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May  2012, 17(3): 759-774. doi: 10.3934/dcdsb.2012.17.759

Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials

Rémi Carles 1, and  Christof Sparber 2,

1. 

CNRS & Univ. Montpellier 2, Mathématiques, CC 051, 34095 Montpellier, France
2. 

Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, 851 South Morgan Street, Chicago, Illinois 60607, United States

Received  March 2011 Revised  November 2011 Published  January 2012

We consider semiclassically scaled Schrödinger equations with an external potential and a highly oscillatory periodic potential. We construct asymptotic solutions in the form of semiclassical wave packets. These solutions are concentrated (both, in space and in frequency) around the effective semiclassical phase-space flow, and involve a slowly varying envelope whose dynamics is governed by a homogenized Schrödinger equation with time-dependent effective mass tensor. The corresponding adiabatic decoupling of the slow and fast degrees of freedom is shown to be valid up to Ehrenfest time scales.
Keywords: Ehrenfest time., periodic potential, semiclassical wave packet, Bloch band, Schrödinger equation.
Mathematics Subject Classification: 81Q20, 35A35, 35Q40, 81Q0.
Citation: Rémi Carles, Christof Sparber. Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 759-774. doi: 10.3934/dcdsb.2012.17.759
References:
[1]

G. Allaire and M. Palombaro, Localization for the Schrödinger equation in a locally periodic medium, SIAM J. Math. Anal., 38 (2006), 127-142 (electronic). doi: 10.1137/050635572.

[2]

G. Allaire and A. Piatnitski, Homogenization of the Schrödinger equation and effective mass theorems, Comm. Math. Phys., 258 (2005), 1-22. doi: 10.1007/s00220-005-1329-2.

[3]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures," Studies in Mathematics and its Applications, 5, North-Holland Publishing Co., Amsterdam-New York, 1978.

[4]

J. M. Bily and D. Robert, The semi-classical Van Vleck formula. Application to the Aharonov-Bohm effect, in "Long Time Behaviour of Classical and Quantum Systems" (Bologna, 1999), Ser. Concr. Appl. Math., 1, World Sci. Publ., River Edge, NJ, (2001), 89-106.

[5]

A. Bouzouina and D. Robert, Uniform semiclassical estimates for the propagation of quantum observables, Duke Math. J., 111 (2002), 223-252. doi: 10.1215/S0012-7094-02-11122-3.

[6]

R. Carles and C. Fermanian-Kammerer, Nonlinear coherent states and Ehrenfest time for Schrödinger equation, Comm. Math. Phys., 301 (2011), 443-472. doi: 10.1007/s00220-010-1154-0.

[7]

R. Carles, P. A. Markowich and C. Sparber, Semiclassical asymptotics for weakly nonlinear Bloch waves, J. Statist. Phys., 117 (2004), 343-375. doi: 10.1023/B:JOSS.0000044070.34410.17.

[8]

M. Combescure and D. Robert, Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow, Asymptot. Anal., 14 (1997), 377-404.

[9]

_____, Quadratic quantum Hamiltonians revisited, Cubo, 8 (2006), 61-86.

[10]

S. De Bièvre and D. Robert, Semiclassical propagation on |logħ| time scales, Int. Math. Res. Not., (2003), 667-696. doi: 10.1155/S1073792803204268.

[11]

M. Dimassi, J.-C. Guillot and J. Ralston, Gaussian beam construction for adiabatic perturbations, Math. Phys. Anal. Geom., 9 (2006), 187-201. doi: 10.1007/s11040-006-9009-9.

[12]

E. Faou, V. Gradinaru and C. Lubich, Computing semiclassical quantum dynamics with Hagedorn wavepackets, SIAM J. Sci. Comput., 31 (2009), 3027-3041. doi: 10.1137/080729724.

[13]

P. Gérard, Mesures semi-classiques et ondes de Bloch, in "Séminaire sur les Équations aux Dérivées Partielles," 1990-1991, Exp. No. XVI, École Polytech., Palaiseau, (1991), 19 pp.

[14]

P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.

[15]

J.-C. Guillot, J. Ralston and E. Trubowitz, Semiclassical asymptotics in solid state physics, Comm. Math. Phys., 116 (1988), 401-415. doi: 10.1007/BF01229201.

[16]

G. A. Hagedorn, Semiclassical quantum mechanics. I. The ħ $\rightarrow 0$ limit for coherent states, Comm. Math. Phys., 71 (1980), 77-93. doi: 10.1007/BF01230088.

[17]

G. A. Hagedorn and A. Joye, Exponentially accurate semiclassical dynamics: Propagation, localization, Ehrenfest times, scattering, and more general states, Ann. Henri Poincaré, 1 (2000), 837-883. doi: 10.1007/PL00001017.

[18]

_____, A time-dependent Born-Oppenheimer approximation with exponentially small error estimates, Comm. Math. Phys., 223 (2001), 583-626. doi: 10.1007/s002200100562.

[19]

E. J. Heller, Frozen Gaussians: A very simple semiclassical approximation, J. Chem. Phys., 75 (1981), 2923-2931. doi: 10.1063/1.442382.

[20]

M. A. Hoefer and M. I. Weinstein, Defect modes and homogenization of periodic Schrödinger operators, SIAM J. Math. Anal., 43 (2011), 971-996. doi: 10.1137/100807302.

[21]

L. Hörmander, Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z., 219 (1995), 413-449. doi: 10.1007/BF02572374.

[22]

F. Hövermann, H. Spohn and S. Teufel, Semiclassical limit for the Schrödinger equation with a short scale periodic potential, Comm. Math. Phys., 215 (2001), 609-629. doi: 10.1007/s002200000314.

[23]

B. Ilan and M. I. Weinstein, Band-edge solitons, nonlinear Schrödinger/Gross-Pitaevskii equations, and effective media, Multiscale Model. Simul., 8 (2010), 1055-1101. doi: 10.1137/090769417.

[24]

H. Kitada, On a construction of the fundamental solution for Schrödinger equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27 (1980), 193-226.

[25]

H. Kitada and H. Kumano-go, A family of Fourier integral operators and the fundamental solution for a Schrödinger equation, Osaka J. Math., 18 (1981), 291-360.

[26]

R. G. Littlejohn, The semiclassical evolution of wave packets, Phys. Rep., 138 (1986), 193-291. doi: 10.1016/0370-1573(86)90103-1.

[27]

J. Lu, W. E and X. Yang, Asymptotic analysis of the quantum dynamics: Bloch-Wigner transform and Bloch dynamics, to appear in Acta Math. Appl. Sin. Engl. Ser., 2011.

[28]

G. Panati, H. Spohn and S. Teufel, Effective dynamics for Bloch electrons: Peierls substitution and beyond, Comm. Math. Phys., 242 (2003), 547-578. doi: 10.1007/s00220-003-0950-1.

[29]

T. Paul, Semi-classical methods with emphasis on coherent states, in "Quasiclassical Methods" (Minneapolis, MN, 1995), IMA Vol. Math. Appl., 95, Springer, New York, (1997), 51-88.

[30]

_____, Échelles de temps pour l'évolution quantique à petite constante de Planck, in "Séminaire: Équations aux Dérivées Partielles," 2007-2008, Exp. No. IV, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, (2009), 21 pp.

[31]

J. V. Ralston, On the construction of quasimodes associated with stable periodic orbits, Comm. Math. Phys., 51 (1976), 219-242. doi: 10.1007/BF01617921.

[32]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics. IV. Analysis of Operators," Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.

[33]

D. Robert, Long time propagation results in quantum mechanics, in "Mathematical Results in Quantum Mechanics" (Taxco, 2001), Contemp. Math., 307, Amer. Math. Soc., Providence, RI, (2002), 255-264.

[34]

_____, Revivals of wave packets and Bohr-Sommerfeld quantization rules, in "Adventures in Mathematical Physics," Contemp. Math., 447, Amer. Math. Soc., Providence, RI, (2007), 219-235.

[35]

_____, On the Herman-Kluk semiclassical approximation, Rev. Math. Phys., 22 (2010), 1123-1145. doi: 10.1142/S0129055X1000417X.

[36]

V. Rousse, Semiclassical simple initial value representations,, to appear in Ark. för Mat., (). 

[37]

C. Sparber, Effective mass theorems for nonlinear Schrödinger equations, SIAM J. Appl. Math., 66 (2006), 820-842 (electronic). doi: 10.1137/050623759.

[38]

H. Spohn, Long time asymptotics for quantum particles in a periodic potential, Phys. Rev. Lett., 77 (1996), 1198-1201. doi: 10.1103/PhysRevLett.77.1198.

[39]

T. Swart and V. Rousse, A mathematical justification for the Herman-Kluk propagator, Comm. Math. Phys., 286 (2009), 725-750. doi: 10.1007/s00220-008-0681-4.

[40]

S. Teufel, "Adiabatic Perturbation Theory in Quantum Dynamics," Lecture Notes in Mathematics, 1821, Springer-Verlag, Berlin, 2003.

[41]

C. H. Wilcox, Theory of Bloch waves, J. Analyse Math., 33 (1978), 146-167. doi: 10.1007/BF02790171.

show all references

References:
[1]

G. Allaire and M. Palombaro, Localization for the Schrödinger equation in a locally periodic medium, SIAM J. Math. Anal., 38 (2006), 127-142 (electronic). doi: 10.1137/050635572.

[2]

G. Allaire and A. Piatnitski, Homogenization of the Schrödinger equation and effective mass theorems, Comm. Math. Phys., 258 (2005), 1-22. doi: 10.1007/s00220-005-1329-2.

[3]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures," Studies in Mathematics and its Applications, 5, North-Holland Publishing Co., Amsterdam-New York, 1978.

[4]

J. M. Bily and D. Robert, The semi-classical Van Vleck formula. Application to the Aharonov-Bohm effect, in "Long Time Behaviour of Classical and Quantum Systems" (Bologna, 1999), Ser. Concr. Appl. Math., 1, World Sci. Publ., River Edge, NJ, (2001), 89-106.

[5]

A. Bouzouina and D. Robert, Uniform semiclassical estimates for the propagation of quantum observables, Duke Math. J., 111 (2002), 223-252. doi: 10.1215/S0012-7094-02-11122-3.

[6]

R. Carles and C. Fermanian-Kammerer, Nonlinear coherent states and Ehrenfest time for Schrödinger equation, Comm. Math. Phys., 301 (2011), 443-472. doi: 10.1007/s00220-010-1154-0.

[7]

R. Carles, P. A. Markowich and C. Sparber, Semiclassical asymptotics for weakly nonlinear Bloch waves, J. Statist. Phys., 117 (2004), 343-375. doi: 10.1023/B:JOSS.0000044070.34410.17.

[8]

M. Combescure and D. Robert, Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow, Asymptot. Anal., 14 (1997), 377-404.

[9]

_____, Quadratic quantum Hamiltonians revisited, Cubo, 8 (2006), 61-86.

[10]

S. De Bièvre and D. Robert, Semiclassical propagation on |logħ| time scales, Int. Math. Res. Not., (2003), 667-696. doi: 10.1155/S1073792803204268.

[11]

M. Dimassi, J.-C. Guillot and J. Ralston, Gaussian beam construction for adiabatic perturbations, Math. Phys. Anal. Geom., 9 (2006), 187-201. doi: 10.1007/s11040-006-9009-9.

[12]

E. Faou, V. Gradinaru and C. Lubich, Computing semiclassical quantum dynamics with Hagedorn wavepackets, SIAM J. Sci. Comput., 31 (2009), 3027-3041. doi: 10.1137/080729724.

[13]

P. Gérard, Mesures semi-classiques et ondes de Bloch, in "Séminaire sur les Équations aux Dérivées Partielles," 1990-1991, Exp. No. XVI, École Polytech., Palaiseau, (1991), 19 pp.

[14]

P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.

[15]

J.-C. Guillot, J. Ralston and E. Trubowitz, Semiclassical asymptotics in solid state physics, Comm. Math. Phys., 116 (1988), 401-415. doi: 10.1007/BF01229201.

[16]

G. A. Hagedorn, Semiclassical quantum mechanics. I. The ħ $\rightarrow 0$ limit for coherent states, Comm. Math. Phys., 71 (1980), 77-93. doi: 10.1007/BF01230088.

[17]

G. A. Hagedorn and A. Joye, Exponentially accurate semiclassical dynamics: Propagation, localization, Ehrenfest times, scattering, and more general states, Ann. Henri Poincaré, 1 (2000), 837-883. doi: 10.1007/PL00001017.

[18]

_____, A time-dependent Born-Oppenheimer approximation with exponentially small error estimates, Comm. Math. Phys., 223 (2001), 583-626. doi: 10.1007/s002200100562.

[19]

E. J. Heller, Frozen Gaussians: A very simple semiclassical approximation, J. Chem. Phys., 75 (1981), 2923-2931. doi: 10.1063/1.442382.

[20]

M. A. Hoefer and M. I. Weinstein, Defect modes and homogenization of periodic Schrödinger operators, SIAM J. Math. Anal., 43 (2011), 971-996. doi: 10.1137/100807302.

[21]

L. Hörmander, Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z., 219 (1995), 413-449. doi: 10.1007/BF02572374.

[22]

F. Hövermann, H. Spohn and S. Teufel, Semiclassical limit for the Schrödinger equation with a short scale periodic potential, Comm. Math. Phys., 215 (2001), 609-629. doi: 10.1007/s002200000314.

[23]

B. Ilan and M. I. Weinstein, Band-edge solitons, nonlinear Schrödinger/Gross-Pitaevskii equations, and effective media, Multiscale Model. Simul., 8 (2010), 1055-1101. doi: 10.1137/090769417.

[24]

H. Kitada, On a construction of the fundamental solution for Schrödinger equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27 (1980), 193-226.

[25]

H. Kitada and H. Kumano-go, A family of Fourier integral operators and the fundamental solution for a Schrödinger equation, Osaka J. Math., 18 (1981), 291-360.

[26]

R. G. Littlejohn, The semiclassical evolution of wave packets, Phys. Rep., 138 (1986), 193-291. doi: 10.1016/0370-1573(86)90103-1.

[27]

J. Lu, W. E and X. Yang, Asymptotic analysis of the quantum dynamics: Bloch-Wigner transform and Bloch dynamics, to appear in Acta Math. Appl. Sin. Engl. Ser., 2011.

[28]

G. Panati, H. Spohn and S. Teufel, Effective dynamics for Bloch electrons: Peierls substitution and beyond, Comm. Math. Phys., 242 (2003), 547-578. doi: 10.1007/s00220-003-0950-1.

[29]

T. Paul, Semi-classical methods with emphasis on coherent states, in "Quasiclassical Methods" (Minneapolis, MN, 1995), IMA Vol. Math. Appl., 95, Springer, New York, (1997), 51-88.

[30]

_____, Échelles de temps pour l'évolution quantique à petite constante de Planck, in "Séminaire: Équations aux Dérivées Partielles," 2007-2008, Exp. No. IV, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, (2009), 21 pp.

[31]

J. V. Ralston, On the construction of quasimodes associated with stable periodic orbits, Comm. Math. Phys., 51 (1976), 219-242. doi: 10.1007/BF01617921.

[32]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics. IV. Analysis of Operators," Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.

[33]

D. Robert, Long time propagation results in quantum mechanics, in "Mathematical Results in Quantum Mechanics" (Taxco, 2001), Contemp. Math., 307, Amer. Math. Soc., Providence, RI, (2002), 255-264.

[34]

_____, Revivals of wave packets and Bohr-Sommerfeld quantization rules, in "Adventures in Mathematical Physics," Contemp. Math., 447, Amer. Math. Soc., Providence, RI, (2007), 219-235.

[35]

_____, On the Herman-Kluk semiclassical approximation, Rev. Math. Phys., 22 (2010), 1123-1145. doi: 10.1142/S0129055X1000417X.

[36]

V. Rousse, Semiclassical simple initial value representations,, to appear in Ark. för Mat., (). 

[37]

C. Sparber, Effective mass theorems for nonlinear Schrödinger equations, SIAM J. Appl. Math., 66 (2006), 820-842 (electronic). doi: 10.1137/050623759.

[38]

H. Spohn, Long time asymptotics for quantum particles in a periodic potential, Phys. Rev. Lett., 77 (1996), 1198-1201. doi: 10.1103/PhysRevLett.77.1198.

[39]

T. Swart and V. Rousse, A mathematical justification for the Herman-Kluk propagator, Comm. Math. Phys., 286 (2009), 725-750. doi: 10.1007/s00220-008-0681-4.

[40]

S. Teufel, "Adiabatic Perturbation Theory in Quantum Dynamics," Lecture Notes in Mathematics, 1821, Springer-Verlag, Berlin, 2003.

[41]

C. H. Wilcox, Theory of Bloch waves, J. Analyse Math., 33 (1978), 146-167. doi: 10.1007/BF02790171.

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