Skyrmions in a ferromagnetic Bose–Einstein condensate

Abstract

Multi-component Bose±Einstein condensates1±3 provide opportunities to explore experimentally the wealth of physics associated with the spin degrees of freedom 4±7 . The ground-state properties 8±11 and line-like vortex excitations 8,12,13 of these quantum systems have been studied theoretically. In principle, nontrivial spin textures consisting of point-like topological excitations, or skyrmions 14,15 , could exist in a multi-component Bose±Einstein condensate, owing to the super¯uid nature of the gas. Although skyrmion excitations are already known in the context of nuclear physics and the quantum-Hall effect, creating these excitations in an atomic condensate would offer an opportunity to study their physical behaviour in much greater detail, while also enabling an ab initio comparison between theory and experiment. Here we investigate theoretically the stability of skyrmions in a ®ctitious spin-1/2 condensate of 87 Rb atoms. We ®nd that skyrmions can exist in such a gas only as a metastable state, but with a lifetime comparable to (or even longer than) the typical lifetime of the condensate itself.An essential feature of a spinor Bose±Einstein condensate is that two or more hyper®ne states of the atoms in the condensate have almost the same energy. As a result, this spin degree of freedom becomes a relevant dynamical variable, which gives rise to new excitations that are not present in the usual single-component Bose±Einstein condensates, where the spins are effectively frozen. One of these excitations is the skyrmion, which is a topological nontrivial spin texture. Roughly speaking, the skyrmion is a pointlike object that can be created out of the ground state, in which all the spins are aligned, by reversing the average spin in a ®nite region of space. Although topological considerations indeed allow for these excitations, which we note are fundamentally different from the topologically trivial coreless vortices discussed in ref. 8, we need to know whether such a con®guration is also energetically stable. However, when solving the appropriate Gross±Pitaevskii equation 8 , we ®nd that, in equilibrium, the energy is always minimized by collapsing the skyrmion to zero size. Fortunately, it turns out that for suf®ciently small sizes of the skyrmion a nonequilibrium stability mechanism starts to work. A number of atoms in the centre of the skyrmion, which we denote here as the core atoms, will be trapped by an effective three-dimensional potential barrier, that is, a repulsive shell with a ®nite radius that is induced by the gradients in the spin texture of the skyrmion itself. As the skyrmion shrinks in size, the barrier height of the repulsive shell increases and the radius decreases. This leads to a squeezing of the core atoms and thus to an increase in their energy, which ultimately stabilizes the skyrmion. As mentioned, this is not an equilibrium state of the condensate because the core atoms will tunnel over the barrier and give the skyrmion a ®nite lifetime. We note that our calculations are for a unifor...

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