Tunable double optomechanically induced transparency in an optomechanical system

We study the dynamics of a driven optomechanical cavity coupled to a charged nanomechanical resonator via Coulomb interaction, in which the tunable double optomechanically induced transparency (OMIT) can be observed from the output field at the probe frequency by controlling the strength of the Coulomb interaction. We calculate the splitting of the two transparency windows, which varies near linearly with the Coulomb coupling strength in a robust way against the cavity decay. Our double-OMIT is much different from the previously mentioned double-EIT or double-OMIT, and might be applied to measure the Coulomb coupling strength.

Figure 1

Schematic diagram of the system. A high-quality Fabry-Pérot cavity consists of a fixed mirror and a movable mirror ${\text{NR}}_1$. ${\text{NR}}_1$ is charged by the bias gate voltage $V_1$ and subject to the Coulomb force due to another charged ${\text{NR}}_2$ with the bias gate voltage $-V_2$. The optomechanical cavity of the length $L$ is driven by two light fields, one of which is the pump field ${\varepsilon }_l$ with frequency ${\omega }_l$ and the other of which is the probe field ${\varepsilon }_p$ with frequency ${\omega }_p$. The output field is represented by ${\varepsilon }_{\text{out}}$. $q_1$ and $q_2$ represent the small displacements of ${\text{NR}}_1$ and ${\text{NR}}_2$ from their equilibrium positions, with $r_0$ the equilibrium distance between the two NRs.

Figure 2

(a) The absorption $\text{Re}\left[{\varepsilon }_T\right]$ and (b) the dispersion $\text{Im}\left[{\varepsilon }_T\right]$ as functions of ${\delta }_{\omega }/{\omega }_1$ under the Coulomb interaction. (c) The absorption $\text{Re}\left[{\varepsilon }_T\right]$ and (d) the dispersion $\text{Im}\left[{\varepsilon }_T\right]$ as functions of ${\delta }_{\omega }/{\omega }_1$ in the absence of the Coulomb interaction. ${\delta }_{\omega }=\delta -{\omega }_1$ is the detuning from the central line of the sideband, ${\lambda }_l=1064$ nm, $L=25$ mm, ${\omega }_1={\omega }_2=2\pi \times 947\times {10}^3$ Hz, the quality factor $Q_1=\frac{{\omega }_1}{{\gamma }_1}(=Q_2=\frac{{\omega }_2}{{\gamma }_2})=6700,m_1=m_2=145$ ng, $\kappa =2\pi \times 215\times {10}^3\mathrm{Hz},{\wp }_l=2$ mW, and $\lambda =8\times {10}^{35}$ $\text{Hz}/{\text{m}}^2$ [27].

Figure 3

Schematic of the energy-level diagram in the cavity optomechanical system, where $|N\rangle ,|n_1\rangle$ and $|n_2\rangle$ denote the number states of the cavity photon, and ${\text{NR}}_1$ and ${\text{NR}}_2$ phonons, respectively. $|N,n_1,n_2\rangle \leftrightarrow |N+1,n_1,n_2\rangle$ transition changes the cavity field, $|N+1,n_1,n_2\rangle \leftrightarrow |N,n_1+1,n_2\rangle$ transition is caused by the radiation pressure coupling, and $|N,n_1+1,n_2\rangle \leftrightarrow |N,n_1,n_2+1\rangle$ transition is induced by the Coulomb coupling [23, 46].

Figure 4

The absorption $\text{Re}\left[{\varepsilon }_T\right]$ as functions of ${\delta }_{\omega }/{\omega }_1$ and $\lambda$ (units of ${\lambda }_0=8\times {10}^{35}$ $\text{Hz}/{\text{m}}^2)$. Other parameters take the same values as in Fig. 2.

Figure 5

The separation $d$ (units of ${\omega }_1)$ between the two minima in the absorption spectrum as a function of the coupling strength $\lambda$ (units of ${\lambda }_0=8\times {10}^{35}\text{Hz}/{\text{m}}^2)$. Other parameters take the same values as in Fig. 2.

Figure 6

The absorption $\text{Re}\left[{\varepsilon }_T\right]$ as a function of ${\delta }_{\omega }/{\omega }_1$ with different cavity decay rates, $\kappa =\pi \times 215\times {10}^3\text{Hz}$ (green dashed line), $\kappa =2\pi \times 215\times {10}^3\text{Hz}$ (red dotted line), $\kappa =3\pi \times 215\times {10}^3\text{Hz}$ (blue solid line). Other parameters take the same values as in Fig. 2.

Figure 7

The absorption $\text{Re}\left[{\varepsilon }_T\right]$ as a function of ${\delta }_{\omega }/{\omega }_1$ for identical and different NR frequencies. Other parameters take the same values as in Fig. 2.