In this work, we perform real time simulations for probing the non-Abelian
fusion of a pair of Majorana zero modes (MZMs). The nontrivial fusion outcomes
can be either a vacuum, or an unpaired fermion, which reflect the underlying
non-Abelian statistics. The two possible outcomes can cause different charge
variations in the nearby probing quantum dot (QD), while the charge occupation
in the dot is detected by a quantum point contact. In particular, we find that
gradual fusion and gradual coupling of the MZMs to the QD (in nearly adiabatic
switching-on limit) provide a simpler detection scheme than sudden coupling
after fusion to infer the coexistence of two fusion outcomes, by measuring the
occupation probability of the QD. For the scheme of sudden coupling (after
fusion), we propose and analyze continuous weak measurement for the quantum
oscillations of the QD occupancy. From the power spectrum of the measurement
currents, one can identify the characteristic frequencies and infer thus the
coexistence of the fusion outcomes.Comment: 7 pages, 5 figure