Skip to main content
SpringerLink
Log in

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

  • Regular Article
  • Published:
The European Physical Journal D Aims and scope Submit manuscript

Abstract

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

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

  2. C.J. Pethick, H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, 2002)

  3. L.P. Pitaevskii, S. Stringari, Bose-Einstein Condensation and Superfluidity (Oxford University Press, Oxford, 2016)

  4. N.N. Bogoliubov, J. Phys. (USSR) 11, 23 (1947)

    Google Scholar 

  5. H. Palevsky, K. Otnes, K.E. Larsson, Phys. Rev. 112, 11 (1958)

    Article  ADS  Google Scholar 

  6. J.L. Yarnell, G.P. Arnold, P.J. Bendt, E.C. Kerr, Phys. Rev. 113, 1379 (1959)

    Article  ADS  Google Scholar 

  7. M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell, Science 269, 198 (1995)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  9. J. Stenger, S. Inouye, A.P. Chikkatur, D.M. Stamper-Kurn, D.E. Pritchard, W. Ketterle, Phys. Rev. Lett. 82, 4569 (1999)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  11. J.M. Vogels, K. Xu, C. Raman, J.R. Abo-Shaeer, W. Ketterle, Phys. Rev. Lett. 88, 060402 (2002)

    Article  ADS  Google Scholar 

  12. J. Steinhauer, R. Ozeri, N. Katz, N. Davidson, Phys. Rev. Lett. 88, 120407 (2002)

    Article  ADS  Google Scholar 

  13. R. Ozeri, J. Steinhauer, N. Katz, N. Davidson, Phys. Rev. Lett. 88, 220401 (2002)

    Article  ADS  Google Scholar 

  14. N. Katz, J. Steinhauer, R. Ozeri, N. Davidson, Phys. Rev. Lett. 89, 220401 (2002)

    Article  ADS  Google Scholar 

  15. J. Steinhauer, N. Katz, R. Ozeri, N. Davidson, C. Tozzo, F. Dalfovo, Phys. Rev. Lett. 90, 060404 (2003)

    Article  ADS  Google Scholar 

  16. R. Ozeri, N. Katz, J. Steinhauer, N. Davidson, Rev. Mod. Phys. 77, 187 (2005)

    Article  ADS  Google Scholar 

  17. I. Carusotto, C. Ciuti, Rev. Mod. Phys. 85, 299 (2013)

    Article  ADS  Google Scholar 

  18. K. Staliunas, V.J. Sanchez-Morcillo, Transverse Patterns in Nonlinear Optical Resonators, Springer Tracts in Modern Physics (Springer, Berlin, 2003), Vol. 183

  19. C. Weisbuch, M. Nishioka, A. Ishikawa, Y. Arakawa, Phys. Rev. Lett. 69, 3314 (1992)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  21. R. Balili, V. Hartwell, D. Snoke, L. Pfeiffer, K. West, Science 316, 1007 (2007)

    Article  ADS  Google Scholar 

  22. H. Deng, G.S. Solomon, R. Hey, K.H. Ploog, Y. Yamamoto, Phys. Rev. Lett. 99, 126403 (2007)

    Article  ADS  Google Scholar 

  23. H. Deng, H. Haug, Y. Yamamoto, Rev. Mod. Phys. 82, 1489 (2010)

    Article  ADS  Google Scholar 

  24. I.A. Shelykh, G. Malpuech, A.V. Kavokin, Phys. Status Solidi A 202, 2614 (2005)

    Article  ADS  Google Scholar 

  25. M.H. Szymańska, J. Keeling, P.B. Littlewood, Phys. Rev. Lett. 96, 230602 (2006)

    Article  ADS  Google Scholar 

  26. M. Wouters, I. Carusotto, Phys. Rev. Lett. 99, 140402 (2007)

    Article  ADS  Google Scholar 

  27. M. Wouters, I. Carusotto, Superlatt. Microstruct. 43, 524 (2008)

    Article  ADS  Google Scholar 

  28. M. Wouters, I. Carusotto, Phys. Rev. B 79, 125311 (2009)

    Article  ADS  Google Scholar 

  29. T. Byrnes, T. Horikiri, N. Ishida, M. Fraser, Y. Yamamoto, Phys. Rev. B 85, 075130 (2012)

    Article  ADS  Google Scholar 

  30. P.-É. Larré, N. Pavloff, A.M. Kamchatnov, Phys. Rev. B 86, 165304 (2012)

    Article  ADS  Google Scholar 

  31. P.-É. Larré, N. Pavloff, A.M. Kamchatnov, Phys. Rev. B 88, 224503 (2013)

    Article  ADS  Google Scholar 

  32. V. Kohnle, Y. Léger, M. Wouters, M. Richard, M.T. Portella-Oberli, B. Deveaud-Plédran, Phys. Rev. Lett. 106, 255302 (2011)

    Article  ADS  Google Scholar 

  33. V. Kohnle, Y. Léger, M. Wouters, M. Richard, M.T. Portella-Oberli, B. Deveaud, Phys. Rev. B 86, 064508 (2012)

    Article  ADS  Google Scholar 

  34. P. Leboeuf, S. Moulieras, Phys. Rev. Lett. 105, 163904 (2010)

    Article  ADS  Google Scholar 

  35. M. Elazar, S. Bar-Ad, V. Fleurov, R. Schilling, Lect. Notes Phys. 870, 275 (2013)

    Article  ADS  Google Scholar 

  36. I. Carusotto, Proc. R. Soc. Lond. Ser. A 470, 20140320 (2014)

    Article  ADS  Google Scholar 

  37. P.-É. Larré, I. Carusotto, Phys. Rev. A 91, 053809 (2015)

    Article  ADS  Google Scholar 

  38. D. Vocke, T. Roger, F. Marino, E.M. Wright, I. Carusotto, M. Clerici, D. Faccio, Optica 2, 484 (2015)

    Article  Google Scholar 

  39. P.-É. Larré, I. Carusotto, Phys. Rev. A 92, 043802 (2015)

    Article  ADS  Google Scholar 

  40. M. Karpov, T. Congy, Y. Sivan, V. Fleurov, N. Pavloff, S. Bar-Ad, Optica 2, 1053 (2015)

    Article  Google Scholar 

  41. P.-É. Larré, I. Carusotto, Eur. Phys. J. D 70, 45 (2016)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  43. A. Chiocchetta, P.-É. Larré, I. Carusotto, Europhys. Lett. 115, 24002 (2016)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  45. C. Noh, D.G. Angelakis, Rep. Prog. Phys. 80, 016401 (2016)

    Article  ADS  Google Scholar 

  46. R.W. Boyd, Nonlinear Optics (Academic Press, San Diego, 1992)

  47. G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 1995)

  48. N.N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, New York, 2002)

  49. U. Bortolozzo, J. Laurie, S. Nazarenko, S. Residori, J. Opt. Soc. Am. B 26, 2280 (2009)

    Article  ADS  Google Scholar 

  50. C. Michel, M. Haelterman, P. Suret, S. Randoux, R. Kaiser, A. Picozzi, Phys. Rev. A 84, 033848 (2011)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  52. A.M. Perego, N. Tarasov, D.V. Churkin, S.K. Turitsyn, K. Staliunas, Phys. Rev. Lett. 116, 028701 (2016)

    Article  ADS  Google Scholar 

  53. N. Tarasov, A.M. Perego, D.V. Churkin, K. Staliunas, S.K. Turitsyn, Nat. Commun. 7, 12441 (2016)

    Article  ADS  Google Scholar 

  54. S. Kumar, A.M. Perego, K. Staliunas, Phys. Rev. Lett. 118, 044103 (2017)

    Article  ADS  Google Scholar 

  55. H.A. Haus, J.G. Fujimoto, E.P. Ippen, J. Opt. Soc. Am. B 8, 2068 (1991)

    Article  ADS  Google Scholar 

  56. A.M. Dunlop, W.J. Firth, E.M. Wright, Opt. Express 2, 204 (1998)

    Article  ADS  Google Scholar 

  57. W. Schöpf, L. Kramer, Phys. Rev. Lett. 66, 2316 (1991)

    Article  ADS  Google Scholar 

  58. F. Genoud, Adv. Nonlinear Stud. 10, 357 (2010)

    Article  MathSciNet  Google Scholar 

  59. N.V. Alexeeva, I.V. Barashenkov, D.E. Pelinovsky, Nonlinearity 12, 103 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  60. V.E. Zakharov, L.A. Ostrovsky, Physica D 238, 540 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  61. S.K. Turitsyn, A.M. Rubenchik, M.P. Fedoruk, Opt. Lett. 35, 2684 (2010)

    Article  ADS  Google Scholar 

  62. M. Conforti, A. Mussot, A. Kudlinski, S. Trillo, Opt. Lett. 39, 4200 (2014)

    Article  ADS  Google Scholar 

  63. R. Paschotta, Field Guide to Lasers (SPIE Press, Bellingham, WA, 2008)

  64. J. Leuthold, C. Koos, W. Freude, Nat. Photon. 4, 535 (2010)

    Article  ADS  Google Scholar 

  65. L. Vivien, L. Pavesi, Handbook of Silicon Photonics (CRC Press, London, 2013)

  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)

  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.1

  68. M. Borghi, M. Mancinelli, F. Merget, J. Witzens, M. Bernard, M. Ghulinyan, G. Pucker, L. Pavesi, Opt. Lett. 40, 5287 (2015)

    Article  ADS  Google Scholar 

  69. D.S. Petrov, Bose-Einstein condensation in low-dimensional trapped gases, Ph.D. Thesis, University of Amsterdam (2003)

  70. D.S. Petrov, D.M. Gangardt, G.V. Shlyapnikov, J. Phys. IV France 116, 5 (2004)

    Article  Google Scholar 

  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)

  72. L.D. Landau, Phys. Rev. 60, 356 (1941)

    Article  ADS  Google Scholar 

  73. L.D. Landau, E.M. Lifshitz, Statistical Physics, Course of Theoretical Physics (Butterworth-Heinemann, London, 1980), Vol. 5

  74. M. Born, V. Fock, Z. Phys. 51, 165 (1928)

    Article  ADS  Google Scholar 

  75. A. Messiah, Quantum Mechanics (Dover Publications, New York, 1999)

  76. C.-P. Sun, Phys. Scripta 48, 393 (1993)

    Article  ADS  Google Scholar 

  77. S. Weinberg, The Quantum Theory of Fields (Cambridge University Press, Cambridge, 1995), Vol. 1

  78. P. Hariharan, Basics of Interferometry (Academic Press, New York, 2007)

  79. M.V. Berry, Sci. Am. 259, 26 (1988)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire Kastler-Brossel, UPMC (Sorbonne Université), CNRS, ENS (Université de Recherche PSL), Collège de France, 4 Place Jussieu, 75005, Paris, France

    Pierre-Élie Larré

  2. Laboratorio di Nanoscienze, Università degli Studi di Trento, CNR, INFM, Via Sommarive 14, 38123, Povo (TN), Italy

    Stefano Biasi, Fernando Ramiro-Manzano & Lorenzo Pavesi

  3. BEC Center, Università degli Studi di Trento, CNR, INO, Via Sommarive 14, 38123, Povo (TN), Italy

    Iacopo Carusotto

Authors
  1. Pierre-Élie Larré
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefano Biasi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fernando Ramiro-Manzano
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lorenzo Pavesi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Iacopo Carusotto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pierre-Élie Larré.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Larré, PÉ., Biasi, S., Ramiro-Manzano, F. et al. Pump-and-probe optical transmission phase shift as a quantitative probe of the Bogoliubov dispersion relation in a nonlinear channel waveguide. Eur. Phys. J. D 71, 146 (2017). https://doi.org/10.1140/epjd/e2017-80208-5

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1140/epjd/e2017-80208-5

Keywords

Search

Navigation