Generation of photonic Cooper pairs in nanoscale optomechanical waveguides

In this paper we theoretically show that counterpart photonic Cooper pairs can be achieved in nanoscale waveguides with radiation pressure effect. We demonstrate that due to Brillouin scattering a Stokes photon and an anti-Stokes one can exchange virtual phonons, leading to an effective photon-photon interaction. The effective interaction Hamiltonian of the two photons is identical to that for electrons in the BCS theory, and the photonic counterpart of the superconducting energy gap can appear. Quite different from the conventional stimulated Brillouin scattering the number of photon pairs can be amplified. The result could be used as a photon-pair laser and might have a possible application in integrated optics and quantum information processing in the future.

Figure 1

(a) Schematic of setup: two photon modes with frequencies ${\omega }_1$ and ${\omega }_2$ and wave numbers $k_1$ and $k_2$ propagate in a nanoscale waveguide and experience an interaction by Brillouin scattering involving a phonon mode with frequency $\mathrm{\Omega }$ and wave number $q$. (b) The normal Brillouin scattering: Stokes process (left), in which due to momentum conservation the photon $k$ emits a phonon $q$ and becomes a photon $k-q$, and Anti-Stokes process (right), in which the photon $k$ transfers to $k+q$ by absorbing a phonon $q$. (c) Schematic level scheme of the combined Stokes–anti-Stokes scattering. The red lines denote the pumping photons ${\omega }_{1\left(2\right)}$ and Stokes (anti-Stokes) scattering photon ${\omega }_S$(${\omega }_{\mathrm{AS}}$), while the phonon is described by the blue lines. Green lines are the frequency difference between two photons (Stokes and anti-Stokes) and the phonon. (d) The combined Stokes–anti-Stokes scattering: two photons $k_1$ and $k_2$ experience simultaneously Brillouin scattering induced by interaction via exchanging a virtual phonon $q$.

Figure 2

(a) The dispersion relation of the $\mathbf{H}{\mathbf{E}}_{11}$ mode in the nanoscale waveguide. ${\omega }_D$ denotes the regime in which a photon pair occurs for photons with $k_0$. (b) The dispersion relation of the lowest two branches of the longitudinal phonon modes denoted by the red dashed line and blue line, respectively.