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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 [H. Zoubi and K. Hammerer, Phys. Rev. A 94, 053827 (2016)]. 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 [R. Van Laer et. al., Nature Phot. 9, 199 (2015) and E. K. Kittlaus, et al., Nature Phot. 10, 463 (2016)]. We explore the possibility of achieving a significant nonlinear phase shift among photons propagating in nanoscale waveguides exploiting interactions among photons that are mediated by vibrational modes and induced through Stimulated Brillouin Scattering (SBS) [H. Zoubi and K. Hammerer, Phys. Rev. Lett. 119, 123602 (2017)].

 

We introduce a configuration that allows slowing down the photons by several orders of magnitude via SBS involving sound waves and two pump fields. We extract the conditions for maintaining vanishing amplitude gain or loss for slowly propagating photons while keeping the influence of thermal phonons to the minimum. The nonlinear phase among two counter-propagating photons can be used to realize a deterministic phase gate.