Abstract
We show how to exchange (braid) Majorana fermions in a network of superconducting nanowires by control over Coulomb interactions rather than tunneling. Even though Majorana fermions are charge-neutral quasiparticles (equal to their own antiparticle), they have an effective long-range interaction through the even–odd electron number dependence of the superconducting ground state. The flux through a split Josephson junction controls this interaction via the ratio of Josephson and charging energies, with exponential sensitivity. By switching the interaction on and off in neighboring segments of a Josephson junction array, the non-Abelian braiding statistics can be realized without the need to control tunnel couplings by gate electrodes.
Highlights
The double sum couples Majoranas in neighboring islands by tunneling. Both the Coulomb and tunnel couplings depend on the fluxes through the Josephson junctions, but in an entirely different Coulomb way: the coupling tunnel coupling varies slowly varies rapidly ∝ exp[−4√(E0/E
The answer is that the state of a pair of Majorana fermions in a superconducting island depends on the parity of the number of electrons on that island, and it is this dependence on the electrical charge modulo 2e that provides an electromagnetic handle on the Majoranas
The Coulomb coupling can be made exponentially small by passing Cooper pairs through a Josephson junction between the island and a bulk superconductor
Summary
The basic building block of the Josephson junction array is the Cooper pair box [23, 24], see figure 1, consisting of a superconducting island (capacitance C) connected to a bulk (grounded) superconductor by a split Josephson junction enclosing a magnetic flux. Since P contains the product of all the Majorana operators on the island, the constraint (5) effectively couples distant Majorana fermions—without requiring any overlap of wave functions. The phase φ (modulo 2π ) has small zero-point fluctuations around the value φmin = 0, which minimizes the energy of the Josephson junction, with occasional 2π quantum phase slips. √ 2EC due to zero-point fluctuations of the phase This offset does not contain the Majorana operators, so it can be ignored. The term −U P due to quantum phase slips depends on the Majorana operators through the fermion parity. This term acquires a dynamics for multiple coupled islands, because the fermion parity of each individual island is no longer conserved
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