate decay rate displayed in Fig.2for atom-surface separations<100µm.Thus,extending long condensate lifetimes much closer to the surface will require further rf shielding and/orﬁltering.In conclusion,we have studied the behavior of Bose-Einstein condensates near a microfabricated surface. Condensates found to fragment while held in a micro-fabricated magnetic trap were observed to remain intact while held at the same position relative to the microchip in an optical dipole trap.A possible explanation is that deviations of the current path from a straight line give rise to corrugations in the longitudinal potential.The origins of such current path deviations are under inves-tigation.Furthermore,our work demonstrates magneti-cally and optically conﬁned condensate lifetimes≥20s at distances≥70µm from a microfabricated surface. The lifetime was measured to be independent of the atom-surface separation and ten times longer than re-sults obtained elsewhere at comparable distances.Spin-ﬂip transitions driven by rf technical noise were directly observed for condensates held in an optical dipole trap, however we found no evidence for fundamental,thermally induced noise driven processes above the level of those attributed to technical noise.Our results demonstrate the extreme sensitivity of Bose-Einstein condensates to small static and dynamic electromagneticﬁelds.This sensitivity provides a challenge for realizing microfabri-cated atom-optical devices,but it also emphasizes the potential for developing new detector and instrumenta-tion technology.We thank T.Pasquini for experimental assistance and M.Crescimanno for a critical reading of the manuscript. This work was funded by ONR,NSF,ARO,NASA,and the David and Lucile Packard Foundation. 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