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Topological energy transfer in an optomechanical system with exceptional points

Abstract

Topological operations can achieve certain goals without requiring accurate control over local operational details; for example, they have been used to control geometric phases and have been proposed as a way of controlling the state of certain systems within their degenerate subspaces1,2,3,4,5,6,7,8. More recently, it was predicted that topological operations can be used to transfer energy between normal modes, provided that the system possesses a specific type of degeneracy known as an exceptional point9,10,11. Here we demonstrate the transfer of energy between two vibrational modes of a cryogenic optomechanical device using topological operations. We show that this transfer arises from the presence of an exceptional point in the spectrum of the device. We also show that this transfer is non-reciprocal12,13,14. These results open up new directions in system control; they also open up the possibility of exploring other dynamical effects related to exceptional points15,16, including the behaviour of thermal and quantum fluctuations in their vicinity.

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Figure 1: The complex eigenvalues of the normal modes of the membrane.
Figure 2: The exceptional point in the spectrum of mechanical modes.
Figure 3: Topological energy transfer.
Figure 4: Non-reciprocal topological dynamics.

References

  1. Simon, B. Holonomy, the quantum adiabatic theorem, and Berry’s phase. Phys. Rev. Lett. 51, 2167–2170 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  2. Berry, M. V. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45–57 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  3. Berry, M. V. Classical adiabatic angles and quantal adiabatic phase. J. Phys. A 18, 15–27 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  4. Hannay, J. H. Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian. J. Phys. A 18, 221–230 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  5. Arovas, D., Schrieffer, J. R. & Wilczek, F. Fractional statistics and the quantum Hall effect. Phys. Rev. Lett. 53, 722–723 (1984)

    Article  ADS  Google Scholar 

  6. Tomita, A. & Chiao, R. Y. Observation of Berry’s topological phase by use of an optical fiber. Phys. Rev. Lett. 57, 937–940 (1986)

    Article  CAS  ADS  Google Scholar 

  7. Kitaev, A. Y. Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003)

    Article  CAS  ADS  MathSciNet  Google Scholar 

  8. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008)

    Article  CAS  ADS  MathSciNet  Google Scholar 

  9. Heiss, W. D. Phases of wave functions and level repulsion. Euro. Phys. J. D 7, 1–4 (1999)

    Article  CAS  ADS  Google Scholar 

  10. Keck, F., Korsch, H. J. & Mossmann, S. Unfolding a diabolic point: a generalized crossing scenario. J. Phys. A 36, 2125–2137 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  11. Berry, M. V. Physics of nonhermitian degeneracies. Czech. J. Phys. 54, 1039–1047 (2004)

    Article  CAS  ADS  Google Scholar 

  12. Berry, M. V. & Uzdin, R. Slow non-Hermitian cycling: exact solutions and the Stokes phenomenon. J. Phys. A 44, 435303 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  13. Uzdin, R., Mailybaev, A. & Moiseyev, N. On the observability and asymmetry of adiabatic state flips generated by exceptional points. J. Phys. A 44, 435302 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  14. Milburn, T. J. et al. General description of quasiadiabatic dynamical phenomena near exceptional points. Phys. Rev. A 92, 052124 (2015)

    Article  ADS  Google Scholar 

  15. Cartarius, H., Main, J. & Wunner, G. Exceptional points in the spectra of atoms in external fields. Phys. Rev. A 79, 053408 (2009)

    Article  ADS  Google Scholar 

  16. Demange, G. & Graefe, E.-M. Signatures of three coalescing eigenfunctions. J. Phys. A 45, 025303 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  17. Arnold, V. I. Mathematical Methods of Classical Mechanics Ch. 10 (Springer, 1989)

  18. Ando, T., Nakanishi, T. & Saito, R. Berry’s phase and absence of back scattering in carbon nanotubes. J. Phys. Soc. Jpn 67, 2857–2862 (1998)

    Article  CAS  ADS  Google Scholar 

  19. Lefebvre, R., Atabek, O., Šindelka, M. & Moiseyev, N. Resonance coalescence in molecular photodissociation. Phys. Rev. Lett. 103, 123003 (2009)

    Article  CAS  ADS  Google Scholar 

  20. Hamamda, M., Pillet, P., Lignier, H. & Comparat, D. Ro-vibrational cooling of molecules and prospects. J. Phys. B 48, 182001 (2015)

    Article  ADS  Google Scholar 

  21. Kaprálová-Žd’ánská, P. R. & Moiseyev, N. Helium in chirped laser fields as a time-asymmetric atomic switch. J. Chem. Phys. 141, 014307 (2014)

    Article  ADS  Google Scholar 

  22. Kim, S. Braid operation of exceptional points. Fortschr. Phys. 61, 155–161 (2013)

    Article  Google Scholar 

  23. Philipp, M., von Brentano, P., Pascovici, G. & Richter, A. Frequency and width crossing of two interacting resonances in a microwave cavity. Phys. Rev. E 62, 1922–1926 (2000)

    Article  CAS  ADS  Google Scholar 

  24. Dembowski, C. et al. Experimental observation of the topological structure of exceptional points. Phys. Rev. Lett. 86, 787–790 (2001)

    Article  CAS  ADS  Google Scholar 

  25. Thompson, J. D. et al. Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane. Nature 452, 72–75 (2008)

    Article  CAS  ADS  Google Scholar 

  26. Aspelmeyer, M., Kippenberg, T. J. & Marquardt, F. Cavity optomechanics. Rev. Mod. Phys. 86, 1391–1452 (2014)

    Article  ADS  Google Scholar 

  27. Jing, H. et al. . -symmetric phonon laser. Phys. Rev. Lett. 113, 053604 (2014)

    Google Scholar 

  28. Graefe, E.-M., Mailybaev, A. A. & Moiseyev, N. Breakdown of adiabatic transfer of light in waveguides in the presence of absorption. Phys. Rev. A 88, 033842 (2013)

    Article  ADS  Google Scholar 

  29. Jalas, D. et al. What is — and what is not — an optical isolator. Nat. Photon. 7, 579–582 (2013)

    Article  CAS  ADS  Google Scholar 

  30. Underwood, M. et al. Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime. Phys. Rev. A 92, 061801 (2015)

    Article  ADS  Google Scholar 

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

    Article  CAS  ADS  Google Scholar 

  32. Choi, Y. et al. Quasieigenstate coalescence in an atom-cavity quantum composite. Phys. Rev. Lett. 104, 153601 (2010)

    Article  ADS  Google Scholar 

  33. Lee, S.-B. et al. Observation of an exceptional point in a chaotic optical microcavity. Phys. Rev. Lett. 103, 134101 (2009)

    Article  ADS  Google Scholar 

  34. Brandstetter, M. et al. Reversing the pump dependence of a laser at an exceptional point. Nat. Commun. 5, 4034 (2014)

    Article  CAS  ADS  Google Scholar 

  35. Peng, B. et al. Loss-induced suppression and revival of lasing. Science 346, 328–332 (2014)

    Article  CAS  ADS  Google Scholar 

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

    Article  CAS  ADS  Google Scholar 

  37. Stehmann, T., Heiss, W. D. & Scholtz, F. G. Observation of exceptional points in electronic circuits. J. Phys. A 37, 7813–7819 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  38. Gao, T. et al. Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard. Nature 526, 554–558 (2015)

    Article  CAS  ADS  Google Scholar 

  39. Heiss, W. D. & Nazmitdinov, R. G. Instabilities, nonhermiticity and exceptional points in the cranking model. J. Phys. A 40, 9475–9481 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  40. Cartarius, H., Main, J. & Wunner, G. Discovery of exceptional points in the Bose-Einstein condensation of gases with attractive 1/r interaction. Phys. Rev. A 77, 013618 (2008)

    Article  ADS  Google Scholar 

  41. Weidenmüller, H. A. Crossing of two Coulomb blockade resonances. Phys. Rev. B 68, 125326 (2003)

    Article  ADS  Google Scholar 

  42. Wu, T.-T. & Huang, Z.-G. Level repulsions of bulk acoustic waves in composite materials. Phys. Rev. B 70, 214304 (2004)

    Article  ADS  Google Scholar 

  43. Günther, U., Stefani, F. & Gerbeth, G. The MHD α2-dynamo, 2-graded pseudo-Hermiticity, level crossings and exceptional points of branching type. Czech. J. Phys. 54, 1075–1089 (2004)

    Article  ADS  Google Scholar 

  44. Michel, N., Nazarewicz, W., Okołowicz, J. & Płoszajczak, M. Open problems in the theory of nuclear open quantum systems. J. Phys. G 37, 064042 (2010)

    Article  ADS  Google Scholar 

  45. Berry, M. V. Optical polarization evolution near a non-Hermitian degeneracy. J. Opt. 13, 115701 (2011)

    Article  ADS  Google Scholar 

  46. Doppler, J. et al. Dynamically encircling an exceptional point for asymmetric mode switching. Nature http://www.dx.doi.org/10.1038/nature18605 (2016)

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Acknowledgements

We thank L. Jiang, D. Lee, T. Milburn, P. Rabl, S. Rotter, A. Shkarin and W. Underwood for discussions. This work was supported by AFOSR Grant FA9550-15-1-0270.

Author information

Authors and Affiliations

Authors

Contributions

H.X., D.M. and L.J. performed the measurements and analysed the data. J.G.E.H. and H.X. wrote the manuscript with input from all the authors. J.G.E.H. directed the research.

Corresponding author

Correspondence to J. G. E. Harris.

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The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 Experimental schematics.

a, Illustration of the optical and electronic components. The measurement laser (‘ML’) is split into a local oscillator (‘LO’ in b) and a probe beam (‘Probe’ in b). The probe-beam frequency is shifted by an acousto-optic modulator (‘AOM1’), and is locked to the cavity using a Pound–Drever–Hall (PDH) scheme and modulation produced by an electro-optic modulator (‘EOM’). The control laser (‘CL’; ‘Control’ in b) is locked to the measurement laser with a frequency offset that is approximately double the free spectral range of the cavity. The control parameters used to access the EP are the power P and detuning Δ of the control laser. P and Δ are set by the amplitude and frequency of a signal generator (‘SG’), which drives another acousto-optic modulator (‘AOM3’). The PDH error signal is used to control the frequency of yet another acousto-optic modulator (‘AOM2’), ensuring that all beams track fluctuations of the cavity. Light is delivered to (and collected from) the cryostat via an optical circulator. Coloured lines, hollow lines and thick black lines show free-space laser beams, optical fibres and electrical circuits, respectively. Triangles, ovals and semicircles show electronics, fibre couplers and photodiodes, respectively. ‘DAQ’ indicates the data acquisition system. The silicon nitride membrane is shown in purple. b, Illustration of the optical frequency domain. Lasers are indicated by coloured arrows and cavity modes by black curves.

Extended Data Figure 2 Lock-in signal at low laser power (Δ = −780 kHz, P = 73 μW).

Left, amplitude (top, red) and phase angle (bottom, blue) of the lock-in signal as a function of drive frequency. Right, the same data shown as a parametric plot of the in-phase and out-of-phase components of the lock-in signal as a function of drive frequency.

Extended Data Figure 3 Lock-in signal at high laser power (Δ = −780 kHz, P = 380 μW).

Left, amplitude (top, red) and phase angle (bottom, blue) of the lock-in signal as a function of drive frequency. Right, the same data shown as a parametric plot of the in-phase and out-of-phase components of the lock-in signal as a function of drive frequency.

Extended Data Figure 4 Magnitudes of propagator matrix elements.

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Xu, H., Mason, D., Jiang, L. et al. Topological energy transfer in an optomechanical system with exceptional points. Nature 537, 80–83 (2016). https://doi.org/10.1038/nature18604

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