  We study theoretically the dipole-dipole interaction and energy transfer in a
hybrid system consisting of a quantum dot and graphene nanodisk embedded in a
nonlinear photonic crystal. In our model a probe laser field is applied to
measure the energy transfer between the quantum dot and graphene nanodisk while
a control field manipulates the energy transfer process. These fields create
excitons in the quantum dot and surface plasmon polaritons in the graphene
nanodisk which interact via the dipole-dipole interaction. Here the nonlinear
photonic crystal acts as a tunable photonic reservoir for the quantum dot, and
is used to control the energy transfer. We have found that the spectrum of
power absorption in the quantum dot has two peaks due to the creation of two
dressed excitons in the presence of the dipole-dipole interaction. The energy
transfer rate spectrum of the graphene nanodisk also has two peaks due to the
absorption of these two dressed excitons. Additionally, energy transfer between
the quantum dot and the graphene nanodisk can be switched on and off by
applying a pump laser to the photonic crystal or by adjusting the strength of
the dipole-dipole interaction. We show that the intensity and frequencies of
the peaks in the energy transfer rate spectra can be modified by changing the
number of graphene monolayers in the nanodisk or the separation between the
quantum dot and graphene. Our results agree with existing experiments on a
qualitative basis. The principle of our system can be employed to fabricate
nano-biosensors, optical nano-switches, and energy transfer devices.
