Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula

[Formules semi-classiques au-delà du temps d’Erhenfest en chaos quantique. (I) La formule des traces.]

Faure, Frédéric

Annales de l'Institut Fourier, Tome 57 (2007), p. 2525-2599 / Harvested from Numdam

Résumé

On considère une application $M$, Anosov non linéaire qui conserve l’aire sur le tore $T^2$. C’est un des exemples les plus simples d’une dynamique chaotique. On s’intéresse à la dynamique quantique pour les temps longs, générée par un opérateur unitaire $\hat{M}$. La formule des traces semi-classique habituelle exprime $\mbox {Tr}\left(\hat{M}^{t}\right)$ pour $t$ fini, dans la limite $\hbar\rightarrow 0$, en termes d’orbites périodiques de $M$ de période $ t$. Des travaux récents atteignent des temps $t \ll t_{E}/6$ où $t_{E} = \log (1/\hbar)/\lambda $ est le temps d’Ehrenfest, et $\lambda $ est le coefficient de Lyapounov. En utilisant une description uniforme de la dynamique au moyen d’une forme normale semi-classique, nous montrons comment étendre la formule des traces pour des temps plus longs, de la forme $t=C.t_{E}$, où $C$ est une constante arbitraire, et avec une erreur arbitrairement petite.

Abstract

We consider a nonlinear area preserving Anosov map $M$ on the torus phase space, which is the simplest example of a fully chaotic dynamics. We are interested in the quantum dynamics for long time, generated by the unitary quantum propagator $\hat{M}$. The usual semi-classical Trace formula expresses $\mbox {Tr}\left(\hat{M}^{t}\right)$ for finite time $t$, in the limit $\hbar\rightarrow 0$, in terms of periodic orbits of $M$ of period $ t$. Recent work reach time $t \ll t_{E}/6$ where $t_{E} = \log (1/\hbar)/\lambda $ is the Ehrenfest time, and $\lambda $ is the Lyapounov coefficient. Using a semi-classical normal form description of the dynamics uniformly over phase space, we show how to extend the trace formula for longer time of the form $t = C.t_{E}$ where $C$ is any constant, with an arbitrary small error.

Détail

Publié le : 2007-01-01

MR 2394550 | Zbl 1145.81035 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2341
Classification:  81Q50,  37D20
Mots clés: chaos quantique, application hyperbolique, formule des traces semi-classique, temps d’Ehrenfest

Bibliographie

[1] Anantharaman, N. Entropy and the localization of eigenfunctions (2004) (Ann. of Math.)

[2] Anantharaman, N.; Nonnenmacher, S. Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold (2006) (ArXiv Mathematical Physics e-prints, math-ph/0610019)

[3] Arnold, V. I. Geometrical methods in the theory of ordinary differential equations, Springer Verlag (1988) | MR 947141 | Zbl 0507.34003

[4] Bambusi, D.; Graffi, S.; Paul, T. Long time semiclassical approximation of quantum flows: A proof of the Ehrenfest time, Asymptot. Anal., Tome 21 (1999) no. 2, pp. 149-160 | MR 1723551 | Zbl 0934.35142

[5] Bohigas, O. Random matrix theories and chaotic dynamics, Chaos and Quantum Physics, Proceedings of the Les Houches Summer School (1989), Tome 45 (1991), pp. 87-199 | MR 1188418

[6] Bohigas, O.; Giannoni, M. J.; Schmit, C. Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett., Tome 52 (1984) no. 1, pp. 1-4 | Article | MR 730191 | Zbl 1119.81326

[7] Bonechi, F.; Debièvre, S. Exponential mixing and ln(h) timescales in quantized hyperbolic maps on the torus, Comm. Math. Phys., Tome 211 (2000), pp. 659-686 | Article | MR 1773813 | Zbl 1053.81032

[8] Bouclet, J. M.; Debièvre, S. Long time propagation and control on scarring for perturbated quantized hyperbolic toral automorphisms, Annales Henri Poincaré, Tome 6 (2005) no. 5, pp. 885-913 | Article | MR 2219861 | Zbl 1088.81049

[9] Bouzouina, A.; Robert, D. Uniform semiclassical estimates for the propagation of quantum observables, Duke Math. J., Tome 111 (2002) no. 2, pp. 223-252 | Article | MR 1882134 | Zbl 1069.35061

[10] Cargo, M.; Gracia-Saz, A.; Littlejohn, R. G.; Reinsch, M. W.; Rios, P. M. Quantum normal forms, moyal star product and bohr-sommerfeld approximation, J. Phys. A: Math. Gen., Tome 38 (2005), pp. 1997-2004 | Article | MR 2124376 | Zbl 1073.81056

[11] Colin De Verdière, Y. Ergodicité et fonctions propres du laplacien. (Ergodicity and eigenfunctions of the Laplacian), Commun. Math. Phys., Tome 102 (1985), pp. 497-502 | MR 818831 | Zbl 0592.58050

[12] Colin De Verdière, Y.; Parisse, B. Équilibre instable en régime semi-classique - I. Concentration microlocale, Communications in Partial Differential Equations, Tome 19 (1994) no. 9–10, pp. 1535-1563 | Article | MR 1294470 | Zbl 0819.35116

[13] Colin De Verdière, Y.; Parisse, B. Équilibre instable en régime semi-classique - II. Conditions de Bohr-Sommerfeld, Annales de l’Institut Henri Poincaré- Physique Théorique, Tome 61 (1994) no. 3, pp. 347-367 | Numdam | Zbl 0845.35076

[14] Colin De Verdière, Y.; Parisse, B. Singular bohr-sommerfeld rules, Commun. Math. Phys, Tome 205 (1999), pp. 459-500 | Article | MR 1712567 | Zbl 01379901

[15] Combescure, M.; Ralston, J.; Robert, D. A proof of the Gutzwiller semiclassical trace formula using coherent states decomposition, Commun. Math. Phys., Tome 202 (1999) no. 2, pp. 463-480 | Article | MR 1690026 | Zbl 0939.58031

[16] Combescure, M.; Robert, D. Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow, Asymptotic Anal., Tome 14 (1997) no. 4, pp. 377-404 | MR 1461126 | Zbl 0894.35026

[17] De Bièvre, S. Recent results on quantum map eigenstates, Mathematical physics of quantum mechanics, Springer, Berlin (Lecture Notes in Phys.) Tome 690 (2006), pp. 367-381 | MR 2234923 | Zbl 1167.81388

[18] De Bièvre, Stephan Quantum chaos: a brief first visit, Second Summer School in Analysis and Mathematical Physics (Cuernavaca, 2000), Amer. Math. Soc. (Contemp. Math.) Tome 289 (2001), pp. 161-218 | MR 1864542 | Zbl 1009.81020

[19] Delatte, D. Nonstationnary normal forms and cocycle invariants, Random and Computational dynamics, Tome 1 (1992), pp. 229-259 | MR 1186375 | Zbl 0778.58058

[20] Delatte, D. On normal forms in hamiltonian dynamics, a new approach to some convergence questions, Ergod. Th. and Dynam. Sys., Tome 15 (1995), pp. 49-66 | Article | MR 1314968 | Zbl 0820.58052

[21] Dimassi, M.; Sjöstrand, J. Spectral Asymptotics in the Semi-Classical Limit, Cambridge University Press, London Mathematical Society Lecture Notes, Tome 268 (1999) | MR 1735654 | Zbl 0926.35002

[22] Eckhardt, B.; Fishman, S.; Keating, J.; Agam, O.; Main, J.; Müller, K. Approach to ergodicity in quantum wave functions, Phys. Rev. E, Tome 52 (1995), pp. 5893-5903 | Article

[23] Evans, L.; Zworski, M. Lectures on semiclassical analysis (2003) (http://math.berkeley.edu/ zworski/)

[24] Faure, F. Semiclassical formula beyond the ehrenfest time in quantum chaos. (II) propagator formula (2006) (in preparation)

[25] Faure, F. Prequantum chaos: Resonances of the prequantum cat map, Journal of Modern Dynamics, Tome 1 (2007) no. 2, pp. 255-285 | Article | MR 2285729 | Zbl pre05238950

[26] Faure, F.; Nonnenmacher, S. On the maximal scarring for quantum cat map eigenstates, Communications in Mathematical Physics, Tome 245 (2004), pp. 201-214 | Article | MR 2036373 | Zbl 1071.81044

[27] Faure, F.; Nonnenmacher, S.; Debièvre, S. Scarred eigenstates for quantum cat maps of minimal periods, Communications in Mathematical Physics, Tome 239 (2003), pp. 449-492 | Article | MR 2000926 | Zbl 1033.81024

[28] Folland, G. B. Harmonic Analysis in phase space, Princeton University Press (1989) | MR 983366 | Zbl 0682.43001

[29] Giannoni, M. J.; Voros, A.; Zinn-Justin, J. Chaos and Quantum Physics, North-Holland, Les Houches Session LII 1989 (1991) | MR 1188415

[30] Gohberg, I.; Goldberg, S.; Krupnik, N. Traces and Determinants of Linear Operators, Birkhauser (2000) | MR 1744872 | Zbl 0946.47013

[31] Gracia-Saz, A. The symbol of a function of a pseudo-differential operator., Annales de l’Institut Fourier, Tome 55 (2005) no. 7, pp. 2257-2284 | Article | Numdam | Zbl 1091.53062

[32] Guillemin, V. Wave-trace invariants, Duke Math. J., Tome 83 (1996) no. 2, pp. 287-352 | Article | MR 1390650 | Zbl 0858.58051

[33] Gutzwiller, M. Periodic orbits and classical quantization conditions, J. Math. Phys., Tome 12 (1971), pp. 343-358 | Article

[34] Gutzwiller, M. Chaos in classical and quantum mechanics, Springer-Verlag (1991) | MR 1077246 | Zbl 0727.70029

[35] Haake, F. Quantum Signatures of Chaos, Springer (2001) | MR 1806548 | Zbl 0985.81038

[36] Hagedorn, G. A.; Joye, A. Exponentially acurrate semiclassical dynamics: Propagation, localization, ehrenfest times, scattering, and more general states, Ann. Henri Poincaré, Tome 1 (2000), pp. 837-883 | Article | MR 1806980 | Zbl 1050.81017

[37] Hasselblatt, B. Hyperbolic dynamics, Handbook of Dynamical Systems, North Holland, Tome 1A (2002), pp. 239-320 | Article | MR 1928520 | Zbl 1047.37018

[38] Heller, E. J. Time dependant approach to semiclassical dynamics, J. Chem. Phys., Tome 62 (1975), pp. 1544-1555 | Article

[39] Hepp, K. The classical limit of quantum mechanical correlation funtions, Comm. Math. Phys., Tome 35 (1974), pp. 265-277 | Article | MR 332046

[40] Hurder, S.; Katok, A. Differentiability, rigidity and Godbillon-Vey classes for Anosov flows, Publ. Math., Inst. Hautes Étud. Sci., Tome 72 (1990), pp. 5-61 | Article | Numdam | MR 1087392 | Zbl 0725.58034

[41] Iantchenko, A. The Birkhoff normal form for a Fourier integral operator. (La forme normale de Birkhoff pour un opérateur intégral de Fourier.), Asymptotic Anal., Tome 17 (1998) no. 1, pp. 71-92 | MR 1632700 | Zbl 01355380

[42] Iantchenko, A.; Sjöstrand, J. Birkhoff normal forms for Fourier integral operators. II, Am. J. Math., Tome 124 (2002) no. 4, pp. 817-850 | Article | MR 1914459 | Zbl 1011.35144

[43] Iantchenko, A.; Sjöstrand, J.; Zworski, M. Birkhoff normal forms in semi-classical inverse problems, Math. Res. Lett., Tome 9 (2002) no. 2-3, pp. 337-362 | MR 1909649 | Zbl 01804060

[44] Joye, A.; Hagedorn, G. Semiclassical dynamics with exponentially small error estimates, Comm. in Math. Phys., Tome 207 (1999), pp. 439-465 | Article | MR 1724830 | Zbl 1031.81519

[45] Katok, A.; Hasselblatt, B. Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press (1995) | MR 1326374 | Zbl 0878.58020

[46] Keating, J. P. Asymptotic properties of the periodic orbits of the cat maps, Nonlinearity, Tome 4 (1991), pp. 277-307 | Article | MR 1107008 | Zbl 0726.58036

[47] Keating, J. P. The cat maps: quantum mechanics and classical motion, Nonlinearity, Tome 4 (1991), pp. 309-341 | Article | MR 1107009 | Zbl 0726.58037

[48] Littlejohn, R. G. The semiclassical evolution of wave-packets, Phys. Rep., Tome 138 (1986) no. 4–5, pp. 193-291 | Article | MR 845963

[49] Martinez, A. An Introduction to Semiclassical and Microlocal Analysis, Springer, New York, Universitext (2002) | MR 1872698 | Zbl 0994.35003

[50] Nonnenmacher, S. Evolution of lagrangian states through pertubated cat maps, Preprint (2004)

[51] Perelomov, A. Generalized coherent states and their applications, Springer-Verlag (1986) | MR 858831 | Zbl 0605.22013

[52] Pollicott, M.; Yuri, M. Dynamical Systems and Ergodic theory, Cambridge University Press (1998) | MR 1627681 | Zbl 0897.28009

[53] Schubert, R. Semi-classical behaviour of expectation values in time evolved lagrangian states for large times, Commun. Math. Phys., Tome 256 (2005), pp. 239-254 | Article | MR 2134343 | Zbl 1067.81040

[54] Sjöstrand, J. Resonances associated to a closed hyperbolic trajectory in dimension 2, Asymptotic Anal., Tome 36 (2003) no. 2, pp. 93-113 | MR 2021528 | Zbl 1060.35096

[55] Sjöstrand, J.; Zworski, M. Quantum monodromy and semi-classical trace formulae, J. Math. Pures Appl., Tome 1 (2002), pp. 1-33 | MR 1994881 | Zbl 1038.58033

[56] Tomsovic, S.; Heller, E. J. Long-time semi-classical dynamics of chaos: the stadium billard, Physical Review E, Tome 47 (1993), pp. 282 | Article | MR 1375006

[57] Zelditch, S. Uniform distribution of the eigenfunctions on compact hyperbolic surfaces, Rev. Mod. Phys., Tome 55 (1987), pp. 919-941 | MR 916129 | Zbl 0643.58029

[58] Zelditch, S. Quantum dynamics from the semi-classical viewpoint, Lectures at I.H.P. (1996) (http://mathnt.mat.jhu.edu/zelditch)

[59] Zelditch, S. Wave invariants at elliptic closed geodesics, Geom. Funct. Anal., Tome 7 (1997) no. 1, pp. 145-213 | Article | MR 1437476 | Zbl 0876.58010

[60] Zelditch, S. Wave invariants for non-degenerate closed geodesics, Geom. Funct. Anal., Tome 8 (1998) no. 1, pp. 179-217 | Article | MR 1601862 | Zbl 0908.58022

[61] Zelditch, S. Quantum ergodicity and mixing of eigenfunctions, Elsevier Encyclopedia of Math. Phys (2005)

[62] Zhang, W. M.; Feng, D. H.; Gilmore, R. Coherent states: theory and some applications, Rev. Mod. Phys., Tome 62 (1990), pp. 867 | Article | MR 1102385