We theoretically study quantum transport through a quantum dot coupled to Majorana bound states confined at the ends of a topological superconductor nanowire. The topological superconductor forms a loop and is threaded by a tunable magnetic flux which allows one to control the electron transport in the system. In particular, we investigate phonon-assisted transport properties in the device when the central quantum dot interacts with a single long-wave optical phonon mode. We find that when the two Majorana bound states are unhybridized, the zero-temperature linear conductance has a $2pi$ periodicity as a function of magnetic flux phase, independent of the electron-phonon interaction, the quantum dot energy, or the finite values of dot-Majorana couplings. For a finite overlap between the Majorana bound states, the linear conductance periodicity generally changes to $4pi$ either due to a finite electron-phonon coupling strength, or a dot level energy which is tuned away from the Fermi level. Additionally, the differential conductance periodicity changes from $2pi$ to $4pi$ when the Majorana bound states hybridize and the electron-phonon coupling is finite. Our results provide insight into transport signatures expected in topological quantum computational platforms which integrate quantum dots as a means for Majorana qubit readout. The energy exchange with an environmental bath, here a single phonon mode, significantly alters the current signatures expected from Majorana modes.
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