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On-chip mechanical exceptional points based on an optomechanical zipper cavity

Exceptional points (EPs) represent a distinct type of spectral singularity in non-Hermitian systems, and intriguing physics concepts have been studied with optical EPs recently. As a system beyond photonics, the mechanical oscillators coupling with many physical systems are expected to be further exploited EPs for mechanical sensing, topology energy transfer, nonreciprocal dynamics etc. In this study, we demonstrated on-chip mechanical EPs with a silicon optomechanical zipper cavity, wherein two near-degenerate mechanical breathing modes are coupled via a single co-localized optical mode. By tailoring the dissipative and coherent couplings between two mechanical oscillators, the spectral splitting with 1/2 order response, a distinctive feature of EP, was observed successfully. Our work provides an integrated platform for investigating the physics related to mechanical EPs on silicon chips and suggests their possible applications for ultrasensitive measurements.


Introduction
The non-Hermitian system, which exchanges energy with the outside environment, is quite different from the conservative Hermitian system. Exceptional points (EPs) are special degenerate points of spectra that exist in the non-Hermitian system. For Nth-order EPs, N eigenvalues and eigenvectors coalesce simultaneously at the EPs, and the theorem of completeness and orthogonality fails in this system 1,2,3 . Thus, owing to this characteristic, intriguing physics concepts are expected. In the past few years, optical EPs have been demonstrated in various platforms including optical and microwave cavities 4,5,6,7,8 , photonic crystal slabs 9 , and multilayered plasmonic structures 10 . Furthermore, certain counterintuitive phenomena have been reported and observed experimentally 1,2,3 . For instance, when a small disturbance of strength ε acts on the EPs, spectral splitting is proportional to ε 1/N . As a result, for a 1/N order response, the spectral splitting near the EPs may be far greater than the normal mode splitting in the Hermitian system, where splitting of degenerate points is proportional to the perturbation ε. This property was utilized to improve the performance of the single-mode operation in multimode laser cavities 11 and enhance the sensitivities of sensors 4,10,12,13,14 . In addition, encircling the EPs in parameter space is nonreciprocal and the counter-clockwise evolution is distinct from the clockwise one. This effect was demonstrated in topology energy transfer 15,16 and asymmetric mode switching 17 . However, studies to date have focused primarily on optical EPs. The EPs in other physical systems are expected to be exploited further.
In cavity optomechanics, mechanical properties such as resonant frequency and dissipation rate can be adjusted effectively via optical modes through radiation pressure.
Moreover, the mechanical and optical modes coupled with the thermal bath constitute a natural non-Hermitian system. Although this system is promising for manipulating mechanical modes and achieving mechanical EPs in multi-physics systems, experimental studies on mechanical EPs are still very limited 15,16,18 . Fully integrated mechanical EPs have remained elusive because of great challenges in on-chip mechanical manipulation.
In this study, we demonstrate on-chip mechanical EPs in an optomechanical zipper cavity at ambient environment, wherein two near-degenerate GHz mechanical breathing modes are coupled via a co-localized single optical mode. In this experiment, the strength of dissipative and coherent couplings between two mechanical oscillators are controlled by adjusting the frequency and power of the pump light to compensate for the difference of complex frequencies between the two mechanical breathing modes. Accordingly, the topology surface near the mechanical EP was mapped and a spectral splitting with 1/2 order response, a distinctive feature of EPs, was observed. As nanomechanical resonators can be coupled with many physical systems 19 and are suitable for detecting quantities such as mass, charge, and torque 20,21,22 , our work paves the way for high-sensitivity measurement with mechanical EPs on integrated platforms. Moreover, this research sets the foundation for studying the related physics of mechanical EPs and other non-Hermitian phenomena based on optomechanical crystals.

Results
Mechanical EPs based on multimode optomechanical coupling can be analyzed as two independent pendulums with different oscillation frequencies (two mechanical modes) interacting with an optical mode in a Fabry-Perot cavity constructed by the two pendulums, as shown schematically in Fig. 1a. The optical mode acts as a bridge that connects the two mechanical modes. The oscillation of each pendulum can modulate the resonant frequency of the optical mode, thus subsequently, the light intensity in the cavity changes.
Accordingly, the oscillation of radiation pressure force that originates from the variation of light intensity in the cavity affects the motion of the other pendulum and thereby enables the coupling between the two mechanical modes. It should be noted that the coupling here is distinct from mode coupling in the Hermitian system. In a Hermitian system, one of the mechanical oscillators exerts a force proportional to its displacement on another oscillator and leads to the coherent coupling 23,24,25 . When enhancing the coherent coupling rate, the normal mode splitting increases. However, for the non-Hermitian optomechanical system, in addition to the coherent coupling, the force exerted by one of the mechanical oscillators on the other also contains the dissipative coupling, 26 which results in a significant change in the physical scenario. The corresponding quantitative description is as follows. Figure 1 a, Schematic of a Fabry-Perot cavity consists of two pendulums as the reflecting mirror that act as mechanical modes b 1 and b 2 with oscillation frequency, Ω 1 and Ω 2 , and are coupled to the thermal bath at the rate of γ 1 and γ 2 , respectively. The optical mode of the Fabry-Perot cavity loses energy via intrinsic loss channel at rate k i and detectable extrinsic coupling channel at rate k ex . b, Scanning electron microscope (SEM) image of silicon optomechanical zipper cavity. Magnified SEM image of (c) periodic structure in P-I region and (d) periodic structure in P-II region. e, Displacement field of the breathing mode simulated with the finite element method (FEM) in one arm of the optomechanical zipper cavity.
FEM simulation of electric field |E| of (f) first order optical mode in a single nanobeam cavity and (g) first order odd optical mode in zipper cavity.
The coupling strength between the mechanical oscillators, including both the coherent and dissipative couplings, is primarily controlled by the optical mode. When a pump laser with frequency L drives the optical mode with resonance frequency cav, detuning Δ=Lcav and intracavity photon number ncav can be adjusted by changing frequency L and power Pin of the pump laser, respectively. To further quantitatively describe the relationship between these parameters and the dynamics of the model shown in Fig. 1a Typically, parameter χeps is a complex number at the EPs, and indicates that the coherent and dissipative coupling mechanisms are both necessary. It can be seen that χeps is an imaginary number when g1=g2 and γ1=γ2, which indicates that the dissipative coupling alone is sufficient to realize the EPs related to anti-parity time symmetry 29,30,31 in this situation. Although the hybridization of two mechanical modes was reported recently in the unresolved-sideband regime (k>>Ω) 23,24,25 , the dissipative coupling is strongly diminishes comparing with the coherent coupling in this regime 27 . As a result, it is crucial to investigate the resolved-sideband regime (k<<Ω) of the EPs wherein the coherent and dissipative couplings can be adjusted independently and effectively to compensate for the difference of resonant frequencies and mechanical dissipation rates simultaneously.
When certain disturbances act on a system at the EPs, parameter χ may deviate from χeps. We define perturbation strength ε as ε=χ-χeps to characterize the deviation from the . In particular, the amplitude of the coefficient A will scale with Ω1-Ω2 if γ1=γ2. This implies that a small perturbation ε results in large mode splitting at EPs owing to the 1/2 order response and the large difference of mechanical resonant frequencies.
Following the multimode optomechanical coupling model introduced above, we propose and design the optomechanical zipper cavity, consisting of two identical silicon nanobeam cavities with a 200 nm gap between them, as shown in Fig. 1b. As one arm of the optomechanical zipper cavity, a nanobeam cavity consists of three parts: two quasiperiodic mirror regions (P-I and P-II) and one defect cavity region. The first quasi-periodic structure P-I (Fig. 1c) can avoid optical energy loss via a waveguide as the optical resonant frequency is on the center of the photonic band gap. The second quasi-periodic structure P-II ( Fig. 1d) is used to regulate the mechanical radiation loss when the phononic bandgap is designed properly. In the defect region, the radius of holes increases gradually from the center moving outward, and both the optical and mechanical modes can be confined 32,33 .
We focus on the mechanical breathing mode (Fig. 1e) and the first-order optical mode (Fig.   1f). When two identical nanobeam cavities approach each other to constitute the zipper cavity, the mechanical breathing mode in each nanobeam cavity remains independent of each other. In contrast, the first-order optical degenerate modes in each nanobeam cavity couple to each other through the evanescent field and result in renormalized odd (Fig.1g) and even optical modes with distinct resonant frequencies. In principle, either of these optical modes can be used to realize the mechanical EPs when it couples to mechanical breathing modes via radiation pressure. In our design, a low optical dissipation rate is required because the system should reach the resolved-sideband regime. Moreover, both high vacuum optomechanical coupling rate gj and low optical dissipation rate k are expected to decrease the intracavity photon number ncav at EPs and diminish the influence of other nonlinear effects related to the light absorption 34,35 . Considering these objectives, the structure parameters such as the radius and pitches of holes were optimized. Finally, the designed optical wavelength was λodd=1542.5 nm (λeven=1551 nm), and the optical linewidth was k/2π=0.04 GHz (0.12 GHz) corresponding to an optical quality factor Qo=4.86×10 6 (1.6×10 6 ). In addition, the designed mechanical frequency was Ω1/2π=Ω2/2π=5.634 GHz and the optomechanical coupling rate was g1/2π=g2/2π=0.63

MHz.
After determining the parameters of the zipper cavity, the designed cavity was fabricated on a silicon-on-insulator wafer having top layer thickness of 220 nm and a 3-μm thick buried-oxide layer (see Supplementary Section III). Figure 2a shows the measurement setup used to characterize the evolution of mechanical spectra with tuned optical parameters. A tapered fiber controlled with nanopositioners was used to evanescently couple the input blue-detuned laser into the zipper cavity. In this experiment, a pump-probe scheme 36,37,38 was used to monitor the detuning Δ and the optical dissipation rate k (see Supplementary Section IV).
The optical and mechanical spectra are illustrated in Fig. 2b-d. In Fig. 2b, the first resonant dip at 1534.9 nm is an odd optical mode while the second resonance dip at 1543 nm corresponds to an even optical mode. In this experiment, we excite the optical odd mode to realize the mechanical EPs, and its Lorentzian fitting is shown in the inset. The corresponding optical linewidth of k/2π=1 GHz (Qo=1.95×10 5 ) indicates that the resolvedsideband regime (Ω>>k) was reached. The derived extrinsic coupling rate of kex/2π= 0.208 GHz from Fig. 2b is assumed to be constant during the experiment and was used to calculate the intracavity photon number ncav. The normalized amplitude response of S12 the optical spring effect. Further, the mechanical peak at higher frequency is weaker than that at lower frequency because the optomechanical coupling rate g2 are lower. In addition, the discrepancy of the intrinsic resonant frequencies between the two mechanical oscillators originates from the fabrication error and it contributes to a large eigenvalue splitting near the EPs. To analyze the evolution of the eigenvalue of the mechanical modes with the variation of the detuning Δ and the intracavity photon number ncav, the mechanical spectra are measured by scanning the optical detuning Δ at a fixed laser power Pin=11 dBm and Pin=13 dBm in Fig 3a and Fig 3b, respectively. For Pin=11 dBm in Fig 3a, we observed a significant increase of the peak of the mechanical mode with lower frequency as the detuning Δ decreases, which indicates the decrease of the mechanical linewidth.
Subsequently, the phonon lasing occurred in this lower frequency mode. In contrast, for Pin=13 dBm, it can be found that the peak (linewidth) of the mechanical mode with higher frequency shown in Fig 3b increases (decreases) dramatically as the detuning Δ decreases.
Thereafter, the phonon lasing occurred in this higher frequency mode. This phenomenon is also an evidence of the coupling between the two mechanical oscillators. Otherwise, the lower frequency mode would always enter the phonon lasing regime first due to its higher optomechanical coupling rate g1.
can be obtained from the mechanical spectra. Figure 3c and 3e (3d and 3f) display the evolution of the real and imaginary part of eigenvalues for different detuning Δ using the spectra in Fig. 3a (Fig. 3b), respectively. Here the experimental data (square markers) are consistent with the theoretical curves (solid lines) with the linear approximation. Additionally, the crossing (anti-crossing) of the dissipation rate γ±/2π and the anti-crossing (crossing) of frequency ω±/2π of the mechanical modes appear as theoretical prediction when sweeping the detuning at Pin=11 (13) dBm. Nevertheless, the system will enter the phonon lasing regime (see Supplementary Section VIII) before the frequency crossing at Pin=13 dBm experimentally, owing to the low intrinsic mechanical dissipation rate of this test sample.  Similarly, when the input optical power is higher than the power required to reach the EPs, the evolution of parameters will cross the black solid line when sweeping the detuning, so that the crossing of mechanical frequency appeared and the phase change occurs at the EPs.
The results in Fig. 3c-d and Fig. 3e-f are in the corresponding two regions, respectively. . As the experimental data are close to the EPs in the parameter space, the perturbation ε of can be very small. This leads to a good agreement between the experimental data and the approximate theoretical response of | | S A  in Fig. 4d. Consequently, the 1/2 order response of the EPs was observed in this test sample.
In addition, as the amplitude of perturbation ε is a nonlinear function of detuning Δ, the 1/2 order response and the nonlinear transduction between perturbation ε and detuning Δ resulted in a sudden change of mechanical frequency splitting near the EPs in Fig. 4a when adjusting the detuning Δ.

Discussion
In our experiment, a larger difference of mechanical frequency |Ω1-Ω2| will further enhance the frequency splitting near the EPs even though the mechanical dissipation rate needs to be increased to avoid the effect of phonon lasing 39,40,41 . In addition, the device also benefits from the reduced mechanical linewidth by using a blue-detuned laser. What's more, owing to the self-reference detection scheme of frequency splitting, this system is robust to the mechanical frequency drifts 42 . In contrast, the sensor proposed by a single optomechanical coupling suffers from frequency drift 43, 44 . Therefore, the device is suitable for realizing high-sensitivity sensors by combining the anti-damping modes and the mechanical EPs. It is noted that this work also offers a reliable, integratable platform for studying and utilizing multimode non-Hermitian physics. For instance, further using the red detuning laser might be suitable for the investigation of multimode optomechanical cooling 45 where  is Planck's constant. â and j b  represent the annihilation operator of the optical mode and the jth mechanical mode, respectively, and j g is the vacuum optomechanical coupling rate between the jth mechanical mode and the optical mode, with where detuning      L cav . In reality, this system is also coupled to both the mechanical and the optical thermal bath. Thus, Hamiltonian ˆb ath H of the thermal bath and the bathsystem interaction term, , must be included 1,2 . As a result, the quantum Langevin equations that describe the system dynamics can be represented as † 1,2ˆ( where the optical vacuum noise ˆi We assume that the deviation of the eigenvalues to the corresponding intrinsic resonant frequency is smaller than the optical dissipation rate (  1 | |   k ). Thus, we obtain Generally, this assumption is satisfied in the weak coupling regime where j cav g n k  . As a result, the eigenvalue of the two-level system can be expressed as The parameter S determines the frequency splitting and  To realize mechanical EPs, a strong optomechanical coupling rate g0 and low optical dissipation coupling rate k are required. Thus, the structure parameters should be designed carefully. In this manuscript, the optomechanical crystal is composed of unit cells with different radius r and lattice constant a. The designed radius r and lattice constant a for the designed cavity are shown in Fig. S1(a), with the width of the nanobeam W=472 nm, the thickness of the device layer t=220 nm, the gap length s=200 nm.
To further verify the effectiveness of the photonic and phononic periodic "mirror," we concentrate on the band structure in one arm of the optomechanical zipper cavity for simplicity. The optical band structure of the unit cell of one arm in the P-I region is shown in Fig. S1(b), wherein the resonant frequency of the x-even and y-odd vector symmetry optical mode is localized at the center of the quasi-bandgap of the guide modes. This indicates that the optical mode is confined without dissipating through the waveguide. The mechanical band structure of the unit cell of one arm in the P-II region is demonstrated in Fig. S1(c), and the mechanical resonant mode with x-and y-symmetry is confined by the bandgap of guide modes with the same symmetry. Furthermore, the control of the mechanical loss can be achieved by changing the radius of the holes in P-II region to shift the frequency range of the bandgap in Fig. S1(c). The fabrication process of the Zipper cavity is shown in Fig. S2. After determining the parameters of the zipper cavity, the designed cavity was fabricated on a silicon-oninsulator wafer having top layer thickness of 220 nm and a 3-μm thick buried-oxide layer.

III. Fabrication process of zipper cavity
Electron-beam lithography (EBL) was used to define the pattern on resist and inductively coupled plasma reactive ion etch (ICP) was employed to transfer the pattern to the top silicon layer. Subsequently, the buried-oxide layer was removed using buffered hydrofluoric acid (BHF) to form the suspended nanobeam structure. The real-time monitoring via VNA is essential because the optical resonant frequency ωcav and the dissipation rate k rely on the intracavity photon number ncav owing to the thermaloptic effect and the nonlinear absorption processes 6 , respectively.

V. Fitting frequency response in pump-probe experiment
In the pump-probe experiment, the pump laser is modulated by driving the phase modulator with RF signal from the VNA. One of the generated sidebands scans across the  This long-term drift may be attributed to the variation of Young's modulus for unknown reasons, which needs to be investigated further. Figure S7. The evolution of the mechanical spectra shape from the linear regime to the phonon lasing regime with high laser power.

VIII. Experimental results in the phonon lasing regime
In the main text, only the experimental results before entering the phonon lasing regime are demonstrated. Figure S7 displays the power spectrum density (PSD) of mechanical spectra in an independent test in this sample with high laser power. As mentioned in the main text, the linewidth of mechanical mode with higher frequency decreases dramatically when increase (decrease) the wavelength (detuning). Subsequently, the phonon lasing of the high-frequency mode happened. It is noted that the mechanical spectra in the phonon lasing regime are beyond the quantitative description in equation (S18). When we increase the laser wavelength further, the peak of the mechanical modes with lower frequency will decrease, and the contrast between these two modes will increase.
Meanwhile, the difference in their mechanical frequencies also changes. Finally, only one peak was observed when further increases the wavelength.