  In this paper, we have formulated a theory for the microscopic interaction of
the asymmetric Laguerre-Gaussian (aLG) beam with the atomic Bose-Einstein
condensate (BEC) in a harmonic trap. Here the asymmetry is introduced to an LG
beam considering a complex-valued shift in the Cartesian plane keeping the axis
of the beam and its vortex states co-axial to the trap axis of the BEC. Due to
the inclusion of the asymmetric nature, multiple quantized circulations are
generated in the beam. We show how these quantized circulations are transferred
to the BEC resulting in a superposition of matter vortex states. The calculated
Rabi frequencies for the dipole as well as quadrupole transitions during the
transfer process show distinct variability with the shift parameters of the
beam. A significant enhancement of the quadrupole Rabi frequency for higher
vorticity states is observed compared to symmetric single orbital angular
momentum (OAM) mode beam at a particular range of the shift parameters. We also
demonstrate the variation of superposition of matter vortex states and observe
its distinct feature compared to the superposition of the LG modes for
different shift parameters. The first order spatial correlation of the
superposed states supports this feature and highlights asymmetry in degree of
transverse coherence along orthogonal directions on the surface.
