Electromechanical control of nitrogen-vacancy defect emission using graphene NEMS

Despite recent progress in nano-optomechanics, active control of optical fields at the nanoscale has not been achieved with an on-chip nano-electromechanical system (NEMS) thus far. Here we present a new type of hybrid system, consisting of an on-chip graphene NEMS suspended a few tens of nanometres above nitrogen-vacancy centres (NVCs), which are stable single-photon emitters embedded in nanodiamonds. Electromechanical control of the photons emitted by the NVC is provided by electrostatic tuning of the graphene NEMS position, which is transduced to a modulation of NVC emission intensity. The optomechanical coupling between the graphene displacement and the NVC emission is based on near-field dipole–dipole interaction. This class of optomechanical coupling increases strongly for smaller distances, making it suitable for nanoscale devices. These achievements hold promise for selective control of emitter arrays on-chip, optical spectroscopy of individual nano-objects, integrated optomechanical information processing and open new avenues towards quantum optomechanics.

contacted graphene membrane of radius R located at z G and suspended in vacuum at a distance δ over a layer of SiO 2 (thickness h). When the device is illuminated by laser radiation incident along the z-axis, an interference pattern is formed by reflection at the material interfaces located at z Si and z SiO 2 . The graphene membrane absorbs a position-dependent fraction of the field intensity and can be electrostatically deflected through the interference pattern towards the Si ++ backgate by a distance ξ static . b Green channel intensity of an optical micrograph of the multilayer graphene flake used in our device. A line section (inset) reveals a contrast of C ≈ 36% between the graphene-covered area and its surroundings.      Supplementary Note 1.

THICKNESS
We record a Raman spectrum (pump laser 532 nm) of the graphene flake in our device to determine its thickness ( Supplementary Figure 1 a). The 2D band profile clearly shows the presence of more than one peak, ruling out the presence of single layer graphene. Even if it is generally accepted that the Raman response of few layer graphene becomes almost indistinguishable for samples with more than five layers, the shape and peak position is consistent with four layer graphene, as discussed for instance by Ferrari et al. 7 . Indeed, the asymmetry of 2D 1 and 2D 2 peak intensity is not as pronounced as it is for a graphite sample, for instance. In addition, optical contrast measurements support that the graphene flake We consider suspended graphene as an absorber with absorption coefficient πα ≈ 2.3% per layer (valid for up to five layers of graphene surrounded by vacuum 12 ). Hence, the presence of graphene modifies the intensity of the optical field but not its phase. The fraction of light absorbed by graphene is proportional to the electric field intensity at its position |E(z G )| 2 .
where I 0 is the measured reflection intensity in the absence of graphene, n g the number of graphene layers and |E max | 2 the maximum field intensity. From Eq. 1 we see that the net reflected intensity is minimum when graphene is located at the position of |E max | 2 .
When the graphene membrane is displaced from its initial position z G by a distance ξ static towards the backgate, the resulting relative change in the reflected intensity is equal to the relative change of the field intensity due to the linearity of Eq. 1: This relation allows us to infer the displacement ξ static from intensity measurements using a model for the electric field intensity of our device.
In the following, we derive the expressions for the electrostatic deflection of a circular graphene drumhead. Here, we adapt this problem to our experimental situation with similar notation to that used by Weber et al. 15 (see also Katsnelson 8 and Landau 9 ). When the graphene membrane is under tension and subjected to an external electrostatic pressure P (x, y) along the z direction, its deformation ξ(x, y, t) obeys the equilibrium condition given by : where ρ 2D = n g ρ g η is the graphene sheet mass density which depends on the number of graphene layers n g , graphene's intrinsic mass density ρ g and a correction factor η ≥ 1 to take into account the adsorbates on the membrane. T = Et n g is the stretching force per unit length at the edge of the membrane, which depends on the Young's modulus E = 1 TPa, the thickness (t=0.3 nm) of a graphene layer, the radial strain and the number n g of graphene layer considered. By applying a constant voltage V dc g between the graphene and the gate electrode, we generate an electrostatic force F el given as: where C eq is the equivalent gate capacitance of the system which contains two dielectric layers: vacuum and SiO 2 . This conservative force arises from an electrical potential U el = 1 2 C eq V dc g 2 . By expansion of the equivalent capacitance, we find: The electrostatic pressure introduced in Eq. 3 fulfills the following equation : where SiO 2 = r 0 =3.9 · 8.85 10 −12 F m −1 is the dielectric constant of the SiO 2 layer, h is the SiO 2 layer thickness and δ is the distance from the non-deformed graphene drum to the vacuum/SiO 2 interface. Reinserting P (x, y) into Eq. 3 yields an analytical solution for the deformation: The graphene drum deflection contains a static term ξ static and a time-dependent term representing the radial mechanical modes. In the steady state, this leads to (cf. Eq. 3): For small deflections, we can consider 2ξ static δ+ h r 1 (in our system, δ+ h r 2 ≈ 65 nm). We now integrate ∇ 2 ξ static in radial coordinates to obtain the lowest-order solution for ξ: At the boundaries of the drum, there is no deflection (ξ static (r = R) = 0), while at the center, the membrane can be considered as flat ( ∂ξ static (r=0) ∂r = 0). Thus, k 1 = 0 and k 2 = − R 2 4 . Therefore, the deflection can be written as: At the center of the drum, the deflection is denoted ξ 0 : In the following, we describe the calibration necessary to read out the graphene membrane's static position by measurement of I r . Maps of reflected intensity I r (x, y) from a membrane at different values of applied static backgate voltage V dc g are shown in Supplementary Figure 2. These measurements reveal that the membrane's deflection may be observed by measuring I r . In addition, these maps display symmetry of the deflection in (V dc g ) 2 as described by Eq. 10.
Supplementary Figure 2-b shows the linear increase of I r with (V dc g ) 2 at the centre of a graphene membrane. We normalize I r by the intensity I 0 reflected from the undeflected drum and obtain an expression which describes our measurements given by Ir Here, A is a device-dependent constant which has dimensions of [V −2 ] and is extracted from a linear fit of the measured data.
We combine Eq. 10 with the expression for Ir I 0 to obtain an expression which allows us to determine the displacement ξ 0 from the reflected intensity: is the device-dependent calibration constant.
For large deflections over 65 nm, our calibration is overestimated and does not hold as the higher terms in Eq. 8 must to be taken into account. For the second order in ξ, the solution becomes: , C 1 and C 2 and constants and J 0 (x) and Y 0 (x) are Bessel functions of the first and second kind, respectively.
For the case of small deflections, we determine the value of the calibration constant β from the model of the normalized electric field intensity. By insertion of measured values of the sample geometry, this model yields the electric field intensity profile |E| 2 (z) in which the graphene membrane lies. To determine β for a given membrane, we assume that the measured change in the normalized reflected intensity ∆Ir I r,0 corresponds to the same relative change of the normalized electric field intensity described by the model: where |E 0 | 2 is the electric field intensity at the initial (undeflected) position of the graphene membrane.
This approximation holds for cases where the maximum deflection is smaller than a period of the standing wave ξ max λ/2 and in the linear region of the electric field intensity away from nodes of |E| 2 (z). Employing this calibration, we also extract the deflection with applied backgate voltage to be to be ξ static The quantity ∆x 2 is then defined by : The Langevin force spectrum is defined by : coth where Γ is the damping rate, χ m is the mechanical susceptibility. This expression associates is the Bolztmann distribution. For T T Q , we define the quantum fluctuation noise spectrum : For T T Q , we define the thermal fluctuations spectrum of the oscillator : Thus, at resonance, we obtain : where the mechanical quality factor is defined as Q = Ωm Γ . When we measure the ac reflection from an thermally driven graphene membrane, we record the power spectral density (PSD) S m of the ac output voltage delivered by the photodiode using a spectrum analyser. S m has two contributions: i) the electrical noise S e , and ii) the mechanical response S z of the thermally driven membrane. Therefore, if the oscillator is only submitted to the thermal fluctuation, we measure S m = S e + S b (P laser ) + (Bχ opt ) 2 S T x . The conversion factor depends on the optical susceptibility χ opt = ∂Ir ∂x , defined as the slope of the standing wave pattern in which the membrane is moving. This interferometric response I r is approximated by a sine function. If the initial position of the membrane is away from any extrema of I r and in the limit of the small displacements (δx λ), we can make the approximation : Bχ opt ∼ βP laser . The quantity S b (P laser ) ∝ P laser represents the background noise of the laser measured by the photodetector.

STATIC DRIVE
To evaluate the oscillation amplitude x d caused by a monochromatic electrostatic drive, we use the following expression : We use the first order expression (cf. Eq. 5) of C to obtain ∂ x C : This leads to the simple expression : Using typical values for a device with d = δ + h SiO 2 = 165 nm, R = 1.5 µm we obtain ∂ x C = 2.07 nF m −1 . This yields x d = 3 pm for a drive of V dc g + V ac g =1 V + 100 µV and Q = 150.
By continuously increasing V dc g , the membrane is deflected to a point where it is in contact with the nano-diamond. Here, we describe how scanning confocal microscopy allows us to determine the value of V dc g for which contact with the nano-diamond occurs. The position of the nano-diamond is determined by AFM scans of the device before graphene transfer, as shown in Supplementary Figure 3  Electrostatic deflection of a graphene resonator gives rise to a tunable mechanical resonance frequency f m , given as 15 : where h is the graphene sheet thickness, E ≈ 1 TPa graphene's Young's modulus, the membrane's intrinsic strain, m ef f the effective mass of the mechanical mode, ef f the effective dielectric constant of the device and R the membrane radius.
For a graphene drum where the intrinsic strain dominates any additional strain induced by electrostatic deflection, we use Eq. 28 to describe electrostatic softening which results in decreasing f m with V dc g .
The measurement of f m (V dc g ) also reveals intrinsic doping of the graphene drum, which is observed as a shift of the maximum value f m,max at non-zero V 0 . Thus, V 0 is an offset voltage which must be taken into account to determine the deflection potential ∆V dc g = V dc g − V 0 applied to a given drum.
We use Eq. 28 to fit typical detuning data shown in Supplementary Figure 3-g to find an effective mass m ef f = 4.6 · 10 −17 kg of a four-layer graphene membrane of radius R = 1.5µm, resonating in its fundamental mode. The offset voltage is found to be V 0 = 0.89 V. As such, our resonator's mass density per layer is a factor 2.13 larger than graphene's intrinsic mass density ρ 2D = 7.6 · 10 −19 kg m − 2, which we attribute to impurities on the membrane surface.
From this, we obtain a spring constant k = m ef f (2πf m (0)) 2 = 0.04 N m −1 in the undeflected state where V dc g = 0 V.
Supplementary Figure 4-b shows a schematic diagram of the experimental setup used to obtain the data described in the main text. Here, our device is placed in a cryostat at 3 K and electrically connected to a voltage source which can actuate the graphene membrane on the device by a combination of DC (V dc g ) and radio-frequency (δV ac g ) voltages. the device is scanned with 532 nm CW laser radiation (green line) by an objective mounted on a translation stage (marked as x,y).
Reflected light (red line) is collected and split into confocal reflection and an emission channels by a dichroic mirror (transmission for λ < 580 nm ). The light reflection channel is read out by splitting it with a beamsplitter (BS) and reading it out with an AC and a DC photodiode. Here, the AC signal is analysed with a spectrum analyser. A combination of a quarter-wave plate ( λ 4 ), a half-wave ( λ 2 ) plate and a polarizing beamsplitter (PBS) is used to control the polarization such that a maximum of reflected light is passed to the reflection channel. Similarily, the emission channel is read out by an avalanche photodiode (APD), where the combination of the dichroic mirror, two 532 nm notch filters and one longpass filter (LP) for λ > 532 nm suppress the reflection component of the signal. Finally, the APD signal is analysed by photon correlator which can count individual photons or perform a synchronous measurement triggered by the radio-frequency drive signal δV ac g .

Supplementary Note 4. NVC EMISSION AS A PREDOMINANT COM-PONENT IN THE MEASURED SIGNAL
To measure NVC emission, we use a combination of dichroic, notch and long-pass filters to suppress the excitation laser component in the reflected signal. We read out the filtered optical signal I AP D with an avalanche photodiode (APD). This signal depends on the position r on the device as well as the excitation laser power P . For a position r where an NVC is excited, I AP D consists of NVC emission I N V C with an added background signal I b : Here, I b includes emission from the substrate, graphene and processing residues as well as a parasitic reflection component and detector noise. For a given laser power, the respective contributions of the NVC signal and background noise to the total APD signal are then given as: In the previous section, we find the governing component of our measured signal to be NVC emission. However, multiple mechanisms may cause a modulation of this signal when the graphene membrane is deflected towards the NVC or driven at radio frequencies.
For instance, the motion of the graphene membrane may change the power absorbed in the graphene, hereby varying the power incident on the NVC and thus its emission.
However, this effect can be ruled out as the measured NVC emission decreases (red curve in Furthermore, we employ off-resonant excitation at 532 nm (linewidth 30 GHz), which implies that the excitation efficiency will not be affected by energy level shifts on the order of few THz. In addition, our collection is insensitive to these shifts as we collect all photons with energies smaller than our excitation energy by employing steep notch filters for the excitation line on the emission collection arm of our setup. Owing to this, we exclude the influence of the Stark effect on our measurements.
We now discuss why I N V C dominates the measured signal I AP D when we record timeresolved emission traces. For a generic, time-varying emission signal, we determine the contribution of the emission variance σ 2 N V C to the total variance σ 2 AP D of the signal, and compare it to the background variance σ 2 b : As I N V C and I b arise from independent processes, cov (I N V C , I b ) = 0 and thus σ 2 Similarly to what has been introduced before, we now define α and β as the relative contributions of the NV emission and background signal to the total measured variance: We recall that σ 2 (kX) = k 2 σ(X), where k is a constant, and σ 2 (X) the variance of the ensemble X.
In Supplementary Figure 4-e, we compare the values of α and β. This comparison shows that NVC emission is the dominant contribution to the periodically modulated emission signal which we attribute to the NVC acting as a transducer of graphene motion by n-RET.
Energy transfer from an emitting dipole to graphene at a separation d G−N V C leads to a modified total decay rate Γ G (d G−N V C ) due to an additional divergent non-radiative decay channel provided by n-RET: where Γ 0 is the decay rate in the absence of graphene, ν ∈ [1, 2] takes into account the emitting dipole orientation, α is the fine structure constant, r is the equivalent relative permittivity of the separating medium and λ is the emission wavelength (e.g. 638 nm for N V − zero phonon line). This causes emission quenching with decreasing d G−N V C : where Φ 0 is the emission in the absence of graphene.

Supplementary Note 5. CALIBRATION OF GRAPHENE-EMITTER SEP-ARATION USING EMISSION
To model NVC ensemble emission, we consider a nano-diamond containing N ∈ [1,4] NVCs emitting simultaneously. We assume an average separation of 15 nm between each NVC, located at positions z i ∈ 1, .., N within a nano-diamond with an average diameter of 40-60 nm. The separation of each NVC to graphene is given as d i = z G − z i , where z G is the position of the graphene. The ensemble emission is then given by the sum of the individual NVC emission: where Φ ensemble,0 = N Φ 0 . Qualitatively, the quenching effect of an ensemble of emitters is thus strongly reduced when compared to a single NVC as is shown in Supplementary Figure   5-a.
For a single NVC near a mobile graphene membrane, Eq. 33 enables the calibration of the absolute value of the static separation d G−N V C by measurement of the decay rate enhancement, e.g. by making excited state lifetime measurements by recording time-resolved emission.
However, such direct decay rate measurements cannot be used to calibrate d i for an emitting ensemble as each NVC experiences an individual and indistinguishable decay rate enhancement. Similarly, broad-band emission measurements of an NVC ensemble using an APD cannot be used to extract information about the NVC distribution within a given nano-diamond, nor about the exact number of emitting NVCs as this information is hidden in the total emission signal.
Experimentally, we have access to the NVC emission intensity, which is related to the decay rate by Eq. 34. We measure the emission rate from the NVC at different applied backgate voltages V dc g . This data is then fitted using Eq. 34, where we substitute d G−N V C = κ(V dc g ) 2 (cf. Eq. 10) and κ is the calibration constant. In addition, we fit our data taking into account an emission background originating from NVCs deeper within the nano-diamond and thus further away from the graphene membrane. We assume that these NVCs are not affected by the graphene and thus contribute a constant background emission signal.
Based on these assumptions, we fit our measured emission data Φ em (V dc g ) with a simple model described by Eq. 34 which takes into account coupling by n-RET to the topmost NVC in the nano-diamond and a constant background emission signal Φ bg : We use this model to fit the measured emission for increasing V dc g in the main text, which also yields a calibration of the graphene membrane's nano-motion.
While our model shows good agreement to the measured data, it neglects n-RET coupling to other NVCs which induces a systematic error in the position detection by emission measurement. This error is enhanced by spectral broadening at 300 K, where NVC emission is broadband and thus the main emission wavelength λ is badly defined. For an optimum displacement measurement by emission, one should therefore use a system where single NVC with a narrow emission linewidth is located close to the top of the nano-diamond.