Quantum double structure in cold atom superfluids

Abstract

The theory of topological quantum computation is underpinned by two important classes of models. One is based on non-abelian Chern-Simons theory, which yields the so-called $\rm{SU}(2)_k$ anyon models that often appear in the context of electrically charged quantum fluids. The physics of the other is captured by symmetry broken Yang-Mills theory in the absence of a Chern-Simons term, and results in the so-called quantum double models. Extensive resources have been invested into the search for $\rm{SU}(2)_k$ anyon quasi-particles; in particular the so-called Ising anyons ($k=2$) of which Majorana zero modes are believed to be an incarnation. In contrast to the $\rm{SU}(2)_k$ models, quantum doubles have attracted little attention in experiments despite their pivotal role in the theory of error correction. Beyond topological error correcting codes, the appearance of quantum doubles has been limited to contexts primarily within mathematical physics, and as such, they are of seemingly little relevance for the study of experimentally tangible systems. However, recent works suggest that quantum double anyons may be found in spinor Bose-Einstein condensates. In light of this, the core purpose of this article is to provide a self-contained exposition of the quantum double structure, framed in the context of spinor condensates, by constructing explicitly the quantum doubles for various ground state symmetry groups and discuss their experimental realisability. We also derive analytically an equation for the quantum double Clebsch-Gordan coefficients from which the relevant braid matrices can be worked out. Finally, the existence of a particle-vortex duality is exposed and illuminated upon in this context.

Figures (10)

• Figure 1: The Aharonov--Bohm experiment. An electron $e$ is encircling a magnetic flux tube $\Phi$, where B stands for magnetic field strength and $\alpha$ is the cross-sectional area.
• Figure 2: Topological equivalence of the spaces $\mathds{R}^3 \backslash \alpha \times \mathds{R}$ and $\mathds{R}^2 \backslash \alpha$: the space $\mathds{R}^3 \backslash \alpha \times \mathds{R}$ may be continuously deformed vertically to the plane $\mathds{R}^2 \backslash \alpha$.
• Figure 3: a)-c) Connection of two loops corresponding to winding $n$ and $m$ resulting in a single loop corresponding to winding $n+m$. d)-f) The reversed process of a)-c) where a single loop corresponding to winding $n+m$ splits into two distinct loops corresponding to windings $n$ and $m$, respectively.
• Figure 4: Phase winding of the wave function as a topological quasi-particle excitation with flux is encircled. The wave function $\psi(r)$ accumulates a $\rm{U}(1)$ phase $e^{i\theta}$ (right) as the topological defect is encircled (left), as a function of the path $\gamma$.
• Figure 5: Braiding three Ising anyons. As the positions of the anyons are permuted in the plane their world lines form a braid in space-time. When the braiding is performed two of the anyons are fused to measure the state of the topological qubit.