Abstract

One of the most fundamental difference between classical and quantum mechanics is observed in the particle tunneling through a localized potential: the former predicts a discontinuous transmission coefficient (T) as a function in incident velocity between one (complete penetration) and zero (complete reflection); while in the latter T always changes smoothly with a wave nature. Here we report a systematic study of the quantum tunneling property for a bright soliton, which behaves as a classical particle (wave) in the limit of small (large) incident velocity. In the intermediate regime, the classical and quantum properties are combined via a finite (but not full) discontinuity in the tunneling transmission coefficient. We demonstrate that the formation of a localized bound state is essential to describe such inelastic collisions, showing a nontrivial nonlinear effect on the quantum transportation of a bright soliton.

© 2012 OSA

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  1. T. Paul, P. Schlagheck, P. Leboeuf, and N. Pavloff, “Superfluidity versus Anderson Localization in a dilute Bose gas,” Phys. Rev. Lett. 98, 210602(1–4) (2007).
    [CrossRef]
  2. L. Fallani, J. E. Lye, V. Guarrera, C. Fort, and M. Inguscio, “Ultracold atoms in a disordered crystal of light: Towards a Bose glass,” Phys. Rev. Lett. 98, 130404(1–4) (2007).
    [CrossRef] [PubMed]
  3. C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, 2008).
    [CrossRef]
  4. C. Fort, L. Fallani, V. Guarrera, J. E. Lye, M. Modugno, D. S. Wiersma, and M. Inguscio, “Effect of optical disorder and single defects on the expansion of a Bose-Einstein condensate in a one-dimensional waveguide,” Phys. Rev. Lett. 95, 170410(1–4) (2005).
    [CrossRef] [PubMed]
  5. Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003).
  6. L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science 296, 1290–1293 (2002).
    [CrossRef] [PubMed]
  7. D. J. Griffiths, Introduction to Quantum Mechanics (Pearson Prentice Hall, 2005).
  8. Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
    [CrossRef]
  9. H. Sakaguchi and M. Tamura, “Scattering of solitons and dark solitons by potential walls in the nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 74, 292–298 (2005).
    [CrossRef]
  10. C. Lee and J. Brand, “Enhanced quantum reflection of matter-wave solitons,” Europhys. Lett. 73, 321–327 (2006).
    [CrossRef]
  11. T. Ernst and J. Brand, “Resonant trapping in the transport of a matter-wave soliton through a quantum well,” Phys. Rev. A 81, 033614 (2010).
    [CrossRef]
  12. J. A. González, A. Bellorín, and L. E. Guerrero, “Soliton tunneling with sub-barrier kinetic energies,” Phys. Rev E 60, R37–R40 (2009).
    [CrossRef]
  13. G. Kälbermann, “Soliton interacting as a particle,” Phys. Lett. A 252, 37–42 (1999).
    [CrossRef]
  14. G. Kälbermann, “Soliton tunneling,” Phys. Rev. E 55, R6360–R6362 (1999).
  15. A. Barak, O. Peleg, C. Stucchio, A. Soffer, and M. Segev, “Observation of soliton tunneling phenomena and soliton ejection,” Phys. Rev. Lett. 100, 153901(1–4) (2008).
    [CrossRef]
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    [CrossRef]
  17. B. Gertjerenken, T. P. Billam, L. Khaykovich, and C. Weiss, “Scattering bright solitons: quantum versus mean-field behavior,” arXiv:1208.2941v2 (2012).
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    [CrossRef]
  19. S. Burtsev, D. J. Kaup, and B. A. Malomed, “Interaction of solitons with a strong inhomogeneity in a nonlinear optical fiber,” Phys. Rev. E 52, 4474–4481 (1995).
    [CrossRef]
  20. R. H. Goodman, P. J. Holmes, and M. I. Weinstein, “Strong NLS soliton defect interactions,” Physica D 192, 215–248 (2004).
    [CrossRef]
  21. Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, “Nonlinear scattering and trapping by local photonic potentials,” Phys. Rev. Lett. 99, 133901(1–4) (2007).
    [CrossRef]
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    [CrossRef]
  24. J. Holmer, J. Marzuola, and M. Zworski, “Fast soliton scattering by delta impurities,” Comm. Math. Phys. 274, 187–216 (2007)
    [CrossRef]
  25. J. Holmer and M. Zworski, “Slow soliton interaction with delta impurities,” J. Modern Dynamics 1, 689–718 (2007).
    [CrossRef]
  26. V. Hakim, “Nonlinear Schrödinger flow past an obstacle in one dimension,” Phys. Rev. E 55, 2835–2845 (1997).
    [CrossRef]
  27. N. Pavloff, “Breakdown of superfluidity of an atom laser past an obstacle,” Phys. Rev. A 66, 013610(1–8) (2002).
    [CrossRef]
  28. B. Seaman, L. D. Carr, and M. J. Holland, “Effect of a potential step or impurity on the Bose-Einstein condensate mean field”, Phys. Rev. A 71, 033609(1–10) (2005).
    [CrossRef]
  29. D. Witthaut, S. Mossmann, and H. J. Korsch, “Bound and resonance states of the nonlinear Schrödinger equation in simple model systems,” J. Phys. A: Math. Gen. 38, 1777–1792 (2005).
    [CrossRef]

2012 (1)

J. L. Helm, T. P. Billam, and S. A. Gardine, “Bright matter-wave soliton collisions at narrow barriers,” Phys. Rev. A 85, 053621 (2012).
[CrossRef]

2011 (1)

2010 (1)

T. Ernst and J. Brand, “Resonant trapping in the transport of a matter-wave soliton through a quantum well,” Phys. Rev. A 81, 033614 (2010).
[CrossRef]

2009 (1)

J. A. González, A. Bellorín, and L. E. Guerrero, “Soliton tunneling with sub-barrier kinetic energies,” Phys. Rev E 60, R37–R40 (2009).
[CrossRef]

2008 (1)

A. Barak, O. Peleg, C. Stucchio, A. Soffer, and M. Segev, “Observation of soliton tunneling phenomena and soliton ejection,” Phys. Rev. Lett. 100, 153901(1–4) (2008).
[CrossRef]

2007 (5)

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, “Nonlinear scattering and trapping by local photonic potentials,” Phys. Rev. Lett. 99, 133901(1–4) (2007).
[CrossRef]

T. Paul, P. Schlagheck, P. Leboeuf, and N. Pavloff, “Superfluidity versus Anderson Localization in a dilute Bose gas,” Phys. Rev. Lett. 98, 210602(1–4) (2007).
[CrossRef]

L. Fallani, J. E. Lye, V. Guarrera, C. Fort, and M. Inguscio, “Ultracold atoms in a disordered crystal of light: Towards a Bose glass,” Phys. Rev. Lett. 98, 130404(1–4) (2007).
[CrossRef] [PubMed]

J. Holmer, J. Marzuola, and M. Zworski, “Fast soliton scattering by delta impurities,” Comm. Math. Phys. 274, 187–216 (2007)
[CrossRef]

J. Holmer and M. Zworski, “Slow soliton interaction with delta impurities,” J. Modern Dynamics 1, 689–718 (2007).
[CrossRef]

2006 (1)

C. Lee and J. Brand, “Enhanced quantum reflection of matter-wave solitons,” Europhys. Lett. 73, 321–327 (2006).
[CrossRef]

2005 (4)

H. Sakaguchi and M. Tamura, “Scattering of solitons and dark solitons by potential walls in the nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 74, 292–298 (2005).
[CrossRef]

C. Fort, L. Fallani, V. Guarrera, J. E. Lye, M. Modugno, D. S. Wiersma, and M. Inguscio, “Effect of optical disorder and single defects on the expansion of a Bose-Einstein condensate in a one-dimensional waveguide,” Phys. Rev. Lett. 95, 170410(1–4) (2005).
[CrossRef] [PubMed]

B. Seaman, L. D. Carr, and M. J. Holland, “Effect of a potential step or impurity on the Bose-Einstein condensate mean field”, Phys. Rev. A 71, 033609(1–10) (2005).
[CrossRef]

D. Witthaut, S. Mossmann, and H. J. Korsch, “Bound and resonance states of the nonlinear Schrödinger equation in simple model systems,” J. Phys. A: Math. Gen. 38, 1777–1792 (2005).
[CrossRef]

2004 (1)

R. H. Goodman, P. J. Holmes, and M. I. Weinstein, “Strong NLS soliton defect interactions,” Physica D 192, 215–248 (2004).
[CrossRef]

2002 (2)

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science 296, 1290–1293 (2002).
[CrossRef] [PubMed]

N. Pavloff, “Breakdown of superfluidity of an atom laser past an obstacle,” Phys. Rev. A 66, 013610(1–8) (2002).
[CrossRef]

1999 (3)

G. Kälbermann, “Soliton interacting as a particle,” Phys. Lett. A 252, 37–42 (1999).
[CrossRef]

G. Kälbermann, “Soliton tunneling,” Phys. Rev. E 55, R6360–R6362 (1999).

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

1997 (1)

V. Hakim, “Nonlinear Schrödinger flow past an obstacle in one dimension,” Phys. Rev. E 55, 2835–2845 (1997).
[CrossRef]

1995 (2)

X. D. Cao and B. A. Malomed, “Soliton-defect collisions in the nonlinear Schrödinger equation,” Phys. Lett. A 206, 177–182 (1995).
[CrossRef]

S. Burtsev, D. J. Kaup, and B. A. Malomed, “Interaction of solitons with a strong inhomogeneity in a nonlinear optical fiber,” Phys. Rev. E 52, 4474–4481 (1995).
[CrossRef]

1989 (1)

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003).

Aimez, V.

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, “Nonlinear scattering and trapping by local photonic potentials,” Phys. Rev. Lett. 99, 133901(1–4) (2007).
[CrossRef]

Alberucci, A.

Ares, R.

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, “Nonlinear scattering and trapping by local photonic potentials,” Phys. Rev. Lett. 99, 133901(1–4) (2007).
[CrossRef]

Assanto, G.

Bar-Ad, S.

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, “Nonlinear scattering and trapping by local photonic potentials,” Phys. Rev. Lett. 99, 133901(1–4) (2007).
[CrossRef]

Barak, A.

A. Barak, O. Peleg, C. Stucchio, A. Soffer, and M. Segev, “Observation of soliton tunneling phenomena and soliton ejection,” Phys. Rev. Lett. 100, 153901(1–4) (2008).
[CrossRef]

Bellorín, A.

J. A. González, A. Bellorín, and L. E. Guerrero, “Soliton tunneling with sub-barrier kinetic energies,” Phys. Rev E 60, R37–R40 (2009).
[CrossRef]

Billam, T. P.

J. L. Helm, T. P. Billam, and S. A. Gardine, “Bright matter-wave soliton collisions at narrow barriers,” Phys. Rev. A 85, 053621 (2012).
[CrossRef]

B. Gertjerenken, T. P. Billam, L. Khaykovich, and C. Weiss, “Scattering bright solitons: quantum versus mean-field behavior,” arXiv:1208.2941v2 (2012).

Bourdel, T.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science 296, 1290–1293 (2002).
[CrossRef] [PubMed]

Brand, J.

T. Ernst and J. Brand, “Resonant trapping in the transport of a matter-wave soliton through a quantum well,” Phys. Rev. A 81, 033614 (2010).
[CrossRef]

C. Lee and J. Brand, “Enhanced quantum reflection of matter-wave solitons,” Europhys. Lett. 73, 321–327 (2006).
[CrossRef]

Burtsev, S.

S. Burtsev, D. J. Kaup, and B. A. Malomed, “Interaction of solitons with a strong inhomogeneity in a nonlinear optical fiber,” Phys. Rev. E 52, 4474–4481 (1995).
[CrossRef]

Cao, X. D.

X. D. Cao and B. A. Malomed, “Soliton-defect collisions in the nonlinear Schrödinger equation,” Phys. Lett. A 206, 177–182 (1995).
[CrossRef]

Carr, L. D.

B. Seaman, L. D. Carr, and M. J. Holland, “Effect of a potential step or impurity on the Bose-Einstein condensate mean field”, Phys. Rev. A 71, 033609(1–10) (2005).
[CrossRef]

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science 296, 1290–1293 (2002).
[CrossRef] [PubMed]

Castin, Y.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science 296, 1290–1293 (2002).
[CrossRef] [PubMed]

Cubizolles, J.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science 296, 1290–1293 (2002).
[CrossRef] [PubMed]

Dalfovo, F.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

Ernst, T.

T. Ernst and J. Brand, “Resonant trapping in the transport of a matter-wave soliton through a quantum well,” Phys. Rev. A 81, 033614 (2010).
[CrossRef]

Fallani, L.

L. Fallani, J. E. Lye, V. Guarrera, C. Fort, and M. Inguscio, “Ultracold atoms in a disordered crystal of light: Towards a Bose glass,” Phys. Rev. Lett. 98, 130404(1–4) (2007).
[CrossRef] [PubMed]

C. Fort, L. Fallani, V. Guarrera, J. E. Lye, M. Modugno, D. S. Wiersma, and M. Inguscio, “Effect of optical disorder and single defects on the expansion of a Bose-Einstein condensate in a one-dimensional waveguide,” Phys. Rev. Lett. 95, 170410(1–4) (2005).
[CrossRef] [PubMed]

Ferrari, G.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science 296, 1290–1293 (2002).
[CrossRef] [PubMed]

Fort, C.

L. Fallani, J. E. Lye, V. Guarrera, C. Fort, and M. Inguscio, “Ultracold atoms in a disordered crystal of light: Towards a Bose glass,” Phys. Rev. Lett. 98, 130404(1–4) (2007).
[CrossRef] [PubMed]

C. Fort, L. Fallani, V. Guarrera, J. E. Lye, M. Modugno, D. S. Wiersma, and M. Inguscio, “Effect of optical disorder and single defects on the expansion of a Bose-Einstein condensate in a one-dimensional waveguide,” Phys. Rev. Lett. 95, 170410(1–4) (2005).
[CrossRef] [PubMed]

Gardine, S. A.

J. L. Helm, T. P. Billam, and S. A. Gardine, “Bright matter-wave soliton collisions at narrow barriers,” Phys. Rev. A 85, 053621 (2012).
[CrossRef]

Gertjerenken, B.

B. Gertjerenken, T. P. Billam, L. Khaykovich, and C. Weiss, “Scattering bright solitons: quantum versus mean-field behavior,” arXiv:1208.2941v2 (2012).

Giorgini, S.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

González, J. A.

J. A. González, A. Bellorín, and L. E. Guerrero, “Soliton tunneling with sub-barrier kinetic energies,” Phys. Rev E 60, R37–R40 (2009).
[CrossRef]

Goodman, R. H.

R. H. Goodman, P. J. Holmes, and M. I. Weinstein, “Strong NLS soliton defect interactions,” Physica D 192, 215–248 (2004).
[CrossRef]

Griffiths, D. J.

D. J. Griffiths, Introduction to Quantum Mechanics (Pearson Prentice Hall, 2005).

Guarrera, V.

L. Fallani, J. E. Lye, V. Guarrera, C. Fort, and M. Inguscio, “Ultracold atoms in a disordered crystal of light: Towards a Bose glass,” Phys. Rev. Lett. 98, 130404(1–4) (2007).
[CrossRef] [PubMed]

C. Fort, L. Fallani, V. Guarrera, J. E. Lye, M. Modugno, D. S. Wiersma, and M. Inguscio, “Effect of optical disorder and single defects on the expansion of a Bose-Einstein condensate in a one-dimensional waveguide,” Phys. Rev. Lett. 95, 170410(1–4) (2005).
[CrossRef] [PubMed]

Guerrero, L. E.

J. A. González, A. Bellorín, and L. E. Guerrero, “Soliton tunneling with sub-barrier kinetic energies,” Phys. Rev E 60, R37–R40 (2009).
[CrossRef]

Hakim, V.

V. Hakim, “Nonlinear Schrödinger flow past an obstacle in one dimension,” Phys. Rev. E 55, 2835–2845 (1997).
[CrossRef]

Helm, J. L.

J. L. Helm, T. P. Billam, and S. A. Gardine, “Bright matter-wave soliton collisions at narrow barriers,” Phys. Rev. A 85, 053621 (2012).
[CrossRef]

Holland, M. J.

B. Seaman, L. D. Carr, and M. J. Holland, “Effect of a potential step or impurity on the Bose-Einstein condensate mean field”, Phys. Rev. A 71, 033609(1–10) (2005).
[CrossRef]

Holmer, J.

J. Holmer and M. Zworski, “Slow soliton interaction with delta impurities,” J. Modern Dynamics 1, 689–718 (2007).
[CrossRef]

J. Holmer, J. Marzuola, and M. Zworski, “Fast soliton scattering by delta impurities,” Comm. Math. Phys. 274, 187–216 (2007)
[CrossRef]

Holmes, P. J.

R. H. Goodman, P. J. Holmes, and M. I. Weinstein, “Strong NLS soliton defect interactions,” Physica D 192, 215–248 (2004).
[CrossRef]

Inguscio, M.

L. Fallani, J. E. Lye, V. Guarrera, C. Fort, and M. Inguscio, “Ultracold atoms in a disordered crystal of light: Towards a Bose glass,” Phys. Rev. Lett. 98, 130404(1–4) (2007).
[CrossRef] [PubMed]

C. Fort, L. Fallani, V. Guarrera, J. E. Lye, M. Modugno, D. S. Wiersma, and M. Inguscio, “Effect of optical disorder and single defects on the expansion of a Bose-Einstein condensate in a one-dimensional waveguide,” Phys. Rev. Lett. 95, 170410(1–4) (2005).
[CrossRef] [PubMed]

Jisha, C. P.

Kälbermann, G.

G. Kälbermann, “Soliton tunneling,” Phys. Rev. E 55, R6360–R6362 (1999).

G. Kälbermann, “Soliton interacting as a particle,” Phys. Lett. A 252, 37–42 (1999).
[CrossRef]

Kaup, D. J.

S. Burtsev, D. J. Kaup, and B. A. Malomed, “Interaction of solitons with a strong inhomogeneity in a nonlinear optical fiber,” Phys. Rev. E 52, 4474–4481 (1995).
[CrossRef]

Khaykovich, L.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science 296, 1290–1293 (2002).
[CrossRef] [PubMed]

B. Gertjerenken, T. P. Billam, L. Khaykovich, and C. Weiss, “Scattering bright solitons: quantum versus mean-field behavior,” arXiv:1208.2941v2 (2012).

Kivshar, Y. S.

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003).

Korsch, H. J.

D. Witthaut, S. Mossmann, and H. J. Korsch, “Bound and resonance states of the nonlinear Schrödinger equation in simple model systems,” J. Phys. A: Math. Gen. 38, 1777–1792 (2005).
[CrossRef]

Leboeuf, P.

T. Paul, P. Schlagheck, P. Leboeuf, and N. Pavloff, “Superfluidity versus Anderson Localization in a dilute Bose gas,” Phys. Rev. Lett. 98, 210602(1–4) (2007).
[CrossRef]

Lee, C.

C. Lee and J. Brand, “Enhanced quantum reflection of matter-wave solitons,” Europhys. Lett. 73, 321–327 (2006).
[CrossRef]

Lee, R.-K.

Linzon, Y.

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, “Nonlinear scattering and trapping by local photonic potentials,” Phys. Rev. Lett. 99, 133901(1–4) (2007).
[CrossRef]

Lye, J. E.

L. Fallani, J. E. Lye, V. Guarrera, C. Fort, and M. Inguscio, “Ultracold atoms in a disordered crystal of light: Towards a Bose glass,” Phys. Rev. Lett. 98, 130404(1–4) (2007).
[CrossRef] [PubMed]

C. Fort, L. Fallani, V. Guarrera, J. E. Lye, M. Modugno, D. S. Wiersma, and M. Inguscio, “Effect of optical disorder and single defects on the expansion of a Bose-Einstein condensate in a one-dimensional waveguide,” Phys. Rev. Lett. 95, 170410(1–4) (2005).
[CrossRef] [PubMed]

Malomed, B. A.

X. D. Cao and B. A. Malomed, “Soliton-defect collisions in the nonlinear Schrödinger equation,” Phys. Lett. A 206, 177–182 (1995).
[CrossRef]

S. Burtsev, D. J. Kaup, and B. A. Malomed, “Interaction of solitons with a strong inhomogeneity in a nonlinear optical fiber,” Phys. Rev. E 52, 4474–4481 (1995).
[CrossRef]

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

Marzuola, J.

J. Holmer, J. Marzuola, and M. Zworski, “Fast soliton scattering by delta impurities,” Comm. Math. Phys. 274, 187–216 (2007)
[CrossRef]

Modugno, M.

C. Fort, L. Fallani, V. Guarrera, J. E. Lye, M. Modugno, D. S. Wiersma, and M. Inguscio, “Effect of optical disorder and single defects on the expansion of a Bose-Einstein condensate in a one-dimensional waveguide,” Phys. Rev. Lett. 95, 170410(1–4) (2005).
[CrossRef] [PubMed]

Morandotti, R.

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, “Nonlinear scattering and trapping by local photonic potentials,” Phys. Rev. Lett. 99, 133901(1–4) (2007).
[CrossRef]

Mossmann, S.

D. Witthaut, S. Mossmann, and H. J. Korsch, “Bound and resonance states of the nonlinear Schrödinger equation in simple model systems,” J. Phys. A: Math. Gen. 38, 1777–1792 (2005).
[CrossRef]

Paul, T.

T. Paul, P. Schlagheck, P. Leboeuf, and N. Pavloff, “Superfluidity versus Anderson Localization in a dilute Bose gas,” Phys. Rev. Lett. 98, 210602(1–4) (2007).
[CrossRef]

Pavloff, N.

T. Paul, P. Schlagheck, P. Leboeuf, and N. Pavloff, “Superfluidity versus Anderson Localization in a dilute Bose gas,” Phys. Rev. Lett. 98, 210602(1–4) (2007).
[CrossRef]

N. Pavloff, “Breakdown of superfluidity of an atom laser past an obstacle,” Phys. Rev. A 66, 013610(1–8) (2002).
[CrossRef]

Peleg, O.

A. Barak, O. Peleg, C. Stucchio, A. Soffer, and M. Segev, “Observation of soliton tunneling phenomena and soliton ejection,” Phys. Rev. Lett. 100, 153901(1–4) (2008).
[CrossRef]

Pethick, C. J.

C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, 2008).
[CrossRef]

Pitaevskii, L. P.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

Sakaguchi, H.

H. Sakaguchi and M. Tamura, “Scattering of solitons and dark solitons by potential walls in the nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 74, 292–298 (2005).
[CrossRef]

Salomon, C.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science 296, 1290–1293 (2002).
[CrossRef] [PubMed]

Schlagheck, P.

T. Paul, P. Schlagheck, P. Leboeuf, and N. Pavloff, “Superfluidity versus Anderson Localization in a dilute Bose gas,” Phys. Rev. Lett. 98, 210602(1–4) (2007).
[CrossRef]

Schreck, F.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science 296, 1290–1293 (2002).
[CrossRef] [PubMed]

Seaman, B.

B. Seaman, L. D. Carr, and M. J. Holland, “Effect of a potential step or impurity on the Bose-Einstein condensate mean field”, Phys. Rev. A 71, 033609(1–10) (2005).
[CrossRef]

Segev, M.

A. Barak, O. Peleg, C. Stucchio, A. Soffer, and M. Segev, “Observation of soliton tunneling phenomena and soliton ejection,” Phys. Rev. Lett. 100, 153901(1–4) (2008).
[CrossRef]

Smith, H.

C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, 2008).
[CrossRef]

Soffer, A.

A. Barak, O. Peleg, C. Stucchio, A. Soffer, and M. Segev, “Observation of soliton tunneling phenomena and soliton ejection,” Phys. Rev. Lett. 100, 153901(1–4) (2008).
[CrossRef]

Stringari, S.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

Stucchio, C.

A. Barak, O. Peleg, C. Stucchio, A. Soffer, and M. Segev, “Observation of soliton tunneling phenomena and soliton ejection,” Phys. Rev. Lett. 100, 153901(1–4) (2008).
[CrossRef]

Tamura, M.

H. Sakaguchi and M. Tamura, “Scattering of solitons and dark solitons by potential walls in the nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 74, 292–298 (2005).
[CrossRef]

Volatier, M.

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, “Nonlinear scattering and trapping by local photonic potentials,” Phys. Rev. Lett. 99, 133901(1–4) (2007).
[CrossRef]

Weinstein, M. I.

R. H. Goodman, P. J. Holmes, and M. I. Weinstein, “Strong NLS soliton defect interactions,” Physica D 192, 215–248 (2004).
[CrossRef]

Weiss, C.

B. Gertjerenken, T. P. Billam, L. Khaykovich, and C. Weiss, “Scattering bright solitons: quantum versus mean-field behavior,” arXiv:1208.2941v2 (2012).

Wiersma, D. S.

C. Fort, L. Fallani, V. Guarrera, J. E. Lye, M. Modugno, D. S. Wiersma, and M. Inguscio, “Effect of optical disorder and single defects on the expansion of a Bose-Einstein condensate in a one-dimensional waveguide,” Phys. Rev. Lett. 95, 170410(1–4) (2005).
[CrossRef] [PubMed]

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D. Witthaut, S. Mossmann, and H. J. Korsch, “Bound and resonance states of the nonlinear Schrödinger equation in simple model systems,” J. Phys. A: Math. Gen. 38, 1777–1792 (2005).
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J. Holmer, J. Marzuola, and M. Zworski, “Fast soliton scattering by delta impurities,” Comm. Math. Phys. 274, 187–216 (2007)
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J. Holmer and M. Zworski, “Slow soliton interaction with delta impurities,” J. Modern Dynamics 1, 689–718 (2007).
[CrossRef]

Comm. Math. Phys. (1)

J. Holmer, J. Marzuola, and M. Zworski, “Fast soliton scattering by delta impurities,” Comm. Math. Phys. 274, 187–216 (2007)
[CrossRef]

Europhys. Lett. (1)

C. Lee and J. Brand, “Enhanced quantum reflection of matter-wave solitons,” Europhys. Lett. 73, 321–327 (2006).
[CrossRef]

J. Modern Dynamics (1)

J. Holmer and M. Zworski, “Slow soliton interaction with delta impurities,” J. Modern Dynamics 1, 689–718 (2007).
[CrossRef]

J. Phys. A: Math. Gen. (1)

D. Witthaut, S. Mossmann, and H. J. Korsch, “Bound and resonance states of the nonlinear Schrödinger equation in simple model systems,” J. Phys. A: Math. Gen. 38, 1777–1792 (2005).
[CrossRef]

J. Phys. Soc. Jpn. (1)

H. Sakaguchi and M. Tamura, “Scattering of solitons and dark solitons by potential walls in the nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 74, 292–298 (2005).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (2)

G. Kälbermann, “Soliton interacting as a particle,” Phys. Lett. A 252, 37–42 (1999).
[CrossRef]

X. D. Cao and B. A. Malomed, “Soliton-defect collisions in the nonlinear Schrödinger equation,” Phys. Lett. A 206, 177–182 (1995).
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Phys. Rev E (1)

J. A. González, A. Bellorín, and L. E. Guerrero, “Soliton tunneling with sub-barrier kinetic energies,” Phys. Rev E 60, R37–R40 (2009).
[CrossRef]

Phys. Rev. A (4)

N. Pavloff, “Breakdown of superfluidity of an atom laser past an obstacle,” Phys. Rev. A 66, 013610(1–8) (2002).
[CrossRef]

B. Seaman, L. D. Carr, and M. J. Holland, “Effect of a potential step or impurity on the Bose-Einstein condensate mean field”, Phys. Rev. A 71, 033609(1–10) (2005).
[CrossRef]

J. L. Helm, T. P. Billam, and S. A. Gardine, “Bright matter-wave soliton collisions at narrow barriers,” Phys. Rev. A 85, 053621 (2012).
[CrossRef]

T. Ernst and J. Brand, “Resonant trapping in the transport of a matter-wave soliton through a quantum well,” Phys. Rev. A 81, 033614 (2010).
[CrossRef]

Phys. Rev. E (3)

G. Kälbermann, “Soliton tunneling,” Phys. Rev. E 55, R6360–R6362 (1999).

S. Burtsev, D. J. Kaup, and B. A. Malomed, “Interaction of solitons with a strong inhomogeneity in a nonlinear optical fiber,” Phys. Rev. E 52, 4474–4481 (1995).
[CrossRef]

V. Hakim, “Nonlinear Schrödinger flow past an obstacle in one dimension,” Phys. Rev. E 55, 2835–2845 (1997).
[CrossRef]

Phys. Rev. Lett. (5)

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, “Nonlinear scattering and trapping by local photonic potentials,” Phys. Rev. Lett. 99, 133901(1–4) (2007).
[CrossRef]

A. Barak, O. Peleg, C. Stucchio, A. Soffer, and M. Segev, “Observation of soliton tunneling phenomena and soliton ejection,” Phys. Rev. Lett. 100, 153901(1–4) (2008).
[CrossRef]

T. Paul, P. Schlagheck, P. Leboeuf, and N. Pavloff, “Superfluidity versus Anderson Localization in a dilute Bose gas,” Phys. Rev. Lett. 98, 210602(1–4) (2007).
[CrossRef]

L. Fallani, J. E. Lye, V. Guarrera, C. Fort, and M. Inguscio, “Ultracold atoms in a disordered crystal of light: Towards a Bose glass,” Phys. Rev. Lett. 98, 130404(1–4) (2007).
[CrossRef] [PubMed]

C. Fort, L. Fallani, V. Guarrera, J. E. Lye, M. Modugno, D. S. Wiersma, and M. Inguscio, “Effect of optical disorder and single defects on the expansion of a Bose-Einstein condensate in a one-dimensional waveguide,” Phys. Rev. Lett. 95, 170410(1–4) (2005).
[CrossRef] [PubMed]

Physica D (1)

R. H. Goodman, P. J. Holmes, and M. I. Weinstein, “Strong NLS soliton defect interactions,” Physica D 192, 215–248 (2004).
[CrossRef]

Rev. Mod. Phys. (2)

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999).
[CrossRef]

Science (1)

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton,” Science 296, 1290–1293 (2002).
[CrossRef] [PubMed]

Other (4)

D. J. Griffiths, Introduction to Quantum Mechanics (Pearson Prentice Hall, 2005).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003).

C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, 2008).
[CrossRef]

B. Gertjerenken, T. P. Billam, L. Khaykovich, and C. Weiss, “Scattering bright solitons: quantum versus mean-field behavior,” arXiv:1208.2941v2 (2012).

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Figures (3)

Fig. 1
Fig. 1

Schematic plots for different tunneling dynamics through a local potential, V(x). The incident, transmission, and reflection velocities are denoted as vi, vt, and vr, respectively. (a) Classical picture with either a total transmission (T = 1) or total reflection (T = 0). (b) Quantum mechanical picture with a partial transmission (0 < T < 1). The notations, Ψi, Ψt, and Ψr represent the incident, transmitted, and reflected wavefunctions, respectively. (c) Inelastic scattering process of a BS through a potential well, V(x) < 0, where a localized bound state, Ψb, appears after scattering.

Fig. 2
Fig. 2

Contour plot of the transmission coefficient as a function of incident soliton velocity vi and potential strength |V0|. Cases of potential barrier V0 > 0 and potential well V0 < 0 are separately shown in (a) and (b). White dashed lines are plotted from Eq. (4) and Eq. (9) for the corresponding analytical results (see the text).

Fig. 3
Fig. 3

A comparison of transmission coefficient between numerical simulations (solid lines) and analytical results from Eq. (9) (dashed points). Note that the parameters are the same as those used in Fig. 2(b).

Equations (10)

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

[ 1 2 2 x 2 + g | Ψ ( x , t ) | 2 + V ( x ) ] Ψ ( x , t ) = i t Ψ ( x , t ) ,
Ψ bs ( x , t ) = β | g | sech [ β ( x x c v i t ) ] e i θ ( x , t ) ,
T ( v i / V 0 ) 2 1 + ( v i / V 0 ) 2 + O ( 1 v i 2 ) .
v ( t ) 2 + β 2 | g | 2 V 0 sech 2 [ x c ( t ) ] = v i 2 + β 2 | g | 2 V 0 sech 2 ( x i ) ,
Ψ b ( x ) = β b | g | sech ( β b | x | + x b ) ,
2 β i | g | = 1 = 2 | g | ( β b | V 0 | ) + T ,
β i 2 2 + v i 2 2 = β b 2 2 ( 1 T ) + ( β t 2 2 + v t 2 2 ) T ,
v i = 2 β t | g | v t = T v t
T = V ˜ 0 ( V ˜ 0 + 1 ) V ˜ 0 2 ( V ˜ 0 + 1 ) 2 ( 4 v ˜ i 2 V ˜ 0 + 3 v ˜ i 2 ) 2 V ˜ 0 + 3 / 2 ,
v c | g | | V 0 / g | ( | V 0 / g | + 1 ) 3 + 4 | V 0 / g | ,

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