Homoclinic motion plays a key role in the organization of classical chaos in
Hamiltonian systems. In this Letter, we show that it also imprints a clear
signature in the corresponding quantum spectra. By numerically studying the
fluctuations of the widths of wavefunctions localized along periodic orbits we
reveal the existence of an oscillatory behavior, that is explained solely in
terms of the primary homoclinic motion. Furthermore, our results indicate that
it survives the semiclassical limit.Comment: 5 pages, 4 figure

D. A. Wisniacki

E. Vergini

E. J. Heller

F. Borondo

J. P. Keating

M. C. Gutzwiller

M. V. Berry

R. M. Benito

English

arXiv

Homoclinic motion plays a key role in the organization of classical chaos in Hamiltonian systems. In this Letter, we show that it also imprints a clear signature in the corresponding quantum spectra. By numerically studying the fluctuations of the widths of wave functions localized along periodic orbits we reveal the existence of an oscillatory behavior that is explained solely in terms of the primary homoclinic motion. Furthermore, our results indicate that it survives the semiclassical limit. © 2005 The American Physical Society.Fil: Wisniacki, Diego Ariel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; ArgentinaFil: Vergini, Eduardo Germán. Universidad Autónoma de Madrid; España. Universidad de Buenos Aires; ArgentinaFil: Benito, R.M.. Escuela Técnica Superior de Ingenieros Agrónomos; ArgentinaFil: Borondo, F.. Universidad Autónoma de Madrid; Españ

Wisniacki, Diego Ariel

Vergini, Eduardo Germán

Benito, R.M.

Borondo, F.

CONICET Digital

Physical Review Letters

Signatures of homoclinic motion in quantum chaos

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PRL 94, 054101 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending11 FEBRUARY 2005Signatures of Homoclinic Motion in Quantum ChaosD. A. Wisniacki,1,2 E. Vergini,1,3 R. M. Benito,4 and F. Borondo1,*1Departamento de Quı́mica C-IX, Universidad Autónoma de Madrid, Cantoblanco, 28049–Madrid, Spain2Departamento de Fı́sica ‘‘J. J. Giambiagi’’, FCEN, Universidad Buenos Aires, 1428 Buenos Aires, Argentina3Departamento de Fı́sica, Comisión Nacional de Energı́a Atómica. Av. del Libertador 8250, 1429 Buenos Aires, Argentina4Departamento de Fı́sica, E.T.S.I. Agrónomos, Universidad Politécnica de Madrid, 28040 Madrid, Spain(Received 24 May 2004; published 7 February 2005)0031-9007=Homoclinic motion plays a key role in the organization of classical chaos in Hamiltonian systems. Inthis Letter, we show that it also imprints a clear signature in the corresponding quantum spectra. Bynumerically studying the fluctuations of the widths of wave functions localized along periodic orbits wereveal the existence of an oscillatory behavior that is explained solely in terms of the primary homoclinicmotion. Furthermore, our results indicate that it survives the semiclassical limit.DOI: 10.1103/PhysRevLett.94.054101 PACS numbers: 05.45.Mt, 03.65.SqPeriodic orbits (POs) are invariant classical objects ofoutmost relevance for the understanding of chaotic dy-namical systems. The pioneering work of Poincaré dem-onstrated the importance of these retracing orbits inorganizing the complexity of the corresponding classicalmotion. Semiclassically, Gutztwiller [1] developed hiscelebrated trace formula, able to quantize chaotic systems,solely in terms of POs information. Quantum mechani-cally, striking manifestations of POs have been reported inthe literature. One of the most important are ‘‘scars’’ [2], anenhanced localization of quantum density along unstableshort POs in certain individual eigenfunctions. This phe-nomenon was first noticed in the Bunimovich stadiumbilliard [3], and subsequently studied systematically byHeller, who constructed a theory of scarring based onwave packet propagation [2,4]. Another important contri-bution to scar theory is due to Bogomolny [5], who derivedan explicit expression for the PO contributions to thesmoothed quantum probability density over small rangesof space and energy (i.e., average over a large number ofeigenfunctions). A corresponding theory for Wigner func-tions was developed by Berry [6]. Recently, there has beena flurry of activity focusing on the influence of bifurcations(mixed systems) on scarring [7].The existence of a scar implies a clear regularity in thecorresponding quantum spectrum, related to the period ofthe PO. In time domain, the dynamics of a packet runningalong a PO induce recurrences in the autocorrelation func-tion, that when Fourier transformed define an envelope inthe spectrum, giving rise to peaks of width proportional tothe Lyapunov exponent, , at energies given by a Bohr-Sommerfeld (BS) quantization condition [1,8].In this Letter, we demonstrate the existence of an addi-tional superimposed spectral regularity also related to scar-ring. It is originated by the associated homoclinic motionand is given by the area enclosed by the stable and unstablemanifolds up to the first crossing. To unveil this regularitywe consider the fluctuations of the spectral widths corre-sponding to localized wave functions along unstable POs.05=94(5)=054101(4)$23.00 05410Our data demonstrate that these fluctuations have a surpris-ingly simple oscillatory behavior essentially governed bythe quantization of only the primary homoclinic dynamics.We provide an explanation of this result in terms of thecoherence of this classical motion [9], which constitutesthe natural global extension of the local hyperbolic struc-ture around the PO. Furthermore, our numerical resultsindicate that the observed oscillatory behavior do notvanish as h ! 0.It should be remarked that, contrary to the previouslydescribed results on scar theory, our study implies dynam-ics beyond the Ehrenfest time (j ln hj), when the POmanifolds start to cross and the homoclinic tangle devel-ops, giving rise to subtle quantum interference effects.Other interesting papers, also considering these longertimes, are due to Tomsovic and Heller [10], who showedhow to construct a valid semiclassical approximation towave functions past this limit; this came as a surprise sincethe underlying chaos has had time to develop much finerstructure than a quantum cell. Similarly to what happens inenergy domain with Gutzwiller trace formula, this theoryunfortunately requires the use of a number of homoclinicexcursions increasing exponentially with time. Our resultindicates that there exist some properties of these long termdynamics that can be understood in terms of a smallnumber of classical invariants.The ideal tool to carry out our investigation are nonsta-tionary wave functions highly localized on POs. The over-lap with the corresponding eigenstates providesinformation on how these structures appear in the spectra,and how they are embedded in the quantum mechanics ofthe system. Such functions, associated to a PO, can beobtained by using the (dynamical) information up to theEhrenfest time, thus including the hyperbolic structureformed by the corresponding unstable and stable mani-folds. In this way, the associated wave functions spreadalong these dynamically relevant regions of phase space(see illustration in Fig. 1). This can be done in differentways [11,12]; in particular, in Ref. [12] a definition of scar1-1  2005 The American Physical Society0 0.5 1 1.5 200.0050.01100 200 300-0.0500.05ISσrelkBSFIG. 2. Relative width for the scar states along the horizontalperiodic orbit as a function of the Bohr-Sommerfeld quantizedwavelength (inset), and its Fourier transform.211 211.5 21200.10.21+   /2π(a) (b)q10p−1(c)kkBS0Iµµqθ=sin(  )p θFIG. 1. Configuration space (a), phase space (b), and wave-length spectrum weights (c) representations of a scar wavefunction along the horizontal periodic orbit of a desymmetrizedstadium billiard corresponding to BS wavelength kBS 211:665. The unstable and stable manifolds of the orbit arealso plotted in panel (b). In panel (a) the Birkhoff coordinateson the boundary of the billiard are shown.PRL 94, 054101 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending11 FEBRUARY 2005functions, based on transversally excited resonances alongPOs at given BS quantized energies with minimum disper-sion, is provided. The width of these functions in theenergy spectrum is given bysc  hj ln hjOhj ln hj2; (1)where the narrowing factor j ln hj1 comes from the inclu-sion of the hyperbolic structure [12]. In this respect, itshould be noticed that although this approach incorporatesthe motion along the manifolds up to the Ehrenfest time,interference effects due to the intersections of the mani-folds are not included.The model used in our calculations is the fully chaoticsystem consisting of a particle of mass M confined in adesymmetrized Bunimovitch stadium billiard, defined bythe radius of the circular part, r  1, and the enclosed area,1 =4. To calculate the corresponding eigenstates,Dirichlet conditions on the stadium boundary andNeumman conditions on the symmetry axes are imposed.We will focus our attention on the dynamics influenced bythe horizontal PO. The corresponding scar wave functions,j	i, are calculated using the method of Ref. [12].In Fig. 1 we show, as an example, the results for the scarstate with wavelength kBS  211:665. This value was ob-tained from the BS rule, kBS  2=LH	n H=4	,where n  134 is the excitation number along the orbit,LH  4 its length, and H  3 the corresponding Maslovindex. In it, all features discussed above are observed.Namely, the Husimi based quantum surface of section[13] spreads along the manifolds structure associatedwith the horizontal PO, and the state appears in the spec-trum distributed among the different eigenstates, ji,within a given range around the corresponding BS quan-tized wavelength, with intensity weights I  jhj	ij2.05410The energy width of the associated envelope can then bedefined as (we take M  h2=2 in such a way that k2 is theenergy of the particle) Xjhj	ij2k2  k2BS	2s: (2)When calculated, this quantity is a decreasing oscillatoryfunction of kBS, that can be more adequately studied bydefining its relative variation with respect to the corre-sponding semiclassical value,rel  scsc: (3)The corresponding results are shown in the inset to Fig. 2.As can be observed, it exhibits an oscillatory behavior,with an amplitude representing only a small fraction of sc.When this signal is Fourier analyzed (see main body ofFig. 2) it is seen to have only two relevant components,corresponding to the peaks appearing at values of theaction S  0:633 and 1.007, respectively. (Notice that, inour case, S has units of length since the total linear mo-mentum of the particle has been set equal to one.) The BSquantization of the wavelength implies that the Fouriertransform of rel is a periodic function with period LH.In this respect, corresponding peaks appear at S  3:367and 2:993.To interpret this result, we consider only the primaryhomoclinic motion corresponding to the horizontal PO.This follows the analysis of Ozorio de Almeida [9], whostudied the quantization of homoclinic orbits, that werethought of as defining an invariant torus, approached byseries of satellite unstable POs. There are two topologicallydistinct types of primary homoclinic points, which appearas the first and second intersections of the associatedmanifolds. The situation is shown in Fig. 3(a), where apicture of the relevant phase space, in Birkhoff coordi-nates, is presented. In it, we have plotted the fixed pointcorresponding to the horizontal PO (labeled A), and theemanating unstable (full line) and stable (dashed line)1-2TABLE I. Exponential convergence of Lm mLH as a func-tion of the number of times the POs intersect our surface ofsection for families h1 and h2 of satellite orbits corresponding tothe primary homoclinic motion. To improve the convergence wehave averaged the length of orbits m and m for the samefamily [see Fig. 3(c)].m Lm mLHFamily h1 Family h23 3:367 727 48 2:990 915 394 3:368 367 57 2:991 131 875 3:368 389 68 2:991 141 816 3:368 390 43 2:991 142 217 3:368 390 45 2:991 142 22h1}h2}h,,21h,,1h,1+ /2πh1h2,h2A10−1qp03− 3+4− 4+3−4+4−3+(c)(b)(a)hS 1hS 2...FIG. 3. (a) Phase space portrait in Birkhoff coordinates relevant to our calculations. Label A indicates the position of the fixed pointcorresponding to the horizontal periodic orbit. The associated unstable and stable manifolds are represented in full and dashed line,respectively. They cross, with different topology, at the primary homoclinic points h1 and h2, which map into h01, h001 , . . . , and h02, h002 ,. . . . (b) Primary homoclinic areas Sh1 and Sh2 associated with the horizontal orbit. (c) Satellite periodic orbits converging to thehomoclinic points h1 and h2. See text for details.PRL 94, 054101 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending11 FEBRUARY 2005manifolds. These manifolds first cross (with different to-pology) at points h1 and h2, and afterwards at the sequen-ces h01, h001 , . . . , and h02, h002 , . . . (and their reflections withrespect to p  0), as fixed point A is approached. Theseinfinite sequences of points, which are the dynamicalimages of h1 and h2, constitute the primary homoclinicorbits that define two relevant areas in phase space, de-noted hereafter as Sh1 and Sh2 [see shaded regions inFig. 3(b)], important for the discussions presented below.Furthermore, these primary homoclinic orbits accumulateinfinite sets of satellite POs with increasingly longer peri-ods, as discussed in Ref. [14]. The first members of thesetwo families, converging to the primary homoclinic orbitsare presented in Fig. 3(c). Accordingly to our notation,based on the number of times that the POs intersect oursurface of section, orbits 3 and 3 consist of fixed pointsnear h1, h01 (or h2, h02), and their reflections with respect top  0; orbits 4 and 4 consists of fixed points near h1,h01, h001 (or h2, h02, h002 ), and their reflections; and so on for thelonger POs in the family. Notice that the reflection withrespect to p  0 of points over q  0 or q  1 =2gives the same point. Moreover, these POs can be groupedin pairs, in such a way that 3 and 3 represent the firstapproximation to the primary homoclinic motion (the indicates the PO with shorter length of the pair), 4 and4 the second, etc., Actually, it is apparent the increasingrelation of these orbits to the horizontal one, since as weprogress in the sequence new segments closer and closer tothe horizontal axis are incorporated into the trajectory.Now, the question arises about under which conditionsthe satellite families of POs reinforce the contribution of05410the central one. As seen before, the horizontal POs satisfythe BS quantization conditionkLH 2H  2nH; (4)being nH an integer. In the same way, the BS condition fororbits crossing m times the surface of section readskLm 2m  2nm: (5)Subtracting m times Eq. (4) to (5), a new quantizationcondition emergeskLm mLH	 2m mH	  2nm mnH	: (6)1-3k∆100 200 30002.5e-05I/ 2(    )k∆FIG. 4. Intensity of the peak at Sh1 divided by k	2 as afunction of k.PRL 94, 054101 (2005) P H Y S I C A L R E V I E W L E T T E R Sweek ending11 FEBRUARY 2005The relevant point here is that Lm mLH convergesexponentially to Sh1  3:368 390 45 and Sh2 2:991 142 22 for the two families h1 and h2, respectively,as shown in Table I. Moreover, m mH is independentof m, and it corresponds to h1  1 and h2  0. In thisway, we are in the position to assess that all satellite orbitsof a given family contribute coherently when two BSquantization rules are simultaneously fulfilled: (a) thequantization of the central (horizontal) orbit [Eq. (4)],and (b) the so-called quantization of the homoclinic torusaccording to Ref. [9],kShi 2hi  2n; i  1; 2: (7)It should be observed that the values of Sh1 and Sh2 agreeextremely well with those obtained numerically for theposition of the two main peaks in the spectrum of Fig. 2.On the other hand, these homoclinic areas correspond inour chosen Poincaré surface of section to the areas indi-cated in Fig. 3(b). The arrows in Fig. 3(b) indicate thecounterclockwise direction of the homoclinic motion, andfor this reason homoclinic areas result negative.This notorious influence of the homoclinic motion in thespectra, and thus in the quantum mechanics of our system,is remarkable, especially when one takes into account thatthe homoclinic dynamics is the origin of the chaotic be-havior in Hamiltonian systems. In order to understand thisbehavior we note that the motion has two components. Thefirst one, close to the primary homoclinic torus, is such thatthe corresponding trajectories interfere coherently, as it hasbeen shown. The second component, associated to thenonprimary homoclinic crosses, gives rise to more com-plicated self-intersecting tori. These excursions are muchlonger and the corresponding interference is much lesscoherent; accordingly, its influence in the spectra is verydiluted. In this respect, it is more efficient to consider theselong excursions, by the interaction with other short POs(heteroclinic connections). This point has been discussedelsewhere [15].Finally, let us consider if the influence of the homoclinicmotion on rel survives the semiclassical limit h ! 0 (k 054101h ! 1 in our units). For this purpose we have performedthe following calculation. Starting from the scaled fluctua-tion curve rel, presented in the inset to Fig. 2, we computethe Fourier transform of the function for increasingly largerintervals of kBS, k0; k1	, keeping fixed the value of thelower limit, k0. The results for the peak at Sh1 are shown inFig. 4, where it is seen that the corresponding Fourierintensity divided by k	2 is an approximately constantfunction of the length of the interval, k  k1  k0. Asimilar behavior is obtained for the peak at Sh2 . This resultimplies that relkBS	 is essentially an oscillatory functionwith frequencies Sh1 and Sh2 and constant coefficients.In conclusion, we have gone one step further the usualunderstanding of the quantum-classical correspondence,by showing how homoclinic areas imprint a clear signaturein the spectra of classically chaotic systems. In particular,we have found that the fluctuations of the relative widthscorresponding to wave functions (with hyperbolic struc-ture) highly localized in the vicinity of a PO oscillate in avery simple way that is clearly controlled by the quantiza-tion of the associated primary homoclinic motions. Thisindicates the importance of classical invariants, other thanthose included in Gutzwiller theory concerning individualPOs, for the systematic description of quantuminformation.This work was supported by MCyT and MCED (Spain)under contract nos. BQU2003-8212, SAB2000-340, andSAB2002-22.1-4*Electronic address: f.borondo@uam.es[1] M. C. Gutzwiller, Chaos in Classical and QuantumMechanics (Springer-Verlag, New York, 1990).[2] E. J. Heller, Phys. Rev. Lett. 53, 1515 (1984).[3] S. W. McDonald, LBL Report No. 14837 (1983).[4] L. Kaplan and E. J. Heller, Ann. Phys. (N.Y.) 264, 171(1998).[5] E. B. Bogomolny, Physica D (Amsterdam) 31, 169 (1988).[6] M. V. Berry, Proc. R. Soc. London A 423, 219 (1989).[7] J. P. Keating and S. D. Prado, Proc. R. Soc. London A 457,1855 (2001).[8] E. J. Heller, in Chaos and Quantum Physics, edited byM. J. Giannoni, A. Voros, and J. Zinn-Justin (Elsevier,Amsterdam, 1991).[9] A. M. Ozorio de Almeida, Nonlinearity 2, 519 (1989).[10] S. Tomsovic and E. J. Heller, Phys. Rev. Lett. 70, 1405(1993); Phys. Rev. E 47, 282 (1993).[11] G. G. de Polavieja, F. Borondo, and R. M. Benito, Phys.Rev. Lett. 73, 1613 (1994).[12] E. G. Vergini and G. Carlo, J. Phys. A 34, 4525 (2001).[13] D. A. Wisniacki, F. Borondo, E. Vergini, and R. M. Benito,Phys. Rev. E 63, 66220 (2001).[14] G. L. Da Silva Ritter, A. M. Ozorio de Almeida, and R.Douady, Physica D (Amsterdam) 29, 181 (1987).[15] D. A. Wisniacki, F. Borondo, E. Vergini, and R. M. Benito,Phys. Rev. E 70, 035202(R) (2004).

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Signatures of Homoclinic Motion in Quantum Chaos

Signatures of homoclinic motion in quantum chaos

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