  Spin-orbit coupled Bose-Einstein condensates (BECs) provide a powerful tool
to investigate interesting gauge-field related phenomena. We study the ground
state properties of such a system and show that it can be mapped to the
well-known Dicke model in quantum optics, which describes the interactions
between an ensemble of atoms and an optical field. A central prediction of the
Dicke model is a quantum phase transition between a superradiant phase and a
normal phase. Here we detect this transition in a spin-orbit coupled BEC by
measuring various physical quantities across the phase transition. These
quantities include the spin polarization, the relative occupation of the nearly
degenerate single particle states, the quantity analogous to the photon field
occupation, and the period of a collective oscillation (quadrupole mode). The
applicability of the Dicke model to spin-orbit coupled BECs may lead to
interesting applications in quantum optics and quantum information science.
