Unveiling nontrivial fusion rule of Majorana zero mode using a fermionic mode

Fusing Majorana zero modes (MZMs) leads to multiple outcomes, a property unique to non-Abelian anyons. The successful demonstration of this nontrivial fusion rule would be a hallmark for the development of topological quantum computation. Here, we propose that such fusion can be achieved by simply attaching a fermionic mode to a single Majorana zero mode. By modulation of the energy level of the fermionic mode (FM) as well as its coupling with MZM in different sequences, we show that a zero or integer charge pumping can be realized when different fusion loops are chosen. Such fusion loops are intimately related with the nontrivial fusion rule of Majorana modes and are solely determined by the crossings at zero energy in the parameter space. Finally, we demonstrate our proposal in a nanowire-based topological superconductor coupled to a quantum dot. We show that charge pumping is robust for MZMs in the realistic system irrespective of the initial condition of FM state. A key feature of Majorana zero modes (MZMs) is their non-Abelian fusion rule, characterized by multiple outcomes. Here, the authors introduce a minimal setup where coupling a tunable fermionic mode to a single MZM allows the control of fusion loops, yielding distinct charge pumping that serves as a direct experimental signature.