Manipulation of decoupled Majorana zero modes (MZMs) could enable topologically protected quantum computing. However, the practical realization of the large number of perfectly decoupled MZMs needed to perform nontrivial quantum computation has proven to be challenging so far. Fortunately, even a small number of imperfect MZMs can be used to qualitatively extend the behavior of standard superconducting qubits, allowing for new approaches for noise suppression, qubit manipulation, and readout. Such hybrid devices take advantage of the interplay of Cooper pair tunneling, coherent single-electron tunneling, and Majorana hybridization. Here we provide a qualitative understanding of this system, give analytical results for its ground state energy spanning the full parameter range, and describe potential sensing applications enabled by the interplay between Majorana and Cooper pair tunneling.

• Received 23 September 2023
• Accepted 22 January 2024

DOI:https://doi.org/10.1103/PhysRevB.109.085410