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  • Letter
  • Open Access

Catastrophic transition between dynamical patterns in a phonon laser

Qing Lin1, Bing He2,*, and Min Xiao3

  • 1Fujian Key Laboratory of Light Propagation and Transformation, College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
  • 2Center for Quantum Optics and Quantum Information, Universidad Mayor, Camino La Pirámide 5750, Huechuraba, Chile
  • 3Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA

  • *bing.he@umayor.cl
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Phys. Rev. Research 3, L032018 – Published 26 July, 2021

DOI: https://doi.org/10.1103/PhysRevResearch.3.L032018

Abstract

The bifurcations or transitions of the classical nonlinear systems with finite degrees of freedom exhibit some dynamical behaviors that are similar to those in the phase transitions of many-body systems. A dynamical type of them in the form of a sudden transition of the amplitudes between two harmonic oscillations, which are stable under periodically driving force, is not so common to see. We show that a sudden transition of this type can be realized in a system of coupling a microresonator that supports a mechanical vibration to another microresonator, and the system was experimentally demonstrated as a phonon laser. If the pump laser power or the intercavity coupling strength is adjusted to a critical value, the cavity field patterns will suddenly change, together with a jump of the stabilized mechanical amplitude. Such transition can be applied to a precise measurement of the optomechanical coupling constant, which is hard to measure due to its proportion to a tiny zero-point mechanical fluctuation amplitude.

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References (74)

  1. Y. A. Kuznetsov, Elements of Applied Bifurcation Theory (Springer, New York, 1998).
  2. T. Ma and S. Wang, Phase Transition Dynamics (Springer, New York, 2014).
  3. J. R. Tredicce, G. L. Lippi, P. Mandel, B. Charasse, A. Chevalier, and B. Picque, Critical slowing down at a bifurcation, Am. J. Phys. 72, 799 (2004).
  4. M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkin, S. R. Carpenter, V. Dakos, H. Held, E. H. van Nes, M. Rietkerk, and G. Sugihara, Early-warning signals for critical transitions, Nature (London) 461, 53 (2009).
  5. E. Garmire, J. H. Marburger, S. D. Allen, and H. G. Winful, Transient response of hybrid bistable optical devices, Appl. Phys. Lett. 34, 374 (1979).
  6. J. Y. Bigot, A. Daunois, and P. Mandel, Slowing down far from the limit points in optical bistability, Phys. Lett. A 123, 123 (1987).
  7. B. A. Huberman and J. Rudnick, Scaling Behavior of Chaotic Flows, Phys. Rev. Lett. 45, 154 (1980).
  8. B.-L. Hao, Universal slowing-down exponent near period-doubling bifurcation points, Phys. Lett. A 86, 267 (1981).
  9. J. Guckenheimer and P. J. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, New York, 1983).
  10. L. Perko, Differential Equations and Dynamical Systems (Springer-Verlag, New York, 2001).
  11. V. A. Gaiko, Global Bifurcation Theory and Hilbert's Sixteenth Problem, Mathematics and its Applications Vol. 559 (Kluwer Academic, Dordrecht, The Netherlands, 2003).
  12. J. Moehlis, K. Josic, and E. Shea-Brown, Periodic orbit, Scholarpedia 1, 1358 (2006).
  13. M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems (Springer, New York, 2007).
  14. F. Dercole and S. Rinaldi, Dynamical systems and their bifurcations, in Advanced Methods of Biomedical Signal Processing, edited by S. Cerutti and C. Marchesi (IEEE-Wiley Press, New York, 2011), pp. 291–325.
  15. A. Capietto, P. Kloeden, J. Mawhin, S. Novo, and R. Ortega, Stability and Bifurcation Theory for Non-Autonomous Differential Equations: Cetraro, Italy 2011 (Springer, New York, 2013).
  16. S. H. Strogatz, Nonlinear Dynamics and Chaos, 2nd ed. (CRC Press/Taylor & Francis, Boca Raton/London, 2018).
  17. H. K. Leung, Critical slowing down in synchronizing nonlinear oscillators, Phys. Rev. E 58, 5704 (1998).
  18. H. K. Leung, Bifurcation of synchronization as a nonequilibrium phase transition, Physica A 281, 311 (2000).
  19. P. Woafo and R. A. Kraenkel, Synchronization: Stability and duration time, Phys. Rev. E 65, 036225 (2002).
  20. J. A. Langa, J. C. Robinson, and A. Suárez, Stability, instability, and bifurcation phenomena in non-autonomous differential equations, Nonlinearity 15, 887 (2002).
  21. M. Marconi, J. Javaloyes, S. Balle, and M. Giudici, How Lasing Localized Structures Evolve out of Passive Mode Locking, Phys. Rev. Lett. 112, 223901 (2014).
  22. A. V. Kovalev and E. A. Viktorov, Class-A mode-locked lasers: Fundamental solutions, Chaos 27, 114318 (2017).
  23. P. Parra-Rivas, D. Gomila, L. Gelens, and E. Knobloch, Bifurcation structure of localized states in the Lugiato-Lefever equation with anomalous dispersion, Phys. Rev. E 97, 042204 (2018).
  24. S. Coen and M. Erkintalo, Universal scaling laws of Kerr frequency combs, Opt. Lett. 38, 1790 (2013).
  25. B. Haegeman, K. Engelborghs, D. Roose, D. Pieroux, and T. Erneux, Stability and rupture of bifurcation bridges in semiconductor lasers subject to optical feedback, Phys. Rev. E 66, 046216 (2002).
  26. J.-P. Francoise and C. C. Pough, Keeping track of limit cycles, J. Diff. Equations 65, 139 (1986).
  27. L. M. Perko, Global family of limit cycles of planar, Trans. Amer. Math. Soc. 322, 627 (1990).
  28. L. M. Perko, Homoclinic loop and multiple limit cycle bifurcation surfaces, Trans. Amer. Math. Soc. 344, 101 (1994).
  29. L. M. Perko, Multiple limit cycle bifurcation surfaces and global families of multiple limit cycles, J. Diff. Equations 122, 89 (1995).
  30. H. Xu, U. Kemiktarak, J. Fan, S. Ragole, J. Lawall, and J. M. Taylor, Observation of optomechanical buckling transitions, Nat. Commun. 8, 14481 (2017).
  31. A. Roy, S. Jahani, C. Langrock, M. Fejer, and A. Marandi, Spectral phase transitions in optical parametric oscillators, Nat. Commun. 12, 835 (2021).
  32. V. M. Bastidas, C. Emary, B. Regler, and T. Brandes, Non-Equilibrium Quantum Phase Transitions in the Dicke Model, Phys. Rev. Lett. 108, 043003 (2012).
  33. J. Klinder, H. Keßler, M. Wolke, L. Mathey, and A. Hemmerich, Dynamical phase transition in the open Dicke model, Proc. Natl. Acad. Sci. USA 112, 3290 (2015).
  34. Z. Zhang, C. H. Lee, R. Kumar, K. J. Arnold, S. J. Masson, A. S. Parkins, and M. D. Barrett, Nonequilibrium phase transition in a spin-1 Dicke model, Optica 4, 424 (2017).
  35. J. M. Fink, A. Dombi, A. Vukics, A. Wallraff, and P. Domokos, Observation of the Photon-Blockade Breakdown Phase Transition, Phys. Rev. X 7, 011012 (2017).
  36. N. Mann, M. R. Bakhtiari, A. Pelster, and M. Thorwart, Nonequilibrium Quantum Phase Transition in a Hybrid Atom-Optomechanical System, Phys. Rev. Lett. 120, 063605 (2018).
  37. C. Gao and Z. Liang, Steady-state phase diagram of quantum gases in a lattice coupled to a membrane, Phys. Rev. A 99, 013629 (2019).
  38. N. Mann, A. Pelster, and M. Thorwart, Tuning the order of the nonequilibrium quantum phase transition in a hybrid atom-optomechanical system, New J. Phys. 21, 113037 (2019).
  39. A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. Walther, Optical Bistability and Mirror Confinement Induced by Radiation Pressure, Phys. Rev. Lett. 51, 1550 (1983).
  40. H. Rokhsari, T. Kippenberg, T. Carmon, and K. Vahala, Radiation-pressure-driven micro-mechanical oscillator, Opt. Express 13, 5293 (2005).
  41. T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity, Phys. Rev. Lett. 95, 033901 (2005).
  42. T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, Temporal Behavior of Radiation-Pressure-Induced Vibrations of an Optical Microcavity Phonon Mode, Phys. Rev. Lett. 94, 223902 (2005).
  43. F. Marquardt, J. G. E. Harris, and S. M. Girvin, Dynamical Multistability Induced by Radiation Pressure in High-Finesse Micromechanical Optical Cavities, Phys. Rev. Lett. 96, 103901 (2006).
  44. K. J. Vahala, Back-action limit of linewidth in an optomechanical oscillator, Phys. Rev. A 78, 023832 (2008).
  45. G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, Collective Dynamics in Optomechanical Arrays, Phys. Rev. Lett. 107, 043603 (2011).
  46. A. G. Krause, J. T. Hill, M. Ludwig, A. H. Safavi-Naeini, J. Chan, F. Marquardt, and O. Painter, Nonlinear Radiation Pressure Dynamics in an Optomechanical Crystal, Phys. Rev. Lett. 115, 233601 (2015).
  47. J. Wen, X. Jiang, L. Jiang, and M. Xiao, Parity-time symmetry in optical microcavity systems, J. Phys. B: At., Mol. Opt. Phys. 51, 222001 (2018).
  48. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non-Hermitian physics and PT symmetry, Nat. Phys. 14, 11 (2018).
  49. I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, Phonon Laser Action in a Tunable Two-Level System, Phys. Rev. Lett. 104, 083901 (2010).
  50. H. Jing, S. K. Özdemir, X. Y. Lü, J. Zhang, L. Yang, and F. Nori, PT-Symmetric Phonon Laser, Phys. Rev. Lett. 113, 053604 (2014).
  51. B. He, L. Yang, and M. Xiao, Dynamical phonon laser in coupled active-passive microresonators, Phys. Rev. A 94, 031802(R) (2016).
  52. H. Lü, S. K. Özdemir, L.-M. Kuang, F. Nori, and H. Jing, Exceptional Points in Random-Defect Phonon Lasers, Phys. Rev. Appl. 8, 044020 (2017).
  53. Y. L. Zhang, C. L. Zou, C. S. Yang, H. Jing, C. H. Dong, G. C. Guo, and X. B. Zou, Phase-controlled phonon laser, New J. Phys. 20, 093005 (2018).
  54. B. Wang, H. Xiong, X. Jia, and Y. Wu, Phonon laser in the coupled vector cavity optomechanics, Sci. Rep. 8, 282 (2018).
  55. Y. F. Xie, Z. Cao, B. He, and Q. Lin, PT-symmetric phonon laser under gain saturation effect, Opt. Express 28, 22580 (2020).
  56. G. Z. Wang, M. M. Zhao, Y. C. Qin, Z. Q. Yin, X. S. Jiang, and M. Xiao, Demonstration of an ultra-low-threshold phonon laser with coupled microtoroid resonators in vacuum, Photonics Res. 5, 73 (2017).
  57. J. Zhang, B. Peng, Ş. K. Özdemir, K. Pichler, D. O. Krimer, G. P. Zhao, F. Nori, Y. X. Liu, S. Rotter, and L. Yang, A phonon laser operating at an exceptional point, Nat. Photonics 12, 479 (2018).
  58. B. He, L. Yang, X. Jiang, and M. Xiao, Transmission Nonreciprocity in a Mutually Coupled Circulating Structure, Phys. Rev. Lett. 120, 203904 (2018).
  59. Z. Cao, Y. F. Xie, B. He, and Q. Lin, Ultra-high optical nonreciprocity with a coupled triple-resonator structure, New J. Phys. 23, 023010 (2021).
  60. B. He, L. Yang, Q. Lin, and M. Xiao, Radiation Pressure Cooling as a Quantum Dynamical Process, Phys. Rev. Lett. 118, 233604 (2017).
  61. W. D. Heiss, The physics of exceptional points, J. Phys. A: Math. Theor. 45, 444016 (2012).
  62. V. V. Konotop, J. Yang, and D. A. Zezyulin, Nonlinear waves in PT-symmetric systems, Rev. Mod. Phys. 88, 035002 (2016).
  63. M.-A. Miri and A. Alù, Exceptional points in optics and photonics, Science 363, eaar7709 (2019).
  64. Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, Paritytime symmetry and exceptional points in photonics, Nat. Mater. 18, 783 (2019).
  65. M. Ludwig, B. Kubala, and F. Marquardt, The optomechanical instability in the quantum regime, New J. Phys. 10, 095013 (2008).
  66. B. He, Q. Lin, M. Orszag, and M. Xiao, Mechanical oscillations frozen on discrete levels by two optical driving fields, Phys. Rev. A 102, 011503(R) (2020).
  67. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.3.L032018, which includes Refs. [68, 69, 70, 71, 72, 73], for more detailed information.
  68. L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Stochastic resonance, Rev. Mod. Phys. 70, 223 (1998).
  69. W. Horsthemke and R. Lefever, Noise-Induced Transitions (Springer-Verlag, Berlin, Heidelberg, 2006).
  70. Q. Lin, B. He, and M. Xiao, Entangling Two Macroscopic Mechanical Resonators at High Temperature, Phys. Rev. Appl. 13, 034030 (2020).
  71. B. He, L. Yang, Z. Zhang, and M. Xiao, Cyclic permutation-time symmetric structure with coupled gain-loss microcavities, Phys. Rev. A 91, 033830 (2015).
  72. B. He, S.-B. Yan, J. Wang, and M. Xiao, Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides, Phys. Rev. A 91, 053832 (2015).
  73. T. F. Roque, F. Marquardt, and O. M. Yevtushenko, Nonlinear dynamics of weakly dissipative optomechanical systems, New J. Phys. 22, 013049 (2020).
  74. M. L. Gorodetksy, A. Schliesser, G. Anetsberger, S. Deleglise, and T. J. Kippenberg, Determination of the vacuum optomechanical coupling rate using frequency noise calibration, Opt. Express 18, 23236 (2010).

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Phys. Rev. Research 3, L032018– Published 26 July, 2021
  • Received 5 January 2021
  • Accepted 12 July 2021
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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