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Chaotic Bloch oscillations in dissipative optical systems driven by a periodic train of coherent pulses

A. Verbitskiy* and A. Yulin

A. G. Balanov

  • School of Physics and Engineering, ITMO University, Kronverksky Prospect 49, Building A, St. Petersburg 197101, Russia

  • Department of Physics, Loughborough University, Loughborough LE11 3TU, United Kingdom

  • *alexey.verbitskiy@metalab.ifmo.ru
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Phys. Rev. A 107, 053519 – Published 24 May, 2023

DOI: https://doi.org/10.1103/PhysRevA.107.053519

Abstract

We study the response of an optical system with the Kerr nonlinearity demonstrating Bloch oscillations to a periodic train of coherent pulses. It has been found out that the intensity of the field excited in the system by pulses resonantly depends on the train period. It is demonstrated numerically and analytically that the response of the system is stronger when the period of the driving pulses is commensurate with the period of the Bloch oscillations. Moreover, large enough pulses are capable to induce the instabilities which eventually lead to onset of chaotic Bloch oscillations of the wave-function envelope bouncing both in time and space. The analysis reveals that these instabilities are associated with period-doubling bifurcations. A cascade of such bifurcations with increase of the pulse amplitude triggers the chaotic behavior.

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References (80)

  1. F. Bloch, Uber die quantenmechanik der elektronen in kristallgittern, Z. Phys. 52, 555 (1929).
  2. C. Zener, A theory of the electrical breakdown of solid dielectrics, Proc. R. Soc. Lond. A 145, 523 (1934).
  3. W. V. Houston, Acceleration of electrons in a crystal lattice, Phys. Rev. 57, 184 (1940).
  4. A. Rabinovitch and J. Zak, Does a Bloch electron in a constant electric field oscillate? Phys. Lett. A 40, 189 (1972).
  5. J. Zak, Comment on “Time evolution of Bloch electrons in a homogeneous electric field”, Phys. Rev. B 38, 6322 (1988).
  6. J. Feldmann, K. Leo, J. Shah, D. B. A. Miller, J. E. Cunningham, S. Schmitt-Rink, T. Meier, G. von Plessen, A. Schulze, and P. Thomas, Optical investigation of Bloch oscillations in a semiconductor superlattice, Phys. Rev. B 46, 7252 (1992).
  7. C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Kohler, Coherent Submillimeter-Wave Emission from Bloch Oscillations in a Semiconductor Superlattice, Phys. Rev. Lett. 70, 3319 (1993).
  8. G. Nenciu, Dynamics of band electrons in electric and magnetic fields: Rigorous justification of the effective Hamiltonians, Rev. Mod. Phys. 63, 91 (1991).
  9. C. Martijn de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, Observation of an optical Wannier-Stark ladder, Phys. Rev. E 57, 2365 (1998).
  10. U. Peschel, T. Pertsch, and F. Lederer, Optical Bloch oscillations in waveguide arrays, Opt. Lett. 23, 1701 (1998).
  11. A. Kavokin, G. Malpuech, A. Di Carlo, P. Lugli, and F. Rossi, Photonic Bloch oscillations in laterally confined Bragg mirrors, Phys. Rev. B 61, 4413 (2000).
  12. G. Malpuech, A. Kavokin, G. Panzarini, and A. Di Carlo, Theory of photon Bloch oscillations in photonic crystals, Phys. Rev. B 63, 035108 (2001).
  13. S. Longhi, Optical Zener-Bloch oscillations in binary waveguide arrays, Europhys. Lett. 76, 416 (2006).
  14. T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays, Phys. Rev. Lett. 83, 4752 (1999).
  15. V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, Photon Bloch Oscillations in Porous Silicon Optical Superlattices, Phys. Rev. Lett. 92, 097401 (2004).
  16. M. Ghulinyan, C. J. Oton, Z. Gaburro, L. Pavesi, C. Toninelli, and D. S. Wiersma, Zener Tunneling of Light Waves in an Optical Superlattice, Phys. Rev. Lett. 94, 127401 (2005).
  17. S. Longhi, M. Lobino, M. Marangoni, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, Semiclassical motion of a multiband Bloch particle in a time-dependent field: Optical visualization, Phys. Rev. B 74, 155116 (2006).
  18. C. Bersch, G. Onishchukov, and U. Peschel, Experimental observation of spectral Bloch oscillations, Opt. Lett. 34, 2372 (2009).
  19. X. Qi, K. G. Makris, R. El-Ganainy, Peng Zhang, Jintao Bai, D. N. Christodoulides, and Zhigang Chen, Observation of accelerating Wannier-Stark beams in optically induced photonic lattices, Opt. Lett. 39, 1065 (2014).
  20. Y.-L. Xu, W. S. Fegadolli, L. Gan, M.-H. Lu, X.-P. Liu, Z.-Y. Li, A. Scherer, and Y.-F. Chen, Experimental realization of Bloch oscillations in a parity-time synthetic silicon photonic lattice, Nat. Commun. 7, 11319 (2016).
  21. I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, Light propagation and localization in modulated photonic lattices and waveguides, Phys. Rep. 518, 1 (2012).
  22. M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, Bloch Oscillations of Atoms in an Optical Potential, Phys. Rev. Lett. 76, 4508 (1996).
  23. S. Wilkinson, C. Bharucha, K. Madison, Q. Niu, and M. Raizen, Observation of Atomic Wannier-Stark Ladders in an Accelerating Optical Potential, Phys. Rev. Lett. 76, 4512 (1996).
  24. H. R. Zhang and C. P. Sun, Bloch oscillations of polaritons of an atomic ensemble in magnetic fields, Phys. Rev. A 81, 063427 (2010).
  25. Z. A. Geiger, K. M. Fujiwara, K. Singh, R. Senaratne, S. V. Rajagopal, M. Lipatov, T. Shimasaki, R. Driben, V. V. Konotop, T. Meier, and D. M. Weld, Observation and Uses of Position-Space Bloch Oscillations in an Ultracold Gas, Phys. Rev. Lett. 120, 213201 (2018).
  26. Z. Pagel, W. Zhong, R. H. Parker, C. T. Olund, N. Y. Yao, and H. Muller, Symmetric Bloch oscillations of matter waves, Phys. Rev. A 102, 053312 (2020).
  27. L. Masi, T. Petrucciani, G. Ferioli, G. Semeghini, G. Modugno, M. Inguscio, and M. Fattori, Spatial Bloch Oscillations of a Quantum Gas in a “Beat-Note” Superlattice, Phys. Rev. Lett. 127, 020601 (2021).
  28. S. Longhi, Dynamic localization and Bloch oscillations in the spectrum of a frequency mode-locked laser, Opt. Lett. 30, 786 (2005).
  29. S. Bahmani and A. N. Askarpour, Bloch oscillations and Wannier-Stark ladder in the coupled LC circuits, Phys. Lett. A 384, 126596 (2020).
  30. G. Monsivais and R. Esquivel-Sirvent, Stark ladder resonances in acoustic waveguides, J. Mech. Mater. Struct. 2, 1585 (2007).
  31. G. Monsivais, R. Mendez-Sanchez, A. de Anda, J. Flores, L. Gutierrez, and A. Morales, Elastic Wannier-Stark ladders in torsional waves, J. Mech. Mater. Struct. 2, 1629 (2007).
  32. N. Lanzillotti-Kimura, A. Fainstein, B. Perrin, B. Jusserand, O. Mauguin, L. Largeau, and A. Lemaitre, Bloch Oscillations of THz Acoustic Phonons in Coupled Nanocavity Structures, Phys. Rev. Lett. 104, 197402 (2010).
  33. Y.-K. Liu, H.-W. Wu, P. Hu, and Z.-Q. Sheng, Spatial Bloch oscillations in acoustic waveguide arrays, Appl. Phys. Express 14, 064501 (2021).
  34. A. R. Davoyan, I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, Plasmonic Bloch oscillations in chirped metal-dielectric structures, Appl. Phys. Lett. 94, 161105 (2009).
  35. V. Kuzmiak, S. Eyderman, and M. Vanwolleghem, Controlling surface plasmon polaritons by a static and/or time-dependent external magnetic field, Phys. Rev. B 86, 045403 (2012).
  36. B. H. Cheng, Y.-C. Lai, and Y.-C. Lan, Plasmonic photonic Bloch oscillations in composite metal-insulator-metal waveguide structure, Plasmonics 9, 137 (2014).
  37. V. Kuzmiak, A. A. Maradudin, and E. R. Mendez, Surface plasmon polariton Wannier-Stark ladder, Opt. Lett. 39, 1613 (2014).
  38. A. Block, C. Etrich, T. Limboeck, F. Bleckmann, E. Soergel, C. Rockstuhl, and S. Linden, Bloch oscillations in plasmonic waveguide arrays, Nat. Commun. 5, 3843 (2014).
  39. H. Wetter, Z. Fedorova, and S. Linden, Observation of the Wannier-Stark ladder in plasmonic waveguide arrays, Opt. Lett. 47, 3091 (2022).
  40. H. Flayac, D. D. Solnyshkov, and G. Malpuech, Bloch oscillations of an exciton-polariton Bose-Einstein condensate, Phys. Rev. B 83, 045412 (2011).
  41. H. Flayac, D. D. Solnyshkov, and G. Malpuech, Bloch oscillations of exciton-polaritons and photons for the generation of an alternating terahertz spin signal, Phys. Rev. B 84, 125314 (2011).
  42. J. Beierlein, O. A. Egorov, T. H. Harder, P. Gagel, M. Emmerling, C. Schneider, S. Hofling, U. Peschel, and S. Klembt, Bloch oscillations of hybrid light-matter particles in a waveguide array, Adv. Opt. Mater. 9, 2100126 (2021).
  43. D. Cai, A. R. Bishop, and N. Gronbech-Jensen, Electric-Field-Induced Nonlinear Bloch Oscillations and Dynamical Localization, Phys. Rev. Lett. 74, 1186 (1995).
  44. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations, Phys. Rev. Lett. 83, 4756 (1999).
  45. O. Morsch, J. H. Muller, M. Cristiani, D. Ciampini, and E. Arimondo, Bloch Oscillations and Mean-Field Effects of Bose-Einstein Condensates in 1D Optical Lattices, Phys. Rev. Lett. 87, 140402 (2001).
  46. M. Cristiani, O. Morsch, J. H. Muller, D. Ciampini, and E. Arimondo, Experimental properties of Bose-Einstein condensates in one-dimensional optical lattices: Bloch oscillations, Landau-Zener tunneling, and mean-field effects, Phys. Rev. A 65, 063612 (2002).
  47. M. Gustavsson, E. Haller, M. J. Mark, J. G. Danzl, G. Rojas-Kopeinig, and H.-C. Nagerl, Control of Interaction-Induced Dephasing of Bloch Oscillations, Phys. Rev. Lett. 100, 080404 (2008).
  48. Y. V. Bludov, V. V. Konotop, and M. Salerno, Dynamical localization of gap-solitons by time periodic forces, Europhys. Lett. 87, 20004 (2009).
  49. V. V. Konotop and M. Salerno, Modulation instability in Bose-Einstein condensates in optical lattices, Phys. Rev. A 65, 021602 (2002).
  50. M. Salerno, V. V. Konotop and Y. V. Bludov, Long-Living Bloch Oscillations of Matter Waves in Periodic Potentials, Phys. Rev. Lett. 101, 030405 (2008).
  51. Y. V. Bludov, V. V. Konotop, and M. Salerno, Linear superpositions of nonlinear matter waves in optical lattices, Europhys. Lett. 93, 30003 (2011).
  52. C. Gaul, R. P. A. Lima, E. Diaz, C. A. Muller, and F. Domnguez-Adame, Stable Bloch Oscillations of Cold Atoms with Time-Dependent Interaction, Phys. Rev. Lett. 102, 255303 (2009).
  53. Y. V. Bludov, V. V. Konotop, and M. Salerno, Long-lived matter wave Bloch oscillations and dynamical localization by time-dependent nonlinearity management, J. Phys. B: At. Mol. Opt. Phys. 42, 105302 (2009).
  54. R. Driben, V. V. Konotop, T. Meier, and A. V. Yulin, Bloch oscillations sustained by nonlinearity, Sci. Rep. 7, 3194 (2017).
  55. A. Yulin, R. Driben and T. Meier, Bloch oscillations and resonant radiation of light propagating in arrays of nonlinear fibers with high-order dispersion, Phys. Rev. A 96, 033827 (2017).
  56. E. Shlizerman and V. Rom-Kedar, Three Types of Chaos in the Forced Nonlinear Schrödinger Equation, Phys. Rev. Lett. 96, 024104 (2006).
  57. E. Shlizerman and V. Rom-Kedar, Classification of solutions of the forced periodic nonlinear Schrodinger equation, Nonlinearity 23, 2183 (2010).
  58. V. Achilleos, A. R. Bishop, S. Diamantidis, D. J. Frantzeskakis, T. P. Horikis, N. I. Karachalios, and P. G. Kevrekidis, Dynamical playground of a higher-order cubic Ginzburg-Landau equation: From orbital connections and limit cycles to invariant tori and the onset of chaos, Phys. Rev. E 94, 012210 (2016).
  59. P. P. Galuzio, S. Benkadda, and S. R. Lopes, Characterization of intermittency at the onset of turbulence in the forced and damped nonlinear Schrodinger equation, Commun. Nonlinear Sci. Numer. Simul. 42, 404 (2017).
  60. K. Koshelev, S. Kruk, E. Melik-Gaykazyan, J-H. Choi, A. Bogdanov, H.-G. Park, and Y. Kivshar, Subwavelength dielectric resonators for nonlinear nanophotonics, Science 367, 288 (2020).
  61. G. P. Zograf, D. Ryabov, V. Rutckaia, P. Voroshilov, P. Tonkaev, D. V. Permyakov, Y. Kivshar, S. V. Makarov, Stimulated Raman scattering from Mie-resonant subwavelength nanoparticles, Nano Lett. 20, 5786 (2020).
  62. S. Jahani, A. Roy, and A. Marandi, Wavelength-scale optical parametric oscillators, Optica 8, 262 (2021).
  63. M.-S. Hwang, K.-Y. Jeong, J.-P. So, K.-H. Kim, and H.-G. Park, Nanophotonic nonlinear and laser devices exploiting bound states in the continuum, Commun. Phys. 5, 106 (2022).
  64. L. Zhang, W. Xie, J. Wang, A. Poddubny, J. Lu, Y. Wang, J. Gu, W. Liu, D. Xu, X. Shen, Y. G. Rubo, B. L. Altshuler, A. V. Kavokin, and Z. Chen, Weak lasing in one-dimensional polariton superlattices, Proc. Natl. Acad. Sci. USA 112, E1516 (2015).
  65. S. S. Gavrilov, Spin oscillations of a single-mode polariton system driven by a plane wave, Phys. Rev. B 106, 045304 (2022).
  66. W. Deering, M. Molina, and G. Tsironis, Directional couplers with linear and nonlinear elements, Appl. Phys. Lett. 62, 2471 (1993).
  67. J. Eilbeck, G. Tsironis, and S. K. Turitsyn, Stationary states in a doubly nonlinear trimer model of optical couplers, Phys. Scr. 52, 386 (1995).
  68. P. G. Kevrekidis, K. O. Rasmussen and A. R. Bishop, The discrete nonlinear Schrödinger equation: A survey of recent results, Int. J. Mod. Phys. B 15, 2833 (2001).
  69. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, M. Segev, Discrete solitons in photorefractive optically induced photonic lattices, Phys. Rev. E 66, 046602 (2002).
  70. A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, Spatial optical solitons in waveguide arrays, IEEE J. Quantum Electron. 39, 31 (2003).
  71. U. Peschel, O. Egorov, and F. Lederer, Discrete cavity solitons, Opt. Lett. 29, 1909 (2004).
  72. D. E. Pelinovsky, P. G. Kevrekidis, and D. J. Frantzeskakis, Stability of discrete solitons in nonlinear Schrödinger lattices, Physica D 212, 1 (2005).
  73. A. Yulin, D. V. Skryabin, and P. S. J. Russell, Dissipative localized structures of light in photonic crystal films, Opt. Express 13, 3529 (2005).
  74. A. V. Yulin, D. V. Skryabin, and A. G. Vladimirov, Modulational instability of discrete solitons in coupled waveguides with group velocity dispersion, Opt. Express 14, 12347 (2006).
  75. H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-Gonzalez, and D. J. Frantzeskakis, Discrete surface solitons in two dimensions, Phys. Rev. E 75, 056605 (2007).
  76. O. A. Egorov and F. Lederer, Lattice-cavity solitons in a degenerate optical parametric oscillator, Phys. Rev. A 76, 053816 (2007).
  77. O. A. Egorov, F. Lederer, and Yu. S. Kivshar, How does an inclined holding beam affect discrete modulational instability and solitons in nonlinear cavities? Opt. Express 15, 4149 (2007).
  78. D. E. Pelinovsky, P. G. Kevrekidis, and D. J. Frantzeskakis, Mechanism for chaos in the Duffing equation, Phys. Lett. A 94, 195 (1983).
  79. M. Wouters and I. Carusotto, Excitations in a Nonequilibrium Bose-Einstein Condensate of Exciton Polaritons, Phys. Rev. Lett. 99, 140402 (2007).
  80. M. Sich, J. K. Chana, O. A. Egorov, H. Sigurdsson, I. A. Shelykh, D. V. Skryabin, P. M. Walker, E. Clarke, B. Royall, M. S. Skolnick, and D. N. Krizhanovskii, Transition from Propagating Polariton Solitons to a Standing Wave Condensate Induced by Interactions, Phys. Rev. Lett. 120, 167402 (2018).

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Phys. Rev. A 107, 053519– Published 24 May, 2023
  • Received 27 December 2022
  • Revised 27 April 2023
  • Accepted 2 May 2023
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