Cascaded optical transparency in multimode-cavity optomechanical systems

Electromagnetically induced transparency has great theoretical and experimental importance in many physics subjects, such as atomic physics, quantum optics, and more recent cavity optomechanics. Optical delay is the most prominent feature of electromagnetically induced transparency, and in cavity optomechanics optical delay is limited by mechanical dissipation rate of sideband-resolved mechanical modes. Here we demonstrate a cascaded optical transparency scheme by leveraging the parametric phonon-phonon coupling in a multimode optomechanical system, where a low damping mechanical mode in the unresolved-sideband regime is made to couple to an intermediate, high frequency mechanical mode in the resolved-sideband regime of an optical cavity. Extended optical delay and higher transmission, as well as optical advancing are demonstrated. These results provide a route to realize ultra-long optical delay, indicating a significant step toward integrated classical and quantum information storage devices.

It is well known that photon is the ideal choice for long-distance communication for its low propagation loss, high speed and large bandwidth. In both quantum and classical domain, optical delay is a favored feature for advanced optical networks, as it offers the ability to buffer and store optical signals 1 . One promising method to produce optical delay is through electromagnetically induced transparency (EIT) 2 , which arises from the quantum interference between different excitation paths as demonstrated in many platforms 3,4,5,6,7 . The optical transmission and delay time are two important parameters to describe the performance of one EIT process. For example, quantum memories have been proposed based on EIT effect 8,9 , where the optical transmission and delay directly determine the information transfer efficiency and storage time 10,11 . EIT effect has also been demonstrated with cavity optomechanical systems 6,7 , which has advanced into the quantum regime, and paved the way to realize integrated quantum devices 12,13,14,15,16 . In spite of great advances in the performance of optomechanical devices, it remains challenging in typical cavity optomechanical systems to realize EIT effect with both high optical transmission and long optical delay. The resolved-sideband condition requires high mechanical frequency which inevitably leads to high mechanical dissipation rate, thus small optical delay 6,7 . Also due to transparency window broadening under large optomechanical cooperativity 6,7 , the optical transmission can only be increased with the expense of further decreased optical delay.
In this article, we propose and experimentally demonstrate the cascaded EIT effect utilizing both optomechanical coupling and parametric phonon-phonon coupling 17 , in which the optical transmission and optical delay can be improved simultaneously. An aluminum nitride (AlN) wheel structure is fabricated to support whispering gallery optical modes, a low frequency breathing mechanical mode and a high frequency pinch mechanical mode. The parametric phonon-phonon coupling is first characterized by electromagnetically induced transparency and absorption effect in the subsystem consisting of the two mechanical modes. Then by combining the optomechanical coupling and parametric phonon-phonon coupling, a secondary narrow transparency window produced by the low frequency breathing mechanical mode is observed on the top of the original wide transparency window produced by the high frequency pinch mechanical mode. In this cascaded EIT scheme, the optical delay is improved by 8 times with simultaneous increase in optical transmission compared with normal EIT configuration.

Results
Theoretical description. As shown in Fig. 1a and b, we start with a regular optomechanical system in the resolved-sideband regime, where an optical cavity (frequency o , linewidth ) is coupled to a high frequency mechanical mode (frequency H , linewidth H ) with coupling rate om . We introduce another sideband-unresolved low frequency mechanical mode (frequency L , linewidth L ) which couples to the high frequency mechanical mode through parametric phononphonon coupling with coupling rate mm . The interaction Hamiltonian of the parametric phononphonon coupling has a similar form as radiation pressure coupling in optomechanical systems (Supplementary Note 1). In cavity optomechanical systems, the photon-phonon coupling can be strongly enhanced by a coherent optical pump, leading to EIT phenomena. Analogously the subsystem consisting of the two parametrically coupled mechanical modes should also exhibit EIT phenomena 17 .
Here , , and are the annihilation operators of the optical cavity, high and low frequency mechanical modes respectively, and om ( mm ) represents the zero-point optomechanical (parametric phonon-phonon) coupling rate. In the presence of an optical pump (which results in a coherent optical field 0 ) at the red side of the optical cavity and a mechanical pump ( 0 ) at the blue side of the high frequency mechanical mode, the interaction Hamiltonian is found by substituting → ( 0 + ), → ( 0 + ) under the rotating wave approximation (RWA) i = ℏ om ( 0 If a weak optical probe is swept across the optical cavity, the first term in equation (2) causes destructive interference between the probe photons and the pump photons scattered by the high frequency mechanical mode phonons, inducing transparency in the transmitted probe light 6,7 . The second term causes constructive interference between the high frequency mechanical mode phonons and the mechanical pump scattered by the low frequency mechanical mode phonons.
The energy level diagram of this cascaded EIT scheme is shown in Fig. 1c. There are three pathways for the optical probe in the cavity: i) directly transmitting through the optical cavity, ii) interfering with the optical pump scattered by the high frequency mechanical mode, iii) interfering with the optical pump scattered by the phonons scattered from the mechanical pump by the low frequency mechanical mode. The total transmitted signal of the optical probe can be written as (Supplementary Note 2) where p and p are the optical and mechanical pump frequencies respectively, ex is the optical coupling loss, and is the frequency difference between the optical pump and probe (Fig.  1c). Beside the transparency window produced due to the high frequency mechanical mode, feature of the low frequency mechanical mode is also imprinted in the optical probe through the parametric phonon-phonon coupling with the high frequency mechanical mode serving as a bridge. To observe the cascaded EIT, the resolved-sideband condition is required individually for the optomechanical coupling ( H > ) and parametric phonon-phonon coupling ( L > H ), but does not restrict the low frequency mechanical mode to be in the resolved-sideband regime with respect to the optical cavity. Therefore, it can be designed to have lower frequency which offers a much smaller linewidth than the high frequency mechanical mode, and leads to prolonged group delay (Fig. 1d).
Device design and characterization. The parametric phonon-phonon coupling is a key part in the cascaded EIT, which we implement in a micro-wheel structure (Fig. 2a). A displacement of one mechanical mode leads to tension and boundary change in the structure, thus modifying the resonant frequencies of other mechanical modes. Unlike the coupling induced by the Duffing effect in flexural beams where the frequency shift is proportional to the square of the displacement 18,19 , the frequency shift for the radial modes in micro-wheel structures is linearly proportional to the displacement because of the broken mirror symmetry in the radial direction.
The coupling rate in equations (1) and (2)  [∇ L + (∇ L ) T ] are the strain tensors, is the fourth-order elasticity tensor, is the material density, and L,zpm is the zero-point motion of the low frequency mode. The first term accounts for the potential energy change of the high frequency mode due to the low frequency mode displacement, and the second term accounts for the fact that the two mechanical modes are not orthogonal when structure deformation is considered. Note that the parametric phonon-phonon coupling is an intrinsic property in multimode mechanical devices, unlike the coupling induced by external electric and optical fields 20,21,22 .
The efficient piezoelectric drive of mechanical resonators makes multimode cavity piezooptomechanical systems an ideal platform to realize our cascaded EIT 23,24 . We use an on-chip AlN wheel structure to obtain high-Q optical and mechanical modes, as well as high Cascaded optical transparency. By combining the parametric phonon-phonon coupling and optomechanical coupling, the cascaded EIT effect can be observed. When the optical probe is swept across the optical cavity over a wide range, we first see the normal EIT which has a transparency window with a linewidth equal to the dissipation rate of the high frequency mechanical mode (Fig. 4a). Upon zooming into the transparency window, a second transparency window sitting on top of the original transparency window can be observed (Fig. 4b). In this case, the linewidth of the narrow transparency window equals to the dissipation rate of the low frequency mechanical mode instead of the much larger linewidth of the high frequency mode.
The high frequency mechanical mode essentially mediates the coupling between the optical cavity and the low frequency mechanical mode which is in the optomechanical unresolvedsideband regime and therefore cannot be used for regular EIT. By fitting the spectrum in Fig. 4b with equation (3), we can obtain both the optomechanical cooperativity om =  Fig. 4 (b) and therefore large optical group delays, which in our case reach 5.0 μs. This corresponds to a more than 8 times improvement compared with 0.6 μs delay of the regular EIT which only involves the high frequency mode (Fig. 4d). Moreover, the optical transmission is increased, and this leads to a 10 times improvement in the bandwidth-delay product 9 . By placing the mechanical pump on the red side of the high frequency mechanical mode, we turn the narrow peak into a dip (Fig. 4c). This gives rise to the advanced light with a leading time of 1.03 μs (Fig. 4d).

Discussion
For optomechanical applications in information processing, one major obstacle is the short phonon lifetime which introduces noise during state transfer and limits the information storage time after state transfer 25,26,27,28 . In our optomechanical system, the phonon lifetime for the high frequency mechanical mode is around 30 . The phonon lifetime can be greatly increased in the cascaded multimode coupling scheme. In this scheme, the low frequency mechanical mode in the optomechanical unresolved-sideband regime functions as if it is in the optomechanical resolved-sideband regime, and the frequency just needs to exceed the linewidth of the high frequency mechanical mode (~33 kHz). For a mechanical resonator with frequency around 200 kHz, a linewidth as low as 0.1 Hz can be realized, corresponding to a 10 s phonon lifetime 29 . This increase in phonon lifetime is directly proportional to the information storage time increase for a classical signal. Also a higher mechanical can be expected for a lower frequency mechanical mode 30 due to the empirical • limit 31 . Thereforethe low frequency mechanical mode can preserve the quantum state much longer due to its longer rethermalization time = ℏ / 32 , thus serving as a better quantum register 33 . In essence, our cascaded coupling scheme provides a viable route to long optical delay and long coherence time by leveraging the parametric phonon-phonon coupling, pushing forward the frontier of optomechanical devices in both classical and quantum regime. In the experiment demonstrating cascaded EIT (Fig. 4), the optical pump ( 0 ) from a laser is delivered to the DUT after passing through an electro-optic modulator (EOM). The RF signal from the NA goes through port 1 of the switch to modulate the strong optical pump, and this generates a weak optical probe ( p+ ). In this case, the only RF signal sent into the device is the mechanical pump ( 0 ) which is provided by the amplified RF signal from SG. The      All the electric displacement components equal zero, because there is no free charge on the device. Also we only consider the radial displacement ( , ), because mechanical modes have rotation symmetry and the wheel thickness is very small 3 . Therefore, the constitutive equations can be simplified to

Methods
where is the material density. Substituting Eq. (2) where = √ (1 − p 2 ) 11 EP , and 1 and 1 are the first-order Bessel functions of the first and second kind, and and are the constants determined by the boundary conditions. The boundary conditions are = 0 at = 1 and = 2 with R1 and R2 representing the inner and outer radius of the wheel.
where G1 and G2 are defined as the following where 0 and 0 are the zero-order Bessel functions of the first and second kind. In order to get non-trivial values for A and B, the equation's determinant must be zero By solving Eq. (8), we can get a series of solutions corresponding to different resonant frequencies (Supplementary Figure 1). And the ratio between and can be obtained from Eq. (6a), thus the mode profile can be plotted from Eq. (5) (Supplementary Figure 1).

Dispersive phonon-phonon coupling
In this section, we use Lagrangian mechanics method to derive the coupling between two mechanical modes when considering the structure deformation induced by the mechanical mode displacement. We first derive the equation for arbitrary structures. We consider two mechanical modes with displacement 1 = 1 ( ) 1 ( ) and 2 = 2 ( ) 2 ( ) , where 1 and 2 are the normalized mode profiles (max(| 1,2 |) = 1). The corresponding strain fields are where (•) T represents a transpose. The total system kinetic energy ( ) is And the potential energy ( ) can be written as where is the fourth-order elasticity tensor, and : ≡ ∑ ij ij ij is the second-order inner product. Here the term (1 + ∇ • 1 1 + ∇ • 2 2 ) accounts for the structure deformation. Then we follow the standard procedure for Lagrangian mechanics We plug Eq. (10), Eq. (11) and Eq. (12) into Eq. (13), and only keep the 1 st and 2 nd order of 1 and 2 . We further ignore the self-coupling terms ( 1 2 , 2 2 ). Then we get If we define the following parameters: And the interaction Hamiltonian is expressed as mi = ℏ mm 2,zpm + ( + + ).
where ( ) is the annihilation operator of the 1 st (2 nd ) mechanical mode, and 2,zpm is the zero-point motion of the 2 nd mechanical mode. When the structure deformation is small, Eq. (15) is simplified to normal harmonic oscillator equation Next we calculate the phonon-phonon coupling rate of two mechanical modes in the AlN wheel. We use Voigt notation in the following derivation for simplicity. Following the assumptions in the last section, we only consider mechanical modes with rotation symmetry, thus the total displacement can be written as ( , , ) = ( 1 ( ) 1 ( ) + 2 ( ) 2 ( ), 0,0) where 1 ( ) and 2 ( ) are the normalized displacement profiles along radial direction, and 1 ( ) and The potential energy ( ), including both the elastic energy and electrical energy, can be written as  Considering the wheel used in the experiment, we have the geometric parameters 1 = 20 μm, 2 = 25 μm, ℎ = 650 nm, and the material parameters are the same as those in Supplementary Figure 1. If we use the 1 st pinch mode as 1 and the breahing mode as 2 whose mode profiles are given in the last section, then we can get = 1 + 2 = −2 × 2.64 × 10 23 Hz 2 /m, corresponding to the coupling rate mm = −2 × 0.019 MHz/nm. With FEM simulation, the dependence between the 1 st pinch mode frequency and breathing mode displacement can be obtained (Supplementary Figure 2), and the simulated phonon-phonon coupling rate is −2 × 0.017 MHz/nm which agrees well with the theoretical calculation.

Supplementary Note 2. Theory of cascaded transparency
We propose to use cascaded EIT to realize ultra-narrow optical transparency window, utilizing both optomechanical coupling and parametric phonon-phonon coupling. The system is modeled with the Hamiltonian = ℏ o + + ℏ H + + ℏ L + + ℏ om + ( + + ) + ℏ mm + ( + + ) With the optical pump ( in ), optical probe ( in+ and in− , generated by modulating the optical pump with electro-optic modulator), and mechanical pump ( in ), the equations of motion can be written as We solve Eq. (25) in the steady state, and make the following substitution.
We assume the optical pump and mechanical pump are very strong so that the amplitudes of 0 and 0 are determined by the optical pump and mechanical pump respectively, and the effect of static displacement s and s can be eliminated by a frequency renormalization. Thus the equations for optical sidebands and mechanical motions are [ ( L + p − ) + L 2 ] 1 = − mm 0 * 1 ( p < ).
Here we ignore the case p = , in which the spectrum is a Dirac delta function because of the mixing between the mechanical and optical pumps 5 . Based on the red or blue detuning of the optical pump and mechanical pump with respect to the optical cavity and the high frequency mechanical mode, there are four different configurations in the system.

Case I:
The optical pump is red detuned and the mechanical pump is red detuned. In this case, only p+ has a significant contribution to the probe transmission, thus the optical probe transmission is Case II: optical pump is red-detuned, and the mechanical pump is blue-detuned. In this case, only p+ has a significant contribution to the probe transmission, thus the optical probe transmission is This equation is plotted in Supplementary Figure 4. An ultra-narrow optical transparency window can be realized with this configuration. Note that with large mechanical cooperativity, the transmission can be larger than unity. This is due to the parametric amplification process with the blue detuned mechanical pump 7 . When the mechanical cooperativity is large enough mm > 1 + om , the low frequency mechanical resonator starts to self-oscillate, and the system enters the unstable regime.Case III: optical pump is bluedetuned, and the mechanical pump is red-detuned. In this case, only p− has a significant contribution to the probe transmission, thus the optical probe transmission is Eq. (32) is plotted in Supplementary Figure 6. In this configuration, we observe a narrow absorption window in the broad absorption window in the weak optomechanical coupling regime. The low frequency mechanical mode starts to self-oscillate when the sum of the optomechanical and phonon-phonon cooperativities exceeds unity ( mm + om > 1), and the system enters unstable regime. One transparency peak can appear in the optical absorption window when the sum of the optomechanical and phonon-phonon cooperativities is large enough (c.f. Supplementary Figure 5), and the dramatic phase modulation can also lead to prolonged optical delay. Compare with Case II, we find both of these two cases can be used for prolonged optical delay, but Case IV has a much higher requirement for the optomechanical and phononphonon cooperativities to realize transparency window.
Supplementary Note 3. Device performance optimization Phonon bandgap engineering for mechanical pinch mode The clamping loss of the mechanical mode is further reduced by fabricating phononic crystal structures on the spokes (Supplementary Figure 7a). The frequency of the high frequency pinch mode is around 1.094 GHz, so a period of = 4.3 μm is chosen, and the unit cell consists of one block (4.3 m long and 1 m wide) with two half ellipse holes (400 nm short axis and 800 nm long axis) on each side as shown in the inset of Supplementary Figure 7b. This structure results in a full phononic bandgap between 0.99 GHz and 1.17 GHz (Supplementary Figure 7b). Because of the fabrication process, the sidewall of the structure is not completely vertical. Our simulation results show little influence of the tilted sidewall on the phononic bandgap structure. This phononic bandgap engineering turns out to be quite effective despite that we only employ three repeating segments in the spokes.
Device Undercut engineering for minimizing clamping loss The material mechanical loss can be greatly decreased by cooling the device to low temperature. As a result the total mechanical loss is determined by the clamping loss. Engineering the undercut of the device to minimize the pedestal is crucial to minimize the mechanical clamping loss 8 . In order to achieve a minimized pedestal, we pattern three devices with slightly different center disk radii (100 nm step) in the same releasing window (Supplementary Figure 8a). During the BOE wet etch step, we monitor these three devices. When the two devices with smaller center disk radiuses disappear, we stop the BOE wet etch. With this method, we can achieve pedestal sizes around 100 nm (Supplementary Figure 8b).

Resist reflow technique for minimizing optical scattering loss
In order to improve the optical performance of the device, a resist reflow technique is utilized. The device is fabricated from AlN-on-insulator wafers with 650 nm AlN on a 3-m layer of buried oxide. A layer of SiO2 with thickness 150 nm is first deposited on the wafer through plasma-enhanced chemical vapor deposition. Then patterns are defined in ma-N 2403 resist by electron-beam lithography. After resist development, the device is placed on the hot-plate to partially melt the ma-N 2403 patterns and form a much smoother boundary in the patterns. Next, the patterns are transferred to the SiO2 hardmask layer by CHF3-based plasma dry etching, and then to AlN layer by Cl2-based plasma dry etching in which only 400 nm out of 650 nm AlN is initially etched. Then a releasing window is defined with a second electron-beam lithography step using ZEP520A resist, followed by dry etching of the residual AlN layer. Last, the underlying oxide in the exposed windows is removed by buffered oxide etchant (BOE). The mechanical structures are dried in a critical point dryer.