Vortex lattices in Bose-Einstein condensates from the Thomas-Fermi to the lowest Landau lev

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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。

Bose-Einstein condensation in a one-dimensional interacting system due to power-law trappin

a r X i v :c o n d -m a t /9810197v 1 [c o n d -m a t .s t a t -m e c h ] 16 O c t 1998Accepted to PHYSICAL REVIEW A for publicationBose-Einstein condensation in a one-dimensional interacting system due to power-lawtrapping potentialsM.Bayindir,B.Tanatar,and Z.GedikDepartment of Physics,Bilkent University,Bilkent,06533Ankara,TurkeyWe examine the possibility of Bose-Einstein condensation in one-dimensional interacting Bose gas subjected to conﬁning potentials of the form V ext (x )=V 0(|x |/a )γ,in which γ<2,by solving the Gross-Pitaevskii equation within the semi-classical two-ﬂuid model.The condensate fraction,chemical potential,ground state energy,and speciﬁc heat of the system are calculated for various values of interaction strengths.Our results show that a signiﬁcant fraction of the particles is in the lowest energy state for ﬁnite number of particles at low temperature indicating a phase transition for weakly interacting systems.PACS numbers:03.75.Fi,05.30.Jp,67.40.Kh,64.60.-i,32.80.PjI.INTRODUCTIONThe recent observations of Bose-Einstein condensation (BEC)in trapped atomic gases [1–5]have renewed inter-est in bosonic systems [6,7].BEC is characterized by a macroscopic occupation of the ground state for T <T 0,where T 0depends on the system parameters.The success of experimental manipulation of externally applied trap potentials bring about the possibility of examining two or even one-dimensional Bose-Einstein condensates.Since the transition temperature T 0increases with decreasing system dimension,it was suggested that BEC may be achieved more favorably in low-dimensional systems [8].The possibility of BEC in one -(1D)and two-dimensional (2D)homogeneous Bose gases is ruled out by the Hohen-berg theorem [9].However,due to spatially varying po-tentials which break the translational invariance,BEC can occur in low-dimensional inhomogeneous systems.The existence of BEC is shown in a 1D noninteracting Bose gas in the presence of a gravitational ﬁeld [10],an attractive-δimpurity [11],and power-law trapping po-tentials [12].Recently,many authors have discussed the possibility of BEC in 1D trapped Bose gases relevant to the magnetically trapped ultracold alkali-metal atoms [13–18].Pearson and his co-workers [19]studied the in-teracting Bose gas in 1D power-law potentials employing the path-integral Monte Carlo (PIMC)method.They have found that a macroscopically large number of atoms occupy the lowest single-particle state in a ﬁnite system of hard-core bosons at some critical temperature.It is important to note that the recent BEC experiments are carried out with ﬁnite number of atoms (ranging from several thousands to several millions),therefore the ther-modynamic limit argument in some theoretical studies [15]does not apply here [8].The aim of this paper is to study the two-body interac-tion eﬀects on the BEC in 1D systems under power-law trap potentials.For ideal bosons in harmonic oscillator traps transition to a condensed state is prohibited.It is anticipated that the external potentials more conﬁn-ing than the harmonic oscillator type would be possible experimentally.It was also argued [15]that in the ther-modynamic limit there can be no BEC phase transition for nonideal bosons in 1D.Since the realistic systems are weakly interacting and contain ﬁnite number of particles,we employ the mean-ﬁeld theory [20,21]as applied to a two-ﬂuid model.Such an approach has been shown to capture the essential physics in 3D systems [21].The 2D version [22]is also in qualitative agreement with the results of PIMC simulations on hard-core bosons [23].In the remaining sections we outline the two-ﬂuid model and present our results for an interacting 1D Bose gas in power-law potentials.II.THEORYIn this paper we shall investigate the Bose-Einstein condensation phenomenon for 1D interacting Bose gas conﬁned in a power-law potential:V ext (x )=V 0|x |κF (γ)G (γ)2γ/(2+γ),(2)andN 0/N =1−TF (γ)=1x 1/γ−1dx1−x,(4)and G (γ)=∞x 1/γ−1/2dxNk B T 0=Γ(1/γ+3/2)ζ(1/γ+3/2)T 01/γ+3/2.(6)Figure 1shows the variation of the critical temperature T 0as a function of the exponent γin the trapping po-tential.It should be noted that T 0vanishes for harmonic potential due to the divergence of the function G (γ=2).It appears that the maximum T 0is attained for γ≈0.5,and for a constant trap potential (i.e.V ext (x )=V 0)the BEC disappears consistent with the Hohenberg theorem.0.00.5 1.0 1.5 2.0γ0.00.20.40.6k B T 0 (A r . U n .)FIG.1.The variation of the critical temperature T 0withthe external potential exponent γ.We are interested in how the short-range interactioneﬀects modify the picture presented above.To this end,we employ the mean-ﬁeld formalism and describe the col-lective dynamics of a Bose condensate by its macroscopictime-dependent wave function Υ(x,t )=Ψ(x )exp (−iµt ),where µis the chemical potential.The condensate wavefunction Ψ(x )satisﬁes the Gross-Pitaevskii (GP)equa-tion [24,25]−¯h 2dx 2+V ext (x )+2gn 1(x )+g Ψ2(x )Ψ(x )=µΨ(x ),(7)where g is the repulsive,short-range interaction strength,and n 1(x )is the average noncondensed particle distribu-tion function.We treat the interaction strength g as a phenomenological parameter without going into the de-tails of actually relating it to any microscopic descrip-tion [26].In the semi-classical two-ﬂuid model [27,28]the noncondensed particles can be treated as bosons in an eﬀective potential [21,29]V eﬀ(x )=V ext (x )+2gn 1(x )+2g Ψ2(x ).(8)The density distribution function is given byn 1(x )=dpexp {[p 2/2m +V eﬀ(x )−µ]/k B T }−1,(9)and the total number of particles N ﬁxes the chemical potential through the relationN =N 0+ρ(E )dE2mgθ[µ−V ext (x )−2gn 1(x )],(12)where θ[x ]is the unit step function.More precisely,the Thomas-Fermi approximation [7,20,30]would be valid when the interaction energy ∼gN 0/Λ,far exceeds the kinetic energy ¯h 2/2m Λ2,where Λis the spatial extent of the condensate cloud.For a linear trap potential (i.e.γ=1),a variational estimate for Λis given by Λ= ¯h 2/2m (π/2)1/22a/V 0 1/3.We note that the Thomas-Fermi approximation would breakdown for tem-peratures close to T 0where N 0is expected to become very small.The above set of equations [Eqs.(9)-(12)]need to be solved self-consistently to obtain the various physical quantities such as the chemical potential µ(N,T ),the condensate fraction N 0/N ,and the eﬀective potential V eﬀ.In a 3D system,Minguzzi et al .[21]solved a simi-lar system of equations numerically and also introduced an approximate semi-analytical solution by treating the interaction eﬀects perturbatively.Motivated by the suc-cess [21,22]of the perturbative approach we consider aweakly interacting system in1D.To zero-order in gn1(r), the eﬀective potential becomesV eﬀ(x)= V ext(x)ifµ<V ext(x)2µ−V ext(x)ifµ>V ext(x).(13) Figure2displays the typical form of the eﬀective po-tential within our semi-analytic approximation scheme. The most noteworthy aspect is that the eﬀective poten-tial as seen by the bosons acquire a double-well shape because of the interactions.We can explain this result by a simple argument.Let the number of particles in the left and right wells be N L and N R,respectively,so that N=N L+N R.The nonlinear or interaction term in the GP equation may be approximately regarded as V=N2L+N2R.Therefore,the problem reduces to the minimization of the interaction potential V,which is achieved for N L=N R.FIG.2.Eﬀective potential V eﬀ(x)in the presence of in-teraction(x0=(µ/V0)1/γa).Thick dotted line represents external potential V ext(x).The number of condensed atoms is calculated to beN0=2γa√ze x−1+ 2µ/k B Tµ/k B TH(γ,µ,xk B T)(2µ/k B T−x)1/γ−1/2dxexp[(E−µ)/k B T]−1=κ(k B T)1/γ+1/2J(γ,µ,T),(18) whereJ(γ,µ,T)= ∞2µ/k B T x1/γ+1/2dxze x−1.and Ecis the energy of the particles in the condensateE c=g(1+γ)(2γ+1)gV1/γ.(19)The kinetic energy of the condensed particles is neglected within our Thomas-Fermi approximation to the GP equa-tion.III.RESULTS AND DISCUSSIONUp to now we have based our formulation for arbitrary γ,but in the rest of this work we shall present our re-sults forγ=1.Our calculations show that the results for other values ofγare qualitatively similar.In Figs. 3and4we calculate the condensate fraction as a func-tion of temperature for various values of the interaction strengthη=g/V0a(at constant N=105)and diﬀerent number of particles(at constantη=0.001),respectively. We observe that as the interaction strengthηis increased, the depletion of the condensate becomes more apprecia-ble(Fig.3).As shown in the correspondingﬁgures,a signiﬁcant fraction of the particles occupies the ground state of the system for T<T0.The temperature depen-dence of the chemical potential is plotted in Figs.5and 6for various interaction strengths(constant N=105) and diﬀerent number of particles(constantη=0.001) respectively.0.00.20.40.60.8 1.0T/T 00.00.20.40.60.81.0N 0/NN 0/N=1−(T/T 0)3/2η=10−5η=10−3η=10−1η=10FIG.3.The condensate fraction N 0/N versus temperature T /T 0for N =105and for various interaction strengths η.Eﬀects of interactions on µ(N,T )are seen as large de-viations from the noninteracting behavior for T <T 0.In Fig.7we show the ground state energy of an interacting 1D system of bosons as a function of temperature for dif-ferent interaction strengths.For small η,and T <T 0, E is similar to that in a noninteracting system.As ηincreases,some diﬀerences start to become noticeable,and for η≈1we observe a small bump developing in E .This may indicate the breakdown of our approxi-mate scheme for large enough interaction strengths,as we can ﬁnd no fundamental reason for such behavior.It is also possible that the Thomas-Fermi approximation em-ployed is violated as the transition to a condensed state is approached.0.00.20.40.60.8 1.0T/T 00.00.20.40.60.81.0N 0/NN 0/N=1−(T/T 0)3/2N=108N=105N=103N=101FIG.4.The condensed fraction N 0/N versus temperature T /T 0for η=0.001and for diﬀerent number of particles N .0.00.20.40.60.8 1.0 1.2T/T 0−100100200300400µ/V 0η=1η=0.1η=0.001η=0.00001FIG.5.The temperature dependence of the chemical potential µ(N,T )for various interaction strength and for N =105particles.Although it is conceivable to imagine the full solution of the mean-ﬁeld equations [Eq.(9)-(12)]may remedy the situation for larger values of η,the PIMC simulations [19]also seem to indicate that the condensation is inhibited for strongly interacting systems.The results for the spe-ciﬁc heat calculated from the total energy curves,i.e.C V =d E /dT ,are depicted in Fig.8.The sharp peak at T =T 0tends to be smoothed out with increasing in-teraction strength.It is known that the eﬀects of ﬁnite number of particles are also responsible for such a be-havior [20].In our treatment these two eﬀects are not disentangled.It was pointed out by Ingold and Lam-brecht [14]that the identiﬁcation of the BEC should also be based on the behavior of C V around T ≈T 0.0.00.20.40.60.8 1.0 1.2T/T 0−5050100µ/V 0N=107N=105N=103N=101FIG.6.The temperature dependence of the chemical po-tential µ(N,T )for diﬀerent number of particles N and for η=0.001.0.00.20.40.60.8 1.0 1.2T/T 00.00.20.40.60.8<E >/N k B T 0η=0η=0.001η=0.1η=1Maxwell−BoltzmannFIG.7.The temperature dependence of the total energy of 1D Bose gas for various interaction strengths ηand N =105particles.Our calculations indicate that the peak structure of C V remains even in the presence of weak interactions,thus we are led to conclude that a true transition to a Bose-Einstein condensed state is predicted within the present approach.0.00.20.40.60.81.01.2T/T 00.00.20.40.60.81.0C V /N k Bη=0η=0.001η=0.1Maxwell−BoltzmannFIG.8.The temperature dependence of the speciﬁc heat C V for various interaction strengths ηand N =105particles.IV.CONCLUDING REMARKSIn this work we have applied the mean-ﬁeld,semi-classical two-ﬂuid model to interacting bosons in 1D power-law trap potentials.We have found that for a range of interaction strengths the behavior of the thermo-dynamic quantities resembles to that of non-interactingbosons.Thus,BEC in the sense of macroscopic occu-pation of the ground state,occurs when the short-range interparticle interactions are not too strong.Our results are in qualitative agreement with the recent PIMC sim-ulations [19]of similar systems.Both 2D and 1D sim-ulation results [19,23]indicate a phase transition for a ﬁnite number system,in contrast to the situation in the thermodynamic limit.Since systems of much larger size can be studied within the present approach,our work complements the PIMC calculations.The possibility of studying the tunneling phenomenon of condensed bosons in spatially diﬀerent regions sepa-rated by a barrier has recently attracted some attention [31–34].In particular,Dalfovo et al .[32]have shown that a Josephson-type tunneling current may exist for bosons under the inﬂuence of a double-well trap potential.Za-pata et al .[34]have estimated the Josephson coupling energy in terms of the condensate density.It is inter-esting to speculate on such a possibility in the present case,since the eﬀective potential in our description is of the form of a double-well potential (cf.Fig.2).In our treatment,the interaction eﬀects modify the single-well trap potential into one which exhibits two minima.Thus if we think of this eﬀective potential as the one seen by the condensed bosons and according to the general ar-guments [31–34]based on two weakly connected systems we should have an oscillating ﬂux of particles when the chemical potential in the two wells is diﬀerent.Any con-ﬁguration with N L =N R which is always the case for odd number of bosons will result in an oscillatory mo-tion.It would be interesting to explore these ideas in future work.ACKNOWLEDGMENTSThis work was supported by the Scientiﬁc and Techni-cal Research Council of Turkey (TUBITAK)under Grant No.TBAG-1736and TBAG-1662.We gratefully ac-knowledge useful discussions with Prof.C.Yalabık and E.Demirel.[5]D.J.Han,R.H.Wynar,Ph.Courteille,and D.J.Heinzen,Phys.Rev.A57,R4114(1998).[6]I.F.Silvera,in Bose-Einstein Condensation,Ed.by A.Griﬃn,D.W.Snoke,and S.Stringari(Cambridge Uni-versity Press,Cambridge,1995).[7]F.Dalfovo,S.Giorgini,L.P.Pitaevskii,and S.Stringari,preprint,cond-mat/9806038(to be published in Reviews of Modern Physics);A.S.Parkins and D.F.Walls,Phys.Rep.303,1(1998).[8]W.Ketterle and N.J.van Druten,Phys.Rev.A54,656(1996).[9]P.C.Hohenberg,Phys.Rev.158,383(1967).[10]A.Widom,Phys.Rev.176,254(1968).[11]L.C.Ioriatti,Jr.,S.G.Rosa,Jr.,and O.Hipolito,Am.J.Phys.44,744(1976).[12]V.Bagnato and D.Kleppner,Phys.Rev.A44,7439(1991).[13]T.Haugset and H.Haugerud,Phys.Rev.A57,3809(1998).[14]G.-L.Ingold and mbrecht,Eur.Phys.J.D1,29(1998).[15]W.J.Mullin,J.Low Temp.Phys.110,167(1998)[16]W.Deng and P.M.Hui,Solid State Commun.104,729(1997).[17]H.Monien,M.Linn,and N.Elstner,preprint,cond-mat/9711178.[18]F.Brosens,J.T.Devreese,and L.F.Lemmens,SolidState Commun.100,123(1996).[19]S.Pearson,T.Pang,and C.Chen,Phys.Rev.A58,1485(1998).[20]S.Giorgini,L.P.Pitaevskii,and S.Stringari,Phys.Rev.A,54,4633(1996);J.Low Temp.Phys.109,309(1997).[21]A.Minguzzi,S.Conti,and M.P.Tosi,J.Phys.Cond.Matter9,L33(1997).[22]M.Bayindir and B.Tanatar,Phys.Rev.A58,3134(1998).[23]S.Heinrichs and W.J.Mullin,preprint,cond-mat/9807331.[24]E.P.Gross,Nuovo Cimento20,454(1961).[25]L.P.Pitaevskii,Zh.Eksp.Teor.Fiz.40,646(1961)[Sov.Phys.JETP13,451(1961)].[26]M.Olshanii,Phys.Rev.Lett.81,938(1998).[27]R.J.Dodd,K.Burnett,M.Edwards,and C.W.Clark,Acta Phys.Pol.A93,45(1998).[28]M.Naraschewski and D.M.Stamper-Kurn,Phys.Rev.A58,2423(1998).[29]V.Bagnato,D.E.Pritchard,and D.Kleppner,Phys.Rev.A35,4354(1987).[30]G.Baym and C.J.Pethick,Phys.Rev.Lett.76,6(1996).[31]S.Grossmann and M.Holthaus,Z.Naturforsch.50a,323(1995).[32]F.Dalfovo,L.P.Pitaevskii,and S.Stringari,Phys.Rev.A54,4213(1996).[33]A.Smerzi,S.Fantoni,S.Giovanazzi,and S.R.Shenoy,Phys.Rev.Lett.79,4950(1997).[34]I.Zapata,F.Sols,and A.J.Leggett,Phys.Rev.A57,28(1998).。

Faraday waves in Bose-Einstein condensates

∗ URL:Leabharlann ing length [10], which is directly related to the prefactor of the eﬀective nonlinearity (due to the mean-ﬁeld interparticle interactions) of the system. This can be achieved by means of Feshbach resonances [11]. These theoretical propositions motivated the very recent experimental implementation of the Faraday waves in Bose-Einstein condensates in the work of Ref. [12]. The actual realization of the spatially modulated patterns arose in a somewhat diﬀerent way than was proposed in the above studies, a way which is very close, however, to the spirit of the theoretical suggestion of Ref. [13] (see also the more recent consideration of a similar problem from a quantum point of view in Ref. [14]). In particular, in Ref. [12], an elongated cigar-shaped condensate was used where the transverse, strong conﬁnement directions were periodically modulated in time, while the weaker longitudinal direction conﬁnement was time-independent. The parametric excitation at the driving frequency was recognized as being responsible for exciting oscillations at half the driving frequency, which is the main resonance also observed in Faraday’s experiments. Subsequently, based on this insight and the dispersion relation of longitudinal collective modes presented in Refs. [15, 16], a relation was derived (and convincingly compared to the experimental results in a quantitative manner) for the resulting pattern periodicity versus the transverse driving frequency. The aim of the present paper is to provide a complete analysis of the instability from ﬁrst principles and to obtain a fully analytical prediction that can be used for a detailed comparison with the experimental results and numerical results obtained from the 1D model reduction as well as full 3D simulations at the mean-ﬁeld level. The principal feature which allows us to provide a detailed quantitative analysis of the system is the fact that for cigar-shaped condensates (such as the ones used in the experiment of Ref. [12]) there is a quantitatively accurate description in the form of the non-polynomial Schr¨ odinger equation (NPSE) derived in Ref. [17] (provided that the transverse direction stays close to its

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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。

Bose-Einstein condensates with vortices in rotating traps

a r X i v :c o n d -m a t /9906144v211J un1999Bose-Einstein condensates with vortices in rotating trapsY.Castin 1and R.Dum 1,21Laboratoire Kastler Brossel ∗,´Ecole normale sup´e rieure,24rue Lhomond,F-75231Paris Cedex 05,France2Institut d’optique,BP 147,F-91403Orsay Cedex,France (16March 1999)We investigate minimal energy solutions with vortices for an interacting Bose-Einstein condensate in a rotating trap.The atoms are strongly conﬁned along the axis of rotation z ,leading to an eﬀective 2D situation in the x −y plane.We ﬁrst use a simple numerical algorithm converging to local minima of energy.Inspired by the numerical results we present a variational Ansatz in the regime where the interaction energy per particle is stronger than the quantum of vibration in the harmonic trap in the x −y plane,the so-called Thomas-Fermi regime.This Ansatz allows an easy calculation of the energy of the vortices as function of the rotation frequency of the trap;it gives a physical understanding of the stabilisation of vortices by rotation of the trap and of the spatial arrangement of vortex cores.We also present analytical results concerning the possibility of detecting vortices by a time-of-ﬂight measurement or by interference eﬀects.In the ﬁnal section we give numerical results for a 3D conﬁguration.I.INTRODUCTION After the achievement of Bose-Einstein condensates in trapped atomic gases [1]many properties of these systems have been studied experimentally and theoretically [2].However a striking feature of superﬂuid helium,quantized vortices [3],[4],has not yet been observed in trapped atomic gases.There is an abundant literature on vortices in helium II,an overview is given in [4].The atomic gases have interesting properties which justify eﬀorts to generate vortices in these systems:the core size of the vortices is adjustable,as in contrast to helium the strength of the interaction can be adjusted through the density;the number of vortices in atomic gases can be in principle well controlled;for a small number of particles in the gas metastability of the vortices can be studied,that is one can watch spontaneous transitions between conﬁgurations with diﬀerent number of vortices.Several ways to create vortices in atomic gases have been suggested.A method inspired from liquid helium consists in rotating the trap conﬁning the atoms [5];at a large enough rotation frequency it becomes energetically favorable at low temperatures to produce vortices;two diﬀerent paths could be in principle followed:(1)producing ﬁrst a condensate then rotating the trap,or (2)cooling the gas directly in a rotating trap.It has been recently proposed in [6]to use quantum topological eﬀects to obtain a vortex.Other methods that do not rely on thermal equilibrium have been suggested [7],[8].Here we study theoretically the minimal energy conﬁgurations of vortices in a rotating trap [9].The model is deﬁned in section II;in sections III to VI we assume a strong conﬁnement of the atoms along the rotation axis z so that we face an eﬀective 2D problem in the transverse plane x −y .We present numerical results for solutions with vortices that are local minima of the Gross-Pitaevskii energy functional (section III).These solutions contain only vortices with a charge ±1,the vortices with a charge larger than or equal to 2are thermodynamically unstable (section IV).We discuss possibilities to get experimental evidence of vortices in atomic gases in section V.Finally,we concentrate on the regime where the interaction energy is much larger than the trap frequenciesωx,y ,the so-called Thomas-Fermi limit [2].This is complementary to the work of [10].We obtain inthis “strong interacting”regime analytical predictions based on a variational Ansatz that reproducesatisfactorily the numerical results (section VI).In section VII we present results for vortices in 3D,that is in a trap with a weak conﬁnement along the rotation axis.II.MODEL CONSIDERED IN THIS PAPERThe atoms are trapped in a potential rotating at angular velocity Ω.In the laboratory frame theHamiltonian of the gas is therefore time dependent.To eliminate this time dependence we introducea rotating frame at the angular velocity Ωso that the trapping potential becomes time independent;this change of frame is achieved by the single-atom unitary transform:U(t)=e i Ω· Lt/¯h(1) where L is the angular momentum operator of a single atom.As the unitary transform is timedependent the Hamiltonian in the rotating frame contains an extra inertial term,given for eachatom byi¯h U†(t)dφ|φ +1φ|φ 2.(3)In this energy functional H0contains the kinetic energy and the trapping potential energy of theparticles:H0=−¯h22mω2αr2α.(5)Furthermore in all but in section VII we will assume that the trapping potential is much strongeralong the z axis than along the x,y axis,with an oscillation frequency much larger than the typicalinteraction energy Ng3D|φ|2per particle.This situation,although not realized experimentally yet,is not out of reach,in particular when one uses optical traps rather than magnetic traps[11].Inthis strong conﬁning regime the motion of the particles along z is frozen in the ground state of thestrong harmonic potential:φ(x,y,z)≃ψ(x,y) mωz2¯hωz:E[ψ,ψ∗]= d2 rψ∗( r)[H⊥−ΩL z]ψ( r)2Ng|ψ|42m ∆x,y+12π¯h 1/2.(9)Most of the results of the paper are dealing with the2D energy functional;a numerical result for a local minimum of the full3D energy functional will be given in the section VII.We concentrate on the so-called Thomas-Fermi regime,where the interaction energy per particle is much larger than ¯hωx,y.The opposite regime has already been studied in[10].III.LOCAL MINIMA OF ENERGY WITH VORTICES In this section we brieﬂy discuss the general problem of minimizing energy functionals of the type Eq.(7).We present the numerical algorithm that we have used and we give numerical results for the2D problem.A.A numerical algorithm toﬁnd local minimaThe algorithm in our numerical calculations is commonly used in the literature to minize energy functionals E[ψ,ψ∗]of the form Eq.(7).The intuitive idea is to start from a randomψand move it opposite to the local gradient of E[ψ,ψ∗]that is along the local downhill slope of the energy. Numerically this is implemented by an evolution ofψparametrized by aﬁctitious timeτ:−dδψ∗[ψ,ψ∗].(10)Assuming aψnormalized to unity we get the following equation of motion forψ:−ddτE[ψ,ψ∗]=−2 d2 r δE2mΩ2r2cannot exceed the trapping potential.Therefore E has to convergeto aﬁnite value forτ→∞.Asymptotically dE/dτ=0andψsatisﬁesδEωx=ω/(1+ǫ)(14)ωy=ω(1+ǫ).(15) In Fig.1we show diﬀerent local minima conﬁgurations obtained forǫ=0.3and a rotation frequencyΩ=0.2ω;each conﬁguration has been obtained for diﬀerent random initialψ’s.The holes observedin the spatial density correspond to the vortex cores.We have always found that the phase ofψchanges by2πaround a vortex core;we have not found vortices with a charge±q,where the integerq is strictly larger than one;this fact will be explained in the next section.Furthermore the senseof circulation is the same for all vortices.To quantify the eﬀect of the non-axisymmetry of the trap we have plotted in Fig.2the dependenceof energy of diﬀerent vortex conﬁgurations onωx/ωy for aﬁxedω;we measure the energies fromE iso,the energy of the zero-vortex solution in the axisymmetric caseǫ=0.The zero-vortex solutionexhibits a signiﬁcant variation of energy withǫ;for a non-zeroǫthe wavefunctionψdevelops a phase proportional toΩfor weakΩ’s,which accounts for the energy change as explained in section VI B.The solutions with vortices experience quasi the same energy shift as function ofǫ.As only theenergy diﬀerence between the various local minima matters we will from now on only consider the axisymmetric caseǫ=0to identify the solution with the absolute minimal energy.Note that the solutionsψwith several vortices obtained in the limiting caseǫ=0are not eigen-vectors of L z;this reﬂects a general property of non-linear equations such as the Gross-Pitaevskiiequations to have symmetry broken solutions;it is explained in[10]how to reconcile this symmetrybreaking with the fact that eigenvectors of the full N-atom Hamiltonian are of well deﬁned angular momentum.IV.STABILITY PROPERTIES OF VORTICESIn this section we recall that a(normalized)wavefunctionψsuch that E[ψ,ψ∗]has a local min-imum inψ,describes a condensate having all the desired properties of stability,that is dynamicaland thermodynamical stability.We then show that a vortex centered at r= 0with an angular mo-mentum strictly larger than¯h is not a local minimum of energy and is therefore thermodynamicallyunstable.A.Stability properties of local minimaLet us express the fact thatψcorresponds to a local minimum of the energy.Aﬁrst condition isthat the energy functional is stationary forψ,that isψsolves the Gross-Pitaevskii equation Eq.(13).To get the second condition,we consider a small variation ofψ,ψ→ψ+δψ(16) preserving the normalization of the condensate wavefunction to unity:||ψ+δψ||2−||ψ||2=0= ψ|δψ + δψ|ψ + δψ|δψ .(17) We expand the energy functional E[ψ,ψ∗]in powers ofδψ,neglecting terms of orderδψ3or higher.Using Eq.(17)and Eq.(13)weﬁnd that terms linear inδψvanish so that1δE=1.Dynamical stabilityConsiderﬁrst the problem of so-called“dynamical stability”:to be a physically acceptable con-densate wavefunction,ψhas to be a stable solution of the time dependent Gross-Pitaevskii equationi¯h∂tψ=H GPψ(21) otherwise any small perturbation ofψ,e.g.the eﬀect of quantumﬂuctuations or experimental noise,may lead to an evolution ofψfar from its initial value.To determine the evolution of a smalldeviationδψas in Eq.(16)we linearize Eq.(21):i¯h∂t |δψ |δψ∗ =L |δψ |δψ∗ (22) where the operator L is related to L c byL c= 100−1 L.(23)Asψis time independent,so is L and dynamical stability is equivalent to the requirement that theeigenvalues of L have all a negative or vanishing imaginary part.As we now show the positivity ofL c leads to a purely real spectrum for L.Consider an eigenvector(u,v)of L with the eigenvalueε.Contracting Eq.(23)between the ket(|u ,|v )and the bra( u|, v|)we getε[ u|u − v|v ]=( u|, v|)L c |u |v .(24)Note that the matrix element of L c is real positive as L c is a positive hermitian operator.We nowface two possible cases for the real quantity u|u − v|v :• u|u − v|v =0.In this case L c has a vanishing expectation value in(|u ,|v );as L c is positive(|u ,|v )has to be an eigenvector of L c with the eigenvalue zero;from Eq.(23)and the fact that 100−1 is invertible weﬁnd that(|u ,|v ) is also an eigenvalue of L with the eigenvalue0,so thatε=0is a real number.• u|u − v|v >0:we getεas the ratio of two real numbers,so thatεis real.2.Thermodynamical stabilityA second criterion of stability is the so-called“thermodynamical”stability.For zero temperature,this condition can be formulated in the Bogoliubov approach[2],where the particles out of thecondensate,which always exist because of the interactions,are described by a set of uncoupledharmonic oscillators with frequenciesεsign[ u|u − v|v ]/¯h,where(u,v)is an eigenvector of L withthe eigenvalueε.In order for a thermal equilibrium to exist for these oscillators,their frequenciesshould be strictly positive,which is the case here in virtue of Eq.(24)[13].If a mode with a negativefrequency were present thermalization by collisions would transfer particles from the condensateψto this mode,leading to a possible evolution of the system far from the initial stateψ[14].What happens for solutionsψof the Gross-Pitaevskii equations that are not local minima of energy?The operator L c has at least an eigenvector with a strictly negative eigenvalue.In thiscase one cannot have thermodynamical stability,that is one cannot haveε[ u|u − v|v ]>0forall modes[13].From the non-positivity of L c one cannot however distinguish between a simplethermodynamically instability or a more dramatic dynamical instability.B.Why not a vortex of angular momentum larger than¯h?For simplicity we consider only a single vortex in the center of an axi-symmetric trap.We show that vortices with a change of phase of2qπare not local minima of energy,that is are(at leastthermodynamically)unstable.We have found numerically a solution of the Gross-Pitaevskii equationEq.(13)by an evolution in complex time,starting from a wavefunctionψwith an angular momentumq¯h along z,as already done in[15];our solution of the Gross-Pitaevskii equation with imposedsymmetry is a local mimimum of energy in the subspace of functions with angular momentum q¯halong z,but not necessarily a local minimum in the whole functional space,as we will see for|q|>1.In the Thomas-Fermi regimeµ≫¯hωweﬁnd that the solutions can be well reproduced bya variational Ansatz of the formψ(x,y)=e iqθ[tanhκq r]|q| ˜µ−1Ng 1/2(25) whereθis the polar angle in the x−y plane and where˜µ,the chemical potential in the lab frame˜µ=µ+q¯hΩ(26) does not depend onΩ.In this Ansatz the vortex core is accounted for by tanh|q|,a function thatvanishes as r|q|in zero as it should,and the condensate density outside the core coincides withthe Thomas-Fermi approximation commonly used for the zero-vortex solution[2].We calculate themean energy Eq.(7)of the variational Ansatz and we minimize it with respect to the variationalparameterκq;we getκq= ˜µmq2 +∞0du u tanh2|q|(u)−1 2(28) is a number(c1=0.7687,c2=0.5349,...).In order for the vortex of charge q to be a local minimum of energy,the operator L c of Eq.(19)has to be positive.This implies that the operator on theﬁrst line,ﬁrst column of L c,the so-calledHartree-Fock Hamiltonian,be positive:H HF=H⊥+2Ng|ψ|2−˜µ+q¯hΩ−ΩL z≥0.(29) To show that this is not the case it is suﬃcient toﬁnd a wavefunction f(x,y)leading to a negative expectation value for H HF.As the potential appearing in H HF has a dip at r=0we have taken fof a form localized around r=0:1f(x,y)=¯h2 1/2r (30) whereγis adjusted to minimize the expectation value.For e.g.q=2we takeγ=1leading tof|H HF|fsection the trap is axi-symmetry with a time dependent frequencyω(t).We consider the evolutionin the laboratory frame,as the detection is performed in this frame:i¯h∂tψlab= −¯h22mω2(t)r2+Ng|ψlab|2 ψlab.(32)As shown in[19,20]the eﬀect of the time dependence ofω(t)can be absorbed by a scaling and gaugetransform of the wavefunction:1ψlab( r,t)=−ω2(t)λ(34)λ3with initial conditionsλ(0)=1,˙λ(0)=0;if the trap in the x−y plane is abruptly switched oﬀatt=0+the scaling parameter is given byλ(t)==dτ(36)λ2(t)weﬁnd that˜ψsolves the same equation asψlab with a constant trap frequency equal toω(0):i¯h∂τ˜ψ= −¯h22mω2(0)r2+Ng|˜ψ|2 ˜ψ.(37)Asψlab rotates in the trap at the frequencyΩin the lab frame,so does˜ψin terms of the renormalizedtimeτ.In the limit of t→∞,τtends to aﬁnite valueτmax,so that˜ψis rotated by aﬁnite angleduring the ballistic expansion:Ωτmax=Ω ∞0dt2ΩVI.INTUITIVE V ARIATIONAL CALCULATIONTo get a better understanding of the numerical results we now proceed to an intuitive Ansatzfor the wavefunction with several vortices.It coincides very well with the numerical results andallows an easy construction of the minimal energy conﬁgurations with vortices.It gives a physical understanding of the stability conditions and of the structure of the solutions:a set of n vortices isequivalent to a gas of interacting particles in presence of an external potential adjusted by the rota-tion frequency of the trap.We restrict to the case of an axi-symmetric trap,a good approximationfor weak(<10%)non-axisymmetries(see section III B).A.Ansatz for the densityTo construct the Ansatz we splitψin a modulus and a phase:ψ(x,y)=|ψ|e iS.(40) In the Thomas-Fermi regime,the modulus in presence of n vortices appears as a slowly varyingenvelope given by the Thomas-Fermi approximation used in the0-vortex case:ψslow= µ−1Ng 1/2(41)with narrow holes digged by the vortices with charge q=±1,represented by tanh functions ofadjustable widths and with zeros at adjustable positions:|ψ|=ψslow×nk=1tanh[κk| r− αk R|].(42)The positions of the vortex cores αk are expressed in units of the Thomas-Fermi radius R of the condensate:R= mω2.(43)From section IV B we expect as typical values for the inverse width of the vortex coresκk≃(mµ/¯h2)1/2.The chemical potential is not an independent variable but is expressed as a functionof the other parameters from the normalization condition ψ|ψ =1;neglecting overlap integralsbetween the holes we getµ=µ0 1+2n k=1(1−α2k)ln2(κR)4) (44)where1/(κR)4∼(¯hω/µ)4≪1and whereµ0is the Thomas-Fermi approximation for the condensatechemical potential without vortices:µ0= mω2Ngcentered on the vortex core and S 0is the single-valued part of the phase.The function S 0can in principle be determined from the modulus of ψfrom the continuity equation:div[|ψ|2 v ]=0.(47)The local velocity ﬁeld v is related to the phase S byv =¯h ¯h [x∂y −y∂x ]|ψ|2=0.(49)This can be turned into an equation for the single-valued part S 0of the phase;because the density |ψ|2in a trap vanishes at the border of the condensate S 0is uniquely determined (up to a constant)by the resulting equation (see Appendix);this is to be contrasted to the case of superﬂuid helium in a container,where the ﬂux,not the density,vanishes at the border,which requires a boundary condition on the gradient of the phase.Eq.(49)can be solved for a non-axisymmetric trap in the absence of vortices.The solution is given byS (x,y )=−m Ωω2x +ω2yxy (50)which leads to a change in the energy per particleδE =−1(ω2x +ω2y )ω2x ω2y(51)where µT F is the Thomas-Fermi approximation for the chemical potential for Ω=0,µT F =(mωx ωy Ng/π)1/2[23].As can be seen in Fig.2this prediction is in good agreement with our numerical results.In presence of vortices the equation for S is more diﬃcult to solve analytically.From now on we consider the case of an axi-symmetric trap,as the energy ordering of the vortices solutions is not aﬀected for weak (<10%)non-axisymmetries (see section III B).For a single vortex at the center of the trap one can see that S 0=0solves Eq.(49).From the spatial dependence of the phase obtained numerically (section III B)for a displaced vortex or several vortices we have identiﬁed the following heuristic Ansatz,obtained in setting ωx =ωy =ωin Eq.(50):S 0(x,y )≡0(52)that we will use in the remaining part of the section.C.Further approximations for the mean energyIn the calculation of the mean energy,we make some further approximations in the spirit of the Ansatz Eq.(42).The reader not interested by these more technical considerations can proceed to the next subsection.The kinetic energy involves an integral of the gradient squared of the wavefunction:| ∇ψ|2=|ψ|2 ( ∇ln |ψ|)2+( ∇S )2 .(53)For the gradient of the modulus of ψwe neglect the variation of the slow envelope ψslow :∇ln |ψ|≃nk =1κktanh ′of ψinvolves holes with a density varying as 1−tanh 2=sech 2.In the following we keep the sech 2for the vortex k only if it is multiplied by ( ∇θk )2,a quantity diverging in the center of the core;the other terms lead toconverging integrals smaller by a factor (µ/¯h ω)2,which is the inverse surface of a vortex core ( d 3 r |ψslow |2sech 2κr ∝1/(κR )2).This ﬁnally leads toE kin ≃¯h 22mω2r 2+13µ0+nk =1W ( αk ,κk )+1µ012ln(1−α2)−q ¯h Ω (1−α2)2 +µ0(κR )2(60)where C =0.495063.The lines inEq.(60)correspond successively to E kin ,E rot and E pot .This can be seen as an eﬀective potential for the vortices.One can check that the part of W independent of Ωexpells the vortex core from the trap center,whereas the part proportional to Ωprovides a conﬁnement of the vortex core (see the following subsection).The vortex interaction potential is given by12m q αq β d 2 r |ψslow |2 ∇θ αR · ∇θ βR .(61)This interaction term is equivalent to the one found in the homogeneous case and describes a repulsive interaction for vortices turning in the same direction (q αq β>0)and is attractive for vortices with opposite charges [9].An attractive interaction will lead to the coalescence and consequently annihilation of vortices with opposite charges.Therefore we ﬁnd in stationary systems always vortices with equal charges.As the interaction potential V ( α, β)does not depend on the parameters κwe can optimize sepa-rately the self-energy part with respect to κand ﬁnd(κR)2=ξ2(1−α2) µ03(4ln2−1) 1/2≃1.08707.By rewriting the above equation as¯h2κ22ξ2 µ−1µ0 13+lnνµ0¯hω2(1−α2) (64) whereν=0.49312.E.Case of a single vortex:critical frequenciesIn Fig.4we have plotted the self-energy of a vortex as a function of the displacement of the corefrom the trap center,for diﬀerent values of the rotation frequencyΩ.The analytical predictioncoincides very well with the numerical value[24].ForΩ=0the position of the vortex at the trap center gives an energy maximum.ForΩ>0therotation of the trap provides an eﬀective conﬁnement of the vortex core at the center of the trap forpositive charges q(see the term proportional toΩin Eq.(64));from now one we therefore take allthe charges q k to be equal to+1.For a large enoughΩwe reach a situation where a vortex at thetrap center corresponds to a local energy minimum,by further increasingΩthe vortex state at thetrap center becomes a global minimum with energy less than the condensate without vortex.The above suggests that we have to distinguish two critical rotation frequencies:Theﬁrst onedeﬁnes the frequencyΩstab above which the vortex is a local minimum of energy.Above the frequencyΩc the single vortex solution has an energy lower than the condensate without vortex.We calculateΩstab from the condition d2W/dα2=0atα=0andΩc from the condition W=0atα=0:Ωc=¯hω2¯hω (65)Ωstab=¯hω2¯hω (66)where C′=e(2ln2+1)/3+1/2ν≃1.8011.As we are in the regimeµ0≫¯hωΩc is approximately twice Ωstab[25].Our prediction forΩc scale as(logµ0)/µ0as in[15],with a coeﬃcient C′leading to better agreement with the numerics.F.Case of several vorticesBy integrating Eq.(61)we get an explicit form for the vortex interaction potential for vortices with equal charges:V( α, β)=(¯hω)2(1− α· β)+1| α− β|4 (67)At short distances between the two vortex cores the logarithmic term in the above expression dominates,leading to a repulsive potential∼−2(1− α· β)log| α− β|(¯hω)2/µ0.In Fig.5we plot the interaction energy between a vortex at the center of the trap and one of equal charge displaced by αR;the interaction is purely repulsive.A conclusion which essentially holds as well for arbitrary vortex positions.In Fig.6we show the total(interaction+self-energy)for two vortices symmetrically displaced from the trap center,as function of the displacement;the analytical prediction coincides again very well with the numerical results[24].To obtain the equilibrium distance between the two vortex cores one minimizes the total energy overαin Fig.6.To get the minimal energy conﬁgurations as function of the rotation frequency of the trap,we minimize our analytical prediction for the energy over the positions of the n=1,2,...vortex cores.The result is shown in Fig.7.Each curve corresponds to aﬁxed value of n;it starts atΩ=Ωstab(n)(forΩ<Ωstab(n)there is no local minima of energy with n vortices);it becomes the global energyminimum forΩ=Ωc(n).We have plotted these two critical frequencies as function of n in Fig.8.We have also given numerical results(circles)in Fig.7.Even if there is good agreement between analytical and numerical results,we still need a numerical calculation to check the stability of thesolutions;our simple analytical Ansatz is indeed not suﬃcient to predict the destabilization of agiven vortex conﬁguration at highΩ,a phenomenon studied with a numerical calculation of theBogoliubov spectrum for a single vortex in[26].For aﬁxed value of the number of vortices n there may exist local minima of energy,in addition to the global minimum plotted in Fig.7,a situation known from superﬂuid helium[4].E.g.for n=6(see Fig.9)the global minimum of energy is given by a conﬁguration with six vortex cores on acircle;there exists also a local minimum of energy with one vortex core at the center of the trap andﬁve vortex cores on a circle.The energy diﬀerence per particle between the two conﬁgurations isvery small,δE≃0.002¯hωfor the parameters of theﬁgure and probably beyond the accuracy of ourvariational Ansatz.For relatively large rotation frequenciesΩone canﬁnd local minima of energyconﬁgurations with many vortices(see[4]for superﬂuid helium);we plot two conﬁgurations with18vortices in Fig.10,with an energy diﬀerenceδE=0.0034¯hω.In estimating the physical relevance of these energy diﬀerences one should keep in mind that NδE matters,rather thanδE,where N is the number of particles in the condensate: e.g.at aﬁnite temperature T the ground energy conﬁguration is statistically favored as compared to themetastable one when NδE≫k B T.VII.VORTICES IN A3D CONFIGURATIONWe have extended the numerical calculation to the case of a3D cigar-shaped trap,that is witha conﬁnement weaker along the rotation axis than in the x−y plane.Even in this case rotationof the trap can stabilize the vortex.We show in Fig.11density cuts of a solution with5vortices;the vortex cores are almost straight lines in the considered Thomas-Fermi regime,except at vicinityof the borders of the condensate.As in section VI the core diameter is determined by the localchemical potential in the gas.This suggests that our2D Ansatz(section VI)can be generalized to3D situations,with αk and κk depending on z.VIII.CONCLUSION AND PERSPECTIVESWe have presented in this paper an eﬃcient numerical algorithm and a heuristic variational Ansatz to determine the local minima energy conﬁgurations for a Bose-Einstein condensate stronglyconﬁned along z and subject to a rotating harmonic trap in the x−y plane.Our results can be used as aﬁrst step towardsﬁnite temperature calculations.Interesting prob-lems are e.g.the critical temperature for the vortex formation and the Magnus forces induced bythe non-condensed particles on the vortex core[27].Acknowledgement:We acknowledge useful discussions with Sandro Stringari and Dan Rokhsar.We thank J.Dalibard for useful comments on the manuscript.We thank the ITP at Santa Barbarafor its hospitality and the NSF for support under grant No.PHY94-07194.This work was partiallysupported by the TMR Network“Coherent Matter Wave Interactions”,FMRX-CT96-0002.∗L.K.B.is an unit´e de recherche de l’Ecole Normale Sup´e rieure et de l’Universit´e Pierre et Marie Curie,associ´e e au CNRS.APPENDIX A:UNIQUENESS OF THE PHASE FROM THE CONTINUITY EQUATION IN A TRAPConsider two solutions S1and S2of the continuity equation:mΩdiv[|ψ|2 ∇S]=。

玻色爱因斯坦凝聚体中暗孤子动力学研究_刘超飞

收稿日期：2016-01-03基金项目：国家自然科学基金资助项目（11304130，11365010，61565007）；江西省教育厅资助项目（GJJ150685）；江西省科技厅资助项目（20151BAB212002）；江西理工大学清江青年英才支持计划资助作者简介：刘超飞（1981-），男，博士，副教授，主要从事玻色爱因斯坦凝聚等方面的研究，E-mail:liuchaofei0809@.江西理工大学学报ＪｏｕｒｎａｌｏｆＪｉａｎｇｘｉＵｎｉｖｅｒｓｉｔｙｏｆＳｃｉｅｎｃｅａｎｄＴｅｃｈｎｏｌｏｇｙ第37卷第5期2016年10月Vol.37,No.5Oct.20160引言在原子凝聚体中，原子间的排斥相互作用导致了许多有趣的非线性现象，一个最典型的例子就是暗孤子的形成[1-8].实验上，通过相印记[1-2]和扰动原子密度[3]方法，暗孤波已在稀释玻色-爱因斯坦凝聚体中产生.在相印记方法中，凝聚体的一个部分被远失谐激光束短时间照射，使它获得了相移，但没有产生重大的密度扰动.根据相映射实验，将出现一个暗孤立波，以及作为副产品的声波.在随后阶段的相映射实验中，由于其固有的不稳定性和横向激发[4]，人们观察到暗孤波退化为涡旋环.另一项实验涉及到通过慢光技术从凝聚体中突然清除一个盘状区域，它将产生暗孤波（暗孤波也将蜕化成涡旋环）和相反传播的声波[3].暗孤子的大小接文章编号：2095-3046（2016）05-0102-10DOI:10.13265/ki.jxlgdxxb.2016.05.016玻色爱因斯坦凝聚体中暗孤子动力学研究刘超飞a ，潘小青a ，张赣源b（江西理工大学，a.理学院；b.应用科学院，江西赣州341000）摘要：文章从侦测暗孤子能量的角度，全面介绍了玻色爱因斯坦凝聚体中暗孤子发生声波辐射以及与声波相互作用的动力学行为.虽然暗孤子-声波相互作用导致暗孤子能量变化，暗孤子在简谐势阱中，以及简谐势阱受到扰动时，都具有类似粒子运动的动力学行为.考虑Rabi 耦合时，暗孤子还可以转化为稳定传播的矢量暗孤子.文章对玻色爱因斯坦凝聚体中暗孤子动力学研究的进行全面总结，将加深人们对暗孤子现象的认识.此外，类似暗孤子-声波相互作用的行为，也将出现在其他孤子动力学研究中.关键词：玻色爱因斯坦凝聚；暗孤子；声波；孤子声波相互作用；简谐势阱中图分类号：O469文献标志码：AKinetic study of dark soliton in Bose-Einstein condensateLIU Chaofei a ,PAN Xiaoqing a ,ZHANG Ganyuan b(a.Faculty of Science;b.Faculty of Applied Science,Jiangxi University of Science and Technology,Ganzhou 341000,China)Abstract ：By exploring the energy of dark soliton,this paper systematically introduces the dynamical behavior of sound-emission of dark soliton and dark soliton-sound interaction in Bose-Einstein condensate.Although the dark soliton-sound interaction leads to the change of the dark soliton ′s energy,dark soliton displays the particle-like behavior very well in the harmonic potential and even in the periodic perturbed harmonic trap.Under the Rabi coupling,dark soliton can transfer into the vector dark soliton,which propagates stably in the condensates.This paper provides a full overview of the kinetic study of dark soliton,and it will greatly increase people ′s knowledge about the dark soliton.Furthermore,similar behaviors of the dark soliton-sound interaction will occur in the dynamical investigation of other soliton.Key words ：Bose-Einstein condensate;dark soliton;sound waves;soliton-sound interaction;harmonic trap刘超飞，等：玻色爱因斯坦凝聚体中暗孤子动力学研究近当前成像技术的极限.在这些实验中，通过一再释放势阱中的凝聚体，让其膨胀，然后采取一个光学吸收图像来实现成像.产生暗孤子的更先进的方法也已经提出[9-10]，包括合并相映射和密度工程方法[5-6].暗孤子也可用两个凝聚碰撞产生[11]，以及用凝聚体的布拉格光学晶格反映[12-13]产生.迄今为止，在几何形状上，暗孤子的实验可以从球对称[4]到高度拉长的情况（长宽比大于30[2]）.这些系统在性质上仍然是三维，致使暗孤子容易由于横向不稳定性而被破坏，从而迅速衰变为涡旋.这是个关键因素，它限制了观察到的孤子寿命，其值约为数十毫秒.然而，最近的实验发现，在准一维凝聚体中，暗孤波将是亚稳的，其寿命可大大延长到直至数秒[14].理论上，根据零温平均场理论，稀释原子形成的玻色爱因斯坦凝聚体可由Gross-Pitaevskii方程描述，它是一个很好的非线性系统.在外势场为零时，Gross-Pitaevskii方程支持暗孤子解.与光学系统的一个重要的区别是，势阱能导致凝聚体有不均匀的背景密度.对于凝聚体中的暗孤波，其性质与光学系统中的暗孤波类似.三维暗孤波由于横向激发，是不稳定的，因此，暗孤子容易衰变弯曲形成涡旋.在实验中，可将凝聚体在横向高度压缩，从而构成所谓的一维体系，而暗孤子的运动则由不均匀的纵向密度控制.总的说来，暗孤子是一个局部的密度缺陷，就像一个凹槽，其周围充满凝聚体.并且，暗孤子的两边存在位相差，它是散焦的色散效应与聚焦的非线性相互作用之间达到平衡的结果.因此，暗孤子的主要特性之一，是能在传播中保持其局域化的形状不变[11,15-17].通常，研究孤子的文章主要是给出新的孤子解，或者探索孤子的不稳定性等.在这篇文章中，系统性的介绍玻色爱因斯坦凝聚体中暗孤子动力学研究.与大多数论文不同，本文的研究重点是暗孤子能量的计算和动力学演化中的能量侦测.暗孤子受到外界环境的干扰后，发生声波辐射，暗孤子能量降低，速度变快，同时，外界的声波又能反作用于暗孤子，使其能量增加.这种研究暗孤子的方式，最初在文献[18-19]上介绍.实际上，这种研究思路完全可以推广到其他孤子解的动力学研究上.1暗孤子的数值解凝聚体背景密度均匀，且为.则含有速度为和位置在的暗孤子的一维的形式的凝聚体的波函数为：ψs（z，t）=n姨exp（-iμ攸t）·i v+1-v2姨tanh[1-v2姨（z-vt）姨姨]（1）这里ξ＝攸n gm姨为凝聚体的愈合长度，它可用来刻画暗孤子的尺度.暗孤子的速度依赖于密度n d和通过其中心的相移S.并且，v/c=1-（n d/n）姨= cos（S/2），孤子速度的最大值由Bogoliubov声速度/ C=ng/m姨决定.这里有两个极限情况：①固定暗孤子完全是黑色的，即有一个零密度节点，以及π相滑移；②孤子速度为c时，无相滑移，也与背景的密度无差异，因此难以分辨.图1展示了各种速度暗孤子的密度和相位.一个重要特点是，固定孤子的能量最高，在v=c时，孤子能量基本上为零，这导致人们认为孤子具有负有效质量的设想.由于不会耗散，孤子往往类似于粒子.事实上，对于一阶弱作用力，暗孤子就像一个有效质量为负的经典粒子[11,20-22].这意味着，例如，在一个谐势阱的凝聚体中，暗孤子将趋于来回震荡.S图1各种速度暗弧子密度和相位（b）暗孤子的相位Sπ/20.0-π/2-505z/ξv/c=0v/c=0.25v/c=0.50v/c=0.751.00.50.0-505z/ξ（a）暗孤子的密度nn/nv/c=0v/c=0.25v/c=0.50v/c=0.75第３７卷第５期103与纵向均匀系统（如非线性光学纤维）不同，原子凝聚体沿孤子的运动方向有束缚.例如，暗孤子在谐势阱束缚下的凝聚体中，将趋于在势阱中来回震荡.由于这一空间束缚，孤子一般会与其他激发共同存在，例如声波.因此，文中说的孤子并非纯数学意义中的孤子，而是指一个空间区域（即“孤子地区”），该区域存在密度凹陷和相位差，以及其他可能的激发.这可以粗略地视为一个受到扰动的孤子[23].2暗孤子的能量在无限大体系中，重整化的一维暗孤子能量（即除去背景流体的贡献部分）由式（2）给出，E s ol ＝4攸n 3/21-vc23/2（2）然而，在非均匀凝聚体中，只有当暗孤子在密度局部均匀区域，该方程式才有效.这将在后面的文章中进一步加以说明.还有一种得出一维暗孤子能量方法，它基于对Gross-Pitaevskii 方程的数值积分，即使在密度不均匀时仍然有效，ε（ψ）=攸22m荦ψ2+V ψ2+12g ψ4（3）暗孤子能量E s 是通过对孤子位置Z s 积分一个距离z ints ，然后减去对应的时间独立性背景密度n TI 的贡献，即E s =Z s +z intsZ s -z ints乙ε（ψ）d z -Z s +z intsZ s -z ints乙ε（n TI 姨）d z （4）“孤子区域”必须足够大，以包含暗孤子能量的绝大部分.实际上，暗孤子的速度和背景密度都影响暗孤子密度凹陷处的宽度.图2说明了积分后的各种速度的孤子能量（实线），分别与该区域的大小和背景密度的函数关系.当z ints >5ξ时，积分得到的能量值几乎与从公式（4）（虚线）渐近预测的数值完全相等，所以我们选择“孤子区域”为（Z s ±5ξ）.在时间依赖性模拟中，“孤子区域”能含有声波.通常，很难能区分孤子能和声波能，但后者的数值，在孤子的速度不是很大时，是非常小的.3简谐势阱中的暗孤子假设凝聚体在一维谐势阱中，现在考虑暗孤子在凝聚体中的动力学行为，简谐势阱为：V （z ）=12ω2z z 2（5）这种势阱通常是由磁场形成.系统在空间上是有限的，这是体系的一个重要特点.因此，凝聚体的大小可用托马斯-费米半径刻画.对于一个束缚频率为ωz 的势阱，托马斯-费米密度分布是一个倒抛物线型，其中n TF =（1-ω2z z 2/2），托马斯-费米半径为R TF =2/ω2z 姨.图3（a ）展示了一个速度为v =0.5c 的暗孤子在势阱中，其初始位置为z=0，凝聚体的密度峰值为n 0，纵向束缚频率为ωz =2姨×10-2（μ/攸）.实际上，除了在边界附近由于小动能贡献导致‘尾巴’状热云外，托马斯-费米密度分布与真实实验状况吻合得很好（图3（a ））.因此，凝聚体的实际大小刚好大于托马斯-费米半径100ξ.在该系统中原子密度的时间演化由图3（b ）显示，其纵坐标为位置，横坐标为时间.暗孤子是一个局部的密度极小.它在向势阱壁移动过程中减速，当其密度极小处触及零密度时，孤子的运动方向改变.众所周知，当孤子远离势阱中心时，其相滑移增加，并在最大振幅处达到π.随后，孤子改变其运动方向，孤子的相滑移变到-π.图2通过积分得到的暗孤子能量E s与积分区间宽度z int 的函数关系1.51.00.50.0012345（a ）固定背景密度n 0时z int /ξv =0.5cv =0.75cv =0.25cv =0c E s /μ（b ）固定孤子速度v =0.5c 时1.00.80.60.40.20.0012345n =n 0n =0.8n 0n =0.6n 0n =0.4n 0n =0.2n 0z int/ξE s /μ江西理工大学学报2016年10月104在谐势阱中的暗孤子，其振荡频率近似为ωz /2姨[14,15,24-28].这是由分析托马斯-费米密度分布得到的，并且人们已用数值模拟证实了这一结论.在图3（b ）中，孤子振荡周期约为T s =630（ξ/c ），而势阱的周期约为T z =444（ξ/c ），这与理论预测结果相同.孤子的运动扰动背景流体，致使流体振荡，其幅度约为2%n 0.可以用下式定义背景凝聚体的偶极振荡，D =乙z ψ（z ）2d z（6）凝聚体的偶极运动D 和孤子路径Z s 由图3（c ）给出.暗孤子的振荡频率为ωz /2姨（实线），它诱发了势阱中背景流体的偶极振荡（虚线），其频率ωz 为[27].在一定程度上，孤子行为就像搅拌器，搅拌着流体.暗孤子将在势阱中加速，孤子由于辐射声波而衰减.在这种情况下，暗孤子的深度将变浅，而其速度将变快，从而更进一步逼近势阱壁，并导致了与反阻尼类似的现象.这与阻尼谐振子相比，结果正好相反.对于阻尼谐振子，其振荡幅度随时间减小.但是，简谐势阱中看不到任何的孤子震荡幅度的净变化，因此，不能推断出孤子的衰变.在图4（a ）中仅仅观察到的孤子振幅的小周期调制，例如，围绕其平均幅度，大约有1％的最大调制幅度变化.类似的结果也出现在了孤子能量的进化中，如图4（b ）所示.孤子能量的平均值保持不变，但有振荡调制.3.1速度对暗孤子运动的影响图5（a ）显示了不同初始速度的暗孤子在一个固定的简谐势阱中的演化路径.增加孤子的速度，其主要结果是孤子振幅增加.但即使孤子速度高达0.7c ，孤子的振荡频率仍保持在预期值ωs =ωz /2姨的周围.但对于运动速度非常快的孤子，如v =0.9c ，如图5（c ）中（点虚线），我们看到其值略有偏差，它趋向于数值更高的振荡频率.这很可能是因为这个快速运动的浅孤子进入了凝聚体的边界造成.在边界处，凝聚体的密度偏离于托马斯-费米密度分布.相比之下，较慢的孤子的振荡束缚在势阱中心，而势阱中心的密度分布几乎与托马斯-费米密度分布相同.对于托马斯-费米密度分布，在简谐势阱中的暗孤子的振荡幅度与孤子的初始速度成正比.为比较各种速度的中心孤子，我们可以使用这一关系，重整化孤子的位置.图5（b ）显示了孤子位置的重整化图.各速度下的重整化位置随孤子速度的增加而幅度增大.对于不同速度的孤子的能量的振荡演化，这种效应也被观察到，如图5（c ）所示.对于快速运动的孤子（例如，v =0.9c ）（点虚线），其能量调制延伸到了最初能量的0.4倍.1.00.50.0-1001001.00.50.0z /ξn /n 0V /μ（a ）凝聚体在简谐势阱中的密度（左轴，实线），其中ωz =2姨×10-2（μ/攸）（右轴，虚线）.速度为v =0.5c 的暗孤子在势阱中心.图4简谐势阱中暗孤子振幅与能量关系E s /μ0.860.850.845000100001500020000t /（ξ·c -1）（b ）暗孤子能量E s 的演化100-100z /ξ（b ）凝聚体随时间演化的重整化图500-50500-505001000150020002500t /（ξ·c -1）（c ）暗孤子位置Z s （实线，左轴）和凝聚体的偶极运动D （虚线，右轴）图3简谐势阱中的暗孤子运动z s /ξD /ξ2n 05001000150020002500t /（ξ·c -1）5150z S/ξ（a ）暗孤子离开势阱中心的距离Z s5000100001500020000t /（ξ·c -1）刘超飞，等：玻色爱因斯坦凝聚体中暗孤子动力学研究第３７卷第５期105刚才我们已经看到，这些小的位置和能量调制，由振荡孤子对背景流体的干扰产生，随后反馈到孤子上.孤子速度越快，有效质量越小，所以背景流体的振荡将会对它们有更大的反馈作用，从而引起更大的调制.3.2束缚势阱强度对暗孤子行为的影响增加势阱的纵向强度，同时保持凝聚体密度峰值固定，这将减少凝聚体的空间范围.对于固定速度的孤子，振荡振幅的绝对值下降，但仍然与托马斯-费米半径形成一个近似的常数比值.图6（a ）显示了各种强度的简谐势阱中，暗孤子位置的变化，其位置已根据托马斯-费米半径重新标度，而时间单位也调整为ω-1z .对于较低的势阱频率，例如，ωz =2姨×10-2（μ/攸）（黑色实线），暗孤子的振荡频率即为其分析预测值ωz /2姨.而对于较高频率的势阱，例如，ωz =62姨×10-2（μ/攸）（点虚线），暗孤子的振荡频率比预测值大.图6（c ）给出了各种初始速度的孤子的振荡频率与势阱频率的函数.对于弱势阱有ωs /ωz ≈2姨，分析值与预测值吻合（虚线）.然而，增加势阱的强度，ωs /ωz 的比值偏离预测值，并随势阱强度的增加而单调增加.这种偏差是不可忽略的，在这里的势阱频率范围里，这个值可高达10％.造成此偏差的原因，是因为对孤子频率的分析预测值，假定了凝聚体为托马斯-费米密度分布.在我们的数值方法里，其密度峰值保持固定，这一假定仅对弱的简谐势阱有效（大量的粒子）.图6（d ）显示了整个凝聚体在各种势频率中的轴向密度分布.随着势阱频率的增加，密度越来越背离倒抛物线型的托马斯-费米密度分布.这偏差在凝聚体的边界处最为明显.甚至可以看到"尾巴"状的低密度伸展通过托马斯-费米半径.为了突出这种偏差，文章还在同一图中绘制了实际密度与托马斯-费米密度分布的差值.文章认为，这一偏差可以解释暗孤子的振荡频率与势阱频率的变化有关.当势阱强度增加时，暗孤子位置（图6（a ））和能量（图6（b ））的调制，由于暗孤子与偶极振荡相互作用的增大而增加.在这里，凝聚体的尺度减小，从而其有效质量降低，而孤子基本保持相同的大小.因此，振荡暗孤子诱发背景凝聚体相对更大的扰动，从而导致孤子的动力学调制更大.4在周期性扰动势阱中的暗孤子通常，人们假设凝聚体在静态简谐势阱中，势阱为V har （x ）=m ω2x 2/2，ω是势阱的频率.在这里，我（c ）孤子能量的演化，该结果经过了由最初的孤子能量E inits 的重整10002000300040005000t /（ξ·c -1）图5不同速度暗孤子在简谐势阱中位置与能量演化1.00.80.60.4E s /E s i n t10610410210098z S /v s /ξ（b ）暗孤子在势阱中的距离（用孤子速度重整后的结果）10002000300040005000t /（ξ·c -1）100500-50-100z s /ξ（a ）暗孤子在无限深简谐势阱（ωz =2姨×10-2（μ/攸））中位置的演化.其中初始速度v /c =0.1（实线），0.5（虚线），0.7（点线）和0.9（点虚线）10002000300040005000t /（ξ·c -1）江西理工大学学报2016年10月106们考虑整个势阱存在扰动，这种势阱可写为V Ext （x ，t ）=m ω2[x +h sin （ωd t ）]2（7）h 和ωd 分别是扰动的幅度和频率[29].我们用数值模拟对以上模型进行研究.图7显示了在各种振幅的扰动下，孤子能量的演变.显然，孤子的运动方向与扰动的运动方向的耦合决定了孤子的演化.在没有扰动的情况下（h =0），该模型将退化为孤子在简谐势阱中的振荡试验.在这种情况下，孤子在背景密度不均匀的凝聚体中传播，其外形变得不对称，同时它还会向相反方向辐射声波[18].众所周知，在简谐势阱中的孤子的振动频率为ωsol =ω/2姨，孤子会发射和重新接收声波.总体而言，孤子不断受到孤子自身带来的流体的扰动，但并不会衰退.当势阱的运动方向与孤子的运动方向相反时（h >0），孤子往往首先获得能量直至达到峰值，然后孤子能量减小到原来的值.这种能量的增益损失周期性的重复.能量变化的周期为1516ξ/c .因此在大量的时间里孤子能量比其初始值大.增加扰动幅度，可以提高孤子在其能量循环中获得和失去能量的能力.相反，当势阱与孤子的移动方向相同时（h <0），孤子往往首先失去能量直至最低值，然后它重新获得能量，恢复其原始值.这个过程构成一个损失获得循环，其周期为1516ξ/c .同样，增加扰动的幅度，孤子失去更多的能量，然后恢复至初始值.因此在大量的时间里孤子能量比其初始值小.实际上，图7比较了在受周期性扰动和不受周期性扰动的简谐势阱中，暗孤子的演变.图8显示了在各种振幅的扰动下，相应的暗孤子的位置的演变.一般来说，凝聚体会伴随势阱运动.由于势阱的振幅和振荡频率都非常小，势阱振荡导致孤子的轨道与没有受扰动的情况（h =0）发生偏离.孤子振荡周期出现波动.整体而言，孤子的振荡频率仍然围绕着ωsol =ω/2姨.因此，这一特征也确保了势阱移动的方向与孤子移动方向的耦合.当势阱与孤子有相同的运动方向时，凝聚体伴随势阱运动.因此，孤子被携带着运动，它偏离振荡中心更远.从而使孤子能量往往比原来的值小（见图8）.但是，当势阱运动与孤子运动方向相反时，凝聚体的运动相对地减小了孤子的震荡幅度，孤子能量出现增益-损失循环.一般来说，如果势阱扰动频率等于天然粒子的振荡频率，很可能引发共鸣.虽然很多研究显示孤子具有粒子状特性，但是，孤子毕竟不是正常的粒子，因此即便势阱振荡与孤子震荡能很好的匹配，也无法造成孤子的共振行为.图6暗孤子在不同频率简谐势阱中位置与能量演化0.90.80.70204060t /ω-1（b ）对应的孤子能量的进化E s /μ（d ）图（a ）和（b ）凝聚体的密度（左轴），以及其与托马斯-费米值的密度偏离（n -n TF ）1.00.750.500.250.000.5 1.0Z /R TFn /n 00.040.020.00-0.02n -n T F /n 00.500.250.00-0.25-0.50z s /R T F（a ）暗孤子位置与托马斯-费米半径R TF 的比值.其中势阱强度ωz =ω0=2姨×10-2（μ/攸）（实线），2ω0（虚线），4ω0（点线）和6ω0（点虚线）204060t /ω-1刘超飞，等：玻色爱因斯坦凝聚体中暗孤子动力学研究第３７卷第５期（c ）孤子振荡频率ωs 与势阱频率ωz 的函数关系，其中.虚线为分析值ωs =ωz /2姨0.800.750.700.10.2ωs /ωzωz /（c ·ξ-1）v /c =0.25v /c =0.50v /c =0.751075暗孤子动力学研究展望实验中，暗孤子的确能展示良好的粒子状特性.例如，暗孤子在简谐势阱中，通常会来回振荡[11,15].如果一个静态的暗孤子最初位置不在势阱中心，它将受到势阱的外力作用，使之加速向势阱中心运动.最近，Parker 和他的同事考虑了对简谐势阱做一些修正，充分展示了可能出现的暗孤子行为[18,19,30].将一个紧束缚的势阱嵌入一个弱束缚的简谐势阱中，这样就可控制声波的逃逸[18].将光晶格势阱加入简谐势阱中，就可用于干扰暗孤子[30].此外，还可以考虑了参数驱动以及阻尼机制[19].而参考文献[31]中，有限温度效应对暗孤子的影响受到了系统性的探讨.类似的，Bilas 和Pavloff 研究了准一维玻色爱因斯坦凝聚体中，随机势对运动暗孤子的影响[32]，还研究了暗孤子在传播途中遇到障碍的情况[33].除了上述单成分凝聚体中暗孤子的工作，随着对BECs 中孤子的深入研究，人们在多元凝聚体混合物中还发现了矢量孤子、如亮-暗矢量孤子[34-37]、亮-亮矢量孤子、暗-暗矢量孤子[38-40].和单分量凝聚体相比，玻色爱因斯坦凝聚体的二元混合物已经被显示具有迷人的宏观量子现象，如复杂的空间结构[41-43]、亚稳态[44-46]、对称破缺不稳定性[47-49].迄今为止，我们已经知道种间相互作用系数对凝聚体混合物的基态结构起决定作用.当不等式g 12≤g 1g 2姨满足时，两种凝聚体是易融合的；当g 12>g 1g 2姨时，由于很强的种间排斥相互作用，凝聚体为不可融合的.就考察孤子来说，多个分量这个自由度的引入给系统带来了丰富的非线性现象，比如：孤子链、孤子对、多模激发等.除此之外，一种新型的孤子，即共生孤子，在两分量87Rb 和85Rb 的凝聚体中被发现.此时，只要分量间原子吸引力足够强，便能够克服各自分量原子内的排斥力而起到一个有效吸引的作用，从而在两分量玻色―爱因斯坦凝聚中形成亮孤子.其实，早在1993年，Kivshar 等[50]通过求图8扰动中的暗孤子震荡（b ）图（a ）的放大图.长度单位为ξ=攸/m μ姨（a ）在各种扰动幅度下，孤子轨迹随时间的变化403020100-10-20-30-40200040006000X /qt /（ξ·c -1）403020100-10-20-30-40300600900h =3ξh =2ξh =1ξh =0h =-1ξh =-2ξh =-3ξt /（ξ·c -1）X /q1.281.241.201.151.121.083000600090001200015000t /（ξ·c -1）（a ）在各种扰动幅度下，孤子能量随时间的变化.孤子初始速度为0.3c 和初始位置为x =0E /μ 1.281.241.201.151.121.0850010001500t /（ξ·c -1）（b ）图（a ）的放大图.体系参数为ω=2姨/100（c /ξ），ωd =ω/2姨h =3ξh =2ξh =1ξh =0h =-1ξh =-2ξh =-3ξ图7孤子能量受扰动幅度的影响E /μ江西理工大学学报2016年10月108解两个耦合非线性薛定谔方程，显示了矢量暗孤子存在的可能性.近来，在耦合的一维非线性薛定谔方程的框架内，双组分凝聚体的矢量暗孤子得到了相应的研究[51].然而，这些研究基于稳定的媒质.无论是凝聚体的种类，还是凝聚体两成分的比率，都是固定的.最近，Rabi 耦合[52]被用于将凝聚体从某一成分向其他成分的凝聚体转化.这就暗示着暗孤子在动态凝聚体媒质中运行是可能的.对于理想情况，即凝聚体种间相互作用与种内相互作用强度相同时，我们可以看到由一种成分构成的暗孤子可以转化为另一成分的暗孤子[53]（如图9所示）.并且，暗孤子转变为动态的矢量暗孤子后，其运动轨迹不受Rabi 耦合强度的影响.而种间相互作用与种内相互作用强度不相同时，矢量暗孤子的运动轨迹受到Rabi 耦合的影响比较明显.但是，这一长时间模拟所说明的最主要的结论为：暗孤子可以在具有Rabi 耦合的凝聚体中存在.这一结果在特定凝聚体比率的矢量暗孤子的设计上，具有非常重要的意义.将来，还可以通过控制Rabi 耦合，如在特定时间终结Rabi 耦合，从而得到特定比率的凝聚体混合物，以及矢量暗孤子.当然，矢量暗孤子稳定存在的内在机制等还有待进一步的探索.相信该研究成果将给实验上认识凝聚体中暗孤子等激发行为提供理论支持.6结论文章根据详细的能量计算分析，对玻色爱因斯坦凝聚体中暗孤子动力学行为的数值研究进行了系统的介绍.从分析暗孤子的数值解、暗孤子的能量计算开始，重点探讨了暗孤子在简谐势阱中的动力学行为，以及简谐势阱出现扰动时的情况.玻色爱因斯坦凝聚体中的暗孤子具有类似粒子的运动行为，在非均匀密度的凝聚体中，暗孤子发生声波图9玻色爱因斯坦凝聚体二元混合物中的矢量暗孤子的震荡行为.Rabi 耦合强度为0.025，g 1=g 2=g 12=1（a ）凝聚体成分1的演化和暗孤子震荡行为100500-50-1006001200180024003000t /（ξ·c -1）X0.13750.27500.41250.55000.58750.82500.96251.100ψ12（b ）凝聚体成分2的演化与暗孤子的震荡行为6001200180024003000X0.13750.27500.41250.55000.58750.82500.96251.100ψ22100500-50-100（c ）两凝聚体的密度和6001200180024003000X0.13750.27500.41250.55000.58750.82500.96251.100ψ12+ψ22100500-50-100t /（ξ·c -1）t /（ξ·c -1）刘超飞，等：玻色爱因斯坦凝聚体中暗孤子动力学研究第３７卷第５期109。

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>。

Bose-Einstein condensation in dense quark matter

arXiv:0809.2892v1 [hep-ph] 17 Sep 2008
Bose-Einstein condensation in dense quark matter
Jens O. Andersen
Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
Preprint submitted to Elsevier 17 September 2008
In this talk, I would like to present some results for Bose condensation of charged pions using the linear sigma model as a low-energy eﬀective for two-ﬂavor QCD. The talk is based on Refs. [8,9] and part of the work is done in collaboration with Tomas Brauner. 2. Interacting Bose gas and 2PI 1/N -expansion The Euclidean Lagrangian for a Bose gas with N species of massive charged scalars is
2 2 2 ωn + m2 4 + ΠLO . The term m4 + ΠLO is thus a local mass term, which we denote by M . The exact propagator D satisﬁes the equation δ Γ/δD = 0. In the present case, it reduces to a local gap equation for M 2 :

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).。

Vortex patterns in a fast rotating Bose-Einstein condensate

arXiv:cond-mat/0410665v1 [cond-mat.other] 26 Oct 2004
Vortex patterns in a fast rotating Bose-Einstein condensate
Amandine Aftalion, Xavier Blanc Laboratoire Jacques-Louis Lions, Universit´e Paris 6,
Vortices are universal features which appear in many macroscopic quantum systems, such as superconductors or superﬂuid liquid helium. Recently, detailed investigations have been performed on rotating atomic gaseous Bose-Einstein condensates. These condensates are usually conﬁned in a harmonic potential, with cylindrical symmetry around the rotation axis z. Two limiting regimes occur depending on the ratio of the rotation frequency Ω and the trap frequency ω in the xy plane. When Ω is notably smaller than ω, only one or a few vortices are present at equilibrium [3, 4]. When Ω approaches ω, since the centrifugal force nearly balances the trapping force, the radius of the rotating gas increases and tends to inﬁnity, and the number of vortices in the condensate diverges [5, 6, 7, 8].

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.

Comparison of mean-field theories for vortices in trapped Bose-Einstein condensates

a r X i v :c o n d -m a t /0108462v 1 [c o n d -m a t .s o f t ] 28 A u g 2001Comparison of mean-ﬁeld theories for vortices in trapped Bose-Einstein condensatesS M M Virtanen,T P Simula and M M SalomaaMaterials Physics Laboratory,Helsinki University of Technology P.O.Box 2200(Technical Physics),FIN-02015HUT,Finland(February 1,2008)We compute structures of vortex conﬁgurations in a har-monically trapped Bose-Einstein condensed atom gas within three diﬀerent gapless self-consistent mean-ﬁeld theories.Outside the vortex core region,the density proﬁles for the condensate and the thermal gas are found to diﬀer only by a few percent between the Hartree-Fock-Bogoliubov-Popov the-ory and two of its recently proposed gapless extensions.In the core region,however,the diﬀerences in the density pro-ﬁles are substantial.The structural diﬀerences are reﬂected in the energies of the quasiparticle states localized near the vortex core.Especially,the predictions for the energy of the lowest quasiparticle excitation diﬀer considerably between the theoretical models investigated.PACS number(s):03.75.Fi,05.30.Jp,67.40.DbThe landmark experiments to realize Bose-Einstein condensation in dilute atomic gases [1]have sparked vig-orous investigation on the physical properties of these novel quantum ﬂuids.Due to the weak interactions,such systems are rare examples of interacting quantum ﬂu-ids amenable to quantitative microscopic analysis,and thus provide unique possibilities to test the fundamental principles and theories of many-body quantum physics.Theoretical approaches yield several quantities,such as density proﬁles for the condensate and the thermal gas component,stability estimates,speciﬁc heats,and prop-erties of various propagating sound modes,to be com-pared with experiments.Experiments also yield detailed information on the energies of the individual excitation modes of these systems [2].Such information provides the most direct and stringent tests for the accuracy of dif-ferent theoretical approaches,as compared to the above-mentioned “collective”quantities which depend on the excitation spectrum as a whole.The Bogoliubov equations [3]are a widely used start-ing point to compute the excitation spectra for dilute Bose-Einstein condensates (BECs).They can be seen as eigenmode equations for the condensate described by the Gross-Pitaevskii equation [4],neglecting eﬀects of the thermal,noncondensed gas component in the sys-tem.The Hartree-Fock-Bogoliubov (HFB)theory [5]takes self-consistently into account the condensate and the thermal gas densities,as well as the lowest order anomalous average of the boson ﬁeld.However,it is plagued by an unphysical gap in the excitation spec-trum,which violates Goldstone’s theorem and invalidatesits value in predicting the lowest collective mode excita-tion frequencies.Goldstone’s theorem can be restored by neglecting the anomalous average mean ﬁeld in the HFB formalism.This yields the gapless Popov version of the HFB theory [6].At low temperatures,predic-tions of the Popov approximation (PA)for the lowest excitation frequencies of irrotational condensates are in good agreement with experimental results,but at tem-peratures T >∼T bec /2(T bec denotes the critical temper-ature of condensation)the deviations become apparent [7].The main inadequacies of the PA are that it neglects the eﬀects of the background gas on atomic collisions and the dynamics of the thermal gas component.As an improvement to overcome the ﬁrst limitation within a computationally manageable formalism,the so-called G1and G2approximations have been suggested [8,9].They are gapless mean-ﬁeld theories which take into account eﬀects of the medium on atomic collisions by allowing the interaction couplings to depend on the correlation mean ﬁelds in a self-consistent manner.The two versions are based on diﬀerent approximations for the momentum de-pendence of the full many-body T -matrix in the homo-geneous gas limit,and their precision for inhomogeneous systems remains to be investigated.To assess the accuracy of the above-mentioned gapless HFB-type approximations,their predictions for the exci-tation frequencies of irrotational,harmonically trapped atomic BECs have been computed and compared with experiments [7,8,10–12].For temperatures T <∼T bec /2,the predictions of the PA,G1and G2for the lowest ex-citation frequencies diﬀer only a few percent [8,9].For higher temperatures,the diﬀerences are larger and ex-ceed the experimental uncertainty estimates for measure-ments,but none of the theories agrees satisfyingly with experiments [8].However,in this temperature range the dynamics of the thermal gas component,which these ap-proximations do not take into account,is expected to have an increasingly important inﬂuence on the excita-tion eigenmodes.Consequently,results for irrotational condensates remain somewhat inconclusive in determin-ing the validity of these theories.Recently,vortex states in dilute atomic BECs have been experimentally realized [13].Furthermore,by ob-serving the precession of vortices,the energy of the low-est excitation,the so-called lowest core localized state (LCLS),has been measured [14].Interestingly enough,the experimental results for this energy agree well with the Bogoliubov approximation [15],while they deﬁnitely1disagree with the picture given by the self-consistent mean-ﬁeld theories:the latter predict the energy of the precession mode to be positive w.r.t.the condensate state [16,17],but experiments imply negative energies.We suggest that this puzzling fact could be due to incomplete thermalization of the(moving)vortex and/or the limita-tions of the quasi-stationary,i.e.,adiabatic HFB formal-ism in describing time-dependent phenomena.The adi-abatic approximation essentially fails if the kinetic rates of the system exceed frequency separations of the tely,we have shown that the requirement of adiabaticity leads to a criterion for the velocity of the moving vortex,which is violated in the precession obser-vations so far[18].However,if the precession radii—and thus the velocities—of the vortices could be reduced,or the physical parameter values appropriately adjusted in order for the system to better fulﬁll the criteria for adia-baticity and thermalization,one should be able to mean-ingfully compare experimental data with the predictions of the self-consistent equilibrium theories for the vortex states.In this paper,we present results of computations for the structures and excitation frequencies of vortex states within the G1and the G2approximations,and compare them with the previously computed predictions of the Popov approximation[16,17].Outside the vortex core region,the density proﬁles for the condensate and the thermal gas component are found to diﬀer by only a few percent between the PA,G1and G2.However,in the core region the diﬀerences are considerably larger.This is reﬂected in substantial diﬀerences in the energy of the LCLS,which is localized in the core region.The gapless HFB-type theories considered in this paper can be expressed in the form of the generalized Gross-Pitaevskii(GP)equation[5,9][H0(r)+U c(r)|φ(r)|2+2U e(r)ρ(r)]φ(r)=µφ(r)(1)for the condensate wavefunctionφ(r),and the eigenvalue equationsL(r)u q(r)+U c(r)φ2(r)v q(r)=E q u q(r),(2a) L(r)v q(r)+U c(r)φ∗2(r)u q(r)=−E q v q(r)(2b)for the quasiparticle amplitudes u q(r),v q(r),and eigenenergies E q.Above,H0(r)=−¯h2∇2/2m+V trap(r) is the bare single-particle Hamiltonian for atoms of mass m conﬁned by a harmonic trapping potential V trap(r)= 1in reference[11].This method allows one to use rather small values for E c with excellent accuracy,thus essen-tially improving the computational eﬃciency.In order to stabilize the iteration,we use underrelaxation in updat-ing the mean-ﬁeld potentials.To facilitate comparison with previously presented re-sults for the Popov approximation,the physical parame-ter values for the gas and the trap were chosen to be the same as those in reference[16].We modelled a sodium gas with the atomic mass m=3.81×10−26kg and the scattering length a=2.75nm in a trap with the radial frequencyνr=ωr/2π=200Hz.The density of the gas was determined by treating N=2×105atoms per length L=10µm in the axial direction.Altogether,these val-ues yield the condensation temperature T bec≈0.8µK. Figures1and2present results of our computations for axisymmetric single-quantum vortex states.The density proﬁles for the condensate,the noncondensate and the anomalous average are displayed inﬁgure1at temper-atures T=50nK and400nK.Outside the vortex core region,the diﬀerences in the density proﬁles between the PA,G1and G2are at most a few percent at temperatures T<∼T bec/2.In the core region,however,the diﬀerences are considerably larger.The total density of the gas is approximately20%larger on the vortex axis in the G2 than within the PA.This squeezing behavior is associated to the“softening”of the repulsive eﬀective interaction in the core region due to many-body eﬀects[9].The diﬀerences in the core densities between the var-ious approximations also suggest diﬀerences in the ener-gies of the quasiparticle excitations localized in the core region.Figure2displays the energies of three such states, the lowest excitations with angular momentum quantum numbers qθ=−1,0,and1,as functions of tempera-ture.For the G1and the G2,the increased core densities are compensated by smaller eﬀective couplings,and the shifts in the excitation energies are generally only a few percent for temperatures T<∼T bec/2;at higher tempera-tures the softening eﬀect of the interaction becomes more pronounced[9,12],also increasing the shifts in the exci-tations.However,the lowest Kelvin mode(consisting of the lowest qθ=−1excitations)state,the LCLS,is espe-cially sensitive to the structure of the core region.The diﬀerences in the energies of the LCLS between the ap-proximations are25–40%even at temperatures for which the predictions of the PA for the excitation frequencies of irrotational condensates diﬀer by less than5%from the experimental data.In addition,the temperature de-pendence of the LCLS is found to be much stronger than for the other states.In fact,the lowest excitation en-ergy vanishes in the zero-temperature limit for all the approximations[17];the remainder of the spectrum is essentially temperature-independent,except in the vicin-ity of T bec.The state displayed inﬁgure2with qθ=1 is the Kohn mode,which should have the exact energy E=¯hωr according to Kohn’s theorem for parabolic con-ﬁnement[21].Kohn’s theorem is satisﬁed to an accuracy of1–6%for all the approximations,suggesting that dy-namical eﬀects of the thermal gas component are small in the temperature range studied.In conclusion,we argue that future measurements of the lowest excitation frequencies of the vortex states could provide stringent tests for the validity of the mean-ﬁeld theories considered.Especially,they could be used to estimate the degree to which the approximations for the many-body T-matrices based on the homogeneous limit remain valid for highly inhomogeneous systems. We thank the Center for Scientiﬁc Computing for com-puter resources,and the Academy of Finland and the Graduate School in Technical Physics for support.[14]Anderson B P,Haljan P C,Wieman C E and Cornell EA2000Phys.Rev.Lett.852857[15]Svidzinsky A A and Fetter A L2000Phys.Rev.Lett.845919[16]Isoshima T and Machida K1999Phys.Rev.A592203[17]Virtanen S M M,Simula T P and Salomaa M M2001Phys.Rev.Lett.862704[18]Virtanen S M M,Simula T P and Salomaa M M2001Preprint cond-mat/0105398[19]Morgan S1999PhD Thesis(Oxford University)[20]Svidzinsky A A and Fetter A L2000Phys.Rev.A62063617Feder D L,Svidzinsky A A,Fetter A L and Clark C W2001Phys.Rev.Lett.86564Fetter A L and SvidzinskyA A2001J.Phys.:Condens.Matter13R135[21]Dobson J1994Phys.Rev.Lett.732244FIG.1.Density proﬁles of the vortex state for the conden-sate(|φ|2),thermal gas component(ρ)and anomalous average (|∆|)in the PA(solid),G1(dashed)and G2(dashed-dotted)at temperatures(a)T=400nK and(b)T=50nK.Axes for the values ofρand|∆|are on the left-hand sides,and for|φ|2on the right-hand sides.Figures(c)and(d)display the vortex core region,where the diﬀerences in the density proﬁles between the approximations are substantially larger than farther from the vortex axis.FIG.2.(a)Energies of the lowest excitation modes with angular momentum quantum numbers qθ=−1,0,and1in the PA(solid),G1(dashed)and G2(dashed-dotted)as func-tions of temperature.The qθ=−1state is the so-called lowest core localized state(LCLS),and the qθ=1excitationis the Kohn mode.(b)Temperature dependence of the energyof the LCLS within the PA,G1and G2.Note the substantial relative diﬀerences between the theories in this energy.4。

Vortex nucleation by collapsing bubbles in Bose-Einstein condensates

a r X i v :c o n d -m a t /0401021v 1 [c o n d -m a t .s o f t ] 3 J a n 2004Vortex nucleation by collapsing bubbles in Bose-Einstein condensatesNatalia G.Berloﬀ1and Carlo F.Barenghi 21Department of Applied Mathematics and Theoretical Physics,University of Cambridge,Wilberforce Road,Cambridge,CB30WA 2School of Mathematics and Statistics,University of Newcastle,Newcastle upon Tyne NE17RU(Dated:January 3,2004)The nucleation of vortex rings accompanies the collapse of ultrasound bubbles in superﬂing the Gross-Pitaevskii equation for a uniform condensate we elucidate the various stages of the collapse of a stationary spherically symmetric bubble and establish conditions necessary for vortex nucleation.The minimum radius of the stationary bubble,whose collapse leads to vortex nucleation,was found to be 28±1healing lengths.The time after which the nucleation becomes possible is determined as a function of bubble’s radius.We show that vortex nucleation takes place in moving bubbles of even smaller radius if the motion made them suﬃciently oblate.PACS numbers:03.75.Lm,05.45.-a,67.40.Vs,67.57.DeIn this Letter we establish a new mechanism of vor-tex nucleation in a uniform condensate.Previously,the nucleation of vortices in a uniform condensate has been connected to critical velocities [1,2,3],instabilities of the initial states [4]or to a transfer of energy among the solitary waves [5].Moving positive [2]and negative [3]ions were shown to generate vortex rings on their surface where the speed of sound was exceeded.Experiments in superﬂuid helium have demonstrated long time ago the production of quantised vortices and turbulence [6]by the collapse of cavitated bubbles [7]generated by ul-trasound in the megahertz frequency range.The aim of this Letter is to analyse theoretically for the ﬁrst time the physics of this process in the context of the Gross-Pitaevskii (GP)equation.Vortex nucleation by collaps-ing bubbles could also be studied in the context of (non-uniform)atomic condensates (BEC),for which the GP equation provides a quantitative model,thus providing experimentalists with a new mechanism to produce vor-tices in BEC systems,alongside rotation [8],the decay of solitons [9]and phase imprinting [10].Moreover,our work illustrates a new aspect of vortex-sound interaction in a Bose-Einstein condensate,a topic which is receiving increasing attention [11].We write the GP equation in dimensionless form as−2i∂ψ2,and the density at inﬁnity is ρ∞=|ψ∞|2=1.To convert the dimensionless units into values applicable to superﬂuid helium-4,we take the number density as ρ=2.18×1028m −3,the quantum of circulation as κ=h/m =9.92×10−8m 2s −1,and the healing length as ξ=0.128nm.This gives a time unit 2πξ2/κ∼1ps.Whereas for a sodium condensate with ξ≈0.14µm,the time unit isabout 8ns.V (x ,t )is the potential of interaction between a boson and a bubble.We will assume that the bubble acts as an inﬁnite potential barrier to the condensate,so that no bosons can be found inside the bubble (ψ=0)before the collapse.This is achieved by setting V to be large inside the bubble and zero outside.First we consider the case of a stationary spherically symmetrical bubble.The spherical symmetry allows us to reduce the problem to dimension one,so that the equa-tion (1)for ψ=ψ(r,t )becomes−2i ψt =ψ′′+2ψ′/r +(1−|ψ|2)ψ,(2)where r 2=x 2+y 2+z 2.Equation (2)is numerically integrated using fourth order ﬁnite diﬀerences discretiza-tion in space and fourth order Runge-Kutta method in time.Before the collapse the ﬁeld around the bubble of radius a is stationary,ψt =0.The boundary con-ditions are ψ(a,t )=0stating that the bubble surface is an inﬁnite potential barrier to the condensate and ψ(∞,t )=1.The stationary solutions for various a were found by the Newton-Raphson iterations.The solutions are ψ(r )=(0,0)if r ≤a and ψ(r )=(R a (r ),0)if r >a ,with the graphs of R a (r +a )for a =1,2,10,30given on Figure 1.If the radius of the bubble,a ,is suﬃ-ciently large,then we can set r =a +ξand to the leading order get R ′′(ξ)+[1−R (ξ)2]R (ξ)=0which has the solution,satisfying the boundary conditions,R (ξ)=tanh(ξ/√2|∇ψ|2dV +13+2π∞a[R ′a (r )2+12FIG.1:(colour online)The plot of the amplitude of the so-lution around the stationary bubbles of radii a =1(red),2(green),10(blue)and 30(black)(the smaller a corresponds to a steeper amplitude).The loglog plot of the energies of the solutions with various a are shown on the inset together with the linear ﬁt.012340.20.40.60.8101234502.557.51012.5Log (a )Log (E )rR a (r +a )From the energy conservation it is clear that after the bubble collapses and the condensate ﬁlls the cavity the necessary (but not suﬃcient)condition for vortex nucle-ation is that the energy has to be greater than that of one vortex ring.The minimal energy of the vortex solution was found in [12]to be about E ∼55±1which corre-sponds to the minimum radius of a =2.2with E =55.7.As the condensate ﬁlls the cavity,most of the energy will be emitted via the sound waves,so the energy of the bubble has to be suﬃciently greater than the energy of a single vortex ring to allow for such an emission.The time-dependent evolution of the condensate after the bubble collapses involves several stages.The overall picture is complicated by a complex interplay between dispersive and nonlinear eﬀects.Dispersive eﬀects be-come important on the wavelengths of order of the heal-ing length with the group velocity approximately given by ∂(3FIG.3:(colour online)The plots of the density per unit vol-ume as function of time for various radii of spheres over which the averaging is performed.The initial radius of the cav-ity in this case is a =128.The radii of averaging spheres are b =4,8,16,32,64.The average density is calculated as ¯ρb =3 b 0ρ(r )r 2dr/b 3.10020030040001234t¯ρ4¯ρ8¯ρ16¯ρ32¯ρ64surfaces to intersect.In this case the instability mech-anism would be somewhat diﬀerent taking longer timeto develop.The radial ’dips’of the density of the ex-panding condensate,noticeable in Figure 2,are unsta-ble to non-spherically symmetrical perturbations,simi-lar to theinstability of the Kadomtsev-Petviashvili 2D solitons in 3D [4].Depending on the energy carried by these ’dips,’they evolve into either vortex solutions or sound waves.From these considerations we expect three possible outcomes after bubble collapses:(1)if the ra-dius of the bubble is smaller then some critical radius a ∗,the density ’dips’generated by the expanding con-densate have rather small amplitude that decreases even further as they travel away from the centre quickly be-coming sound waves before the instability has time to develop;(2)after the collapse of a bubble of an inter-mediate size,say,of the radius a ∗<a <ˆa ,the waves of suﬃciently large amplitudes are generated and the in-stability of these waves develops in time inversely pro-portional to the radius a ;(3)if the radius is suﬃciently large,a >ˆa ,the time of the ﬁrst vortex nucleation is approximately given by the moment of the start of the outward ﬂux of the particles t ∗approximated above.As a condensate continues to expand the instability mecha-nism described in (2)is further facilitated by the broken symmetry resulting from the previous nucleation events which leads to even more vortex rings being nucleated.To conﬁrm the scenario outlined above we performed full three-dimensional calculations for cavities of vari-ous radii in a computational box of the side 200healing lengths [13].We determined that there is a critical radius of the bubble for which vortex ring nucleate a ∗∼28±1.The borderline radius between regime (2)and (3)was found as ˆa ∼45.Figure 4shows the density isoplotsFIG.4:(colour online)Time snapshots of the density isoplotsρ=0.1after the collapse of the bubble of radius a =50.The side of the computational box is 200healing lengths and the distance between ticks on the side corresponds to 20healing lengths.The vortex rings nucleate at about t ∼40(compared with t ∗∼38).at the various time snapshots after the bubble of radiusa =50collapsed.Notice,that the non-symmetry of the ﬁeld that was created at the time of nucleation (t ≈40)continues to produce even more vortex rings as conden-sate expands.Each vortex ring at the moment of its birth has zero radius,as the surfaces of zero real and imagi-nary parts touch each other,and gradually evolve into a vortex ring of increasingly larger and radius,as clearly seen on Figure 4.The process in which solitary waves evolve into states of a higher energy was elucidated in [5].A ﬁnite amplitude sound wave that moves behind a vortex ring transfers its energy to it,allowing the vor-tex ring to grow in size.The radius of the vortex ring stabilises only when it travelled suﬃciently far from the center of the collapsing bubble,where the ﬂow became almost uniform.The larger the radius of the bubble,the more ﬁnite amplitude sound waves will be generated at shorter distances,the larger the size of the ﬁnal ring is going to be.So far we considered the collapse of the stationary bub-ble,where the vortex nucleation is connected to the in-stabilities developed in the spherically symmetric ﬂow.There are situations when the nucleation is facilitated by an initial lack of the symmetry in the ﬂow as in the case of a moving bubble or a bubble in the nonuniform (trapped)condensate.The surrounding helium exerts a net inward pressure across the surface,which is balanced4 by the pressure inside the bubble.In was shown in[3]by asymptotic analysis of the GP equation coupled withthe equation of the motion for the wavefunction of anelectron that a moving bubble becomes oblate in the di-rection of its motion.Thisﬂattening is created by thediﬀerence in pressure between the poles and equator asso-ciated with the greater condensate velocity at the latterthan at the former.How oblate the bubble becomes dur-ing its motion will depend on the velocity and pressureinside the bubble.The non-uniformity of theﬂow in thecollapsing oblate bubble leads to vortex ring nucleationfor bubble sizes much smaller than in the case of a sta-tionary spherically symmetric bubble.Figure5showssnapshots of the density plots of the cross-section of thecollapsing bubble that prior to t=0was moving with aconstant velocity U=0.2and acquired an oblate formgiven by x2+1。

Phase separation of Bose-Einstein condensates

a rXiv:c ond-ma t/979301v221Oct1997Phase Separation of Bose-Einstein Condensates E.Timmermans Institute for Atomic and Molecular Physics Harvard-Smithsonian Center for Astrophysics 60Garden Street Cambridge,MA 02138(February 1,2008)Abstract The zero-temperature system of two dilute overlapping Bose-Einstein con-densates is unstable against long wavelength excitations if the interaction strength between the distinguishable bosons exceeds the geometric mean of the like-boson interaction strengths.If the condensates attract each other,the instability is similar to the instability of the negative scattering length con-densates.If the condensates repel,they separate spatially into condensates of equal pressure.We estimate the boundary size,surface tension and energy of the phase separated condensate system and we discuss the implications fordouble condensates in atomic traps.PACS numbers(s):03.75.Fi,05.30.Jp,32.80Pj,67.90.+zTypeset using REVT E XAs dilute gases,the atomic trap Bose-Einstein condensates[1]occupy a unique position among the superﬂuid systems.One intriguing consequence of their dilute gas nature is the prospect of studying condensate mixtures,which are theﬁrst experimentally realizable bosonic superﬂuid mixtures[2]-[6](the only other mixture of superﬂuids is the fermion-boson3He–4He system).Understandably,this prospect has attracted interest[7]-[10],and recently theﬁrst observation of overlapping condensates was reported[8].In this paper,we study if and when zero temperature dilute condensates overlap.Weﬁnd that the homogeneous overlapping condensate system is unstable against long wavelength excitations if the strength of the interaction between the distinguishable bosons exceeds the geometric mean of the like-boson interaction strengths.In that case,two repelling condensates spatially separate into single condensates of equal pressure.The condensates still partially overlap in the boundary region that separates them.We estimate the size of the boundary region,as well as the corresponding surface tension.We show how the homogenous treatment may be generalized to describe phase-separated large double condensate systems in a trap.The instability of the overlapping condensates manifests itself in the energy dispersion of the elementary excitations[10],as we show below.The wavefunctions,φ1andφ2,of two interacting condensates satisfy coupled Gross-Pitaevski equations:i¯h˙φ1= −¯h2∇2φ2+λ|φ1|2φ2,(1)2m2−µ2+λ2|φ2|2whereµj(j=1,2)represents the chemical potential of the j-bosons.The interaction strength values,λj andλ,are determined by the scattering lengths for binary collisions of distinguishable bosons:λj=4π¯h2a j/m j andλ=2π¯h2a/m red,where m−1red=m−11+m−12. The excitations of the static homogeneous condensates,φj(r,t)=φ(0)j,are described by ﬂuctuations of theﬁelds,φj(r,t)=φ(0)j+δφj,which evolve according to the Gross-Pitaevski equations(1),linearized inδφandδφ∗[11].Decomposing theﬁeldﬂuctuations into Fourier components,δφj= k c j,k exp(i k·r),we obtain the equations of motion for the c-amplitudes,i ¯h ˙c 1,k = k 2/2m 1+λ1n 1 c 1,k +λ1φ(0)1c ∗1,−k+λφ(0)1 φ(0)∗2c 2,k +φ(0)2c ∗2,−k ,(2)where n j =|φ(0)j |2and where we have used that ˙φ0j=0.A second equation for i ¯h ˙c 2is obtained by exchanging the 1and 2subscripts.Alternatively,we can introduce the phaseand density of the condensate ﬁeld,φ=√2 ¯h 2k 2/2m 1+2λ1n 1 δρ1,k −λn 1δρ2,k .(3)Thus,the phase ﬂuctuations of one condensate couple to the density ﬂuctuations of the other.We cancel out the dependence on the phase ﬂuctuations by taking the derivative ofthe ﬁrst equation in Eqs.(3)with respect to time and by substituting δ˙Πk from the second equation.With δρk (t )=δρk cos(Ωk t ),we ﬁnd the normal mode equations for the coupled density ﬂuctuations,−Ω2k δρ1,k =−ω21,k δρ1,k −λn 1k 2m 2δρ1,k ,(4)where ¯h ωj,k =2± 2.(5)where c j is the sound velocity of the j-condensate,c j=[c21−c22]2+4(λ2/λ1λ2)c21c222n21(r)+λ2When v j(r)=0,the Thomas-Fermi condensate densities of Eq.(8)are homogeneous.How-ever,equatingﬁrst-order derivatives to zero,only gives a minimum provided the second-order derivatives satisfy(∂2F/∂n2j)>0and(∂2F/∂n21)(∂2F/∂n22)−(∂2F/∂n1∂n2)2>0.The lat-ter condition implies that the Thomas-Fermi equations(8)only gives a minimum provided the stability criterion,λ2<λ1λ2,is satisﬁed.To see that‘strongly’repulsive condensates,λ>√2 n1(r)+n2(r)λ1λ2 n1(r)n2(r)−µ1n1(r)−µ2n2(r).(9) Starting from the homogeneous overlapping condensate system,redistributing bosons1and 2spatially while keeping[n1(r)+n2(r)the equilibrium condition of equal pressures exerted by both condensates.With the pressure P j=λj n2j,s/2,where n j,s denotes the density within the separated condensates,we are lead to the equivalent condition for the condensate densities n1,s=n2,sλ1/λ2(m1/m2)],where we used that(n2,s/n1,s)=λ1λ2)n1,s n2,s b/6.To estimate the actual boundary sizeb=2l1√[1+(m1/m2)λ/√4m1n1,sλ1.The boundary con-tribution to the energy is a surface energy E b( [1+(m1/m2) λ1λ2−1]/√from a single condensate of type2and replacing condensate2bosons in the droplet volume V1by condensate1bosons.The energy∆E required in the replacement is equal to∆E= [λ1,s n21/2−λ2n22,s/2]V1+E b.Minimizing the‘replacement energy’∆E with respect to V1 and realizing that E b∝V2/31,weﬁndλ1n21,s2+2V1.(11)The previous result,λ1n21,s/2=λ2n22,s/2,obtained by ignoring the boundary energy,is ac-curate provided the size of the droplet exceeds R s=2σ/P1=2l1Σ1.The energy per droplet particle,∆E/N1,with Eq.(11)is equal to∆E/N1=5E b/V1=5σ/[Rn1,s],a function that decreases monotonically as N1increases.Consequently,splitting up the droplet into smaller droplets further increases the free energy and it is energetically favorable for condensate1 to gather in a single region of space(i.e.real space condensation).To describe separated double condensates in traps,we subtract the overlap term,λn1n2, and include the boundary surface energy in the expression of the free energy(Eq.(7)).The validity of this description rests on two conditions: 1.the local coherence length within each condensate is much less than the length scale on which the condensates vary spatially and2.the change of the potential energy across the inter-condensate boundary,|f j|b<<λj n j,s(j=1,2).If these conditions are satisﬁed,the physics of the phase separation is similar to the above v j(r)=0-case,and we can answer interesting questions regarding trapped phase separated condensates.For instance:if we add a droplet of condensate1to a trapped condensate2,does it‘sink’to the middle of the trap,or does it remain‘ﬂoating’on the surface of condensate2?For the sake of simplicity, we assume that the size of the droplet is large enough to neglect the boundary surface energy and small enough to neglect the spatial variation of the density inside the droplet.Then, the previously deﬁned‘replacement energy’∆E depends on the center of mass position R of the droplet through the external potentials,v1(r)=v(r),and v2(r)=αv(r).Since the pressures inside and outside the droplet are equal,λ1n21,2/2−λ2n22,s/2≈0,weﬁnd that∆E(R)= V1[n1(r)v1(r)−n2(r)v2(r)]d3r≈N1v(R)[1−αλ1/λ2>1,the force on the droplet,−N1[1−ααλ2/λ1we thenﬁnd that αbetween the separated condensates is a region of partial overlap.We have estimated the size of this region,as well as the resulting surface tension.Finally,we brieﬂy discussed phase separation of large condensates in atomic traps.The phase separation suggests many experimental applications.For instance,one can use a condensate to spatially conﬁne a droplet of a diﬀerent condensate in the middle of a ing light that is resonant with the droplet atoms,the droplet can be displaced and its subsequent motion inside the conﬁning condensate can be observed.If the motion is undamped,we have a direct observation of superﬂuidity.To further motivate such ex-periments,we mention that damping generally could occur at velocities less than the sound velocity of the conﬁning condensate because of the creation of superﬂuid vortices etc...Fur-thermore,it is interesting to note that schemes have been proposed to continuously alter the interaction strengths,either by using light[13],or by varying the biasﬁeld of the magnetic traps[14],so that it might be possible in the future to observe phase separation in real time as the interaction strengths are altered.The author gratefully acknowledges fruitful interactions with Dr.P.Tommasini,Prof.E.Heller,Prof. A.Dalgarno and Prof.K.Huang.The work of the author is supported by the NSF through a grant for the Institute for Atomic and Molecular Physics at Harvard University and Smithsonian Astrophysical Observatory.REFERENCES[1]K.B.Davis et al.,Phys.Rev.Lett.75,3969(1995);M.H.Anderson et al.,Science,269,198(1995);C.C.Bradley et al.,Phys.Rev.Lett.75,1687(1995).[2]Previously,the only candidate for creating interpenetrating bosonic superﬂuids seemedto be a mixture of4He and6He.Unfortunately,the short lifetime of the6He-isotope (∼0.8seconds)did not allow cooling to suﬃciently low temperatures.Nevertheless,the prospect motivated theoretical work that is relevant to our paper[3],[4],[5]and[6].[3]I.M.Khalatnikov,Zh.Eksp.Teor.Fiz.32,653(1957)[Sov.Phys.–JETP5,542(1957)].[4]W.H.Bassichis,Phys.Rev.A134,543(1964).[5]Y.A.Nepomnnyashchii,Zh.Eksp.Teor.Fiz.70,1070(1976)[Sov.Phys.-JETP43,559(1976)];Teor.Mat.Fiz.20,399(1974).[6]W.B.Colson and Alexander L.Fetter,J.Low Temp.Phys.33,231(1978).[7]The overlapping double atomic condensate systems wereﬁrst discussed by T.-L.Ho andV.B.Shenoy,Phys.Rev.Lett.77,3276(1996).[8]C.J.Myatt et al.,Phys.Rev.Lett.78,586(1997).[9]B.D.Esry et al.,Phys.Rev.Lett.78,3594(1997).[10]The dispersion for overlapping condensate systems wasﬁrst obtained for boson particlesof the same mass by R.Graham and D.Walls,cond-mat/9611111.[11]The same equations can be obtained from the Heisenberg equations of motion,neglectingthe interactions between excited particles.[12]E.Timmermans,P.Tommasini and K.Huang,Phys.Rev.A,55,3645(1997).[13]P.O.Fedichev,Yu.Kagan,G.V.Shlyapnikov,and J.T.M.Walraven,Phys.Rev.Lett.77,2913(1996).[14]E.Tiesinga,B.J.Verhaar,and H.T.Stoof,Phys.Rev.A,47,4114(1993).11Figure Captions Fig.1。

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a r X iv :c o n d -m a t /0509559v 1 [c o n d -m a t .m e s -h a l l ] 21 S e p 2005Vortex lattices in Bose-Einstein condensates:from the Thomas-Fermi to the lowest Landau level regimeM.Cozzini,S.Stringari,and C.TozzoDipartimento di Fisica,Universit`a di Trento and BEC-INFM,I-38050Povo,Italy(Dated:February 2,2008)We consider a periodic vortex lattice in a rotating Bose-Einstein condensed gas,where the cen-trifugal potential is exactly compensated by the external harmonic trap.By introducing a gauge transformation which makes the Hamiltonian periodic,we solve numerically the 2D Gross-Pitaevskii equation ﬁnding the exact mean ﬁeld ground state.In particular,we explore the crossover between the Thomas-Fermi regime,holding for large values of the coupling constant,and the lowest Landau level limit,corresponding to the weakly interacting case.Explicit results are given for the equation of state,the vortex core size,as well as the elastic shear modulus,which is crucial for the calculation of the Tkachenko frequencies.PACS numbers:03.75.Kk,03.75.Lm,67.40.VsIn the last years,a signiﬁcant eﬀort has been to the study of vortex lattices in harmonically rotating Bose-Einstein condensates.Striking been obtained experimentally,leading to the of large vortex lattices in fast rotating condensates to the measurement of their dynamical Theoretically,several predictions have been 5,6]in agreement with experiments and more regimes,related to the quantum Hall eﬀect,been proposed [7].In this paper we study the rotating analogue form Bose-Einstein condensate,where a periodic lattice is present.For dilute atomic gases,highly compressible,this can only be obtained centrifugal potential is exactly cancelled by the harmonic conﬁnement,i.e.,when the harmonic trapping frequency is precisely equal to the angular velocity Ω.Under this condition,in spite of the non-periodicity of the Gross-Pitaevskii (GP)equation governing the system,the den-sity n (see Fig.1)and the velocity ﬁeld v ′in the rotating frame are periodic in the plane of rotation [8,9],as can be proved by a proper gauge transformation.Analyt-ical predictions are available in the literature for both the weakly interacting regime [6,10,11,12,13],where the system is in the lowest Landau level (LLL),and the strongly interacting one [9],where the density is basically constant and the Thomas-Fermi (TF)approximation ap-plies.Solving numerically the GP equation,we recover these limiting cases and compute the ground state for any value of the interaction strength.The validity of the mean-ﬁeld approach requires that the number of atoms per vortex be very large.If this condition is not satisﬁed the system enters a strongly correlated regime related to the quantum Hall eﬀect [7].In the rotating frame,the external conﬁnement V ext combines with the centrifugal potential,giving rise to the eﬀective trapping V eﬀ=V ext −m Ω2(x 2+y 2)/2,where m is the atomic mass.As anticipated above,we impose the compensation V eﬀ=0and decouple the motion inplane computed by solving Eq.(2)for (a)g =0,(b)g =300.In (b)the dashed hexagon is the Wigner-Seitz cell,while the solid parallelogram is the computation box (see text).Darker regions correspond to lower density.the axial direction from the radial one,therefore reduc-ing to a 2-dimensional problem with an eﬀective cou-pling constant g 2D [14].In addition,our discussion can be conveniently reformulated in terms of dimensionless quantities by using harmonic oscillator units,i.e., Ω,Ω,and l Ω=2+g |ψ|2ψ,(1)where µis the chemical potential,e z is the unit vec-tor along z,and the dimensionless coupling constant g=πg2D n / Ω=g2D N cell m/ 2is the only parameter governing the equation.The normalization is |ψ|2=1, where ≡ A cell d r means integration over a single cell. Due to the presence of the rotational term e z∧r, Eq.(1)is not spatially periodic.However,in the spirit of the magneticﬁeld analogy discussed in Refs.[7,10], one can perform a gauge transformation which turns the symmetric eﬀective vector potential A=e z∧r into a periodic function A′=A−∇Λ.We chooseΛ=S T, where S T is the(non-periodic)phase associated with the Tkachenko(non-periodic)velocityﬁeld v T=∇S T used by Tkachenko to describe the vortex lattice of an incom-pressibleﬂuid[9].With such a choice,indeed,oneﬁnds A′=e z∧r−v T=−v′T,where the Tkachenko velocity in the rotating frame is now periodic[9].We there-v′Tfore writeψ=˜ψe iS T and substitute this expression into Eq.(1),ﬁnally obtainingµ˜ψ= (−i∇+v′T)2. FIG.3:Fractional vortex core area A as function ofΓ−1LLL Solid line:numerical solution of Eq.(2).Dashed line:LLL (Γ−1LLL→∞)limit from Eq.(2).Dash-dotted line:TF (Γ−1LLL→0)limit A≃4ξ2.Points with error bars:experi-mental results from Ref.[22](see text).The behaviour ofµ(g)(g∝ n )can be used to evalu-ate the coarse grained density proﬁle n of the trapped rotating gas in terms of the residual eﬀective potential V eﬀ,through the local density relationshipµ( n )+V eﬀ= const.In particular,for the experimentally relevant case of harmonic trapping,one notices that the quasi-linear behaviour ofµ(g)implies the density proﬁle to be an inverted parabola[20].Another important quantity to study is the vortex coresize.In the TF limit this length scale isﬁxed by the heal-√ing lengthξ=1/4tions in harmonically trapped condensates.Indeed,one can average the predicted bulk values for C2over the TF proﬁle of the trapped condensate,using a local density approximation(see for example Ref.[29]).This proce-dure allows to estimate the Tkachenko frequencies also in the intermediate regime between the TF and the LLL limit.In practice,for the highest rotation rate realized in the experiments,Ω/ω⊥=0.99,where the gas is in a 2D regime andω⊥is the radial trapping frequency,we ﬁnd that C2is lowered by only about30%with respect to the TF value n /8.This means that the experimental data are still far from the LLL limit of the Tkachenko fre-quency,where the shear modulus would depend linearly on g[11].Furthermore,the predicted reduction of C2is not suﬃcient to explain the sizable discrepancy between the experimental measurements and the TF prediction [29].Possible explanations of the remaining discrepancies are:(i)inadequacy of the local density approximation in the calculation of the Tkachenko frequencies,(ii)eﬀects associated with the strong observed damping,(iii)de-viations from linearity in the experimental excitation of vortex modes,and(iv)occurrence of anharmonic eﬀects in the trapping potential.In conclusion,we have investigated the bulk proper-ties of a rotating condensate in the presence of exact compensation between the centrifugal potential and the harmonic trapping.To this purpose,the meanﬁeld GP equation has been solved by introducing a gauge trans-formation which explicitly exploits the periodic structure of the density distribution and of the velocityﬁeld in the rotating frame.We have systematically investigated the role of interactions exploring the transition between the Thomas-Fermi and the LLL regime.We have found that, while the size of the vortex core exhibits a smooth mono-tonic behaviour as a function of the interaction parame-ter,the elastic shear modulus is characterized by an in-termediate maximum.The comparison with experiments reveals a good agreement as concerns the size of the vor-tex core and the static properties of the lattice,whereas we proved that the remaining discrepancies regarding the Tkachenko oscillations measured at the highest angular velocities cannot be attributed to LLL eﬀects alone. We are indebted with L.P.Pitaevskii, A.L.Fetter, J.Dalibard,and N.H.Lindner for fruitful discussions. We would like to thank E.Cornell and V.Schweikhard for valuable comments and sharing their data. C.T.would like to warmly thank M.Paolini for introducing him to theﬁnite element method.[1]K.W.Madison et al.,Phys.Rev.Lett.84,806(2000);J.R.Abo-Shaeer et al.,Science292,476(2001);P.C.Hal-jan et al.,Phys.Rev.Lett.87,210403(2001).[2]P.Engels et al.,Phys.Rev.Lett.89,100403(2002).[3]I.Coddington et al.,Phys.Rev.Lett.91,100402(2003).[4]M.Cozzini and S.Stringari,Phys.Rev.A67,041602(R)(2003).[5]G.Baym and C.J.Pethick,Phys.Rev.A69,043619(2004).[6]E.J.Mueller and Tin-Lun Ho,Phys.Rev.Lett.88,180403(2002).[7]N.K.Wilkin and J.M.F.Gunn,Phys.Rev.Lett.84,6(2000);N.R.Cooper,N.K.Wilkin,and J.M.F.Gunn, Phys.Rev.Lett.87,120405(2001).[8]A.L.Fetter,J.A.Sauls,and D.L.Stein,Phys.Rev.B28,5061(1983).[9]achenko,Zh.Eksp.Teor.Fiz.49,1875(1965)[Sov.Phys.JETP22,1282(1966)].[10]Tin-Lun Ho,Phys.Rev.Lett.87,060403(2001).[11]E.B.Sonin,Phys.Rev.A72,021606(R)(2005).[12]A.Aftalion,X.Blanc,and J.Dalibard,Phys.Rev.A71,023611(2005).[13]G.Watanabe,G.Baym,and C.J.Pethick,Phys.Rev.Lett.93,190401(2004).[14]By considering a system uniform along the z-axis witha vertical length Z,the eﬀective2D coupling constantis given by g2D=g3D/Z,where g3D is the3D coupling constant.If instead the motion along z is squeezed into the lowest oscillator state,one has g2D=g3D/a z√。