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1

Optomechanical ground-state cooling in a continuous

and efficient electro-optic transducer

B. M. Brubaker,1, 2, ∗ J. M. Kindem,1, 2, ∗ M. D. Urmey,1, 2, ∗ S. Mittal,1, 2

R. D. Delaney,1, 2 P. S. Burns,1, 2 M. R. Vissers,3 K. W. Lehnert,1, 2, 3 and C. A. Regal1, 2

1JILA, National Institute of Standards and Technology and the University of Colorado, Boulder, Colorado 80309, USA2Department of Physics, University of Colorado, Boulder, Colorado 80309, USA3National Institute of Standards and Technology, Boulder, Colorado 80305, USA

(Dated: December 28, 2021)

The demonstration of a quantum link between microwave and optical frequencies would be an im-portant step towards the realization of a quantum network of superconducting processors. A majorimpediment to quantum electro-optic transduction in all platforms explored to date is noise added bythermal occupation of modes involved in the transduction process, and it has proved difficult to re-alize low thermal occupancy concurrently with other desirable features like high duty cycle and highefficiency. In this work, we present an efficient and continuously operating electro-optomechanicaltransducer whose mechanical mode has been optically sideband-cooled to its quantum ground state.The transducer achieves a maximum efficiency of 47% and minimum input-referred added noiseof 3.2 photons in upconversion. Moreover, the thermal occupancy of the transducer’s microwavemode is minimally affected by continuous laser illumination with power more than two orders ofmagnitude greater than that required for optomechanical ground-state cooling.

I. INTRODUCTION

The past two decades have seen remarkable progressin the control of increasingly complex quantum systems,towards the ultimate goal of realizing a general-purposequantum computer [1]. Superconducting electrical cir-cuits operating at microwave frequencies and tempera-tures below 100 mK have emerged as a leading platformfor quantum computing [2, 3], but microwave photonshave energies much smaller than thermal energy at am-bient temperature, and thus cannot be used to mediatethe exchange of quantum information between distantquantum processors. Optical photons, conversely, areuniquely well-suited for the transmission of quantum in-formation over long distances at room temperature [4, 5].A quantum state-preserving interface linking microwaveand optical frequencies, or quantum transducer, is thusa critical element in a future quantum internet [6].Of the various figures of merit for quantum transduc-

tion [7], perhaps the most important is the input-referredadded noise Nadd = Nout/ηt, where Nout is the noisemeasured at the transducer output in photon units and ηtis the transducer efficiency. The utility of Nadd as a per-formance metric stems from the fact that it can be com-pared directly to the signal to be transduced. Althoughhigh efficiency is not strictly a requirement for all trans-duction protocols [7], achieving Nadd . 1 photon requireshigh efficiency insofar as Nout cannot be made extremelysmall. Thermal occupancy of any mode involved in thetransduction process will contribute to Nadd, and thus anideal transducer should operate with every participatingmode in its quantum ground state.

∗ These authors contributed equally; direct correspondence to

maxwell.urmey@colorado.edu

In all systems that have been explored as potentialplatforms for quantum electro-optic transduction [8, 9],ground-state operation is made challenging by the pres-ence of laser light required to bridge the energy gapbetween microwave and optical photons. Laser illumi-nation can lead to bulk heating of various materialsused in transducers [10, 11], and the superconductingmicrowave circuits integral to most transducer architec-tures are especially susceptible. High-energy optical pho-tons impinging on the superconductor will break Cooperpairs, increasing the thermal occupancy of the microwavemode and adding noise to the upconverted quantum state[10, 12].One approach to mitigating the detrimental effects of

laser illumination is to pulse transducer operation, at thecost of a reduced duty cycle. Using this method, severalrecent experiments have demonstrated ground-state op-eration [11, 13–15] and even added noise Nadd,up < 1in upconversion [10, 16]. However, achieving this perfor-mance has generally required low duty cycles < 10−3 andlow efficiency ηt < 10−5, with two exceptions with betterperformance in only one of these metrics [11, 16]. Dutycycle and efficiency are both important figures of meritfor certain applications, such as the generation of entan-gled microwave/optical photon pairs [17–19], but thusfar continuous operation with high efficiency has beendemonstrated only in an electro-optomechanical platformthat used a low-frequency mechanical mode to mediatetransduction [20]. This transducer suffered from multi-ple sources of added noise including the residual thermaloccupancy of the mechanical mode.In this work, we demonstrate continuous and effi-

cient electro-optic transduction mediated by a mechani-cal mode that has been optomechanically sideband cooledto its quantum ground state, reaching a minimum occu-pancy of 0.34 phonons. We operate this transducer inupconversion, and achieve input-referred added noise as

2

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row

ave

pu

mp

upconversiond

ow

nco

nve

rsio

n

(b) (c)

(a)

ΓoΓe

ωe ωo frequency

ωm ωm

optical pump

FIG. 1. Transducer design and operation. (a) Illus-tration of transducer (not to scale). A vibrational mode ofa silicon nitride membrane (purple) couples to a microwave-frequency inductor-capacitor circuit (dark blue) and a Fabry-Perot optical cavity mode (red) confined between two high-reflectivity mirrors (light blue). The membrane, microwavecircuit, and one of the cavity mirrors are integrated into astack of three silicon chips. Microwave and optical pumps(blue and red arrows) enhance the electromechanical and op-tomechanical interactions, respectively, and allow upconver-sion of a microwave signal (green arrow) or downconversion ofan optical signal (orange arrow). The inset shows the mem-brane (false-colored purple) suspended on a phononically en-gineered silicon substrate (gray). (b) Transducer frequencyscales. The microwave and optical pumps are red-detuned bythe mechanical frequency ωm from microwave and optical res-onances ωe and ωo, respectively. The electromechanical andoptomechanical damping rates Γe and Γo scale with the corre-sponding pump powers. (c) Simplified diagram in which thetransducer is represented as a two-port network.

low as Nadd,up = 3.2 photons and a maximum efficiencyηt = 47%. The residual added noise is dominated byfluctuations introduced by the strong microwave pumpthat couples the mechanical mode to a superconductingmicrowave circuit, and could be reduced by increasingthe vacuum electromechanical coupling rate. Notably,however, laser light near the superconducting circuit isnot a significant contribution to the added noise, evenwith power more than two orders of magnitude greaterthan that required for optomechanical ground-state cool-ing. Our results lay the groundwork for quantum trans-duction and continuous entanglement generation usinga membrane-based electro-optic interface with improvedmicrowave circuit performance.

II. TRANSDUCER DESIGN AND OPERATION

The transducer, illustrated in Fig. 1(a), consists ofmicrowave and optical resonators dispersively coupled

to a single vibrational mode of a silicon nitride mem-brane suspended on a silicon chip [20, 21]. The sili-con chip substrate is patterned into a phononic shield([22, 23], inset in Fig. 1(a)) with a bandgap centered onthe resonant frequency of the membrane mode of interest,ωm/2π = 1.45 MHz. Our phononic shield has relativelyfew unit cells to enable integration into the transducer,but it nonetheless reduces acoustic radiation from themembrane into the substrate. The improved isolationresults in a lower density of substrate modes near themembrane mode used for transduction and a low intrin-sic energy dissipation rate γm/2π = 113 mHz.The microwave resonator is a superconducting niobium

titanium nitride (NbTiN) inductor-capacitor (LC) circuitin a flip-chip configuration, whose capacitance is modu-lated by the motion of the membrane [21]. The circuit hasresonant frequency ωe/2π = 7.938 GHz and is coupled toa transmission line at rate κe,ext/2π = 1.42 MHz. Itstotal linewidth κe increases from 1.64 MHz to 2.31 MHzas the microwave power incident on the circuit increases.The optical resonator is a single-sided Fabry-Perot cav-

ity (resonant frequency ωo/2π = 277 THz) formed by acurved mirror on a superpolished fused-silica substrateopposite a Bragg mirror stack deposited on a silicon chipbonded to the flip-chip assembly. This cavity architectureensures robust alignment of the membrane to the cavityaxis, suppressing scattering losses. As a result, the to-tal cavity linewidth κo/2π = 2.68 MHz is dominated byits external coupling κo,ext/2π = 2.12 MHz through thecurved mirror.The transducer is operated in an optical-access dilu-

tion refrigerator with a 40 mK base plate temperature.Device design and fabrication details are provided in Ap-pendix B, and device characterization is discussed furtherin Appendix D.To operate the transducer, we apply microwave and

optical pumps red-detuned by ωm from the respectiveresonators (Fig. 1(b)). The microwave pump generatesan electromechanical beamsplitter interaction whereinphonon excitations of the membrane mode can decayinto propagating microwave photons at a rate Γe pro-portional to the incident microwave pump power Pe, andthe optical pump generates an analogous optomechanicaldamping rate Γo proportional to the incident power Po

(see Ref. [24] and Appendix C). These beamsplitter in-teractions enable bidirectional transduction between mi-crowave and optical frequencies over a bandwidth givenby ΓT = Γe + Γo + γm [21]. The transmission efficiencyon resonance between the external ports of the microwaveand optical resonators (Fig. 1(c)) is given by

ηt = ηM4ΓeΓo

Γ2T

, (1)

where the matched efficiency ηM = ǫκo,ext

κo

κe,ext

κeis the

maximum efficiency achievable in the presence of imper-fect optical cavity modematching ǫ and loss in the electro-magnetic resonators. The transducer achieves this max-imum efficiency when Γe = Γo ≫ γm.

3

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out,e N~

out,o

Nout,e,m Nout,o,m

ηt

Γe /2π = 100 Hz e

Γe /2π = 100 Hz e

(a) (c)

(d)

(e)

(b)

Γo nm

(ph

on

on

s)

nm

(ph

on

on

s)

γmγm

Γo / 2π (Hz)

Γe << Γo

Nadd,up (

ph

oto

ns/s

/Hz)

ΓoΓeγmγm

nm

nm

(Γe + Γo) / 2π (Hz)

(Γe + Γo) / 2π (Hz)

FIG. 2. Ground-state cooling of electro-optomechanical transducer. (a) Membrane mode occupancy nm vs. optome-chanical damping rate Γo with electromechanical damping Γe = 0. Points are data, solid gray line is a fit, and dotted gray lineshows the predicted behavior in the absence of technical noise. Orange dashed line indicates nm = 1, and yellow dashed lineindicates the backaction limit (increase at low Γo due to backaction from auxiliary optical cavity lock beam). Inset illustratesoptomechanical sideband cooling, where external blurred discs represent baths (with size indicating occupancy) to which themembrane mode (purple circle) is coupled by the corresponding damping rates (arrows, with width indicating magnitude).The membrane mode occupancy (purple blurred disk) is determined by the weighted average of bath occupancies. (b) nm

vs. Γe + Γo, sweeping Γo at fixed electromechanical damping Γe/2π = 100 Hz, with curves color-coded as in panel (a). Insetillustrates relative bath occupancies and damping rates for electro-optomechanical sideband cooling in the gray shaded regionwhere Γe ≪ Γo. (c) Transducer efficiency ηt vs. Γe+Γo, sweeping Γo as in (b), with fit. Orange dashed line marks the dampingat which nm = 1. (d) Upconversion added noise Nadd,up vs. Γe +Γo, with theory curve. (e) Schematic representation of trans-ducer output noise in the Γe ≪ Γo regime shaded gray in panels (b), (c), and (d). The box represents the transducer, and wavyarrows represent contributions to the microwave and optical output noise around mechanical resonance, from membrane modeoccupancy (purple) and technical noise (red/blue), with thickness indicating noise density. Photodetector indicates opticalreadout. All error bars represent one standard deviation.

In thermal equilibrium at temperature Teq, the mem-brane mode is coupled to a bath with occupancy nth =kBTeq/~ωm by its intrinsic decay rate γm. Optomechani-cal damping couples the membrane mode to a bath withoccupancy no = nmin,o + neff,o ≪ nth, where the firstterm represents the quantum backaction limit [25] fromimperfect sideband resolution and the second term is aneffective optical mode thermal occupancy due to pumppower-dependent technical noise (see Appendix C). Anal-ogous effects contribute to the electromechanical bath oc-cupancy ne. The membrane mode occupancy nm is givenby the weighted average of the baths to which it is cou-pled,

nm =γmnth + Γene + Γono

ΓT

, (2)

leading to sideband cooling of the membrane mode(Fig. 2(a) and (b), insets).

The optomechanical interaction enables readout ofmembrane motion in addition to sideband cooling andtransduction. Random motion of the membrane is im-printed on the reflected optical pump in the form of am-plitude and phase fluctuations. These fluctuations man-ifest as a pair of sidebands with Lorentzian profiles cen-tered at detunings ±ωm in the noise spectral density atthe transducer’s optical output port (see Appendix C),which we measure using balanced heterodyne detection.We can then infer the membrane mode occupancy nm

from the ratio of upper and lower sideband amplitudes[25, 26] and measure the transducer’s added noise inupconversion. The analogous electromechanical readoutcan also be used to measure nm as well as added noise indownconversion, albeit with lower signal-to-noise ratio.

Although the transducer is inherently bidirec-tional [21], upconversion is likely to be more useful in

4

practice, because superconducting processors enable thesynthesis of arbitrary quantum states in the microwavedomain [27], and heralded entanglement of remote quan-tum processors can be achieved using upconversion andoptical detection only [28]. In this work we measure thetransducer’s added noise in upconversion only, and alsouse optomechanical readout for all measurements of themembrane mode occupancy.

III. OPTOMECHANICAL GROUND-STATE

COOLING

We first demonstrate optomechanical ground-statecooling of the membrane mode with the microwave pumpabsent (Fig. 2(a)), to show that our transducer realizesthe successes of past membrane optomechanical systems[25, 29] even with the additional design complexity re-quired for electromechanical coupling to superconductingcircuitry. We measure the optomechanical sideband ratiowhile varying the damping Γo, and correct for squashingeffects due to laser phase noise (see Refs. [30, 31] andAppendices C and E) to obtain the membrane mode oc-cupancy nm at each value of Γo. We then fit this datato Eq. (2) with Γe = 0 and the equilibrium occupancynth as a fit parameter. We also introduce an additionalfit parameter ao = neff,o/Γo to model the optical modeoccupancy due to phase noise, and include a fixed termto account for the effects of a weak near-resonant beamused to lock the optical cavity (see Appendix A).The fit (solid gray line) yields nth = 1000± 90, imply-

ing that the membrane mode equilibrates to Teq = 70 mKin the presence of laser light, consistent with an inde-pendent calibration based on sweeping the temperatureof the cryostat base plate (see Appendix E). Phase noiseparameterized by ao = (2.8±1.6)×10−6 Hz−1 causes thefit to deviate slightly at high damping from the behav-ior expected from thermal noise with the best-fit valueof nth and imperfect sideband resolution alone (dottedgray line). The good agreement with the fit, without ad-ditional power-dependent terms, indicates that the laser-induced heating of the membrane mode responsible for itselevated equilibrium temperature Teq saturates quicklyat very low power and does not preclude cooling be-low nm = 1 (orange dashed line) for Γo/2π > 190 Hz.With large Γo, we reach a minimum mode occupancy ofnm = 0.32, within a factor of 1.5 of the backaction limit(yellow dashed line).To enable transduction, we now introduce a microwave

pump with fixed electromechanical damping Γe/2π =100 Hz, and again sweep Γo in order to ground-state coolthe membrane mode (Fig. 2(b)). We observe that nm

deviates further from the expected behavior from ther-mal noise and imperfect sideband resolution alone (dot-ted gray line) than in the purely optomechanical case,indicating an additional source of technical noise. Fix-ing nth and ao at the best-fit values from the pure op-tomechanical ground-state cooling data, we find good

agreement with Eq. (2) with effective thermal occupancyneff,e = 0.8± 0.1, consistent with neff,e = 0.77± 0.01 ob-tained from an independent heterodyne measurement ofthe noise in the transducer’s microwave output spectrum(see Appendix F). This good agreement, with neff,e in-dependent of laser power, indicates that the additionalsource of technical noise is coupled in via the microwavepump. With sufficiently high Γo, the membrane modecan again be cooled close to the backaction limit, reach-ing a minimum occupancy of nm = 0.34.

IV. TRANSDUCER CHARACTERIZATION

We next consider how the electro-optomechanical de-vice performs as a transducer when the optomechani-cal damping Γo is varied at constant electromechanicaldamping Γe, as in Fig. 2(b). We measure the transducerefficiency ηt using the method developed in Ref. [21], andfit the data to Eq. (1) with the matched efficiency ηM asthe only fit parameter (Fig. 2(c)). The fit yields ηM,fit =43 ± 1%, in tension with ηM,exp = ǫ

κo,ext

κo

κe,ext

κe= 55%

expected from independent measurements of the opticalcavity modematching ǫ and the overcoupling fractionsκext/κ of the two electromagnetic resonators. We sus-pect the measured values of ηt were artificially suppressed(see Appendix E) and that the true matched efficiencyduring these measurements was ηM,exp rather than ηM,fit.Nonetheless, adopting the fit value as a conservative mea-sure of transducer performance, we find ηt = 38% at thedamping for which the membrane mode occupancy is re-duced below nm = 1.

We can infer the transducer’s input-referred upconver-sion added noise Nadd,up from the above measurementsof the membrane mode occupancy nm and the trans-ducer efficiency ηt. We also take into account an ad-ditional contribution to Nadd,up from white noise Nout,o

measured at the transducer’s optical output port, whicharises from the technical noise responsible for the effec-tive optical mode occupancy neff,o (see Appendix C),and plot the results in Fig. 2(d). To understand theobserved increase in added noise when Γo ≫ Γe, it ishelpful to consider a schematic representation (Fig. 2(e))of the noise emerging from the transducer in this regime,shaded gray in Figs. 2(b), (c), and (d). When the op-tomechanical damping Γo dominates, almost all the noisearising from thermal occupancy of the membrane modeis routed to the transducer’s optical output port. Thiscontribution Nout,o,m to the optical output noise densityasymptotes at high Γo. But to obtain the input-referredadded noise Nadd,up we must divide by the transducerefficiency, which decreases with increasing Γo, resultingin increased added noise. The fact that Nout,o increaseswith increasing Γo further exacerbates the added noise.

More precisely, the transducer’s input-referred upcon-version added noise is given in units of photons/s/Hz (or

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(a)

(c)

Γo /2π = 85 Hz Γo /2π = 85 Hz

N~

out,e N~

out,o

Nout,e,m Nout,o,m

Γe ≈ Γo

Nad

d,u

p (

ph

oto

ns/s

/Hz)

ηt

(Γe + Γo) / 2π (Hz)

FIG. 3. Minimizing input-referred added noise. (a)Transducer efficiency ηt vs. Γe +Γo, sweeping electromechan-ical damping Γe at fixed optomechanical damping Γo/2π =85 Hz, with fit. (b) Upconversion added noise Nadd,up vs.Γe + Γo at Γo/2π = 85 Hz, with fit. (c) Schematic repre-sentation of transducer output noise in the Γe ≈ Γo regimeshaded yellow in panels (a) and (b), where the box repre-sents the transducer, wavy arrows represent contributions tothe microwave and optical output noise near mechanical reso-nance, and the photodetector indicates optical readout, as inFig. 2(e). Error bars representing one standard deviation aresmaller than the size of the points.

more simply photons) by

Nadd,up =Nout,o,m + Nout,o

AeAo ηt

=γmnth + Γene + Γono

Aeκe,ext

κeΓe

+Nout,o

AeAo ηt, (3)

where Ae = 1 + nmin,e is the transducer gain from im-perfect electromechanical sideband resolution, and Ao isdefined analogously. The electromechanical and optome-chanical bath occupancies ne and no depend implicitlyon the damping rates Γe and Γo, respectively, because oftechnical noise. In the absence of technical noise in theoptomechanical system, no → nmin,o and the last termvanishes. Then it is clear that increasing the optome-chanical damping Γo can only ever increase Nadd,up, andthat quantum-enabled upconversion would require elec-tromechanical ground-state cooling.As the added noise is lowest at small Γo in Fig. 2(d),

we now fix Γo/2π = 85 Hz and vary Γe to minimizeNadd,up. We begin by measuring transducer efficiencyas a function of Γe (Fig. 3(a)). We observe that themicrowave mode linewidth κe increases with increasing

microwave pump power Pe, an effect likely related to theelevated thermal occupancy neff,e observed in the electro-optomechanical experiment in Fig. 2(b). We account forthis power-dependence in fitting the efficiency measure-ments (see Appendix E), and obtain a peak efficiencyηt,max = 47 ± 1% at Γe/2π = 75 Hz, close to the ex-pected vale of 49%.

We then measure noise in the transducer’s opticaloutput spectrum while sweeping the electromechanicaldamping Γe over this same range (Fig 3(b)), and combinethe results with the efficiency measurements to obtainNadd,up as a function of electromechanical damping. Fit-ting the data, we find that at Γe/2π = 135 Hz the addednoise reaches a minimum value ofNadd,up = 3.2±0.1 pho-tons/s/Hz, an order of magnitude improvement relativeto previously reported measurements on a similar de-vice [20]. In this fit, we parameterize the effective mi-crowave mode occupancy as neff,e = aeΓe + be and againincluding an additional fixed term to account for the op-tical cavity lock beam (see Appendix E). The fit yieldsnth = 980±30 and ae and be consistent with independentmicrowave noise measurements (see Appendix F).

The dominant contributions to the added noise at theminimum are 1.0 photons/s/Hz from residual thermalmotion of the membrane and 1.4 photons/s/Hz from theeffective occupancy of the microwave mode, with sev-eral smaller sources responsible the remaining 0.8 pho-tons/s/Hz (see Appendix E). Fig. 3(c) shows a schematicrepresentation of transducer noise around the added noiseminimum, where Γe ≈ Γo (yellow shaded region inFigs. 3(a) and (b)). The noise arising from membranemode thermal occupancy is divided roughly equally be-tween the two transducer ports, but the mode is not in itsquantum ground state due to the smaller total damping:nm = 1.5 phonons when Nadd,up is minimized. Eq. (3) in-dicates that in the absence of technical noise, the addednoise would continue to decrease with increasing elec-tromechancial damping even in the Γe ≫ Γo regimewhere the efficiency is small. In our experiment, thepower-dependent effective microwave mode occupancyneff,e causes nm to increase at large Γe, resulting in anoptimum value of Γe at which Nadd,up is minimized.

Finally, we consider how the transducer’s downconver-sion added noise Nadd,down (not measured in this work)would scale with microwave and optical pump powers.Added noise is in general not the same in downconversionas in upconversion. In the absence of microwave techni-cal noise, optomechanical ground-state cooling would bea sufficient condition for quantum-enabled downconver-sion, independent of the strength of the microwave pump.However, the technical noise responsible for the effectivemicrowave mode thermal occupancy neff,e also generates

a white noise contribution Nout,e to the microwave noisedensity at the external port of the microwave circuit. Asindicated in Figs. 2(e) and 3(c), this excess noise is signif-icantly larger than the analogous optical technical noiseNout,o, and it precludes quantum-enabled downconver-sion with this transducer.

6

7.93 7.940

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0.3 mW

74 mW

1 10 100

0.01

0.1

(b)

(a)

nef

f,e

(ph

oto

ns)

Γo /2π = 3 kHz Γo /2π = 3 kHz

nm = 1

Circulating power (mW)

Frequency (GHz)

2

FIG. 4. Effects of laser illumination on microwave cir-

cuit. (a) Microwave circuit power reflection measurementswith 0.3 mW of circulating lock light (gray) and with 74 mWof circulating pump light (black). This pump power is a fac-tor of 10 greater than that required for the highest damp-ing rate Γo shown in Fig. 2, and two orders of magnitudegreater than that required for optomechanical ground-statecooling. (b) Laser-induced microwave circuit noise vs. circu-lating power in the optical cavity. The noise remains negligi-ble up to very large circulating power.

V. EFFECTS OF LASER ILLUMINATION

As indicated by the data presented in Figs. 2 and3, the transducer performance is chiefly limited by thepump power-dependent effective thermal occupancy neff,e

of the superconducting microwave circuit. The continu-ously applied optical pump, however, had no discernibleeffect on the microwave circuit during the experimentsdescribed above, in striking contrast to other platformsin which pulsed operation is required to avoid microwavecircuit heating [10, 14–16]. Indeed, laser illumination hasvery little effect on the microwave circuit even with muchlarger optical pump power. The gray curve in Fig. 4(a)

shows the circuit’s power reflection coefficient |See(ω)|2when the only light in the optical cavity is 0.3 mW of cir-culating power from the lock beam, while the black curveshows |See(ω)|2 in the presence of 74± 8 mW of circulat-ing pump power. The laser-induced frequency shift andincrease in internal loss are barely perceptible.

By measuring the noise emerging from the externalport of the superconducting circuit over a bandwidthmuch broader than ΓT, we can directly probe laser-induced heating of the microwave mode as a function ofthe circulating power in the optical cavity (Fig. 4(b)),with no microwave pump present. In the transducer

ground-state cooling measurements shown in Fig. 2(b),the maximum optomechanical damping of 3 kHz was ob-tained with 7 mW of circulating power, indicating thatdamping (and thus transducer bandwidth) could be fur-ther increased by more than an order of magnitude whilemaintaining neff,e < 0.15. This insensitivity of the su-perconducting circuit to optical illumination is a conse-quence of our modular transducer design, and has provenadvantageous in work integrating superconducting qubitswith electro-optic transducers [32].

VI. CONCLUSION

The device improvements necessary for achiev-ing quantum-enabled operation of a membrane-basedelectro-optomechanical transducer are readily attainable.In particular, the effects of the noise introduced by themicrowave pump were exacerbated by an atypically smallvacuum electromechanical coupling rate ge = 2π×1.6 Hzfor the transducer used in this work, arising from modi-fications to the flip-chip fabrication procedure to accom-modate the phononic shield (see Appendix D). Simply re-producing couplings realized in other devices [33] wouldreduce the microwave pump power required to obtain agiven electromechanical damping Γe by more than an or-der of magnitude.

With this improvement alone, we expect Nadd,up < 1for transduction with Γe = Γo ≈ 2π × 1 kHz. Thetransducer bandwidth ΓT/2π ≈ 2 kHz would be smallcompared to the bandwidth potentially available in inte-grated transducers, but nonetheless compatible with su-perconducting qubit lifetimes [34]. Moreover, the trans-ducer would operate continuously, requiring no deadtime to recover from laser illumination, and reductionof microwave pump power-dependent noise would enablebandwidth to be further increased by at least an order ofmagnitude. Integrating such a transducer with an opticalmeasurement setup that enables the detection of singleMHz-frequency phonons [35] would put the heralded up-conversion of non-Gaussian quantum states within reach.

Our electro-optomechanical device also has other ap-plications in which its high efficiency and continuous op-eration would compensate for its low bandwidth. For in-stance, with reduced technical noise such a device couldbe operated as a source of entangled microwave/opticalphoton pairs [17–19], in which case an important figureof merit is the rate at which entanglement is generated,proportional to the product of bandwidth, efficiency, andduty cycle. By this metric a device with 2 kHz bandwidthand 50% efficiency operated continuously would performas well as a device with 20 MHz bandwidth, 10% effi-ciency and a duty cycle of 6× 10−4 [16].

7

ACKNOWLEDGEMENTS

We thank Hans Green, Kim Hagen, Jim Beall andKat Cicak for technical and fabrication assistance, LucaTalamo and Kazemi Adachi for helpful discussions, andJosh Combes, Tasshi Dennis, and Akira Kyle for com-ments on the manuscript. This work was supported byfunding from ARO CQTS Grant No. 67C1098620, NSFGrant No. PHYS 1734006, and AFOSR MURI Grant No.FA9550-15-1-0015.

Appendix A: Experimental setup

The transducer is operated in a CryoConcept Hori-zontal 200 dilution refrigerator with free-space opticalaccess and 4He precooling to minimize vibrations (pre-viously used in Ref. [20]). Optical access to the de-vice is provided by fused silica windows that filter room-temperature thermal radiation to avoid heating of thecryostat base plate [36], and the cryostat reaches a basetemperature of Tbp = 40 mK. A back port enables cav-ity transmission measurements for beam alignment andmembrane imaging. The transducer device shares spaceon the cryostat base plate with a circuit QED moduleused in Ref. [32] but not in this work.The overall experimental setup is shown in Fig. 5. The

microwave pump, sourced by an a signal generator withultralow phase noise (Rohde & Schwartz SMA100B withoption SMAB-B711), is injected into the cryostat alongwith a phase-coherent cancellation tone. The amplitudeand phase of the cancellation tone are adjusted to mini-mize the power routed towards the microwave measure-ment chain, to prevent saturation of the cryogenic pream-plifier (a LNF LNC4 8C HEMT) by the high-power mi-crowave pump. Demodulation of microwave signals isperformed by an effective heterodyne detector compris-ing a homodyne measurement followed by AC-coupledamplification and modulation of the baseband signal upto 10 MHz.The most significant change relative to the optical con-

trol and measurement system of Ref. [20] is the use of awidely tunable external cavity diode laser (Toptica Pho-tonics CTL) in place of an Innolight Mephisto Nd:YAGlaser to source the optical pump. Wavelength tuning isnecessary to compensate for the absence of piezoelectricactuators to adjust the length of the optomechanical cav-ity. We operate the laser at a wavelength of 1084.4 nm,where there is a local maximum in the transducer’s vac-uum optomechanical coupling go (see Appendix B2).The CTL is placed in an acoustically isolated enclosureto reduce its sensitivity to environmental disturbances,and locked to a filter cavity with 40 kHz linewidth [37]via feedback to the diode current and piezo actuators onthe laser cavity (not shown in Fig. 5) using the Pound-Drever-Hall (PDH) technique. Transmission through thefilter cavity reduces the laser phase noise at detuningωm from the carrier by 37 dB. We use a single-frequency

fiber amplifier (Nufern Nuamp NUA-1064-PB-0005-C1)to amplify the CTL output beam before the filter cavity.The beam transmitted through the filter cavity is then

frequency-shifted by a double-pass acousto-optic modu-lator (AOM) and split three ways: one beam (red inFig. 5) provides the optical pump to operate the trans-ducer, a second (yellow) is used to lock the frequencyof the optomechanical cavity to that of the laser, anda third (maroon) provides a local oscillator (LO) forbalanced heterodyne detection. The relative detuningsof the three beams are controlled by three additionalAOMs. The incident pump beam and lock beam are or-thogonally polarized to route the beams emerging fromthe cryostat to separate detectors. The detuning ofthe pump beam from the cavity mode is thus given by∆o = (ωpump − ωlock +∆lock) ± ∆B, where ∆lock is thedetuning of the locking beam from the cavity mode, ∆B isthe cavity birefringence, and the sign depends on whetherthe lock beam addresses the higher- or lower-frequencycavity mode. The LO beam is detuned from the pumpbeam by (ωpump − ωLO) /2π = 12.9 MHz.The frequency of the incident lock beam is locked to

a TEM00 mode of the optical cavity using PDH feed-back. An electro-optic modulator (EOM) imprints phasemodulation sidebands on the lock beam, and the PDHerror signal obtained by demodulating the detected lockbeam at the sideband frequency is used to apply feedbackin parallel to the double-pass AOM and a piezoelectricactuator on one of the filter cavity mirrors. The AOMchannel has high bandwidth but limited range, while theslow feedback to the filter cavity length eliminates drift ofthe laser frequency relative to the frequency of the cavitymode. The incident lock beam power is Plock = 20 nW,and the PDH error signal bias is adjusted to keep the lockbeam slightly red-detuned from the cavity, with ∆lock/2πranging from −30 kHz to −80 kHz, to avoid optomechan-ical instability.A Zurich Instruments HF2LI lock-in amplifier is used

for all data acquisition. To characterize the transducerand measure its efficiency, we use the HF2LI to syn-thesize swept MHz-frequency tones for single-sideband(SSB) modulation of the transducer pumps and demod-ulate the heterodyne detector output signals to recoverthese tones (see Appendix D3). For noise measurements,we use the HF2LI to digitize time traces of the noise atthe heterodyne detector outputs, and then compute thenoise variance or power spectral density. See Ref. [20] forfurther details of signal processing with the HF2LI.

Appendix B: Transducer device

1. Design and Layout

As shown in Fig. 1(a), the membrane, the microwavecircuit, and one of the optical cavity mirrors are inte-grated into a stack of three 380 µm-thick silicon chips.We refer to the chips hosting the membrane, the greater

8

50 Ω termination

outin

Variable attenuator

Variable phase shifter

180° hybrid

Fiber coupler Directional coupler

Polarizing Beamsplitter

Photodetector

40 mK

HEMT

800 mK

4 K

Microwave pump

(R & S SMA100B)

7.937 GHz

SSB

mod

Transducer PDH Lock

infast outslow out

ca

nce

l

pump

lock

LO

λ/2λ/2λ/4

20 dB

pu

mp

ωpump

= ωo – ω

m

ωlock

= ωo

ωLO

= ωpump

– 2π × 12.9 MHz

λ/4

Toptica CTL

1084 nm

piezo-actuated

filter cavity

AOM

AOM AOM

AOM EOM

20

dB

20

dB

ZI Lock-in Amplifier

(Zurich Instruments HF2LI)

to circuit

QED system

to rear-port

optics

to ZI in

10 MHz

74 MHz

IL Q

R

ILQ

R

SSB

mod

Faraday

rotator

Fa

rad

ay

rota

tor

FIG. 5. Experimental setup. Diagram illustrating the system used to control and probe the transducer, comprisingmicrowave (top) and optical (bottom) subsystems along with the lock-in amplifier used for data acquisition (left). Elementsused to lock the optomechanical cavity are shown in yellow.

part of the circuit, and the mirror as the top, middle, andbottom chips, respectively. The fabrication procedure isdescribed in Appendix B3. Here we discuss the devicedesign at a higher level.

The silicon substrate of the top chip is patterned intoa few-cell phononic shield, made up of 900 µm × 900 µmmasses joined by 100 µm × 550 µm tethers, with the100 nm-thick silicon nitride membrane supported on thecentral mass (Fig. 6(a)). The transverse dimensions ofthe membrane (500 µm × 500 µm) determine the res-onant frequency ωm/2π = 1.45 MHz of the vibrationalmode with two antinodes in each direction (Fig. 6(b)),

which we label the (2,2) mode [38]. The mass and tetherdimensions are chosen to center the phononic bandgapon ωm, and finite-element simulations of the shield withperiodic boundary conditions predict a bandgap between1.10 and 1.98 MHz.

The overall size of the top chip is constrained by theflip-chip electromechanical circuit architecture: largerchips make it difficult to control the capacitor pad separa-tion d that determines Ge, the coefficient relating the mi-crowavemode frequency shift to membrane displacement.This constraint limits the number of unit cells in thephononic shield, but the design nevertheless yields a vi-

9

0 0.2 0.4 0.6 0.8 10

1

2

3

1 1.5 2 2.5

100

102

104

1

-1

Po

we

r sp

ectr

um

(a

.u.)

(a)

(c)

(b)

(d)

(Γe+

γm

) /2π

(H

z)

Frequency (MHz)

Relative microwave power

FIG. 6. Mechanical design and characterization.

(a) Phononic shield design. Black circles (not to scale) in-dicate location of 20 µm diameter posts that determine thenominal capacitor pad separation. (b) Normalized displace-ment of the 1.45 MHz vibrational mode from finite-elementsimulation of the membrane and capacitor pad (dashed line).(c) Optically measured mechanical spectrum. The shaded re-gion indicates the bandgap obtained from a simulation withperiodic boundary conditions. Expected frequencies of mem-brane modes (dashed lines) are obtained from finite-elementsimulation. (d) Total damping obtained from electromechani-cal ring-down measurements vs. microwave pump power. Ex-trapolation to zero pump power yields an intrinsic mechanicaldamping rate of γm/2π = 108 mHz.

brational spectrum with relatively few substrate modes inthe vicinity of the (2,2) mode (Fig. 6(c)). Membranes em-bedded in similar phononic crystal structures with moreunit cells have been shown to thermalize to T . 100 mKin the presence of laser light [39].

A NbTiN pad on one corner of the membrane togetherwith a split pad on the middle chip form a parallel platecapacitor, and the two halves of the split pad are con-nected through a loop inductor to complete the elec-tromechanical circuit. The middle chip portion of thecircuit contributes some parasitic capacitance that di-lutes the sensitivity of the circuit’s resonant frequencyto membrane motion. The electromechanical frequency

shift per unit membrane displacement is given by

Ge = pωe

2d, (B1)

where p is the ratio of the circuit’s motionally modulatedcapacitance to its total capacitance, which decreases withincreasing pad separation d. Four posts arrayed in a3.6 mm square on the middle chip set the target capac-itor pad separation d0 = 300 nm and support the topchip, which is affixed to the middle chip using cryogenicepoxy. The actual capacitor gap d can differ from d0because the epoxy contracts upon thermal cycling. Forthis device, we infer d = 830 nm from electromechanicalmeasurements (see Appendix D3 for further discussion).Regions of the top chip and middle chip substrates arerecessed to limit the area with small separation betweenthe top chip and middle chip.The bottom chip serves as a substrate for a high-

reflectivity Bragg mirror (7 ppm power transmission co-efficient). The top and bottom layers of this Bragg mirrorare both SiO2, which results in a strong bond betweenthe mirror coating and the substrate and enables bond-ing of the mirror chip to the underside of the middle chip(see Appendix B3). A hole in the middle chip oppositethe non-metalized portion of the membrane allows laserlight to pass through to the mirror. A higher-throughputcurved mirror (190 ppm power transmission coefficient,1 cm radius of curvature) is positioned opposite the mir-ror chip at a distance of L = 2.3 mm to form a stablesingle-sided Fabry-Perot optical cavity (Fig. 7(a)), whosemode intersects the corner of the membrane opposite thecapacitor pad.The bonding of the bottom chip to the middle chip

fixes the distance between the flat mirror and the mem-brane at the middle chip thickness t = 380 µm and en-sures that the mirror and the membrane do not signifi-cantly misalign as the transducer is cooled to cryogenictemperatures. The coefficientGo relating the optical cav-ity frequency shift to membrane displacement is deter-mined by the longitudinal position of the membrane inthe standing wave of the cavity mode (see Appendix B2).The vacuum electromechanical and optomechanical

couplings that enter into the system Hamiltonian aregiven respectively by ge = Gexzp,e and go = Goxzp,o,where xzp denotes the zero-point motion of the mem-brane mode. The zero-point motion xzp,o for the op-tomechanical interaction is modified relative to the naıvevalue xzp =

√

~/2ωmm for a harmonic oscillator withmass m by a factor that accounts for the transverse dis-placement profile of the membrane mode (Fig. 6(b)) in-tegrated over the spatial extent of the beam spot. Thezero-point motion xzp,e for the electromechanical inter-action is likewise modified by the integral of membranemotion over the area of the capacitor pad. For the per-turbed (2,2) mode we use for transduction, xzp,e < xzp,o

because motion of the membrane quadrant occupied bythe capacitor pad is reduced relative to the motion atthe optical spot due to the extra mass of the NbTiN

10

0

50

100

0 1 2 3 4

Wavelength offset (nm)

0

1

2

(o,ext

+ o,back

) / 2

o, ext / 2

0

50

100

0 1 2 3 4

Wavelength offset (nm)

0

1

2

(o,ext

+ o,back

) / 2

o, ext / 2

t = 380 μmt = 380 μm

(a) (b)

Lin

ew

idth

(M

Hz)

(κo,ext + κo,back) / 2π

κo,ext / 2π

|Go

/2π

|(H

z/fm

)

Wavelength offset (nm)

L = 2.3 mm

FIG. 7. Optical cavity simulation. (a) Optical cavitylength scales. (b) Simulated optomechanical cavity parame-ters. The free spectral range associated with the t = 380 µmspacing between the membrane and mirror chip sets the peri-odicity of the optical frequency shift per membrane displace-ment Go (top) and cavity mirror loss rates κo,ext and κo,back

(bottom) that would be obtained if the optical cavity had apiezoelectrically actuated curved mirror. Gray dotted linesindicate operating parameters available to a cavity withoutpiezoelectric actuation like the one used in this work.

film. Using a finite-element simulation of the membranewith the normalization convention of Ref. [38], we obtainxzp,e = 0.5 fm and xzp,o = 0.9 fm.

2. Fixed membrane-mirror cavity architecture

Previous optomechanical cavity designs [20, 21] usedtwo piezoelectric actuators to provide the requisite twodegrees of freedom for in situ control of the frequencyshift per membrane displacement Go and continuous tun-ing of the perturbed cavity frequency to match that ofa fixed-frequency laser. The presence of two planar sur-faces susceptible to misalignment in such cavities resultedin large scattering losses when the membrane was posi-tioned at regions of higher intracavity field intensity, lim-iting operation to membrane positions with low Go [20].The piezoelectric actuators on these cavities also pro-vided a channel through which electrical noise could cou-ple in to the optomechanical system. In this work, we in-stead used a cavity with two fixed mirrors together witha tunable laser.To understand the operation of an optomechanical cav-

ity with two fixed mirrors, it is illustrative to first con-sider the case of an otherwise identical cavity with piezo-electric actuation of the curved mirror opposite the sil-icon chip stack. Because the distance between the flatmirror and the membrane is fixed, the only way to trans-late the membrane relative to the cavity mode stand-ing wave (and thus tune Go in situ) is to vary thelaser wavelength. The curved mirror can then be ac-tuated to maintain cavity resonance and probe Go con-

tinuously as a function of laser wavelength. Tuning thelaser also changes the optical cavity decay rates κo,ext,κo,back, and κo,loss associated with the input mirror, theweakly coupled back mirror, and absorption/scatteringlosses, respectively. Go, κo,ext, and κo,back can be calcu-lated using a one-dimensional transfer matrix model. Allthree quantities are periodic in the free spectral range∆λFSR,mem = λ2/2t = 1.55 nm associated with the dis-tance t = 380 µm between the membrane and flat mirror.The loss rate κo,loss is difficult to model because it de-pends on the alignment between the membrane and thecavity mirrors, the membrane-perturbed frequencies ofhigher-order modes, and the location of lossy scatterers.

Fig. 7(b) shows the results of the transfer matrix modelsimulation using the distances t and L and the mirrorcoating transmission coefficients of our cavity. In thecourse of tuning the resonant wavelength of the cavityby ∆λFSR,mem, the optomechanical frequency shift Go

exhibits two nulls, corresponding to the membrane sit-ting at the node and antinode of the optical field inten-sity, and two local maxima, where the intensity gradientis maximized. The larger local maximum correspondsto light building up preferentially in the shorter sectionof the cavity, between the membrane and the mirror de-posited on the bottom chip, while at the lower local max-imum, the intracavity light is concentrated in the longercavity section between the membrane and curved mir-ror. The features have different widths, with the widerone corresponding to light building up on the side of themirror that is actuated (the curved mirror in the simu-lation shown in Fig. 7). The decay rate κo,ext throughthe higher-throughput curved mirror is maximized whenintracavity light is concentrated in the longer cavity sec-tion, while the decay rate κo,back through the lower-throughput flat mirror is maximized when light is con-centrated in the shorter section.

We chose to deposit the higher-transmission coatingon the curved mirror farther from the membrane, des-ignating it the input mirror, in order keep the sumκo,ext + κo,back (equal to the cavity linewidth κo in theabsence of loss) relatively constant as a function of wave-length. Depending on the laser wavelength, the opticalfield will preferentially build up on either one side of themembrane or the other, and the change between the twocavity section lengths counteracts the change between themirror throughputs. This insensitivity of κo,ext + κo,back

to wavelength allows flexibility in operating the trans-ducer, because in the absence of additional loss the op-tomechanical sideband resolution is similar at the twolocal maxima in Go. The option to use the lower localmaximum is important because there the cavity mode isless susceptible to loss from any misalignment betweenthe membrane and flat mirror on the bottom chip [37].

Eliminating piezoelectric actuation of the curved mir-ror restricts operation to a discrete set of wavelengths atwhich the laser is resonant with a longitudinal mode ofthe optical cavity, separated by roughly the unperturbedoptical cavity free spectral range ∆λFSR = λ2/2L =

11

260 pm (gray dashed lines in Fig. 7(b)). Because themembrane sits at a different position in the intracavitystanding wave at each of these wavelengths, we retain theability to choose an operating point with high couplingand low loss in situ. In this work we operate our trans-ducer at a wavelength near the lower of the two localmaxima in Go.

3. Device Fabrication

Three separate silicon wafers are processed in order toconstruct the device. The bottom chip wafer has a high-reflectivity SiO2/TaO5 Bragg mirror coating depositedusing ion beam sputtering by FiveNine Optics. In orderto ensure the wafer is flat at room temperature, a re-quirement for bonding the bottom chip and middle chipwafers, a 3 µm SiO2 stress compensation layer is de-posited on the back of the wafer as well. The portionof the electromechanical circuit on the middle chip waferis fabricated from a 200 nm layer of sputtered NbTiN andpatterned using a fluorine etch. The posts that define thetarget capacitor pad separation are then patterned froma 300 nm aluminum film, and the optical access hole inthe middle chip is etched using a deep reactive ion etch(DRIE).The bottom chip and middle chip wafers are bonded

using a hydrophilic Si-SiO2 bonding process [40]. Surfacepreparation of the bonding interfaces is done with two at-mospheric plasma treatments: one oxygen-rich plasma todescum the surface and one hydrogen-rich plasma to cre-ate the desired surface chemistry. The wafers are thenpressed together and annealed at 150 C. After bond-ing, the SiO2 stress-compensation layer is removed witha HF vapor etch to make a symmetric silicon/mirror coat-ing/silicon stack. The bonded wafers are then diced tocreate 8 mm × 8 mm chips.A 100 nm high-stress layer of stoichiometric Si3N4 is

grown on the top chip wafer using low-pressure chemicalvapor deposition (LPCVD). The capacitor pad is thenpatterned from a 30 nm film of sputtered NbTiN using a5:2:1 H2O:H2O2:NH4OH solution. A DRIE step followedby immersion in potassium hydroxide (KOH) is used todefine and release the membrane. Lastly, a final DRIEstep patterns the phononic shield (see Appendix B1) anddices the wafer into 6 mm × 6 mm chips. The membranesare cleaned in nanostrip at 60 C and then in HF. A topchip is then flipped over onto a middle chip/bottom chipstack using a Westbond manual die bonder, and smallamounts of Stycast 2850 cryogenic epoxy are applied toaffix the top chip and middle chip together.A superpolished fused silica substrate for the curved

mirror is fabricated by Perkins Precision Developments.A high-reflectivity SiO2/TaO5 Bragg mirror is then de-posited on this substrate by FiveNine Optics. The fusedsilica substrate and three-chip stack are supported by anInvar enclosure in order to minimize thermal contractionupon cryogenic cooling. External coupling to the mi-

crowave circuit is achieved through its mutual inductancewith a loop coupler at the end of a coplanar waveguideon the middle chip. This coplanar waveguide is wired-bonded to a flexible printed circuit board with an SMAconnector at the opposite end supported by the invarstructure.

Appendix C: Theory

1. Hamiltonian formalism

The electro-optomechanical system is described by theHamiltonian

H/~ = ωoa†a+ ωeb

†b+ ωmc†c

+(

goa†a+ geb

†b)

(

c+ c†)

, (C1)

where a, b and c are respectively the optical, microwaveand mechanical mode annihilation operators, and go andge are the vacuum optomechanical and electromechanicalcoupling rates [24].Optical and microwave pumps detuned from the corre-

sponding electromagnetic resonances by ∆o and ∆e pro-duce phase space displacements a and b for the opticaland microwave modes, respectively. Linearizing Eq. (C1)about a and b yields

Hlin/~ = −∆oa†a−∆eb

†b+ ωmc†c+ goa(a

† + a)(c+ c†)

+ geb(b† + b)(c+ c†), (C2)

in the rotating frame of the pump fields. Hlin containsboth beamsplitter terms of the form a†c and two-modesqueezing terms of the form a†c†. The beamsplitter termscan be resonantly enhanced by red-detuning the pumpsby approximately the mechanical mode frequency (∆o ≈∆e ≈ −ωm). Then Eq. (C2) becomes the Hamiltonianfor three harmonic oscillators at the same frequency thatcan exchange quanta via the beamsplitter interactions atrates go|a| and ge|b|.We then extend the Hamiltonian model to include de-

cay rates represented by ports on the three resonators.Specifically, we add one port to the mechanical mode(representing energy dissipation at rate γm), two ports tothe microwave mode (representing coupling to a trans-mission line at a rate κe,ext and internal loss at rateκe,int), and two ports to the optical mode (represent-ing coupling through the higher-throughput curved mir-ror at rate κo,ext and the sum κo,int + κo,back of scat-tering/absorption losses and transmission through thelower-throughput back mirror). In the weak couplingregime (go ≪ κo = κo,ext + κo,int + κo,back), the optome-chanical interaction then contributes net damping to themechanical mode given by

Γo = g2o |a|2(

κo

κ2o/4 + (ωm +∆o)

2− κo

κ2o/4 + (ωm −∆o)

2

)

,

(C3)

12

where |a| =√

(Po/~ωp,o) ǫPC κo,ext/ (κ2o/4 + ∆2

o), ωp,o =ωo − ∆o is the optical pump frequency, and ǫPC is themodematching of the incident pump beam to the op-tical cavity mode. The second term in Eq. (C3) rep-resents anti-damping from imperfect suppression of thetwo-mode squeezing terms a†c† and ac in Eq. (C2); it issuppressed relative to the first term at optimal detuning∆o = −ωm in the resolved sideband limit κo ≪ 4ωm.Likewise, the electromechanical damping rate for ge ≪κe = κe,ext + κe,int is

Γe = g2e∣

∣b∣

∣

2

(

κe

κ2e/4 + (ωm +∆e)

2− κe

κ2e/4 + (ωm −∆e)

2

)

,

(C4)

where∣

∣b∣

∣ =√

(Pe/~ωp,e)κe,ext/ (κ2e/4 + ∆2

e) and ωp,e =ωe −∆e is the microwave pump frequency.

2. Transducer Efficiency

The transmission of a narrowband signal from the ex-ternal port of the microwave resonator through the trans-ducer to the external port of the optical cavity is governedby the scattering parameter

Soe(ω) =

√

AeAo

κe,ext

κe

κo,ext

κo

√ΓeΓo

ΓT/2− i(ω − ωm), (C5)

where ω is the detuning of the input signal from the mi-crowave pump, Ao = −

(

(κo/2)2 + (∆o − ωm)

2)

/4∆oωm

is a gain factor arising from the optomechanical two-mode squeezing terms in Eq. (C2), and the definitionof Ae is analogous [21]. These factors are undesirable be-cause any transducer gain is necessarily accompanied byadded noise [41]. With optimally red-detuned pumps,the gain factors reduce to Ao = 1 + (κo/4ωm)

2 andAe = 1+(κe/4ωm)

2, and the transducer gain approachesunity in the resolved sideband limit. The transmission ofa narrowband signal modematched to the optical cavitythrough the transducer to the external port of the mi-crowave resonator, Seo(ω), is formally identical to Soe(ω).Following Ref. [21], we use a conservative definition

of the transducer efficiency that accounts for energy lostin each of the three modes involved in the transductionprocess as well as imperfect optical cavity modematch-ing (not included in Eq. (C5)). To properly account formodematching, we must adapt the formalism of Ref. [21]to our heterodyne detection scheme. In downconversionthe relevant modematching factor is the modematchingǫPC of the incident pump beam to the cavity, while in up-conversion the relevant factor is the modematching ǫCL ofthe light emerging from the cavity to the LO beam. Wethus redefine the optical input and output ports to absorbmodematching factors, so that Soe(ωm) →

√ǫCLSoe(ωm)

and Seo(ωm) → √ǫPCSeo(ωm). The upconversion effi-

ciency for an incident signal on resonance is then de-fined as ηt,up = |Soe(ωm)|2 /AoAe and the downconver-

sion efficiency is defined as ηt,down = |Seo(ωm)|2 /AoAe.

We take our figure of merit to be the bidirectional effi-ciency ηt =

√ηt,upηt,down, given by Eq. (1) in the main

text. For Γe = Γo ≫ γm, ηt achieves its maximum valueηM = ǫ

κo,ext

κo

κe,ext

κe, where we have defined the bidirec-

tional modematching ǫ =√ǫPCǫCL.

3. Optomechanical sideband cooling

As noted in the main text, the same beamsplitterprocesses that mediate transduction also cool the mem-brane mode. For definiteness, we consider the optome-chanical interaction in isolation. Optomechanical coolingarises from the imbalance between the rates of Stokesand anti-Stokes scattering processes when the pump isred-detuned from the cavity mode. The Stokes process,wherein a pump photon is scattered to lower energy byemitting a phonon into the membrane mode, is sup-pressed relative to the complementary anti-Stokes pro-cess by the reduced density of states away from cavity res-onance. The rates of the anti-Stokes and Stokes processesare given respectively by nmΓo,+ and (nm+1)Γo,−, wherenm is the phonon occupancy of the membrane mode, andΓo,+ and Γo,− are the anti-Stokes and Stokes contribu-tions to the net optomechanical damping rate Γo (i.e.,the positive and negative terms in Eq. (C3)) [24].

If the mechanical mode were completely decoupledfrom its thermal environment, it would reach equilibriumwhen the Stokes and anti-Stokes scattering rates becomeequal, resulting in a final occupancy

nmin,o =ro

1− ro=

(κo/2)2+ (∆o + ωm)

2

−4∆oωm

, (C6)

where ro = Γo,−/Γo,+. To generalize to the case wherethe mechanical mode is coupled to more than one bath,note that Γo,− = Γonmin,o: thus heating by Stokes scat-tering can be described as a coupling to a bath nmin,o atrate Γo, to which the membrane mode equilibrates whenΓo overwhelms all other contributions to the total me-chanical mode damping ΓT. This lower bound on themode occupancy arising from residual Stokes scatteringis called the quantum backaction limit.

The derivation of the electromechanical backactionlimit is entirely analogous to the above optomechanicalderivation. In the absence of technical noise, the compe-tition between the membrane mode’s coupling to its ther-mal environment through its intrinsic dissipation rate γmand its coupling to optomechanical and electromechani-cal backaction baths results in Eq. (2) for the final oc-cupancy of the membrane mode. The transducer gainfactors introduced in Appendix C 2 are related to thebackaction bath occupancies by Ao = 1 + nmin,o andAe = 1 + nmin,e.

13

4. Optical output spectrum

To infer the membrane mode occupancy and studythe transducer’s noise performance, we make heterodynemeasurements of the noise density at the optical cavityoutput port. Here we describe the relevant features ofthe optical output spectrum in the absence of technicalnoise, and in Appendix C 5 we generalize the expressionsto include the effects of technical noise. The scatter-ing processes described above generate Lorentzian peaksin the optical output spectral density at detunings ±ωm

from the pump frequency. In units of photons/s/Hz re-ferred to the transducer’s optical output port, the spec-trum around the upper (anti-Stokes) sideband has theform

So,+(ω) = 1 + ǫCL

κo,ext

κo

ΓoΓT (1 + nmin,o)nm

Γ2T/4 + (ω − ωm)

2, (C7)

and the spectrum around the lower (Stokes) sideband is

So,−(ω) = 1 + ǫCL

κo,ext

κo

ΓoΓT nmin,o (nm + 1)

Γ2T/4 + (ω + ωm)

2, (C8)

where the background terms correspond to the sum ofvacuum noise and the added noise of an ideal hetero-dyne detector. The upper and lower sideband ampli-tudes Nout,o,± = So,±(±ωm) − 1 are proportional to theoptomechanical anti-Stokes and Stokes scattering rates,respectively, and thus the ratio of measured sideband am-plitudes ro = Nout,o,−/Nout,o,+ can be used to infer themode occupancy by inverting

ro =nmin,o

nmin,o + 1

nm + 1

nm

. (C9)

At high temperatures, nm ≈ nm + 1, and comparingEq. (C9) to Eq. (C6) reveals that ro → ro. As nm de-creases, ro increases, asymptoting towards 1 as nm ap-proaches its minimum value nmin,o.When both pumps are present, the beamsplitter in-

teractions transduce incident microwave signals detunedby +ωm from the microwave pump to outgoing signalsat detuning +ωm from the optical pump, and vice versa.Thus the transducer noise that competes with an upcon-verted signal at the optical output port is Nout,o,+. Theinput-referred upconversion added noise is then given by

Nadd,up =Nout,o,+

AeAo ηt,up=

nm

Aeκe,ext

κe

ΓT

Γe

, (C10)

This equation indicates that, up to small effects fromimperfect sideband resolution and microwave mode in-ternal loss, the ability to ground-state cool the mechan-ical mode with purely electromechanical damping is anecessary and sufficient condition for Nadd,up < 1. As-suming comparable electromechanical and optomechan-ical sideband resolution, nm will remain approximatelyconstant at constant total damping ΓT, but trading off

electromechanical damping for optomechanical dampingwhile holding ΓT constant can only increase the addednoise. Moreover, Nadd,up is independent of all other pa-rameters of the optomechanical system. Qualitatively,this is because after the signal to be upconverted en-ters the mechanical mode, it copropagates with any noiserouted to the optical output port, so the effects of the op-tical cavity parameters on the signal and noise cancel.The form of the upper and lower electromechanical

sideband amplitudes Nout,e,+ and Nout,e,− in the mi-crowave output spectrum is analogous to the optome-chanical case described above. In the absence of techni-cal noise, the input-referred downconversion added noiseis given by

Nadd,down =Nout,e,+

AeAo ηt,down

=nm

ǫPCAoκo,ext

κo

ΓT

Γo

, (C11)

independent of electromechanical system parameters.Without technical noise, the ability to ground-state withthe purely optomechanical damping is a necessary andsufficient condition for Nadd,down < 1.

5. Effects of technical noise

We define technical noise broadly to encompass allpump power-dependent effects on the transducer noiseperformance other than heating of the thermal bath nth

to which the mechanical mode is coupled. Noise sourcesthat fall into this category include amplitude and phasefluctuations of the incident microwave or optical pumpsat detuning ωm, fluctuations of resonator parameters atfrequency ωm (e.g., fluctuations of the superconductingcircuit resonant frequency due to two-level systems [42]),and heating of the superconducting circuit by either ofthe pumps. All of these noise sources present either as aneffective microwave mode thermal occupancy neff,e cou-pled to the membrane mode at rate Γe or as an effectiveoptical mode occupancy neff,o coupled to the membranemode at rate Γo [24, 30]. Thus the formalism presentedin the main text can be used to model the effects of tech-nical noise on sideband cooling without regard to thephysical origin of the noise.When readout of mechanical motion is performed using

a pump coupled to a bath with nonvanishing effectiveoccupancy neff, two other effects of technical noise onthe output spectrum must be considered. First, technicalnoise will generate an additional contribution to Nadd

distinct from its direct effect on nm. Second, technicalnoise from pump or parameter fluctuations can modifythe amplitudes of the Stokes and anti-Stokes peaks inthe output spectrum, and to correctly infer nm we mustcorrect for this effect.In the remainder of this section, we restrict our fo-

cus to technical noise arising from amplitude and phasefluctuations of the incident optical pump, following theformalism of Ref. [30], as optomechanical readout is usedfor all the data presented in Figs. 2 and 3. The optical

14

mode effective thermal occupancy due to amplitude and phase fluctuations is given by

neff,o =1

4ǫPC

κo,ext

κo

Ao

κ2o

4

(

|Bx(ωm)|2 Cxx + |By(ωm)|2 Cyy + 2Im[

Bx(ωm)B∗y(ωm)

]

Cxy

)

, (C12)

where Bx(ω) = e−iφχo(ω) + eiφχ∗o(−ω) and By(ω) =

e−iφχo(ω) − eiφχ∗o(−ω) are amplitude and phase noise

susceptibilities, φ = tan−1 (2∆o/κo) is the optomechan-ical phase shift, χo(ω) = 1/ (κo/2− i (ω +∆o)) is theoptical cavity susceptibility, and Cxx, Cyy, and Cxy aretechnical noise spectral densities. Specifically, Cxx andCyy are two-sided amplitude and phase noise spectraldensities, respectively, while Cxy is the two-sided ampli-tude/phase cross-correlation spectral density, subject tothe constraint C2

xy ≤ CxxCyy. All three spectral densitiesare evaluated at detuning ωm from the pump, and ex-

pressed in units of photons/s/Hz normalized to the shotnoise of the beam incident on the optical cavity. Theyare thus all proportional to pump power, motivating theparameterization neff,o = aoΓo used in the main text (seeAppendix E 3 for further discussion).Cxx, Cyy, and Cxy can all be modeled as white noise

spectral densities in the narrow spectral window of in-terest around ±ωm. They thus generate frequency-independent excess noise So,+

(

So,−

)

over the back-ground in the optical output spectrum around the upper(lower) sideband. These excess noise spectral densitiesare given by

So,± =1

4ǫCLǫPC

([

|ρ|2 + |κo,extχo(±ωm)− 1|2]

(Cxx + Cyy)− 2Re [ρ∗ (κo,extχo(±ωm)− 1) (Cxx + 2iCxy − Cyy)])

(C13)

where ρ = 1 − κo,ext/ (κo/2− i∆o) is the attenuation ofthe reflected pump due to absorption in the cavity. Am-plitude noise Cxx would contribute to So,± even in theabsence of a cavity, while Cyy and Cxy contribute be-cause the cavity rotates phase fluctuations into the am-plitude quadrature. For an overcoupled cavity at opti-mal detuning in the resolved sideband limit, phase noisecontributes primarily to So,+ and amplitude noise con-

tributes primarily to So,−. In the discussion of added

noise in the main text, for which only noise in the uppersideband is relevant, we introduce the symbol Nout,o =

So,+(+ωm) = So,+, to parallel the notation used for theLorentzian contribution to the noise.The final effect we must account for is a modification

of the Stokes and anti-Stokes peaks in the optical out-put spectrum due to correlations between membrane mo-tion and amplitude and phase fluctuations of the incidentlight. Eqs. (C7) and (C8) become

So,+(ω) = 1 + So,+ + ǫCL

κo,ext

κo

Ao

Γo

Γ2T/4 + (ω − ωm)

2

×[

ΓT

(

nm − ǫPC

κ2o

4Re[

B+

]

)

− 2ǫPC (ω − ωm)κ2o

4Im[

B+

]

]

(C14)

and

So,−(ω) = 1 + So,− + ǫCL

κo,ext

κo

Ao

Γo

Γ2T/4 + (ω + ωm)

2

×[

ΓT

(

nmin,o

Ao

(nm + 1) + ǫPC

κ2o

4Re[

B−

]

)

− 2ǫPC (ω + ωm)κ2o

4Im[

B−

]

]

, (C15)

where

B± =e−iφ

4

(

κo,ext |χo(±ωm)|2 [Bx(∓ωm) (Cxx + iCxy) +By(∓ωm) (iCxy − Cyy)]

− χ∗o(±ωm) [(Bx(∓ωm)Cxx + iBy(∓ωm)Cxy) (1 + ρ) + (iBx(∓ωm)Cxy −By(∓ωm)Cyy) (1− ρ)]

)

. (C16)

Eqs. (C14) and (C15) reveal that these correlations both change the peak amplitudes Nout,o,± and add an

15

anti-symmetric contribution to the lineshape of eachpeak, where the sign of these extra terms depends onthe relative values of Cxx, Cyy, and Cxy. Eq. (C10) forthe transducer’s upconversion added noise is then mod-ified to account for amplitude and phase noise by thesubstitution Nout,o,+ → Nout,o,+ + Nout,o.

Appendix D: Transducer characterization

Transducer device parameters are summarized in Ta-ble I. Measurements of these parameters are detailed be-low.

1. Microwave circuit characterization

We characterize the microwave circuit by measuringthe reflection scattering parameter See(ω) encoding itsamplitude and phase response as a function of detuningfrom resonance. When normalized to the off-resonancelevel, this scattering parameter is

See(ω) =κe,ext

κe/2− i(ω − ωe)− 1. (D1)

We use a vector network analyzer to generate a weakswept probe tone and inject it into the fridge togetherwith the strong pump tone at fixed detuning ∆e = −ωm

from ωe. Sweeping the microwave pump power Pe andfitting See(ω) then allows us to measure the power-dependence of the microwave circuit’s internal loss κe,int

(plotted in Fig. 10(a) in Appendix F) as well as thepower-independent microwave circuit external couplingrate κe,ext/2π = 1.42 MHz.For the data shown in Fig. 3, the incident microwave

pump power was swept over the range 0.2 nW <Pe < 10 nW, and the circuit’s total linewidthκe/2π = (κe,ext + κe,int) /2π varied between 1.64 MHzand 2.31 MHz over this range.

2. Optical cavity characterization

We also characterize the optical cavity by measuringits reflection response, and the relevant scattering pa-rameter is formally analogous to See(ω). We tune thelaser wavelength to put the lock beam close to resonancewith the optomechanical cavity, and then sweep the fre-quency of the laser relative to that of the optomechanicalcavity by ramping the voltage applied to a piezoelectricactuator on the filter cavity to which the laser is locked(see Appendix A). This scheme measures |Soo(ω)|2, withthe horizontal axis calibrated using the phase modulationsidebands on the lock beam. Fitting the data then yieldsκo/2π = 2.68±0.05 MHz for the optical cavity linewidth.To optimize and measure the modematching of the

cavity mode to the pump or LO beams (ǫPC and ǫCL,respectively; see Appendix C 2), we inject an additional

probe beam into the cavity through the low-transmissionback mirror and steer the transmitted mode to a detectorwhere it interferes with the beam of interest. We thenequalize the power of the beams, measure the visibilityV of the interference fringes that result from a relativedetuning between the beams, and define the associatedmodematching as V2.The modematching factors could change slightly over

the course of a cooldown as beam alignments drifted.The data shown in Fig. 2 in the main text was acquiredimmediately after the modematchings were reoptimizedand measured to be ǫPC = 0.86 and ǫCL = 0.91. Both be-fore and after the acquisition of the data shown in Fig. 3,the modematchings were measured to be ǫPC = 0.80 andǫCL = 0.79. Once ǫPC and the cavity linewidth κo areknown, the external coupling rate κo,ext/2π = 2.12 MHzis inferred from the depth of dip on resonance in a mea-surement of |Soo(ω)|2 in which the pump beam is used toprobe the cavity. The modematching of the lock beamto the cavity mode was not directly measured. We in-fer ǫlock = 0.85 using the depth of dip in the lock beam|Soo(ω)|2 measurement together with the measured valueof κo,ext.One final modematching factor that does not involve

the cavity mode is also relevant to the calibration of theefficiency (see Appendix E 1). We define ǫPL as the mode-matching of the LO beam to the promptly reflected pumpbeam with the cavity unlocked. We measured this mode-matching to be ǫPL = 0.75 for the data shown in Fig. 2and ǫPL = 0.79 for the data shown in Fig. 3.For the data shown in Fig. 2, the optical pump power

incident on the cavity was swept over the range 20 nW <Po < 1.7 µW. For the data shown in Fig. 4, the relevantquantity is not the incident power Po but rather the in-tracavity circulating power

Pcirc = ∆νFSR κo,ext

(

ǫPCPo

κ2o/4 + ∆2

o

+ǫlockPlock

κ2o/4 + ∆2

lock

)

,

(D2)where ∆νFSR = c/2L is the free spectral range of theoptical cavity in frequency units. Uncertainty in Pcirc

comes from uncertainty in the cavity length L and un-certainty in the measured value of the optical insertionloss within the cryostat.

3. Optomechanical and electromechanical

characterization

To study the vibrational modes of the membrane,we exploit the phenomenon of optomechanically inducedtransparency (OMIT), wherein Soo(ω) is modified by op-tomechanical interference effects for a swept probe tonecoherent with the pump field [24], as well as the analo-gous phenomenon of electromechanically induced trans-parency (EMIT). We generate these coherent probe tonesusing SSB modulation of either the microwave pump (forEMIT measurements) or the RF drive to the AOM that

16

TABLE I. Electro-optic transducer parameters. The range of values for the microwave circuit linewidth κe and the electrome-chanical transducer gain Ae correspond to the range of pump powers used in Fig. 3. For the optical cavity modematchings andtransducer efficiency, the first (second) numbers correspond to the data shown in Fig. 2 (Fig. 3)

Parameter Symbol Value

Optical cavity frequency ωo ωo/2π = 277 THzOptical cavity external coupling κo,ext κo,ext/2π = 2.12 MHzOptical cavity linewidth κo κo/2π = 2.68 MHzMicrowave circuit frequency ωe ωe/2π = 7.938 GHzMicrowave circuit external coupling κe,ext κe,ext/2π = 1.42 MHzMicrowave circuit linewidth κe κe/2π = 1.64− 2.31 MHzMechanical mode frequency ωm ωm/2π = 1.451 MHzIntrinsic mechanical dissipation rate γm γm/2π = 113 mHzVacuum optomechanical coupling go go/2π = 60 HzVacuum electromechanical coupling ge ge/2π = 1.6 HzModematching of pump beam to cavity mode ǫPC ǫPC = 0.86 (0.80)Modematching of cavity mode to LO beam ǫCL ǫCL = 0.91 (0.79)Bidirectional modematching ǫ =

√ǫPCǫCL ǫ = 0.88 (0.79)

Matched efficiency (peak efficiency) ηM (ηt,max) ηM = 55% (ηt,max = 49%)Optomechanical transducer gain Ao Ao = 1.22Electromechanical transducer gain Ae Ae = 1.08 − 1.16

controls the optical pump beam detuning (for OMITmeasurements; see Appendix A), with the pump fre-quency fixed in both cases. These measurements revealthe vibrational spectrum of the membrane over a widefrequency range (see Fig. 6(c)); we then identify the (2,2)mode and sweep over a narrower range for further charac-terization. The mechanical features in OMIT and EMITgenerally have Fano lineshapes, with linewidth ΓT. Wecan also obtain ΓT by transducing a swept signal to mea-sure the mechanical mode’s Lorentzian transmission pro-file (see Appendix C 2), using SSB modulation of the mi-crowave pump and optical heterodyne detection to mea-sure Soe(ω) or SSB modulation of the optical pump andmicrowave heterodyne detection to measure Seo(ω).

The membrane mode quality factor Qm = ωm/γm ofthe (2,2) mode is sufficiently high that the frequency-domain characterization described above becomes un-wieldy, so we interrogate the mechanical mode at lowdamping using a purely electromechancial time-domainmeasurement (Γo = 0). We apply a low-power microwavepump tone at fixed red detuning ∆e = −ωm from ωe, turnon a temporary SSB probe tone at detuning +ωm fromthe pump to ring up the mechanical mode, then turn offthe probe tone and demodulate the microwave hetero-dyne signal at frequency ωm to monitor the exponentialdecay of the membrane oscillations at rate ΓT = Γe+γm.By repeating this measurement at different microwavepump powers Pe we map out ΓT vs. Pe (Fig. 6(d)). Ex-trapolating to Pe = 0 then yields the bare mechanicallinewidth γm/2π = 108 mHz, corresponding to a qual-ity factor Qm = 1.3 × 107. Performing these ringdownmeasurements electromechanically rather than optome-chanically allows us to avoid optomechanical dampingfrom the lock beam (See Appendix D4). Measurementsof the temperature-dependence of γm, discussed further

in Appendix E 2, indicate that γm/2π = 113 mHz is actu-ally the relevant value for the data presented in the maintext because the membrane equilibrates to an elevatedtemperature in the presence of laser light.

We use this same low-power measurement of ΓT vs.Pe and Eq. (C4) to infer the vacuum electromechanicalcoupling ge (the electromechanical damping Γe does notscale linearly with Pe at high power because of power-dependent microwave circuit loss). The slope of the lin-ear fit in Fig. 6(d) yields ge = 1.6 ± 0.2 Hz, with theuncertainty dominated by uncertainty in the cryostat in-sertion loss. Together with the zero-point motion ob-tained from simulation (see Appendix B1), this mea-surement implies that the electromechanical frequencyshift per unit membrane displacement is Ge = 3.2 Hz/fm.Solving Eq. (B1) self-consistently for the capacitor padseparation d and the participation ratio p(d) then yieldsd = 830 nm and p = 0.67. An analogous linear fit to ΓT

vs. Po at ∆o = −ωm in a purely optomechanical measure-ment yields go/2π = 60 ± 6 Hz, with uncertainty againdue to insertion loss. The corresponding frequency shiftGo/2π = 70± 10 Hz/fm is consistent with operation onthe lower local maximum of optomechanical coupling inthe top panel of Fig. 7(b), and the associated value ofκo,ext from the simulation (bottom panel) matches themeasured value κo,ext/2π = 2.12 MHz.

The capacitor pad separation inferred from the elec-tromechanical damping sweep is substantially larger thanthose previously observed in similar flip-chip devices.The device used in Ref. [20] had d = 300 nm, and purelyelectromechanical devices tested since then have regu-larly achieved gaps in the 150 − 200 nm range [33]. Thelarge gap in this device was likely a consequence of mod-ifications to the fabrication procedure required to inte-grate the phononic shield into the flip-chip architecture.

17

In past devices without phononic shielding, the top chipwas supported by two inner posts near the membrane aswell as the four outer posts described in Appendix B 1above, but these inner posts reduced the effectiveness ofthe phononic shield in simulations.We are presently exploring the effects of aligning inner

posts with nodal lines of the membrane mode’s transversedisplacement profile to mitigate their impact on phononicshield performance. Another cause of unreliable pad sep-aration is the thermal contraction of the epoxy used to af-fix the top chip to the middle chip, and we are investigat-ing the Si-SiO2 bonding technique used to affix the bot-tom and middle chips as an alternative to epoxy in futuredevices. Achieving d = 200 nm would boost ge by a factorof 5, corresponding to an enhancement of the dampingper incident microwave photon by a factor of 25 at lowpump power (the improvement would be larger at higherpower because the internal loss of the microwave circuitwould be smaller at the power required to yield a givendamping). Without any other device improvements, thisincreased electromechanical coupling would yield a mem-brane mode occupancy of nm = 0.4 phonons and up-conversion added noise Nadd,up = 0.9 photons/s/Hz forΓe = Γo = 2π × 1 kHz.

4. Characterization of other parameters

For all the measurements described above, the mi-crowave pump detuning ∆e is set to the optimal value−ωm by stepping up the pump frequency until it beginsto ring up the membrane mode for ∆e > 0, and thenstepping the pump back down by ωm. We use a similarthough slightly more involved procedure to set the pumpand lock beam detunings ∆o and ∆lock to the desired val-ues. First, with the pump beam blocked, we adjust thePDH error signal bias while monitoring ringing of thelock to set ∆lock = 0. We then unblock the pump beamand adjust the AOM frequency that controls ωpump (seeAppendix A) while monitoring the ringing of the lock toset ∆o = −ωm. In doing so we measure the cavity bire-fringence to be ∆B/2π = 2.4 MHz. Finally, we adjustthe PDH error signal bias to slightly red-detune the lockbeam. The lock beam detuning can be set to ∼ 10 kHzprecision by measuring how far ωpump must be adjustedto maintain ∆o = −ωm. The lock beam optomechanicaldamping γlock is determined by comparing the mechan-ical mode linewidth obtained from an Soe measurementat low damping and ∆lock = 0 to the value obtained froma measurement with the same Γe and Γo but ∆lock 6= 0.The lock beam damping was γlock/2π = 5 Hz during themeasurements shown in Fig. 2 and γlock/2π = 2 Hz dur-ing the measurements shown in Fig. 3.The values of γlock and ∆lock are important because

any contribution to the optomechanical or electrome-chanical damping of the membrane mode also couples themode to a backaction bath as described in Appendix C 3above. This coupling cannot be neglected despite the

low damping γlock because the small detuning ∆lock leadsto a large backaction bath occupancy nmin,lock. Indeed,for ∆lock ≪ ωm, the product γlocknmin,lock is indepen-dent of detuning, and can only be reduced by reduc-ing the incident lock power Plock. To fit the data inFigs. 2 and 3 we thus modify Eqs. (2) and (3) to in-clude a γlocknmin,lock term in the numerator, fixed atγlocknmin,lock = 2π× 40 Hz by the independent measure-ments described above. This term results in the increasein the backaction limit at low damping in Fig. 2.With optimally detuned pumps, the measured optical

cavity linewidth κo above yields Ao = 1.22 for the op-tomechanical transducer gain (see Appendix C 2), and abackaction-limited membrane mode occupancy nmin,o =0.22 for optomechanical ground-state cooling. Likewise,power-dependence of the microwave mode linewidth im-plies an electromechanical transducer gain Ae that variesbetween 1.08 and 1.16 over the range of microwave pumppower Pe used in this work. The net transducer gainAeAo thus varies between 1.32 and 1.42.For the data shown in Fig. 2, with fixed electrome-

chanical damping Γe/2π = 100 Hz, the microwave modelinewidth was κe/2π = 1.79 MHz, yielding ηM,exp = 0.55for the expected value of the matched efficiency. For thedata shown in Fig. 3(a), ηM was not constant as a resultof the power-dependent microwave loss. In these mea-surements the transducer efficiency ηt was maximized atΓe/2π = 75 Hz, where κe/2π = 1.75 MHz. The expectedmaximum efficiency ηt,max = 49% comes from evaluat-ing Eq. (1) at Γe/2π = 75 Hz and Γo/2π = 85 Hz usingthis value of the microwave circuit linewidth and otherindependently measured parameters.

Appendix E: Analysis Details

1. Transducer efficiency

We measure the transducer efficiency ηt using themethod developed in Ref. [21], wherein measurementsof upconversion, downconversion, and off-resonance re-flection from both microwave and optical resonators canbe used to calibrate out unknown path losses and gains.More precisely, a set of network analyzer measurementsat fixed Γe and Γo yield

√

√

√

√

√

(

α |Soe(ωm)|2 δ)(

γ |Seo(ωm|2 β)

(

α |See,off|2 β)(

ǫPLγ |Soo,off|2 δ) =

AeAo√ǫPL

ηt, (E1)

where α is the insertion loss of the microwave path fromthe HF2LI lock-in amplifier output to the external portof the electromechanical circuit, β is the gain of the mi-crowave path from the electromechanical circuit outputto the HF2LI input, γ is the insertion loss of the RF andoptical path from the HF2LI output to the external portof the optical cavity, δ is the efficiency of the optical pathfrom the optical cavity output to the HF2LI input. With

18

path losses and gains defined this way, the “e” and “o”ports coincide with the external ports of the microwaveand optical resonators, so |See,off|2 = |Soo,off|2 = 1 byconstruction. In our heterodyne detection scheme, a fac-tor of the pump beam/LO modematching ǫPL appearsin the measurement of the prompt reflection off the op-tical cavity, as also noted in Ref. [20], and must be cal-ibrated out with an independent measurement (see Ap-pendix D2).During the acquisition of the data shown in Fig 2(c), a

mixer in the RF chain used for single sideband modula-tion of the optical pump (Fig. 5) was underdriven, whichmay have resulted in variation of γ between measure-ments of |Seo|2 and |Soo,off|2, and thus a miscalibrationof the efficiency at each data point. We suspect thiswas the origin of the discrepancy between the expectedmatched efficiency ηM,exp and the value ηM,fit obtainedfrom fitting the data in Fig. 2(c). The alternative expla-nation is miscalibration of the parameters that determineηM,exp, but this seems unlikely in light of the good agree-ment between the measured and expected values of ηt,max

obtained from the data in Fig. 3(a).The fit to efficiency vs. damping must be modified

when the electromechanical damping Γe is swept, as inFig. 3(a), because of the power-dependent microwavecircuit loss. To process this data we define ζt =ηt/(κe,ext/κe) and fit ζt vs. damping to Eq. (1), thencompare ζM = 0.59 obtained from the fit to the expectedvalue ζM = ǫκo,ext/κo = 0.63. The theory curve shownin Fig. 3(a) is obtained by multiplying the fit to ζt vs.damping data by a polynomial fit to the power-dependentmicrowave mode overcoupling ratio κe,ext/κe. Because ofthe power-dependent loss, the efficiency is maximized atΓe = 2π × 75 Hz rather that at Γe = Γo = 2π × 85 Hz.

2. Calibration of spectra and temperature sweeps

The measured optical heterodyne noise spectra arenormalized to the sum of LO beam shot noise and hetero-dyne detector dark noise, obtained by blocking the pumpbeam incident on the heterodyne detector. These spec-tra are related to the transducer output-referred spec-tra (Eqs. (C14) and (C15)) by Sdet,o,±(ω) = (1− ξo) +ξoSo,±(ω), where ξo is the efficiency of the optical mea-surement chain, modeled as an effective beamsplitter be-tween the transducer output and the input of an idealheterodyne detector. Measured microwave heterodynenoise spectra are likewise normalized to the noise mea-sured in the absence of the pump tone, given by the sumof vacuum noise, residual thermal noise at the base platetemperature Tbp = 40 mK, and the added noise of themicrowave measurement chain. As in the optomechanicalcase, the spectra normalized this way are related to spec-tra normalized at the transducer’s microwave output portby Sdet,e,±(ω) = (1− ξe) + ξeSe,±(ω), where ξe is the ef-ficiency of the microwave measurement chain. The mea-surement chain efficiencies ξe and ξo appearing in these

expressions can equivalently be expressed in terms of theadded noise of the respective measurement chains re-ferred to the transducer output, via ξe = 1/ (Nξ,e + 1/2)and ξo = 1/ (Nξ,o + 1/2).As a cross-check on Stokes/anti-Stokes sideband ratio

thermometry using Eq. (C9), we can use Eq. (C7) to in-fer the membrane mode occupancy nm from the uppersideband spectrum Sdet,o,+(ω) given independent mea-surements of ξo and parameters in Table I. To determineξo, we sweep the cryostat base temperature Tbp whilemeasuring the optical noise spectrum, at low dampingsuch that technical noise is negligible. At sufficientlyhigh temperature, the local thermal environment of themembrane will equilibrate to Tbp, and thus we can re-place the base-temperature equilibrium occupancy nth

in Eq. (2) with nbp = kBTbp/~ωm. Then, if other pa-rameters are independent of temperature, the measure-ment chain efficiency ξo can be obtained from the slope ofNdet,o,+(Tbp), where Ndet,o,+ is the optomechanical up-per sideband amplitude normalized at the detector input(Ndet,o,+ = ξoNout,o,+ in the absence of technical noise).As Tbp decreases, the membrane mode may decouplefrom the fridge base plate and equilibrate instead to athermal bath at some elevated temperature Teq > Tbp.The temperature sweep enables us to identify Teq as thetemperature below which Ndet,o,+(Tbp) deviates from theexpected linear behavior, and the corresponding thermaloccupancy nth = kBTeq/~ωm can be compared to thevalues obtained from fitting the data in Fig. 2(a) andFig. 3(b).While sweeping the base plate temperature Tbp, we

repeated sets of electromechanical ringdown measure-ments of the sort shown in Fig. 6(d) and observed thatthe mechanical dissipation rate γm increased linearlywith temperature (Fig. 8(a)). Fitting to this data toγm = aγTbp + bγ yields aγ/2π = 176 mHz/K andbγ/2π = 101 mHz. This behavior will generate quadraticterms in Ndet,o,+(Tbp) and Ndet,e,+(Tbp), as the mem-brane mode’s coupling to the temperature-dependentbath is itself temperature-dependent.The optomechanical temperature sweep data is shown

in Fig. 8(b). We obtain peak amplitudes Ndet,o,+ andpeak widths ΓT from Lorentzian fits to the spectra at dif-ferent temperatures, and fit the temperature-dependenceof the peak area rather than the peak amplitude to con-trol for fluctuations in damping. We parameterize thetemperature-dependence as Ndet,o,+ΓT = aξ(aγT

2bp +

bγTbp) + bξ with aγ and bγ fixed by the fit to the data inFig. 8(a), where the theoretical values of the coefficientsare

aξ = 4ξoǫCL

κo,ext

κo

Ao

Γo

ΓT

kB~ωm

(E2)

and

bξ = 4ξoǫCL

κo,ext

κo

Ao

Γo

ΓT

(γlocknmin,lock + Γonmin,o) .

(E3)

19

0 0.2 0.40

100

200

0 0.2 0.40

500

1000

1500

2000

0 0.2 0.40

50

100

150

200(a) (b) (c)

Tbp (K) Tbp (K) Tbp (K)

γ m/2π

(m

Hz)

Op

tom

ech

an

ica

l p

ea

k a

rea

(Hz ×

ph

oto

ns/s

/Hz)

Ele

ctr

om

ech

an

ica

l p

ea

k a

rea

(Hz ×

ph

oto

ns/s

/Hz)

FIG. 8. Temperature sweeps. (a) Intrinsic mechanical dissipation rate γm vs. cryostat base plate temperature Tbp, withlinear fit. (b) Area under mechanical peak in optical output spectrum vs. Tbp. The quadratic fit (gray) calibrates the opticalmeasurement chain efficiency. Blue squares indicate data excluded from the fit, as the membrane mode thermalizes poorly tothe base plate at low temperatures in the presence of the lock beam. (c) Area under mechanical peak in microwave outputspectrum vs. Tbp in the absence of optical illumination. All error bars represent one standard deviation.

At low temperature, the data begins to deviate fromquadratic temperature-dependence, indicating an ele-vated membrane mode equilibration temperature. Weobtain the best fit when excluding the three lowest-temperature data points, marked in blue in Fig. 8(b).Then, since all other parameters are independently mea-sured, the best-fit value of aξ yields ξo = 0.276± 0.007,while bξ, whose fractional uncertainty is much larger,yields ξo = 0.27 ± 0.07. The mode equilibration tem-perature Teq, defined as the temperature at which thearea predicted by the fit is equal to the measured base-temperature peak area, is inferred to be Teq = 70 ±10 mK, with uncertainty again dominated by the frac-tional error in bξ.Finally, we explore how the values of the measurement

chain efficiency ξo and the equilibration temperature Teq

depend on the decision to exclude low-temperature pointsfrom the fit. Excluding three points yields the best agree-ment between the values of ξo inferred from Eqs. (E2) and(E3). When only the lowest point is excluded, Eq. (E3)yields ξo = 0.37 ± 0.05, two standard deviations awayfrom the value obtained from aξ, which is very insen-sitive to the low-temperature data. Excluding two orfour points does not change the fit very much, and theinferred equilibration temperature remains within therange quoted above. We thus take ξo = 0.276 ± 0.007and Teq = 70 ± 10 mK as our final values for the mea-surement chain efficiency and equilibration temperature.The latter corresponds to nth = 1000±140, which agreeswell with the values obtained from independent fits tothe data in Figs. 2(a) and 3(b).We follow an analogous procedure to measure the

microwave measurement chain efficiency ξe via thetemperature-dependence of the electromechanical up-per sideband amplitude Ndet,e,+. The analysis of theelectromechanical temperature sweep data, shown inFig. 8(c), is very similar to the optomechanical analy-sis and will not be discussed in detail. The fit yieldsξe = 0.029 ± 0.001, with no evidence for equilibrationat an elevated temperature. We use this value of ξe to

calibrate broadband measurements of the noise emerg-ing from the external port of the microwave circuit, withwhich we study the dependence of neff,e on microwaveand optical power (Figs. 10(b) and 4(b), respectively).The electromechanical temperature sweep data indi-

cates that laser light is responsible for the elevatedequilibration temperature inferred from Figs. 8(b), 2(a),and 3(b). As noted in the main text, the fact thatthe optomechanical ground-state cooling data plotted inFig. 2(a) agrees well with Eq. (2) without additionalpower-dependent terms indicates that membrane modeheating saturates at the low power used to lock the cavity,with no further scaling with the power of the pump beam.Similar low-power saturation was observed in membraneoptomechanical systems in Refs. [20] and [25]. The ele-vated Teq implies that the appropriate value of the intrin-sic mechanical dissipation rate γm for analyzing the datain the main text is the 70 mK value γm/2π = 113 mHzrather than the 40 mK value γm/2π = 108 mHz.We can compare the temperature sweep calibration of

the optical measurement chain efficiency ξo to a moredirect but less precise measurement. From power metermeasurements we infer the efficiency of the optical pathbetween the optical cavity output and the heterodynedetector input to be ξpath = 0.4. The fractional contri-bution of LO beam shot noise (at power PLO = 1.3 mW)to the sum of shot noise and heterodyne detector darknoise was measured to be ξdark = 0.79. The nominal pho-todetector quantum efficiency of σq = 0.87 then yieldsξo = ξpath ξdark σq = 0.27, consistent with the tempera-ture sweep measurements.On the microwave side, we measured the added noise of

the microwave chain referred to the input of the HEMTamplifier at the 4 K stage of the cryostat by sweeping Tbp

and measuring the thermal noise of the effective 50Ω loadseen by the HEMT. From this measurement we obtainedNHEMT = 8.5 photons/s/Hz, equivalent to an efficiencyof 1/(NHEMT+1/2) = 0.11 referred to the HEMT input.The discrepancy between this number and the measure-ment chain efficiency ξe referred to the transducer’s mi-

20

100 1000

1

2

1.445 1.45 1.455

Frequency (MHz)

-1.455 -1.45 -1.445

Frequency (MHz)

1

1.2

1.4

1.6

1.8o/2 = 200

o/2 =3k

100 10000

1

2

3

(a)

(b)

nm

(p

ho

no

ns)

Sdet,o,±

(p

ho

ton

s/s

/Hz)

Γo / 2π (Hz)

(c)

nm

(p

ho

no

ns)

Γo / 2π = 200 Hz

Γo / 2π = 3 kHz

(Γe+ Γo) / 2π (Hz)

FIG. 9. Analysis of optomechanical spectra. (a) Twoexamples of optomechanical spectra normalized to shot noiseat the optical heterodyne detector input. The left (right)panel displays the lower (upper) sideband spectra. At highdamping, we fit the spectra to the squared sum of two com-plex Lorentzian amplitudes (yellow line) to account for the1.448 MHz mode of the silicon substrate. (b) Comparing twomethods of inferring membrane mode occupancy nm frompurely optomechanical damping sweep, with correspondingfits. Circles and solid line (also shown in Fig. 2(a)) are ob-tained using sideband asymmetry thermometry, while trian-gles and dashed line are obtained from the upper-sidebandspectra Sdet,o,+ and optical measurement chain efficiency ξo.Red and blue data points correspond to the spectra shown inpanel (a). (c) Comparing the two methods of inferring nm

for optomechanical damping sweep at fixed electromechani-cal damping (sideband asymmetry data and fit also shown inFig. 2(b)). All error bars represent one standard deviation.

crowave output port implies a 5.8 dB loss between thetransducer’s microwave output port and the HEMT in-put. This is a rather large value, but the value inferredfrom measurements in Ref. [20] using the same cryostatwas only slightly smaller, and more microwave connec-tors were present in the signal path here than in that

earlier work.

3. Ground-state cooling measurements

To process the data shown in Figs. 2(a) and (b), wefirst normalize the optomechanical upper- and lower-sideband spectra as described in Appendix E 2 above.Two such normalized spectra from the pure optomechan-ical data set are shown in Fig. 9(a). We observe thatthe white noise background around both sidebands in-creases linearly with optical pump power, as expectedin the presence of amplitude and phase noise (see Ap-pendix C 5). We also observe a feature at frequencyωs/2π = 1.448 MHz emerging from the noise floor aroundthe upper sideband at high power. We attribute this fea-ture to a relatively massive vibrational mode of the siliconchip substrate, and account for it by fitting spectra to thesquared sum of two complex Lorentzians with the ampli-tude of the second mode and the relative phase as addi-tional fit parameters. This fit was motivated by obser-vations of destructive interference between a membranemode and a substrate mode in unpublished data froma previous device. For the present device, this coherentdouble-Lorentzian fit differs by less than 10% from a fitto the incoherent sum of two squared Lorentzian ampli-tudes. The fits are not appreciably altered by includingantisymmetric terms as in Eqs. (C14) and (C15).The upper and lower sideband amplitudes Ndet,o,± ob-

tained from these fits must then be corrected to undothe effects of amplitude and phase noise (Appendix C5)before we can infer the membrane mode occupancy. Cor-recting the peak amplitudes in this way in turn re-quires knowledge of the relative magnitudes of the shotnoise-normalized technical noise spectral densities Cxx,Cyy, and Cxy, which are related to the spectral densi-ties of fractional amplitude and phase fluctuations byCxx = 4No (SδA,δA(ωm)/2), Cyy = 4No (Sδφ,δφ(ωm)/2),

and Cxy = 4No (SδA,δφ(ωm)/2), where No = Po/~ωp,o

is the photon flux of the pump incident on the opticalcavity. Here SδA,δA(ω), Sδφ,δφ(ω), and SδA,δφ(ω) are de-fined as one-sided double-sideband spectral densities inunits of rad2/Hz. The factor of 1/2 in each expressionaccounts for the transition from the two-sided spectraldensities assumed by the theoretical framework in Ap-pendix C 5 to the one-sided convention more commonlyused by experimentalists.We made independent measurements of SδA,δA(ω) and

Sδφ,δφ(ω) shortly before the cooldown in which we ac-quired the data presented in this work, using a verysimilar optomechanical cavity in the same setup as anAM/ΦM transducer. We measured amplitude noise bycomparing the sum and difference photocurrents in bal-anced direct detection of the pump beam promptly re-flected off the cavity far from resonance [43]. We then re-peated this measurement with the laser locked to the cav-ity and the pump beam near resonance to transduce am-plitude noise to phase noise, and calibrated the cavity’s

21

AM/ΦM transduction coefficient using a phase modu-lation sideband of known modulation depth [31]. Fromthese measurements we obtained 10 log10 (SδA,δA(ω)) ≈−155 dBc/Hz and 10 log10 (Sδφ,δφ(ω)) ≈ −136 dBc/Hzat frequencies near ωm. Using these numbers in Eq. (C13)at the highest-power data point in Fig. 2, we reproducethe measured values of the off-resonance excess noise So,±

to within a factor of 1.5 if we assume maximal positivecorrelations Cxy =

√

CxxCyy, lending credence to theconclusion that Cyy ≫ Cxx, Cxy. The results obtained

from the subsequent analysis assuming phase noise onlydiffer negligibly from the results obtained assuming themeasured ratio between SδA,δA(ωm) and Sδφ,δφ(ωm) andmaximal positive correlations, so we discuss the formeranalysis here for simplicity.For Cyy ≫ Cxx, Cxy, the anti-symmetric contribution

to both the Stokes and the anti-Stokes peak lineshapes issuppressed and the Lorentzian contribution is negativefor both peaks, resulting in squashing. We undo thisphase noise squashing by taking

Ndet,o,± → Ndet,o,± ±∂(

ǫCLǫPCκo,ext

κoκ2oAo

Γo

ΓTRe[

B±

]

)

∂Cyy

(

∂So,±

∂Cyy

)−1

Sdet,o,±, (E4)

where the two partial derivatives can be evaluated us-ing independently measured transducer parameters, andSdet,o,± is the measured excess over the white noise back-ground level of 1 photon/Hz/s. This method exploitsthe fact that the phase noise is responsible for both thesquashing of the peak amplitudes and the increase in thewhite noise background to correct for squashing with-out assuming a specific value of Cyy. We then use thecorrected peak amplitudes to infer the membrane modeoccupancy nm via Eq. (C9), and fit the data as shownin Figs. 2(a) and (b), with an additional γlocknmin,lock

term in Eq. (2) to account for the lock backaction effectsdiscussed in Appendix D4.As a final check on the consistency of the sideband

asymmetry analysis, we can use the value of the coeffi-cient ao obtained from the fit shown in Fig. 2(a) to getan independent estimate of the phase noise spectral den-sity. Setting the effective optical mode occupancy neff,o =aoΓo in Eq. (C12) and dropping the Cxx and Cxy terms,

we obtain 10 log10 (Sδφ,δφ(ωm)) = −135 +2−3 dBc/Hz, con-

sistent with the independent measurement.As discussed in Appendix E 2, we can also obtain

the membrane mode occupancy nm from the normalizedupper-sideband spectrum Sdet,o,+, given the optical mea-surement chain efficiency ξo and independent measure-ments of other transducer parameters. The nm valuesobtained this way and the corresponding fit are over-laid on the values obtained from the sideband asymme-try analysis in Fig. 9(b) and (c) for the purely optome-chanical and electro-optomechanical damping sweeps re-spectively. Fitting to the purely optomechanical data toEq. (2) yields nth = 750 ± 50 and ao = 5 ± 1, and fix-ing these values in the fit to the electro-optomechanicaldata yields neff,e = 1.02±0.07 for the effective microwavemode thermal occupancy. The origin of the discrepancybetween these values and those obtained from the side-band asymmetry analysis is unknown, but the analysisusing only the upper sideband data is more susceptibleto miscalibration of parameters. The equilibrium occu-pancy nth inferred from this fits is very sensitive to the

first data point, and the discrepancy between the valuesof neff,e obtained from the two electro-optomechanical fitsis strongly correlated with the discrepancy between thevalues of nth.

4. Added noise measurements

The values of Nadd,up plotted in Fig. 2(d) are obtainedfrom the inferred phonon occupancy data in Fig. 2(b) viaEq. (C10), with an additional term accounting for the off-

resonance level So,± of each spectrum. This off-resonancelevel (normalized at the transducer output) can be ob-tained from the directly measured spectra without needfor independent measurement of the optical measurementchain efficiency ξo, because a measurement of nm cali-brates the optical output spectra.The analysis of the data shown in Fig. 3(b) is sim-

pler than the ground-state cooling analysis discussedabove, as both optical pump phase noise and interfer-ence from the substrate mode shown in Fig. 9(a) werenegligible with the optomechanical damping fixed atΓo/2π = 85 Hz. However, the lower sideband amplitudeNdet,o,− was a very small fractional excess over the whitenoise background at high microwave pump power, whereΓo ≪ Γe. We thus calibrate the added noise using the op-tical measurement chain efficiency ξo obtained from thetemperature sweep rather than sideband asymmetry. Wefit the upper sideband amplitude Nout,o,+ = Ndet,o,+/ξoas a function of damping to the expected behavior ob-tained by substituting Eq. (2) into Eq. (C7) and evalu-ating at frequency ωm:

Nout,o,+ = 4ǫCLAo

κo,ext

κo

Γo

γmnth + Γene + Γono

Γ2T

. (E5)

As in the analysis of the data shown in Fig. 2, we alsoinclude a contribution from lock beam backaction in thefit. To obtain the data and the theory curve shown inFig. 3(b), we divide by the product of efficiency ηt and

22

transducer gain AeAo. Calibrating the data with thesideband asymmetry analysis instead yields slightly loweradded noise, so our choice to present the data calibratedusing ξo is also conservative.At the optimal electromechanical damping rate

Γe/2π = 135 Hz where added noise is minimized,the total added noise arises from the weighted sumof couplings to the various baths that determine nm:1.4 photons/s/Hz from neff,e, 1.0 photons/s/Hz from nth,0.4 photons/s/Hz from nmin,lock, 0.2 photons/s/Hz fromnmin,o, and 0.1 photons/s/Hz from nmin,e. The final0.1 photons/s/Hz comes from excess white noise referredto the transducer’s microwave input.

107 1080

0.2

0.4

0.6

0.8

1

1.2

1 10 1000.01

0.1

1

10

Γe / 2π (Hz)

κe,

int / 2

π (

MH

z)

nef

f,e

(ph

oto

ns)

(a)

(b)

n-p,e (photons)

FIG. 10. Microwave pump effects on superconducting

circuit. (a) Internal loss of the microwave circuit κe,int vs.np,e, the resonator coherent state mean photon number. (b)Effective microwave mode occupancy neff,e vs. electromechan-ical damping Γe, with linear fit. All error bars represent onestandard deviation.

Appendix F: Microwave pump effects on

superconducting circuit

We observed two adverse effects of the microwavepump on the superconducting microwave circuit, whoseorigins are likely related. The power-dependence of thecircuit’s internal loss κe,int is plotted in Fig. 10(a) as afunction of np,e, the mean photon number of the coherentstate induced in the microwave mode by the microwavepump, over the range of power used in Fig. 3. As noted

in Appendix D, this power-dependence results in power-dependent electromechanical transducer gain Ae, modi-fies the expected behavior of transducer efficiency ηt vs.electromechanical damping, and leads to sublinear scal-ing of damping vs. microwave pump power.

A more significant constraint on the transducer perfor-mance is the power-dependent microwave mode effectiveoccupancy neff,e. To calibrate neff,e, we first refer themeasured microwave noise spectral density Sdet,e to thetransducer output by dividing by the measurement chainefficiency ξe, then divide by 4κe,ext/κe to translate thespectral density into a mode occupancy. Operationally,we are interested in the behavior of neff,e as a functionof electromechanical damping Γe, plotted in Fig. 10(b).The noise is seen to scale as neff,e = aeΓe + be, with

ae = 1.1 × 10−3 Hz−1 and be = 0.077, with 1% and5% fractional uncertainty in the slope and offset, respec-tively. Evaluated at Γe = 2π × 100 Hz, as in the datashown in Fig. 2(b), the fit yields neff,e = 0.77 ± 0.01,consistent with the value obtained from the Fig. 2(b) fit.Fitting the data in Fig. 3(b), in which Γe is swept, yieldsae = (1.17± 0.05)× 10−3 Hz−1 and be = 0.1± 0.1 , con-sistent with the values inferred from direct measurementsof the noise described above.

One possible contribution to neff,e is the phase noiseof the microwave pump. We used a Rohde & SchwartzSMA100B signal generator to source the pump specif-ically for the ultra-low phase noise afforded by optionSMAB-B711, and measured its DSB noise in situ to be10 log10 (Sδφ,δφ,e(ω)) = −149 dBc/Hz at frequencies nearωm. Following the procedure used to infer the phase noisespectral density of the optical pump from the coefficientao, we find that microwave pump phase noise can accountfor about 25% of the measured value of ae. The remain-der of the power-dependent noise must be attributed tothe behavior the superconducting circuit itself.

At present, the physical origin of this excess power-dependent noise is unknown, though it is likely relatedto the power-dependent internal loss shown in Fig. 10(a).Possibilities include bulk heating of the circuit facilitatedby low electron-phonon thermal conductivity, fluctua-tions of the circuit’s resonant frequency or loss due tothe participation of dielectric substrates in its electricfield distribution, or trapped flux in the circuit, whichis necessarily in close proximity to the optical cavity’sinvar support structure. The low-power asymptote be isa surprising feature of the data that need not have thesame physical origin as the high-power linear behavior.A spectral density of circuit parameter fluctuations thatscales inversely with microwave pump power over somerange is one possible origin for this nonzero intercept.

Finally, we note that microwave power-dependentnoise was also observed in the device described inRef. [20], and motivated our decision to fabricate the su-perconducting circuit from NbTiN rather than Nb as inpast work. In test circuits without released membranesthat were otherwise identical to the flip-chip circuits usedin our transducer, we observed reduction of the power-

23

dependent noise by a factor of 4 in circuits fabricatedfrom NbTiN. This improvement was largely preservedin our integrated electro-optomechanical device, and themicrowave power-dependent noise remained a chief limi-tation primarily because of the large capacitor pad sepa-ration d of the device used in this work. As noted in Ap-

pendix D 3, reproducing the values of d obtained in pre-vious devices would reduce the power required to obtaina given electromechanical damping by at least a factorof 25, which would greatly reduce the impact of power-dependent noise on transducer performance even withoutfurther reduction of the noise itself.

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