Optics and Spectroscopy

, Volume 118, Issue 5, pp 781–793 | Cite as

Quantum fluctuations of one- and two-dimensional spatial dissipative solitons in a nonlinear interferometer: I. One-dimensional dark solitons

  • L. A. Nesterov
  • N. A. Veretenov
  • N. N. Rosanov
Nonlinear and Quantum Optics


Quantum fluctuations of one-dimensional dark dissipative solitons sustained by an external radiation in an interferometer with a Kerr nonlinearity are analyzed theoretically. The stability region of classical solitons in this interferometer is studied. The boundaries of this region are determined, and types of excited solitons are classified. Quantum fluctuations of solitons are analyzed in an approximation linear in fluctuations. This problem was solved by linearizing the quantum Langevin equation in a neighborhood of a classical solution for the main type of a soliton from the obtained stability region. The main attention has been paid to studying quantum fluctuations of collective variables of dissipative solitons, namely, the coordinate of the center and momentum of the soliton. Based on the expansion of solutions of the linearized equation in eigenfunctions of the discrete spectrum of this equation, a solution describing quantum fluctuations of these variables is constructed. Using this expansion scheme made it possible to give a rigorous definition of the dissipative soliton position fluctuation operator. The study performed based on this scheme has made it also possible to construct a solution for a one-dimensional dark relaxing dissipative soliton. This soliton generalizes the stationary soliton with allowance for the shift of its center and deformation of its profile followed by the recovery of its initial shape. Average squares of quantum fluctuations of collective variables are calculated. A domain of parameters in which there exist quantum states of solitons with an initially high degree of squeezing with respect to the momentum is found. It is shown that such states are in correspondence with significantly higher velocities of soliton center drift. An experiment that could detect the relative squeezing with respect to the momentum due to the soliton center drift is discussed.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, Amsterdam, 2003; Fizmatlit, Moscow, 2005).Google Scholar
  2. 2.
    N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, Berlin, 2002).CrossRefMATHGoogle Scholar
  3. 3.
    Dissipative Solitons in Lecture Notes in Physics, Ed. by N. Akhmediev and A. Ankiewicz (Springer, Berlin, 2005), Vol. 661.Google Scholar
  4. 4.
    Dissipative Solitons: From Optics to Biology and Medicine in Lecture Notes in Physics, Ed. by N. Akhmediev and A. Ankiewicz (Springer, Berlin, 2008), Vol. 751.Google Scholar
  5. 5.
    N. N. Rosanov, Dissipative Optical Solitons: From Micro- to Nano- and Atto- (Fizmatlit, Moscow, 2011) [in Russian].Google Scholar
  6. 6.
    Y. Lai and H. A. Haus, Phys. Rev. A 40(2), 844 (1989).CrossRefADSGoogle Scholar
  7. 7.
    Y. Lai and H. A. Haus, Phys. Rev. A 40(2), 854 (1989).CrossRefADSGoogle Scholar
  8. 8.
    M. Rosenbluh and R. M. Shelby, Phys. Rev. Lett. 66(2), 153 (1991).CrossRefADSGoogle Scholar
  9. 9.
    S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, Phys. Rev. Lett. 58(18), 1841 (1987).CrossRefADSGoogle Scholar
  10. 10.
    P. D. Drummond and S. J. Carter, J. Opt. Soc. Am. B 4(10), 1565 (1987).CrossRefADSGoogle Scholar
  11. 11.
    P. D. Drummond, S. J. Carter, and R. M. Shelby, Opt. Lett. 14(7), 373 (1990).CrossRefADSGoogle Scholar
  12. 12.
    H. A. Haus and Y. Lai, J. Opt. Soc. Am. B 7(3), 386 (1990).CrossRefADSGoogle Scholar
  13. 13.
    P. D. Drummond et al., Nature 365, 307 (1993).CrossRefADSGoogle Scholar
  14. 14.
    S. R. Friberg et al., Phys. Rev. Lett. 77(18), 3775 (1996).CrossRefADSGoogle Scholar
  15. 15.
    S. Spalter et al., Europhys. Lett. 38(5), 335 (1997).CrossRefADSGoogle Scholar
  16. 16.
    S. Spalter et al., Opt. Express 2, 77 (1998).CrossRefADSGoogle Scholar
  17. 17.
    M. J. Werner, Phys. Rev. A 54(4), R2567 (1996).CrossRefADSGoogle Scholar
  18. 18.
    A. Mecozzi, Opt. Lett. 22(16), 1232 (1997).CrossRefADSGoogle Scholar
  19. 19.
    D. Levandovsky, M. Vasilyev, and P. Kumar, Opt. Lett. 24(1), 43 (1999).CrossRefADSGoogle Scholar
  20. 20.
    E. Lantz et al., J. Opt. B 6, 295 (2004).CrossRefADSGoogle Scholar
  21. 21.
    E. M. Nagasako, R. W. Boyd, and G. S. Agarwal, Opt. Express 3(5), 171 (1998).CrossRefADSGoogle Scholar
  22. 22.
    A. Mecozzi and P. Kumar, Quantum Semiclass. Opt. 10, L21 (1998).CrossRefADSGoogle Scholar
  23. 23.
    J.-L. Oppo and J. Jeffers, in Quantum Image, Ed. by M. I. Kolobov (Fizmatlit, Moscow, 2009) [in Russian].Google Scholar
  24. 24.
    W. J. Firth and C. O. Weiss, Opt. and Photon. News, No. 13, 54 (2002).Google Scholar
  25. 25.
    I. Rabbiosi, A. J. Scroggie, and J.-L. Oppo, Phys. Rev. Lett. 89, 254102 (2002).CrossRefADSGoogle Scholar
  26. 26.
    I. Rabbiosi, A. J. Scroggie, and J.-L. Oppo, Eur. Phys. J. D 22, 453 (2003).CrossRefADSGoogle Scholar
  27. 27.
    R. Zambrini et al., Eur. Phys. J. D 22, 460 (2003).CrossRefADSGoogle Scholar
  28. 28.
    L. A. Nesterov, Al. S. Kiselev, An. S. Kiselev, and N. N. Rosanov, Opt. Spectrosc. 106(4), 570 (2009).CrossRefADSGoogle Scholar
  29. 29.
    L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58(2), 2209 (1987).CrossRefADSGoogle Scholar
  30. 30.
    N. N. Rosanov, Optical Bistability and Hysteresis in Distributed Nonlinear Systems (Nauka, Moscow, 1997) [in Russian].Google Scholar
  31. 31.
    N. N. Rosanov and G. V. Khodova, Opt. Spektrosk. 65, 1375 (1988).Google Scholar
  32. 32.
    N. N. Rosanov, A. V. Fedorov, and G. V. Khodova, Phys. Stat. Sol. (b) 150(2), 545 (1988).CrossRefADSGoogle Scholar
  33. 33.
    N. N. Rosanov and G. V. Khodova, J. Opt. Soc. Am. B 8(7), 1471 (1991).CrossRefADSGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  • L. A. Nesterov
    • 1
    • 2
  • N. A. Veretenov
    • 1
    • 2
  • N. N. Rosanov
    • 1
    • 2
    • 3
  1. 1.St. Petersburg ITMO UniversitySt. PetersburgRussia
  2. 2.Vavilov State Optical InstituteScientific Research Institute for Laser PhysicsSt. PetersburgRussia
  3. 3.Ioffe Physicotechnical InstituteSt. PetersburgRussia

Personalised recommendations