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 of ${}^{23} \textrm{Na}$ atoms, we study the energetical stability of a singly quantum vortex injected into the center of a quasi-two-dimensional gas with zero total spin against dissocation into a pair of half-quantum vortices. We find that the critical dissociation point of this confinement-deconfinement type phase 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 $(c_2 / c_0)_{\textrm{cr}}$ 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 $c_2 / c_0 < (c_2 / c_0)_{\textrm{cr}}$, 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 $(c_2 / c_0)_{\textrm{cr}}$. Tuning the coupling ratio $c_2/c_0$ by using microwave control techniques, the dependence of the deconfinement phase transition on coupling parameters, density, and system size we predict, can be verified in experiments with ultracold spinor gases.