Optomechanically Induced Transparency and Cooling in Thermally


Jan 22, 2018 - (1, 4) Additionally, large C is important in other coherent optomechanical processes such as light pulse storage(34) and wavelength con...
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Optomechanically induced transparency and cooling in thermally stable diamond microcavities David P. Lake, Matthew Mitchell, Yasmeen Kamaliddin, and Paul E Barclay ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.7b01516 • Publication Date (Web): 22 Jan 2018 Downloaded from http://pubs.acs.org on January 23, 2018

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Optomechanically induced transparency and cooling in thermally stable diamond microcavities David P. Lake,†,‡,¶ Matthew Mitchell,†,‡,¶ Yasmeen Kamaliddin,† and Paul E. Barclay∗,†,‡ †Department of Physics and Astronomy and Institute for Quantum Science and Technology, University of Calgary, Calgary, AB, T2N 1N4, Canada ‡National Institute for Nanotechnology, Edmonton, AB, T6G 2M9, Canada ¶Contributed equally to this work E-mail: [email protected]

Abstract Diamond cavity optomechanical devices hold great promise for quantum technology based on coherent coupling between photons, phonons and spins. These devices benefit from the exceptional physical properties of diamond, including its low mechanical dissipation and optical absorption. However the nanoscale dimensions and mechanical isolation of these devices can make them susceptible to thermo-optic instability when operating at the high intracavity field strengths needed to realize coherent photon–phonon coupling. In this work, we overcome these effects through engineering of the device geometry, enabling operation with large photon numbers in a previously thermally unstable regime of red-detuning. We demonstrate optomechanically induced transparency with cooperativity > 1 and normal mode cooling from 300 K to 60 K, and predict that these device will enable coherent optomechanical manipulation of diamond spin systems.

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Keywords: optomechanics, diamond, optical resonator, nanophotonics, diamond Nanophotonic cavity optomechanical devices localize light within nanostructures supporting both optical and mechanical resonances, creating large optical forces that can coherently couple light to phonons of a mechanical mode. These devices provide a testbed for fundamental studies of quantum science, 1,2 with hallmark experiments demonstrating phenomena such as optomechanically induced transparency, 3–5 optomechanical cooling, 6 observation of a mechanical resonator’s zero point motion, 7 quantum optical–mechanical correlations, 8–12 and entanglement between mechanical resonators. 13 Within the realm of quantum technology, the ability of these systems to coherently interface GHz frequency phonons with optical photons has sparked efforts to create transducers 14 that optomechanically convert quantum information between photonic channels and solid state 15 or superconducting microwave 16–19 qubits via a shared mechanical coupling. Diamond cavity optomechanical devices 20,21 are poised to advance experiments in quantum optomechanics, in part thanks to device performance improvements from diamond’s best-in-class Young’s modulus, low intrinsic mechanical dissipation, high thermal conductivity, and large optical transparency window from ∼ 230 nm to far–IR. 22 In addition, these devices offer an interface between highly coherent diamond colour centre spins and optically controlled phonons via strain coupling, 15 which could enable quantum transducers between spins and quantum phononic and photonic states, 23–25 as well as platforms for entangling remote spins via nanomechanical coupling. 26,27 To date, nanomechanical devices used for spin manipulation have relied on piezo actuated phonon-strain coupling. 28–33 Incorporating coherent cavity optomechanics would enable optical control of phonon-spin interactions with sensitivity necessary for operation at the single phonon level and enable new photon-spin interfaces that are independent of the spin optical properties. A requirement for realizing the large-amplitude mechanical oscillations required to couple ex , where C ≡ 4N g02 /κo Γm > 1, to colour centres is the phonon lasing criterion, C > 1 − κoκ+κ ex

is the optomechanical cooperativity, N is the intracavity photon number, g0 is the single–

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photon optomechanical coupling rate, and κo and Γm are the optical cavity and mechanical resonator energy decay rates, and κex is the cavity external coupling rate. 1,4 Additionally, large C is important in other coherent optomechanical processes such as light pulse storage 34 and wavelength conversion, 5,35 where the spectral response and conversion efficiency are dependent on C, respectively. This should not be confused with the much more stringent √ case of quantum–coherent interactions which requires that 2g0 N > (κo , γ), where γ = Γm (nth + 1) is the mechanical decoherence rate and nth is the thermal equilibrium phonon occupation of the mechanical mode. 1,36 We have developed single–crystal diamond microdisks ex that achieve C > 1 − κoκ+κ by coupling optical whispering gallery modes to GHz frequency ex

(ωm ) mechanical radial breathing mode resonances. 20 The microdisk mechanical resonances have a large mechanical quality factor Qm ≡ ωm /Γm that is enhanced by restriction of phonon leakage into the substrate by a nanoscale pedestal, while the optical modes have optical quality factor Qo ≡ ωo /κo ∼ 105 , where ωo /2π ∼ 200 THz is the optical mode frequency, and can support N > 106 photons without suffering from nonlinear absorption that degrades device performance in other less transparent materials such as silicon. 37 However, in previous work, at high N the device performance becomes limited by linear optical absorption and accompanying heating. The resulting change in microdisk temperature is exacerbated by the ∼ 100 nm waist of the microdisk pedestal, in which thermal conductivity is reduced at room temperature due to size effects by approximately an order of magnitude compared to in bulk diamond, 20 resulting in thermo-optic instability 38 for red laser–cavity detunings needed for applications such as coherent phonon–photon coupling and optomechanical cooling. Here we overcome this limitation through modification of the microdisk pedestal shape to improve its thermal conductivity without affecting Qm , enabling demonstration of optomechanically induced transparency, a hallmark of coherent phononphoton coupling, as well as stable optomechanical cooling in diamond microdisks for the first time.

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Device characterization: optical, mechanical and thermal properties An example of the diamond microdisk devices studied here is shown in the scanning electron micrograph image in Fig. 1(a). These devices were fabricated by modifying the diamond undercutting process demonstrated in Ref., 20 as described in the Supporting Information, to shape the pedestal supporting the microdisk into the flared profile shown in Fig. 1(b). Note that the device in Fig. 1(b) was broken as result of being over–undercut, and its minimum pedestal dimension is smaller than that of the devices studies here, which are ∼ 350 nm. The optical and mechanical modes of the microdisks were characterized by monitoring the transmission of a tuneable diode laser (Newport TLB 6700B) through a dimpled optical fiber taper evanescently coupled to the microdisk. Scans of transmission for varying laser wavelength (λp ) when coupled to the 5 µm diameter microdisk considered in the remainder of this work revealed resonances such as those shown in Fig. 1(c). The doublet resonance structure indicates the presence of backscattering induced standing wave optical modes, which are either a symmetric or anti-symmetric combination of the degenerate travelling wave modes in the microdisk (see Supporting Information). The modes studied here have a central wavelength of λo ≡ 2πc/ωo ∼ 1550 nm, splitting of (s,a)

the backscattering rate, and intrinsic (unloaded) Qo (s,a)

where κo

λ2o κ 2πc bs (s,a)

= ωo /κo

∼ 70 pm, where κbs is

= 8.7 × 104 and 7.4 × 104 ,

are the optical energy decay rates of the symmetric and antisymmetric (red and

blue shifted) modes of the doublet, respectively. This places the system near the resolved (s,a)

sideband regime, with ωm /κo

∼ 1.0 and 0.84. Typically these devices are operated in the

under–coupled regime with κex /κo ∼ 0.58, where κex is the external energy coupling loss rate. The microdisk’s mechanical resonances were probed by fixing λp slightly off-resonance from the cavity modes and monitoring fluctuations in optical transmission due to mechanical motion using a high-speed photodetector (Newport 1554-B). Initial measurements were per-

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formed at low optical input power Pin to avoid modifying the mechanical mode dynamics via optomechanical back action. 1 The power spectral density (PSD) of this signal, as analyzed on a real time spectrum analyzer (Tektronix RSA5106A) and shown in Fig. 1(d), reveals the thermal motion of a mechanical resonance at ωm /2π ∼ 2.2 GHz, with a mechanical quality factor Qm = ωm /Γm ∼ 8, 400. Comparison to COMSOL finite element simulations of the microdisk suggest that this mechanical mode is the fundamental radial breathing mode (RBM) of the microdisk, as the measured ωm is within 5% of the simulated value. The RBM studied here has an effective mass predicted from simulation of meff ∼ 45 pg, corresponding p to a quantum zero point motion amplitude, xzpm ∼ 0.30 fm, where xzpm = ~/2meff ωm . Reaching the regime of coherent optomechanical coupling can in principle always be achieved by operating with high enough Pin to increase N so that C > 1. In practice, even in absence of nonlinear absorption, N is limited by linear absorption and thermooptic dispersion, particularly in small optical mode volume devices such as microdisks. The microdisk pedestal shape plays a critical role in determining whether C > 1 can be reached, as it influences Qm (∝ C) 20 as well as the device’s ability to conduct thermal energy away from the microdisk and mitigate optical heating. The importance of the pedestal’s thermal conductance can be seen by considering the threshold for N above which the microdisk becomes bistable due to the thermo-optic effect, 39,40 for λp red-detuned from λo as required for optomechanically induced transparency 3–5 and cooling: 6 

ηPin Qo ~ωo2



Cp ≡N < |β|τth ~ωo2



Qabs Qo

 (1)

(see Supporting Information). Here τth and Cp are the microdisk thermal time constant and the heat capacity, respectively, and β = dλo /dT is the microdisk thermo-optic coefficient that accounts for thermal expansion and refractive index temperature dependence. 38 The on-resonance fiber taper waveguide-microdisk power coupling efficiency is defined as η, and 1/Qabs is the contribution to 1/Qo from linear absorption. Equation (1) illustrates the inverse

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relationship between τth and maximum N . Note that in the anomalous case of β < 0, the cavity becomes bistable for blue instead of red detuning. To determine the impact of the pedestal shape on the properties of the microdisks, we compare the flared pedestal devices from this work with hourglass pedestal devices studied previously. 20 In Ref. 20 it was found that increasing pedestal width degraded Qm . However, the Qm of the flared pedestal device measured here is similar to the best valued reported for the hourglass microdisks, despite the larger width of the flared pedestals. However, as shown by measurements of τth in Fig. 1(e-f), the highest Qm flared and hourglass pedestal microdisks have τth ∼ 0.5 µs and 6 µs, respectively, indicating that the flared pedestal devices studied here can support over an order of magnitude larger N in the red-detuned regime compared to previous hourglass pedestal devices. Here τth was measured by monitoring the response of the microdisk to an optical pulse that causes λo to shift via the photothermal effect, as described in the Supporting Information.

Optomechanical spring effect The optomechanical parameters of the system were further probed by means of the optical spring effect. 1,41,42 For this measurement, Pin was increased via an erbium doped fiber amplifier (EDFA: Pritel LNHPFA-30) connected to the tunable laser output. The laser wavelength was then discretely stepped across the optical cavity resonances, and the PSD of the transmitted signal was acquired at each step. By fitting a Lorentzian lineshape to the 0 PSD, both ωm (∆; Pin ) = ωm + δωm (∆; Pin ) and Γm (∆; Pin ) = Γ0m + Γopt (∆; Pin ) as a function

of pump-cavity detuning, ∆ = ωp − ωo , were extracted. The results of this measurement for intermediate input power (Pin ∼ 4.7 mW) are shown in Figs. 2(a) and 2(b). Here the data is fit to analytic expressions for the predicted values of the optomechanical spring effect δωm and optomechanical damping Γopt , taking into account the doublet nature of the optical mode (see Supporting Information). Using measurements of N (∆) = N s (∆) + N a (∆) shown

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in Fig. 2(c) determined from the power dropped into the microdisk and κs,a we are able to extract the sole fitting parameter, g0 /2π ∼ 17 kHz. The optomechanical coupling rate is not expected to vary for each mode of the doublet, which was confirmed in our analysis by allowing g0 to vary for the symmetric and antisymmetric modes, which gave the same result as a single g0 value. In general, as the coupling rates to each of the modes will depend on the phase of fields compared to the point of fiber taper coupling, κext can differ for each mode. 43 However, for the modes considered in this work κext was approximately equal for each mode, and a single value was used. Unlike previous measurements of the optical spring effect in diamond microdisks, 20 δωm here was dominated by optomechanical back-action owing to the device’s reduced τth and low optical heating. The cavity optomechanical damping, Γopt , modifies the mechanical normal-mode temperature, Teff , as in experiments of ground state cooling, 6 or generation of self–oscillations that drive stress fields for coupling to diamond colour center spins. 20 The normal-mode temperature can be measured as a function of ∆ from the area under the PSD normalized by the wavelength dependent optomechanical transduction (see Supporting Information) and assuming that for large |∆| the cavity is in thermal equilibrium with the room-temperature environment (see Supporting Information). Figure 3(a) shows the normal mode temperature Teff and corresponding phonon occupation nm measured using this technique. Also shown is a prediction of Teff obtained by inputting the fit of Γopt (∆) from Fig. 2(b) to the optomechanical back-action cooling expression,

nm = nth ×

Γ0m , Γ0m + Γopt (∆)

(2)

where nm is the final phonon number, nth = kB T /~ωm is the equilibrium thermal phonon occupation. 1 This expression is valid in the high–T limit nth  nmin applicable here, where nmin = 0.088 is the minimum backaction limited phonon number achievable through optomechanical damping, which in the resolved sideband regime is given by nmin = (κ/4ωm )2 ,

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which holds for the device studied here. 1 The good agreement indicates that the ∆ dependent normalization of the PSD is accurate, and that optical absorption and heating is small compared with changes to nm from optomechanical backaction. At higher Pin , microdisk heating and modal thermo-optic dispersion become significant, and the transduction calibration could not be readily applied to measurements of PSD area for varying ∆. However, measurement of optomechanical cooling using Eq. (2) with ∆ optimized to maximize Γopt was possible: for N ∼ 1 × 106 (Pin ∼ 20 mW), Teff = 60 K (nm = 588 phonons) was measured for the symmetric mode, as shown in Fig. 3(b). Here Teff includes an increase in bath temperature (i.e. nth ) of 4 K due to optical heating, inferred from the shift in λo calibrated by its independently measured temperature dependence (see Supporting Information). This optimized cooling was obtained when red-detuned by ωm from the higher-Qo doublet mode, as expected for a sideband-resolved cavity optomechanical device. This detuning was not achievable in previous work with hourglass pedestal microdisks due to an inability to operate at red-detuning with Pin large enough to significantly reduce nm because of thermal instability. 20

Optomechanically induced transparency Optomechanically induced transparency (OMIT) is a signature of coherent coupling between optical and mechanical resonances, and has been demonstrated in cavity optomechanical systems such as microtoroids, 3 optomechanical crystals (OMC’s), 4 and microdisks. 5 OMIT occurs when a strong control field (ωc ) is red–detuned from the microdisk such that ∆oc = ωo − ωc = ωm , resulting in destructive interference between anti-Stokes photons scattered from the control field and a weak probe field (ωp ). This creates a transparency window (dip) in the probe transmission (reflection) spectrum when ∆pc = ωp − ωc = ωm (corresponding to ωp = ωo if the control field detuning condition is ideally satisfied) whose amplitude and width depends on the cooperativity. 1,3,4

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To characterize OMIT in the diamond microdisks, the laser output was amplified to Pin ∼ 40 mW, and its wavelength was slowly stepped across the cavity resonance, creating a control field with varying ∆oc . At each ∆oc , a phase electro–optic modulator (EOM) driven by a vector network analyzer (VNA:Keysight E5036A) was used to create a sideband on the control field that serves as the probe, and whose frequency can be swept accross the cavity resonance, varying ∆pc . The probe field reflected by the microdisk back into the fiber taper was measured using a high–bandwidth photoreceiver connected to an optical circulator, and analyzed by the VNA. The symmetric mode of the microdisk doublet (ωo −

κbs ) 2

was

used for all of the measurements described below. Figure 4(a) shows the results of these measurements for several ∆oc , with each exhibiting a sharp OMIT feature when ∆pc = ωm . Here R is the reflectivity normalized by its maximum value in absence of OMIT. When ∆oc is tuned away from ωm the OMIT feature exhibits a Fano shape due to the phase difference between the scattered control field and the probe field. At the OMIT condition ∆oc = ωm the dip amplitude reaches a maximum, as shown in detail in Fig. 4(b). From the dip amplitude ∼ 0.8 = 1 − 1/(1 + C)2 , 5 cooperativity C ∼ 1.2 was extracted. This C was achieved with an intracavity photon number of N ∼ 2.7 × 106 , and corresponds to g0 /2π = 18 kHz, in excellent agreement with the value predicted from the optomechanical spring effect fits in Figs. 3(a) and (b). Compared with previous demonstrations of OMIT in microdisks, 5 the diamond devices demonstrated here support nearly two orders of magnitude higher N and 2.6 times higher C, despite their lower Qo . Furthermore, this performance is realized without thermal stabilization or cryogenic cooling. This enables optomechanical cooling with large N in the red-detuned regime not accessible in previous studies. 20 Additionally, the microdisk geometry combined with the broadband transparency of diamond allows these devices to simultaneously support high-Qo optical modes spanning a broad wavelength range, for example at both the 637 nm range of diamond NV centre emission and in the 1550 nm telecommunications wavelength band. 20 The C > 1 OMIT shown here will allows these multiwavelength

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cavities to be used for optomechanical wavelength conversion, 5,35 and the broad microdisk mode spectrum will allow conversion over a larger range than diamond OMC. 21 Finally, any improvement to Qm , either through low-temperature operation 44 or engineering of the microdisk connection to the pedestal 45 would greatly increase the maximum achievable cooperativity.

Conclusion In summary we have demonstrated optomechanically induced transparency, cooling, and full characterization of the optical spring effect in single-crystal diamond microdisks. Access to the red–detuned sideband, and the associated aforementioned phenomena was enabled by improving the thermal stability of the microdisk by modify the pedestal geometry. Future work will seek to further enhance Qo and Qm by improving the fabrication process and investigating post-fabrication surface treatments. Furthermore, recent work has demonstrated that the diamond undercutting fabrication technique utilized here may be applied to other geometries such as photonic crystals, 46 which holds promise for the fabrication of future optomechanical devices. Finally, implantation of NV’s or SiV’s in these devices will allow the study of the interaction of solid state qubits with coherently driven cavity–optomechanics, as spin-photon coupling rates between the RBM and NV ground state are predicted to reach 0.6 MHz. 20

Supporting Information Description of modified fabrication process; Thermal response and properties of the cavity; Optical spring effect and transduction for an optical doublet.

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Funding This work was supported by NRC, CFI, iCORE/AITF, and NSERC.

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(27) Albrecht, A.; Retzker, A.; Jelezko, F.; Plenio, M. B. Coupling of nitrogen vacancy centres in nanodiamonds by means of phonons. New Journal of Physics 2013, 15, 083014. (28) MacQuarrie, E. R.; Gosavi, T. A.; Jungwirth, N. R.; Bhave, S. A.; Fuchs, G. D. Mechanical Spin Control of Nitrogen-Vacancy Centers in Diamond. Phys. Rev. Lett. 2013, 111, 227602. (29) Ovartchaiyapong, P.; Lee, K. W.; Myers, B. A.; Jayich, A. C. B. Dynamic strainmediated coupling of a single diamond spin to a mechanical resonator. Nat. Commun. 2014, 5, 4429. (30) Teissier, J.; Barfuss, A.; Appel, P.; Neu, E.; Maletinsky, P. Strain Coupling of a Nitrogen-Vacancy Center Spin to a Diamond Mechanical Oscillator. Phys. Rev. Lett. 2014, 113, 020503. (31) Meesala, S.; Sohn, Y.-I.; Atikian, H. A.; Kim, S.; Burek, M. J.; Choy, J. T.; Lonˇcar, M. Enhanced Strain Coupling of Nitrogen-Vacancy Spins to Nanoscale Diamond Cantilevers. Phys. Rev. Applied 2016, 5, 034010. (32) Ali Momenzadeh, S.; de Oliveira, F. F.; Neumann, P.; Bhaktavatsala Rao, D. D.; Denisenko, A.; Amjadi, M.; Chu, Z.; Yang, S.; Manson, N. B.; Doherty, M. W.; Wrachtrup, J. Thin Circular Diamond Membrane with Embedded Nitrogen-Vacancy Centers for Hybrid Spin-Mechanical Quantum Systems. Phys. Rev. Applied 2016, 6, 024026. (33) Golter, D. A.; Oo, T.; Amezcua, M.; Stewart, K. A.; Wang, H. Optomechanical Quantum Control of a Nitrogen-Vacancy Center in Diamond. Phys. Rev. Lett. 2016, 116, 143602. (34) Fiore, V.; Dong, C.; Kuzyk, M. C.; Wang, H. Optomechanical light storage in a silica microresonator. Phys. Rev. A 2013, 87, 023812. 15

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(35) Hill, J. T.; Safavi-Naeini, A. H.; Chan, J.; Painter, O. Coherent optical wavelength conversion via cavity optomechanics. Nat. Commun. 2012, 3, 1196. (36) Verhagen, E.; Del´eglise, S.; Weis, S.; Schliesser, A.; Kippenberg, T. J. Quantumcoherent coupling of a mechanical oscillator to an optical cavity mode. Nature 2012, 482, 63–67. (37) Barclay, P. E.; Srinivasan, K.; Painter, O. Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and a fiber taper. Opt. Express 2005, 13, 801–820. (38) Carmon, T.; Yang, L.; Vahala, K. J. Dynamical thermal behavior and thermal selfstability of microcavities. Opt. Express 2004, 12, 4742–4750. (39) Gibbs, H. M. Optical bistability: controlling light with light; Academic Press: Orlando, FL, 1985. (40) Almeida, V. R.; Lipson, M. Optical bistability on a silicon chip. Opt. Lett. 2004, 29, 2387–2389. (41) Braginsky, V.; Manukin, A. Ponderomotive effects of electromagnetic radiation. JETP 1967, 25, 653. (42) Braginsky, V.; Manukin, A.; Tikhonov, M. Investigation of dissipative ponderomotive effects of electromagnetic radiation. JETP 1970, 31, 829. (43) Borselli, M. High-Q Microresonators as Lasing Elements for Silicon Photonics. Ph.D. thesis, California Institute of Technology, 2006. (44) Khanaliloo, B.; Jayakumar, H.; Hryciw, A. C.; Lake, D. P.; Kaviani, H.; Barclay, P. E. Single-Crystal Diamond Nanobeam Waveguide Optomechanics. Phys. Rev. X 2015, 5, 041051.

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(45) Anetsberger, G.; Riviere, R.; Schliesser, A.; Arcizet, O.; Kippenberg, T. Ultra-Low Dissipation Optomechanical Resonators on a Chip. Nature Photon. 2008, 2, 627–633. (46) Mouradian, S.; Wan, N. H.; Schr¨oder, T.; Englund, D. Rectangular photonic crystal nanobeam cavities in bulk diamond. Appl. Phys. Lett. 2017, 111, 021103.

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ACS Photonics

(a)

(b)

5µm

(c)

(d)

PSD [dBm/Hz]

Transmission

s

Qo~87,000

a

Qo~75,000 1551.25

(e)

1.10

1.03 0

1551.35

1551.45

λ [nm]

τth ~ 0.5 μs

Data Fit 0.2

0.4

0.6

Time [s] x10

-5

0.8

-110

(f )

Qm~ 8,400

2.1995

0.79

Response [V]

0.4

3 µm

-96

1.0

Response [V]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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2.2005

2.2015

Frequency [GHz]

τth ~ 6 μs

0.72 1.0 0

Data Fit 1.0

2.0

Time [s] x10

-5

3.0

Figure 1: Characterization of single–crystal diamond microdisk structures fabricated using modified plasma undercutting method. (a) Scanning electron micrograph (SEM) of a “flared” microdisk structure. (b) Top down SEM view of the modified pedestal shape using the modified undercutting method described above. (c) Typical high–Qo doublet optical mode for the device studied here. (d) RBM of microdisk measured at low input power exhibiting a Qm ∼ 8, 400. (e,f) Response of the optical transmission for an input step function for microdisks with similar sized flared and hourglass shaped pedestals, respectively. The exponential rise is fit to extract the thermal time constant. The negatively sloped feature in (e) is an EDFA artifact, resultant from the large input powers required to achieve thermal bistability. 18

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Page 19 of 22

(a)

40

δωm/2π [kHz]

20 0 Measured

-20

Predicted

-40 -60 -5

-4

-3

-2

-1

0

1

2

3

4

5

Γopt /2π [kHz]

(b) 200

(c)

100

Measured Predicted

0 -100 -200 -5

Photons (×105)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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-4

-3

-2

-1

0

1

2

3

4

5

4 Total

3

Symmetric Antisymmetric

2 1 0 -5

-4

-3

-2

-1

0

∆ / ωm

1

2

3

4

5

Figure 2: Characterization of optomechanical coupling via the spring effect for the device shown in Fig. 1, for Pin ∼ 4.7 mW. Shift in (a) mechanical resonance frequency δωm and (b) optomechanical damping Γopt as a function of laser detuning. The fits are from the optomechanical spring effect calculation with a single–photon optomechanical coupling rate of g0 /2π ∼ 17 kHz as the sole free parameter, and the measured N shown in (c). Dashed lines indicate ∆ ± κbs /2 = 0 (corresponding to the resonance frequency of each doublet mode), and illustrate that δωm and Γopt = 0 when the laser is on-resonance with the cavity mode.

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(a)

4

0 80 60

0

Measured Predicted Thermal

0

6

40

Phonons (×103)

8

0 -5

20 0

2 -4

-2

-1

0

∆/ω m

1

2

3

4

5

0

PSD- [dBm/Hz] -

10

4

(b)

-3

Temperature [K]

10

8

300 K

11 2

60 K

16

-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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2.200

2.201 Frequency [GHz]

2.202

Figure 3: Optomechanical heating and cooling. (a) Measured phonon occupation and temperature of the RBM, and the corresponding values predicted from the fit to Γopt in Fig. 2(b). (b) PSD at the operating point ∆ ∼ −κbs /2 − ωm of maxiumum cooling (blue), and at ∆ tuned to the point of zero damping (red). Small peaks at intervals of 500 kHz are from technical noise.

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(a)

(b) 1.0

1.4

0.8

1.0

R

R

0.6

0.6

0.4

0.2 2.208

0.2

06

2 2.

02

]

z π [GH ∆ oc/2

2 2.

1

8 19

4

2

2.

6

5

3

94

∆ / 2.204 pc 2π [GH 2.20 2.196 z]

C ~ 1.2

1 2.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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∆pc/2π [GHz]

¯ as a function of Figure 4: Optomechanically induced transparency. (a) Probe reflection,R, ∆oc and ∆pc shown for OMIT using the symmetric mode of the doublet. (b) OMIT response corresponding to C ∼ 1.2 for N ∼ 2.7 × 106 , with ∆pc = ∆oc = ωm .

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Page 22 of 22 -ωm

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0

ωm