  Vortices in a one-component dilute atomic ultracold Bose-Einstein condensate
(BEC) usually arise as a response to externally driven rotation. Apart from a
few special situations, these vortices are singly quantized with unit
circulation. Recently, the NIST group has constructed a two-component BEC with
a spin-orbit coupled Hamiltonian involving Pauli matrices, and I here study the
dynamics of a two-component vortex in such a spin-orbit coupled condensate.
These spin-orbit coupled BECs use an applied magnetic field to split the
hyperfine levels. Hence they rely on a focused laser beam to trap the atoms. In
addition, two Raman laser beams create an effective (or synthetic) gauge
potential. The resulting spin-orbit Hamiltonian is discussed in some detail.
The various laser beams are fixed in the laboratory, so that it is not feasible
to nucleate a vortex by an applied rotation that would need to rotate all the
laser beams and the magnetic field. In a one-component BEC, a vortex can also
be created by a thermal quench, starting from the normal state and suddenly
cooling deep into the condensed state. I propose that a similar method would
work for a vortex in a spin-orbit coupled BEC. Such a vortex has two
components, and each has its own circulation quantum number. If both components
have the same circulation, I find that the composite vortex should execute
uniform precession, like that observed in a single-component BEC. In contrast,
if one component has unit circulation and the other has zero circulation, then
some fraction of the dynamical vortex trajectories should eventually leave the
condensate, providing clear experimental evidence for this unusual vortex
structure. In the context of exciton-polariton condensates, such a vortex is
known as a "half-quantum vortex".
