超晶格输运与光格子中BEC的量子相变
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摘要
本文主要对半导体超晶格中的Bloch电子动力学以及光格子中玻色-爱因斯坦凝聚的相变问题进行研究。首先综述了过去三十年低维量子器件与半导体超晶格的发展与相关研究，介绍了Bloch振荡、Wannier-Stark台阶、Zener隧穿等关键理论以及相关实验方面的进展，并引入简化模型：紧束缚模型与单带模型。接着讨论了在双模交流场驱动下Bloch电子的动力学特性，发现了使Bloch电子处于局域态的条件，并把这个结论推广到了任意多个模的交流电场驱动的情况。接着引入更为复杂的格点能交替变化的超晶格模型，并在此基础上考察了Hamiltonian中交替变化的对角项对准能谱与动力学的影响。应用Floquet定理，从解析与数值计算两个角度计算了该模型的准能谱与时间演化，指出动力学局域化与准能谱的塌缩的对应关系。接下来我们讨论了光格子中玻色-爱因斯坦凝聚的相变问题。首先简单介绍了在实验上实现光格子囚禁超冷原子的一些方法，以及相关的基本理论与相应的研究工作。从多体理论出发，得到玻色-哈伯德模型(BHM)，并利用波格留波夫变换计算了光格子中超冷原子的激发谱。在长波极限下，色散关系是线性的，这与无外场时的色散关系相同。从激发谱分析可得到superfluid-Mott绝缘相变条件，给出了光格子中超流速度，并指出在实验上可以通过控制光格子参数来改变超流速度。最后利用格林函数方法讨论了光格子中超冷原子的能带结构，根据Mott相存在能隙的判据我们在平均场近似下重新得到superfluid-Mott相变条件，该结论与相关文献一致。
This paper is mainly devoted to study of the quantum transport in semiconductor superlattice and phase transition of BEC in optical lattice. Firstly, I provide a brief review of the previous achievements and investigations on the low-dimensional quantum devices and semiconductor superlattice, in which some principal theories such as Bloch Oscillations, Wannier-Stark ladder, Zener tunneling and related progress in experiments are introduced. Secondly, based on the discussion of the dynamical characteristics of Bloch electron driven by two-mode AC electrical fields we find a localization condition of the Bloch electron on a localized state and then extend this result to the multi-field case. Thirdly we introduce the superlattice model with alternating site energies which is more complicated than the simple tight-binding model, and moreover we explore the influence of the alternating diagonal term in Hamiltonian on the quasi-energy bands and dynamics. According to Floquet theorem, the relations between the dyna
mical localization and quasi-energy bands are pointed out with both numerical and analytical calculations on quasi-energy bands and their corresponding time evolution. Then we investigate the phase transition of Bose-Einstein condensation in optical lattice. After a brief introduction of some experimental methods required in condensing ultracold bosons, and of the relevant fundamental theories, starting from the Bose-Hubbard model we study the excitation
spectrum with Bogliubov transformation. The dispersion relation within the long-wave limit is linear, which is the same as that for the Bose gas of superfluid phase in the absence of the optical lattice. Based on the analysis of the excitation spectrum, we obtain the superfluid-Mott-insulator phase transition condition, and superfluid velocity of ultracold dilute gas of bosonic atoms in an optical lattice, and point out that the velocity of superfluid can be experimentally adjusted by controlling the parameters of optical lattice. Lastly, we discuss the energy-band structure of ultracold atoms in optical lattice by means of Green function method and in addition, procure the superfluid-Mott phase transition condition in mean-field approximation which is in agreement with the result in the literature.
This paper is mainly devoted to study of the quantum transport in semiconductor superlattice and phase transition of BEC in optical lattice. Firstly, I provide a brief review of the previous achievements and investigations on the low-dimensional quantum devices and semiconductor superlattice, in which some principal theories such as Bloch Oscillations, Wannier-Stark ladder, Zener tunneling and related progress in experiments are introduced. Secondly, based on the discussion of the dynamical characteristics of Bloch electron driven by two-mode AC electrical fields we find a localization condition of the Bloch electron on a localized state and then extend this result to the multi-field case. Thirdly we introduce the superlattice model with alternating site energies which is more complicated than the simple tight-binding model, and moreover we explore the influence of the alternating diagonal term in Hamiltonian on the quasi-energy bands and dynamics. According to Floquet theorem, the relations between the dyna
mical localization and quasi-energy bands are pointed out with both numerical and analytical calculations on quasi-energy bands and their corresponding time evolution. Then we investigate the phase transition of Bose-Einstein condensation in optical lattice. After a brief introduction of some experimental methods required in condensing ultracold bosons, and of the relevant fundamental theories, starting from the Bose-Hubbard model we study the excitation
spectrum with Bogliubov transformation. The dispersion relation within the long-wave limit is linear, which is the same as that for the Bose gas of superfluid phase in the absence of the optical lattice. Based on the analysis of the excitation spectrum, we obtain the superfluid-Mott-insulator phase transition condition, and superfluid velocity of ultracold dilute gas of bosonic atoms in an optical lattice, and point out that the velocity of superfluid can be experimentally adjusted by controlling the parameters of optical lattice. Lastly, we discuss the energy-band structure of ultracold atoms in optical lattice by means of Green function method and in addition, procure the superfluid-Mott phase transition condition in mean-field approximation which is in agreement with the result in the literature.
引文
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[14] X.-G. Zhao, Phys. Lett. A 155 (1991) 209.
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[22] Mendez E E, Agullo-Rueda F and Hong J M Phys. Rev. Lett. 60 (1988) 2426; Voisin P, Bleuse J, Bouche C, Gaillard S, Alibert C and Regreny A Phys. Rev. Lett. 61 (1988) 1639.
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[26] P. S. S. Guimaraes, B. J. Keay, J. P. Kaminski, S. J. Allen, P. F. Hopkins, A. C. Gossard, L.T. Florez, and J. P. Harbison, Phys. Rev. Lett. 70, 3792(1993).
[27] C. Waschke, H.G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Kohler, ibid. 70(1993)3319.
[28] K. H. Schmidt, N. Linder, G.. H. Dohler, H. T. Grab.n, K. Ploog, and H. Schneider, ibid. 72 (1994) 2769.
[29] A. Sibille, J. F. Palmier, C. Minor, and F. Mollet, Appl. Phys. Lett. 54 (1989) 165.
[30] A. Sibille, J. F. Palmier, F. Mollet, H. Wang, and J. C. Esnault, Phys. Rev. B 39 (1989) 6272.
[31] A. Sibille, J. F. Palmier, H. Wang, and F. Mollet, Phys. Rev. Lett. 64 (1990) 52.
[32] B. J. Keay, Zeuner, S. J. Allen, K. D. Maranowskim, A. C. Gossard, U. Hattacharya, and M.J.W. Rodwell, Phys. Rev. Lett. 75 (1995) 4102.
[33] K. Unterrainer, B. J. Keay, M. C. Wanke, S. J. Allen, D. Leonard, G. Medeiros-Ribeiro, U. Bhattacharya, and M.J.W. Rodwell, Phys. Rev. Lett. 76 (1996) 2973.
[34] B. S. Monozon, J. L. Dunn, and C. A. Bates, Phys. Rev. B 50 (1994) 17907.
[35] Q. Niu, X.-G. Zhao, G. A. Georgakis, and M.G. Raizen, Phys. Rev. Lett. 76 (1996) 4504.
[36] S. R.Wilkinson, C. F. Bharucha, K.W. Madison, Qian Niu, and M.G. Raizen phys. Rev. Lett. 76 (1996) 4512.
[37] M. C. Fischer, K. W. Madison, Qian Niu, and M.G. Raizen, Phys. Rev. A 58 (1998) R2648.
[38] K. W. Madison, M. C. Fischer, R. B. Diener, Qian Niu, and M. G. Raizen, Phys. Rev. Lett. 81 (1998) 5093.
[39] K. W. Madison, M. C. Fischer, and M. G. Raizen, Phys. Rev. A 60 (1999) R1767.
[40] P. A. Lebwohl and R. Tsu, J. Appl. Phys. 41 (1970) 2664; D. Miller and B.
Laikhtman, Phys. Rev. B 50 (1994)18426.
[41] K. N. Alekseev, G.P. Berman, D. K. Campbell, E. H. Cannon, and Me. Cargo, Phys. Rev. B 54 (1996) 10625.
[42] R.Tsu and G.H. Dohler, Phys. Rev. B 12 (1975) 680; R. Tsu and L. Esaki, Phys. Rev. B 43 (1991) 5204; Q. Niu, Phys. Rev. B 40 (1989) 3625.
[43] M. M. Dignam and J. E. Sipe, Phys. Rev. Lett. 64 (1990)1797; N. H. Shon and H. N. Nazareno, J. Phys. Condens. Matter 4 (1992) L611.
[44] Holthaus Phys. Rev. Lett. 69 (1992) 351; Z. Phys. B 89 (1992) 251.
[45] X. G. Zhao, J. Phys. Condens. Matter 6 (1994)2751.
[46] X. G. Zhao, R Jahnke and Q. Niu Phys. Lett. A202 (1995) 297.
[47] T. Meler, G. von Plessen, P. Thomas. and S. W. Koch, Phys. Rev. Lett.73 (1994) 902; Phvs. Rev. B 51 (1995) 14490; Phvs. Rev. Lett. 75 (1995) 2558.
[2] W. Kohn, Phys. Rev. 115 (1959) 809.
[3] G. Nenciu, Com. Math. Phys. 91 (1983) 81.
[4] H. Fukuyama, B.A. Bari, and H.C. Fogedby, Phys. Rev. B 8 (1973) 5579.
[5] J. Rotvig, A.-P. Jauho, and H. smith, Phys. Rev. Lett. 74 (1995) 1831; Phys. Rev. B 54 (1996)17691.
[6] H. Schneier, H. T. Grahn, K. Von Klitzing, and K. Ploog, Phys. Rev. Lett. 65 (1990) 2720; M. Nakayama, I. Tanaka, N. Nishimura, K. Kawashima, and K. Fujiwara, Phys. Rev. B 44 (1991) 5935.
[7] X.G. Zhao, J. Phys. Condens. Matter 4 (1992) L383; D. W. Hone and X.-G. Zhao, Phys. Rev. B 53 (1996) 4834; M Holthaus and D. W. Hone, Phys. Rev. B 49 (1994) 16605.
[8] Y. Gefen, E. Ben-Jacob, and A.O. Caldeira, Phys. Rev. B 36. (1987) 2770.
[9] X.-G Zhao and Q. Niu, Phys. Lett. A222 (1996) 435; Duan Suqing, W. Zhang, X.-G Zhao, Physica E 9 (2001) 659; Phys. Rev. B 62 (2000) 9943.
[10] H. Drexler, J. S. Scott, S.J.Allen, K.L. Campman, and A.C. Gossard, Appl. Phys. Lett. 67 (1995) 2816.
[12] F. Bloch, Z. Phys. 52 (1928) 555.
[13] G. H. Wannier Phys. Rev. 117 (1960) 432.
[14] X.-G. Zhao, Phys. Lett. A 155 (1991) 209.
[15] F. Bentosela, R. Carmona, P. duclos, B. Simon, B. souillard, and R. Weder, Com. Math. Phys. 88 (1983) 387.
[16] W. V. Houston, Phys. Rev. 57 (1940) 184.
[17] L. D. Landau, Phys. Z. Sov. 1 (1932) 46.
[18] G. H. Wannier Rev. Mod. Phys. 34 (1962) 645; Phys. Rev. 151 (1969) 1364.
[19] A Nencin and G Nencin. Phys. Lett. A78 (1980) 101
[20] J. B Krieger and GJ Iafrate Phvs. Rev B 33 (1986) 5494.
[21] L. Esaki and R. Tsu, IBM J. Res. Dev. 14 (1970) 61.
[22] Mendez E E, Agullo-Rueda F and Hong J M Phys. Rev. Lett. 60 (1988) 2426; Voisin P, Bleuse J, Bouche C, Gaillard S, Alibert C and Regreny A Phys. Rev. Lett. 61 (1988) 1639.
[23] E. E. Mendez. and G.Bastard, Phys. Today 46 (1993) 34.
[24] P. Voisin, J. Bleuse, C. Bouche, S. Gaillard, C. Alibert, A. Regreny, Phys. Rev. Lett. 61 (1988) 1639.
[25] J. Feldmann, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunninghan, T. Meier, G. Von Plessen, A. Schulze, P. Thomas and S. Schmitt-Rink, Phys. Rev. B 46 (1992) 7252.
[26] P. S. S. Guimaraes, B. J. Keay, J. P. Kaminski, S. J. Allen, P. F. Hopkins, A. C. Gossard, L.T. Florez, and J. P. Harbison, Phys. Rev. Lett. 70, 3792(1993).
[27] C. Waschke, H.G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Kohler, ibid. 70(1993)3319.
[28] K. H. Schmidt, N. Linder, G.. H. Dohler, H. T. Grab.n, K. Ploog, and H. Schneider, ibid. 72 (1994) 2769.
[29] A. Sibille, J. F. Palmier, C. Minor, and F. Mollet, Appl. Phys. Lett. 54 (1989) 165.
[30] A. Sibille, J. F. Palmier, F. Mollet, H. Wang, and J. C. Esnault, Phys. Rev. B 39 (1989) 6272.
[31] A. Sibille, J. F. Palmier, H. Wang, and F. Mollet, Phys. Rev. Lett. 64 (1990) 52.
[32] B. J. Keay, Zeuner, S. J. Allen, K. D. Maranowskim, A. C. Gossard, U. Hattacharya, and M.J.W. Rodwell, Phys. Rev. Lett. 75 (1995) 4102.
[33] K. Unterrainer, B. J. Keay, M. C. Wanke, S. J. Allen, D. Leonard, G. Medeiros-Ribeiro, U. Bhattacharya, and M.J.W. Rodwell, Phys. Rev. Lett. 76 (1996) 2973.
[34] B. S. Monozon, J. L. Dunn, and C. A. Bates, Phys. Rev. B 50 (1994) 17907.
[35] Q. Niu, X.-G. Zhao, G. A. Georgakis, and M.G. Raizen, Phys. Rev. Lett. 76 (1996) 4504.
[36] S. R.Wilkinson, C. F. Bharucha, K.W. Madison, Qian Niu, and M.G. Raizen phys. Rev. Lett. 76 (1996) 4512.
[37] M. C. Fischer, K. W. Madison, Qian Niu, and M.G. Raizen, Phys. Rev. A 58 (1998) R2648.
[38] K. W. Madison, M. C. Fischer, R. B. Diener, Qian Niu, and M. G. Raizen, Phys. Rev. Lett. 81 (1998) 5093.
[39] K. W. Madison, M. C. Fischer, and M. G. Raizen, Phys. Rev. A 60 (1999) R1767.
[40] P. A. Lebwohl and R. Tsu, J. Appl. Phys. 41 (1970) 2664; D. Miller and B.
Laikhtman, Phys. Rev. B 50 (1994)18426.
[41] K. N. Alekseev, G.P. Berman, D. K. Campbell, E. H. Cannon, and Me. Cargo, Phys. Rev. B 54 (1996) 10625.
[42] R.Tsu and G.H. Dohler, Phys. Rev. B 12 (1975) 680; R. Tsu and L. Esaki, Phys. Rev. B 43 (1991) 5204; Q. Niu, Phys. Rev. B 40 (1989) 3625.
[43] M. M. Dignam and J. E. Sipe, Phys. Rev. Lett. 64 (1990)1797; N. H. Shon and H. N. Nazareno, J. Phys. Condens. Matter 4 (1992) L611.
[44] Holthaus Phys. Rev. Lett. 69 (1992) 351; Z. Phys. B 89 (1992) 251.
[45] X. G. Zhao, J. Phys. Condens. Matter 6 (1994)2751.
[46] X. G. Zhao, R Jahnke and Q. Niu Phys. Lett. A202 (1995) 297.
[47] T. Meler, G. von Plessen, P. Thomas. and S. W. Koch, Phys. Rev. Lett.73 (1994) 902; Phvs. Rev. B 51 (1995) 14490; Phvs. Rev. Lett. 75 (1995) 2558.