We study quantum localization phenomena in chaotic systems with a parameter.
The parametric motion of energy levels proceeds without crossing any other and
the defined avoided crossings quantify the interaction between states. We
propose the elimination of avoided crossings as the natural mechanism to
uncover localized structures. We describe an efficient method for the
elimination of avoided crossings in chaotic billiards and apply it to the
stadium billiard. We find many scars of short periodic orbits revealing the
skeleton on which quantum mechanics is built. Moreover, we have observed strong
interaction between similar localized structures.Comment: RevTeX, 3 pages, 6 figures, submitted to Phys. Rev. Let

Vergini, E.

Wisniacki, D. A.

English

arXiv

Fil:Vergini, E. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.Fil:Wisniacki, D.A. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina

Wisniacki, D.A.

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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/1781396Localized Structures Embedded in the Eigenfunctions of Chaotic HamiltonianSystemsArticle  in  Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics · April 1998DOI: 10.1103/PhysRevE.58.R5225 · Source: arXivCITATIONS11READS142 authors, including:D. A. WisniackiUniversity of Buenos Aires80 PUBLICATIONS   814 CITATIONS   SEE PROFILEAll content following this page was uploaded by D. A. Wisniacki on 29 September 2013.The user has requested enhancement of the downloaded file.arXiv:chao-dyn/9804011v1  6 Apr 1998Localized Structures Embedded in the Eigenfunctions of Chaotic Hamiltonian SystemsE. Verginia and D.A. Wisniackib1 Departamento de F́ısica, Comisión Nacional de Enerǵıa Atómica. Av. del Libertador 8250, 1429 Buenos Aires, Argentina.2Departamento de F́ısica “J.J. Giambiagi”, FCEN, UBA, Pabellón 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina(February 5, 2008)We study quantum localization phenomena in chaotic sys-tems with a parameter. The parametric motion of energylevels proceeds without crossing any other and the definedavoided crossings quantify the interaction between states. Wepropose the elimination of avoided crossings as the naturalmechanism to uncover localized structures. We describe anefficient method for the elimination of avoided crossings inchaotic billiards and apply it to the stadium billiard. Wefind many scars of short periodic orbits revealing the skeletonon which quantum mechanics is built. Moreover, we have ob-served strong interaction between similar localized structures.PACS numbers: 05.45.+b, 03.65.Sq, 03.20.+iThe connection between individual eigenstates andclassical invariant sets in chaotic systems is far from be-ing fully understood at present. A first step was given byBerry [1] and Voros [2] twenty years ago. They conjec-tured that the eigenstates are associated with the wholeenergy surface explored ergodically by the orbits andthen, the eigenstates would be locally like random super-positions of plane waves. That picture is supported byShnirelman’s theorem [3] and the pioneering numericalwork by McDonald and Kaufman [4] on the Bunimovichstadium billiard - an ergodic system [5]. However, thepaper by Heller [6] modified that point of view. He em-phasized that a large number of high excited eigenfunc-tions for the stadium billiard has an enhancement alongthe shorter periodic trajectories (scars). In 1988, Bogo-molny [7] developed the semiclassical theory of scars andexplained the extra density near unstable periodic trajec-tories. He represented the probability density as a sumover a finite number of periodic orbits (see also [8]).Another unexpected localization phenomenon was ob-served in the stadium billiard for the first time; the fa-mous bouncing ball (b.b) states [9]. They correspondto the family of neutral periodic orbits living in the sta-dium. Localization is stronger than in the case of scarsand survives the semiclassical limit [10]. The semiclassi-cal theory of b.b was developed recently by Tanner [11].It is easy to observe b.b states in the stadium billiard,but it is practically impossible to find complete seriesof b.b states. On the other hand, it is very difficult tofind a family of scars. That is, given a scar with wavenumber k which is associated to a particular periodicorbit of lenght L, we expect to see a family of scars ofthe same orbit at k +2πn/L [6], with n an integer. Butin general, a sequence of similar scars is not observed.Such a phenomenon appears only for some short periodicorbits in some regions of the spectrum but it is impossibleto define a systematic of scars [12].We believe that in order to study the mechanisms giv-ing rise to localization it is necessary first to clean thespectrum. The individual properties of two well definedstates depending on a parameter are mixed when theycollide in an avoided crossing [13]. Then, we expect thatthe elimination of avoided crossings (a.c) in systems gov-erned by a parameter dependent Hamiltonian providesa mechanism for localization. But in a chaotic systemit is impossible to remove all a.c because an individualeigenstate interacts strongly with other states and theinteraction is of long range. However, we will extractonly the a.c with highest curvatures. Recently, it wasstressed [14] that the distribution of highest curvatureshas a universal character and the tail of low curvaturesis a particular property of the system possibly associatedwith scarring phenomena. Precisely, we want to removethe universal property of chaotic eigenfunctions relatedto the Shnirelman’s theorem, and to retain fluctuationsrevealed like phase space localization. On the same foot-ing, Takami [15] has suggested that a.c are contributedby long periodic orbits. In the light of this, by removinga.c, scars of short periodic orbits would arise.We have developed a simple method to remove a.c inchaotic billiards governed by a shape parameter. Let’sφµ(r) and φν(r) be normalized eigenfunctions of a pla-nar billiard with Dirichlet conditions on the boundary C.In ref. [16] the following quasiorthogonality relation wasshown∮C∂φµ∂n∂φν∂nrn ds2kµkν= δµ,ν +(kµ − kν)(kµ + kν)O(1) , (1)where kµ and kν are the corresponding eigenwave num-bers, rn ≡ r.n with n the outward normal unit vec-tor to C, and s is arc lenght round C. Relation (1)defines an effective Hilbert space whose dimension isof order k, and where an eigenstate is represented byϕ(s) ≡ ∂φ/∂n(r(s)) ; the eigenfunction in this space.Now we deform the boundary by changing a shape pa-rameter ℓ (ℓ=ℓ0 for C). Using equation (1) it is possibleto show [17] that eigenenergies (k2) and eigenfunctions atdifferent values of the parameter are connected throughthe following parameter dependent Hamiltonian writtenin the basis {ϕµ} of eigenfunctions at ℓ0,Hµν(ℓ0 + δℓ) ≃ k2µ δµ,ν + δℓ H′µν , (2)with1H ′µν = −(∮Cr′n ϕµ ϕν ds) × [1 + (kµ−kν)O(k−1)] (3)where primes indicate derivation with respect to ℓ. Re-lation (3) was obtained previously [18] for the case whenkµ = kν . By diagonalizing H(ℓ1) we obtain the eigenen-ergies k̃2µ and the eigenfunctions ϕ̃µ at ℓ1. Suppose thatat ℓ1 the states µ and ν collide in an a.c. In ref. [19] anefficient way of determining a.c was described in terms ofthe coefficients Cµν(ℓ) ≡< ϕµ|∂ϕν/∂ℓ > which define adriven evolution of the system. Using equation (2) andperturbation theory we obtain Cµν(ℓ) = H′µν/(k2µ−k2ν) ,where µ and ν are associated to eigenstates of H(ℓ)(the adiabatic basis). Around ℓ1, Cµν behaves like theLorentzian function (ℓint/2)/[ℓ2int + (ℓ − ℓ1)2] , withℓint = 1/2Cµν(ℓ1) . The area in the energy spec-trum, where the a.c is relevant, can be estimated byAµν = ∆k2 × ℓint . Then, if the area is lower than aprescribed value vAµν = (k2µ − k2ν)2/|H ′µν | < v ≡ m ℓ̇ ln 24/πh̄ , (4)we eliminate the a.c by transforming the diagonal matrizof H(ℓ1) as follows: k̃2µ(new) = k̃2ν(new) = (k̃2µ + k̃2ν)/2 .All a.c which satisfy (4) are eliminated in the increasingorder of their area. Criterion (4) has also a simple dy-namic interpretation. If the billiard contains a particleof mass m and the boundary is driven with a velocity ℓ̇,the a.c is eliminated when the Landau-Zener probabilitytransition is greater than one half.The new Hamiltonian is less chaotic than the previ-ous one, and the adiabatic basis of the new Hamilto-nian is more adequate to study nonadiabatic effects [20].For nearly integrable billiards, this new adiabatic basisgoes to the diabatic one of the original Hamiltonian. Forchaotic billiards, the elimination of a.c between nearestneighboring levels gives rise to a.c between distant levels(see Fig. 2(a)).We have studied the desymmetrized stadium billiardwith radius r and straight line of length a [19]. Theboundary only depends on the shape parameter ℓ = a/r(the area is fixed to the value 1+π/4). Fig. 1 comparesthe approximated spectrum [21] obtained from eq. (2)with the exact one; the agreement is excellent. We findstructures surviving parametric variation (straight linesinterrupted by small a.c). The most evident being b.bstates, e.g. states 6, 14 and 18 of Fig. 3(a). But in gen-eral any localized eigenfunction has the same behavior.In Fig. 2(a) we show the spectrum of the transformedHamiltonian by elimination of all a.c satisfying eq.(4),with v = 1.5. At this stage all b.b structures embeddedin the spectrum emerge clearly. b.b states are identifiedby the quantum numbers (nx, ny) counting excitationsin the horizontal and vertical direction respectively; nylabels b.b series. The states 14, 18 and 23 of Fig. 3(b)correspond to (nx, 15), with nx = 1, 2 and 3 resp., andstates 6, 16 and 28 to (nx, 14), with nx = 6, 7 and 8. Atthis stage also many scars of short periodic orbits clearlyappear. They are the states 1, 2, 4, 7, 8, 9, 12, 17, 20, 21,22, 26, 28, 29 and 30 of Fig. 3(b); the associated periodicorbits are identified in Fig. 4. In Fig. 2(b) we have elim-inated avoided crossings up to v = 5 giving rise to newlocalized structures; states 3, 5, 10, 11, 13, 24 and 27 ofFig. 3(b). We have not identified the states 15 and 25(probably contributed by the whispering gallery familyof periodic orbits) and the state 19.The structures in Fig. 3(b) are obtained from the spec-trum (Fig. 3(a)) by an orthogonal transformation whichreduces the parametric interaction among states; that is, the non-diagonal elements of the deformation matrixH ′ are reduced considerably in the new basis. The newstates are characterized by the mean energy < k2 > andthe dispersion σ measured in units of the mean energylevels. If σ is lower than one, the new state has highprobability of appearing in the spectrum. For σ = 1 theassociated structure lives a time equal to the Heisenbergtime. For scars of periodic trajectories we expect σ toincrease with k according to the Shnirelman’s theorem.For b.b states with fixed nx, we expect σ to go to zerowhen ny go to infinite.The non-diagonal elements of the original Hamiltonianin the new basis give us the interaction between localizedstructures. We have observed strong interaction betweensimilar structures. For example, b.b states (1, 15) and(2, 15) have low interaction with the other states (a.c arevery small -see Fig. 1), except with the b.b state (7, 14)and with the states 10 and 24 of Fig. 3(b). Strong inter-action was also observed between scars of short periodicorbits which are close in phase space. For example, states4 and 22 have strong interaction with states 2 and 17 re-spectively, and states 10 and 24 with states 11 and 27respectively. In some situations, this strong interactionprobably decides the occurrence of a defined structure.For example, states 4 and 22 are built principally withorbit (j), but the Bohr-Sommerfeld quantization rule forthis orbit (k = 0.6999 (n + 1/4)) predicts a scar at k ∼47.07 which is not observed. However, the second contri-bution, given by orbit (d) (k = 1.3013 (n + 1/2)), doesnot predict such scar. In fact, in the range 35 < k < 55we have observed structures like 4 and 22 only for thosevalues of k which satisfy approximatelly quantization for(j) and (d) simultaneously. A deep study of this beat-ing phenomenon and the interaction between periodic or-bits, which appear very strong and can introduce newinsight into the semiclassical theory of chaotic systems,is presently under way [17].We transform the eigenfunctions in order to reduce theparametric interaction between states. From a semiclas-sical point of view, the new structures are not contributedby long periodic orbits. The study of the new states interms of short periodic orbits is amenable to the tech-nique developed recently in ref. [12].Finally, we mention that the large parametric corre-2lation observed by Tomsovic [22] in the stadium billiard(random matrix theory predicts no correlations) is simplyunderstood in terms of these localized structures. On theother hand, diffusion and dissipation in mesoscopic sys-tems are affected by these localized structures and thetheory developped by Wilkinson [23] based on randommatrix theory needs to be revised.We would like to thank M. Saraceno for useful sugges-tions. E.V. acknowledges the hospitality of U. Smilanskyand The Weizmann Institute of Science where part of thiswork has been done.[1] M. V. Berry, J. Phys. A 10, 2083 (1977).[2] A. Voros, in Stochastic Behaviour in Classical and Quan-tum Hamiltonian Systems, eds. G. Casati and G. Ford,Lectures Notes in Physics 93, (Springer, Berlin, 1979) p.326.[3] A. I. Shnirelman, Usp. Mat. Nauk 29, 181 (1974).[4] S. W. McDonald and A. N. Kaufman, Phys. Rev. Lett.42, 1189 (1979).[5] L. A. Bunimovich, Funct. Anal. Appl. 8, 254 (1974).[6] E. J. Heller, Phys. Rev. Lett. 53, 1515 (1984).[7] E. B. Bogomolny, Physica D 31, 169 (1988).[8] M. V. Berry, Proc. R. Soc. London A 423, 219 (1989).O. Agam and S. Fishman, J. Phys. A 26, 2113 (1993).[9] S. W. McDonald, Ph. D. Thesis UC berkeley LBL14837(1983).[10] P. W. O’Connor and E. J. Heller, Phys. Rev. Lett. 61,2288 (1988).[11] G. Tanner, J. Phys. A 30, 2863 (1997).[12] F. Simonotti, E. Vergini and M. Saraceno, Phys. Rev. E56, 3859 (1997).[13] C. E. Zener, Proc. R. Soc. London Ser. A 137, 696 (1932).[14] J. Zakrzewski and D. Delande, Phys. Rev. E 47, 1650(1993).[15] T. Takami, Phys. Rev. Lett. 68, 3371 (1992).[16] E. Vergini and M. Saraceno, Phys. Rev. E 52, 2204(1995).[17] E. Vergini and D. A. Wisniacki (to be submitted).[18] M. V. Berry and M. Wilkinson, Proc. R. Soc. London A392, 15 (1984).[19] M. J. Sánchez, E. Vergini, and D. A. Wisniacki, Phys.Rev. E 54, 4812 (1996).[20] W. Lichten, Phys. Rev. 131, 229 (1963); F. T. Smith,ibid. 179, 111 (1969).[21] We have multiplied the deformation matrix (eq. (3)) bythe Lorentzian function [1 + Area(k2µ − k2ν)]−1 to haveinto account the finite character of the effective Hilbertspace.[22] S. Tomsovic, Phys. Rev. Lett. 77, 4158 (1996).[23] M. Wilkinson, Phys. Rev. A 41, 4645 (1990).FIG. 1. The approximated spectrum (solid lines) obtainedfrom eq. (2) is compared with the exact spectrum (dots).FIG. 2. (a) The spectrum of the tranformed Hamiltonianby elimination of all avoided crossing satisfying eq. (4) withv = 1.5. The scarred states corresponding to Fig.3 (b) areshown. (b) Idem (a) for v = 5.FIG. 3. (a) Linear density plots of the eigenfunctions forthe stadium billiar with ℓ = 1 and area 1+π/4. The numbersbelow each plot are the label (left) and the wave number k. (b)Linear density plots of the eigenfunctions for the transformedHamiltonian by elimination of all avoided crossing satisfyingeq.(4), with v = 5. Letters on the top identify periodic orbitsof Fig. 4 with higher contribution to each localized state. Thelabel,√< k2 > and the dispersion σ are showm below eachplot. σ is measured in units of the mean energy levels.FIG. 4. Several short periodic orbits of the desymmetrizedstadium billiard with ℓ=1.325(a)286 1 423 18 14162930172220268972112215(b)19272410115133This figure "fig3a.gif" is available in "gif" format from:http://arXiv.org/ps/chao-dyn/9804011v1This figure "fig3b.gif" is available in "gif" format from:http://arXiv.org/ps/chao-dyn/9804011v1View publication stats

Localized structures embedded in the eigenfunctions of chaotic Hamiltonian systems

E. Vergini

D. A. Wisniacki

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Localized Structures Embedded in the Eigenfunctions of Chaotic
  Hamiltonian Systems

Localized Structures Embedded in the Eigenfunctions of Chaotic Hamiltonian Systems

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