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利用Bogoliubov理论研究了自由空间中可调自旋-轨道耦合玻色-爱因斯坦凝聚体(Bose-Einstein condensates, BECs)的激发谱. 通过高频近似得到具有两体相互作用时与时间无关的有效Floquet哈密顿量, 从而获得一种可调的自旋-轨道耦合和一种可由周期驱动拉曼耦合调控的有效两体相互作用. 基于系统有效的Floquet哈密顿量, 得到凝聚体具有相互作用时的色散关系, 发现周期驱动强度可以有效地调控色散关系的结构, 即周期驱动的拉曼耦合可以调控系统在零动量相与平面波相之间的相变. 进一步利用Bogoliubov理论得到系统的Bogoliubov-de-Gennes (BdG)方程, 分别研究了凝聚体在零动量相和平面波相中的激发谱. 发现零动量相中的激发谱均为声子激发, 且激发谱随周期驱动强度的增加表现出贝塞尔函数的行为; 平面波相中的激发谱存在声子激发和旋子激发, 当周期驱动强度增加时, 旋子模出现软化现象. 因此, 可以通过周期驱动拉曼耦合实时地调控自旋-轨道耦合BECs激发谱中的声子激发和旋子激发.
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关键词:
- 玻色-爱因斯坦凝聚体 /
- 可调自旋-轨道耦合 /
- 激发谱
In a recent experiment, the excitation spectrum of spin-orbit (SO) coupled Bose-Einstein condensates (BECs) of $^{87}{\rm{Rb}}$ atoms was studied by using Bragg spectroscopy, and the roton-maxon structure was found to exist in the excitation spectrum of magnetized phase. In addition, the roton-mode and its softening phenomenon are obtained by using various artificial SO couplings such as Rashba SO coupling and spin-orbital-angular-momentum coupling. However, the SO coupling strength in previous studies could not be controlled in real time, which limits the further study and precise regulation of the excitation spectrum of condensate. Thus, it is still an important topic to study how to regulate the SO coupling strength of the system through an external driving field, and further regulate the excitation spectrum of SO coupled BECs.In this work, the excitation spectrum of a tunable SO coupled BECs in free space is studied by using Bogoliubov theory. The time-independent effective Floquet Hamiltonian with two-body interaction is obtained in the high frequency approximation, and then a tunable SO coupling and an effective two-body interaction that can be regulated by the periodic driving of Raman coupling are obtained. Based on the effective Floquet Hamiltonian of the system, the dispersion relation of the BECs with interactions is numerically calculated. It is found that the periodic driving can effectively regulate the structure of the dispersion relation, which indicates that the periodic driving can regulate the phase transition between the zero-momentum phase and the plane wave phase. Then, the Bogoliubov-de-Gennes (BdG) equation of the system is obtained by using Bogoliubov theory. Moreover, the excitation spectrum of the BECs in the zero momentum phase and the plane wave phase are studied, respectively. Only the phonon excitation exists in the excitation spectrum of the zero momentum phase, and the excitation spectrum behaves as a Bessel function with the increase of the periodic driving strength. The phonon and roton excitations exist in the excitation spectrum of the plane wave phase, and the roton mode gradually softens with the increase of periodically driving strength. Therefore, the phonon and roton excitations in the excitation spectrum of SO coupled BECs can be regulated in real time by periodically driving Raman coupling. [1] Sinova J, Valenzuela S O, Wunderlich J, Back C H, Jungwirth T 2015 Rev. Mod. Phys. 87 1213
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图 1 (a1), (a2)不同
$ \chi $ 和$ \delta $ 时最低能带结构; (b1), (b2)不同$ g_{12} $ 和$ \delta $ 时最低能量对应的准动量与$ \chi $ 的关系. 其他参数为$ \varOmega_{0}=2.0 $ ,$ k_{0}=1.0 $ 和$ g=1.0 $ Fig. 1. (a1), (a2) The lowest band structure for different
$ \chi $ and$ \delta $ ; (b1), (b2) relationship between the quasi-momentum corresponding to the lowest energy and$ \chi $ for different$ g_{12} $ and$ \delta $ . The other parameters are$ \varOmega_{0}=2.0 $ ,$ k_{0}=1.0 $ and$ g=1.0 $ 
图 2 不同
$ \chi $ ,$ g_{12} $ ,$ \delta $ 和$ k_{0} $ 时零动量相的激发谱(较低的一支). 其他参数为$ g=1.0 $ ,$ \varOmega_{0}=2.0 $ Fig. 2. Excitation spectrum of zero-momentum phase (lower branch) for different
$ \chi $ ,$ g_{12} $ ,$ \delta $ and$ k_{0} $ . The other parameters are$ g=1.0 $ ,$ \varOmega_{0}=2.0 $ 
图 3 不同参数下周期驱动强度
$ \chi $ 对零动量相激发谱(较低的一支)的调控. 其他参数为q = 0.01,$ g=1.0 $ 和$ \varOmega_{0}=2.0 $ Fig. 3. Regulation of the excitation spectrum (lower branch) in zero-momentum phase by periodic driving strength
$ \chi $ under different parameters. The other parameters are q = 0.01,$ g=1.0 $ and$ \varOmega_{0}=2.0 $ 
图 4 取不同
$ \chi $ 和$ \delta $ 时平面波相的激发谱(较低的一支). 其他参数为$ g=1.0 $ ,$ g_{12}=1.5 $ ,$ \varOmega_{0}=2.0 $ 和$ k_{0}=2.0 $ Fig. 4. Excitation spectrum (lower branch) of the plane wave phase for different
$ \chi $ and$ \delta $ . The other parameters are$ g=1.0 $ ,$ g_{12}=1.5 $ ,$ \varOmega_{0}=2.0 $ and$ k_{0}=2.0 $ -
[1] Sinova J, Valenzuela S O, Wunderlich J, Back C H, Jungwirth T 2015 Rev. Mod. Phys. 87 1213
Google Scholar
[2] Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045
Google Scholar
[3] Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057
Google Scholar
[4] Lin Y J, Jiménez-García K, Spielman I B 2011 Nature 471 7336
Google Scholar
[5] Zhang J Y, Ji S C, Chen Z, Zhang L, Du Z D, Yan Bo, Pan G S, Zhao B, Deng Y J, Pan J W 2012 Phys. Rev. Lett. 109 115301
Google Scholar
[6] Qu C L, Hamner C, Gong M, Zhang C W, Engels P 2013 Phys. Rev. A 88 021604
Google Scholar
[7] LeBlanc L J, Beeler M C, Jiménez-García K, Perry A R, Sugawa S, Williams R A, Spielman I B 2013 New J. Phys. 15 073011
Google Scholar
[8] Ji S C, Zhang L, Xu X T, Wu Z, Deng Y J, Chen S, Pan J W 2015 Phys. Rev. Lett. 114 105301
Google Scholar
[9] Hauke P, Cucchietti F M, Tagliacozzo L, Deutsch I, Lewenstein M 2012 Rep. Prog. Phys. 75 082401
Google Scholar
[10] Galitski V, Spielman I B 2013 Nature 494 7345
Google Scholar
[11] Zheng W, Yu Z Q, Cui X L, Zhai H 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134007
Google Scholar
[12] Goldman N, Juzeliünas G, Öhberg P, Spielman I B 2014 Rep. Prog. Phys. 77 126401
Google Scholar
[13] Li Y, Pitaevskii L P, Stringari S 2012 Phys. Rev. Lett. 108 225301
Google Scholar
[14] Martone G I, Pepe F V, Facchi P, Pascazio S, Stringari S 2016 Phys. Rev. Lett. 117 125301
Google Scholar
[15] Li S, Wang H, Li F, Cui X L, Liu B 2020 Phys. Rev. A 102 033328
Google Scholar
[16] Zhang D W, Fu L B, Wang Z D, Zhu S L 2012 Phys. Rev. A 85 043609
Google Scholar
[17] Ji A C, Sun Q, Xie X C, Liu W M 2009 Phys. Rev. Lett. 102 023602
Google Scholar
[18] Qi R, Yu X L, Li Z B, Liu W M 2009 Phys. Rev. Lett. 102 185301
Google Scholar
[19] Xu Y, Zhang Y P, Wu B 2013 Phys. Rev. A 87 013614
Google Scholar
[20] Liang Z X, Zhang Z D, Liu W M 2005 Phys. Rev. Lett. 94 050402
Google Scholar
[21] Wang L X, Dai C Q, Wen L, Liu T, Jiang H F, Saito H, Zhang S G, Zhang X F 2018 Phys. Rev. A 97 063607
Google Scholar
[22] Merkl M, Jacob A, Zimmer F E, Öhberg P, Santos L 2010 Phys. Rev. Lett. 104 073603
Google Scholar
[23] Zhang Y P, Mossman M E, Busch T, Engels P, Zhang C W 2016 Front. Phys. 11 118103
Google Scholar
[24] Zhu Q Z, Zhang C W, Wu B 2012 Eur. Phys. Lett. 100 50003
Google Scholar
[25] Zhu Q Z, Wu B 2015 Chin. Phys. B 24 050507
Google Scholar
[26] Landau L 1941 Phys. Rev. 60 356
[27] Khamehchi M A, Zhang Y P, Hamner C, Busch T, Engels P 2014 Phys. Rev. A 90 063624
Google Scholar
[28] Steinhauer J, Ozeri R, Katz N, Davidson N 2002 Phys. Rev. Lett. 88 120407
Google Scholar
[29] Santos L, Shlyapnikov G V, Lewenstein M 2003 Phys. Rev. Lett. 90 250403
Google Scholar
[30] O’Dell D H J, Giovanazzi S, Kurizki G 2003 Phys. Rev. Lett. 90 110402
Google Scholar
[31] Chen K J, Wu F, Hu J S, He L Y 2020 Phys. Rev. A 102 013316
Google Scholar
[32] Ravisankar R, Fabrelli H, Gammal A, Muruganandam P, Mishra P K 2021 Phys. Rev. A 104 053315
Google Scholar
[33] Chen Y Y, Lyu H, Xu Y, Zhang Y P 2022 New J. Phys. 24 073041
Google Scholar
[34] Zhang Y P, Chen G, Zhang C W 2013 Sci. Rep. 3 1
Google Scholar
[35] Jiménez-García K, LeBlanc L J, Williams R A, Beeler M C, Qu C, Gong M, Zhang C, Spielman I B 2015 Phys. Rev. Lett. 114 125301
Google Scholar
[36] Dalfovo F, Giorgini S, Pitaevskii L P, Stringari S 1999 Rev. Mod. Phys. 71 463
Google Scholar
[37] Lin Y J, Compton R L, Jiménez-García K, Phillips W D, Porto J V, Spielman I B 2011 Nat. Phys. 7 531534
[38] Achilleos V, Frantzeskakis D J, Kevrekidis P G, Pelinovsky D E 2013 Phys. Rev. Lett. 110 264101
Google Scholar
[39] Bukov M, D’Alessio L, Polkovnikov A 2015 Adv. Phys. 64 139226
Google Scholar
[40] Salerno M, Abdullaev F Kh, Gammal A, Tomio L 2016 Phys. Rev. A 94 043602
Google Scholar
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