Nonlinear effects for Bose Einstein condensates in optical lattices

Anderson localization of a non-interacting Bose-Einstein condensateG. Roati, C. D’Errico, L. Fallani, M. Fattori1, C. Fort, M. Zaccanti, G. Modugno, M. Modugno2 & M. Inguscio LENS and Physics Department,Università di Firenze, and INFM-CNR, Via Nello Carrara 1,50019 Sesto Fiorentino, Italy1Museo Storico della Fisica e Centro Studi e Ricerche “E. Fermi”, Roma, Italy2Dipartimento di Matematica Applicata, Università di Firenze, Italy – BEC-INFM Center, Univ. Trento, ItalyOne of the most intriguing phenomena in physics is the localization of waves in disordered media1. This phenomenon was originally predicted by Anderson, fifty years ago, in the context of transport of electrons in crystals2. Anderson localization is actually a much more general phenomenon3, and it has been observed in a large variety of systems, including light waves4,5. However, it has never been observed directly for matter waves. Ultracold atoms open a new scenario for the study of disorder-induced localization, due to high degree of control of most of the system parameters, including interaction6. Here we employ for the first time a non-interacting Bose-Einstein condensate to study Anderson localization. The experiment is performed with a one-dimensional quasi-periodic lattice, a system which features a crossover between extended and exponentially localized states as in the case of purely random disorder in higher dimensions. Localization is clearly demonstrated by investigating transport properties, spatial and momentum distributions. We characterize the crossover, finding that the critical disorder strength scales with the tunnelling energy of the atoms in the lattice. Since the interaction in the condensate can be controlled at will, this system might be employed to solve open questions on the interplay of disorder and interaction7and to explore exotic quantum phases8,9.The transition between extended and localized states originally studied by Anderson for non-interacting electrons has actually never been observed in crystals due to the high electron-electron and electron-phonon interactions2. Researchers have therefore turned their attention to other systems, where interactions or nonlinearities are almost absent. Evidence of the Anderson localization for light waves in disordered media has been provided by an observed modification of the classical diffusive regime, featuring a conductor-insulator transition4,5. However, clear understanding of the interplay between disorder and nonlinearity is considered a crucial task in contemporary physics. First effects of weak nonlinearities have been recently shown in experiments with light waves in photonic lattices10,11. The combination of ultracold atoms and optical potentials is offering a novel platform to study disorder-related phenomena, where most of the relevant physical parameters, including interaction, can be controlled6,8. The introduction of laser speckles12and quasi-periodic optical lattices9 have made possible the investigation of the physics of disorder. The investigations reported so far have explored either quantum phases induced by interaction9or regimes of weak interaction where however the observation of Anderson localization was precluded either by the size of disorder or by delocalizing effects of nonlinearity12-16.Figure 1 | The quasi-periodic optical lattice. a, Sketch of the quasi-periodic potential realized in the experiment. The hopping energy J describes the tunnelling between different sites of the primary lattice and 2∆ is the maximum shift of the on-site energy induced by the secondary lattice. The lattice constant is 516 nm. b, Typical density plot of an eigenstate of the bichromatic potential, as a function of ∆/J (vertical axis). For small values of ∆/J the state is delocalized over many lattice sites. For ∆/J≥7 the state becomes exponentially localized on lengths smaller than the lattice constant.In this work, we employ for the first time a Bose-Einstein condensate where the interaction can be tuned independently from the other parameters17, to study localization purely due to disorder. We study localization in a one-dimensional (1D) lattice perturbed by a second, weak incommensurate lattice, which constitutes the first experimental realization of the non-interacting Harper18or Aubry-André model19. This quasi-periodic system displays a transition from extended to localized states analogous to the Anderson transition, already in 1D20,21whereas in case of pure random disorder, dimensions higher than two would be needed22. We clearly observe this transition by studying transport, and both spatial and momentum distributions, and we verify the scaling behaviour of the critical disorder strength.Figure 2 | Probing the localization with transport . a , In-situ absorption images of the BEC diffusing along the quasi-periodic lattice for different values of ∆ and J/h=153 Hz. For ∆/J t 7 the size of the BEC remains stacked to its original value, reflecting the onset of localization. b , Rms size of the condensate for three different values of J, at a fixed diffusion time of τ =750 ms, vs the rescaled disorder strength ∆/J. The dashed line indicates the initial size of the condensate. The onset of localization appears in all three cases in the same range of values of ∆/J.Our system is described by the Aubry-André hamiltonian:(1)||)2cos(|)||(|11∑∑〉〈+∆+〉〈+〉〈=++mmm m m m m m w w m w w w w J H φπβwhere |w m Ú is the Wannier state localized at the lattice site m , J is the site-to-site tunnelling energy, ∆ is the strength of disorder and β=k 2/k 1 is the ratio between the two lattice wavevectors. In the experiment, the two relevant energies J and ∆, can be controlled independently, by changing the height of the primary and secondary lattice potential, respectively. For a maximally incommensurate ratio β=(√5−1)/2, the model exhibits a sharp transition from extended to localized states at ∆/J =2 18,19,21. For the actual experimental parameters, β=1.1972…, the transition is broadened, and shifted towards larger ∆/J, see Fig.1. Due to the quasi-periodic nature of the potential, these localized states appear approximately every 5 sites (2.6µm).The non-interacting Bose-Einstein condensate is prepared by sympathetically cooling a cloud of interacting 39K atoms in an optical trap, and then by tuning the s-wave scattering length almost to zero by means of a Feshbach resonance 17,23 (see Methods Summary). The spatial size of the condensate can be controlled by changing the harmonic confinement provided by the trap. For most of the measurements the size along the direction of the lattice is σ~5 µm. The quasi-periodic potential is realized by using two lasers in standing-wave configuration16. The Gaussian shape of the laser beams forming the primary lattice provides also radial confinement of the condensate in absence of the harmonic trap.In a first experiment we have investigated transport, by abruptly switching off the main harmonic confinement and letting the atoms expand along the 1D bichromatic lattice. We detect the spatial distribution of the atoms after increasing evolution times by absorption imaging, see Fig.2a. In a regular lattice (∆/J=0) the eigenstates of the potential are extended Bloch states, and the system expands ballistically. In the limit of large disorder (∆/J t 7) we observe no diffusion, since in this regime the condensate can be described as the superposition of several localized eigenstates, whose individual extension is smaller than the initial size of the condensate. In the crossover between these two regimes we observe a ballistic expansion with reduced speed. This crossover is summarized in Fig.2b, where we report the width of the atomic distribution at a fixed evolution time of 750 ms vs the rescaled disorder strength ∆/J, for three different values of J. In all three cases, the system enters the localized regime at the same disorder strength, providing a compellingevidence of the scaling behaviour intrinsic in the model (1).Figure 3 | Observing the nature of the localized states. a , b Experimental profiles and fitting function f α(x) (red line) for ∆/J=1 (a) and ∆/J=9 (b). Note the vertical log scale. The blue line represents a Gaussian fit, α=2. c , Dependence of the fitting parameter α on ∆/J, indicating a transition from a Gaussian to an exponential distribution.In this regime, the eigenstates of the hamiltonian (1) are exponentially localized. We have therefore analyzed the tails of the spatial distributions with an exponential function of the form f α(x)=Aexp(-|(x-x 0)/l|α) +B, with the exponent α being a fitting parameter. Two examples of this analysis for weak and strong disorder are shown in Fig. 3a. The exponent α, shown in Fig. 3b, features a smooth crossover from 2 to 1 for increasing ∆/J, signalling the onset of an exponential localization. We note instead that in the radial direction, where the system is just harmonically trapped, the spatial3distribution is always well fitted by a Gaussian function (α=2).Information on the eigenstates of the system can also be extracted from the analysis of the momentum distribution of the stationary atomic states in the presence of the harmonic confinement. The width of the axial momentum distribution P(k) is inversely proportional to the spatial extent of the condensate in the lattice. We measure it by releasing the atoms from the lattice and imaging them after a ballisticexpansion.Figure 4 | Momentum distribution. a,b Experimental and theoretical momentum distributions P(k) of the BEC released from the quasi-periodic potential for increasing values of ∆/J (0, 1.1, 7.2, 25, from top to bottom). When ∆=0, we observe the typical interference pattern of a regular lattice. By increasing ∆, we observe peaks at the beating between the two lattices. At the onset of localization ∆/J ≅6 the width of the momentum distribution becomes of the order of the Brillouin zone, corresponding to the localization of wavefunction on a single lattice site. The modulation on the top of the density profiles is due to the interference between several localized states. c , Rms size of central peak of P(k) for three different values of J, vs ∆/J. The experimental data follow a unique scaling behaviour, in good agreement with the theoretical prediction (continuous line). d , Visibility of the interference pattern vs ∆/J. The visibility suddenly drops in correspondence of the localization of the eigenstates on distances smaller that the lattice period at ∆/J ≈6.In Fig. 4, we show examples of the experimental momentum distributions with the model predictions, in excellent agreement. Without disorder, we observe the typical grating interference pattern with three peaks at k=0,±2k 1 reflecting theperiodicity of the primary lattice. The tiny width of the peak at k=0 indicates that the wavefunction is spread over many lattice sites 24. For weak disorder strength, the eigenstates of (1) are still extended, and additional momentum peaks appear at a distance ±2(k 1±k 2) around the main peaks, corresponding to the beating of the two lattices. By further increasing ∆/J, P(k) broadens and its width eventually becomes comparable with that of the Brillouin zone, k 1, signalling that the extension of the localized states becomes comparable with the lattice spacing. From the theoretical analysis of the Aubry-André model, we have a clear indication that in this regime the eigenstates are exponentially localized on individual lattice sites. Note that the side peaks in the two bottom profiles of Fig. 4a-b indicate that the localization is non-trivial, i.e. the tails of the eigenstates extend over several lattice sites even for large disorder. In Fig. 4c, we present the rms width of the central peak of P(k) as a function of ∆/J for three different values of J. The three data sets lie on the same line, confirming the scaling behaviour of the system. A visibility of the interference pattern, V=(P(2k 1)-P(k 1))/(P(2k 1)+P(k 1)), can be defined to highlight the appearance of a finite population of the momentum states ±k 1, and therefore the onset of exponential localization with an extension comparable with the lattice spacing. In Fig. 4d we show the visibility extracted from the same data above. Experiment and theory are again in good agreement, and feature a sudden drop of the visibility for ∆/J ≈6.Figure 5 | Interference of localized states. Momentum distribution of the condensate prepared in a disordered lattice with ∆/J~10 for different values of the harmonic confinement. a, Profile of a single localized state (initial spatial size of the condensate, σ =1.2 µm), b, Interference of two localized states (σ =1.2 µm), c, Three states (σ =2.1 µm), d, Dislocated interference pattern resulting from the interference between two localized states one of which containing a thermally activated vortex.Further information on the localized states can be extracted from the interference of a small number of them. This regime can be reached in the experiment by simply reducing the spatial extent of the condensate through an increase the4harmonic confinement. Typical profiles of P(k) are reported in Fig.5a-c. Depending on the confinement, we can observe one, two or three states, featuring a smooth distribution or a clear multiple-slit interference pattern. The spacing of the fringes yields a spatial separation between the localized states of about 5 sites, as expected. The independent localized states have a quasi-2D geometry, being their axial extent much smaller that the radial one. This feature makes our system an excellent playground to study the physics of quasi-2D systems25, recently investigated with widely spaced optical lattices26. We also observe (Fig. 5d) interference patterns which present a dislocation, possibly produced by thermal activation of a vortex in one of the two localized states26, in our case for non interacting atoms.In this work we have observed Anderson localization of coherent non-interacting matter-waves. Future studies might reveal how a weak, controllable interaction affects the observed localization transition. More in general, the high theoretical and experimental control of this system opens a novel scenario for the study of exotic quantum phases arising from the interplay between interaction and disorder 6,8,27.Note. A related work is being carried out in the group of A. Aspect.Acknowledgements.We thank J. Dalibard for stimulating discussions, S. Machluf for contributions and all the colleagues of the Quantum Gases group at LENS. This work has been supported by MIUR, EU (MEIF-CT-2004-009939), INFN, Ente CRF, and IP SCALA.Author information.Correspondence and requests for materials should be addressed to M. I. (e-mail:inguscio@lens.unifi.it). MethodsNon-interacting Bose-Einstein condensate. A sample of laser-cooled 39K atoms is further cooled by thermal contact with 87Rb atoms in a magnetic trap to about 1µK. The potassium sample is and transferred into an optical trap generated by crossing on the horizontal plane two focused laser beams, and the polarized in the absolute ground state |F=1,m F=1〉. A further evaporation stage is then performed in presence of a homogenous magnetic field to access a broad Feshbach resonance17,28. In this phase the s-wave scattering length is large and positive (a≈180a0, a0=0.529×10-10 m) allowing the formation of a stable condensate composed by about 105atoms at T/Tc<0.1 (Tc=100nK). Once the condensate is produced, we adiabatically bring the magnetic field to 350.0 G, where the residual scattering length is of the order of 0.1 a0 23. This corresponds to an atom-atom interaction energy U/J~10-5.Quasi-periodic optical lattice.The lattice is created by superimposing two standing waves at incommensurate wavelengths. The primary lattice is generated by a single-mode Yb-YAG laser at 1032nm, whose linewidth and intensity are actively stabilized. The secondary lattice is obtained by a single-mode Ti:Sa laser at 862nm. The bichromatic lattice is mildly focused on the condensate with a beam-waist of about 150 µm. The lattice depths are independently adjusted by means of acousto-optical modulators, and are calibrated by means of Bragg diffraction. The estimated relative uncertainty at 3σ level on the depths is 10%. To prepare the condensate in the quasi-periodic potential, we raise the intensity of the two lattices from zero to the final value in about 100 ms, using s-shaped ramps.Analyzing the atomic distributions. The atomic samples are imaged on a CCD camera through a lens system with a spatial resolution of the order of 5 µm. The images analyzed in Figs.2-3 were recorded in situ, i.e. immediately after release from the trap. The analysis of the profiles shown in Fig. 3, reporting exponential localization, was done by integrating the atomic density distributions along the radial direction. The central 20 % of the signal was then dropped and only the remaining tails were fitted. This procedure reduces possible Gaussian broadening effects due to the transfer function of the imaging system and to the fact that we populate a few localized states. The profiles in Figs. 4-5 were obtained after a long ballistic expansion (25.5 ms) to reduce the contribution of the initial spatial distribution. For the quantitative analysis in Fig. 4c,d, also the width of the spatial distribution was however measured and subtracted from the data. Vertical error bars in Figs.2-5 represent the standard deviation of three to four independent measurements.1. Anderson, P. W. Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492-1505 (1958).2. Lee, P. A. & Ramakrishnan, T. V. Disordered electronic systems, Rev. Mod. Phys.57, 287-337 (1985).3. Kramer, B. & MacKinnon, A. Localization: theory and experiment, Rep. Prog. Phys. 56, 1469-1564 (1993).4. Van Albada, M. P. & Lagendijk, A. 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Effect of optical disorder and single defects on the expansion of a Bose-Einstein condensate in a one-dimensional waveguide, Phys. Rev. Lett.95, 170410 (2005).15. Schulte, T., Drenkelforth, S., Kruse, J., Ertmer, W., Arlt, J., Sacha, K. Zakrzewski, J. & Lewenstein, M. Routes towards Anderson-like localization of Bose-Einstein condensates in disordered optical lattices, Phys. Rev. Lett.95, 170411 (2005).16. Lye, J. E., Fallani, L., Fort, C., Guarrera, V., Modugno, M., Wiersma, D. S. & Inguscio, M. Effect of interactions on the localization of a Bose-Einstein condensate in a quasi-periodic lattice, Phys. Rev. A75, 061603 (2007).17. Roati, G., Zaccanti, M., D'Errico, C., Catani, J., Modugno, M., Simoni, A., Inguscio, M. & Modugno, G. 39K Bose-Einstein condensate with tunable interactions, Phys. Rev. Lett.99, 010403 (2007).18. Harper, P. G. Single band motion of conduction electrons in a uniform magnetic field, Proc. Phys. Soc. A68, 874-978 (1955).19. Aubry, S. & André, G. 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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 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在玻色—爱因斯坦凝聚态中类Dicke模型的相变赵秀琴【摘要】自旋和轨道耦合为中性的超冷原子在玻色—爱因斯坦凝聚态(BEC)中的玻色系统提供了研究的机会.文章研究此类系统的相变和基态性质.首先将它映射到著名的量子光学中的Dicke模型,Dicke模型描述了一个原子系综和单模光场之间的相互作用.Dicke模型的中心问题是预测了超辐射相和一个正常相之间的量子相变.我们研究在自旋和轨道耦合中的类似Dicke模型的量子相变.采用平均场自旋相干态法,特别是考虑原子之间的相互作用,计算出描述系统的相变点和基态性质的物理量如平均光子数,平均基态能量、两种自旋激化等物理量的解析表达式,得到在相变前后物理量变化的趋势图并与实验结果相比较.【期刊名称】《太原师范学院学报（自然科学版）》【年(卷),期】2015(014)003【总页数】5页(P58-62)【关键词】自旋和轨道耦合;相变;基态特性【作者】赵秀琴【作者单位】太原师范学院物理系,山西太原030031【正文语种】中文【中图分类】O48自旋和轨道耦合是量子粒子的自旋和它的动量的一种相互作用,在物理系统中是普遍存在的．在玻色-爱因斯坦凝聚态系统中,实验可以精确地控制超冷原子来研究自旋和轨道耦合相互作用量子多体系统．提出了在玻色-爱因斯坦凝聚态中用中性原子通过控制外场激光场来实现不同类型的自旋轨道耦合[1]．NIST的I. B. Spellman 小组通过一对耦合的激光在超冷87Rb原子实现了Rashba 和Dresselhaus自旋轨道耦合[2].在玻色-爱因斯坦凝聚态中,所有的原子都占据同一个量子态,因此基态性质具有特殊性,许多不曾发现过的多体现象有可能发生．例如,在上述自旋和轨道耦合下,玻色-爱因斯坦凝聚态通过调节由两组不同动量的非正交原子缀饰自旋态之间的相互作用,可以实现从自旋相分离态(简写为SP)到单个最小值相(简写为SMP)之间的量子相变[3-5]．潘建伟小组在2012年通过测量自旋和动量振荡的振幅比从实验上观察到了理论所预测的量子相变[6]．在本文中,首先,根据实验,获得类似于Dicke模型的哈密顿量．通过改变拉曼耦合强度,系统可以从一个自旋极化相,发生了非零准动量SP,与零准动量自旋平衡相SMP 的量子相变,类似于在Dicke模型中从超辐射过渡到正常相的量子相变．利用平均场自旋相干态法,计算相变点,每个物理量在基态时的解析表达式,并研究物理量的变化趋势．图1是超冷87Rb原子被囚禁在xy平面中,ωz是强囚禁势在z方向的频率, 一对Raman入射的激光和x轴成π/4 角, 其拉比(Raman)频率分别是Ω1和Ω2．在拉比激光的作用下,两个超精细基态|F=1,mF=-1〉(|↑〉)和|F=1,mF=0〉(|↓〉)之间就会形成对动量敏感的耦合作用．在缀饰态基矢和下(其中和分别为两束拉比激光的波矢),可以构建相当于凝聚态物理中一维Rashba和Dresselhaus自旋轨道耦合的耦合项,即有效自旋轨道耦合项．相关的非线性Gross-Pitsevskii动力学方程(GP 方程)是[7]在方程(1)中,ψ=(ψ↑,ψ↓)T表示在缀饰态表象中的一对正交波函数．代表谐振子囚禁原子的势能项,m是原子的质量,ωx,ωy 分别是x和y方向的囚禁频率．其自旋轨道耦合项写成：HSOC=2γ0pxσz+ħΩσx其中γ0=ħħ/(mλ)是自旋轨道耦合强度,λ为激光的波长,有效的拉比频率Ω=Ω1Ω2/Δ,σx(z)是泡利矩阵．原子相互碰撞的平均作用为其中相同和不同自旋之间相互作用常数分别为g↑↑=g↑↓=4πħ2N(c0+c2)/(maz)和g↓↓=4πħ2Nc0/(maz),c0和c2为s波散射长度是原子数．所有的超冷原子在强非线性碰撞相互作用下都被限制在相同的基态上,且每个原子具有完全相同的动量．引入玻色算符,不考虑超冷原子与y方向的玻色子模之间的相互作用,通过简单计算,可以获得类似于单模Dicke[8]类型的描述玻色-爱因斯坦凝聚态中自旋和轨道耦合的有效哈密顿量其中a†a是谐振子模†和是集体自旋算符．ψ↑和ψ↓是自旋组分中不同的场算符, q=(g↑↑+g↓↓-2g↑↓)/4ħ是原子间的有效相互作用,很明显,哈密顿量(4)中的〈Jz〉代表自旋不同组分间的原子布局数,在实验中可测得．该系统的性质可以用单模Dicke类型哈密顿量来描述,取自然单位ħ=1,其中ω=Nωx为与原子数相关的囚禁频率是有效的自旋轨道耦合强度．在当前的实验条件下,囚禁频率ωx在NIST实验中可调为10 Hz的数量级,当原子数为N=1.8×105,囚禁频率ω的数量级可调为MHz,拉比激光的波长λ=804.1 nm,所以,参量γ2为kHz的数量级．有效的拉比频率Ω的可调范围可从0到MHz 量级．另外,由于c0=100.86aB,c2=-0.46aB(aB为玻尔半径),所以g↑↑=g↓↓=g↑↓,在这种情况下有效原子相互作用q=0．因此在NIST的实验中原子间有效相互作用q不影响系统的能级结构．但在Feshbach共振时,可通过调节有效原子相互作用的强度在Feshbach共振点附近,其大小甚至可达MHz的量级．本文讨论在Feshbach共振情况下,讨论原子之间的相互作用,对相变和物理量的影响．为方便起见,以作为能量的自然单位,其数量级为kHz.用平均场理论来求相变的关键点,假设基态波函数为[9-11]这里定义相干态a|α〉=α|α〉,自旋相干态定义为对于自旋为1/2的原子j=N/2和θ∈[0,2π]对于原子的平均值方程(5)的哈密顿量的基态能量u和v分别是α的实部和虚部．对应于E(θ,α)的最小值,分别对u和v求偏微分,并令其为零,可得将(10)式和(11)式代入(9)式得,每个原子的平均基态能量变为对于E(θ)最小值有可得cosθ=0,定义Ωc=γ2-q可得两个不同的区域．1)Ω>Ωc,平均场能量只有一个最小值,属于单个最小值相SMP区域．这时,光子数np=v2=0．对(13)式再次求导:并将(16)式代入并且cos2θ=2cos2θ-1=-1,代入得：态是稳定态与Dicke模型中的自旋平衡正常相对应．2)Ω<Ωc,能量最小值对应于(15)式,对应的有两个可能带入取得,对应于自旋相分离态SP范围．对应于凝聚态会有两个最小值的带．对(13)式再次求导,并将(19)式代入得:显然态属于稳定态与Dicke模型中的超辐射相相对应．在图2中当有效的自旋轨道耦合强度(a)γ2=1.8EL(b) γ2=2.6EL是定值时,相变点是一条直线,超辐射相的区域与原子间相互作用力有关,当原子间的相互作用力是排斥力时,即q>0时,区域将减少,当原子间的相互作用力是吸引力即q<0,超辐射的区域增加,并随着有效的自旋轨道耦合强度γ2的增大,如图2(b)相变点向右移．3.1 基态能量的二阶导数和每个原子的平均基态能量每个原子的平均基态能量的二阶导数的分布为:每个原子的平均基态能量分布为:在图3中取γ2=1.8EL(a)可看出基态能量的分布随着有效的Rabi频率Ω的增加而增加．基态能量随着有效的原子之间的相互作用q的增大而增大．当q>0时基态能量较高,当q<0,基态能量较低,在有效的Rabi频率Ω较大时基态能量不受影响．(b)基态能量的泛函的二阶导数在两个区域内都是大于零,说明在这两个区域都是稳定态．3.2 两个方向的自旋极化为在图4中取γ2=2.6EL(a)中〈Jx〉/N随Ω的变化,并且随着γ2的增大,相变点向右移,特别是在(b)中〈Jz〉/N有两个可能值,对应于凝聚态中的两个最小值．总之,我们将凝聚态中的自旋和轨道相互作用中的量子相变和标准的Dicke模型中的量子相变相类比,得出了一维自旋和轨道相互作用,特别是考虑原子之间的相互作用时,用平均场理论可得出类似于标准Dicke模型的量子相变点,光子数分布,基态能量分布和自旋极化的分布情况,并用图表示出来,这种方法非常简洁明了,这与参考文献[10]是一致的．当然我们还有待于考虑失谐的情况,类似于参考文献[12]．[1] LIN Y J,GARCIA K J, SPIELMAN I B.Spin-orbit coupled Bose-Einstein condensates[J].Nature,20114,71:83-86[2] KATO Y K,MYERS R C,GOSSARD A C,et al.Observation of the spin Hall effect in semi-conductors[J].Science,2004,306,1910-1913[3] GALITSKI V,SPIELMAN I B.Spin{orbit coupling in quantumgases[J].Nature,2013,494:11841[4] JI S C, ZHANG Jinyi, ZHANG Long,et al.Experimental determination of the finite temperature phase diagram of a spin-orbit coupled Bosegas[J].Nat.Phys,2014(10):314[5] HO T L,ZHANG S Z.Bose-Einstein condensates with spin-orbit interaction[J].Phys. Rev.Lett,2011,107:150403[6] ZHOU X F,LI Y,CAI Z,et al.Unconventional states of bosons with the synthetic spin-orbit coupling[J].J.Phys.B:At.Mol.Opt.Phys,2013,46:134001 [7] LIAN Jinling,ZHANG Yuanwei,LIANG J Q,et al.Thermodynamics of spin-orbit coupled Bose-Einstein condensates[J].Phys.Rev.A,2012,86:063620 [8] DICKE R H.Coherence in Spontaneous RadiationProcesses[J].Phys.Rev,1954,93:99[9] ZHANG J Y,JI S C,CHEN Z,ZHANG L,et al.Collective Dipole Oscillations ofa Spin-Orbit Coupled Bose-EinsteinCondensate[J].Phys.Rev.Lett,2012,109:115301[10] CHRIS Hamner, QU Chunlei, ZHANG Yongping,et al.Dicke-type phase transition in a spin-orbit-coupled Bose-Einsteincondensate[J].Ncomms,2014,5023:1-8[11] ALVERMANN L,ALVERMANN,FEHSKE A,Quantum H.phase transition in the Dicke model with critical and non-criticalentanglement[J].Phys.Rev.A,2012,85:043821[12] ZHAO X Q,LIU N,LIANG J Q.Nonlinear atom-photon-interaction-induced population inversion and inverted quantum phase transition of Bose-Einstein condensate in an optical cavity[J].Phys.Rev.A,2014,90:023622。

多体相互作用下非线性双能级体系的薛定谔方程魏秀芳;张国恒【摘要】能级间的Landau-Zener隧穿是量子力学中的一个基本过程.在非线性双能级体系的基础上,进一步讨论非线性双能级体系在考虑多体相互作用下的薛定谔方程.利用考虑多体相互作用下的薛定谔方程,可以进一步研究同时考虑两体和多体相互作用时玻色-爱因斯坦凝聚体的隧穿率.【期刊名称】《北华大学学报（自然科学版）》【年(卷),期】2013(014)005【总页数】3页(P539-541)【关键词】非线性;Landau-Zener隧穿;多体相互作用;薛定谔方程【作者】魏秀芳;张国恒【作者单位】兰州城市学院培黎工程技术学院,甘肃兰州730070;西北民族大学电子材料国家民委重点实验室,甘肃兰州730030【正文语种】中文【中图分类】O469玻色-爱因斯坦凝聚体(BEC)的许多特性已做了大量的理论和实验研究，如在光晶格中BEC的Landau-Zener隧穿[1-3]、布朗赫振荡[4]、自俘获现象[5-8]、非线性两能级体系中的Rose-Zener转换[9]、BEC在加速光晶格中的动力学隧穿特性[10]、光晶格中BEC的自俘获现象[11]、D维光晶格中的双组分BEC等[12].能级间的Landau-Zener隧穿是量子力学中的一个基本过程，依赖于外部参数的一个双能级体系有一个明显的特征，即对于足够大的相互作用力，在绝热情况下出现了非零的隧穿概率.而且相互作用有助于跃迁概率的增大，并存在一个甚至在绝热情况下使跃迁概率变为非零的强相互作用的临界值.对多能级体系也有类似的特性.本文主要分析了BEC能级间的Landau-Zener隧穿在考虑多体相互作用时，描述非线性双能级体系的薛定谔方程，以便进一步研究多体相互作用参数对隧穿概率的影响.在一个光晶格中，两个Bloch带之间的BEC的Landau-Zener隧穿可以用非线性双能级体系模型来解释.在低密度极限情况下，原子间的相互作用可以被忽略，这个问题在本质上相同于一个超冷但非凝聚原子体系.非线性双能级体系可以用无量纲的薛定谔方程描述[13]其中总概率a2+b2=1，哈密顿矩阵为其中：γ是能级间距；V是能级间的耦合参数；c是描述相互作用的非线性参数. 在足够低的温度下，一个BEC的运动可以被一维非线性薛定谔方程模拟：其中：m是原子的质量；kL是激光的波数；as是原子间的s-波散射长度；V0是正比于激光强度的周期势；φ2是原子在点x，t时刻的数密度；力mal表示在加速晶格框架结构中的惯性力，或表示重力场中的引力.经过下列变量代换其中：n0是BEC的平均密度.则方程(2)变成一个无量纲形式：得到假定非线性项不会打破周期性的对称性，所以能带结构不变.在k=1/2附近，即布里渊区的边缘处，波函数可以近似为其中a2+b2=1.将式(4)代入式(3)，并比较等式两边eikx和ei(k-1)x的系数，得到式(5)在k=1/2，且忽略不变量c[1+(a2+b2)/2]时，对二次动力学项线性化后与式(1)是等价的.因此，我们看到非线性二能级模型为理解在光晶格中凝聚体的隧穿提供了基础.在上述情况中，若考虑多体相互作用，则非线性双能级体系仍可以用无量纲的薛定谔方程(1)来描述，但其中的哈密顿H(γ)与式(1)中的不同.薛定谔方程(3)的形式可以表示为其中：λφ4φ项表示多体相互作用，λ是表示多体相互作用的非线性参数.下面主要计算λφ4φ项.因为φ(x，t)=a(t)eikx+b(t)ei(k-1)x，所以有式(6)右边为式(6)左边为ieikx+iei(k-1)x.比较等式两边的系数得因为所以考虑两体和多体相互作用时的哈密顿H为其中γ=αt.考虑两体和多体相互作用时，非线性双能级体系的无量纲的薛定谔方程仍然为式(1)，但其中的哈密顿H变为式(7).考虑多体相互作用时的哈密顿H相对于只考虑两体相互作用情况下的哈密顿H，出现了反映多体相互作用的参数λ.利用上述非线性双能级体系模型，结合该Hamiltonian和薛定谔方程(1)可以用来分析两个Bloch带之间考虑多体相互作用时BEC的Landau-Zener隧穿特性.【相关文献】[1] Wu Biao，Niu Qian.Nonlinear Landau-Zener Tunneling[J].Phys Rev A，2000，61：023402-1-4.[2] Jona L M，Morsch O，Cristiani M，et al.Asymmetric Landau-Zener Tunneling in a Periodic Potential[J].Phys Rev Lett，2003，91：230406-1-4.[3] 王沙，杨志安.二维周期光子晶格中的非线性Landau-Zener隧穿[J].物理学报，2009，58(2)：729-733.[4] Choi D I，Niu Q.Bloch-Oscillations and Mean-Field Effects of Bose-Einstein Condensates in 1D Optical Lattices[J].Phys Rev Lett，1999，82：2022-2025.[5] Oliver Morsch，Markus Oberthaler.Dynamics of Bose-Einstein Condensates in Optial Lattices[J].Rev Mod Phys，2006，78：179-215.[6] Liu B，Fu L B，Yang S P，et al.Josephson Oscillation and Transition to Self-Trapping for Bose-Einstin Condensates in a Triple-Well Trap[J].Phys Rev，2007，75：033601-1-9.[7] Wang G F，Ye D F，Fu L B，et ndau-Zener Tunneling in a Nonlinear Three-level System[J].Phys Rev，2006，74：033414-1-7.[8] Wang Guan-fang，Fu Li-bin，Zhao Hong，et al.Self-Trapping and Its Periodic Modulation of Bose-Einstein Condensate in Double-Well Trap[J].Acta Physica Sinica，2005，54：5003-5013.[9] Ye Di-fa，Fu Li-bin，Liu Jie.Rosen-Zener Transition in a Nonlinear Two-level System[J].Phys Rev ，2008，77：013402-1-7.[10] Tie Lu，Xue Ju-kui.Tunneling Dynamics of Bose-Einstein Condensates with Higher-Order Interactions in Optical Lattice[J].Chin Phys B，2011，20：120311-1-6.[11] Xue Ju-kui，Zhang Ai-xia，Liu Jie.Self-trapping of Bose-Einstein Condensates in Optical Lattices：the Effect of the Lattice Dimension[J].Phys Rev A，2008，77：013602-1-7.[12] Jian-Jun Wang，Ai-Xia Zhang，Ju-Kui Xue.Two-component Bose-Einstein Condensates in D-dimensional Optical Lattices[J].Phys Rev A，2010，81：033607-1-6. [13] Liu Jie，Fu Li-bin，Ou Bi-Yiao，et al.Theory of Nonlinear Landau-Zener Tunneling[J].Phys Rev，2002，66：023404-1-7.。

非线性薛定谔-基尔霍夫型系统的基态解袁瑞银; 刘艳琪【期刊名称】《《南华大学学报（自然科学版）》》【年(卷),期】2019(033)005【总页数】4页(P68-71)【关键词】薛定谔系统; 基尔霍夫项; Nehari流形; 基态解【作者】袁瑞银; 刘艳琪【作者单位】南华大学数理学院湖南衡阳421001【正文语种】中文【中图分类】O290 引言本文主要研究以下非线性薛定谔-基尔霍夫型系统基态解的存在性(1)其中参数a>0,b≥0,λi,μi>0(i=1,2)，b12=b21>0，q∈(2,3)。

若a=1,b=0时，方程组(1)变成如下形式Oliverira和Tavares在文献[1]中运用Nehari流形方法证明了以上系统存在基态解，并把该系统推广到n个方程组的情形，其中基态解的每个分量都是非平凡的。

受文献[1]的启发，本文研究一类更加复杂的带有基尔霍夫项的耦合薛定谔系统基态解的存在性问题。

基尔霍夫型方程产生于弹性力学当中,以及在人口动力学、非牛顿力学、宇宙物理、血浆问题和弹性理论等诸多领域都有广泛应用，具有深刻的物理意义，见文献[4-6]。

由于算子的出现，这类问题经常被视作为非局部问题，具体体现在系统(1)不再是一个点态等式，这不同于经典的椭圆问题。

最近，越来越多的学者通过使用变分技术等来研究基尔霍夫方程解的存在性以及解的各种性态，可以参看文献[7-9]等。

在文献[2]中,Liu和Guo利用了山路引理等方法，证明了渐近线性薛定谔-基尔霍夫型方程正解的存在性。

近年来，方程组解的存在性研究也取得了丰富的成果。

在文献[3]中，Li和Zhang研究了带有扰动项的薛定谔系统，利用扰动法证明了系统的正解的存在性。

这些对薛定谔系统解的研究为本文奠定了研究基础。

利用变分法证明系统(1)解的存在性问题实质是转化为一个能量泛函的临界点问题，关键是证明能量泛函的紧性。

由于本文在全空间R3中研究薛定谔-基尔霍夫型系统，标准的Sobolev空间缺乏相应的紧性，给研究带来了很大的困难。

B.2.1. Symmetrized Momentum Eigenstates for Bose-Einstein Particles A symmetrized state can be constructed as()11,,,,NN Pk k P k k +=∑L L= Sum of all N ! permutations of the i k's in 1,,N k k L (B.19)For example, 123123231312,,,,,,,,PP k k kk k k k k k k k k =++∑213321132,,,,,,k k k k k k k k k +++(B.20)Now, if there are n α particles with momentum k α, there'll be only !!N n αα∏ distinct terms in (B.19) so that()11,,,,NN P k k P k k +=∑L L1!,,N distinctn P k k αα⎛⎫=⨯ ⎪⎝⎭∏∑L For example, setting 12k k = in (B.20) gives 0n α= except for 12n = and 31n =, so that only 3!3!32!0!1!0!2!1!==L terms are distinct:113113131311,,,,,,,,PP k k kk k k k k k k k k =++∑113311131,,,,,,k k k k k k k k k +++()1133111312,,,,,,k k k k k k k k k =++The orthonormality of the 1-particle states implies'''''',,,,,,a b l a b l aa bb ll k k k k k k δδδ=L L L(B.21)Thus, for a given permutation ()1,,N P k k L , we have()()11,,,,!N NP k k k k n αα+=∏L Lso that()()11,,,,!!N Nk k k k N n αα++=∏L LAn orthonormal set of symmetrized states is therefore()()11,,,,S NNk k k +=L L1,,NPP k k =L (B.22)with()()11,,,,1S S N Nk k k k =L L (B.23)Now, in a sum()11,,NNk k k k ∑L L , only distinct permutations are included. Forexample, let ,,k a b c = and 3N =, we have 3327= terms,()()123123,,,,k k k a b c k k k aaa aab aac baa bab bac caa cab cac ==++++++++∑aba abb abc bba bbb bbc cba cbb cbc +++++++++ aca acb acc bca bcb bcc cca ccb ccc +++++++++which are all distinct.On the other hand, for a fixed 1,,N k k L , all the!!N n αα∏ distinct permutations ()1,,N P k k L give rise to only 1 distinct symmetrized state, i.e.,()()11,,,,S S NNk k P k k =L L . Therefore, in a sum()11,,,,NS Nk k k k ∑L L , each distinctsymmetrized state will appear!!N n αα∏ times. Since the completeness relation involves a sum with each distinct orthonormal state counted once, we have()()()111,,1!,,,,1!N S S S NNk k n k k k k N αα⎛⎫= ⎪⎝⎭∑∏L $L L(B.24)。

a r X i v :n u c l -t h /9609022v 2 17 M a r 1997NBI–96–48March,1996Bose-Einstein Correlations from Opaque SourcesHenning HeiselbergNORDITA,Blegdamsvej 17,DK-2100Copenhagen Ø.,Denmarkand Axel P.VischerNiels Bohr Institute,DK-2100,Copenhagen Ø,Denmark.AbstractBose-Einstein correlations in relativistic heavy ion collisions are very different for opaque sources than for transparent ones.The Bose-Einstein radius parameters measured in two-particle correlation functions depend sensitively on the mean free path of the particles.In particular we find that the outward radius parameter for an opaque source is smaller than the sideward radius parameter for sufficiently short duration of emission.A long duration of emission can compensate the opacity reduction of the longitudinal radius parameter and explain the experimental measurements of very similar side-and outward radius parameters.I.INTRODUCTIONBose-Einstein interference of identical particles(pions,kaons,etc.),emitted from the collision zone in relativistic heavy ion collisions shows up in correlation functions and is an important tool for determining the source at freeze-out.It is commonly assumed that the source is cylindrical symmetric around the beam axis and transparent,meaning that the detector receives particles from all over the source.The radius parameter outwards or towards the detector,R o,is found to be larger[1,2]than the radius parameter,R s, sidewards or perpendicular to the detectorR2o=R2s+β2oδτ2,(1)for particles of zero rapidity.The excess is due to the duration of emission,δτ,of the source in which particles with transverse momentum p⊥and outward velocityβo=p⊥/m⊥travel a distanceβoδτtowards the detector.Distances perpendicular to this velocity like,for example,R s are not affected by the duration of emission and thus reflect the“true”transverse size of the source,i.e.,the region over which the source can be considered homogeneous.Whenflow is included the relation(1)is still valid in the analyses of[1,2] whereas strongflow coupled to surface emission can reduce R o more than R s[3]as well as an ellipsoidal source[4,5].Experimentally the HBT radius parameters,R s,o,l,ol,are extracted from measurements of the two-pion and two-kaon correlation function by parametrizing it with the common gaussian formC(q s,q o,q l)=1+λexp(−q2s R2s−q2o R2o−q2l R2l−2q o q l R2ol).(2) Surprisingly,in relativistic heavy ion collisions the outward and sideward radius parameters are measured to be similar[6–10](and in a few cases the outward size is even measured to be smaller than the sideward size[6,7]contradicting equation(1))within experimental uncertainty.According to equation(1)this implies that particles freeze–out suddenly,δτ≪R i,as in a“flash”[11],in particular when resonance life-times are included[4,12].We want to point out that an opaque source emitting away from its surface naturally leads to R o≪R s unless the duration of emission,δτ,is very long.There is therefore a strong correlation between the direction of the emitted particles and the emission zone-contrary to cylindrically symmetric transparent sources withoutflow.In our model the measured sideward radius parameter samples the full source size,while the measured outward radius parameter sample only a small surface region of the opaque source in the direction of the emitted particles.The opaque source is inspired by hydrodynamical as well as cascade calculations of particle emission in relativistic heavy ion collisions.In hydrodynamical calculations particles freeze-out at a hypersurface that generally does not move very much transversally until the very end of the freeze–out [13].However,most hydrodynamical freeze–out mechanisms as well as the analysis in[3]do not include the directional condition that particles can only be emitted away from the surface,though strongflow at the surface has a similar effect.In cascade codes the last interaction points are also found to be distributed in transverse direction around a mean value that does not change much with time[14–16].The thickness of the emission layer is related to the particle mean free pathλmfp,and is zero in hydrodynamical calculations but several fm’s in cascade codes.We shall in section2consider the parameters R andλmfp as constants and in section3discuss effects of moving surfaces,temporally dependent emission layers and transverseflow.II.SURF ACE EMISSION AND HBT RADIUS PARAMETERSsymmetric but longitudinally expanding source.However,we add the important requirement that the source should be opaque emitting away from a surface layer of thickness∼λmfp.For the correlation function analysis of Bose-Einstein interference from a source of size R we consider two particles emitted a distance∼R apart with relative momentum q=(k1−k2)and average momentum, K=(k1+k2)/2.Typical heavy ion sources in nuclear collisions are of size R∼5fm,so that interference occurs predominantly when q∼¯h/R∼40MeV/c.Since typical particle momenta are k i≃K∼300MeV, the particles escape almost parallel(see Figure),i.e.,k1≃k2≃K≫q.The correlation function due to Bose-Einstein interference of identical particles from an incoherent source is(see,e.g.,[1])C(q,K)=1+| d4x S(x,K)e iqxd4x S(x,K),σ(O)≡ O2 − O 2.(5) The reduction factorλin equation(2)may be due to long lived resonances[12,4],coherence effects,incorrect Gamov corrections[17]or other effects.It is found to beλ∼0.5for pions andλ∼0.9for kaons.FIG.1.Cross section of the interaction region perpendicular to the longitudinal or z–direction.Particles have to penetrate a distance l out to the surface of the interaction region in order to escape and reach the detector.The bulk part of the emitted particles comes from a surface region of widthλmf p,the mean free path of the particle.Generally a particle is emitted and escapes after it has made itsfinal interaction with any of the other particles in the source.If the source is dense it is more probable to emit from the surface than from the center.In Glauber theory particles are emitted from a surface layer with a depth of order of the mean free path,λmfp,with probabilityS(x,K)∼exp(− ∞x dx′σn(x′))=exp(−l/λmfp).(6) Here the integral runs over the particle trajectory from emission point x to the detector.Assuming an average density n in the emission layer,the distance l,that the particle has to pass through is of order of the mean free path,λmfp=1/σn(see Fig.1).A similar surface layer was introduced by Grassi et al.[18] in order to improve the standard Cooper-Frye freeze-out in hydrodynamical calculations and to study the effect on transverse momentum spectra.In the following subsection we assume that the source size R and thickness of emission layerλmfp are constants in time.In section3will will generalize these results including temporal dependences in connection with thefinal freeze-out of the source.A.Spherical SourcesFirst we investigate a spherically symmetric source of radius R.Including the temporal evolution of the source emission function by S t(t),the phase space density of the source will have the approximate formS(x,K)∼Θ(R2−x2−y2−z2)e−l/λmfp S t(t).(7) For strict surface emission∗(λmfp=0)the source reduces toS(x,K)∼cosθΘ(cosθ)δ(R2−x2−y2−z2)S t(t).(8) whereθis the polar angle with respect to the outward direction along K;we will in the following orient our cartesian axes such that the x-axis is in the outward direction.The geometric factor cosθ=x/R suppresses the peripheral zones and theΘ(cosθ)=Θ(x)factor insures that particles are only emitted away from the surface,i.e.,only particles from the surface layer of the half hemisphere directed towards the detector will reach it whereas particles from the other hemisphere will interact on their passage through the source. The temporal emission is determined by S t(t).It is commonly approximated by a gaussian,S t(t)∼exp(−(t−t0)2/2δt2),around the source mean life-time,τ0with width orfluctuation,δτ,which is the duration of emission.These gaussian parameters approximate the average emission time, t and the variance orfluctuation,σ(t),for a general source,respectively.Orienting our coordinate system with outward or x-axis along K we haveβK=K/E K−(βo,0,0)(see Figure1)when the pair rapidity has been boosted longitudinally to their center-of-mass system,Y=0.Inthat case l=(18R2+294R2−1x2 .For comparison,a transparent(λmfp=∞)and spherically symmetric source has the same extent in all directions and onefinds R s=R o=R l=R/√∗A spherical source similar to(8)with strict surface emission,λmf p=0,was applied to proton emission from excited nuclei[19].As proton decay times are long only thefinite duration of emission contribution was considered (i.e,the last term in equation(9).)S (x,K )∼Θ(R 2−x 2−y 2)exp −m ⊥cosh(Y −η)λmfp S τ(τ).(12)Here,τ=√R 2−y 2−x).Actually,when η=Y the particles pass a distance longitudinally,which will lead to a minor reduction of R l ,but as we concentrate on the transverse radius parameters we will ignore this effect in the following.From equations (3-5)we obtain when λmfp ≪R R 2o = 24)2 R 2+ 732 λ2mfp +β2o σ(τ),(13)R 2s =16λ2mfp ,(14)R 2l≃ τ2 T0.000.250.500.75 1.00(λ/R)20.00.10.20.3R i 2/R 2R o 2/R2R s 2/R2∆R 2/R 2 at τfFIG.2.Sideward and outward HBT radius parameters as function of the relative thickness of surface layer thickness to source size.The small mean free path limits are given in equations (9)and (10)and for λmf p ≫R theyapproach R s =R o =R/2.The duration of emission contribution β2o σ(τ)is not included in R 2o .The strength of the opacity effect,∆R 2/R 2,is estimated from equation (18)at the freeze–out time τf .The freeze-out can be described by allowing the layer of emission or λmfp to increase with time after the collision.According to the Bjorken scaling model the mean free path increases linearly with proper time τλmfp (τ)=1¯σdN/dy τ≡λmfp (τf )ττ2f τf 0R 2i (λmfp (τ))τdτ= λ2mfp (τf )0R 2i (λmfp (τ))dλ2mfpdλ2mfp R2o−R2s= λ2mfp(τf)0 R2o(λmfp)−R2s(λmfp)τ2 =τf/√as measured in several experiments.In the case of central P b+P b collisions at160A·GeV the opacity reduction was estimated to be a significant fraction of the outward HBT radius parameter and larger than the contribution from a source emitting during all its life-time.Otherfluctuations like moving surfaces, short lived resonances,and other effects do,however,also add to R2o and we expect that this is the reason that the sideward and outward HBT radius parameters are measured to be very similar in relativistic heavy ion collisions.Wefind that pions do not appear as in a“flash”.In fact a long duration of emission of order the life-time of the source is possible and consistent with an outward HBT radius parameter smaller or comparable to the sideward due to the opacity effect.In contrast,a simple transparent source would necessarily have a very short duration of emission as implied by equation(1).The Cooper-Frye freeze-out condition in hydrodynamic models does not take the opacity effect into account and generally onefinds considerably larger outward than sideward source radius parameters due to the long freeze–out time[13].Cascade codes have implicitly opacities build in through rescatterings and dofind a directional effect[14],but also long duration of emission and mean free paths[15]are found leading to larger outward radius parameters than sideward.Other effects like,e.g.,transverseflow,can also reduce the outward HBT radius parameter more than the sideward.At larger transverse momenta the sideward and outward HBT radius parameters are reduced by transverseflow and the longitudinal scales by the factor1/m⊥.This may explain the decrease of all the HBT radius parameters with increasing p⊥as found in the NA44experiments[6].By studying the rapidity and p⊥dependence of the HBT radius parameters the duration of emission contribution,β2oσ(τ),may be separated from the opacity effect,∆R2,as well as otherfluctuations.However, this may be non-trivial if the mean free path,duration of emission and resonance contributions are rapidity and p⊥dependent.Transverseflow may also add a strong p⊥dependence of the radius parameters but the magnitude of transverseflow can approximately be determined from transverse momentum spectra[20]. HBT radius parameters and transverseflow increase for heavier ions colliding.The nuclear A-dependence can therefore provide additional information on freeze-out times,sizes,transverseflow and opacity of the sources.ACKNOWLEDGEMENTSWe would like to thank Larry McLerran and Scott Pratt for stimulating discussions.[8]T.Alber et al.(NA35and NA49collaborations),Nucl.Phys.A590(1995)453c;Z.Phys.C66(1995)77;K.Kadija et al.(NA49collaboration),Nucl.Phys.A610(1996)248c;D.Ferenc et al.,Nucl.Phys.A544(1992) 531c.[9]T.C.Awes et al.(WA80collaboration),Z.Phys.C65(1995)207;ibid.C69(1995)67.T.Abbott et al.(E802collaboration),Phys.Rev.Lett.69(1992)1030.[10]J.Barrette et al.(E877collaboration),Nucl.Phys.A590(1995)259c.[11]T.Cs¨o rg˝o and L.P.Csernai,Phys.Lett.B333,494(1994);[12]H.Heiselberg,Phys.Lett.B379,27(1996).[13]J.Bolz,U.Ornik,M.Pl¨u mer,B.R.Schlei,and R.M.Weiner,Phys.Rev.D47(1993)3860;J.Sollfrank et al.,hep-ph/9607029.[14]T.J.Humanic,Phys.Rev.C53(1996)901.[15]J.P.Sullivan,M.Berenguer,B.V.Jacak,M.Sarabura,J.Simon–Gillo,H.Sorge,H.van Hecke,S.Pratt,Phys.Rev.Lett.70(1993)3000.[16]L.V.Bravina,I.N.Mishustin,N.S.Amelin,J.P.Bondorf,L.P.Csernai,Phys.Lett B354(1995)196.[17]G.Baym and P.Braun-Munzinger,Nucl.Phys.A610(1996)286c.[18]F.Grassi,Y.Hama and T.Kodama,Phys.Lett.B355(1995)9.[19]G.I.Kopylov and M.I.Podgoretskii,Sov.J.Nucl.Phys.15(1972)219;ibid.18(1974)336.[20]I.G.Bearden et al.(NA44collaboration),CERN Preprint CERN-PPE/96-163,submitted to Phys.Rev.Lett.[21]H.Heiselberg and A.P.Vischer,to be published.[22]H.W.Barz et al.,Proc.of Hirschegg meeting,jan.13-17,1997.。

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>。