Abstract
Majorana zero-modes bound to vortices in a topological superconductor have a non-Abelian exchange statistics expressed by a non-deterministic fusion rule: When two vortices merge they may or they may not produce an unpaired fermion with equal probability. Building on a recent proposal to inject edge vortices in a chiral mode by means of a Josephson junction, we show how the fusion rule manifests itself in an electrical measurement. A \bm {2\pi}2𝛑 phase shift at a pair of Josephson junctions creates a topological qubit in a state of even-even fermion parity, which is transformed by the chiral motion of the edge vortices into an equal-weight superposition of even-even and odd-odd fermion parity. Fusion of the edge vortices at a second pair of Josephson junctions results in a correlated charge transfer of zero or one electron per cycle, such that the current at each junction exhibits shot noise, but the difference of the currents is nearly noiseless.
Highlights
Vortices in a two-dimensional topological superconductor contain a midgap state, or zeromode, that can be used to store quantum mechanical information in a nonlocal way, protected from local sources of decoherence [1,2,3,4,5]
The Majorana fermion modes in the topological superconductor are not self-conjugate, instead creation and annihilation operators a†, a are related by the particle-hole symmetry relation (3.8)
We have shown how the method of time-resolved and “on-demand” injection of edge vortices proposed in Ref. 15 can be used to demonstrate the non-Abelian fusion rule of Majorana zeromodes
Summary
Vortices in a two-dimensional topological superconductor contain a midgap state, or zeromode, that can be used to store quantum mechanical information in a nonlocal way, protected from local sources of decoherence [1,2,3,4,5]. The fusion of two vortices σ produces a quantum superposition of states ψ and with and without a quasiparticle excitation This is the Majorana fusion rule of nonAbelian anyons, symbolically written as σ ⊗ σ = ψ ⊕. Neither the braiding nor the fusion of vortices has been realized in the laboratory This has motivated a variety of theoretical proposals for methods to demonstrate the appearance of nonAbelian anyons in a topological superconductor [10,11,12,13,14]. The obstacle that these proposals seek to remove, is the need to physically move the zero-modes around.
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