Abstract
We study individual eigenstates of quantized area-preserving maps on the 2-torus which are classically chaotic. In order to analyze their semiclassical behavior, we use the Bargmann–Husimi representations for quantum states as well as their stellar parametrization, which encodes states through a minimal set of points in phase space (the constellation of zeros of the Husimi density). We rigorously prove that a semiclassical uniform distribution of Husimi densities on the torus entails a similar equidistribution for the corresponding constellations. We deduce from this property a universal behavior for the phase patterns of chaotic Bargmann eigenfunctions which is reminiscent of the WKB approximation for eigenstates of integrable systems (though in a weaker sense). In order to obtain more precise information on “chaotic eigenconstellations,” we then model their properties by ensembles of random states, generalizing former results on the 2-sphere to the torus geometry. This approach yields statistical predictions for the constellations which fit quite well the chaotic data. We finally observe that specific dynamical information, e.g., the presence of high peaks (like scars) in Husimi densities, can be recovered from the knowledge of a few long-wavelength Fourier coefficients, which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, Cambridge, UK, 1965).
A. Erdélyi (ed), Higher transcendental functions (Bateman Manuscript Project, McGraw-Hill, 1953).
R. P. Boars, Jr., Entire Functions (Academic Press, New York, 1954).
M. Nakahara, Geometry, Topology and Physics (Institute of Physics Publishing, Bristol, 1992).
S. Klimek and A. Lesniewski, Quantum Riemann surfaces II: the discrete series, Lett. Math. Phys. 24:125–139 (1992).
S. Nonnenmacher, États propres de systèmes classiquement chaotiques dans l'espace des phases, Thèse de l'Université Paris XI, Orsay, France, 1998 (unpublished).
M. V. Berry, N. L. Balazs, M. Tabor, and A. Voros, Quantum maps, Ann. Phys. N.Y. 122:26–63 (1979).
M. Tabor, A semiclassical quantization of area-preserving maps, Physica D 6:195–210 (1983).
S. Zelditch, Index and dynamics of quantized contact transformations, Ann. Inst. Fourier (Grenoble) 47:305–363 (1997).
A. Schnirelman, Ergodic properties of eigenfunctions, Usp. Math. Nauk. 29:181–182 (1974).
A. Bouzouina and S. De Bièvre, Equipartition of the eigenfunctions of quantized ergodic maps on the torus, Commun. Math. Phys. 178:83–105 (1996); S. Klimek, A. Leśniewski, N. Maitra, and R. Rubin, Ergodic properties of quantized toral automorphisms, J. Math. Phys. 38:67–83 (1997).
M. Degli Esposti, S. Graffi, and S. Isola, Classical limit of the quantized hyperbolic toral automorphisms, Commun. Math. Phys. 167:471–507 (1995).
S. De Bièvre and M. Degli Esposti, Egorov Theorems and equidistribution of eigenfunctions for the quantized sawtooth and Baker maps, Ann. Inst. Henri Poincaré 69, vol. 1, 1–130 (1998).
F. Faure, Approche géométrique de la limite semi-classique par les états cohérents et mécanique quantique sur le tore, Thèse de l'Université Joseph Fourier, Grenoble, France, 1993 (unpublished).
A. Bouzouina, Comportement semi-classique de symplectomorphismes du tore quantifiés, Thèse de l'Université París IX, Paris, France, 1997 (unpublished).
T. Prosen, Quantum surface of section method: eigenstates and unitary quantum Poincaré evolution, Physica D 91:244–277 (1996).
A. Wehrl, On the relation between classical and quantum entropy, Rep. Math. Phys. 16:353–358 (1979).
E. Lieb, Proof of an entropy conjecture of Wehrl, Commun. Math. Phys. 62:35–41 (1978); C.-T. Lee, Wehrl's entropy of spin states and Lieb's conjecture, J. Phys. A 21:3749–3761 (1988).
W. Slomezyński and K. Zyczkowski, Mean dynamical entropy of quantum maps on the sphere diverges in the semiclassical limit, Phys. Rev. Lett. 80:1880–1883 (1998).
P. Leboeuf and A. Voros, Chaos-revealing multiplicative representation of quantum eigenstates, J. Phys. A 23:1765–1774 (1990) and Quantum Nodal Points as Fingerprints of Classical Chaos, in Quantum Chaos: Between order and disorder, G. Casati and B. V. Chirikov, eds. (Cambridge University Press, UK, 1995), pp. 507–533.
A. Voros, Thèse d'Etat (ch. V), Université Paris-Sud, Orsay, France, 1977 (unpublished); A. Voros, Wentzel-Kramers-Brillouin method in the Bargmann representation, Phys. Rev. A 40:6814–6825 (1989).
P. Leboeuf, Phase space approach to quantum dynamics. J. Phys. A 24:4575–4586 (1991).
P. Leboeuf and P. Shukla, Universal fluctuations of zeros of chaotic wavefunctions, J. Phys. A 29:4827–4835 (1996).
P. Shukla, On the distribution of zeros of chaotic wavefunctions, J. Phys. A 30:6313–6326 (1997).
B. Schiffman and S. Zelditch, Distribution of zeros of random and quantum chaotic sections of positive line bundles, preprint, Dept. of Mathematics, Johns Hopkins University, Baltimore, MD (1998).
E. Bogomolny, O. Bohigas, and P. Leboeuf, Quantum chaotic dynamics and random polynomials. J. Stat. Phys. 85:639–679 (1996).
J. H. Hannay, Chaotic analytic zero points: exact statistics for those of a random spin state, J. Phys. A 29:L101-L105 (1996).
T. Prosen, Exact statistics of complex zeros for Gaussian random polynomials with real coefficients, J. Phys. A 29:4417–4423 (1996).
A. Edelman and E. Kostlan, How many zeros of a random polynomial are real?, Bull. Amer. Math. Soc. 32:1–37 (1995).
G. A. Mezincescu, D. Bessis, J.-D. Fournier, G. Mantica, and F. D. Aaron, Distribution of roots of random real generalized polynomials, J. Stat. Phys. 86:675–705 (1996).
P. Bleher and X. Di, Correlations between zeros of a random polynomial, J. Stat. Phys. 88:269–305 (1997).
O. Bohigas, Random matrix theories and chaotic dynamics, in Les Houches, session LII, M.-J. Giannoni, A. Voros, and J. Zinn-Justin, eds. (North-Holland, 1991).
G. Le Caër and J. S. Ho, The Voronoi tessellation generated from eigenvalues of complex random matrices, J. Math. Phys. A 23:3279–3295 (1990).
H. Iwaniec and P. Sarnak, L ∞ norms of eigenfunctions of arithmetic surfaces, Ann. Math. 141:301–320 (1995).
E. Bogomolny, Semiclassical quantization of multidimensional systems, Nonlinearity 5:805–866 (1992).
T. Prosen, General quantum surface-of-section method, J. Phys. A 28:4133–4155 (1995).
M. V. Berry, Semiclassical theory of spectral rigidity, Proc. R. Soc. A 400:229–251 (1985), and Some quantum-to-classical asymptoties, in Les Houches, session LII, M.-J. Giannoni, A. Voros, and J. Zinn-Justin, eds. (North-Holland, 1991).
J. H. Hannay and M. V. Berry, Quantisation of linear maps on a torus—Fresnel diffraction by a periodic grating, Physica D 1:267–291 (1980).
N. L. Balazs and A. Voros, The quantized baker's transformation, Ann. Phys. 190:1–31 (1989).
M. Saraceno and A. Voros, Towards a semiclassical theory of the quantum baker's map, Physica D 79:206–268 (1994).
P. W. O'Connor, S. Tomsovic, and E. J. Heller, Accuracy of semiclassical dynamics in the presence of chaos, J. Stat. Phys. 68:131–152 (1992).
S. Nonnenmacher, Crystal properties of eigenstates for quantum cat maps, Nonlinearity 107:1569–1598 (1997).
O. Agam and S. Fishman, Semiclassical criterion for sears in wave functions of chaotic systems, Phys. Rev. Lett. 73:806–809 (1994).
W. Rudin, Real and complex analysis (Mc Graw-Hill, 1970).
M. Reed and B. Simon, Methods of modern mathematical physics, I: Functional Analysis (Academic Press, 1972).
W. K. Hayman and P. B. Kennedy, Subharmonic Functions (Academic Press, 1976).
P. Lelong and L. Gruman, Entire functions of several complex variables (Springer, 1986).
J. L. Doob, Classical potential theory and its probabilistic counterpart (Springer, 1984).
E. B. Saff and A. B. J. Kuijlaars, Distributing many points on a sphere, The Mathematical Intelligencer 19(1):5–11 (1997).
B. Osgood, R. Phillips, and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80:148–211 (1988).
M. Kierlanczyk, Determinants of Laplacians (PhD thesis, MIT, 1986).
T. Miyake, Modular forms (Springer, 1980).
H. Cohn, Advanced number theory (Dover, 1980).
V. Kač and D. H. Peterson, Infinite dimensional Lie algebras, theta functions and modular forms, Adv. in Math. 53:125–264 (1984).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nonnenmacher, S., Voros, A. Chaotic Eigenfunctions in Phase Space. Journal of Statistical Physics 92, 431–518 (1998). https://doi.org/10.1023/A:1023080303171
Issue Date:
DOI: https://doi.org/10.1023/A:1023080303171