Tunable induced transparency and Fano-resonance in double cavity optomechanical system

We analyze optomechanically induced Transparency and asymmetric Fano-line shape Profi le in a two-mode cavity system, coupling at weak and strong coupling regimes. The model system consists of one mechanical mode and two optical modes. The transmission shows nonreciprocal behavior. Both the forward transmission and backward refl ection for the system are analyzed for both optic-optic and mechanical-optic cavities by considering various system parameters. The output spectra lead to sharp asymmetric Fano-resonance and tunable transparency. Double line-shape profi le is observed in the output Spectrum. Our proposal provides a new platform for application in quantum telecommunications and a photonic device like optical Switches.


Introduction
Induced transparency with detuning parameters for optical systems has the impact to be used for potential applications such as power switching, quantum computing, etc. Different types of induced transparency are electromagnetically, optomechanically, optically, coupled mode induced transparency, etc. In this regard, we have discussed the tunable line-shaped fano pro ile in a double cavity optomechanical system.
In cavity optomechanics, the investigation of Controlled induced transparency and Fano-resonance clearly describe the controlled behavior of Electromagnetically induced transparency (EIT) for a hybrid Bose-Einstein condensate (BEC) and signi icantly uses in quantum, engineering [1]. Electromagnetically induced transparency a phenomenon of direct development of quantum systems exclusively observed [2,3] and inds tremendous application [4,5]. Tunable asymmetric Fano property and optically induced transparency (OIT) for weak ield clearly show the application in telecom system [6], in double EIT phenomena with lab-application [7] switches, sensors, and selection of single-mode laser [8]. A Fano-like asymmetric resonance is induced with time [9] and shows a description of Ugo-Fano [10], nanoscale Structure [11,12]. The purpose of this type of research has many applications in ultra-slow wave propagation [13], Single-photon source [14], Photo Switch [15], Quantum wavelength conversion [16], Optical narrow cavity system like microresonator, four-wave mixing [17], Quantum Trampling [18], ultra-high-precision assessment [19,20]. Squeezing of a mechanical observation in the sideband state [21,22] can't be solved in the traditional Opto-mechanical way.
It requires a new technique. Chen, et al. [23] build a new technique in which a slow light is realized in BEC. A tunable Fano-resonance can be achieved via the Passive -Passive regime [6]. In recent times the Fano-resonance property has been shown for double cavity con iguration [24]. The induced transparency and Fano-resonance have been achieved by the study of absorption and dispersion pro iles for output ield probe frequency [25]. The correlation function has been calculated [26]. Controlled behavior of EIT has been shown for transverse ields [1]. Fano-resonance has also been presented in a hybrid cavity system for different parameters [27,28]. However, to the best of our knowledge multiple Fanoresonances useful in three-dimensional Plasmon ruler [29], highly directive antenna [30], Humble Waveguide [31]. EIT is a result of the Interference effect which is known as Fano interference [32,33]. Fano resonance and EIT Occurs in single silica -iber taper [34]. In Quantum optics it has also been shown that by using a microcavity it is possible to design an Kerr type nonlinear strength induced by optomechanical coupling.
Considering the effect of quantum noise term the Heisenberg-Langevin equations for the ield modes are written as † 2 1 The perturbation operators satisfy the following correlations Where the equation [3] satis ies the following equation.
For this purpose, we use the following parameters To examine the expectation values of small quantum luctuations of the above equation we use the matrix formation to calculate the luctuations components of Optic and mechanical ield's modes for solving the Eigenvalue equation. We obtain the luctuation parameters which help us to calculate the forward and backward re lection pro ile.
After matrix transformations, we get For this calculation, we have used the constants Here α β represents the input, output signal information respectively.
Applying standard input-output theory for this model So in equation (10) both the input and output ields have their frequency components. The irst two indicate input and output ield frequency and the third one is for additional frequency component 2ω c -ω p called stokes ield frequency [37]. To study forward power transmission and backward power re lection we are interested in the component of output ield frequency.
For the probe ield frequency, the ratio of output to input ield amplitude for forward transmission and backward re lection is given by

Results and discussions
In this section, we have discussed the transmission of the probe ield with respect to system parameters. We present different situations by using forward and backward re lection rates as a function of normalized probe detuning and different ield parameters. The tunable external laser having 1350 nm band is used as control or pump laser and other with 1450 nm band is used as probe laser. A cavity resonator can be designed from silica. This setup can emit photons in the wavelength band 1500 nm band for a laser driving wavelength 900 nm or 1350 nm bands. By slit change of the resonator gap the total loss rate can be tuned via external coupling loss rate. The effective gain rate in the cavity is ≈ -2π × 10.5 MHz. The photon tunneling rate between the two cavities is ≈ -2π × 5.5 MHz, and it can be timbre by changing the gap between the cavities. The optical kerr medium strength ≈ -2π × 1.05 MHz. The gain to loss ratio for this cavity system varies from -3 to +3 [40][41][42][43][44][45][46]. We propose the transmission and re lection pro iles over critical region 1 μ 2  . Figures 2-4 represent photon tunneling strength on the forward transmission and backward re lection rate for the double cavity mechanical system. It is seen from igure 2 that forward transmission rate T F has a single transparent peak around ∆ p /k a = 0 and two dips are there on both sides of it. So it represents symmetric dip peak -dips spectral structure and con irms the IT effect. In igure 3 if we increase the cavity coupling strength λ = 0.4, and Hence one may tune the IT by increasing detuning parameter and coupling strength (here re lection rate is very small due to the small gain to the loss ratio). If we choose coupling strength 1.67 ( Figure 3) the re lection pro ile splits into two peaks and expiates the forward rate and the separation between two peaks depends on the tunneling strength and weak coupling parameters. Next, we have studied the variation of forward transmission rate with strength.     Figures 4-7 show the variation of forward transmission rate with probe strength for the Optic-Optic system. Now we have plotted line shape transmission spectra for a double Optomechanical system. In igure 5 we have taken Optic cavity detuning ∆ a = -0.01k a; the plot of T F has a sharp peak around the center i.e. ∆ p = 0 and two symmetric dips around both sides of the peak. In igure-6 all the parameters remain the same as in igure 5 except the passive cavity detuning ∆ a = -0.02k a . There is an increase in transparency peak and the dips nature are asymmetric. It is also called a two-sided coupled cavity system. The frequency of this line always is close to the cavity mode which indicates the origin of the Fano [39]. For the same cavity detuning if we decrease the value of effective gain k c = -1.0k a then the peak height also reduces as shown in igure 7. At k c = -2.0k a with all other parameters remain unchanged the transparency peak disappears. So the transmission rate is controlled by changing the gain to lose ratio. In igure 8 if we use λ = 0.067 then the pro ile is highly asymmetric. This pro ile shows an asymmetric Fano line Shape.
Now we calculate the forward transmission and backward re lection in the over-coupling region by taking μ = 2. The interference between two optical systems excited simultaneously in a resonator creates EIT otherwise Fanoresonance. The several transmission line shape is manipulated by γ j + ω j Σ(j = 1, 2) with a c k k between two optical cavities which affect the transmission re lection pro ile. In our calculation, both the cavity photon and incident laser source have the same polarization. When Σ >> γ j + ω j i.e no interaction between these cavities then there are two separate dips in transmission spectrum but if Σ < γ j + ω j then two modes interact, and the Fano line shape is formed. If Σ (Kerr-type nonlinear strength) is decreased further i. e. the interaction between two optic modes is stronger than the Fano lineshape has more contrast as shown in igures 8-11 and when it becomes zero that means Fano line Shape became EIT. It also clear from igure 12, the case zero asymmetries, which shows symmetrical resonance which is referred to as anti-resonance [39].      Figures 12-15 shows the variation of forward transmission rate with probe detuning in over coupling regime for Optic-mechanical System. It is to be noted that if the coupling parameters are 0.1 then there is no effect in output transmission. Figure 12 shows absorptive properties of the system if we change the coupling parameter ratio from  Figure  13 indicates a sharp Fano line with a resonance peak at ∆ p = 0.00341k a and dip at ∆ p = -0.00567k a, the Fano spectral width is of 0.00908k a. The forward transmission contrast with Fano resonance by approximately 60% which satis ies the minimum criterion for a telecommunication system [35].
Again if we decrease the non linear coupling parameters then the transmission pro ile almost remain same but its peak become sharp and increases height. From this study the Fano line shape is ruled out and transmission pro ile is absorptive.

Conclusion
In conclusion, we have investigated the tunneling effect and Fano resonance for double cavity Optomechanical system via Forward transmission and backward re lection pro ile.      These results support how Fanoresonance is controlled in various optical systems. The results are very interesting for destructive and constructive line shape with anti-symmetric tunable Fano-resonance. we have focused here on how Fano resonance in light, propagates through the optical device and how we can use this type of research in quantum optical communications and can enrich our optical device for using line optical switches. We model this for two coupling schemes i.e for critical and over coupling. Here we have shown that both the pro ile continuously changes due to gain to loss ratio and coupling strength with coupling parameter. For the Optic-Optic cavity con iguration, the transmission spectra show asymmetric behavior which can be controlled by an input/ output probe. In our study, we use coupling parameters opposite sign for an active-passive and the same sign for the passive-passive system. This study is useful for different optical switching systems and also telecom systems.

Author contribution statement
This worked was discussed by three authors and careful reading veri ied the output pro ile. AS performed theoretical calculations and wrote the paper.