Microwave spectroscopy of semiconductor-nanowire-based superconducting qubits with Majorana bound states

Recent experimental efforts have focused on replacing the weak link in the Josephson Junction (JJ) of a superconducting qubit by electrostatically-gateable technologies compatible with high magnetic fields. Such alternatives are crucial in order to reach a regime relevant for readout of topological qubits based on Majorana bound states (MBS). We here focus on a system where a semiconducting nanowire forming the JJ is driven to a topological superconductor phase with MBS which coherently interact with the superconducting qubit degrees of freedom. Our fully microscopic theoretical description of this nanowire-based superconducting qubit allows to unveil new physics. This includes the magnetic field dependence of the qubit frequency, which follows the gap closing and the emergence of MBS in the topological phase. In this phase, the periodicity of the qubit spectrum with respect to n_g can be either that of Cooper pairs 2e or single electrons e, depending on microscopic parameters such as the degree of Majorana energy splitting or the ratio of Josephson to charging energy E_J/E_C. Overall, the corresponding microwave spectroscopy presents nontrivial features, including a full mapping of zero energy crossings and fermionic parity switches in the nanowire owing to Majorana oscillations.