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应用哈特里-福克-博戈留波夫平均场理论近似和基于托马斯-费米近似的解析方法, 研究碟形玻色-爱因斯坦凝聚体中(0, 0, 2)剪刀模的朗道阻尼和频移, 计算阻尼系数和频移大小以及它们的温度依赖. 计算中, 在集体激发本征频移微扰关系中考虑元激发弛豫及其弛豫之间的正交关系以获得阻尼和频移的计算公式, 把凝聚体基态波函数取为高斯分布函数的一级近似以消除托马斯-费米近似中三模耦合矩阵元的发散. 采用与相关实验研究相同的粒子数、囚禁频率和各向异性参量, 理论计算结果与相关实验测量结果相符合. 由于理论的复杂性和计算的困难性, 在大多数基于平均场理论的单分量和两分量玻色-爱因斯坦凝聚集体激发阻尼和频移的研究中采用半经典近似, 把准粒子激发能谱看成是连续的来积分计算各个准粒子跃迁对阻尼和频移的贡献, 而本文和本文前期工作按分立的准粒子激发频谱计算阻尼或频移, 并在研究过程中提出了考虑元激发弛豫及弛豫之间正交关系的改进方法, 希望这种方法对今后的工作有一定参考价值.
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关键词:
- 玻色-爱因斯坦凝聚 /
- 朗道阻尼和频移 /
- 哈特里-福克-博戈留波夫近似 /
- 托马斯-费米近似
By using the Hartree-Fock-Bogoliubov approximation of mean-field theory and the analytic method based on Thomas-Feimi approximation, the Landau damping and frequency-shift of (0, 0, 2) scissors mode in a disc-shaped Bose-Einstein condensate are investigated and the damping rate and frequency-shift magnitude as well as their temperature dependence are calculated. In the calculation, the practical relaxations of the elementary excitations and the orthometric relation among the relaxations are considered in the relation for the perturbed eigenfrequency of mean-field theory to obtain the calculation formula of damping and frequency-shift, and the first-order approximation of Gaussian distribution function is employed for the ground-state wavefunction to eliminate the divergence of the three-mode coupling matrix elements in Thomas-Fermi approximation. Taking the same parameters of particle number, trapping frequency and anisotropy as those in relevent experiment research, our theoretical calculation results accord with the relevent experimental measurement results. Because of the complexity of the theory and the difficulty of calculation, most of mean-field theory researches on damping and frequency shift of collective excitation in one and two component Bose-Einstein condensates adopt semi-classical approximation, the quasi-particle excitation spectrum is regarded as continuously integrating each quasi-particle transition contribution to damping and frequency shift. In this paper, the damping and frequency shift are calculated according to the discrete quasi-particle excitation spectrum, and in the course of the study the improving of method of considering the practical relaxations of the elementary excitations and the orthometric relation among the relaxations is put forward. It is hoped that the method will have some reference value in the future work.-
Keywords:
- Bose-Einstein condensate /
- Landau damping and frequency-shift /
- Hartree-Fock-Bogoliubov approximation /
- Thomas-Fermi approximation
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图 1 以跃迁频率
$\gamma_{ij}$ 为变量的剪刀模阻尼强度$\omega_{ij}$ 函数线状图Fig. 1. Histogram of damping strength
$\gamma_{ij}$ as a function of the transition$\omega_{ij}$ for the scissors mode in the condensate.
图 2 以跃迁频率
$ \gamma_{ij} $ 为变量的剪刀模阻尼强度$ \omega_{ij} $ 函数线状图 ($ \omega_{ij}/\omega_{\rm ho} $ 取值范围为0—1.6)Fig. 2. Histogram of damping strength
$ \gamma_{ij} $ as a function of the transition$ \omega_{ij} $ for the scissors mode in the condensate, ($ \omega_{ij}/\omega_{\rm ho} $ = 1–1.6).
图 3 朗道阻尼系数
$ \gamma_{0} $ 随$ \gamma $ 变化函数图Fig. 3. The
$ \gamma_{0} $ as a function of$ \gamma $ for the Landau damping rate.
图 4 凝聚体集体模的频移(a)和朗道阻尼系数(b)随温度T变化
Fig. 4. The frequency-shift (a) and the Landau damping rate
$ \gamma_{0} $ (b) of the collective mode in the condensate as a function of the temperature T. -
[1] Pethick C J, Smith H 2008 Bose-Einstein Condensation in Dilute Gases (2nd Ed.) (Cambridge: Cambridge University Press) p23
[2] Dalfovo F, Minniti C, Pitaevskii L P 1997 Phys. Rev. A 56 4855
Google Scholar
[3] Morgan S A, Choi S, Burnett K, Edwards M 1998 Phys. Rev. A 57 3818
Google Scholar
[4] Hechenblaikner G, Maragò O M, Hodby E, Arlt J, Hopkins S, Foot C J 2000 Phys. Rev. Lett. 85 692
Google Scholar
[5] Hodby E, Maragò O M, Hechenblaikner G, Foot C J 2001 Phys. Rev. Lett. 86 2196
Google Scholar
[6] Edwards M, Dodd R J, Chark C W, Ruprecht P A, Burnett K 1996 Phys. Rev. A 53 R1950
Google Scholar
[7] Maragò O M, Hopkins S A, Arlt J, Hodby E, Hechenblaikner G, Foot C J 2000 Phys. Rev. Lett. 84 2056
Google Scholar
[8] Khawaja U A, Stoof H T C 2001 Phys. Rev. A 65 013605
Google Scholar
[9] Hechenblaikner G, Morgan S A, Hodby E, Maragò O M, Foot C J 2002 Phys. Rev. A 65 033612
Google Scholar
[10] Bijlsma M J, Stoof H T C 1999 Phys. Rev. A 60 3973
Google Scholar
[11] Öhberg P, Stenholm S 1998 Phys. Rev. A 57 1272
Google Scholar
[12] Stringari S 1996 Phys. Rev. Lett. 77 2360
Google Scholar
[13] Fetter A L 1996 Phys. Rev. A 53 4245
Google Scholar
[14] Shchedrin G, Jaschke D, Carr L D 2018 Sci. Rep. 8 11523
Google Scholar
[15] Ota M, Larcher F, Dalfovo F, Pitaevskii L, Proukakis N P, Stringari S 2018 Phys. Rev. Lett. 121 145302
Google Scholar
[16] Mendonca J T, Tercas H, Gammal A 2018 Phys. Rev. A 97 063610
Google Scholar
[17] Cappellaro A, Toigo F, Salasnich L 2018 Phys. Rev. A 98 043605
Google Scholar
[18] 席忠红, 杨雪滢, 唐娜, 宋琳, 李晓霖, 石玉仁 2018 67 230501
Google Scholar
Xi Z H, Yang X Y, Tang N, Song L, Li X L, Shi Y R 2018 Acta Phys. Sin. 67 230501
Google Scholar
[19] Zhu K Q, Yu Z F, Gao J M, Zhang A X, Xu H P, Xue J K 2019 Chin. Phys. B 28 010307
Google Scholar
[20] 李吉, 刘伍明 2018 67 110302
Google Scholar
Li J, Liu W M 2018 Acta Phys. Sin. 67 110302
Google Scholar
[21] Zhou W Y, Wu Y J, Kou S P 2018 Chin. Phys. B 27 050302
Google Scholar
[22] Ma Y L, Chui S T 2002 Phys. Rev. A 65 053610
Google Scholar
[23] Hu B, Huang G, Ma Y L 2004 Phys. Rev. A 69 063608
Google Scholar
[24] Maragò O, Hechenblaikner G, Hodby E, Foot C 2001 Phys. Rev. Lett. 86 3938
Google Scholar
[25] Stamper-Kurn D M, Miesner H J, Inouye S, Andrews M R, Ketterle W 1998 Phys. Rev. Lett. 81 500
Google Scholar
[26] Chevy F, Bretin V, Rosenbusch P, Madison K W, Dalibard J 2002 Phys. Rev. Lett. 88 250402
Google Scholar
[27] Jin D S, Matthews M R, Ensher J R, Wieman C E, Cornell E A 1997 Phys. Rev. Lett. 78 764
Google Scholar
[28] Zaremba E, Griffin A, Nikuni T 1998 Phys. Rev. A 57 4695
Google Scholar
[29] Zaremba E, Nikuni T, Griffin A 1999 J. Low. Temp. Phys. 116 277
Google Scholar
[30] Jackson B, Zaremba E 2002 Phys. Rev. Lett. 88 180402
Google Scholar
[31] Jackson B, Zaremba E 2002 Phys. Rev. Lett. 89 150402
Google Scholar
[32] Guilleumas M, Pitaevskii L P 1999 Phys. Rev. A 61 013602
Google Scholar
[33] Das K, Bergeman T 2001 Phys. Rev. A 64 013613
Google Scholar
[34] Pitaevskii L P, Stringari S 1997 Phys. Lett. A 235 398
Google Scholar
[35] Fedichev P O, Shlyapnikov G V, Walraven J T M 1998 Phys. Rev. Lett. 80 2269
Google Scholar
[36] Reidl J, Csordás A, Graham R, Szépfalusy P 2000 Phys. Rev. A 61 043606
Google Scholar
[37] Mizushima T, Ichioka M, Machida K 2003 Phys. Rev. Lett. 90 180401
Google Scholar
[38] Morgan S A, Rusch M, Hutchinson D A W, Burnett K 2003 Phys. Rev. Lett. 91 250403
Google Scholar
[39] Giorgini S 1998 Phys. Rev. A 57 2949
Google Scholar
[40] Giorgini S 2000 Phys. Rev. A 61 063615
Google Scholar
[41] Ma X, Ma Y L, Huang G 2007 Phys. Rev. A 75 013628
Google Scholar
[42] 柴兆亮, 周昱, 马晓栋 2013 62 130307
Google Scholar
Chai Z L, Zhou Y, Ma X D 2013 Acta Phys. Sin. 62 130307
Google Scholar
[43] Rahmut A, Peng S Q, Ma X D 2014 Chin. Phys. B 23 090311
Google Scholar
[44] 彭胜强, 阿孜古丽·马合木提, 马晓栋 2015 原子与分子 32 1018
Google Scholar
Peng S Q, Rahmut A, Ma X D 2015 J. At. Mol. Phys. 32 1018
Google Scholar
[45] Bhattacherjee A B 2014 Mode. Phys. Lett. B 28 1450029
Google Scholar
[46] Natu S S, Wilson R M 2013 Phys. Rev. A 88 063638
Google Scholar
[47] Moniri S M, Yavari H, Darsheshdar E 2016 Eur. Phys. J. Plus. 131 363
Google Scholar
[48] Moniri S M, Yavari H, Darsheshdar E 2016 Chin. Phys. B 25 126701
Google Scholar
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