Mesoscopics of half-quantum vortex pair deconfinement in a trapped spin-one condensate

Motivated by a recent experiment in an antiferromagnetic spin-1 Bose-Einstein condensate (BEC) of Na-23 atoms, we study the energetic stability of a singly quantized vortex injected into the center of a quasi-twodimensional gas with zero total spin against dissociation into a pair of half-quantum vortices, by using a variational ansatz. We find that the critical dissociation point of this confinement-deconfinement-type transition can be expressed in terms of the ratio of density-density (c(0)) and spin-spin (c(2)) coupling constants. The transition of bound to unbound vortices, in particular, sensitively depends on (1) the ratio of system size (R) to density healing length (xi(d)), and (2) the trap potential. Specifically, the critical ratio (c(2)/c(0))c(r) increases when R/xi(d) decreases and is relatively larger in a harmonic trap than in a box trap. Dissociation is energetically generally favored for c(2)/c(0) < (c(2)/c(0)) cr, which as a corollary implies that vortex dissociation is observed as well for negative c(2) < 0, e.g., in a rubidium spin-1 BEC, whereas in a sodium spin-1 BEC (c(2) > 0) it is energetically blocked above the critical ratio (c(2)/c(0)) cr. Tuning the coupling ratio c(2)/c(0) by using microwave control techniques, the dependence of the deconfinement transition on coupling parameters, density, and system size we predict can be verified in experiments with ultracold spinor gases.