Surface-acoustic-wave-controlled optomechanically induced transparency in a hybrid piezo-optomechanical planar distributed Bragg-reflector-cavity system
Shi-Chao Wu Li Zhang Jian Lu Li-Guo Qin Zhong-Yang Wang


apsrev4-1



Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai, 201210, China
University of Chinese Academy of Sciences, Beijing, 100000,China
Jiangsu Ocean University, School of Sciences, Lianyungang, 320700, China

Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai, 201210, China
University of Chinese Academy of Sciences, Beijing, 100000, China

Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai, 201210, China

Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai, 201210, China
Department of Physics, Shanghai University Of Engineering Science, Shanghai, 201620, China

wangzy@sari.ac.cn
Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai, 201210, China


 We propose a scheme that can generate tunable optomechanical induced transparency in a hybrid piezo-optomechanical cavity system. The system is composed of a high-quality planar distributed Bragg reflector cavity modified with an embedded Gaussian-shaped defect. Moreover, interdigitated transducers are fabricated on the surface of the cavity to generate surface acoustic waves. Under the actuation of the surface acoustic wave, the upper Bragg mirrors can be vibrated as a bulk acoustic resonator, and the distributed Bragg reflector cavity becomes a standard three-level optomechanical system. In this situation, we show that when a strong pump optical field and a weak probe optical field are simultaneously applied to the hybrid optomechanical cavity system, optomechanically induced transparency occurs under the quantum interference between different energy level pathways. Our scheme can be applied in the fields of optical switches and quantum information processing in solid-state quantum systems.
 


42.50.pq,425.50.wk,42.50.Ar

Surface-acoustic-wave-controlled optomechanically induced transparency in a hybrid piezo-optomechanical planar distributed Bragg-reflector-cavity system
    Zhong-Yang Wang
    
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§ INTRODUCTION


In the field of quantum information, many types of artificial atomic systems have been proposed to regulate and manipulate the processing of optical or microwave information, including superconducting quantum circuits <cit.>, N-V centers <cit.>, semiconductor quantum dots <cit.>, and distributed Bragg reflector (DBR) cavities <cit.>. Furthermore, since most of these artificial atomic systems can be coupled with both photon and phonon modes, the optical information can also be operated by modulating the phonon modes. There are many methods to control the phonon modes; for example, phonon modes can be driven by the radiation pressure of the optical field or microwave field <cit.>. Moreover, in recent experiments, the phonon modes can be driven by different types of external mechanical fields, such as the Lorentz force and piezoelectric force <cit.>. A representative mechanical field is the surface acoustic wave (SAW) <cit.>, which is generated by the piezoelectric effect. Since piezoelectric materials are widely used in optically DBR cavity systems, SAW can be applied to modulate the optical properties of DBR cavity systems by controlling the phonon modes <cit.>.

The SAW is a type of phonon-like excitation containing mechanical vibration modes <cit.>. It is electrically generated on the surface of the piezoelectric substrate using interdigital transducers (IDTs), which are driven by the radio frequency voltage source (RF). Furthermore, SAW contains both transverse and longitudinal vibration modes; when it propagates along the surface of the substrate, it simultaneously propagates downward. The velocity of the SAW propagating along the surface is determined by the material performance of the substrate. When the SAW propagates downward, its vibration frequency can be adjusted with the design of the IDTs, and the extension depth below the surface is approximately one wavelength of the SAW. Accordingly, the surface of the substrate in the vertical direction is vibrated as a bulk acoustic resonator (BAR), which can be considered a standard mechanical resonator <cit.>. Moreover, the intrinsic damping rate of the BAR is determined by the SAW mode, which is equal to the line width of the SAW mode. In the related research, the quality factor of the BAR can exhibit the order of 10^5. Correspondingly, its frequency can exceed the order of GHz, and its intrinsic damping rate can exceed the order of 10 kHz <cit.>.

Planar DBR cavity systems have been applied to realize light-sound interactions in related experiments <cit.>. Generally, Generally, the quality of the DBR cavities can be improved by increasing the number of mirror pairs or optimizing the design of the structure <cit.>, and a representative type of cavity structures is the DBR cavity structure modified with the embedded sub-micrometer Gaussian-shaped defect. This structure is proposed by F. Ding et al.; relative to traditional DBR structures with the same number of mirror pairs, its quality factor can be effectively improved by nearly 2 orders of magnitude <cit.>. Physically, when the Gaussian-shaped defect is sandwiched between the upper and the lower Bragg mirrors, due to the local change in cavity length and the deformation of the photonic defect band, the photons are confined in a small modal volume in the vertical direction. Moreover, the light scattering induced by the lateral dielectric discontinuities is minimized. As a result, the quality of this structure dramatically improves. Based on the finite difference time domain (FDTD) calculation, the quality factors Q of the designed DRR cavity can exceed the order of 10^5 <cit.>.

Here, we propose a hybrid planar DBR cavity system, where IDTs are fabricated on the surface to generate the SAW. Moreover, the quality factor of the DBR cavity is optimized by the embedded Gaussian-shaped defect, and the DBR cavity is composed of AlAs and GaAs alternating layers, which are piezoelectric materials to support the SAW generation. When the SAW is applied to the system, we can estimate that the thickness of the upper Bragg mirrors is within the wavelength scope of the SAW. As a result, the upper Bragg mirrors in the vertical direction vibrate as a BAR, which can be considered a mechanical resonator. Accordingly, our system is turned into a standard three-level optomechanical cavity system <cit.>, which is formed by the energy levels of the DBR cavity and BAR.

The traditional three-level optomechanical cavity system is composed of an optical cavity and a mechanical resonator; it has been applied in many fields, and a representative system is the optomechanically induced transparency (OMIT) <cit.>. OMIT is a special optical phenomenon; when it occurs, the susceptibility of the optomechanical cavity system changes; then, the resonance optical field can be modulated from opacity to transparency. Physically, it arises from the quantum interference among different energy-level pathways <cit.>. Similar to the electromagnetically induced transparency (EIT) observed in three-level atomic systems <cit.>, OMIT has been applied in many fields including quantum information processing <cit.>, fast and slow light <cit.> and quantum optical storage <cit.>.

In our system, we show that when a strong pump optical field and a weak probe optical field are simultaneously applied to the hybrid optomechanical cavity system, in the presence of the SAW, a transmission window can be obtained in the weak output probe field. This phenomenon arises because under the driving of the SAW, the upper Bragg mirrors vibrate as a BAR; then, the DBR cavity becomes a lambda-type three-level optomechanical system. Under quantum interference among different energy level pathways, OMIT occurs; as a result, the transmission window is observed in the weak output probe field. Conversely, without the actuation of the SAW, the transmission window disappears. 

Compared to traditional micro-nano optomechanical cavity systems, we propose an artificial DBR optomechanical cavity system that can be applied in optical quantum information processing, such as optical switches and information storage. Moreover, our system has many advantages: (i) The system parameter settings of our system are flexible; in particular, both frequencies of the optical cavity and nanomechanical resonator can be autonomously and accurately designed according to the application requirements. (ii) Our system combines radio-frequency and optical signals to realize the electrically controlled quantum optical information processing. (iii) The relevant manufacturing technology of the DBR microcavity our system adopted is mature and feasible, and our system can be combined with quantum dots and quantum traps to expand to the field of quantum device integration. 

The paper is organized as follows. In Sec. II, we describe the proposed model and derive the system Hamiltonian. In Sec. III, we present the dynamical process of the system. In Sec. IV, we discuss the detailed physical mechanism of the OMIT and study the change in output field controlled by the SAW. Finally, we make a brief conclusion in Sec. V.



§ MODEL AND HAMILTONIAN OF THE SYSTEM






The hybrid piezo-optomechanical planar DBR cavity system that we propose is illustrated in Fig. 1, where IDTs are fabricated on the surface of the cavity to generate the SAW. The IDTs are driven by the RF, which can convert the RF voltage signal to SAW via the piezoelectric effect. The high-Q cavity planar DBR cavity is modified with the embedded Gaussian-shaped defect, as designed by F. Ding et al. to improve the quality factor of the cavity <cit.>. The optomechanical cavity is driven by a strong optical pump field and a weak optical probe field. In detail, in our system, the quarter-wavelength design for the DBR mirrors is used to maximize the photon confinement. The DBR cavity is composed of AlAs and GaAs alternating layers. The GaAs spacer is sandwiched between the upper (10 pairs) and the lower (15 pairs) Bragg mirrors, and the same structure is applied in the recent experiment <cit.>. 

Correspondingly, based on the design of the embedded Gaussian-shaped defect DBR cavity structure, the photons on resonance are confined in a small modal volume on the order of (λ/n)^3 (λ is the spacer optical thickness, and n is the refractive index of the spacer material), and the light scattering induced by the lateral dielectric discontinuities is minimized. As a result, the quality factor is improved by nearly 2 orders of magnitude relative to the traditional DBR cavity structure <cit.>. 

First, we consider the situation where only a pump optical field and a probe optical field are simultaneously applied to the DBR cavity, and the frequencies of optical fields are referred to as ω_pu and ω_pr, respectively. Moreover, we assume that the phase difference between the probe and pump fields is zero, and the phase difference is locked. In this situation, the DBR cavity is a standard two-level energy system, whose Hamiltonian is given by

    H_I=ω_aâ^†â+  
       iħε_pu(â^†e^-iω_put-âe^iω_put)+iħε_pr(â^†e^-iω_prt-âe^iω_prt).

Here, the first term of H_I describes the energy of the cavity mode, where ω_a is the frequency of the DBR cavity mode. â^† and â are the creation and annihilation operators of the cavity mode, respectively. The second and third terms describe the energies of the input optical fields, and ε_pu=√(P_puκ_a/(ħω_pu)) and ε_pr=√(P_prκ_a/(ħω_pr)) are the amplitudes of the optical pump field and probe field, respectively. P_pu and P_pr are the input powers of the optical pump field and optical probe field, respectively. κ_a is the decay rate of the optical DBR cavity.

When the RF is also applied to the IDTs, the SAW is generated, and the upper Bragg mirrors vibrate as a BAR. Then, the Hamiltonian of the BAR can be written as 



    H_b=p̂^2/2m_b+1/2m_bω_b^2q̂^2,



where q̂ and p̂ are the position and momentum operators of the BAR, m_b is the mass of the BAR, and ω_b is the frequency of the BAR, which is equal to the frequency of RF. When the upper Bragg mirrors are vibrated as a BAR, the thickness of the DBR cavity L is changed accordingly. 

In our system, the driving power of the RF transmitted by the IDTs is represented as P_rf.Referring to the relevant study <cit.>, we also assume that the SAW energy is uniformly distributed between the surface and one wavelength deep; then, the cross-sectional area through which the SAW travels is determined as A=l_IDTsλ_s=l_IDTsv_SAW/ω_b2π, where l_IDTs is the length of the IDTs fingers, v_SAW is the velocity of the SAW. According to the relevant research <cit.>, the vibration amplitude of the BAR is given as

    q_0=√(P_rf/4 π l_IDTs v_SAW^2 ρ_bω_b),

where ρ_b is the average density of the BAR. Based on a recent experiment <cit.>, the amplitude of the BAR in our system can exceed 1 pm.

For the BAR whose vibration amplitude is q_0, referring to the relevant study <cit.>, the total energy of it is given as E_b=2m_bω_b^2q_0^2. Taking the derivative of energy E_b with respect to position, we can obtain the amplitude of driving force induced by the RF, which is

    F_rf=4m_bω_b^2q_0.

Since the frequency of RF is also ω_b, according to the relevant research <cit.>, the Hamiltonian of the system can be written as

    H_O=(1-q̂/L)ω_aâ^†â+
    iħε_pu(â^†e^-iω_put-âe^iω_put)
       +iħε_pr(â^†e^-iω_prt-âe^iω_prt)
    +p̂^2/2m_b+1/2m_bω_b^2q̂^2
       -F_rfq_0sin(ω_bt).

Here, we assume that the phase difference between RF and optical fields is zero, and the phase difference is locked.

Now, we introduce the creation and annihilation operators for the BAR, which are

    q̂=√(ħ/2ω_b m_b)(b̂+b̂^†),
    p̂=i√(ħω_b m_b/2)(b̂-b̂^†),

where b̂^† and b̂ are the creation and annihilation operators of the BAR, respectively. Then the Hamiltonian is rewritten as

    H_O=ħω_aâ^†â+ħω_bb̂^†b̂-ħg_omâ^†â(b̂+b̂^†) 
       +iħε_pu(â^†e^-iω_put-âe^iω_put) +iħε_pr(â^†e^-iω_prt-âe^iω_prt)
       -iħε_rf(b̂^†+b̂)(e^-iω_prt-e^iω_prt),

where g_om=w_a/L√(ħ/2ω_b m_b) is the single-photon coupling strength, and ε_rf=F_rf√(1/8ħω_b m_b) is the amplitude of RF. As a result, the initial two-level energy cavity system becomes a standard three-levelenergy optomechanical system, as shown in Fig. 2. In the frame that rotates at the frequency of the pump field, with respect to H^'=ħω_puâ^†â-ħω_bb̂^†b̂ and neglecting the fast oscillating terms at the frequencies ± 2ω_a, ± 2ω_b, the Hamiltonian of the system is rewritten as

    H_O=ħΔ_aâ^†â+ħω_bb̂^†b̂-ħg_omâ^†â(b̂^†+b̂)
       +iħε_pu(â^†-â)+iħε_pr(â^†e^-iδt-âe^iδt)+iħε_rf(b̂^†-b̂),

where Δ_a=ω_a-ω_pu is the frequency detuning of the optical cavity from the pump field, and δ=ω_pr-ω_pu is the frequency detuning of the optical probe field from the pump field.

By combining the amplitude q_0 of the BAR with the creation and annihilation operators, the mean values of the creation and annihilation operators of the BAR are ⟨b̂⟩=⟨b̂^†⟩=b_0 <cit.>, and we can obtain

    q_0=√(ħ/2ω_b m_b)2b_0=√(ħ/2ω_b m_b)2√(n_0),

 b_0=q_0√(ω_b m_b/2 ħ) is the average amplitude of the BAR phonon operator, and n_0 is the initial mean phonon number of the BAR produced by the SAW. Moreover, we can obtain the relation of the average BAR phonon amplitude and RF amplitude, which is shown as b_0=ε_rf/2 ω_b.



§ DYNAMICS OF THE SYSTEM

In our system, we consider the situation where the intensities of the optical pump field and the probe field satisfy the condition ε_pr≪ε_pu. The system is operated in a resolved sideband regime, which satisfies the condition ω_b ∼κ_a <cit.>. Now, we adopt the quantum Langevin equations (QLEs) for the operators, where the damping and noise terms are supplemented <cit.>. More importantly, the average amplitude of the BAR phonon operator b_0 is supplemented in the quantum Langevin equations. Then, Heisenberg-Langevin equations of the cavity mode and BAR mode can be obtained as

    â̇=-(iΔ_a+κ_a/2)â+ig_omâ(b̂^†+b̂)+ε_pu+ε_pre^-iδt+f̂,
       ḃ̂̇=-(iω_b+γ_b/2)b̂+ig_omâ^†â+ε_rf+ξ̂.

Here, f̂ and ξ̂ are the quantum and thermal noise operators, respectively <cit.>. γ_b is the intrinsic damping rate of the BAR.

Then, we linearize the dynamical equations of the operators by assuming a=a_s+δa, b=b_s+δb, which are both composed of an average amplitude and a fluctuation term. Here, a_s and b_s are the steady-state values of the operators when only the strong driving field is applied to the system. By assuming ε_pr→0 and setting all time derivatives to zero, we obtain

    a_s=ε_pu/iΔ_a^'+κ_a/2,
       b_s=ig_om|a_s|^2+ε_rf/iω_b+γ_b/2,

where Δ_a^'=Δ_a-g_om( b_s^∗+b_s ) denotes the effective frequency detuning of the optical pump field from the optical cavity, including the frequency shift caused by the mechanical motion. In our system, to ensure a strong-strength coupling between the optical pump field and the BAR, we consider the situation where the cavity is driven near the red sideband with  Δ_a^'∼ω_b <cit.>.

Next, by substituting the assumptions â=a_s+δâ and b̂=b_s+δb̂ into the nonlinear QLEs and dropping the small nonlinear terms, we can obtain the linearized QLEs, which are

    δ̇â̇=-(iΔ_a^'+κ_a/2)δâ+iG_om(δb^†+δb̂)+ε_pe^-iδt+f̂,
       δ̇ḃ̂̇=-(iω_b+γ_b/2)δb̂+i(G_0^∗δa+G_0δa^†)+ξ̂,

where G_om=g_oma_s is the total coupling strength between the optical mode and BAR mode.

Furthermore, the fluctuation terms can be rewritten as

    δâ=δâ_+e^-iδt+δâ_-e^iδt,
    δb̂=δb̂_+e^-iδt+δb̂_-e^iδt,
    δf̂=δf̂_+e^-iδt+δf̂_-e^iδt,
    δξ̂=δξ̂_+e^-iδt+δξ̂_-e^iδt,

where δÔ_+ and δÔ_- (with O=a,b) correspond to the components at the original frequencies of ω_pr and 2ω_pu-ω_pr, respectively <cit.>. Next we substitute Eq. (13) into Eq. (12) and ignore the second-order small terms by equating coefficients of terms with the same frequency, and the components at the frequencies ω_pr can be obtained as

    δ̇â̇_+=(iλ_a-κ_a/2)δâ_++iG_omδb̂_++ε_pr+δf̂_+,
       δ̇ḃ̂̇_+=(iλ_b-γ_b/2)δb̂_++iG_0^∗δâ_++δξ̂_+,

where λ_a=δ-Δ_a^', λ_b=δ-ω_b.

Next, we take the expectation values of the operators in our system. The noise terms obey the following fluctuations in correlation 

    ⟨f̂_in(t) f̂_in^†(t^')⟩=[N(ω_a)+1]δ(t-t^'),
    ⟨f̂_in^†(t) f̂_in(t^')⟩=[N(ω_a)]δ(t-t^'),
    ⟨ξ̂_in(t) ξ̂_in^†(t^')⟩=[N(ω_c)+1]δ(t-t^'),
    ⟨ξ̂_in^†(t) ξ̂_in(t^')⟩=[N(ω_c)]δ(t-t^'),

where N(ω_a)=[exp(ħω_a/k_BT)-1]^-1 is the equilibrium mean thermal photon numbers of the optical fields and N(ω_b)=[exp(ħω_b/k_BT)-1]^-1 is the equilibrium mean thermal phonon number of the BAR. 

We can safely assume that the optical field satisfies the condition ħω_a/k_BT≫1 at room temperature. For the BAR driven by the SAW, whose frequency is in the GHz regime, the environment temperature in the mK regime, which can be reached inside a dilution refrigerator, is sufficient to ensure that ħω_b/k_BT≫1 <cit.>. We assume that our system is operated in the mK temperature regime, and the quantum and thermal noise terms can be ignored. 

Under the mean-field steady-state condition ⟨δ̇â̇⟩=⟨δ̇ḃ̂̇⟩=0, we can get the solution of ⟨δâ_+⟩, which is

    ⟨δâ_+⟩=(γ_b/2-iλ_b)ε_pr/(κ_a/2-iλ_a)(γ_b/2-iλ_b)+G_om^2.


Based on the input-output relation, the output field at the probe frequency ω_pr can be expressed as <cit.>

    ε_out=κ_a ⟨δâ_+⟩-ε_pr.

Then the transmission coefficient t_pr of the probe field is given by <cit.>

    t_pr=ε_out/ε_pr=κ_a ⟨δâ_+⟩/ε_pr-1.

By defining ε_T=κ_a⟨δâ_+⟩/ε_pr, we can obtain the quadrature ε_T of the output field at frequency ω_pr, which is

    ε_T=κ_a(γ_b/2-iλ_b)/(κ_a/2-iλ_a)(γ_b/2-iλ_b)+G_om^2,

which has the standard form for the OMIT <cit.>. The real part Re[ε_T] and imaginary part Im[ε_T] describe the absorptive and dispersive behaviors of the system, respectively. Correspondingly, the power transmission coefficient is further defined as 

    T_pr=|t_pr|^2.


Moreover, the phase ϕ_T of the output field can be given as <cit.>

    ϕ_T=arg[ε_T]=1/2iIm(ε_T/ε_T^∗).

In the red sideband region, the rapid phase dispersion can cause the group delay τ_T of the probe field, which can be expressed as

    τ_T=∂ϕ_T/∂ω_pr=Im[1/ε_T∂ε_T/∂ω_pr].




§ SAW-CONTROLLED OMIT AND PHYSICAL MECHANISM OF THE SYSTEM




We present below a discussion of the feasibility of the tunable OMIT in the hybrid piezo-optomechanical cavity system. In our system, the parameters that we used are described above, all of which are based on realistic systems. For the high-Q DBR cavity, the thicknesses of the GaAs and AlAs layers are λ/4n_ GaAs∼ 64.8 nm (refractive index n_ GaAs=3.57) and λ/4n_ AlAs∼ 77.6 nm (refractive index n_ AlAs=2.98), respectively. Correspondingly, the GaAs spacer optical thickness of the DBR cavity is λ=925 nm, the corresponding frequency is ω_a = 324 THz, and the actual thickness is L = λ/n_ GaAs = 259.1 nm. To improve the quality of the DBR cavity, referring to the same structure proposed by F. Ding et al., the shape of the defect is defined by Gaussian function G(x) =  L e^-x^2/2w^2, where L and W are the thickness and half-width, respectively, as shown in Fig. 1. In our system, we assume that L = L/10 = 25.9 nm and W = L = 259.1 nm. We can estimate that the quality factor of our system is increased to 10^5, and the corresponding cavity linewidth is approximately 3.5 GHz, so our system satisfies the condition of the sideband-resolved regime ω_b∼κ_a <cit.>.

The BAR is driven by the SAW, which is generated by the IDTs, and the length of the IDTs fingers is l_IDTs=400μ m. The thickness of the upper 10 pairs of Bragg mirrors is approximately 1.42 μm; as a result, it is within the wavelength scope of the SAW, whose wavelength is λ_s=2.9 μm. In our system, the selected frequency of the BAR ω_b /2 π= 1.05 GHz, which is used in the recent experiment. Based on the related studies, the quality factor of BAR exhibits the order of 10^5. Accordingly, we can estimate that the intrinsic damping rate of the BAR is γ_b /2 π= 10.5 kHz, which satisfies the OMIT occurring condition κ_a ≫γ_b <cit.>. Moreover, their material densities are ρ_ GaAs=5.37 g/cm^3 and ρ_ AlAs=3.72 g/cm^3, we can estimate that the average density ρ_b∼ 4.47 g/cm^3 and the effective motional mass of the BAR is m_b=0.33 pg. Then, based on the parameters that we use, the single-photon coupling strength can be estimated as g_om/2π = 1.54× 10^7 Hz.

Furthermore, to ensure that OMIT can be generated, the total optomechanical coupling strength should satisfy the condition G_om≥√(κ_aγ_b)/2, where a typical transmission window can be obtained <cit.>. Combining this condition with equations (3) and (11) and dropping the small terms, we obtain the maximum power of the RF power, which satisfies the relationship

    P_rf≤ (8 ħπ l_IDTs v_SAW^2 ρ_b/m_b)(ε_pu√(1/κ_aγ_b)+▵_a/2 g_om)^2,

where the relevant parameters are discussed and provided above. To ensure b_0 ≥ 1, by combining equations (3) and (9), we obtain the minimum power of the RF power, which satisfies the relationship

    P_rf≥8 ħπ l_IDTs v_SAW^2 ρ_b/m_b.


Next, we discuss the tunable OMIT behaviors of our system. First, based on the selected parameters, Re [ε_T], Im [ε_T] and the power transmission coefficient of the optical probe field as a function of (δ-ω_b)/ω_b are plotted in Fig. 3, and the blue-solid line and black-dotted line represent the situations with and without the SAW mode, respectively. When the SAW mode is absent, no transparency window can be obtained in the transmission spectrum curve. However, when the SAW mode is present, a transparency window is obtained in the transmission spectrum curve, whose center position is determined by the frequency point δ-ω_b=0. As a result, our system can be applied in the fields of optical switches, high-resolution spectroscopy and quantum information processing, which have been extensively studied <cit.>.

The phenomenon in Fig. 3 arises as follows: when the SAW is absent, the initial system is a two-level cavity system, as shown in Fig. 2(a). It also corresponds to the situation where g_om=0, referring to equation (3). As a result, the quantum interference between the energy levels cannot induce the occurrence of OMIT, and no transparency window appears. However, when the system is driven by the SAW, the upper Bragg mirrors vibrate as a BAR. Then, the initial two-level cavity system becomes a standard three-level optomechanical system, which is formed by the energy levels of the DBR cavity and BAR, as shown in Fig. 2(b). Under the effect of the optical radiation pressure, the destructive quantum interference between different energy level pathways can be generated. Then, the OMIT phenomenon occurs, and a transparency window is obtained in the transmission spectrum curve. The relevant mechanism has been extensively studied <cit.>.







Fig. 4 presents the power transmission coefficient T_pr with respect to (δ-ω_b)/ω_b for different strengths P_pu of the optical pump field. In the presence of the SAW, when P_pu increases, the transmission window at the frequency position δ=ω_b widens. The reason is that in the OMIT situation, the width Γ of the transmission window is proportional to the optical pump field power with the relations Γ=γ_b+4G_om^2/κ_a and G_om=g_oma_s, and the relevant mechanisms have been extensively studied <cit.>. As a result, this phenomenon can be applied in the fields of optical switches and optical quantum information process.

Fig. 5 presents the power transmission coefficient T_pr with respect to (δ-ω_b)/ω_b for different powers P_rf of RF. When the SAW is generated by the IDTs, which is driven by the RF, with the increase in P_rf, the power transmission coefficient at frequency position δ=ω_b decreases. The reason is that in the OMIT situation, the power transmission coefficient T_pr is proportional to the total coupling strength G_om=g_oma_s. Based on equations (3), (9) and (11), the average phonon number a_s of the pump field can be determined by P_rf. With the increase in P_rf, the total coupling strength is diminished; hence, the power transmission coefficient at frequency position δ=ω_b decreases. The minimum and maximum values of the RF power approximately correspond to transmission coefficients of 1 and 0, respectively, as presented in equations (23) and (24). The relevant mechanisms have also been extensively studied <cit.>. This phenomenon can also be applied in the field of optical quantum information processing.

To further explore the characteristics of our system, when OMIT occurs, the phase ϕ_T as functions of (δ-ω_b)/ω_b and group delay τ_T as functions of pump field power strength P_pu are plotted in Fig. 6. Fig. 6 (a) shows that when OMIT occurs, the phase at frequency position δ=ω_b is excessively modulated, which indicates that the group velocity of the probe field is altered. This situation generates a slow-light effect. Fig. 6 (b) shows that with increasing pump field power, the group delay of the probe field decreases. When the power of the pump field is sufficiently weak, the maximal time of the group delay can reach 0.03 ms. As a result, our system can be applied in the field of optical quantum information memory in solid-state systems.



§ CONCLUSION


In conclusion, we propose a scheme to generate tunable OMIT in a hybrid piezo-optomechanical cavity system. The system is composed of a high-quality planar DBR cavity modified with an embedded Gaussian-shaped defect. Moreover, IDTs are fabricated on the surface of the cavity and can generate the SAW under RF driving. We show that when a strong pump optical field and a weak probe optical field are applied to the hybrid optomechanical cavity system, in the presence of the SAW, a transmission window can be obtained in the weak output probe field. In the presence of the SAW, the upper Bragg mirrors vibrate as a BAR, and the DBR cavity becomes a lambda-type three-level optomechanical system. As a result, the destructive quantum interference between different energy level pathways generates OMIT, which makes a transmission window appear in the weak output probe field. Conversely, without the SAW, the transmission window in the weak output probe field disappears. Our scheme can be applied in the fields of optical switches, quantum information memory, high-resolution spectroscopy and quantum information processing in solid-state systems.


This work is supported by the Strategic Priority Research Program (Grant No. XDB01010200), the National Natural Sciences Foundation of China (Grants No. 11674337, No. 61605225, and No. 11547035), and the Natural Science Foundation of Shanghai (Grants No. 16ZR1448400), the General Project of National Laboratory Foundation of Radar Signal Processing (Grant No. 61424010106), and the Natural Science Foundation of the Jiangsu Higher Education Institutions
of China (Grant 20KJB140010).





apsrev4-1


