Fractional spin Hall effect in atomic Bose gases

We propose fractional spin Hall effect (FSHE) by coupling pseudospin states of cold bosonic atoms to optical fields. The present scheme is an extension to interacting bosonic system of the recent work [X.-J. Liu, X. Liu, L. C. Kwek, and C. H. Oh, Phys. Rev. Lett. 98, 026602 (2007) and S.-L. Zhu, H. Fu, C.-J. Wu, S.-C. Zhang, and L.-M. Duan, Phys. Rev. Lett. 97, 240401 (2006)] on optically induced spin Hall effect in noninteracting atomic system. The system has two different types of ground states. The first type of ground state is a 1/3-factor Laughlin function and has the property of chiral-antichiral interchange antisymmetry, while the second type is shown to be a 1/4-factor wave function with chiral-antichiral symmetry. The fractional statistics corresponding to the fractional spin Hall states are studied in detail and are discovered to be different from that corresponding to the fractional quantum Hall (FQH) states. Therefore the present FSHE can be distinguished from FQH regime in the measurement.

Figure 1

(Color online) (a) Four-level bosonic atoms interacting with two light fields; this case can be experimentally realized with, e.g., ${}^{87}\text{R}\text{b}$ atoms [(b) and (c)]; (d) Initial condition achieved by pumping the atoms into $|S_{\pm }⟩$ with $\left|{\Omega }_{+}\right|=\left|{\Omega }_{-}\right|$ through the $\Lambda$-type configuration.

Figure 2

(Color online) (a) Double $\Lambda$-type bosonic atoms interacting with two light fields $\left({\Omega }_{1,2}\right)$ with orbital angular momentum and one strong field ${\Omega }_c$; (b) this situation can be experimentally realized with, e.g., ${}^{87}\text{R}\text{b}$ atoms. The Zeeman splitting is considered.