We report the realization of quantum microwave circuits using hybrid superconductor-semiconductor Josephson elements comprised of InAs nanowires contacted by NbTiN. Capacitively-shunted single elements behave as transmon qubits with electrically tuna
ble transition frequencies. Two-element circuits also exhibit transmon-like behavior near zero applied flux, but behave as flux qubits at half the flux quantum, where non-sinusoidal current-phase relations in the elements produce a double-well Josephson potential. These hybrid Josephson elements are promising for applications requiring microwave superconducting circuits operating in magnetic field.
We show how the superconducting phase difference in a Josephson junction may be used to split the Kramers degeneracy of its energy levels and to remove all the properties associated with time reversal symmetry. The superconducting phase difference is
known to be ineffective in two-terminal short Josephson junctions, where irrespective of the junction structure the induced Kramers degeneracy splitting is suppressed and the ground state fermion parity must stay even, so that a protected zero-energy Andreev level crossing may never appear. Our main result is that these limitations can be completely avoided by using multi-terminal Josephson junctions. There the Kramers degeneracy breaking becomes comparable to the superconducting gap, and applying phase differences may cause the change of the ground state fermion parity from even to odd. We prove that the necessary condition for the appearance of a fermion parity switch is the presence of a discrete vortex in the junction: the situation when the phases of the superconducting leads wind by $2pi$. Our approach offers new strategies for creation of Majorana bound states as well as spin manipulation. Our proposal can be implemented using any low density, high spin-orbit material such as InAs quantum wells, and can be detected using standard tools.
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. This is a solution to the problem how to operate on topological qubits when gate voltages are screened by the superconductor.
The Josephson energy of two superconducting islands containing Majorana fermions is a 4pi-periodic function of the superconducting phase difference. If the islands have a small capacitance, their ground state energy is governed by the competition of
Josephson and charging energies. We calculate this ground state energy in a ring geometry, as a function of the flux -Phi- enclosed by the ring, and show that the dependence on the Aharonov-Bohm phase 2ePhi/hbar remains 4pi-periodic regardless of the ratio of charging and Josephson energies - provided that the entire ring is in a topologically nontrivial state. If part of the ring is topologically trivial, then the charging energy induces quantum phase slips that restore the usual 2pi-periodicity.
