Diamond optomechanical crystals

Cavity-optomechanical systems realized in single-crystal diamond are poised to benefit from its extraordinary material properties, including low mechanical dissipation and a wide optical transparency window. Diamond is also rich in optically active defects, such as the nitrogen-vacancy (NV) and silicon-vacancy (SiV) centers, which behave as atom-like systems in the solid state. Predictions and observations of coherent coupling of the NV electronic spin to phonons via lattice strain has motivated the development of diamond nanomechanical devices aimed at realization of hybrid quantum systems, in which phonons provide an interface with diamond spins. In this work, we demonstrate diamond optomechanical crystals (OMCs), a device platform to enable such applications, wherein the co-localization of ~ 200 THz photons and few to 10 GHz phonons in a quasi-periodic diamond nanostructure leads to coupling of an optical cavity field to a mechanical mode via radiation pressure. In contrast to other material systems, diamond OMCs operating in the resolved-sideband regime possess large intracavity photon capacity (>10$^5$) and sufficient optomechanical coupling rates to reach a cooperativity of ~ 20 at room temperature, allowing for the observation of optomechanically induced transparency and the realization of large amplitude optomechanical self-oscillations.


INTRODUCTION
Optomechanical crystals (OMCs), first demonstrated in silicon 1 , and later in other materials like silicon nitride 2,3 , aluminum nitride 4,5 , and gallium arsenide 6,7 , have emerged as a fruitful optomechanics platform, wherein radiation pressure effects provide exquisitely sensitive optical control of mechanical vibrations. Such systems have enabled demonstrations of quantum ground state cooling 8 , optomechanically induced transparency (OMIT) 9 , squeezed light 10 , and wavelength conversion 11 . Highly coherent photon-phonon interactions in OMCs are a direct result of the ability to engineer a large singlephoton optomechanical coupling rate (g o ), while retaining sufficiently small optical (κ) and intrinsic mechanical (γ i ) dissipation rates. Similar structures realized in single-crystal diamond -which features a unique combination of superior mechanical, thermal, and optical properties 12 -are expected to exhibit pronounced optomechanical interactions, quantified by the cooperativity parameter precludes multi-photon absorption over a wide wavelength range (from visible to infrared). This, combined with its high thermal conductivity and small thermal expansion, enables monolithic diamond optical cavities that can withstand significant optical power densities, while avoiding degradation in optical linewidth or drifts in resonance wavelength due to thermal lensing. The large intracavity photon capacity of diamond can thus result in high cooperativities necessary for either strong mechanical driving or effective laser cooling 8 . Moreover, diamond is among the stiffest materials known and possesses extremely low thermoelastic mechanical damping, with recently demonstrated monolithic diamond cantilevers exhibiting mechanical Q-factors in excess of 10 6 at room temperature 13 . In what follows, we make use of these features to demonstrate OMCs in single-crystal diamond with unique performance. Our diamond OMCs support an optical mode at ω o /2π ~ 200 THz, co-resonant with two localized acoustic phonon modes at ω m /2π ~ 5.5 GHz and ~ 9.5 GHz. Both mechanical resonances are well coupled to the optical cavity, with vacuum optomechanical coupling rates of g o /2π ~ 120 kHz and ~ 220 kHz, respectively. With a measured optical linewidth of κ/2π ~ 1.1 GHz, our diamond OMC system operates in the so-called resolved sideband regime (ω m /κ >> 1), necessary for efficient radiationpressure driven dynamic backaction. This enables our diamond OMCs to be optically driven to C >> 1 at room temperature, highlighted by the observations of "phonon lasing" 14 and OMIT 9 in our structures.

DIAMOND OPTOMECHANICAL CRYSTAL DESIGN AND FABRICATION
The OMCs of this work consist of a one dimensional nanobeam photonic crystal cavity fabricated in synthetic single-crystal diamond 15 using previously developed 'angled-etching' techniques 16,17 . The nanobeam cavity is based on a diamond waveguide with a triangular cross-section that is perforated with a periodic lattice of elliptically shaped air holes. One unit cell of the waveguide and corresponding photonic bandstructure are shown in Figure 1 (a) and Figure 1 (b), respectively. The latter includes both transverse electric (TE-like, solid black lines) and transverse magnetic (TM-like, dashed blue lines) guided modes, while the grey shaded region indicates the continuum of radiation and leaky modes that exist above the light line for the structure. In this work, we focus on TE-like modes (see Figure 1 (b) inset), near the X-point frequency of ω o /2π ~ 200 THz (λ ~ 1550 nm), since they can lead to the realization of very high Q-factor optical cavities 15 . Importantly, our photonic crystal waveguide also supports acoustic guided modes that spatially overlap with optical modes, and can couple to them via radiation pressure. The corresponding mechanical bandstructure (Figure 1 (c)) reveals a rich library of guided acoustic modes in the few to ~ 12 GHz frequency range (see Supplementary Information for extended discussions 18 ). The guided modes, categorized by even (solid black lines) and odd (dashed blue lines) vector symmetries about the xz-plane, yield symmetry based quasi-bandgaps. Following OMC design rules 19,20 , we identified the guided modes derived from the Γ-point of the 4 th and 7 th y-symmetric bands (frequency of ω m /2π ~ 6.9 GHz and ~ 11.5 GHz) -referred to hereafter as the "flapping" and "swelling" acoustic guided modes (Figure 1 (d) and (e)), respectively -as the mechanical modes of interest for large optomechanical coupling. To produce an optimized diamond OMC design, we focus on the acoustic flapping mode due to the large quasi-bandgap below its native band -indicated by the shaded pink region in Figure 1 (c).
To realize a diamond OMC cavity from the aforementioned OMC waveguide, the lattice of air holes is chirped 19 such as to transition from a "mirror" region formed by the base unit cell in Figure 1 (a) to a "defect" cell. The selected defect cell dimensions simultaneously raise and lower the frequencies of the target optical and mechanical modes, respectively, into their corresponding quasi-bandgaps. Gradually reducing the unit cell lattice constant while also decreasing the air hole aspect ratio (h y /h x ) achieves the necessary band edge tuning (see right and left panels of Figure 1  With our final diamond OMC design optimized for the acoustic flapping mode, we also observe a localization of the previously mentioned acoustic swelling mode (displacement profile shown in Figure   1 (h)) at a mechanical frequency of ω m /2π = 9.01 GHz, with a zero-point motion of x zpf = 2.2 fm. The simulated optomechanical coupling rate for this design was g o /2π = 234 kHz, which includes a moving boundary and photo-elastic contribution of g o,MB /2π = 50 kHz and g o,PE /2π = 184 kHz, respectively. We attribute the overall greater optomechanical coupling rate of the acoustic swelling mode to its crosssectional strain profile, which more favorably overlaps with the TE-like optical mode. While this mode is better coupled to the localized optical cavity, its predicted mechanical resonance frequency is not localized within a symmetry-based quasi-bandgap (see Figure 1 (c)), which may ultimately limit its mechanical Q-factor in fabricated structures 1,20 .
As previously mentioned, fabrication of diamond OMCs utilized angled-etching techniques [15][16][17][18] (as illustrated in Figure 2 (a)), which employ anisotropic oxygen-based plasma etching at an oblique angle to the substrate surface resulting in suspended structures with a triangular cross-section. The final fabricated structures, displayed in Figure 2 (b) -(d), reveal excellent reproduction of the intended design. A unique feature of angled-etched structures is their triangular cross-sectional symmetry 18 . The high-resolution SEM image shown in Figure 2 (e) reveals a fabricated diamond OMC (oriented upside down), with insets displaying a tilted cross-sectional view.

OPTICAL AND MECHANICAL SPECTROSCOPY
The fiber-optical characterization set up 18 used to perform both optical and mechanical spectroscopy of diamond OMCs is schematically displayed in Figure 3 (a). Briefly, light from a tunable laser source (TLS) was evanescently coupled to the device under test via a dimpled fiber taper. A small portion of laser signal fed to a wavemeter enabled continuous monitoring of the laser frequency. An erbium doped 7 fiber amplifier (EDFA) was used in certain experiments to increase the maximum input laser power, and a variable optical attenuator (VOA) was used to set the final laser power delivered to the device. The optical cavity transmission spectrum was collected by a low-speed (125 MHz) photodetector, while a high-speed (12 GHz) photoreceiver monitored the radio frequency (RF) response of the mechanical cavity via a real-time spectrum analyzer (RSA). For OMIT measurements discussed later in this work, an electro-optic phase modulator (EOPM), placed in the input fiber path, was used to create a weak tunable probe signal on the pump laser control field. Port 1 of a high frequency vector network analyzer (VNA) supplied the RF input to the EOPM, while port 2 of the VNA collected the RF output of the high-speed photoreceiver. All measurements were performed at room temperature and ambient pressure.
A transmission spectrum of a representative diamond OMC, displayed in Figure 3 (b), reveals the optical cavity resonance centered at λ o = 1529.2 nm, with a measured total and intrinsic optical Q-factor of Q t ~ 1.76 x 10 5 and Q i ~ 2.70 x 10 5 , respectively. The corresponding total cavity decay rate, fiber taper coupling rate, and intrinsic optical decay rate are κ/2π = 1.114 GHz, κ e /2π = 399 MHz, and κ i /2π = 715 MHz, respectively. With the input laser slightly detuned from the optical cavity, the broadband RF spectrum of thermally excited motion at room temperature (i.e., thermal Brownian motion) reveals a series of mechanical resonances 18 , as shown in the normalized power spectral density (NPSD) in Figure   3 (c). Specifically, we attribute the sharp resonance observed at ~ 5.5 GHz to the diamond OMC acoustic flapping mode. A high-resolution RF spectrum (shown in Figure 3 (d)) of this feature reveals a Lorentzian mechanical resonance of the diamond OMC centered at ω m /2π = 5.52 GHz with a room temperature mechanical Q-factor of Q m ~ 4100.
Given the measured optical cavity decay rate, our diamond OMC operates in the resolved sideband regime, with ω m /κ ~ 4.86. In this regime, while the input laser is either red-or blue-detuned from the optical cavity by a mechanical frequency (Δ = (ω o -ω l ) = ± ω m ), mechanical motion of the acoustic mode phase-modulates the transmitted light, giving rise to a sideband of the input laser resonant with the 8 optical cavity. The other first-order motional sideband, which is not resonant with the optical cavity, is suppressed in this scenario. As a result, the mechanical motion produces an intensity modulation in the radio frequency (RF) power spectrum of the photoreceiver signal. To observe this effect directly, a weak input laser was tuned across the optical cavity at a constant power, while simultaneously monitoring the RF spectrum near the diamond OMC acoustic flapping mode. Figure 3 (e) displays the collected spectra as a function of laser detuning, with the simultaneously collected optical transmission spectrum also plotted. A clear increase in optomechanical transduction is observed as the laser is tuned off-resonance from the optical cavity by ± ~ 45 pm, corresponding to a detuning of approximately a mechanical frequency. Additionally, strong transduction occurs with the laser tuned within the cavity bandwidth, and a clear optical bistability is present in the optical cavity transmission spectrum. We attribute both observations to non-linear optical absorption (likely due to surface contamination), which cause a thermo-optic red shift in the optical resonance wavelength and an increase in thermal Brownian motion of the mechanical cavity. To mitigate such thermal effects, a similar measurement was performed, however now with the input laser power continually adjusted via the VOA to maintain a constant intracavity photon number at each laser detuning (Figures 3 (f) and (g)). From the measured optical cavity resonance frequency and linewidth, n c is calculated by the relation: where P i is the input laser power set by the VOA. In the resolved sideband limit 22 , optomechanical backaction causes additional mechanical damping (γ OM ) and springing (δω m = |ω m -ω m,o |) rates, respectively, of: Under optimal detuning, with Δ = ± ω m , a maximum optomechanically induced damping rate of is expected. Figure 3 (f) and (g) display the experimentally derived damping and springing curves (grey circles) for the diamond OMC acoustic flapping mode, respectively. A weak intracavity power, corresponding to n c ~ 10,000 photons, was used for this measurement to avoid any thermal drifts in the cavity resonance. Indeed, the optomechanically induced damping is maximized (minimized) when the laser is detuned a mechanical frequency red (blue) of the optical cavity. Fits to these data sets following Eq. (2) and Eq. (3) (solid red lines), gave an estimate for the intrinsic mechanical damping of γ i /2π ~ 1.37 MHz and the single-photon optomechanical coupling rate of g o /2π ~ 118 kHz. This estimate differs only slightly from our design, which we attribute to uncertainty in the photo-elastic constants of diamond at telecom frequencies, as well as fabrication imperfections. 10 The inset of Figure 4 (a) displays the optomechanically induced damping (γ OM = γ red -γ i , black squares), plotted versus n c . A linear fit to the γ OM data yields g o /2π ~ 123 +/-6 kHz, which agrees well with simulations, and is consistent with previous estimates from the data plotted in Figure 3 (f) and (g). With the laser blue-detuned by a mechanical frequency, a threshold where γ blue ~ 0 is reached at approximately n c , thr ~ 27,000, exciting the diamond OMC mechanical cavity into large amplitude optomechanical self-oscillations, so-called "phonon lasing" 14 . Mechanical spectra of the diamond OMC taken below, at, and above this phonon lasing threshold (shown in Figure 4  With the demonstration of C >> 1, optomechanical transduction in our diamond OMC acoustic flapping mode occurs at a substantially faster rate than energy loss of the system. This enables observation of the optomechanical analog to electromechanically induced transparency, so-called OMIT 9 . To observe OMIT in our diamond OMC structures, the input laser is red-detuned from the optical cavity and fixed as a strong driving control field (ω c ), while a weak probe field (ω p , realized as sidebands created by an EOPM) is swept in frequency across the optical cavity resonance. Under optimal detuning conditions, whereby the control laser detuning equals a mechanical frequency (Δ oc ≡ (ω o -ω c ) = ω m ) and the probe-control detuning satisfies a two-photon resonance condition (Δ pc ≡ (ω pω c ) = Δ oc ), destructive interference of probe photons with control photons scattered by the mechanical resonator occurs. This yields a transparency window on the optical cavity transmission spectra, with its bandwidth set by the mechanical damping rate. A central requirement for this scattering phenomenon is that the probe and phonon-scattered photons are phase coherent, which demonstrates a coherent interaction of the mechanical resonator with the optical cavity. As previously mentioned, OMIT in our diamond OMC structures is observed via an |S 21 | measurement with a VNA (Figure 3(a)), where port 1 of the VNA drives the EOPM input to create the weak probe field which sweeps across the optical cavity, and port 2 collects the RF output of the high-speed photoreceiver.  Figure 4 (d) display zoomed-in spectra of this fine feature). Fits to the normalized OMIT spectra 18 , which followed the methodology reported previously 2,3 , estimate a cooperativity of C ~ 1.9 for data collected with optimal Δ oc ~ ω m detuning, in good agreement with the cooperativity value measured in Figure 4 (c) under similar input laser power.
In addition to the resonance feature at ~ 5.5 GHz, two sharp features are also observed in the diamond OMC broadband thermal Brownian motion RF spectrum (Figure 3 (c)) near ~ 9.5 GHz. Figure 5 (a) displays a zoomed-in RF spectrum around these features, collected with a weak laser signal slightly detuned from the optical cavity resonance. Four clear resonances are present in this span, with the central feature at ~ 9.5 GHz most strongly transduced by the optical cavity field. A high-resolution RF spectrum (shown in Figure 5 (b)) of this feature reveals a Lorentzian mechanical resonance of the diamond OMC centered at ω m /2π = 9.45 GHz with a mechanical Q-factor of Q m ~ 7700. This corresponds to a f·Q product of ~ 7.3 x 10 13 Hz, which is among the highest demonstrated for either a bulk or small-scale single-crystal diamond mechanical oscillator at room temperature 23,24 .
As before, we extract the optomechanical coupling rate for this mode by tuning the laser across the optical cavity resonance, while maintaining a constant intracavity photon number of n c ~ 6000, and simultaneously monitoring the mechanical resonance at 9.45 GHz. Fitting Eq. (2) and (3)   optimal blue-detuning was only n c,thr ~ 7,600. Under optimal red-detuned laser conditions, the increased laser power afforded by the input EDFA enabled us to reach a room temperature mechanical linewidth γ red /2π ~ 26.7 MHz ( Figure 5 (e)), corresponding to a maximum observed cooperativity of C ~ 19.9 ( Figure 5 (f)). With previous estimates of κ, γ i , and g o for this acoustic swelling mode, an intracavity photon number of only n c ~ 162,000 was inferred at this cooperativity level. As before, higher cooperativities were not observed due to instabilities in the measurement under the high optical input power. OMIT was also observed for this acoustic swelling mode 18 .

CONCLUSIONS
In summary, we have demonstrated resolved sideband cavity-optomechanics in single-crystal diamond, operating in the few to ~ 10 GHz range, where optomechanical coupling via radiation pressure was sufficient to reach a room temperature cooperativity of nearly ~ 20 for an intracavity photon population on the order of 10 5 . Present devices also offer a promising platform for reaching much larger cooperativities when, for instance, operated at cryogenic temperatures, where mechanical Q-factors of diamond resonators have been shown to improve significantly 13 (ω p -ω c )). Right inset panels of (d) display zoomed-in OMIT spectra of the transparency window induced by coherent interaction of the mechanical and optical cavities. Fits to OMIT spectra 18 (solid red and blue lines), estimate a cooperativity of C ~ 1.9 for data collected with Δ oc ~ ω m .

SUPPLEMENTARY INFORMATION i) Guided acoustic phonon modes in diamond optomechanical crystals
To supplement our discussion of the guided acoustic phonon modes supported by diamond optomechanical crystals (OMCs), we present normalized displacement profiles of the nominal unit cell at the Γ (k x = 0) and X (k x = π/a) points of its mechanical bandstructure (originally displayed in Figure 1  While the mechanical bandstructures reveal a rich library of guided acoustic modes in the few to 16 GHz frequency range, only guided modes originating from y-symmetric bands ultimately couple to the optical cavity 2 . Additionally, modes originating from the Γ-point ensure large optomechanical coupling rates in the final design 3 . With this in mind, two modes from the Γ-point of y-symmetric bands enable design of diamond OMCs with large single-photon optomechanical coupling rates, g o . Specifically, the Γ-point modes from the 4 th and 7 th y-symmetric bands, referred to as the "flapping" and "swelling" modes, respectively, were both investigated.

ii) Optimized diamond optomechanical crystal design
As discussed in the main text, the final diamond OMC design relies on transitioning from a "mirror" region formed by the base unit cell in Figure 1 (a) to a "defect" cell, which localizes the target optical and mechanical guided modes into their respective quasi-bandgaps. Out-of-plane scattering losses in the optical cavity are minimized by transitioning from the mirror region to defect cell over seven lattice periods. This "defect region" is parameterized by the maximum change in lattice constant in the defect region, d = (1 -a defect /a nominal ), the aspect ratio of the center hole, and curvature of the transition. Figure   S3 illustrates the mirror to defect cell transition of our optimized diamond OMC design. The optimized design was determined via previously described numerical optimization methods 3 , based upon finite element method (FEM) simulations (COMSOL) to calculate the optical and mechanical cavity resonance frequencies, ω o and ω m , the optical Q-factor, Q o , and the single-photon optomechanicalcoupling rate, g o . In the optimization, the mirror region unit cell geometry (w, a, h x, h y ) and the aforementioned defect region parameters were varied (within suitable fabrication tolerances), and a 5 fitness function for the optimization was set such as to converge on a design with the largest g o .

iii) Calculation of single-photon optomechanical coupling rate
Both moving boundary (g o,MB ) and photo-elastic contributions (g o,PE ) to the single-photon optomechanical coupling rate were considered 3,4 , with g o,MB given by: where Q is the normalized displacement field, n  is the outward facing surface normal, E and D are the electric and displacement fields respectively, the subscripts || and  subscripts designate field components parallel and perpendicular to the surface respectively, ε is the material permittivity, , and . The photo-elastic contribution to the optomechanical coupling rate, g o,PE , for a cubic crystal with m3m point symmetry and the x-axis and y-axis aligned to the [100] and [010] crystal directions, respectively, is given by: where Σ is a summation, according to Einstein notation x  y  z  x. S ij are the strain tensor components, and p ij are the photoelastic coefficients of diamond 5 : (p 11 , p 12 , p 44 ) = (-0.25, 0.043, -0.172).
As mentioned previously and in the main text, diamond OMCs were fabricated with their x-axis aligned with the [110] crystallographic direction. In the calculation of g o,MB this was taken into account by using a rotated version of the elasticity matrix 4  where: The angled-etching fabrication procedure used in this work, schematically depicted in Figure 2 (a) of the main text, is illustrated in Figure S4 (a), with corresponding SEM images displayed in Figure S4 subpanels (b) to (e). Fabrication began with single-crystal diamond substrates grown using microwave-  Following this, an angled-etching step was performed to realize the final free-standing diamond OMCs.
Our angled-etching approach 7-9 employs anisotropic oxygen-based plasma etching at an oblique angle to the substrate surface, resulting in suspended structures with triangular cross-section. Angled-etching was achieved using the same ICP-RIE parameters as the initial top down etch, but included housing the sample inside a specifically designed aluminum Faraday cage [7][8][9] to direct the plasma ions to the substrate surface at the intended angle. Following the oxygen-based plasma etching, the remaining etch mask was removed in concentrated hydrofluoric acid. Diamond OMCs were cleaned in piranha solution, then annealed at 450 o C in a high-purity oxygen environment for 8 hours prior to optical and mechanical mode spectroscopy.

v) Characterization of fabricated diamond OMC cross-sectional symmetry
Evidently, a unique consideration of angled-etched structures is their triangular cross-sectional symmetry. For instance, uneven sample mounting within the Faraday cage, diamond substrate wedge tolerances, and non-ideal Faraday cage construction 9 will lead to a distribution of effective etch angles across the sample, breaking the symmetry in the final device cross-section. Because of such asymmetry, localized mechanical and optical cavity modes will inevitably couple to anti-symmetric guided modes, which exist in their respective quasi-bandgaps, bringing about potentially significant losses. To circumvent this, periodic sample rotation was implemented during angled-etching to average the effective etch angle across the substrate, with the goal of retaining a symmetric cross-section.
To investigate the symmetry of the fabricated diamond OMCs, we used a stamping technique to transfer angled-etch diamond nanobeams from their bulk diamond substrate, onto a smooth silver thin film supported by a silicon wafer. While this technique is ultimately destructive, it ensured simultaneous removal of many diamond OMCs, with most ending up on their backside to expose the angled-etched surfaces. High-resolution SEM images shown in Figure S5 (a) and (b), respectively, reveal diamond OMCs (oriented upside down) fabricated without and with sample rotation during angled-etching, with insets displaying a tilted cross-sectional view. Sample rotation appears to reduce the degree of asymmetry (defined as the offset in the bottom apex of the triangular cross-section from its centerline) considerably. However, even minimal asymmetry significantly reduces the simulated optical and mechanical Q-factors, as illustrated in Figure S5

vi) Optical and mechanical spectroscopy characterization setup
Characterization of fabricated diamond OMCs, performed under ambient conditions at room temperature, used a dimpled fiber taper (setup illustrated in Figure 3 (a) of the main text) to evanescently couple to the device under test (DUT). A tunable laser source (TLS, Santec telecom c-Band TSL-510, 1480-1580 nm tuning bandwidth) was used to locate the optical cavity resonance. A small percentage of the input laser sent to a wavelength meter (λ-meter, EXFO WA-1650) via a 99:1 coupler (BS) enabled a stabilized laser frequency position. For general optical and radio frequency (RF) mechanical spectroscopy performed at low input powers, the laser pump was sent directly to a variable optical attenuator (VOA, EXFO FA-3150), before coupling into the DUT. In the case of measurements which required greater input laser power, the laser pump was first amplified by a c-Band erbium-doped fiber amplifier (EDFA, Amonics AEDFA-27-B), then coupled to the VOA. This is indicated in Figure 3 (a) of the main text as switch point 1 (SW1). In addition, at SW1, the input laser was coupled into a fiber polarization controller (FPC) and electro-optic phase modulator (EOPM, EOSpace Inc.) for optomechanically induced transparency (OMIT) measurements.
After the VOA, laser light was first sent through a FPC to maximize coupling with the device under test, and then into the dimpled fiber taper. The dimpled fiber taper position was precisely controlled with respect to the DUT via motorized stages with 50 nm encoder resolution. Optical scans of the DUT were initially collected with the dimpled fiber taper hovering above the device, to evaluate the optical cavity parameters under weak fiber coupling. For final measurements, the dimpled fiber was placed in direct contact with the DUT, in a position such as to maximize coupling while minimizing parasitic losses due to dielectric loading of the cavity by the silica fiber taper. A 2 x 2 fiber switch (SW2) was used to control the direction of light through the device region, which allowed for precise calibration of insertion and bidirectional coupling losses. From this calibration, the input laser power directly coupled to the DUT was calculated, and thus also the intracavity photon number for a given laser detuning.
After passing the DUT, the transmitted laser signal was split (via a 90:10 coupler) between a low speed and high speed path, in order to collect the optical cavity spectrum and mechanical cavity spectrum, respectively. In the high speed path, transmitted laser light is optically amplified by a second EDFA, with any amplified spontaneous emission (ASE) removed by a band pass filter (JDS Uniphase TB9) centered on the optical cavity resonance wavelength, and then detected by a high-bandwidth photoreceiver (D1, New Focus 1554-B, 12 GHz bandwidth). The high-bandwidth detector was threshold power for the observation of optomechanical self-oscillations under optimal blue-detuning was estimated to be n c,thr ~ 7,600 (see Figure 5 (e) of the main text). Here, we display mechanical spectra of the diamond OMC acoustic swelling mode taken below, at, and above this phonon lasing threshold (shown in Figure S7). An increase of more than 60 dB increase in peak mechanical amplitude ( Figure S7 inset) is observed. Normalized power spectral densities (PSD) collected below, at, and above the phonon lasing threshold input power. The inset plots the peak PSD amplitude versus n c , with a ~ 64 dB increase in peak amplitude observed above threshold.
Additionally, OMIT was also observed for this diamond OMC acoustic swelling mode. Figure S8 displays a representative series of normalized OMIT spectra (|S 21 |/max{|S 21 |}), collected with the control laser detuned approximately Δ oc ~ [(ω m -440 MHz), ω m , (ω m + 500 MHz)] and an intracavity photon number of n c ~ 26,000. In these broadband OMIT spectra, we observe a series of clear dips in the spectra, representing transparency windows originating from the several mechanical resonances coupled to the optical cavity field. The largest dip corresponds to the transparency window attributed to the diamond acoustic swelling mode (right inset panels of Figure S8 display zoomed-in spectra of this fine feature). Fits to the normalized OMIT spectra, which followed the methodology reported in the previous section, estimate a cooperativity of C ~ 2.7 for data collected with optimal Δ oc ~ ω m detuning, in good agreement with the cooperativity value measured in Figure 5 (f) of the main text under similar input laser power. , ω m , (ω m -440 MHz)], plotted versus probe laser (ω p ) detuning (Δ pc ≡ (ω p -ω c )). Several fine transparency window features are observed in the broadband OMIT spectra, originating from the four mechanical resonances coupled to the optical cavity observed in this spectral range (see Figure 5 (a) of the main text). Right inset panels display zoomed-in OMIT spectra of the transparency window induced by coherent interaction of the mechanical and optical cavities. Fits to OMIT spectra (solid red and blue lines), estimate a cooperativity of C ~ 2.7 for data collected with Δ oc ~ ω m .