  We develop a systematic method for deriving a quantum optical multi-mode
Hamiltonian for the interaction of photons and phonons in nanophotonic
dielectric materials by applying perturbation theory to the electromagnetic
Hamiltonian. The Hamiltonian covers radiation pressure and electrostrictive
interactions on equal footing. As a paradigmatic example, we apply our method
to a cylindrical nanoscale waveguide, and derive a Hamiltonian description of
Brillouin quantum optomechanics. We show analytically that in nanoscale
waveguides radiation pressure dominates over electrostriction, in agreement
with recent experiments. The calculated photon-phonon coupling parameters are
used to infer gain parameters of Stokes Brillouin scattering in good agreement
with experimental observations.
