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Synthetic gauge fields for phonon transport in a nano-optomechanical system

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Gauge fields in condensed matter physics give rise to nonreciprocal and topological transport phenomena and exotic electronic states1. Nanomechanical systems are applied as sensors and in signal processing, and feature strong nonlinearities. Gauge potentials acting on such systems could induce quantum Hall physics for phonons at the nanoscale. Here, we demonstrate a magnetic gauge field for nanomechanical vibrations in a scalable, on-chip optomechanical system. We induce the gauge field through multi-mode optomechanical interactions, which have been proposed as a resource for the necessary breaking of time-reversal symmetry2,3,4. In a dynamically modulated nanophotonic system, we observe how radiation pressure forces mediate phonon transport between resonators of different frequencies. The resulting controllable interaction, which is characterized by a high rate and nonreciprocal phase, mimics the Aharonov–Bohm effect5. We show that the introduced scheme does not require high-quality cavities, such that it allows exploring topological acoustic phases in many-mode systems resilient to realistic disorder.

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Fig. 1: Optomechanical system for synthetic nanomechanical gauge fields.
Fig. 2: Optically mediated phonon conversion.
Fig. 3: Nonreciprocal phase imprint.
Fig. 4: Extended optomechanical nanobeam lattice.

Data availability

The data that support the plots within this paper and other findings of this study are available from the open-access repository Zenodo, with assigned digital object identifier (DOI) 10.5281/zenodo.3554024.

Code availability

The computer codes that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

Change history

  • 07 April 2022

    In the version of this article initially published, a conversion error led to “>” symbols appearing as “gt”, once in the paragraph above Eq. (2a) and twice in the paragraph below Eq. (3). The errors have been corrected in the HTML and PDF versions of the article.


  1. Laughlin, R. B. Quantized Hall conductivity in two dimensions. Phys. Rev. B 23, 5632 (1981).

    Article  Google Scholar 

  2. Peano, V., Brendel, C., Schmidt, M. & Marquardt, F. Topological phases of sound and light. Phys. Rev. X 5, 031011 (2015).

    Google Scholar 

  3. Ruesink, F., Miri, M.-A., Alù, A. & Verhagen, E. Nonreciprocity and magnetic-free isolation based on optomechanical interactions. Nat. Commun. 7, 13662 (2016).

    Article  CAS  Google Scholar 

  4. Fang, K. et al. Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering. Nat. Phys. 13, 465–471 (2017).

    Article  CAS  Google Scholar 

  5. Aharonov, Y. & Bohm, D. Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115, 485–491 (1959).

    Article  Google Scholar 

  6. Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).

    Article  CAS  Google Scholar 

  7. Goldman, N., Budich, J. & Zoller, P. Topological quantum matter with ultracold gases in optical lattices. Nat. Phys. 12, 639–645 (2016).

    Article  CAS  Google Scholar 

  8. Huber, S. D. Topological mechanics. Nat. Phys. 12, 621–623 (2016).

    Article  CAS  Google Scholar 

  9. Lindner, N. H., Refael, G. & Galitski, V. Floquet topological insulator in semiconductor quantum wells. Nat. Phys. 7, 490–495 (2011).

    Article  CAS  Google Scholar 

  10. Dalibard, J., Gerbier, F., Juzeliuūnas, G. & Öhberg, P. Colloquium: Artificial gauge potentials for neutral atoms. Rev. Mod. Phys. 83, 1523–1543 (2011).

    Article  CAS  Google Scholar 

  11. Bermudez, A., Schaetz, T. & Porras, D. Synthetic gauge fields for vibrational excitations of trapped ions. Phys. Rev. Lett. 107, 150501 (2011).

    Article  Google Scholar 

  12. Fang, K., Yu, Z. & Fan, S. Photonic Aharonov–Bohm effect based on dynamic modulation. Phys. Rev. Lett. 108, 153901 (2012).

    Article  Google Scholar 

  13. Fang, K., Yu, Z. & Fan, S. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nat. Photon. 6, 782–787 (2012).

    Article  CAS  Google Scholar 

  14. Nash, L. M. et al. Topological mechanics of gyroscopic metamaterials. Proc. Natl Acad. Sci. USA 112, 14495–14500 (2015).

    Article  CAS  Google Scholar 

  15. Wang, Y. et al. Observation of nonreciprocal wave propagation in a dynamic phononic lattice. Phys. Rev. Lett. 121, 194301 (2018).

    Article  CAS  Google Scholar 

  16. Cha, J., Kim, K. W. & Daraio, C. Experimental realization of on-chip topological nanoelectromechanical metamaterials. Nature 564, 229–233 (2018).

    Article  CAS  Google Scholar 

  17. Shen, Z. et al. Experimental realization of optomechanically induced non-reciprocity. Nat. Photon. 10, 657–661 (2016).

    Article  CAS  Google Scholar 

  18. Peterson, G. A. et al. Demonstration of efficient nonreciprocity in a microwave optomechanical circuit. Phys. Rev. X 7, 031001 (2017).

    Google Scholar 

  19. Bernier, N. R. et al. Nonreciprocal reconfigurable microwave optomechanical circuit. Nat. Commun. 8, 604 (2017).

    Article  CAS  Google Scholar 

  20. Xu, H., Mason, D., Jiang, L. & Harris, J. Topological energy transfer in an optomechanical system with exceptional points. Nature 537, 80–83 (2016).

    Article  CAS  Google Scholar 

  21. Xu, H., Jiang, L., Clerk, A. A. & Harris, J. G. E. Nonreciprocal control and cooling of phonon modes in an optomechanical system. Nature 568, 65–69 (2019).

    Article  CAS  Google Scholar 

  22. Walter, S. & Marquardt, F. Classical dynamical gauge fields in optomechanics. New J. Phys. 18, 113029 (2016).

    Article  Google Scholar 

  23. Shkarin, A. et al. Optically mediated hybridization between two mechanical modes. Phys. Rev. Lett. 112, 013602 (2014).

    Article  CAS  Google Scholar 

  24. Weaver, M. J. et al. Coherent optomechanical state transfer between disparate mechanical resonators. Nat. Commun. 8, 824 (2017).

    Article  Google Scholar 

  25. Ockeloen-Korppi, C. et al. Stabilized entanglement of massive mechanical oscillators. Nature 556, 478–482 (2018).

    Article  CAS  Google Scholar 

  26. Leijssen, R., La Gala, G. R., Freisem, L., Muhonen, J. T. & Verhagen, E. Nonlinear cavity optomechanics with nanomechanical thermal fluctuations. Nat. Commun. 8, 16024 (2017).

    Article  Google Scholar 

  27. Okamoto, H. et al. Coherent phonon manipulation in coupled mechanical resonators. Nat. Phys. 9, 480–484 (2013).

    Article  CAS  Google Scholar 

  28. Faust, T., Rieger, J., Seitner, M. J., Kotthaus, J. P. & Weig, E. M. Coherent control of a classical nanomechanical two-level system. Nat. Phys. 9, 485–488 (2013).

    Article  CAS  Google Scholar 

  29. Tzuang, L. D., Fang, K., Nussenzveig, P., Fan, S. & Lipson, M. Non-reciprocal phase shift induced by an effective magnetic flux for light. Nat. Photon. 8, 701–705 (2014).

    Article  CAS  Google Scholar 

  30. Roushan, P. et al. Chiral ground-state currents of interacting photons in a synthetic magnetic field. Nat. Phys. 13, 146–151 (2017).

    Article  CAS  Google Scholar 

  31. Celi, A. et al. Synthetic gauge fields in synthetic dimensions. Phys. Rev. Lett. 112, 043001 (2014).

    Article  CAS  Google Scholar 

  32. Hatanaka, D., Mahboob, I., Onomitsu, K. & Yamaguchi, H. Phonon waveguides for electromechanical circuits. Nat. Nanotechnol. 9, 520–524 (2014).

    Article  CAS  Google Scholar 

  33. Cha, J. & Daraio, C. Electrical tuning of elastic wave propagation in nanomechanical lattices at MHz frequencies. Nat. Nanotechnol. 13, 1016–1020 (2018).

    Article  CAS  Google Scholar 

  34. Groth, C. W., Wimmer, M., Akhmerov, A. R. & Waintal, X. Kwant: a software package for quantum transport. New J. Phys. 16, 063065 (2014).

    Article  Google Scholar 

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This work is part of the research programme of the Netherlands Organisation for Scientific Research (NWO). The authors acknowledge support from the Office of Naval Research (grant no. N00014-16-1-2466), the European Research Council (ERC starting grant no. 759644-TOPP) and the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 732894 (FET Proactive HOT).

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Authors and Affiliations



J.P.M. performed the experiments and analysed the data. J.d.P. developed the theoretical model. E.V. supervised the project. All authors contributed to the interpretation of results and writing of the manuscript.

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Correspondence to Ewold Verhagen.

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Extended data

Extended Data Fig. 1 Detuning dependent phonon transfer.

a,b,c, Mode transfer measurements for a detuning of \(\Delta /k\ \simeq\) 0.25, 0.01, and \(-0.23\) respectively. The left two panels show the driven response of mode 1 and simultaneously measured transferred response to mode 2 for a fixed \({\omega }_{{\rm{m}}}\) chosen at each detuning. The phase imprint on mode 2 as a function of the modulation phase is shown in the rightmost panels. There is no clear phase pickup for near-zero detuning of the drive laser. The dashed line in the transferred response corresponds to the average magnitude of the transfer signal measured during the phase transfer measurement.

Extended Data Fig. 2 Robust phononic edge states.

a, Optical spring shift measured at higher optical powers for a separate device shows large tunability of the mechanical modes. The dashed, white line shows the drive laser detuning used in b. The unusual features between 193.0 and 193.2 THz are due to a dynamically unstable regime of potential photothermoelastic origin. b, The modulated coupling strength, geff, is higher for higher optical powers. The largest modulated coupling strength measured in our experiments is marked by the dashed line and seen to be \(\sim 2\pi \times 200\) kHz. Here the detection laser was absent and the drive laser response was directly demodulated using the lock-in amplifier leading to a Fano shaped feature for the driven response. c,d,e, (top panels) Band structure for a ribbon geometry (width \(L=20\)) for increasing values of the direct mechanical coupling from left to right (0, 10, and 20 kHz, respectively), displaying the driving modulation frequency. The steady-state phononic amplitude in the absence of disorder is displayed in the middle row, while the result with a phononic frequency disorder with standard deviation \({\sigma }_{\Omega }/2\pi\) = 20 kHz (averaged over 100 realizations) is shown at the bottom. For these plots, \(p/q=1/3\).

Supplementary information

Supplementary Information

Supplementary equations, Figs. 1–4 and refs. 1–16.

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Mathew, J.P., Pino, J.d. & Verhagen, E. Synthetic gauge fields for phonon transport in a nano-optomechanical system. Nat. Nanotechnol. 15, 198–202 (2020).

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