Interaction of atom with nonparaxial Laguerre-Gaussian beam: Forming superposition of vortex states in Bose-Einstein condensates

The exchange of orbital angular momentum (OAM) between paraxial optical vortex and a Bose-Einstein condensate (BEC) of atomic gases is well known. In this paper, we develop a theory for the microscopic interaction between matter and an optical vortex beyond paraxial approximation. We show how superposition of vortex states of BEC can be created with a focused optical vortex. Since the polarization or spin angular momentum (SAM) of the optical field is coupled with OAM of the field, in this case, these angular momenta can be transferred to the internal electronic and external center-of-mass motion of atoms provided both the motions are coupled. We propose a scheme for producing the superposition of matter-wave vortices using Gaussian and a focused Laguerre-Gaussian beam. We study how two-photon Rabi frequencies of stimulated Raman transitions vary with focusing angles for different combinations of OAM and SAM of optical states. We demonstrate the formation of vortex-antivortex structure and discuss interference of three vortex states in a BEC.

Figure 1

Energy-level scheme of the two-photon transitions. The atomic states show the ${}^{23}\mathrm{Na}$ hyperfine states. Atoms are initially in $|3s_{\frac{1}{2}}F=1,m_f=-1\rangle$. $\mathrm{\Delta }$ represents two-photon detuning.

Figure 2

Single LG and three Gaussian (${\mathrm{G}}_1$, ${\mathrm{G}}_2$, ${\mathrm{G}}_3$) pulses are applied to BEC.

Figure 3

Variations of dipole Rabi frequency (in ${\sec }^{-1}$) with focusing angles (in deg) are plotted on a semilog scale. Red solid line refers to electronic transition $|F=1,m_f=-1\rangle$ to $|F^{\prime }=2,m_f=-2\rangle$, blue dashed line is for $|F=1,m_f=-1\rangle$ to $|F^{\prime }=2,m_f=0\rangle$, and dotted line represents $|F=1,m_f=-1\rangle$ to $|F^{\prime }=2,m_f=-1\rangle$.

Figure 4

Plot of the density distribution of vortex antivortex states for focusing angles (a) ${70}^{\circ }$, (b) ${60}^{\circ }$, (c) ${50}^{\circ }$, and (d) ${40}^{\circ }$. All quantities are in dimensionless units.