## Abstract

We study the controllable optical response in a three-mode optomechanical system comprised of two indirectly coupled cavity modes and an intermediate mechanical mode. The two cavity modes are assumed to have different frequencies and driven by two control fields on the red and blue sidebands, respectively. When the system is perturbed by two probe fields satisfying the specific matching condition, a series of intriguing phenomena can be observed by adjusting phases and amplitudes of the control fields, such as absorption-amplification switching, ultra-narrow response windows, frequency-independent perfect reflection, and ultralong optical group delay. We also compare our system with conventional optomechanical systems to highlight its distinct features. Our results may have potential applications in optical communication and quantum information processing.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In the past decades, due to the rapid development of nanotechnology, optomechanical systems have become a powerful platform for exploring and testing quantum sciences at the macroscopic scale [1, 2]. As an interdiscipline, optomechanics primarily studies the nonlinear interactions between electromagnetic fields and mechanical objects. Such interactions, arising from the radiation pressure of optical modes, have led to a host of important applications, such as ground-state cooling of mechanical oscillators [3,4], quantum synchronization [5–7], ultralow-threshold chaos [8], and tunable optical nonreciprocity [9–15]. Similar to multilevel atomic systems, some intriguing phenomena have been observed in multimode optomechanical systems due to quantum interference between different excitation paths [1,2]. This has sparked an upsurge in the study of multimode optomechanical systems with diverse driving schemes [9–23]. For instance, optomechanically induced transparency (OMIT) [24–26] and absorption (OMIA) [27,28] have been well explored as the optomechanical analogues of electromagnetically induced transparency (EIT) [29,30] and absorption (EIA) [31,32], respectively. We note that OMIT is usually realized in optomechanical systems driven on the red sidebands, whose beam-splitter interactions can result in the required destructive interference. On the other hand, the realization of OMIA depends on constructive interference, which typically arises from mode-squeezing interactions in the case of a blue-sideband driving [1,2,33]. Quite recently, it has been shown that optomechanical systems driven on multiple sidebands can provide a promising platform for many important applications, such as strong entanglement [34–36] and directional amplifier [33,37]. Exploiting optomechanical interference skillfully, it is also viable to realize other coherent phenomena like coherent perfect absorption (CPA), transmission (CPT), and synthesis (CPS) [18,20,38], or attain phase-dependent optical responses as a closed-loop transition structure is formed [21, 39, 40]. In spite of a few seminal works [33–37], combining the techniques of phase modulation and multi-sidebands excitation is still a less cultivated land in optomechanical studies.

In this paper, we consider a three-mode optomechanical system, whose left and right cavity modes interact linearly with an intermediate mechanical mode and are driven by two control fields on the red and blue sidebands, respectively. Our model is different from a similar one [19] where two cavity modes are coupled with a mechanical mode via quadratic interactions and driven by different red-detuned fields. We also employ two probe fields to perturb our optomechanical system by assuming that they are near resonant with corresponding cavities modes. When the probe frequencies satisfy the specific matching condition, our system is found to exhibit distinct features as compared to conventional three-mode optomechanical systems [16–19], including ultra-narrow response windows and significant amplification. It is also viable to realize other intriguing phenomena like absorption-amplification switching, frequency-independent reflection, and ultraslow light propagation in our model.

## 2. Model and equations

We consider in Fig. 1(a) a three-mode optomechanical system based on Fabry-Pérot cavities with two optical cavities separated by a mechanical membrane. The two cavities are assumed to have different mode frequencies so that they cannot interact directly even if the membrane is not a perfect mirror. Here the left (right) cavity mode is driven by a control field of amplitude *ε*_{c1} (*ε*_{c2}) and frequency *ω*_{c1} (*ω*_{c2}), and also perturbed by a probe field of amplitude *ε*_{p1} (*ε*_{p2}) and frequency *ω*_{p1} (*ω*_{p2}). Fig. 1(b) shows a four-level configuration with *n*_{a1} (*n*_{a2}) and *n _{b}* being the excitation numbers of the left (right) cavity mode and the mechanical mode, respectively. It is clear that the couplings between different levels form a closed-loop transition so that the optical response should be phase-dependent. In addition, our results do not depend on the quality factors, or more precisely, the frequencies of cavity and mechanical modes as shown below, thus our model can also be realized with other platforms like microring optomechanical systems [41], superconducting circuit systems [42], and optomechanical crystal systems [43]. The Hamiltonian of our optomechanical system can be written as

*a*(${a}_{j}^{\u2020}$) is the annihilation (creation) operator of the

_{j}*jth*cavity mode with frequency

*ω*,

_{j}*b*(

*b*

^{†}) is the annihilation (creation) operator of the mechanical mode with frequency

*ω*, and

_{m}*g*is the single-photon optomechanical coupling strength between cavity mode

_{j}*a*and mechanical mode

_{j}*b*. The factor (−1)

*stands for the different dependence of the left and right cavity mode frequencies on the mechanical motion. We have taken*

^{j}*ϑ*

_{1}and

*ϑ*

_{2}as phases of the two control fields while assuming vanishing phases for both probe fields because it is the relative phase of all external fields that determines the optical response in a closed-loop transition. We have also set

*ε*

_{p1}=

*ε*

_{p2}=

*ε*for simplicity.

_{p}In the rotating frame with respect to ${\mathscr{H}}_{0}={\sum}_{j=1,2}{\omega}_{cj}{a}_{j}^{\u2020}{a}_{j}$, the Hamiltonian can be rewritten as

*=*

_{j}*ω*−

_{j}*ω*is the detuning between cavity mode

_{cj}*a*and control field

_{j}*ε*;

_{cj}*δ*=

_{j}*ω*−

_{pj}*ω*is the detuning between probe field

_{cj}*ε*and control field

_{pj}*ε*. Taking relevant dissipation and noise terms into account, one can attain with

_{cj}*ℋ′*the following Heisenberg-Langevin equations

*κ*represents the identical decay rate of both cavity modes (we consider here the overcoupled regime [44,45] where the total cavity loss almost roots in the external coupling loss) while

*γ*is the damping rate of the mechanical mode. Moreover, ${a}_{j}^{\mathit{in}}$ and

_{m}*b*denote respectively the input vacuum noise of cavity mode

^{in}*a*and the input thermal noise of mechanical mode

_{j}*b*, both of which have zero mean values. For strong pumping, each mode operator can be expressed as a sum of its steady-state value and quantum fluctuation,

*i.e.*,

*a*=

_{j}*α*+

_{js}*δa*and

_{j}*b*=

*β*+

_{s}*δb*.

Neglecting the weak probe fields and all time derivatives, we attain from Eq. (3) the following steady-state values

*= Δ*

_{j}*+ (−1)*

_{j}*2*

^{j}*g*Re(

_{j}*β*) being the effective detuning between cavity mode

_{s}*a*and control field

_{j}*ε*. Meanwhile, the linearized equations for the quantum fluctuations can be written as

_{cj}*G*=

_{j}*g*is an enhanced (effective) optomechanical coupling strength. To determine dynamics of the quantum fluctuations, we need to simultaneously solve Eq. (5) and its complex conjugate counterparts

_{j}α_{js}Now we assume that (*i*.) the two cavity modes are driven on the red and blue sidebands respectively with Δ′_{1} = −Δ′_{2} = *ω _{m}* and (

*ii*.) our optomechanical system works in the resolved sideband regime with

*ω*≫ {

_{m}*κ*,

*γ*,

_{m}*G*

_{1,2}}. In this case, we can perform the transformation

*σ*=

_{j}*δ*− Δ′

_{j}*≈*

_{j}*ω*−

_{pj}*ω*is the detuning between cavity mode

_{j}*a*and probe field

_{j}*ε*. Since it is the relative phase of all external fields that determines the optical response, we can further simplify Eq. (8) into

_{pj}*G*

_{1}=

*G*and

*G*

_{2}=

*ηGe*with

^{iθ}*η*(

*θ*) being the strength ratio (phase difference) of the two coupling strengths. For convenience, one can express Eq. (9) in the matrix form as where $u={\left(\delta {a}_{1},\delta {a}_{2}^{\u2020},\delta b\right)}^{T}$,

*f*= (

*ε*

_{p}e^{−iσ1t},

*ε*

_{p}e^{iσ2t}, 0)

^{T}, and $n={\left(\sqrt{2\kappa}{a}_{1}^{\mathit{in}},\sqrt{2\kappa}{a}_{2}^{\mathit{in}\u2020},\sqrt{2{\gamma}_{m}}{b}^{\mathit{in}}\right)}^{T}$ are the vectors of quantum fluctuations, probe fields, and system noises in order, while the coefficient matrix

*M*can be written as

Note our system may be unstable due to the introduction of a blue-detuned control field with its stability typically examined via the Routh-Hurwitz criterion [46,47]. Since the analytic formula is too cumbersome, we adopt here an alternative method to judge the stability: the system is stable when the real parts of all eigenvalues of matrix *M* are negative. As shown in Fig. 2, the system stability is independent of the phase difference *θ* and the stable regime can be attained only in case of 0 ⩽ *η* ⩽ 1 (|*G*_{2}| ≤ |*G*_{1}|).

Our aim is to realize controllable optical response for the left probe field with the help of other fields. To achieve this aim, we further assume *σ*_{1} = −*σ*_{2} = *σ*, which implies in fact *ω*_{p1} + *ω*_{p2} = *ω*_{1} + *ω*_{2}. Assuming that all steady-state solutions to Eq. (9) have the form 〈*δo*〉 = *O* +*e*^{−iσt} + *O* −*e ^{iσt}*, it is viable to attain the following two terms

*δa*

_{1}and

*δa*

_{2}, respectively, with

*f*

_{1}=

*κ*−

*iσ*and

*f*

_{2}=

*γ*−

_{m}*iσ*. We can find from Eq. (12) that only the left cavity responds to the left probe field while the right cavity just serves as an auxiliary cavity. According to the input-output relation [48], the output amplitude of the left cavity can be written as where the oscillating term can be removed, if we also assume

*ε*=

_{outL}*ε*

_{L+}

*e*

^{−iσt}+

*ε*

_{L−}

*e*, to yield which describes the left output component oscillating at frequency

^{iσt}*ω*

_{p1}. In this way, we can define the left reflection coefficient

*r*= 2

_{L}*κA*

_{1+}/

*ε*− 1 and the left reflectivity

_{p}*R*= |

_{L}*r*|

_{L}^{2}.

For a conventional three-mode optomechanical system driven by two red-sideband control fields (Δ′_{1} = Δ′_{2} = *ω _{m}*) and perturbed by two identical probe fields (

*σ*

_{1}=

*σ*

_{2}=

*σ*) [18, 20], Eq. (12) turns out to be

*R′*= |

_{L}*r′*|

_{L}^{2}with

*r′*= 2

_{L}*κA′*

_{1+}/

*ε*− 1.

_{p}## 3. Optomechanical switching with ultra-narrow linewidth

We first consider that the two control fields are in phase (*θ* = 0). Fig. 3(a) shows the influence of coupling strength *G* and detuning *σ* on the reflectivity *R _{L}* with an absorption window found near the resonance position (

*σ*= 0). Clearly, for a fixed strength ratio

*η*= |

*G*

_{2}/

*G*

_{1}| = 0.9, this absorption window becomes narrower and narrower as

*G*decreases. For comparison, we make a similar calculation on

*R′*in Fig. 3(b) and find that the absorption window is much wider for the same parameters in a conventional three-mode system. Reflectivities

_{L}*R*and

_{L}*R′*are also plotted against

_{L}*η*and

*σ*, respectively, in Figs. 3(c) and 3(d). We find that the absorption window deepens and narrows gradually as

*η*increases in our system. In the conventional three-mode system, however, the window depth (width) exhibits a fast (slow) non-monotonic variation, i.e., first increases and then reduces, with the increase of

*η*. Moreover,

*R*and

_{L}*R′*can approach zero for

_{L}*η*→ 1 and

*η*→ 0.4, respectively, implying the existence of CPA phenomenon [18, 38] in both three-mode systems. The CPA window in our system is certainly much narrower than that in the conventional three-mode system. As is well known, one major advantage of optomechanical systems is that they can serve as interferometers to allow a direct measurement of mechanical displacements via the phase shifts of transmitted or reflected fields [1,49]. In general, a narrower window enables a more precise measurement, so our three-mode system may have potential applications in,

*e.g.*, weak force sensing [50–53] and mechanical Fock-state detection [54–56].

To gain a deeper insight, we further examine in Fig. 4 the profiles of reflectivity *R _{L}* and relevant effective mechanical damping rates. Figs. 4(a) and 4(b) show that the absorption window becomes much narrower with the decrease of

*G*and/or the increase of

*η*, and a larger

*η*typically corresponds a stronger absorption. In particular, the green dot-dashed line in Fig. 4(b) verifies that our three-mode system supports an ultra-narrow CPA window. For

*η*= 0, the right cavity is decoupled with the mechanical mode so that our system reduces to a two-mode optomechanical system driven on the red sideband. In this case, as shown in Fig. 4(c), the left probe field is almost unabsorbed with a very shallow window in reflectivity. On the other hand, for

*η*= 1, we may observe a frequency-independent perfect reflection (FIPR) phenomenon with

*R*(

_{L}*σ*) ≡ 1, as shown by the yellow dotted line in Fig. 4(b). This is because we have

*r*=

_{L}*ε*

_{L+}/

*ε*= (

_{p}*κ*+

*iσ*)/(

*κ*−

*iσ*) when

*η*= 1 and

*θ*= 0 according to Eq. (12). Thus FIPR is a result of perfect destructive interference between the input probe field and the corresponding reflected field. This phenomenon may also attributed to the phononic dark mode [20], an optomechanical analogue of the atomic dark state [57,58], in which the mechanical excitation is completely suppressed (

*β*=

_{s}*B*

_{±}= 0).

Considering that the linewidth of an absorption, transparency, or amplification window in optomechanics is determined by the effective mechanical damping rate [1,59,60], we calculate the steady-state solutions to Eq. (9) to attain

*δb*. It is clear that the damping rate

*γ*and the detuning

_{m}*σ*are modified by a complex quantity (|

*G*

_{1}|

^{2}− |

*G*

_{2}|

^{2})/(

*κ*−

*iσ*). In this case, the effective mechanical damping rate of our system reads

*η*= 0 [38] and

*G*and/or the decrease of

*η*will yield a widened window in our system, but also show that Γ

_{opt,3}> Γ

_{opt,2}> Γ

_{opt,1}always holds as long as

*η*and

*G*are nonvanishing, as shown intuitively in Fig. 4(d). We find in particular from the three red lines that Γ

_{opt,1}increases at a much lower rate as compared to Γ

_{opt,2}and Γ

_{opt,3}as

*G*becomes larger, while from the three blue lines that only Γ

_{opt,1}decreases as

*η*becomes larger and can approach zero for

*η*→ 1.

Now, we turn to consider that the two control fields are out of phase (*θ* = *π*). Fig. 5 is plotted in a way similar to Fig. 3 by examining the dependence of *R _{L}* and

*R′*on

_{L}*G*and

*η*for comparison. In this case, it is an amplification (instead of absorption) window that appears near the resonant position

*σ*= 0. We also find that (

*i*.) the amplification window in our system is much narrower than that in the conventional three-mode optomechanical system and (

*ii*.) the corresponding linewidths are sensitive to

*G*and

*η*in different ways just as found in the case of

*θ*= 0. Figs. 5(c) and 5(d) show, in particular, that the amplification effect in our system can be exponentially enhanced as

*η*is increased in the stable regime while that in the conventional three-mode system first increases and then reduces in a roughly linear way with the maximum

*R′*= 2 located at

_{L}*η*≈ 0.42. Therefore, our three-mode system may have potential application in precise optical amplification owing to the tunable ultra-narrow window.

Heretofore our calculations have been restricted to the cases of *θ* = 0 and *θ* = *π*. Next, we examine in Fig. 6 the reflectivity profiles of our system for different values of *θ*. It is clear that we can realize an absorption-amplification transition by adjusting *θ* with the optimal absorption (amplification) corresponding to *θ* = 0 (*θ* = *π*), and the reflectivity profiles are asymmetric with respect to *σ* = 0 as long as we have *θ* ≠ *nπ* (*n* = 0, ±1, ±2, ...).

## 4. Slow light effect

It is well known that an optical signal with its carrier frequency falling into an OMIT window experience a rapid phase dispersion and thus exhibit a largely reduced group velocity, similar to what happens in an EIT window of atomic systems. Such a slow light effect is essential for optical communications and quantum information process and have been widely studied in various optomechanical systems [24–26]. This urges us to further examine the slow light effect of our system by evaluating the optical group delay

where*ϕ*=arg[

*r*(

_{L}*ω*

_{p1})] is the phase of the left output field at frequency

*ω*

_{p1}. Similarly, we can define for the left output field of the conventional three-mode optomechanical system [18, 20] with

*ϕ′*=arg[

*r′*(

_{L}*ω*)]. As usual, the slow (fast) light effect will be attained in case that the optical group delay is positive (negative). The group refractive index

_{p}*n*(

_{g}*n′*), another typical figure of merit for evaluating the slow and fast light effects, can be easily attained because it is proportional to the optical group delay

_{g}*τ*(

_{L}*τ′*) [61,62].

_{L}We plot in Fig. 7 the profiles of optical group delays in the amplification regime (*θ* = *π*). Figs. 7(a) and 7(b) show that both *τ _{L}* and

*τ′*can be largely modified as

_{L}*G*is slightly changed and a smaller

*G*always yields a larger optical group delay. It is also clear that

*τ*increases at a larger rate than

_{L}*τ′*as

_{L}*G*gradually decreases, indicating that our system is more promising than the conventional three-mode optomechanical system in achieving ultralong optical group delays. Fig. 7(c) further shows that in our system the slow light effect can be greatly enhanced by slightly increasing

*η*, which is especially true when

*η*→ 1. This is however impossible in the conventional three-mode system because

*τ′*seems not sensitive to

_{L}*η*as shown in Fig. 7(d), which is more evident when

*η*→ 1.

Finally, we examine in Fig. 8 how the slow light effect depends on the phase difference of control fields. Fig. 8(a) shows that a deviation of *θ* from *nπ* (*n* = 0, ±1, ±2, ...) can not change the optical group delay near the resonance position (*σ* = 0), but may lead to an additional slow light window with its position, width, and height being *θ*-dependent. This is consistent with the results described by output phase *ϕ* in Fig. 8(b), where the slope of phase dispersion is not sensitive to *θ* near *σ* = 0 but exhibits an evident difference near *σ* = *κ* for different values of *θ*. It is worth noting that *η* should not be too large in order to avoid working near the unstable regime, and *G* should not be too small in order to ensure the strong-pump assumption. That is, there is a tradeoff between the amplification degree, the slow light effect, and the stability condition, as discussed in [12]. In spite of this tradeoff, it is still viable to realize remarkable optical amplification accompanied by ultralong group delay in our system.

## 5. Conclusions

In summary, we have proposed a three-mode optomechanical system, comprising two indirectly coupled cavity modes and an intermediate mechanical mode. The two cavity modes are driven on the red and blue sidebands respectively by two control fields, and are also perturbed by two probe fields of identical amplitudes and matching frequencies. Compared to optomechanical systems driven only on the red sidebands, our system exhibits some distinct advantages, such as ultra-narrow response windows and significant amplification effect. In particular, the linewidth of the response window and the amplification effect can be flexibly controlled by adjusting the amplitudes and phase difference of the control fields. It is also viable to realize phase-dependent absorption-amplification transition, frequency-independent perfect reflection, and ultralong optical group delay in our optomechanical system.

## Funding

National Natural Science Foundation of China (NSFC) (10534002, 11674049, 11774024, and 11704063); Jilin Scientific and Technological Development Program (20180520205JH); Program of State Key Laboratory of Quantum Optics and Quantum Optics Devices (KF201807).

## Acknowledgments

Thanks go to Z.-H. Wang, Y. Zhang, and Y.-M. Liu for helpful discussions.

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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