Search

Article

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

One-dimensional spin-orbit coupling Bose gases with harmonic trapping

Li Zhi-Qiang Wang Yue-Ming

One-dimensional spin-orbit coupling Bose gases with harmonic trapping

Li Zhi-Qiang, Wang Yue-Ming
PDF
HTML
Get Citation
  • Rabi model is a popular model in quantum optics and describes a two-level system coupling to a quantum resonator. The fruitful physics appears when the coupling strength is comparable to the frequency of the resonator. We investigate the Bose gases of Raman induced spin-orbit coupling with an external harmonic trapping. Using the displacement Fock state in quantum optics we seek for an approximate ground state. We find the superposition state of left and right displaced oscillator state with odd parity has lower energy than the displaced state itself. Besides, we study the time evolution of both the momentum and the position of the system at single particle level to demonstrate the Zitterbewegung oscillating characteristics, which present an intuitive physical picture and are in qualitative agreement with the relevant experimental results. The results are useful to study the Rabi model in deep-strong coupling regime, the model that is difficult to realize in today’s experiment based on the high controllability property of laser, and these results are also instructive for the cold atom physics field.
      Corresponding author: Wang Yue-Ming, wang_ym@sxu.edu.cn
    [1]

    Xiao D, Chang M C, Niu Q 2010 Rev. Mod. Phys. 82 1959

    [2]

    Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045

    [3]

    Zutic I, Fabian J, Sarma S D 2004 Rev. Mod. Phys. 76 323

    [4]

    Lin Y J, Garcıa K J, Spielman I B 2011 Nature 471 8386

    [5]

    Wang P, Yu Z Q, Fu Z, Miao J, Huang L, Chai S, Zhai H, Zhang J 2012 Phys. Rev. Lett. 109 095301

    [6]

    Cheuk L W, Sommer A T, Hadzibabic Z, Yefsah T, Bakr W S, Zwierlein M W 2012 Phys. Rev. Lett. 109 095302

    [7]

    Zhang J Y, Ji S C, Chen Z, Zhang L, Du Z D, Yan B, Pan G S, Zhao B, Deng Y J, Zhai H, Chen S, Pan J W 2012 Phys. Rev. Lett. 109 115301

    [8]

    Galitski V, Spielman I B 2013 Nature 494 4954

    [9]

    Lin Y J, Compton R L, Garcia K J, Porto J V, Spielman I B 2009 Nature 462 628

    [10]

    Lin Y J, Compton R L, Perry A R, Phillips W D, Porto J V, Spielman I B 2009 Phys. Rev. Lett. 102 130401

    [11]

    Qu C, Hamner C, Gong M, Zhang C, Engels P 2013 Phys. Rev. A 88 021604

    [12]

    LeBlanc L J, Beeler M C, Jimenez-Garcia K, Perry A R, Sugawa S, Williams R A, Spielman I B 2013 New J. Phys. 15 073011

    [13]

    Wu C J, Mondragon-Shem I, Zhou X F 2011 Chin. Phys. Lett. 28 097102

    [14]

    Hu H, Ramachandhran B, Pu H, Liu X J 2012 Phys. Rev. Lett. 108 010402

    [15]

    Sinha S, Nath R, Santos L 2011 Phys. Rev. Lett. 107 270401

    [16]

    Ghosh S K, Vyasanakere J P, Shenoy V B, 2011 Phys. Rev. A 84 053629

    [17]

    Larson J, Anderson B M, Altland A 2013 Phys. Rev. A 87 013624

    [18]

    Rabi I I 1936 Phys. Rev. 49 324

    [19]

    Rabi I I 1937 Phys. Rev. 51 652

    [20]

    Braak D 2011 Phys. Rev. Lett. 107 100401

    [21]

    Chen Q H, Wang C, He S, Liu T, Wang K L 2012 Phys. Rev. A 86 023822

    [22]

    Solano E 2011 Physics 4 68

    [23]

    Gardas, Dajka J 2013 arXiv: 1301.5660[quant-ph]

    [24]

    Wolf F A, Kollar M, Braak D 2012 Phys. Rev. A 85 053817

    [25]

    Walther H, Varcoe B, Englert B, Becker T 2006 Rep. Prog. Phys. 69 1325

    [26]

    Raimond J M, Brune M, Haroche S 2001 Rev. Mod. Phys. 73 565

    [27]

    Holstein T 1959 Ann. Phys. 8 325

    [28]

    Mabuchi H, Doherty A C 2002 Science 298 1372

    [29]

    Niemczyk T, Deppe F, Huebl H, Menzel E P, Hocke F, Schwarz M J, Garcia-Ripoll J J, Zueco D, Hümmer T, Solano E, Marx A, Gross R 2010 Nature Phys. 6 772

    [30]

    Forn-Diaz P, Lisenfeld J, Marcos D, Garcia-Ripoll J J, Solano E, Harmans C J P M, Mooij J E 2010 Phys. Rev. Lett. 105 237001

    [31]

    Fu Z, Wang P, Chai S, Huang L, Zhang J 2011 Phys. Rev. A 84 043609

    [32]

    Cahill K K, Glauber R J 1969 Phys. Rev. 177 1857

    [33]

    Graham R, Hohnerbach M 1984 Phys. B: Condens. Matter 57 233

  • 图 1  简并量子态$\left| {{\psi _{N{\rm{,L}}\left( {\rm{R}} \right)}}} \right\rangle $能量${E_{N,{\rm{L/R}}}}$与左右平移奇宇称叠加态$\left| {{\psi _{ - ,N}}} \right\rangle $能量${E_{ - ,N}}$随SO耦合强度$\lambda $的变化 可见$N = 0$叠加态$\left| {{\psi _{ - ,0}}} \right\rangle $能量最低, 更接近基态; 而对于激发态$N \ne 0$, 二者能量随参数变化出现交叉; 相关参数取值为$\varOmega \; = \;{\rm{1}}{\rm{.4}}\omega $, 与文献[19]精确解的结果基本一致

    Figure 1.  The energies of degenerate quantum states $\left| {{\psi _{N{\rm{,L}}\left( {\rm{R}} \right)}}} \right\rangle $ and the superposition state of odd parity $\left| {{\psi _{ - ,N}}} \right\rangle $ of left(right)-displaced number states varies as the spin-orbit coupling strength $\lambda $. It is seen that for $N = 0$, the superposition state has the lowest energy which is the best approximation for the ground state in our interest. And for the cases of $N \ne 0$, the energies of the two quantum states have pitchforks.The relevant parameters is Ω=1.4 and the results are in agreement with those in Ref.[19].

    图 2  原子动量分布概率的粗粒动力学演化 (3D, 左侧; 2D, 右侧) 相关参数取值为$\varOmega \; = \;{\rm{3}}\omega $, $ \lambda \; = \;{\rm{2}}\omega $, 初态为$\varPsi (t = 0) = {\psi _{0,{\rm L}}}$, 动量$ \tilde p = \sqrt {1/m\hbar \omega } p $

    Figure 2.  The coarse dynamics evulution of momentum distribution of single particle (left for 3D; right for 2D) with $\varOmega \; = \;{\rm{3}}\omega $ and $ \lambda \; = \;{\rm{2}}\omega $. The initial state is set as $\varPsi (t = 0) = {\psi _{0,{\rm{L}}}}$. Momentum $\tilde p$ is defined by $ \tilde p = \sqrt {1/m\hbar \omega } p $.

    图 3  原子空间位置分布概率的粗粒动力学演化(3D, 左侧; 2D, 右侧)  相关参数取值及初态同图2, 位置$ \tilde q = \sqrt {m\omega /\hbar } q $

    Figure 3.  The coarse dynamics evolution of position distribution of single particle (left for 3D; right for 2D) with the same parameters and the initial state in Fig. 2 and $ \tilde q = \sqrt {m\omega /\hbar } q $.

    图 4  原子极化$\left\langle {{\sigma _z}} \right\rangle $随时间演化初态为$\varPsi \left( {t = 0} \right) = {\psi _{0,{\rm{L}}}}$, 参数取值为$\varOmega \; = \;{\rm{3}}\omega $$ \lambda \; = \;{\rm{2}}\omega $, 时间以因子${{2{\text{π}}}/{\Delta \omega }}$标度

    Figure 4.  Time evolution of $\left\langle {{\sigma _z}} \right\rangle $ with the initial state being $\varPsi \left( {t = 0} \right) = {\psi _{0,{\rm{L}}}}$ and the parameters $\varOmega \; = \;{\rm{3}}\omega $ and $ \lambda \; = \;{\rm{2}}\omega $. The time is scaled by the tunneling period $2{\text{π}}/\Delta\omega $.

  • [1]

    Xiao D, Chang M C, Niu Q 2010 Rev. Mod. Phys. 82 1959

    [2]

    Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045

    [3]

    Zutic I, Fabian J, Sarma S D 2004 Rev. Mod. Phys. 76 323

    [4]

    Lin Y J, Garcıa K J, Spielman I B 2011 Nature 471 8386

    [5]

    Wang P, Yu Z Q, Fu Z, Miao J, Huang L, Chai S, Zhai H, Zhang J 2012 Phys. Rev. Lett. 109 095301

    [6]

    Cheuk L W, Sommer A T, Hadzibabic Z, Yefsah T, Bakr W S, Zwierlein M W 2012 Phys. Rev. Lett. 109 095302

    [7]

    Zhang J Y, Ji S C, Chen Z, Zhang L, Du Z D, Yan B, Pan G S, Zhao B, Deng Y J, Zhai H, Chen S, Pan J W 2012 Phys. Rev. Lett. 109 115301

    [8]

    Galitski V, Spielman I B 2013 Nature 494 4954

    [9]

    Lin Y J, Compton R L, Garcia K J, Porto J V, Spielman I B 2009 Nature 462 628

    [10]

    Lin Y J, Compton R L, Perry A R, Phillips W D, Porto J V, Spielman I B 2009 Phys. Rev. Lett. 102 130401

    [11]

    Qu C, Hamner C, Gong M, Zhang C, Engels P 2013 Phys. Rev. A 88 021604

    [12]

    LeBlanc L J, Beeler M C, Jimenez-Garcia K, Perry A R, Sugawa S, Williams R A, Spielman I B 2013 New J. Phys. 15 073011

    [13]

    Wu C J, Mondragon-Shem I, Zhou X F 2011 Chin. Phys. Lett. 28 097102

    [14]

    Hu H, Ramachandhran B, Pu H, Liu X J 2012 Phys. Rev. Lett. 108 010402

    [15]

    Sinha S, Nath R, Santos L 2011 Phys. Rev. Lett. 107 270401

    [16]

    Ghosh S K, Vyasanakere J P, Shenoy V B, 2011 Phys. Rev. A 84 053629

    [17]

    Larson J, Anderson B M, Altland A 2013 Phys. Rev. A 87 013624

    [18]

    Rabi I I 1936 Phys. Rev. 49 324

    [19]

    Rabi I I 1937 Phys. Rev. 51 652

    [20]

    Braak D 2011 Phys. Rev. Lett. 107 100401

    [21]

    Chen Q H, Wang C, He S, Liu T, Wang K L 2012 Phys. Rev. A 86 023822

    [22]

    Solano E 2011 Physics 4 68

    [23]

    Gardas, Dajka J 2013 arXiv: 1301.5660[quant-ph]

    [24]

    Wolf F A, Kollar M, Braak D 2012 Phys. Rev. A 85 053817

    [25]

    Walther H, Varcoe B, Englert B, Becker T 2006 Rep. Prog. Phys. 69 1325

    [26]

    Raimond J M, Brune M, Haroche S 2001 Rev. Mod. Phys. 73 565

    [27]

    Holstein T 1959 Ann. Phys. 8 325

    [28]

    Mabuchi H, Doherty A C 2002 Science 298 1372

    [29]

    Niemczyk T, Deppe F, Huebl H, Menzel E P, Hocke F, Schwarz M J, Garcia-Ripoll J J, Zueco D, Hümmer T, Solano E, Marx A, Gross R 2010 Nature Phys. 6 772

    [30]

    Forn-Diaz P, Lisenfeld J, Marcos D, Garcia-Ripoll J J, Solano E, Harmans C J P M, Mooij J E 2010 Phys. Rev. Lett. 105 237001

    [31]

    Fu Z, Wang P, Chai S, Huang L, Zhang J 2011 Phys. Rev. A 84 043609

    [32]

    Cahill K K, Glauber R J 1969 Phys. Rev. 177 1857

    [33]

    Graham R, Hohnerbach M 1984 Phys. B: Condens. Matter 57 233

  • [1] Yang Xiao-Yong, Xue Hai-Bin, Liang Jiu-Qing. Spin coherent-state transformation and analytical solutions of ground-state based on variational-method for spin-Bose models. Acta Physica Sinica, 2013, 62(11): 114205. doi: 10.7498/aps.62.114205
    [2] Xiong Zhuang, Wang Zhen-Xin, Naoum C. Bacalis. Accuracy study for excited atoms (ions):A new variational method. Acta Physica Sinica, 2014, 63(5): 053104. doi: 10.7498/aps.63.053104
    [3] Zhang Lei, Li Hui-Wu, Hu Liang-Bin. Study of stability of persistent spin helix in two-dimensional electron gases with spin-orbit coupling. Acta Physica Sinica, 2012, 61(17): 177203. doi: 10.7498/aps.61.177203
    [4] Yu Li-Xian, Liang Qi-Feng, Wang Li-Rong, Zhu Shi-Qun. Photon squeezing of the Rabi model. Acta Physica Sinica, 2013, 62(16): 160301. doi: 10.7498/aps.62.160301
    [5] Lu Ya-Xin, Ma Ning. The coupled electromagnetic field effects on quantum magnetic oscillations of graphene. Acta Physica Sinica, 2016, 65(2): 027502. doi: 10.7498/aps.65.027502
    [6] Tang Nai-Yun. Bonding-antibonding ground state transition in coupled quantum dots. Acta Physica Sinica, 2013, 62(5): 057301. doi: 10.7498/aps.62.057301
    [7] Li Gui-Xia, Jiang Yong-Chao, Ling Cui-Cui, Ma Hong-Zhang, Li Peng. The characteristics of excited states for HF+ ion under spin-orbit coupling. Acta Physica Sinica, 2014, 63(12): 127102. doi: 10.7498/aps.63.127102
    [8] Gong Shi-Jing, Duan Chun-Gang. Recent progress in Rashba spin orbit coupling on metal surface. Acta Physica Sinica, 2015, 64(18): 187103. doi: 10.7498/aps.64.187103
    [9] Shi Ting-Ting, Wang Liu-Jiu, Wang Jing-Kun, Zhang Wei. Some recent progresses on the study of ultracold quantum gases with spin-orbit coupling. Acta Physica Sinica, 2020, 69(1): 016701. doi: 10.7498/aps.69.20191241
    [10] Li Qun, Chen Qian, Chong Jing. Variational study of the 2DEG wave function in InAlN/GaN heterostructures. Acta Physica Sinica, 2018, 67(2): 027303. doi: 10.7498/aps.67.20171827
    [11] Yu Zhi-Qiang, Xie Quan, Xiao Qing-Quan. Effects of the spin-orbit coupling on X-ray spectrum in special relativity. Acta Physica Sinica, 2010, 59(2): 925-931. doi: 10.7498/aps.59.925
    [12] Chen Guang-Ping. Ground state of a rotating spin-orbit-coupled Bose-Einstein condensate in a harmonic plus quartic potential. Acta Physica Sinica, 2015, 64(3): 030302. doi: 10.7498/aps.64.030302
    [13] Yang Jie, Dong Quan-Li, Zhang Jie, Jiang Zhao-Tan. Electronic energy band structures of carbon nanotubeswith spin-orbit coupling interaction. Acta Physica Sinica, 2011, 60(7): 075202. doi: 10.7498/aps.60.075202
    [14] Chen Dong-Hai, Yang Mou, Duan Hou-Jian, Wang Rui-Qiang. Electronic transport properties of graphene pn junctions with spin-orbit coupling. Acta Physica Sinica, 2015, 64(9): 097201. doi: 10.7498/aps.64.097201
    [15] Yang Yuan,  Chen Shuai,  Li Xiao-Bing. Topological phase transitions in square-octagon lattice with Rashba spin-orbit coupling. Acta Physica Sinica, 2018, 67(23): 237101. doi: 10.7498/aps.67.20180624
    [16] Liang Tao, Li Ming. Integer quantum Hall effect in a spin-orbital coupling system. Acta Physica Sinica, 2019, 68(11): 117101. doi: 10.7498/aps.68.20190037
    [17] Dong Cheng-Wei. Periodic orbits of diffusionless Lorenz system. Acta Physica Sinica, 2018, 67(24): 240501. doi: 10.7498/aps.67.20181581
    [18] Chen Yuan-Yuan, Yang Pan-Jie, Zhang Wei-Zhi, Yan Xiao-Na. A powerful method to analyze of photonic crystals: mixed variational method. Acta Physica Sinica, 2016, 65(12): 124206. doi: 10.7498/aps.65.124206
    [19] Liu Sheng-Li, Li Jian-Zheng, Cheng Jie, Wang Hai-Yun, Li Yong-Tao, Zhang Hong-Guang, Li Xing-Ao. Doping and Raman scattering of strong spin-orbit-coupling compound Sr2-xLaxIrO4. Acta Physica Sinica, 2015, 64(20): 207103. doi: 10.7498/aps.64.207103
    [20] Xing Wei, Sun Jin-Feng, Shi De-Heng, Zhu Zun-Lüe. Theoretical study of spectroscopic properties of 5 -S and 10 states and laser cooling for AlH+ cation. Acta Physica Sinica, 2018, 67(19): 193101. doi: 10.7498/aps.67.20180926
  • Citation:
Metrics
  • Abstract views:  210
  • PDF Downloads:  7
  • Cited By: 0
Publishing process
  • Received Date:  25 January 2019
  • Accepted Date:  28 May 2019
  • Available Online:  26 November 2019
  • Published Online:  01 September 2019

One-dimensional spin-orbit coupling Bose gases with harmonic trapping

    Corresponding author: Wang Yue-Ming, wang_ym@sxu.edu.cn
  • 1. Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China
  • 2. School of Physics and Electronic Engineer, Shanxi University, Taiyuan 030006, China
  • 3. State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China

Abstract: Rabi model is a popular model in quantum optics and describes a two-level system coupling to a quantum resonator. The fruitful physics appears when the coupling strength is comparable to the frequency of the resonator. We investigate the Bose gases of Raman induced spin-orbit coupling with an external harmonic trapping. Using the displacement Fock state in quantum optics we seek for an approximate ground state. We find the superposition state of left and right displaced oscillator state with odd parity has lower energy than the displaced state itself. Besides, we study the time evolution of both the momentum and the position of the system at single particle level to demonstrate the Zitterbewegung oscillating characteristics, which present an intuitive physical picture and are in qualitative agreement with the relevant experimental results. The results are useful to study the Rabi model in deep-strong coupling regime, the model that is difficult to realize in today’s experiment based on the high controllability property of laser, and these results are also instructive for the cold atom physics field.

    • 自旋轨道耦合(spin-orbit coupling, SOC)是粒子自旋内禀自由度与其外部运动自由度之间的相互作用, 在凝聚态物理许多重要的现象中扮演着重要的角色[13]. 近几年来, 冷原子物理学的一个重要进展是实现了光诱导合成规范场的中性原子的SOC[47], 目前在实验上可以在玻色子和费米子超冷原子气体中实现各种SOC的哈密顿量[812]. 虽然大多理论工作都集中在Rashba耦合的SOC的均匀系统上, 但是谐振子束缚势下具有SOC的冷原子系统也受到关注和研究[1317].

      量子Rabi模型[18,19]是量子光学中重要的基础模型之一, 模型形式简单, 但精确求解并不容易. 直到2011年, Braak[20]提出Rabi模型具有Z2对称性保证了其可积性, 因而可以通过求解超越方程得到系统的精确能谱. 该工作重新激起了人们对量子Rabi模型研究的兴趣[2124]. Rabi模型被广泛应用于不同的物理领域, 包括量子光学[25]、量子信息[26]、凝聚态物理[27]和腔量子电动力学[2831], 也为研究受限空间中的冷原子系统提供了模型和方法.

      在量子光学中考虑原子的动能大大超过了相互作用能, 通常采用绝热近似不考虑原子质心动能. 然而, 利用激光冷却原子运动降低原子动能可以制备出1 K量级的冷原子气体, 这个能量与腔量子电动力学实验的相互作用能相当, 从哈密顿量中排除动能项不再合理. 本文研究一维(1D)谐波势阱中具有SOC的Bose气体(考虑稀薄原子气体, 忽略相互作用), 采用量子光学中的方法求解系统的本征能态及可观测物理量的动力学演化, 与目前相关的实验结果定性一致[11,12].

    2.   谐振子势束缚自旋轨道耦合的Bose气体
    • 三维空间谐振子束缚势中具有SOC的单原子Bose气体的哈密顿量形式为

      其中$\hat V\left( {\hat r} \right) = m\left( {\omega _x^2{x^{\rm{2}}} + \omega _y^2y{}^{\rm{2}} + \omega _z^2{z^{\rm{2}}}} \right){\rm{/2}}$是束缚势, m为原子质量, $\hat P$为原子动量; ${\hat V_{{\rm{SO}}}}$为SOC项, 对于x方向具有SOC两组分的Bose气体(视为赝自旋1/2)可以写为[4,30]

      其中${\kappa _r}$是双光子反冲动量, $\varOmega $是拉曼耦合强度, $\delta $是双光子失谐, $\hat \sigma $为泡利矩阵算符. 三维哈密顿量${\hat H_{{\rm{3d}}}} = \hat H(x) + {\hat H_{{\rm{2d}}}}(y,z)$可以通过维度塌缩方法变为一维系统, 这样一维系统的哈密顿量为

      ${\omega _x}$简记为$\omega $. (3)式的模型看似简单, 除了特殊情况$\omega = 0$$\varOmega = {\rm{0}}$之外, 该模型不容易解析求解.

    3.   映射为量子Rabi模型
    • 为了将谐波势阱中的SOC模型变换到量子Rabi模型, 将原子质心运动量子化用${a^\dagger }(a)$产生(湮灭)算符来表示, 方程(3)改写为

      其中${p_x} = {\rm{i}}\sqrt {m\hbar \omega /2} ({a^\dagger } - a)$$\lambda = {k_r}\sqrt {2m\hbar \omega } $. 略去常数项哈密顿量简化为

      做一旋转变换${a^\dagger } \to {a^\dagger }{{\rm{e}}^{{\rm{i}}\theta }},\;a \to a{{\rm{e}}^{ - {\rm{i}}\theta }}\left( {\theta = {\text{π}}/2} \right)$, 方程(5)变为标准的Rabi模型

      量子光学中Rabi模型描述了量子化光场与二能级原子内态的耦合, 此处则描述了原子赝自旋(此后简称为量子比特)与质心动量的耦合. 这样就可以利用量子光学中的方法来求解该系统. 当$\delta = {\rm{0}}$时, 系统进一步简化为

      其中$\varOmega $是原子内部能级的能量差, $\omega $为谐振子束缚势频率, $\lambda $为自旋-轨道耦合强度.

    4.   变分法求解基态
    • 系统的哈密顿量(7)式在${\hat \sigma _z}$表象中可以表示为

      其中${H_{\rm{L}}}$${H_{\rm{R}}}$分别是依赖于量子比特的振子左右平移哈密顿量[32],

      $\left| { \pm z} \right\rangle $${\hat \sigma _z}$的本征态. 如果忽略$\varOmega $项, 则能量本征态就是与量子比特相关的振子的平移Fock态($\left| { + z} \right\rangle $对应左平移态, $\left| { - z} \right\rangle $对应右平移态). 通常人们将$\varOmega $项视为微扰(绝热近似), 但本文主要研究$\varOmega {\rm{/}}\omega > 1$的情况. 下面我们采用试探波函数$\left| {{\psi _0}} \right\rangle $方法, 将振子平移量和量子比特内态的相干叠加参数分别作为变分参数:

      其中$\left| \alpha \right\rangle $为相干态, $\left| { \pm x} \right\rangle $${\hat \sigma _x}$的本征态. 在坐标和动量表象中表示为

      其中m是原子的质量. 假定$\alpha $为实数, 则能量期待值表示为

      故此, 可以通过对含参数$\alpha $$\theta $的能量泛函最小化以求得基态能量. 根据二元函数极值判据可知当${{{\rm{4}}{\lambda ^{\rm{2}}}}/{\left( {\varOmega \omega } \right)}} < 1$时, 系统能量只有一个局域最小值, 在$\alpha = {\rm{0}}$$\theta = {\rm{0}}$时取得; 但如果条件

      成立, 则能量有两个局域极小值, 分别由$\alpha = \pm {\alpha _0}$$\theta = \pm {\theta _0}$给出, 且有

      此后我们假设条件(13)始终成立, 另外我们假定选择${\theta _{\rm{0}}}$使得${\rm{sin}}{\theta _0} \geqslant 0$, 这使得${\alpha _{\rm{0}}} \leqslant {\rm{0}}$, 分别对应两个简并态$\left| {{\psi _{0,{\rm{L}}}}} \right\rangle = \left| {{\alpha _0}} \right\rangle \left( {{\rm{cos}}\dfrac{{{\theta _0}}}{2}\left| { + x} \right\rangle + {\rm{sin}}\dfrac{{{\theta _0}}}{2}\left| { - x} \right\rangle } \right)$(左平移)和$\left| {{\psi _{0,{\rm{R}}}}} \right\rangle = \left| { - {\alpha _0}} \right\rangle \left( {{\rm{cos}}\dfrac{{{\theta _0}}}{2}\left| { + x} \right\rangle - {\rm{sin}}\dfrac{{{\theta _0}}}{2}\left| { - x} \right\rangle } \right)$(右平移), 相应的能量为

      此处$\varepsilon=\varOmega \omega/(4\lambda^2) $. 这两个简并态均不是宇称算符$\hat P = \exp [{\rm i}{\text{π}}({a^ + }a + {\sigma _x}/$$2 - 1/2)]$的本征态[33], 通过相干叠加可以得到具有确定宇称的量子态

      这里${\eta _{ \pm ,0}} = \sqrt {1 \mp \varepsilon {{\rm{e}}^{ - 2\alpha _0^2}}} $是归一化系数, $\left| {{\psi _{{\rm{0,}}\,{\rm{L}}}}} \right\rangle \,$$\left| {{\psi _{0,{\rm{R}}}}} \right\rangle $在一般不正交(见(20), (22)式). 计算证明可得若(13)式成立, 就有$\left\langle {{\varPhi _{ - ,0}}} \right|H\left| {{\varPhi _{ - ,0}}} \right\rangle < E\left( {{\alpha _0},{\theta _0}} \right)$成立(见图1), 因此, $\left| {{\varPhi _{ - ,0}}} \right\rangle $是系统基态更好的近似. 为保证波函数具有确定的宇称, 定义

      Figure 1.  The energies of degenerate quantum states $\left| {{\psi _{N{\rm{,L}}\left( {\rm{R}} \right)}}} \right\rangle $ and the superposition state of odd parity $\left| {{\psi _{ - ,N}}} \right\rangle $ of left(right)-displaced number states varies as the spin-orbit coupling strength $\lambda $. It is seen that for $N = 0$, the superposition state has the lowest energy which is the best approximation for the ground state in our interest. And for the cases of $N \ne 0$, the energies of the two quantum states have pitchforks.The relevant parameters is Ω=1.4 and the results are in agreement with those in Ref.[19].

      这里左右平移态分别为:

      归一化系数为

      在(19)式中, $\hat D\left[ \alpha \right] \equiv \exp \left[ {\alpha \left( {{a^ + } - a} \right)} \right]$是平移算符, $\hat D\left[ { \pm \alpha } \right]\left| N \right\rangle $是平移Fock态, 简记为$ \left| { \pm\alpha ,N} \right\rangle $, 左右平移Fock态的内积可用Laguerre多项式给出:

      平移Fock态$\left| {{\psi _{N,{\rm{L}}\left( {\rm{R}} \right)}}} \right\rangle $$\left| {{\psi _{0,{\rm{L}}\left( {\rm{R}} \right)}}} \right\rangle $源于相同的变分计算, 对于不同声子数N均具有相同的${\alpha _0}$${\theta _0}$, 并且对于不同N , $\left| {{\psi _{N,{\rm{L/R}}}}} \right\rangle $彼此正交, 但左右平移振子态并不正交; 对于相同的声子数且当N较大时, 内积以${\rm{1/}}\left( {{N^{{1/4}}}{\alpha ^{{1/2}}}} \right)$衰变. $\left| {{\psi _{N,{\rm{L}}}}} \right\rangle $$\left| {{\psi _{N,{\rm{R}}}}} \right\rangle $在能量上简并, 考虑宇称自然构造出(18)式的叠加态. 容易看出, $\left| {{\varPhi _{ \pm ,N}}} \right\rangle $为一完备非正交的基. $\left| {{\varPhi _{ \pm ,N}}} \right\rangle $的宇称是$ \pm {\left( { - 1} \right)^N}$, 并且相反宇称态正交, 相同宇称态一般不正交; 对于大声子数NM, 交叠积分$\left\langle { - \alpha ,N|\alpha ,M} \right\rangle $至少以${\alpha ^{ - 1/2}}$的速度衰减. 考虑(15)式, 在极限$\lambda \gg \omega $的参数区域可以得到近似的正交基. 此外交叠积分也为因子$\cos {\theta _0} = - \varepsilon $所抑制, 对于大的耦合常数λ积分也是小量. 这些条件允许我们用期望值近似作为H的能量本征值, 也就是哈密顿量在近似正交基(18)式下的对角元. 对于足够大的${\alpha _0}$非对角元很小. 近似能量${E_{ \pm ,N}} = \left\langle {{\varPhi _{ \pm ,N}}} \right|H\left| {{\varPhi _{ \pm ,N}}} \right\rangle $表示为

      这个近似结果解有助于我们直观地理解系统的动力学.

    5.   系统动力学演化
    • 考虑系统的动力学演化, 对于稀薄原子气体可以忽略原子间相互作用. 我们取初态$\left| {{\psi _{{\rm{0,L}}}}} \right\rangle $, 在初始时刻时开启拉曼诱导的SOC, 则系统的初始波函数为

      其时间演化近似为

      其中$\Delta\omega $是频率差,

      在(25)式中忽略了${{\rm{e}}^{ - 2\alpha _0^2}}$的高阶幂次. 可以看出在初始时刻$t = 0$时, 初态动量分布主要位于左侧(振子相干态$\left| {{\alpha _0}} \right\rangle $); 而在时刻$t = {{\text{π}}/{\Delta\omega }}$时, 动量分布主要位于右侧(振子相干态$\left| { - {\alpha _0}} \right\rangle $); 在时刻$t = {{\text{π}}/({2\Delta \omega })}$, 原子动量概率分布呈双峰分布, 对应于两个相干态的叠加, 这是标准的隧穿运动, 与经典双势阱完全类似.

      图2图3分别给出了粒子在动量空间和坐标空间概率分布的动态特性, 由(23)式的近似值计算得出, 在这里我们取$\varOmega \; = \;{\rm{3}}\omega $$ \lambda \; = \;{\rm{2}}\omega $. 可以清楚地看到原子质心动量和空间位置分布的特征隧穿行为, 即所谓 Zitterbewegung振荡.

      Figure 2.  The coarse dynamics evulution of momentum distribution of single particle (left for 3D; right for 2D) with $\varOmega \; = \;{\rm{3}}\omega $ and $ \lambda \; = \;{\rm{2}}\omega $. The initial state is set as $\varPsi (t = 0) = {\psi _{0,{\rm{L}}}}$. Momentum $\tilde p$ is defined by $ \tilde p = \sqrt {1/m\hbar \omega } p $.

      Figure 3.  The coarse dynamics evolution of position distribution of single particle (left for 3D; right for 2D) with the same parameters and the initial state in Fig. 2 and $ \tilde q = \sqrt {m\omega /\hbar } q $.

      另外两组分原子布居差${\sigma _z}$的期望值$\left\langle {{\sigma _z}} \right\rangle = \sin {\theta _0}\cos \left( {\Delta \omega t} \right)$描述了原子的极化率的动力学. 图4显示了原子极化$\left\langle {{\sigma _z}} \right\rangle $随时间的演化, 可以看到$\left\langle {{\sigma _z}} \right\rangle $在1和–1之间周期振荡.

      Figure 4.  Time evolution of $\left\langle {{\sigma _z}} \right\rangle $ with the initial state being $\varPsi \left( {t = 0} \right) = {\psi _{0,{\rm{L}}}}$ and the parameters $\varOmega \; = \;{\rm{3}}\omega $ and $ \lambda \; = \;{\rm{2}}\omega $. The time is scaled by the tunneling period $2{\text{π}}/\Delta\omega $.

    6.   结 论
    • 综上, 我们求解了谐波势阱中拉曼诱导自旋轨道耦合的Bose气体, 通过将系统完全映射到量子Rabi模型, 将辐射场变为声子场, 运用量子光学中平移Fock态的方法得到了强耦合区域谐波势阱中自旋轨道耦合的Bose气体模型的基态解及系统的动力学演化, 直观地给出了原子质心空间坐标和动量及原子极化随时间的振荡图像, 与相关的实验结果定性相符.

      传统量子光学中二能级系统与振子系统的耦合强度受到很大限制, 但在本系统中原子自旋轨道耦合强度可以通过Raman耦合来调节, 冷原子的质心动能很小, 不能像量子光学中惯常采用绝热近似忽略掉原子的质心动能, 这使得本模型科学合理. 冷原子系统具有良好的可调控性, 通过改变束缚势阱的频率以及Raman激光的波长和强度, 可以实现量子光学中Rabi模型目前无法达到的参数区域—深度强耦合区域. 本文的研究也为自旋轨道耦合的冷原子系统提供了一个新的方法和视角.

Reference (33)

Catalog

    /

    返回文章
    返回