One of the most basic consequences of topological superconductivity is the fractional Josephson effect, which arises due to the coherent tunneling of single electrons between two superconductors, leading to an ac Josephson effect with half the usual Josephson frequency. In this paper we will review the theoretical and experimental foundations of topological superconductivity and the fractional ...

Time-reversal invariance places strong constraints on the properties of the quantum spin Hall edge. One such restriction is the inevitability of dissipation in a Josephson junction between two superconductors formed on such an edge without the presence of interaction. Interactions and spinconservation breaking are key ingredients for the realization of dissipationless ac Josephson effect on suc...

We discuss how to control dc Josephson current by influencing the structure and nonequilibrium population of Andreev levels via external electrostatic gates, current injection and electromagnetic radiation. In particular we will consider the ”giant” Josephson critical current in ”long” SIS tunnel junctions and the regular and anomalous nonequilibrium Josepson currents in three terminal SNS junc...

We investigate the non-Abelian Josephson effect in F=2 spinor Bose-Einstein condensates with double optical traps. We propose a real physical system which incorporates the non-Abelian Josephson effect and has very different density and spin tunneling characteristics compared with the Abelian case. We calculate the frequencies of the pseudo Goldstone modes in different phases between two traps, ...

Using the pseudospin representation and the SU(2) phase operators we introduce a complex parameter to characterize both infinite and finite superconducting systems. While in the bulk limit the parameter becomes identical to the conventional order parameter, in the nanoscopic limit its modulus reduces to the number parity effect parameter and its phase takes discrete values. We evaluate the Jose...

The macroscopic quantum interference effects in ballistic Josephson microstructures are investigated. The studied system consists of four bulk superconductors (terminals) which are weakly coupled through the mesoscopic rectangular normal metal (two dimensional electron gas). We show that nonlocal electrodynamics of ballistic systems leads to specific current-phase relations for the mesoscopic m...

It has been shown previously that the coupling between two Majorana end states in single-channel superconducting quantum wires leads to the fractional Josephson effect. However, in realistic experimental conditions, multiple bands of the wires are occupied and the Majorana end states are accompanied by other fermionic end states. This raises the question concerning the robustness of the fractio...