Engineering and manipulating the interaction between the optical and mechanical modes is an active research area, which has been studied theoretically and experimentally in many systems,^[1–15] such as optomechanical systems.^[9–15] A traditional optomechanical system is composed of an optical cavity and a mechanical resonator. Following the development of micro- and nano-fabrication techniques, it is now feasible to integrate the traditional optomechanical system with other systems, such as additional mechanical resonators,^[16,17] superconducting microwave cavities,^[18] phononic^[19] or photonic crystal cavities,^[4] piezoelectric systems,^[20] and charged systems.^[21] Compared with the traditional optomechanical system, the photon–phonon interaction in these hybrid optomechanical systems can be controlled by the optical radiation pressure, piezoelectric forces,^[4,5] Lorentz forces,^[22,23] and Coulomb forces.^[17,24] The interaction between the optical and mechanical modes can generate many interesting phenomena in the optomechanical systems, such as optomechanically induced transparency (OMIT). OMIT is a phenomenon in which a photon cavity can be changed from opacity to transparency. It arises from the quantum interference effect between different energy levels.^{[5,11,25–29]} Similar to electromagnetically induced transparency (EIT) observed in three-level atomic systems,^[30–33] OMIT can also be applied to many fields, including quantum ground-state cooling,^[34,35] fast and slow light,^[36,37] and quantum information processing.^[38,39]

Compared with the traditional optomechanical system, in hybrid optomechanical systems the single-OMIT has been extended to the double-OMIT.^{[14,17,40–45]} In contrast to the single-window transmission spectrum observed in the OMIT, the double-window transmission spectrum can be observed in the double-OMIT. This phenomenon is similar to the two-photon absorption observed in four-level energy-structure atomic systems.^[46,47] In double-OMIT hybrid optomechanical systems, which combine a traditional optomechanical system with an additional two-level energy system, the original three-level energy system is converted into a four-level system. The double-OMIT is generated by quantum interference between the pathways of different energy levels. The double-OMIT has many applications, including optical switches,^[45] temperature measurement,^[24] high-resolution spectroscopy, and double-channel quantum information processing.^[17,42]

In this study, we propose a hybrid optomechanical system in which the mechanical resonator of the optomechanical cavity is coupled with an additional mechanical resonator, and the additional mechanical resonator can be driven by a weak external coherently mechanical driving field. In this system, the tunable double-OMIT can be generated. We also show that both the intensity and the phase of the external coherently mechanical driving field can control the propagation of the probe field, including changing the transmission spectrum from double windows to single window.

The phenomenon of the tunable double-OMIT arises because the energy levels of the system can be modulated by the external mechanical driving field. In our system, a four-level energy structure can be formed. Under the coupling effect between the mechanical resonators, the energy level of the original mechanical resonator is dressed into an empty state. When the external driving field is applied to the additional mechanical resonator, the empty state level is replenished to yield an occupied state. Consequently, the absorption and the dispersion of the probe field are modulated by the interference of the optical pumped field and the external mechanical driving field.

Many devices can be applied to implement our scheme, both in theory and in experimentation. The two mechanical resonators can be coupled to each other via the common coupling overhang or Coulomb interaction.^[40,48–54] More importantly, the strength of coupling between them can be modulated flexibly.^[53,54] With regard to the mechanical driving field, many forms of driving forces can be applied. For example, the mechanical resonators can be driven by piezoelectric forces if they are fabricated with piezoelectric materials,^[55–58] and by Lorentz forces where a current-carrying resonator is placed in the magnetic field.^[22,23]

Hybrid optomechanical cavity systems with coherently mechanical driving have been studied in many fields, including the implementation of single^[29,59,60] and double^[13,14] optomechanically induced opacity and amplification, and weak force measurement.^[61] Compared to these systems, our scheme has the following features: (i) it can realize switching between the double-OMIT and the single-OMIT, which can be controlled by many forms of mechanical driving fields, such as piezoelectric forces and Lorentz forces; and (ii) our scheme also provides an effective way to generate an intensity-controllable narrow-bandwidth transmission spectrum, with the probe field modulated from excessive opacity to remarkable amplification.

The remainder of this paper is organized as follows. In Section 2, we describe the theoretical model and derive the dynamical equation of the proposed system. In Section 3, we discuss the experimental feasibility and physical mechanism of the double-OMIT. In Section 4, the external mechanical driving field-controlled double-OMIT is presented. In Section 5, we discuss the tunable double-OMIT controlled by other parameters of the system. The last section offers the conclusions of this paper.

A schematic diagram of the proposed hybrid optomechanical cavity system is shown in Fig. 1, in which the mechanical resonator b of a traditional optomechanical cavity is coupled with an additional mechanical resonator c, and the additional mechanical resonator c is driven by a weak external coherently mechanical driving field. The frequency of the optical cavity is , and the frequencies of the two mechanical resonators are and , respectively. The mechanical resonators b and c can be coupled via the common coupling overhang or Coulomb interaction.^[40,48–54] The mechanical driving field can be applied to the system in the form of piezoelectric forces^[55–58] and Lorentz forces.^[22,23] Consequently, the hybrid optomechanical cavity system can be driven by both the optical fields and the mechanical driving field. We assume that the optomechanical cavity is driven by a strong optical pump field with frequency and a weak optical probe field with frequency . Mechanical resonator c is driven by a weak external coherently mechanical driving field with frequency . The strength of the optomechanical coupling between the optical fields and the mechanical resonator b is referred to as G_om, and the coupling strength between the mechanical resonators b and c is referred to as J.

Fig. 1.

Figure Option ViewDownloadNew Window

Fig. 1. Schematic diagram of the hybrid optomechanical system, in which the mechanical resonator b of a traditional optomechanical cavity is coupled with an additional mechanical resonator c, and the additional mechanical resonator c is driven by a weak external coherently mechanical driving field. The optomechanical cavity is driven by a strong optical pump field and a weak optical probe field. The coupling strength between the mechanical resonators is denoted by J.

In the frame rotating at the frequency of the pump field , the Hamiltonian of the total system is given as

By adopting the quantum Langevin equations (QLEs) for the operators, in which the damping and noise terms are supplemented,^[29,62] we obtain

Relative to the intensity of the optical pump field, we assume that both the intensities of the optical probe field and the external mechanical driving field satisfy the conditions and . We then linearize the dynamical equations of the system by assuming , , and , all of which are composed of an average amplitude and a fluctuation term. a_s, c_s, and b_s are the steady-state values when only the strong optical pump field is applied. Assuming , and setting all the time derivatives to zero, we have

We assume that the cavity is driven by the optical pump field at the red sideband and the system is operated in a resolved sideband regime in which . The mechanical resonator also has a high-quality factor for . For simplicity, we assume that the frequencies of the mechanical resonators b and c are equal, and the frequency of the weak external mechanical driving field is also equal to c, where . With , the fluctuation terms , , and noise terms , , can be rewritten as

The noise terms obey the following fluctuations in correlation:

The solution of can be obtained as

Defining ,^[11,29] we obtain the quadrature of the output field at frequency

If we assume that no external mechanical driving field is applied to the additional mechanical resonator c, defining , then after the simplification, the term becomes

We now discuss the feasibility of the tunable double OMIT for the hybrid optomechanical cavity system. To ensure that the double-OMIT can be generated, the total optomechanical coupling strength should meet the condition , in which a typical single-OMIT phenomenon can be obtained when only the pump and the probe fields are applied.^[29] Moreover, in our system, the strength of coupling between the two mechanical resonators meets the condition , otherwise, the new dressed energy-levels will be covered by the line-width of the mechanical resonators. For the sake of simplicity, we assume that the system is over-coupled; i.e., the coupling parameter .

Correspondingly, we choose the power of the pump field , . The optical cavity is driven by the optical pump field on the red sideband, where . The parameters of the system we chose are shown below, and are all based on the realistic system.^[66] The frequency and decay rate of the optical cavity are THz and . For simplicity, we assume that the frequencies of the mechanical resonators b and c are equal, which are set at . The two mechanical resonators have high-quality factors for , which strictly meet the condition of the sideband-resolved regime . The qualities of the two mechanical resonators are equal, ng. The strength of the single-photon optomechanical coupling is . Furthermore, we assume that the strength of coupling between the mechanical resonators b and c is .^[16]

We first consider the situation that there is no external driving applied to the mechanical resonator c, as shown in Figs. 2(a) and 2(b). The real part Re and the imaginary part Im of the optical probe field as a function of are plotted, which exhibit the absorptive and the dispersive behaviors of the probe field, respectively. When the pump field is applied to the optical system, in the transmission spectrum, a double-transparency window can be obtained and the positions of two minima are determined by the imaginary parts of , as shown by the solid curve. The distance between the minima is 2J, which is closely related to the strength of coupling between the mechanical resonators b and c. If there is no coupling interaction between the mechanical resonators b and c, as shown by the dashed curve, a single-transparency window in the transmission spectrum curve can be obtained. In this situation, the minima point is determined by the frequency point , and the relevant mechanisms have been studied extensively.^[26] Moreover, if no pump field is applied to the optical cavity a, then no transparency window appears, as shown by the dotted curve.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. (a) The real part Re and (b) imaginary part Im as a function of for the cavity system under the condition of no external mechanical driving field applied, with parameters , σ=1, , , THz, , , ng, and .

We now consider the situation where a weak external coherently mechanical driving field is applied to the system. For simplicity, we assume that the frequency of the mechanical driving field is equal to c, where , then we have . We also assume that the amplitude-ratio (η meets the condition . In Figs. 3(a)–3(d), we plot the real part Re and the imaginary part Im of the optical probe field as a function of for different phase differences (ϕ. It is shown that when ϕ=0 and , thee absorptive and the dispersive behavior curves are symmetric and antisymmetric, respectively. The difference between the situations ϕ=0 and is that when ϕ=0, the destructive interference between the two terms in Eq. (12) suppresses the absorptive behavior at frequency when , the constructive interference between the terms in Eq. (12) amplifies the absorptive behavior at frequency . It is shown that when and , the absorptive and the dispersive behavior curves are both anomalous, and the maxima of the absorptive and the dispersive behavior curves appear in the red- or blue-detuned regions, respectively. In particular, comparing the situations of and , it is evident that the absorptive behavior curves are mirror symmetric between them. For comparison, in the absence of the optical pump field, the real part Re and imaginary part Im of the optical probe field are also shown in Fig. 3(e).

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. The real part Re (solid line) and imaginary part Im (dashed line) as a function of for different phase factors: (a) η=1, ϕ=0; (b) η=1, (c) η=1, (d)η=1, (e) η=0, Ppu=0. The other parameters are identical to those in Fig. 2.

The standard double-transparency window shown in Fig. 2(a) originates from the quantum interference effect between different energy level pathways. In the hybrid optomechanical system, a four-level energy configuration is formed by the energy levels of the optical cavity and the mechanical resonators. Under the coupling effect between the mechanical resonators b and c, the original energy level of the mechanical resonator level is split into two new dressed levels. With , the two new dressed levels are , and the disparity between them is 2J, as shown in Fig. 4(b). Under the effects of optical radiation pressure, quantum interference between different energy level pathways occurs, the third-order nonlinear absorption is enhanced by a constructive quantum interference, and the linear absorption is inhibited by a destructive quantum interference. Consequently, the double-OMIT window appears, and the relevant mechanisms have been studied extensively.^[67,68]

Fig. 4.

Figure Option ViewDownloadNew Window

Fig. 4. (a) Energy-level structure of the hybrid optomechanical system, where the states of the levels are (i = 1, 2, 3, 4). The amplitude of the external mechanical driving field is . The energy differences between and , and , and and are the frequencies of cavity a, mechanical resonator b, and mechanical resonator c, which are , , and , respectively. is equal to , and the detuning events between them are . (b) Energy-level structure of the hybrid optomechanical system in the dressed-state picture, with the relation . are the new dressed levels generated by the coupling effect between the mechanical resonators, with an energy difference of 2J.

Comparing Fig. 3 with Fig. 2, it is shown that when a weak external mechanical driving field is applied to the mechanical resonator c, the absorptive and dispersive behavior curves at frequency are modulated prominently by the phase of the external mechanical driving field. This phenomenon is consistent with Eq. (11), which is composed of two different terms, the first term is dominated by the external mechanical driving field and the second term is dominated by the optical fields. When the optical pump and the optical probe fields are applied to the system, in the absence of the external mechanical driving field, the value of the first term is zero and a standard double-OMIT can be generated. When the external mechanical driving field is also applied to the system, the first term turns into a nonzero value, then the standard double-OMIT is modulated, as shown in Figs. 3(a)–3(d). Typically, under the situation ϕ=0 or , equation (11) becomes

Physically, this phenomenon arises because the energy levels of the system are modulated by the external mechanical driving field. As shown in Fig. 4(b), under the coupling between the mechanical resonators b and c, the energy level of the original mechanical resonator turns into an empty-level state. When the external mechanical driving field is applied to the mechanical resonator c, with the condition that the frequency of the mechanical driving field is equal to the frequency of the mechanical resonator, the number of phonons with frequency is increased, then the empty mechanical resonator level is replenished into an occupied state by the mechanical driving field. Meanwhile, when the optical pump field is applied on the red-detuned sideband of the cavity, the number of phonons with frequency is decrease by the sideband-cooling effect of the pump field. As the population of phonons with frequency can be controlled by both the mechanical field and the optical fields, under the destructive or constructive quantum interference between the mechanical-phonons and optical-photons, the absorptive and dispersive behavior curves near frequency are modulated, as shown in Eq. (11). Consequently, the optical probe field is absorbed excessively or amplified remarkably, which is consistent with Eq. (11). A similar mechanism has also been studied in the single-OMIT systems.^[29]

To explore the effect of the external mechanical driving field, we plot the real part Re as a function of and (η, where . We consider the situation in which the phase difference between the mechanical driving field and the optical fields meets the condition ϕ=0, as shown in Fig. 5. In the absence of the mechanical driving field, when η=0, under interference between the optical pump field and the optical probe field, a standard double OMIT window appears. With the enhancement of the mechanical driving intensity, the absorption rate at frequency gradually decreases from positive to negative, where the narrow probe curve at frequency is modulated from full opacity to remarkable amplification. More importantly, when η=0.5, the transmission curve is converted into a standard single-OMIT window, where the probe field at frequency is transmitted perfectly. This phenomenon arises from the destructive quantum interference between the optical fields and the mechanical driving field, which is consistent with Eq. (11).

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. The real part Re as a function of and (η under the condition ϕ=0. The values of (η are 1, 0.25, 0.5, 0.75, 1.0; the forces corresponding to them are F = 9×10−9 N, 18×10−9 N, 27×10−9 N, 36×10−9 N, 45 ×10−9 N, respectively. The other parameters are the same as those in Fig. 2.

Furthermore, we discuss the situation that the phase difference between the mechanical driving field and the optical fields meets the condition . As shown in Fig. 6, which shows the real part Re as a function of and (η, in the absence of the mechanical driving field, the transmission spectrum has a standard double-OMIT window. With the enhancement of the mechanical driving intensity, the absorption rate at frequency is increased, and the narrow probe curve at frequency is modulated from full opacity to excessive opacity. Consequently, the intensity of the probe field in the cavity is enhanced. This phenomenon is opposite to the situation shown in Fig. 5, which originates from the constructive quantum interference between the optical fields and the mechanical driving field, which is also consistent with Eq. (11).

Fig. 6.

	Figure Option ViewDownloadNew Window
	Fig. 6. The real part Re as a function of and (η under the condition . The values of (η are 1, 0.25, 0.5, 0.75, 1.0; the forces corresponding to them are F = 9×10−9 N, 18×10−9 N, 27×10−9 N, 36×10−9 N, 45 ×10−9 N, respectively. The other parameters are the same as those in Fig. 2.

In the case of ϕ=0 or , the real part Re of the optical probe field at the frequency can be obtained. As , Re can be obtained as

In addition, as shown in Fig. 2(a), in the double OMIT, the bandwidth of the probe field at frequency is determined by the imaginary parts of , where . In case of the single OMIT, the bandwidth of the probe field at frequency is determined by the delay rates of cavity a and mechanical resonator b, the total optomechanical coupling strength is G_om, where .^[9] In the absence of the pump field, the bandwidth of the probe field is equal to the delay rate of cavity a, where .^[9] Consequently, the double-OMIT can be applied to generated the high-resolution spectroscopy, in the case that the coupling strength meets the conditions and .

Based on Figs. 5 and 6, when the phase of the external mechanical driving field meets either condition ϕ=0 or , the rate of absorption at frequency is proportional to the intensity of the external driving field. Consequently, the curve of the narrow probe can be modulated from excessive opacity to remarkable amplification. This phenomenon can be applied to many fields. For example, as the transmission spectrum can be changed from the double windows to the single window, our system can be applied to implement double-channel quantum information processing and optical switching. Similar phenomena have been studied in natural atomic systems.^[69,70] Moreover, the line width of the curve of narrow transmission at frequency is approximately equal to J, which is small enough relative to the line width of the optical cavity. Thus, our system can be applied to tunable high-resolution spectroscopy, which is similar to the sub-Doppler spectral resolution observed in natural atomic systems.^[71]

We further explore the tunable double-OMIT controlled by other parameters of the system, under the assumption that and . In Fig. 7, we plot the real part Re as a function of and P_pu with different powers of the mechanical driving fields. It is shown that when the system is driven by both the optical fields and the mechanical field, with the increasing power of the optical pump field, both the absorptive (when ) and gain (when ϕ = 0) behavior curves of the optical probe field are increased firstly and then decreased at the frequency . Based on Eq. (16), by setting , (when ), or (when ϕ = 0), we can obtain the maximum point (when ) and the minimum point (when ϕ = 0) of the absorptive behavior curves, which are . Moreover, when we assume , based on the power transmission coefficient , the transmission coefficient at the frequency can be expressed as

Fig. 7.

Figure Option ViewDownloadNew Window

Fig. 7. The real part Re as a function of and Ppu (Ppu was 0.04 ×10−3 mW) with different powers of optical pump fields under different situations: η = 1, (red dashed curve); η = 0 (blue solid curve); η = 1, ϕ = 0 (green dotted curve). The powers of the optical pump fields are Ppu = 0.004 mW, 0.008 mW, 0.04 mW, 0.08 mW, 0.4 mW, 0.8 mW, 4 mW, 8 mW, respectively. The other parameters are the same as those in Fig. 2.

Physically, in Fig. 7, with the driving of the external mechanical field, the number of phonons with frequency is increased by the excitation of the mechanical field. This progress can enhance the photon–phonon interaction strength between the external mechanical field and the optical fields. On the other hand, when the optical pump field is applied on the red-detuned sideband, the mechanical resonator b is cooled by the pump field, then the number of phonons with frequency is decreased. This progress can decrease the photon-sphonon interaction strength between the external mechanical field and the optical fields. Consequently, under the combined effects of the mechanical field and the optical fields, a maximum amplitude exists for the probe field, as shown in Fig. 7.

Figure 8 presents the real part Re as a function of and J with different powers of the mechanical driving fields. It shows that when both the optical fields and the weak mechanical field are applied to the system, with the increasing strength of the coupling J, both the rates of absorption (when ) and gain (when ϕ = 0 ) are also increased firstly and then decreased. Meanwhile, the line-width of the optical probe field at the frequency is broadened constantly. Based on Eq. (16), by setting , (when ), or (when ϕ = 0), we can obtain the maximum point (when ) and the minimum point (when ϕ = 0) of the absorptive behavior curves, which are . Physically, in Fig. 8, with the enhancement of the coupling strength, the number of photons exchanged between the two mechanical resonators is increased. This progress can enhance the photon–phonon interaction strength between the external mechanical field and the optical fields. On the other hand, with the enhancement of the coupling strength, the line-width of the optical probe field is broadened constantly, the number of phonons excited by the external mechanical field exactly with the frequency is reduced. This progress can decrease the photon–phonon interaction strength between the external mechanical field and the optical fields. Consequently, under the combined effects of these two progresses, a maximum amplitude also exists for the probe field in this situation.

Fig. 8.

	Figure Option ViewDownloadNew Window
	Fig. 8. The real part Re as a function of and J (J was kHz) with different mechanical driving fields under different situations: η = 1, (red dashed curve); (blue solid curve); η = 1, ϕ = 0 (green dotted curve). The coupling strength is , 0.14 kHz, 0.7 kHz, 1.4 kHz, 7 kHz, 14 kHz, 70 kHz, 140 kHz, respectively. The other parameters are the same as those in Fig. 2.

In general, we consider the situation in which the frequencies of two mechanical resonators b and c are slightly different. In Fig. 9, the real part Re as a function of for the two mechanical resonators with different frequencies is illustrated. When the system is pumped only by the optical field, relative to the case of , the absorptive behavior curves move leftward (rightward) in case of ( ). When the mechanical field is also applied to the system, the rates of absorption (when ) and gain (when ϕ = 0) of the optical probe field are both increased. Relative to the case of , the frequencies of the absorption and gain peaks move leftward (rightward) when ( ). In addition, relative to the case of , the absorptive behavior curves (when ) and the gain curves (when ϕ = 0) are both asymmetry, the rates of absorption peaks and gain peaks of the optical probe field are also larger than those in the case of . This phenomenon shows that in the situation of two slightly different mechanical resonators, the phase-dependent double OMIT driven by the mechanical field can still be generated.

Fig. 9.

	Figure Option ViewDownloadNew Window
	Fig. 9. The real part Re as a function of for the mechanical resonators with different frequencies. The mechanical driving fields are (a) (b) η = 1, and (c) η = 1, . The frequencies of the two mechanical resonators are (red dashed curves); (blue solid curves); (green dotted curves). The other parameters are the same as those in Fig. 2.

The hybrid optomechanical cavity system proposed here provides a feasible way to control the double OMIT by a weak external mechanical driving field. In this system, the empty state dressed by the coupling effect between the mechanical resonators can be replenished into an occupied state by an external coherently mechanical driving field. Under interference between the optical pumped field and the external mechanical driving field, the absorption and dispersion of the probe field are modulated. It is shown that both the intensity and the phase of the external coherently mechanical driving field can control the propagation of the probe field, including changing the transmission spectrum from double window to single window. Our system can be used to implement tunable, high-resolution spectroscopy and optical switching. More importantly, our system can be extended to other hybrid, solid-state systems to explore new quantum phenomena.