Quantum transport through a quantum dot side-coupled to a Majorana bound state pair in the presence of electron-phonon interaction

We theoretically study quantum transport through a quantum dot coupled to\nMajorana bound states confined at the ends of a topological superconducting\nnanowire. The topological superconductor forms a loop and is threaded by a\ntunable magnetic flux, which allows one to control the electron transport in\nthe system. In particular, we investigate phonon-assisted transport properties\nin the device when the central quantum dot interacts with a single long-wave\noptical phonon mode. We find that when the two Majorana bound states are\nunhybridized, the zero-temperature linear conductance has a $2\\pi$ periodicity\nas a function of magnetic flux phase, independent of the electron-phonon\ninteraction, the quantum dot energy, or the finite values of dot-Majorana\ncouplings. For a finite overlap between the Majorana bound states, the linear\nconductance periodicity generally changes to $4\\pi$ either due to a finite\nelectron-phonon coupling strength, or a dot energy level that is tuned away\nfrom the Fermi level. Additionally, the differential conductance periodicity\nchanges from $2\\pi$ to $4\\pi$ when the Majorana bound states hybridize and the\nelectron-phonon coupling is finite. Our results provide insight into transport\nsignatures expected in topological quantum computational platforms that\nintegrate quantum dots as a means for Majorana qubit readout. The energy\nexchange with an environmental bath, here a single phonon mode, significantly\nalters the current signatures expected from Majorana modes.\n