  We investigate the ground-state phase diagram and excitation spectrum of an
interacting spinor Bose-Einstein condensate with spin-orbital-angular-momentum
(SOAM) coupling realized in recent experiments by introducing atomic Raman
transition with a pair of copropagating Laguerre-Gaussian laser beams that
carry different orbital angular momenta (OAM) [Chen et al., Phys. Rev. Lett.
121, 113204 (2018) and Zhang et al., Phys. Rev. Lett. 122, 110402 (2019)].
Because of the ground-state degeneracy of the single-particle Hamiltonian at
vanishing detuning, several angular-stripe phases, which are superposition of
states with different angular quantum numbers, appear in the phase diagram.
However, these phases normally exist at small detuning, which makes them hard
to be probed in experiments. We show that for a large OAM difference of the
laser beams, an asymmetric kind of angular-stripe phase can exist even at large
detuning. The excitation spectra in different phases exhibit distinct features:
In the angular-stripe phase there exist two gapless bands corresponding to the
broken U(1) and rotational symmetries, while in the half-skyrmion phase the
gapless band exhibits a roton-like structure. Our predictions of the
angular-stripe phases and the low-energy excitations can be examined in
recently realized BECs with SOAM coupling.
