Coupling graphene mechanical resonators to superconducting microwave cavities

P. Weber∗, J. Guttinger∗†, I. Tsioutsios, D. E. Chang, A. BachtoldICFO-Institut de Ciencies Fotoniques,Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain

Graphene is an attractive material for nanomechanical devices because it allows for exceptionalproperties, such as high frequencies and quality factors, and low mass. An outstanding challenge,however, has been to obtain large coupling between the motion and external systems for efficientreadout and manipulation. Here, we report on a novel approach, in which we capacitively couplea high-Q graphene mechanical resonator (Q ∼ 105) to a superconducting microwave cavity. Theinitial devices exhibit a large single-photon coupling of ∼ 10 Hz. Remarkably, we can electrostat-ically change the graphene equilibrium position and thereby tune the single photon coupling, themechanical resonance frequency and the sign and magnitude of the observed Duffing nonlinearity.The strong tunability opens up new possibilities, such as the tuning of the optomechanical couplingstrength on a time scale faster than the inverse of the cavity linewidth. With realistic improvements,it should be possible to enter the regime of quantum optomechanics.

Keywords: Optomechanics, graphene, mechanical resonator, NEMS, cavity readout, Duffing oscillator

Mechanical resonators based on individual nanotubesand graphene flakes have outstanding properties. Theirmasses are ultra-low, their quality factors can be remark-ably high, the resonance frequencies are widely tunable,and their equilibrium positions can be varied by a largeamount. As a result, the resonators can be used as sen-sors of mass [1, 2] and force [3–5] with unprecedentedsensitivities, and they can be employed as parametricamplifiers [6] and as tunable oscillators [6–8]. Thus far,all these scientific applications are accomplished in theclassical regime.

Reaching the quantum regime with mechanical res-onators has attracted considerable interest [9, 10]. Thusfar, three groups have been successful in this quest bydemonstrating that the number of vibrational quantacan be lowered below one [11–13]. These three groupswere using different resonators, namely, a piezoelectricresonator, a superconducting resonator, and an opto-mechanical crystal. There is now an intense effort fromthe community to develop new types of opto-mechanicaland electro-mechanical devices, the goal being to explorenew scientific and technological applications when thesedevices will enter the quantum regime. This includes levi-tating particles [14–16], optically trapped cantilevers [17],and heavy pillars [18] to test the foundations of quan-tum mechanics; metal coated silicon nitride membranesto coherently convert radio-frequency photons to visiblephotons [19, 20]; microdisks and nanopillars to boost thesingle-photon coupling and to enter the ultra strong cou-pling regime [21, 22]. In this context, the unique proper-ties of nanotube and graphene resonators are very inter-esting.

Although nanotubes and graphene have exceptionalproperties, an outstanding challenge in approaching thequantum regime has been the development of efficientcoupling to external elements, which would enable mo-

[∗] These authors contributed equally to this work.[†] Corresponding author. E-mail johannes.guettinger@icfo.es.

tional readout and manipulation. For example, whilegraphene has been coupled to an optical cavity [23], the2.3% optical absorption of graphene makes it extremelychallenging to reach the quantum regime, due to heat-ing of the graphene and quenching of the optical cav-ity finesse. Here, we employ a different strategy, whichis to couple the mechanical resonator capacitively to asuperconducting cavity [12, 24–28]. This is a promis-ing approach with graphene resonators, because the two-dimensional shape of graphene is ideal for large capacitivecoupling.

In this work we report on the integration of a circulargraphene resonator with a superconducting microwavecavity. We use a transfer technique to precisely posi-tion a high-quality exfoliated graphene flake with respectto a predefined superconducting cavity. We develop areliable method to reduce the separation between thegraphene membrane and the cavity by tightly clampingthe graphene sheet in between a support electrode anda cross-linked Polymethyl methacrylate (PMMA) struc-ture. We show that this technique allows us to improvethe mechanical stability and to achieve high mechani-cal quality factors. By pumping the cavity on a mo-tional sideband, we are able to sensitively readout thegraphene motion. Importantly, by applying a constantvoltage V DC

g to the graphene, the properties of the op-tomechanical device can be dramatically tuned. Namely,large static forces can be produced, allowing to tune thesteady-state displacement, the mechanical resonance fre-quency, the optomechanical coupling, and the mechanicalnonlinearities. Such a tunability cannot be achieved inother opto-mechanical systems.

Our device (see Fig. 1a-d) consists of a superconduct-ing microwave cavity, modeled as a LC-circuit with an-gular frequency ωc = 1/

√LCtot ≈ 6.7 GHz, capacitance

Ctot ≈ 90 fF, inductance L ≈ 6.3 nH, and characteristicimpedance Zc =

√L/Ctot ≈ 260 Ω. The total capaci-

tance Ctot = C + Cext + Cm(z) effectively consists of acavity capacitance C ≈ 85 fF, a contribution Cext ≈ 5 fFfrom the external feedline, and importantly, a contribu-

arX

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200μm

c)((b)

100μm

(a)

cavity electrode

PMMA cross-linked

graphene

1μm

(d)

transmission line

(e)

graphene electrodes

cavityDC

Vg30mK

300K

CL Cm

Cext

ωp ωd

PMMA cross-linked

cavity electrodegrapheneelectrode

graphene

transmission line

substrate

Z0

Z0

4K

R

FIG. 1: (a) False color SEM image of a circular graphene resonator capacitively coupled to a cavity electrode. The graphenesheet is clamped in between cross-linked PMMA and graphene support electrodes. (b,c) Optical microscope images of the su-perconducting cavity, two electrodes contacting the graphene flake, and a capacitively coupled transmission line. (d) Schematiccross-section of the mechanical resonator and the cavity counter electrode. (e) Schematic of the measurement circuit. A coher-ent pump field at ωp is applied to the transmission line. The graphene mechanical resonator is driven by a field at ωd and aconstant voltage V DC

g . The microwave signal from the cavity is amplified at 4 K with a HEMT amplifier and recorded at roomtemperature with a spectrum analyzer. The impedance Z0 is 50 Ω.

tion Cm(z) ≈ 0.3− 0.4 fF that depends on the grapheneposition z, which arises from the graphene acting as amoving capacitor plate. A small displacement z thereforeproduces a shift in ωc quantified by the optomechanicalcoupling G0 = ∂ωc

∂z . As a result, the interaction betweenthe mechanical resonator and the superconducting cavitycan be described by the Hamiltonian Hint = hG0npz [12]with np the number of pump photons in the cavity. Thecharacteristic coupling at the level of the zero-point mo-tion zzp =

√h/2meffωm is given by the so-called single-

photon coupling g0 = G0zzp, with meff the effective massand ωm/2π the resonance frequency of the mechanicalmode of interest. Central to this work is (i) that the lowmass of graphene boosts zzp and thus g0, and (ii) thatCm and g0 can be tuned electrostatically with V DC

g .

We start with engineering considerations in order tomaximize the coupling g0. When describing Cm by aplate capacitor and noting that C Cext Cm(z) in

our device, we have g0 ≈ ωc

2C∂Cm(z)∂z zzp ∝

√Aωm

ωc

Cd2 us-

ing ∂Cm(z)∂z ∝ A/d2 and zzp ∝ 1/

√Aωm. Here A is the

area of the suspended graphene region and d is the sep-aration between the graphene membrane and its cavitycounter electrode. In order to optimize the coupling g0, itis crucial to minimize both C and d. To this end, we uti-

lize a narrow cavity conductor structured in a meanderto increase L, while minimizing the capacitance to theground for a given ωc. In order to be able to tune d withV DC

g , we use a cavity that is shorted to ground on oneside, allowing for a well defined electrical DC potential.The fundamental mode of the cavity is a quarter wave-length standing wave, with a voltage node at the shortedend and the largest voltage oscillation amplitudes at theopen end. The graphene membrane is coupled close tothe open end of the cavity to harness the largest cavityfields (see Fig. 1b,c) [24, 29, 30]. Using this geometry,we achieve a cavity capacitance of C ≈ 90 fF. This com-pares favorably with C = 18 fF-1 pF in previous stud-ies [12, 24–28]. Note that the lowest values for C havebeen achieved in closed-loop cavities, where the mechani-cal capacitance is incorporated between the two ends of ahalf-wavelength cavity [12, 27]. In this geometry the twoelectrodes of the mechanical capacitance are shorted overthe cavity, so that no static DC potential can be applied.Compared to the capacitance of a gated half-wavelengthcavity [31–33], the capacitance of a quarter wavelengthcavity is lowered by a factor of two.

In order to detect the vibrations of the graphene res-onator, we couple the open-end of the superconductingcavity to a microwave transmission line through the ca-

3

pacitance Cext. The transmission line is used to pump thesuperconducting cavity at frequency ωp/2π with inputpower Pp,in. The transmission line is also employed tomeasure the output power Pout of the cavity at frequencyωc/2π. Pout is amplified at 4 K by a high-electron-mobility transistor (HEMT) with a noise temperatureof about 2 K and measured in a spectrum analyzer (seeschematic in Fig. 1e and SI).

We use a graphene resonator with a circular shape.This geometry improves the attachment of the graphenesheet to its support when compared to the doubly-clamped resonator geometry. As further discussed below,a strong attachment of the graphene to its support is cru-cial to be able to lower d. Another advantage of circulargraphene resonators over doubly-clamped resonators isthat the quality factor tends to be larger [34]. In ad-dition, the mechanical eigenmodes of circular resonatorsare well defined [34, 35]. In particular, it avoids the for-mation of modes localized at the edges, which were ob-served in doubly-clamped resonators [36].

(c)

(a)

(d)

d

Rz

xy

Rg

Si

Nb

PMMA

(b)

FIG. 2: Fabrication process for PMMA-clamped graphenemechanical resonator. (a) Transfer of graphene with PMMA(blue) onto predefined structure (yellow/green, gray). (b)Cross-linking part of the transferred PMMA by electron-beamoverexposure (red). (c) Schematic of the final device. (d)Cross-section of the device.

To fabricate the devices, we start by carving out thesuperconducting cavity structure from a 200 nm thicksputtered Niobium (Nb) film by ion milling and reac-tive ion etching (see SI). We employ a PMMA sup-

ported transfer technique pioneered at Columbia [37] toposition graphene flakes on the superconducting cavitystructure. For this, we exfoliate graphene sheets fromlarge graphite crystals onto a silicon (Si) chip coveredby a polymer film consisting of 100 nm polyvinyl alcohol(PVA) and 200 nm PMMA 495K. The thickness of thePVA/PMMA film is optimized to give the largest opticalcontrast of graphene flakes in an optical microscope. Inparticular, it allows to calibrate the number of layers ofthe graphene flake [38, 39]. The solvability of PVA inwater is used to separate the Si chip from the PMMAwith the graphene. Using a brass slide with a volcano-shaped hole, the membrane is fished from the water anddried on a hotplate. When drying, the PMMA mem-brane gets uniformly stretched across the volcano hole.By mounting the slide upside down into a micromanipu-lator, the graphene sheet can be aligned and transferredonto the pre-patterned superconducting cavity structure,as illustrated in Fig. 2a. To improve the attachment ofthe graphene flake to its support, it was shown that itis important to clamp the graphene membrane on thetwo sides of its surface [40]. For this, we crosslink partof the transferred PMMA with a 10, 000 µC/cm2 elec-tron beam dose (Fig. 2b). The unexposed PMMA is re-moved in 80C hot N-Methyl-2-pyrrolidone (NMP), fol-lowed by critical point drying of the device. As a result,the graphene is firmly sandwiched between the supportelectrode and the crosslinked PMMA (Fig. 2c,d). Usingthis technique the graphene sheet is less likely to collapseagainst its counter electrode. This allows us to increasethe success yield of the device fabrication. We have suc-cessfully lowered the separation to d = 85 nm for a 3.5 µmdiameter graphene resonator, which is equal to the bestdiameter-separation ratio of 2R/d = 40 reported so farfor graphene resonators [41]. In addition, the strong at-tachment between the graphene and its support allowsus to electrostatically tune the equilibrium position by alarge amount (see below).

In this letter we present results measured at 30 mKfor two different graphene devices, hereafter called de-vices A and B. Device A is a three layer graphene res-onator with radius R = 1.75 µm and with d = 95 nm.The number of layers is determined from optical con-trast measurements [38, 39]. The radius of the counterelectrode is Rg = 1.1 µm (see Fig. 2d). Device B is afour layer graphene resonator with the same membraneradius, d = 135 nm and Rg = 1.25 µm.

The principle of mechanical vibration readout is anal-ogous to Stokes and anti-Stokes Raman scattering. Bypumping the cavity at ωp, sidebands in energy are cre-ated at ωp ±ωm due to the coupling of the photons withthe mechanical motion. If the pump is detuned such thatthe upper sideband frequency is matched with the cav-ity resonance frequency ωc = ωp + ωm (see Fig. 3a), theanti-Stokes scattering is resonantly enhanced. Then, therate of the anti-Stokes scattering per phonon is given byΓopt ≈ 4npg

20/κ, with np ∝ Pp,in(ωp) the number of pho-

tons in the cavity. We drive the graphene resonator by

4

applying a constant voltage V DCg and an oscillating volt-

age with amplitude V ACg at a frequency ωd/2π close to

ωm/2π so that ωd = ωc − ωp. As a result, the grapheneresonator vibrates at z(t) = z cos (ωdt+ φ) with φ thephase difference between the displacement and the driv-ing force. The output power at ωc is

Pout = Pp,inκ2ext

κ2 + 4(ωc − ωp)24g2

0

κ2

⟨z(t)2

⟩2z2

zp

. (1)

From a transmission measurement of the feedline wereadily get the resonance frequency of the cavity ωc/2π =6.73 GHz and the total linewidth κ/2π = κext/2π +κint/2π = 15.2 MHz with κext/2π = 2 MHz the cou-pling rate of the superconducting cavity to the feedlineand κint/2π = 13.2 MHz the internal loss rate of thecavity. A detailed analysis of the circuit, which includesa resistance to describe the losses in the graphene flakeand the DC connections, shows that this additional resis-tance contributes roughly 20% to κint (see SI). The highvalue of κint is attributed to the contamination and im-perfections of the cavities. Indeed, we tested the cavity ofdevices A and B at T = 4.2 K before the transfer of thegraphene flakes, and we observed larger κint than whatwe usually observe in devices processed in the same way.

Figures 3b,c show the resonance of the driven vibra-tions for the fundamental modes of device A and B.Modes at higher frequencies are observed as well, butthey are hardly detectable. For device A we extractthe mechanical quality factor Qm = ωm/γm ≈ 100, 000from the linewidth of the resonance γm/2π = 575 Hz.This Qm is comparable to the largest values reportedthus far for graphene resonators [42], showing that ourfabrication process does provide us with mechanical res-onators of excellent quality. We used np = 8000 pho-tons for this measurement, so that Γopt/2π ≈ 0.12 Hz.With these parameters, the measurement imprecision, es-timated to be 2.5 pm/

√Hz, is limited by the noise of the

low-temperature HEMT amplifier. For comparison, theheight of the resonance in the power spectral density ofthe thermal motion at 30 mK is (7 fm)2/Hz (see SI). Indevice B we measure a quality factor of Qm = 17, 700.We attribute this lower Qm to the fact that the devicewas imaged in a scanning electron microscope (SEM) be-fore the measurements, where the graphene surface gotcontaminated by amorphous carbon. This measurementwas done with np = 4500 photons, corresponding toΓopt/2π ≈ 0.01 Hz. If we further increase the pumppower we observe a reduction of the quality factor. Weattribute this reduction of Qm to Joule heating in thegraphene flake. A rough estimate of the heating can bemade by measuring the quality factor as a function ofthe temperature of the cryostat. From this comparison,np = 106 corresponds for instance to a temperature ofabout 200 mK (see SI).

The resonance frequency decreases upon increasing|V DC

g | (see Fig. 3d,e). This reduction of the resonancefrequency has been observed previously in graphene res-onators under tension [42–44]. This softening of the res-

0-4 42-2

2.5

3

3.5

33

33.4

120

80

40

057.5657.555

0

10

20

30

40

z (

pm

)rm

s

ω/2

π (

MH

z)

mg

/2π

(H

z)

0

ω /2π (MHz)d ω /2π (MHz)d

DCV (V)g

2

device Bdevice A

(a)

(b)

(d)

(f)

60

55

50

0 2

5

10

15

DCV (V)g

(e)

33.8

(c)

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15

5

10

0 0

2

46

Γoptω ~ωd m

ωp ωcωd

ω -ω ~ωc p m

(g)

FIG. 3: (a) Measurement scheme: If the pump frequency isdetuned such that ωp = ωc − ωm, anti-Stokes scattering withphonons at rate Γopt leads to a detectable photon populationat ωc. (b),(c) Sideband measurement of the mechanical mo-tion for device A with V DC

g = −2.894 V and V ACg = 190 nV,

and for device B with V DCg = 3.405 V and V AC

g = 4.3 µV. Redlines are Lorentzian fits to the data which yield a mechanicalquality factor of Qm = 100, 000 in device A and Qm = 17, 700in device B. The calculated motional rms amplitude z is plot-ted on the right scale. (d),(e) Mechanical resonance frequencyas a function of V DC

g . We have compensated V DCg by an off-

set of 0.434 V for device A and 0.395 V for device B. Inaddition to capacitive softening, the static deflection zs of theresonator towards the cavity counter electrode is consideredin order to account for the measurement (red line). (f),(g)Single-photon coupling rate g0 = G0zzp. By including thestatic displacement zs we are able to model the single-photoncoupling as a function of V DC

g (red line).

onator is attributed to the change of the restoring po-tential of the resonator by the capacitive energy [42–45].We model the mechanical resonator with a circular mem-brane under tension [46] to quantify the observed depen-dence. When neglecting static deflection, the frequencydependence is given by

ωm(V DCg ) =

√4.92Ehε

meff− 0.271

meff

ε0πR2g

d3(V DC

g )2, (2)

with ε the strain in the graphene sheet, E ≈ 1 TPathe Young’s modulus of graphite, h = ng × 0.34 nmthe graphene thickness, ng the number of graphene lay-ers [3, 43] and meff = 0.27πR2ρ2D the effective massof the fundamental mode (see SI). The two dimensional

5

mass density ρ2D = ηngρgraphene includes the graphenemass density ρgraphene = 7.6× 10−19 kg/µm2 and a cor-rection factor η ≥ 1 to account for contamination onthe graphene surface. From a fit of Eq. (2) to the mea-surements around V DC

g = 0 (in Fig. 3d,e), we extract

meff = 13 · 10−18 kg and ε = 0.036% for device A,and meff = 36 · 10−18 kg and ε = 0.024% for deviceB. The obtained mass is η = 2.2 times larger than thetotal graphene mass for device A and η = 4.5 timeslarger for device B. The larger η for device B mightbe attributed to the amorphous carbon deposited duringSEM inspection. The tension is intermediate comparedto previous measurements, where ε ranges from 0.002%to 1% [3, 34, 42, 47].

(a) (b)

10

20

30

0 8-8 -4 4

ω/2

π (

MH

z)m

DCV (V)g

15

25

35

device B

0 2 4 60

10

20

5

15

z (

nm

)s

DCV (V)g

device B

device A

FIG. 4: (a) Static displacement of the center of the membranecalculated from Eq. (3) with constants cs = 0.405 nm/V2 fordevice A and cs = 0.287 nm/V2 for device B. (b) Mechanicalresonance frequency as a function of V DC

g for device B.

In order to account for the variation of ωm for largeV DC

g in Fig. 3d,e, the static deflection of the graphenesheet towards the cavity counter electrode has to be con-sidered. The static displacement of the center of themembrane zs is given by

zs =ε0R

2g

8Ehεd2(V DC

g )2 = cs(VDCg )2 (3)

for small displacement compared to d. Although therenormalization of the mechanical frequency due to staticdisplacement cannot be solved exactly, as an approxima-tion we can include zs in Eq. (2) using d = d0 − zs, withd0 the separation for V DC

g = 0. We get a good agree-

ment for ωm(V DCg ) between the measurements and the-

ory without any fitting parameter over the V DCg range

shown in Fig. 3d,e. The effect of zs on the shift in ωm is42% at V DC

g = −6 V for device A and 10% at V DCg = 4 V

for device B. The expected variation of zs is plotted inFig. 4a.

The softening of the graphene resonator becomes enor-mous upon further increasing V DC

g , with a reduction ofωm by a factor of three down to ≈ 10 MHz as shownin Fig. 4b for device B. This reduction of ωm is largecompared to that measured in previous works [43–45].Such a large reduction is expected when the capacitive

force becomes comparable to the restoring force of theresonator. When the two forces are equal, ωm drops tozero and the resonator collapses against the counter elec-trode [48]. Even though further work is needed to under-stand the quantitative dependence of ωm on V DC

g , it re-veals that the graphene resonators we fabricate can bendby a large amount without being ripped apart due to thelarge induced strain and without sliding with respect tothe anchor electrodes.

The static displacement of the graphene sheet alsochanges the resonance frequency of the microwave cav-ity upon varying V DC

g . As the graphene moves closer tothe cavity counter electrode, the total capacitance of thecavity increases, so that the cavity frequency decreases.For ∆V DC

g = 6 V the decrease is ∆ωc/2π = 2 MHz indevice A. The measured ∆ωc agrees well with the shiftexpected from the static displacement (see SI).

Our device layout allows us to get large couplingsg0 between the mechanical resonator and the super-conducting cavity (Fig. 3f,g). We extract g0 from themeasurements of the response of driven vibrations atωd = ωm using Eq. (1) where

⟨z(t)2

⟩= [∂zCm ·

V DCg V AC

g Qm/(meffω2m)]2. Remarkably, g0 gets larger

upon increasing |V DCg | for device A. This tunability of

g0 is attributed to the static deflection of the graphenesheet. When incorporating the effect of the static dis-placement into Cm, we get a good agreement betweenthe expected g0 = ωc/(2C)·∂zCm and the measurements,using C = 75 fF and 100 fF for devices A and B, respec-tively (red lines in Fig. 3f,g). These values of C agreewell with C = 90 fF estimated from simulations. Theobtained coupling rates g0 compare favourably with pre-vious experiments carried out with mechanical resonatorsmade from other materials. Indeed, the coupling wasg0/2π ∼ 1 Hz in works with cavity geometries similarto ours [26, 28, 49]. Larger values were achieved withclosed-loop cavities (g0/2π = 40 and 210 Hz) but this ge-ometry does not allow one to apply V DC

g between the me-chanical resonator and a counter electrode as discussedabove [12, 27].

Now, we investigate how the strong tunability of thegraphene equilibrium position affects the nonlinear re-sponse of the mechanical resonator. For this, we measurePout as a function of ωd as in Fig. 3b,c in order to obtainthe response of the vibrational amplitude z(ωd) for largedriving forces at different V DC

g (Fig. 5a-c). Interestingly,we are able to tune the sign of the Duffing nonlinear-ity from a hardening behavior at low V DC

g (Fig. 5a) to

a softening behavior at high V DCg (Fig. 5c). At an in-

termediate V DCg of about 3.4 V, we are able to cancel

the Duffing nonlinearity, that is, the resonant frequencyremains roughly constant upon varying the driving force(Fig. 5b). We quantify the Duffing nonlinearity from thecritical displacement amplitude zcrit above which the re-sponse gets bistable. For a Duffing resonator with lineardamping, the effective Duffing constant αeff is relatedto zcrit by αeff = 8

3√

3meffω

2m/(Qmzcrit) [50]. Figure 5d

6

z (n

m)

-8 -4 0 4 8 -4 0 4 -8 -4 0 4 8

0.4

0.8

1.2

1.6

(ω -ω ) / 2π (kHz)d m

DCV = 2.605 Vg

DCV = 3.605 VgDCV = 3.405 Vg

2

0

-2

-40 1 2 3 4 5

DCV (V)g

15

-2-2

α(1

0 k

g m

s)

eff

(a) (b) (c)

(d)

<

B

FIG. 5: (a)-(c) Dependence of the vibrational amplitudez in device B on the drive frequency for different V AC

g in

each plots. In (a) and (c), V ACg = 3 µV - 31 µV. In (b)

V ACg = 1.9µV − 31µV . The onset of bistability is determined

to be at zcrit = 360 pm for (a) and at zcrit = 900 pm for (c).(d) Effective Duffing parameter αeff as a function of V DC

g .The red line is a plot of Eq. (4) with cs = 0.65 nm/V2 andα0 = 1.9 · 1015 kg·m−2s−2.

shows that αeff is positive at low V DCg and becomes nega-

tive at large V DCg . This dependence can be attributed to

the symmetry breaking of the mechanical motion inducedby static deflection [51, 52], which reads

αeff ≈ α0 −10

meffω2m

α20z

2s (4)

where α0 is the Duffing constant when V DCg = 0; α0

could have a geometrical origin [50]. The fit of Eq. (4)to the measurement yields cs = 0.65 nm/V2 and α0 =1.9 ·1015 kgm−2s−2 (red line in Fig. 5d). This value of csis consistent with that expected from Eq. (3). The signchange of the Duffing nonlinearity due to static defor-mation is a unique property of graphene and nanotuberesonators [53].

The prospects to reach the quantum regime withgraphene resonators are promising. For this, it is illus-trative to compare the figures of merit achieved here tothose reported by Teufel et al. [12], which demonstratedground-state cooling with a superconducting cavity. Inthe device A of our work, we measure g0/2π ≈ 15 Hz,np = 8000, Qm = 100, 000 and κint/2π = 13 MHz, whilethe parameters of Teufel et al. are g0/2π ≈ 200 Hz,np = 4000, Qm = 350, 000 and κint/2π = 40 kHz. Asdiscussed above, an obvious way to improve κint is to fab-ricate cavities with less contamination and imperfections.

κint can then be further reduced by lowering the resis-tance of the graphene flake. This can be achieved for in-stance by selecting thicker graphene flakes or electrostat-ically doping the graphene. Minimizing the graphene re-sistance, together with increasing the area of the interfacebetween the graphene and the electrodes, is beneficial fordiminishing Joule heating at high pump power. In orderto increase g0, we will reduce d further by fabricationand graphene pulling. We should reach g0/2π ≈ 250 Hzwith d = 30 nm. An alternative route to increase g0 is toenhance the coupling using a cooper-pair box [54, 55].

In conclusion, we have reported devices where agraphene resonator is coupled to a superconductingcavity. The tunability of these devices, in combinationwith the large graphene-cavity coupling, constitutes apromising approach to study quantum motion. Thelarge reduction of the resonance frequency of thegraphene resonator observed here is interesting toenhance the zero-point motion and to increase the effectof mechanical nonlinearities [56–58]. The tunabilityof the resonance frequency with V DC

g is suitable forparametric amplification and quantum squeezing ofmechanical states [59]. In these graphene-cavity devices,the opto-mechanical coupling can be varied not onlywith the number of pump photons but also with V DC

g .

Interestingly, the tuning of the coupling with V DCg can

be made faster than that with np, since the inverse ofthe cavity linewidth poses an upper limit on how fastthe photon number inside the cavity can be changed.Because the mass of graphene is ultra-low, its motion isextremely sensitive to changes in the environment. Itwill be interesting to couple the quantum vibrations ofmotion to other degrees of freedom, such as electronsand spins.

Author contributionsPW, JG and IT developed the fabrication process.PW fabricated the devices with support from JG. Theexperimental setup was built up by JG with supportfrom PW. PW and JG carried out the measurements.JG and PW analyzed the data with support from ABand DEC. JG and AB wrote the manuscript with criticalcomments from all authors. AB and JG conceived theexperiment and supervised the work.

AcknowledgementsWe would like to thank Joel Moser, Gabriel Puebla,Christopher Eichler, Andreas Isacsson, Martin Eriksson,Sara Hellmuller and Andreas Wallraff for helpful discus-sions. We gratefully acknowledge Gustavo Ceballos andthe ICFO mechanical and electronic workshop for sup-port. We acknowledge support from the European Unionthrough the RODIN-FP7 project, the ERC-carbonNEMSproject, and the Graphene Flagship (grant agreement604391), the Spanish state (MAT2012-31338), the Cata-lan government (AGAUR, SGR).

7

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1

Supplementary material to: Coupling graphene mechanical resonators tosuperconducting microwave cavities

P. Weber, J. Guttinger, I. Tsioutsios, D.E. Chang, A. Bachtold

ICFO-Institut de Ciencies Fotoniques,Mediterranean Technology Park, 08860 Castelldefels(Barcelona), Spain

I. FABRICATION OF THE SUPERCONDUCTING STRUCTURE

We use a highly resistive silicon substrate (6 kΩcm) with a 295 nm thick, dry chlorinated thermal oxide from NOVAwafers. The wafers are sputtered with 200 nm Nb, followed by optical lithography and ion-milling to define thesuperconducting cavity, the feedline and the graphene contacts. These process steps, and the subsequent wafer dicing,are carried out by STAR cryoelectronics. We use Nb as a cavity material because of the high critical temperatureTc = 9.2 K that allows the cavity to be tested at liquid helium temperature and to sustain large pump fields. The finestructure of the device, shown in Fig. 1a of the letter, consists of the cavity counter electrode and the support electrodesused later on to anchor the graphene flake. The fabrication of this fine structure is carried out with electron-beamlithography (EBL) and reactive-ion etching (RIE). In a first EBL/RIE step, the cavity counter electrode is separatedfrom the support electrodes. As a mask for etching, we use 50 nm aluminium (Al). The Al-mask is structured withEBL using PMMA and etched in 0.2% Tetra-Methyl-Ammonium-Hydroxide (TMAH) diluted in H2O. Unmaskedareas are cleaned from Al-residues with 30 s ion-milling in an argon (Ar) atmosphere. The Nb is etched with RIEin a 10 mTorr SF6/Ar atmosphere with a radio frequency (RF) power of 100 W. In a second EBL/RIE step thecavity counter electrode is thinned down, such that the height difference between the cavity counter electrode andthe support electrodes equals d.

Here we would like to comment as well on the gap between the two support electrodes, which contact the graphene(Fig. 1a). On the one hand this gap allows measuring electrical transport through the graphene, on the otherhand it helps in preventing the collapse of the graphene against the cavity counter electrode during critical pointdrying. The openings did not show a significant influence on the mechanical behaviour in numeric simulations(private communication with Andreas Isacsson and Martin Eriksson).

II. CHARACTERIZATION OF THE ELECTRICAL SETUP AND THE CAVITY

A. Calibration of loss and gain in the input and output lines of the cryostat

To relate the externally applied RF power and the measured RF power to the actual fields at the sample, acareful calibration of the attenuation and gain in the setup is needed. The RF-input lines are attenuated at differenttemperature stages in the cryostat to shield the device from electromagnetic noise and to thermalize the lines. Theattenuation is 10 dB at T = 47 K, 20 dB at T = 4 K, 6 dB at T = 700 mK and 20 dB at T = 30 mK, where we usefor the last attenuation step a directional coupler to physically interrupt the central part of the coaxial line [S1]. Thetotal loss in the lines is the sum of the contributions from the attenuators and the loss in cables and connectors. Inthe input lines of the cold cryostat we measure a total attenuation of loss(ωd) = 57 dB in the 10-100 MHz range andloss(ωc) = 64 dB around ωc/2π = 6.7 GHz. The output of the cavity is shielded by two QUINSTAR CTH0408KCcirculators that are operated as isolators at 30 mK, and then amplified by a low-noise amplifier LNF-LNC4 8A fromLow Noise Factory at 4 K with gain G(ωc) = 43 dB and noise temperature Tnoise ≈ 2 K measured by the factory at10 K.

We measure a detection limit of SN,SA = −157 dBm/Hz in our spectrum analyzer (SA). This noise floor is limited bythe input noise of the amplifier. From kBTnoise(G− loss4K−SA) = −157 dBm we can extract G− loss4K−SA = 38.5 dBand loss4K−SA = 4.5 dB. The total measured gain of the output line is gain = G− loss4K−SA− losssample−4K ≈ 35 dBwith losssample−4K ≈ 3.5 dB which is reasonable considering the losses in the two circulators and the line at thelevel of the sample. Hence we are able to resolve noise powers of SN > −157 dBm/Hz−35 dB= −192 dBm/Hz inthe transmission line of the sample. We measure a total transmitted power of |S21|2 = −29.5 dB in device A and|S21|2 = −28.5 dB in device B, from the output of the RF source to the input of the spectrum analyzer around ωc.This values agree well with our calibration, |S21|2 = −loss+ gain = −29 dB.

2

B. Analyzing the cavity lineshape

To characterize the external coupling and the internal loss of the cavity, we measure the lineshape of the resonanceof the cavity. The normalized transmission is given by [S2]

S21(δωc) = 1− κext/κ

1 + 2iδωc/κ. (S1)

with δωc the detuning from the cavity resonance frequency ωc, κ the total cavity decay rate and κext the externalcoupling rate of the cavity to the feedline. At resonance (δωc = 0) the transmission is

S21,min = −κext

κint, (S2)

with κint the internal cavity decay due to cavity internal losses. By measuring the depth and the width of thetransmission dip, κ, κext and κint are extracted. In Fig. S1 we show the measured transmission spectrum of device A.

6.7 6.72 6.74 6.76

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

S2

1 (

dB

)

Δωc

ω /2π (GHz)p

DC V = -0.434 VgDC V = -6.434 Vg

FIG. S1: Transmission through the feedline around the cavity resonance frequency.

For this plot we subtract the background of the measurement containing contributions from the input and the outputlines. From a fit of the spectrum to Eq. (S1) we extract for V DC

g = −0.434 V the resonance frequency of the cavity

ωc/2π = 6.73 GHz and the external and internal decay rates κext = 2 MHz and κint = 13.2 MHz. By changing V DCg

to −6.434 V we observe a decrease in the resonance frequency ∆ωc/2π ≈ 2 MHz, which is equivalent to a change incavity capacitance of ∆C ≈ 50 aF. In addition, the internal decay rate of the cavity increases to κint = 14 MHz.The change in resonance frequency can be well explained with an increased graphene-cavity capacitance due to staticdisplacement

∆Cm =

∫ 2π

0

dφ

∫ Rg

0

rdrε0

d− ξs(r)− Cm0, (S3)

where Cm0 is the capacitance of the graphene without static displacement at V DCg ≈ 0 V and ξs(r) = zs(V

DCg ) ·

(r2/R2 − 1) (see S.4.2) the static mode shape of the pulled down graphene with zs(VDCg ) the deflection of the center

point of the membrane. If we calculate the capacitive change using zs(VDCg ) = 15 nm for V DC

g ≈ 6 V (Fig. 4a, maintext) we obtain ∆Cm = 49 aF. This value is in excellent agreement with the value of ∆C estimated from the changein ωc.

C. Modeling the dissipation of the cavity

In Fig. S2a we show a detailed equivalent circuit of our measurement setup. In order to model dissipation weincluded (i) an input and output impedance of Z0 = 50 Ω in the RF source and the cryogenic amplifier, (ii) a resistorRm for the loss in the graphene and the DC connection and (iii) a resistor R for the internal loss in the cavity. Byusing a Norton equivalent circuit [S3] we can convert all contributions into a parallel equivalent RLC circuit (Fig. S2b)

3

DCVg

CL Cm

Cext

ωp ωd

HEMT

R

Rm

Z0

Z0

CL CmCext R R´mRextIN

(a) (b)

FIG. S2: (a) Equivalent circuit of the measurement scheme. (b) Norton equivalent circuit where all the contributions areconverted into a parallel RLC circuit.

with 1/Rtot = 1/Rext + 1/R + 1/R′m and Ctot = Cext + C + Cm. We obtain 1/Rext ≈ ω2C2extZ0/2, IN = VpiωCext,

1/R′m ≈ ω2C2gRm and we have made use of the fact that in our circuit ωCextZ0 1 and ω2R2

mC2g 1. The linewidth

of the equivalent parallel RLC-circuit is then given by

κ =1

CtotRtot=

1

CtotRext︸ ︷︷ ︸κext

+1

CtotR︸ ︷︷ ︸κcavity

+1

CtotR′m︸ ︷︷ ︸κg

(S4)

By substituting the equivalent resistances we get

κext =ω2cC

2extZ0

2Ctotand κg =

ω2cC

2gRm

Ctot.

From the measured external linewidth we estimate the coupling capacitance using the above expression to be

Cext =

√2Ctotκext

ω2cZ0

.

Using Ctot = 90 fF, κext/2π = 2 MHz, ωc = 6.7 GHz and Z0 = 50 Ω we get Cext = 5 fF in good agreement with thesimulated values of Cext = 4 fF for device A and Cext = 6 fF for device B.

Furthermore, the measured increase of κ with V DCg in Fig. S1 allows to estimate the resistance Rm in device A. If

we assume that the whole change of the linewidth is due to the static displacement of the resonator (∆κ = ∆κg) wehave

Rm =∆κgCtot

ω2c (C2

m − C2m0)

.

Using ∆κ = ∆κg, ∆κg/2π = 0.8 MHz, Cm = 0.4 fF and Cm0 = 0.35 fF (from section S.2.2) we obtain Rm ≈ 6 kΩ.Here, the change in the capacitance Cm is derived from the measured change of the cavity resonance frequency∆ωc. By inserting the value for Rm in equation (S4) we obtain κcavity/2π = 10.8 MHz and κg/2π = 2.4 MHz withκint = κcavity + κg. The high value of κint is therefore mainly attributed to the contamination and imperfections ofthe cavities. Indeed, we have tested the cavity of devices A and B at T = 4.2 K before the transfer of the grapheneflakes, and we observed larger κint than what we usually observe in devices processed in the same way.

III. COUPLING OF THE GRAPHENE RESONATOR TO THE SUPERCONDUCTING CAVITY

A. Estimation of the coupling parameter g0

By applying a pump power Pp,in at frequency ωp at the feedline, we create a cavity photon population of

np =1

hωpPp,in ·

2

κext· κ2

ext

κ2 + 4(ωp − ωc)2.

Here, κext/2 is the coupling of the input mode of the feedline to the cavity and κ2ext/(κ

2 +4(ωp−ωc)2) is the lineshapeof the cavity resonance. The photon population in the cavity interacts by Stokes (-) and anti-Stokes (+) scattering

4

with the mechanical resonator. The scattering rates are given by

Γ± = 4npg20

κ

κ2 + 4(ωp − ωc ± ωm)2,

with g0 the single-photon coupling and ωm/2π the mechanical resonance frequency. In the case of ωp − ωc = −ωmanti-Stokes scattering is resonantly enhanced and we have Γopt ≈ 4npg

20/κ in the so-called resolved sideband limit

where ωm κ. Here we introduced Γopt, the opto-mechanical coupling rate.The anti-Stokes scattering leads to an equilibrium cavity population nc at ωc determined by ncκ ≈ Γoptnm. Here,

we assumed negligible thermal population of the superconducting cavity (at 30 mK nc,th = 1/(exp hωc/kBT −1) 1)and negligible direct population due to the phase noise of the pump. The number of phonons nm is related to thezero-point motion zzp by nm ≈ 〈z〉2 /2z2

zp, where 〈z〉 is the time averaged deflection of the effective mass motion. Thecavity mode leaks into the output mode of the feedline with a rate κext/2 and results in the detectable output powerPout = nchωcκext/2 or

Pout(ωc) = Pp,in ·κ2

ext

κ2 + 4(ωp − ωc)2·(

1

κ

∂ωc

∂x

)2

· 2⟨z2⟩.

From the measured output power we can then estimate g0 as

g0 = zzp

√Pout(ωc)

κ2

nphωcκext 〈z2〉. (S5)

To model the dependence of g0 = G0zzp on the voltage V DCg between the graphene and the cavity counter electrode,

we have to account for the V DCg dependence of both G0 and zzp. To estimate G0(V DC

g ) we use the calculated valueof the equilibrium position zs (Fig. 4a of the letter) to substitute d by d = d0 − zs in the calculated graphene-cavitycapacitance Cm

G0(V DCg ) =

ωc2Ctot

∂Cm(V DCg )

∂z≈ ωc

2C

0.433πR2g

[d0 − zs(V DCg )]2

,

where d0 is the value of d at V DCg = 0, Ctot is the total cavity capacitance approximated by the cavity capacitance

C and Rg is the radius of the cavity counter electrode. The factor 0.433 is a correction due to the effective massmodeling (see section S.4.4). For the calculation of the capacitance Cm see below the section about the effective massmodelling. The increase of the zero-point motion is accounted for by calculating zzp as a function of V DC

g from the

measurement of the resonance frequency ωm as a function of V DCg in Fig. 3c,d of the letter

zzp(V DCg ) =

√h

2meffωm(V DCg )

.

B. Displacement sensitivity

The detection limit of our readout circuit SN = −192 dBm/Hz (see section S.2.1) imposes a limit on the measurementimprecision

√Sz,imp with

Sz,imp =SNκ

2z2zp

nphωcκextg20

.

For the parameters in device A we get√Sz,imp = 2.55 pm/

√Hz at np = 8000 and

√Sz,imp = 230 fm/

√Hz at

np = 106. For comparison, the height of the resonance in the power spectral density of the thermal motion at 1 K

is (42 fm/√

Hz)2 ((7 fm/√

Hz)2 at 30 mK). We will improve our displacement resolution (i) by reducing the lossin the cavity (up to a factor 8 improvement in

√Sz,imp), (ii) by using a quantum limited amplifier [S4] (up to a

factor 10 improvement in√Sz,imp) and (iii) by increasing the coupling (with a factor 10 improvement in

√Sz,imp for

g0/2π = 70 Hz).

5

IV. MECHANICAL MODELLING

A. Circular graphene resonator in the membrane limit

We model the deflection ξ(t, x, y) of the graphene resonator as a thin plate subject to large external stretching(membrane limit) [S5]

ρ2D∂2ξ

∂t2= T∇2ξ + P (x, y), (S6)

with ρ2D the sheet mass density, P (x, y) the local pressure in z-direction and T a stretching force per unit lengthat the edge of the membrane. If we consider radially symmetric modes ξ(t, r), the stretching force T is related to aradial strain ε = (R′ −R)/R with the elongated radius R′ by

T = Ehε = Etngε, (S7)

with the Young’s modulus of graphite E ≈ 1 TPa or the two dimensional graphene Young’s modulus Et = 340 N/m[S6], ng the number of graphene layers and t = 0.335 nm [S7] the interlayer spacing in graphite. The total sheet massdensity ρ2D = ηngρgraphene includes the mass from the graphene layers, with the graphene mass density ρgraphene =7.6× 10−19 kg/µm2, and a correction factor η ≥ 1 to account for additional adsorbents on the graphene.

The electrostatic pressure due to the gate voltage is modelled in a parallel plate approximation with the capacitiveenergy given by U = 1

2CmV2g . If we expand the capacitance in terms of ξ we get

U ≈∫dxdy

ε0V2g

2

1

d− ξ(x, y)

≈∫dxdy

ε0V2g

2d

(1 +

ξ(x, y)

d+ξ(x, y)2

d2+ξ(x, y)3

d3+ . . .

)∂U

∂z≈∫dxdy

ε0V2g

2d2

(1 +

2ξ(x, y)

d+

3ξ(x, y)2

d2+

4ξ(x, y)3

d3+ . . .

)≈∫dxdyP (x, y).

The differential equation for the deflection is then given by

ρ2D∂2ξ

∂t2= T∇2ξ −

ε0V2g

2d2

(1 +

2ξ(x, y)

d+

3ξ(x, y)2

d2+

4ξ(x, y)3

d3+ . . .

)(S8)

To solve the equation, we decompose the deflection ξ(r, t) into a static displacement ξs(r) and time-dependent(radial) modes k with amplitude ξk(r)

ξ(r, t) ≈ ξs(r) +∑k

ξk(r)e−iωt.

B. Static displacement as a function of DC voltage

For the static displacement we have

0 = T∇2ξs(r)−ε0(V DC

g )2

2d2

(1 +

2ξs(r)

d+ . . .

)by assuming 2ξs(r)/d 1. The solution at lowest order in ξs(r)/d is given by

ξs(r) =ε0(V DC

g )2

8Td2

(r2 −R2

)with the normalized center deflection

zs =ε0R

2g

8Td2(V DC

g )2 = cs(VDCg )2. (S9)

6

For device A, the approximation of small static deflections is well valid up to V DCg ≈ 3.5 V where zs = 5 nm and

2zs/d = 0.1 1. At large V DCg we underestimate the static displacement by not including higher order corrections

of the electric force. On the other hand we also underestimate the mechanical force when neglecting nonlinear effectsas described below. In device B zs ≈ 7.5 nm with 2zs/d = 0.1 1, which corresponds to V DC

g ≈ 5 V.The assumption of constant strain at moderate gate voltages is justified by analysing the strain induced by the

static deflection. At V DCg ≈ 6 V the additional strain induced by the static deflection of 10 nm (in device B) is given

by εs = 2 · 10−5 εinit, significantly smaller than the initial strain.

C. Mechanical resonance frequency as a function of gate voltage

If we assume orthogonal modes and neglect mode coupling, we can project Eq. (S8) on the fundamental mode

−ρ2Dω2ξf(r) = T∇2ξf(r)−

ε0(V DCg )2

d3ξf(r) (S10)

and solve for the mode amplitude ξf(r). Considering a clamped boundary with ξf(R) = 0 we get

ξf(r) = zJ0

(2.4

Rr

), (S11)

where z = ξf(0) is the deflection amplitude at the center of the membrane and J0 is the 0th Bessel function withJ0(2.4) = 0. The resonance frequency as a function of gate voltage is then given by

ωm(V DCg ) =

√2.42T

R2ρ2D− ε0d3ρ2D

(V DCg )2. (S12)

By taking into account the reduced radius of the cavity counter electrode Rg with respect to the membrane radius R,the electrical force gets reduced by a factor R2

g/R2 and we obtain

ωm(V DCg ) =

√2.42T

R2ρ2D−R2g

R2

ε0d3ρ2D

(V DCg )2 (S13)

for the resonance frequency as a function of V DCg . At V DC

g = 0 V we get in agreement with Ref. [S8]

ωm(0) =2.404

R

√Ehε

ρ2D.

D. Harmonic oscillator model with effective mass

It is instructive and useful to analyze the dynamics of the resonator by a harmonic oscillator model with an effectivemass. From the total kinetic energy

Ekin =1

2ρ2D2π

∫rξ2

f (r)dr =1

2meff z

2 (S14)

we obtain for the effective mass

meff = 0.27ρ2DπR2, (S15)

with

2π

∫ R

0

drJ0

(2.4

Rr

)r = 2π

R2

2.42

∫ 2.4

0

dr′J0(r′)r′ = 0.27πR2.

7

We multiply all the terms of Eq. (S8) by J0

(2.4R r)

and integrate over the area. As a result, we get the normalizedequation of motion with higher order corrections for the capacitive force

meffω2z =

(0.271πR22.42T − 0.271

ε0πR2gV

2g

d3

)z (S16)

+0.1963ε0πR

2gV

2g

2d4z2

+0.1252ε0πR

2gV

2g

d5z3 + . . . .

Note that we obtain the same expression as Eq. S13 for the resonance frequency

ωm(V DCg ) =

√4.92Ehε

meff− 0.271

meff

ε0πR2g

d3(V DC

g )2 =

√2.42T

R2ρ2D−R2

g

R2

ε0d3ρ2D

(V DCg )2.

E. Induced motion over capacitive drive

First we are interested in the mechanical response under a weak electrostatic drive, such that nonlinear motionaleffects can be neglected:

meff z(t) + γmmeff z(t) +meffω2mz(t) = Fd cos (ωdt). (S17)

The damping is characterized by the linewidth γm = ωm/Qm with Qm the quality factor of the mechanical resonator

and ωm the mechanical resonance frequency. The electrostatic drive amplitude is given by Fd = ∂zCmVDCg

√2V AC

g ,

with V ACg the root-mean-square amplitude of the drive voltage. Including the capacitive correction for the modeshape

from above we have Fd = 0.433 · ε0πR2g/d

2V DCg

√2V AC

g . With the ansatz z(t) = zeiωdt we get

−meffω2dz + iγmmeffωdz +meffω

2mz = Fd (S18)

and hence for the amplitude

z(ωd) =Fd/meff√

(ω2m − ω2

d)2 + γ2mω

2d

. (S19)

When driving at resonance ωd = ωm, we have

z(ωm) =Fd

meffγmωd. (S20)

F. Nonlinear lineshape

We include the cubic nonlinear force in the driven equation of motion

meff z(t) + iγmmeff z(t) +meffω2mz(t) + αeffz

3(t) = Fd cos (ωdt). (S21)

with αeff a constant.In analogy to the linear lineshape in Eq.(S19) we get for the amplitude of the motion

z(ωd) ≈ Fd/2meffω2m√(

ωd−ωm

ωm− 3

8αeff

meffω2mz2

0

)2

+ (2Q)−2

in the limit of small oscillations where αeff z3 < kz/Qm (see Ref. [S9] Eq. 1.31a). The onset of bistability is given

by [S9]

zcrit =

√8

3√

3

meffω2m

Qαeff= 1.24

√meffω2

m

Qαeff. (S22)

8

Thus the Duffing nonlinearity can be calculated from the critical deflection amplitude

αeff =8

3√

3

meffω2m

Qmz2crit

. (S23)

G. Harmonic oscillator with nonlinear contributions and static displacement

We consider quadratic and cubic nonlinear terms in the equation of motion (without dissipation and drive)

meff z(t) = −meffω2mz(t)− β0z

2(t)− α0z3(t) + Fel, (S24)

with β0 and α0 two constants. With the separation ansatz z(t) = zs + zf(t) we get

meff zf(t) = −[meffω

2m,0zs + β0z

2s + α0z

3s −

1

2∂zCm(zs)V

2g

]−[meffω

2m,0 + 2β0zs + 3α0z

2s −

1

2∂2zCm(zs)V

2g

]︸ ︷︷ ︸

ktot

zf(t)

−[β0 + 3α0zs −

1

4∂3zCm(zs)V

2g

]︸ ︷︷ ︸

βtot

z2f (t)

−[α0 −

1

12∂4zCm(zs)V

2g

]︸ ︷︷ ︸

αtot

z3f (t)

From the first bracket we can estimate the static displacement by neglecting the nonlinear contributions andassuming a similar deflection profile as the fundamental oscillation

zs ≈1

2meffω2m,0

∂zCm(V DCg )2

≈ 0.433

2meffω2m,0

ε0πR2g

d2(V DC

g )2

=ε0R

2g

7.21Td2(V DC

g )2.

Compared to the result of the direct calculation with the static modeshape in Eq. (S9) there is a small difference witha factor 7.21 instead of 8 in the denominator. For device B, it is possible to analyze the nonlinear contribution inktot. For zs = 17 nm the nonlinear contribution equals the linear contribution as 3α0z

2s = meffω

2m = 1.6 kg·s−2 with

α0 = 1.9× 1015 kg·s−2m−2.For small nonlinear amplitudes we transform the quadratic and cubic nonlinear terms in a single cubic term [S10]

αeff ≈ αtot −10

9

β2tot

meffω2m

≈ α0 −1

12∂4zCm(V DC

g )2 − 10

9meffω2m

(3αzs −

1

4∂3zCm(V DC

g )2

)2

≈ α0 −1

12∂4zCm(V DC

g )2 − 10

144meffω2m

∂3zC

2m(V DC

g )4 − 10

meffω2m

α20z

2s .

We assumed that ∂nz Cm(zs) ≈ ∂nz Cm(zs = 0) and that β is small (no symmetry breaking visible at small V DCg ). With

α0 = 1.9 × 1015 kg s−2m−2 and Vg = 5 V, the second and the third terms of the last equation are ≈ −6 × 1012kgs−2m−2 and ≈ −6×1011kg s−2m−2 respectively. This is much smaller than the fourth term (≈ −4×1015kg s−2m−2).Hence we can write

αeff ≈ α0 −10

meffω2m

α20z

2s . (S25)

9

The measured values for the Duffing nonlinearities are within the range of αeff = 1.74 · 1012 kg·m−2s−2 to7.16 · 1017 kg·m−2s−2 observed in other graphene resonators [S11, S12] and are compatible with the observationof intermediate strain.

H. Heating of the mechanical resonator by large pump fields

device B

(a) (b)

pump photons n p

210 310410

510 6100

10

5

15

20

25

3Q

(1

0)

m

0

10

5

15

20

25

3Q

(1

0)

m

T (K)cryostat

0.1 1

device B

0.01 10

FIG. S3: Heating of the mechanical mode by increasing the pump field in device B.

In Fig. S3a we show a measurement of the quality factor in device B as a function of the number of pump photonsin the cavity. While the quality factor is roughly constant for np < 6000, Qm decreases for higher pump fields. Uponincreasing the temperature of our cryostat Qm also decreases, as shown in Fig. S3b. From the comparison betweenthe two figures we conclude that a pump power of np = 106 has the same influence on the mechanical resonatoras heating the cryostat to 200 mK. In order to minimize the heating it is beneficial to reduce the resistance of thegraphene and to improve the heat flow away from the mechanical resonator.

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[S2] B. A. Mazin, Microwave kinetic inductance detectors, Ph.D. thesis, California Institute of Technology (2005).[S3] M. Goppl, A. Fragner, M. Baur, R. Bianchetti, S. Filipp, J. M. Fink, P. J. Leek, G. Puebla, L. Steffen, and A. Wallraff,

Journal of Applied Physics 104, 113904 (2008).[S4] M. Castellanos-Beltran, K. Irwin, G. Hilton, L. Vale, and K. Lehnert, Nature Physics 4, 929 (2008).[S5] L. D. Landau, E. Lifshitz, J. Sykes, W. Reid, and E. H. Dill, Theory of elasticity: Vol. 7 of course of theoretical physics,

2nd ed. (Pergamon Press, 1970) pp. 61,114.[S6] C. Lee, X. Wei, J. W. Kysar, and J. Hone, Science 321, 385 (2008).[S7] R. Al-Jishi and G. Dresselhaus, Phys. Rev. B 26, 4514 (1982).[S8] S. Timoshenko, Vibration problems in engineering (1974).[S9] R. Lifshitz and M. Cross, Reviews of nonlinear dynamics and complexity 1, 1 (2008).

[S10] A. H. Nayfeh and D. T. Mook, Nonlinear oscillations (John Wiley & Sons, 2008).[S11] A. Eichler, J. Moser, J. Chaste, M. Zdrojek, I. Wilson-Rae, and A. Bachtold, Nature Nanotechnology 6, 339 (2011).[S12] X. Song, M. Oksanen, M. A. Sillanpaa, H. Craighead, J. Parpia, and P. J. Hakonen, Nano Letters 12, 198 (2011).