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# Multimode optomechanics with a two-dimensional optomechanical crystal

Guilhem Madiot  Catalan Institute of Nanoscience and Nanotechnology (ICN2),
CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain  Marcus
Albrechtsen  Department of Electrical and Photonics Engineering, DTU Electro,
Technical University of Denmark, Building 343, DK-2800 Kgs. Lyngby, Denmark
Clivia M. Sotomayor-Torres  Catalan Institute of Nanoscience and
Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona,
Spain  ICREA - Institució Catalana de Recerca i Estudis Avançats, 08010
Barcelona, Spain  Søren Stobbe  Department of Electrical and Photonics
Engineering, DTU Electro, Technical University of Denmark, Building 343,
DK-2800 Kgs. Lyngby, Denmark  NanoPhoton - Center for Nanophotonics, Technical
University of Denmark, Ørsteds Plads 345A, DK-2800 Kgs. Lyngby, Denmark
Guillermo Arregui  Department of Electrical and Photonics Engineering, DTU
Electro, Technical University of Denmark, Building 343, DK-2800 Kgs. Lyngby,
Denmark

###### Abstract

Chip-scale multimode optomechanical systems have unique benefits for sensing,
metrology and quantum technologies relative to their single-mode counterparts.
Slot-mode optomechanical crystals enable sideband resolution and large
optomechanical couplings of a single optical cavity to two microwave-frequency
mechanical modes. Still, previous implementations have been limited to
nanobeam geometries, whose effective quantum cooperativity at ultralow
temperatures is limited by their low thermal conductance. In this work, we
design and experimentally demonstrate a two-dimensional mechanical-optical-
mechanical (MOM) platform that dispersively couples a slow-light slot-guided
photonic-crystal waveguide mode and two slow-sound $\sim 7$ GHz phononic wire
modes localized in physically distinct regions. We first demonstrate
optomechanical interactions in long waveguide sections, unveiling acoustic
group velocities below 800 m/s, and then move on to mode-gap adiabatic
heterostructure cavities with a tailored mechanical frequency difference.
Through optomechanical spectroscopy, we demonstrate optical quality factors
$Q\sim 10^{5}$, vacuum optomechanical coupling rates, $g_{o}/2\pi$, of 1.5 MHz
and dynamical backaction effects beyond the single-mode picture. At larger
power and adequate laser-cavity detuning, we demonstrate regenerative
optomechanical oscillations involving a single mechanical mode, extending to
both mechanical modes through modulation of the input laser drive at their
frequency difference. This work constitutes an important advance towards
engineering MOM systems with nearly degenerate mechanical modes as part of
hybrid multipartite quantum systems.

##  I Introduction

The study of the interaction between an electromagnetic and a mechanical
resonator in a cavity-optomechanical system has led to scientific and
technological advances in quantum physics, nonlinear optics, and condensed
matter physics [1, 2, 3]. However, the canonical theoretical description of
the one-to-one interaction [1] fails to describe certain phenomena observed in
realistic devices which naturally host multiple optical and mechanical modes
[4], i.e., multimode optomechanical systems, which can cause quantum
decoherence when the collective interaction is uncontrolled [5, 6]. In the
usual experimental setting with a single laser drive, undriven optical modes
are usually unimportant. However, the individual parametric optomechanical
couplings of the various mechanical modes to the driven optical mode lead to
an effective coupling between them. The case of two mechanical modes, i.e.,
mechanical-optical-mechanical (MOM) systems, has witnessed particular
attention due to its potential impact in the quantum regime, e.g. to probe
decoherence processes [7, 8, 9], to introduce nonreciprocity [10], or to
enhance robustness to thermal noise via mode squeezing [11]. In addition, its
linearized Hamiltonian description is particularly suited for the study of
exceptional points [12, 13, 14], while strongly-driven MOM devices exhibit
collective nonlinear dynamics, including mode competition [15, 16],
synchronization [17, 18], and bistability control [19]. At the frontier
between these two fields of study — namely the linear and nonlinear regimes —
lies the so far largely unexplored paradigm of nonlinear non-Hermitian
physics, where phenomena like topological mode transfer or unidirectional
phonon emission could be realized based on high-frequency coherent phonon
self-oscillations [20, 21, 22].

The optically mediated coupling between mechanical modes in MOM systems scales
inversely with their frequency difference and becomes symmetrical only when
their respective vacuum optomechanical coupling rates to the common optical
cavity mode, $g_{\mathrm{o,1}}$ and $g_{\mathrm{o,2}}$, are identical. This
has fostered research on physical implementations that exhibit (nearly)
degenerate mechanical modes, where the modes coherently mix into
optomechanically dark and bright dressed states [23] and can transfer energy
efficiently [24]. These include Fabry-Pérot microcavities with two identical
membranes [25], membrane-in-the-middle cavities with symmetry-enforced (high-
order) mechanical mode degeneracies [24], or parallel evanescently coupled
optical resonators. The latter category exploits the strong dependence of the
frequencies of the resulting optical supermodes on the distance between the
resonators and embraces double-disk microcavities [26], bilayer photonic-
crystal cavities [27], and photonic-crystal zipper cavities [28].
Interestingly, the electromagnetic boundary conditions across material
interfaces lead to a strong local field enhancement, i.e., a slot-mode effect,
when such distances are deep sub-wavelength and the field polarization is
adequate [29]. Such an effect is non-resonant; therefore, the formation of
supermodes is not required. This has allowed integrated MOM devices with large
optomechanical couplings by using triple nanobeam geometries separated by
tenths of nanometers, each supporting one of the excitations [30]. These
quasi-one-dimensional (1D) MOM slot-mode optomechanical crystals (OMCs) can
display microwave-frequency phononic-crystal cavity modes and reach the
sideband-resolved regime, which has recently allowed the observation of
mechanical exceptional points [31]. However, their geometries may be
unsuitable to study emergent macroscopic quantum phenomena in millikelvin MOM
systems because of inefficient thermalization [32]. In the presence of
residual absorption, the limited heat dissipation pathways of 1D OMCs lead to
a phononic hot bath which can destroy the prepared quantum states. In
addition, the poor stiffness of nanobeams makes them prone to surface-force-
induced collapses during and after fabrication [33], limiting how narrow the
gap between them can be, i.e., how large $g_{\text{o}}$ can be [30, 34], and
potentially require stress-release management [35, 30].

Two-dimensional (2D) optomechanical structures can sustain even larger optical
quality factors [36, 37], large $g_{\text{o}}$ to long-lived hypersonic
mechanical modes [32, 38], and have the additional benefits of enhanced heat
dissipation [39, 32] and convenient stiffness. Nonetheless, to the best of our
knowledge, no experimental work has focused so far on MOM experiments in 2D
OMCs. Here we propose a novel waveguide and cavity optomechanics platform that
enables the coupling of a slot-guided optical mode to two independent, nearly
degenerate, microwave-frequency mechanical modes. By building mode-gap
adiabatic heterostructure cavities [40], we demonstrate a sideband-resolved
system with $g_{\text{o}}$ as high as
$1.5\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ between C-band telecom photons in
a cavity with a $Q\sim 10^{5}$ and two
$\sim$7\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ acoustic resonators. Passive
control over the frequency difference of the latter two via a geometrical
parameter is achieved, which may enable tailored MOM systems adaptable to
specific experimental requirements. By performing wavelength and power-
dependent optomechanical spectroscopy of a device with mechanical modes only
differing in frequency by 6 MHz, we provide evidence of multimode dynamical
backaction in good agreement with the linearized optomechanical equations of
motion of a MOM system. Finally, we demonstrate simultaneous self-oscillatory
dynamics of two mechanical resonators stimulated by an intermodulation tone,
consistent with recent demonstrations using two mechanical modes of a single
nanobeam OMCC [41].

##  II Slot-mode multimode optomechanical crystal waveguides

Figure 1: Multimode optomechanical crystal waveguides (OMCWs). (a) Scanning
electron micrograph of an OMCW that couples two phononic waveguide modes — one
on each side of the air slot — to a slot-guided optical mode. Insets: cross-
section of the slot (top) and a cleaved shamrock. (b) Optical (left) and
mechanical (right) dispersion diagrams of the structure in (a). The Bloch
modes of interest are represented at $k=\pi/a_{\text{x}}$ and $q=0$. (c)
Optical transmission spectrum of a mirror-terminated OMCW ($L$ = 350$a_{x}$)
through a tapered fiber loop. (d) Radio-frequency (RF) spectrum measured with
a laser drive on an optical mode at
$\lambda=$1543\text{\,}\mathrm{n}\mathrm{m}$$, and comparison of the simulated
(line) and experimentally reconstructed (blue dots) acoustic group velocity.

The geometry of the multimode OMC waveguides (OMCWs) we explore here consists
of a line defect waveguide composed of two rows of circular holes with
(optionally) different radii $R_{1}$ and $R_{2}$, a slot of width $s$, and two
triangular lattices of shamrock-shaped [42, 43] holes around them, with the
shamrocks facing each other (Fig. 1(a)). The structures are fabricated in
$220\text{\,}\mathrm{nm}$ silicon-on-insulator (SOI) and the
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ buried oxide is undercut to release
the suspended structures with hydrofluoric vapour-phase etching. The patterns
are defined in chemically semi-amplified resist with electron-beam lithography
[29] and etched into the silicon using dry-etching based on a modified CORE
process [44, 45]. The fabrication process is tailored to yield an excellent
design-to-realized pattern fidelity, including smooth and vertical sidewalls
(Fig. 1(a), insets) [46]. The presence of the slot mechanically decouples the
two membrane sides, making the system a MOM optomechanical waveguide with a
geometry-controlled mechanical frequency difference provided by $\Delta
R=R_{1}-R_{2}$. The employed shamrock crystal enables large near-infrared
electromagnetic [43] and GHz mechanical band gaps [47], and a similar geometry
has been exploited for the in-situ generation of coherent acoustic phonons
using Anderson-localized optical modes resulting from residual roughness in
the etched sidewalls [45]. However, the quality factors, $Q$, observed in Ref.
[45] were limited due to a multimode optical dispersion of the slot-guided
mode and much below those measured in a slot photonic-crystal waveguide solely
based on circular holes [48]. The geometry we propose leverages the best
features of both shamrock-shaped and circular holes as it exhibits single-
moded dispersions with zero group velocity, i.e., simultaneous slow light and
sound, at the mechanical and optical Bloch wavevectors where forward-type
intra-modal Brillouin interactions are phase-matched [49, 50], respectively at
$q=0$ and $k=\pi/a_{\text{x}}$ (Fig. 1(b)). In addition, the vector parities
of the Bloch optical ($y$\- and $z$-symmetric $\mathbf{E}$ field) and
mechanical ($z$-symmetric) modes make them optomechanically bright. The
electric field amplitude of the optical mode, $|\mathbf{E}(\mathbf{r})|$, and
the displacement amplitude, $|\mathbf{u}(\mathbf{r})|$, and deformation
profile of the mechanical mode of interest are shown in Fig. 1(b). The former
exhibits subwavelength light confinement in the etched air slot, making the
band edge frequency very sensitive to slot-width variations [30, 29], while
the latter is reminiscent of the in-plane breathing mode of a nanobeam
waveguide, where one of the lateral free boundary conditions is replaced by a
full-gap phononic crystal [51]. The independent breathing motion of the two
phononic waveguides (the mechanical mode is only represented for the bottom
mechanical waveguide in Fig. 1(b)) strongly modulates the slot width, which
leads to large unit-cell vacuum optomechanical coupling rates,
$g_{\text{o,cell}}/2\pi$, of 4.8 MHz, between the optical and the two
mechanical Bloch modes. Departure from mechanical-mode degeneracy when the
radii difference is $\Delta R\neq 0$ has a negligible effect on the respective
values of $g_{\text{o,cell}}/2\pi$ (see Supplementary Section 1).

We probe the optical and mechanical properties of the OMCWs using a tunable
diode laser connected to an optical fiber circuit that leads to a tapered
fiber loop placed in contact with the slot waveguide. We terminate long OMCWs
with short 32 unit cells waveguide segments within which the horizontal pitch
is expanded from $a_{\text{x,1}}$ = 484 nm to $a_{\text{x,2}}$ = 510 nm. These
sections behave simultaneously as optical and acoustic mirrors forming
standing waves in the central waveguide region [45] (see Supplementary Section
1 for dispersion diagrams as a function of $a_{\text{x}}$). Sharp spectral
dips in the transmitted optical signal (Fig. 1(c)) evidence evanescent
coupling to resonant optical modes of the waveguide. We observe three spectral
regions with distinct features: First, a region above
$\lambda\sim$1580\text{\,}\mathrm{n}\mathrm{m}$$ made of Fabry-Pérot optical
modes with large on-resonance coupling fraction and a free spectral range
(FSR) determined by the group index, $n_{\text{g}}$, and the length, $L$, of
the waveguide. Second, the wavelength region
$$1560\text{\,}\mathrm{n}\mathrm{m}$<\lambda<$1580\text{\,}\mathrm{n}\mathrm{m}$$,
within which the coupling fraction is also large, but the resonant wavelengths
are seemingly random. We attribute these resonances also to Fabry-Pérot modes
whose wavelengths are perturbed by the presence and exact position of the loop
(see next section for a discussion of the dispersive effects of the loop on a
single resonant optical mode). Third, a region close to the band edge
($\lambda>$1560\text{\,}\mathrm{n}\mathrm{m}$$) where strong slow-light-
induced backscattering from sidewall roughness leads to Anderson-localized
modes at random spectro-spatial locations, leading to a random FSR and strong
mode-to-mode fluctuations in the coupling fraction [48, 45].

We characterize the mechanical properties of the OMCWs by moderate power and
blue-detuned driving of optical modes in the first and second regions. We
detect the thermal motion of the Fabry-Pérot standing mechanical modes using a
fast photoreceiver and an electronic spectrum analyzer. A characteristic
radio-frequency (RF) spectrum for an Anderson-localized optical mode at
$\lambda=$1543\text{\,}\mathrm{n}\mathrm{m}$$ is shown in Fig. 1(d). For the
particular case shown ($\Delta R$ = 0), no degeneracy-lifting between the two
independent phononic waveguide modes is observed. The spectrum is composed of
many overlapping and regularly spaced Lorentzian-shaped mechanical resonances
with estimated linewidths, $\Gamma$, in the range 2–5 MHz and a transduction
amplitude envelope that grows as the mechanical band edge at
$7.2\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ is approached. We attribute this
overall shape to a relaxed sinc-like phase-matching condition between an
optical mode with $k$-components dominated by $k=\pi/a_{\text{x,1}}$ and
mechanical modes in the vicinity of $q$ = 0 for forward-type Brillouin
scattering interactions in a finite waveguide [50]. This remains true even if
the optical mode employed is Anderson-localized because its $k$-space
representation is often centered around the wavevector of the underlying
waveguide at that frequency [52], i.e. $k\sim\pi/a_{\text{x,1}}$. While Fig.
1(d) only exhibits peaks within the single-mode regime, RF spectra obtained by
driving other optical modes also reveal peaks in the multi-mode propagation
regime, in which the FSR is ill-defined due to inter-modal mechanical mixing.
Using five additional RF spectra (see Supplementary Section 3), we reconstruct
the acoustic group velocity, $v_{\text{g}}$, of the top and bottom phononic
waveguides for $\Delta R$ = 0. The reconstructed $v_{\text{g}}$ is shown at
the bottom of Fig. 1(d) and compared to the simulated $v_{\text{g}}$. The
simulation curve has been rigidly offset by only
$-$90\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$ ($\sim$1\text{\,}\mathrm{\char
37\relax}$$ of $\Omega$) to account for potential systematic errors on the
SEM-extracted contour of the geometric features, illustrating the good
agreement between simulations and measurements. We measure slow propagation of
$\sim$ 7 GHz acoustic waves down to a group velocity below 800 m/s,
constituting a 7-fold reduction relative to the transverse speed of sound in
bulk silicon.

##  III Slot-mode multimode optomechanical crystal cavities

Figure 2: Optomechanical spectroscopy of a 2D slot-mode mechanical-optical-
mechanical (MOM) system. (a) Electric field amplitude, $|\mathbf{E}|$, of the
optical cavity mode and (b) displacement amplitude, $|\mathbf{u}|$, and
deformation profiles of the two mechanical cavity modes. (c) Normalized
transmission spectrum at different positions along the waveguide axis, $x$.
The spectra are offset for clarity and the red curves are Lorentzian fits to
the resonances. Insets: Schematic of the configuration with the loop on top of
the cavity (top) and a close-up of a normalized spectrum (bottom). (d)
Resonant wavelength as a function of the loop position, extracted from the
fits in (a) and through a convolution of the calculated electric field
intensity, $|\mathbf{E}|^{2}$, with a Gaussian
($\sigma=$0.99\text{\,}\mathrm{\SIUnitSymbolMicro m}$$). (e) Total, external,
and intrinsic decay rates as a function of the loop position. (f) Normalized
radiofrequency (RF) spectra measured for devices with decreasing circular hole
radius $R_{2}$. The spectra are offset for clarity. Blue solid lines are
Lorentzian fits to the individual peaks. The inset shows a characteristic
power spectral density of and fit to a single mechanical peak. (g) Extracted
mechanical frequencies of the two plates (dots), linear fit (solid line) and
confidence interval of the fit (shaded background). (h) Same as (e) for their
frequency difference.

The efficient coupling of a multitude of Fabry-Pérot mechanical modes and
Anderson-localized optical modes in the mirror-enclosed 2D OMCWs presented
above provides an interesting platform to explore collective effects in
multimode optomechanics, such as light squeezing [4]. However, some of the
strongest optomechanical effects beyond the canonical one-to-one interaction
occur in the MOM configuration. To explore such a setting, we engineer a MOM
OMC cavity (OMCC) that couples two mechanical resonators and a high-$Q$
optical mode based on the waveguide modes of Fig. 1 and the respective partial
band gaps above their band-edge frequencies. We adiabatically tune the
horizontal pitch, $a_{\text{x}}$, along the waveguide axis from a central
defect unit cell ($a_{\text{x,1}}$ = 484 nm) to mirror unit cells
($a_{\text{x,2}}$ = 510 nm) on both sides. The full defect region is formed by
$N_{\text{c}}$ = 15 unit cells, and additional invariant sections made of
$N_{\text{m}}$ = 32 mirror unit cells are included at the edges of the defect
to prevent in-plane losses. Fig. 2(a) shows the amplitude of the electric
field, $|\mathbf{E}|$, of the cavity mode for $\Delta R$ = 0, whose
theoretical resonant wavelength and quality factor are $\lambda_{\text{o}}$ =
1556 nm and $Q_{\text{i}}$ = 1.05 $\times 10^{8}$. The deformation profile and
displacement amplitude, $|\mathbf{u}|$, of the two mechanical modes are shown
in Fig. 2(b), and both have mechanical frequency $\Omega_{1,2}/2\pi$ = 7.15
GHz in the degenerate case. The frequency difference between the two
mechanically decoupled mechanical cavity modes is controlled by decreasing
$R_{2}$ relative to $R_{1}$, which in turn slightly redshifts the resonant
wavelength of the optical cavity mode. More details on the cavity-
optomechanical figures of merit are provided in Supplementary Section 2.

We characterize the optical and mechanical properties of the MOM OMCCs using
the same fiber-loop evanescent technique as in Fig. 1. To account for the
strong perturbative effect of the fiber loop on the resonant wavelength and
losses of the optical cavity mode and infer their unperturbed parameters,
i.e., in the absence of the loop, we systematically study the optical response
as a function of the loop position. Figure 2(c) shows the optical transmission
spectra across the cavity resonance for 24 different loop positions, $x$, with
the loop in contact with the sample and approximately aligned to the slot axis
(Fig. 2(c) top inset). The position is extracted via analysis of microscope
images acquired with a $100\times$ objective imaging the probed structure from
above. By moving the sample under the loop while in contact, the loop slides
along the slot and changes its overlap with the optical cavity mode. We
extract the resonant wavelength, $\lambda_{\text{o}}$, and the extrinsic,
$\kappa_{\text{e}}/2\pi$, and intrinsic, $\kappa_{\text{i}}/2\pi$, decay rates
of the cavity mode for each loop position by fitting the cavity resonance with
a Lorentzian lineshape (Fig. 2(c) bottom inset) and using that the on-
resonance transmission is given by $T_{\text{0}}$ =
$(1-\kappa_{\text{e}}/\kappa_{\text{t}})^{2}$, with
$\kappa_{\text{t}}=\kappa_{\text{e}}+\kappa_{\text{i}}$. Figure 2(d) shows the
extracted $\lambda_{\text{o}}$ along with a theoretical prediction obtained
from a convolution between the calculated $|\mathbf{E}|^{2}$ and a Gaussian
envelope representing the loop [29]. The Gaussian has a standard deviation,
$\sigma=$0.99\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, identified with a
least-mean-square optimization. When the loop is centered on top of the
cavity, the dispersive perturbation is maximal, shifting the wavelength by as
much as $\sim$15\text{\,}\mathrm{n}\mathrm{m}$$, i.e.,
$1\text{\,}\mathrm{\char 37\relax}$ of the cavity wavelength. We identify the
center position, $x$ = 0, as the position causing the largest redshift of the
resonance wavelength. The evolution of the total, extrinsic and intrinsic
decay rates as a function of the loop position are shown in Fig. 2(e). Several
interesting properties can be observed. First, $\kappa_{\text{i}}$ changes
considerably with the loop position, which implies that the loop not only
loads the cavity but also adds additional undetected loss pathways. Second, we
observe a decrease of both $\kappa_{\text{i}}$ and $\kappa_{\text{e}}$ when
the loop is centered on the cavity. We hypothesize that this is due to the
additional symmetry of this configuration. Third, when the center of the loop
is far from the geometric cavity center, the behaviour of $\kappa_{\text{i}}$
may indicate that the value is still unconverged, contrary to the wavelength.
This indicates that the employed technique may not allow the measurement of
the unperturbed optical linewidth. Therefore, we identify the unperturbed
cavity parameters to be $\lambda_{\text{o}}$ = 1562 nm and
$\kappa_{\text{i}}/2\pi<$2\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ (for the
$R_{2}=$193\text{\,}\mathrm{n}\mathrm{m}$$ here).

Figure 3: Dynamical back-action in the nearly degenerate case. (a) Measured
radiofrequency (RF) spectra as a function of the laser wavelength $\lambda$
for a laser power $P_{\mathrm{in}}=131$ $\mu$W. The dashed lines indicate the
value of $\lambda$ for the spectra in (b). (b) Individual RF spectra (black
dots) fitted with a double-Lorentzian function (grey solid lines) to extract
the mechanical frequencies and damping rates. The curves are vertically offset
for clarity. (c-e) Experimental (left) and theoretical (right) evolution of
(c) the mechanical frequencies $\Omega_{1}$ and $\Omega_{2}$, and the
mechanical damping rates (d) $\Gamma_{1}$ and (e) $\Gamma_{2}$. These
parameters are plotted at four different powers (see legend with
$P_{\mathrm{in}}$ indicated in units of $\mu$W) and as a function of $\lambda$
or the laser-cavity detuning, $\delta\lambda$, respectively, for experiment
and theory. The blue dashed lines indicate the wavelength (or detuning) at
which the minimum of the damping rates occur for $P_{\mathrm{in}}=219$ $\mu$W.

We systematically characterize the mechanical cavity modes for OMCCs with
different values of $R_{2}$ as in Fig. 1(d), with the fiber loop positioned to
achieve a minimal perturbation and critical coupling to the optical cavity
mode, i.e., $T_{\text{0}}=1/2$. The value of $R_{2}$ is nominally reduced in
$1\text{\,}\mathrm{n}\mathrm{m}$ steps from
$R_{1}=R_{2}=$194\text{\,}\mathrm{n}\mathrm{m}$$ to
$R_{2}=$184\text{\,}\mathrm{n}\mathrm{m}$$. Figure 2(f) shows the RF spectra
with Lorentzian fits to both mechanical modes. Following the prediction of the
finite-element simulations (see Supplementary Sections 1 and 2), we identify
the lower (higher) frequency mechanical mode, fitted in dark (light) blue, as
that of the top (bottom) membrane. This probably holds true except for the
smaller values of $\Delta R$, where the effect of disorder-induced dispersion
might overcome the as-designed deterministic frequency difference. We note
that the relative amplitude of the transduced signals is determined by the
exact loop position transverse to the slot axis, which determines to what
extent the modes are dampened, and by the presence of dynamical backaction
effects, which may occur preferentially for one of the modes at a fixed
detuning [15]. Despite this, the signal-to-noise ratio (SNR) remains
sufficient to extract the central frequencies accurately. Figure 2(g) shows
the extracted frequencies as well as linear fits predicting $\Omega_{1}(\Delta
R)/2\pi=7.052(2)\,$\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$+0.5(4)\Delta R\
$\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{n}\mathrm{m}$$ and
$\Omega_{2}(\Delta
R)/2\pi=7.055(3)\,$\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$+11.8(5)\Delta R\
$\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{n}\mathrm{m}$$. The
non-zero slope found for $\Omega_{1}$ likely originates from short-range
proximity effects [29, 46], i.e., the smaller value for $R_{2}$ leads to a
smaller effective electron-beam dose on the other membrane side, which is not
accounted for with standard long-range proximity effect correction [47].
Meanwhile, the slope of $\Omega_{2}$ agrees well with the simulation
prediction (10.19(7) MHz/nm). Note that the mechanical frequencies are
approximately 90 MHz lower than the simulated values, which is in very good
agreement to the theory-experiment offset found for the acoustic waveguide
band of Fig. 1(d). Figure 2(h) shows $\delta\Omega(\Delta R)/2\pi$
illustrating its linear dependence with an intercept
$\delta\Omega(0)/2\pi=(3\pm 4)\,$\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$.
This non-zero intercept corroborate that we consistently observe a separation
of a few MHz between the nominally degenerate mechanical modes, which is
caused by inherent fabrication imperfections. The values of
$\kappa_{\text{t}}$ under the measurement conditions of Fig. 2(f) are all in
reasonable agreement with those reported in Fig. 2(e), i.e., we observe no
pronounced effect of $\Delta R$ on the optical $Q$ (as expected from
simulations, see Supplementary Figure S3), and therefore the MOM OMCs we
demonstrate are all in the sideband-resolved regime
($\kappa_{\text{t}}$<$\Omega_{1,2}$). In addition, the values of
$g_{\mathrm{o,1}}/2\pi$ and $g_{\mathrm{o,2}}/2\pi$ are measured to be in the
range of 1.2–1.5 MHz, which is in excellent agreement with simulations (see
Supplementary Sections 2 and 4 for details on the simulated and measured
$g_{\mathrm{o,1}}$ and $g_{\mathrm{o,2}}$). In the next section, we explore
the implications of the reported cavity-optomechanical figures of merit and
the MOM nature of the system on the detuning- and power-dependence of the
dynamical backaction effects on a laser-driven device with nearly degenerate
mechanical modes.

##  IV Optomechanical spectroscopy of a mechanical-optical-mechanical system

We consider a structure with $R_{1}=194$ nm and $R_{2}=193$ nm and place the
loop in the position indicated previously, leading to loss rates of
$\kappa_{\text{i}}/2\pi\approx 3.5$ GHz and $\kappa_{\text{e}}/2\pi\approx
0.76$ GHz, and a resonance wavelength (at low power) of
$\lambda_{\text{o}}=1559.35$ nm. Using a laser power of $P_{\mathrm{in}}=131$
$\mu$W, we step-scan the laser wavelength, $\lambda$, from the blue-detuned
side of the optical resonance and measure the RF spectrum. In Fig. 3(a), we
show the resulting power spectral density as a colormap, highlighting the two
mechanical resonances separated by approximately 6 MHz. To track the
mechanical frequencies and linewidths as a function of $\lambda$, we fit each
spectrum with a sum of two Lorentzians. Examples of the fitted spectra are
presented in Fig. 3(b), with experimental data extracted from the above map as
indicated by the white dashed lines. The same procedure is applied for
$P_{\mathrm{in}}=$ 87, 131, 175 and 219 $\mu$W, and the extracted parameters
are summarized in Fig. 3(c-e) (left column). For increasing input power, the
mechanical resonances experience an increasing displacement around their
natural value (Fig. 3(c)) due to optomechanical dynamical backaction.
Meanwhile, the mechanical damping rates tend to decrease down to a minimum, as
expected in the blue-detuned regime. For sufficiently high input power (here
at $P_{\mathrm{in}}=219$ $\mu$W), the damping rate saturates around $\sim 10$
kHz, which indicates that the mode is self-oscillating [1]. In Fig. 3(c-e)
(right column), we qualitatively compare the extracted parameters as a
function of laser wavelength to the values predicted by the linearized
optomechanical equations of motion of a MOM system [13, 53]. In addition to
the optical parameters given above, the model uses
$\Omega_{1,2}/2\pi=7.061\text{ GHz }\pm 3.05$ MHz,
$\Gamma_{1}/2\pi=\Gamma_{2}/2\pi=3.2$ MHz, $g_{\mathrm{o,1}}/2\pi=1.25$ MHz
and $g_{\mathrm{o,2}}/2\pi=1.5$ MHz. Details on the theoretical model are
given in Supplementary Section 6. We note that the model ignores the observed
residual-absorption-mediated thermal non-linearities, so the horizontal axis
in the theoretical plots of Fig. 3(c-e), which use the laser detuning
$\delta\lambda=\lambda-\lambda_{0}$, cannot be directly mapped to the
horizontal axis of the experimental plots as the true detuning scales non-
linearly with the laser wavelength due to the thermo-optic drag, i.e.
$\lambda_{0}=f(\lambda)$. Nevertheless, the latter produces a nearly
asymptotic decrease of $\delta\lambda$ towards zero (see Supplementary Section
5), followed by a sudden jump in the red-detuned regime ($\delta\lambda>0$).
Therefore, theory and experiment can be qualitatively compared by using a
logarithmic scale for the horizontal axis of the theoretical plots. We observe
that the model captures the overall evolution of the mechanical frequencies
and damping rates. In particular, the damping rates reach their minimum at
different values of $\lambda$ (or $\delta\lambda$), as highlighted with the
dashed lines for the case $P_{\mathrm{in}}=219$ $\mu$W. This feature, which
does not emerge in a model with two independent optomechanical oscillators
(see Supplementary Section 5), suggests that the mechanical modes start
hybridizing despite their non-identical mechanical frequencies. The same
applies to any asymmetrical feature in the relative evolution of the
parameters of both mechanical modes. We note that below $\lambda\approx
1559.4$ nm, the SNR of the transduced mechanical resonances is very low, which
prevents determining the frequencies and damping rates. Furthermode, above
$\lambda\approx 1559.7$, the properties of the modes stabilize, and their
transduction slowly decreases until the laser exits the thermo-optic
resonance.

Figure 4: Simultaneous Floquet mechanical lasing of two nearly degenerate
mechanical modes. (a) Measured RF spectra as a function of the modulation
frequency, $\Omega_{\mathrm{VNA}}$. (b) Individual RF spectra extracted from
(a) and vertically offset for clarity. (c) Measured RF peak amplitude as a
function of the modulation depth and modulation frequency for mechanical modes
1 (left) and 2 (right). The green lines are the theoretical amplification
threshold obtained using $\Delta=1.024\overline{\Omega}$ (with
$\overline{\Omega}=6.984$ GHz) and the model in Ref. [41].

##  V Stimulated two-mode optomechanical amplification via intermodulation

The simultaneous parametric amplification of two thermally excited mechanical
oscillators coupled to a common optical mode is typically prevented by mode
competition [15] leading to anomalous cooling [54], except for the case of two
mechanical resonators of disparate frequency [30, 53]. Nevertheless, such two-
mode amplification can be stimulated by modulating the input laser intensity
at the intermodal frequency [41]. To demonstrate such a feature in the
investigated MOM OMCC, we focus on a device with $R_{1}=194$ nm and
$R_{2}=190$ nm, which results in a frequency difference of
$\delta\Omega/2\pi\approx 38$ MHz. We fix the optical loading conditions as
before, with a blue-detuned driving at a power $P_{\mathrm{in}}=90$ $\mu$W and
considerably below self-oscillation threshold. We then apply direct intensity
modulation of the laser with modulation depth $d$ and frequency
$\Omega_{\mathrm{VNA}}$. We report in Fig. 4(a) the recorded RF spectrum for
$d=30\%$ while step-scanning the modulation frequency. We observe the two
mechanical modes at $\Omega_{1}/2\pi=6.965$ GHz and $\Omega_{2}/2\pi=7.003$
GHz with respective linewidths —when the modulation is off—
$\Gamma_{1}/2\pi=2.417$ MHz and $\Gamma_{2}/2\pi=1.234$ MHz. Note that the
mechanical frequencies are slightly lower than those reported in Fig. 2(f-g-h)
because the device we analyze here used a lithography mask with exposed (void)
features shrunk by 5 nm uniformly. Each mechanical peak is surrounded by a
pair of sidebands at a distance of $\pm\Omega_{\mathrm{VNA}}$. When the high-
frequency (low-frequency) sideband of the peak at $\Omega_{1}$ ($\Omega_{2}$)
crosses the mode at $\Omega_{2}$ ($\Omega_{1}$), i.e., when
$\Omega_{\mathrm{VNA}}=\delta\Omega$, the amplitude of the mechanical peaks
increases significantly, far beyond (by at least 20 dB) the value given by the
sum of the thermal transduction and the sideband peak. In Fig. 4(b), we show
three spectra extracted from the upper map, corresponding to
$\Omega_{\mathrm{VNA}}<\delta\Omega$, in which case the two modes are
thermally excited (bottom), $\Omega_{\mathrm{VNA}}\approx\delta\Omega$, where
the modes amplify by nearly 40 dB (middle), and
$\Omega_{\mathrm{VNA}}>\delta\Omega$, for which the amplitudes of the modes
reduce down to that of the sub-threshold regime (top).

In order to identify the dynamical range enabling stimulated multimode lasing
to occur, we record the transduced amplitudes of both mechanical modes as a
function of both modulation parameters $d$ and $\Omega_{\mathrm{VNA}}$. Figure
4(c) represents the SNR evaluated based on the mode amplitude in the absence
of modulation, i.e. in its unaltered thermal regime. For each mode, we plot
the lasing threshold calculated from the theory in Ref. [41] using
$g_{\mathrm{o,1}}/2\pi=g_{\mathrm{o,2}}/2\pi=1.5$ MHz, $\lambda_{0}=1579.895$
nm, $\kappa_{\text{i}}=2.074$ GHz, and $\kappa_{\text{e}}=0.664$ GHz. The
theory is calculated using the optical detuning $\Delta$ as a fitting
parameter, since the presence of thermo-optic effect prevents its independent
determination. However, we note that the asymmetry of the amplification area
about $\Omega_{\mathrm{VNA}}$ is finely determined by the laser detuning
$\Delta$, and is symmetric when the optomechanical amplification is optimal,
i.e. at $\Delta=\overline{\Omega}$, where $\overline{\Omega}=6.984$ GHz is the
mean mechanical frequency. This allows an unambiguous fit using
$\Delta=1.024\overline{\Omega}$, and leads to a qualitative agreement of the
theory with the experimental findings. Note that the second mechanical mode
reaches the lasing threshold at lower modulation depth because of its
significantly lower damping rate ($\Gamma_{2}<\Gamma_{1}$).

##  VI Conclusion and outlook

In summary, we have demonstrated a 2D OMC design that can operate as a
sideband-resolved multimode MOM cavity-optomechanical system with two $\sim 7$
GHz mechanical modes at a close spectral distance,
$\delta\Omega_{1,2}\ll\Omega_{1,2}$, and a high-$Q$ optical cavity ($Q\sim
10^{5}$). The measured optomechanical coupling rates reaching $g_{o}/2\pi\sim$
1.5 MHz are amongst the largest values observed in OMCCs with microwave-
frequency mechanical modes, enabling low-power self-oscillations of the
individual mechanical modes and simultaneous lasing upon modulation of the
laser drive at their frequency difference. The latter is shown to be
controlled, starting from the nominally degenerate case, by slightly breaking
a structural symmetry, which does not significantly degrade the aforementioned
properties over the 120 MHz passive tuning range. By performing a wavelength
and power-dependent spectroscopic analysis of a structure with nominally
degenerate mechanical modes, we show that features of multimode optomechanical
backaction are observed. The combination of stochastic and deterministic
deviations of the fabricated structure from the nominal design introduces a
built-in frequency difference that we observe to be lower-bounded to around 4
MHz, preventing stronger optically-induced hybridization and the exploration
of exceptional points. We foresee that introducing thermal tuning elements on
both sides of the waveguide [55], e.g., metallic bonding pads, will enable
independent and low-cross-talk tuning of the mechanical frequencies, which
will allow the proposed system to reach the degenerate case and explore the
physics across the frequency-crossing. Given the investigated limitations
imposed by the physical presence of the fiber loop on the optical and
mechanical losses, we expect further improvement by incorporating butt-coupled
[32] or side-coupled input/output waveguides [56]. Another necessary avenue
required to leverage the improved heat dissipation of the 2D OMCs is the
passivation, via termination chemistry [57] or encapsulation layers [58], of
the slot-sidewall defect states that lead to the thermo-optical bistability we
observe and that generally prevents us from performing red-detuned
optomechanical cooling. In addition, the prospects of self-assembling gaps way
below the limits of top-down nanofabrication [59] may allow reaching
$g_{\text{o}}$ rates about four times larger than the ones we already report.
Finally, the strong transduction of Fabry-Pérot mechanical modes we observe on
the mirror-terminated waveguides and the demonstrated slow-down of sound to
$\sim$ 700 m/s, finds applications in injection-locked optomechanical
oscillators [60], on-chip phonon networks for information processing [51], or
phononic quantum memories [61]. Conversely, the sensitivity of slow sound to
fabrication imperfection [62] and large ensemble measurements on mirror-
terminated waveguides may allow the observation of spectral features resulting
from Anderson localization of hypersonic acoustic waves [63].

###### Acknowledgements.

The authors aknowledge Karl Pelka for useful discussion. G.M. and C.M.S.T.
acknowledge the support from the project LEIT funded by the European Research
Council, H2020 Grant Agreement No. 885689. ICN2 is supported by the Severo
Ochoa program from the Spanish Research Agency (AEI, Grant No. SEV–2017-0706)
and by the CERCA Programme/Generalitat de Catalunya. M.A. and S.S. gratefully
acknowledge funding from the Villum Foundation Young Investigator Program
(Grant No. 13170). S.S. additionally acknowledges funding from the Danish
National Research Foundation (Grant No. DNRF147–NanoPhoton), the Innovation
Fund Denmark (Grant No. 0175-00022–NEXUS), the Independent Research Fund
Denmark (Grant No. 0135-00315–VAFL), and the European Research Council (Grant.
No. 101045396 – SPOTLIGHT). G.A. acknowledges financial support from the
European Union’s Horizon 2021 research and innovation programme under the
Marie Skłodowska-Curie Action (Grant No. 101067606 - TOPEX).

##  Appendix A Two-dimensional optomechanical crystal waveguides: band
structures

In the waveguides we explore, shown in Fig. 1(a) in the main text, we vary
several geometrical parameters. The most relevant ones are the horizontal
pitch, $a_{x}$, and the radius of the circular hole in one of the membrane
sides, $R_{2}$. The rest of the parameters are fixed, such as the vertical
pitch, $a_{y}$ = 484 nm, or the different parameters defining the shamrock-
shaped holes. We reconstruct the underlying band structures of the relevant
waveguide unit cells (both for the waveguides themselves and for the
heterostructure cavities) by extracting the contour of the fixed features from
analysis of scanning electron microscopy (SEM) images. Figures 5(a) and (b)
show a high-magnification SEM image of a shamrock-shaped hole and a circular
hole along with the outline (in green) extracted after averaging that of many
holes of the same shape. The geometry of the shamrock is kept as extracted,
while the circle is found to have a radius $R_{\text{1,fab}}$ = 196.5 nm. This
evidences a slight overgrowth of the exposed and etched areas compared to the
lithography mask ($R_{\text{1,mask}}$ = 194 nm). The fabricated samples also
include varying slot widths, $s$, but we mainly focus here on MOM devices with
extracted $s$ of 50 nm. Using the extracted geometric features and the
centroid positions given by the triangular lattice parameters ($a_{x}$ and
$a_{y}$), we calculate the dispersion diagrams using a commercial finite-
element solver (COMSOL Multiphysics). Note that for the mechanics, we consider
the anisotropy of the silicon stiffness tensor and its particular orientation
with respect to the waveguide coordinate system [47, 38]. Figures 5(c) and (d)
show the optical and mechanical band diagrams for the case where
$R_{1}=R_{2}$, $a_{y}$ = 484 nm and varying $a_{x}$. We see that, as expected,
the frequencies of the optical and mechanical bands of interest decrease as
$a_{x}$ grows, which we use to build a rectangular confinement potential in
the mirror-terminated long waveguides or a smooth potential well in the mode-
gap adiabatic heterostructure cavities. We also provide in Fig. 5(e) the unit
cell vacuum optomechanical coupling rate, $g_{\text{o,cell}}/2\pi$, between
the Bloch modes respectively at $q=0$ and $k=\pi/a_{x}$ for the varying
$a_{x}$. We observe nearly no variation of $g_{\text{o,cell}}/2\pi$ as a
moving boundary contribution dominates it. The latter is mainly determined by
the slot width, $s$, which is kept fixed.

Figures 5(f) and (g) depict the effect of reducing $R_{2}$ on the optical and
mechanical band structure for the case $a_{x}$ = $a_{y}$ = 484 nm. We see that
the reduced air-filling fraction as $R_{2}$ is reduced relative to $R_{1}$,
which lowers the effective refractive index of the mode, leads to a redshift
of the optical band. However, we note that the band edge of the mode is always
well within the telecom C-band. In the case of the mechanical band structures,
$\Delta R=R_{1}-R_{2}\neq 0$, breaks the $y$-symmetry and lifts the degeneracy
between the frequencies of the modes on the two membrane sides. While the
bands on one side are independent of $R_{2}$, the frequency of the mechanical
mode on the other membrane side increases as $R_{2}$ is reduced. In the
following section, we see how this effect is used in the case of the MOM
system to control the frequency difference between the two cavity phonon
modes.

Figure 5: Optical and mechanical dispersion diagrams of the fabricated
structures. (a) Scanning electron microscopy (SEM) image of a representative
shamrock-shaped hole and extracted average outline (green line). (b) Same as
(a) for a circular hole. (c) Optical and (d) mechanical band structures for
the waveguides with $R_{1}=R_{2}$, $a_{y}$ = 484 nm and varying $a_{x}$. (e)
The coupling rate $g_{\text{o,cell}}$ between the Bloch modes respectively at
$q=0$ and $k=\pi/a_{x}$ for the varying $a_{x}$. (f-h) Same as (c-e) for
waveguides with $a_{x}$ = $a_{y}$ = 484 nm, $R_{1}$ = 196.5 nm and varying
$R_{2}$.

##  Appendix B Two-dimensional optomechanical crystal cavities: cavity details

Figure 6: Cavity-optomechanical figures of merit as a function of the number
of mirror unit cells. Vacuum optomechanical coupling rates, $g_{\text{o}}$,
optical wavelength, $\lambda_{\text{o}}$, optical quality factor,
$Q_{\text{o}}$, mechanical frequencies, $\Omega_{\text{m}}$, and mechanical
quality factors, $Q_{\text{m}}$ as a function of $N_{m}$. The mechanical and
optomechanical figures of merit are the same for both degenerate mechanical
modes.

The mode-gap adiabatic heterostructure cavities are built by leveraging the
band-structure properties depicted in Figs. 5(c) and (d). The horizontal pitch
of the waveguide, $a_{x}$, is changed adiabatically from $a_{1}$ = 484 nm in
the central unit cell to $a_{N_{c}}$ = 510 nm at the edges of the defect
region, which comprises $N_{c}=8$ unit cells, following a cubic interpolation

$a_{i}=\Bigg{\lfloor}a_{N_{c}}\Bigg{(}1-\Big{(}1-\frac{a_{1}}{a_{N_{c}}}\Big{)}\Big{(}2\big{(}\frac{i}{N_{c}}\big{)}^{3}-3\big{(}\frac{i}{N_{c}}\big{)}^{2}+1\Big{)}\Bigg{)}\Bigg{\rceil}$

where $i\in[0,N_{c}]$ and the pitch $a_{i}$ is rounded to integer values to
conform with requirements for the lithographic mask/process. Such adiabatic
tapering of $a_{x}$ is fixed for all the cavities we explore. Beyond the
defect region, two mirror sections comprised of $N_{\text{m}}$ unit cells with
pitch $a_{N_{c}}$ are included. To investigate the intrinsic frequencies and
losses of both the optical and mechanical cavity modes, we explore the
behaviour of the different cavity parameters as a function of $N_{\text{m}}$
(Fig. 6). Note that the simulated cavities use the geometry of the electron-
beam lithography mask instead of the outlines of the fabricated holes, as in
Fig 5, because the smaller mesh elements required to capture the SEM-extracted
contours would otherwise lead to prohibitive computational memory
requirements. We see that nearly all cavity-optomechanical figures of merit
except for the radiation-limited mechanical quality factor, $Q_{\text{m}}$,
are converged —with some numerical fluctuations but no trend— for
$N_{\text{m}}>10$. For the saturation of $Q_{\text{m}}$, 32 mirror unit cells
are necessary. This large number is linked to the low effective mass
associated with the dispersion at the band edge, which results in a low mirror
strength [64]. In addition, we observe that the value of $Q_{\text{m}}$
saturates, which implies that the size of the phononic-crystal cladding
transverse to the waveguide axis is the ultimate limiting factor in our
simulation setting. Our fabricated cavities use a cladding twice as wide as
the simulations; therefore, the radiation-limited $Q_{\text{m}}$, i.e. at very
low temperatures, can be expected to be much larger.

Figure 7: Cavity-optomechanical figures of merit as a function of the radius
of the circular holes in the bottom membrane, $R_{2}$. Vacuum optomechanical
coupling rates, $g_{\text{o}}$, optical wavelength, $\lambda_{\text{o}}$,
optical quality factor, $Q_{\text{o}}$, mechanical frequencies,
$\Omega_{\text{m}}$, and mechanical quality factors, $Q_{\text{m}}$ as a
function of $R_{2}$.

Similar to Fig. 6, we investigate the role of varying $R_{2}$ on the MOM
system. In this case, the two mechanical modes are not degenerate anymore. The
resulting cavity-optomechanical figures of merit for the case $N_{\text{m}}$ =
16 are shown in Fig. 7. We observe the following: 1) the coupling rates,
$g_{\text{o,1}}$ and $g_{\text{o,2}}$, which only coincide for the case
$R_{1}$ = $R_{2}$ = 194 nm, depend very weakly on the value of $R_{2}$; 2) the
resonant wavelength $\lambda_{\text{o}}$ redshifts by 1.63 nm per 1 nm
decrease in $R_{2}$; 3) the optical $Q$ is not clearly affected by $R_{2}$; 4)
the mechanical frequency $\Omega_{\text{m,2}}/2\pi$ increases by 10.19 MHz per
1 nm decrease in $R_{2}$; and 5) the mechanical quality factor grows steadily
as $R_{2}$ increases. In connection with the discussion of the dependence of
$Q_{\text{m}}$ with $N_{\text{m}}$, we attribute the behaviour with $R_{2}$ to
the increase of the effective mass as $R_{2}$ decreases (see Fig. 5(g)).

##  Appendix C Group velocity reconstruction

In Fig. 1(d) in the main text, we compare the simulated group velocity,
$v_{\text{g}}$, of the mechanical mode of interest with that extracted via the
free spectral range (FSR) of the transduced peaks in a waveguide of length $L$
= 350$a_{x}$. Although a single radiofrequency (RF) spectrum is provided in
the main text, the reconstruction uses more RF spectra—albeit with poorer
transduction amplitudes—to either confirm the modes observed in the spectrum
given in the main text or to detect the presence of additional peaks that are
difficult to identify in that single RF spectrum. Figure 8 reproduces the
spectra used and also evidences that in the region where the acoustic
waveguide is multimoded, the peaks can not be easily resolved, and there is no
well-defined FSR [51]. The persistent presence of the peak at the highest
frequency might indicate that it is a tightly-localized acoustic mode, i.e. an
Anderson-localized acoustic mode, but confirming such a hypothesis is beyond
the scope of this work.

Figure 8: Radiofrequency spectra measured using different Anderson-localized
modes on a waveguide made of 350 unit cells. For reference, the simulated
group velocity, $v_{\text{g}}$, shifted down by 90 MHz, is included as the top
panel .

##  Appendix D Measurement of the vacuum optomechanical coupling rate

We measure the vacuum optomechanical coupling rate, $g_{0}$, of the two
mechanical modes of the MOM system following the method of Gorodetksy et al.
[65]. The thermally excited displacement of the two mechanical modes is
probed. Their power spectral density is compared to that transduced by the
optical cavity for a phase-modulation tone induced by an electro-optic
modulator (EOM) driven by a vector network analyzer (VNA) at
$\omega_{\text{VNA}}=$7.01\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ and fixed
power $-10\text{\,}\mathrm{d}\mathrm{B}\mathrm{m}$. This power corresponds to
a modulation voltage $V_{\text{VNA}}=$0.0707\text{\,}\mathrm{V}$$ relative to
the voltage required for a full phase-shift, i.e.,
$V_{\pi}($7\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$)=$5.29\text{\,}\mathrm{V}$$.
The corresponding modulation depth is $d=\pi\frac{V_{\mathrm{VNA}}}{V_{\pi}}$
The two vacuum optomechanical coupling rates, $g_{o,1}$ and $g_{o,2}$, are
evaluated as, [65]

Figure 9:  Measurement of the vacuum optomechanical coupling,
$\mathbf{g_{0}}$. The green lines represent Gaussian (Lorentzian) fits to the
VNA (opto-mechanical) mode(s). The inset shows a magnification of the phase-
modulation peak and the corresponding fit to a Gaussian.

$\frac{g_{o,i}}{2\pi}=\frac{d\times\omega_{\mathrm{VNA}}}{4\sqrt{n}}\sqrt{\frac{A(\Omega_{\mathrm{m,i}})\Gamma_{\mathrm{m,i}}}{A(\omega_{\mathrm{VNA}})\gamma_{\mathrm{mod}}}}.$
(1)

Here, $A(\Omega_{\mathrm{m,i}})$ and $A(\omega_{\mathrm{VNA}})$ are the
measured amplitudes of the power spectral density at the mechanical peak and
at the modulation peak, respectively. $\Gamma_{\mathrm{m,i}}$ and
$\gamma_{\mathrm{mod}}$ are the mechanical and modulation peak widths,
repectively provided by the Lorentzian and Gaussian fits.
$n=k_{\text{B}}T/(h\Omega_{i})$ is the phonon thermal occupancy, with
$k_{\text{B}}$, $T$ and $h$ the Boltzmann constant, the temperature and
Planck’s constant, respectively. Figure 9 shows the measured power spectral
density, as well as two independent Lorentzian fits to the mechanical modes
and a Gaussian fit to the VNA tone for a MOM system with $R_{2}$ = 192 nm.
Table 1 summarizes the results obtained from the fits and the evaluation of
Eq.( 1).

Table 1: Frequency ($\Omega$), linewidth ($\Gamma$), quality factor ($Q=\Omega/\Gamma$) and vacuum optomechanical coupling ($g_{\text{o}}/(2\pi)$) for the two mechanical modes. Mode |  $\Omega/2\pi$ (GHz) |  $\Gamma/2\pi$ (MHz) | $Q$ |  $g_{0}/2\pi$ (MHz)
---|---|---|---|---
1 (left) | $7.043\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ | $2.04\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ | $3464$ |  $1.543\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$\pm$ $39\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$
2 (right) | $7.065\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ | $2.42\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ | $2969$ |  $1.373\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$$\pm$ $45\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$

##  Appendix E Effective laser detuning in a thermo-optic cavity

Due to the thermo-optic nonlinearity, the laser wavelength $\lambda$ does not
straightforwardly control the laser detuning to the cavity $\Delta$. Within a
linear case, the optical cavity can be modelled using coupled mode theory and
the transmission through the fiber loop is given by,

$T=\frac{\Delta^{2}+\kappa_{i}^{2}/4}{\Delta^{2}+\kappa_{t}^{2}/4}$ (2)

with $\kappa_{i}$ and $\kappa_{e}$ the internal and external decay rates and
$\kappa_{t}=\kappa_{i}+\kappa_{e}$. The laser-cavity detuning, $\Delta$, is
given by $\Delta=\omega-\omega_{c}$, with $\omega$ the laser frequency and
$\omega_{c}$ the cavity resonance frequency. In the presence of a thermo-optic
nonlinearity, $\beta$, the former is power-dependent and writes
$\omega_{c}=\omega_{0}+\beta E$ with $E$ the intracavity energy and
$\omega_{0}$ the cavity natural frequency (at zero input power). The value of
$\beta$ depends on several optical and thermal properties of the host material
and is also affected by the optical field profile [66]. The independent
extraction of such parameters is cumbersome and subject to numerous sources of
uncertainty.

Figure 10: Extraction of the laser-cavity detuning as a function of the laser
wavelength. For increasing optical power launched to the cavity, we measure
the normalized transmission (top) from which we infer the laser detuning
$\Delta$ as a function of the laser wavelength $\lambda$. Blue-sideband
excitation occurs at $\Delta=+\overline{\Omega}$, as indicated with the dashed
lines. The evaluated detuning is fitted with a power law (green lines).

On the contrary, it is possible to evaluate the detuning from the measured
transmission $T$ using Eq. 2,

$|\Delta|=\sqrt{\frac{1}{4}\frac{\kappa_{t}^{2}T-\kappa_{i}^{2}}{1-T}}$ (3)

This enables us to obtain the detuning as a function of, e.g., the laser
wavelength $\lambda$ and establish a correspondence between the control
parameter ($\lambda$) and the physically relevant quantity ($\Delta$). For
several measurements of the transmission spectrum $T(\lambda)$ at increasing
laser power (see Fig. 10, top), we show the corresponding laser detuning
$\Delta$ deduced from Eq. 3 (bottom). The data are plotted as a function of
$\lambda-\lambda_{0}$ with $\lambda_{0}=2\pi c/\omega_{0}$ the natural cavity
resonance wavelength evaluated to be $\lambda_{0}\sim 1559.35$ nm. In absence
of thermo-optic nonlinearity, the detuning would scale linearly with the laser
wavelength in the case of the nearly-resonant excitations used here:
$\Delta\approx\frac{2\pi c(\lambda_{0}-\lambda)}{\lambda_{0}^{2}}$. Instead,
we observe with this representation how $\Delta$ scales asymptotically with
$\lambda$. We fit the data with a power law
$\Delta(\lambda)=A+B(\lambda-\lambda_{0})^{C}$ (green lines). The fits are
rather satisfying which justifies the use of semi-logarithmic scales in Fig.
3(c) of the main text for comparing the experiment to theory.

This method assumes that the cavity decay rates remain unchanged with the
input power, which remains an approximation due to the emergence of two-photon
absorption at higher power that tends to increase the intrinsic loss rate
$\kappa_{i}$. However, we note an irregularity observed in the transmission
curves that becomes stronger when the power is increased. This “kink”
typically indicates optomechanical self-oscillations, which occurs when the
laser reaches the blue sideband at $\Delta=+\overline{\Omega}$, where
$\overline{\Omega}=\frac{1}{2}(\Omega_{1}+\Omega_{2})$. Here, it appears in
agreement with the determined laser detuning as shown with the dashed lines at
$\Delta/\overline{\Omega}\approx+1$, where we use
$\overline{\Omega}/2\pi=7.060$ GHz. It is also in agreement with the
measurements presented in Fig.3d-e of the manuscript, where at least one
mechanical mode becomes very close to its lasing threshold $\Gamma_{i}=0$ for
input powers above 131 $\mu$W.

##  Appendix F Linearized optomechanical susceptibility

We consider a Mechanical-Optical-Mechanical configuration (MOM), i.e. two
mechanically independent mechanical modes with displacements $x_{1}$ and
$x_{2}$, coupled to a single optical mode with amplitude $a$ via their
respective optomechanical coupling rates $g_{\mathrm{o,1}}$ and
$g_{\mathrm{o,2}}$. The classical equations of motion of this system are given
by,

$\displaystyle\dot{a}$
$\displaystyle=\Big{(}j\big{(}\Delta+g_{\mathrm{o,1}}x_{1}+g_{\mathrm{o,2}}x_{2}\big{)}-\frac{\kappa_{t}}{2}\Big{)}a+\sqrt{\frac{\kappa_{e}}{2}}s_{\mathrm{in}}$
(4) $\displaystyle\ddot{x_{1}}$
$\displaystyle=-\Gamma_{\mathrm{m,1}}\dot{x_{1}}-\Omega_{\mathrm{m,1}}^{2}x_{1}+2\Omega_{\mathrm{m,1}}g_{\mathrm{o,1}}|a|^{2}$
$\displaystyle\ddot{x_{2}}$
$\displaystyle=-\Gamma_{\mathrm{m,2}}\dot{x_{2}}-\Omega_{\mathrm{m,2}}^{2}x_{2}+2\Omega_{\mathrm{m,2}}g_{\mathrm{o,2}}|a|^{2}$

where $\Delta=\omega_{\ell}-\omega_{0}$ is the detuning between the laser
frequency $\omega_{\ell}=2\pi c/\lambda$ and the cavity resonance frequency
$\omega_{0}$, and $s_{\mathrm{in}}$ is the input laser field amplitude, such
that $P_{\mathrm{in}}=\hbar\omega_{\ell}|s_{\mathrm{in}}|^{2}$ is the laser
incident power and $\overline{n}_{\mathrm{cav}}=|a|^{2}$ the cavity photon
number. Note that the displacements are normalized by the respective zero-
point fluctuations of the mechanical modes such that the mechanical masses can
be eliminated.

Each mechanical resonator has a natural mechanical susceptibility
$\chi_{\mathrm{m,i}}^{-1}(\omega)=\Big{[}(\Omega_{\mathrm{m,i}}^{2}-\omega^{2})-j\omega\Gamma_{\mathrm{m,i}}\Big{]}$,
that is perturbed by radiation pressure forces. The perturbation is described
by a self-coupling term [1]
$\Sigma_{i}(\omega)=2\Omega_{\mathrm{m,i}}g_{\mathrm{o,i}}^{2}\beta(\omega)$
where
$\beta(\omega)=\frac{\overline{n}_{\mathrm{cav}}}{(\Delta+\omega)+j\kappa_{t}/2}+\frac{\overline{n}_{\mathrm{cav}}}{(\Delta-\omega)-j\kappa_{t}/2}$.
The presence of a second mechanical mode leads to a deviation in the effective
susceptibility of each mechanical oscillator. Applying a linearization to Eq.
4, we obtain:

$\displaystyle\big{[}\chi_{\mathrm{eff,1}}^{-1}(\omega)+\Sigma_{1}(\omega)\big{]}x_{1}=2\Omega_{\mathrm{m,1}}g_{\mathrm{o,1}}g_{\mathrm{o,2}}\beta(\omega)x_{2}$
(5)
$\displaystyle\big{[}\chi_{\mathrm{eff,2}}^{-1}(\omega)+\Sigma_{2}(\omega)\big{]}x_{2}=2\Omega_{\mathrm{m,2}}g_{\mathrm{o,1}}g_{\mathrm{o,2}}\beta(\omega)x_{1}$

This expression highlights that the coupling between the mechanical modes
scales with $g_{1}g_{2}$, which is expected since they do not interact
directly but through the optical field. Note that taking
$\Gamma_{1}\approx\Gamma_{2}$ provides the Hamiltonian formulation presented
in Ref. [13]. Solving Eq. 5 provides an expression for the effective
mechanical coupling between the mechanical modes:

$K_{i}(\omega)=-\frac{\Sigma_{i}(\omega)\Sigma_{j}(\omega)}{\chi_{\mathrm{m,j}}^{-1}(\omega)+\Sigma_{j}(\omega)}$
(6)

Figure 11: The role of the optically-induced mechanical coupling in the
observed dynamical back-action effects. Comparison of the theory curves
presented in Fig. 3 of the main text when accounting for a MOM with effective
coupling (right) compared to a MOM with no optically-mediated coupling between
the two mechanical modes (left). The mechanical frequencies and linewidths are
shown for different input laser power indicated in units of $\mu$W.

Therefore we can write the total mechanical susceptibility of mode i as
follow:

$\chi_{\mathrm{eff,i}}^{-1}(\omega)=\chi_{\mathrm{m,i}}^{-1}(\omega)+\Sigma_{i}(\omega)+K_{i}(\omega)$
(7)

From Eq. 7, we evaluate the effective mechanical frequencies and damping rates
using
$\Omega_{\mathrm{i}}=\Omega_{\mathrm{m,i}}+\mathrm{Re}[\Sigma_{i}(\omega)+K_{i}(\omega)]/(2\Omega_{i})$,
and $\Gamma_{\mathrm{i}}=-\mathrm{Im}[\chi_{\mathrm{eff,i}}^{-1}]/\Omega_{i}$,
respectively. These are the theoretical quantities plotted as a function of
$\lambda$ and $P_{\mathrm{in}}$ in Fig. 3 in the main text.

In Fig. 11, we compare the theoretical evolutions of $\Omega_{1}$,
$\Omega_{2}$, $\Gamma_{1}$, and $\Omega_{2}$ as a function of the detuning
($\delta\lambda$, expressed in terms of wavelength). We compare the situation
where the mechanical oscillator do not interact through the optical mode
(i.e., by imposing $K_{i}=0$) with the MOM configuration (with $K_{i}$ given
by Eq. 6). We note a significant change in the evolution of the linewidths. In
particular, accounting for the coupling enables the recovery of two features:
1) the mode linewidths reach a minimum at different spectral locations
(indicated by dashed lines) which is not the case when considering single
optomechanical cavities, and 2) the decrease of $\Gamma_{1}$ is more
pronounced, enabling the mode to pass its lasing threshold ($\Gamma_{1}=0$) at
the power $P_{\mathrm{in}}=219$ $\mu$W, while the linewidth of the other mode
reduces as much, but within a shorter spectral range. In summary, in the
absence of coupling the frequency and linewidth of mode 1 and 2 evolve
identically, with only a scaling factor and an offset. However accounting for
the full MOM mechanical susceptibility reveals that the modes follow different
spectral evolutions (i.e., along the $x$-axis).

##  Appendix G Stimulated multimode lasing: methodology

Here we provide more details on the theoretical curves included in Figure 4(c)
in the main text. We present, first, the methodology used to determine the
lasing threshold from the measurement of the mechanical peak amplitudes as a
function of the modulation depth, $d$, and frequency, $\omega_{\mathrm{VNA}}$.
Secondly, we detail how the extracted thresholds are fitted using the laser
detuning $\Delta$ as a fitting parameter and the theory described in Ref.
[41].

The amplitude modulation of the input laser leads to sidebands of the
mechanical modes in the radiofrequency spectrum [67]. When
$\delta\omega\equiv\omega_{\mathrm{VNA}}-(\Omega_{2}-\Omega_{1})<\Gamma_{1,2}$,
a sideband overlaps with the mechanical modes. This situation complicates the
fitting of the mechanical modes with a Lorentzian lineshape, and the
mechanical linewidth cannot be accurately determined. In order to determine
the lasing thresholds – given by $\Gamma_{i}=0$, by definition – we use the
fact that the peak amplitude grows quickly at the threshold. For each map
presented in the Figure 4(c) in the main text, i.e., for each mechanical mode,
we plot the statistics of the peak amplitude within the full map (see Fig.
12(a)). The statistics display a minimum (dashed lines) that we interpret as
the value of the oscillator amplitude that indicates the threshold. For each
mode, we then determine the positions in the parameter space,
$\\{\omega_{\mathrm{VNA}}^{\mathrm{th}},d^{\mathrm{th}}\\}$, where the
amplitude passes this threshold. This provides the data points in Fig. 12(b).

Now, we theoretically determine the threshold condition in the same parameter
space. For a given mode i, it corresponds to the ensemble of pairs
$\\{\omega_{\mathrm{VNA}},d\\}$ fulfilling the condition $\Gamma_{i}=0$. The
model in Ref. [41] leads to

$\Gamma_{i}=-\frac{1}{\Omega_{i}}\mathrm{Im}\Big{[}\chi_{\mathrm{m,i}}^{-1}+\Sigma_{i}(\Omega_{\mathrm{m,1}})-\frac{2\Omega_{i}\sigma^{2}(\Omega_{i})}{\delta\omega+j\frac{\Gamma_{\mathrm{m,i}}}{2}+\sigma^{\prime}(\Omega_{i})}\Big{]}$

with

$\displaystyle\sigma(\omega)$
$\displaystyle=g_{\mathrm{o,1}}g_{\mathrm{o,2}}\overline{n}_{\mathrm{cav}}d\beta(\omega)$
$\displaystyle\sigma^{\prime}(\omega)$
$\displaystyle=g_{\mathrm{o,1}}g_{\mathrm{o,2}}\overline{n}_{\mathrm{cav}}d^{2}\beta(\omega)$

and where $\chi_{\mathrm{m,i}}$, $\Sigma_{i}(\omega)$ and $\beta(\omega)$ are
defined in the previous section,.

The fit is performed using $\Delta$ as the only fitting parameter, as it can
be noted that both $\beta(\omega)$ and $\overline{n}_{\mathrm{cav}}$ depends
on $\Delta$. The thresholds for mode 1 and 2 are fitted at once (see red line
in Fig. 12(b)) and provides $\Delta=1.024\times\overline{\Omega}$ using the
mean mechanical frequency $\overline{\Omega}=6.984$ GHz. The residual is shown
in the inset.

Figure 12: Lasing threshold determination and fitting a. Distribution signal-
to-noise ratio of mechanical modes 1 (left) and 2 (right) over the colormaps
presented in Fig. 4 of the manuscript. The minimum of each distribution is
interpreted as the lasing threshold amplitude. b. Lasing threshold of each
mode (1:left and 2:right) plotted in the parameter space
$\\{\omega_{\mathrm{VNA}}^{\mathrm{th}},d^{\mathrm{th}}\\}$ (black dots) and
simultaneously fitted with the theoretical model from Ref.[41] (red lines).
The residuals (inset) yield a minimum at $\Delta/\overline{\Omega}=1.024$.

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