  Motivated by a goal of realizing spin-orbit coupling (SOC) beyond
one-dimension (1D), we propose and analyze a method to generate an effective 2D
SOC in bilayer BECs with laser-assisted inter-layer tunneling. We show that an
interplay between the inter-layer tunneling, SOC and intra-layer atomic
interaction can give rise to diverse ground state configurations. In
particular, the system undergoes a transition to a new type of stripe phase
which spontaneously breaks the time-reversal symmetry. Different from the
ordinary Rashba-type SOC, a fractionalized skyrmion lattice emerges
spontaneously in the bilayer system without external traps. Furthermore, we
predict the occurrence of a tetracritical point in the phase diagram of the
bilayer BECs, where four different phases merge together. The origin of the
emerging different phases is elucidated.
