Abstract

We theoretically and numerically investigate the effect of focusing and defocusing nonlinearities on Anderson localization in highly nonlocal media. A perturbative approach is developed to solve the nonlocal nonlinear Schrödinger equation in the presence of a random potential, showing that nonlocality stabilizes Anderson states.

© 2012 Optical Society of America

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References

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  1. P. W. Anderson, Phys. Rev. 109, 1492 (1958).
    [CrossRef]
  2. P. Sheng, ed., Scattering and Localization of Classical Waves in Random Media (World Scientific, 1990).
  3. D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
    [CrossRef]
  4. S. John, Phys. Rev. Lett. 58, 2486 (1987).
    [CrossRef]
  5. C. Conti and A. Fratalocchi, Nat. Phys. 4, 794 (2008).
    [CrossRef]
  6. J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
    [CrossRef]
  7. G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
    [CrossRef]
  8. I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
    [CrossRef]
  9. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 77, 051802 (2008).
    [CrossRef]
  10. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
    [CrossRef]
  11. V. Folli and C. Conti, Opt. Lett. 36, 2830 (2011).
    [CrossRef]
  12. O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
    [CrossRef]
  13. P. Rasmussen, O. Bang, and W. Królikowski, Phys. Rev. E 72, 066611 (2005).
    [CrossRef]
  14. F. Maucher, W. Krolikowski, and S. Skupin, http://arxiv.org/abs/1008.1891 (2010).
    [CrossRef]
  15. C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 91, 073901 (2003).
    [CrossRef]
  16. V. Folli and C. Conti, Phys. Rev. Lett. 104, 193901(2010).
    [CrossRef]
  17. S. Batz and U. Peschel, Phys. Rev. A 83, 033826 (2011).
    [CrossRef]
  18. C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, Phys. Rev. Lett. 105, 263902 (2010).
    [CrossRef]
  19. C. Conti, E. D’Asaro, S. Stivala, A. Busacca, and G. Assanto, Opt. Lett. 35, 3760 (2010).
    [CrossRef]
  20. C. Conti and L. Leuzzi, Phys. Rev. B 83, 134204 (2011).
    [CrossRef]

2011 (3)

S. Batz and U. Peschel, Phys. Rev. A 83, 033826 (2011).
[CrossRef]

C. Conti and L. Leuzzi, Phys. Rev. B 83, 134204 (2011).
[CrossRef]

V. Folli and C. Conti, Opt. Lett. 36, 2830 (2011).
[CrossRef]

2010 (4)

C. Conti, E. D’Asaro, S. Stivala, A. Busacca, and G. Assanto, Opt. Lett. 35, 3760 (2010).
[CrossRef]

C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

V. Folli and C. Conti, Phys. Rev. Lett. 104, 193901(2010).
[CrossRef]

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef]

2008 (4)

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 77, 051802 (2008).
[CrossRef]

C. Conti and A. Fratalocchi, Nat. Phys. 4, 794 (2008).
[CrossRef]

J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

2007 (1)

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

2005 (1)

P. Rasmussen, O. Bang, and W. Królikowski, Phys. Rev. E 72, 066611 (2005).
[CrossRef]

2003 (1)

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef]

2002 (1)

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
[CrossRef]

1997 (1)

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

1987 (1)

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef]

1958 (1)

P. W. Anderson, Phys. Rev. 109, 1492 (1958).
[CrossRef]

Anderson, P. W.

P. W. Anderson, Phys. Rev. 109, 1492 (1958).
[CrossRef]

Aspect, A.

J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

Assanto, G.

C. Conti, E. D’Asaro, S. Stivala, A. Busacca, and G. Assanto, Opt. Lett. 35, 3760 (2010).
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef]

Bang, O.

P. Rasmussen, O. Bang, and W. Królikowski, Phys. Rev. E 72, 066611 (2005).
[CrossRef]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
[CrossRef]

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Bartolini, P.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Batz, S.

S. Batz and U. Peschel, Phys. Rev. A 83, 033826 (2011).
[CrossRef]

Bernard, A.

J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

Biancalana, F.

C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

Billy, J.

J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

Bliokh, K. Y.

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef]

Bliokh, Y. P.

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef]

Bouyer, P.

J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

Busacca, A.

Clement, D.

J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

Conti, C.

C. Conti and L. Leuzzi, Phys. Rev. B 83, 134204 (2011).
[CrossRef]

V. Folli and C. Conti, Opt. Lett. 36, 2830 (2011).
[CrossRef]

C. Conti, E. D’Asaro, S. Stivala, A. Busacca, and G. Assanto, Opt. Lett. 35, 3760 (2010).
[CrossRef]

V. Folli and C. Conti, Phys. Rev. Lett. 104, 193901(2010).
[CrossRef]

C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

C. Conti and A. Fratalocchi, Nat. Phys. 4, 794 (2008).
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef]

D’Asaro, E.

D’Errico, C.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

Fallani, L.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

Fattori, M.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Folli, V.

V. Folli and C. Conti, Opt. Lett. 36, 2830 (2011).
[CrossRef]

V. Folli and C. Conti, Phys. Rev. Lett. 104, 193901(2010).
[CrossRef]

Fort, C.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

Fratalocchi, A.

C. Conti and A. Fratalocchi, Nat. Phys. 4, 794 (2008).
[CrossRef]

Freilikher, V.

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef]

Hambrecht, B.

J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

Inguscio, M.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

John, S.

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef]

Josse, V.

J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

Kartashov, Y. V.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 77, 051802 (2008).
[CrossRef]

Kivshar, Y. S.

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef]

Krolikowski, W.

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
[CrossRef]

Królikowski, W.

P. Rasmussen, O. Bang, and W. Królikowski, Phys. Rev. E 72, 066611 (2005).
[CrossRef]

Lagendijk, A.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Leuzzi, L.

C. Conti and L. Leuzzi, Phys. Rev. B 83, 134204 (2011).
[CrossRef]

Lugan, P.

J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

Modugno, G.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

Modugno, M.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

Peccianti, M.

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef]

Peschel, U.

S. Batz and U. Peschel, Phys. Rev. A 83, 033826 (2011).
[CrossRef]

Rasmussen, J. J.

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
[CrossRef]

Rasmussen, P.

P. Rasmussen, O. Bang, and W. Królikowski, Phys. Rev. E 72, 066611 (2005).
[CrossRef]

Righini, R.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Roati, G.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

Sanchez-Palencia, L.

J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

Schmidt, M. A.

C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

Schwartz, T.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Segev, M.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Shadrivov, I. V.

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef]

St. J. Russell, P.

C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

Stivala, S.

Torner, L.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 77, 051802 (2008).
[CrossRef]

Vysloukh, V. A.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 77, 051802 (2008).
[CrossRef]

Wiersma, D. S.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Wyller, J.

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
[CrossRef]

Zaccanti, M.

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

Zuo, Z.

J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

Nat. Phys. (1)

C. Conti and A. Fratalocchi, Nat. Phys. 4, 794 (2008).
[CrossRef]

Nature (4)

J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature 453, 891 (2008).
[CrossRef]

G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature 453, 895 (2008).
[CrossRef]

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. (1)

P. W. Anderson, Phys. Rev. 109, 1492 (1958).
[CrossRef]

Phys. Rev. A (2)

S. Batz and U. Peschel, Phys. Rev. A 83, 033826 (2011).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. A 77, 051802 (2008).
[CrossRef]

Phys. Rev. B (1)

C. Conti and L. Leuzzi, Phys. Rev. B 83, 134204 (2011).
[CrossRef]

Phys. Rev. E (2)

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
[CrossRef]

P. Rasmussen, O. Bang, and W. Królikowski, Phys. Rev. E 72, 066611 (2005).
[CrossRef]

Phys. Rev. Lett. (5)

C. Conti, M. A. Schmidt, P. St. J. Russell, and F. Biancalana, Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef]

V. Folli and C. Conti, Phys. Rev. Lett. 104, 193901(2010).
[CrossRef]

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef]

I. V. Shadrivov, K. Y. Bliokh, Y. P. Bliokh, V. Freilikher, and Y. S. Kivshar, Phys. Rev. Lett. 104, 123902 (2010).
[CrossRef]

Other (2)

P. Sheng, ed., Scattering and Localization of Classical Waves in Random Media (World Scientific, 1990).

F. Maucher, W. Krolikowski, and S. Skupin, http://arxiv.org/abs/1008.1891 (2010).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

Left axis: critical power for exponential (Pc=|β0|(2σ+l), dashed–dotted line), Gaussian (Pc=l|β0|exp(σ2/l2)/Erfc(σ/l), continuous line), quadratic (Pc=|β0|/|χ(0)|(1l2/2σ2), dots) and rectangular (Pc=2σ|β0|/(1e2σ/l), diamonds) response functions, for a single realization of the disorder; right, residual coupling δβ versus degree of nonlocality, averaged over 10 disorder realizations (V0=1).

Fig. 2.
Fig. 2.

Evolution of the ground-state intensity for a fixed disorder realization, for σ=10, V0=10, the focusing (attractive) case s=1, for (a) P=0.04Pc and for (b) P=4Pc, the defocusing (repulsive) case s=1, for (c) P=0.04Pc and for (d) P=4Pc. The superimposed dashed line in the panels represents the ground state at the initial time.

Fig. 3.
Fig. 3.

Wave function waist for (a) focusing and (b) defocusing cases, for P=0.04Pc (dashed–dotted line), P=0.64Pc (continuous line), P=2Pc (dotted line), P=4Pc (dashed line) (σ=10, V0=10). The results are averaged over 10 disorder realizations.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

iψt+ψxx=V(x)ψsψ+χ(xx)|ψ(x)|2dx,
ψ+χ(xx)|ψ(x)|2dx=Pχ(x).
β0(P)=β0sPχ00+P2n0|χn0|2β0βn,
Pc=|β0|+|ψ0(x)|2χ(x)dx.
l(P)l1+sP/Pc,
δβ(σ)=β0(Pc)β0=n0|χn0|2|χ00|2(1βn/β0).

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