Bose-Einstein condensation of magnons

合集下载

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).。

二维吸引Bose—Einstein凝聚的临界值和稳定性

ｍ，具有负色散长度的 Ⅳ个原子在径向限制调和势下的吸引Ｂｓ．ｉｔｎ聚（Ｅ）－］ｏｅｎｅ凝ＥｓｉＢＣｕ３．
ＢＣ已成为现代物理的一个重要研究领域【．Ｅ４］吸引ＢＣ具有一个令人瞩目的物理性质，Ｅ即凝聚原子数 Ⅳ存在一个临界值 Ⅳ 当Ｎ＜Ｎ，聚是稳．时凝定的；当 Ⅳ ≥ Ⅳ ，聚由于有坍塌发生而变得而时凝不稳定．引ＢＣ的临界值研究已引起一大批物理吸Ｅ学家的兴趣．理理论和实验的研究已提供了一系物列的结果～２．对Ｇｔ针］Ｐ方程（）从偏微分方程的１，
中图分类号：１５２０７．７文献标识码：Ａ文章编号：０１８９（０２Ｏ－５３０１０．３５２ｏ）６０６．６
考虑二维ＧｏｓＰｔｅｓｉＧ）程ｒｓ．ｉｖｋｉＰ方本文的思想是建立ＧＰ方程与一个经典的数量场方程的对应关系，重地借鉴并使用了Ｍ．Ｉ着．（）１
变分法的思想［及孤立子的概念为数学１与物理的结合以及静态与动态的关系给出了一个完善的方式和路子．
下面首先讨论ＧＰ方程的Ｃｕｈａｃｙ问题的局部解的适定性；在第二节推导临界值的严格公式及其计算结
ｉ
＝一△ ＋Ｉ～＋口Ｉ～，

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).。

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 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 :

Magnon Bose condensation in symmetry breaking magnetic field

a r X i v :c o n d -m a t /0607711v 2 [c o n d -m a t .s t r -e l ] 1 S e p 2006Magnon Bose condensation in symmetry breaking magnetic ﬁeld S.V.Maleyev,V.P.Plakhty,S.V.Grigoriev,A.I.Okorokov and A.V.Syromyatnikov Petersburg Nuclear Physics Institute,Gatchina,Leningrad District 188300,Russia E-mail:maleyev@.spb Abstract.Magnon Bose condensation (BC)in the symmetry breaking magnetic ﬁeld is a result of unusual form of the Zeeman energy that has terms linear in the spin-wave operators and terms mixing excitations which momenta diﬀer in the wave-vector of the magnetic structure.The following examples are considered:simple easy-plane tetragonal antiferromagnets (AFs),frustrated AF family R 2CuO 4,where R=Pr,Nd etc.,and cubic magnets with the Dzyaloshinskii-Moriya interaction (MnSi etc.).In all cases the BC is important when the magnetic ﬁeld is comparable with the spin-wave gap.The theory is illustrated by existing experimental results.1.Introduction Magnon Bose condensation (BC)in magnetic ﬁeld was intensively studied in spin singlet materials (see for example [1]and references therein).In this case magnons condens in the ﬁeld just above the triplet gap.In this paper we consider magnon BC that appears in the symmetry breaking magnetic ﬁeld.The theoretical discussion is illustrated by experimental observation of this BC in frustrated antiferromagnet (AF)Pr 2CuO 4and cubic helimagnets MnSi and FeGe.To clarify our idea we begin with consideration of conventional AFs.In textbooks two limiting cases are considered.First,the magnetic ﬁeld is directed along the sublattices.In this case the system remains stable up to the critical ﬁeld H C =∆,where ∆is the spin-wave gap.Then the ﬁrst order transition occurs to the state in which the ﬁeld is perpendicular to sublattices (spin-ﬂoptransition).Second,the ﬁeld is perpendicular to initial staggered magnetization.The system remains stable but the spins are canted toward the ﬁeld by the angle determined by sin ϑ=−H/(2SJ 0),where J 0=Jz ;J and z are the exchange interaction and the number of nearest neighbors,respectively.At H +2SJ 0the spin-ﬂip transition occurs to the ferromagnetic state.To the best of our knowledge the ﬁrst consideration of the symmetry breaking ﬁeld was performed theoretically in [2]in connection with experimental study of the magnetic structure of the frustrated AF R 2CuO 4,where R=Pr,Nd,Sm and Eu [3,4].In these papers the non-collinear structure was observed using the neutron scattering in the ﬁeld directed at angle of δ=450to the sublattices.It was found in[2]that in inclinedﬁeld the Zeeman energy has unusual form with terms which are linear in the spin-wave operators and term mixing magnons which momenta diﬀer in the AF vector k0.As a result the BC arises of the spin-waves with momenta equal to zero and±k0.Similar situation exists in cubic helimagnets MnSi etc.[5].If theﬁeld is directed along the helix wave-vector k the plain helix transforms into conical structure and then the ferromagnetic spin state occurs at criticalﬁeld H C.But if H⊥k the magnon condense with momenta zero,±k,±2k etc.This leads to the following observable phenomena:i)a transition to the state with k directed along theﬁeld at H⊥∼H C1=∆√S/2(a l−a+l),l=1,2 and a l(a+l)are Bose operators.As a result the Zeeman energy has unusual formH Z=H a+q+k0a q+iϑN,i.e. these operators has to be considered as classical variables as in the Bogoliubov theory of the BC in dilute Bose gas.Due to theﬁrst term in(2)we must consider the operatorsa±k0and a+∓kas classical variables too.Minimizing the full Hamiltonian with respectto these variables we obtainE=(∆2sin22ϕ)/(16J0)−S2J0ϑ2−(H H⊥)2/[4J0(∆2(ϕ,H)],(3)where theﬁrst term is the energy of the square anisotropy.In cuprates with S=1/2ithas quantum origin and arises due to pseudodipolar in-plane interaction[9].The secondterm is the energy of the spin canting in perpendicularﬁeld.The last term is the BCenergy and∆2(ϕ,H)=∆2cos4ϕ+H2⊥−H2 is the spin-wave gap in theﬁeld[2].This contribution becomes important at H∼∆.The spin conﬁguration is determined byd E/dϕ=0and equilibrium condition d2E/dϕ2≥0.This theory was veriﬁed by neutrons scattering[10,11].In diagonalﬁeld H (1,1,0)the spin conﬁguration in frustrated Pr2CuO4is governed by Eq.(3)and the intensityof the(1/2,1/2,−1)is given by I∼1+sin2ϕ[2].Neglecting the BC term we getsin2ϕ=−(H/H C)2,where H C=∆.As a result at H→H C we obtain I∼H C−H. But very close to H C the BC term becomes important and we have a crossover to I∼(H C−H)1/2.It is clearly seen inﬁgure2.This crossover was observed in[10,11].3.Frustrated AFsIn frustrated R2CuO4AFs there are two copper spins in unit cell belonging to diﬀerent CuO2planes(see inset inﬁgure1).From symmetry considerations these spins do not interact in the exchange approximation.The orthogonal spin structure is a result of the interplane pseudodipolar interaction(PDI)[2,3]and the ground state energy is given by∆2E=on T we obtain H C≃7.8T andγC≃5.30that is in stronger disagreement wit the experiment.The experimentally obtained anglesαandγat T=18K andδ=9.50are shown inﬁgure4[12].The transition to the collinear state withα∼−450andγC∼200 was observed.Again the non-BC theory can not explain the experimental data.For example it givesγC≃2.50.Explanation of all these experimental data using the BC theory will be given elsewhere.4.BC in helimagnetsIn helimagnets MnSi etc.Dzyaloshinskii-Moriya interaction(DMI)stabilizes the helical structure and the helix wave-vector has the form k=SD[ˆa×ˆb]/A,where D is the strength of the DMI,A is the spin-wave stiﬀness at momenta q≫k,ˆa andˆb are unit orthogonal vectors in the plane of the spin rotation.The classical energy depends on theﬁeld component H along the vector k and the cone angle of the spin rotation is given by sinα=−H/H C,where H C=Ak2is the criticalﬁeld of the transition into ferromagnetic state[5].However at H⊥≪H C rotation of the helix axis toward theﬁeld direction and the second harmonic2k of the spin rotation were observed[6-8].Both phenomena are related to the magnon BC in perpendicularﬁeld[5].The linear and mixing terms appear in the Zeeman energy in much the same way as it was discussed above:H Z=(H a−i H b)2the real form of the BC energy is not so simple.It is determined by nonlinear interactions but consideration of this problem is out of the scope of this paper.As a result the perpendicular susceptibility is proportional to1/(∆2−H2⊥/2)and 2k harmonic appears.The last was observed by neutron scattering[6-8].Intensities of corresponding Bragg satellites have the formI±∼[∆2/(∆2−H2⊥/2)]2[1∓(kP)]δ(q∓2k),(7) where P is the neutron polarization.If H⊥→∆√5.ConclusionsWe discuss a few examples of the magnon BC in symmetry breaking magneticﬁeld.BC appears due to unusual terms in Zeeman energy.Obviously this phenomenon is very general and can be observed in other ordered magnetic systems.Eﬀects related to the BC has to be more pronounced in theﬁeld of order of the sin-wave gap.6.AcknowledgmentsThis work is partly supported by RFBR(Grants03-02-17340,06-02-16702and00-15-96814),Russian state programs”Quantum Macrophysics”,”Strongly Correlated electrons in Semiconductors,Metals,Superconductors and Magnetic Materials””Neutron Research of Solids”,Japan-Russian collaboration05-02-19889-JpPhysics-RFBR and Russian Science Support Foundation(A.V.S.).References1Crisan M,Tifrea I,Bodea D and Grosu I2005Phys.Rev.B721644142Petitgrand D,Maleyev S,Bourges P and Ivaniv A1999Phys.Rev.B5910793D.Petitgrand,Moudden A,Galez P and Boutrouille P1990J.Less-Common Metals164-165768 4ChattpodhayaT,Lynn J,Rosov N,Grigereit T,Barilo S and Zigunov D1994Phys.Rev.B49 99445Maleyev S2006Phys.Rev.B731744026Lebech B,Bernard J and Feltfoft1989J.Phys.:Condens.Matter161057Okorokov A,Grigoriev S,Chetverikov Yu,Maleyev S,Georgii R,B¨o ni P,Lamago D,EckerslebeH and Pranzas P2005Physica B3562598Grigoriev S,Maleyev S,Chetverikov Yu,Georgii R,B¨o ni P,Lamago D,Eckerlebe H and Pranzas P2005Phys.Rev B721344029Yildirim T,Harris A,Aharony A and Entin-Wohlman O1995Phys.Rev.B521023910Plakhty V,Maleyev S,Burlet P,Gavrilov S and Smirnov O1998Phys.Lett.A25020111Ivanov A and Petitgrand D2004J.Mag.Mag Materials272-27622012Plakhty V,Maleyev S,Gavrilov S,Bourdarot F,Pouget S and Barilo S2003Europhys.Lett.61 53413Ivanov A,Bourges P and Petitgrand D1999Physica B259-261879FiguresaFigure 1.Spin conﬁguration in the ﬁeld.Full and dashed arrows correspond to zero and nonzero ﬁeld,respectively.Addition spin canting in H ⊥is shown by broken arrows.Inset:spin conﬁguration in neighboring planes of frustrated AF.Figure 2.Log-Log plot of the (1/2,1/2,−1)Bragg intensity in diagonal ﬁeld,h ∼(H C −H ).Figure3.Theﬁrst order transition in theﬁeld directed along b axis.Calculated intensities for the spinﬂop conﬁgurations when spins are perpendicular to theﬁeld (white arrows).Figure4.Field dependence of anglesαandγatδ=9.50.)10a H mT =)50b H mT =)150c H mT=Figure 5.Bragg reﬂections in the ﬁeld along (1,1,0).a)Four strong spots corresponds to ±(1,1,1)and ±(1,1,−1)reﬂections.Weak spots are the double Bragg scattering.b)The 2k satellites appear.c)The helix vector is directed along the ﬁeld.。

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

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

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

a r X i v :c o n d -m a t /0405619v 1 [c o n d -m a t .s u p r -c o n ] 26 M a y 2004Bose-Einstein condensation of magnonsM.Crisan,D.Bodea,I.Tifrea and I.Grosu Department of Theoretical Physics,University of Cluj,3400Cluj,Romania (Dated:February 2,2008)Abstract We use the Renormalization Group method to study the Bose-Einstein condensation of the interacting dilute magnons which appears in three dimensional spin systems in magnetic ﬁeld.The obtained temperature dependence of the critical ﬁeld H c (T )−H c (0)∼T 2is diﬀerent from the recent self-consistent Hartree-Fock-Popov calculations (cond-matt 0405422)in which a T 3/2dependence was reported .The origin of this diﬀerence is discussed in the framework of quantum critical phenomena.I.INTRODUCTIONRecently,the quantum spin system in magneticﬁeld became of great interest because the experimental data on the ladder systems showed the possibility of a quantum phase transition(QPT)driven by the magneticﬁeld[1].The QPT in the spin systems have been treated using the quantum rotor model[2].However,the transition driven by a magnetic ﬁeld is diﬃcult to be described in such a formalism.The occurrence of the magnetic order in the spin-gap magnetic compounds has been interpreted as a Bose-Einstein condensation(BEC)of magnons.At the present time it is known that the BEC has been discovered in ultra-cooled dilute atomic gases,but these experiments present various limitations.The analogy between a quantum spin system which presents long-range order and an interacting Bose gas which presents BEC is well known for a long time[3].In the case of quantum spin system is possible to tune the density of magnons by magneticﬁeld[11,12]or pressure[20]to observe BEC in these systems.In this paper we formulate the theory of magnon condensation using the Renormalization Group method(RNG)applied for bosonic systems[4,5,6,7,8,9]and recently reconsidered for the spin systems[10].The outline of this paper is as follows.In Sect.II we present the basic experimental evidence obtained on the material class XCuCl3(X=T l,K,NH4)and the main results obtained with the meanﬁeld theory.The model based on these data will be presented in Sect.III.The Sect.IV is devoted to the RNG formulation using results from Ref.[10] in order to calculate the low temperature quantum critical properties of the system.Sec. V will present the main results of the paper,the relevant thermodynamic quantities next to the quantum critical point.Finally,in the last section we will discuss the results and compare with other theoretical approaches and experimental evidence.II.EXPERIMENTAL EVIDENCESIn this section we present the basic experimental data obtained on the XCuCl3com-pounds which gave us the possibility to elaborate a model for the critical behavior of this QPT as a BEC driven by the magneticﬁeld.The compound which oﬀered the possibility of performing many measurements isT lCuCl3which is composed of a chemical double chain Cu2Cl6.The magnetic suscep-tibility,measured as function of temperature T for three diﬀerent directions of the magnetic ﬁeld,exhibit a broad maxima at T=38K,decreasing to zero with the temperature de-creasing[11].This result indicates that the compound has an excitation gap∆.µB gAn important result has been obtained by Sherman et al.[15]by sound attenuation in T lCuCl3in magneticﬁeld at low temperatures.The occurence of the sharp peak in the sound attenuation near the BEC critical temperature and the Drude form of the sound dumping suggested a constant weak magnon-magnon coupling.The sound attenuation suggested a magnons dispersion likeω2=∆2+J2k2.The temperature dependence of the criticalﬁeld has the form[H c(T)−H c(0)]∼TΦ2whereΦ=3III.MICROSCOPIC MODELWe introduce the description of the magnon condensation in magnetic ﬁeld using the action:S eff =S (2)eff +S (4)eff(1)where:S (2)eff =116k 1... k 4φ(k 1)...φ(k 4)δ(k 1+...+k 4)(3)Here we introduced the notations k =(k ,ωn ),ωn being the bosonic Matsubara frequency and k =T n d d kω2n +k 2+r 0(4)where r 0=∆−µB gH and H is the external magnetic ﬁeld.IV.RENORMALIZATION GROUP EQUATIONSThe model has the dynamical critical exponent z =1and for this model d =3.The Renormalizationgroupequations in one-loop approximation at ﬁnite temperature T for d =3have the form [10]:dT (l )dl =−(n +8)K 3dl =2r (l )+(n +2)K 32π2.We denoteby u (l =0)=u 0,and r (l =0)=r 0and the functions F 1,2are characteristic functions for the model [10].We approximate F 2[r (l ),T (l )]≃1C 0(l +l 0)(8)where C0=(n+8)K32π2.The physics near the QCP is described by the scaling ﬁeld t r(l)deﬁned as:t r(l)=r(l)+n+24K d l0dx e−2x u(x)n+8ln(lλr(13)whereλr is the eigenvalue of the relevant parameter r and is given by the expression:λr=2,ǫ=0(d=3)(14)and t r(T)is given by the relation[10]:t r0(T)=r0−r0c+n+22ln1V.THERMODYNAMIC QUANTITIESUsing these results we will calculate the relevant thermodynamic quantities near the quantum critical point,but in the disordered state.First,we deﬁne the critical line by t(T)=0and we get:H c(T)−H c(0)=C0T2(17) where C0∝u0and H c=∆(2π)3f B[T(l∗)](18) This equation gives for n(T)a temperature dependence of the form:n(T)∝T3128u0T2(20) and for r0≤r0c we get the general equationT c(r0)= 12832π2u0(22) we obtainT c(H)∼|H−˜H c|1/α(23) withα=2and˜H c =H c+(n+2)n+8](24)and it give us a logarithmic behavior:C v(T)n+8](25) We will discuss these results in connection with the existent theoretical approaches and the experimental data for T lCuCl3.VI.DISCUSSIONSThe occurence of BEC induced by the magneticﬁeld in a spin-gap system has been predicted by Giamarki and Tsvelik[16]in order to explain the three-dimensional ordering in coupled ladders.Nikumi et al.[17]applied Popov theory(see for example[18])for the BEC obtaining a temperature dependence for magnetization in agreement with the experimental data,of the form T3/2.However,as it was mentioned in Ref.[17]the critical exponentΦof the criticalﬁeld [H c(T)−H c(0)]∼TΦwas obtained asΦ=3/2,but the experimental data showsΦ=2.2.Recently Misguich and Oshikawa[19]reconsidered the self-consistent Hartree-Fock-Popov (HPF)method using a realistic dispersion for magnons[13]and they reobtainedΦ=3/2. They also calculated the speciﬁc heat and obtained aλ-shape.We would like to mention several basic points related to this problem:•in the HPF the eﬀect ofﬂuctuations have been neglected and it appears normal that thermodynamic quantities on the critical line have wrong critical exponents.•the changing of the dispersion law for the magnon can improve the HFP results,but does not aﬀect the basic approach.•both papers[17]and[18]considered the QPT as driven by the temperature and in fact it is induced by the magneticﬁeld.These observations justify the application of the RNG method in the version proposed by Caramico et al.[10]to study the magnon condensation.Our main result can be summarized as:•we obtained the critical exponentΦ=2,which is in a better agreement with experi-mental data(Φ=2.2).•the magnetization calculated in our approach is also T3/2dependent.•the critical temperature T c(H)has a dependence given by Eq.(23)withα=2in agreement with[12].The numerical results have been analyzed in[20].•the speciﬁc heat calculated by RNG method has a logarithmic correction as was ex-pected.The T dependence of C v is not relevant if we take the approach of a magnetic ﬁeld driven QPT,but we have it to the dependence of H(T).Such an experiment has been done[12]and we can notice a small anomaly in this dependence.But aﬁnal decision about the H(T)dependence of our Eq.(25)cannot be taken.The agreement between the experimental data and the theory presented in[19]appears in our opinion due to the fact that the authors considered,on a very narrow interval a linear dependence between the criticalﬁeld H c(T)and the critical density n c.This approximation can be valid for n c<0.002which describes a part of the real temperature dependence of H c(T).VII.ACKNOWLEDGMENTSOne of the authors(M.C.)thanks Luigi De Cesare for numerous correspondences helping him to understand the RG methods developed by the Salerno group(A.Caramico D’Auria, L.De Cesare and I.Rabuﬀo)for this problem.[1]S.Sachdev,Science288,475(2000).[2]S.Sachdev,Quantum Phase transitions,(Cambridge Univ.Press,1999).[3]T.Matsubara and H.Matsuda Prog.Theor.Phys.16569(1956).[4] D.S.Fisher and P.C.Hohenberg Phy.Rev.B374936(1988).[5] A.I.Uzumov,Introduction to the Theory of Critical Phenomena(World Scien-tiﬁc,Singapore,1993).[6]S.Sachdev,T.Sentil,and R.Sankar Phys.Rev.B50,258(1994).[7] A.Caramico D’Auria,L.De Cesare,and I.Rabufo Physica A152(1997).[8]M.Crisan,D.Bodea,I.Grosu and I.Tifrea J.Phys.A;Math.Gen.35,239(2002).[9] A.A.J.Schakel,cond-mat0301050.[10] A.Caramico D’Auria,L.De Cesare,and I.Rabufo Phys.Lett.A308,208(2003);PhysicaA327,442(2003).[11]K.Takasu,W.Shimramura and H.Tanaka,J.Phys.Soc.Japan.66,1611(1997).[12] A.Oosawa,M.Ishi and H.Tanaka J.Phys;Condens.Matter11265,(1999).[13]M.Matsumoto,B.Normand,T.M.Rice and M.Sigrist Phys.Rev.Lett.89,077203(2003).[14]N.Cavadini,G.Heigold,W.Henggler,A.Furrer,H-U Gudel,K.Kramer and H.Mutka Phys.Rev.B63,172414(2001)[15] E.Ya.Sherman,P.Lemmens,B.Busse,A.Oosawa,and H.Tanaka Phys.Rev.Lett.91,057201(2003).[16]T.Giamarchi and A.M.Tsvelik Phys.Rev.B59,11398(1999).[17]T.Nikuni,M.Oshikawa,A.Oosawa and H.Tanaka Phys.Rev.Lett.84,5868(2000).[18]H.Si and A.Griﬃn Phys.Rep.304,1(1978)[19]G.Misguich and M.Oshikawa cond-mat/00405422[20]O.Nohamadani S.Wessel B.Normand and S.Haas,cond-mat/0307126;B.Normand,M.Matsumoto,O.Nohadani,S.Weesel,S.Haas,T.M.Rice,and M.Sigrist J.Phy:Condens Matter,16,S867(2004)。