精确非线性布洛赫解及其应用
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摘要
处于周期势阱中的物质波(玻色—爱因斯坦凝聚),因其所处外势易受到人为控制且与电子处于晶格中的动力学有很多共同之处,而在近些年来引起理论与实验物理学家的普遍兴趣。由于超冷原子之间存在非线性相互作用,在该系统中也呈现出许多前所未见的物理现象。然而由于非线性相互作用对精确求解所造成的困难,关于该系统的研究多是利用数值求解的方法。因而发展求解处于周期势阱中玻色爱因斯坦凝聚的精确解,将有助于人们对数值结果的进一步理解。这就是本文的研究目的。
本文将Kronig-Penny模型推广到非线性情况,研究处于一维周期量子阱中的非线性薛定谔方程定态解,并在此基础上研究与布洛赫理论相关的物理问题。具体地讲有以下几个方面:
首先,我们首先找到一组可以正确描述处于Kronig-Penny势中非线性薛定谔方程(或Gross-Pitaevskii方程)的精确解。由于该精确解可以在非线性参数为零时,退化到线性薛定谔方程的解而使我们可以方便地研究非线性对线性布洛赫理论的影响。利用该精确解,我们详细研究了Bloch能带、压缩率、有效质量以及声速与势阱深度和相互作用强度的变化关系。结果显示:随着非线性参数的增加,Bloch能带宽度增加,而随着势阱深度的增加,Bloch能带宽度变窄。当势阱足够深时,Bloch波函数局域到势阱中,此时可以用紧束缚模型来描述。非线性参数很小时,压缩率的倒数κ~(-1)与其呈线性关系;当非线性参数比较大时,κ~(-1)的增加与其有非线性依赖关系。有效质量随势阱深度的增加而显著增大,同时随着原子间相互作用的增大,有效质量逐渐减小。由于压缩率和有效质量之间的竞争关系使得当势阱深度增大时,声速减小。
其次,利用Wannier函数与Bloch函数之间的关系,我们得到了一维周期量子阱中的非线性Wannier函数。在此基础上,研究了非线性Wannier函数的性质以及Bose-Hubbard模型中的格点相互作用项U和近邻隧穿项J。发现非线性相互作用的增加使得Wannier的指数衰减性变差,同时使相邻格点间的隧穿耦合强度增加。另一方面,格点间原子相互作用U与非线性相互作用gn的比值随gn的增加而单调递减。当势阱深度大于23E_R时,U/J大于相变临界点,此时系统处于绝缘态。
接下来,我们对该系统的稳定性进行了分析。在紧束缚极限下,可以得到Landau不稳定性和动力学不稳定性的解析表达式以及其在第一布里渊区内的稳定性相图。从相图中可以看到随着势阱深度和非线性参数的增加,Landau不稳定性的区域是逐渐减小的。动力学不稳定在Landau不稳定区域中占的越来越多。对于任何势阱深度和非线性参数,动力学不稳定性区域的左边界一直在k=π/2处。
最后,我们还对一维周期量子阱中玻色—爱因斯坦凝聚的集体激发进行了研究。我们得到了紧束缚极限下的元激发谱以及动力学结构因子中激发强度的解析表达式,发现这两个解析解是依赖于有效质量和压缩率的。我们将激发谱的最低能带和最低Bloch能带做比较,可以得出相互作用对元激发谱的影响要比对Bloch能带的影响大。当动量比较小时,最低能带中的激发满足线性色散关系。最低能带的激发强度与动量转移(transfering momentum)呈现某种周期关系,并且在偶数倍的Bragg动量处为零。
Recently,matter waves(Bose-Einstein Condensates) in periodic potential have attracted the attention both experimentist and theoretist,due to controllable potential and sharing many features with electron in solid.Since the nonlinear interaction between ultra-cold atoms,many novel phenomena have been reported.So far,most of the theretical works are obtained by numerical calculation for this nonlinear interaction term.Therefore,it is interesting to look for the exact solution for this system,which will be helpful to deeply understand the reported numerical works.This is goal of this thesis.
We introduced the nonlinear terms into the Kronig-Penny model and found the exact stationary solutions for nonlinear Schrodinger equation in a periodic array of quantum wells.The relative problems related with the Bloch theory are investigated,based on the exact solutions.The detail contents are following:
Firstly,we found a full set of exact nonlinear Bloch-like solutions to the nonlinear Scr(o|¨)dinger equation(or Gross-Pitaevskii equation) in a periodic array of quantum wells.Since these solutions can be reduced to the linear solutions for Schrodinger equation in the case of zero nonlinear interaction,it is convenient to study the effect of the nonlinear interaction on the Bolch theory.We comprehensively studied the Bloch band,the compressibility, effective mass and the sound speed as functions of both the potential depth and interatomic interaction,based on these exact nonlinear Bloch solutions. Our results shown that increasing the nonlinear parameter induces the Bloch band become wider,in contrast increasing the potential depth makes it narrow. When the potential well is high enough,the Bloch function localized in the well,and then the tight-banding approximation is availabe.The inverse of the compressibilityΚ~(-1) increases linearly only for small gn,while in the case of large gn,the relation betweenΚ~(-1) and gn is nonlinear.The effective mass significantly increases with the increase of potential depth and gradually decreases with the enhancement of the nonlinear interaction.The sound velocity is decreasing with increasing the potential depth,for the competition betweenΚ~(-1) and gn.
Secondly,we obtained the nonlinear Wannier function for this system by considering the relation between the Wannier function and the Bloch function. Further more,we investigated the properties of nonlinear Wannier functions and the basic parameters U and J of Bose-Hubbard model are calculated. The nonlinear interaction makes the Wannier functions slowly fall off as exponential law with distance compared with the linear case,and enhances the tunneling coupling between the neighbor wells.On the other hand the on-site interaction U/gn is monotonously decreased with increasing gn,but increasing the potential depth makes U increase.When the potential depth large than23E_R,the ratio U/J exceed the critical value of phase transition, the system is in Mott-insulator state.
Thirdly,we have comprehensively investigated the Landau and dynamical instabilities for a Bose-Einstein condensate in a periodic array of quantum wells.In the limit of tight-binding,the analytical expressions for both Landau and dynamical instabilities are obtained and then the stability phase diagrams are shown.The region of Landau instability decreases with the increasing of the potential depth and the nonlinear parameter.The dynamical instability spreads more in the area of Landau instability.For any nonlinear parameter and potential depth,the left borderline of dynamical instability is at k=π/2.
Finally,we investigated the collective excitation of Bose-Einstein condensates in a periodic array of quantum wells.The analytic expression of spectrum of the collective excitations and the excitation strength in terms of the effective mass and compressibility,are obtained in the limit of tight-binding.Comparing with the lowest energy band of the collective excitation and the lowest Bloch band,we found that the effect of the nonlinear interaction on the collective excitation is larger than on the Bloch energy band.All the excitations in the lowest band acquire the linear phonon dispersion with a finite slope at small quasi-momenta.The excitation strength towards the first band develops an oscillating behavior as a function of the momentum transfer,vanishing at even multiples of the Bragg momentum due to the presence of a phononic regime.
本文将Kronig-Penny模型推广到非线性情况,研究处于一维周期量子阱中的非线性薛定谔方程定态解,并在此基础上研究与布洛赫理论相关的物理问题。具体地讲有以下几个方面:
首先,我们首先找到一组可以正确描述处于Kronig-Penny势中非线性薛定谔方程(或Gross-Pitaevskii方程)的精确解。由于该精确解可以在非线性参数为零时,退化到线性薛定谔方程的解而使我们可以方便地研究非线性对线性布洛赫理论的影响。利用该精确解,我们详细研究了Bloch能带、压缩率、有效质量以及声速与势阱深度和相互作用强度的变化关系。结果显示:随着非线性参数的增加,Bloch能带宽度增加,而随着势阱深度的增加,Bloch能带宽度变窄。当势阱足够深时,Bloch波函数局域到势阱中,此时可以用紧束缚模型来描述。非线性参数很小时,压缩率的倒数κ~(-1)与其呈线性关系;当非线性参数比较大时,κ~(-1)的增加与其有非线性依赖关系。有效质量随势阱深度的增加而显著增大,同时随着原子间相互作用的增大,有效质量逐渐减小。由于压缩率和有效质量之间的竞争关系使得当势阱深度增大时,声速减小。
其次,利用Wannier函数与Bloch函数之间的关系,我们得到了一维周期量子阱中的非线性Wannier函数。在此基础上,研究了非线性Wannier函数的性质以及Bose-Hubbard模型中的格点相互作用项U和近邻隧穿项J。发现非线性相互作用的增加使得Wannier的指数衰减性变差,同时使相邻格点间的隧穿耦合强度增加。另一方面,格点间原子相互作用U与非线性相互作用gn的比值随gn的增加而单调递减。当势阱深度大于23E_R时,U/J大于相变临界点,此时系统处于绝缘态。
接下来,我们对该系统的稳定性进行了分析。在紧束缚极限下,可以得到Landau不稳定性和动力学不稳定性的解析表达式以及其在第一布里渊区内的稳定性相图。从相图中可以看到随着势阱深度和非线性参数的增加,Landau不稳定性的区域是逐渐减小的。动力学不稳定在Landau不稳定区域中占的越来越多。对于任何势阱深度和非线性参数,动力学不稳定性区域的左边界一直在k=π/2处。
最后,我们还对一维周期量子阱中玻色—爱因斯坦凝聚的集体激发进行了研究。我们得到了紧束缚极限下的元激发谱以及动力学结构因子中激发强度的解析表达式,发现这两个解析解是依赖于有效质量和压缩率的。我们将激发谱的最低能带和最低Bloch能带做比较,可以得出相互作用对元激发谱的影响要比对Bloch能带的影响大。当动量比较小时,最低能带中的激发满足线性色散关系。最低能带的激发强度与动量转移(transfering momentum)呈现某种周期关系,并且在偶数倍的Bragg动量处为零。
Recently,matter waves(Bose-Einstein Condensates) in periodic potential have attracted the attention both experimentist and theoretist,due to controllable potential and sharing many features with electron in solid.Since the nonlinear interaction between ultra-cold atoms,many novel phenomena have been reported.So far,most of the theretical works are obtained by numerical calculation for this nonlinear interaction term.Therefore,it is interesting to look for the exact solution for this system,which will be helpful to deeply understand the reported numerical works.This is goal of this thesis.
We introduced the nonlinear terms into the Kronig-Penny model and found the exact stationary solutions for nonlinear Schrodinger equation in a periodic array of quantum wells.The relative problems related with the Bloch theory are investigated,based on the exact solutions.The detail contents are following:
Firstly,we found a full set of exact nonlinear Bloch-like solutions to the nonlinear Scr(o|¨)dinger equation(or Gross-Pitaevskii equation) in a periodic array of quantum wells.Since these solutions can be reduced to the linear solutions for Schrodinger equation in the case of zero nonlinear interaction,it is convenient to study the effect of the nonlinear interaction on the Bolch theory.We comprehensively studied the Bloch band,the compressibility, effective mass and the sound speed as functions of both the potential depth and interatomic interaction,based on these exact nonlinear Bloch solutions. Our results shown that increasing the nonlinear parameter induces the Bloch band become wider,in contrast increasing the potential depth makes it narrow. When the potential well is high enough,the Bloch function localized in the well,and then the tight-banding approximation is availabe.The inverse of the compressibilityΚ~(-1) increases linearly only for small gn,while in the case of large gn,the relation betweenΚ~(-1) and gn is nonlinear.The effective mass significantly increases with the increase of potential depth and gradually decreases with the enhancement of the nonlinear interaction.The sound velocity is decreasing with increasing the potential depth,for the competition betweenΚ~(-1) and gn.
Secondly,we obtained the nonlinear Wannier function for this system by considering the relation between the Wannier function and the Bloch function. Further more,we investigated the properties of nonlinear Wannier functions and the basic parameters U and J of Bose-Hubbard model are calculated. The nonlinear interaction makes the Wannier functions slowly fall off as exponential law with distance compared with the linear case,and enhances the tunneling coupling between the neighbor wells.On the other hand the on-site interaction U/gn is monotonously decreased with increasing gn,but increasing the potential depth makes U increase.When the potential depth large than23E_R,the ratio U/J exceed the critical value of phase transition, the system is in Mott-insulator state.
Thirdly,we have comprehensively investigated the Landau and dynamical instabilities for a Bose-Einstein condensate in a periodic array of quantum wells.In the limit of tight-binding,the analytical expressions for both Landau and dynamical instabilities are obtained and then the stability phase diagrams are shown.The region of Landau instability decreases with the increasing of the potential depth and the nonlinear parameter.The dynamical instability spreads more in the area of Landau instability.For any nonlinear parameter and potential depth,the left borderline of dynamical instability is at k=π/2.
Finally,we investigated the collective excitation of Bose-Einstein condensates in a periodic array of quantum wells.The analytic expression of spectrum of the collective excitations and the excitation strength in terms of the effective mass and compressibility,are obtained in the limit of tight-binding.Comparing with the lowest energy band of the collective excitation and the lowest Bloch band,we found that the effect of the nonlinear interaction on the collective excitation is larger than on the Bloch energy band.All the excitations in the lowest band acquire the linear phonon dispersion with a finite slope at small quasi-momenta.The excitation strength towards the first band develops an oscillating behavior as a function of the momentum transfer,vanishing at even multiples of the Bragg momentum due to the presence of a phononic regime.
引文
[1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science (1995) 269, 198.
[2] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. (1995) 75, 3969.
[3] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett. (1995) 75, 1687.
[4] Inouye S, Anderson M R, Stenger J, et al., Nature (1998) 392, 151.
[5] C. J. Myatt et al., Phys. Rev. Lett. (1997) 78, 586; M. R. Matthews et al., ibid. (1998) 81, 243; G. Modugno et al., Science (2001) 294, 1320;
[6] D. M. Stamper-Kurn et al., Phys. Rev. Lett. (1998) 80, 2027 ; H. J. Miesner et al., ibid. (1999) 82, 2228 ; D. M. Stamper-Kurn et al., ibid. (1999) 83, 661;
[7] Z. Hadzibabic, S. Stock, B. Battelier, V. Bretin, and J. Dalibard, Phys. Rev. Lett. (2004)93,180403.
[8] S. Peil, J. V. Porto, B. L. Tolra, J. M. Obrecht, B. E. King, M. Subbotin, S. L. Rolston, and W. D. Phillips, Phys. Rev. A (2003) 67, 051603.
[9] A. Hemmerich and T.W. Hansch, Phys. Rev. Lett. (1993) 70, 410.
[10]K.I. Petsas, A.B. Coates and G. Grynberg, Phys. Rev. A (1994) 50, 5173.
[11]M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch and I. Bloch, Nature (London) (2002) 415, 39.
[12]Biao Wu and Qian Niu, Phys. Rev. A (2001) 64, 061603(R); Biao Wu and Qian Niu, New J. Phys. (2003)5, 104.
[13]D. Diakonov, L. M. Jensen, C. J. Pethick, and H. Smith, Phys. Rev. A (2002) 66, 013604; M. Machholm, C. J. Pethick, and H. Smith, Phys. Rev. A (2003) 67, 053613.
[14] G. Birkl, M. Gatzke, I. H. Deutsch, S. L. Rolston, and W. D. Phillips, Phys. Rev. Lett. (1995) 75, 2823.
[15]M.Weidem uller,A.Hemmerich,A.G.orlitz,T.Esslinger,and T.W.Hansch,Phys.Rev.Lett.(1995) 75,4583.
[16]M.Ben Dahan,E.Peik,Y.Castin,and C.Salomon,Phys.Rev.Lett.(1996) 76,4508.
[17]M.Cristiani,O.Morsch,J.H.M(u|¨)ller,D.Ciampini,and E.Arimondo,Phys.Rev.A (2002) 65,063612.
[18]M.Jona-Lasinio,O.Morsch,M.Cristiani,N.Malossi,J.H.Muller,E.Courtade,M.Anderlini and E.Arimondo,Phys.Rev.Lett.(2003) 91,230406.
[19]F.S.Cataliotti,S.Burger,C.Fort,P.Maddaloni,F.Minardi,A.Trombettoni,A.Smerzi and M.Inguscio,Josephson junction arrays with Bose-Einstein condensates,Science(2001) 293,843.
[20]Yongping Zhang and Biao Wu,Phys.Rev.Lett.(2009) 102,093905.
[21]Bingbing Wang,Panming Fu,Jie Liu and Biao Wu,Phys.Rev.A(2006) 74,063610
[22]T.Anker,M.Albiez,R.Gati,S.Hunsmann,B.Eiermann,A.Trombettoni,and M.K.Oberthaler,Phys.Rev.Lett.(2005) 94,020403
[23]Ju-Kui Xue,Ai-Xia Zhang and Jie Liu,Phys.Rev.A(2008) 77,013602.
[24]G.K.Brennen,C.M.Caves,P.S.Jessen,and I.H.Deutsch,Phys.Rev.Lett.(1999)82,1060.
[25]D.Jaksch,H.-J.Briegel,J.I.Cirac,C.W.Gardiner,and P.Zoller,Phys.Rev.Lett.(1999) 82,1975.
[26]F.Schmidt-Kaler,H.H(a|¨)ffner,M.Riebe,S.Gulde,G.P.T.Lancaster,T.Deuschle,C.Becher,C.F.Roos,J.Eschner,and R.Blatt,Nature(2003) 422,408.
[27]D.Leibfried,B.DeMarco,V.Meyer,D.Lucas,M.Barrett,J.Brit-ton,W.M.Itano,B.Jelenkovic,C.Langer,T.Rosenband and D.J.Wineland,Nature(2003) 422,412.
[28]A.S.rensen and K.Mlmer,Phys.Rev.Lett.(1996) 76,4508.
[29]B.T.Seaman,L.D.Can-,and M.J.Holland,Phys.Rev.A(2005) 71,033622;B.T.Seaman,L.D.Carr,and M.J.Holland,Phys.Rev.A(2005) 72,033602.
[30]W.Krauth,Phys.Rev.Lett.(1996) 77,3695.
[31]F.Dalfovo,S.Giorgini,L.P.Pitaevskii,and S.Stringari,Rev.Mod.Phys.(1999) 71,463.
[32]M.Naraschewski,H.Wallis,A.Schenzle,J.I.Cirac,and P.Zoller,Phys.Rev.Lett.(1996) 54,2185.
[33]P.Navez,D.Bitouk,M.Gajda,Z.Idziaszele,and K.Rzazewski,Phys.Rev.Lett.(1997) 79,1789.
[34]S.L.Cornish,N.R.Claussen,J.L.Roberts,E.A.Comell,and C.E.Wieman,Phys.Rev.Lett.(2000) 85,1795.
[35]S.Burger,K.Bongs,S.Dettmer,W.Ertmer,and K.Sengstock,Phys.Rev.Lett.(1999)83,5198.
[36]M.Ben Dahan,E.peik,J.Reichel,Y.Castin and C.Salomon,Phys.Rev.Lett.(1996)76,4508.
[37]P.G..Kevrekidis,D.J.Frantzeskakis,R.Can.etero-Gonzalez,B.A.Malomed,G.Herring,and A.R.Bishop,Phys.Rev.A(2005) 71,023614.
[38]N.Bogoliubov,J.Phys.(Moscow)(1947) 11,23.
[39]E.P.Gross,Nuovo Ciment(1961) 20,454;J.Math,Phys.(1963) 4,195.
[40]L.P.Pitaevskii,Zh.Eksp.Teor.Fiz.(1961),40,646.
[41]H.M.J.M.Bosesten,et.al.,Phys.Rev.A(1997) 55,636.
[42]Inouye S,Anderson M R,Stenger J,et al.,Nature(1998) 392,151.
[43]M.J.Holland,Jin D.M.L.Chiofalo and J.Cooper,Pyhs.Rev.Lett.(1997) 78,3801.
[44]L.V.Hau,B.D.Buseh,C.Liu,Z.Dutton,M.M.Burus and J.A.Golovehenko,Phys. Rev.A(1998) 58,R54.
[45]D.S.Jin,J.R.Enser,M.R.Matthews,C.E.Wieman,and E.A.Cornell,Phys.Rev.Lett.(1996) 77,420.
[46]Dyson,F.J.,J.Math.Phys.(1967) 8,1538.
[47]Lieb,E.H.and J.Yngvason,Phys.Rev.Lett.(1998) 80,2504.
[48]L.Tonks,Phvs.Rev.(1936) 50,955;M.Girardeau,J.Math.Phys(1960) 1,516.
[49]Eugene B.Kolomeisky and Joseph P.Straley,Phys.Rev.B,(1992),46,11749
[50]Eugene B.Kolomeisky,T.J.Newman,Joseph P.Straley and Xiaoya Qi,Phys.Rev.Lett.(2000) 85,1146.
[51]S.Burger,F.S.Cataliotti,C.Fort,F.Minardi,and M.Inguscio,Phys.Rev.Lett.(2001) 86,4447.
[52]M.LewenStein,L.Stantos,M.A.Baranov and H.Fehrmann,Phys.Rev.Lett.(2004)92,050401.
[53]R.Grimm,M.Weidem(u|¨)ller,and Y.B.OvchinnikovAdv.At.,Mol.,Opt.Phys.(2000)42,95.
[54]J.Dalibard and C.Cohen-Tannoudji,J.opt.Soc.Am.B(1989) 6,2023.
[55]C.Cohen-Tannoudji and W.D.Phillips,Phys.Today(1990) 43,33.
[56]Biao Wu and Qian Niu,Phys.Rev.A(2000) 61,023402.
[57]Biao Wu,Qian Niu,New Journal of Physics(2003) 5,104.
[58]Biao Wu,Roberto Diener,and Qian Niu,Phys.Rev.A(2002) 65,025601.
[59]D.Diakonov,L.M.Jensen,C.J.Pethick and Smith H,Phys.Rev.A(2002) 66,013604.
[60]N.Elstner and H.Monien,Phys.Rev.B(1999) 59,12184.
[61]E.J.Mueller,Phys.Rev.A(2002) 66,063603.
[62]M.Machholm,C.J.Pethick and H.Smith,Phys.Rev.A(2003) 67,053613.
[63]W.D.Li and A.Smerzi,Phys.Rev.E.(2004) 70,016605.
[64]R.Ya.Kapitza,Nature(1937) 141,74.
[65]F.London,Phys.Rev.(1938) 54,947.
[66]L.Tisza,Nature(1938) 171,913.
[67]L.D.Landau and E.M.Lifshitz,Statistical Physics(Part 2) New York:Pergamon Press,1980 85-93.
[68]A.J.Leggett,Phys.Rev.Mod.(1999) 71,318.
[69]Jongchul Mun,Patrick Medley,Gretchen K.Campbell,Luis G.Marcassa,David E.Pritchard and Wolfgang Ketterle,Phys.Rev.Lett.(2007) 99,150604.
[70]S.Burger,F.S.Cataliotti,C.Fort,F.Minardi,M.Inguscio,M.L.Chiofalo,and M.P.Tosi,Phys.Rev.Lett.(2001) 86,4447.
[71]B.Wu and Q.Niu,Phys.Rev.A(2000) 61,023402;B.Wu and Q.Niu,Phys.Rev.A (2001) 64,061603(R);B.Wu and Q.Niu,Phys.Rev.Lett.(2002) 89,088901;B.Wu and Q.Niu,NewJ.Phys.(2003) 5,104.
[72]S.Burger,F.S.Cataliotti,C.Fort,F.Minardi,M.Ignuscio,M.L.Chiofalo,and M.P.Tosi,Phys.Rev.Lett.(2002) 89,088902.
[73]M.Machholm,C.J.Pethick,and H.Smith,Phys.Rev.A(2003) 67,053613.
[74]C.Menotti,A.Smerzi and A.Yrombettoni,NewJ.Phys.(2003) 5,112.1.
[75]M.Modugno,C.Tozzo,and F.Dalfovo,Phys.Rev.A(2004) 70,043625.
[76]L.Fallani,L.De Sarlo,J.E.Lye,M.Modugno,R.Saers,C.Fort and M.Inguscio,Phys.Rev.Lett.(2004) 93,140406.
[77]Qian Niu,Xian-Geng Zhao,G.A.Georgakis,and M.G.Raizen,Phys.Rev.Lett (1996) 76,4504.
[78]Maxime Ben Dahan,Ekkehard Peik,Jakob Reichel,Yvan Castin,and Christophe Salomon,Phys.Rev.Lett(1996) 76,4508.
[79]M.Salerno,V.V.Konotop and Yu.V.Bludov,Phys.Rev.Lett(2005) 101,030405.
[80]B.P.Anderson and M.A.Kasevich,Science(1998) 282,1686
[81]A.Smerzi,and A.Trombettoni,Phys.Rev.A(2003) 68,023613.
[82]A.Smerzi,A.Trombettoni,P.G.Kevrekidis and A.R.Bishop,Phys.Rev.Lett.(2002)89,170402.
[83]A.Trombettoni and A.Smerzi,Phys.Rev.Lett.(2001) 86,2353.
[84]Min-Chul Cha,M.P.A.Fischer,S.M.Girvin,M.Wallin,and A.P.Young,Phys.Rev.B(1991) 44,6883.
[85]M.P.A.Fischer,G.Grinstein,S.M.Girvin,Phys.Rev.Lett.(1990) 64,587.
[86]M.P.A.Fischer,Phys.Rev.Lett.(1990) 65,923.
[87]S.Sachdev,Quantum Phase Transitions,Cambridge Univ.Press(2001).
[88]D.Jaksch,C.Binder,J.I.Cirac,C.W.Gardiner and P.Zoller,Phys.Rev.Lett.(1998)81,3108.
[89]J.-Q.Liang,J.-L.Liu,W.-D.Li and Z.-J.Li,Phys.Rev.A(2009) 79,033617.
[90]S.Folling,S.Trotzky,P.Cheinet,M.Feld,R.Saers,A.Widera,T.Muller,and I.Bloch,Nature(London)(2009) 448,1029.
[91]S.Sachdev,Quantum Phase Transitions,Cambridge Univ.Press(2001).
[92]D.van Oosten,P.van der Straten,and H.Y.C.Stool,Phys.Rev.A(2001) 63,0536O1.
[93]K.Sheshadri,H.R.Krishnamurthy,R.Pandit,and T.V.Ramakrishnan,Europhys.Lett.(1993) 22,257.
[94]I.K.Freericks and H.Monien,Europhys.Lett.(1995) 26,545.
[95]T.D.Kuhner and H.Monien,Phys.Rev.B(1998) 58,14741.
[96]N.Elstner and H.Monien,Phys.Rev.B(1999) 59,12184.
[97]S.Rapsch,U.Schollwock,and W.Zwerger,Europhys.Lett.(1999)46,559.
[98]C.Orzel,A.K.Tuchman,M.L.Fenselau,et al.,Science(2001) 291,2386.
[99]M.P.A.Fisher,P.B.Weichman,G.Grinstein,and D.S.Fisher,Phys.Rev.B (1989) 40,546.
[100]G.-H.Chen and Y.-S.Wu,Phys.Rev.A(2003) 67,013606.
[101]J.-J.Liang,J.-Q.Liang,and W.-M.Liu,Phys.Rev.A(2003) 68,043605.
[102]T.Schulte,S.Drenkelforth,J.Kruse,W.Ertmer,J.Arlt,K.Sacha,J.Zakrzewski,and M.Lewenstein,Phys.Rev.Lett(2005) 95,170411.
[103]S.R.Wilkinson,C.F.Bharucha,K.W.Madison,Qian Niu,and M.G.Raizen,Phys.Rev.Lett.(1996) 76,4512.
[104]V.V.Ivanov,A.Alberti,M.Schioppo,G.Ferrari,M.Artoni,M.L.Chiofalo and G.M.Tinol,Phys.Rev.Lett.(2008) 100,043602.
[1]O.Morsch and M.Oberthaler,Rev.Mod.Phys.(2006) 78,179.
[2]I.Bloch,J.Dalibard,and W.Zwerger,Rev.Mod.Phys.(2008) 80,885.
[3]Biao Wu and Qian Niu,Phys.Rev.A(2001) 64,061603(R);Biao Wu and Qian Niu,New J.Phys.(2003) 5,104.
[4]D.Diakonov,L.M.Jensen,C.J.Pethick,and H.Smith,Phys.Rev.A(2002) 66,013604;M.Machholm,C.J.Pethick,and H.Smith,Phys.Rev.A(2003) 67,053613.
[5]Dae-I1 Choi and Qian Niu,Phys.Rev.Lett.(1999) 82,2022.
[6]O.Morsch,J.H.MAuller,M.Cristiani,D.Ciampini,and E.Arimondo,Phys.Rev.Lett.(2001)87,140402.
[7]Biao Wu and Qian Niu,Phys.Rev.A(2000) 61,023402.
[8]O.Zobay and B.M.Garraway,Phys.Rev.A(2000) 61,033603.
[9]Dae-I1 Choi and Biao Wu,Phys.Lett.A(2003) 318,558.
[10]M.Jona-Lasinio,O.Morsch,M.Cristiani,N.Malossi,J.H.M(A|¨)uller,E.Courtade,M.Anderlini,and E.Arimondo,Phys.Rev.Lett.(2003) 91,230406.
[11]B.P.Anderson and M.A.Kasevich,Science(1998) 282,1686.
[12]F.S.Cataliotti,C.Fort,P.Maddaloni,F.Minardi,A.Trombeffoni,A.Smerzi,and H.Inguscio,Science(2001) 293,843.
[13]I.Bloch,Nat.Phys.(2005) 1,23.
[14]D.Jaksch,C.Bruder,J.I.Cirac,C.W.Gardiner,and P.Zoller,Phys.Rev.Lett.(1998) 81,3108.
[15]M.Greiner,O.Mandel,T.Esslinger,T.W.HAansch,and I.Bloch,Nature(2002)415,39.
[16]W.D.Li and A.Smerzi,Phys.Rev.E.(2004) 70,016605.
[17]B.T.Seaman,L.D.Carr,and M.J.Holland,Phys.Rev.A(2005) 71,033622;B.T.Seaman,L.D.Carr,and M.J.Holland,Phys.Rev.A(2005) 72,033602.
[18]I.Danshita,S.Kurihara,and S.Tsuchiya,Ph.vs.Rev.A(2005) 72,053611;I.Danshita,and S.Tsuchiya,Phys.Rev.A(2007) 75,033612.
[19]Kronig R de L and Penney W G,Proc.R.Soc.London(1931) 130 499.
[20]Kittel C,(2005) Introduction to Solid State Physics,8th ed.(Wiley,New York).
[21]G.Ritt,C.Geckeler,T.Salger,G.Cermini,and M.Weitz,Phys.Rev.A(2006) 74,063622.
[22]E.M.Lifshitz and L.P.Pitaevskii,Statistical Physics(Butterworth-Heinemann,Oxford,1999).
[23]N.W.Ashcroft and N.D.Mermin,Solid State Physics(Harcourt Brace College Publishers,1976).
[24]Zener,G.,Proc.R.Soc.London,Ser.A(1932) 137,696.
[25]M.Olshanii,Phys.Rev.Lett.(1998) 81,938.
[26]M.KrAamer,C.Menotti,L.Pitaevskii,and S.Stringari,Eur.Phys.J.D(2003) 27,247.
[27]Machholm M,Nicolin A,Pethick C J and Smith H,Phys.Rev.A(2004) 69,043604.
[28]Z.X.Liang,Xi Dong,Z.D.Zhang and Biao Wu,Phys.Rev.A(2008) 78,023622.
[29]J.C.Bronski,L.D.Carr,B.Deconinck,and J.N.Kutz,Phys.Rev.Lett.(2001) 86,1402.
[30]J.C.Bronski,L.D.Cart,B.Deconinck,J.N.Kutz,and K.Promislow,Phys.Rev.E (2001) 63,036612.
[31]Wei-Dong Li,Phys.Rev.A(2006) 74,063612.
[32]E.Zaremba,Phys.Rev.A(1998) 57,518;G.M.Kavoulakis and C.J.Pethick,Phys.Rev.A(1998) 58,1563;S.Stringari,Phys.Rev.A(1998) 58,2385;P.O.Fedichev,and G.V.Shlyapnikov,Phys.Rev.A(2001) 63,045601;E.Taylor and E.Zaremba,Phys.Rev.A(2003) 68,053611.
[33]C.Menotti,M.KrAamer,A.Smerzi,L.Pitaevskii,and S.Stringari,Phys.Rev.A (2004) 70,023609.
[34]D.Boers,C.Weiss and M.Holthaus,Europhys.Lett.(2004) 67,887.
[35]M.R.Andrews,D.M.Kum,H.J.Miesner,D.S.Durfee,C.G.Townsend,S.Inouye,and W.Ketterle,Phys.Rev.Lett.(1997) 79,553;M.R.Andrews,D.M.Stamper-Kurn,H.J.Miesner,D.S.Durfee,C.G.Townsend,S.Inouye,and W.Ketterle,Phys.Rev.Lett.(1998) 80,2967(E).
[36]C.Raman,M.Kohl,R.Onofrio,D.S.Durfee,C.E.Kuklewicz,Z.Hadzibabic,and W.Ketterle,Phys.Rev.Lett.(1999) 83,2502.
[37]D.S.Petrov,G.V.Shlyapnikov,and J.T.M.Walraven,Phys.Rev.Lett.(2001) 87,050404.
[38]Meret Kramer,Ph.D.thesis,INFM-BEC and Dipartimento di Fisica,Universita di Yrento,(2004)
[1]M.Greiner,O.Mandel,T.Esslinger,T.W.Hansch,and I.Bloch,Nature(London)(2002) 415,39.
[2]F.S.Cataliotti,S.Burger,C.Fort,P.Maddaloni,F.Minardi,A.Trombettoni,A.Smerzi,and M.Inguscio,Science(2001) 293,843.
[3]B.Eiermann,Th.Anker,M.Albiez,M.Taglieber,P.Treutlein,K.-P.Marzlin,and M.K.Oberthaler,Phys.Rev.Lett.(2004) 92,230401.
[4]Biao Wu,Qian Niu,Phys.Rev.A(2000) 61,023402;Biao Wu,Roberto B.Diener,and Qian Niu,Phys.Rev.A(2002) 65,025601;E.J.Mueller,Phys.Rev.A(2002) 66,063603.
[5]Biao Wu and Qian Niu,Phys.Rev.A(2001) 64,061603(R).
[6]K.Rapedius,D.Witthaut,and H.J.Korsch,Phys.Rev.A(2006) 73,033608.
[7]D.Jaksch,C.Bruder,J.I.Cirac,C.W.Gardiner,and P.Zoller,Phys.Rev.Lett.(1998)81,3108.
[8]A.Trombettoni and A.Smerzi,Phys.Rev.Lett.(2001) 86,2353.
[9]W.Kohn,Phys.Rev.(1973) 7,4388.
[10]Joydeep Bhattacharjee and Umesh V.Waghmare,Phys.Rev.B(2005) 71,045106.
[11]F.B.Pedersen,G.T.Einevoll and P.C.Hemmer,Phys.Rev.B(1991) 44,5470.
[12]I.Talanina and C.Martijn de Sterke,Phys.Rev.A(2000) 62,043802.
[13]A.Bruno-Alfonso and D.R.Nacbar,Phys.Rev.B(2007) 75,115428.
[14]G.H.Wannier,Phys.Rev.(1937) 52,191.
[15]D.Hennig and G.P.Tsironis,Phys.Rep.(1999) 307,333.
[16]C.Kittel,Introduction to Solid State Physics,8th ed.(Wiley,New York,2005).
[17]W.-D.Li and A.Smerzi,Phys.Rev.E(2004) 70,016605.
[18]B.T.Seaman,L.D.Carr,and M.J.Holland,Phys.Rev.A(2005) 71,033622.
[19]I.Danshita and S.Tsuchiya,Phys.Rev.A(2007) 75,033612.
[20]N.W.Ashcroft and N.D.Mermin,Solid State Physics(Saunders College,Philadelphia,1976)
[21]F.Dalfovo,S.Giorgini,L.P.Pitaevskii and S.Stringari,Rev.Mod.Phys.(1999) 71,463.
[22]M.KrAamer,C.Menotti,L.Pitaevskii,and S.Stringari,Eur.Phys.J.D(2003) 27,247.
[23]Z.X.Liang,B.B.Hu and Biao Wu,(2009) cond-mat:0903.4058vl;Jinbin Li,Yue Yu,Artem M Dudarev and Qian Niu,New J.Phys.(2003) 8,154.
[24]Xin Yan Jia,WeiDong Li and J.Q.Liang,Phys.Rev.A(2008) 78,023613.
[25]Dae-I1 Choi and Qian Niu,Phys.Rev.Lett.(1999) 82,2022.
[26]Wilhelm Zwerger,J.Opt.B:Quantum Semiclass.Opt.(2003) 5,$9
[27]D.van Oosten,E van der Straten and H.T.C.Stool,Phys.Rev.A(2003) 67,033606
[1]Burger,S.,F.S.Cataliotti,C.Fort,P.Maddaloni,F.Minardi and M.Inguscio,Europhys.Lett.(2002) 57,1.
[2]Burger,S.,F.S.Cataliotti,C.Fort,F.Minardi,M.Inguscio,M.L.Chiofalo,and M.P.Tosi,Phys.Rev.Lett.(2001) 86,4447.
[3]Burger,S.,F.S.Cataliotti,C.Fort,F.Minardi,M.Ignuscio,M.L.Chiofalo,and M.P.Tosi,Phys.Rev.Lett.(2002) 89,088902.
[4]B.Wu,and Q.Niu,Phys.Rev.A(2001) 64,061603.
[5]B.Wu,and Q.Niu,Phys.Rev.Lett.(2002) 89,088901.
[6]B.Wu,andQ.Niu,New J.Phys.(2003) 5,104.
[7]Machholm,M.,C.J.Pethick,and H.Smith,Phys.Rev.A(2003) 67,053613.
[8]Smerzi,A.,S.Fantoni,S.Giovanazzi,and S.R.Shenoy,Phys.Rev.Lett.(1997) 79,4950.
[9]Smerzi,A.,and A.Trombettoni,Phys.Rev.A(2003) 68,023613.
[10]Smerzi,A.,A.Trombettoni,P.G.Kevrekidis,and A.R.Bishop,Phys.Rev.Lett.(2002) 89,170402.
[1]N.Bogoliubov,J.Phys.(Moscow)(1947) 11,23.
[2]A.Fetter L.Non-uniform states of an imperfect Bose gas.Ann Phys(1972) 70 67-10
[3]D.S.Jin,J.R.Ensher,M.R.Matthews,C.E.Wieman,and E.A.Comell,Phys.Rev.Lett.77,420(1996).
[4]M.-O.Mewes,M.R.Andrews,N.J.van Druten,D.M.Kum,D.S.Durfee,C.G.Townsend,and W.Ketterle,Phys.Rev.Lett.(1996) 77,988.
[5]L.Pitaevskii and S.Stringari,Bose-Einstein Condensation(Oxford University Press,Oxford,2003).
[6]K.Berg-Sφrensen and K.Mφlmer,Phys.Rev.A(1998) 58,1480.
[7]E.Taylor and E.Zaremba,Phys.Rev.A(2003) 68,053611.
[8]M.Ichioka and K.Machida,J.Phys.Soc.Jpn.(2003) 72,2137.
[9]B.Wu and Q.Niu,Phys.Rev.A(2001) 64,061603(R).
[10]J.Javanainen,Phys.Rev.A(1999) 60,4902.
[11]A.Smerzi and A.Trombettoni,Phys.Rev.A(2003) 68,023613.
[12]M.Kramer,C.Menotti,L.Pitaevskii,and S.Stringari,Eur.Phys.J.D(2003) 27,247.
[13]S.Tsuchiya and A.Griffin,Phys.Rev.A(2004) 70,023611.
[14]L.D.Landau J.Phys.(USSR)(1941) 5,71.
[15]Meret Kramer,Ph.D.thesis,INFM-BEC and Dipartimento di Fisica,Universita di Trento,(2004)
[16]M.L.Chiofalo,M.Polini and M.P.Tosi,Eur.Phys.J.D(2000) 11,371.
[17]A.Smerzi and A.Trombettoni,Phys.Rev.A(2003) 68,023613.
[18] J. Stenger, S. Inouye, A.P. Chikkatur, D.M. Stamper-Kurn, D.E. Pritchard, and W. Ketterle, Phys. Rev. Lett. (1999) 82,4569.
[19] D.M. Stamper-Kurn, A.P. Chikkatur, A. Gorlitz, S. Inouye, S. Gupta, D.E. Pritchard, and W. Ketterle, Phys. Rev. Lett. (1999) 83, 2876.
[20] J. M. Vogels, K. Xu, C. Raman, J.R. Abo-Shaeer, and W. Ketterle, Phys. Rev. Lett. (2002)88,060402.
[21] J. Steinhauer, R. Ozeri, N. Katz, and N. Davidson, Phys. Rev. Lett. (2002) 88, 120407.
[22] R. Ozeri, J. Steinhauer, N. Katz, and N. Davidson, Phys. Rev. Lett. (2002) 88, 220401.
[23] J. Steinhauer, N. Katz, R. Ozeri, N. Davidson, C. Tozzo, and F. Dalfovo, Phys. Rev. Lett. (2003) 90, 060404.
[24] A.L. Fetter and D. Rokshar, Phys. Rev. A (1998) 57, 1191.
[25] F. Zambelli, L. Pitaevskii, D.M. Stamper-Kurn and S. Stringari, Phys. Rev. A (2000) 61, 063608.
[26] A. Brunello, F. Dalfovo, L. Pitaevskii, S. Stringari and F. Zambelli, Phys. Rev. A (2001)64,063614.
[27] P.B. Blakie, R.J. Ballagh, and C.W. Gardiner, Phys. Rev. A (2002) 65, 033602.
[28] C. Menotti, M. Kramer, L. Pitaevskii, and S. Stringari, Phys. Rev. A (2003) 67, 053609.
[29] C. Orzel, A.K. Tuchmann, M.L. Fenselau, M. Yasuda, and M.A. Kasevich, Science (2001)291,2386.
[30] M. Greiner, O. Mandel, T. Esslinger, T.W. Hansch and I. Bloch, Nature (2002) 415, 39
[31] T. Stoferle, H. Moritz, C. Schori, M. Koh, and T. Esslinger, Phys. Rev. Lett. (2004) 92,130403
[32] D. van Oosten, P. van der Straten, and H. T. C. Stoof,, Phys. Rev. A (2001) 63, 053601.
[33] A.M. Rey, K. Burnett, R. Roth, M. Edwards, C.J. Williams, and C.W. Clark, J. Phys. B (2003) 36, 825.
[2] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. (1995) 75, 3969.
[3] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett. (1995) 75, 1687.
[4] Inouye S, Anderson M R, Stenger J, et al., Nature (1998) 392, 151.
[5] C. J. Myatt et al., Phys. Rev. Lett. (1997) 78, 586; M. R. Matthews et al., ibid. (1998) 81, 243; G. Modugno et al., Science (2001) 294, 1320;
[6] D. M. Stamper-Kurn et al., Phys. Rev. Lett. (1998) 80, 2027 ; H. J. Miesner et al., ibid. (1999) 82, 2228 ; D. M. Stamper-Kurn et al., ibid. (1999) 83, 661;
[7] Z. Hadzibabic, S. Stock, B. Battelier, V. Bretin, and J. Dalibard, Phys. Rev. Lett. (2004)93,180403.
[8] S. Peil, J. V. Porto, B. L. Tolra, J. M. Obrecht, B. E. King, M. Subbotin, S. L. Rolston, and W. D. Phillips, Phys. Rev. A (2003) 67, 051603.
[9] A. Hemmerich and T.W. Hansch, Phys. Rev. Lett. (1993) 70, 410.
[10]K.I. Petsas, A.B. Coates and G. Grynberg, Phys. Rev. A (1994) 50, 5173.
[11]M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch and I. Bloch, Nature (London) (2002) 415, 39.
[12]Biao Wu and Qian Niu, Phys. Rev. A (2001) 64, 061603(R); Biao Wu and Qian Niu, New J. Phys. (2003)5, 104.
[13]D. Diakonov, L. M. Jensen, C. J. Pethick, and H. Smith, Phys. Rev. A (2002) 66, 013604; M. Machholm, C. J. Pethick, and H. Smith, Phys. Rev. A (2003) 67, 053613.
[14] G. Birkl, M. Gatzke, I. H. Deutsch, S. L. Rolston, and W. D. Phillips, Phys. Rev. Lett. (1995) 75, 2823.
[15]M.Weidem uller,A.Hemmerich,A.G.orlitz,T.Esslinger,and T.W.Hansch,Phys.Rev.Lett.(1995) 75,4583.
[16]M.Ben Dahan,E.Peik,Y.Castin,and C.Salomon,Phys.Rev.Lett.(1996) 76,4508.
[17]M.Cristiani,O.Morsch,J.H.M(u|¨)ller,D.Ciampini,and E.Arimondo,Phys.Rev.A (2002) 65,063612.
[18]M.Jona-Lasinio,O.Morsch,M.Cristiani,N.Malossi,J.H.Muller,E.Courtade,M.Anderlini and E.Arimondo,Phys.Rev.Lett.(2003) 91,230406.
[19]F.S.Cataliotti,S.Burger,C.Fort,P.Maddaloni,F.Minardi,A.Trombettoni,A.Smerzi and M.Inguscio,Josephson junction arrays with Bose-Einstein condensates,Science(2001) 293,843.
[20]Yongping Zhang and Biao Wu,Phys.Rev.Lett.(2009) 102,093905.
[21]Bingbing Wang,Panming Fu,Jie Liu and Biao Wu,Phys.Rev.A(2006) 74,063610
[22]T.Anker,M.Albiez,R.Gati,S.Hunsmann,B.Eiermann,A.Trombettoni,and M.K.Oberthaler,Phys.Rev.Lett.(2005) 94,020403
[23]Ju-Kui Xue,Ai-Xia Zhang and Jie Liu,Phys.Rev.A(2008) 77,013602.
[24]G.K.Brennen,C.M.Caves,P.S.Jessen,and I.H.Deutsch,Phys.Rev.Lett.(1999)82,1060.
[25]D.Jaksch,H.-J.Briegel,J.I.Cirac,C.W.Gardiner,and P.Zoller,Phys.Rev.Lett.(1999) 82,1975.
[26]F.Schmidt-Kaler,H.H(a|¨)ffner,M.Riebe,S.Gulde,G.P.T.Lancaster,T.Deuschle,C.Becher,C.F.Roos,J.Eschner,and R.Blatt,Nature(2003) 422,408.
[27]D.Leibfried,B.DeMarco,V.Meyer,D.Lucas,M.Barrett,J.Brit-ton,W.M.Itano,B.Jelenkovic,C.Langer,T.Rosenband and D.J.Wineland,Nature(2003) 422,412.
[28]A.S.rensen and K.Mlmer,Phys.Rev.Lett.(1996) 76,4508.
[29]B.T.Seaman,L.D.Can-,and M.J.Holland,Phys.Rev.A(2005) 71,033622;B.T.Seaman,L.D.Carr,and M.J.Holland,Phys.Rev.A(2005) 72,033602.
[30]W.Krauth,Phys.Rev.Lett.(1996) 77,3695.
[31]F.Dalfovo,S.Giorgini,L.P.Pitaevskii,and S.Stringari,Rev.Mod.Phys.(1999) 71,463.
[32]M.Naraschewski,H.Wallis,A.Schenzle,J.I.Cirac,and P.Zoller,Phys.Rev.Lett.(1996) 54,2185.
[33]P.Navez,D.Bitouk,M.Gajda,Z.Idziaszele,and K.Rzazewski,Phys.Rev.Lett.(1997) 79,1789.
[34]S.L.Cornish,N.R.Claussen,J.L.Roberts,E.A.Comell,and C.E.Wieman,Phys.Rev.Lett.(2000) 85,1795.
[35]S.Burger,K.Bongs,S.Dettmer,W.Ertmer,and K.Sengstock,Phys.Rev.Lett.(1999)83,5198.
[36]M.Ben Dahan,E.peik,J.Reichel,Y.Castin and C.Salomon,Phys.Rev.Lett.(1996)76,4508.
[37]P.G..Kevrekidis,D.J.Frantzeskakis,R.Can.etero-Gonzalez,B.A.Malomed,G.Herring,and A.R.Bishop,Phys.Rev.A(2005) 71,023614.
[38]N.Bogoliubov,J.Phys.(Moscow)(1947) 11,23.
[39]E.P.Gross,Nuovo Ciment(1961) 20,454;J.Math,Phys.(1963) 4,195.
[40]L.P.Pitaevskii,Zh.Eksp.Teor.Fiz.(1961),40,646.
[41]H.M.J.M.Bosesten,et.al.,Phys.Rev.A(1997) 55,636.
[42]Inouye S,Anderson M R,Stenger J,et al.,Nature(1998) 392,151.
[43]M.J.Holland,Jin D.M.L.Chiofalo and J.Cooper,Pyhs.Rev.Lett.(1997) 78,3801.
[44]L.V.Hau,B.D.Buseh,C.Liu,Z.Dutton,M.M.Burus and J.A.Golovehenko,Phys. Rev.A(1998) 58,R54.
[45]D.S.Jin,J.R.Enser,M.R.Matthews,C.E.Wieman,and E.A.Cornell,Phys.Rev.Lett.(1996) 77,420.
[46]Dyson,F.J.,J.Math.Phys.(1967) 8,1538.
[47]Lieb,E.H.and J.Yngvason,Phys.Rev.Lett.(1998) 80,2504.
[48]L.Tonks,Phvs.Rev.(1936) 50,955;M.Girardeau,J.Math.Phys(1960) 1,516.
[49]Eugene B.Kolomeisky and Joseph P.Straley,Phys.Rev.B,(1992),46,11749
[50]Eugene B.Kolomeisky,T.J.Newman,Joseph P.Straley and Xiaoya Qi,Phys.Rev.Lett.(2000) 85,1146.
[51]S.Burger,F.S.Cataliotti,C.Fort,F.Minardi,and M.Inguscio,Phys.Rev.Lett.(2001) 86,4447.
[52]M.LewenStein,L.Stantos,M.A.Baranov and H.Fehrmann,Phys.Rev.Lett.(2004)92,050401.
[53]R.Grimm,M.Weidem(u|¨)ller,and Y.B.OvchinnikovAdv.At.,Mol.,Opt.Phys.(2000)42,95.
[54]J.Dalibard and C.Cohen-Tannoudji,J.opt.Soc.Am.B(1989) 6,2023.
[55]C.Cohen-Tannoudji and W.D.Phillips,Phys.Today(1990) 43,33.
[56]Biao Wu and Qian Niu,Phys.Rev.A(2000) 61,023402.
[57]Biao Wu,Qian Niu,New Journal of Physics(2003) 5,104.
[58]Biao Wu,Roberto Diener,and Qian Niu,Phys.Rev.A(2002) 65,025601.
[59]D.Diakonov,L.M.Jensen,C.J.Pethick and Smith H,Phys.Rev.A(2002) 66,013604.
[60]N.Elstner and H.Monien,Phys.Rev.B(1999) 59,12184.
[61]E.J.Mueller,Phys.Rev.A(2002) 66,063603.
[62]M.Machholm,C.J.Pethick and H.Smith,Phys.Rev.A(2003) 67,053613.
[63]W.D.Li and A.Smerzi,Phys.Rev.E.(2004) 70,016605.
[64]R.Ya.Kapitza,Nature(1937) 141,74.
[65]F.London,Phys.Rev.(1938) 54,947.
[66]L.Tisza,Nature(1938) 171,913.
[67]L.D.Landau and E.M.Lifshitz,Statistical Physics(Part 2) New York:Pergamon Press,1980 85-93.
[68]A.J.Leggett,Phys.Rev.Mod.(1999) 71,318.
[69]Jongchul Mun,Patrick Medley,Gretchen K.Campbell,Luis G.Marcassa,David E.Pritchard and Wolfgang Ketterle,Phys.Rev.Lett.(2007) 99,150604.
[70]S.Burger,F.S.Cataliotti,C.Fort,F.Minardi,M.Inguscio,M.L.Chiofalo,and M.P.Tosi,Phys.Rev.Lett.(2001) 86,4447.
[71]B.Wu and Q.Niu,Phys.Rev.A(2000) 61,023402;B.Wu and Q.Niu,Phys.Rev.A (2001) 64,061603(R);B.Wu and Q.Niu,Phys.Rev.Lett.(2002) 89,088901;B.Wu and Q.Niu,NewJ.Phys.(2003) 5,104.
[72]S.Burger,F.S.Cataliotti,C.Fort,F.Minardi,M.Ignuscio,M.L.Chiofalo,and M.P.Tosi,Phys.Rev.Lett.(2002) 89,088902.
[73]M.Machholm,C.J.Pethick,and H.Smith,Phys.Rev.A(2003) 67,053613.
[74]C.Menotti,A.Smerzi and A.Yrombettoni,NewJ.Phys.(2003) 5,112.1.
[75]M.Modugno,C.Tozzo,and F.Dalfovo,Phys.Rev.A(2004) 70,043625.
[76]L.Fallani,L.De Sarlo,J.E.Lye,M.Modugno,R.Saers,C.Fort and M.Inguscio,Phys.Rev.Lett.(2004) 93,140406.
[77]Qian Niu,Xian-Geng Zhao,G.A.Georgakis,and M.G.Raizen,Phys.Rev.Lett (1996) 76,4504.
[78]Maxime Ben Dahan,Ekkehard Peik,Jakob Reichel,Yvan Castin,and Christophe Salomon,Phys.Rev.Lett(1996) 76,4508.
[79]M.Salerno,V.V.Konotop and Yu.V.Bludov,Phys.Rev.Lett(2005) 101,030405.
[80]B.P.Anderson and M.A.Kasevich,Science(1998) 282,1686
[81]A.Smerzi,and A.Trombettoni,Phys.Rev.A(2003) 68,023613.
[82]A.Smerzi,A.Trombettoni,P.G.Kevrekidis and A.R.Bishop,Phys.Rev.Lett.(2002)89,170402.
[83]A.Trombettoni and A.Smerzi,Phys.Rev.Lett.(2001) 86,2353.
[84]Min-Chul Cha,M.P.A.Fischer,S.M.Girvin,M.Wallin,and A.P.Young,Phys.Rev.B(1991) 44,6883.
[85]M.P.A.Fischer,G.Grinstein,S.M.Girvin,Phys.Rev.Lett.(1990) 64,587.
[86]M.P.A.Fischer,Phys.Rev.Lett.(1990) 65,923.
[87]S.Sachdev,Quantum Phase Transitions,Cambridge Univ.Press(2001).
[88]D.Jaksch,C.Binder,J.I.Cirac,C.W.Gardiner and P.Zoller,Phys.Rev.Lett.(1998)81,3108.
[89]J.-Q.Liang,J.-L.Liu,W.-D.Li and Z.-J.Li,Phys.Rev.A(2009) 79,033617.
[90]S.Folling,S.Trotzky,P.Cheinet,M.Feld,R.Saers,A.Widera,T.Muller,and I.Bloch,Nature(London)(2009) 448,1029.
[91]S.Sachdev,Quantum Phase Transitions,Cambridge Univ.Press(2001).
[92]D.van Oosten,P.van der Straten,and H.Y.C.Stool,Phys.Rev.A(2001) 63,0536O1.
[93]K.Sheshadri,H.R.Krishnamurthy,R.Pandit,and T.V.Ramakrishnan,Europhys.Lett.(1993) 22,257.
[94]I.K.Freericks and H.Monien,Europhys.Lett.(1995) 26,545.
[95]T.D.Kuhner and H.Monien,Phys.Rev.B(1998) 58,14741.
[96]N.Elstner and H.Monien,Phys.Rev.B(1999) 59,12184.
[97]S.Rapsch,U.Schollwock,and W.Zwerger,Europhys.Lett.(1999)46,559.
[98]C.Orzel,A.K.Tuchman,M.L.Fenselau,et al.,Science(2001) 291,2386.
[99]M.P.A.Fisher,P.B.Weichman,G.Grinstein,and D.S.Fisher,Phys.Rev.B (1989) 40,546.
[100]G.-H.Chen and Y.-S.Wu,Phys.Rev.A(2003) 67,013606.
[101]J.-J.Liang,J.-Q.Liang,and W.-M.Liu,Phys.Rev.A(2003) 68,043605.
[102]T.Schulte,S.Drenkelforth,J.Kruse,W.Ertmer,J.Arlt,K.Sacha,J.Zakrzewski,and M.Lewenstein,Phys.Rev.Lett(2005) 95,170411.
[103]S.R.Wilkinson,C.F.Bharucha,K.W.Madison,Qian Niu,and M.G.Raizen,Phys.Rev.Lett.(1996) 76,4512.
[104]V.V.Ivanov,A.Alberti,M.Schioppo,G.Ferrari,M.Artoni,M.L.Chiofalo and G.M.Tinol,Phys.Rev.Lett.(2008) 100,043602.
[1]O.Morsch and M.Oberthaler,Rev.Mod.Phys.(2006) 78,179.
[2]I.Bloch,J.Dalibard,and W.Zwerger,Rev.Mod.Phys.(2008) 80,885.
[3]Biao Wu and Qian Niu,Phys.Rev.A(2001) 64,061603(R);Biao Wu and Qian Niu,New J.Phys.(2003) 5,104.
[4]D.Diakonov,L.M.Jensen,C.J.Pethick,and H.Smith,Phys.Rev.A(2002) 66,013604;M.Machholm,C.J.Pethick,and H.Smith,Phys.Rev.A(2003) 67,053613.
[5]Dae-I1 Choi and Qian Niu,Phys.Rev.Lett.(1999) 82,2022.
[6]O.Morsch,J.H.MAuller,M.Cristiani,D.Ciampini,and E.Arimondo,Phys.Rev.Lett.(2001)87,140402.
[7]Biao Wu and Qian Niu,Phys.Rev.A(2000) 61,023402.
[8]O.Zobay and B.M.Garraway,Phys.Rev.A(2000) 61,033603.
[9]Dae-I1 Choi and Biao Wu,Phys.Lett.A(2003) 318,558.
[10]M.Jona-Lasinio,O.Morsch,M.Cristiani,N.Malossi,J.H.M(A|¨)uller,E.Courtade,M.Anderlini,and E.Arimondo,Phys.Rev.Lett.(2003) 91,230406.
[11]B.P.Anderson and M.A.Kasevich,Science(1998) 282,1686.
[12]F.S.Cataliotti,C.Fort,P.Maddaloni,F.Minardi,A.Trombeffoni,A.Smerzi,and H.Inguscio,Science(2001) 293,843.
[13]I.Bloch,Nat.Phys.(2005) 1,23.
[14]D.Jaksch,C.Bruder,J.I.Cirac,C.W.Gardiner,and P.Zoller,Phys.Rev.Lett.(1998) 81,3108.
[15]M.Greiner,O.Mandel,T.Esslinger,T.W.HAansch,and I.Bloch,Nature(2002)415,39.
[16]W.D.Li and A.Smerzi,Phys.Rev.E.(2004) 70,016605.
[17]B.T.Seaman,L.D.Carr,and M.J.Holland,Phys.Rev.A(2005) 71,033622;B.T.Seaman,L.D.Carr,and M.J.Holland,Phys.Rev.A(2005) 72,033602.
[18]I.Danshita,S.Kurihara,and S.Tsuchiya,Ph.vs.Rev.A(2005) 72,053611;I.Danshita,and S.Tsuchiya,Phys.Rev.A(2007) 75,033612.
[19]Kronig R de L and Penney W G,Proc.R.Soc.London(1931) 130 499.
[20]Kittel C,(2005) Introduction to Solid State Physics,8th ed.(Wiley,New York).
[21]G.Ritt,C.Geckeler,T.Salger,G.Cermini,and M.Weitz,Phys.Rev.A(2006) 74,063622.
[22]E.M.Lifshitz and L.P.Pitaevskii,Statistical Physics(Butterworth-Heinemann,Oxford,1999).
[23]N.W.Ashcroft and N.D.Mermin,Solid State Physics(Harcourt Brace College Publishers,1976).
[24]Zener,G.,Proc.R.Soc.London,Ser.A(1932) 137,696.
[25]M.Olshanii,Phys.Rev.Lett.(1998) 81,938.
[26]M.KrAamer,C.Menotti,L.Pitaevskii,and S.Stringari,Eur.Phys.J.D(2003) 27,247.
[27]Machholm M,Nicolin A,Pethick C J and Smith H,Phys.Rev.A(2004) 69,043604.
[28]Z.X.Liang,Xi Dong,Z.D.Zhang and Biao Wu,Phys.Rev.A(2008) 78,023622.
[29]J.C.Bronski,L.D.Carr,B.Deconinck,and J.N.Kutz,Phys.Rev.Lett.(2001) 86,1402.
[30]J.C.Bronski,L.D.Cart,B.Deconinck,J.N.Kutz,and K.Promislow,Phys.Rev.E (2001) 63,036612.
[31]Wei-Dong Li,Phys.Rev.A(2006) 74,063612.
[32]E.Zaremba,Phys.Rev.A(1998) 57,518;G.M.Kavoulakis and C.J.Pethick,Phys.Rev.A(1998) 58,1563;S.Stringari,Phys.Rev.A(1998) 58,2385;P.O.Fedichev,and G.V.Shlyapnikov,Phys.Rev.A(2001) 63,045601;E.Taylor and E.Zaremba,Phys.Rev.A(2003) 68,053611.
[33]C.Menotti,M.KrAamer,A.Smerzi,L.Pitaevskii,and S.Stringari,Phys.Rev.A (2004) 70,023609.
[34]D.Boers,C.Weiss and M.Holthaus,Europhys.Lett.(2004) 67,887.
[35]M.R.Andrews,D.M.Kum,H.J.Miesner,D.S.Durfee,C.G.Townsend,S.Inouye,and W.Ketterle,Phys.Rev.Lett.(1997) 79,553;M.R.Andrews,D.M.Stamper-Kurn,H.J.Miesner,D.S.Durfee,C.G.Townsend,S.Inouye,and W.Ketterle,Phys.Rev.Lett.(1998) 80,2967(E).
[36]C.Raman,M.Kohl,R.Onofrio,D.S.Durfee,C.E.Kuklewicz,Z.Hadzibabic,and W.Ketterle,Phys.Rev.Lett.(1999) 83,2502.
[37]D.S.Petrov,G.V.Shlyapnikov,and J.T.M.Walraven,Phys.Rev.Lett.(2001) 87,050404.
[38]Meret Kramer,Ph.D.thesis,INFM-BEC and Dipartimento di Fisica,Universita di Yrento,(2004)
[1]M.Greiner,O.Mandel,T.Esslinger,T.W.Hansch,and I.Bloch,Nature(London)(2002) 415,39.
[2]F.S.Cataliotti,S.Burger,C.Fort,P.Maddaloni,F.Minardi,A.Trombettoni,A.Smerzi,and M.Inguscio,Science(2001) 293,843.
[3]B.Eiermann,Th.Anker,M.Albiez,M.Taglieber,P.Treutlein,K.-P.Marzlin,and M.K.Oberthaler,Phys.Rev.Lett.(2004) 92,230401.
[4]Biao Wu,Qian Niu,Phys.Rev.A(2000) 61,023402;Biao Wu,Roberto B.Diener,and Qian Niu,Phys.Rev.A(2002) 65,025601;E.J.Mueller,Phys.Rev.A(2002) 66,063603.
[5]Biao Wu and Qian Niu,Phys.Rev.A(2001) 64,061603(R).
[6]K.Rapedius,D.Witthaut,and H.J.Korsch,Phys.Rev.A(2006) 73,033608.
[7]D.Jaksch,C.Bruder,J.I.Cirac,C.W.Gardiner,and P.Zoller,Phys.Rev.Lett.(1998)81,3108.
[8]A.Trombettoni and A.Smerzi,Phys.Rev.Lett.(2001) 86,2353.
[9]W.Kohn,Phys.Rev.(1973) 7,4388.
[10]Joydeep Bhattacharjee and Umesh V.Waghmare,Phys.Rev.B(2005) 71,045106.
[11]F.B.Pedersen,G.T.Einevoll and P.C.Hemmer,Phys.Rev.B(1991) 44,5470.
[12]I.Talanina and C.Martijn de Sterke,Phys.Rev.A(2000) 62,043802.
[13]A.Bruno-Alfonso and D.R.Nacbar,Phys.Rev.B(2007) 75,115428.
[14]G.H.Wannier,Phys.Rev.(1937) 52,191.
[15]D.Hennig and G.P.Tsironis,Phys.Rep.(1999) 307,333.
[16]C.Kittel,Introduction to Solid State Physics,8th ed.(Wiley,New York,2005).
[17]W.-D.Li and A.Smerzi,Phys.Rev.E(2004) 70,016605.
[18]B.T.Seaman,L.D.Carr,and M.J.Holland,Phys.Rev.A(2005) 71,033622.
[19]I.Danshita and S.Tsuchiya,Phys.Rev.A(2007) 75,033612.
[20]N.W.Ashcroft and N.D.Mermin,Solid State Physics(Saunders College,Philadelphia,1976)
[21]F.Dalfovo,S.Giorgini,L.P.Pitaevskii and S.Stringari,Rev.Mod.Phys.(1999) 71,463.
[22]M.KrAamer,C.Menotti,L.Pitaevskii,and S.Stringari,Eur.Phys.J.D(2003) 27,247.
[23]Z.X.Liang,B.B.Hu and Biao Wu,(2009) cond-mat:0903.4058vl;Jinbin Li,Yue Yu,Artem M Dudarev and Qian Niu,New J.Phys.(2003) 8,154.
[24]Xin Yan Jia,WeiDong Li and J.Q.Liang,Phys.Rev.A(2008) 78,023613.
[25]Dae-I1 Choi and Qian Niu,Phys.Rev.Lett.(1999) 82,2022.
[26]Wilhelm Zwerger,J.Opt.B:Quantum Semiclass.Opt.(2003) 5,$9
[27]D.van Oosten,E van der Straten and H.T.C.Stool,Phys.Rev.A(2003) 67,033606
[1]Burger,S.,F.S.Cataliotti,C.Fort,P.Maddaloni,F.Minardi and M.Inguscio,Europhys.Lett.(2002) 57,1.
[2]Burger,S.,F.S.Cataliotti,C.Fort,F.Minardi,M.Inguscio,M.L.Chiofalo,and M.P.Tosi,Phys.Rev.Lett.(2001) 86,4447.
[3]Burger,S.,F.S.Cataliotti,C.Fort,F.Minardi,M.Ignuscio,M.L.Chiofalo,and M.P.Tosi,Phys.Rev.Lett.(2002) 89,088902.
[4]B.Wu,and Q.Niu,Phys.Rev.A(2001) 64,061603.
[5]B.Wu,and Q.Niu,Phys.Rev.Lett.(2002) 89,088901.
[6]B.Wu,andQ.Niu,New J.Phys.(2003) 5,104.
[7]Machholm,M.,C.J.Pethick,and H.Smith,Phys.Rev.A(2003) 67,053613.
[8]Smerzi,A.,S.Fantoni,S.Giovanazzi,and S.R.Shenoy,Phys.Rev.Lett.(1997) 79,4950.
[9]Smerzi,A.,and A.Trombettoni,Phys.Rev.A(2003) 68,023613.
[10]Smerzi,A.,A.Trombettoni,P.G.Kevrekidis,and A.R.Bishop,Phys.Rev.Lett.(2002) 89,170402.
[1]N.Bogoliubov,J.Phys.(Moscow)(1947) 11,23.
[2]A.Fetter L.Non-uniform states of an imperfect Bose gas.Ann Phys(1972) 70 67-10
[3]D.S.Jin,J.R.Ensher,M.R.Matthews,C.E.Wieman,and E.A.Comell,Phys.Rev.Lett.77,420(1996).
[4]M.-O.Mewes,M.R.Andrews,N.J.van Druten,D.M.Kum,D.S.Durfee,C.G.Townsend,and W.Ketterle,Phys.Rev.Lett.(1996) 77,988.
[5]L.Pitaevskii and S.Stringari,Bose-Einstein Condensation(Oxford University Press,Oxford,2003).
[6]K.Berg-Sφrensen and K.Mφlmer,Phys.Rev.A(1998) 58,1480.
[7]E.Taylor and E.Zaremba,Phys.Rev.A(2003) 68,053611.
[8]M.Ichioka and K.Machida,J.Phys.Soc.Jpn.(2003) 72,2137.
[9]B.Wu and Q.Niu,Phys.Rev.A(2001) 64,061603(R).
[10]J.Javanainen,Phys.Rev.A(1999) 60,4902.
[11]A.Smerzi and A.Trombettoni,Phys.Rev.A(2003) 68,023613.
[12]M.Kramer,C.Menotti,L.Pitaevskii,and S.Stringari,Eur.Phys.J.D(2003) 27,247.
[13]S.Tsuchiya and A.Griffin,Phys.Rev.A(2004) 70,023611.
[14]L.D.Landau J.Phys.(USSR)(1941) 5,71.
[15]Meret Kramer,Ph.D.thesis,INFM-BEC and Dipartimento di Fisica,Universita di Trento,(2004)
[16]M.L.Chiofalo,M.Polini and M.P.Tosi,Eur.Phys.J.D(2000) 11,371.
[17]A.Smerzi and A.Trombettoni,Phys.Rev.A(2003) 68,023613.
[18] J. Stenger, S. Inouye, A.P. Chikkatur, D.M. Stamper-Kurn, D.E. Pritchard, and W. Ketterle, Phys. Rev. Lett. (1999) 82,4569.
[19] D.M. Stamper-Kurn, A.P. Chikkatur, A. Gorlitz, S. Inouye, S. Gupta, D.E. Pritchard, and W. Ketterle, Phys. Rev. Lett. (1999) 83, 2876.
[20] J. M. Vogels, K. Xu, C. Raman, J.R. Abo-Shaeer, and W. Ketterle, Phys. Rev. Lett. (2002)88,060402.
[21] J. Steinhauer, R. Ozeri, N. Katz, and N. Davidson, Phys. Rev. Lett. (2002) 88, 120407.
[22] R. Ozeri, J. Steinhauer, N. Katz, and N. Davidson, Phys. Rev. Lett. (2002) 88, 220401.
[23] J. Steinhauer, N. Katz, R. Ozeri, N. Davidson, C. Tozzo, and F. Dalfovo, Phys. Rev. Lett. (2003) 90, 060404.
[24] A.L. Fetter and D. Rokshar, Phys. Rev. A (1998) 57, 1191.
[25] F. Zambelli, L. Pitaevskii, D.M. Stamper-Kurn and S. Stringari, Phys. Rev. A (2000) 61, 063608.
[26] A. Brunello, F. Dalfovo, L. Pitaevskii, S. Stringari and F. Zambelli, Phys. Rev. A (2001)64,063614.
[27] P.B. Blakie, R.J. Ballagh, and C.W. Gardiner, Phys. Rev. A (2002) 65, 033602.
[28] C. Menotti, M. Kramer, L. Pitaevskii, and S. Stringari, Phys. Rev. A (2003) 67, 053609.
[29] C. Orzel, A.K. Tuchmann, M.L. Fenselau, M. Yasuda, and M.A. Kasevich, Science (2001)291,2386.
[30] M. Greiner, O. Mandel, T. Esslinger, T.W. Hansch and I. Bloch, Nature (2002) 415, 39
[31] T. Stoferle, H. Moritz, C. Schori, M. Koh, and T. Esslinger, Phys. Rev. Lett. (2004) 92,130403
[32] D. van Oosten, P. van der Straten, and H. T. C. Stoof,, Phys. Rev. A (2001) 63, 053601.
[33] A.M. Rey, K. Burnett, R. Roth, M. Edwards, C.J. Williams, and C.W. Clark, J. Phys. B (2003) 36, 825.