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Non-reciprocal transmission of microwave acoustic waves in nonlinear parity–time symmetric resonators


Acoustic waves are versatile on-chip information carriers that can be used in applications such as microwave filters and transducers. Nonreciprocal devices, in which the transmission of waves is non-symmetric between two ports, are desirable for the manipulation and routing of phonons, but building acoustic non-reciprocal devices is difficult because acoustic systems typically have a linear response. Here, we report non-reciprocal transmission of microwave surface acoustic waves using a nonlinear parity–time symmetric system based on two coupled acoustic resonators in a lithium niobate platform. Owing to the strong piezoelectricity of lithium niobate, we can tune the gain, loss and nonlinearity of the system using electric circuitry. Our approach can achieve 10 dB of non-reciprocal transmission for surface acoustic waves at a frequency of 200 MHz, and we use it to demonstrate a one-way circulation of acoustic waves in cascading non-reciprocal devices.

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Fig. 1: Non-reciprocal SAW transmission using nonlinear PT-symmetric resonators.
Fig. 2: Characterization of SAW gain and nonlinearity.
Fig. 3: SAW transmission of the nonlinear PT-symmetric resonators.
Fig. 4: One-way circulation of SAWs.

Data availability

Source data are available for the graphs plotted in Figs. 14 and Extended Data Fig. 2. All other data and findings of this study are available from the corresponding author upon reasonable request.


  1. 1.

    Chu, Y. et al. Quantum acoustics with superconducting qubits. Science 358, 199–202 (2017).

    MathSciNet  Article  Google Scholar 

  2. 2.

    Satzinger, K. J. et al. Quantum control of surface acoustic-wave phonons. Nature 563, 661–665 (2018).

    Article  Google Scholar 

  3. 3.

    Whiteley, S. J. et al. Spin–phonon interactions in silicon carbide addressed by Gaussian acoustics. Nat. Phys. 15, 490–495 (2019).

    Article  Google Scholar 

  4. 4.

    Maity, S. et al. Coherent acoustic control of a single silicon vacancy spin in diamond. Nat. Commun. 11, 193 (2020).

    Article  Google Scholar 

  5. 5.

    Arrangoiz-Arriola, P. et al. Resolving the energy levels of a nanomechanical oscillator. Nature 571, 537–540 (2019).

    Article  Google Scholar 

  6. 6.

    Otterstrom, N. T., Behunin, R. O., Kittlaus, E. A., Wang, Z. & Rakich, P. T. A silicon Brillouin laser. Science 360, 1113–1116 (2018).

    MathSciNet  Article  Google Scholar 

  7. 7.

    Campbell, C. Surface Acoustic Wave Devices and their Signal Processing Applications (Academic Press, 1989).

  8. 8.

    Fleury, R., Sounas, D. L., Sieck, C. F., Haberman, M. R. & Alù, A. Sound isolation and giant linear nonreciprocity in a compact acoustic circulator. Science 343, 516–519 (2014).

    Article  Google Scholar 

  9. 9.

    Liang, B., Guo, X. S., Tu, J., Zhang, D. & Cheng, J. C. An acoustic rectifier. Nat. Mater. 9, 989–992 (2010).

    Article  Google Scholar 

  10. 10.

    Liang, B., Yuan, B. & Cheng, J. C. Acoustic diode: rectification of acoustic energy flux in one-dimensional systems. Phys. Rev. Lett. 103, 104301 (2009).

    Article  Google Scholar 

  11. 11.

    Li, Y. et al. Tunable asymmetric transmission via lossy acoustic metasurfaces. Phys. Rev. Lett. 119, 035501 (2017).

    Article  Google Scholar 

  12. 12.

    Popa, B. I. & Cummer, S. A. Non-reciprocal and highly nonlinear active acoustic metamaterials. Nat. Commun. 5, 3398 (2014).

    Article  Google Scholar 

  13. 13.

    Walker, E. et al. Nonreciprocal linear transmission of sound in a viscous environment with broken P symmetry. Phys. Rev. Lett. 120, 204501 (2018).

    Article  Google Scholar 

  14. 14.

    Devaux, T., Cebrecos, A., Richoux, O., Pagneux, V. & Tournat, V. Acoustic radiation pressure for nonreciprocal transmission and switch effects. Nat. Commun. 10, 3292 (2019).

    Article  Google Scholar 

  15. 15.

    Sasaki, R., Nii, Y., Iguchi, Y. & Onose, Y. Nonreciprocal propagation of surface acoustic wave in Ni/LiNbO3. Phys. Rev. B 95, 020407 (2017).

    Article  Google Scholar 

  16. 16.

    Nomura, T. et al. Phonon magnetochiral effect. Phys. Rev. Lett. 122, 145901 (2019).

    Article  Google Scholar 

  17. 17.

    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  Google Scholar 

  18. 18.

    Bender, C. M. & Boettcher, S. Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998).

    MathSciNet  Article  Google Scholar 

  19. 19.

    El-Ganainy, R. et al. Non-Hermitian physics and PT symmetry. Nat. Phys. 14, 11–19 (2018).

    Article  Google Scholar 

  20. 20.

    Konotop, V. V., Yang, J. & Zezyulin, D. A. Nonlinear waves in PT-symmetric systems. Rev. Mod. Phys. 88, 035002 (2016).

    Article  Google Scholar 

  21. 21.

    Li, Y. et al. Anti-parity–time symmetry in diffusive systems. Science 364, 170–173 (2019).

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Feng, L., El-Ganainy, R. & Ge, L. Non-Hermitian photonics based on parity–time symmetry. Nat. Photon. 11, 752–762 (2017).

    Article  Google Scholar 

  23. 23.

    Peng, B. et al. Parity–time-symmetric whispering-gallery microcavities. Nat. Phys. 10, 394–398 (2014).

    Article  Google Scholar 

  24. 24.

    Feng, L. et al. Experimental demonstration of a unidirectional reflectionless parity–time metamaterial at optical frequencies. Nat. Mater. 12, 108–113 (2013).

    Article  Google Scholar 

  25. 25.

    Chang, L. et al. Parity–time symmetry and variable optical isolation in active–passive-coupled microresonators. Nat. Photon. 8, 524–529 (2014).

    Article  Google Scholar 

  26. 26.

    Feng, L., Wong, Z. J., Ma, R.-M., Wang, Y. & Zhang, X. Single-mode laser by parity–time symmetry breaking. Science 346, 972–975 (2014).

    Article  Google Scholar 

  27. 27.

    Klauck, F. et al. Observation of PT-symmetric quantum interference. Nat. Photon. 13, 883–887 (2019).

    Article  Google Scholar 

  28. 28.

    Zhen, B. et al. Spawning rings of exceptional points out of Dirac cones. Nature 525, 354–358 (2015).

    Article  Google Scholar 

  29. 29.

    Özdemir, Ş. K., Rotter, S., Nori, F. & Yang, L. Parity–time symmetry and exceptional points in photonics. Nat. Mater. 18, 783–798 (2019).

    Article  Google Scholar 

  30. 30.

    Fleury, R., Sounas, D. & Alu, A. An invisible acoustic sensor based on parity–time symmetry. Nat. Commun. 6, 5905 (2015).

    Article  Google Scholar 

  31. 31.

    Shi, C. et al. Accessing the exceptional points of parity–time symmetric acoustics. Nat. Commun. 7, 11110 (2016).

    Article  Google Scholar 

  32. 32.

    Auregan, Y. & Pagneux, V. PT-symmetric scattering in flow duct acoustics. Phys. Rev. Lett. 118, 174301 (2017).

    Article  Google Scholar 

  33. 33.

    Zhu, X., Ramezani, H., Shi, C., Zhu, J. & Zhang, X. PT-symmetric acoustics. Phys. Rev. X 4, 031042 (2014).

    Google Scholar 

  34. 34.

    Zhang, J. et al. Giant nonlinearity via breaking parity–time symmetry: a route to low-threshold phonon diodes. Phys. Rev. B 92, 115407 (2015).

    Article  Google Scholar 

  35. 35.

    Xu, X.-W., Liu, Y.-x, Sun, C.-P. & Li, Y. Mechanical PT symmetry in coupled optomechanical systems. Phys. Rev. A 92, 013852 (2015).

    Article  Google Scholar 

  36. 36.

    Bender, C. M., Berntson, B. K., Parker, D. & Samuel, E. Observation of PT phase transition in a simple mechanical system. Am. J. Phys. 81, 173–179 (2013).

    Article  Google Scholar 

  37. 37.

    Assawaworrarit, S., Yu, X. & Fan, S. Robust wireless power transfer using a nonlinear parity–time-symmetric circuit. Nature 546, 387–390 (2017).

    Article  Google Scholar 

  38. 38.

    Morgan, D. & Paige, E. G. S. Surface Acoustic Wave Filters with Applications to Electronic Communications and Signal Processing (Academic Press, 2007).

  39. 39.

    Shao, L. et al. Phononic band structure engineering for high-Q gigahertz surface acoustic wave resonators on lithium niobate. Phys. Rev. Appl. 12, 014022 (2019).

    Article  Google Scholar 

  40. 40.

    Jiang, X. et al. On-chip optical nonreciprocity using an active microcavity. Sci. Rep. 6, 38972 (2016).

    Article  Google Scholar 

  41. 41.

    Wen, J. et al. Modeling of on-chip optical nonreciprocity with an active microcavity. Photonics 2, 498–508 (2015).

    Article  Google Scholar 

  42. 42.

    Shi, Y., Yu, Z. & Fan, S. Limitations of nonlinear optical isolators due to dynamic reciprocity. Nat. Photon. 9, 388–392 (2015).

    Article  Google Scholar 

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We thank S. Bogdanovic, M. Yu, M. Zhang, C. Chia, B. Machielse and Y.-F. Xiao for fruitful discussions. This work is supported by the STC Center for Integrated Quantum Materials, NSF grant no. DMR-1231319, NSF CQIS grant no. ECCS-1810233, ONR MURI grant no. N00014-15-1-2761 and AFOSR MURI grant no. FA9550-14-1-0389. N.S. acknowledges support by the Natural Sciences and Engineering Research Council of Canada (NSERC), the AQT Intelligent Quantum Networks and Technologies (INQNET) research programme and the DOE/HEP QuantISED programme grant and QCCFP (Quantum Communication Channels for Fundamental Physics) award no. DE-SC0019219. W.M. acknowledges support from the undergraduate overseas internship programme of Nankai University supported by the National Science Fund for Talent Training in the Basic Sciences, grant no. J1103208. This work was performed in part at the Center for Nanoscale Systems (CNS), Harvard University.

Author information




L.S. conceptualized, designed, fabricated and measured the devices. W.M. and Y.H. analysed the system theoretically, with discussion from other authors. W.M. and L.S. performed numerical simulations. All authors analysed and interpreted the results. L.S. and W.M. prepared the manuscript with contributions from all authors. M.L. and L.Y. supervised the project.

Corresponding authors

Correspondence to Linbo Shao or Marko Lončar.

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Competing interests

M.L. is involved in developing lithium niobate technologies at HyperLight Corporation. The other authors declare no competing interests.

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Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Photo of the printed circuit board (PCB) used for non-reciprocal SAW device measurements.

This PCB supports simultaneous measurements of two devices. Signal from Port 1 to Port 2 is defined as the forward direction.

Extended Data Fig. 2 Measured non-reciprocity of the broken-PT-symmetric SAW resonators versus various input powers.

The non-reciprocity is defined in Equ. (1).

Source data

Supplementary information

Supplementary Information

Supplementary Notes 1–5 and Figs. 1–12.

Source data

Source Data Fig. 1

Source data of the plots.

Source Data Fig. 2

Source data of the plots.

Source Data Fig. 3

Source data of the plots.

Source Data Fig. 4

Source data of the plots.

Source Data Extended Data Fig. 2

Source data of the plots.

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Shao, L., Mao, W., Maity, S. et al. Non-reciprocal transmission of microwave acoustic waves in nonlinear parity–time symmetric resonators. Nat Electron 3, 267–272 (2020).

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