Pump-and-probe optical transmission phase shift as a quantitative probe of the Bogoliubov dispersion relation in a nonlinear channel waveguide

  • Pierre-Élie LarréEmail author
  • Stefano Biasi
  • Fernando Ramiro-Manzano
  • Lorenzo Pavesi
  • Iacopo Carusotto
Regular Article


We theoretically investigate the dispersion relation of small-amplitude optical waves superimposing upon a beam of polarized monochromatic light propagating along a single-mode channel waveguide characterized by an instantaneous and spatially local Kerr nonlinearity. These small luminous fluctuations propagate along the waveguide as Bogoliubov elementary excitations on top of a one-dimensional dilute Bose quantum fluid evolve in time. They consequently display a strongly renormalized dispersion law, of Bogoliubov type. Analytical and numerical results are found in both the absence and the presence of one- and two-photon losses. Silicon and silicon-nitride waveguides are used as examples. We finally propose an experiment to measure this Bogoliubov dispersion relation, based on a stimulated four-wave mixing and interference spectroscopy techniques.

Graphical abstract


Optical Phenomena and Photonics 


  1. 1.
    Y. Castin, Bose-Einstein condensates in atomic gases: simple theoretical results, in Coherent Atomic Matter Waves, Proceedings of Les-Houches Summer School, edited by R. Kaiser, C.I. Westbrook, F. David (Springer, New York, 2001)Google Scholar
  2. 2.
    C.J. Pethick, H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, 2002)Google Scholar
  3. 3.
    L.P. Pitaevskii, S. Stringari, Bose-Einstein Condensation and Superfluidity (Oxford University Press, Oxford, 2016)Google Scholar
  4. 4.
    N.N. Bogoliubov, J. Phys. (USSR) 11, 23 (1947)Google Scholar
  5. 5.
    H. Palevsky, K. Otnes, K.E. Larsson, Phys. Rev. 112, 11 (1958)ADSCrossRefGoogle Scholar
  6. 6.
    J.L. Yarnell, G.P. Arnold, P.J. Bendt, E.C. Kerr, Phys. Rev. 113, 1379 (1959)ADSCrossRefGoogle Scholar
  7. 7.
    M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell, Science 269, 198 (1995)ADSCrossRefGoogle Scholar
  8. 8.
    K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995)ADSCrossRefGoogle Scholar
  9. 9.
    J. Stenger, S. Inouye, A.P. Chikkatur, D.M. Stamper-Kurn, D.E. Pritchard, W. Ketterle, Phys. Rev. Lett. 82, 4569 (1999)ADSCrossRefGoogle Scholar
  10. 10.
    D.M. Stamper-Kurn, A.P. Chikkatur, A. Görlitz, S. Inouye, S. Gupta, D.E. Pritchard, W. Ketterle, Phys. Rev. Lett. 83, 2876 (1999)ADSCrossRefGoogle Scholar
  11. 11.
    J.M. Vogels, K. Xu, C. Raman, J.R. Abo-Shaeer, W. Ketterle, Phys. Rev. Lett. 88, 060402 (2002)ADSCrossRefGoogle Scholar
  12. 12.
    J. Steinhauer, R. Ozeri, N. Katz, N. Davidson, Phys. Rev. Lett. 88, 120407 (2002)ADSCrossRefGoogle Scholar
  13. 13.
    R. Ozeri, J. Steinhauer, N. Katz, N. Davidson, Phys. Rev. Lett. 88, 220401 (2002)ADSCrossRefGoogle Scholar
  14. 14.
    N. Katz, J. Steinhauer, R. Ozeri, N. Davidson, Phys. Rev. Lett. 89, 220401 (2002)ADSCrossRefGoogle Scholar
  15. 15.
    J. Steinhauer, N. Katz, R. Ozeri, N. Davidson, C. Tozzo, F. Dalfovo, Phys. Rev. Lett. 90, 060404 (2003)ADSCrossRefGoogle Scholar
  16. 16.
    R. Ozeri, N. Katz, J. Steinhauer, N. Davidson, Rev. Mod. Phys. 77, 187 (2005)ADSCrossRefGoogle Scholar
  17. 17.
    I. Carusotto, C. Ciuti, Rev. Mod. Phys. 85, 299 (2013)ADSCrossRefGoogle Scholar
  18. 18.
    K. Staliunas, V.J. Sanchez-Morcillo, Transverse Patterns in Nonlinear Optical Resonators, Springer Tracts in Modern Physics (Springer, Berlin, 2003), Vol. 183Google Scholar
  19. 19.
    C. Weisbuch, M. Nishioka, A. Ishikawa, Y. Arakawa, Phys. Rev. Lett. 69, 3314 (1992)ADSCrossRefGoogle Scholar
  20. 20.
    J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J.M.J. Keeling, F.M. Marchetti, M.H. Szymańska, R. André, J.L. Staehli, V. Savona, P.B. Littlewood, B. Deveaud, L.S. Dang, Nature 443, 409 (2006)ADSCrossRefGoogle Scholar
  21. 21.
    R. Balili, V. Hartwell, D. Snoke, L. Pfeiffer, K. West, Science 316, 1007 (2007)ADSCrossRefGoogle Scholar
  22. 22.
    H. Deng, G.S. Solomon, R. Hey, K.H. Ploog, Y. Yamamoto, Phys. Rev. Lett. 99, 126403 (2007)ADSCrossRefGoogle Scholar
  23. 23.
    H. Deng, H. Haug, Y. Yamamoto, Rev. Mod. Phys. 82, 1489 (2010)ADSCrossRefGoogle Scholar
  24. 24.
    I.A. Shelykh, G. Malpuech, A.V. Kavokin, Phys. Status Solidi A 202, 2614 (2005)ADSCrossRefGoogle Scholar
  25. 25.
    M.H. Szymańska, J. Keeling, P.B. Littlewood, Phys. Rev. Lett. 96, 230602 (2006)ADSCrossRefGoogle Scholar
  26. 26.
    M. Wouters, I. Carusotto, Phys. Rev. Lett. 99, 140402 (2007)ADSCrossRefGoogle Scholar
  27. 27.
    M. Wouters, I. Carusotto, Superlatt. Microstruct. 43, 524 (2008)ADSCrossRefGoogle Scholar
  28. 28.
    M. Wouters, I. Carusotto, Phys. Rev. B 79, 125311 (2009)ADSCrossRefGoogle Scholar
  29. 29.
    T. Byrnes, T. Horikiri, N. Ishida, M. Fraser, Y. Yamamoto, Phys. Rev. B 85, 075130 (2012)ADSCrossRefGoogle Scholar
  30. 30.
    P.-É. Larré, N. Pavloff, A.M. Kamchatnov, Phys. Rev. B 86, 165304 (2012)ADSCrossRefGoogle Scholar
  31. 31.
    P.-É. Larré, N. Pavloff, A.M. Kamchatnov, Phys. Rev. B 88, 224503 (2013)ADSCrossRefGoogle Scholar
  32. 32.
    V. Kohnle, Y. Léger, M. Wouters, M. Richard, M.T. Portella-Oberli, B. Deveaud-Plédran, Phys. Rev. Lett. 106, 255302 (2011)ADSCrossRefGoogle Scholar
  33. 33.
    V. Kohnle, Y. Léger, M. Wouters, M. Richard, M.T. Portella-Oberli, B. Deveaud, Phys. Rev. B 86, 064508 (2012)ADSCrossRefGoogle Scholar
  34. 34.
    P. Leboeuf, S. Moulieras, Phys. Rev. Lett. 105, 163904 (2010)ADSCrossRefGoogle Scholar
  35. 35.
    M. Elazar, S. Bar-Ad, V. Fleurov, R. Schilling, Lect. Notes Phys. 870, 275 (2013)ADSCrossRefGoogle Scholar
  36. 36.
    I. Carusotto, Proc. R. Soc. Lond. Ser. A 470, 20140320 (2014)ADSCrossRefGoogle Scholar
  37. 37.
    P.-É. Larré, I. Carusotto, Phys. Rev. A 91, 053809 (2015)ADSCrossRefGoogle Scholar
  38. 38.
    D. Vocke, T. Roger, F. Marino, E.M. Wright, I. Carusotto, M. Clerici, D. Faccio, Optica 2, 484 (2015)CrossRefGoogle Scholar
  39. 39.
    P.-É. Larré, I. Carusotto, Phys. Rev. A 92, 043802 (2015)ADSCrossRefGoogle Scholar
  40. 40.
    M. Karpov, T. Congy, Y. Sivan, V. Fleurov, N. Pavloff, S. Bar-Ad, Optica 2, 1053 (2015)CrossRefGoogle Scholar
  41. 41.
    P.-É. Larré, I. Carusotto, Eur. Phys. J. D 70, 45 (2016)ADSCrossRefGoogle Scholar
  42. 42.
    D. Vocke, K.E. Wilson, F. Marino, I. Carusotto, E.M. Wright, T. Roger, B.P. Anderson, P. Öhberg, D. Faccio, Phys. Rev. A 94, 013849 (2016)ADSCrossRefGoogle Scholar
  43. 43.
    A. Chiocchetta, P.-É. Larré, I. Carusotto, Europhys. Lett. 115, 24002 (2016)ADSCrossRefGoogle Scholar
  44. 44.
    M.J. Gullans, J.D. Thompson, Y. Wang, Q.-Y. Liang, V. Vuletić, M.D. Lukin, A.V. Gorshkov, Phys. Rev. Lett. 117, 113601 (2016)ADSCrossRefGoogle Scholar
  45. 45.
    C. Noh, D.G. Angelakis, Rep. Prog. Phys. 80, 016401 (2016)ADSCrossRefGoogle Scholar
  46. 46.
    R.W. Boyd, Nonlinear Optics (Academic Press, San Diego, 1992)Google Scholar
  47. 47.
    G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 1995)Google Scholar
  48. 48.
    N.N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, New York, 2002)Google Scholar
  49. 49.
    U. Bortolozzo, J. Laurie, S. Nazarenko, S. Residori, J. Opt. Soc. Am. B 26, 2280 (2009)ADSCrossRefGoogle Scholar
  50. 50.
    C. Michel, M. Haelterman, P. Suret, S. Randoux, R. Kaiser, A. Picozzi, Phys. Rev. A 84, 033848 (2011)ADSCrossRefGoogle Scholar
  51. 51.
    E.G. Turitsyna, S.V. Smirnov, S. Sugavanam, N. Tarasov, X. Shu, S.A. Babin, E.V. Podivilov, D.V. Churkin, G. Falkovich, S.K. Turitsyn, Nat. Photon. 7, 783 (2013)ADSCrossRefGoogle Scholar
  52. 52.
    A.M. Perego, N. Tarasov, D.V. Churkin, S.K. Turitsyn, K. Staliunas, Phys. Rev. Lett. 116, 028701 (2016)ADSCrossRefGoogle Scholar
  53. 53.
    N. Tarasov, A.M. Perego, D.V. Churkin, K. Staliunas, S.K. Turitsyn, Nat. Commun. 7, 12441 (2016)ADSCrossRefGoogle Scholar
  54. 54.
    S. Kumar, A.M. Perego, K. Staliunas, Phys. Rev. Lett. 118, 044103 (2017)ADSCrossRefGoogle Scholar
  55. 55.
    H.A. Haus, J.G. Fujimoto, E.P. Ippen, J. Opt. Soc. Am. B 8, 2068 (1991)ADSCrossRefGoogle Scholar
  56. 56.
    A.M. Dunlop, W.J. Firth, E.M. Wright, Opt. Express 2, 204 (1998)ADSCrossRefGoogle Scholar
  57. 57.
    W. Schöpf, L. Kramer, Phys. Rev. Lett. 66, 2316 (1991)ADSCrossRefGoogle Scholar
  58. 58.
    F. Genoud, Adv. Nonlinear Stud. 10, 357 (2010)MathSciNetCrossRefGoogle Scholar
  59. 59.
    N.V. Alexeeva, I.V. Barashenkov, D.E. Pelinovsky, Nonlinearity 12, 103 (1999)ADSMathSciNetCrossRefGoogle Scholar
  60. 60.
    V.E. Zakharov, L.A. Ostrovsky, Physica D 238, 540 (2009)ADSMathSciNetCrossRefGoogle Scholar
  61. 61.
    S.K. Turitsyn, A.M. Rubenchik, M.P. Fedoruk, Opt. Lett. 35, 2684 (2010)ADSCrossRefGoogle Scholar
  62. 62.
    M. Conforti, A. Mussot, A. Kudlinski, S. Trillo, Opt. Lett. 39, 4200 (2014)ADSCrossRefGoogle Scholar
  63. 63.
    R. Paschotta, Field Guide to Lasers (SPIE Press, Bellingham, WA, 2008)Google Scholar
  64. 64.
    J. Leuthold, C. Koos, W. Freude, Nat. Photon. 4, 535 (2010)ADSCrossRefGoogle Scholar
  65. 65.
    L. Vivien, L. Pavesi, Handbook of Silicon Photonics (CRC Press, London, 2013)Google Scholar
  66. 66.
    S. Biasi,Simulation and measurement of the optical phase in integrated linear and nonlinear photonics, M.Sc. Thesis, Università degli Studi di Trento (2016)Google Scholar
  67. 67.
    J. Ong, R. Kumar, X. Luo, G.-Q. Lo, S. Mookherjea, Improved four-wave mixing with free-carrier removal in silicon coupled microring waveguides, in CLEO: 2014, OSA Technical Digest (Online) (Optical Society of America, 2014), Paper SW3M.1Google Scholar
  68. 68.
    M. Borghi, M. Mancinelli, F. Merget, J. Witzens, M. Bernard, M. Ghulinyan, G. Pucker, L. Pavesi, Opt. Lett. 40, 5287 (2015)ADSCrossRefGoogle Scholar
  69. 69.
    D.S. Petrov, Bose-Einstein condensation in low-dimensional trapped gases, Ph.D. Thesis, University of Amsterdam (2003)Google Scholar
  70. 70.
    D.S. Petrov, D.M. Gangardt, G.V. Shlyapnikov, J. Phys. IV France 116, 5 (2004)CrossRefGoogle Scholar
  71. 71.
    S. Gao, S. He, Four-wave mixing in silicon nanowire waveguides and its applications in wavelength conversion, in Nanowires – Fundamental Research, Section “Nanotechnology and Nanomaterials”, edited by A. Hashim (InTech, Croatia, 2011)Google Scholar
  72. 72.
    L.D. Landau, Phys. Rev. 60, 356 (1941)ADSCrossRefGoogle Scholar
  73. 73.
    L.D. Landau, E.M. Lifshitz, Statistical Physics, Course of Theoretical Physics (Butterworth-Heinemann, London, 1980), Vol. 5Google Scholar
  74. 74.
    M. Born, V. Fock, Z. Phys. 51, 165 (1928)ADSCrossRefGoogle Scholar
  75. 75.
    A. Messiah, Quantum Mechanics (Dover Publications, New York, 1999)Google Scholar
  76. 76.
    C.-P. Sun, Phys. Scripta 48, 393 (1993)ADSCrossRefGoogle Scholar
  77. 77.
    S. Weinberg, The Quantum Theory of Fields (Cambridge University Press, Cambridge, 1995), Vol. 1Google Scholar
  78. 78.
    P. Hariharan, Basics of Interferometry (Academic Press, New York, 2007)Google Scholar
  79. 79.
    M.V. Berry, Sci. Am. 259, 26 (1988)CrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Pierre-Élie Larré
    • 1
    Email author
  • Stefano Biasi
    • 2
  • Fernando Ramiro-Manzano
    • 2
  • Lorenzo Pavesi
    • 2
  • Iacopo Carusotto
    • 3
  1. 1.Laboratoire Kastler-Brossel, UPMC (Sorbonne Université), CNRS, ENS (Université de Recherche PSL), Collège de FranceParisFrance
  2. 2.Laboratorio di Nanoscienze, Università degli Studi di Trento, CNR, INFMPovo (TN)Italy
  3. 3.BEC Center, Università degli Studi di Trento, CNR, INOPovo (TN)Italy

Personalised recommendations