Optomechanically induced transparency and possible application in a quadratic-coupling optomechanical system with an induced electric eld

Tunable optomechanically induced transparency (OMIT) with an induced electric field (IEF) in a quadratic-coupling optomechanical system is theoretically investigated. The system transmission rate under different controlling parameters has been discussed. It is revealed that both phase and group delay of the probe field can be adjusted by the IEF and pump field. Such a system may be used in tunable optical buffer, IEF detector, modulator or other optical devices.

On the other hand, many interesting phenomena have been studied thoroughly, such as quantum entanglement [6,[23][24][25], high-order sidebands [26][27][28][29][30][31], optical bistability and multibility [16,[32][33][34][35][36][37][38], OMIT [14,, and so on. One interesting review of OMIT is by Xiong et al. [48] which has examined the fundamentals and utilities of OMIT. Many different physical mechanisms of OMIT allow it potential application in future quantum networks such as filter, optical buffering, amplification, and so on. Especially, for quadratic coupling between the optomechanical cavity and mechanical resonator [10,[50][51][52][53][54], OMIT has also been explored involving two-photon process. Zhang et al. [54] presented double OMIT in a compound optomechanical system due to the linear and quadratic coupling between the cavity and the membrane. Khan et al. [58] recently reported controllable OMIT in a two-cavity optomechanical system due to the interference between the two possible routs for excitations. Liao et al. [59] investigated OMIT in a hybrid optomechanical system with Kerr medium which could be tuned from slow light to fast light. In addition, Huang et al. [63] studied the Fano resonance in a quadratic coupling optomechanical system with a nonlinear Kerr medium.
In comparison with previous work, coulomb-interaction effect [28] and magnetic-field sensing [62] in an optomechanical system are demonstrated. To the best of our knowledge, few research has been carried out on the IEF interacting with the optomechanical system. In this paper, we investigate OMIT with IEF which originates form an energized solenoid in a quadratic-coupling optomechanical system. We also discuss its transmission rate and possible applications such as optical buffer, IEF detector, and so on. R dB E D dt = . In our system, assuming the membrane position is located exactly at an antinode of the intracavity field, we can ignore the linear coupling between the cavity field and the membrane displacement. And we just need consider the probe field optical response to the quadratic interaction by the radiation pressure in the optomechanical system [53,63]. In the rotating frame at the frequency of the pump frequency l  , the system Hamiltonian can be written as [28,53,63

Model and theory
is the detuning between the optical cavity (probe laser) and the pump laser. The second term denotes the free Hamiltonian of the mechanical resonator with m  , p and q being the resonator frequency, momentum and position operator respectively. The third term denotes the coupling between the IEF and NR [28]. The fourth term denotes the coupling between optical cavity and the mechanical resonator with quadratic coupling strength g . The fifth (last) term denotes the interaction between the optical cavity and the pump (probe) laser.
Considering the decay terms, one can obtain the Heisenberg-Langevin equations of the operators as follows, where the decay rate for the mechanical resonator (  ) and the cavity (  ) are introduced phenomenologically.
To study OMIT, the steady-state solutions of the expectation values we can obtain the following mean value equations from Eqs. 2 [10,50,51], And Eqs. (3)can be transformed into the following, where, In addition, we can also calculate the system group delay by [54], where arg[ ( )] p t  is the phase of the transmitted field.

Results and discussions
In the following, we will investigate the OMIT of our system. The system parameters are chose experimentally as follows: 10 10 QC − =

Possible Application
OMIT, known as the destructive interference between the probe field and the anti-Stokes scattering field, can be applied in many ways such as photon blockade, ground state cooling of the mechanical resonator and so on. In the following, we will discuss possible application in the slow light，IEF detector and light modulator for our system.  down with the increasing of k E . Such a trend also can be found in Fig. 5(b). So, we can adjust the system group delay by changing the IEF strength and the coupling field. Such a result may be used to design tunable optical buffer.
To design an IEF detector and optical modulator, Fig. 7   ). So, we can get a linear modulator in this scope, and such a system may be attached an electrostatic field firstly to set up an operating point. And the field strength may be 0.24 / NC (correspondingly =0.56 T ) seeing Fig. 7. In Fig.  7, when we input a sine IEF field, the output is a sine light intensity. So it may be an optical modulator. What is more, the sine IEF field can be produced by applying a cosine current in the energized solenoid. So, it may be a differentiator for the input cosine current.

Conclusion
In summary, we have theoretically investigated the transmission rate, group delay and t Operating point Input IEF signal Output light intensity t possible application in an quadratic-coupling optomechanical system with an IEF. Solving the nonlinear Heisenberg-Langevin equations approximately, the expression of the probe field and group delay can be obtained. The system transmission rate under different controlling parameters, such as the IEF strength k E , the pump field l E , the quadratic coupling strength g , has been studied. The phase and group delay can be adjusted by the IEF and the coupling field. Such a system may be used as an optical buffer, IEF detector and optical modulator. (see Manuscript le for gure legend)