Over the past two decades cavity quantum electrodynamics (QED) has the blooming developments and become one of the most promising setups to investigate the interactions between light and matter at the quantum level.^{[1, 2]} The progress has shown that cavity QED offers an interesting tool to implement quantum manipulation technology. For example, cavity-induced transparency^[3] is an electromagnetically induced transparency (EIT)-like effect.^[4] The EIT-like effect provided large valuable contributions and important roles in a variety of quantum processes, such as nonlinear susceptibility modulation,^[5] light switch,^[6] quantum memory,^[7] and quantum communication.^[8]

As a novel extension of the cavity, cavity opto-mechanics system is usually composed of Fabry–Pérot (FP) cavity with one fixed mirror replaced by a mechanical oscillator as a movable mirror, where the optical field in the cavity can couple to the mechanical oscillator via the radiation pressure.^[9] Recently such cavity opto-mechanics has become one of the most promising platform to study the interactions between optical and mechanical modes or between optical fields and atoms, and has attracted increasing interest in both theory and experiment.^{[9, 10]} More importantly, some theories in QED can be further extended to the cavity opto-mechanics,^[10] where many interesting phenomena and effects have been theoretically predicted then experimentally demonstrated, such as opto-mechanically induced transparency (OMIT, EIT-like),^[11–13] frequency transfer,^[14] and normal-mode splitting.^[15] In addition, this system can be applied to studying the cooling of mechanical oscillator,^{[16, 17]} precision measurement,^[18–21] quantum information transfer,^{[22, 23]} etc. When the combination of opto-mechanical system with atoms or the nitrogen–vacancy centers in various hybrid systems allows for light switch,^[24] electro-optic waveform interconnect,^[25] the strong coupling of the mechanical oscillations to the atomic spin^[26] and phonon-induced spin–spin interactions.^[27] Therefore cavity opto-mechanics system has shown significant advantages and may be applicable to future quantum networks.^[10]

The transmission spectroscopy of the cavity is of crucial importance to control the propagation and absorption of optical field in practical application. Therefore, the output of cavity opto-mechanics is widely studied, especially the OMIT effect caused by the destructive interference.^[28] However, the OMIT with a single transparency window in the frequency domain limits the manifold functionalities for nonlinear optics and optical communication.^[10] In recent studies, the single-OMIT has been extended to the double transparency windows in the divers opto-mechanical modes. The double transparency can be induced by two charged nano-mechanical resonators with Coulomb interaction,^[29] the coherent oscillations of the two mechanical oscillators,^[30] two tunneling-coupled resonators,^[31] two coupling opto-mechanical cavities,^[32] and coupling the mechanical resonator to a two-level system.^[33] Further, multiple-OMIT was also found in multiple-cavity opto-mechanical system coupled to one two-level atom.^[34] It is worth noting that the multiple-OMIT in the nanostructure system can be applied in all-optical communication,^[31] all optical quantum network, and multi-channel quantum information processing.^[35] Therefore it is very valuable and significant to research the manipulation of the multiple-OMIT.

In order to break frequency limitation of the transparency window and realize multiple-channel optical communication,^[10] this device should satisfy the following conditions: (i) the multiple transparent windows; (ii) controllable transparent windows; (iii) advantageous integrability on the chip. For this purpose we propose a scheme to manipulate the opto-electromechanically induced transparency (OEMIT) with multi-windows in a hybrid opto-electromechanical device consisted of two tunneling coupled cavities including one FP cavity and one opto-mechanical cavity. The charged mechanical mirror of the opto-mechanical cavity is capacitively coupled to another charged mechanical oscillator by the Coulomb interaction, i.e., a mechanically tunable capacitor. By combining the coherent cavity,^[31] opto-mechanical,^[11] and Coulomb couplings^[29] into a micro-fabricated hybrid opto-electromechanical module, we successively show the single, double, and triple transparency windows, i.e., a controllable multiple-OMIT, whose width can be adjusted by changing the parameters of the system. Here the transparency is a hybrid opto-electromechanically induced transparency. This integrated device including functions of three of induced transparency will excite a development trend and implementation prospect towards the integration quantum module device.

The paper is organized as follows. In Section 2, we introduce the hybrid opto-electromechanical model and theoretical equations. We will discuss OEMIT in our system and propose a scheme to modulate the transparency windows of OEMIT in Section 3. Finally we give the conclusion.

The hybrid opto-electromechanical system (HOEMS) under consideration is shown schematically in Fig. 1. The system consists of an FP cavity tunneling-connected to a cavity opto-electromechanical system, which includes a typical opto-mehcanical cavity and a variable capacitor. As an interface, the nanomechanical oscillator (NMO) not only couples to the cavity field by the radiation pressure but also capacitively interacts with another mechanical oscillator by the Coulomb interaction. The FP cavity is separately driven by a strong pump field with frequency and a weak probe field with frequency . In the rotating frame, the total Hamiltonian of the HOEMS can be written as

Fig. 1.

Figure Option ViewDownloadNew Window

Fig. 1. The schematic diagram of the HOEMS. The FP cavity is tunneling coupled to an opto-mechanical cavity , where the charged NMO as a mechanical mirror of the opto-mechanical cavity is capacitively coupled to another charged NMO by the Coulomb interaction. J is the tunneling coupling strength between the two cavities. The HOEMS is driven by a strong pump field and a weak probe field . The output field at the probe frequency of the HOEMS is .

Considering the effects of noise terms on the cavity fields and NMOs, the Langevin equations of the HOEMS are given by the following expressions^[37]

In order to find the response of the system to the probe frequency, by the output field based on the input–output theory,^{[11, 31]} we can obtain

Now we present the opto-electromechanically induced multi-transparency in the HOEMS and study the manipulation of OEMIT in the different cases. We find that the number and width of the OEMIT windows can be modulated and some interesting phenomena can be observed. Here we employ the parameters from the recent experiment,^{[12, 31]} where the mass m = 20 ng, the frequency , the damping rate , and the wavelength of the cavity field , and the power the pump laser and the single photon coupling strength .

If the tunneling-coupling strength between two cavity fields J = 0, i.e., the system only includes an FP cavity driven by the pump and probe fields, Figure 2(a) shows the usual absorption and dispersion of the optical field with no transparency window. The output field becomes , which only depends the first cavity decay and the detuning .

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. The absorption Re (blue-solid line) and dispersion Im (red-dotted line) as a function of the detuning with no opto-mechanical coupling g = 0 in the different cases (a) , J = 0; (b) , (c) , (d) , .

When the opto-mechanical coupling g is negligible with in Eq. (10), the output field can be rewritten as

Further considered the opto-mechanical coupling and ignoring the Coulomb coupling, the output field can be given by

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. The absorption Re (blue-solid line) and dispersion Im (red-dotted line) as a function of the detuning in the case of and the other parameters are the same as those in Fig. 2(c).

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. Energy level structure of the HOEMS. The number states of photons and phonons are denoted by Ni and ni, respectively. The tunneling coupling strength between and is J, and the effective opto-mechanical coupling strength between and is G. The Coulomb coupling strength between and is λ.

Moreover, when the NMO capacitively couples to the NMO by the Coulomb interaction, the output field is shown in Eq. (12). The absorption of the probe field versus the detuning and the Coulomb coupling strength λ is plotted in Fig. 5(a). With no the Coulomb coupling, the absorption of the probe field is just as Fig. 3 with the double OMIT. Considering the electrical coupling, we find that the triple-OMIT appears. The two-dimensional elevation diagram given in Fig. 5(b) with the corresponding to Fig. 5(a) shows clearly the change of the transparency window as the Coulomb coupling increases, i.e., the width of the central transparency window is significantly widened by the Coulomb coupling. The origin of the triple-OMIT can be explained by the level configuration in Fig. 4. We know that the double-OMIT arising from quantum interference is sensitive to phase disturbances due to the first NMO .^{[29, 31]} The capacitive coupling between both NMOs not only gives an additionally degenerate level for identical NMOs, but also breaks down the symmetry of the double OMIT interference, and thereby produces a spectrally sharp bright resonance within the OMIT line shape.^[29] Then the double OMIT transparency windows are split into triple transparency windows, which is an opto-electromechanically induced transparency, i.e., OEMIT. Moreover, for two different NMOs, the results will be slightly different from those above for identical NMOs.^[29] Therefore, our scheme of the transparency window combines the advantages of the coupling-cavity, opto-mechanically and electromechanically induced transparency.

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. The absorption as a function of the detuning and λ is plotted in panel (a) and the corresponding two-dimensional elevation diagram is given in panel (b), when the Coulomb interaction is added into the system. The other parameters are the same as those in Fig. 3.

In summary, we proposed a scheme to implement the manipulation of the opto-electromechanically induced transparency based on OMIT in a hybrid opto-electromechanical device composed of an FP cavity tunneling-coupled to a hybrid opto-electromechanical system. By studying the probe output field of the FP cavity, we found the OEMIT with the tunable multi-transparency windows. When the tunneling coupling is considered and the opto-mechanical coupling is negligible (g = 0) with the high quality opto-mechanical cavity , we can find a single coupled cavity-induced transparency, where the strong J and weak can enhance the width of the transparency window. Further, the Coulomb coupling is not added into the system considering the case of and , the output field gives the double OMIT with the single window split into two transparency windows due to the opto-mechanical coupling. Finally, the tunneling, opto-mechanical and Coulomb couplings are considered, the triple-OMIT appears with electromechanical induced an additional transparency window due to a degenerate level induced a new interference channel, where the width of transparency window can been widen by increasing λ. Therefore, by studying the effects of the different coupling parameters on the output field, we gave a scheme to modulate the number of transparency windows and its width. We believe that our results would be helpful for further understanding and applying quantum interference effects, especially, in the aspect of the integration quantum module device.