Annales Henri Poincaré

, Volume 17, Issue 12, pp 3321–3360 | Cite as

The NLS Limit for Bosons in a Quantum Waveguide



We consider a system of N bosons confined to a thin waveguide, i.e. to a region of space within an \({\epsilon}\)-tube around a curve in \({\mathbb{R}^3}\). We show that when taking simultaneously the NLS limit \({N \to \infty}\) and the limit of strong confinement \({\epsilon \to 0}\), the time-evolution of such a system starting in a state close to a Bose–Einstein condensate is approximately captured by a non-linear Schrödinger equation in one dimension. The strength of the non-linearity in this Gross–Pitaevskii type equation depends on the shape of the cross-section of the waveguide, while the “bending” and the “twisting” of the waveguide contribute potential terms. Our analysis is based on an approach to mean-field limits developed by Pickl (On the time-dependent Gross–Pitaevskii-and Hartree equation. arXiv:0808.1178, 2008).


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  1. 1.
    Adami R., Golse F., Teta A.: Rigorous derivation of the cubic NLS in dimension one. J. Stat. Phys. 127(6), 1193–1220 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ben Abdallah N., Méhats F., Schmeiser C., Weishäupl R.: The nonlinear Schrödinger equation with a strongly anisotropic harmonic potential. SIAM J. Math. Anal. 37(1), 189–199 (2005)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Benedikter N., De Oliveira G., Schlein B.: Quantitative derivation of the Gross–Pitaevskii equation. Commun. Pure Appl. Math. 68(8), 1399–1482 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Benedikter, N., Porta, M., Schlein, B.: Effective evolution equations from quantum dynamics (2015). arXiv:1502.02498
  5. 5.
    Chen X., Holmer J.: On the rigorous derivation of the 2d cubic nonlinear Schrödinger equation from 3d quantum many-body dynamics. Arch. Ration. Mech. Anal. 210(3), 909–954 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen, X., Holmer, J.: Focusing quantum many-body dynamics II: the rigorous derivation of the 1d focusing cubic nonlinear Schrödinger equation from 3d (2014). arXiv:1407.8457
  7. 7.
    Erdős L., Schlein B., Yau H.-T.: Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems. Invent. Math. 167, 515–614 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fortágh J., Zimmermann C.: Magnetic microtraps for ultracold atoms. Rev. Mod. Phys. 79(1), 235–289 (2007)ADSCrossRefGoogle Scholar
  9. 9.
    Golse, F.: On the dynamics of large particle systems in the mean field limit (2013). arXiv:1301.5494
  10. 10.
    Görlitz A., Vogels J.M., Leanhardt A.E., Raman C., Gustavson T.L., Abo-Shaeer J.R., Chikkatur A.P., Gupta S., Inouye S., Rosenband T., Ketterle W.: Realization of Bose–Einstein condensates in lower dimensions. Phys. Rev. Lett. 87, 130402 (2001)CrossRefGoogle Scholar
  11. 11.
    Grillakis M., Machedon M.: Pair excitations and the mean field approximation of interacting Bosons. Commun. Math. Phys. 324, 601–636 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Haag S., Lampart J., Teufel S.: Generalised quantum waveguides. Ann. Henri Poincaré 16, 2535–2568 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Henderson K., Ryu C., MacCormick C., Boshier M.G.: Experimental demonstration of painting arbitrary and dynamic potentials for Bose–Einstein condensates. New J. Phys. 11(4), 043030 (2009)ADSCrossRefGoogle Scholar
  14. 14.
    Knowles A., Pickl P.: Mean-field dynamics: singular potentials and rate of convergence. Commun. Math. Phys. 298(1), 101–138 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Krejčiřík, D.: Twisting versus bending in quantum waveguides. In: Analysis on Graphs and its Applications: Proceedings of the Symposium on Pure Mathematics, pp. 617–636. American Mathematical Society, Providence (2008)Google Scholar
  16. 16.
    Lampart, J., Teufel, S.: The adiabatic limit of Schrödinger operators on fibre bundles. Math. Ann. (2014). arXiv:1402.0382
  17. 17.
    Lewin M., Nam P.T., Rougerie N.: The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases. Trans. Am. Math. Soc. 368, 6131–6157 (2016)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lieb E.H., Seiringer R., Solovej J.P., Yngvason J.: The mathematics of the Bose gas and its condensation. In: Oberwolfach Seminars, vol. 34. Birkhäuser, Boston (2005)MATHGoogle Scholar
  19. 19.
    Lieb E.H., Seiringer R., Yngvason J.: One-dimensional behaviour of dilute, trapped Bose gases. Commun. Math. Phys. 244(2), 347–393 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Méhats, F., Raymond, N.: Strong confinement limit for the nonlinear Schrödinger equation constrained on a curve (2014). arXiv:1412.1049
  21. 21.
    Nam, P.T., Rougerie, N., Seiringer, R.: Ground states of large bosonic systems: the Gross–Pitaevskii limit revisited (2015). arXiv:1503.07061
  22. 22.
    Nam, P.T., Napiórkowski, M.: Bogoliubov correction to the mean-field dynamics of interacting bosons (2015). arXiv:1509.04631
  23. 23.
    Pickl, P.: On the time dependent Gross–Pitaevskii-and Hartree equation (2008). arXiv:0808.1178
  24. 24.
    Pickl, P.: Derivation of the time dependent Gross–Pitaevskii equation with external fields. Rev. Math. Phys. 27, 1550003 (2015). arXiv:1001.4894
  25. 25.
    Pickl P.: Derivation of the time dependent Gross–Pitaevskii equation without positivity condition on the interaction. J. Stat. Phys. 140(1), 76–89 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Pickl P.: A simple derivation of mean field limits for quantum systems. Lett. Math. Phys. 97(2), 151–164 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Rodnianski I., Schlein B.: Quantum fluctuations and rate of convergence towards mean field dynamics. Commun. Math. Phys. 291(1), 31–61 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Rougerie, N.: De finetti theorems, mean-field limits and Bose–Einstein condensation (2015). arXiv:1506.05263
  29. 29.
    Schnee K., Yngvason J.: Bosons in disc-shaped traps: from 3D to 2D. Commun. Math. Phys. 269(3), 659–691 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Schlein, B.: Derivation of effective evolution equations from microscopic quantum dynamics (2008). arXiv:0807.4307
  31. 31.
    Sparber C.: Weakly nonlinear time-adiabatic theory. Ann. Henri Poincaré 17(4), 913–936 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Tao, T.: Nonlinear dispersive equations. Local and global analysis. In: CBMS Regional Conference Series in Mathematics, vol. 106. American Mathematical Society (2006)Google Scholar
  33. 33.
    Wachsmuth, J., Teufel, S.: Effective Hamiltonians for constrained quantum systems. Mem. AMS 230(1083) (2014). doi:10.1090/memo/1083

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© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität TübingenTübingenGermany

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