Parafermionic Zero Modes in Ultracold Bosonic Systems

Exotic topologically protected zero modes with parafermionic statistics (also called fractionalized Majorana modes) have been proposed to emerge in devices fabricated from a fractional quantum Hall system and a superconductor. The fractionalized statistics of these modes takes them an important step beyond the simplest non-Abelian anyons, Majorana fermions. Building on recent advances towards the realization of fractional quantum Hall states of bosonic ultracold atoms, we propose a realization of parafermions in a system consisting of Bose-Einstein-condensate trenches within a bosonic fractional quantum Hall state. We show that parafermionic zero modes emerge at the end points of the trenches and give rise to a topologically protected degeneracy. We also discuss methods for preparing and detecting these modes.

Figure 1

A quasi-one-dimensional finite trench, i.e., a potential dip, is created by spatially modulating the intensity of the laser used to create the dipole trapping potential confining the atoms to the plane (a cut midway along one trench is shown). Within the trench, the two-dimensional density of trapped bosons deviates from the FQH filling fraction and gives rise to a BEC.

Figure 2

The top view of a BEC trench within a FQH state. The trapping potential is engineered such that the boson density is uniform across the shaded area but vanishes in a small region of size $l$ near the end points. Quantum fluctuations of the BEC couple opposite edges in the shaded region.