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Skyrmions are topological solitons. While a structure with the topology of a line vortex can still exist in two dimensions, Skyrmions are intrinsically 3D objects. Further they require a two component wave function, which can be realised in a BEC by simultaneously trapping two different hyperfine states (section 1.2.1). An important property of a Skyrmion is that the atomic field approaches a constant value at spatial infinity: for a trapped BEC this means sufficiently far from the trap centre. Then we can identify all points at infinity, turning the domain of our wave function into the compact spaceS3_{, the three dimensional spherical surface}

in four dimensions. The two component wavefunction of the BEC (φ1, φ2)T at a

particular position can be represented by the SU(2) matrix U(x) which maps a standard state, such as (1,0), into it:

φ1(x) φ2(x) =n(x)U†(x) 1 0 . (4.1)

Here the total density of atoms is denoted by n(x) = |φ1(x)|2+|φ2(x)|2. The

function Λ defined as Λ = U(x) now represents a mapping Λ : S3 → SU(2) due to the above mentioned identification of all points at infinity. In this case, homotopy theory again predicts an integer winding number [27, 146].

Any SU(2) matrix U(x) can be parametrised as U = exp (iσ·ω(x)), where

x [a.u.] x [a.u.] x [a.u.]

z [a.u.] z [a.u.] y [a.u.]

a) b) c)

Figure 4.1: Skyrmion topology given by Eq. (4.3). For simplicity we assume a step function density profile n with radius Rsph = 8. (a) Relative magnitude of

the ring component (component one) nr=|φ1|/(|φ1|+|φ2|) in the xz plane. The

3D structure is rotationally symmetric around the z axis. The dashed black line indicates a circumference along which nr changes twice fully from 0 to 1. This

demonstrates the equivalence of spatial rotations and component space rotation. (b) Phase ϑ1 of the ring component in the xz plane in units of π. Regions with

nr < 50% are shown in white. (c) Phase ϑ2 of the line component (component

two) in the xy plane in units of π. Regions with nr > 50% are shown in white

and the plotted domain differs from (a) and (b). The dashed black line indicates a spatial rotation which is equivalent to a line component phase rotation. In (b) and (c) the BEC flow associated with the phase structure via vi = ∇ϑi/m is

indicated by white arrows.

ω(x) =λ(x)ˆx(m)_{, for some function λ and the unit vector}

ˆ

x(m) = (cos (mϕ) sinθ,sin (mϕ) sinθ,cos (θ))T. (4.2) ˆ

x(m) _{is given in spherical polar coordinatesr,ϕ,θ and reduces to the radial unit}

vector form = 1. In any case, we can see that physical rotations as well as radial translations are now equivalent to rotations in “component space”.

Explicitly the wave function (4.1) can be written in our coordinates as: φ1(x) φ2(x) =n(x)

cos[λ(x)]−isin[λ(x)] cosθ

−isin[λ(x)] sinθexp(imφ)

. (4.3)

λ determines the detailed shape of the wavefunction, but as long as it meets the boundary conditions λ(0) = 0 and λ(∞) = nπ, the topology will be that of a Skyrmion. The resulting wavefunctions are visualised for n = 1, m = 1

in Fig. 4.1. If we consider the BEC flow, associated via Eq. (2.11) with the phase profiles shown in Fig. 4.1, we can see that a Skyrmion consists of a line vortex carried by component two, which is threaded into the low density region of component one, which carries a ring vortex. Component one hence circulates around component two and through the line vortex core, see Fig. 4.2 for a 3D illustration.

We will therefore refer to component one as the ring component and com- ponent two as the line component. It can be seen in Fig. 4.1 (a) that a spatial rotation on the dashed line is equivalent to a rotation in the two dimensional component space. Other rotations, such as indicated in Fig. 4.1 (c), correspond to phase rotations of one of the components. In general it can be seen that any

SU(2) transformation of the state Eq. (4.3) corresponds to another Skyrmion with rotated symmetry axis (which was the z-axis for the case in Fig. 4.1) [139]. In this sense for a Skyrmion, spatial andSU(2) rotations are equivalent.

If we vary the position x over all space, the wave function (φ1, φ2)T “winds”

over the whole ofSU(2). Similar to the example in the previous section, it is thus not possible to continuously deform the Skyrmion wave function into one with a simpler topology, such as a coreless vortex (A line vortex, whose core is filled by a second, nonflowing Hyperfine component). Our quest for stable Skyrmions later in this chapter is essentially a search for functionsλ(x), for which Eq. (4.3) corresponds to a local minimum of energy.

The Skyrmion winding number can be most conveniently written as [27]:

W = αβν 24π2

d3xTrU(∂αU†)U(∂βU†)U(∂νU†) , (4.4)

where αβν is the completely antisymmetric tensor and repeated indices have to

be summed. For the practical implementation it is more useful to write this expression in terms of the individual components, however the result is unwieldy and hence banished into appendix C.

For the topological structure given by Eq. (4.3) with the appropriate boundary conditions for λ(x), W evaluates to W = nm. This is just the product of the charge m of the line vortex in component two with the charge n of the ring vortex in component one. Higher winding number Skyrmions are thus obtained by increasing the vortex charge in either component.

There are more topological structures that can be realised in multi-component BECs besides line vortices, ring vortices and Skyrmions. Further examples are Dirac monopoles [147] and Alice rings [148], but these are not studied in this thesis.

ring component (1) line component (2)

Figure 4.2: 3D illustration of a Skyrmion in a 87_{Rb BEC (taken from Ref. [27]).}

The central (blue) torus is an isosurface of the line component. An isosurface of the ring component (red) is shown for x < 0: on the y-z plane between the isosurface sections the ring component density is indicated by a colormap from red (lowest) to purple (highest). The circulation associated with each vortex is indicated by the arrows. Each component consists of 4.5×106 _{atoms within a}

7.8 Hz spherical harmonic trap.