Advertisement

Anderson localization in the quintic nonlinear Schrödinger equation

  • Wesley B. Cardoso
  • Salviano A. Leão
  • Ardiley T. Avelar
Article
  • 119 Downloads

Abstract

In the present paper we consider the quintic defocusing nonlinear Schrödinger equation in presence of a disordered random potential and we analyze the effects of the quintic nonlinearity on the Anderson localization of the solution. The main result shows that Anderson localization requires a cutoff on the value of the parameter that controls the quintic nonlinearity, with the cutoff depending on the amplitude of the random potential.

Keywords

Anderson localization Nonlinear Schrödinger equation Random potential Quintic nonlinearity 

Notes

Acknowledgments

We thank CNPq (National Counsel of Technological and Scientific Development) and INCT-IQ (National Institute of Science and Technology of Quantum Information), Brazilian Agencies, for financial support.

References

  1. Adhikari, S.K.: Localization of a Bose–Einstein-condensate vortex in a bichromatic optical lattice. Phys. Rev. A 81(4), 043636 (2010). doi: 10.1103/PhysRevA.81.043636 ADSCrossRefGoogle Scholar
  2. Adhikari, S.K., Salasnich, L.: Localization of a Bose–Einstein condensate in a bichromatic optical lattice. Phys. Rev. A 80(2), 023606 (2009). doi: 10.1103/PhysRevA.80.023606 ADSCrossRefGoogle Scholar
  3. Agrawal, G.P.: Nonlinear Fiber Optics, 5a edn. Academic Press, San Diego (2012)MATHGoogle Scholar
  4. Albert, M., Leboeuf, P.: Localization by bichromatic potentials versus Anderson localization. Phys. Rev. A 81(1), 013614 (2010). doi: 10.1103/PhysRevA.81.013614 ADSCrossRefGoogle Scholar
  5. Anderson, P.W.: Absence of diffusion in certain random lattices. Phys. Rev. 109(5), 1492–1505 (1958). doi: 10.1103/PhysRev.109.1492 ADSCrossRefGoogle Scholar
  6. Aubry, S., André, G.: Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Isr. Phys. Soc. 3(18), 133–164 (1979)MATHGoogle Scholar
  7. Avelar, A.T., Bazeia, D., Cardoso, W.B.: Solitons with cubic and quintic nonlinearities modulated in space and time. Phys. Rev. E 79(2), 025602 (2009). doi: 10.1103/PhysRevE.79.025602 ADSCrossRefGoogle Scholar
  8. Avelar, A.T., Bazeia, D., Cardoso, W.B.: Modulation of breathers in the three-dimensional nonlinear Gross–Pitaevskii equation. Phys. Rev. E 82(5), 057601 (2010). doi: 10.1103/PhysRevE.82.057601 ADSMathSciNetCrossRefGoogle Scholar
  9. Belmonte-Beitia, J., Calvo, G.F.: Exact solutions for the quintic nonlinear Schrödinger equation with time and space modulated nonlinearities and potentials. Phys. Lett. A 373(4), 448–453 (2009). doi: 10.1016/j.physleta.2008.11.056. http://linkinghub.elsevier.com/retrieve/pii/S037596010801712X
  10. Belmonte-Beitia, J., Cuevas, J.: Solitons for the cubic–quintic nonlinear Schrödinger equation with time- and space-modulated coefficients. J. Phys. A Math. Theor. 42(16), 165201 (2009). doi: 10.1088/1751-8113/42/16/165201. http://stacks.iop.org/1751-8121/42/i=16/a=165201?key=crossref.8d7be66afcc0684e833dc80e341f1ea2
  11. Belmonte-Beitia, J., Pérez-García, V.M., Vekslerchik, V., Konotop, V.V.: Localized nonlinear waves in systems with time- and space-modulated nonlinearities. Phys. Rev. Lett. 100(16), 164102 (2008). doi: 10.1103/PhysRevLett.100.164102 ADSCrossRefGoogle Scholar
  12. Billy, J., Josse, V., Zuo, Z., Bernard, A., Hambrecht, B., Lugan, P., Clément, D., Sanchez-Palencia, L., Bouyer, P., Aspect, A.: Direct observation of Anderson localization of matter waves in a controlled disorder. Nature 453(7197), 891–894 (2008). doi: 10.1038/nature07000 ADSCrossRefGoogle Scholar
  13. Bouyer, P.: Quantum gases and optical speckle: a new tool to simulate disordered quantum systems. Rep. Prog. Phys. 73(6), 062401 (2010). doi: 10.1088/0034-4885/73/6/062401. http://stacks.iop.org/0034-4885/73/i=6/a=062401?key=crossref.da2611daaf2d6402a7bd652b0a3925f8
  14. Burger, S., Bongs, K., Dettmer, S., Ertmer, W., Sengstock, K., Sanpera, A., Shlyapnikov, G.V., Lewenstein, M.: Dark solitons in Bose–Einstein condensates. Phys. Rev. Lett. 83(25), 5198–5201 (1999). doi: 10.1103/PhysRevLett.83.5198 ADSCrossRefGoogle Scholar
  15. Cai, X., Chen, S., Wang, Y.: Superfluid-to-Bose-glass transition of hard-core bosons in a one-dimensional incommensurate optical lattice. Phys. Rev. A 81(2), 023626 (2010). doi: 10.1103/PhysRevA.81.023626 ADSCrossRefGoogle Scholar
  16. Cardoso, W., Avelar, A., Bazeia, D.: Bright and dark solitons in a periodically attractive and expulsive potential with nonlinearities modulated in space and time. Nonlinear Anal. Real World Appl. 11(5), 4269–4274 (2010a). doi: 10.1016/j.nonrwa.2010.05.013. http://linkinghub.elsevier.com/retrieve/pii/S1468121810000805
  17. Cardoso, W., Avelar, A., Bazeia, D.: Modulation of breathers in cigar-shaped Bose–Einstein condensates. Phys. Lett. A 374(26), 2640–2645 (2010b). doi: 10.1016/j.physleta.2010.04.050. http://linkinghub.elsevier.com/retrieve/pii/S0375960110004895
  18. Cardoso, W.B., Avelar, A.T., Bazeia, D.: One-dimensional reduction of the three-dimenstional Gross-Pitaevskii equation with two- and three-body interactions. Phys. Rev. E 83(3), 036604 (2011). doi: 10.1103/PhysRevE.83.036604 ADSCrossRefGoogle Scholar
  19. Cardoso, W., Avelar, A., Bazeia, D.: Anderson localization of matter waves in chaotic potentials. Nonlinear Anal. Real World Appl. 13(2), 755–763 (2012). doi: 10.1016/j.nonrwa.2011.08.014. http://linkinghub.elsevier.com/retrieve/pii/S1468121811002410
  20. Cestari, J.C.C., Foerster, A., Gusmão, M.A.: Finite-size effects in Anderson localization of one-dimensional Bose–Einstein condensates. Phys. Rev. A 82(6), 063634 (2010). doi: 10.1103/PhysRevA.82.063634 ADSCrossRefGoogle Scholar
  21. Chabanov, A.A., Stoytchev, M., Genack, A.Z.: Statistical signatures of photon localization. Nature 404(6780), 850–853 (2000). doi: 10.1038/35009055 ADSCrossRefGoogle Scholar
  22. Cheng, Y., Adhikari, S.K.: Matter-wave localization in a random potential. Phys. Rev. A 82(1), 013631 (2010). doi: 10.1103/PhysRevA.82.013631 ADSCrossRefGoogle Scholar
  23. Cherroret, N., Skipetrov, S.E.: Effect of interactions on the diffusive expansion of a Bose–Einstein condensate in a three-dimensional random potential. Phys. Rev. A 79(6), 063604 (2009). doi: 10.1103/PhysRevA.79.063604 ADSCrossRefGoogle Scholar
  24. Deng, X., Citro, R., Orignac, E., Minguzzi, A.: Superfluidity and Anderson localisation for a weakly interacting Bose gas in a quasiperiodic potential. Eur. Phys. J. B 68(3), 435–443 (2009). doi: 10.1140/epjb/e2009-00069-7 ADSCrossRefGoogle Scholar
  25. Denschlag, J.: Generating solitons by phase engineering of a Bose–Einstein condensate. Science (80-. ) 287(5450), 97–101 (2000). doi: 10.1126/science.287.5450.97 ADSCrossRefGoogle Scholar
  26. Flach, S., Krimer, D.O., Skokos, C.: Universal spreading of wave packets in disordered nonlinear systems. Phys. Rev. Lett. 102(2), 024101 (2009). doi: 10.1103/PhysRevLett.102.024101 ADSCrossRefGoogle Scholar
  27. García-Mata, I., Shepelyansky, D.L.: Delocalization induced by nonlinearity in systems with disorder. Phys. Rev. E 79(2), 026205 (2009). doi: 10.1103/PhysRevE.79.026205 ADSCrossRefGoogle Scholar
  28. Grémaud, B., Wellens, T.: Speckle instability: coherent effects in nonlinear disordered media. Phys. Rev. Lett. 104(13), 133901 (2010). doi: 10.1103/PhysRevLett.104.133901 ADSCrossRefGoogle Scholar
  29. Hu, H., Strybulevych, A., Page, J.H., Skipetrov, S.E., van Tiggelen, B.A.: Localization of ultrasound in a three-dimensional elastic network. Nat. Phys. 4(12), 945–948 (2008). doi: 10.1038/nphys1101 CrossRefGoogle Scholar
  30. Karbasi, S., Hawkins, T., Ballato, J., Koch, K.W., Mafi, A.: Transverse Anderson localization in a disordered glass optical fiber. Opt. Mater. Express 2(11), 1496 (2012). doi: 10.1364/OME.2.001496. https://www.osapublishing.org/ome/abstract.cfm?uri=ome-2-11-1496
  31. Khaykovich, L., Schreck, F., Ferrari, G., Bourdel, T., Cubizolles, J., Carr, L.D., Castin, Y., Salomon, C.: Formation of a matter-wave bright soliton. Science (80-. ) 296(5571), 1290–1293 (2002). doi: 10.1126/science.1071021 ADSCrossRefGoogle Scholar
  32. Kivshar, Y.S., Agrawal, G.: Optical Solitons: From Fibers to Photonic Crystals. Elsevier, Amsterdam (2003). https://books.google.com.br/books?id=zzWgibj4ypsC
  33. Kopidakis, G., Komineas, S., Flach, S., Aubry, S.: Absence of wave packet diffusion in disordered nonlinear systems. Phys. Rev. Lett. 100(8), 084103 (2008). doi: 10.1103/PhysRevLett.100.084103 ADSCrossRefGoogle Scholar
  34. Lahini, Y., Avidan, A., Pozzi, F., Sorel, M., Morandotti, R., Christodoulides, D.N., Silberberg, Y.: Anderson localization and nonlinearity in one-dimensional disordered photonic lattices. Phys. Rev. Lett. 100(1), 013906 (2008). doi: 10.1103/PhysRevLett.100.013906 ADSCrossRefGoogle Scholar
  35. Larcher, M., Dalfovo, F., Modugno, M.: Effects of interaction on the diffusion of atomic matter waves in one-dimensional quasiperiodic potentials. Phys. Rev. A 80(5), 053606 (2009). doi: 10.1103/PhysRevA.80.053606 ADSCrossRefGoogle Scholar
  36. Lye, J.E., Fallani, L., Modugno, M., Wiersma, D.S., Fort, C., Inguscio, M.: Bose–Einstein condensate in a random potential. Phys. Rev. Lett. 95(7), 070401 (2005). doi: 10.1103/PhysRevLett.95.070401 ADSCrossRefGoogle Scholar
  37. Modugno, M.: Exponential localization in one-dimensional quasi-periodic optical lattices. New J. Phys. 11(3), 033023 (2009). doi: 10.1088/1367-2630/11/3/033023 ADSMathSciNetCrossRefGoogle Scholar
  38. Muruganandam, P., Kumar, R.K., Adhikari, S.K.: Localization of a dipolar Bose–Einstein condensate in a bichromatic optical lattice. J. Phys. B At. Mol. Opt. Phys. 43(20), 205305 (2010). doi: 10.1088/0953-4075/43/20/205305 ADSCrossRefGoogle Scholar
  39. Nattermann, T., Pokrovsky, V.L.: Bose–Einstein condensates in strongly disordered traps. Phys. Rev. Lett. 100(6), 060402 (2008). doi: 10.1103/PhysRevLett.100.060402 ADSCrossRefGoogle Scholar
  40. Paul, T., Albert, M., Schlagheck, P., Leboeuf, P., Pavloff, N.: Anderson localization of a weakly interacting one-dimensional Bose gas. Phys. Rev. A 80(3), 033615 (2009). doi: 10.1103/PhysRevA.80.033615 ADSCrossRefGoogle Scholar
  41. Pérez-García, V.M., Torres, P.J., Konotop, V.V.: Similarity transformations for nonlinear Schrödinger equations with time-dependent coefficients. Phys. D Nonlinear Phenom. 221(1), 31–36 (2006). doi: 10.1016/j.physd.2006.07.002. http://linkinghub.elsevier.com/retrieve/pii/S0167278906002405
  42. Pethick, C.J., Smith, H.: Bose–Einstein Condensation in Dilute Gases. Cambridge University Press, Cambridge (2002)Google Scholar
  43. Pikovsky, A.S., Shepelyansky, D.L.: Destruction of Anderson localization by a weak nonlinearity. Phys. Rev. Lett. 100(9), 094101 (2008). doi: 10.1103/PhysRevLett.100.094101 ADSCrossRefGoogle Scholar
  44. Pitaevskii, L.P., Stringari, S.: Bose–Einstein Condensation. International Series of Monographs on Physics. Clarendon Press, Oxford (2003). https://books.google.com.br/books?id=rIobbOxC4j4C
  45. Roati, G., D’Errico, C., Fallani, L., Fattori, M., Fort, C., Zaccanti, M., Modugno, G., Modugno, M., Inguscio, M.: Anderson localization of a non-interacting Bose–Einstein condensate. Nature 453(7197), 895–898 (2008). doi: 10.1038/nature07071 ADSCrossRefGoogle Scholar
  46. Roscilde, T.: Bosons in one-dimensional incommensurate superlattices. Phys. Rev. A 77(6), 063605 (2008). doi: 10.1103/PhysRevA.77.063605 ADSCrossRefGoogle Scholar
  47. Roux, G., Barthel, T., McCulloch, I.P., Kollath, C., Schollwöck, U., Giamarchi, T.: Quasiperiodic Bose–Hubbard model and localization in one-dimensional cold atomic gases. Phys. Rev. A 78(2), 023628 (2008). doi: 10.1103/PhysRevA.78.023628 ADSCrossRefGoogle Scholar
  48. Salasnich, L., Parola, A., Reatto, L.: Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates. Phys. Rev. A 65(4), 043614 (2002). doi: 10.1103/PhysRevA.65.043614 ADSCrossRefGoogle Scholar
  49. Sanchez-Palencia, L., Clément, D., Lugan, P., Bouyer, P., Shlyapnikov, G.V., Aspect, A.: Anderson localization of expanding Bose–Einstein condensates in random potentials. Phys. Rev. Lett. 98(21), 210401 (2007). doi: 10.1103/PhysRevLett.98.210401 ADSCrossRefGoogle Scholar
  50. Sanchez-Palencia, L., Clément, D., Lugan, P., Bouyer, P., Aspect, A.: Disorder-induced trapping versus Anderson localization in Bose–Einstein condensates expanding in disordered potentials. New J. Phys. 10(4), 045019 (2008). doi: 10.1088/1367-2630/10/4/045019. http://stacks.iop.org/1367-2630/10/i=4/a=045019?key=crossref.f8ea4aef7cd1b60db1dadc2134f7e648
  51. Sanchez-Palencia, L., Lewenstein, M.: Disordered quantum gases under control. Nat. Phys. 6(2), 87–95 (2010). doi: 10.1038/nphys1507 CrossRefGoogle Scholar
  52. Schwartz, T., Bartal, G., Fishman, S., Segev, M.: Transport and Anderson localization in disordered two-dimensional photonic lattices. Nature 446(7131), 52–55 (2007). doi: 10.1038/nature05623 ADSCrossRefGoogle Scholar
  53. Shechtman, D., Blech, I., Gratias, D., Cahn, J.W.: Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53(20), 1951–1953 (1984). doi: 10.1103/PhysRevLett.53.1951 ADSCrossRefGoogle Scholar
  54. Skokos, C., Krimer, D.O., Komineas, S., Flach, S.: Delocalization of wave packets in disordered nonlinear chains. Phys. Rev. E 79(5), 056211 (2009). doi: 10.1103/PhysRevE.79.056211 ADSMathSciNetCrossRefGoogle Scholar
  55. Srinivasan, G., Aceves, A., Tartakovsky, D.M.: Nonlinear localization of light in disordered optical fiber arrays. Phys. Rev. A 77(6), 063806 (2008). doi: 10.1103/PhysRevA.77.063806 ADSCrossRefGoogle Scholar
  56. Störzer, M., Gross, P., Aegerter, C.M., Maret, G.: Observation of the critical regime near anderson localization of light. Phys. Rev. Lett. 96(6), 063904 (2006). doi: 10.1103/PhysRevLett.96.063904 ADSCrossRefGoogle Scholar
  57. Strecker, K.E., Partridge, G.B., Truscott, A.G., Hulet, R.G.: Formation and propagation of matter-wave soliton trains. Nature 417(6885), 150–153 (2002). doi: 10.1038/nature747 ADSCrossRefGoogle Scholar
  58. Wiersma, D.S., Bartolini, P., Lagendijk, A., Righini, R.: Localization of light in a disordered medium. Nature 390(6661), 671–673 (1997). doi: 10.1038/37757 ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Instituto de FísicaUniversidade Federal de GoiásGoiâniaBrazil

Personalised recommendations