Photon propagation in a one-dimensional optomechanical lattice

We consider a one-dimensional optomechanical lattice where each site is strongly driven by a control laser to enhance the basic optomechanical interaction. We then study the propagation of photons injected by an additional probe laser beam; this is the lattice generalization of the well-known optomechanically induced transparency (OMIT) effect in a single optomechanical cavity. We find an interesting interplay between OMIT-type physics and geometric, Fabry-Perot-type resonances. In particular, phononlike polaritons can give rise to high-amplitude transmission resonances which are much narrower than the scale set by internal photon losses. We also find that the local photon density of states in the lattice exhibits OMIT-style interference features. It is thus far richer than would be expected by just looking at the band structure of the dissipation-free coherent system.

Figure 1

Schematic picture of the 1D optomechanical array studied in this work. On each site, the blue (red) circle represents a localized mechanical (photonic) mode; they interact locally via a standard radiation pressure coupling. Photons can hop between neighboring site with hopping strength $J$. A uniform control laser is applied to each site at frequency ${\omega }_L={\omega }_c+\Delta$, where $\Delta$ is the detuning between the laser frequency and cavity photon frequency ${\omega }_c$. For transport, external waveguides are attached to sites 1 and $N+1$; these are used to send and extract probe light at frequency ${\omega }_p$.

Figure 2

(a) Band structure of the bare photons (red dashed curve) and mechanical modes (purple solid curve) for red sideband detuning $\Delta =-{\Omega }_{\mathrm{M}}$ and $G\to 0$. (b) Band structure of the polaritons, same parameters in (a) but now $G=0.1J$. The upper (dashed) curve represents the $+$ polariton band and the lower (solid) curve represents the $-$ polariton band. (c) Variation of the factors ${sin}^2{\theta }_k$, ${cos}^2{\theta }_k$ which describe the photon and phonon parts of the polariton wave functions [c.f. Eq. (9)].

Figure 3

(a) The local cavity photon DOS in the optomechanical array [c.f. Eq. (17)] for frequencies near the energy gap in the coherent band structure; the detuning $-\Delta ={\Omega }_{\mathrm{M}}$ and optomechanical coupling $G=0.01J$ for all the curves (see Fig. 2 for a representative coherent band structure). ${\kappa }_0=0.1J$ for all the curves except the purple curve (dot-dashed) which corresponds to ${\kappa }_0=\gamma =0$. The different cooperativities in the figure are achieved by tuning the mechanical decay rate $\gamma$. One sees that if the effective cooperativity $C_J$ is larger than $\sim 1$, the gap in the coherent-system density of states persist, even if its magnitude is smaller than ${\kappa }_0$. (b) The local reflection probability $R\left[\omega \right]={\left|r\left[\omega \right]\right|}^2$ [Eq. (23)] at frequencies near ${\Omega }_{\mathrm{M}}$ (center of the coherent-system gap) for a single weak probe on site 1 with ${\kappa }_{\text{ex,1}}=0.1J$. Curves correspond to the legend in panel (a). $R\left[\omega \right]$ directly mirrors the behavior of the DOS in (a).

Figure 4

Transmission probability of probe light from sites 1 to 11 as a function of probe frequency, for the case of a relatively weak coupling to probe waveguides (${\kappa }_{\text{ex,1}}={\kappa }_{\text{N,ex}}=0.01J$), and for relatively large internal cavity loss (${\kappa }_0=0.1J$). We have picked parameters such that the bare photon and phonon bands just touch at $k=0$ (i.e., ${\delta }_0=-\Delta -{\Omega }_{\mathrm{M}}-2J=0$), and taken $G=0.1{\kappa }_0,\gamma ={10}^{-6}{\kappa }_0$. (a) Shows the contribution from the lower, phononlike polariton band (using a logarithmic $\omega$ axis); (b) shows on the same scale the variation of the photonic component of the $-$ polariton as a function of frequency; (c) shows the transmission probability from the photonlike upper polariton band. Surprisingly, both polariton bands can make comparable contributions to the photon transmission in this limit; this is due to the fact that the optical damping of the lower polariton band ${\Gamma }_{\mathrm{opt}}$ is much larger than $\gamma$.

Figure 5

Transmission probability for probe light from sites $j=1$ to $j=21$, as a function of the probe beam frequency $\omega$, in the regime of a strong coupling to the external probe waveguides: ${\kappa }_{\mathrm{ex}}=5J$. We also take a control laser detuning yielding ${\delta }_0=0$ [c.f. Eq. (8)], implying that the bare photon band just touches the bare phonon band at $k=0$. (a) (Blue curve) Transmission spectrum without optomechanical coupling, $G=0$, and with ${\kappa }_0=0.01J$. One sees well-developed Fabry-Perot style resonances associated with the bare photon band. (Red curve) Same, but now with $G={\kappa }_0/2$ and $\gamma ={10}^{-5}{\kappa }_0$. The main effect of increasing the optomechanical coupling is the emergence of an extremely sharp transmission feature near $\omega \sim {\Omega }_{\mathrm{M}}$, associated with the phononlike $-$ polariton band. (b) and (c) Zoom-in of the $-$ polariton band features in (a). One again sees Fabry-Perot style resonances, leading to features in the transmission that are much narrower than ${\kappa }_0$.

Figure 6

Transmission probability for probe light from sites $j=1$ to $j=21$ for a finite optomechanical array of 21 cells, as a function of the probe beam frequency $\omega$. (a) (Black curve) Transmission spectrum without optomechanical coupling, $G=0$. (Red curve) Transmission spectrum with $G={\kappa }_0/2$. All the other parameters are the same as in Fig. 5. (b) and (c) Zoom-in of the $-$ polariton band features in (a). One again sees Fabry-Perot style resonances. The location and width of the Fabry-Perot resonances are almost the same as in the infinite array in Fig. 5, whereas the resonance heights are different.