Abstract
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 singleelectron 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 g_{0} = 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 quantumconfined single electron devices.
Introduction
The technically challenging integration of suspended singlewall cabon nanotubes into complex qantum devices has recently made significant advances^{1,2,3,4,5,6}, as has also the integration of nanotube quantum dots into coplanar microwave cavities^{7,8,9}. Both regarding their nanomechanical^{10,11} as well as their electronic properties^{12,13}, carbon nanotubes are a prototypical experimental system. However, small vibrational deflection and length have made their optomechanical coupling to microwave fields^{14} so far impossible.
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 fingerlike 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 halfwavelength coplanar microwave cavity with a suspended carbon nanotube quantum dot. Near the coupling capacitor, the center conductor of the niobiumbased 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 wellknown 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} ~ 10^{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. 2a, b: a strong, reddetuned drive field (ω_{d} ≃ ω_{c} − ω_{m}) pumps the microwave cavity; the transmission of a weak, superimposed probe signal ω_{p} near ω_{c} is detected. A distinct, sharp OMIT absorption feature within the transmission 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 twophoton 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 \(g=\sqrt{{n}_{{\rm{c}}}}(\partial {\omega }_{{\rm{c}}}/\partial x){x}_{{\rm{zpf}}}\), 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 singlephoton coupling on the order of \({g}_{0}=g/\sqrt{{n}_{{\rm{c}}}} \sim 2\pi \cdot 100\ {\rm{Hz}}\).
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 zeropoint 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 \({g}_{0}^{* }=2\pi \cdot 2.9\ {\rm{mHz}}\), 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 displacementdependent 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 wellknown characteristic of suspended carbon nanotube quantum dots^{18,24}. More interestingly, the resulting gatedependent 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 \(\left\langle N\right\rangle ({V}_{{\rm{g}}})\) 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 \(\left\langle N\right\rangle ({V}_{{\rm{g}}})\) 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 \(\left\partial {C}_{{\rm{CNT}}}/\partial x\right\), becomes proportional to the derivative ∂C_{CNT}/∂V_{g} and thus the second derivative of \(\left\langle N\right\rangle ({V}_{{\rm{g}}})\), 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 can be found in the Supplementary Information, Supplementary Note 12, and Supplementary Table 1. The parameter entering the optomechanical coupling in Eq. (1), the derivative of the quantum capacitance ∂C_{CNT}/∂x, is found to be
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 \({\partial }^{2}\,\left\langle N\right\rangle /\partial {V}_{{\rm{g}}}^{2}\)) 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 \({g}_{0}^{{{\exp }}}/{g}_{0}^{{\rm{th}}}=5.77\). 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 twotone 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 (antiStokes 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} ~ 10^{6}, allowing for two orders of magnitude improvement and a line width of κ_{m} ~ 2π · 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}.
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 sideband cooling of the vibration mode and potentially its quantum control^{37} all the way to realtime 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.
References
Wu, C. C., Liu, C. H. & Zhong, Z. Onestep direct transfer of pristine singlewalled carbon nanotubes for functional nanoelectronics. Nano Lett. 10, 1032–1036 (2010).
Pei, F., Laird, E. A., Steele, G. A. & Kouwenhoven, L. P. Valleyspin blockade and spin resonance in carbon nanotubes. Nat. Nanotechnol. 7, 630–634 (2012).
Ranjan, V. et al. Clean carbon nanotubes coupled to superconducting impedancematching circuits. Nat. Commun. 6, 7165 (2015).
Gramich, J., Baumgartner, A., Muoth, M., Hierold, C. & Schönenberger, C. Fork stamping of pristine carbon nanotubes onto ferromagnetic contacts for spinvalve devices. Phys. Stat. Solidi B 252, 2496–2502 (2015).
Waissman, J. et al. Realization of pristine and locally tunable onedimensional electron systems in carbon nanotubes. Nat. Nanotechnol. 8, 569–574 (2013).
Blien, S., Steger, P., Albang, A., Paradiso, N. & Hüttel, A. K. Quartz tuningfork based carbon nanotube transfer into quantum device geometries. Phys. Stat. Solidi B 255, 1800118 (2018).
Delbecq, M. R. et al. Photonmediated interaction between distant quantum dot circuits. Nat. Commun. 4, 1400 (2013).
Viennot, J. J., Dartiailh, M. C., Cottet, A. & Kontos, T. Coherent coupling of a single spin to microwave cavity photons. Science 349, 408–411 (2015).
Cubaynes, T. et al. Highly coherent spin states in carbon nanotubes coupled to cavity photons. npj Quant. Inf. 5, 47 (2019).
Witkamp, B., Poot, M. & van der Zant, H. Bendingmode vibration of a suspended nanotube resonator. Nano Lett. 6, 29042908 (2006).
Sazonova, V. et al. A tunable carbon nanotube electromechanical oscillator. Nature 431, 284–287 (2004).
Laird, E. A. et al. Quantum transport in carbon nanotubes. Rev. Mod. Phys. 87, 703–764 (2015).
Margańska, M. et al. Shaping electron wave functions in a carbon nanotube with a parallel magnetic field. Phys. Rev. Lett. 122, 086802 (2019).
Aspelmeyer, M., Kippenberg, T. J. & Marquardt, F. Cavity optomechanics. Rev. Mod. Phys. 86, 1391–1452 (2014).
Poot, M. & van der Zant, H. S. J. Mechanical systems in the quantum regime. Phys. Rep. 511, 273–335 (2012).
Weig, E. M. et al. Singleelectronphonon interaction in a suspended quantum dot phonon cavity. Phys. Rev. Lett. 92, 046804 (2004).
Hüttel, A. K. et al. Carbon nanotubes as ultrahigh quality factor mechanical resonators. Nano Lett. 9, 2547–2552 (2009).
Steele, G. A. et al. Strong coupling between singleelectron tunneling and nanomechanical motion. Science 325, 1103–1107 (2009).
Götz, K. J. G. et al. Nanomechanical characterization of the Kondo charge dynamics in a carbon nanotube. Phys. Rev. Lett. 120, 246802 (2018).
Wu, C. C. & Zhong, Z. Capacitive spring softening in singlewalled carbon nanotube nanoelectromechanical resonators. Nano Lett. 11, 1448–1451 (2011).
Stiller, P. L., Kugler, S., Schmid, D. R., Strunk, C. & Hüttel, A. K. Negative frequency tuning of a carbon nanotube nanoelectromechanical resonator under tension. Phys. Stat. Solidi B 250, 2518–2522 (2013).
Weis, S. et al. Optomechanically induced transparency. Science 330, 1520–1523 (2010).
Regal, C. A., Teufel, J. D. & Lehnert, K. W. Measuring nanomechanical motion with a microwave cavity interferometer. Nat. Phys. 4, 555–560 (2008).
Lassagne, B., Tarakanov, Y., Kinaret, J., GarciaSanchez, D. & Bachtold, A. Coupling mechanics to charge transport in carbon nanotube mechanical resonators. Science 325, 1107–1110 (2009).
Duty, T. et al. Observation of quantum capacitance in the Cooperpair transistor. Phys. Rev. Lett. 95, 206807 (2005).
Ares, N. et al. Resonant optomechanics with a vibrating carbon nanotube and a radiofrequency cavity. Phys. Rev. Lett. 117, 170801 (2016).
Pirkkalainen, J.M. et al. Hybrid circuit cavity quantum electrodynamics with a micromechanical resonator. Nature 494, 211–215 (2013).
Rimberg, A. J., Blencowe, M. P., Armour, A. D. & Nation, P. D. A cavityCooper pair transistor scheme for investigating quantum optomechanics in the ultrastrong coupling regime. New J. Phys. 16, 055008 (2014).
Heikkilä, T. T., Massel, F., Tuorila, J., Khan, R. & Sillanpää, M. A. Enhancing optomechanical coupling via the Josephson effect. Phys. Rev. Lett. 112, 203603 (2014).
Abdi, M., Pernpeintner, M., Gross, R., Huebl, H. & Hartmann, M. J. Quantum state engineering with circuit electromechanical threebody interactions. Phys. Rev. Lett. 114, 173602 (2015).
Lecocq, F., Teufel, J. D., Aumentado, J. & Simmonds, R. W. Resolving the vacuum fluctuations of an optomechanical system using an artificial atom. Nat. Phys. 11, 635–639 (2015).
Pirkkalainen, J.M. et al. Cavity optomechanics mediated by a quantum twolevel system. Nat. Commun. 6, 6981 (2015).
Xue, Z.Y., Yang, L.N. & Zhou, J. Circuit electromechanics with single photon strong coupling. Appl. Phys. Lett. 107, 023102 (2015).
Santos, J. T., Li, J., Ilves, J., OckeloenKorppi, C. F. & Sillanpää, M. Optomechanical measurement of a millimetersized mechanical oscillator approaching the quantum ground state. New J. Phys. 19, 103014 (2017).
Shevchuk, O., Steele, G. A. & Blanter, Y. M. Strong and tunable couplings in fluxmediated optomechanics. Phys. Rev. B 96, 014508 (2017).
Moser, J., Eichler, A., Güttinger, J., Dykman, M. I. & Bachtold, A. Nanotube mechanical resonators with quality factors of up to 5 million. Nat. Nanotechnol. 9, 1007–1011 (2014).
O’Connell, A. D. et al. Quantum ground state and singlephonon control of a mechanical resonator. Nature 464, 697–703 (2010).
Barnard, A. W., Zhang, M., Wiederhecker, G. S., Lipson, M. & McEuen, P. L. Realtime vibrations of a carbon nanotube. Nature 566, 89–93 (2019).
Reinhardt, S. et al. Lab::Measurement—a portable and extensible framework for controlling lab equipment and conducting measurements. Comput. Phys. Commun. 234, 216–222 (2019).
Kouwenhoven, L. P. et al. Electron Transport in Quantum Dots (Kluwer, 1997).
Nazarov, Y. V. & Blanter, Y. M. Quantum Transport: Introduction to Nanoscience (Cambridge University Press, Cambridge, 2009).
Acknowledgements
The authors acknowledge funding by the Deutsche Forschungsgemeinschaft via Emmy Noether grant Hu 1808/1, SFB 631, SFB 689, SFB 1277, and GRK 1570. We would like to thank G. Rastelli, F. Marquardt, E. A. Laird, Y. M. Blanter, and D. Weiss for insightful discussions, O. Vavra for experimental help, and Ch. Strunk and D. Weiss for the use of the experimental facilities. The data have been recorded using Lab::Measurement^{39}.
Author information
Authors and Affiliations
Contributions
A.K.H. and S.B. conceived and designed the experiment. P.S. and R.G. developed and performed the nanotube growth and transfer; N.H. and S.B. developed and fabricated the coplanar waveguide device. The low temperature measurements were performed jointly by all authors. Data evaluation and writing of the paper was done jointly by S.B., N.H., and A.K.H. The project was supervised by A.K.H.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Peer review information Nature Communications thanks Joel Moser and the other anonymous reviewer(s) for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Blien, S., Steger, P., Hüttner, N. et al. Quantum capacitance mediated carbon nanotube optomechanics. Nat Commun 11, 1636 (2020). https://doi.org/10.1038/s41467020154333
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467020154333
This article is cited by

Nonlinear nanomechanical resonators approaching the quantum ground state
Nature Physics (2023)

Visualizing nonlinear resonance in nanomechanical systems via singleelectron tunneling
Nano Research (2021)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.