  Orbital angular momentum (OAM) of light represents a fundamental optical
freedom that can be exploited to manipulate quantum state of atoms. In
particular, it can be used to realize spin-orbital-angular-momentum (SOAM)
coupling in cold atoms by inducing an atomic Raman transition using two laser
beams with differing OAM. Rich quantum phases are predicted to exist in
many-body systems with SOAM coupling. Their observations in laboratory,
however, are often hampered by the limited control of the system parameters. In
this work we report, for the first time, the experimental observation of the
ground-state quantum phase diagram of the SOAM coupled Bose-Einstein condensate
(BEC). The discontinuous variation of the spin polarization as well as the
vorticity of the atomic wave function across the phase boundaries provides
clear evidence of first-order phase transitions. Our results open up a new way
to the study of phase transitions and exotic quantum phases in quantum gases.
