† Corresponding author. E-mail:

‡ Corresponding author. E-mail:

We theoretically investigate the optomechanical induced transparency (OMIT) phenomenon in a two-cavity system which is composed of two optomechanical cavities. Both of the cavities consist of a fixed mirror and a high-*Q* mechanical resonator, and they couple to each other via a common waveguide. We show that in the presence of a strong pump field applied to one cavity and a weak probe field applied to the other, a triple-OMIT can be observed in the output field at the probe frequency. The two mechanical resonators in the two cavities are identical, but they lead to different quantum interference pathways. The transparency windows are induced by the coupling of the two cavities and the optical pressure radiated to the mechanical resonators, which can be controlled via the power of the pump field and the coupling strength of the two cavities.

Electromagnetically induced transparency (EIT) is an interesting phenomenon which was first observed in atomic vapors.^{[1]} It allows a transparency window of the weak probe field at the resonant frequency of a three-level system in the presence of a strong pump field, while the transparency window will disappear in the absence of the pump field. EIT is caused by the destructive quantum interference between different pathways of the internal quantum states.^{[2]} It has led to many unexpected applications, such as the realization of slow light,^{[3]} vibration cooling,^{[4]} quantum memory,^{[5]} frequency transfer,^{[6]} and normal-mode splitting.^{[7]}

Recently, EIT has been studied in many novel systems. Some new phenomena were observed, such as, the coupled-cavity-induced transparency appeared in the double-cavity system^{[8,9]} and the optomechanical induced transparency appeared in the optomechanical system.^{[10–13]} Double optomechanically induced transparency has also been investigated, which has been achieved in the coupled-disks structure,^{[14]} charged-resonators optomechanical cavity system,^{[15]} double-resonators optomechanical system,^{[16,17]} and couple-cavity optomechanical system.^{[18]} Furthermore, the multiple-OMIT has been achieved in the atomic-media assisted optomechanical system^{[19]} and multiple-resonators optomechanical system.^{[20]} Double- and multiple-OMIT have extend the single EIT to those with double or triple transparency windows, which have led some new applications, such as the all-optical quantum network and multi-channel quantum information processing.^{[21]}

In this work, we consider a two-cavity optomechanical system, in which the two cavities couple to each other via the common waveguide. A strong pump field is applied to one cavity and a weak probe field is applied to the other. Our results illustrate that with the excitation of the strong pump field, triple-OMIT appears in the spectrum of the output probe field. This optomechanically induced transparency is associated with the coupling between the two cavities, as well as the coupling between the cavities and the mechanical resonators.

Nowadays, the strong coupling between two photonic nanocavities has been observed in the structure based on a two-dimensional photonic crystal,^{[22]} and the coupling strength can be controlled by tuning the distance between them. Moreover, strong optomechanical coupling between femtogram nanomechanical resonators and high-*Q* photonic nanocavities has also been realized in the two-dimensional photonic crystal.^{[23]} Therefore, the two-dimensional photonic crystal is a feasible experimental scheme for our system to be realized.

Different from the other double-resonators optomechanical system which consists of two different resonators in a single cavity,^{[16]} our model has two identical mechanical resonators, but they lead to different quantum interference pathways. We provide a new approach to produce multi-OMIT in multi-cavities optomechanical systems.

The paper is organized as follows. In Section 2, we introduce the model and the analytical expressions of the two-cavity optomechanical system, and with the standard derivation, we obtain the general analytical expression for the component of the output field at the probe frequency. In Section 3, we present the numerical results for the triple-OMIT, with the parameters chosen based on recent experiments. Then we give a detailed physical explanation for the optical phenomenon and analyze the significant factors of the system. Finally, we give a brief conclusion in Section 4.

The scheme investigated is illustrated in Fig. *a* and *c* are coupled to each other via the common waveguide with a coupling strength *J*. The mechanical resonator *b* (*d*) couples to the cavity *a* (*c*) with the coupling strength *g*_{a} (*g*_{c}) via the radiation pressure. The annihilation (creation) operators for cavities *a* and *c* are defined as *â* (*â*^{†}) and *ĉ* (*ĉ*^{†}). Moreover, *b* and *d*. The frequencies of the two resonators are *ω*_{b} and *ω*_{d}. We assume that a strong pump field with frequency *ω*_{pu} is applied to the cavity *a* and a weak probe field with frequency *ω*_{pr} is applied to the cavity *c*.

The total Hamiltonian of the whole system in the rotating frame at the pump laser frequency *ω*_{pu} reads

*Δ*

_{a}=

*ω*

_{a}−

*ω*

_{pu},

*Δ*

_{c}=

*ω*

_{c}−

*ω*

_{pu}, and

*δ*=

*ω*

_{pr}−

*ω*

_{pu}. The

*ω*

_{a}and

*ω*

_{c}are the resonance frequencies of cavity modes

*a*and

*c*, respectively.

Here, *H*_{0} is the free Hamiltonian of the system. The first two terms describe the free Hamiltonian of the two optical modes, and the last two terms describe the free Hamiltonian of the two mechanical modes. *H*_{I} is the interaction Hamiltonian of the system. The first term describes the interaction between the two cavities, the second and the third terms describe the optomechanical interactions in cavities *a* and *c*. *g*_{a} = (*ω*_{pu}/*L*)*x*_{b} and *g*_{c} = (*ω*_{pu}/*L*)*x*_{d} are the single-photon coupling rates, where *b* and *d*, and *m*_{b} and *m*_{d} are the masses of the two mechanical resonators. The last three terms describe the input laser Hamiltonian with driving strength *P*_{pu} denotes the input power of the pump field and *P*_{pr} denotes the power of the probe field. *κ*_{a} (*κ*_{c}) is the decay rate of photons leaving optical cavity *a* (*c*) into the associated outgoing modes, and *γ*_{b} (*γ*_{d}) is the intrinsic mechanical damping rate of resonator *b* (*d*).

In order to describe the coupling dynamics of the system, we use the formalism of the quantum Langevin equations.^{[24]} Omitting the hat symbol for simplicity, the Heiserberg–Langevin equations of the intracavity modes and mechanical modes can be written as

*f*and

*ξ*are the quantum and thermal noise operators, respectively. These are the nonlinear quantum Langevin equations and the dynamics is complicated.

In the following, we linearize the operators around the steady-state values by assuming *a* = 〈*a*〉_{s} + *δa*, *c* = 〈*c*〉_{s} + *δc*, *b* = 〈*b*〉_{s} + *δb*, *d* = 〈*d*〉_{s} + *δd*. We also assume that the fluctuation operators *δa*, *δc*, *δb*, and *δd* have zero mean. Here 〈*a*〉_{s}, 〈*c*〉_{s}, 〈*b*〉_{s}, and 〈*d*〉_{s} are the steady-state values of the system when only the strong driving field is applied, as *ɛ*_{pr} ≪ *ɛ*_{pu}, they can be obtained from Eqs. (

*a*(

*c*) and the control field, including the frequency shift caused by the mechanical motion.

Furthermore, ignoring the small nonlinear terms, we can obtain the following linearized quantum Langevin equations:

*G*

_{a}=

*g*

_{a}〈

*a*〉

_{s}is the total coupling strength between the optical mode

*a*and the mechanical mode

*b*, and

*G*

_{c}=

*g*

_{c}〈

*c*〉

_{s}is the total coupling strength between the optical mode

*c*and the mechanical mode

*d*.

We assume that the cavity is driven by a control field at the mechanical red sideband with *ω*_{b(d)}/*κ* ≫ 1. The mechanical resonator *b* (*d*) has a high quality factor *ω*_{b(d)}/*γ*_{m} ≫ 1. As *a*_{s} and *c*_{s} can be obtained by Eqs. (

Transforming the equations into the interaction picture by introducing *δa* → *δa*e^{−iδt}, *δc* → *δc *e^{iδt}, *δb* → *δb*e^{−iδt}, *δd* → *δd* e^{−iδt}, *δɛ* → *δɛ* e^{−iδt}, *δ f* → *δ f*e^{−iδt}, and ignoring the fast oscillating terms e^{2iδt}, we obtain the following equations:

*λ*

_{d}=

*δ*−

*ω*

_{d}. Note that the mean values of the quantum and thermal noise terms are zero, under the steady-state condition

*δc*,

*λ*=

*λ*

_{a}=

*λ*

_{c}=

*λ*

_{b}=

*λ*

_{d}=

*ω*

_{pr}−

*ω*

_{pu}−

*ω*

_{b}, after some simplification, we can write Eq. (

*λ*

_{±}and

*A*

_{±}are

By applying the input–output relation of the intracavity field,^{[25]} which is

*ɛ*

_{T}] and imaginary part Im[

*ɛ*

_{T}] describe the absorption and dispersion of the optomechanical system, respectively.

For numerical work, the parameters of the optical cavity and the mechanical resonator we used are based on the realistic cavity optomechanical system.^{[26]} The coupling strength between the cavities is chosen based on the experimental double-cavity system,^{[22]} which can reach the strong coupling region, and it can be controlled by adjusting the distance between the cavities.

Triple-OMIT based on the coupled-cavity optomechanical system we considered is illustrated in Fig. *ɛ*_{T}] and dispersion Im[*ɛ*_{T}] of the output field are plotted as a function of (*δ* − *ω*_{b})/*ω*_{b}. Three dips of the absorption line correspond to the three transparency frequencies of the probe field. They are determined by the imaginary parts *λ*_{±} and real part *γ*_{d}/2, which have been given in Eq. (^{[19]} or composed of many mechanical resonators in a single cavity,^{[18]} the two mechanical resonators in our scheme are identical.

If we ignore the optomechanical coupling *G*_{a} and *G*_{c}, the term *ɛ*_{T} becomes

^{[8]}Its real and imaginary parts are shown in Figs.

If we only ignore the optomechanical coupling *G*_{c} in cavity *c*, *ɛ*_{T} becomes

^{[18]}Its real and imaginary parts are shown in Figs.

The origin of the triple-OMIT can be explained by the quantum interference effects, the energy level configuration is presented in Fig. ^{[8]} If the resonator *b* is coupled to the cavity *a* via optomechanical interaction, the additional optomechanical coupling will break down the symmetry of the EIT-like level. Then the single transparency window is split into two windows, which induces the occurrence of double OMIT.^{[18]} Moreover, when the coupling between the resonator *d* and the cavity *c* occurs, another pathway of quantum interference forms via the optomechanical interaction, which will lead to another transparency frequency of the probe field with the condition *ω*_{pr} − *ω*_{pu} ∼ *ω*_{d}. As a result, the triple-OMIT occurs, which yields three transparency windows of the probe field. The mechanical resonators *b* and *d* are identical, but they lead to different quantum interference pathways.

To further explore the characteristic of the tunable triple-OMIT, we plot the absorption Re[*ɛ*_{T}] as a function of (*δ* − *ω*_{b})/*ω*_{b} and *ɛ*_{pu} in Fig. *G*_{a} and *G*_{c} are both proportional to the power of the pump field. The parameters *κ*_{a} and *κ*_{c} we used can be ignored compared to *Δ*′, and the ratio of the total coupling strength in the two cavities can be obtained as *G*_{a}/*G*_{c} = *Δ*′/*J*, with *Δ*′ = *ω*_{b}. As the parameter we used *J* (= 2 GHz) < *Δ*′ (= 3.68 GHz), we have *G*_{a} < *G*_{c}. In the absence of the driving field, the single transparency window appears, which is induced by the coupling between the cavities.^{[8]} When a weak pumping field is applied, the coupling strength *G*_{a} between resonator *b* and cavity *a* is strong enough to break down the symmetry of the EIT-like level, which induces the double OMIT, while the coupling strength *G*_{c} between resonator *d* and cavity *c* is not strong enough to lead to another pathway of quantum interference. With the increase of the power of the pump field, the coupling strength *G*_{c} will gradually become strong enough to induce the appearance of triple-OMIT. The width of the transparency window is proportional to the power of the pump field *ɛ*_{pu}, as shown in the inset. It is in accord with Eqs. (^{[27]}

Figure *ɛ*_{T}) with (*δ* − *ω*_{b})/*ω*_{b} for different coupling strengths between the cavities. In the absence of the coupling between the two cavities, the total optomechanical coupling strength *G*_{c} in cavity *c* is zero. As a result, no interference pathway forms between the pump field and the probe field. With the enhancing of the coupling strength *J*, the interference strength between the pump field and the probe field is increased, which induces the enlarging of both the transparency depth and the widths of the three windows, and the result also accords with Eqs. (

For a more general case, we discuss the absorption of the triple-OMIT with two mechanical resonators slightly different in frequency. As illustrated in Fig. *ω*_{d} = *ω*_{b} case, the absorption curves move rightward (leftward) in the case of *ω*_{d} < *ω*_{b} (*ω*_{d} > *ω*_{b}). As the transparency frequency of the probe field is determined by the condition *ω*_{pr} − *ω*_{pu} ∼ *ω*_{d}, when enlarging (diminishing) the frequency of the mechanical resonator *d*, the transparency of the probe field will move rightward (leftward). The principal frequency of the pumping field in cavities *a* and *c* is determined by the condition *ω*_{a} − *ω*_{pu} ∼ *ω*_{b}, with *ω*_{a} ∼ *ω*_{c}, when *ω*_{d} ≠ *ω*_{b}, the windows will split into three windows asymmetrically.

We have demonstrated the generation of a triple-OMIT for the probe field. The system we considered is a two-cavity optomechanical system, and the two cavities couple to each other via a common waveguide. Based on the quantum interference effects, under the exciting of a strong pump field, three transparency windows of the output field will appear at the probe frequency. It can be analyzed as a combinatorial effect, which is composed of the coupled-cavity-induced transparency effect and the optomechanically induced transparency effect. We also find that the power of the pump field and the coupling strength of the two cavities are two significant factors for the triple-OMIT. The width of the transparency windows is proportional to both of them. Moreover, different from other multiple optomechanically induced transparency systems, the two mechanical resonators in our scheme are identical, and they lead to different quantum interference pathways. Our work provides a new approach to produce multi-OMIT in multi-cavities optomechanical systems.

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