低温玻色系统的量子动力学研究
|
摘要
1924年印度科学家S. N.玻色完成一篇关于光子气体的统计性质的文章,接着在第二年A.爱因斯坦指出在一定临界温度下,无相互作用的原子可以在基态存在宏观的凝聚。在之后的70多年中,玻色-爱因斯坦凝聚成为物理学中具有重大吸引力的课题之一,许多伟大的物理学家,包括A.爱因斯坦,伦敦兄弟,L.D.朗道,N. N.波戈留波夫,R. P.费曼……,都在这方面做出过贡献。实验的突破发生在1995年,美国科学家在碱金属的稀薄原子气体中实现了几乎纯净的玻色-爱因斯坦凝聚体;通过调节外囚禁势的频率,一维、二维凝聚体跟着制备出来,其它一些重要的进展包括宏观凝聚体的相干现象,在光学格子势中实现超流-绝缘体相变,在玻色凝聚体中制备孤子以及通过Feshbach技术调节原子之间的相互作用强度等等。理论方面的突破始于宏观波函数概念的提出,它极大的推动了玻色-爱因斯坦凝聚理论的发展,并且成为分析凝聚体性质的重要方法。然而即使在接近零温的凝聚体中,原子之间的相互作用引起的量子波动也是一个不可忽略的现象,是分析凝聚体的稳态和动力学行为的一个重要因素。本文从理论的角度出发,着重分析超低温有限粒子数目玻色系统的多体动力学,内容主要集中在一维吸引相互作用凝聚体的多体动力学和双阱势中两分量凝聚体的纠缠现象上。
1.在一维环状势中的玻色凝聚体,如果原子之间是吸引的相互作用,平均场理论预测在较弱的相互作用强度时系统的基态是均匀分布的状态,当相互作用强度超过特定临界值后,基态局域化为亮孤子状态。两个状态对应的化学势连续但是一阶导数不连续,存在量子相变。在系统是亮孤子状态时,原子之间的相互作用强度较大,量子波动必然具有明显的影响。基于这样的考虑,我们从多体动力学的角度,通过计算系统的仙农(Shannon)熵、动量分布等物理量,分析了量子波动对系统的动力学演化在不同参数区间的影响。我们的计算结果不仅是对平均场理论的补充,同时表明在强相互作用下,高角动量态显著的激发,平均场理论不能全面描述系统的特性。
2.分析了双阱势中两分量凝聚体的能级和纠缠特性。从两模近似出发,我们首先计算了系统的能谱。发现在增加同种原子和不同种原子之间的相互作用强度时,能谱存在明显的反交叉。接着分析了系统的动力学演化,如果初态选择两种分量分别处在两个不同的势阱中,已有的理论分析显示在弱和强相互作用极限下,原子在势垒间的隧穿是以不同原子之间耦合成对的形式进行。我们的计算结果表明,当系统的对隧穿形式占主要的时候,二阶交叉关联函数违背Cauthy-Schwarz不等式,纠缠的程度可以用Von-Neumann熵度量。
In 1924, S. N. Bose wrote a paper on the statistics of photons. Following thiswork, A. Einstein considered a gas of non-interacting atoms and concluded thata macroscopic fraction of the total atoms will occupy the lowest-energy single-particle state below a certain critical temperature. In the following seventy years,Bose-Einstein condensation becomes one of the most exciting fields in modernphysics. Many famous physicists have contributed to this field. The list of thenames includes, for example, A. Einstein, F. London and his brother H. Lon-don, L. D. Landau, N. N. Bogoliubov, R. P. Feynman... ... The most importantadvancement of experiment happened in 1995. Three individual groups in USAgenerated almost purified Bose condensates of dilute alkalis gases in harmonictraps. By adjusting the frequencies of external traps, one- and two-dimensionBose condensates are generated too. Other important and interest experimen-tal results include the interference between two independent condensates, thesuper?uid-insulator transition of Bose gas in optical lattice, dark and brightsolitons in the condensates and tuning the interaction strength of atoms withFeshbach resonance,and so on. The breakthrough on the theory of Bose-Einsteincondensation started from the combination of condensates and macroscopic wavefunction. It is convenient and promotes the development of theory of condensates.Nowadays, it becomes one of the important tools on analyzing the condensates.On the other hand, quantum correlation e?ects plays an unique role in the studyof condensates, which in?uences the ground state and dynamics of the systemeven at almost zero temperature. In this thesis, we study the many-body dy-namics of finite Bose gas at low temperature. The work concentrates on thequantum dynamics of one dimensional Bose gas with attractive interaction andthe entanglement of two-component Bose condensates in a double-well.
1. We investigate the many-body dynamics of an e?ectively attractive one-dimensional Bose system confined in a toroidal trap. The mean-field theorypredicts that a bright-soliton state will be formed when the interparticle inter- action increases over a critical point. The chemical potential is continuous whilethe first order derivation of the chemical potential is discontinuous. Thus thereis a quantum phase transition at the critical point. Due to the strong interac-tion strength of atoms in a soliton, quantum correlation has to be considered.The study of quantum many-body dynamics in this thesis reveals that thereis a modulation instability in a finite Bose system correspondingly. We showthat Shannon entropy becomes irregular near and above the critical point due toquantum correlations. We also study the dynamical behavior of the instabilityby exploring the momentum distribution and the fringe visibility, which can beverified experimentally by releasing the trap.
2. We consider a novel system of two-component atomic Bose–Einsteincondensate in a double-well potential. Based on the well-known two-mode ap-proximation, we demonstrate that there are obvious avoided level crossings whenboth interspecies and intraspecies interactions of two species are increased. Thequantum dynamics of the system exhibits revised oscillating behaviors comparedwith a single component condensate. We also examine the entanglement of twospecies. Our numerical calculations show that the onset of paired-tunneling canbe signalled by a violation of the Cauchy–Schwarz inequality of a second-ordercross-correlation function. Consequently, we use Von Neumann entropy to quan-tify the degree of entanglement.
1.在一维环状势中的玻色凝聚体,如果原子之间是吸引的相互作用,平均场理论预测在较弱的相互作用强度时系统的基态是均匀分布的状态,当相互作用强度超过特定临界值后,基态局域化为亮孤子状态。两个状态对应的化学势连续但是一阶导数不连续,存在量子相变。在系统是亮孤子状态时,原子之间的相互作用强度较大,量子波动必然具有明显的影响。基于这样的考虑,我们从多体动力学的角度,通过计算系统的仙农(Shannon)熵、动量分布等物理量,分析了量子波动对系统的动力学演化在不同参数区间的影响。我们的计算结果不仅是对平均场理论的补充,同时表明在强相互作用下,高角动量态显著的激发,平均场理论不能全面描述系统的特性。
2.分析了双阱势中两分量凝聚体的能级和纠缠特性。从两模近似出发,我们首先计算了系统的能谱。发现在增加同种原子和不同种原子之间的相互作用强度时,能谱存在明显的反交叉。接着分析了系统的动力学演化,如果初态选择两种分量分别处在两个不同的势阱中,已有的理论分析显示在弱和强相互作用极限下,原子在势垒间的隧穿是以不同原子之间耦合成对的形式进行。我们的计算结果表明,当系统的对隧穿形式占主要的时候,二阶交叉关联函数违背Cauthy-Schwarz不等式,纠缠的程度可以用Von-Neumann熵度量。
In 1924, S. N. Bose wrote a paper on the statistics of photons. Following thiswork, A. Einstein considered a gas of non-interacting atoms and concluded thata macroscopic fraction of the total atoms will occupy the lowest-energy single-particle state below a certain critical temperature. In the following seventy years,Bose-Einstein condensation becomes one of the most exciting fields in modernphysics. Many famous physicists have contributed to this field. The list of thenames includes, for example, A. Einstein, F. London and his brother H. Lon-don, L. D. Landau, N. N. Bogoliubov, R. P. Feynman... ... The most importantadvancement of experiment happened in 1995. Three individual groups in USAgenerated almost purified Bose condensates of dilute alkalis gases in harmonictraps. By adjusting the frequencies of external traps, one- and two-dimensionBose condensates are generated too. Other important and interest experimen-tal results include the interference between two independent condensates, thesuper?uid-insulator transition of Bose gas in optical lattice, dark and brightsolitons in the condensates and tuning the interaction strength of atoms withFeshbach resonance,and so on. The breakthrough on the theory of Bose-Einsteincondensation started from the combination of condensates and macroscopic wavefunction. It is convenient and promotes the development of theory of condensates.Nowadays, it becomes one of the important tools on analyzing the condensates.On the other hand, quantum correlation e?ects plays an unique role in the studyof condensates, which in?uences the ground state and dynamics of the systemeven at almost zero temperature. In this thesis, we study the many-body dy-namics of finite Bose gas at low temperature. The work concentrates on thequantum dynamics of one dimensional Bose gas with attractive interaction andthe entanglement of two-component Bose condensates in a double-well.
1. We investigate the many-body dynamics of an e?ectively attractive one-dimensional Bose system confined in a toroidal trap. The mean-field theorypredicts that a bright-soliton state will be formed when the interparticle inter- action increases over a critical point. The chemical potential is continuous whilethe first order derivation of the chemical potential is discontinuous. Thus thereis a quantum phase transition at the critical point. Due to the strong interac-tion strength of atoms in a soliton, quantum correlation has to be considered.The study of quantum many-body dynamics in this thesis reveals that thereis a modulation instability in a finite Bose system correspondingly. We showthat Shannon entropy becomes irregular near and above the critical point due toquantum correlations. We also study the dynamical behavior of the instabilityby exploring the momentum distribution and the fringe visibility, which can beverified experimentally by releasing the trap.
2. We consider a novel system of two-component atomic Bose–Einsteincondensate in a double-well potential. Based on the well-known two-mode ap-proximation, we demonstrate that there are obvious avoided level crossings whenboth interspecies and intraspecies interactions of two species are increased. Thequantum dynamics of the system exhibits revised oscillating behaviors comparedwith a single component condensate. We also examine the entanglement of twospecies. Our numerical calculations show that the onset of paired-tunneling canbe signalled by a violation of the Cauchy–Schwarz inequality of a second-ordercross-correlation function. Consequently, we use Von Neumann entropy to quan-tify the degree of entanglement.
引文
[1] S. N. Bose, Plancks Gesetz und Lichtquantenhypothese. Z. Phys., 26:178,1924.
[2] A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzber. Kgl.Preuss. Akad. Wiss., 261, 1924, and 3, 1925.
[3] S. T. Beliaev, Sov. Phys. JETP, 7:289, 1958.
[4] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A.Cornell. Observation of Bose-Einstein condensation in a dilute atomic vapor.Science, 269:198, 1995.
[5] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet. Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions. Phys. Rev.Lett., 75:1687, 1995.
[6] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee,D. M. Kurn, and W. Ketterle. Bose-Einstein condensation in a gas of sodiumatoms. Phys. Rev. Lett., 75:3969, 1995.
[7] F. London. On the Bose-Einstein condensation. Phys. Rev., 54:947, 1938
[8] Fritz London. The λ-phenomenon of liquid helium and the Bose-Einsteindegeneracy. Nature, 141:643, 1938.
[9] A. Gri?n, D. W. Snoke, and S. Stringari, editors, Bose-Einstein Condensa-tion, ( Cambridge University Press, 1995).
[10] Charles E. Hecht. The possible super?uid behaviour of hydrogen atom gasesand liquids. Physica, 25:1159, 1959.
[11] Dale G. Fried, Thomas C. Killian, Lorenz Willmann, David Landhuis,Stephen C. Moss, Daniel Kleppner, and Thomas J. Greytak. Bose-Einsteincondensation of atomic hydrogen. Phys. Rev. Lett., 81:3811, 1998.
[12] C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases(Cambridge University, Cambridge, 2002).
[13] J. R. Anglin and W. Ketterle, Bose-Einstein condensation of atomic gases,Nature, 416:211, 2002.
[14] J. Dunningham, K. Burnett and W. D. Phillips, Bose–Einstein condensatesand precision measurements, Phil. Trans. R. Soc. A 363:2165–2175, 2005.
[15] L. D. Landau and E. M. Lifshitz, Statistical physics (third edition, Perga-mon, Oxford, 1980).
[16] Huang Kerson, Introduction to statistical physics (Taylor and Fran-cis,London, 2001).
[17] 周世勋,量子力学教程(高等教育出版社,北京,1979).
[18] Tin-Lun Ho. Spinor Bose condensates in optical traps. Phys. Rev. Lett.,81:742, 1998.
[19] 王竹溪,郭敦仁, 特殊函数概论(北京大学出版社,北京,2000).
[20] A. G¨olitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J.R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W.Ketterle, Realization of Bose-Einstein condensates in low dimensions, Phys.Rev. Lett. 87:130402, 2001.
[21] P. C. Hohenberg, Existence of Long-Range Order in One and Two Dimen-sions,Phys. Rev. 158:383,1967.
[22] W. J. Mullin. Bose-Einstein condensation in a harmonic potential. J. LowTemp. Phys., 106:615, 1997.
[23] William J. Mullin. A study of Bose-Einstein condensation in a two-dimensional trapped gas. J. Low Temp. Phys., 110(1/2):167, 1998.
[24] Franco Dalfovo, Stefano Giorgini, Lev P. Pitaevskii, and Sandro Stringari.Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys.,71:463, 1999.
[25] Wolfgang Ketterle and N. J. van Druten. Bose-Einstein condensation of afinite number of particles trapped in one or three dimenstions. Phys. Rev. A,54(1):656, July 1996.
[26] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics (parttwo),(Butterwoth-Heinemann, Oxford, 1980)
[27] 徐在新,高等量子力学(华东师范大学出版社,上海,1994).
[28] M. Girardeau and R. Arnowitt, Theory of Many-Boson Systems: Pair The-ory, Phys. Rev. 113:755, 1959.
[29] M. D. Girardeau, Comment on “Particle-number-conserving Bogoli-ubov method which demonstrates the validity of the time-dependent Gross-Pitaevskii equation for a highly condensed Bose gas”, Phys. Rev. A 58:775,1998.
[30] V. N. Popov, Functional integrals and collective excitations, (CambridgeUniv. Press, Cambridge, 1987).
[31] A. Gri?n, Conserving and gapless approximations for an inhomogeneousBose gas at finite temperatures, Phys. Rev. B 53:9341 , 1996.
[32] L. D. Landau and E. M. Lifshitz, Quantum mechanics(Pergaman, New York,1977).
[33] L. P. Pitaevskii,Vortex lines in an imperfect Bose gas, Sov. Phys. JETP13:451 1961.
[34] E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo Ci-mento 20:454 ,1961; Hydrodynamics of a Super?uid Condensate , J. Math.Phys. (N.Y.) 4: 195, 1963.
[35] Herman Feshbach. A unified theory of nuclear reactions. II. Ann. Phys.,19:287, 1962.
[36] S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper- Kurn,and W. Ketterle. Observation of Feshbach resonances in a Bose-Einstein con-densate. Nature, 392:151, 1998.
[37] Pierre Meystre,Atom optics(Springer, New York, 2001).
[38] Mark Edwards and K. Burnett. Numerical solution of the nonlinearSchr¨odinger equation for small samples of trapped neutral atoms. Phys. Rev.A, 51(2):1382, February 1995.
[39] P. A. Ruprecht, M. J. Holland, and K. Burnett, Time-dependent solution ofthe nonlinear Schr¨odinger equation for Bose-condensed trapped neutral atoms,Phys. Rev. A 51: 4704 ,1995.
[40] M. Holland and J. Cooper. Expansion of a Bose-Einstein condensate in aharmonic potential. Phys. Rev. A, 53(4):R1954, April 1996.
[41] Mark Edwards, R. J. Dodd, C. W. Clark, P. A. Ruprecht, and K. Burnett.Properties of a Bose-Einstein condensate in an anisotropic harmonic potential.Phys. Rev. A, 53(4):R1950, April 1996.
[42] Mark Edwards, P. A. Ruprecht, K. Burnett, R. J. Dodd, and Charles W.Clark. Collective excitations of atomic Bose-Einstein condensates. Phys. Rev.Lett., 77(9):1671, August 1996.
[43] P. A. Ruprecht, Mark Edwards, K. Burnett, and Charles W. Clark, Probingthe linear and nonlinear excitations of Bose-condensed neutral atoms in a trap,Phys. Rev. A 54:4178, 1996.
[44] L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne,J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips. Four-wavemixing with matter waves. Nature, 398:218, 1999.
[45] Kozuma, M. et al. Coherent splitting of Bose-Einstein condensed atoms withoptically induced Bragg di?raction. Phys. Rev. Lett. 82, 871 ,1999.
[46] Hagley, E. W. et al.Measurement of the coherence of a Bose-Einstein con-densate. Phys. Rev. Lett. 83: 3112,1999.
[47] J. Denschlag, J. E. Simsarian, D. L. Feder, Charles W. Clark, L. A. Collins,J. Cubizolles, L. Deng, E. W. Hagley, K. Helmerson, W. P. Reinhardt, S. L.Rolston, B. I. Schneider, and W. D. Phillips. Generating solitons by phaseengineering of a Bose-Einstein condensate. Science, 287:97, 2000.
[48] S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G.V. Shlyapnikov, and M. Lewenstein. Dark solitons in Bose-Einstein conden-sates. Phys. Rev. Lett., 83:5198, 1999.
[49] D. L. Feder, M. S. Pindzola, L. A. Collins, B. I. Schneider, and C. W. Clark.Dark-soliton states of Bose-Einstein condensates in anisotropic traps. Phys.Rev. A, 62:053606, 2000.
[50] Kevin E. Strecker, Guthrie B. Partridge, Andrew G. Truscott, Randall G.Hulet,Formation and Propagation of Matter Wave Soliton Trains, Nature,417:150, 2002.
[51] Raman, C. et al. Evidence for a critical velocity in a Bose-Einstein condensedgas. Phys. Rev. Lett. 83:2502, 1999.
[52] Onofrio, R. et al.Observation of super?uid ?ow in a Bose-Einstein condensedgas. Phys. Rev. Lett. 85: 2228,2000.
[53] S. Burger, F. S. Cataliotti, C. Fort, F. Minardi, M. Inguscio, M. L. Chio-falo, and M. P. Tosi. Super?uid and dissipative dynamics of a Bose-Einsteincondensate in a periodic optical potential. Phys. Rev. Lett., 86(20):4447, 2001.
[54] J. E. Williams, M. J. Holland, Preparing topological states of aBose–Einstein condensate. Nature 401, 568–572 (1999).
[55] Matthews, M. R. et al. Vortices in a Bose-Einstein condensate. Phys. Rev.Lett. 83, 2498–2501 (1999).
[56] K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard. Vortex formationin a stirred Bose-Einstein condensate. Phys. Rev. Lett., 84(5):806, January2000.
[57] J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle. Observationof vortex lattices in Bose-Einstein condensates. Science, 292(5516):476, April2001.
[58] P. C. Haljan, I. Coddington, P. Engels, and E. A. Cornell. Driving Bose-Einstein-condensate vorticity with a rotating normal cloud. Phys. Rev. Lett.,87(21):210403, November 2001.
[59] B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C.W. Clark, and E. A. Cornell. Watching dark solitons decay into vortex ringsin a Bose-Einstein condensate. Phys. Rev. Lett., 86(14):2926, April 2001.
[60] A. Minguzzi, S. Conti, and M. P. Tosi. The internal energy and condensatefraction of a trapped interacting Bose gas. J. Phys., Condens. Matter, 9:L33,1997.
[61] S. Giorgini, L. P. Pitaevskii, and S. Stringari. Condensate fraction and criti-cal temperature of a trapped interacting Bose gas. Phys. Rev. A, 54(6):R4633,December 1996.
[62] Hualin Shi and Wei-Mou Zheng, Critical temperature of a trapped inter-acting Bose-Einstein gas in the local-density approximation ,Phys. Rev. A 56:1046,1997.
[63] H. T. C. Stoof, Macroscopic quantum tunneling of a bose condensate, Jour-nal of Statistical Physics, 87:1353,1997.
[64] A. D. Jackson, G. M. Kavoulakis, and C. J. Pethick. Solitary waves in cloudsof Bose-Einstein condensed Phys. Rev. A, 58(3):2417, September 1998.
[65] U. Al Khawaja, J. O. Andersen, N. P. Proukakis, and H. T. C. Stoof. Lowdimensional Bose gases. Phys. Rev. A, 66:013615, July 2002.
[66] D. S. Petrov, M. Holzmann, and G. V. Shlyapnikov,Bose-Einstein Conden-sation in Quasi-2D Trapped Gases ,Phys. Rev. Lett. 84:2551 2000.
[67] D.S. Petrov, D.M. Gangardt, and G.V. Shlyapnikov, Low-dimensionaltrapped gases, J. Phys. IV France 116:5, 2004.
[68] Harald F. Hess. Evaporative cooling of magnetically trapped and compressedspinpolarized hydrogen. Phys. Rev. B, 34(5):3476, September 1986.
[69] Wolfgang Ketterle and N. J. van Druten. Evaporative cooling of trappedatoms. Adv. At. Mol. Opt. Phys., 37(0):181, 1996.
[70] T. W. H¨ansch and A. L. Schawlow. Cooling of gases by laser radiation. Opt.Commun., 13(1):68, 1975.
[71] Steven Chu, L. Hollberg, J. E. Bjorkholm, Alex Cable, and A. Ashkin. Three-dimensional viscous confinement and cooling of atoms by resonance radiationpressure. Phys. Rev. Lett., 55(1):48, July 1985.
[72] Paul D. Lett, Richard N. Watts, Christoph I. Westbrook, William D.Phillips, Phillip L. Gould, and Harold J. Metcalf. Observation of atoms lasercooled below the Doppler limit. Phys. Rev. Lett., 61(2):169, 1988.
[73] J. Dalibard and C. Cohen-Tannoudji. Laser cooling below the Doppler limitby polarization gradients: simple theoretical-models. J. Opt. Soc. Am. B,6(11):2023, November 1989.
[74] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, tle Laser cooling below the one-photon recoil by velocity-selectivecoherent population trapping, Phys. Rev. Lett. 61:826,1988.
[75] Naoto Masuhara, John M. Doyle, Jon C. Sandberg, Daniel Kleppner,Thomas J. Greytak, Harald Hess, and Gregory P. Kochanski. Evaporativecooling of spin-polarized atomic hydrogen. Phys. Rev. Lett., 61(8):935, Au-gust 1988.
[76] C. C. Bradley, C. A. Sackett, and R. G. Hulet. Bose-Einstein condensa-tion of lithium: Observation of limited condensate number. Phys. Rev. Lett.,78(6):985, 1997.
[77] S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell, and C. E. Wie-man. Stable 85Rb Bose-Einstein condensates with widely tunable interactions.Phys. Rev. Lett., 85(9):1795, August 2000.
[78] Jordan M. Gerton, Dmitry Strekalov, Ionut Prodan, and Randall G. Hulet.Direct observation of growth and collapse of a Bose-Einstein condensate withattractive interactions. Nature, 408:692, December 2000.
[79] M. Girardeau, Relationship between systems of impenetrable bosons andfermions in one dimension. J. Math. Phys. 1, 51:523 ,1960.
[80] E.H. Lieb and W. Liniger,Exact analysis of an interacting Bose gas. Thegeneral solution and the ground state. Phys. Rev. 130, 160:1616 ,1963.
[81] E. Tiesinga, B. J. Verhaar and H. T. C. Stoof,Threshold and resonancephenomena in ultracold groundstate collisions. Phys. Rev. A 47: 4114,1993.
[82] Elizabeth A. Donley, Neil R. Claussen, Simon L. Cornish, Jacob L. Roberts,Eric A. Cornell, and Carl E. Wieman. Dynamics of collapsing and explodingBose-Einstein condensates. Nature, 412(6844):295, July 2001。
[83] L. D. Carr, C. W. Clark, and W. P. Reinhardt, Stationary solutions of theone-dimensional nonlinear Schro¨dinger equation. II. Case of attractive nonlin-earity ,Phys. Rev. A, 62:63611, 2000.
[84] R. Kanomoto, H. Saito, and M. Ueda, Quantum phase transition in one-dimensional Bose-Einstein condensates with attractive interactions ,Phys. Rev.A, 67:13608,2003.
[85] Weibin Li, Xiaotao Xie, Zhiming Zhan, and Xiaoxue Yang, Many-body dy-namics of a Bose system with attractive interactions on a ring, Phys. Rev. A,72: 043615 2005.
[86] M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn,and W. Ketterle. Observation of interference between two Bose-Einstein con-densates. Science, 275(0):637, January 1997.
[87] C. Orzel, A. K. Tuchman, M. L. Fenselau, M. Yasuda, and M. A. Kasevich,Squeezed states in a Bose-Einstein Condensate, Science 291:2386, 2001.
[88] M. Greiner, O. Mandel, T. Esslinger, T. W. H ”ansch and I. Bloch, Quantumphase transition from a super?uid to a Mott insulator in a gas of ultracoldatoms, Nature, 415:39,2002.
[89] F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trom-bettoni, A. Smerzi, and M. Inguscio, Josephson Junction arrays with Bose-Einstein condensates, Science, 293:843, 2001.
[90] B. P. Anderson and M. A. Kasevich. Macroscopic quantum interference fromatomic tunnel arrays. Science, 282:1686, November 1998.
[91] Michael Albiez, Rudolf Gati, Jonas F¨olling, Stefan Hunsmann, Matteo Cris-tiani, and Markus K. Oberthaler, Direct Observation of Tunneling and Non-linear Self-Trapping in a Single Bosonic Josephson Junction, Phys. Rev. Lett.,95:010402, 2005.
[92] D. S. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E. A. Cor-nell. Dynamics of component separation in a binary mixture of Bose-Einsteincondensates. Phys. Rev. Lett., 81(8):1539, August 1998.
[93] D. S. Hall, M. R. Matthews, C. E. Wieman, and E. A. Cornell. Measure-ments of relative phase in two-component Bose-Einstein condensates. Phys.Rev. Lett., 81(8):1543, August 1998.
[1] P.S. Julienne,A.M. Smith and K. Burnett, Theory of collisions between lasercooled atoms Adv. At. Mol. Opt. Phys. 30:141, 1993.
[2] J.T.M. Walraven,In: G.-L. Oppo, S.M. Barnett, E. Riis, M. Wilkinson,(Eds.), Quantum Dynamics of Simple Systems( Institute of Physics, Bristol,1996).
[3] C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases(Cambridge University, Cambridge, 2002).
[4] Franco Dalfovo, Stefano Giorgini, Lev P. Pitaevskii, and Sandro Stringari.Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys.,71(3):463, April 1999.
[5] A. S. Parkins and D. F. Walls, The physics of trapped dilute-gas Bose-Einsteincondensates, Physics Reports, 303:1, 1998.
[6] F. Dalfovo and S. Stringari. Bosons in anisotropic traps: Ground state andvortices. Phys. Rev. A, 53(4):2477, April 1996.
[7] E. M. Lifshitz and L. P. Pitaevskii,Statistical physics (Pergamon Press Ox-ford; New York,1980).
[8] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet. Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions. Phys. Rev.Lett., 75:1687, 1995.
[9] P. A. Ruprecht, M. J. Holland, and K. Burnett, Time-dependent solution ofthe nonlinear Schr¨odinger equation for Bose-condensed trapped neutral atoms,Phys. Rev. A 51: 4704 ,1995.
[10] Mark Edwards and K. Burnett. Numerical solution of the nonlinearSchr¨odinger equation for small samples of trapped neutral atoms. Phys. Rev.A, 51(2):1382, February 1995.
[11] Yu. Kagan, G. V. Shlyapnikov, and J. T. M. Walraven. Bose-Einstein con-densation in trapped atomic gases. Phys. Rev. Lett., 76(15):2670, April 1996.
[12] Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov. Evolution of a Bose-condensed gas under variations of the confining potential. Phys. Rev. A,54(3):R1753, September 1996.
[13] Gordon Baym and C. J. Pethick. Ground-state properties of magneticallytrapped Bose-condensed rubidium gas. Phys. Rev. Lett., 76(1):6, January 1996.
[14] N. Bogolubov. On the theory of super?uidity. J. Phys., 11(1):23, 1947.
[15] A. L. Fetter, Nonuniform states of an imperfect bose gas, Annals of Physics,70:67,1972.
[16] A. Gri?n. Conserving and gapless approximations for an inhomogeneousBose gas at finite temperatures. Phys. Rev. B, 53(14):9341, April 1996.
[17] Pierre Meystre, Atom optics(Springer, New York, 2001).
[18] V. N. Popov, Functional Integrals and Collective Modes (Cambridge Uni-versity Press, New York, 1987).
[19] H. Shi, G. Verechaka, and A. Gri?n, Phys. Rev. B, 50:1119, 1994.
[20] C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, and C. E. Wieman.Production of two overlapping Bose-Einstein condensates by sympathetic cool-ing. Phys. Rev. Lett., 78(4):586, January 1997.
[21] D. S. Hall, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Measurementsof Relative Phase in Two-Component Bose-Einstein Condensates , Phys. Rev.Lett. 81:1543, 1998.
[22] Tin-Lun Ho and V. B. Shenoy. Binary mixtures of Bose condensates of alkaliatoms. Phys. Rev. Lett., 77(16):3276, October 1996.
[23] Th. Busch, J. I. Cirac, V. M. P′erez-Garc′?a, and P. Zoller. Stability andcollective excitations of a two-component Bose-Einstein condensed gas: A mo-ment approach. Phys. Rev. A, 56(4):2978, October 1997.
[24] Elena V. Goldstein and Pierre Meystre. Quasiparticle instabilities in multi-component atomic condensates. Phys. Rev. A, 55(4):2935, April 1997.
[25] A. G¨olitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J.R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W.Ketterle, Realization of Bose-Einstein condensates in low dimensions, Phys.Rev. Lett. 87:130402, 2001.
[26] A. D. Jackson, G. M. Kavoulakis, and C. J. Pethick. Solitary waves in cloudsof Bose-Einstein condensed atoms. Phys. Rev. A, 58(3):2417, September 1998.
[27] L. Salasnich, A. Parola, and L Reatto, E?ective wave equations for thedynamics of cigar-shaped and disk-shaped Bose condensates, Phys. Rev. A,65:43614, 2002.
[28] D. S. Petrov, M. Holzmann, and G. V. Shlyapnikov,Bose-Einstein Conden-sation in Quasi-2D Trapped Gases ,Phys. Rev. Lett. 84:2551 2000.
[29] D.S. Petrov, D.M. Gangardt, and G.V. Shlyapnikov, Low-dimensionaltrapped gases, J. Phys. IV France 116:5, 2004.
[30] Lewi Tonks, The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres , Phys. Rev. 50: 955,1936.
[31] M. Girardeau, Relationship between Systems of Impenetrable Bosons andFermions in One Dimension, J. Math. Phys, 1:516, 1960.
[32] Elliott H. Lieb and Werner Liniger, Exact Analysis of an Interacting BoseGas. I. The General Solution and the Ground State, Phys. Rev. 130:1605, 1963.
[33] D. S. Petrov1,G. V. Shlyapnikov, and J. T. M. Walraven,Regimes ofQuantum Degeneracy in Trapped 1D Gases,Phys. Rev. Lett. 85:3745,2000.
[34] Toshiya Kinoshita, Trevor Wenger, David S. Weiss, Observation of a One-Dimensional Tonks-Girardeau Gas, Science, 305:1125,2004. Toshiya Kinoshita,Trevor Wenger, David S. Weiss
[1] Lewi Tonks, The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres , Phys. Rev. 50: 955,1936.
[2] M. Girardeau, Relationship between Systems of Impenetrable Bosons andFermions in One Dimension, J. Math. Phys, 1:516, 1960.
[3] Elliott H. Lieb and Werner Liniger, Exact Analysis of an Interacting BoseGas. I. The General Solution and the Ground State, Phys. Rev. 130:1605,1963.
[4] A. G¨olitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J.R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W.Ketterle, Realization of Bose-Einstein condensates in low dimensions, Phys.Rev. Lett. 87:130402, 2001.
[5] Toshiya Kinoshita, Trevor Wenger, David S. Weiss, Observation of a One-Dimensional Tonks-Girardeau Gas, Science, 305:1125,2004.
[6] A. D. Jackson, G. M. Kavoulakis, and C. J. Pethick. Solitary waves in cloudsof Bose-Einstein condensed Phys. Rev. A, 58(3):2417, September 1998.
[7] L. D. Carr1, Charles W. Clark, and W. P. Reinhardt, Stationary solutionsof the one-dimensional nonlinear Schro¨dinger equation. I. Case of repulsivenonlinearity,Phys. Rev. A 62:063610, 2000.
[8] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet. Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions. Phys. Rev.Lett., 75(9):1687, August 1995. ibid. 79, 1170 (1997).
[9] L. D. Carr1, Charles W. Clark, and W. P. Reinhardt, Stationary solutionsof the one-dimensional nonlinear Schro¨dinger equation. II. Case of attractivenonlinearity,Phys. Rev. A 62:063611, 2000.
[10] Pierre Meystre, Atom optics(Springer, New York, 2001).
[11] Kevin E. Strecker, Guthrie B. Partridge, Andrew G. Truscott, RandallG. Hulet, Formation and propagation of matter-wave soliton trains, Nature,417:150, 2002.
[12] L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr,Y. Castin, and C. Salomon. Formation of a matter-wave bright soliton. Science,296(5571):1290, May 2002.
[13] Rina Kanamoto, Hiroki Saito, and Masahito Ueda, Quantum phase transi-tion in one-dimensional Bose-Einstein condensates with attractive interactions, Phys. Rev. A 67: 013608 ,2003.
[14] G. M. Kavoulakis, Bose-Einstein condensates with attractive interactions ona ring ,Phys. Rev. A 67: 011601(R) ,2003.
[15] S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper- Kurn,and W. Ketterle. Observation of Feshbach resonances in a Bose-Einstein con-densate. Nature, 392(0):151, March 1998.
[16] S. Gupta, K. W. Murch, K. L. Moore, T. P. Purdy, and D. M. Stamper-Kurn, Bose-Einstein Condensation in a Circular Waveguide, Phys. Rev. Lett.95: 143201 ,2005.
[17] 王竹溪,郭敦仁编著,特殊函数概论(北京大学出版社,北京,2004).
[18] LD Landau, EM Lifshitz,Statistics Physics ( Butterworth-Heinemann, Ox-ford, 1984).
[19] Subir Sachdev, Quantum Phase Transitions, (Yale University, Connecticut,2001).
[20] Weibin Li, Xiaotao Xie, Zhiming Zhan and Xiaoxue Yang, Many-body dy-namics of a Bose system with attractive interaction on a ring, Phys.Rev. A,72:043615 (2005).
[21] G.P.Berman, A. Smerzi, and A. R. Bishop, Irregular Dynamics in a One-Dimensional Bose System, Phys. Rev.Lett. 92:30404, 2004.
[22] Y. Wu, X.Yang, and Y. Xiao, Analytical Method for Yrast Line States inInteracting Bose-Einstein Condensates, Phys. Rev. Lett. 86: 2200, 2001.
[23] V. V. Flambaum1 and F. M. Izrailev, Entropy production and wave packetdynamics in the Fock space of closed chaotic many-body systems , Phys. Rev.E 64: 036220 ,2001.
[24] K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods forPhysics and Engineering: A comprehensive Guide, 2nd ed., P574. (CambridgeUniversity Press, Cambridge, Uk, 2002).
[1] I. Bloch, Ultracold quantum gases in optical lattices, Nature Physics, 1:23,2005.
[2] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller. Cold bosonicatoms in optical lattices. Phys. Rev. Lett., 81(15):3108, October 1998.
[3] Markus Greiner, Olaf Mandel, Tilman Esslinger, Theodor W. H¨ansch, andImmanuel Bloch. Quantum phase transition from a super?uid to a Mott insu-lator in a gas of ultracold atoms. Nature, 415:39, January 2002.
[4] C. Orzel, A. K. Tuchman, M. L. Fenselau, M. Yasuda, M. A. Kasevich,Squeezed States in a Bose-Einstein Condensate, Science,291: 2386, 2001.
[5] I Bloch,Quantum gases in optical lattices, Physics World, 17:25, 2004.
[6] M. J¨aaskel¨ainen and P. Meystre, Dynamics of Bose-Einstein condensates indouble-well potentials,Phys. Rev. A,71: 043603,2005.
[7] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Bosonlocalization and the super?uid-insulator transition , Phys. Rev. B, 40:546,1989.
[8] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller. Cold bosonicatoms in optical lattices. Phys. Rev. Lett., 81(15):3108, October 1998.
[9] G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls. Quantum dynamicsof an atomic Bose-Einstein condensate in a doublewell potential. Phys. Rev.A, 55(6):4318, June 1997.
[10] M. J. Steel and M. J. Collet, Quantum state of two trapped Bose-Einsteincondensates with a Josephson coupling, Phys. Rev. A 57:2920, 1997.
[11] R. W. Spekkens and J. E. Sipe,Spatial fragmentation of a Bose-Einsteincondensate in a double-well potential, Phys. Rev. A 59:3868, 1999.
[12] Juha Javanainen and Misha Yu. Ivanov. Splitting a trap containing a Bose-Einstein condensate: Atom number ?uctuations. Phys. Rev. A, 60(3):2351,September 1999.
[13] A. J. Leggett, Bose-Einstein condensation in the alkali gases: Some funda-mental concepts, Rev. Mod. Phys. 73:307,2001.
[14] S. Raghavan1, A. Smerzi, S. Fantoni, and S. R. Shenoy, Coherent oscillationsbetween two weakly coupled Bose-Einstein condensates: Josephson e?ects,πoscillations, and macroscopic quantum self-trapping, Phys. Rev. A 59:620,1999.
[15] A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, Quantum CoherentAtomic Tunneling between Two Trapped Bose-Einstein Condensates, Phys.Rev. Lett. 79:4950, 1997.
[16] Michael Albiez, Rudolf Gati, Jonas F?lling, Stefan Hunsmann, Matteo Cris-tiani, and Markus K. Oberthaler, Direct Observation of Tunneling and Non-linear Self-Trapping in a Single Bosonic Josephson Junction, Phys. Rev. Lett.95:010402,2005.
[17] J. E. Williams, Optimal conditions for observing Josephson oscillations in adouble-well Bose-Einstein condensate, Phys. Rev. A 64: 013610,2001.
[18] R. W. Robinett, Quantum wave packet revivals, Phys. Rep., 392:1, 2004.
[19] G. Kalosakas and A. R. Bishop, Small-tunneling-amplitude boson-Hubbarddimer: Stationary states, Phys. Rev. A 65:043616, 2002.
[20] Y. Wu and X. Yang, Analytical results for energy spectrum and eigen-states of a Bose-Einstein condensate in a Mott insulator state,Phys. Rev.A, 68:13608, 2003.
[21] D. M. Stamper-Kurn, M. R. Andrews, A. P. Chikkatur, S. Inouye, H.-J.Miesner, J. Stenger, and W. Ketterle, Optical Confinement of a Bose-EinsteinCondensate, Phys. Rev. Lett. 80:2027,1998.
[22] D. S. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E. A. Cor-nell. Dynamics of component separation in a binary mixture of Bose-Einsteincondensates. Phys. Rev. Lett., 81(8):1539, August 1998.
[23] M. Modugno, F. Dalfovo, C. Fort, P. Maddaloni, and F. Minardi, Dynam-ics of two colliding Bose-Einstein condensates in an elongated magnetostatictrap,Phys. Rev. A 62:063607,2000.
[24] H. T. Ng, C. K. Law, and P. T. Leung, Quantum-correlated double-well tun-neling of two-component Bose-Einstein condensates, Phys. Rev. A 68:013604,2003.
[25] Olaf Mandel, Markus Greiner, Artur Widera, Tim Rom, Theodor W.H?nsch, and Immanuel Bloch, Coherent Transport of Neutral Atoms in Spin-Dependent Optical Lattice Potentials, Phys. Rev. Lett. 91: 010407,2003.
[26] Guang-Hong Chen and Yong-Shi Wu, Quantum phase transition in amulticomponent Bose-Einstein condensate in optical lattices, Phys. Rev. A67:013606, 2003.
[27] Eugene Demler and Fei Zhou, Spinor Bosonic Atoms in Optical Lattices:Symmetry Breaking and Fractionalization , Phys. Rev. Lett. 88: 163001 ,2002.
[28] L. You, Creating Maximally Entangled Atomic States in a Bose-EinsteinCondensate , Phys. Rev. Lett. 90: 030402, 2003.
[29] S. K. Yip, Dimer State of Spin-1 Bosons in an Optical Lattice,Phys. Rev.Lett. ,90:250402 , 2003.
[30] B. Damski,L. Santos, E. Tiemann, M. Lewenstein, S. Kotochigova, P. Juli-enne,and P. Zoller, Creation of a Dipolar Super?uid in Optical Lattices, Phys.Rev. Lett. 90:110401, 2003.
[31] M. G. Moore and H. R. Sadeghpour,Controlling two-species Mott-insulatorphases in an optical lattice to form an array of dipolar molecules ,Phys. Rev.A ,67:041603(R),2003.
[32] Sahel Ashhab and Carlos Lobo, External Josephson e?ect in Bose-Einsteincondensates with a spin degree of freedom,Phys. Rev. A,66:13609,2002.
[33] H. T. Ng and P. T. Leung,Two-mode entanglement in two-component Bose-Einstein condensates, Phys. Rev. A,71:013601 , 2005.
[34] T. Guhr, A. Mu¨ller-Groeling, and H. A. Weidenmu¨er, Random-matrix the-ories in quantum physics: common concepts, Phys. Rep.,299:189, 1998.
[35] M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn,and W. Ketterle. Observation of interference between two Bose-Einstein con-densates. Science, 275(0):637, January 1997.
[36] L. D. Landau and E. M. Lifshitz, Quantum mechanics (Pergaman, NewYork, 1977).
[37] J C Eilbeck, P S Lomdahl, A C Scott, The discrete self-trapping equation ,Physica D, 16:318, 1985.
[38] J. Javanainen, Oscillatory exchange of atoms between traps containing Bosecondensates , Phys. Rev. Lett. 57:3164 ,1986.
[39] S GROSSMANN, M HOLTHAUS, BOSE-EINSTEIN CONDENSATIONAND CONDENSATE TUNNELING, Zeitschrift für Naturforschung, 50:323,1995.
[40] B. D. Josephson, Possible new e?ects in superconductive tunnelling, Phys.Lett. A, 1:251, 1962.
[41] J. Chen, Y. Guo, H. Gao and H. Song, An E?ective Two-component Entan-glement in Double-well Condensation, arxiv:quant-ph/050719, 2005.
[42] V. A. Andreev and O. A. Ivanova, Symmetries and reduced systems of equa-tions for three-boson and four-boson interactions, J. Phys. A,35:8587,2002.
[43] Y. Wu, X. Yang and Y. Xiao,Analytical Method for Yrast Line States inInteracting Bose-Einstein Condensates , Phys. Rev. Lett. ,86:2200,2001.
[44] Here we list the MATHEMATICA code:< [45] J.M. Arias, J. Dukelsky, and J. E. Garc′?a-Ramos, Quantum Phase Transi-tions in the Interacting Boson Model: Integrability, Level Repulsion, and LevelCrossing , Phys. Rev. Lett.,91:162502, 2003.
[46] G. Modugno, M. Modugno, F. Riboli, G. Roati, and M. Inguscio, TwoAtomic Species Super?uid , Phys. Rev. Lett. ,89:190404, 2002.
[47] A. Simoni, F. Ferlaino, G. Roati, G. Modugno, and M. Inguscio., Mag-netic Control of the Interaction in Ultracold K-Rb Mixtures , Phys. Rev. Lett.,90:163202, 2003.
[48] A. Micheli, D. Jaksch, J. I. Cirac, and P. Zoller, Many-particle entanglementin two-component Bose-Einstein condensates , Phys. Rev. A,67: 013607, 2003.
[49] K. W. Mahmud, H. Perry, and W. P. Reinhardt, Quantum phase-spacepicture of Bose-Einstein condensates in a double well, Phys. Rev. A ,71:023615,2005.
[50] A. Peres, Quantum Theory: Conceps and Methods (Kluwer Academic, Dor-drecht, 1995).
[51] D. F. Walls and G. J. Milburn, Quantum Optics. (Springer Verlag, Berlin,1994).
[52] Ying Wu and Xiaoxue Yang, Magic numbers and erratic level crossings ofdouble-well Bose-Einstein condensates, Optics Letters, 31:519, 2006.
[53] M. Latka, P. Grigolini, and B. J. West, Chaos-induced avoided level crossingand tunneling , Phys. Rev. A ,50:1071, 1994.
[1] J. Dunningham, K. Burnett and W. D. Phillips, Bose–Einstein condensatesand precision measurements, Phil. Trans. R. Soc. A 363:2165–2175, 2005.
[2] Y. Shin, M. Saba, T. A. Pasquini,W. Ketterle, D. E. Pritchard, and A. E.Leanhardt, Atom Interferometry with Bose-Einstein Condensates in a Double-Well Potential, Phys. Rev. Lett., 92:050405, 2004.
[3] Y. Shin, G.-B. Jo, M. Saba, T. A. Pasquini, W. Ketterle, and D. E. Pritchard,Optical Weak Link between Two Spatially Separated Bose-Einstein Conden-sates, Phys. Rev. Lett., 95:170402, 2005.
[4] G.-B. Jo, Y. Shin, S. Will, T. A. Pasquini, M. Saba, W. Ketterle, and D. E.Pritchard, Long Phase Coherence Time and Number Squeezing of Two Bose-Einstein Condensates on an Atom Chip, Phys. Rev. Lett., 98:030407, 2007.
[5] S. Gupta, K. W. Murch, K. L. Moore, T. P. Purdy, and D. M. Stamper-Kurn, Bose-Einstein Condensation in a Circular Waveguide, Phys. Rev. Lett.,95:143201 ,2005.
[2] A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzber. Kgl.Preuss. Akad. Wiss., 261, 1924, and 3, 1925.
[3] S. T. Beliaev, Sov. Phys. JETP, 7:289, 1958.
[4] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A.Cornell. Observation of Bose-Einstein condensation in a dilute atomic vapor.Science, 269:198, 1995.
[5] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet. Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions. Phys. Rev.Lett., 75:1687, 1995.
[6] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee,D. M. Kurn, and W. Ketterle. Bose-Einstein condensation in a gas of sodiumatoms. Phys. Rev. Lett., 75:3969, 1995.
[7] F. London. On the Bose-Einstein condensation. Phys. Rev., 54:947, 1938
[8] Fritz London. The λ-phenomenon of liquid helium and the Bose-Einsteindegeneracy. Nature, 141:643, 1938.
[9] A. Gri?n, D. W. Snoke, and S. Stringari, editors, Bose-Einstein Condensa-tion, ( Cambridge University Press, 1995).
[10] Charles E. Hecht. The possible super?uid behaviour of hydrogen atom gasesand liquids. Physica, 25:1159, 1959.
[11] Dale G. Fried, Thomas C. Killian, Lorenz Willmann, David Landhuis,Stephen C. Moss, Daniel Kleppner, and Thomas J. Greytak. Bose-Einsteincondensation of atomic hydrogen. Phys. Rev. Lett., 81:3811, 1998.
[12] C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases(Cambridge University, Cambridge, 2002).
[13] J. R. Anglin and W. Ketterle, Bose-Einstein condensation of atomic gases,Nature, 416:211, 2002.
[14] J. Dunningham, K. Burnett and W. D. Phillips, Bose–Einstein condensatesand precision measurements, Phil. Trans. R. Soc. A 363:2165–2175, 2005.
[15] L. D. Landau and E. M. Lifshitz, Statistical physics (third edition, Perga-mon, Oxford, 1980).
[16] Huang Kerson, Introduction to statistical physics (Taylor and Fran-cis,London, 2001).
[17] 周世勋,量子力学教程(高等教育出版社,北京,1979).
[18] Tin-Lun Ho. Spinor Bose condensates in optical traps. Phys. Rev. Lett.,81:742, 1998.
[19] 王竹溪,郭敦仁, 特殊函数概论(北京大学出版社,北京,2000).
[20] A. G¨olitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J.R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W.Ketterle, Realization of Bose-Einstein condensates in low dimensions, Phys.Rev. Lett. 87:130402, 2001.
[21] P. C. Hohenberg, Existence of Long-Range Order in One and Two Dimen-sions,Phys. Rev. 158:383,1967.
[22] W. J. Mullin. Bose-Einstein condensation in a harmonic potential. J. LowTemp. Phys., 106:615, 1997.
[23] William J. Mullin. A study of Bose-Einstein condensation in a two-dimensional trapped gas. J. Low Temp. Phys., 110(1/2):167, 1998.
[24] Franco Dalfovo, Stefano Giorgini, Lev P. Pitaevskii, and Sandro Stringari.Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys.,71:463, 1999.
[25] Wolfgang Ketterle and N. J. van Druten. Bose-Einstein condensation of afinite number of particles trapped in one or three dimenstions. Phys. Rev. A,54(1):656, July 1996.
[26] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics (parttwo),(Butterwoth-Heinemann, Oxford, 1980)
[27] 徐在新,高等量子力学(华东师范大学出版社,上海,1994).
[28] M. Girardeau and R. Arnowitt, Theory of Many-Boson Systems: Pair The-ory, Phys. Rev. 113:755, 1959.
[29] M. D. Girardeau, Comment on “Particle-number-conserving Bogoli-ubov method which demonstrates the validity of the time-dependent Gross-Pitaevskii equation for a highly condensed Bose gas”, Phys. Rev. A 58:775,1998.
[30] V. N. Popov, Functional integrals and collective excitations, (CambridgeUniv. Press, Cambridge, 1987).
[31] A. Gri?n, Conserving and gapless approximations for an inhomogeneousBose gas at finite temperatures, Phys. Rev. B 53:9341 , 1996.
[32] L. D. Landau and E. M. Lifshitz, Quantum mechanics(Pergaman, New York,1977).
[33] L. P. Pitaevskii,Vortex lines in an imperfect Bose gas, Sov. Phys. JETP13:451 1961.
[34] E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo Ci-mento 20:454 ,1961; Hydrodynamics of a Super?uid Condensate , J. Math.Phys. (N.Y.) 4: 195, 1963.
[35] Herman Feshbach. A unified theory of nuclear reactions. II. Ann. Phys.,19:287, 1962.
[36] S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper- Kurn,and W. Ketterle. Observation of Feshbach resonances in a Bose-Einstein con-densate. Nature, 392:151, 1998.
[37] Pierre Meystre,Atom optics(Springer, New York, 2001).
[38] Mark Edwards and K. Burnett. Numerical solution of the nonlinearSchr¨odinger equation for small samples of trapped neutral atoms. Phys. Rev.A, 51(2):1382, February 1995.
[39] P. A. Ruprecht, M. J. Holland, and K. Burnett, Time-dependent solution ofthe nonlinear Schr¨odinger equation for Bose-condensed trapped neutral atoms,Phys. Rev. A 51: 4704 ,1995.
[40] M. Holland and J. Cooper. Expansion of a Bose-Einstein condensate in aharmonic potential. Phys. Rev. A, 53(4):R1954, April 1996.
[41] Mark Edwards, R. J. Dodd, C. W. Clark, P. A. Ruprecht, and K. Burnett.Properties of a Bose-Einstein condensate in an anisotropic harmonic potential.Phys. Rev. A, 53(4):R1950, April 1996.
[42] Mark Edwards, P. A. Ruprecht, K. Burnett, R. J. Dodd, and Charles W.Clark. Collective excitations of atomic Bose-Einstein condensates. Phys. Rev.Lett., 77(9):1671, August 1996.
[43] P. A. Ruprecht, Mark Edwards, K. Burnett, and Charles W. Clark, Probingthe linear and nonlinear excitations of Bose-condensed neutral atoms in a trap,Phys. Rev. A 54:4178, 1996.
[44] L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne,J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips. Four-wavemixing with matter waves. Nature, 398:218, 1999.
[45] Kozuma, M. et al. Coherent splitting of Bose-Einstein condensed atoms withoptically induced Bragg di?raction. Phys. Rev. Lett. 82, 871 ,1999.
[46] Hagley, E. W. et al.Measurement of the coherence of a Bose-Einstein con-densate. Phys. Rev. Lett. 83: 3112,1999.
[47] J. Denschlag, J. E. Simsarian, D. L. Feder, Charles W. Clark, L. A. Collins,J. Cubizolles, L. Deng, E. W. Hagley, K. Helmerson, W. P. Reinhardt, S. L.Rolston, B. I. Schneider, and W. D. Phillips. Generating solitons by phaseengineering of a Bose-Einstein condensate. Science, 287:97, 2000.
[48] S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G.V. Shlyapnikov, and M. Lewenstein. Dark solitons in Bose-Einstein conden-sates. Phys. Rev. Lett., 83:5198, 1999.
[49] D. L. Feder, M. S. Pindzola, L. A. Collins, B. I. Schneider, and C. W. Clark.Dark-soliton states of Bose-Einstein condensates in anisotropic traps. Phys.Rev. A, 62:053606, 2000.
[50] Kevin E. Strecker, Guthrie B. Partridge, Andrew G. Truscott, Randall G.Hulet,Formation and Propagation of Matter Wave Soliton Trains, Nature,417:150, 2002.
[51] Raman, C. et al. Evidence for a critical velocity in a Bose-Einstein condensedgas. Phys. Rev. Lett. 83:2502, 1999.
[52] Onofrio, R. et al.Observation of super?uid ?ow in a Bose-Einstein condensedgas. Phys. Rev. Lett. 85: 2228,2000.
[53] S. Burger, F. S. Cataliotti, C. Fort, F. Minardi, M. Inguscio, M. L. Chio-falo, and M. P. Tosi. Super?uid and dissipative dynamics of a Bose-Einsteincondensate in a periodic optical potential. Phys. Rev. Lett., 86(20):4447, 2001.
[54] J. E. Williams, M. J. Holland, Preparing topological states of aBose–Einstein condensate. Nature 401, 568–572 (1999).
[55] Matthews, M. R. et al. Vortices in a Bose-Einstein condensate. Phys. Rev.Lett. 83, 2498–2501 (1999).
[56] K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard. Vortex formationin a stirred Bose-Einstein condensate. Phys. Rev. Lett., 84(5):806, January2000.
[57] J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle. Observationof vortex lattices in Bose-Einstein condensates. Science, 292(5516):476, April2001.
[58] P. C. Haljan, I. Coddington, P. Engels, and E. A. Cornell. Driving Bose-Einstein-condensate vorticity with a rotating normal cloud. Phys. Rev. Lett.,87(21):210403, November 2001.
[59] B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C.W. Clark, and E. A. Cornell. Watching dark solitons decay into vortex ringsin a Bose-Einstein condensate. Phys. Rev. Lett., 86(14):2926, April 2001.
[60] A. Minguzzi, S. Conti, and M. P. Tosi. The internal energy and condensatefraction of a trapped interacting Bose gas. J. Phys., Condens. Matter, 9:L33,1997.
[61] S. Giorgini, L. P. Pitaevskii, and S. Stringari. Condensate fraction and criti-cal temperature of a trapped interacting Bose gas. Phys. Rev. A, 54(6):R4633,December 1996.
[62] Hualin Shi and Wei-Mou Zheng, Critical temperature of a trapped inter-acting Bose-Einstein gas in the local-density approximation ,Phys. Rev. A 56:1046,1997.
[63] H. T. C. Stoof, Macroscopic quantum tunneling of a bose condensate, Jour-nal of Statistical Physics, 87:1353,1997.
[64] A. D. Jackson, G. M. Kavoulakis, and C. J. Pethick. Solitary waves in cloudsof Bose-Einstein condensed Phys. Rev. A, 58(3):2417, September 1998.
[65] U. Al Khawaja, J. O. Andersen, N. P. Proukakis, and H. T. C. Stoof. Lowdimensional Bose gases. Phys. Rev. A, 66:013615, July 2002.
[66] D. S. Petrov, M. Holzmann, and G. V. Shlyapnikov,Bose-Einstein Conden-sation in Quasi-2D Trapped Gases ,Phys. Rev. Lett. 84:2551 2000.
[67] D.S. Petrov, D.M. Gangardt, and G.V. Shlyapnikov, Low-dimensionaltrapped gases, J. Phys. IV France 116:5, 2004.
[68] Harald F. Hess. Evaporative cooling of magnetically trapped and compressedspinpolarized hydrogen. Phys. Rev. B, 34(5):3476, September 1986.
[69] Wolfgang Ketterle and N. J. van Druten. Evaporative cooling of trappedatoms. Adv. At. Mol. Opt. Phys., 37(0):181, 1996.
[70] T. W. H¨ansch and A. L. Schawlow. Cooling of gases by laser radiation. Opt.Commun., 13(1):68, 1975.
[71] Steven Chu, L. Hollberg, J. E. Bjorkholm, Alex Cable, and A. Ashkin. Three-dimensional viscous confinement and cooling of atoms by resonance radiationpressure. Phys. Rev. Lett., 55(1):48, July 1985.
[72] Paul D. Lett, Richard N. Watts, Christoph I. Westbrook, William D.Phillips, Phillip L. Gould, and Harold J. Metcalf. Observation of atoms lasercooled below the Doppler limit. Phys. Rev. Lett., 61(2):169, 1988.
[73] J. Dalibard and C. Cohen-Tannoudji. Laser cooling below the Doppler limitby polarization gradients: simple theoretical-models. J. Opt. Soc. Am. B,6(11):2023, November 1989.
[74] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, tle Laser cooling below the one-photon recoil by velocity-selectivecoherent population trapping, Phys. Rev. Lett. 61:826,1988.
[75] Naoto Masuhara, John M. Doyle, Jon C. Sandberg, Daniel Kleppner,Thomas J. Greytak, Harald Hess, and Gregory P. Kochanski. Evaporativecooling of spin-polarized atomic hydrogen. Phys. Rev. Lett., 61(8):935, Au-gust 1988.
[76] C. C. Bradley, C. A. Sackett, and R. G. Hulet. Bose-Einstein condensa-tion of lithium: Observation of limited condensate number. Phys. Rev. Lett.,78(6):985, 1997.
[77] S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell, and C. E. Wie-man. Stable 85Rb Bose-Einstein condensates with widely tunable interactions.Phys. Rev. Lett., 85(9):1795, August 2000.
[78] Jordan M. Gerton, Dmitry Strekalov, Ionut Prodan, and Randall G. Hulet.Direct observation of growth and collapse of a Bose-Einstein condensate withattractive interactions. Nature, 408:692, December 2000.
[79] M. Girardeau, Relationship between systems of impenetrable bosons andfermions in one dimension. J. Math. Phys. 1, 51:523 ,1960.
[80] E.H. Lieb and W. Liniger,Exact analysis of an interacting Bose gas. Thegeneral solution and the ground state. Phys. Rev. 130, 160:1616 ,1963.
[81] E. Tiesinga, B. J. Verhaar and H. T. C. Stoof,Threshold and resonancephenomena in ultracold groundstate collisions. Phys. Rev. A 47: 4114,1993.
[82] Elizabeth A. Donley, Neil R. Claussen, Simon L. Cornish, Jacob L. Roberts,Eric A. Cornell, and Carl E. Wieman. Dynamics of collapsing and explodingBose-Einstein condensates. Nature, 412(6844):295, July 2001。
[83] L. D. Carr, C. W. Clark, and W. P. Reinhardt, Stationary solutions of theone-dimensional nonlinear Schro¨dinger equation. II. Case of attractive nonlin-earity ,Phys. Rev. A, 62:63611, 2000.
[84] R. Kanomoto, H. Saito, and M. Ueda, Quantum phase transition in one-dimensional Bose-Einstein condensates with attractive interactions ,Phys. Rev.A, 67:13608,2003.
[85] Weibin Li, Xiaotao Xie, Zhiming Zhan, and Xiaoxue Yang, Many-body dy-namics of a Bose system with attractive interactions on a ring, Phys. Rev. A,72: 043615 2005.
[86] M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn,and W. Ketterle. Observation of interference between two Bose-Einstein con-densates. Science, 275(0):637, January 1997.
[87] C. Orzel, A. K. Tuchman, M. L. Fenselau, M. Yasuda, and M. A. Kasevich,Squeezed states in a Bose-Einstein Condensate, Science 291:2386, 2001.
[88] M. Greiner, O. Mandel, T. Esslinger, T. W. H ”ansch and I. Bloch, Quantumphase transition from a super?uid to a Mott insulator in a gas of ultracoldatoms, Nature, 415:39,2002.
[89] F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trom-bettoni, A. Smerzi, and M. Inguscio, Josephson Junction arrays with Bose-Einstein condensates, Science, 293:843, 2001.
[90] B. P. Anderson and M. A. Kasevich. Macroscopic quantum interference fromatomic tunnel arrays. Science, 282:1686, November 1998.
[91] Michael Albiez, Rudolf Gati, Jonas F¨olling, Stefan Hunsmann, Matteo Cris-tiani, and Markus K. Oberthaler, Direct Observation of Tunneling and Non-linear Self-Trapping in a Single Bosonic Josephson Junction, Phys. Rev. Lett.,95:010402, 2005.
[92] D. S. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E. A. Cor-nell. Dynamics of component separation in a binary mixture of Bose-Einsteincondensates. Phys. Rev. Lett., 81(8):1539, August 1998.
[93] D. S. Hall, M. R. Matthews, C. E. Wieman, and E. A. Cornell. Measure-ments of relative phase in two-component Bose-Einstein condensates. Phys.Rev. Lett., 81(8):1543, August 1998.
[1] P.S. Julienne,A.M. Smith and K. Burnett, Theory of collisions between lasercooled atoms Adv. At. Mol. Opt. Phys. 30:141, 1993.
[2] J.T.M. Walraven,In: G.-L. Oppo, S.M. Barnett, E. Riis, M. Wilkinson,(Eds.), Quantum Dynamics of Simple Systems( Institute of Physics, Bristol,1996).
[3] C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases(Cambridge University, Cambridge, 2002).
[4] Franco Dalfovo, Stefano Giorgini, Lev P. Pitaevskii, and Sandro Stringari.Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys.,71(3):463, April 1999.
[5] A. S. Parkins and D. F. Walls, The physics of trapped dilute-gas Bose-Einsteincondensates, Physics Reports, 303:1, 1998.
[6] F. Dalfovo and S. Stringari. Bosons in anisotropic traps: Ground state andvortices. Phys. Rev. A, 53(4):2477, April 1996.
[7] E. M. Lifshitz and L. P. Pitaevskii,Statistical physics (Pergamon Press Ox-ford; New York,1980).
[8] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet. Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions. Phys. Rev.Lett., 75:1687, 1995.
[9] P. A. Ruprecht, M. J. Holland, and K. Burnett, Time-dependent solution ofthe nonlinear Schr¨odinger equation for Bose-condensed trapped neutral atoms,Phys. Rev. A 51: 4704 ,1995.
[10] Mark Edwards and K. Burnett. Numerical solution of the nonlinearSchr¨odinger equation for small samples of trapped neutral atoms. Phys. Rev.A, 51(2):1382, February 1995.
[11] Yu. Kagan, G. V. Shlyapnikov, and J. T. M. Walraven. Bose-Einstein con-densation in trapped atomic gases. Phys. Rev. Lett., 76(15):2670, April 1996.
[12] Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov. Evolution of a Bose-condensed gas under variations of the confining potential. Phys. Rev. A,54(3):R1753, September 1996.
[13] Gordon Baym and C. J. Pethick. Ground-state properties of magneticallytrapped Bose-condensed rubidium gas. Phys. Rev. Lett., 76(1):6, January 1996.
[14] N. Bogolubov. On the theory of super?uidity. J. Phys., 11(1):23, 1947.
[15] A. L. Fetter, Nonuniform states of an imperfect bose gas, Annals of Physics,70:67,1972.
[16] A. Gri?n. Conserving and gapless approximations for an inhomogeneousBose gas at finite temperatures. Phys. Rev. B, 53(14):9341, April 1996.
[17] Pierre Meystre, Atom optics(Springer, New York, 2001).
[18] V. N. Popov, Functional Integrals and Collective Modes (Cambridge Uni-versity Press, New York, 1987).
[19] H. Shi, G. Verechaka, and A. Gri?n, Phys. Rev. B, 50:1119, 1994.
[20] C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, and C. E. Wieman.Production of two overlapping Bose-Einstein condensates by sympathetic cool-ing. Phys. Rev. Lett., 78(4):586, January 1997.
[21] D. S. Hall, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Measurementsof Relative Phase in Two-Component Bose-Einstein Condensates , Phys. Rev.Lett. 81:1543, 1998.
[22] Tin-Lun Ho and V. B. Shenoy. Binary mixtures of Bose condensates of alkaliatoms. Phys. Rev. Lett., 77(16):3276, October 1996.
[23] Th. Busch, J. I. Cirac, V. M. P′erez-Garc′?a, and P. Zoller. Stability andcollective excitations of a two-component Bose-Einstein condensed gas: A mo-ment approach. Phys. Rev. A, 56(4):2978, October 1997.
[24] Elena V. Goldstein and Pierre Meystre. Quasiparticle instabilities in multi-component atomic condensates. Phys. Rev. A, 55(4):2935, April 1997.
[25] A. G¨olitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J.R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W.Ketterle, Realization of Bose-Einstein condensates in low dimensions, Phys.Rev. Lett. 87:130402, 2001.
[26] A. D. Jackson, G. M. Kavoulakis, and C. J. Pethick. Solitary waves in cloudsof Bose-Einstein condensed atoms. Phys. Rev. A, 58(3):2417, September 1998.
[27] L. Salasnich, A. Parola, and L Reatto, E?ective wave equations for thedynamics of cigar-shaped and disk-shaped Bose condensates, Phys. Rev. A,65:43614, 2002.
[28] D. S. Petrov, M. Holzmann, and G. V. Shlyapnikov,Bose-Einstein Conden-sation in Quasi-2D Trapped Gases ,Phys. Rev. Lett. 84:2551 2000.
[29] D.S. Petrov, D.M. Gangardt, and G.V. Shlyapnikov, Low-dimensionaltrapped gases, J. Phys. IV France 116:5, 2004.
[30] Lewi Tonks, The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres , Phys. Rev. 50: 955,1936.
[31] M. Girardeau, Relationship between Systems of Impenetrable Bosons andFermions in One Dimension, J. Math. Phys, 1:516, 1960.
[32] Elliott H. Lieb and Werner Liniger, Exact Analysis of an Interacting BoseGas. I. The General Solution and the Ground State, Phys. Rev. 130:1605, 1963.
[33] D. S. Petrov1,G. V. Shlyapnikov, and J. T. M. Walraven,Regimes ofQuantum Degeneracy in Trapped 1D Gases,Phys. Rev. Lett. 85:3745,2000.
[34] Toshiya Kinoshita, Trevor Wenger, David S. Weiss, Observation of a One-Dimensional Tonks-Girardeau Gas, Science, 305:1125,2004. Toshiya Kinoshita,Trevor Wenger, David S. Weiss
[1] Lewi Tonks, The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres , Phys. Rev. 50: 955,1936.
[2] M. Girardeau, Relationship between Systems of Impenetrable Bosons andFermions in One Dimension, J. Math. Phys, 1:516, 1960.
[3] Elliott H. Lieb and Werner Liniger, Exact Analysis of an Interacting BoseGas. I. The General Solution and the Ground State, Phys. Rev. 130:1605,1963.
[4] A. G¨olitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J.R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W.Ketterle, Realization of Bose-Einstein condensates in low dimensions, Phys.Rev. Lett. 87:130402, 2001.
[5] Toshiya Kinoshita, Trevor Wenger, David S. Weiss, Observation of a One-Dimensional Tonks-Girardeau Gas, Science, 305:1125,2004.
[6] A. D. Jackson, G. M. Kavoulakis, and C. J. Pethick. Solitary waves in cloudsof Bose-Einstein condensed Phys. Rev. A, 58(3):2417, September 1998.
[7] L. D. Carr1, Charles W. Clark, and W. P. Reinhardt, Stationary solutionsof the one-dimensional nonlinear Schro¨dinger equation. I. Case of repulsivenonlinearity,Phys. Rev. A 62:063610, 2000.
[8] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet. Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions. Phys. Rev.Lett., 75(9):1687, August 1995. ibid. 79, 1170 (1997).
[9] L. D. Carr1, Charles W. Clark, and W. P. Reinhardt, Stationary solutionsof the one-dimensional nonlinear Schro¨dinger equation. II. Case of attractivenonlinearity,Phys. Rev. A 62:063611, 2000.
[10] Pierre Meystre, Atom optics(Springer, New York, 2001).
[11] Kevin E. Strecker, Guthrie B. Partridge, Andrew G. Truscott, RandallG. Hulet, Formation and propagation of matter-wave soliton trains, Nature,417:150, 2002.
[12] L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr,Y. Castin, and C. Salomon. Formation of a matter-wave bright soliton. Science,296(5571):1290, May 2002.
[13] Rina Kanamoto, Hiroki Saito, and Masahito Ueda, Quantum phase transi-tion in one-dimensional Bose-Einstein condensates with attractive interactions, Phys. Rev. A 67: 013608 ,2003.
[14] G. M. Kavoulakis, Bose-Einstein condensates with attractive interactions ona ring ,Phys. Rev. A 67: 011601(R) ,2003.
[15] S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper- Kurn,and W. Ketterle. Observation of Feshbach resonances in a Bose-Einstein con-densate. Nature, 392(0):151, March 1998.
[16] S. Gupta, K. W. Murch, K. L. Moore, T. P. Purdy, and D. M. Stamper-Kurn, Bose-Einstein Condensation in a Circular Waveguide, Phys. Rev. Lett.95: 143201 ,2005.
[17] 王竹溪,郭敦仁编著,特殊函数概论(北京大学出版社,北京,2004).
[18] LD Landau, EM Lifshitz,Statistics Physics ( Butterworth-Heinemann, Ox-ford, 1984).
[19] Subir Sachdev, Quantum Phase Transitions, (Yale University, Connecticut,2001).
[20] Weibin Li, Xiaotao Xie, Zhiming Zhan and Xiaoxue Yang, Many-body dy-namics of a Bose system with attractive interaction on a ring, Phys.Rev. A,72:043615 (2005).
[21] G.P.Berman, A. Smerzi, and A. R. Bishop, Irregular Dynamics in a One-Dimensional Bose System, Phys. Rev.Lett. 92:30404, 2004.
[22] Y. Wu, X.Yang, and Y. Xiao, Analytical Method for Yrast Line States inInteracting Bose-Einstein Condensates, Phys. Rev. Lett. 86: 2200, 2001.
[23] V. V. Flambaum1 and F. M. Izrailev, Entropy production and wave packetdynamics in the Fock space of closed chaotic many-body systems , Phys. Rev.E 64: 036220 ,2001.
[24] K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods forPhysics and Engineering: A comprehensive Guide, 2nd ed., P574. (CambridgeUniversity Press, Cambridge, Uk, 2002).
[1] I. Bloch, Ultracold quantum gases in optical lattices, Nature Physics, 1:23,2005.
[2] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller. Cold bosonicatoms in optical lattices. Phys. Rev. Lett., 81(15):3108, October 1998.
[3] Markus Greiner, Olaf Mandel, Tilman Esslinger, Theodor W. H¨ansch, andImmanuel Bloch. Quantum phase transition from a super?uid to a Mott insu-lator in a gas of ultracold atoms. Nature, 415:39, January 2002.
[4] C. Orzel, A. K. Tuchman, M. L. Fenselau, M. Yasuda, M. A. Kasevich,Squeezed States in a Bose-Einstein Condensate, Science,291: 2386, 2001.
[5] I Bloch,Quantum gases in optical lattices, Physics World, 17:25, 2004.
[6] M. J¨aaskel¨ainen and P. Meystre, Dynamics of Bose-Einstein condensates indouble-well potentials,Phys. Rev. A,71: 043603,2005.
[7] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Bosonlocalization and the super?uid-insulator transition , Phys. Rev. B, 40:546,1989.
[8] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller. Cold bosonicatoms in optical lattices. Phys. Rev. Lett., 81(15):3108, October 1998.
[9] G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls. Quantum dynamicsof an atomic Bose-Einstein condensate in a doublewell potential. Phys. Rev.A, 55(6):4318, June 1997.
[10] M. J. Steel and M. J. Collet, Quantum state of two trapped Bose-Einsteincondensates with a Josephson coupling, Phys. Rev. A 57:2920, 1997.
[11] R. W. Spekkens and J. E. Sipe,Spatial fragmentation of a Bose-Einsteincondensate in a double-well potential, Phys. Rev. A 59:3868, 1999.
[12] Juha Javanainen and Misha Yu. Ivanov. Splitting a trap containing a Bose-Einstein condensate: Atom number ?uctuations. Phys. Rev. A, 60(3):2351,September 1999.
[13] A. J. Leggett, Bose-Einstein condensation in the alkali gases: Some funda-mental concepts, Rev. Mod. Phys. 73:307,2001.
[14] S. Raghavan1, A. Smerzi, S. Fantoni, and S. R. Shenoy, Coherent oscillationsbetween two weakly coupled Bose-Einstein condensates: Josephson e?ects,πoscillations, and macroscopic quantum self-trapping, Phys. Rev. A 59:620,1999.
[15] A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, Quantum CoherentAtomic Tunneling between Two Trapped Bose-Einstein Condensates, Phys.Rev. Lett. 79:4950, 1997.
[16] Michael Albiez, Rudolf Gati, Jonas F?lling, Stefan Hunsmann, Matteo Cris-tiani, and Markus K. Oberthaler, Direct Observation of Tunneling and Non-linear Self-Trapping in a Single Bosonic Josephson Junction, Phys. Rev. Lett.95:010402,2005.
[17] J. E. Williams, Optimal conditions for observing Josephson oscillations in adouble-well Bose-Einstein condensate, Phys. Rev. A 64: 013610,2001.
[18] R. W. Robinett, Quantum wave packet revivals, Phys. Rep., 392:1, 2004.
[19] G. Kalosakas and A. R. Bishop, Small-tunneling-amplitude boson-Hubbarddimer: Stationary states, Phys. Rev. A 65:043616, 2002.
[20] Y. Wu and X. Yang, Analytical results for energy spectrum and eigen-states of a Bose-Einstein condensate in a Mott insulator state,Phys. Rev.A, 68:13608, 2003.
[21] D. M. Stamper-Kurn, M. R. Andrews, A. P. Chikkatur, S. Inouye, H.-J.Miesner, J. Stenger, and W. Ketterle, Optical Confinement of a Bose-EinsteinCondensate, Phys. Rev. Lett. 80:2027,1998.
[22] D. S. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman, and E. A. Cor-nell. Dynamics of component separation in a binary mixture of Bose-Einsteincondensates. Phys. Rev. Lett., 81(8):1539, August 1998.
[23] M. Modugno, F. Dalfovo, C. Fort, P. Maddaloni, and F. Minardi, Dynam-ics of two colliding Bose-Einstein condensates in an elongated magnetostatictrap,Phys. Rev. A 62:063607,2000.
[24] H. T. Ng, C. K. Law, and P. T. Leung, Quantum-correlated double-well tun-neling of two-component Bose-Einstein condensates, Phys. Rev. A 68:013604,2003.
[25] Olaf Mandel, Markus Greiner, Artur Widera, Tim Rom, Theodor W.H?nsch, and Immanuel Bloch, Coherent Transport of Neutral Atoms in Spin-Dependent Optical Lattice Potentials, Phys. Rev. Lett. 91: 010407,2003.
[26] Guang-Hong Chen and Yong-Shi Wu, Quantum phase transition in amulticomponent Bose-Einstein condensate in optical lattices, Phys. Rev. A67:013606, 2003.
[27] Eugene Demler and Fei Zhou, Spinor Bosonic Atoms in Optical Lattices:Symmetry Breaking and Fractionalization , Phys. Rev. Lett. 88: 163001 ,2002.
[28] L. You, Creating Maximally Entangled Atomic States in a Bose-EinsteinCondensate , Phys. Rev. Lett. 90: 030402, 2003.
[29] S. K. Yip, Dimer State of Spin-1 Bosons in an Optical Lattice,Phys. Rev.Lett. ,90:250402 , 2003.
[30] B. Damski,L. Santos, E. Tiemann, M. Lewenstein, S. Kotochigova, P. Juli-enne,and P. Zoller, Creation of a Dipolar Super?uid in Optical Lattices, Phys.Rev. Lett. 90:110401, 2003.
[31] M. G. Moore and H. R. Sadeghpour,Controlling two-species Mott-insulatorphases in an optical lattice to form an array of dipolar molecules ,Phys. Rev.A ,67:041603(R),2003.
[32] Sahel Ashhab and Carlos Lobo, External Josephson e?ect in Bose-Einsteincondensates with a spin degree of freedom,Phys. Rev. A,66:13609,2002.
[33] H. T. Ng and P. T. Leung,Two-mode entanglement in two-component Bose-Einstein condensates, Phys. Rev. A,71:013601 , 2005.
[34] T. Guhr, A. Mu¨ller-Groeling, and H. A. Weidenmu¨er, Random-matrix the-ories in quantum physics: common concepts, Phys. Rep.,299:189, 1998.
[35] M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn,and W. Ketterle. Observation of interference between two Bose-Einstein con-densates. Science, 275(0):637, January 1997.
[36] L. D. Landau and E. M. Lifshitz, Quantum mechanics (Pergaman, NewYork, 1977).
[37] J C Eilbeck, P S Lomdahl, A C Scott, The discrete self-trapping equation ,Physica D, 16:318, 1985.
[38] J. Javanainen, Oscillatory exchange of atoms between traps containing Bosecondensates , Phys. Rev. Lett. 57:3164 ,1986.
[39] S GROSSMANN, M HOLTHAUS, BOSE-EINSTEIN CONDENSATIONAND CONDENSATE TUNNELING, Zeitschrift für Naturforschung, 50:323,1995.
[40] B. D. Josephson, Possible new e?ects in superconductive tunnelling, Phys.Lett. A, 1:251, 1962.
[41] J. Chen, Y. Guo, H. Gao and H. Song, An E?ective Two-component Entan-glement in Double-well Condensation, arxiv:quant-ph/050719, 2005.
[42] V. A. Andreev and O. A. Ivanova, Symmetries and reduced systems of equa-tions for three-boson and four-boson interactions, J. Phys. A,35:8587,2002.
[43] Y. Wu, X. Yang and Y. Xiao,Analytical Method for Yrast Line States inInteracting Bose-Einstein Condensates , Phys. Rev. Lett. ,86:2200,2001.
[44] Here we list the MATHEMATICA code:<
[46] G. Modugno, M. Modugno, F. Riboli, G. Roati, and M. Inguscio, TwoAtomic Species Super?uid , Phys. Rev. Lett. ,89:190404, 2002.
[47] A. Simoni, F. Ferlaino, G. Roati, G. Modugno, and M. Inguscio., Mag-netic Control of the Interaction in Ultracold K-Rb Mixtures , Phys. Rev. Lett.,90:163202, 2003.
[48] A. Micheli, D. Jaksch, J. I. Cirac, and P. Zoller, Many-particle entanglementin two-component Bose-Einstein condensates , Phys. Rev. A,67: 013607, 2003.
[49] K. W. Mahmud, H. Perry, and W. P. Reinhardt, Quantum phase-spacepicture of Bose-Einstein condensates in a double well, Phys. Rev. A ,71:023615,2005.
[50] A. Peres, Quantum Theory: Conceps and Methods (Kluwer Academic, Dor-drecht, 1995).
[51] D. F. Walls and G. J. Milburn, Quantum Optics. (Springer Verlag, Berlin,1994).
[52] Ying Wu and Xiaoxue Yang, Magic numbers and erratic level crossings ofdouble-well Bose-Einstein condensates, Optics Letters, 31:519, 2006.
[53] M. Latka, P. Grigolini, and B. J. West, Chaos-induced avoided level crossingand tunneling , Phys. Rev. A ,50:1071, 1994.
[1] J. Dunningham, K. Burnett and W. D. Phillips, Bose–Einstein condensatesand precision measurements, Phil. Trans. R. Soc. A 363:2165–2175, 2005.
[2] Y. Shin, M. Saba, T. A. Pasquini,W. Ketterle, D. E. Pritchard, and A. E.Leanhardt, Atom Interferometry with Bose-Einstein Condensates in a Double-Well Potential, Phys. Rev. Lett., 92:050405, 2004.
[3] Y. Shin, G.-B. Jo, M. Saba, T. A. Pasquini, W. Ketterle, and D. E. Pritchard,Optical Weak Link between Two Spatially Separated Bose-Einstein Conden-sates, Phys. Rev. Lett., 95:170402, 2005.
[4] G.-B. Jo, Y. Shin, S. Will, T. A. Pasquini, M. Saba, W. Ketterle, and D. E.Pritchard, Long Phase Coherence Time and Number Squeezing of Two Bose-Einstein Condensates on an Atom Chip, Phys. Rev. Lett., 98:030407, 2007.
[5] S. Gupta, K. W. Murch, K. L. Moore, T. P. Purdy, and D. M. Stamper-Kurn, Bose-Einstein Condensation in a Circular Waveguide, Phys. Rev. Lett.,95:143201 ,2005.