  Ultracold atoms endowed with tunable spin-orbital-angular-momentum coupling
(SOAMC) represent a promising avenue for delving into exotic quantum phenomena.
Building on recent experimental advancements, we propose the generation of
synthetic gauge fields ,and by including exotic vortex phases within spinor
Bose-Einstein condensates, employing a combination of a running wave and
Laguerre-Gaussian laser fields. We investigate the ground-state characteristics
of the SOAMC condensate, revealing the emergence of exotic vortex states with
controllable orbital angular momenta. It is shown that the interplay of the
SOAMC and conventional spin-linear-momentum coupling induced by the running
wave beam leads to the formation of a vortex state exhibiting a phase stripe
hosting single multiply quantized singularity. The phase of the ground state
will undergo the phase transition corresponding to the breaking of rotational
symmetry while preserving the mirror symmetry. Importantly, the observed
density distribution of the ground-state wavefunction, exhibiting broken
rotational symmetry, can be well characterized by the synthetic magnetic field
generated through light interaction with the dressed spin state. Our findings
pave the way for further exploration into the rotational properties of stable
exotic vortices with higher orbital angular momenta against splitting in the
presence of synthetic gauge fields in ultracold quantum gases.
