超冷原子体系中一些新奇量子现象的研究
|
摘要
玻色-爱因斯坦凝聚(BEC),作为与许多物理分支交叉的研究领域,已经呈现出诸多备受关注的新奇量子现象,为人们在宏观尺度内研究量子现象提供了有效途径。本学位论文首先回顾BEC的重要实验进展,然后概述BEC的相关基础理论；接着分别详细介绍我们对旋转BEC及分子BEC中一些新奇量子现象的研究成果。
在平均涡旋近似下,采用变分法研究了非简谐振子势下快速旋转BEC中一些与涡旋态有关的现象。研究结果很好地解释了实验中所呈现出的许多令人迷惑的新奇现象；同时,证明了在非简谐振子势下,快速旋转的BEC中能够形成带多个单位角动量的涡旋。首次研究了外加电场对快速旋转BEC的影响,发现外加电场会使体系的能量降低,从而会使体系变得更加稳定；同时,电场会使涡旋格点和粒子数密度分布发生平移。当体系的旋转频率接近于简谐振子的自然频率时,我们预期实验上也许可以探测到电场导致的粒子数密度分布平移。
研究了光诱导的非阿贝尔规范势在冷原子中的效应。指出振幅随时间和空间变化的激光能够直接同时诱导非阿贝尔规范矢势和标势而不需要一投影过程。通过构造特殊的激光,在旋转BEC中诱导产生了不随时间变化且空间结构较简单的规范势；在该规范势下,对Landau能级的性质及旋转BEC的基态相图进行了研究,指出即使结构如此简单的规范势也能够使旋转BEC呈现出更加丰富的现象。
研究了双原子体系中原子-分子转化的问题。对于玻色-费米混合物到分子的转化问题,以40K和87Rb组成的混合物为例,指出Feshbach共振协助的受激Raman绝热通道技术能有效地将玻色-费米混合物转化为分子而单纯的Feshbach共振技术则不能。对于处在不同自旋态上的费米混合物到分子的转化问题,发现对于6Li原子系统,Feshbach共振协助的受激Raman绝热通道技术要比单纯的Feshbach共振技术更有效,而对于40K原子系统,情况则恰好与6Li原子系统相反。指出通过引入相干布局数囚禁(CPT)态的保真度可有效刻画体系的绝热演化性质,并阐明原子-分子的转化率与体系绝热性之间的联系,揭示6Li原子-分子的转化率高而40K原子-分子的转化率低的原因。
采用平均场近似下的单粒子密度矩阵方法,初步研究了双阱中部分相干玻色体系的动力学性质。在忽略粒子间相互作用时,通过解析的方法证明随相干度的减小Rabi振荡的振幅也会减小,当相干度为零时,Rabi振荡消失；在考虑粒子间相互作用的情况下,通过引入混合态的保真度刻画了体系的绝热演化性质,并采用数值计算的方法研究了部分相干BEC的基态的绝热演化问题。
Bose-Einstein condensates (BECs) exist at the confluence of several physical fields, which have been exhibiting many novel phenomena in the last decade. The realization of BECs in experiment have opened up the exploration of novel quan-tum phenomena in a qualitatively new regime. In this dissertation, we first review the main experimental developments and the elementary theory approaches on BECs. Then we present our studies on several novel quantum phenomena in the rotating and molecular BECs.
In the averaged vortex approximation, a fast rotating BEC in an anharmonic potential is studied in variational method. We prove that the multiply-quantized vortex can be generated in such a system. Many puzzling phenomena presented in experiment are also explained. Then we study the effect of an applied electric field on a rotating BEC in a harmonic trap. We find that the system becomes more stable in the presence of electric field because it suppresses the energy of the system. The electric field can also bring out a shift to both the vortex lattice and the particle distribution. When the rotation frequency of the system is near the frequency of the harmonic potential, the shift induced by the electric field may be detected according to our advice.
We study the effects of an optically induced gauge field in cold atoms show-ing that the laser with time-dependent amplitudes can induce non-Abelian gauge fields, where the scalar fields are obtained directly rather than through a pro-jecting process. Some features of the Landau levels and the ground-state phase diagram of a rotating system for a concrete gauge field are also discussed. We show that such a simple gauge field can make the picture of rotating BECs more fascinating.
We theoretically study how to convert two-species atoms into molecules through the technique of Stimulated Raman adiabatic passage (STIRAP) aided by Feshbach resonance. Taking the mixture of 40K and 87Rb as an example, we show that the technique of STIRAP aided by Feshbach resonance can efficiently convert Bose-Fermi mixtures into molecules, which is contrast to the case of the bare Feshbach resonance. We also study the conversion of fermionic atoms in two different hyperfine states into molecules showing that the technique of STIRAP aided by Feshbach resonance is more effective than the bare Feshbach resonance for 6Li atoms rather than 40K atoms. We introduce a adiabatic fidelity for a coherent population trapping (CPT) state to characterize the adiabatic property of the system and explain why the atom-to-molecule conversion efficiencies for 40K and 6Li are distinctly different with the help of the adiabatic fidelity of CPT state.
Based on the single-particle density matrix, we investigate some dynamical properties of a partly coherent BEC in double wells. We show that the amplitude of Rabi oscillation decreases with the reduction of coherence degree for a non-interacting system. Considering the interaction between atoms, we also study the adiabatic evolution of the ground state for the partly coherent BEC by the numerical method.
在平均涡旋近似下,采用变分法研究了非简谐振子势下快速旋转BEC中一些与涡旋态有关的现象。研究结果很好地解释了实验中所呈现出的许多令人迷惑的新奇现象；同时,证明了在非简谐振子势下,快速旋转的BEC中能够形成带多个单位角动量的涡旋。首次研究了外加电场对快速旋转BEC的影响,发现外加电场会使体系的能量降低,从而会使体系变得更加稳定；同时,电场会使涡旋格点和粒子数密度分布发生平移。当体系的旋转频率接近于简谐振子的自然频率时,我们预期实验上也许可以探测到电场导致的粒子数密度分布平移。
研究了光诱导的非阿贝尔规范势在冷原子中的效应。指出振幅随时间和空间变化的激光能够直接同时诱导非阿贝尔规范矢势和标势而不需要一投影过程。通过构造特殊的激光,在旋转BEC中诱导产生了不随时间变化且空间结构较简单的规范势；在该规范势下,对Landau能级的性质及旋转BEC的基态相图进行了研究,指出即使结构如此简单的规范势也能够使旋转BEC呈现出更加丰富的现象。
研究了双原子体系中原子-分子转化的问题。对于玻色-费米混合物到分子的转化问题,以40K和87Rb组成的混合物为例,指出Feshbach共振协助的受激Raman绝热通道技术能有效地将玻色-费米混合物转化为分子而单纯的Feshbach共振技术则不能。对于处在不同自旋态上的费米混合物到分子的转化问题,发现对于6Li原子系统,Feshbach共振协助的受激Raman绝热通道技术要比单纯的Feshbach共振技术更有效,而对于40K原子系统,情况则恰好与6Li原子系统相反。指出通过引入相干布局数囚禁(CPT)态的保真度可有效刻画体系的绝热演化性质,并阐明原子-分子的转化率与体系绝热性之间的联系,揭示6Li原子-分子的转化率高而40K原子-分子的转化率低的原因。
采用平均场近似下的单粒子密度矩阵方法,初步研究了双阱中部分相干玻色体系的动力学性质。在忽略粒子间相互作用时,通过解析的方法证明随相干度的减小Rabi振荡的振幅也会减小,当相干度为零时,Rabi振荡消失；在考虑粒子间相互作用的情况下,通过引入混合态的保真度刻画了体系的绝热演化性质,并采用数值计算的方法研究了部分相干BEC的基态的绝热演化问题。
Bose-Einstein condensates (BECs) exist at the confluence of several physical fields, which have been exhibiting many novel phenomena in the last decade. The realization of BECs in experiment have opened up the exploration of novel quan-tum phenomena in a qualitatively new regime. In this dissertation, we first review the main experimental developments and the elementary theory approaches on BECs. Then we present our studies on several novel quantum phenomena in the rotating and molecular BECs.
In the averaged vortex approximation, a fast rotating BEC in an anharmonic potential is studied in variational method. We prove that the multiply-quantized vortex can be generated in such a system. Many puzzling phenomena presented in experiment are also explained. Then we study the effect of an applied electric field on a rotating BEC in a harmonic trap. We find that the system becomes more stable in the presence of electric field because it suppresses the energy of the system. The electric field can also bring out a shift to both the vortex lattice and the particle distribution. When the rotation frequency of the system is near the frequency of the harmonic potential, the shift induced by the electric field may be detected according to our advice.
We study the effects of an optically induced gauge field in cold atoms show-ing that the laser with time-dependent amplitudes can induce non-Abelian gauge fields, where the scalar fields are obtained directly rather than through a pro-jecting process. Some features of the Landau levels and the ground-state phase diagram of a rotating system for a concrete gauge field are also discussed. We show that such a simple gauge field can make the picture of rotating BECs more fascinating.
We theoretically study how to convert two-species atoms into molecules through the technique of Stimulated Raman adiabatic passage (STIRAP) aided by Feshbach resonance. Taking the mixture of 40K and 87Rb as an example, we show that the technique of STIRAP aided by Feshbach resonance can efficiently convert Bose-Fermi mixtures into molecules, which is contrast to the case of the bare Feshbach resonance. We also study the conversion of fermionic atoms in two different hyperfine states into molecules showing that the technique of STIRAP aided by Feshbach resonance is more effective than the bare Feshbach resonance for 6Li atoms rather than 40K atoms. We introduce a adiabatic fidelity for a coherent population trapping (CPT) state to characterize the adiabatic property of the system and explain why the atom-to-molecule conversion efficiencies for 40K and 6Li are distinctly different with the help of the adiabatic fidelity of CPT state.
Based on the single-particle density matrix, we investigate some dynamical properties of a partly coherent BEC in double wells. We show that the amplitude of Rabi oscillation decreases with the reduction of coherence degree for a non-interacting system. Considering the interaction between atoms, we also study the adiabatic evolution of the ground state for the partly coherent BEC by the numerical method.
引文
[1]S. N. Bose, A. Phys.26,178 (1924).
[2]A. Einstein, Sitzungsberichte der Preussischen Akademie der Wis-senschaften, Physikalisch-mathematische Klasse p.261 (1924); p.3 (1925).
[3]M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269,198 (1995).
[4]K. B. Davis, M.O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, Phys. Rev. Lett.75,3969 (1995).
[5]C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett.75,1687(1995).
[6]S. Chu, L. Hollberg, J. E. Bjorkholm, Alex Cable, and A. Ashkin, Phys. Rev. Lett.55,48-51 (1985).
[7]C. E. Hecht, Physica 25,1159 (1959).
[8]D. G. Fried, T. C. Killian, et al., Phys. Rev. Lett.81,3811 (1998).
[9]W. Ertmer, R. Blatt, J. L. Hall, M. Zhu, Phys. Rev. Lett.54,996 (1985).
[10]C. E. Wieman, L. Hollberg, Rev. Sci. Instrum.62,1 (1991).
[11]W. D. Phillips, H. Metcalf, Phys. Rev. Lett.48,596 (1982); J. Prodan, A. Migdall, W. D. Phillips, I. So, H. Metcalf, J. Dalibard, Phys. Rev. Lett. 54,992 (1985).
[12]P. A. Molenaar, P. van der Straten, H. G. M. Heideman, H. Metcalf, Phys. Rev. A 55,605 (1997).
[13]S. V. Andreev, V. I. Balykin, V. S. Letokhov, V. G. Minogin, JETP Lett. 34,442 (1981).
[14]P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, H. J. Metcalf, Phys. Rev. Lett.61,169 (1988).
[15]W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell, Phys. Rev. Lett,74,3352 (1995).
[16]H. F. Hess, Phys. Rev. B 34,3476 (1986).
[17]W. Ketterle and N. J. van Druten, Adv. At. Mol. Opt. Phys.37,181 (1996).
[18]H. Feshbach, Ann. Phys.19,287 (1962).
[19]J. E. Lye, L. Fallani, M. Modugno, D. S. Wiersma, C. Fort, M. Inguscio, Phys. Rev. Lett,95,070401 (2005).
[20]F. S. Levin, H. Feshbach, Reaction Dynamics, Gordon and Breach, New York,1973.
[21]W. C. Stwalley, Phys. Rev. Lett,37,1628 (1976).
[22]E. Tiesinga, B. J. Verhaar, H. T. C. Stoof, Phys. Rev. A 47,4114 (1993).
[23]A. J. Moerdijk, B. J. Verhaar, A. Axelsson, Phys. Rev. A 51,4852 (1995).
[24]A. J. Moerdijk, B. J. Verhaar, T. M. Nagtegaal, Phys. Rev. A 53,4343 (1996).
[25]H. M. J. M. Boesten, J. M. Yogels, J. G. C. Tempelars, B. J. Verhaar, phys. Rev. A 54, R3726 (1996).
[26]J. M. Vogels, C. C. Tsai, R. S. Freeland, et al., Phys. Rev. A,56, R1067 (1997).
[27]N. R. Newbury, C. J. Myatt, C. E.Wieman, Phys. Rev. A 51, R 2680 (1995).
[28]J. Stenger, S. Inouye, M. R. Andrews, H. J. Miesner, D. M. Stamper-Kurn,W. Ketterle, Phys. Rev. Lett.82,2422 (1999).
[29]S. D. Gensemer, P. L. Gould, Phys. Rev. Lett.80,936 (1998).
[30]K. M. Jones, P. D. Lett, E. Tiesinga, P. S. Julienne, Phys. Rev. A 61, 012501 (1999).
[31]J. Ortigoso, M. Rodriguez, M. Gupta, B. Friedrich, J. Chem. Phys.110, 3870 (1999).
[32]F. H. Mies, E. Tiesinga, P. S. Julinne, Phys. Rev. A 61,022721 (2000).
[33]V. Kokoouline, J. Vala, R. Kosloff, J. Chem. Phys.114,3046 (2001).
[34]E. G. M. van Kempen, S. J. J. M. F. Kokkelmans, D. J. Heinzen, B. J. Verhaar, Phys. Rev. Lett.88,093201 (2002).
[35]C. Ticknor,C. A. Regal, D. S. Jin, J. L. Bohn, Phys. Rev. A 69,042712 (2004).
[36]J. L. Roberts, N. R. Claussen, J. P. Burke Jr., C. H. Greene, E. A. Cornell, C. E. Wieman, Phys. Rev. Lett.81,5109 (1998).
[37]V. Vuletic, A. J. Kerman, C. Chin, S. Chu, Phys. Rev. Lett.82,1406 (1999).
[38]J. L. Roberts, N. R. Claussen, S. L. Cornish, E. A. Donley, E. A. Cornell, C. E. Wieman, Phys. Rev. Lett.86,4211 (2001).
[39]A. Marte, T. Volz, J. Schuster, S. Durr, G. Rempe, E. G. M. van Kempen, B. J. Verhaar, Phys. Rev. Lett.89,283202 (2002).
[40]N. R. Claussen, E. A. Donley, S. T. Thompson, C. E. Wieman, Phys. Rev. Lett.89,010401 (2002).
[41]T. Volz, S. Durr, S. Ernst, A. Marte, G. Rempe, Phys. Rev. A 68,010702 (2003).
[42]C. A. Stan, M. W. Zwierlein, C. H. Schunck, S. M. F. Raupach, W. Ket-terle, Phys. Rev. Lett.93,143001 (2004).
[43]I. D. Prodan, M. Pichler, M. Junker, G. Hulet, Phys. Rev. Lett.91,080402 (2003).
[44]M. Houbiers, H. T. C. Stoof, W. I. McAlexander, R. G. Hulet, Phys. Rev. Lett. A 57, R 1497 (1998).
[45]S. Jochim, M. Bartenstein, G. Hendl, J. H. Denschlag, R. Grimm, Phys. Rev. Lett.89,273202 (2002).
[46]C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Nature 424,47 (2003).
[47]C. H. Schunck, M. W. Zwierlein, C. A. Stan, S. M. F. Raupach, W. Ket-terle, A. Simoni, E. Tiesinga, C. J. Williams, P. S. Julienne, Phys. Rev. A 71,045601 (2005).
[48]A. Simoni, F. Ferlaino, G. Roati, G. Modugn, M. Inguscio, Phys. Rev. Lett.90,163202 (2003).
[49]S. Inouye, J. Goldwin, M. L. Olsen, C. Ticknor, J. L. Bohn, D. S. Jin, Phys. Rev. Lett.93 183201 (2004).
[50]E. Timmermans, P. Tommasini, M. Hussein, A. Kerman, Phys. Rep.315, 199 (1999).
[51]S. Inouye, M. R. Andrews, J. Stenger, D. M. Stamper-Kurn, and W. Ket-terle, Nature,392,151 (1998).
[52]G. Taubes, Science 269,152 (1995).
[53]C. C. Bradley, C. A. Sackett, R. G. Hulet, Phys. Rev. Lett.78,985 (1997).
[54]T.Weber, J. Herbig, M. Mark, H.C. Nagert, R. Grimm, Science 299 (2003) 232.
[55]F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cu-bizolles, C. Salomon, Phys. Rev. Lett.87,080403 (2001).
[56]G. Modugno, G. Ferrari, G. Roati, R. J. Brecha, A. Simoni, M. Inguscio, Science 294,1320 (2001).
[57]S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell, C. E. Wieman, Phys. Rev. Lett.85,1795 (2000).
[58]Y. Takasu, K. Maki, K. Komori, T. Takano, K. Honda, M. Kumakura, T.Yabuzaki,Y. Takahashi, Phys. Rev. Lett.91,040404 (2003).
[59]A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, T. Pfau, Phys. Rev. Lett. 94,160401 (2005).
[60]A. Robert, O. Sirjean, A. Browaeys, J. Poupard, S. Nowak, D. Boiron, C. I. Westbrook, A. Aspect, Science 292,461 (2001).
[61]F. Pereira, D. Santos, J. Leonard, J. Wang, C. J. Barrelet, F. Rasel, C. S. Unnikrishnan, M. Leduc, C. Cohen-Tannoudji, Phys. Rev. Lett.86,3459 (2001)(61).
[62]M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell. Phys. Rev. Lett.83,2498 (1999).
[63]K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett.84,806 (2000).
[64]J. R. Abo-Shaeer, C. Raman, J. M Vogels, W. Ketterle, Science 292,476 (2001).
[65]D. R. Scherer, C. N. Weiler, T. W. Neely, and B. P. Anderson, Phys. Rev. Lett.98,110402 (2007).
[66]M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn and W. Ketterle, Science 275,637 (1997).
[67]A. J. Leggett, Rev. Mod. Phys.73,307 (2001).
[68]H. W. Xiong, S. J. Liu, and M. S. Zhan, New J. Phys.8,245 (2006).
[69]R. G. Hulet, McGraw-HillYearbook of Science and Technology, McGraw-Hill, NewYork, pp.19-21 (2005).
[70]A. Cho, Science 301,750 (2003).
[71]J. E. Thomas, M. Gehm, Amer. Scientist 92,238 (2004).
[72]M. Greiner, C. A. Regal, D. S. Jin, cond-mat/0502539 (2005).
[73]B. DeMarco, D. S. Jin, Science 285,1703 (1999).
[74]A. G. Truscott, K. E. Strecker,W. I. McAlexander, G. B. Partridge, R. G. Hulet, Science 291,2570 (2001).
[75]T. Bourdel, J. Cubizolles, L. Khaykovich, K. M. F. Magalhaes, S. J.J. M. E. Kokkelmans, G. V. Shlyapnikov, C. Salomon, Phys. Rev. Lett.91, 020402 (2003).
[76]S. R. Granade, M. E. Gehm, K. M. O'Hara, J. E. Thomas, Phys. Rev. Lett.88,120405 (2002).
[77]Z. Hadzibabic, C. A. Stan, K. Dieckmann, S. Gupta, M. W. Zwierlein, A. Gorlitz, W. Ketterle, Phys. Rev. Lett.88,160401 (2002).
[78]G. Roati, F. Riboli, G. Modugno, M. Inguscio, Phys. Rev. Lett.89,150403 (2002).
[79]S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. H. Denschlag, R. Grimm, Science 302,2101 (2003).
[80]G. Modugno, F. Ferlaino, R. Heidemann, G. Roati, M. Inguscio, Phys. Rev. A 68, R011601 (2003).
[81]T. Loftus, C. A. Regal, C. Ticknor, J. L. Bohn, D. S. Jin, Phys. Rev. Lett. 88,173201 (2002).
[82]C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett,92,083201 (2004).
[83]J. Cubizolles, T. Bourde, S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov, C. Salomon, Phys. Rev. Lett.91,240401 (2003).
[84]S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, C. Chin, J. Hecker Denschlag, R. Grimm, Phys. Rev. Lett.91,240402 (2003).
[85]M.Greiner, C.A.Regal, D. S. Jin, Nature 426,537 (2003).
[86]M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J.H. Denschlag, R. Grimm, Phys. Rev. Lett.92,120401 (2004).
[87]M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Den-schlag, and R. Crimm, Phys. Rev. Lett,92,203201 (2004).
[88]M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta, Z. Hadzibabic,W. Ketterle, Phys. Rev. Lett.91,250401 (2003).
[89]M. W. Zwierlein, A. Schirotzek, C. H. Schunck, and W. Ketterle, Science 311,492 (2006).
[90]K. Machida, T. Mizushima, and M. Ichioka, Phys. Rev. Lett.97,120407 (2006).
[91]C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett.92,040403 (2004).
[92]M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Kerman,W. Ketterle, Phys. Rev. Lett.92,120403 (2004).
[93]M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo, R. Walser, Phys. Rev. Lett.87,120406 (2001).
[94]E. Timmermans, K. Furuya, P. W. Milonni, A. K. Kerman, Phys. Lett. A 285,228 (2001).
[95]Y. Ohashi, A. Griffin, Phys. Rev. Lett.89,130402 (2002).
[96]J. Stajic, J. N. Milstein, Q. Chen, M. L. Chiofalo, M. J. Holland, K. Levin, Phys. Rev. A 69,063610 (2004).
[97]R. A. Duine, H.T. C. Stoof, J. Opt. B Quantum Semiclassical Opt.5, S 212 (2003).
[98]M. Greiner, C. A. Regal, C.Ticknor, J. L. Bohn, D. S. Jin, Phys. Rev. Lett.92,150405 (2004).
[99]C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. H. Den-schlag, R. Grimm, Science 305,1128 (2004).
[100]M. W Zwierlein, J. R. Abo-shaeer, A. Schirotzek, C. H. Schunck, and W. Ketterle, Nature 435,1047 (2005).
[101]L. D. Dandau and E. M. Lifshitz, Statistical Physics, Part 1, Third editin, (Pergamon, Oxford,1980),§62.
[102]C. J. Pethick and H. Smith:Bose-Einstein Condensation in Dilute Gass, Cambridge University press, United Kingdom,2002.
[103]L. P. Pitaevskii, Zh. Eksp. Thor. Fiz.40,646 (1961) [Sov. Phys.-JEYP 13, 451 (1961)]; E. P. Gross, Nuovo Cimento 20,454 (1961); J. Math. Phys. 4,195 (1963).
[104]P. S. Julienne, A. M. Smith, and K. Burnett, Adv. At. Mol. Opt. Phys. 30,141 (1993).
[105]E. M. Lifshitz, L. P. Pitaevskii, Statistical Physics, Part Ⅱ. Pergamon, Oxford,1980.
[106]A. S. Parkins, and D. F. Walls, Phys. rep.303,1-80 (1998).
[107]V. Schweikhard, I. Coddington, P. Engels, V. P. Mogendorff, and E. A. Cornell, Phys. Rev. Lett.92,040404 (2004).
[108]G. Baym, and C. J. Pethick, Phys. Rev. A 69,043619 (2004).
[109]U. R. Fischer and G. Baym, Phys. Rev. Lett.90,140402 (2003).
[110]J. Sinova, C. B. Hanna, and A. H. MacDonald, Phys. Rev. Lett.90,120401 (2003).
[111]M. Edwards, and K. Burnett, Phys. Rev. A 51,1382 (1995).
[112]P. A. Ruprecht, M. J. Holland, K. Burnett, and M. Edwards, Phys. Rev. A 51,4704 (1995).
[113]Y. Kagan, G. V. Shlyapnikov, and J. T. M. Walraven, Phys. Rev. Lett. 76,2670 (1996); Y. Kagan, E. L. Surkov, and G. V. Shlyapnikov, Phys. Rev. A 54, R1753 (1996).
[114]L. You, M. Lewenstein, R. J. Glauber, and J. Cooper, Phys. Rev. A 53, 329 (1996).
[115]M. Edwards, R. J. Dodd, C. W. Clark, P. A. Ruprecht, and K. Burnett, Phys. Rev. A 53, R1950 (1996).
[116]F. Dalfovo, and S. Stringari, Phys. Rev. A 53,2477 (1996).
[117]M. Holland, and J. Cooper, Phys. Rev. A 53, R1954 (1996).
[118]D. S. Jin, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett.77,420 (1996).
[119]M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, and W. Ketterle, Phys. Rev. Lett.77,416 (1996).
[120]C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett.75,1687 (1995).
[121]C. C. Bradley, C. A. Sackett, and R. G. Hulet, Phys. Rev. Lett.78,985 (1997).
[122]T. Bergeman, Phys. Rev. A 55,3658 (1997).
[123]R. J. Dodd, M. Edwards, C. J. Williams, and C. W. Clark, Phys. Rev. A 54,661 (1996).
[124]A. L. Fetter, arxiv. org/cond-mat/9510037.
[125]R. J. Donnelly, Quantized Vortices in Liquid He II, (Cambridge University Press, Cambridge,1991).
[126]A. L. Fetter and A. A. Svidzinsky, J. Phys.:Condens. Matter 13,135 (2001).
[127]J. A. Dunningham, M. J. Collett, and D. F. Walls, Phys. Lett. A 245,49 (1998).
[128]L. Onsager, Nuovo Cimento 6, Suppl.2,249 (1949).
[129]R. P. Feynman, Progress in Low Temperature Physics, Vol.1, ed. C. J. Gorter, (North-Holland, Amsterdam,1955), Chapter 2.
[130]E. Lundh, C. J. Pethick, H. Smith, Phys. Rev. A 55,2126 (1997).
[131]L. D. Landau and E. M. Lifshitz, Mechanics, Third edition, (Pergamon, Oxford,1976), §39.
[132]E. Hodby, G. Hechenblaikner, S. A. Hopkins,0. M. Maraggraveo, and C.J. Foot, Phys. Rev. Lett.88,010405 (2002).
[133]P. C. Haljan, I. Coddington, P. Engels, and E. A. Cornell, Phys. Rev. Lett. 87,210403 (2001).
[134]T.-L. Ho, Phys. Rev. Lett.87,060403 (2001).
[135]Vincent Bretin, Sabine Stock, Yannick Seurin, and Jean Dalibard, Phys. Rev. Lett.92,050403 (2004).
[136]Kenichi Kasamatsu, Makoto Tsubota, and Masahito Ueda, Phys. Rev. A 66,053606 (2002).
[137]E. Lundh, Phys. Rev. A 65,043604 (2002).
[138]A. L. Fetter, Phys. Rev. A 64,063608 (2001).
[139]E. Lundh, A. Collin and K.-A. Suominen, Phys. Rev. Lett.92,070401 (2004).
[140]T. P. Simula, A. A. Penckwitt and R. J. Ballagh, Phys. Rev. Lett.92, 060401 (2004).
[141]A. Aftalion and I. Danaila, Phys. Rev. A 69,033608 (2004).
[142]A. L. Fetter, B. Jackson, and S. Stringari, Phys. Rev. A 71,013605 (2005).
[143]A. D. Jackson and G. M. Kavoulakis, Phys. Rev. A 70,023601 (2004).
[144]V. Subrahmanyam, Phys. Rev. A 67,035601 (2003).
[145]T. K. Ghosh, Phys. Rev. A 69,043606 (2004).
[146]A. Aftalion, X. Blanc and J. Dalibard, Phys. Rev. A 71,023611 (2005).
[147]Gentaro Watanabe, Gordon Baym, and C. J. Pethick, Phys. Rev. Lett. 93,190401 (2004).
[148]E. J. Mueller and T.-L. Ho, Phys. Rev. Lett.88,180403 (2002).
[149]A. D. Jackson, G. M. Kavoulakis and E. Lundh, Phys. Rev. A 69,053619 (2004).
[150]A. Aftalion and I. Danaila, (2003), arxiv.org/abs/cond-mat/0309668.
[151]Ionut Danaila, arxiv.org/abs/cond-mat/0503122.
[152]M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett.83,2498 (1999).
[153]C. Raman, J. R. Abo-Shaeer, J. M. vogels, K. Xu and W. Ketterle, Phys Rev. Lett.87,210402 (2001).
[154]P. Engels, I. Coddington, P. C. Haljan, and E. A. Cornell, Phys. Rev. Lett.89,100403 (2002).
[155]J. D. Jackson, Classical Electrodynamics (Wiley, New York,1975),2nd ed., Chap.11.
[156]D. A. Butts and D. S. Rokhsa, Nature 397,327 (1999).
[157]N. R. Cooper, S. Komineas and N. Read, Phys. Rev. A 70,033604, (2004).
[158]A. Aftalion, X. Blanc and F. Nier, arxiv.org/abs/cond-mat 0508521.
[159]E. J. Mueller and T. L. Ho, Phys. Rev. A 67,063602 (2003).
[160]A. L. Fetter, B. Jackson and S. Stringari, Phys. Rev. A 71,013605 (2005).
[161]S. G. Benka, Physics today 12 (2006).
[162]M. V. Berry, Proc. R. Soc. A 392,45 (1984).
[163]F. Wilczek and A. Zee, Phys. Rev. Lett.52,2111 (1984).
[164]G. Juzeliunas and P. Ohberg, Phys. Rev. Lett.93,033602 (2004).
[165]G. Juzeliunas and P. Ohberg, J. Ruseckas, and A. Klein, Phys. Rev. A 71, 053614 (2005).
[166]P. Ohberg, G. Juzeliunas, J. Ruseckas, and M. Fleischhauer, Phys. Rev. A 72,053632 (2005)
[167]D. Jaksch and P. Zoller, New. J. Phys.5,56 (2003).
[168]P. Zhang, Y. Li and C. P. Sun, Eur. Phys. J. D 36,229 (2005)
[169]R. Dum and M. Olshanii, Phys. Rev. Lett.76,1788 (1996).
[170]K. Osterloh, M. Baig, L. Santos, P. Zoller, and M. Lewenstein, Phys. Rev. Lett.95,010403 (2005).
[171]J. Ruseckas, G. Juzeliunas, P. Ogberg and M. Fleischhauer, Phys. Rev. Lett.95,010404 (2005).
[172]M. Fleischhauer and M. D. Lukin, Phys. Rev. Lett.84,5094 (2000); Phys. Rev. A 65,022314 (2002).
[173]P. Q. Jin, Y. Q. Li and F. C. Zhang, J. Phys. A:Math and Gen,39,7115 (2006).
[174]L.-A. Wu, M. S. Sarandy, and D. A. Lidar, Phys. Rev. Lett.93,250404 (2004).
[175]A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature 416,608 (2002).
[176]S. J. Gu, H. Q. Lin and Y. Q. Li, Phys. Rev. A 68,042330 (2003).
[177]G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett.90,227902 (2003);
[178]P. Zanardi, M. Cozzini, and P. Giorda, arxiv.org/abs/cond-mat/0606130.
[179]B. DeMarco and D. S. Jin, Science 285,1703 (1999).
[180]K. M. O'Hara, S. L. Hemmer, M. E. Gehm, S. R. Granade, and J. E. Thomas, Scinece 298,2179 (2002).
[181]A. G. Truscou, K. E. Strecker, W. I. McAlexander, G. B. Partridge, and R. G. Hulet, Science 291,2570 (2001).
[182]Z. Hadzibabic, S. Guptam, C. A. Stan, C. H. Schunck, M. W. Zwierlein, K. Dieckmann, and W. Ketterle, ibid,91,160401 (2003).
[183]G. Modugno, G. Roati, F. Riboli, F. Ferlaino, R. J. Brecha, and M. In-guscio, Science 297,2240 (2002).
[184]G. Roati, F. Riboli, G. Modugno, and M. Inguscio, Phys. Rev. Lett.89, 150403 (2002).
[185]K. E. Strecker, G. B. Partridge, and R. G. Hulet, Phys. Rev. Lett.91, 080406 (2003).
[186]C. Ospelkaus, S. Ospelkaus, K. Sengstock, and K. Bongs, Phys. Rev. Lett. 96,020401 (2006).
[187]M.W.Zwierlein, C.A.Stan, et al., Phys.Rev.Lett.91,250401 (2003).
[188]S.Jochim, M.Bartenstein, et al., Science 302,2101 (2003).
[189]A. Vardi et al., J. Chem. Phys.107,6166 (1997); U. Gaubatz et al., J. Chem. Phys.92,5363 (1990).
[190]K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod. Phys.70,1003 (1998).
[191]M. Mackie, R. Kowalski and J. Javanainen, Phys. Rev. Lett.84,3803 (2000).
[192]G. Alzetta et al., Nuovo Cimento B 36,5 (1976); G. Alzetta, L. Moi, and G. Orriols, Nuovo Cimento 52,209 (1979).
[193]S. J. J. M. F. Kokkelmans, H. M. J. Vissers, and B. J. Verhaar, Phys. Rev. A 63,031601 (2001).
[194]M. Mackie, Phys. Rev. A 66,043613 (2002).
[195]H. Y. Ling, H. Pu, and B. Seaman, Phys. Rev. Lett.93,250403 (2004).
[196]M. W. Zwierlein, et al., Science 311,492 (2006).
[197]Y. Shin, et al., Phys. Rev. Lett.97,030401 (2006).
[198]C. H. Schunck, et al., Science 316,867 (20007).
[199]M. Salerno, Phys. Rev. A 72,063602 (2005).
[200]P. Capuzzi, A. Minguzzi, and M. P. Tosi, Phys. Rev. A 67,053605 (2003); P. Capuzzi, A. Minguzzi, and M. P. Tosi, Phys. Rev. A 68,033605 (2003).
[201]S. K. Adhikari, Phys. Rev. A 70,043617 (2004); S. K. Adhikari, Phys. Rev. A 72,053608 (2005); S. K. Adhikari, Phys. Rev. A 73,043619 (2006).
[202]F. Ferlaino, C. Derrico, G. Roatim, M. Zaccanti, M. Inguscio, G. Modugno, and A. Simoni, Phys. Rev. A 73,040702 (2006).(215)
[203]M. Zaccanti, C. Derrico, F. Ferlaino, G. Roati, M. Inguscio, and G. Mod-ugno, Phys. Rev. A 74,041605 (2006).
[204]M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England,2000), pp.399-424.
[205]J. Cubizolles, T. Bourdel, S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov, and C. Salomon, Phys. Rev. Lett.91,240401 (2003).
[206]C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Phys. Rev. Lett.90, 053201 (2003).
[207]S. Y. Meng, L. B. Fu and J. Liu, e-print arxiv.org/abs/0709.0359.
[208]R. Jozsa, J. Mod. Opt.41,2315, (1994).
[209]B. D. Josephson, Phys. Rev. Lett.1,251 (1962).
[210]K. K. Likharev, Rev. Mod. Phys.51,101 (1979).
[211]S. V. Pereverzev, A. Loshak, S. Backhaus, J. C. Davis, and R. E. Packard, Nature,388,449 (1997).
[212]K. Sukhatme, Y. Mukharsky, T. Chui, and D. Pearson, Nature 411,280 (2001).
[213]M. Albiez, R. Gati, J. Foiling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, Phys. Rev. Lett.95,010402 (2005).
[214]J. Liu, L. B. Fu, B. Y. Ou, S. G. Chen, D. Choi, B. Wu, and Q. Niu, Phys. Rev. A 66,023404 (2002).
[215]B. Wu and J. Liu, phys. Rev. Lett.96,020405 (2006).
[216]G. F. Wang, L. B. Fu and J. Liu, phys. Rev. A 73,013619 (2007).
[217]T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth, I. Bar-Joseph, J. Schmiedmayer, and P. Kruger, Nature 1,57 (2005).
[218]S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmied-mayer, Nature 449,324 (2007).
[219]E. J. Mueller, T. L. Ho, M. Ueda and G. Baym 74,0.33612 (2006).
[220]R. Jozsa, J. Mod. Opt.41,2315, (1994).
[221]M. Hiibner, Phys. Lett. A 163,239 (1992).
[222]M. Hiibner, Phys. Lett. A 179,226 (1993).
[223]B. R. D. Cunha and M. C. D. Oliveira, Phys. Rev. A 75,063615 (2007).
[224]N. Syassen, D. M. Bauer, M. Lettner, D. Dietze, T. Volz, S. Durr and G. Rempe, Phys. Rev. Lett.99,033201 (2007).
[225]J. M. Choi, G. N. Kim and D. Cho, Phys. Rev. A 77,040501 (2008).
[226]D. S. Hall, M. R. Matthews, C. E. Wieman and E. A. Cornell, Phys. Rev. Lett.81,1543 (1998).
[2]A. Einstein, Sitzungsberichte der Preussischen Akademie der Wis-senschaften, Physikalisch-mathematische Klasse p.261 (1924); p.3 (1925).
[3]M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269,198 (1995).
[4]K. B. Davis, M.O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, Phys. Rev. Lett.75,3969 (1995).
[5]C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett.75,1687(1995).
[6]S. Chu, L. Hollberg, J. E. Bjorkholm, Alex Cable, and A. Ashkin, Phys. Rev. Lett.55,48-51 (1985).
[7]C. E. Hecht, Physica 25,1159 (1959).
[8]D. G. Fried, T. C. Killian, et al., Phys. Rev. Lett.81,3811 (1998).
[9]W. Ertmer, R. Blatt, J. L. Hall, M. Zhu, Phys. Rev. Lett.54,996 (1985).
[10]C. E. Wieman, L. Hollberg, Rev. Sci. Instrum.62,1 (1991).
[11]W. D. Phillips, H. Metcalf, Phys. Rev. Lett.48,596 (1982); J. Prodan, A. Migdall, W. D. Phillips, I. So, H. Metcalf, J. Dalibard, Phys. Rev. Lett. 54,992 (1985).
[12]P. A. Molenaar, P. van der Straten, H. G. M. Heideman, H. Metcalf, Phys. Rev. A 55,605 (1997).
[13]S. V. Andreev, V. I. Balykin, V. S. Letokhov, V. G. Minogin, JETP Lett. 34,442 (1981).
[14]P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, H. J. Metcalf, Phys. Rev. Lett.61,169 (1988).
[15]W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell, Phys. Rev. Lett,74,3352 (1995).
[16]H. F. Hess, Phys. Rev. B 34,3476 (1986).
[17]W. Ketterle and N. J. van Druten, Adv. At. Mol. Opt. Phys.37,181 (1996).
[18]H. Feshbach, Ann. Phys.19,287 (1962).
[19]J. E. Lye, L. Fallani, M. Modugno, D. S. Wiersma, C. Fort, M. Inguscio, Phys. Rev. Lett,95,070401 (2005).
[20]F. S. Levin, H. Feshbach, Reaction Dynamics, Gordon and Breach, New York,1973.
[21]W. C. Stwalley, Phys. Rev. Lett,37,1628 (1976).
[22]E. Tiesinga, B. J. Verhaar, H. T. C. Stoof, Phys. Rev. A 47,4114 (1993).
[23]A. J. Moerdijk, B. J. Verhaar, A. Axelsson, Phys. Rev. A 51,4852 (1995).
[24]A. J. Moerdijk, B. J. Verhaar, T. M. Nagtegaal, Phys. Rev. A 53,4343 (1996).
[25]H. M. J. M. Boesten, J. M. Yogels, J. G. C. Tempelars, B. J. Verhaar, phys. Rev. A 54, R3726 (1996).
[26]J. M. Vogels, C. C. Tsai, R. S. Freeland, et al., Phys. Rev. A,56, R1067 (1997).
[27]N. R. Newbury, C. J. Myatt, C. E.Wieman, Phys. Rev. A 51, R 2680 (1995).
[28]J. Stenger, S. Inouye, M. R. Andrews, H. J. Miesner, D. M. Stamper-Kurn,W. Ketterle, Phys. Rev. Lett.82,2422 (1999).
[29]S. D. Gensemer, P. L. Gould, Phys. Rev. Lett.80,936 (1998).
[30]K. M. Jones, P. D. Lett, E. Tiesinga, P. S. Julienne, Phys. Rev. A 61, 012501 (1999).
[31]J. Ortigoso, M. Rodriguez, M. Gupta, B. Friedrich, J. Chem. Phys.110, 3870 (1999).
[32]F. H. Mies, E. Tiesinga, P. S. Julinne, Phys. Rev. A 61,022721 (2000).
[33]V. Kokoouline, J. Vala, R. Kosloff, J. Chem. Phys.114,3046 (2001).
[34]E. G. M. van Kempen, S. J. J. M. F. Kokkelmans, D. J. Heinzen, B. J. Verhaar, Phys. Rev. Lett.88,093201 (2002).
[35]C. Ticknor,C. A. Regal, D. S. Jin, J. L. Bohn, Phys. Rev. A 69,042712 (2004).
[36]J. L. Roberts, N. R. Claussen, J. P. Burke Jr., C. H. Greene, E. A. Cornell, C. E. Wieman, Phys. Rev. Lett.81,5109 (1998).
[37]V. Vuletic, A. J. Kerman, C. Chin, S. Chu, Phys. Rev. Lett.82,1406 (1999).
[38]J. L. Roberts, N. R. Claussen, S. L. Cornish, E. A. Donley, E. A. Cornell, C. E. Wieman, Phys. Rev. Lett.86,4211 (2001).
[39]A. Marte, T. Volz, J. Schuster, S. Durr, G. Rempe, E. G. M. van Kempen, B. J. Verhaar, Phys. Rev. Lett.89,283202 (2002).
[40]N. R. Claussen, E. A. Donley, S. T. Thompson, C. E. Wieman, Phys. Rev. Lett.89,010401 (2002).
[41]T. Volz, S. Durr, S. Ernst, A. Marte, G. Rempe, Phys. Rev. A 68,010702 (2003).
[42]C. A. Stan, M. W. Zwierlein, C. H. Schunck, S. M. F. Raupach, W. Ket-terle, Phys. Rev. Lett.93,143001 (2004).
[43]I. D. Prodan, M. Pichler, M. Junker, G. Hulet, Phys. Rev. Lett.91,080402 (2003).
[44]M. Houbiers, H. T. C. Stoof, W. I. McAlexander, R. G. Hulet, Phys. Rev. Lett. A 57, R 1497 (1998).
[45]S. Jochim, M. Bartenstein, G. Hendl, J. H. Denschlag, R. Grimm, Phys. Rev. Lett.89,273202 (2002).
[46]C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Nature 424,47 (2003).
[47]C. H. Schunck, M. W. Zwierlein, C. A. Stan, S. M. F. Raupach, W. Ket-terle, A. Simoni, E. Tiesinga, C. J. Williams, P. S. Julienne, Phys. Rev. A 71,045601 (2005).
[48]A. Simoni, F. Ferlaino, G. Roati, G. Modugn, M. Inguscio, Phys. Rev. Lett.90,163202 (2003).
[49]S. Inouye, J. Goldwin, M. L. Olsen, C. Ticknor, J. L. Bohn, D. S. Jin, Phys. Rev. Lett.93 183201 (2004).
[50]E. Timmermans, P. Tommasini, M. Hussein, A. Kerman, Phys. Rep.315, 199 (1999).
[51]S. Inouye, M. R. Andrews, J. Stenger, D. M. Stamper-Kurn, and W. Ket-terle, Nature,392,151 (1998).
[52]G. Taubes, Science 269,152 (1995).
[53]C. C. Bradley, C. A. Sackett, R. G. Hulet, Phys. Rev. Lett.78,985 (1997).
[54]T.Weber, J. Herbig, M. Mark, H.C. Nagert, R. Grimm, Science 299 (2003) 232.
[55]F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cu-bizolles, C. Salomon, Phys. Rev. Lett.87,080403 (2001).
[56]G. Modugno, G. Ferrari, G. Roati, R. J. Brecha, A. Simoni, M. Inguscio, Science 294,1320 (2001).
[57]S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell, C. E. Wieman, Phys. Rev. Lett.85,1795 (2000).
[58]Y. Takasu, K. Maki, K. Komori, T. Takano, K. Honda, M. Kumakura, T.Yabuzaki,Y. Takahashi, Phys. Rev. Lett.91,040404 (2003).
[59]A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, T. Pfau, Phys. Rev. Lett. 94,160401 (2005).
[60]A. Robert, O. Sirjean, A. Browaeys, J. Poupard, S. Nowak, D. Boiron, C. I. Westbrook, A. Aspect, Science 292,461 (2001).
[61]F. Pereira, D. Santos, J. Leonard, J. Wang, C. J. Barrelet, F. Rasel, C. S. Unnikrishnan, M. Leduc, C. Cohen-Tannoudji, Phys. Rev. Lett.86,3459 (2001)(61).
[62]M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell. Phys. Rev. Lett.83,2498 (1999).
[63]K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett.84,806 (2000).
[64]J. R. Abo-Shaeer, C. Raman, J. M Vogels, W. Ketterle, Science 292,476 (2001).
[65]D. R. Scherer, C. N. Weiler, T. W. Neely, and B. P. Anderson, Phys. Rev. Lett.98,110402 (2007).
[66]M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn and W. Ketterle, Science 275,637 (1997).
[67]A. J. Leggett, Rev. Mod. Phys.73,307 (2001).
[68]H. W. Xiong, S. J. Liu, and M. S. Zhan, New J. Phys.8,245 (2006).
[69]R. G. Hulet, McGraw-HillYearbook of Science and Technology, McGraw-Hill, NewYork, pp.19-21 (2005).
[70]A. Cho, Science 301,750 (2003).
[71]J. E. Thomas, M. Gehm, Amer. Scientist 92,238 (2004).
[72]M. Greiner, C. A. Regal, D. S. Jin, cond-mat/0502539 (2005).
[73]B. DeMarco, D. S. Jin, Science 285,1703 (1999).
[74]A. G. Truscott, K. E. Strecker,W. I. McAlexander, G. B. Partridge, R. G. Hulet, Science 291,2570 (2001).
[75]T. Bourdel, J. Cubizolles, L. Khaykovich, K. M. F. Magalhaes, S. J.J. M. E. Kokkelmans, G. V. Shlyapnikov, C. Salomon, Phys. Rev. Lett.91, 020402 (2003).
[76]S. R. Granade, M. E. Gehm, K. M. O'Hara, J. E. Thomas, Phys. Rev. Lett.88,120405 (2002).
[77]Z. Hadzibabic, C. A. Stan, K. Dieckmann, S. Gupta, M. W. Zwierlein, A. Gorlitz, W. Ketterle, Phys. Rev. Lett.88,160401 (2002).
[78]G. Roati, F. Riboli, G. Modugno, M. Inguscio, Phys. Rev. Lett.89,150403 (2002).
[79]S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. H. Denschlag, R. Grimm, Science 302,2101 (2003).
[80]G. Modugno, F. Ferlaino, R. Heidemann, G. Roati, M. Inguscio, Phys. Rev. A 68, R011601 (2003).
[81]T. Loftus, C. A. Regal, C. Ticknor, J. L. Bohn, D. S. Jin, Phys. Rev. Lett. 88,173201 (2002).
[82]C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett,92,083201 (2004).
[83]J. Cubizolles, T. Bourde, S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov, C. Salomon, Phys. Rev. Lett.91,240401 (2003).
[84]S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, C. Chin, J. Hecker Denschlag, R. Grimm, Phys. Rev. Lett.91,240402 (2003).
[85]M.Greiner, C.A.Regal, D. S. Jin, Nature 426,537 (2003).
[86]M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J.H. Denschlag, R. Grimm, Phys. Rev. Lett.92,120401 (2004).
[87]M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Den-schlag, and R. Crimm, Phys. Rev. Lett,92,203201 (2004).
[88]M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta, Z. Hadzibabic,W. Ketterle, Phys. Rev. Lett.91,250401 (2003).
[89]M. W. Zwierlein, A. Schirotzek, C. H. Schunck, and W. Ketterle, Science 311,492 (2006).
[90]K. Machida, T. Mizushima, and M. Ichioka, Phys. Rev. Lett.97,120407 (2006).
[91]C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett.92,040403 (2004).
[92]M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Kerman,W. Ketterle, Phys. Rev. Lett.92,120403 (2004).
[93]M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo, R. Walser, Phys. Rev. Lett.87,120406 (2001).
[94]E. Timmermans, K. Furuya, P. W. Milonni, A. K. Kerman, Phys. Lett. A 285,228 (2001).
[95]Y. Ohashi, A. Griffin, Phys. Rev. Lett.89,130402 (2002).
[96]J. Stajic, J. N. Milstein, Q. Chen, M. L. Chiofalo, M. J. Holland, K. Levin, Phys. Rev. A 69,063610 (2004).
[97]R. A. Duine, H.T. C. Stoof, J. Opt. B Quantum Semiclassical Opt.5, S 212 (2003).
[98]M. Greiner, C. A. Regal, C.Ticknor, J. L. Bohn, D. S. Jin, Phys. Rev. Lett.92,150405 (2004).
[99]C. Chin, M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, J. H. Den-schlag, R. Grimm, Science 305,1128 (2004).
[100]M. W Zwierlein, J. R. Abo-shaeer, A. Schirotzek, C. H. Schunck, and W. Ketterle, Nature 435,1047 (2005).
[101]L. D. Dandau and E. M. Lifshitz, Statistical Physics, Part 1, Third editin, (Pergamon, Oxford,1980),§62.
[102]C. J. Pethick and H. Smith:Bose-Einstein Condensation in Dilute Gass, Cambridge University press, United Kingdom,2002.
[103]L. P. Pitaevskii, Zh. Eksp. Thor. Fiz.40,646 (1961) [Sov. Phys.-JEYP 13, 451 (1961)]; E. P. Gross, Nuovo Cimento 20,454 (1961); J. Math. Phys. 4,195 (1963).
[104]P. S. Julienne, A. M. Smith, and K. Burnett, Adv. At. Mol. Opt. Phys. 30,141 (1993).
[105]E. M. Lifshitz, L. P. Pitaevskii, Statistical Physics, Part Ⅱ. Pergamon, Oxford,1980.
[106]A. S. Parkins, and D. F. Walls, Phys. rep.303,1-80 (1998).
[107]V. Schweikhard, I. Coddington, P. Engels, V. P. Mogendorff, and E. A. Cornell, Phys. Rev. Lett.92,040404 (2004).
[108]G. Baym, and C. J. Pethick, Phys. Rev. A 69,043619 (2004).
[109]U. R. Fischer and G. Baym, Phys. Rev. Lett.90,140402 (2003).
[110]J. Sinova, C. B. Hanna, and A. H. MacDonald, Phys. Rev. Lett.90,120401 (2003).
[111]M. Edwards, and K. Burnett, Phys. Rev. A 51,1382 (1995).
[112]P. A. Ruprecht, M. J. Holland, K. Burnett, and M. Edwards, Phys. Rev. A 51,4704 (1995).
[113]Y. Kagan, G. V. Shlyapnikov, and J. T. M. Walraven, Phys. Rev. Lett. 76,2670 (1996); Y. Kagan, E. L. Surkov, and G. V. Shlyapnikov, Phys. Rev. A 54, R1753 (1996).
[114]L. You, M. Lewenstein, R. J. Glauber, and J. Cooper, Phys. Rev. A 53, 329 (1996).
[115]M. Edwards, R. J. Dodd, C. W. Clark, P. A. Ruprecht, and K. Burnett, Phys. Rev. A 53, R1950 (1996).
[116]F. Dalfovo, and S. Stringari, Phys. Rev. A 53,2477 (1996).
[117]M. Holland, and J. Cooper, Phys. Rev. A 53, R1954 (1996).
[118]D. S. Jin, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett.77,420 (1996).
[119]M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, and W. Ketterle, Phys. Rev. Lett.77,416 (1996).
[120]C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett.75,1687 (1995).
[121]C. C. Bradley, C. A. Sackett, and R. G. Hulet, Phys. Rev. Lett.78,985 (1997).
[122]T. Bergeman, Phys. Rev. A 55,3658 (1997).
[123]R. J. Dodd, M. Edwards, C. J. Williams, and C. W. Clark, Phys. Rev. A 54,661 (1996).
[124]A. L. Fetter, arxiv. org/cond-mat/9510037.
[125]R. J. Donnelly, Quantized Vortices in Liquid He II, (Cambridge University Press, Cambridge,1991).
[126]A. L. Fetter and A. A. Svidzinsky, J. Phys.:Condens. Matter 13,135 (2001).
[127]J. A. Dunningham, M. J. Collett, and D. F. Walls, Phys. Lett. A 245,49 (1998).
[128]L. Onsager, Nuovo Cimento 6, Suppl.2,249 (1949).
[129]R. P. Feynman, Progress in Low Temperature Physics, Vol.1, ed. C. J. Gorter, (North-Holland, Amsterdam,1955), Chapter 2.
[130]E. Lundh, C. J. Pethick, H. Smith, Phys. Rev. A 55,2126 (1997).
[131]L. D. Landau and E. M. Lifshitz, Mechanics, Third edition, (Pergamon, Oxford,1976), §39.
[132]E. Hodby, G. Hechenblaikner, S. A. Hopkins,0. M. Maraggraveo, and C.J. Foot, Phys. Rev. Lett.88,010405 (2002).
[133]P. C. Haljan, I. Coddington, P. Engels, and E. A. Cornell, Phys. Rev. Lett. 87,210403 (2001).
[134]T.-L. Ho, Phys. Rev. Lett.87,060403 (2001).
[135]Vincent Bretin, Sabine Stock, Yannick Seurin, and Jean Dalibard, Phys. Rev. Lett.92,050403 (2004).
[136]Kenichi Kasamatsu, Makoto Tsubota, and Masahito Ueda, Phys. Rev. A 66,053606 (2002).
[137]E. Lundh, Phys. Rev. A 65,043604 (2002).
[138]A. L. Fetter, Phys. Rev. A 64,063608 (2001).
[139]E. Lundh, A. Collin and K.-A. Suominen, Phys. Rev. Lett.92,070401 (2004).
[140]T. P. Simula, A. A. Penckwitt and R. J. Ballagh, Phys. Rev. Lett.92, 060401 (2004).
[141]A. Aftalion and I. Danaila, Phys. Rev. A 69,033608 (2004).
[142]A. L. Fetter, B. Jackson, and S. Stringari, Phys. Rev. A 71,013605 (2005).
[143]A. D. Jackson and G. M. Kavoulakis, Phys. Rev. A 70,023601 (2004).
[144]V. Subrahmanyam, Phys. Rev. A 67,035601 (2003).
[145]T. K. Ghosh, Phys. Rev. A 69,043606 (2004).
[146]A. Aftalion, X. Blanc and J. Dalibard, Phys. Rev. A 71,023611 (2005).
[147]Gentaro Watanabe, Gordon Baym, and C. J. Pethick, Phys. Rev. Lett. 93,190401 (2004).
[148]E. J. Mueller and T.-L. Ho, Phys. Rev. Lett.88,180403 (2002).
[149]A. D. Jackson, G. M. Kavoulakis and E. Lundh, Phys. Rev. A 69,053619 (2004).
[150]A. Aftalion and I. Danaila, (2003), arxiv.org/abs/cond-mat/0309668.
[151]Ionut Danaila, arxiv.org/abs/cond-mat/0503122.
[152]M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett.83,2498 (1999).
[153]C. Raman, J. R. Abo-Shaeer, J. M. vogels, K. Xu and W. Ketterle, Phys Rev. Lett.87,210402 (2001).
[154]P. Engels, I. Coddington, P. C. Haljan, and E. A. Cornell, Phys. Rev. Lett.89,100403 (2002).
[155]J. D. Jackson, Classical Electrodynamics (Wiley, New York,1975),2nd ed., Chap.11.
[156]D. A. Butts and D. S. Rokhsa, Nature 397,327 (1999).
[157]N. R. Cooper, S. Komineas and N. Read, Phys. Rev. A 70,033604, (2004).
[158]A. Aftalion, X. Blanc and F. Nier, arxiv.org/abs/cond-mat 0508521.
[159]E. J. Mueller and T. L. Ho, Phys. Rev. A 67,063602 (2003).
[160]A. L. Fetter, B. Jackson and S. Stringari, Phys. Rev. A 71,013605 (2005).
[161]S. G. Benka, Physics today 12 (2006).
[162]M. V. Berry, Proc. R. Soc. A 392,45 (1984).
[163]F. Wilczek and A. Zee, Phys. Rev. Lett.52,2111 (1984).
[164]G. Juzeliunas and P. Ohberg, Phys. Rev. Lett.93,033602 (2004).
[165]G. Juzeliunas and P. Ohberg, J. Ruseckas, and A. Klein, Phys. Rev. A 71, 053614 (2005).
[166]P. Ohberg, G. Juzeliunas, J. Ruseckas, and M. Fleischhauer, Phys. Rev. A 72,053632 (2005)
[167]D. Jaksch and P. Zoller, New. J. Phys.5,56 (2003).
[168]P. Zhang, Y. Li and C. P. Sun, Eur. Phys. J. D 36,229 (2005)
[169]R. Dum and M. Olshanii, Phys. Rev. Lett.76,1788 (1996).
[170]K. Osterloh, M. Baig, L. Santos, P. Zoller, and M. Lewenstein, Phys. Rev. Lett.95,010403 (2005).
[171]J. Ruseckas, G. Juzeliunas, P. Ogberg and M. Fleischhauer, Phys. Rev. Lett.95,010404 (2005).
[172]M. Fleischhauer and M. D. Lukin, Phys. Rev. Lett.84,5094 (2000); Phys. Rev. A 65,022314 (2002).
[173]P. Q. Jin, Y. Q. Li and F. C. Zhang, J. Phys. A:Math and Gen,39,7115 (2006).
[174]L.-A. Wu, M. S. Sarandy, and D. A. Lidar, Phys. Rev. Lett.93,250404 (2004).
[175]A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature 416,608 (2002).
[176]S. J. Gu, H. Q. Lin and Y. Q. Li, Phys. Rev. A 68,042330 (2003).
[177]G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett.90,227902 (2003);
[178]P. Zanardi, M. Cozzini, and P. Giorda, arxiv.org/abs/cond-mat/0606130.
[179]B. DeMarco and D. S. Jin, Science 285,1703 (1999).
[180]K. M. O'Hara, S. L. Hemmer, M. E. Gehm, S. R. Granade, and J. E. Thomas, Scinece 298,2179 (2002).
[181]A. G. Truscou, K. E. Strecker, W. I. McAlexander, G. B. Partridge, and R. G. Hulet, Science 291,2570 (2001).
[182]Z. Hadzibabic, S. Guptam, C. A. Stan, C. H. Schunck, M. W. Zwierlein, K. Dieckmann, and W. Ketterle, ibid,91,160401 (2003).
[183]G. Modugno, G. Roati, F. Riboli, F. Ferlaino, R. J. Brecha, and M. In-guscio, Science 297,2240 (2002).
[184]G. Roati, F. Riboli, G. Modugno, and M. Inguscio, Phys. Rev. Lett.89, 150403 (2002).
[185]K. E. Strecker, G. B. Partridge, and R. G. Hulet, Phys. Rev. Lett.91, 080406 (2003).
[186]C. Ospelkaus, S. Ospelkaus, K. Sengstock, and K. Bongs, Phys. Rev. Lett. 96,020401 (2006).
[187]M.W.Zwierlein, C.A.Stan, et al., Phys.Rev.Lett.91,250401 (2003).
[188]S.Jochim, M.Bartenstein, et al., Science 302,2101 (2003).
[189]A. Vardi et al., J. Chem. Phys.107,6166 (1997); U. Gaubatz et al., J. Chem. Phys.92,5363 (1990).
[190]K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod. Phys.70,1003 (1998).
[191]M. Mackie, R. Kowalski and J. Javanainen, Phys. Rev. Lett.84,3803 (2000).
[192]G. Alzetta et al., Nuovo Cimento B 36,5 (1976); G. Alzetta, L. Moi, and G. Orriols, Nuovo Cimento 52,209 (1979).
[193]S. J. J. M. F. Kokkelmans, H. M. J. Vissers, and B. J. Verhaar, Phys. Rev. A 63,031601 (2001).
[194]M. Mackie, Phys. Rev. A 66,043613 (2002).
[195]H. Y. Ling, H. Pu, and B. Seaman, Phys. Rev. Lett.93,250403 (2004).
[196]M. W. Zwierlein, et al., Science 311,492 (2006).
[197]Y. Shin, et al., Phys. Rev. Lett.97,030401 (2006).
[198]C. H. Schunck, et al., Science 316,867 (20007).
[199]M. Salerno, Phys. Rev. A 72,063602 (2005).
[200]P. Capuzzi, A. Minguzzi, and M. P. Tosi, Phys. Rev. A 67,053605 (2003); P. Capuzzi, A. Minguzzi, and M. P. Tosi, Phys. Rev. A 68,033605 (2003).
[201]S. K. Adhikari, Phys. Rev. A 70,043617 (2004); S. K. Adhikari, Phys. Rev. A 72,053608 (2005); S. K. Adhikari, Phys. Rev. A 73,043619 (2006).
[202]F. Ferlaino, C. Derrico, G. Roatim, M. Zaccanti, M. Inguscio, G. Modugno, and A. Simoni, Phys. Rev. A 73,040702 (2006).(215)
[203]M. Zaccanti, C. Derrico, F. Ferlaino, G. Roati, M. Inguscio, and G. Mod-ugno, Phys. Rev. A 74,041605 (2006).
[204]M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England,2000), pp.399-424.
[205]J. Cubizolles, T. Bourdel, S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov, and C. Salomon, Phys. Rev. Lett.91,240401 (2003).
[206]C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Phys. Rev. Lett.90, 053201 (2003).
[207]S. Y. Meng, L. B. Fu and J. Liu, e-print arxiv.org/abs/0709.0359.
[208]R. Jozsa, J. Mod. Opt.41,2315, (1994).
[209]B. D. Josephson, Phys. Rev. Lett.1,251 (1962).
[210]K. K. Likharev, Rev. Mod. Phys.51,101 (1979).
[211]S. V. Pereverzev, A. Loshak, S. Backhaus, J. C. Davis, and R. E. Packard, Nature,388,449 (1997).
[212]K. Sukhatme, Y. Mukharsky, T. Chui, and D. Pearson, Nature 411,280 (2001).
[213]M. Albiez, R. Gati, J. Foiling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, Phys. Rev. Lett.95,010402 (2005).
[214]J. Liu, L. B. Fu, B. Y. Ou, S. G. Chen, D. Choi, B. Wu, and Q. Niu, Phys. Rev. A 66,023404 (2002).
[215]B. Wu and J. Liu, phys. Rev. Lett.96,020405 (2006).
[216]G. F. Wang, L. B. Fu and J. Liu, phys. Rev. A 73,013619 (2007).
[217]T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth, I. Bar-Joseph, J. Schmiedmayer, and P. Kruger, Nature 1,57 (2005).
[218]S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmied-mayer, Nature 449,324 (2007).
[219]E. J. Mueller, T. L. Ho, M. Ueda and G. Baym 74,0.33612 (2006).
[220]R. Jozsa, J. Mod. Opt.41,2315, (1994).
[221]M. Hiibner, Phys. Lett. A 163,239 (1992).
[222]M. Hiibner, Phys. Lett. A 179,226 (1993).
[223]B. R. D. Cunha and M. C. D. Oliveira, Phys. Rev. A 75,063615 (2007).
[224]N. Syassen, D. M. Bauer, M. Lettner, D. Dietze, T. Volz, S. Durr and G. Rempe, Phys. Rev. Lett.99,033201 (2007).
[225]J. M. Choi, G. N. Kim and D. Cho, Phys. Rev. A 77,040501 (2008).
[226]D. S. Hall, M. R. Matthews, C. E. Wieman and E. A. Cornell, Phys. Rev. Lett.81,1543 (1998).