Optomechanically engineered phononic mode resonance

Yong-Pan Gao; Zhong-Xiao Wang; Tie-Jun Wang; Chuan Wang

doi:10.1364/OE.25.026638

1. Introduction

Optomechanics describes the interaction of the optical modes with the mechanical modes via radiation pressure. The optical microcavities provides an ideal platform of the optomechanical interactions which could be used in single photon and single phonon devices. The study of optomechanics [1–10] becomes a hot topic with the improvement of quality factors (Q) of the microcavities [11–14]. The microcavities can be precisely fabricated with picometer precision in resonance wavelength [15], various controlling means are available for microcavity, such as three-level electromagneticlly induced transparency [16], gain competition induced mode evolution [17, 18]. It is also of great importance in entanglement manipulation [19–23]. The optomechanics could achieve the strong intracavity electromagnetic field which ensures the interaction between the optical mode and the mechanical vibrator in optomechanics. Based on this strong interaction, the optomechanically induced transparency (OMIT) could be achieved [24–27] based on the concept of electromagneticlly induced transparency(EIT) [28–31]. Recently, the experimental implementation of OMIT indicates that the electromagnetic controlling of the photon and phonon interaction becomes reality. Both OMIT and EIT indicate that the characteristics of the system can be controlled by electromagnetic fields [24]. The optomechanics has been used to experimentally controlling the multi-mode mechanical resonators [32], and the tunable topological properties in the reduced subspace of the system is also reported [33]. On the other hand, both the optical and mechanical modes could be excited by pumping field in high-Q microresonators which exhibit various new and peculiar properties. For example, the phonon dark state could be generated by using the electromagnetic field couples with the multimode mechanical resonators [32]. Moreover, the resonance of the mechanical resonators could be synchronized by using optomechanics. Through a common or cascaded electromagnetic field, the slight detuning mechanical resonators departured from a few micrometers to several kilometers have been experimentally synchronized [34, 35]. Due to the low factors of the mechanical mode quantity, the number of mechanical oscillators can be synchronized has reached nine in experiment [36]. Furthermore, by lowering the temperature of the environment, the optomechanics could approach quantum regime which exhibits quantum mechanical behaviors [37–41]. Also the studies of electromagnetic tuning of mechanical resonator is multifarious in recent studies [42–44], but how to understand the role of the electromagnetic field in the process of its modulation phonon interaction is lack of discussion.

Traditionally, the propagation of phonons in the electromagnetic field is usually explained as an optical spring effect [45], here we believe that the electromagnetic field can also work as the phonon transmission medium. So the parameters of the electromagnetic field that affects on the transport properties of the phonon becomes significant. Here in this paper, we first give an analytical model of a single-cavity dual mechanical vibrator optomechanical system. Then the propagation characteristics of vibration between two mechanical oscillators under different electromagnetic fields are studied. The relationship between the electromagnetic field frequency and the phonon transmission coefficient are obtained. Meanwhile, the resonance of phononic mode is studied when only one mechanical resonator is excited.

The article is structured as follows: In the Sec. 2, we introduce the structure of the system and present its Hamiltonian. Based on the Hamiltonian, we deduce the relationship between the phonon transmission and the electromagnetic field of it. Then, the characteristics of the electromagnetic field is illustrated as a phonon transport medium in Sec. 3. Finally, the effects of the frequency of the mechanical resonators and the electromagnetic field on the phonon transmission is discussed in Sec. 4.

2. The Hamiltonian of the system

The structure of system is depicted Fig. 1 which describes the mechanical resonators interaction through the electromagnetic field: Two mechanical resonators are inserted in the Fabry-Pérot cavity. The additional continuous laser pump enters the optical cavity at one side. The electromagnetic field within the cavity works as a transmission medium that can transfer the vibration between the mechanical resonators. To further study the mode properties, the mechanical driven force is added to the right side of the resonator as shown in the Fig.

Fig. 1 The structure of the system. Here an input optical field works as a medium to connect the detached mechanical resonators. The CW pumping denote a continuous laser pump. The extra driven force acts on the mechanical resonator is symboled as EF.

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The Hamiltonian of the system can be written as [24]

\hat{H} = {\hat{H}}_{m e c h} + {\hat{H}}_{o p t} + {\hat{H}}_{i n t} + {\hat{H}}_{d r i v e},

{\hat{H}}_{m e c h} = \frac{{\hat{p}}_{1}^{2}}{2 m_{e f f}} + \frac{1}{2} m_{e f f} Ω_{m, 1} {\hat{x}}_{1}^{2} + \frac{{\hat{p}}_{2}^{2}}{2 m_{e f f}} + \frac{1}{2} m_{e f f} Ω_{m, 2} {\hat{x}}_{2}^{2},

{\hat{H}}_{o p t} = ℏ ω_{c a v} ({\hat{a}}^{†} \hat{a} + \frac{1}{2}),

{\hat{H}}_{i n t} = ℏ (g_{1} {\hat{x}}_{1} + g_{2} {\hat{x}}_{2}) {\hat{a}}^{†} \hat{a},

{\hat{H}}_{d r i v e} = i ℏ \sqrt{η_{o} κ_{o}} (s_{i n, o} (t) {\hat{a}}^{†} - s_{i n, o}^{*} (t) \hat{a}) + i ℏ \sqrt{η_{r} κ_{r}} (s_{i n, r} (t) {\hat{b}}_{1}^{†} - s_{i n, r}^{*} (t) {\hat{b}}_{1}) .

Here, we write the eigen-frequency of the optical mode of the micro-cavity as ω_cav, while for the mechanical resonator on the right(left) sideband the eigen-frenquency is Ω_m_,1(Ω_m_,2). The

{\hat{p}}_{1}

and

{\hat{x}}_{1}

(

{\hat{p}}_{2}

and

{\hat{x}}_{2}

) are the momentum and position operator of the right (left) handside mechanical resonator. m_eff is the effective mass of the mechanical resonator. The position and momentum operator is denoted as

{\hat{x}}_{i}

and

{\hat{p}}_{i}

. The interaction Hamiltonian H_int is related to the interaction strength g₁ and g₂, g₁ (g₂) corresponds to the coupling strength between the electromagnetic field and the mechanical resonator in the right (left) side. The coherent drive of the environment could be written as

{\hat{H}}_{d r i v e}

, the pumping amplitude for the optical mode

\hat{a}

and the mechanical mode

{\hat{b}}_{1}

are s_in_,_a and s_in_,_r. κ_o denotes the dissipation of the electromagnetic field, while κ_r shows the dissipation the mechanical resonator, η_o (η_r) is the coupling parameter of the electromagnetic(mechanical) mode. In this article we choose the critical coupling condition (η_o = η_r = 1/2), the coupling loss is equal to the intrinsic loss in this condition.

The perturbation term under the c-number form of the Heisenberg-Langevin equations with the perturbation hypothesis can be deduced from Eq. (1) (a detail derivation can be find in the appendix),

δ \dot{a} = (i (Ω - δ Ω) - \frac{κ}{2}) δ a - i α (g_{1} x_{1, z p f} (δ b_{1}^{*} + δ b_{1}) + g_{2} x_{2, z p f} (δ b_{2}^{*} + δ b_{2})),

δ {\dot{b}}_{1} = (i Ω - \frac{κ_{r, 1}}{2}) δ b_{1} - i g_{1} x_{1, z p f} (α δ a^{*} + α^{*} δ a) + \sqrt{η_{r} κ_{r, i}} s_{r} e^{- i Ω_{p} * t},

δ {\dot{b}}_{2} (i Ω - \frac{κ_{r, 2}}{2}) δ b_{2} - i g_{2} x_{2, z p f} (α δ a^{*} + α^{*} a) .

In these equations, the pumping electromagnetic field supports the steady state of the optomechanical system. And the transmission of the phonons are assumed to be the perturbation here. The terms with δ represent the perturbed field of corresponding parts. And

α = \sqrt{η_{o} κ_{o}} s_{i n, o} / (- i \bar{Δ} - κ / 2)

is the steady state amplitude of the intracavity electromagnetic field induced by the coherent drive. The

\bar{Δ} ≃ Δ = ω_{c a v} - ω l

term indicates the frequency shift of the effective detuning of the cavity x_i_,_zpf, (i = 1, 2) represents the zero point fluctuation which corresponds to the minimum displacement of the resonator. Then by rotating the electromagnetic part with the input frequency ω_l, and we set

Ω_{1} = Ω_{2} = \bar{Δ} + δ Ω = Ω

to reduced complexity of the symbolic expression.

When the phonon is pumped with the frequency of Ω_p, we can set the ansatzs:

δ a = A^{-} e^{- i Ω_{p} * t} + A^{+} e^{i Ω_{p} * t},

δ b_{1} = B_{1}^{-} e^{- i Ω_{p} * t} + B_{1}^{+} e^{i Ω_{p} * t},

δ b_{2} = B_{2}^{-} e^{- i Ω_{p} * t} + B_{2}^{+} e^{i Ω_{p} * t} .

By neglecting the high order feedback of this system, we can get

χ_{r}^{- 1} (- Ω_{p}) B_{1}^{-} = Ξ (Ω_{p}, δ Ω) B_{1}^{-} + s_{r},

χ_{r}^{- 1} (- Ω_{p}) B_{2}^{-} = Ξ (Ω_{p}, δ Ω) B_{1}^{-},

χ_{r}^{- 1} (Ω_{p}) B_{2}^{+} = Ξ^{*} (Ω_{p}, δ Ω) B_{1}^{-} .

with G = g₁x_1,_zpf = g₂x_2,_zpf, and

χ_{O} (Ω_{P}, δ Ω) = {(i Ω_{p} - i (Ω - δ Ω) - κ / 2)}^{- 1}

χ_{r} (Ω_{1 p}) = {(i Ω_{p} - i Ω - κ_{r, i})}^{- 1},

Ξ (Ω_{p}, δ Ω) = - {| α |}^{2} G^{2} [χ_{O}^{*} (Ω_{P}, δ Ω) + χ_{O} (- Ω_{P}, δ Ω)] .

3. The influence of electromagnetic field frequency on phonon transmission

Based on the expression of the coefficient Ξ(Ω_p, δΩ) in Eq. (5c), we can conclude that it is related to the transmission coefficient of the electromagnetic field to the phonon. Under the realistic conditions, we take the experimental achievable parameters in the optomechanical system. The mechanical resonators have the same mechanical frequency Ω_i = 1GHz, the cavity detuning is also set as Ω = 1GHz. And the intrinsic loss of the cavity is κ_o = 20MHz, while the mechanical resonators have the loss κ_r_,_i = 1MHz. The optomechanical coupling parameter is denoted as g_i = 2GHz/nm. The effective mass of the resonators are 5pg. And the cavity is pumped by a 1550nm CW-laser with the power 1mW.

Here we show the characteristics of the phonon propagation through the electromagnetic medium in Figs. 2. In Fig. 2(a), we show the real part of Ξ which reveals that it could be tuned by both the frequency of electromagnetic fields and mechanical resonator. And the phonon transmission factor is sensitive to the frequency of the intracavity electromagnetic field. Especially, when the frequency detuning δΩ is approaching −1GHz, the detuning of the electromagnetic field is close to zero. Meanwhile, the intracavity field is strong while the optomechanical is weakly excited. On the other hand, when the phonon is excited, the frequency of the cavity changes and the strong electromagnetic field in the cavity changes drastically, so that the vibration of the mechanical resonator can be efficiently propagated through the electromagnetic field.

Fig. 2 The real and imaginary part of the optical material parameter Ξ. Fig. (a) shows the real part of this transmission parameter. While Fig. (b) corresponds to the imaginary part. In Fig. (c) we zoom the extremum value part of the imaginary part.

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The imaginary part of Ξ is shown in Figs. 2(b) and 2(c). Here the negative value means the energy of mechanical vibration is absorbed by the electromagnetic field, i.e., phonon transmission loss. In these figures, we can conclude that the loss approaches the maximal value in the zero detuning region. In fact, there is phonon absorption in the relatively narrow band, and the loss of the amplitude is small when compared with the transmission shown in Figs. 2(a). Likewise, the transmission of the phonon could also been influenced by its own frequency Ω_p. When the frequency is appropriately selected, the loss can reach 10¹⁰. In this sense, the absorption of the phonon is also wide range controllable. Even more, we find that the imaginary part can be positive from Figs. 2(b) and 2(c). When its value is positive, the electromagnetic field is equivalent to the phonon gain medium as the energy of the gain comes from the CW laser pump. This gain-loss adjustable structure can be used to construct phonon Parity-Time symmetric(PT-symmetric) systems [46, 47]. The gain region can also be used to implement the phonon “repeater” system. Moreover, the topological phase change of the system can also be achieved by the gain-loss control.

4. The collective mechanical resonance under unbalanced driven

To characterize the phonon propagation in the practical optical mechanical system, we calculate the transmission of the phonon mode by performing a positive mode phonon drive on the right side of the mechanical resonator. By using Eq. (5), the transmittance of the phonon is given by

t_{12}^{-} = \frac{Ξ (Ω_{p}, δ Ω) \sqrt{κ_{r, 1} κ_{r, 2}}}{χ_{r}^{- 2} (- Ω_{p}) - χ_{r}^{- 1} (- Ω_{p}) Ξ (Ω_{p}, δ Ω)},

t_{12}^{+} = \frac{Ξ (Ω_{p}, δ Ω) \sqrt{κ_{r, 1} κ_{r, 2}}}{χ_{r}^{- 1} (- Ω_{p}) - χ_{r}^{- 1} (Ω_{p}) - χ_{r}^{- 1} (Ω_{p}) Ξ (Ω_{p}, δ Ω)} .

Here the subscript “12” denotes that the phonon spreads from right to left, while “−” and “+” corresponds to the positive and negative modes of the left oscillator, respectively. It is clear that the transmittance of the phonon is related to both the electromagnetic field phonon transmission parameters Ξ and the intrinsic susceptibility χ_r of the mechanical resonator, i.e. both the frequency of the pumped electromagnetic field and the frequency of the phonon to be transmitted.

The transmission coefficients of both the positive and negative modes are shown in Fig. 3. As the transmission system is relatively small in most of the areas, the high order feedback in previous could be neglected. Figure 3(a) describes the negative frequency part and we can conclude that the system has a very low transmittance in the strong absorption region of Fig. 2, that is because the energy of the phonon is absorbed by the electromagnetic field in this region. The low transmission region can also be find in Fig. 3(b) in which the positive transmission is plotted. The main difference between the negative and positive part is when the frequency of the phonon resonates with the mechanical resonators, the system has a strong transmission band. This is mainly because when the input phonon and mechanical vibrator resonance, the peaks of the intrinsic susceptibility of the two resonators coincidence. Another question is in Fig. 2(b) when the δΩ is −1GHz the phonon absorption of the electromagnetic field is very strong, but we can find in Fig. 3(a) the transmission is also very strong in this region. This is not inconsistent, its because the strong absorption comes from the strong excitation of the mechanical mode of the electromagnetic field, when the δΩ = −1Ghz, the energy of the mechanical resonator was strongly transported into the mechanical oscillate of the electromagnetic field, then the energy absorption will also increase. In fact, the strong phonon absorption makes the strong mechanical energy transmission in the electromagnetic field.

Fig. 3 The module of the transmission rate from right hand excition to the postive and negative model of the left side resonator.(a) shows the absolute value of positive transmission coefficient $| t_{12}^{-} |$ , (b) shows the absolute value of positive transmission coefficient $| t_{12}^{+} |$ .

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4.1. The phonon transmission under the electromagnetic modulation

By fixing the frequency of the transmitted phonons, how the electromagnetic field frequency affects the transmission properties of the phonons is studied.

Figure 4(a) plots the transmission coefficient vs. the frequency of the pump electromagnetic field. In the frequency domain, the phonon mode could be excited on its eigen-frequency 1GHz. The negative mode t⁺ mode is much strong than the positive mode t⁻. As the frequency of the electromagnetic field is approaching the frequency of the mechanical resonators, i.e. on condition that the pump electromagnetic field is resonant with the cavity mode, the transmission approaches the maximal value. The transmission cofficient can exceed one unit, the redundant energy comes from the optical field pumping. It is worth noting that the right mechanical resonator is excited in the positive phonon mode, and t⁺ shows the transmission coefficient for the negative mode of the left side mechanical resonator. Here the negative value of Ξ corresponds to the gain for the transmission coefficient $t_{12}^{+}$ .

Fig. 4 The absolute value of the transmission coefficient for both positive and negative mode. (a) The transmission coefficient for the phonon with frequency 1GHz under different electromagnetic frequency, (b) The transmission coefficient for the phonon with frequency 0.5GHz under different electromagnetic frequency, (c) The transmission coefficient for the phonon with frequency 0.1GHz under different electromagnetic frequency.

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In Figs. 4(b) and 4(c), the system exhibits two well-separated peaks in the transmission spectra of the pumping frequency of the mechanical resonator is 0.5GHz in Fig. 4(b), while it is 0.1GHz in Fig. 4(c). Here the amplitudes of the t⁺ part are always stronger than that of t⁻. This phenomena is attributed to the reason that the negative value of Ξ means gain for $t_{12}^{+}$ transmission while loss for $t_{12}^{-}$ . Comparing figure (b) with (c), by neglecting the changes in the amplitude, the main difference is the splitting width of the two bands. The splitting width increases with the decrement of the phonon frequency. This phenomena is similar to the Autler-Townes splitting effect (ATS) [48]. Both the ATS effect and our system shown in this subsection illustrate the phenomenon that the distance between the two transmission bands changes with the pump frequency. The main difference is, ATS effect describes the band-splitting with the pump frequency of the electromagnetic field, while this study shows the band-splitting under different frequency of the transmitted phonon.

4.2. The transmission rate for different phonon frequency

Here, the transmission properties of phonons with different frequencies is studied under the same electromagnetic field pumping. In order to achieve this, we fix the frequency of the electromagnetic field and study how the transmission coefficient various with the phonon frequency.

This transmission features is shown in Fig. 5. In these Fig. 5(a) is corresponding to the δΩ = −1.0GHz, Fig. 5(b) with δΩ = −0.99GHz while in Fig. 5(c) the detuning δΩ = −0.8GHz. We can find a split for the t⁺ mode and a single extremal curve for the t₋ mode. Here we first study $t_{12}^{+}$ model, we can find the split increase with the increase of the electromagnetic frequency. Even more, we find that only the location of the low frequency band changes with the frequency of the electromagnetic field. This is different from the energy split progress of ATS and the progress shown in last section. Actually here, the transmission bands correspond to the phonon level structure of the whole system. Under the electromagnetic medicated, the shift of the transmission band shows the movement of phonon level. The fixed band in these Figs come from the system resonance, so it is always site in the position of 1GHz.

Fig. 5 The transmission coefficient for both positive and negative part under unbalcanced driven. (a) The transmission coefficient for the phonon under different frequency with electromagnetic frequency detune δΩ = −1GHz, (b) The transmission coefficient for the phonon under different frequency with electromagnetic frequency detune δΩ = −0.99GHz, (c) The transmission coefficient for the phonon under different frequency with electromagnetic frequency detune δΩ = −0.8GHz,.

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The blue dashed line in Fig. 5 shows how the transmission coefficient $| t_{12}^{-} |$ varies with the pumping mechanical frequency. This is a single extremum value curve, it is because the mode pattern of two identical oscillators has exactly the same intrinsic susceptibility, the same intrinsic susceptibility produce produce the same transmission position, then this curve single valued. Both the positive and the negative transmission coefficient provide us an effective method to tune the position of the transmission band. Based on this feature, we can build an adjustable phonon filter device.

In fact, the detune of frequency of the mechanical resonators will also influence the result. But, there is no doubt that the detuning will reduce the transmission rate. And, the synchronization of the mechanical resonator in the optomechanics have been experimentally achieved [49, 50]. So we will not discuss the influence of the detuning of the mechanical resoantors.

5. Summary

In summary, we propose to tune the phononic mode resonance of the mechanical resoantors using the electromagnetic field as phonon propagation media. By constructing the analytic model to describe the double mechanical resonators and a single Fabry-Pérot cavity interaction, we find that the propagation of phonons could be controlled by the electromagnetic fields. Moreover, the mode-splitting phenomena similar to ATS is observed during the process of phonon transportation. We envision that the multi-mechanical resonator system could hold great potential for the further study of phononic device and quantum manipulation of single phonon.

A. The electromagnetic field induced photon-phonon coupling

Here the full Hamiltonian of the system in the main text could be written as

\hat{H} = {\hat{H}}_{m e c h} + {\hat{h}}_{o p t} + {\hat{H}}_{i n t} + {\hat{H}}_{d r i v e}

{\hat{H}}_{m e c h} = \frac{{\hat{p}}_{1}^{2}}{2 m_{e f f}} + \frac{1}{2} m_{e f f} Ω_{m, 1} {\hat{x}}_{1}^{2} + \frac{{\hat{p}}_{2}^{2}}{2 m_{e f f}} + \frac{1}{2} m_{e f f} Ω_{m, 2} {\hat{x}}_{2}^{2}

{\hat{H}}_{o p t} = ℏ ({\hat{a}}^{†} \hat{a} + \frac{1}{2})

{\hat{H}}_{i n t} = ℏ (g_{1} \hat{x} + g_{2} {\hat{x}}_{2}) {\hat{a}}^{†} \hat{a}

{\hat{H}}_{d r i v e} = i ℏ \sqrt{η_{r} κ_{r}} (s_{i n, o} (t) {\hat{a}}^{†} - s_{i n, o}^{*} (t) \hat{a}) + i ℏ \sqrt{η_{r} κ_{r}} (s_{i n, r} (t) {\hat{b}}_{1}^{†} - s_{i n, r}^{*} (t) {\hat{b}}_{1})

We assume that the electromagnetic field and right mechanical mode could be excited by the external field. The Heisenberg-Langevin equations of the photon and phonon representation could be described by the form

\frac{d \hat{a}}{d t} = (i Δ - \frac{κ}{2}) \hat{a} - i (g_{1} \sqrt{\frac{ℏ}{2 m_{e f f} Ω_{m, 1}}} ({\hat{b}}_{1}^{†} + {\hat{b}}_{1}) + g_{2} \sqrt{\frac{ℏ}{2 m_{e f f} Ω_{m, 1}}} ({\hat{b}}_{2}^{†} + {\hat{b}}_{2})) \hat{a} + \sqrt{η_{r} κ_{r}} s_{i n, o} (t)

\frac{d {\hat{b}}_{1}}{d t} = (i Ω_{m, 1} - \frac{κ_{r, 1}}{2}) {\hat{b}}_{1} - i g_{1} \sqrt{\frac{ℏ}{2 m_{e f f} Ω_{m, 1}}} {\hat{a}}^{†} \hat{a} + \sqrt{η_{r} κ_{r, i}} s_{i n, r} (t)

\frac{d {\hat{b}}_{2}}{d t} = (i Ω_{m, 2} - \frac{κ_{r, 2}}{2}) {\hat{b}}_{2} - i g_{1} \sqrt{\frac{ℏ}{2 m_{e f f} Ω_{m, 2}}} {\hat{a}}^{†} \hat{a}

Under the strong optical pumping condition assumptions, we can solve the steady state solution which is defined by the electromagnetic field. When neglecting the driving field of the mechanical part, The steady state equations are given by

0 = (i Δ - \frac{κ}{2}) \hat{a} - i (g_{1} ({\hat{b}}_{1}^{†} + {\hat{b}}_{1}) + g_{2} ({\hat{b}}_{2}^{†} + {\hat{b}}_{2})) {\hat{a}}^{†} \hat{a} + \sqrt{η_{o} κ_{o}} s_{i n, o} (t)

0 = (i Ω_{i} - \frac{κ_{r, i}}{2}) {\hat{b}}_{i} - i g_{i} {\hat{a}}^{†} \hat{a} i = 1, 2 .

Then we can get the steady state solutions of the optical field and mechanical resontors as

α = \frac{\sqrt{η_{o} κ_{o}}}{- i \bar{Δ} - κ / 2} s_{i n, o}

{\bar{x}}_{i} = \frac{{\bar{a}}^{2}}{m_{e f f, i} Ω_{i}^{2}} i = 1, 2 .

B. The perturbation theory of the phonons transmission

By neglecting the average field of mechanical resonator displacement, we can eliminate it by slight tuning of the frequency of the input field. Then we can rewrite the field of the system in the first order perturbation $\hat{a} (t) = α + δ \hat{a} (t)$ and ${\hat{b}}_{i} (t) = β_{i} + δ {\hat{b}}_{i} (t)$ . The perturbation terms obey the kinetic equation,

δ \dot{\hat{a}} = (i \bar{Δ} - \frac{κ}{2}) δ \hat{a} - i α (g_{1} (δ {\hat{b}}_{1}^{†} + δ {\hat{b}}_{1}) + g_{2} (δ {\hat{b}}_{2}^{†} + δ {\hat{b}}_{2}))

δ {\dot{\hat{b}}}_{1} = (i Ω_{1} - \frac{κ_{r, 1}}{2}) δ {\hat{b}}_{1} - i g_{1} (α δ {\hat{a}}^{†} + α^{*} δ \hat{a}) + \sqrt{η_{r} κ_{r, i}} s_{i n, r} (t)

δ {\dot{\hat{b}}}_{2} = (i Ω_{2} - \frac{κ_{r, 2}}{2}) δ {\hat{b}}_{2} - i g_{2} (α δ {\hat{a}}^{†} + α^{*} δ \hat{a})

Also by setting the parameters of the system as

\bar{Δ} ≃ Δ ≃ ω_{c a v} - ω_{l}

and

Ω_{1} = Ω_{2} = \bar{Δ} + δ Ω = Ω

. When the mechanical resonator is harmonic drive, we write

s_{i n, r} = s_{r} e^{- i Ω_{p} * t}

. The s_r is the mechanical pumping strength, while Ω_p is its frequency, Then the c-number form of this equation could be expressed as

δ \dot{a} = (i (Ω - δ Ω) - \frac{κ}{2}) δ a - i α (g_{1} (δ b_{1}^{*} + δ b_{1}) + g_{2} (δ b_{2}^{*} + δ b_{2}))

δ {\dot{b}}_{1} = (i Ω - \frac{κ_{r, 1}}{2}) δ b_{1} - i g_{1} (α δ a^{*} + α^{*} δ a) + \sqrt{η_{r} κ_{r, i}} s_{r} e^{- i Ω_{p} * t}

δ {\dot{b}}_{2} = (i Ω - \frac{κ_{r, 2}}{2}) δ b_{2} - i g_{2} (α δ a^{*} + α^{*} δ a)

With the assumption that the pump frequency of the phonon is Ω_p, we can get the ansatz

δ a = A^{-} e^{- i Ω_{p} * t} + A^{+} e^{i Ω_{p} * t}

δ b_{1} = B_{1}^{-} e^{- i Ω_{p} * t} + B_{1}^{+} e^{i Ω_{p} * t}

δ b_{2} = B_{2}^{-} e^{- i Ω_{p} * t} + B_{2}^{+} e^{i Ω_{p} * t}

Putting Eq. (13) into Eq. (12), and taking the terms with same frequency, we get

χ_{O}^{- 1} (- Ω_{P}, δ Ω) A^{-} = - i α {g_{1} [B_{1}^{+ *} + B_{1}^{-}] + g_{2} [B_{2}^{+ *} + B_{2}^{-}]}

χ_{O}^{- 1} (Ω_{P}, δ Ω) A^{+} = - i α {g_{1} [B_{1}^{- *} + B_{1}^{+}] + g_{2} [B_{2}^{- *} + B_{2}^{+}]}

χ_{r}^{- 1} (- Ω_{p}) B_{1}^{-} = - i g_{1} [α A^{+ *} + α^{*} A^{-}] + s_{r}

χ_{r}^{- 1} (Ω_{p}) B_{1}^{+} = - i g_{1} [α A^{-} + α^{*} A^{- *}]

χ_{r}^{- 1} (- Ω_{p}) B_{2}^{-} = - i g_{2} [α A^{+ *} + α^{*} A^{-}]

χ_{r}^{- 1} (Ω_{p}) B_{2}^{+} = - i g_{2} [α A^{-} + α^{*} A^{- *}]

Here we have the the intrinsic susceptibility of this hybrid system as

χ_{O} (Ω_{P}, δ Ω) = {(i Ω_{p} - i (Ω - δ Ω) - κ / 2)}^{- 1}

χ_{r} (Ω_{1 p}) = {(i Ω_{p} - i Ω - κ_{i, i})}^{- 1}

If we assume the feedback of the electromagnetic filed is weak enough, then the high order feedback (the photon-phonon-photon) process can be neglected. So we have

χ_{r}^{- 1} (- Ω_{p}) B_{1}^{-} = - | α |^{a} g^{a} [χ_{O}^{*} (Ω_{P}, δ Ω) + χ_{O} (- Ω_{P}, δ Ω)] B_{1}^{-} + s_{r}

χ_{r}^{- 1} (- Ω_{p}) B_{2}^{-} = - | α |^{a} g^{a} [χ_{O}^{*} (Ω_{P}, δ Ω) + χ_{O} (- Ω_{P}, δ Ω)] B_{1}^{-}

and the optical phonon transmit material parameter could be written as

Ξ (Ω_{p}, δ Ω) = - | α |^{2} g^{2} [χ_{O}^{*} (Ω_{P}, δ Ω) + χ_{O} (- Ω_{P}, δ Ω)]

Then we can get the transmission coefficients as

t_{12}^{-} = \frac{Ξ (Ω_{p}, δ Ω) \sqrt{κ_{r, 1} κ_{r, 2}}}{χ_{r}^{- 2} (- Ω_{p}) - χ_{r}^{- 1} (- Ω_{p}) Ξ (Ω_{p}, δ Ω)}

t_{12}^{+} = \frac{Ξ (Ω_{p}, δ Ω) \sqrt{κ_{r, 1} κ_{r, 2}}}{χ_{r}^{- 1} (- Ω_{p}) χ_{r}^{- 1} (Ω_{p}) - χ_{r}^{- 1} (Ω_{p}) Ξ (Ω_{p}, δ Ω)}

Funding

Ministry of Science and Technology of the People's Republic of China (MOST) (2016YFA0301304); National Natural Science Foundation of China (61622103, 61471050 and 61671083); the Fok Ying-Tong Education Foundation for Young Teachers in the Higher Education Institutions of China (Grant No. 151063); and the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), P. R. China.

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Optomechanically engineered phononic mode resonance

Abstract

1. Introduction

2. The Hamiltonian of the system

3. The influence of electromagnetic field frequency on phonon transmission

4. The collective mechanical resonance under unbalanced driven

4.1. The phonon transmission under the electromagnetic modulation

4.2. The transmission rate for different phonon frequency

5. Summary

A. The electromagnetic field induced photon-phonon coupling

B. The perturbation theory of the phonons transmission

Funding

References and links

Cited By

Figures (5)

Equations (53)

Optics Express