Photon and phonon spectral functions for continuum quantum optomechanics

We study many-particle phenomena of propagating multimode photons and phonons interacting through a Brillouin scattering type Hamiltonian in nanoscale waveguides. We derive photon and phonon retarded Green's functions and extract their spectral functions in applying the factorization approximation of the mean-field theory. The real part of the self-energy provides renormalization energy shifts for the photons and the phonons. Besides the conventional leaks, the imaginary part gives effective photon and phonon damping rates induced due to many-particle phenomena. The results extend the simple spectral functions of quantum optomechanics into continuum quantum optomechanics. We present the influence of thermal phonons on the photon effective damping rates and consider cases of specific photon fields to be excited within the waveguide and which are of importance for phonon cooling scenarios.

Figure 1

Nanoscale waveguide of circular cross section with radius $a$, made, e.g., of silicon or silica and embedded in free space. The length can be of several centimeters.

Figure 2

The photon dispersion ${\omega }_k={\omega }_0+v_{g}k$ vs $ka$ for the lowest branch in the linear zone. Here, ${\omega }_0/\left(2\pi \right)\approx {10}^{14}$ Hz, with $a=250\mathrm{nm}$, and the group velocity is $v_g\approx c/5$.

Figure 3

The acoustic phonon dispersion ${\mathrm{\Omega }}_q=v_{a}q$ vs $qa$. Here, the sound velocity in silicon is $v_a=8433$ m/s. The horizontal dashed line is for the vibrational mode of frequency ${\mathrm{\Omega }}_v/\left(2\pi \right)\approx 10$ GHz.

Figure 4

The photon frequency shift ${\mathrm{\Delta }}_k^M\left(\omega \right)$ vs $(\omega -{\omega }_{k+q})$ for $ka=2$.

Figure 5

The photon effective damping rate ${\mathrm{\Lambda }}_k^M\left(\omega \right)$ vs $(\omega -{\omega }_{k+q})$ for $ka=2$.

Figure 6

The photon SF, ${\mathcal{S}}_k^{\mathrm{phot}}\left(\omega \right)$, vs $(\omega -{\omega }_k)$ for $ka=2$.

Figure 7

The photon frequency shift ${\mathrm{\Delta }}_k^{EM}\left(\omega \right)$ vs $(\omega -{\omega }_{k_0})$ for $ka=k_{0}a=2$.

Figure 8

The photon effective damping rate ${\mathrm{\Lambda }}_k^{EM}\left(\omega \right)$ vs $(\omega -{\omega }_{k_0})$ for $ka=k_{0}a=2$.