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PROPAGATION OF SEMICLASSICAL WAVE PACKETS THROUGH AVOIDED EIGENVALUE CROSSINGS IN NONLINEAR SCHRÖDINGER EQUATIONS

Part of: Equations of mathematical physics and other areas of application Groups and algebras in quantum theory General mathematical topics and methods in quantum theory

Published online by Cambridge University Press:  20 October 2014

Lysianne Hari
Lysianne Hari*
Affiliation:
University of Cergy-Pontoise, UMR CNRS 8088, F-95000 Cergy-Pontoise, France (Lysianne.Hari@u-cergy.fr)
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Abstract

We study the propagation of wave packets for a one-dimensional system of two coupled Schrödinger equations with a cubic nonlinearity, in the semiclassical limit. Couplings are induced by the nonlinearity and by the potential, whose eigenvalues present an avoided crossing: at one given point, the gap between them reduces as the semiclassical parameter becomes smaller. For data which are coherent states polarized along an eigenvector of the potential, we prove that when the wave function propagates through the avoided crossing point there are transitions between the eigenspaces at leading order. We analyze the nonlinear effects, which are noticeable away from the crossing point, but see that in a small time interval around this point the nonlinearity’s role is negligible at leading order, and the transition probabilities can be computed with the linear Landau–Zener formula.

Keywords

nonlinear Schrödinger equationssemiclassical limitquantum propagation of coherent statesavoided crossing

MSC classification

Primary: 35Q40: PDEs in connection with quantum mechanics
Secondary: 35Q55: NLS-like equations (nonlinear Schrödinger) 81Q05: Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 81R20: Covariant wave equations 81R30: Coherent states
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 15 , Issue 2 , April 2016 , pp. 319 - 365
Copyright
© Cambridge University Press 2014 

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References

Aftalion, A., Blanc, X. and Nier, F., Lowest Landau level functional and Bargmann spaces for Bose–Einstein condensates, J. Funct. Anal. 241(2) (2006), 661–702.CrossRefGoogle Scholar
Athanassoulis, A., Paul, T., Pezzotti, F. and Pulvirenti, M., Semiclassical propagation of coherent states for the Hartree equation, Ann. Henri Poincaré 12(8) (2011), 1613–1634.Google Scholar
Biao, W. and Qian, N., Nonlinear Landau–Zener tunneling, Phys. Rev. A 61 (2000), 023402.Google Scholar
Cao, P. and Carles, R., Semi-classical wave packet dynamics for Hartree equations, Rev. Math. Phys. 23(9) (2011), 933–967.Google Scholar
Carles, R., Nonlinear Schrödinger equation with time dependent potential, Comm. Math. Sci. 9(4) (2011), 937–964.Google Scholar
Carles, R. and Fermanian Kammerer, C., A nonlinear adiabatic theorem for coherent states, Nonlinearity 24(8) (2011), 2143–2164.Google Scholar
Carles, R. and Fermanian Kammerer, C., Nonlinear coherent states and Ehrenfest time for Schrödinger equation, Comm. Math. Phys. 301(2) (2011), 443–472.CrossRefGoogle Scholar
Carles, R. and Fermanian Kammerer, C., A nonlinear Landau–Zener formula, J. Stat. Phys. 152(4) (2013), 619–656.CrossRefGoogle Scholar
Carles, R. and Sparber, C., Nonlinear dynamics of semiclassical coherent states in periodic potentials, J. Phys. A 45 (2012), 244032.Google Scholar
Carles, R. and Sparber, C., Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials, Discrete Contin. Dyn. Syst. Ser. B 17(3) (2012), 759–774.Google Scholar
Colin de Verdière, Y., The level crossing problem in semi-classical analysis. I. The symmetric case, in Proceedings of the International Conference in Honor of Frédéric Pham (Nice, 2002), Volume 53, pp. 1023–1054. (2003).Google Scholar
Colin de Verdière, Y., The level crossing problem in semi-classical analysis. II. The Hermitian case, Ann. Inst. Fourier (Grenoble) 54(5) (2004), 1423–1441, xv, xx–xxi.Google Scholar
Fermanian Kammerer, C., Semi-classical analysis of a Dirac equation without adiabatic decoupling, Monatsh. Math. 142(4) (2004), 281–313.Google Scholar
Fermanian Kammerer, C. and Gérard, P., Mesures semi-classiques et croisement de modes, Bull. Soc. Math. France 130(1) (2002), 123–168.Google Scholar
Fermanian Kammerer, C. and Gérard, P., A Landau–Zener formula for non-degenerated involutive codimension 3 crossings, Ann. Henri Poincaré 4(3) (2003), 513–552.CrossRefGoogle Scholar
Fermanian Kammerer, C. and Lasser, C., Propagation through generic level crossings: a surface hopping semigroup, SIAM J. Math. Anal. 40(1) (2008), 103–133.Google Scholar
Fujiwara, D., A construction of the fundamental solution for the Schrödinger equation, J. Anal. Math. 35 (1979), 41–96.Google Scholar
Fujiwara, D., Remarks on the convergence of the Feynman path integrals, Duke Math. J. 47(3) (1980), 559–600.CrossRefGoogle Scholar
Ginibre, J. and Velo, G., Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133(1) (1995), 50–68.CrossRefGoogle Scholar
Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products, 6th ed. (Academic Press Inc., San Diego, CA, 2000). Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger.Google Scholar
Hagedorn, G. A., Proof of the Landau–Zener formula in an adiabatic limit with small eigenvalue gaps, Comm. Math. Phys. 136(3) (1991), 433–449.Google Scholar
Hagedorn, G. A., Molecular propagation through electron energy level crossings, Mem. Amer. Math. Soc. 111(536) (1994),. vi+130.Google Scholar
Hagedorn, G. A. and Joye, A., Landau–Zener transitions through small electronic eigenvalue gaps in the Born–Oppenheimer approximation, Ann. Inst. H. Poincaré Phys. Théor. 68(1) (1998), 85–134.Google Scholar
Hagedorn, G. A. and Joye, A., Molecular propagation through small avoided crossings of electron energy levels, Rev. Math. Phys. 11(1) (1999), 41–101.CrossRefGoogle Scholar
Hagedorn, G. A. and Joye, A., Determination of non-adiabatic scattering wave functions in a Born–Oppenheimer model, Ann. Henri Poincaré 6(5) (2005), 937–990.CrossRefGoogle Scholar
Hagedorn, G. A., Classification and normal forms for avoided crossings of quantum-mechanical energy levels, J. Phys. A 31(1) (1998), 369–383.Google Scholar
Hall, D. S., Matthews, M. R. , Ensher, J. R. , Wieman, C. E. and Cornell, E. A., Dynamics of component separation in a binary mixture of Bose–Einstein condensates, Phys. Rev. Lett. 81 (1998), 1539–1542.Google Scholar
Hall, D. S., Matthews, M. R. , Wieman, C. E. and Cornell, E. A., Measurements of relative phase in two-component Bose–Einstein condensates, Phys. Rev. Lett. 81 (1998), 1543–1546.Google Scholar
Hall, D. S., Matthews, M. R. , Wieman, C. E. and Cornell, E. A., Measurements of relative phase in two-component Bose–Einstein condensates [Phys. Rev. Lett. 81 (1998), 1543], Phys. Rev. Lett. 81 (1998), 4532–4532.CrossRefGoogle Scholar
Hari, L., Coherent states for systems of L 2 -supercritical nonlinear Schrödinger equations, Comm. Partial Differential Equations 38(3) (2013), 529–573.Google Scholar
Jona-Lasinio, M., Morsch, O., Cristiani, M., Malossi, N., Müller, J. H., Courtade, E., Anderlini, M. and Arimondo, E., Asymmetric Landau–Zener tunneling in a periodic potential, Phys. Rev. Lett. 91 (2003), 230406.Google Scholar
Jona-Lasinio, M., Morsch, O., Cristiani, M., Malossi, N., Müller, J. H., Courtade, E., Anderlini, M. and Arimondo, E., Erratum: Asymmetric Landau–Zener tunneling in a periodic potential [Phys. Rev. Lett. prltao0031-9007 91 (2003), 230406], Phys. Rev. Lett. 93 (2004), 119903.Google Scholar
Joye, A., Proof of the Landau–Zener formula, Asymptot. Anal. 9(3) (1994), 209–258.Google Scholar
Joye, A. and Marx, M., Semiclassical determination of exponentially small intermode transitions for 1 + 1 spacetime scattering systems, Comm. Pure Appl. Math. 60(8) (2007), 1189–1237.Google Scholar
Khomeriki, R. and Ruffo, S., Nonadiabatic Landau–Zener tunneling in waveguide arrays with a step in the refractive index, Phys. Rev. Lett. 94 (2005), 113904.CrossRefGoogle ScholarPubMed
Landau, L. D., Collected Papers of L. D. Landau, Second printing (Gordon and Breach Science Publishers, New York, 1967). Edited and with an introduction by D. ter Haar.Google Scholar
Lasser, C. and Swart, T., Single switch surface hopping for a model of pyrazine, J. Chem. Phys. 129 (2008), 034302, 1–8.Google Scholar
Martinez, A. and Sordoni, V., Twisted pseudodifferential calculus and application to the quantum evolution of molecules, Mem. Amer. Math. Soc. 200(936) (2009),. vi+82.Google Scholar
Roman, S., Formula of Faa di Bruno, Amer. Math. Monthly 87 (1980), 805–809.Google Scholar
Rousse, V., Landau–Zener transitions for eigenvalue avoided crossings in the adiabatic and Born–Oppenheimer approximations, Asymptot. Anal. 37(3–4) (2004), 293–328.Google Scholar
Sacchetti, A., Universal critical power for nonlinear Schrödinger equations with a symmetric double well potential, Phys. Rev. Lett. 103 (2009), 194101.CrossRefGoogle ScholarPubMed
Segal, I., Space-time decay for solutions of wave equations, Adv. Math. 22(3) (1976), 305–311.Google Scholar
Spohn, H. and Teufel, S., Adiabatic decoupling and time-dependent Born–Oppenheimer theory, Comm. Math. Phys. 224(1) (2001), 113–132. Dedicated to Joel L. Lebowitz.CrossRefGoogle Scholar
Strichartz, R. S., Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44(3) (1977), 705–714.Google Scholar
Triebel, H., Spaces of distributions with weights multipliers in L p-spaces with weights, Math. Nachr. 78 (1977), 339–355.Google Scholar
Zener, C., Non-adiabatic crossing of energy levels, Proc. R. Soc. Lond. Ser. A 137(833) (1932), 696–702.Google Scholar