Controllable photon-phonon conversion via the topologically protected edge channel in an optomechanical lattice

We propose a scheme to achieve the periodically modulated Su-Schrieffer-Heeger model based on a one-dimensional optomechanical lattice. We show the energy-eigenvalue spectrum and the winding number to demonstrate two topologically distinct phases of the Su-Schrieffer-Heeger model. Specifically, we realize the photon-phonon conversion process via the topologically protected edge channel with a controllable conversion efficiency. By calculating the fidelities of the photon-phonon conversion, we find that our system is more robust against the on-site defect potential throughout the overall lattice sites than the edge lattice sites. Interestingly, the large defect added into the edge sites can induce additional quantum channels to achieve the photon-photon transfer and the phonon-phonon transfer. Our scheme opens an alternative avenue to investigate the topological phases and the quantum state transfer in optomechanical lattice systems.

Figure 1

Schematic of the 1D optomechanical lattice, which is composed by $N$ unit cells. Each unit cell contains an optical cavity, $a_n$, and a mechanical resonator, $b_n$. The coupling between the mechanical resonator $b_n$ and two adjacent cavity fields is $g_n$.

Figure 2

(a) The energy spectrum of the system. The red line and the blue line represent the $N\mathrm{th}$ and $(N+1)\mathrm{th}$ eigenvalue versus the periodic parameter $\theta$, respectively. (b) Winding numbers for different parameters $\theta$, including black solid line with $\theta =0$, red asterisk with $\theta =0.1\pi$, blue pentagram with $\theta =0.5\pi$, and green dotted line with $\theta =0.7\pi$. (c)–(f) The state distributions of the zero-energy modes are defined as $\left|\right|{\psi }_n{\rangle |}^2$, where $|{\psi }_n\rangle$ is the $n\mathrm{th}$ eigenstate of the system. (c) $\theta =0.001\pi$. (d) $\theta =0.02\pi$. (e) $\theta =0.007\pi$. (f) $\theta =0.002\pi$. The lattice size of the system is $N=5$.

Figure 3

(a) The photon distribution in the first cavity and the phonon distribution in the last mechanical resonator for the evolved final state versus the varying rate $\mathrm{\Omega }$. (b) The detailed patterns of the photon distribution and phonon distribution in the two end sites. The evolution time is $t=2\pi /\mathrm{\Omega }$. The lattice size of the system is $N=5$.

Figure 4

(a)–(f) The different conversion efficiencies between photons and phonons for various varying rates $\mathrm{\Omega }$. (a) $100\%:0\%$ with $\mathrm{\Omega }=0.019049$. (b) $80\%:20\%$ with $\mathrm{\Omega }=0.019172$. (c) $60\%:40\%$ with $\mathrm{\Omega }=0.019231$. (d) $50\%:50\%$ with $\mathrm{\Omega }=0.019259$. (e) $20\%:80\%$ with $\mathrm{\Omega }=0.019347$. (f) $0\%:100\%$ with $\mathrm{\Omega }=0.019473$. (g) and (h) The photon-phonon conversion with the larger varying rate. (g) $\mathrm{\Omega }=3$. (h) $\mathrm{\Omega }=30$.

Figure 5

(a) The photon distribution in $a_1$ and the phonon distribution in $b_n$ for the evolved final state versus the varying rate $\mathrm{\Omega }$. (b) The detailed patterns of the photon distribution and the phonon distribution in the two edge sites. The evolution time is $t=2\pi /\mathrm{\Omega }$. (c) The efficiencies of photon-phonon conversion are $50\%:50\%$ for $\mathrm{\Omega }=0.000833189$. (d) The efficiencies of photon-phonon conversion are $0\%:100\%$ for $\mathrm{\Omega }=0.0008353$. The lattice size of the system is $N=30$.

Figure 6

(a) The fidelity versus the disorder strength with three different conversion efficiencies between photons and phonons. (b) The energy spectra of the SSH model vs $\theta$ for the disorder strength $W=0.6g_0$.

Figure 7

(a) Energy spectrum of the system when the defect is added into the first site with $V=0.3$. (b) The fidelity versus the defect potential with three different conversion efficiencies between photons and phonons. (c) Energy spectrum of the system when the defects are added into the overall sites with $V=0.3$. (d) The fidelity versus the defect potential with three different conversion efficiencies. Other parameter takes $N=5$.

Figure 8

(a) The distribution of gap state versus $\theta$ and the lattice site when the defect is added into the first site with $V=8$. (b) The phonon-phonon transfer when $\mathrm{\Omega }=0.005$ and $V=8$. (c) The distribution of the gap state versus $\theta$ and the lattice site when the defect is added into the last site with $V=8$. (d) The photon-photon transfer when $\mathrm{\Omega }=0.005$ and $V=8$.