Spin-orbit coupled weakly interacting Bose-Einstein condensates in harmonic traps.

We investigate theoretically the phase diagram of a spin-orbit coupled Bose gas in two-dimensional harmonic traps. We show that at strong spin-orbit coupling the single-particle spectrum decomposes into different manifolds separated by ℏω{⊥}, where ω{⊥} is the trapping frequency. For a weakly interacting gas, quantum states with Skyrmion lattice patterns emerge spontaneously and preserve either parity symmetry or combined parity-time-reversal symmetry. These phases can be readily observed in a spin-orbit coupled gas of ^{87}Rb atoms in a highly oblate trap.