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Dynamical characteristics of internal and external states of a Bose-Einstein condensate are generally different and independent, thus requiring different experimental manipulation techniques. The spin-orbit coupling recently achieved in Bose-Einstein condensates essentially connects spin and motion degree of freedom, endowing spin states with the ability to respond to orbital manipulation, and vice versa. In this work, a dynamical response effect, induced by simultaneously manipulating the internal and external states of a spin-orbit-coupled Bose-Einstein condensate, is predicted. Here, the “simultaneously manipulating the internal and external states” means that the driving field combines the Zeeman field applied to the internal state of the atom and the orbital potential affecting the external states of the atom. Specifically, the Bose-Einstein condensate is assumed to be activated by an abruptly applied Zeeman field and a sudden shake of the trapping potential. After some reasonable simplification and approximation of the model (i.e. neglecting the inter-atomic interactions and modelling the shake of the trapping potential by a short time-dependent pulse), an analytical relationship connecting spin frequency spectrum and the parameters of the driving fields is derived. The numerical calculations based on directly integrating the Gross-Pitaevskii equation are in good agreement with the results from the analytical relationship. The physical origin of the predicted spin dynamical response can be traced back to the quantum interference among different spin-orbit states. Due to the fact that a series of characteristic parameters of the condensate can be manifested in the spin frequency spectrum, the dynamical response effect predicted here provides a candidate method for determining and calibrating various system parameters by measuring the spin frequency spectrum.
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Keywords:
- Bose-Einstein condensate /
- spin-orbit coupling /
- ultracold quantum gases /
- spin frequency spectrum
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图 1 (a)产生自旋-轨道耦合BEC的实验示意图; (b)原子的能级结构以及激光诱导的跃迁
Figure 1. (a) Schematic illustration of the experimental setup to generate spin-orbit coupled BEC; (b) the atomic level structure and their transitions.
图 2 $ \left\langle {{\sigma _x}(t)} \right\rangle $在不同驱动参数下以$ \left| {\varPsi (0)} \right\rangle = \left( {\left| {{\chi _ + }(0)} \right\rangle \left| + \right\rangle + \left| {{\chi _ - }(0)} \right\rangle \left| - \right\rangle } \right)/\sqrt 2 $为初态随时间的演化, 其他参数固定为$ \alpha = 0.2\sqrt {\hbar \omega /m} , $$ \tau = 0.01/\omega , \;\;\varOmega = 0 $.
Figure 2. Time evolution of $ \left\langle {{\sigma _x}(t)} \right\rangle $ under the initial state $ \left| {\varPsi (0)} \right\rangle = \left( {\left| {{\chi _ + }(0)} \right\rangle \left| + \right\rangle + \left| {{\chi _ - }(0)} \right\rangle \left| - \right\rangle } \right)/\sqrt 2 $ for different driving parameters. The other parameters are set by $ \alpha = 0.2\sqrt {\hbar \omega /m} , \, \, \tau = 0.01/\omega , \;\;\varOmega = 0 $.
图 3 固定其余参数, 变化$ \zeta $(上图)和变化$ {\eta _0} $(下图)时自旋频谱$ \left| {{S_x}(\nu )} \right| $的取值, 其他参数固定为$ \alpha = 0.2\sqrt {\hbar \omega /m} $, $ \tau = 0.01/\omega $, $ {T_{\text{L}}} = 400/\omega $, $ \varOmega = 0 $, 初态选择为$ \left| {\varPsi (0)} \right\rangle = $$ \left( {\left| {{\chi _ + }(0)} \right\rangle \left| + \right\rangle + \left| {{\chi _ - }(0)} \right\rangle \left| - \right\rangle } \right)/\sqrt 2 $
Figure 3. Spin frequency spectrum $ \left| {{S_x}(\nu )} \right| $ for varying $ \zeta $ (up panel) and $ {\eta _0} $ (bottom panel). The other parameters are set by $ \alpha = 0.2\sqrt {\hbar \omega /m} $, $ \tau = 0.01/\omega $, $ {T_{\text{L}}} = 400/\omega $, and $ \varOmega = $$ 0 $. The initial state is chosen as $ \left| {\varPsi (0)} \right\rangle = ( \left| {{\chi _ + }(0)} \right\rangle \left| + \right\rangle + $$ \left| {{\chi _ - }(0)} \right\rangle \left| - \right\rangle )/\sqrt 2 $.
图 4 自旋频谱$ \left| {{S_x}(\nu )} \right| $在(a) $ \zeta {\text{ - }}\nu $和(b) $ {\eta _0}{\text{ - }}\nu $ 平面上的取值 (a) $ {\eta _0} = 0.2\omega $; (b) $ \zeta = 100\sqrt {\hbar /\left( {{\text{π}}m\omega } \right)} $; 其他参数以及初态选择与图3一致
Figure 4. Spin frequency spectrum $ \left| {{S_x}(\nu )} \right| $ in the (a) $ \zeta {\text{ - }}\nu $ plane and the (b) $ {\eta _0}{\text{ - }}\nu $ plane: (a) $ {\eta _0} = 0.2\omega $; (b) $ \zeta = $$ 100\sqrt {\hbar /\left( {{\text{π}}m\omega } \right)} $. The other parameters and the initial state are the same as those in Fig.3.
图 5 多自旋-轨道态量子干涉的示意图, 动力学调控图像与量子干涉图像具有对应关系, 图中不同颜色的模块表示干涉过程的不同组成部分
Figure 5. Schematic description of the multiple spin-orbit-states interference. The dynamical control image corresponds to the quantum interference image, where different colored modules in the diagram represent different components of the interference process.
图 6 $ \varOmega = 0 $(蓝色实线)和$ \varOmega = 0.15\omega $(黑色虚线)时自旋频谱$ \left| {{S_x}(\nu )} \right| $的取值, 驱动参数固定为$ \zeta = 100\sqrt {\hbar /({\text{π}}m\omega )} , $$ \eta = 0 $, 其他参数以及初态选择与图3一致
Figure 6. Spin frequency spectrum $ \left| {{S_x}(\nu )} \right| $ for $ \varOmega = 0 $ (blue solid curve) and $ \varOmega = 0.15\omega $ (green dashed curve). The driving parameters are set by $ \zeta = 100\sqrt {\hbar /({\text{π}}m\omega )} , \;\eta = 0 $. The other parameters and the initial state are the same as those in Fig.3.
图 7 自旋频谱$ \left| {{S_x}(\nu )} \right| $在$ \nu = \tilde \nu = 0.38\omega $处随驱动参数(a) $ \zeta $和(b) $ {\eta _0} $的变化, 其他参数以及初态选择与图3一致
Figure 7. Value of spin frequency spectrum $ \left| {{S_x}(\nu )} \right| $ at $ \nu = \tilde \nu = 0.38\omega $ as a function of (a) $ \zeta $ and (b) $ {\eta _0} $. The other parameters and the initial state are the same as those in Fig.3.
图 8 $ \left| {{S_x}(\nu )} \right| $对于不同脉冲宽度$ \tau $的取值 (a1)—(c1)对应于(3)式描述的高斯型脉冲; (a2)—(c2)对应于(21)式描述的方波型脉冲; $ \zeta = 100\sqrt {\hbar /\left( {\varGamma m\omega } \right)} $, 其中$ \varGamma = \sqrt {\text{π}} $对应于高斯型脉冲, $ \varGamma = 1 $对应于方波型脉冲, 其他参数以及初态选择与图3一致
Figure 8. Value of spin frequency spectrum $ \left| {{S_x}(\nu )} \right| $ for different $ \tau $: (a1)–(c1) correspond to the Gaussian pulse defined in Eq.(3); (a1)–(c1) correspond to the square pulse defined in Eq.(21). In these figures, $ \zeta = 100\sqrt {\hbar /\left( {\varGamma m\omega } \right)} $ , where $ \varGamma = \sqrt {\text{π}} $ for the Gaussian pulse and $ \varGamma = 1 $ for the square pulse. The other parameters and the initial state are the same as those in Fig.3.
图 9 (a) $ \left| {{S_x}(\nu )} \right| $在不同积分时间$ T $下的取值; (b)图(a)中两个谱峰$ {\nu _1} $和$ {\nu _2} $的半高宽度随积分时间$ T $的变化; (c)图(a)中两个谱峰$ {\nu _1} $和$ {\nu _2} $的峰值位置随积分时间$ T $的变化; 所有图中$ \zeta = 250\sqrt {\hbar /\left( {{\text{π}}m\omega } \right)} $, $ {\eta _0}/\omega = 0 $, 其他参数以及初态选择与图3一致
Figure 9. (a) Value of spin frequency spectrum $ \left| {{S_x}(\nu )} \right| $ for different $ T $; (b) the half-height width of the spectrum peak $ {\nu _1} $ and $ {\nu _2} $ appearing in panel (a) as functions of $ T $; (c) the peak position of $ {\nu _1} $ and $ {\nu _2} $ appearing in panel (a) as functions of $ T $. In these figures, $ \zeta = 250\sqrt {\hbar /\left( {{\text{π}}m\omega } \right)} $ and $ {\eta _0}/\omega = 0 $. The other parameters and the initial state are the same as those in Fig.3.
图 10 (a)相互作用$ g $不同时, 自旋频谱$ \left| {{S_x}(\nu )} \right| $的取值; (b)自旋频谱$ \left| {{S_x}(\nu )} \right| $在$ \nu = \tilde \nu = \omega $处随驱动参数$ \zeta $的变化, 不同相互作用$ g $用不同线型表示;其他参数以及初态选择与图3一致
Figure 10. (a) Value of spin frequency spectrum $ \left| {{S_x}(\nu )} \right| $ for different interaction coefficient $ g $; (b) the value of spin frequency spectrum $ \left| {{S_x}(\nu )} \right| $ at $ \nu = \tilde \nu = \omega $ as a function of $ \zeta $, and the results of different $ g $ are labelled by different linetypes. The other parameters and the initial state are the same as those in Fig.3.
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