Effects of quantum dot-continuum coupling in topological superconductor-quantum dot-topological superconductor system: implications for quantum dot-based measurements of Majorana qubits and the 0-π transition

Quantum dot coupled to Majorana Zero Modes (MZMs) in one-dimensional topological superconductor-quantum dot-topological superconductor junctions can potentially serve two purposes for the needs of topological quantum computing. First, it is believed that absence of the 0-π transition in the junction can indicate presence of the MZMs and the topological phase. Second, quantum dot can be used as a tool to measure the state encoded by the MZMs. Considering proximitized Rashba nanowire with Zeeman energy as a model for a topological superconductor, here we investigate effect of continuum nanowire modes coupled to a quantum dot in topological superconductor-quantum dot-topological superconductor junctions. Using perturbation theory in the strength of the coupling between the quantum dot and the continuum modes, we find that for realistic parameters coupling to continuum modes does not considerably alter visibility of the quantum dot-based measurements of Majorana qubits. However, we also find that presence of this coupling can lead to a π-junction behavior even in the topological phase which would limit capabilities of the 0-π transition as a diagnostic tool for the presence/absence of MZMs.