  Realistic methods to create vortices in spin-orbit-coupled Bose-Einstein
condensates are discussed. It is shown that, contrary to common intuition,
rotation of the trap containing a spin-orbit condensate does not lead to an
equilibrium state with static vortex structures, but gives rise instead to
non-equilibrium behavior described by an intrinsically time-dependent
Hamiltonian. We propose here the following alternative methods to induce
thermodynamically stable static vortex configurations: (1) to rotate both the
lasers and the anisotropic trap; and (2) to impose a synthetic Abelian field on
top of synthetic spin-orbit interactions. Effective Hamiltonians for spin-orbit
condensates under such perturbations are derived for most currently known
realistic laser schemes that induce synthetic spin-orbit couplings. The
Gross-Pitaevskii equation is solved for several experimentally relevant
regimes. The new interesting effects include spatial separation of left- and
right-moving spin-orbit condensates, the appearance of unusual vortex
arrangements, and parity effects in vortex nucleation where the topological
excitations are predicted to appear in pairs. All these phenomena are shown to
be highly non-universal and depend strongly on a specific laser scheme and
system parameters.
