  In this work, we present a systematic study of the plasmon modes in a system
of vertically stacked pair of graphene discs. Quasistatic approximation is used
to model the eigenmodes of the system. Eigen-response theory is employed to
explain the spatial dependence of the coupling between the plasmon modes and a
quantum emitter. These results show a good match between the semi-analytical
calculation and full-wave simulations. Secondly, we have shown that it is
possible to engineer the decay rates of a quantum emitter placed inside and
near this cavity, using Fermi level tuning, via gate voltages and variation of
emitter location and polarization. We highlighted that by coupling to the
bright plasmon mode, the radiative efficiency of the emitter can be enhanced
compared to the single graphene disc case, whereas the dark plasmon mode
suppresses the radiative efficiency.
