Microwave-controlled optical double optomechanically induced transparency in a hybrid piezo-optomechanical cavity system

We propose a scheme that can generate a microwave-controlled optical double optomechanical induced transparency (OMIT) in a hybrid piezo-optomechanical cavity system in which a piezoelectric optomechanical crystal AlN-nanobeam resonator is placed in a superconducting coplanar microwave cavity. We show that when an intense microwave field is applied to the superconducting microwave cavity, with a strong pump optical field and a weak probe optical field applied to the optomechanical crystal cavity through the optical waveguide, a double transmission window can be obtained in the weak output probe field. This phenomenon arises because an $N$-type four-level system can be formed in our scheme, because under the effects of the radiation pressure and the piezoelectric interaction, the quantum interference between different energy-level pathways induces the occurrence of the double-OMIT. Our scheme can be applied in the fields of optical switches, high-resolution spectroscopy, coherent population trapping, and quantum information processing in solid-state quantum systems.

Figure 1

(a) Schematic diagram of the hybrid optical and microwave piezo-optomechanical cavity system in which an optomechanical crystal AlN-nanobeam resonator is placed in a superconducting coplanar microwave cavity without contact. The superconducting coplanar microwave cavity is driven by an intense microwave field. The optomechanical crystal AlN-nanobeam resonator is driven by a strong pump field and a weak probe field via the optical waveguide. (b) Schematic diagram for the experimental configuration, where the superconducting coplanar microwave cavity comprises a top electrode (purple), bottom electrodes (yellow), ${\mathrm{SiO}}_2$ substrate (cyan), and silicon substrates (blue).

Figure 2

(a) The absorption $\mathrm{Re}\left[{\varepsilon }_T\right]$ and (b) the dispersion $\mathrm{Im}\left[{\varepsilon }_T\right]$ as a function of $(\delta -{\omega }_b)/{\omega }_b$. The parameters we chose are $g_{\mathrm{om}}/2\pi =1.1$ MHz, $g_{\mathrm{em}}/2\pi =12.3$ MHz, ${\kappa }_a/2\pi =5.2$ GHz, ${\kappa }_c/2\pi =0.025$ MHz, ${\gamma }_b/2\pi =0.096$ MHz, ${\omega }_a/2\pi =194$ THz, ${\omega }_c/2\pi =10$ GHz, ${\omega }_b/2\pi =2.4$ GHz, $P_{\mathrm{pu}}=0.12$ mW, $P_k=0.1$ mW, and ${\mathrm{\Delta }}_a={\mathrm{\Delta }}_c={\omega }_b$. The insets in (a) and (b) show the magnified $\mathrm{Re}\left[{\varepsilon }_T\right]$ and $\mathrm{Im}\left[{\varepsilon }_T\right]$ for the same parametric values, respectively.

Figure 3

(a) The absorption $\mathrm{Re}\left[{\varepsilon }_T\right]$ and (b) the dispersion $\mathrm{Im}\left[{\varepsilon }_T\right]$ as a function of $(\delta -{\omega }_b)/{\omega }_b$, with eliminating the piezomechanical interaction, i.e., $g_{\mathrm{em}}=0$. The other parameters are the same as those in Fig. 2. The insets in (a) and (b) show the magnified $\mathrm{Re}\left[{\varepsilon }_T\right]$ and $\mathrm{Im}\left[{\varepsilon }_T\right]$ for the same parametric values, respectively.

Figure 4

(a) Energy-level structure of the hybrid piezo-optomechanical cavity system, where the states of the levels are designated $|i\rangle (i=1,2,3,4)$. The energy differences between $|1\rangle$ and $|4\rangle$, $|1\rangle$ and $|3\rangle$, and $|1\rangle$ and $|2\rangle$ are the frequencies of cavity $a$, cavity $c$, and mechanical resonator $b$, respectively, and are designated ${\omega }_a$, ${\omega }_c$, and ${\omega }_b$, respectively. ${\omega }_{\mathrm{pr}}$ is equal to ${\omega }_a$, and the detuning events between them are ${\omega }_a-{\omega }_{\mathrm{pu}}={\omega }_c-{\omega }_k={\omega }_b$. (b) Energy-level structure of the hybrid piezo-optomechanical cavity system in the dressed-state picture. $|{\lambda }_{\pm }\rangle$ are the new dressed levels generated by the piezomechanical coupling, with the energy difference of $2g_{\mathrm{em}}$.

Figure 5

The absorption $\mathrm{Re}\left[{\varepsilon }_T\right]$ as functions of $(\delta -{\omega }_b)/{\omega }_b$ and piezomechanical coupling strength $g_{\mathrm{em}}$. The units of $g_{\mathrm{em}}$ are $2\pi \times 3$ MHz, the values of them are $g_{\mathrm{em}}/2\pi =0,3,6,9,12$ MHz, the blue (deep-dark gray) curve, green (dark gray) curve, red (medium gray) curve, cyan (light gray) curve, and purple (shallow-light gray) curve correspond to them, respectively. The other parameters are the same as those in Fig. 2.

Figure 6

The absorption $\mathrm{Re}\left[{\varepsilon }_T\right]$ as functions of $(\delta -{\omega }_b)/{\omega }_b$ and optical pumping power $P_{\mathrm{pu}}$. The units of $P_{\mathrm{pu}}$ are $1\times {10}^{-6}$ W, the values of them are $P_{\mathrm{pu}}=1.0\times {10}^{-9},1.001\times {10}^{-6},2.001\times {10}^{-6},3.001\times {10}^{-6},4.001\times {10}^{-6}$ W, the blue (deep-dark gray) curve, green (dark gray) curve, red (medium gray) curve, cyan (light gray) curve, and purple (shallow-light gray) curve correspond to them, respectively. The other parameters are the same as those in Fig. 2.

Figure 7

The absorption $\mathrm{Re}\left[{\varepsilon }_T\right]$ as functions of $(\delta -{\omega }_b)/{\omega }_b$ and the detuning ${\mathrm{\Delta }}_c$. The units of ${\mathrm{\Delta }}_c$ are $0.05\times {\omega }_b$, the values of them are ${\mathrm{\Delta }}_c/{\omega }_b=0.9,0.95,1,1.05,1.1$, the blue (deep-dark gray) curve, green (dark gray) curve, red (medium gray) curve, cyan (light gray) curve, and purple (shallow-light gray) curve correspond to them, respectively. The other parameters are the same as those in Fig. 2.

Figure 8

The absorption $\mathrm{Re}\left[{\varepsilon }_T\right]$ as functions of $(\delta -{\omega }_b)/{\omega }_b$ and microwave field strength $P_k$, with ${\mathrm{\Delta }}_c=0$. The units of $P_k$ are $5\times {10}^{-7}$ W, the values of them are $P_k=1.0\times {10}^{-9},5.01\times {10}^{-7},1.001\times {10}^{-6},1.501\times {10}^{-6},2.001\times {10}^{-6}$ W, the blue (deep-dark gray) curve, green (dark gray) curve, red (medium gray) curve, cyan (light gray) curve, and purple (shallow-light gray) curve correspond to them, respectively. The other parameters are the same as those in Fig. 2.