-
We review some recent theoretical and experimental developments of one-dimensional few-body problems in ultracold atomic system. The experiments have so far realized the deterministic loading of few atoms in the ground state of a potential well, the observation of tunneling dynamics out of the metastable trap controlled by a magnetic gradient for a repulsively or attractively interacting system, the preparation of two fermionic atoms in an isolated double-well potential with a full control over the quantum state of the system, the formation of a Fermi sea by studying quasi-one-dimensional systems of ultracold atoms consisting of a single impurity interacting with an increasing number of identical fermions, and the deterministic preparation of antiferromagnetic Heisenberg spin chains consisting of up to four fermionic atoms in a one-dimensional trap. These achievements make the ultracold atoms an ideal platform to study many-body physics in a bottom-up approach, i.e., one starts from the fundamental building block of the system and observes the emergence of many-body effects by adding atoms one by one into the system. Corresponding theoretical models have been developed to explain the experimental data, to tackle the crossover boundary between few and many particles, and even explore the solvability and integrability of the models, especially the energy spectrum of interacting few atoms such as two atoms in a harmonic trap, two heteronuclear atoms of unequal mass in a ring trap, and two atoms in a
$\delta$ -barrier split double well potential. After a brief review of Bethe-Ansatz method, a theory for the tunneling of one atom out of a trap containing two interacting cold atoms is developed based on the calculation of the quasiparticle wave function, and the tunneling dynamics of two atoms starting from the NOON state is explored from the exactly solved model of$\delta$ -barrier split double well based on a Bethe ansatz type hypothesis of the wave functions. It was shown that the spectroscopy and spin dynamics for strongly interacting few atoms of spin-1/2 and spin-1 can be described by effective spin chain Hamiltonians, which serves as a useful and efficient tool to study the quantum magnetism with clod atoms.-
Keywords:
- ultracold atom physics /
- one dimensioanl system /
- few-body problem /
- exactly solvable model
[1] Pitaevskii L, Stringari S 2016 Bose-Einstein Condensation and Superfluidity(2nd Ed.) (New York: Oxford University Press) p.42
[2] Pethick C J, Smith H 2008 Bose-Einstein Condensation in Dilute Gases (2nd Ed.) (Cambridge: Cambridge University Press) p.159
[3] Gross E P 1961 Nuovo Cimento 20 454
Google Scholar
[4] Gross E P 1963 J. Math. Phys. 4 195
Google Scholar
[5] Pitaevskii L P 1961 J. Exp. Theor. Phys. 40 646
[6] Dalfovo F, Giorgini S, Pitaevskii L P, Stringari S 1999 Rev. Mod. Phys. 71 463
Google Scholar
[7] Baranov M A 2008 Phys. Rep. 464 71
Google Scholar
[8] Lahaye T, Menotti C, Santos L, Lewenstein M, Pfau T 2009 Rep. Prog. Phys. 72 126401
Google Scholar
[9] Serwane F, Zürn G, Lompe T, Ottenstein T B, Wenz A N, Jochim S 2011 Science 332 236
[10] Greiner M, Mandel O, Esslinger T, Hänsch T W, Bloch I 2002 Nature 415 39
Google Scholar
[11] Kohl M, Moritz H, Stöferle T, Günter K, Esslinger T 2005 Phys. Rev. Lett. 94 080403
Google Scholar
[12] Chin C, Grimm R, Julienne P, Tiesinga E 2010 Rev. Mod. Phys. 82 1225
Google Scholar
[13] Inouye S, Andrews M R, Stenger J, Miesner H J, Stamper-Kurn D M, Ketterle W 1998 Nature 392 151
Google Scholar
[14] Courteille P, Freeland R S, Heinzen D J, van Abeelen F A, Verhaar B J 1998 Phys. Rev. Lett. 81 69
Google Scholar
[15] Roberts J L, Claussen N R, Burke J P, Greene C H, Cornell E A, Wieman C E 1998 Phys. Rev. Lett. 81 5109
Google Scholar
[16] Efimov V 1970 Phys. Lett. B 33 563
Google Scholar
[17] Efimov V 1970 Sov. J. Nucl. Phys. 12 598
[18] Efimov V 1973 Nucl. Phys. A 210 157
Google Scholar
[19] Blume D, Daily K M 2010 Phys. Rev. Lett. 105 170403
Google Scholar
[20] Naidon P, Endo S 2017 Rep. Prog. Phys. 80 056001
Google Scholar
[21] Kraemer T, Mark M, Waldburger P, Danzl J G, Chin C, Engeser B, Lange A D, Pilch K, Jaakkola A, Nägerl H C, Grimm R 2006 Nature 440 315
Google Scholar
[22] 刘彦霞 2017 博士学位论文(太原: 山西大学)
Liu Y X 2017 Ph. D. Dissertation (Taiyuan: Shanxi University) (in Chinese)
[23] Blume D 2012 Rep. Prog. Phys. 75 046401
Google Scholar
[24] Zürn G, Serwane F, Serwane F, Lompe T, Wenz A N, Ries M G, Bohn J E, Jochim S 2012 Phys. Rev. Lett. 108 075303
Google Scholar
[25] Zürn G, Wenz A N, MurmannS, Bergschneider A, Lompe T, Jochim S 2013 Phys. Rev. Lett. 111 175302
Google Scholar
[26] Wenz A N, Zürn G, Murmann S, Brouzos I, Lompe T, Jochim S 2013 Science 342 457
Google Scholar
[27] Murmann S, Bergschneider A, Klinkhamer V M, Zürn G, Lompe T, Jochim S 2015 Phys. Rev. Lett. 114 080402
Google Scholar
[28] Murmann S, Deuretzbacher F, Zürn G, Bjerlin J, Reimann S M, Santos L, Lompe T, Jochim S 2015 Phys. Rev. Lett. 115 215301
Google Scholar
[29] Kaufman A M, Lester B J, Regal C A 2012 Phys. Rev. X 2 041014
[30] Kaufman A M, Lester B J, Reynolds C M, Wall M L, Foss-Feig M, Hazzard K R A, Rey A M, Regal C A 2014 Science 345 306
Google Scholar
[31] Busch T, Englert B G, Rzazewski K, Wilkens M 1998 Found. Phys. 28 549
Google Scholar
[32] Olshanii M 1998 Phys. Rev. Lett. 81 938
Google Scholar
[33] Rontani M 2012 Phys. Rev. Lett. 108 115302
Google Scholar
[34] Rontani M 2013 Phys. Rev. A 88 043633
Google Scholar
[35] Bethe H A 1931 Z. Phys. 71 205
Google Scholar
[36] Lieb E H, Liniger W 1963 Phys. Rev. 1963 130
[37] McGuire J B 1965 J. Math. Phys. 6 432
Google Scholar
[38] Gaudin M 1967 Phys. Lett. A 24 55
Google Scholar
[39] Yang C N 1967 Phys. Rev. Lett. 19 1312
Google Scholar
[40] Baxter R J 1972 Ann. Phys. 70 193
Google Scholar
[41] Lieb E H, Wu F Y 1968 Phys. Rev. Lett. 20 1445
Google Scholar
[42] Korepin V E, Essler F H L 1994 Exactly Solvable Models of Strongly Correlated Electrons (Singapore: World Scientific)
[43] Mattis D C 1993 The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension (Singapore: World Scientific)
[44] Takahashi M 1999 Thermodynamics of One-Dimensional Solvable models (Cambridge: Cambridge University Press)
[45] Baxter R J 1989 Exactly solved models in statistical mechanics (London: Academic Press)
[46] Sutherland B 2004 Beautiful Models-70 years of exactly solved quantum many-body problems (Singapore: World Scientific)
[47] Hao Y, Zhang Y, Liang J, Chen S 2006 Phys. Rev. A 73 053605
Google Scholar
[48] Hao Y, Zhang Y, Chen S 2009 Phys. Rev. A 79 043633
Google Scholar
[49] Hao Y, Zhang Y, Liang J, Chen S 2006 Phys. Rev. A 73 063617
Google Scholar
[50] Hao Y, Zhang Y, Guan X, Chen S 2009 Phys. Rev. A 79 033607
Google Scholar
[51] Hao Y, Chen S 2009 Phys. Rev. A 80 043608
Google Scholar
[52] Hao Y, Guo H, Zhang Y, Chen S 2011 Phys. Rev. A 83 053632
Google Scholar
[53] Hao Y, Zhang Y, Chen S 2008 Phys. Rev. A 78 023631
Google Scholar
[54] Guan X 2014 Int. J. Mod. Phys. B 28 1430015
Google Scholar
[55] Guan X, Batchelor M, Lee C 2013 Rev. Mod. Phys. 85 1633
Google Scholar
[56] Jiang Y, Chen Y, Guan X 2015 Chinese Phys. B 24 050311
Google Scholar
[57] Yin X, Chen S, Zhang Y 2009 Phys. Rev. A 79 053604
Google Scholar
[58] Yin X, Guan X, Batchelor M, Chen S 2011 Phys. Rev. A 83 013602
Google Scholar
[59] Yin X, Guan X, Chen S, Batchelor M 2011 Phys. Rev. A 84 011602(R)
Google Scholar
[60] Yin X, Guan X, Zhang Y, Chen S 2012 Phys. Rev. A 85 013608
Google Scholar
[61] Yin X, Guan X, Zhang Y, Su H, Zhang S 2018 Phys. Rev. A 98 023605
Google Scholar
[62] Wang H, Hao Y, Zhang Y 2012 Phys. Rev. A 85 053630
Google Scholar
[63] Wang H, Zhang Y 2013 Phys. Rev. A 88 023626
Google Scholar
[64] Paredes B, Widera A, Murg V, Mandel O, Fölling S, Cirac I, Shlyapnikov G, Hänsch T, Bloch I 2004 Nature 429 277
Google Scholar
[65] Kinoshita, T, Wenger T, Weiss D 2004 Science 305 1125
Google Scholar
[66] Haller E, Gustavsson M, Mark M, Danzl J, Hart R, Pupillo G, Nägerl H 2009 Science 325 1224
Google Scholar
[67] Liao Y, Rittner A, Paprotta T, Li W, Partridge G, Hulet R, Baur S, Mueller E 2010 Nature 467 567
Google Scholar
[68] Guan L, Chen S, Wang Y, Ma Z 2009 Phys. Rev. Lett. 102 160402
Google Scholar
[69] Chen S, Guan X, Yin X, Guan L, Batchelor M 2010 Phys. Rev. A 81 031608(R)
Google Scholar
[70] Chen S, Guan L, Yin X, Hao Y, Guan X 2010 Phys. Rev. A 81 031609(R)
Google Scholar
[71] Guan L, Chen S 2010 Phys. Rev. Lett. 105 175301
Google Scholar
[72] Yang B, Chen Y, Zheng Y, Sun H, Dai H, Guan X, Yuan Z, Pan J 2017 Phys. Rev. Lett. 119 165701
Google Scholar
[73] Chen Y, Jiang Y, Guan X, Zhou Q 2014 Nat. Commun. 5 5140
Google Scholar
[74] Goold J, Busch T 2008 Phys. Rev. A 77 063601
Google Scholar
[75] Rubeni D, Foerster A, Roditi I 2012 Phys. Rev. A 86 043619
Google Scholar
[76] Liu Y, Ye J, Li Y, Zhang Y 2015 Chin. Phys. B 24 086701
Google Scholar
[77] Stan C A, Zwierlein M W, Schunck C H, Raupach S M F, Ketterle W 2004 Phys. Rev. Lett. 93 143001
Google Scholar
[78] Ospelkaus C, Ospelkaus S, Humbert L, Ernst P, Sengstock K, Bongs K 2006 Phys. Rev. Lett. 97 120402
Google Scholar
[79] Wille E, Spiegelhalder F M, Kerner G, Naik D, Trenkwalder A, Hendl G, Schreck F, Grimm R, Tiecke T G, Walraven J T M, Kokkelmans S J J M F, Tiesinga E, Julienne P S 2008 Phys. Rev. Lett. 100 053201
Google Scholar
[80] Iskin M, Sa de Melo C A R 2006 Phys. Rev. Lett. 97 100404
Google Scholar
[81] Deuretzbacher F, Plassmeier K, Pfannkuche D, Werner F, Ospelkaus C, Ospelkaus S, Sengstock K, Bongs K 2008 Phys. Rev. A 77 032726
Google Scholar
[82] Massignan P, Zaccanti M, Bruun G M 2014 Rep. Prog. Phys. 77 034401
Google Scholar
[83] Chen X, Guan L M, Chen S 2011 Eur. Phys. J. D 64 459
Google Scholar
[84] Will S, Best T, Schneider U, Hackermüller L, Lühmann D S, Bloch I 2010 Nature 465 201
[85] Bardeen J 1961 Phys. Rev. Lett. 6 57
Google Scholar
[86] Simmonds R W, Lang K M, Hite D A, Nam S, Pappas D P, Martinis J M 2004 Phys. Rev. Lett. 93 077003
Google Scholar
[87] Albiez M, Gati R, Fölling J, Hunsmann S, Cristiani M, Oberthaler M K 2005 Phys. Rev. Lett. 95 010402
Google Scholar
[88] Shin Y, Saba M, Pasquini T A, Ketterle W, Pritchard D E, Leanhard A E 2004 Phys. Rev. Lett. 92 050405
Google Scholar
[89] Saba M, Pasquini T A, Sanner C, Shin Y, Ketterle W, Pritchard D E 2005 Science 307 1948
Google Scholar
[90] Milburn G J, Corney J, Wright E M, Walls D F 1997 Phys. Rev. A 55 4318
Google Scholar
[91] Smerzi A, Fantoni S, Giovanazzi S, Shenoy S R 1997 Phys. Rev. Lett. 79 4950
Google Scholar
[92] Zöllner S, Meyer H D, Schmelcher P 2008 Phys. Rev. Lett. 100 040401
Google Scholar
[93] Zöllner S, Meyer H D, Schmelcher P 2007 Phys. Rev. A 75 043608
Google Scholar
[94] Zöllner S, Meyer H D, Schmelcher P 2008 Phys. Rev. A 78 013621
Google Scholar
[95] Murphy D S, McCann J F, Goold J, Busch T 2007 Phys. Rev. A 76 053616
Google Scholar
[96] Yin X, Hao Y, Chen S, Zhang Y 2008 Phys. Rev. A 78 013604
Google Scholar
[97] Lü X, Yin X, Zhang Y 2010 Phys. Rev. A 81 043607
Google Scholar
[98] Liu Y, Zhang Y 2015 Phys. Rev. A 91 053610
Google Scholar
[99] Deuretzbacher F, Fredenhagen K, Becker D, Bongs K, Sengstock K, Pfannkuche K 2008 Phys. Rev. Lett. 100 160405
Google Scholar
[100] Deuretzbacher F, Becker D, Bjerlin J, Reimann S, Santos L 2014 Phys. Rev. A 90 013611
Google Scholar
[101] Yang L, Cui X 2016 Phys. Rev. A 93 013617
Google Scholar
[102] Yang L, Guan L, Pu H 2015 Phys. Rev. A 91 043634
Google Scholar
[103] Yang L, Pu H 2016 Phys. Rev. A 94 033614
Google Scholar
[104] Lindgren E, Rotureau J, Forssen C, Volosniev A, Zinner N 2014 New J. Phys. 16 063003
Google Scholar
[105] Volosniev A, Petrosyan D, Valiente M, Fedorov D, Jensen A, Zinner N. 2015 Phys. Rev. A 91 023620
Google Scholar
[106] Deuretzbacher F, Becker D, Santos L 2016 Phys. Rev. A 94 023606
Google Scholar
[107] Dehkharghani A, Volosniev A, Lindgren E, Rotureau J, Forssn C, Fedorov D, Jensen A, Zinner N 2015 Sci. Rep. 5 16075
Google Scholar
[108] Barfknecht R, Foerster A, Zinner N 2017 Phys. Rev. A 95 023612
Google Scholar
[109] Volosniev A, Fedorov D, Jensen A, Valiente M, Zinner N 2014 Nat. Commun. 5 5300
Google Scholar
[110] Volosniev A, Petrosyan D, Valiente M, Fedorov D, Jensen A, Zinner N 2015 Phys. Rev. A 91 023620
[111] Levinsen J, Massignan P, Bruun G, Parish, M 2015 Sci. Adv. 1 e1500197
Google Scholar
[112] Hu H, Pan L, Chen S 2016 Phys. Rev. A 93 033636
Google Scholar
[113] Yang L, Guan X, Cui X 2016 Phys. Rev. A 93 051605
Google Scholar
[114] Massignan P, Levinsen J, Parish M 2015 Phys. Rev. Lett. 115 247202
Google Scholar
[115] Hu H, Guan L, Chen S 2016 New J. Phys. 18 025009
Google Scholar
[116] Deuretzbacher F, Becker D, Bjerlin J, Reimann S, Santos L 2017 Phys. Rev. A 95 043630
Google Scholar
[117] Volosniev A 2017 Few-body Syst. 58 54
Google Scholar
[118] Pan L, Liu Y, Hu H, Zhang Y, Chen S 2017 Phys. Rev. B 96 075149
Google Scholar
[119] Liu Y, Chen S, Zhang Y 2017 Phys. Rev. A 95 043628
Google Scholar
-
图 2 三种相互作用的两原子系统的能量E与相互作用强度g的关系: (a)谐振子势阱中的两原子, 黑实线和红虚线分别表示相对运动奇宇称和偶宇称波函数所对应的能量; (b)周期边界条件下两异核原子, 黑实线和红虚线分别表示质量相等两原子和质量比为2.175情况下对应的能量; (c)双势阱中相互作用两原子的奇宇称态, 一维无限深方势阱中心为一个强度为d = 0.5的
$ \delta $ 势垒劈开, 黑实线和红虚线分别表示准动量为实数的原子本征态和准动量为复数的分子态对应的能量. 这里能量的单位分别是$ \hbar \omega, 8\hbar^2 /m_1 L^2, 2\hbar^2 /m L^2 $ , 相互作用强度g的单位分别是$ \sqrt{\hbar^3\omega/\mu}, 8\hbar^2 /m_1 L, 2\hbar^2 /m L $ Figure 2. Energy spectrum of three types of interacting two-atom system: (a)Two atoms in a harmonic oscillator potential. Black solid lines and red dashed lines are odd parity and even parity energy level respectively; (b)two heteronuclear atoms in a ring trap. Black solid lines and red dashed lines are energy levels for equal mass and mass ratio
$\alpha $ = 2.175 respectively; (c) two atoms in a$\delta $ -split hard-wall double well. Five lowest odd parity levels for barrier height d = 0.5. Black solid lines and red dashed lines are the bound states for atoms with real-valued quasimomentum and the molecule states with complex-valued quasimomentum respectively.
图 3 (a)衰减时间随磁场的变化. 黑色虚线和实线分别是WKB和考虑准粒子波动方程的修正的结果. 绿色线表示通过微扰论考虑非谐项贡献的结果; (b)相互作用强度和隧穿能量随磁场的变化.本图摘自参考文献[33]
Figure 3. (a) Decay time
$ \tau $ vs magnetic field B. The points with error bars are the experimental data[24], the dashed and solid lines are, respectively, the WKB$ (\tau_0) $ and QPWF predictions. The green light gray lines include the perturbation theory correction to the tunneling energy$ \varepsilon $ ; (b) interaction strength (red gray curve) and tunneling energy (black and green light gray curve) vs B (Reproduced with permission from Ref. [33]).
图 4 (a)两个原子在同一个阱中的占据概率随时间的演化图. 黑实线, 蓝虚线, 红点线分别表示不同的相互作用强度g = 0.5, 1.5, 20. 在t = 0时刻势垒高度突然从d = 300降到d =0.5. (b)g = 1.5和20时一个周期里不同时刻的密度分布图, 这里坐标
$ x_1 $ 和$ x_2 $ 的单位是$ L/2 $ . 本图摘自参考文献[98]Figure 4. (a) Tunneling dynamics of the occupation probability
$ P_2(t) $ of finding both atoms in the same well for g = 0.5 (black solid line), g = 1.5 (blue dashed line), and g = 20 (red dotted line). The$ \delta $ barrier is abruptly lowered from a height d = 300 to 0.5 at time t = 0. (b) the two-body density functions$ \rho (x_1, x_2, t) $ at different times t, for g = 1.5 and 20, respectively. Here the coordinates$ x_1 $ and$ x_2 $ are in units of$ L/2 $ (Reproduced with permission from Ref. [98]). -
[1] Pitaevskii L, Stringari S 2016 Bose-Einstein Condensation and Superfluidity(2nd Ed.) (New York: Oxford University Press) p.42
[2] Pethick C J, Smith H 2008 Bose-Einstein Condensation in Dilute Gases (2nd Ed.) (Cambridge: Cambridge University Press) p.159
[3] Gross E P 1961 Nuovo Cimento 20 454
Google Scholar
[4] Gross E P 1963 J. Math. Phys. 4 195
Google Scholar
[5] Pitaevskii L P 1961 J. Exp. Theor. Phys. 40 646
[6] Dalfovo F, Giorgini S, Pitaevskii L P, Stringari S 1999 Rev. Mod. Phys. 71 463
Google Scholar
[7] Baranov M A 2008 Phys. Rep. 464 71
Google Scholar
[8] Lahaye T, Menotti C, Santos L, Lewenstein M, Pfau T 2009 Rep. Prog. Phys. 72 126401
Google Scholar
[9] Serwane F, Zürn G, Lompe T, Ottenstein T B, Wenz A N, Jochim S 2011 Science 332 236
[10] Greiner M, Mandel O, Esslinger T, Hänsch T W, Bloch I 2002 Nature 415 39
Google Scholar
[11] Kohl M, Moritz H, Stöferle T, Günter K, Esslinger T 2005 Phys. Rev. Lett. 94 080403
Google Scholar
[12] Chin C, Grimm R, Julienne P, Tiesinga E 2010 Rev. Mod. Phys. 82 1225
Google Scholar
[13] Inouye S, Andrews M R, Stenger J, Miesner H J, Stamper-Kurn D M, Ketterle W 1998 Nature 392 151
Google Scholar
[14] Courteille P, Freeland R S, Heinzen D J, van Abeelen F A, Verhaar B J 1998 Phys. Rev. Lett. 81 69
Google Scholar
[15] Roberts J L, Claussen N R, Burke J P, Greene C H, Cornell E A, Wieman C E 1998 Phys. Rev. Lett. 81 5109
Google Scholar
[16] Efimov V 1970 Phys. Lett. B 33 563
Google Scholar
[17] Efimov V 1970 Sov. J. Nucl. Phys. 12 598
[18] Efimov V 1973 Nucl. Phys. A 210 157
Google Scholar
[19] Blume D, Daily K M 2010 Phys. Rev. Lett. 105 170403
Google Scholar
[20] Naidon P, Endo S 2017 Rep. Prog. Phys. 80 056001
Google Scholar
[21] Kraemer T, Mark M, Waldburger P, Danzl J G, Chin C, Engeser B, Lange A D, Pilch K, Jaakkola A, Nägerl H C, Grimm R 2006 Nature 440 315
Google Scholar
[22] 刘彦霞 2017 博士学位论文(太原: 山西大学)
Liu Y X 2017 Ph. D. Dissertation (Taiyuan: Shanxi University) (in Chinese)
[23] Blume D 2012 Rep. Prog. Phys. 75 046401
Google Scholar
[24] Zürn G, Serwane F, Serwane F, Lompe T, Wenz A N, Ries M G, Bohn J E, Jochim S 2012 Phys. Rev. Lett. 108 075303
Google Scholar
[25] Zürn G, Wenz A N, MurmannS, Bergschneider A, Lompe T, Jochim S 2013 Phys. Rev. Lett. 111 175302
Google Scholar
[26] Wenz A N, Zürn G, Murmann S, Brouzos I, Lompe T, Jochim S 2013 Science 342 457
Google Scholar
[27] Murmann S, Bergschneider A, Klinkhamer V M, Zürn G, Lompe T, Jochim S 2015 Phys. Rev. Lett. 114 080402
Google Scholar
[28] Murmann S, Deuretzbacher F, Zürn G, Bjerlin J, Reimann S M, Santos L, Lompe T, Jochim S 2015 Phys. Rev. Lett. 115 215301
Google Scholar
[29] Kaufman A M, Lester B J, Regal C A 2012 Phys. Rev. X 2 041014
[30] Kaufman A M, Lester B J, Reynolds C M, Wall M L, Foss-Feig M, Hazzard K R A, Rey A M, Regal C A 2014 Science 345 306
Google Scholar
[31] Busch T, Englert B G, Rzazewski K, Wilkens M 1998 Found. Phys. 28 549
Google Scholar
[32] Olshanii M 1998 Phys. Rev. Lett. 81 938
Google Scholar
[33] Rontani M 2012 Phys. Rev. Lett. 108 115302
Google Scholar
[34] Rontani M 2013 Phys. Rev. A 88 043633
Google Scholar
[35] Bethe H A 1931 Z. Phys. 71 205
Google Scholar
[36] Lieb E H, Liniger W 1963 Phys. Rev. 1963 130
[37] McGuire J B 1965 J. Math. Phys. 6 432
Google Scholar
[38] Gaudin M 1967 Phys. Lett. A 24 55
Google Scholar
[39] Yang C N 1967 Phys. Rev. Lett. 19 1312
Google Scholar
[40] Baxter R J 1972 Ann. Phys. 70 193
Google Scholar
[41] Lieb E H, Wu F Y 1968 Phys. Rev. Lett. 20 1445
Google Scholar
[42] Korepin V E, Essler F H L 1994 Exactly Solvable Models of Strongly Correlated Electrons (Singapore: World Scientific)
[43] Mattis D C 1993 The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension (Singapore: World Scientific)
[44] Takahashi M 1999 Thermodynamics of One-Dimensional Solvable models (Cambridge: Cambridge University Press)
[45] Baxter R J 1989 Exactly solved models in statistical mechanics (London: Academic Press)
[46] Sutherland B 2004 Beautiful Models-70 years of exactly solved quantum many-body problems (Singapore: World Scientific)
[47] Hao Y, Zhang Y, Liang J, Chen S 2006 Phys. Rev. A 73 053605
Google Scholar
[48] Hao Y, Zhang Y, Chen S 2009 Phys. Rev. A 79 043633
Google Scholar
[49] Hao Y, Zhang Y, Liang J, Chen S 2006 Phys. Rev. A 73 063617
Google Scholar
[50] Hao Y, Zhang Y, Guan X, Chen S 2009 Phys. Rev. A 79 033607
Google Scholar
[51] Hao Y, Chen S 2009 Phys. Rev. A 80 043608
Google Scholar
[52] Hao Y, Guo H, Zhang Y, Chen S 2011 Phys. Rev. A 83 053632
Google Scholar
[53] Hao Y, Zhang Y, Chen S 2008 Phys. Rev. A 78 023631
Google Scholar
[54] Guan X 2014 Int. J. Mod. Phys. B 28 1430015
Google Scholar
[55] Guan X, Batchelor M, Lee C 2013 Rev. Mod. Phys. 85 1633
Google Scholar
[56] Jiang Y, Chen Y, Guan X 2015 Chinese Phys. B 24 050311
Google Scholar
[57] Yin X, Chen S, Zhang Y 2009 Phys. Rev. A 79 053604
Google Scholar
[58] Yin X, Guan X, Batchelor M, Chen S 2011 Phys. Rev. A 83 013602
Google Scholar
[59] Yin X, Guan X, Chen S, Batchelor M 2011 Phys. Rev. A 84 011602(R)
Google Scholar
[60] Yin X, Guan X, Zhang Y, Chen S 2012 Phys. Rev. A 85 013608
Google Scholar
[61] Yin X, Guan X, Zhang Y, Su H, Zhang S 2018 Phys. Rev. A 98 023605
Google Scholar
[62] Wang H, Hao Y, Zhang Y 2012 Phys. Rev. A 85 053630
Google Scholar
[63] Wang H, Zhang Y 2013 Phys. Rev. A 88 023626
Google Scholar
[64] Paredes B, Widera A, Murg V, Mandel O, Fölling S, Cirac I, Shlyapnikov G, Hänsch T, Bloch I 2004 Nature 429 277
Google Scholar
[65] Kinoshita, T, Wenger T, Weiss D 2004 Science 305 1125
Google Scholar
[66] Haller E, Gustavsson M, Mark M, Danzl J, Hart R, Pupillo G, Nägerl H 2009 Science 325 1224
Google Scholar
[67] Liao Y, Rittner A, Paprotta T, Li W, Partridge G, Hulet R, Baur S, Mueller E 2010 Nature 467 567
Google Scholar
[68] Guan L, Chen S, Wang Y, Ma Z 2009 Phys. Rev. Lett. 102 160402
Google Scholar
[69] Chen S, Guan X, Yin X, Guan L, Batchelor M 2010 Phys. Rev. A 81 031608(R)
Google Scholar
[70] Chen S, Guan L, Yin X, Hao Y, Guan X 2010 Phys. Rev. A 81 031609(R)
Google Scholar
[71] Guan L, Chen S 2010 Phys. Rev. Lett. 105 175301
Google Scholar
[72] Yang B, Chen Y, Zheng Y, Sun H, Dai H, Guan X, Yuan Z, Pan J 2017 Phys. Rev. Lett. 119 165701
Google Scholar
[73] Chen Y, Jiang Y, Guan X, Zhou Q 2014 Nat. Commun. 5 5140
Google Scholar
[74] Goold J, Busch T 2008 Phys. Rev. A 77 063601
Google Scholar
[75] Rubeni D, Foerster A, Roditi I 2012 Phys. Rev. A 86 043619
Google Scholar
[76] Liu Y, Ye J, Li Y, Zhang Y 2015 Chin. Phys. B 24 086701
Google Scholar
[77] Stan C A, Zwierlein M W, Schunck C H, Raupach S M F, Ketterle W 2004 Phys. Rev. Lett. 93 143001
Google Scholar
[78] Ospelkaus C, Ospelkaus S, Humbert L, Ernst P, Sengstock K, Bongs K 2006 Phys. Rev. Lett. 97 120402
Google Scholar
[79] Wille E, Spiegelhalder F M, Kerner G, Naik D, Trenkwalder A, Hendl G, Schreck F, Grimm R, Tiecke T G, Walraven J T M, Kokkelmans S J J M F, Tiesinga E, Julienne P S 2008 Phys. Rev. Lett. 100 053201
Google Scholar
[80] Iskin M, Sa de Melo C A R 2006 Phys. Rev. Lett. 97 100404
Google Scholar
[81] Deuretzbacher F, Plassmeier K, Pfannkuche D, Werner F, Ospelkaus C, Ospelkaus S, Sengstock K, Bongs K 2008 Phys. Rev. A 77 032726
Google Scholar
[82] Massignan P, Zaccanti M, Bruun G M 2014 Rep. Prog. Phys. 77 034401
Google Scholar
[83] Chen X, Guan L M, Chen S 2011 Eur. Phys. J. D 64 459
Google Scholar
[84] Will S, Best T, Schneider U, Hackermüller L, Lühmann D S, Bloch I 2010 Nature 465 201
[85] Bardeen J 1961 Phys. Rev. Lett. 6 57
Google Scholar
[86] Simmonds R W, Lang K M, Hite D A, Nam S, Pappas D P, Martinis J M 2004 Phys. Rev. Lett. 93 077003
Google Scholar
[87] Albiez M, Gati R, Fölling J, Hunsmann S, Cristiani M, Oberthaler M K 2005 Phys. Rev. Lett. 95 010402
Google Scholar
[88] Shin Y, Saba M, Pasquini T A, Ketterle W, Pritchard D E, Leanhard A E 2004 Phys. Rev. Lett. 92 050405
Google Scholar
[89] Saba M, Pasquini T A, Sanner C, Shin Y, Ketterle W, Pritchard D E 2005 Science 307 1948
Google Scholar
[90] Milburn G J, Corney J, Wright E M, Walls D F 1997 Phys. Rev. A 55 4318
Google Scholar
[91] Smerzi A, Fantoni S, Giovanazzi S, Shenoy S R 1997 Phys. Rev. Lett. 79 4950
Google Scholar
[92] Zöllner S, Meyer H D, Schmelcher P 2008 Phys. Rev. Lett. 100 040401
Google Scholar
[93] Zöllner S, Meyer H D, Schmelcher P 2007 Phys. Rev. A 75 043608
Google Scholar
[94] Zöllner S, Meyer H D, Schmelcher P 2008 Phys. Rev. A 78 013621
Google Scholar
[95] Murphy D S, McCann J F, Goold J, Busch T 2007 Phys. Rev. A 76 053616
Google Scholar
[96] Yin X, Hao Y, Chen S, Zhang Y 2008 Phys. Rev. A 78 013604
Google Scholar
[97] Lü X, Yin X, Zhang Y 2010 Phys. Rev. A 81 043607
Google Scholar
[98] Liu Y, Zhang Y 2015 Phys. Rev. A 91 053610
Google Scholar
[99] Deuretzbacher F, Fredenhagen K, Becker D, Bongs K, Sengstock K, Pfannkuche K 2008 Phys. Rev. Lett. 100 160405
Google Scholar
[100] Deuretzbacher F, Becker D, Bjerlin J, Reimann S, Santos L 2014 Phys. Rev. A 90 013611
Google Scholar
[101] Yang L, Cui X 2016 Phys. Rev. A 93 013617
Google Scholar
[102] Yang L, Guan L, Pu H 2015 Phys. Rev. A 91 043634
Google Scholar
[103] Yang L, Pu H 2016 Phys. Rev. A 94 033614
Google Scholar
[104] Lindgren E, Rotureau J, Forssen C, Volosniev A, Zinner N 2014 New J. Phys. 16 063003
Google Scholar
[105] Volosniev A, Petrosyan D, Valiente M, Fedorov D, Jensen A, Zinner N. 2015 Phys. Rev. A 91 023620
Google Scholar
[106] Deuretzbacher F, Becker D, Santos L 2016 Phys. Rev. A 94 023606
Google Scholar
[107] Dehkharghani A, Volosniev A, Lindgren E, Rotureau J, Forssn C, Fedorov D, Jensen A, Zinner N 2015 Sci. Rep. 5 16075
Google Scholar
[108] Barfknecht R, Foerster A, Zinner N 2017 Phys. Rev. A 95 023612
Google Scholar
[109] Volosniev A, Fedorov D, Jensen A, Valiente M, Zinner N 2014 Nat. Commun. 5 5300
Google Scholar
[110] Volosniev A, Petrosyan D, Valiente M, Fedorov D, Jensen A, Zinner N 2015 Phys. Rev. A 91 023620
[111] Levinsen J, Massignan P, Bruun G, Parish, M 2015 Sci. Adv. 1 e1500197
Google Scholar
[112] Hu H, Pan L, Chen S 2016 Phys. Rev. A 93 033636
Google Scholar
[113] Yang L, Guan X, Cui X 2016 Phys. Rev. A 93 051605
Google Scholar
[114] Massignan P, Levinsen J, Parish M 2015 Phys. Rev. Lett. 115 247202
Google Scholar
[115] Hu H, Guan L, Chen S 2016 New J. Phys. 18 025009
Google Scholar
[116] Deuretzbacher F, Becker D, Bjerlin J, Reimann S, Santos L 2017 Phys. Rev. A 95 043630
Google Scholar
[117] Volosniev A 2017 Few-body Syst. 58 54
Google Scholar
[118] Pan L, Liu Y, Hu H, Zhang Y, Chen S 2017 Phys. Rev. B 96 075149
Google Scholar
[119] Liu Y, Chen S, Zhang Y 2017 Phys. Rev. A 95 043628
Google Scholar
DownLoad:
Catalog
Metrics
- Abstract views: 11350
- PDF Downloads: 235
- Cited By: 0
