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# Real-time vibrations of a carbon nanotube

## Abstract

The field of miniature mechanical oscillators is rapidly evolving, with emerging applications including signal processing, biological detection1 and fundamental tests of quantum mechanics2. As the dimensions of a mechanical oscillator shrink to the molecular scale, such as in a carbon nanotube resonator3,4,5,6,7, their vibrations become increasingly coupled and strongly interacting8,9 until even weak thermal fluctuations could make the oscillator nonlinear10,11,12,13. The mechanics at this scale possesses rich dynamics, unexplored because an efficient way of detecting the motion in real time is lacking. Here we directly measure the thermal vibrations of a carbon nanotube in real time using a high-finesse micrometre-scale silicon nitride optical cavity as a sensitive photonic microscope. With the high displacement sensitivity of 700 fm Hz−1/2 and the fine time resolution of this technique, we were able to discover a realm of dynamics undetected by previous time-averaged measurements and a room-temperature coherence that is nearly three orders of magnitude longer than previously reported. We find that the discrepancy in the coherence stems from long-time non-equilibrium dynamics, analogous to the Fermi–Pasta–Ulam–Tsingou recurrence seen in nonlinear systems14. Our data unveil the emergence of a weakly chaotic mechanical breather15, in which vibrational energy is recurrently shared among several resonance modes—dynamics that we are able to reproduce using a simple numerical model. These experiments open up the study of nonlinear mechanical systems in the Brownian limit (that is, when a system is driven solely by thermal fluctuations) and present an integrated, sensitive, high-bandwidth nanophotonic interface for carbon nanotube resonators.

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## Data availability

The data that support the findings of this study are available from the corresponding author on reasonable request.

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## Acknowledgements

We thank A. Bachtold for discussions. This work was supported in part by the National Science Foundation under grant number 0928552. It was also supported by the Cornell Center for Materials Research with funding from the NSF MRSEC programme (DMR-1719875) and funding from IGERT (DGE-0654193). This work was performed in part at the Cornell NanoScale Facility, a member of the National Nanotechnology Coordinated Infrastructure (NNCI), which is supported by the National Science Foundation (Grant ECCS-1542081). G.S.W. acknowledges FAPESP (grant 2012/17765-7) and CNPq for financial support in Brazil.

## Author information

### Author notes

• Arthur W. Barnard

Present address: Physics Department, Stanford University, Stanford, CA, USA

• Mian Zhang

Present address: John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA

• Michal Lipson

Present address: Electrical Engineering, Columbia University, New York, NY, USA

1. These authors contributed equally: Arthur W. Barnard, Mian Zhang

### Affiliations

1. #### School of Applied and Engineering Physics, Cornell University, Ithaca, NY, USA

• Arthur W. Barnard
•  & Mian Zhang
2. #### Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA

• Arthur W. Barnard
•  & Paul L. McEuen
3. #### School of Electrical and Computer Engineering, Cornell University, Ithaca, NY, USA

• Mian Zhang
• , Gustavo S. Wiederhecker
•  & Michal Lipson
4. #### Gleb Wataghin Physics Institute, University of Campinas, Campinas, Brazil

• Gustavo S. Wiederhecker
5. #### Kavli Institute at Cornell for Nanoscale Science, Cornell University, Ithaca, NY, USA

• Michal Lipson
•  & Paul L. McEuen

### Contributions

A.W.B. and M.Z. conceived the experiment, designed and fabricated the devices and performed the measurements. A.W.B., M.Z. and G.S.W. performed data analysis. All authors contributed to the writing of the manuscript. M.L. and P.L.M. supervised the project.

### Competing interests

The authors declare no competing interests.

### Corresponding authors

Correspondence to Michal Lipson or Paul L. McEuen.

## Extended data figures and tables

1. ### Extended Data Fig. 1 Tweezer scattering.

a, Spatial dependence of optical transmission at a fixed laser detuning as tweezers are scanned across the optical mode. Both tines (blue and yellow dots) are distinguishable and the gap region (orange dot) shows negligible perturbation. b, Cavity transmission spectra at select points in a along with a far-field spectrum. The dashed line corresponds to the fixed detuning used in a. The small differences in the gap spectrum and the far-field spectrum are attributed to slow thermal drifts between measurements.

2. ### Extended Data Fig. 2 Photocurrent mapping of the optical modes.

a, Principle of the near-field photocurrent mapping of the cavity field. The suspended CNT is positioned close to the optical cavity and slowly scanned across the perimeter of the optical cavity. Modes 1 and 2 are two standing waves spatially π out of phase. b, Photocurrent signal (Iphoto) from the CNT as the laser wavelength is rapidly swept across the two optical standing wave modes. The alternating photocurrent strength corresponds to the spatial geometry of the two optical standing waves.

3. ### Extended Data Fig. 3 Optical polarizability measurements.

a, Schematic of the measurement. The CNT is touched to the surface of the cavity (dotted line) and then the tweezers are moved downward in the plane of the cavity, moving the CNT over several optical nodes. b, The resulting transmission data as a function of displacement (plotted from black to red). Resonance spectra are plotted as the difference between the off-resonance power (P0) and the transmitted power (Ptrans) and are displaced for clarity. c, Relationship between shifts in damping γ1 and frequency f1 (referenced to their respective far-field quantities γ01 and f01) for the higher-frequency mode. d, Relationship between the damping rates of both cavities. Vertical and horizontal lines denote the far-field damping rates γ01 and γ02 respectively and the blue spectra correspond to the maximum damping condition for each mode. The linear fit specifies the maximum damping rate γ ≈ 450 MHz and is due to the orthogonality of the two spatial modes.

4. ### Extended Data Fig. 4 Broadband analysis of pseudo-periodic resonances.

a, Spectrogram of 145 ms of continuous data, revealing correlations of amplitude and frequency variation among several resonance modes. The power spectrum is plotted on the right with modes labelled based on analysis in Supplementary section 5. b, Correlations between resonance mode amplitudes A and frequency shifts Δf. The frequency shifts of $${f}_{3z}$$ is plotted above (blue), with dotted lines corresponding to local minima. The (normalized) amplitudes of five modes are plotted below.

## Supplementary information

1. ### Supplementary Information

This file contains Supplementary Discussions 1–16, and Supplementary Figures 1–17. The discussions detail: (1) photocurrent mapping of cavity fields, (2) simulations of optical cavity field, (3) measurements of CNT polarizability, (4) electrostatic tuning of CNT resonances, (5) optical tomography of mechanical resonances, (6) simulations of mechanical resonances, (7) spectrographic analysis of CNT time traces, (8) measurements of nonlinearity in thermally driven resonances, (9) theory of optical mode perturbation, (10) calculation of optical scattering length, (11) calculation of optomechanical displacement sensitivity, (12) calibration of displacement signal, (13) evidence that spectral diffusion is intrinsic to CNT motion, (14) analysis of spectral diffusion, (15) additional simulation data and (16) analysis of CNT thermal statistics.

2. ### Supplementary Video 1

Audio representation of thermally driven spectral diffusion in a CNT. The measured real-time time trace (a segment of Fig. 3 data) is slowed down 1,300 times to form an audible signal.

### DOI

https://doi.org/10.1038/s41586-018-0861-0