Control and manipulation of electromagentically induced transparency in a nonlinear optomechanical system with two movable mirrors
aa r X i v : . [ qu a n t - ph ] J un Control and manipulation of electromagentically induced transparency in a nonlinearoptomechanical system with two movable mirrors
S.Shahidani , ∗ M. H. Naderi , , and M. Soltanolkotabi , Department of Physics, Faculty of Science, University of Isfahan, Hezar Jerib, 81746-73441, Isfahan, Iran Quantum Optics Group, Department of Physics, Faculty of Science,University of Isfahan, Hezar Jerib, 81746-73441, Isfahan, Iran (Dated: November 10, 2018)We consider an optomechanical cavity made by two moving mirrors which contains a Kerr-downconversion nonlinear crystal. We show that the coherent oscillations of the two mechanical oscil-lators can lead to splitting in the electromagnetically induced transparency (EIT) resonance, andappearance of an absorption peak within the transparency window. In this configuration the coher-ent induced splitting of EIT is similar to driving a hyperfine transition in an atomic Lambda-typethree-level system by a radio-frequency or microwave field. Also, we show that the presence of non-linearity provides an additional flexibility for adjusting the width of the transparency windows. Thecombination of an additional mechanical mode and the nonlinear crystal suggests new possibilitiesfor adjusting the resonance frequency, the width and the spectral positions of the EIT windows aswell as the enhancement of the absorption peak within the transparency window.
PACS numbers: 37.30.+i, 03.67.Bg, 42.50.Wk, 42.50.Pq
I. INTRODUCTION
The coherent interaction of laser radiation with multi-level atoms can induce interesting phenomena suchas electromagnetically induced transparency (EIT) andelectromagnetically induced absorption (EIA). EIT is atechnique for turning an opaque medium into a trans-parent one and EIA is a technique for enhancement ofabsorption of light around resonance. These techniqueshave been used widely to manipulate the group velocityof light[1–3], for storage of quantum information [4–6],and for enhancement of nonlinear processes[7–10].Theoretical studies and technological advances innanofabrication, laser cooling and trapping [11–15] havemade it possible to reach a considerable control overthe light-matter interaction in an optomechanical system.Optomechanically induced transparency (OMIT) and ab-sorption (OMIA) are notable examples of light beam con-trol in optomechanical systems. In OMIT which hasbeen predicted theoretically[16, 17] and demonstratedexperimentally[18–20], the anti-Stokes scattering of anintense red-detuned optical ”control” field brings abouta modification in the optical response of the optomechan-ical cavity making it transparent in a narrow bandwidtharound the cavity resonance for a probe beam. In analogyto the atomic EIT, the happening of OMIT is accompa-nied by a sharp negative derivative of the dispersion pro-file of the cavity near the resonance and subluminal groupvelocity for the probe field [21, 22]. In the atomic EIT thepossibility of modification of the probe laser absorption,splitting and reshaping of the EIT peak and reduction ofthe EIT linewidth have been studied widely[23–29].In this work we are interested in the engineering andcontrol of the probe response, specially OMIT resonance, ∗ [email protected] in the presence of an additional mechanical mode andthe Kerr-down conversion nonlinearity. To this end, weconsider a cavity with two moving mirrors, driven by astrong coupling and a weak probe field which containsa nonlinear crystal consisting of a Kerr medium and adegenerate optical parametric amplifier (OPA). This ex-ploration is motivated by the following reasons. First, ina recent theoretical work [30] it has been shown that thecoherent coupling between the two cavity modes and themechanical mode of a moving mirror in a double cavityconfiguration of optomechanical system leads to the ap-pearance of an absorption peak within the transparencywindow. In this configuration by changing the powerof the electromagnetic field the switching between EITand EIA is possible. This model is quite general and avariety of systems, which can be modeled by three cou-pled oscillators, can make the same response. A drivenFabry-Perot cavity with two vibrating mirrors can be ef-fectively described by three coupled oscillators whenevera slightly difference in the mechanical frequencies leads tothe center-of-mass-relative-motion coupling [31]. In thisconfiguration the two mechanical oscillators are coupledto a single cavity mode.Second, an OPA inside a cavity can considerably im-prove the optomechanical coupling, the normal modesplitting (NMS), and the cooling of the mechanicalmirror[32]. This kind of cooling process which is accom-panied by the enhancement of the effective damping rateof the mirror can be used to increase the width of thetransparency window and reduce the group velocity of apropagating probe pulse.Third, it has been predicted[33] that by tuning theKerr nonlinearity in an optomechanical cavity one canuse the cavity energy shift to reduce the photon numberfluctuation and provide a coherently-controlled dynamicsfor the mirror.Based on these reasons, we first investigate the effectof the additional mechanical oscillator on the OMIT res-onance. We will show that if the coupling field oscillatesclose to the mechanical resonance frequencies there aretwo occasions of two-photon resonance for the probe andcoupling lasers. Consequently, the coherent oscillationsof the two mechanical oscillators give rise to splittingof the OMIT resonance, and appearance of an absorp-tion peak within the transparency window. The coher-ent induced splitting of OMIT resonance in this config-uration is similar to driving a hyperfine transition in anatomic Λ-type three-level system by a radio-frequency ormicrowave field.Then we explore how EIT and EIA resonances respondto the presence of a Kerr-down conversion nonlinearity inthe cavity. We will show that in the presence of Kerr-down conversion nonlinearity one can effectively controlthe width of the transparency window. Also, we demon-strate that to achieve a desirable control over the OMITresonance the presence of both nonlinearities is needed.In addition, for the three-mode nonlinear optomechan-ical system we show that the coherent oscillation of thecenter-of-mass mode which is responsible for the absorp-tion peak and splitting in the transparency window, in-creases. This results in the increment of the central peakabsorption.Briefly, the combination of an additional mechanicalmode and the nonlinear crystal suggests new possibilitiesfor ”engineering” the OMIT resonance. II. THE PHYSICAL MODEL
The model we consider is an optomechanical cavitywith two vibrating mirrors which contains a Kerr-downconversion nonlinear crystal (Fig.1). The vibrating mir-rors are treated as two independent quantum mechani-cal harmonic oscillator with resonance frequency Ω k , ef-fective mass m k , and energy decay rate γ k ( k = 1 , ω . The nonlinear crystal is composed of a de-generate OPA and a nonlinear Kerr medium. The cav-ity mode is coherently driven by a strong input couplinglaser field with frequency ω c and amplitude ε c as wellas a weak probe field with frequency ω p and amplitude ε p through the left mirror. Furthermore, the system ispumped by a coupling beam to produce parametric os-cillation and induce the Kerr nonlinearity in the cavity.When the detection bandwidth is chosen such that it in-cludes only a single, isolated, mechanical resonance andmode-mode coupling is negligible we can restrict to asingle mechanical mode for each mirror so that the me-chanical Hamiltonian of the mirrors is given by H m = X k =1 ( p k m k + 12 m k Ω k q k ) . (1)Furthermore, in the adiabatic limit, in which the mirrorfrequencies are much smaller than the cavity free spectral range c/ L ( c is the speed of light in vacuum and L is thecavity length in the absence of the intracavity field) thephoton scattering into the other modes can be neglectedand we can restrict the model to the case of single-cavitymode[34, 35]. We also assume that the induced reso-nance frequency shift of the cavity and the nonlinear pa-rameter of the Kerr medium are much smaller than thelongitudinal-mode spacing in the cavity. It should benoted that in the adiabatic limit, the number of photonsgenerated by the Casimir, retardation, and Doppler ef-fects is negligible [36–38]. Under this condition, the total FIG. 1. (Color online) Schematic of the setup studied inthe text. The cavity that consists of two movable mirrorscontains a Kerr-down conversion system which is pumped bya coupling beam to produce parametric oscillation and induceKerr nonlinearity in the cavity. The cavity mode is coherentlydriven by a strong input coupling laser field and a weak probefield through the left mirror.
Hamiltonian of the system can be written as H = H + H , (2)where H = ~ ω a † a + H m + ~ g m a † a ( q − q )+ i ~ ( s in ( t ) a † − s ∗ in ( t ) a ) , (3a) H = i ~ G ( e iθ a † − e − iθ a ) + ~ ηa † a . (3b)The first term in H is the free Hamiltonian of the cav-ity field with the annihilation (creation) operator a ( a † ),frequency ω and decay rate κ , H m is the free Hamil-tonian of the mirrors given by Eq.(1), the third termdescribes the optomechanical coupling between the cav-ity field and the mechanical oscillators due to the radi-ation pressure force, and the last term in H describesthe driving of the intracavity mode with the input laseramplitude s in ( t ). Also, the two terms in H describe,respectively, the coupling of the intracavity field withthe OPA and the Kerr medium; G is the nonlinear gainof the OPA which is proportional to the pump powerdriving amplitude, θ is the phase of the field drivingthe OPA, and η is the anharmonicity parameter propor-tional to the third order nonlinear susceptibility χ (3) ofthe Kerr medium. We will solve this problem for the to-tal driving field s in ( t ) = ( ε c + ε p e − i ( ω p − ω c ) t ) e − iω c t , where ε c = p κP c / ~ ω c ( ε p = p κP p / ~ ω p ) and P c ( P p ) are, re-spectively, the amplitude and power of the input coupling(probe) field. The dynamics of the system is described bya set of nonlinear Langevin equations. Since we are in-terested in the mean response of the system to the probefield we write the Langevin equations for the mean val-ues. In a frame rotating at the coupling laser frequency ω c , neglecting quantum and thermal noises we obtain h ˙ a i = − [ i ( ω − ω c ) + κ ] h a i − ig m h a i ( h q i − h q i ) − iη h a † ih a i + 2 Ge iθ h a † i + s in ( t ) , (4a) h ˙ q k i = h p k i /m k , ( k = 1 , , (4b)˙ h p k i = − m k Ω k h q k i + ( − k ~ g m h a † ih a i− γ k h p k i , ( k = 1 , . (4c)Under the assumption that the input coupling laser fieldis much stronger than the probe field( ε c ≫ ε p ), we obtainthe steady-state mean values of p , q and a as p sk = 0 , ( k = 1 , , (5a) q sk = ( − k ~ g m m k Ω k | a s | ( k = 1 , , (5b) a s = ε c p (∆ − G sin( θ )) + ( κ − G cos( θ )) , (5c)where q sk denotes the new equilibrium position of themovable mirrors and ∆ = ω − ω c + g m ( q s − q s ) +2 η | a s | = ∆ + 2 η | a s | is the effective detuning of thecavity which includes both the radiation pressure andthe Kerr medium effects. It is obvious that the opticalpath and hence the cavity detuning are modified in anintensity-dependent way. Since the effective detuning ∆satisfies a fifth-order equation, it can have five real so-lutions and hence the system may exhibit multistabilityfor a certain range of parameters. In our work we choosethe parameters such that only one solution exists and thesystem has no bistability. Now we consider the pertur-bation made by the probe field. The quantum Langevinequations for the fluctuations are given by δ ˙ a = − ( i ∆ + κ ) δa − ig m a s ( δq − δq )+(2 Ge iθ − iηa s ) δa † + s in ( t ) , (6a) δ ˙ q k = δp k /m k , ( k = 1 , , (6b)˙ δp k = − m k Ω k δq k + ( − k ~ g m a s ( δa † + δa ) − γ k δp k , ( k = 1 , , (6c)where ∆ = ∆ +4 ηa s . It is evident that the cavity modeis coupled only to the relative motion of the two mir-rors, and it is therefore convenient to rewrite the aboveequations in terms of the fluctuations of the relative andcenter-of-mass coordinates: δQ = m M δq + m M δq , δP = δp + δp , (7) δq = δq − δq , δpµ = δp m − δp m , (8)where M = m + m and µ = m m /M are the effectivemasses of the relative and center-of-mass modes, respec-tively. The linearized quantum Langevin equations for the fluctuation operators of these coordinates take theforms h δ ˙ a i = − ( i ∆ + κ ) h δa i + ig m a s h δq i +(2 Ge iθ − iηa s ) h δa † i + ε p e − i ( ω p − ω c ) t , (9a) h δ ˙ q i = h δp i /µ, (9b) h δ ˙ p i = − µ Ω r h δq i − γ r h δp i − µ (Ω − Ω ) h δQ i− µM ( γ − γ ) h δP i + ~ g m a s ( h δa † i + h δa i ) , (9c) h δ ˙ Q i = h δP i /M, (9d) h δ ˙ P i = − M Ω cm h δQ i − γ cm h δP i − µ (Ω − Ω ) h δq i− ( γ − γ ) h δp i , (9e)where we have defined the relative motion frequencyΩ r = ( m Ω + m Ω ) /M , damping rate γ r = ( m γ + m γ ) /M and also the center-of-mass frequency Ω cm =( m Ω + m Ω ) /M and damping rate γ cm = ( m γ + m γ ) /M . The above equations show that even thoughthe cavity mode interacts only with the relative motionmode, there is a coupling between the center-of-mass andrelative motion modes when Ω = Ω or γ = γ . Wewill show that the presence of this coupling makes theswitching from EIT to EIA possible. Now we use a fairlystandard procedure for the investigation of the probe re-sponse. Defining δ = ω p − ω c , we use the following ansatz h δa i = A − e − iδt + A + e iδt , (10a) h δa † i = A ∗− e − iδt + A ∗ + e iδt , (10b) h δq i = qe − iδt + q ∗ e iδt , (10c) h δQ i = Qe − iδt + Q ∗ e iδt . (10d)In the original frame A − and A + oscillate at ω p and 2 ω c − ω p , respectively. Using the input-output relation[39], weobtain ε out + ε c e − iω c t + ε p e − iω p t = 2 κ ( a s + δa ) e − iω c t . (11)Substituting Eq.(10) into Eq.(9) we obtain the followingequations(Θ + iδ ) A − + Γ( A + ) ∗ + ig m a s q + ε p = 0 , (12a)Γ ∗ A − + (Θ ∗ + iδ )( A + ) ∗ − ig m a s q = 0 , (12b) ~ ga s ( A − + ( A + ) ∗ ) + µχ r ( δ ) q + Λ Q = 0 , (12c) M χ cm ( δ ) Q + Λ q = 0 , (12d)where we have definedΘ = − ( κ + i ∆ ) , (13a)Γ = 2 Ge iθ − iηa s , (13b)Λ = µ (Ω − Ω + iδ ( γ − γ )) , (13c) χ r ( δ ) = δ − Ω r + iδγ r , (13d) χ cm ( δ ) = δ − Ω cm + iδγ cm . (13e)From the Eq.(13c) we find that when Ω = Ω and γ = γ , Λ = 0 and thus the center-of-mass motion is fullydecoupled from the cavity mode and the relative motion.While whenever Λ = 0 the three modes are all coupled.The total output field ε t , at the probe frequency isgiven by ε t = 2 κA − /ε p = 2 κd ( δ ) { κ − i (∆ + δ ) − if ( δ ) } , (14)where f ( δ ) = ~ g m a s /χ ( δ ) , (15a) χ ( δ ) = µχ r ( δ ) − Λ M χ cm ( δ ) , (15b) d ( δ ) = ( κ − iδ ) + ∆ − | Γ | +2(∆ + Im (Γ)) f ( δ ) . (15c)The real part ( ε R ) and imaginary part ( ε I ) of the fieldamplitude ε t , respectively, show the absorptive and dis-persive behavior of the output field at the probe fre-quency. These quantities can be measured by homodynetechnique [40].The structure of the output field has some main char-acteristics, arising from the nonlinearity of the systemand the freedom in choosing equal or unequal mechan-ical frequencies and damping rates. To understand thecoupling-field-induced modification of the probe responseand its structure we present the results and numericalcalculations in the next section. III. RESULTS AND DISCUSSIONS
In this section, we first consider the bare cavity op-tomechanical system and investigate the condition inwhich the coherent coupling between the mechanical andoptical modes leads to OMIT and OMIA. Then we exam-ine the effects of the Kerr-down conversion nonlinearityon these phenomena.
A. Bare cavity
To simplify our treatment for the bare cavity we canuse the reasonable rotating wave approximation (RWA)to neglect the far off-resonance lower sideband ( A + ≃ κ ≪ Ω k , k = 1 , ε t is simplifiedto the following form ε t ≃ κκ + i (∆ − δ ) + i ~ g m a s χ ( δ ) . (16)In what follows we investigate the two cases of equal anddifferent mechanical frequencies separately.
1. Equal mechanical frequencies and damping rates (
Λ = 0 ) First, we consider the case in which the frequencies anddamping rates of the two mechanical oscillators are thesame, i.e., Ω = Ω = ω m and γ = γ = γ m . As men-tioned before, in this condition the radiation pressure isonly coupled to the relative position of the two mirrorsand the center-of-mass becomes an isolated quantum os-cillator. Therefore χ ( δ ) = µχ r ( δ ). When ω p is close tothe cavity frequency ( ω p ∼ ω ) and the coupling field ω c drives the cavity on its red sideband (∆ ∼ ω m ) thestructure of the resonance response of the output field ε t is simplified to that of a cavity with one movable mirrorand effective mass 2 µ : ε t ≃ κκ − ix + { β/ ( γ m / − ix ) } , (17)where β = ~ g m a s / µ and x = δ − ω m is the detuningfrom the line center. Therefore the denominator of theresponse function is quadratic in x .
2. Different mechanical frequencies and equal dampingrates ( Λ = 0 ) Now we consider the case in which the frequency ofthe mechanical oscillators is different Ω = Ω but theirdamping rates are equal γ = γ = γ m . The new aspectof this condition is the coupling between the center-of-mass and the relative motion modes which results in theanomalous EIA in the optomechanical cavity. When ω p isclose to the cavity frequency ( ω p ∼ ω ) and the couplingfield ω c is red tuned by an amount ω m = (Ω + Ω ) / ε t ≃ κκ − ix + 2 βδ x + b − Λ /µMδ x + b , (18)where δ = ω m + Ω r , b = ω m − Ω r + iω m γ m and δ = ω m + Ω cm , b = ω m − Ω cm + iω m γ m . Therefore thedenominator of the response function is cubic in x .To illustrate the numerical results we show the probefield absorption and dispersion profiles for the bare cav-ity in Fig.2 for the two cases of equal and different me-chanical frequencies. We use the following set of exper-imentally realizable parameters [44]: P c = 6 mW, λ =2 πc/ω c = 1064 nm,Ω = 2 π × Hz, m = m = 12 ng, κ/ Ω = 0 . γ / π = γ / π = 200 Hz, and L = 6 mm.The figure clearly shows the splitting of the transparencywindow due to an additional coherency in the system.Physically, in the two mode optomechanical system(Λ = 0) when the coupling field ω c is red detuned byan amount ω m (∆ ∼ ω m ) and ω p is close to the cav-ity frequency the optomechanical system behaves like athree-level Lambda medium for the probe field as shownin Fig.(3). The intense coupling laser field ”dresses” the (a) - - x (cid:144) Ω m ¶ R H b L - - - - x (cid:144) Ω m ¶ R FIG. 2. (Color online)(a)The real and (b) the imaginaryparts of the field amplitude ε t versus the normalized fre-quency x/ω m for the bare cavity with equal mechanical fre-quencies Ω = Ω = 2 π × Hz(red solid line) and with dif-ferent mechanical frequencies Ω = 2 π × Hz,Ω = 1 . (blue dashed line).The coupling field ω c is red detuned by anamount ω m = (Ω + Ω ) / γ and γ are equal. mechanical mode. In this view , the OMIT can be seenas a level splitting like an Autler-Towns doublet [45], asshown in Fig.(3).The coherent cancellation of the two res-onances in the middle of the doublet, at the two-photonresonance, provides the system transmittive in a narrow-band around the cavity resonance for the probe field.Similarly, for the three-mode system (Λ = 0) we candescribe the happening of anomalous EIA based on alevel diagram structure. In Fig.4 the | i ↔ | i transi-tion is the excitation at cavity frequency and the cou-pling laser is red tuned by an amount ω m = (Ω + Ω ) / ∼ ω m ) forming a Λ-type three-level system produc-ing OMIT. But the radiation pressure induces an addi-tional coherency between the mechanical modes givingrise to OMIT splitting. The coherent induced splittingof OMIT due to the radiation pressure is similar to driv-ing a hyperfine transition in an atomic Λ-type three-levelsystem by a radio-frequency or microwave field[46–50].Figure 5 shows how the OMIT splitting varies linearly as FIG. 3. Level diagram structure for the OMIT. The | i ↔ | i transition is the excitation at cavity frequency andthe | i ↔ | i transition is the excitation of the mechanicaloscillator. Coherent coupling of the mechanical and opti-cal modes generates the destructive interference of excitationpathways in the middle of the doublet of dressed states | d i and | d i for the probe beam.FIG. 4. Level diagram structure for the OMIA. The | i ↔ | i transition is the excitation at cavity frequency; The couplinglaser is red tuned by an amount ω m = (Ω + Ω ) / a function of the strength of radiation pressure coupling g m . The splitting in OMIT is due to the fact that thereare two occasions of two-photon resonance for the probeand coupling lasers at δ = Ω and δ = Ω . B. Nonlinear cavity
Now we investigate the effect of the Kerr-down con-version nonlinearity on the total output field amplitude ε t . Although the nonlinearity does not alter the leveldiagram structure of the OMIT, it manifests itself inthe steady-state response of the system (Eq.(5)), in theoptomechanical coupling rate g m a s (Eq.(6)), and in theparameter Γ + which is responsible for a direct couplingbetween A − and A + (Eq.(12a)). g m (cid:144) g - - x (cid:144) Ω m ¶ R FIG. 5. (Color online)The probe field absorption profileversus the normalized frequency x/ω m showing a linear OMITsplitting as a function of the normalized radiation pressurecoupling g m /g where g = ω c /L . The mechanical frequenciesare Ω = 2 π × Hz and Ω = 1 . . Other parameters arethe same as those in Fig.2. - - x (cid:144) Ω m ¶ R FIG. 6. (Color online)The absorption profile of the probefield versus the normalized frequency x/ω m for a bare cavity( G = η = 0) (green line) and a nonlinear cavity with G =4 × Hz, η = 0 . θ = 3 π/ G = 4 × Hz, η = 0 . θ = π/ P c = 8mW, m = m = 15 ng, Ω = Ω =2 π × Hz, λ = 512 nm, L = 2mm, and κ = 0 . . Otherparameters are the same as those in Fig.2. In the OMIT condition the optomechanical couplingrate g m a s is equivalent to the Rabi frequency in theatomic EIT[18]. The dependence of a s on the nonlin-earity can be used to control the width of the trans-parency window which is related to the effective mechan-ical damping rate γ eff . This parameter is approximatelygiven by[18, 19, 51] γ eff = γ m (1 + C ) , (19)where C = 2 ~ ( g m a s ) /mω m κγ m denotes the optome-chanical cooperativity of the cavity[14, 18, 52]. In Fig. 6we have plotted the absorption profile for different val-ues of G , θ and η . It shows that by controlling theseparameters the width of the transparency window can - - x (cid:144) Ω m Κ A + ¤ (cid:144) ¶ p FIG. 7. (Color online)The parameter 2 κ | A + | /ε p versus thenormalized frequency x/ω m for a bare cavity ( G = η = 0) (redsolid line) and a nonlinear cavity ( G = 1 . κ , η = 0 . θ = π/ =Ω = 2 π × Hz. Other parameters are the same as those inFig.2. be increased or decreased in comparison with that of abare cavity. It should be noted that in the presence ofonly one of the two nonlinearities we cannot control thetransparency window desirably. This can be explainedby the fact that according to Eq.(14), in the absence ofoptomechanical coupling ( g m = 0) there would be an ab-sorption peak near the modified resonance condition ofthe cavity δ = p ∆ − | Γ | . Therefore the nonlinear pa-rameters should be choosen such that p ∆ − | Γ | ≃ ∆,otherwise the control and probe fields induce a radiation-pressure force oscillating at the frequency δ , which is notclose enough to the resonance frequency of the movingmirrors to induce coherent oscillations in them. Thisfeature leads to disappearance of OMIT in the outputprobe field.Also, according to Eq.(12a), in the presence of nonlin-earity there is a direct coupling between A − and A + be-cause of the factor Γ. Therefore it seems that in contrastto the bare optomechanical cavity the Stokes scatteringof the light from the strong intracavity coupling field isno longer negligible. In Fig.7 we have plotted the param-eter 2 κ | A + | /ε p as a function of the normalized frequency x/ω m for a bare cavity and a cavity with Kerr-down con-version nonlinearity. As is seen, in the dip of the trans-parency window 2 κ | A + | /ε p reaches its local minimum fora nonlinear cavity and its local maximum for a bare cav-ity. Hence even though in the presence of nonlinearityoutside the OMIT window the lower sideband can alsobe tuned by the strong coupling field, but the contribu-tion of the Stokes scattering around the cavity resonanceis more negligible for a nonlinear cavity.Now we consider the probe response in the presenceof Kerr-down conversion nonlinearity for the second case(Ω = Ω ). As stated before, the OMIT splitting andappearance of the central absorption peak are due to anadditional coherent oscillation in the system which is pro-vided by the fluctuations in the center-of-mass mode, i.e., H a L - - - - x (cid:144) Ω m g m a s Re @ q D(cid:144) ¶ p H b L - - - - x (cid:144) Ω m g m a s Re @ Q D(cid:144) ¶ p H c L - - x (cid:144) Ω m ¶ R FIG. 8. (Color online)The real parts of (a) the normalizedparameter g m a s q/ε p , (b) the normalized parameter g m a s Q/ε p and (c) the field amplitude ε t versus the normalized frequency x/ω m for a bare cavity ( G = η = 0) (red line) and a nonlinearcavity with G = 10 Hz, η = 0 . θ = π/ = 2 π × Hz andΩ = 1 . . Other parameters are the same as those inFig.6. h δQ i . Figures 8(a) and 8(b) illustrate the effect of thenonlinearity on q and Q , respectively. They show a shiftin the coherent oscillations of q which leads to the broad-ening of the width of the transparency windows and anincrease in the coherent oscillations of the center-of-massmode Q which results in the enhancement of central peakabsorption (Fig.8(c)).In conclusion, we have studied theoretically the effectof an additional mechanical mode and a Kerr-down con-version nonlinear crystal on the EIT resonance in an op-tomechanical system with two movable mirrors. We haveshown that the coherent oscillations of the two mechani-cal oscillators can lead to splitting in the EIT resonance,and appearance of an absorption peak within the trans-parency window. This configuration is similar to driv-ing a hyperfine transition in an atomic Λ-type three-levelsystem by a radio-frequency or microwave field. Also, wehave shown that in the presence of Kerr-down conversionnonlinearity by controlling the nonlinear parameters G , η and θ the width of transparency can be adjusted tobe greater or smaller than that of a bare cavity. Thecombination of an additional mechanical mode and non-linear crystal suggests new possibilities for manipulatingand controlling the EIT resonance in the optomechanicalsystems. ACKNOWLEDGEMENT
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