Dynamical strong coupling and parametricamplification of mechanical modes ofgraphene drumsJohn P. Mathew1, Raj N. Patel1,2, Abhinandan Borah1, R. Vijay1 and Mandar M. Deshmukh1*

Mechanical resonators are ubiquitous in modern informationtechnology. With the possibility of coupling them to electro-magnetic and plasmonic modes, they hold promise as the keybuilding blocks in future quantum information technology.Graphene-based resonators are of interest for technological appli-cations due to their high resonant frequencies, multiple mechan-ical modes and low mass1–7. The tension-mediated nonlinearcoupling between various modes of the resonator can be excitedin a controllable manner8–11. Here we engineer a graphene resona-tor with large frequency tunability at low temperatures, resultingin a large intermodal coupling strength. We observe the emer-gence of new eigenmodes and amplification of the coupledmodes using red and blue parametric excitation, respectively.We demonstrate that the dynamical intermodal coupling istunable. A cooperativity of 60 between two resonant modes of∼100 MHz is achieved in the strong coupling regime. The abilityto dynamically control the coupling between the high-frequencyeigenmodes of a mechanical system opens up the possibilityof quantum mechanical experiments at low temperatures12,13.

Experiments in cavity optomechanics14,15, where a low-frequencymechanical oscillator parametrically modulates the resonantfrequency of an electromagnetic mode of a cavity, have demonstratedthe ability to prepare a mechanical system in its ground state16–19 aswell as entanglement between the propagating photons andphonons20. As mechanical resonators support a large number ofvibrational modes, a natural extension of the optomechanicalscheme is to use coupled modes of a mechanical resonator wherea high-frequency mode plays the role of the cavity21,22. There hasbeen a considerable interest in exploring coupling among the eigen-modes of a mechanical system, demonstrating coherent couplingbetween low-frequency modes21–24. The eventual goal of such exper-iments is to demonstrate coupling in the quantum regime12,13. Toachieve a quantum coherent coupling, graphene mechanical resona-tors offer many advantages. First, the high frequencies of the gra-phene drum resonators yield fewer phonons once cooled down tolow temperatures. Second, the large frequency dispersion withgate voltage results in large intermodal coupling, offering advan-tages when using higher-frequency modes as a phononic cavity.

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Figure 1 | Graphene drum electromechanics in the low-tension regime. a, False-coloured scanning electron microscope image of the graphene drumresonator. The region shaded green shows the suspended part of the graphene. S, source; D, drain. Scale bar, 3 μm. b, Schematic of the circuit used toactuate and detect the mechanical modes of the drum. c, The large frequency tunability of the modes with a d.c. gate voltage. The modes near 95 MHzshow an avoided crossing (highlighted by the dashed rectangle).

1Department of Condensed Matter Physics and Materials Science, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India.2Physics Department, Birla Institute of Technology and Science Pilani – K. K. Birla Goa Campus, Goa 403726, India. *e-mail: deshmukh@tifr.res.in

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Furthermore, the large quantum zero-point motion of grapheneresonators offers the advantage of efficient coupling with electro-magnetic cavities4–7.

Here we fabricate graphene drum resonators1–3 by a new methodwith low built-in tension, yielding large frequency tunability at lowtemperatures (see Methods for details on the fabrication). Theactuation and detection of the mechanical modes of graphene areimplemented using an all-electrical scheme. The drums we studyconsist of a graphene flake in contact with Cr/Au electrodes thathave a central region with a diameter 3.5 μm suspended 300 nmabove a local gate electrode of Ti/Pt (Fig. 1a,b). The local gate ison a sapphire substrate and the graphene is transferred ontoelectrodes on SiO2. The region of SiO2 over the gate electrode isetched out before placing the flake. The suspended region of the gra-phene and the gate electrode form an electromechanical systemwhere the capacitance between the gate and graphene is used totransduce the mechanical motion of the drum. The gate electrodeserves the dual purpose of actuating the membrane’s motion aswell as tuning the resonance frequencies of various modes.

All measurements were carried out with the sample kept under ahigh vacuum at 5 K in a 4He cryostat. Weak radio frequency (RF)signals are applied to the gate to drive the resonator, and themotion is measured using a lock-in detection of the RF signal trans-mitted through the graphene flake. Figure 1b shows the schematic ofthe circuit used for the actuation and detection of the resonatormodes. The driving RF signal (of amplitude Vg and frequency ωd)is combined with an appropriate RF pump signal (of amplitude Vp

and frequency ωp) and applied to the gate electrode. A d.c. voltage(applied using a bias tee) on the gate electrode induces tension inthe suspended graphene due to the electrostatic force; this tensionprovides tunability of various modes of the resonator with the d.c.gate voltage (Fig. 1c). Moreover, for a membrane with low built-intension, the gate voltage induced static deflection can be significantand induce a stress that varies spatially over the membrane. Thisinhomogeneous, deflection-induced tension couples various flexuralmodes of the resonator and can lead to frequency crossings8.The intermodal coupling strength can be increased in graphenedue to the high elastic modulus and large static deflection (seeSupplementary Information Section II for the device specific,tunable intermodal coupling strength). Our device geometry thusenables us to dynamically tune the intermodal coupling under theaction of a detuned pump using an all electrical configuration21.

Figure 2a shows the electrostatic tunability with the gate voltageof two coupled modes, which allows them to be brought into aregion of avoided crossing. We refer to the lower (higher) frequencymode as mode 1 (2) in the remainder of the text. The separationbetween modes 1 and 2 decreases with decreasing |Vdc

g | achievingperfect tuning at Vdc

g = −35V (see the Supplementary Informationfor detailed measurements on the gate tunability of the coupling).The dynamics of the system can be studied under the influence ofa red (ωp ≈ (ω2 − ω1) = Δω) or blue (ωp ≈ (ω1 + ω2)) detuned pumpsignal that parametrically modulates the coupling. This is analogousto optomechanical systems where the parametric coupling of amechanical mirror to an optical cavity can be manipulated to coolor amplify the mechanical motion by using red or blue detunedpumps, respectively25. In our system the first mode (oscillator) isparametrically coupled to the higher-frequency second mode,which plays the role of a cavity. This allows for experiments wheretwo coupled modes with a large frequency separation can be usedto cool the low-frequency mode, equivalent to optomechanicalsystems21. In the Supplementary Information we show strongcoupling between modes separated by a frequency ratio of ∼2,however, we now focus on the dynamics of the two nearby modesin the presence of a red detuned pump.

By applying Vdcg = −36V on the gate electrode, we tune the

modes to ω1 = 2π × 94.65MHz and ω2 = 2π × 96.94MHz such

that Δω = 2π × 2.29MHz. We measure the response of mode 1 asa function of the pump frequency at various pump amplitudes.Here, the response of the mode is detected with a weak drivingsignal of amplitude ∼1 mV at ωd. Figure 2c shows the response ofmode 1 as a function of the pump frequency at Vp = 0V. Atzero pump power, mode 1 remains unperturbed. When the pumpstrength is increased to 1.5 V, an avoided crossing in the responseof mode 1 is observed in regions where the pump frequencyapproaches Δω (Fig. 2d). This avoided crossing signifies a regimeof strong coupling, where mixing between the modes gives riseto new eigenmodes in the system21,22,26. In the strong couplingregime we expect coherent energy transfer between the twomodes. This is demonstrated by the measurements given in theinset of Fig. 2d, where the response is simultaneously demodulatedat ωd + ωp with no driving signal at ω2. Energy transfer to thesecond mode is observed as a non-zero signal in the region whereωd + ωp ≈ ω2.

The amount of splitting characterizes the strength of the intermodalcoupling g and it is tunable with the amplitude of the pump signal.

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Figure 2 | Strong coupling between electromechanical modes. a, Zoomed-in scan of the frequency response of the coupled modes as a function of thed.c. gate voltage, showing avoided crossing behaviour with no parametricpumping. Red pump experiments are done at the gate voltage (Vdc

g = −36V)indicated by the white dashed line. b, The red-detuned pump is shownalongside the two coupled modes on a frequency axis. c,d, The response ofmode 1 as a function of the red-pump detuning when Vp = 0V (c) andVp = 1.5V (d). For non-zero pump amplitudes, mode 1 is seen to split in thevicinity of ωp≈Δω. Inset, the response detected at ωd +ωp. Energy transferto the second mode is seen when ωd +ωp≈ω2.

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Figure 3a,b shows the response of mode 1 when pumped at a frequencyof Δω = 2π × 2.29MHz. As the pump voltage is increased, the reson-ance peak of mode 1 can be seen to split into two peaks. At moderatepowers this peak splitting is equivalent to an optomechanically inducedtransparency21,27. As the pump voltage is further increased, theresponse of mode 1 splits into well-resolved peaks with the separationgiven by the coupling rate equal to 2g. The region of splitting ischaracterized by the coupling rate of the modes becoming largerthan their individual dissipation rates (γi). At the highest pumpamplitude of Vp = 1.5V the splitting (2g ≈ 2π × 450 kHz) exceedsboth γ1 ≈ 2π × 64 kHz and γ2 ≈ 2π × 51 kHz. The intermodal coup-ling can be quantified by a figure of merit called cooperativitydefined as28 C = (4g2)/(γ1γ2). Figure 3c shows the cooperativity asa function of the pump amplitude; C is seen to be as high as 60,

indicating that many cycles of energy transfer can be achievedbetween the modes before dissipation to the bath. (Further datashowing strong coupling between modes that have a frequencyratio of ∼2 is included in the Supplementary Information.)

The dynamics of our system (the possible microscopic origin isdiscussed in Section II of the Supplementary Information) can bedescribed using the equations of motion for two coupled vibrationalmodes22, given by:

x + γ1x + (ω21 + Γ1 cos(ωpt))x +Λ cos(ωpt)y = F1 cos(ωdt +f) (1)

y + γ2y + (ω22 + Γ2 cos(ωpt))y + Λ cos(ωpt)x = F2 cos(ωdt +f) (2)

where x and y are the displacements of the two modes, Γi are theparametric drives that modulate the stiffness of the modes, Λ is

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Figure 3 | Normal mode splitting and large cooperativity. a,b, With an increase in the red-detuned pump amplitude, mode 1 is seen to split into twowell-resolved peaks with a separation given by 2g. The plots show the response of mode 1 at coarse (a) and fine (b) intervals of the pump voltage,respectively. A constant offset has been added to the response at various pump amplitudes in a. Dotted lines are guides to the eye. c, Cooperativity of themodes is seen to be as high as 60 at the largest pump amplitudes. The solid line is a quadratic fit of the data (circles) to the equation C = αV2

p . d, Mode 1response is probed as a function of the pump detuning over a larger frequency range with Vp = 1.5V. Apart from normal mode splitting at ωp≈Δω, an extrasplitting at ωp≈ (Δω)/2 can be seen, suggesting the onset of higher-order intermodal coupling. e,f, Schematics to explain the normal mode splitting andavoided crossing due to higher-order coupling. e, The coupled modes undergo mode splitting in the presence of a red detuned pump when the coupling ratecompensates the individual mechanical losses. f, The coupled modes further interact near ωp =Δω/2 due to the second-order coupling. Hereω1 = 2π × 94.65MHz and ω2 = 2π × 96.94MHz. g, The simulated response of mode 1 closely resembles the experimental data, showing first- andsecond-order coupling. A frequency-dependent background was added to the simulated response.

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the coefficient of mode coupling, and Fi are the forces at drivefrequency ωd. The terms involving Λ are responsible for the transferof energy between the two modes, whereas the terms with Γi, whencombined with Λ, excite higher-order coupling between the modes.The mode splitting, 2g, is related to Λ by22 2g ∼ Λ/(2

������

ω1ω2√

).Equations (1)–(2) are used to solve the dynamics of the systemnumerically. The above model captures all of the experimentallyobserved features. The coupling coefficients Λ and Γi are pro-portional to the pump amplitude Vp. The experimentally observedsplitting follows a linear trend (Fig. 3a) with increasing pumpvoltage. This is indicative of the Vp dependence of Λ.

The terms involving Γi result in higher-order coupling betweenthe modes where a second-order coupling is characterized byΓiΛ. In Fig. 3d, the response of mode 1 over a large range ofpump frequencies shows the emergence of an extra splittingwhen ωp ≈ Δω/2. The simulations and experimental results showvery good agreement, as seen in Fig. 3g. (The SupplementaryInformation shows detailed measurements and comparisonsbetween the first- and second-order coupling rates.)

The nature of the dynamical intermodal coupling can bechanged under the action of a blue (ωp ≈ ω1 + ω2) detuned pump.A red detuned pump swaps the energy between modes, whereasthe blue detuned pump amplifies both modes. Figure 4a showsthe frequency spectrum of the modes along with the blue pump(ω1 = 2π × 94.90MHz, ω2 = 2π × 102.03MHz at Vdc

g = 44V). Insimilar optomechanical experiments, a blue-detuned pump laseris introduced into a cavity to amplify the motional amplitude of amechanical resonator where the mechanical loss of the resonatoris compensated by the action of the pump, schematically shownin Fig. 4b. Here we demonstrate the non-degenerate parametricamplification of a mechanical mode by introducing a blue pumpsignal of frequency ωp = 2π × 196.93MHz on the gate electrode.Figure 4c shows the response of mode 1 as a function of thepump amplitude. Figure 4d plots the linewidth of mode 1 withincreasing blue-detuned pump strength. The narrowing of linewidthcan be understood from the non-degenerate parametric

down-conversion of the pump signal, which compensates thedissipation to the thermal bath leading to mechanical amplification.The effective dissipation rate can be tuned by a factor of fourat the highest pump amplitude before reaching the region ofinstability. The non-degenerate parametric amplification of thesehigh-frequency modes could possibly lead to the realization oftwo mode-squeezed states at sufficiently low temperatures13. Theexperiments with the blue pump can be treated as a generalizedform of parametric amplification of a single mode with a 2ωpump, where the gain is dependent on the phase differencebetween the drive and pump signals29. In our system the gatevoltage-induced tension can be used to parametrically excite amode at twice its resonant frequency and amplify the mechanicalresponse to a regime of self-oscillations4. In the SupplementaryInformation we show the parametric amplification of the modenear 2π × 50MHz with a gain factor of nearly three, limited bynonlinearities in the system30–32.

In summary, by fabricating a graphene resonator with a largefrequency tunability, we demonstrate tunable strong couplingbetween mechanical modes using red- and blue-detuned pumps.We demonstrate a large cooperativity of ∼60 between mechanicalmodes having the highest frequency-quality factor (fQ) product(∼2 × 1011Hz) among similar studies on other systems thusfar21,22,26. This, combined with the high mechanical frequencies(h−ω/kB∼5mK; where kB is Boltzmann’s constant), will facilitatethe study of mechanics in the quantum regime at millikelvin temp-eratures. We further demonstrate strong coupling between modeswith frequencies that differ by a factor of two, suggesting that byreducing the drum diameter to increase the resonant frequencies,one can realize efficient cooling mechanisms using two coupledphonon modes16,17. Recent experiments5–7 demonstrating largecoupling between a graphene drum’s electromechanical modesand an electromagnetic cavity opens up the possibility of realizingthe entanglement of several graphene modes mediated by the elec-tromagnetic cavity bus. Furthermore, the high frequencies attainablein this system could help to realize the vacuum-squeezed states of

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Figure 4 | Amplification of motion using blue pump. a, Illustration of the spectrum showing the two modes along with the blue pump. b, The action of theblue pump is understood as a non-degenerate parametric amplification of the mechanical modes. Here ω1 = 2π × 94.90MHz and ω2 = 2π × 102.03MHz.c, The response of mode 1 is shown as a function of the blue pump amplitude at Vdc

g = 44V. The response is seen to narrow with increasing pump amplitude,indicative of the amplification of the mode. d, The effective dissipation rate, γeff, of the mode is seen to decrease with the pump amplitude. The data (circles)are fitted to the theoretical response given by the equation γ(Vp) = γ1(1 − βV2

p ) (ref. 28). At Vdcg = 44V the initial dissipation rate of mode 1 is

γ1 ∼ 2π × 89 kHz. Error bars represent the standard deviation of the dissipation as inferred from the fits to experimental data.

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mechanical motion at low temperatures13. Considering the suit-ability of graphene-based NEMS devices as sensors, the ability toamplify the motional amplitude in the device also makes it attractivefor a wide variety of sensing applications.

MethodsMethods and any associated references are available in the onlineversion of the paper.

Received 24 November 2015; accepted 9 May 2016;published online 13 June 2016

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AcknowledgementsWe thank V. Singh, A. A. Clerk, A. Bhushan and A. Naik for discussions and comments onthe manuscript. We acknowledge funding from the Department of Atomic Energy, theDepartment of Science and Technology (Swarnajayanti Fellowship for M.M.D) of theGovernment of India and ITC-PAC Grant No. FA5209-15-P-0092.

Author contributionsJ.P.M performed the experiments, simulations and analysed the data. R.N.P fabricated thedevices and contributed to experiments. A.B. contributed to the fabrication andexperiments. R.V. provided input for the measurements. J.P.M. and M.M.D co-wrote themanuscript. All authors provided input on the manuscript. M.M.D supervised the project.

Additional informationSupplementary information is available in the online version of the paper. Reprints andpermissions information is available online at www.nature.com/reprints. Correspondence andrequests for materials should be addressed to M.M.D.

Competing financial interestsThe authors declare no competing financial interests.

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MethodsDevice fabrication. The devices are fabricated on sapphire substrates to suppressthe effects of parasitic capacitance. A local gate electrode of evaporated Ti/Pt isdrawn on the sapphire substrate using e-beam lithography (EBL). A 20 nmthick layer of chromium is evaporated before all EBL steps to avoid chargingeffects. 300 nm of SiO2 is deposited by plasma-enhanced chemical vapourdeposition followed by selective reactive ion etching above the gate electrode toform the hole that suspends the graphene. Further EBL is carried out to define thesource/drain electrodes on top of the SiO2 near the hole region. Cr/Au isevaporated for the contact electrodes. Finally the graphene flake, exfoliated on apolydimethylsiloxane stamp, is located and draped over the source/drain

electrodes and the hole. The van der Waal’s forces along the edge of the grapheneand SiO2 ensure a sealed drum geometry for the resonator. As the flake is notclamped by metal electrodes, free expansion of the graphene flake relative to thesubstrate and electrodes results in the membrane sagging when it is cooled to lowtemperatures. This leads to a low built-in tension, giving rise to largeelectrostatic tunability.

Simulations. Coupled equations (1)–(2) are numerically solved using Mathematicawith experimentally obtained parameters as inputs to the simulation. The steady-state response is obtained and compared with experiments. Details on the relevantparameters are given in the Supplementary Information.

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