Angular Momentum Josephson Effect between Two Isolated Condensates
Wei-Feng Zhuang1, Yue-Xin Huang1, and Ming Gong1,2*
1CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China 2Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China
Abstract：We demonstrate that the two degenerate energy levels in spin–orbit coupled trapped Bose gases, coupled by a quenched Zeeman field, can be used for angular momentum Josephson effect. In a static quenched field, we can realize a Josephson oscillation with a period ranging from millisecond to hundreds of milliseconds. Moreover, by a driven Zeeman field, we realize a new Josephson oscillation, in which the population imbalance may have the same expression as the current in the direct-current Josephson effect. When the dynamics of the condensate cannot follow up the modulation frequency, it is in the self-trapping regime. This new dynamic is understood from the time-dependent evolution of the constant-energy trajectory in the phase space. This model has several salient advantages compared to the previous ones. The condensates are isolated from their excitations by a finite gap, thus can greatly suppress the damping effect induced by thermal atoms and Bogoliubov excitations. The oscillation period can be tuned by several orders of magnitude without influencing other parameters. In experiments, the dynamics can be mapped out from spin and momentum spaces, thus it is not limited by the spatial resolution in absorption imaging. This system can serve as a promising platform for matter wave interferometry and quantum metrology.
The integrals for these three parameters are: $V_{m} = \int g\big(|\varphi_{m-2\uparrow}|^{4}+|\varphi_{m-2\downarrow}|^{4}+2c_{12}| \varphi_{m-2\uparrow}|^{2}|\varphi_{m-2\downarrow}|^{2}\big)d\boldsymbol{r}$ for $m= 1$, $2$, and $V_{12} = \frac{1}{2} \int g\big(2|\varphi_{0\uparrow}|^{2}|\varphi_{-1\uparrow}|^{2} +2|\varphi_{-1\downarrow}|^{2}|\varphi_{0\downarrow}|^{2}+ c_{12}|\varphi_{-1\uparrow}|^{2}|\varphi_{0\downarrow}|^{2} +c_{12}|\varphi_{-1\downarrow}|^{2}|\varphi_{0\uparrow}|^{2}+ c_{12}\varphi_{-1\uparrow}^{\ast}\varphi_{-1\downarrow}^{\ast} \varphi_{0\downarrow}\varphi_{0\uparrow}+\textrm{H.c.}\big)d\boldsymbol{r} $.
In the Josepshson junction considered by Feynman (Ref.[3]), the current and phase are described by $I = \dot{z} = K \sin(\theta)$ and $\dot{\theta} = [\mu_0 + A \cos(\omega t)]$. From the second equation, $\theta(t) = \theta_0 + \mu_0 t + A \sin(\omega t)/\omega$. The current $I$ in this case has exactly the same form as $z(t)$ considered in Eq. (10).
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In the strong modulating limit when $\omega_2 > \omega$, the excited bands may also be excited. We find that when $\omega_2 = 0.8\omega$ ($\eta = 80\%$), the excitation of the states are still negligible for the parameters used in Fig. 4(b).
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Using the transverse confinement frequency in one-dimensional BEC (Refs.[64, 65]) with $\omega = 2\pi \cdot 2.0$ kHz, $\alpha_J = 0.9$, $h_x = 2\pi\cdot 0.8$ kHz ($\eta = 40\%$), we estimate $T = 1.4$ ms.
2021, Vol. 38 
