Abstract

We show, numerically and experimentally, that the presence of weak disorder results in an enhanced energy distribution of an initially localized wave-packet, in one- and two-dimensional finite lattices. The addition of a focusing nonlinearity facilitates the spreading effect even further by increasing the wave-packet effective size. We find a clear transition between the regions of enhanced spreading (weak disorder) and localization (strong disorder).

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  3. B. Deissler, M. Zaccantu, G. Roati, C. D’Errico, M. Fattori, M. Modugno, G. Modugno, and M. Inguscio, “Delocalization of a disordered bosonic system by repulsive interactions,” Nature Phys.6, 354–358 (2010).
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  4. Y. S. Kivshar, S. A. Gredeskul, A. Sánchez, and L. Vázquez, “Localization decay induced by strong nonlinearity in disordered systems,” Phys. Rev. Lett.64, 1693–1696 (1990).
    [CrossRef] [PubMed]
  5. M. I. Molina, “Transport of localized and extended excitations in a nonlinear Anderson model,” Phys. Rev. B58, 12547–12550 (1998).
    [CrossRef]
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  13. R. Bruinsma and S. N. Coppersmith, “Anderson localization and breakdown of hydrodynamics in random ferromagnets,” Phys. Rev. B33, 6541–6544 (1986).
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    [CrossRef] [PubMed]
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  16. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1–126 (2008).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  24. R. A. Vicencio and S. Flach, “Control of wave packet spreading in nonlinear finite disordered lattices,” Phys. Rev. E79, 016217 (2009).
    [CrossRef]
  25. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett.13, 794–796 (1988).
    [CrossRef] [PubMed]
  26. M. I. Molina and G. P. Tsironis, “Dynamics of self-trapping in the discrete nonlinear Schrödinger equation,” Phys. D65, 267–273 (1993).
    [CrossRef]
  27. P. G. Kevrekidis, J. A. Espinola-Rocha, Y. Drossino, and A. Stefanov, “Dynamical barrier for the formation of solitary waves in discrete lattices,” Phys. Lett. A372, 2247–2253 (2008).
    [CrossRef]
  28. U. Naether, A. J. Martínez, D. Guzmán-Silva, M. I. Molina, and R. A. Vicencio, “Self-trapping transition in nonlinear cubic lattices,” in preparation (2012).
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    [CrossRef]

2012 (1)

2011 (2)

M. V. Ivanchenko, T. V. Laptyeva, and S. Flach, “Anderson localization or nonlinear waves? A matter of probability,” Phys. Rev. Lett.107, 240602 (2011).
[CrossRef]

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science332, 1541–1544 (2011).
[CrossRef] [PubMed]

2010 (4)

S. Flach, “Wave propagation in nonlinear disordered media,” Phys. Part. Nucl.41, 1020 (2010).
[CrossRef]

A. Szameit and S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B43, 163001 (2010).
[CrossRef]

B. Deissler, M. Zaccantu, G. Roati, C. D’Errico, M. Fattori, M. Modugno, G. Modugno, and M. Inguscio, “Delocalization of a disordered bosonic system by repulsive interactions,” Nature Phys.6, 354–358 (2010).
[CrossRef]

L. Sanchez-Palencia and M. Lewenstein, “Disordered quantum gases under control,” Nature Phys.6, 87–95 (2010).
[CrossRef]

2009 (2)

S. Flach, D. O. Krimer, and Ch. Skokos, “Universal spreading of wave packets in disordered nonlinear systems,” Phys. Rev. Lett.102, 024101 (2009).
[CrossRef] [PubMed]

R. A. Vicencio and S. Flach, “Control of wave packet spreading in nonlinear finite disordered lattices,” Phys. Rev. E79, 016217 (2009).
[CrossRef]

2008 (7)

P. G. Kevrekidis, J. A. Espinola-Rocha, Y. Drossino, and A. Stefanov, “Dynamical barrier for the formation of solitary waves in discrete lattices,” Phys. Lett. A372, 2247–2253 (2008).
[CrossRef]

G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, “Absence of wave packet diffusion in disordered nonlinear systems,” Phys. Rev. Lett.100, 084103 (2008).
[CrossRef] [PubMed]

A. S. Pikovsky and D. L. Shepelyansky, “Destruction of Anderson localization by a weak nonlinearity,” Phys. Rev. Lett.100, 094101 (2008).
[CrossRef] [PubMed]

S. Flach and A. Gorbach, “Discrete breathers: Advances in theory and applications,” Phys. Reps.467, 1–116 (2008).
[CrossRef]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1–126 (2008).
[CrossRef]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett.100, 013906 (2008).
[CrossRef] [PubMed]

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, “Direct observation of Anderson localization of matter waves in a controlled disorder,” Nature453, 891–894 (2008).
[CrossRef] [PubMed]

2007 (1)

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446, 52–55 (2007).
[CrossRef] [PubMed]

2004 (1)

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett.93, 053901 (2004).
[CrossRef] [PubMed]

1998 (1)

M. I. Molina, “Transport of localized and extended excitations in a nonlinear Anderson model,” Phys. Rev. B58, 12547–12550 (1998).
[CrossRef]

1993 (2)

B. Kramer and A. MacKinnon, “Localization: theory and experiment,” Rep. Prog. Phys.56, 1469 (1993).
[CrossRef]

M. I. Molina and G. P. Tsironis, “Dynamics of self-trapping in the discrete nonlinear Schrödinger equation,” Phys. D65, 267–273 (1993).
[CrossRef]

1991 (1)

R. Dalichaouch, J. P. Armstrong, S. Schulz, P. M. Platzman, and S. L. McCal, “Microwave localization by two-dimensional random scattering,” Nature354, 53–55 (1991).
[CrossRef]

1990 (2)

Y. S. Kivshar, S. A. Gredeskul, A. Sánchez, and L. Vázquez, “Localization decay induced by strong nonlinearity in disordered systems,” Phys. Rev. Lett.64, 1693–1696 (1990).
[CrossRef] [PubMed]

R. L. Weaver, “Anderson localization of ultrasound,” Wave Motion12, 129–142 (1990).
[CrossRef]

1988 (1)

1986 (1)

R. Bruinsma and S. N. Coppersmith, “Anderson localization and breakdown of hydrodynamics in random ferromagnets,” Phys. Rev. B33, 6541–6544 (1986).
[CrossRef]

1977 (1)

D. Weaire and A. R. Williams, “The Anderson localization problem. I. A new numerical approach,” J. Phys. C: Solid State Phys.10, 1239 (1977).
[CrossRef]

1958 (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev.109, 1492–1505 (1958).
[CrossRef]

Anderson, P. W.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev.109, 1492–1505 (1958).
[CrossRef]

Armstrong, J. P.

R. Dalichaouch, J. P. Armstrong, S. Schulz, P. M. Platzman, and S. L. McCal, “Microwave localization by two-dimensional random scattering,” Nature354, 53–55 (1991).
[CrossRef]

Aspect, A.

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, “Direct observation of Anderson localization of matter waves in a controlled disorder,” Nature453, 891–894 (2008).
[CrossRef] [PubMed]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1–126 (2008).
[CrossRef]

Aubry, S.

G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, “Absence of wave packet diffusion in disordered nonlinear systems,” Phys. Rev. Lett.100, 084103 (2008).
[CrossRef] [PubMed]

Avidan, A.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett.100, 013906 (2008).
[CrossRef] [PubMed]

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446, 52–55 (2007).
[CrossRef] [PubMed]

Bartelt, H.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett.93, 053901 (2004).
[CrossRef] [PubMed]

Bernard, A.

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, “Direct observation of Anderson localization of matter waves in a controlled disorder,” Nature453, 891–894 (2008).
[CrossRef] [PubMed]

Billy, J.

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, “Direct observation of Anderson localization of matter waves in a controlled disorder,” Nature453, 891–894 (2008).
[CrossRef] [PubMed]

Bouyer, P.

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, “Direct observation of Anderson localization of matter waves in a controlled disorder,” Nature453, 891–894 (2008).
[CrossRef] [PubMed]

Bruinsma, R.

R. Bruinsma and S. N. Coppersmith, “Anderson localization and breakdown of hydrodynamics in random ferromagnets,” Phys. Rev. B33, 6541–6544 (1986).
[CrossRef]

Christodoulides, D. N.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1–126 (2008).
[CrossRef]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett.100, 013906 (2008).
[CrossRef] [PubMed]

D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett.13, 794–796 (1988).
[CrossRef] [PubMed]

Clement, C.

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, “Direct observation of Anderson localization of matter waves in a controlled disorder,” Nature453, 891–894 (2008).
[CrossRef] [PubMed]

Coppersmith, S. N.

R. Bruinsma and S. N. Coppersmith, “Anderson localization and breakdown of hydrodynamics in random ferromagnets,” Phys. Rev. B33, 6541–6544 (1986).
[CrossRef]

D’Errico, C.

B. Deissler, M. Zaccantu, G. Roati, C. D’Errico, M. Fattori, M. Modugno, G. Modugno, and M. Inguscio, “Delocalization of a disordered bosonic system by repulsive interactions,” Nature Phys.6, 354–358 (2010).
[CrossRef]

Dalichaouch, R.

R. Dalichaouch, J. P. Armstrong, S. Schulz, P. M. Platzman, and S. L. McCal, “Microwave localization by two-dimensional random scattering,” Nature354, 53–55 (1991).
[CrossRef]

Deissler, B.

B. Deissler, M. Zaccantu, G. Roati, C. D’Errico, M. Fattori, M. Modugno, G. Modugno, and M. Inguscio, “Delocalization of a disordered bosonic system by repulsive interactions,” Nature Phys.6, 354–358 (2010).
[CrossRef]

Drossino, Y.

P. G. Kevrekidis, J. A. Espinola-Rocha, Y. Drossino, and A. Stefanov, “Dynamical barrier for the formation of solitary waves in discrete lattices,” Phys. Lett. A372, 2247–2253 (2008).
[CrossRef]

Espinola-Rocha, J. A.

P. G. Kevrekidis, J. A. Espinola-Rocha, Y. Drossino, and A. Stefanov, “Dynamical barrier for the formation of solitary waves in discrete lattices,” Phys. Lett. A372, 2247–2253 (2008).
[CrossRef]

Fattori, M.

B. Deissler, M. Zaccantu, G. Roati, C. D’Errico, M. Fattori, M. Modugno, G. Modugno, and M. Inguscio, “Delocalization of a disordered bosonic system by repulsive interactions,” Nature Phys.6, 354–358 (2010).
[CrossRef]

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446, 52–55 (2007).
[CrossRef] [PubMed]

S. Fishman, Y. Krivolapov, and A. Soffer, “The nonlinear Schrödinger equation with a random potential: Results and puzzles,” eprint: arXiv 1108.2956v1 [math-ph].

Flach, S.

M. V. Ivanchenko, T. V. Laptyeva, and S. Flach, “Anderson localization or nonlinear waves? A matter of probability,” Phys. Rev. Lett.107, 240602 (2011).
[CrossRef]

S. Flach, “Wave propagation in nonlinear disordered media,” Phys. Part. Nucl.41, 1020 (2010).
[CrossRef]

R. A. Vicencio and S. Flach, “Control of wave packet spreading in nonlinear finite disordered lattices,” Phys. Rev. E79, 016217 (2009).
[CrossRef]

S. Flach, D. O. Krimer, and Ch. Skokos, “Universal spreading of wave packets in disordered nonlinear systems,” Phys. Rev. Lett.102, 024101 (2009).
[CrossRef] [PubMed]

S. Flach and A. Gorbach, “Discrete breathers: Advances in theory and applications,” Phys. Reps.467, 1–116 (2008).
[CrossRef]

G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, “Absence of wave packet diffusion in disordered nonlinear systems,” Phys. Rev. Lett.100, 084103 (2008).
[CrossRef] [PubMed]

Freedman, B.

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science332, 1541–1544 (2011).
[CrossRef] [PubMed]

Gorbach, A.

S. Flach and A. Gorbach, “Discrete breathers: Advances in theory and applications,” Phys. Reps.467, 1–116 (2008).
[CrossRef]

Gredeskul, S. A.

Y. S. Kivshar, S. A. Gredeskul, A. Sánchez, and L. Vázquez, “Localization decay induced by strong nonlinearity in disordered systems,” Phys. Rev. Lett.64, 1693–1696 (1990).
[CrossRef] [PubMed]

Guzmán-Silva, D.

U. Naether, A. J. Martínez, D. Guzmán-Silva, M. I. Molina, and R. A. Vicencio, “Self-trapping transition in nonlinear cubic lattices,” in preparation (2012).

Hambrecht, B.

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, “Direct observation of Anderson localization of matter waves in a controlled disorder,” Nature453, 891–894 (2008).
[CrossRef] [PubMed]

Inguscio, M.

B. Deissler, M. Zaccantu, G. Roati, C. D’Errico, M. Fattori, M. Modugno, G. Modugno, and M. Inguscio, “Delocalization of a disordered bosonic system by repulsive interactions,” Nature Phys.6, 354–358 (2010).
[CrossRef]

Ivanchenko, M. V.

M. V. Ivanchenko, T. V. Laptyeva, and S. Flach, “Anderson localization or nonlinear waves? A matter of probability,” Phys. Rev. Lett.107, 240602 (2011).
[CrossRef]

Joseph, R. I.

Josse, V.

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, “Direct observation of Anderson localization of matter waves in a controlled disorder,” Nature453, 891–894 (2008).
[CrossRef] [PubMed]

Kartashov, Y.

Kevrekidis, P. G.

P. G. Kevrekidis, J. A. Espinola-Rocha, Y. Drossino, and A. Stefanov, “Dynamical barrier for the formation of solitary waves in discrete lattices,” Phys. Lett. A372, 2247–2253 (2008).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar, S. A. Gredeskul, A. Sánchez, and L. Vázquez, “Localization decay induced by strong nonlinearity in disordered systems,” Phys. Rev. Lett.64, 1693–1696 (1990).
[CrossRef] [PubMed]

Kobelke, J.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett.93, 053901 (2004).
[CrossRef] [PubMed]

Komineas, S.

G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, “Absence of wave packet diffusion in disordered nonlinear systems,” Phys. Rev. Lett.100, 084103 (2008).
[CrossRef] [PubMed]

Kopidakis, G.

G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, “Absence of wave packet diffusion in disordered nonlinear systems,” Phys. Rev. Lett.100, 084103 (2008).
[CrossRef] [PubMed]

Kramer, B.

B. Kramer and A. MacKinnon, “Localization: theory and experiment,” Rep. Prog. Phys.56, 1469 (1993).
[CrossRef]

Krimer, D. O.

S. Flach, D. O. Krimer, and Ch. Skokos, “Universal spreading of wave packets in disordered nonlinear systems,” Phys. Rev. Lett.102, 024101 (2009).
[CrossRef] [PubMed]

Krivolapov, Y.

S. Fishman, Y. Krivolapov, and A. Soffer, “The nonlinear Schrödinger equation with a random potential: Results and puzzles,” eprint: arXiv 1108.2956v1 [math-ph].

Lahini, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett.100, 013906 (2008).
[CrossRef] [PubMed]

Laptyeva, T. V.

M. V. Ivanchenko, T. V. Laptyeva, and S. Flach, “Anderson localization or nonlinear waves? A matter of probability,” Phys. Rev. Lett.107, 240602 (2011).
[CrossRef]

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1–126 (2008).
[CrossRef]

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett.93, 053901 (2004).
[CrossRef] [PubMed]

Levi, L.

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science332, 1541–1544 (2011).
[CrossRef] [PubMed]

Lewenstein, M.

Lugan, P.

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, “Direct observation of Anderson localization of matter waves in a controlled disorder,” Nature453, 891–894 (2008).
[CrossRef] [PubMed]

MacKinnon, A.

B. Kramer and A. MacKinnon, “Localization: theory and experiment,” Rep. Prog. Phys.56, 1469 (1993).
[CrossRef]

Manela, O.

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science332, 1541–1544 (2011).
[CrossRef] [PubMed]

Martínez, A. J.

U. Naether, A. J. Martínez, D. Guzmán-Silva, M. I. Molina, and R. A. Vicencio, “Self-trapping transition in nonlinear cubic lattices,” in preparation (2012).

McCal, S. L.

R. Dalichaouch, J. P. Armstrong, S. Schulz, P. M. Platzman, and S. L. McCal, “Microwave localization by two-dimensional random scattering,” Nature354, 53–55 (1991).
[CrossRef]

Modugno, G.

B. Deissler, M. Zaccantu, G. Roati, C. D’Errico, M. Fattori, M. Modugno, G. Modugno, and M. Inguscio, “Delocalization of a disordered bosonic system by repulsive interactions,” Nature Phys.6, 354–358 (2010).
[CrossRef]

Modugno, M.

B. Deissler, M. Zaccantu, G. Roati, C. D’Errico, M. Fattori, M. Modugno, G. Modugno, and M. Inguscio, “Delocalization of a disordered bosonic system by repulsive interactions,” Nature Phys.6, 354–358 (2010).
[CrossRef]

Molina, M. I.

M. I. Molina, “Transport of localized and extended excitations in a nonlinear Anderson model,” Phys. Rev. B58, 12547–12550 (1998).
[CrossRef]

M. I. Molina and G. P. Tsironis, “Dynamics of self-trapping in the discrete nonlinear Schrödinger equation,” Phys. D65, 267–273 (1993).
[CrossRef]

U. Naether, A. J. Martínez, D. Guzmán-Silva, M. I. Molina, and R. A. Vicencio, “Self-trapping transition in nonlinear cubic lattices,” in preparation (2012).

Morandotti, R.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett.100, 013906 (2008).
[CrossRef] [PubMed]

Naether, U.

U. Naether, A. J. Martínez, D. Guzmán-Silva, M. I. Molina, and R. A. Vicencio, “Self-trapping transition in nonlinear cubic lattices,” in preparation (2012).

Nolte, S.

S. Stützer, Y. Kartashov, V. Vysloukh, A. Tünnermann, S. Nolte, M. Lewenstein, L. Torner, and A. Szameit, “Anderson cross-localization,” Opt. Lett.37, 1715–1717 (2012).
[CrossRef] [PubMed]

A. Szameit and S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B43, 163001 (2010).
[CrossRef]

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett.93, 053901 (2004).
[CrossRef] [PubMed]

Pertsch, T.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett.93, 053901 (2004).
[CrossRef] [PubMed]

Peschel, U.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett.93, 053901 (2004).
[CrossRef] [PubMed]

Pikovsky, A. S.

A. S. Pikovsky and D. L. Shepelyansky, “Destruction of Anderson localization by a weak nonlinearity,” Phys. Rev. Lett.100, 094101 (2008).
[CrossRef] [PubMed]

Platzman, P. M.

R. Dalichaouch, J. P. Armstrong, S. Schulz, P. M. Platzman, and S. L. McCal, “Microwave localization by two-dimensional random scattering,” Nature354, 53–55 (1991).
[CrossRef]

Pozzi, F.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett.100, 013906 (2008).
[CrossRef] [PubMed]

Rechtsman, M.

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science332, 1541–1544 (2011).
[CrossRef] [PubMed]

Roati, G.

B. Deissler, M. Zaccantu, G. Roati, C. D’Errico, M. Fattori, M. Modugno, G. Modugno, and M. Inguscio, “Delocalization of a disordered bosonic system by repulsive interactions,” Nature Phys.6, 354–358 (2010).
[CrossRef]

Sánchez, A.

Y. S. Kivshar, S. A. Gredeskul, A. Sánchez, and L. Vázquez, “Localization decay induced by strong nonlinearity in disordered systems,” Phys. Rev. Lett.64, 1693–1696 (1990).
[CrossRef] [PubMed]

Sanchez-Palencia, L.

L. Sanchez-Palencia and M. Lewenstein, “Disordered quantum gases under control,” Nature Phys.6, 87–95 (2010).
[CrossRef]

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, “Direct observation of Anderson localization of matter waves in a controlled disorder,” Nature453, 891–894 (2008).
[CrossRef] [PubMed]

Schulz, S.

R. Dalichaouch, J. P. Armstrong, S. Schulz, P. M. Platzman, and S. L. McCal, “Microwave localization by two-dimensional random scattering,” Nature354, 53–55 (1991).
[CrossRef]

Schuster, K.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett.93, 053901 (2004).
[CrossRef] [PubMed]

Schwartz, T.

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science332, 1541–1544 (2011).
[CrossRef] [PubMed]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446, 52–55 (2007).
[CrossRef] [PubMed]

Segev, M.

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science332, 1541–1544 (2011).
[CrossRef] [PubMed]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1–126 (2008).
[CrossRef]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446, 52–55 (2007).
[CrossRef] [PubMed]

Sheng, P.

P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Springer, 2006).

Shepelyansky, D. L.

A. S. Pikovsky and D. L. Shepelyansky, “Destruction of Anderson localization by a weak nonlinearity,” Phys. Rev. Lett.100, 094101 (2008).
[CrossRef] [PubMed]

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1–126 (2008).
[CrossRef]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett.100, 013906 (2008).
[CrossRef] [PubMed]

Skokos, Ch.

S. Flach, D. O. Krimer, and Ch. Skokos, “Universal spreading of wave packets in disordered nonlinear systems,” Phys. Rev. Lett.102, 024101 (2009).
[CrossRef] [PubMed]

Soffer, A.

S. Fishman, Y. Krivolapov, and A. Soffer, “The nonlinear Schrödinger equation with a random potential: Results and puzzles,” eprint: arXiv 1108.2956v1 [math-ph].

Sorel, M.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett.100, 013906 (2008).
[CrossRef] [PubMed]

Stefanov, A.

P. G. Kevrekidis, J. A. Espinola-Rocha, Y. Drossino, and A. Stefanov, “Dynamical barrier for the formation of solitary waves in discrete lattices,” Phys. Lett. A372, 2247–2253 (2008).
[CrossRef]

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1–126 (2008).
[CrossRef]

Stützer, S.

Szameit, A.

Torner, L.

Tsironis, G. P.

M. I. Molina and G. P. Tsironis, “Dynamics of self-trapping in the discrete nonlinear Schrödinger equation,” Phys. D65, 267–273 (1993).
[CrossRef]

Tünnermann, A.

S. Stützer, Y. Kartashov, V. Vysloukh, A. Tünnermann, S. Nolte, M. Lewenstein, L. Torner, and A. Szameit, “Anderson cross-localization,” Opt. Lett.37, 1715–1717 (2012).
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett.93, 053901 (2004).
[CrossRef] [PubMed]

Vázquez, L.

Y. S. Kivshar, S. A. Gredeskul, A. Sánchez, and L. Vázquez, “Localization decay induced by strong nonlinearity in disordered systems,” Phys. Rev. Lett.64, 1693–1696 (1990).
[CrossRef] [PubMed]

Vicencio, R. A.

R. A. Vicencio and S. Flach, “Control of wave packet spreading in nonlinear finite disordered lattices,” Phys. Rev. E79, 016217 (2009).
[CrossRef]

U. Naether, A. J. Martínez, D. Guzmán-Silva, M. I. Molina, and R. A. Vicencio, “Self-trapping transition in nonlinear cubic lattices,” in preparation (2012).

Vysloukh, V.

Weaire, D.

D. Weaire and A. R. Williams, “The Anderson localization problem. I. A new numerical approach,” J. Phys. C: Solid State Phys.10, 1239 (1977).
[CrossRef]

Weaver, R. L.

R. L. Weaver, “Anderson localization of ultrasound,” Wave Motion12, 129–142 (1990).
[CrossRef]

Williams, A. R.

D. Weaire and A. R. Williams, “The Anderson localization problem. I. A new numerical approach,” J. Phys. C: Solid State Phys.10, 1239 (1977).
[CrossRef]

Zaccantu, M.

B. Deissler, M. Zaccantu, G. Roati, C. D’Errico, M. Fattori, M. Modugno, G. Modugno, and M. Inguscio, “Delocalization of a disordered bosonic system by repulsive interactions,” Nature Phys.6, 354–358 (2010).
[CrossRef]

Zuo, Z. C.

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, “Direct observation of Anderson localization of matter waves in a controlled disorder,” Nature453, 891–894 (2008).
[CrossRef] [PubMed]

J. Phys. B (1)

A. Szameit and S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B43, 163001 (2010).
[CrossRef]

J. Phys. C: Solid State Phys. (1)

D. Weaire and A. R. Williams, “The Anderson localization problem. I. A new numerical approach,” J. Phys. C: Solid State Phys.10, 1239 (1977).
[CrossRef]

Nature (3)

J. Billy, V. Josse, Z. C. Zuo, A. Bernard, B. Hambrecht, P. Lugan, C. Clement, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, “Direct observation of Anderson localization of matter waves in a controlled disorder,” Nature453, 891–894 (2008).
[CrossRef] [PubMed]

R. Dalichaouch, J. P. Armstrong, S. Schulz, P. M. Platzman, and S. L. McCal, “Microwave localization by two-dimensional random scattering,” Nature354, 53–55 (1991).
[CrossRef]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446, 52–55 (2007).
[CrossRef] [PubMed]

Nature Phys. (2)

L. Sanchez-Palencia and M. Lewenstein, “Disordered quantum gases under control,” Nature Phys.6, 87–95 (2010).
[CrossRef]

B. Deissler, M. Zaccantu, G. Roati, C. D’Errico, M. Fattori, M. Modugno, G. Modugno, and M. Inguscio, “Delocalization of a disordered bosonic system by repulsive interactions,” Nature Phys.6, 354–358 (2010).
[CrossRef]

Opt. Lett. (2)

Phys. D (1)

M. I. Molina and G. P. Tsironis, “Dynamics of self-trapping in the discrete nonlinear Schrödinger equation,” Phys. D65, 267–273 (1993).
[CrossRef]

Phys. Lett. A (1)

P. G. Kevrekidis, J. A. Espinola-Rocha, Y. Drossino, and A. Stefanov, “Dynamical barrier for the formation of solitary waves in discrete lattices,” Phys. Lett. A372, 2247–2253 (2008).
[CrossRef]

Phys. Part. Nucl. (1)

S. Flach, “Wave propagation in nonlinear disordered media,” Phys. Part. Nucl.41, 1020 (2010).
[CrossRef]

Phys. Rep. (1)

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1–126 (2008).
[CrossRef]

Phys. Reps. (1)

S. Flach and A. Gorbach, “Discrete breathers: Advances in theory and applications,” Phys. Reps.467, 1–116 (2008).
[CrossRef]

Phys. Rev. (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev.109, 1492–1505 (1958).
[CrossRef]

Phys. Rev. B (2)

M. I. Molina, “Transport of localized and extended excitations in a nonlinear Anderson model,” Phys. Rev. B58, 12547–12550 (1998).
[CrossRef]

R. Bruinsma and S. N. Coppersmith, “Anderson localization and breakdown of hydrodynamics in random ferromagnets,” Phys. Rev. B33, 6541–6544 (1986).
[CrossRef]

Phys. Rev. E (1)

R. A. Vicencio and S. Flach, “Control of wave packet spreading in nonlinear finite disordered lattices,” Phys. Rev. E79, 016217 (2009).
[CrossRef]

Phys. Rev. Lett. (7)

M. V. Ivanchenko, T. V. Laptyeva, and S. Flach, “Anderson localization or nonlinear waves? A matter of probability,” Phys. Rev. Lett.107, 240602 (2011).
[CrossRef]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett.100, 013906 (2008).
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett.93, 053901 (2004).
[CrossRef] [PubMed]

G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, “Absence of wave packet diffusion in disordered nonlinear systems,” Phys. Rev. Lett.100, 084103 (2008).
[CrossRef] [PubMed]

A. S. Pikovsky and D. L. Shepelyansky, “Destruction of Anderson localization by a weak nonlinearity,” Phys. Rev. Lett.100, 094101 (2008).
[CrossRef] [PubMed]

S. Flach, D. O. Krimer, and Ch. Skokos, “Universal spreading of wave packets in disordered nonlinear systems,” Phys. Rev. Lett.102, 024101 (2009).
[CrossRef] [PubMed]

Y. S. Kivshar, S. A. Gredeskul, A. Sánchez, and L. Vázquez, “Localization decay induced by strong nonlinearity in disordered systems,” Phys. Rev. Lett.64, 1693–1696 (1990).
[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

B. Kramer and A. MacKinnon, “Localization: theory and experiment,” Rep. Prog. Phys.56, 1469 (1993).
[CrossRef]

Science (1)

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science332, 1541–1544 (2011).
[CrossRef] [PubMed]

Wave Motion (1)

R. L. Weaver, “Anderson localization of ultrasound,” Wave Motion12, 129–142 (1990).
[CrossRef]

Other (3)

U. Naether, A. J. Martínez, D. Guzmán-Silva, M. I. Molina, and R. A. Vicencio, “Self-trapping transition in nonlinear cubic lattices,” in preparation (2012).

S. Fishman, Y. Krivolapov, and A. Soffer, “The nonlinear Schrödinger equation with a random potential: Results and puzzles,” eprint: arXiv 1108.2956v1 [math-ph].

P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Springer, 2006).

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Figures (4)

Fig. 1
Fig. 1

(a) Sketch of light propagation through a 1D waveguide array, when only the center waveguide is initially excited. (b)–(d) Simulated intensity output distribution for lattices with on-site disorder level Wε = 0 (b), Wε = 0.36 (single realization) (c), and Wε = 1 (single realization) (d), respectively.

Fig. 2
Fig. 2

(a) Simulated disorder-averaged PR vs. on-site disorder width for different levels of inter-site disorder after linear (low power) propagation. (b) Simulated disorder-averaged PR vs. power P and onsite-disorder Wε after nonlinear propagation (thick contour denotes R = R0). In all cases, the PR is normalized to R0R(Wε = Wc = 0).

Fig. 3
Fig. 3

(a) Microscope images of an ordered (upper row) and disordered (bottom row) 1D array. (b) Experimental linear intensity output patterns: ordered array - averaged (upper row), weak disorder δd = 1.5μm - averaged (middle row), and strong disorder δd = 6μm - single realization (bottom row). (c) Experimental averaged R versus disorder strengths for different levels of input power, normalized to R0,lin. Errors bars indicate the standard deviation of R.

Fig. 4
Fig. 4

(a) Microscope image of the fabricated ordered square lattice. (b) Output intensity profile for a single-site excitation. (c) Numerical simulation of the PR for different mixed disorder widths: Wc = 0 (black-solid line), Wc = Wε/4 (red-dashed line), Wc = Wε/2 (blue-dotted line). (d) Experimental results for single-site (black-solid line) and gaussian (red-dashed line) input excitations, averaged over 25 realizations. All data is normalized to R0,lin. Errors bars indicate the standard deviation of R.

Equations (1)

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i d u n d z = ε n u n + m n V n , m u m + γ | u n | 2 u n .

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