Quantum capacitance mediated carbon nanotube optomechanics

Cavity optomechanics allows the characterization of a vibration mode, its cooling and quantum manipulation using electromagnetic fields. Regarding nanomechanical as well as electronic properties, single wall carbon nanotubes are a prototypical experimental system. At cryogenic temperatures, as high quality factor vibrational resonators, they display strong interaction between motion and single-electron tunneling. Here, we demonstrate large optomechanical coupling of a suspended carbon nanotube quantum dot and a microwave cavity, amplified by several orders of magnitude via the nonlinearity of Coulomb blockade. From an optomechanically induced transparency (OMIT) experiment, we obtain a single photon coupling of up to g0 = 2π ⋅ 95 Hz. This indicates that normal mode splitting and full optomechanical control of the carbon nanotube vibration in the quantum limit is reachable in the near future. Mechanical manipulation and characterization via the microwave field can be complemented by the manifold physics of quantum-confined single electron devices.

In this work, we demonstrate large optomechanical coupling of a suspended carbon nanotube quantum dot and a microwave cavity. The nanotube is deposited onto source and drain electrodes close to the coplanar waveguide cavity; a finger-like extension of the cavity center conductor, passing below the suspended nanotube, serves as capacitively coupling gate. We find that the optomechanical coupling of the transversal nanotube vibration and the cavity mode is amplified by several orders of magnitude via the inherent nonlinearity of Coulomb blockade. With this, full optomechanical control of the carbon nanotube vibration in the quantum limit 15 is reachable in the near future. A unique experimental system becomes accessible, where the nanomechanically active part directly incorporates a quantumconfined electron system 16 .

Results
Device precharacterization. Our device, depicted in Fig. 1a, combines a half-wavelength coplanar microwave cavity with a suspended carbon nanotube quantum dot. Near the coupling capacitor, the center conductor of the niobium-based cavity is connected to a thin gate electrode, buried between source and drain contacts of the carbon nanotube, see the sketch of Fig. 1b. At the cavity center, i.e., the location of the voltage node of its fundamental mode, a bias connection allows additional application of a dc voltage V g to the gate. The device is mounted at the base temperature stage (T ≃ 10 mK) of a dilution refrigerator; for details see Supplementary Note 4 and Supplementary Fig. 4.
At cryogenic temperatures, electronic transport through the carbon nanotube is dominated by Coulomb blockade, with the typical behavior of a small band gap nanotube 12 . Near the electronic band gap, sharp Coulomb oscillations of conductance can be resolved; measurements are shown in Fig. 1c and Supplementary Fig. 3. A well-known method to detect the transversal vibration resonance of a suspended nanotube quantum dot is to apply a rf signal and measure the timeaveraged dc current [17][18][19] . On resonance, the oscillating geometric capacitance, effectively broadening the Coulomb oscillations, leads to an easily recognizable change in current. This was used to identify the transversal vibration resonances of the device; Fig. 1d plots the resonance frequencies over a wide gate voltage range. Two coupled vibration modes are observed (see also Supplementary Note 5), one of which clearly displays electrostatic softening 20,21 . At low gate voltages, |V g | ≤ 1.2 V, where subsequent experiments are carried out, the resonance which we will utilize in the following is at ω m ≃ 2π ⋅ 502.5 MHz, with typical quality factors around or exceeding Q m~1 0 4 observed in timeaveraged dc current detection 17 .
The combined suspended nanotube-cavity device forms a dispersively coupled optomechanical system 14 . The cavity has a resonance frequency of ω c = 2π ⋅ 5.74 GHz with a decay rate of κ c = 2π ⋅ 11.6 MHz, dominated by internal losses. Nevertheless, due to the large mechanical resonance frequency ω m of the carbon nanotube, the coupled system is far in the resolved sideband regime ω m ≫ κ c , the most promising parameter region for a large number of optomechanical protocols including ground state cooling and quantum control.
Optomechanically induced transparency (OMIT). To probe for optomechanical coupling, we perform an OMIT type experiment 22 , cf.  Fig. 1 Integrating a suspended carbon nanotube into a microwave cavity. a Optical micrograph showing a niobium-based λ/2 coplanar waveguide cavity for transmission measurement, with carbon nanotube deposition areas and dc contact structures (see the red dashed squares) near the coupling capacitors. For fabrication redundancy, two deposition areas exist on the device, but only one is used here. Bond wires visible as dark lines connect different segments of the ground plane to avoid spurious resonances. b Simplified sketch of the nanotube deposition area, including source and drain electrodes, a carbon nanotube deposited on them, and the buried gate connected to the cavity center conductor. c dc transport characterization of the carbon nanotube at T base ≃ 10 mK. The plot of the absolute value of currrent I j j as function of gate voltage V g and bias voltage V sd displays the typical diamond-shaped Coulomb blockade regions of suppressed conductance 12,40,41 . d Using rf excitation with an antenna and dc measurement 17,18 , two transversal vibration modes can be traced across a large gate voltage range; the figure plots the detected resonance frequencies. The corresponding raw data as well as a fit can be found in Supplementary Figs. 6 and 7.
resonance of the cavity becomes visible in the measurements of Fig. 2c-e. It occurs due to destructive interference of the probe field with optomechanically upconverted photons of the drive field, when the two-photon resonance condition ω p − ω d = ω m is fulfilled 22 , and shifts in frequency as expected when ω d is detuned from the precise red sideband condition, see Fig. 2d, e. Fitting the OMIT feature allows to extract the optomechanical coupling parameter g ¼ ffiffiffiffi ffi n c p ð∂ω c =∂xÞx zpf , describing the cavity detuning per displacement of the mechanical harmonic oscillator 14,22 , see Supplementary Note 9 for details. Surprisingly, from Fig. 2c, one obtains a single-photon coupling on the order of Such a value of g 0 strongly exceeds expectations from the device geometry 23 . For a mechanical oscillator dispersively coupled to a coplanar waveguide resonator, the coupling is given by where C c is the total capacitance of the cavity, x is the mechanical displacement, and x zpf the mechanical zero-point fluctuation length scale. Assuming a metallic wire over a metallic plane and inserting device parameters 23 , the coupling calculated from the change in geometric gate capacitance C g (x) becomes ∂C g /∂x~10 −12 F m −1 . This leads to g Ã 0 ¼ 2π Á 2:9 mHz, more than four orders of magnitude smaller than the measured g 0 . To explain this discrepancy, we need to focus on the properties of the carbon nanotube as a quantum dot, with a strongly varying quantum capacitance C CNT (x) as the displacement-dependent component of C c dominating g 0 . Figure 2f depicts OMIT measurements for similar parameters as in Fig. 2c-e, however, we now keep the drive frequency ω d constant and vary the gate voltage V g across a Coulomb oscillation of conductance. The mechanical resonance frequency ω m shifts to lower frequencies in the vicinity of the charge degeneracy point. This electrostatic softening is a well-known characteristic of suspended carbon nanotube quantum dots 18,24 . More interestingly, the resulting gate-dependent coupling g(V g ) (along with g 0 (V g )) is plotted in Fig. 2g. It is maximal at the edges of the finite conductance peak, whereas at its center and on the outer edges, the coupling vanishes; the enhancement of g 0 is intrinsically related to Coulomb blockade.
Mechanism of enhanced coupling. Figure 3 explores the nature of this enhanced coupling mechanism. We treat the nanotube as a single quantum dot; see Supplementary Note 3 for a discussion of the validity of this assumption. Further, we assume a full separation of time scales ω m ≪ ω c ≪ Γ, where Γ describes the tunnel rates of the quantum dot. We can then introduce the quantum capacitance 25,26 where N h iðV g Þ is the number of charge carriers (here holes) on the quantum dot averaged over the tunneling events, and C dot is the total quantum dot capacitance; see Supplementary Note 12 for a derivation. In a quantum dot, each Coulomb oscillation corresponds to the addition of one electron or hole. The charge occupation N h iðV g Þ resembles a step function, with the sharpness of the step given for zero bias voltage by lifetime and temperature broadening. This is plotted in Fig. 3a, for the limit of k B T ≪ Γ. The quantum capacitance C CNT (V g ) becomes a Lorentzian, as plotted in Fig. 3b.
Any motion δx modulates the geometric capacitance C g (x). It thus shifts the position of the Coulomb oscillations in gate voltage, acting equivalent to an effective modulation of the gate voltage δV g . With this, the optomechanical coupling g, scaling with ∂C CNT =∂x j j , becomes proportional to the derivative ∂C CNT /∂V g and thus the second derivative of N h iðV g Þ, as is illustrated in Fig. 3c. The functional dependence has been fitted to the data points of Fig. 2g, here again shown in the background.
The three key situations depending on the gate voltage are sketched in Fig. 3d-f: away from the conductance peak, the charge on the nanotube is constant, and only geometric capacitances change, see Fig. 3d. On the flank of the conductance resonance, a small change δx (∝δC g ) strongly modulates C CNT , see Fig. 3e. At the center of the conductance resonance, the charge adapts to x, but the derivative ∂C CNT /∂V g and with it g ∝ |∂C CNT / ∂x| is approximately zero.
The detailed derivation and the full expressions and values for Fig. 3 indicating that for significant optomechanical coupling a sharp Coulomb oscillation (i.e., low temperature and low intrinsic line width Γ, leading to large values of ∂ 2 N h i=∂V 2 g ) and a large V g are required. From device data, we obtain an amplification factor η~10 4 . The experimental gate voltage dependence g 0 (V g ) is qualitatively reproduced very well. To obtain the quantitative agreement of Fig. 3c, we have introduced an additional scaling prefactor as free fit parameter, resulting in g exp 0 =g th 0 ¼ 5:77. Given the uncertainties of input parameters, this is a good agreement; see Supplementary Note 15 for a discussion of error sources.

Discussion
In literature, many approaches have been pursued to enhance optomechanical coupling [26][27][28][29][30][31][32][33][34][35] . Resonant coupling, with ω m = ω c , has been demonstrated successfully for a carbon nanotube quantum dot 26 , but does not provide access to the wide set of experimental protocols developed for the usual case of dispersive coupling and the "good cavity limit" ω m ≫ κ c . The mechanism presented here is most closely related to those where a superconducting charge qubit was coherently introduced between mechanical resonator and cavity 27 . However, the impact of single electron tunneling and shot noise on the optomechanical system shall require careful analysis.
Given the sizeable coupling in the good cavity limit κ c ≪ ω m , many experimental techniques for future experiments are at hand. First steps are demonstrated in Fig. 4 in a two-tone spectroscopy experiment: a mechanical drive signal ω a is applied simultaneously to a cavity pump signal at ω d = ω c − ω a ; the plotted cavity output power at ω c clearly shows the optmechanical upconversion (anti-Stokes scattering) at mechanical resonance ω a = ω m . In Fig. 4a, the dc bias across the nanotube is set to zero, and the antenna drive kept at a minimum. In Fig. 4b, both antenna drive and bias voltage have been increased. A background signal independent of device parameters emerges; at the same time, the upconverted signal displays a phase shift and destructive interference with the background for parts of the gate voltage range, meriting further measurements and analysis.
Future improvements of the optomechanical coupling via drive power and device geometry and of the detection sensitivity via the output amplifier chain shall allow detection of the thermal motion of the carbon nanotube and subsequently motion amplitude calibration.
The observation of strong optomechanical coupling and the corresponding normal mode splitting requires a coupling g exceeding both mechanichal linewidth κ m and cavity line width κ c . Clean carbon nanotubes have reached mechanical quality factors up to 36 Q m~1 0 6 , allowing for two orders of magnitude improvement and a line width of κ m~2 π · 500 Hz. Regarding microwave resonators we have reached up to Q c = 10 5 in our setup so far, corresponding to κ c = 2π · 57 kHz. This means that strong coupling should be reachable already at moderate increase of our so far rather low cavity photon number n c .  Solid lines correspond to the model of a Lorentz-broadened quantum dot level at k B T ≪ Γ. The Coulomb oscillation center V Ã g ¼ À1:18841 V, the line width ℏΓ = 0.673 meV, and a scaling prefactor a = 5.77 (see text) have been obtained by fitting to the OMIT data g(V g ). a Time-averaged charge occupation N h iðV g Þ of the quantum dot (note that we are in the hole conduction regime). b Conductance dI/dV sd (V g ) (left axis) and quantum capacitance C CNT (right axis), cf. Supplementary Fig. 12. c Coulombblockade enhanced optomechanical coupling g(V g ) (left axis) and single photon coupling g 0 (V g ) (right axis). The data points are identical to Fig. 2g; the calculation result has been scaled with 5.77 to fit the data. d-f Schemata for the situations corresponding to the dashed lines in a-c, see the text.
Regarding the cooperativity C = 0.0042 of our experiment (cf. Supplementary Table 1), already an improvement of the nanotube Q m by a factor 100 brings it into the same order of magnitude as the thermal mode occupation n m = 0.4, with significant further and independent room for improvement via the cavity photon number n c .
With this, a wide range of physical phenomena becomes experimentally accessible, ranging from side-band cooling of the vibration mode and potentially its quantum control 37 all the way to real-time observation of its interaction with single electron tunneling phenomena 38 .

Data availability
The datasets generated during and/or analyzed during this study are available from the corresponding author on reasonable request.