**Majorana States in S-TI-S Junctions**

In superconductor-topological insulator-superconductor (S-TI-S) Josephson junctions, Majorana fermion (MF) modes are stable in the TI when there is a phase difference of π across the junction. The phase difference is induced by a perpendicular magnetic field, which penetrates the TI in the form of flux vortices. MFs are localized at the Josephson vortices, and we can control the spacing of them by varying the magnetic field.

We fabricate S-TI-S junctions in the form of hexagonal networks. Via application of currents and voltages to the network, we are able to manipulate the positions of the Josephson vortices, and therefore the positions of the MFs.

**Superconducting Pairing Symmetry **

We study unconventional superconductors (SC) to determine the symmetry of the superconducting order parameter. We fabricate Josephson junctions between an unconventional SC and an s-wave SC, e.g. in the form of corner junctions which are sensitive to the order parameter in orthogonal directions. Using Josephson interferometry, we can detect changes in the junction current-phase relation which is a result of pairing symmetry.

**Methods: Josephson Interferometry**

Applying a perpendicular magnetic field to a Josephson junction induces a phase gradient across the junction which modulates the critical current. We observe a Fraunhofer diffraction pattern in critical current as a result of the junction current-phase relation.

The current-phase relation for an S-TI-S junction contains an added sin(ϕ/2) term that results in even-odd node variation and entering features of Majorana fermions in the diffraction pattern.

**Methods: Scanning SQUID Microscopy**

Pioneered with the help of the Van Harlingen group, scanning SQUID microscopy (SSM) is a technique that involves scanning a SQUID (Superconducting QUantum Interference Device) detector or a pickup coil over a surface to detect local magnetic fields. The magnetic field sensitivity allows us to image magnetic flux vortices and other structures in superconducting materials and arrays. We are building an SSM to image vortices in S-TI-S Josephson junctions which will track the positions of MFs that are bound to the vortices.

**Braiding Schemes for Majorana States**

Exchange Braiding: Interchange Majorana States

We can design multiply-connected junction circuits to exchange MF states in functional quantum operations. An exchange is a topologically-protected operation that results in a change of parity. These operations are critical for achieving topological quantum computing.

We propose two schemes for braiding MFs by exchanging the position of the Josephson vortices that carry them. The first approach involves applying a series of phase pulses in a Y-junction S-TI-S device to exchange vortices. The second uses magnetic field pulse asymmetries to control the separation of vortices, using induced interactions to effectively braid the MFs.

Hybridization: Interacting Majorana States

Hybridization braiding involves moving two MF states close enough together to

interact and evolve their parity.

Hybridization is effected by creating an inhomogeneous magnetic field that brings two vortices and the associated Majorana states closer together.