Topological superconductors are novel phases of matter, that are not only exciting from the purely scientific point of view, but could also pave the way to a fault-tolerant quantum computation. These states are known to host exotic low-energy excitations termed Majorana zero modes, which are inherently non-local, and thus would allow to store and manipulate quantum information in a highly robust manner. While these states of matter can, in principle, emerge intrinsically in a bulk of a material, the dominant fraction of both theoretical and experimental effort in the field is directed at their realizations in semiconducting-superconducting heterostructures. Such hybrid structures could allow for a high level of control in device fabrication.

### Realizations of topological superconductivity in one-dimensional systems

In a number of works, my collaborators and I have proposed possible experimental setups in which one-dimensional topological superconducting states can form, and studied their unique features and experimental signatures. Among these, we have proposed an entirely novel route to topological superconductivity, using a planar Josephson junction geometry. In this geometry, two conventional s-wave superconductors are placed along the two sides of a two-dimensional semiconducting system, in a way that allows for the relative phase between the superconductors to be tuned. This extra knob allows the topological phases to be accessed in a more straightforward manner with less fine-tuning of other experimental parameters. Following our proposal, signatures of topological superconductivity in such devices have been observed experimentally.

**Read more:**

- Falko Pientka, Anna Keselman, Erez Berg, Amir Yacoby, Ady Stern, and Bertrand I. Halperin, Topological superconductivity in a planar josephson junction, Phys. Rev. X, 7:021032, May 2017
- Anna Keselman, Liang Fu, Ady Stern, and Erez Berg, Inducing time-reversal-invariant topological superconductivity and fermion parity pumping in quantum wires, Phys. Rev. Lett., 111:116402, Sep 2013

### Spectral response of a Majorana-transmon qubit

Theoretical schemes proposed for probing and manipulation of topological qubits (qubits based on topologically protected Majorana modes) often require their embedding in standard superconducting qubit circuits. It is therefore important to understand how the dynamics of the superconducting circuit is modified when such semiconductor-superconductor heterostructures are used. Studying the dynamics of a capacitively shunted semiconductor-based Josephson junction, we found that the response of such a device can be significantly altered compared to a conventional superconducting qubit, both due to the trivial low energy fermionic degrees of freedom naturally present in these devices as well as to the topological degrees of freedom comprising the topological qubit.

**Read more:**

- Anna Keselman, Chaitanya Murthy, Bernard van Heck, and Bela Bauer, Spectral response of Josephson junctions with low-energy quasiparticles, SciPost Phys., 7:50, 2019

### Gapless symmetry-protected topological phases in one dimension

The topological protection of the Majorana modes present at the ends of one-dimensional topological superconductors relies on the bulk of the system being gapped. This is indeed the case in semiconductor-superconductor heterostructures in which a one-dimensional semiconducting system is proximity coupled to a bulk superconductor. A natural question to ask is whether topological phases can exist in purely one-dimensional systems in which the tendency towards superconductivity is intrinsic to the one-dimensional material. Since superconductivity cannot exist in a purely one-dimensional system, due to the large quantum phase fluctuations that destroy the superconducting order parameter, one could assume that the topological properties would be lost.

We have shown that although quantum fluctuations destroy the gap in the system, in presence of symmetries, topologically protected end states can still prevail.

Read more:

- Anna Keselman, Erez Berg, and Patrick Azaria, From one-dimensional charge conserving superconductors to the gapless Haldane phase, Phys. Rev. B, 98:214501, Dec 2018
- Anna Keselman and Erez Berg, Gapless symmetry-protected topological phase of fermions in one dimension, Phys. Rev. B, 91:235309, Jun 2015