  We consider a many-body system of pseudo-spin-1/2 bosons with spin-orbit
interactions, which couple the momentum and the internal pseudo-spin degree of
freedom created by spatially varying laser fields. The corresponding single-
particle spectrum is generally anisotropic and contains two degenerate minima
at finite momenta. At low temperatures, the many-body system condenses into
these minima generating a new type of entangled Bose-Einstein condensate. We
show that in the presence of weak density-density interactions the many-body
ground state is characterized by a twofold degeneracy. The corresponding
many-body wave function describes a condensate of ``left-'' and
``right-moving'' bosons. By fine-tuning the parameters of the laser field, one
can obtain a bosonic version of the spin-orbit coupled Rashba model. In this
symmetric case, the degeneracy of the ground state is very large, which may
lead to phases with nontrivial topological properties. We argue that the
predicted new type of Bose-Einstein condensates can be observed experimentally
via time-of-flight imaging, which will show characteristic multipeak structures
in momentum distribution.
