Magnon Bose condensation in symmetry breaking magnetic field

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 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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).。

a r X i v :n u c l -t h /9308015v 1 19 A u g 1993Bose-condensation through resonance decayU.Ornik 1∗,M.Pl¨u mer 2†and D.Strottman 3‡1Theory Group,Gesellschaft f¨u r Schwerionenforschung (GSI),Darmstadt,F.R.Germany 2Physics Department,University of Marburg,FRG 3Theoretical Division,Los Alamos National Laboratory,Los Alamos,USA 87545February 9,2008Abstract We show that a system described by an equation of state which contains a high number of degrees of freedom (resonances)can create a considerable amount of super-fluid (condensed)pions through the decay of short-lived resonances,if baryon number and entropy are large and the dense matter decouples from chemical equilibrium ear-lier than from thermal equilibrium.The system cools down faster in the presence of a condensate,an effect that may partially compensate the enhancement of the lifetime expected in the case of quark-gluon-plasma formation.M@VAX.HRZ.UNI-MARBURG.DE‡E.Mail:DDS@1The investigation of hot hadronic matter through heavy ion collisions has shown that an understanding of medium effects andfinal state interactions is of great importance for the interpretation of the particle spectra in order to distinguish new physical phenomena like the formation of quark-gluon plasma(QGP)from background phenomena caused by “conventional”physical effects.We refer here,e.g.,to the strangeness enhancement,the J/Ψsuppression and the soft-pion puzzle,which can in principle be explained either by the formation of a QGP or from purely hadronic origin(for a recent review,see ref.[1]).An interesting explanation for the soft pion puzzle was proposed in[2].The authors argued that the baryonic and mesonic resonances may decouple from chemical equilibrium but still remain in thermal equilibrium.Since the resonance production rates fall drastically after their decoupling from chemical equilibrium,the remaining short-lived resonances would then rapidly decay into pions.As a consequence,the pions may acquire a non-zero chemical potentialµπwhich leads to a softer p⊥-spectrum.A different possible explanation for the absence of local chemical equilibrium of pions in thefinal(decoupling)stage,which should not be confused with the scenario discussed in ref.[2],was proposed in refs.[3,4]where it is assumed that the pions are initially created out of local chemical(and even thermal!)equilibrium and it turns out that the system never reaches complete local chemical equilibrium[3,4].In the present letter,we investigate the scenario of ref.[2],where it is assumed that the dense matter reaches local thermodynamic equilibrium in an early stage of the expansion. The authors of[2]considered only the case that the chemical potential stays below the pion mass.Moreover,they did not attempt to determine the value of the pion chemical potential in terms of the temperature and baryon number density at the point of decou-pling from local chemical equilibrium.Below,we address the problem if and under what conditions the decay of short-lived resonances can lead to the formation of a pionic Bose condensate,and what happens to the condensate in the subsequent expansion until the system decouples from local thermal equilibrium and thefinal state particles are emitted. To do so,we explicitly consider the transition from a resonance gas to a gas of stable and long-lived particles for a hadronic equation of state which contains the known resonances up to masses of2GeV,imposing constraints from the conservation of energy-momentum,2baryon number and strangeness.In particular,we shall show that for a system rich in entropy and baryon numberµπmay reach the pion mass and,consequently,the pions may form a Bose condensate in the central region in relativistic heavy ion collisions.For a hydrodynamically expanding system,the formation of such a condensate implies the presence of a superfluid component.The idea that the hadronic matter might be superfluid was put forward a long time ago in a somewhat different context[5].This could explain in a natural way the presence of a coherent component in multiparticle production.The implications for pion interferometry measurements will be discussed in a separate paper[6].We note that although the creation of a remarkable amount of condensed pions is interesting by itself,these effects could also be used to answer the question of whether pions and resonances are in chemical equilibrium before they decouple from thermal equilibrium.Because thefinal stage in the hadronic phase can have a significant influence on the particle spectrum,the question of whether the freeze-out occurs in chemical and thermal equilibrium,as assumed in[7,8]or only in thermal equilibrium[2]is of great importance.In order to investigate the conditions necessary to form a condensate,let usfirst consider a system in local thermodynamic(i.e.,thermal and chemical)equilibrium.The medium is then completely described in terms of its equation of state(EOS)which can be expressed in the formεfl=f(T fl,µk fl)(1)whereεfl is the energy-density of thefluid,T fl the temperature,and the quantitiesµk fl are the chemical potentials related to conserved charges Q k(such as baryon number B and strangeness S).The index fl refers to quantities which describe thefluid.During a heavy ion collision we expect under certain conditions the formation of a system of hot hadronic matter or a QGP described by eq.(1)which is in chemical and thermal equilibrium.The system then undergoes a phase of hydrodynamic expansion until it has become so dilute that the interactions are no longer strong enough to maintain local equi-3librium.Thisfinal stage,when the collective behaviour terminates and particles decouple from the dense matter,is referred to as freeze-out.The scenario of ref.[2]differs from most of the other hydrodynamic descriptions of heavy ion collisions in so far as in[2]the dense matter is assumed to go out of chemical equilibrium but remain in thermal equilibrium for some time.Or,to put it differently,there will be a“chemical freeze-out”independent of and prior to the“thermal freeze-out”.As was mentioned above,the transition out of chemical equilibrium leads to the decay of short-lived resonances and the appearance of a non-zero chemical potential of the pions.Below,we shall assume that the chemical freeze-out,just like the thermal freeze-out,occurs “instantaneously”(i.e.,on a three-dimensional hypersurface characterized by a condition such as T(x,t)=const.).This seems a reasonable assumption since the relevant time scales are the decay times of the short-lived resonances which are on the order of∼1 fm/c.To illustrate the rapidness of such a transition,we consider the simplified case of a system of pions and rho-mesons in local thermal equilibrium that undergoes a longitudinal scaling[9]expansion.In addition to the decayρ→ππwe also take into account the inverse processππ→ρ+X.The equations which describe the time evolution of the system aredετ(2)dnπτ+2nρdτ=−nρτρ+ σv n2π(4)whereε=ερ+επand P=Pρ+Pπare the energy density and pressure,n i(i=ρ,π) are the number densities andτ≡√transition is even faster.Thefigure also shows that the decay of rho-mesons alone is not sufficient to drive the pions into the condensate(for that,one needs to take into account the decay contributions of all the other short-lived resonances as well).We now return to the description of the expanding hot and dense matter at the point of chemical freeze-out.It will be assumed that thefluid freezes out into“stable”π’s,K’s, nucleons,Λ’s and the long living resonancesω’s andη’s.The term“stable”here means that the decay of the particle or resonance occurs only after the complete(thermal and chemical)freeze-out of the system.In particular,the contribution ofω’s andη’s to the pion chemical potential is zero.Nevertheless,their later decay leads to a non-thermal component of the pion spectrum which is determined by the decay kinematics[7].The system of chemically frozen out particles can be described by a chemical decoupling temperature T ch.f.,chemical potentials for baryons and strange particles,µB andµS, and a pion chemical potential,µπ,which describes the overpopulation of pions due to the decoupling of resonances.The chemical potentials are determined by the requirement that the energy densityεfl,the baryonic density b fl and the strangeness density s fl=0of the fluid are equal to those of the system after chemical freeze-out.We obtain the following system of equations:εfl=εthermπ(µπ,T ch.f.)+mπn conπΘ(mπ−µπ)+i=K,N,Λ,ω,η,...εthermi(µS,µB,T ch.f.)(5)b fl= k B k n therm k(µS,µB,µπ,T ch.f.)(6)s fl= k S k n therm k(µS,µB,µπ,T ch.f.)=0(7) whereεtherm i (µS,µB,µπ,T)=g iexp E i−˜µi2π2p2dp1T ±1(9)5are the thermal parts of the energy density and the number density of particle species i, and where we have used the notation˜µi= k Q k iµk(10) for the chemical potentials.Note that on the r.h.s.of eq.(10)the charges Q k include the pion number Nπwhich is a conserved quantity after chemical freeze-out.In eq.(5)we have introduced a term n conπthat describes the condensed component of pions which will appear if the chemical equilibrium for pions reaches the pion mass.In this case an overpopulation of pion states occurs and the remaining pions will be forced into the ground state,thereby forming a Bose condensate.Thefluid then consists of a thermal component and a superfluid component.The latter moves with the rapidity of thefluid, whereas the thermal pions have on top of thefluid rapidity a thermal distribution.The resultant single inclusive distribution of pions emitted from a hydrodynamically expanding source1dydp⊥=gπexp p iµuµ−˜µπresonances and a treatment of compression effects,thus retaining the essential features of nuclear matter near the ground state[11].In addition to the work presented in[11]we take into account the effects offinite baryonic and mesonic masses,use Fermi statistics for the baryons and include mesons up to masses of2GeV.Fig.2shows the thermal and the condensate component of pions at chemical freeze-out, and the chemical potentialsµπ,µb andµs,as functions of the temperature T ch.f.that characterizes the decoupling from chemical equilibrium,for three different values of the baryon number density b.It can be seen how the baryonic and strange chemical potentials,µb andµs,decrease with increasing temperature T ch.f..Clearly,the strange chemical potential induced by thefinite baryon number never execeeds150MeV.As the amount of heavier resonances increases with temperature,the pion chemical potentialµπgrows with T ch.f..It becomes equal to the pion mass in the temperature region170-200MeV;the value of T ch.f.whereµπ=mπdepends on the baryonic density.This behaviour is also reflected in the dependence of the thermal and condensate densities of pions on the temperature and baryon number density at chemical freeze-out.It can be seen that at sufficiently high baryonic densities and temperatures the condensate contribution to the chemically decoupled pions becomes significantly large.It is noteworthy that the condensation effect is enhanced if the density of the strange particles has not yet reached its equilibrium value. In this case the remaining energy is distributed among the non strange resonances,and this leads to an enhanced production of particles in the condensate.Finally,we need to discuss the question whether or not the condensate survives the time period between the chemical and the thermal freeze-out.This is of particular importance if the dense matter decouples from chemical equilibrium at temperatures T ch.f.∼170−200 MeV which are considerably higher than those usually associated with the thermal freeze-out,T th.f.∼mπ.After decoupling from chemical equilibrium,the system of stable and long-lived particles continues to expand until it has cooled down to the temperature T th.f..For a one-dimensional scaling expansion,the evolution of the system is determined by the equations7s τ=const .(12)b τ=const .(13)s e τ=const .(14)n πτ=const .(15)n ατ=const .(16)which describe the conservation of entropy S e ,baryon number B ,strangeness S and pion number N π(s e ,b ,s and n πbeing the corresponding densities,respectively).The index αlabels those of the stable particle species (K,N,Λ,ω,η,...)which also have decoupled from chemical equilibrium.For the gas of stable particles and long-lived resonances,the thermodynamic quantities are given by the expressions on the r.h.s.of eqs.(5)–(10),with s e =i (εi +P i −˜µi n i )/T .Energy density and number density of the pions consistof a thermal and a condensate component,επ=εtherm π+m πn con πand n π=n therm π+n con π,respectively.Note that the condensate does not contribute to the entropy.Consequently,the cooling of the system –i.e.,the function T (τ)–does not depend on the superfluid component.Fig.3shows the cooling curves T (τ)of a pion gas,for different values of the pion chemical potential.For comparison we have also included the results for an ideal gas of massless pions.It can be seen that the cooling rate increases with increasing pion chemical potential and becomes maximal in the presence of a condensate.This implies that Bose condensates may reduce the lifetime of the fireball.In particular,we are interested in the time dependence of the fraction of pions in thecondensate,f con ≡n con π/(n therm π+n con π).Eqs.(12,15)imply that for the fraction of thermal pions,f therm =1−f con ,f th.f.therm =f ch.f.therm n therm π/s e ) th.f.the pion gas,it turns out that in the temperature range between200MeV and150MeV, f therm varies by less than10%.To summarize,we have shown that the excited hadronic matter created in ultrarelativistic heavy ion collisions could form a pion condensate if(i)the system is rich in baryon number and entropy density,(ii)the hadrons decouple from local chemical equilibrium earlier than from thermal equilibrium,as suggested in[2],and(iii)the loss of chemical equilibrium occurs at temperatures of∼180MeV or higher.The effect depends on the chemical decoupling temperature,on the baryonic density and,rather sensitively,on the resonance contribution to the EOS.1The effect is increased if the chemical equilibrium for strange particles is not complete,i.e.if the amount of strangeness which is initially zero and starts to increase during the collision and expansion process has not yet reached its equilibrium value.It is not the purpose of the present study to prove that such a decoupling process occurs. To do so would require a detailed knowledge of the density and temperature dependence of the inelastic and elastic cross sections of hadrons in the dense matter.Rather,we would like to point out here that,if this decoupling happens in a certain temperature region(like it was quoted in[2]),then a pion condensate should appear.If a pion condensate is created in a heavy ion collision,it may be distinguishable from the case of a resonance gas in thermal and chemical equilibrium essentially by the following effects.•It should lead to a coherent component which then would appear in the two-particle correlation functions of identical pions through a reduction of the intercept of the correlation function and the appearance of a two exponent behaviour of the correla-tion function[13].•The condensate component is moving with thefluid velocity and has no additional thermal component which smears out the distribution.This might lead to character-istic bumps and shoulders in the rapidity and transverse momentum distributions of pions2.One also can expect that the coherent component disappears if one considers particles of velocities that exceed the maximumfluid velocities.•The lifetime of thefireball may be reduced.This could at least partially compensate the enhancement of the lifetime expected[15]in the case that quark-gluon-plasma is formed in the intial stage.Instructive and helpful discussions with R.M.Weiner,F.Navarra and H.Leutwyler are gratefully acknowledged.This work was supported in part by the Deutsche Forschungsge-meinschaft(DFG),the Federal Minister for Research and Technology(BMFT)under the contract no.06MR731,the Gesellschaft f¨u r Schwerionenforschung(GSI)and the Los Alamos National Laboratory.References[1]Proceedings of Quark Matter91,eds.T.C.Awes,F.E.Obenshain,F.Plasil,M.R.Strayer and C.Y.Wong,Nucl.Phys.A544(1990)513.[3]S.Gavin and P.V.Ruuskanen,Phys.Lett.B262(1992)1403.[5]A.Mann and R.M.Weiner,Nouvo Cim.10A(1972)383;S.Eliezer and R.M.Weiner,Phys.Rev D13(1991)503.[8]J.Sollfrank,P.Koch and U.Heinz,Phys.Lett.B252,(1982)140.[10]F.Cooper and G.Frye,Phys.Rev.D10(1989)3735.[12]U.Ornik,F.W.Pottag and R.M.Weiner,Phys.Rev.Lett.63(1989)278;B242(1990)181.[15]G.Bertsch,M.Gong,L.McLerran,V.Ruuskanen and E.Sarkkinen,Phys.Rev.D37(1988)1896;G.Bertsch,Nucl.Phys.A498FIGURE CAPTIONSFigure1:Left column:Dependence of the chemical potentials for baryons(dashed), strangeness(dotted)and pions(solid)on the chemical freeze-out temperature T ch.f.,for three different values of the baryon number density b.Right column:The thermal(solid) and condensate(dashed)components of the pion number densities at chemical freeze-out.Figure3:。

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 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In the July 14, 1995 issue of Science magazine, researchers from JILA reported achieving a temperature far lower than had ever been produced before and creating an entirely new state of matter predicted decades ago by Albert Einstein and Indian physicist Satyendra Nath Bose. Cooling rubidium atoms to less than 170 billionths of a degree above absolute zero caused the individual atoms to condense into a "superatom" behaving as a single ent ity. The graphic shows three-dimensional successive snap shots in time in which the atoms condensed from less dense red, yellow and green areas into very dense blue to white areas. JILA is jointly operated by NIST and the University of Colorado at Boulder.A Bose–Einstein condensate (BEC) is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very near to absolute zero (0 K, −273.15 °C, or −459.67 °F). Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, and all wave functions overlap each other, at which point quantum effects become apparent on a macroscopic scale.This state of matter was first predicted by Satyendra Nath Bose and Albert Einstein in 1924–25. Bose first sent a paper to Einstein on the quantum statistics of light quanta (now called photons). Einstein was impressed, translated the paper himself from English to German and submitted it for Bose to the Zeitschrift für Physik which published it. Einstein then extended Bose's ideas to material particles (or matter) in two other papers.[1]Seventy years later, the first gaseous condensate was produced by Eric Cornell and Carl Wieman in 1995 at the University of Colorado at Boulder NIST-JILA lab, using a gas of rubidium atoms cooled to 170 nanokelvin (nK) [2](1.7×10−7 K). For their acheivments Cornell, Wieman, and Wolfgang Ketterle at MIT received the 2001 Nobel Prize in Physics. [3]TheoryThe slowing of atoms by use of cooling apparatus produces a singular quantum state known as a Bose condensate or Bose–Einstein condensate. This phenomenon was predicted in 1925 by generalizing Satyendra Nath Bose's work on the statistical mechanics of (massless) photons to (massive) atoms. (The Einstein manuscript, once believed to be lost, was found in a library at Leiden University in 2005.[4]) The result of the efforts of Bose and Einstein is the concept of a Bose gas, governed by Bose–Einstein statistics, which describes the statistical distribution of identical particles with integer spin, now known as bosons. Bosonic particles, which include the photon as well as atoms such as helium-4, are allowed to share quantum states with each other. Einstein demonstrated that cooling bosonic atoms to a very low temperature would cause them to fall (or "condense") into the lowest accessible quantum state, resulting in a new form of matter.This transition occurs below a critical temperature, which for a uniform three-dimensional gas consisting of non-interacting particles with no apparent internal degrees of freedom is given by:where:is t he critical temperature,is t he particle density,is t he mass per boson,is t he reduced Planck constant,is t he Boltzmann constant, andthe Riemann zeta function; (sequenceA078434 in OEIS)Einstein's argumentConsider a collection of N noninteracting particles which can each be in one of two quantum states, and . If the two states are equal in energy, each different configuration is equally likely.If we can tell which particle is which, there are 2N differentconfigurations, since each particle can be in or independently. In almost all the configurations, about half the particles are in and the other half in . The balance is a statistical effect, the number of configurations is largest when the particles are divided equally.If the particles are indistinguishable, however, there are only N+1 different configurations. If there are K particles in state , there are N-K particles in state . Whether any particular particle is in state or in state cannot be determined, so each value of K determines a unique quantum state for the whole system. If all these states are equally likely, there is no statistical spreading out; it is just as likely for all the particles to sit in as for the particles to be split half and half.Suppose now that the energy of state is slightly greater than the energy of state by an amount E. At temperature T, a particle will have a lesser probability to be in state by exp(-E/T). In the distinguishable case, the particle distribution will be biased slightly towards state and the distribution will be slightly different from half and half. But in the indistinguishable case, since there is no statistical pressure toward equal numbers, the most likely outcome is that most of the particles will collapse into state .In the distinguishable case, for large N, the fraction in state can be computed. It is the same as coin flipping with a coin which has probability p = exp(-E/T) to land tails. The fraction of heads is 1/(1+p), which is a smooth function of p, of the energy.In the indistinguishable case, each value of K is a single state, which has its own separate Boltzmann probability. So the probability distribution is exponential:For large N, the normalization constant C is (1-p). The expected total number of particles which are not in the lowest energy state, in the limitthat , is equal to . It doesn't grow when N islarge, it just approaches a constant. This will be a negligible fraction of the total number of particles. So a collection of enough bose particles in thermal equilibrium will mostly be in the ground state, with only a few in any excited state, no matter how small the energy difference.Consider now a gas of particles, which can be in different momentum states labelled . If the number of particles is less than the number of thermally accessible states, for high temperatures and low densities, the particles will all be in different states. In this limit the gas is classical. As the density increases or the temperature decreases, the number of accessible states per particle becomes smaller, and at some point more particles will be forced into a single state than the maximum allowed for that state by statistical weighting. From this point on, any extra particle added will go into the ground state.To calculate the transition temperature at any density, integrate over all momentum states the expression for maximum number of excited particles p/(1-p):When the integral is evaluated with the factors of k B and ℏ restored by dimensional analysis, it gives the critical temperature formula of the preceding section. Therefore, this integral defines the critical temperature and particle number corresponding to the conditions of zero chemical potential (μ = 0 in the Bose–Einstein statistics distribution).Gross–Pitaevskii equationMain article: Gross–Pitaevskii equationThe state of the BEC can be described by the wavefunction of the condensate . For a system of this nature, is interpreted as the particledensity, so the total number of atoms isProvided essentially all atoms are in the condensate (that is, have condensed to the ground state), and treating the bosons using mean fieldtheory, the energy (E) associated with the state is:Minimising this energy with respect to infinitesimal variations in ,and holding the number of atoms constant, yields the Gross-Pitaevski equation (GPE) (also a non-linear Schrödinger equation):where:is the mass of the bosons,is the external potential,is representative of the inter-particle interactions.The GPE provides a good description of the behavior of BEC's and is thus often applied for theoretical analysis.DiscoveryIn 1938, Pyotr Kapitsa, John Allen and Don Misener discovered that helium-4 became a new kind of fluid, now known as a superfluid, at temperatures less than 2.17 K (the lambda point). Superfluid helium has many unusual properties, including zero viscosity (the ability to flow without dissipating energy) and the existence of quantized vortices. It was quickly realized that the superfluidity was due to partialBose–Einstein condensation of the liquid. In fact, many of the properties of superfluid helium also appear in the gaseous Bose–Einstein condensates created by Cornell, Wieman and Ketterle (see below). Superfluid helium-4 is a liquid rather than a gas, which means that the interactions between the atoms are relatively strong; the original theory of Bose–Einstein condensation must be heavily modified in order to describe it. Bose–Einstein condensation remains, however, fundamental to the superfluid properties of helium-4. Note that helium-3, consisting of fermions instead of bosons, also enters a superfluid phase at lowtemperature, which can be explained by the formation of bosonic Cooper pairs of two atoms each (see also fermionic condensate).The first "pure" Bose–Einstein condensate was created by Eric Cornell, Carl Wieman, and co-workers at JILA on June 5, 1995. They did this by cooling a dilute vapor consisting of approximately two thousand rubidium-87 atoms to below 170 nK using a combination of laser cooling (a technique that won its inventors Steven Chu, Claude Cohen-Tannoudji, and William D. Phillips the 1997 Nobel Prize in Physics) and magnetic evaporative cooling. About four months later, an independent effort led by Wolfgang Ketterle at MIT created a condensate made of sodium-23. Ketterle's condensate had about a hundred times more atoms, allowing him to obtain several important results such as the observation of quantum mechanical interference between two different condensates. Cornell, Wieman and Ketterle won the 2001 Nobel Prize in Physics for their achievement.[5]The Bose–Einstein condensation also applies to quasiparticles in solids.A magnon in an antiferromagnet carries spin 1 and thus obeysBose–Einstein statistics. The density of magnons is controlled by an external magnetic field, which plays the role of the magnon chemical potential. This technique provides access to a wide range of boson densities from the limit of a dilute Bose gas to that of a strongly interacting Bose liquid. A magnetic ordering observed at the point of condensation is the analog of superfluidity. In 1999 Bose condensation of magnons was demonstrated in the antiferromagnet TlCuCl3.[6] The condensation was observed at temperatures as large as 14 K. Such a high transition temperature (relative to that of atomic gases) is due to the greater density achievable with magnons and the smaller mass (roughly equal to the mass of an electron). In 2006, condensation of magnons in ferromagnets was even shown at room temperature,[7]where the authors used pumping techniques.Velocity-distribution data graph（速度分布数据图）Velocity-distribution data of a gas of rubidium atoms, confirming the discovery of a new phase of matter, the Bose–Einstein condensate. Left: just before the appearance of a Bose–Einstein condensate. Center: just after the appearance of the condensate. Right: after further evaporation, leaving a sample of nearly pure condensate.In the image accompanying this article, the velocity-distribution data indicates the formation of a Bose–Einstein condensate out of a gas of rubidium atoms. The false colors indicate the number of atoms at each velocity, with red being the fewest and white being the most. The areas appearing white and light blue are at the lowest velocities. The peak is not infinitely narrow because of the Heisenberg uncertainty principle: since the atoms are trapped in a particular region of space, their velocity distribution necessarily possesses a certain minimum width. This width is given by the curvature of the magnetic trapping potential in the given direction. More tightly confined directions have bigger widths in the ballistic velocity distribution. This anisotropy of the peak on the right is a purely quantum-mechanical effect and does not exist in the thermal distribution on the left. This famous graph served as the cover-design for 1999 textbook Thermal Physics by Ralph Baierlein.[8]Vortices（涡旋）As in many other systems, vortices can exist in BECs. These can be created, for example, by 'stirring' the condensate with lasers, or rotating the confining trap. The vortex created will be a quantum vortex. Thesephenomena are allowed for by the non-linear term in the GPE. As the vortices must have quantised angular momentum, the wavefunction will be of the form where ρ,z and θ are as in thecylindrical coordinate system, and is the angular number. To determine φ(ρ,z), the energy of must be minimised, according to the constraint . This is usually done computationally, however in a uniform medium the analytic formwhere:is d ensity far from the vortex,is h ealing length of the condensate.demonstrates the correct behavior, and is a good approximation.A singly-charged vortex () is in the ground state, with its energy εv given bywhere:is t he farthest distance from the vortex considered.(To obtain an energy which is well defined it is necessary to include this boundary b.)For multiply-charged vortices () the energy is approximated bywhich is greater than that of singly-charged vortices, indicating that these multiply-charged vortices are unstable to decay. Research has,however, indicated they are metastable states, so may have relatively long lifetimes.Closely related to the creation of vortices in BECs is the generation of so-called dark solitons in one-dimensional BECs. These topological objects feature a phase gradient across their nodal plane, which stabilizes their shape even in propagation and interaction. Although solitons carry no charge and are thus prone to decay, relativelylong-lived dark solitons have been produced and studied extensively.[9] Unusual characteristicsFurther experimentation by the JILA team in 2000 uncovered a hitherto unknown property of Bose–Einstein condensates. Cornell, Wieman, and their coworkers originally used rubidium-87, an isotope whose atoms naturally repel each other, making a more stable condensate. The JILA team instrumentation now had better control over the condensate so experimentation was made on naturally attracting atoms of another rubidium isotope, rubidium-85 (having negative atom-atom scattering length). Through a process called Feshbach resonance involving a sweep of the magnetic field causing spin flip collisions, the JILA researchers lowered the characteristic, discrete energies at which the rubidium atoms bond into molecules, making their Rb-85 atoms repulsive and creating a stable condensate. The reversible flip from attraction to repulsion stems from quantum interference among condensate atoms which behave as waves.When the scientists raised the magnetic field strength still further, the condensate suddenly reverted back to attraction, imploded and shrank beyond detection, and then exploded, blowing off about two-thirds of its 10,000 or so atoms. About half of the atoms in the condensate seemed to have disappeared from the experiment altogether, not being seen either in the cold remnant or the expanding gas cloud.[10] Carl Wieman explained that under current atomic theory this characteristic of Bose–Einstein condensate could not be explained because the energy state of an atom near absolute zero should not be enough to cause an implosion; however, subsequent mean field theories have been proposed to explain it.Because supernova explosions are also preceded by an implosion, the explosion of a collapsing Bose–Einstein condensate was named "bosenova", a pun on the musical style bossa nova.The atoms that seem to have disappeared almost certainly still exist in some form, just not in a form that could be accounted for in that experiment. Most likely they formed molecules consisting of two bonded rubidium atoms.The energy gained by making this transition imparts a velocity sufficient for them to leave the trap without being detected.Current researchCompared to more commonly encountered states of matter, Bose–Einstein condensates are extremely fragile. The slightest interaction with the outside world can be enough to warm them past the condensation threshold, eliminating their interesting properties and forming a normal gas. It is likely to be some time before any practical applications are developed.Nevertheless, they have proven useful in exploring a wide range of questions in fundamental physics, and the years since the initial discoveries by the JILA and MIT groups have seen an explosion in experimental and theoretical activity. Examples include experiments that have demonstrated interference between condensates due to wave-particle duality,[11]the study of superfluidity and quantized vortices,[12]and the slowing of light pulses to very low speeds using electromagnetically induced transparency.[13]Vortices in Bose–Einstein condensates are also currently the subject of analogue gravity research, studying the possibility of modeling black holes and their related phenomena in such environments in the lab. Experimentalists have also realized "optical lattices", where the interference pattern from overlapping lasers provides a periodic potential for the condensate. These have been used to explore the transition between a superfluid and a Mott insulator,[14] and may be useful in studying Bose–Einstein condensation in fewer than three dimensions, for example the Tonks-Girardeau gas.Bose–Einstein condensates composed of a wide range of isotopes have been produced.[15]Related experiments in cooling fermions rather than bosons to extremely low temperatures have created degenerate gases, where the atoms do not congregate in a single state due to the Pauli exclusion principle. To exhibit Bose–Einstein condensation, the fermions must "pair up" to form compound particles (e.g. molecules or Cooper pairs) that are bosons. The first molecular Bose–Einstein condensates were created in November 2003 by the groups of Rudolf Grimm at the University of Innsbruck, Deborah S. Jin at the University of Colorado at Boulder and Wolfgang Ketterle at MIT. Jin quickly went on to create the first fermionic condensate composed of Cooper pairs.[16]In 1999, Danish physicist Lene Vestergaard Hau led a team from Harvard University which succeeded in slowing a beam of light to about 17 metresper second and, in 2001, was able to momentarily stop a beam. She was able to achieve this by using a superfluid. Hau and her associates at Harvard University have since successfully transformed light into matter and back into light using Bose–Einstein condensates: details of the experiment are discussed in an article in the journal Nature, 8 February 2007.[17]Subtleties(微妙之处)Up to 2004, using the above-mentioned "ultralow temperatures",Bose–Einstein condensates had been obtained for a multitude of isotopes involving mainly alkaline and alkaline earth atoms (7Li, 23Na, 41K, 52Cr,84St, 85Rb, 87Rb, 133Cs and 174Yb). Not astonishingly, condensation research was finally successful even with hydrogen, although with the aid of special methods. In contrast, the superfluid state of the bosonic 4He at temperatures below the temperature of 2.17 K is not a good example of Bose–Einstein condensation, because the interaction between the 4He bosons is simply too strong, so that at zero temperature, contrary to Bose–Einstein theory, only 8% rather than 100% of the atoms are in the ground state. Even the fact that the above-mentioned alkaline gases show bosonic, rather than fermionic behaviour, as solid state physicists or chemists would expect, is based on a subtle interplay of electronic and nuclear spins: at ultralow temperatures and corresponding excitation energies, the half-integer (in units of ) total spin of the electronic shell and the also half-integer total spin of the nucleus of the atom are coupled by the (very weak) hyperfine interaction to the integer (!) total spin of the atom. Only the fact that this last-mentioned total spin is integral causes the ultralow-temperature behaviour of the atom to be bosonic, whereas the "chemistry" of the systems at room temperature is determined by the electronic properties, i.e. is essentially fermionic, since at room temperature thermal excitations have typical energies which are much higher than the hyperfine values. (Here one should remember the spin-statistics theorem of Wolfgang Pauli, which states that half-integer spins lead to fermionic behaviour, e.g., the Pauli exclusion principle forbidding that more than two electrons possess the same energy, whereas integer spins lead to bosonic behaviour, e.g., condensation of identical bosonic particles in a common ground state.)The ultralow temperature requirement of Bose–Einstein condensates of alkali metals does not generalize to all types of Bose–Einstein condensates. In 2006, physicists under S. Demokritov in Münster, Germany,[18]found Bose–Einstein condensation of magnons (i.e. quantized spinwaves) at room temperature, admittedly by the application of pump processes。