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We study many-particle phenomena of propagating multi-mode photons andphonons interacting through Brillouin scattering-type Hamiltonian in nanoscalewaveguides. We derive photon and phonon retarded Green's functions and extracttheir spectral functions in applying the factorization approximation of themean-field theory. The real part of the self-energy provides renormalizationenergy shifts for the photons and the phonons. Besides the conventional leaks,the imaginary part gives effective photon and phonon damping rates induced dueto many-particle phenomena. The results extend the simple spectral functions ofquantum optomechanics into continuum quantum optomechanics. We present theinfluence of thermal phonons on the photon effective damping rates, andconsider cases of specific photon fields to be excited within the waveguide andwhich are of importance for phonon cooling scenarios.

Photon and Phonon Spectral-Functions for Continuum Quantum Optomechanics