Classical bifurcations and entanglement in smooth Hamiltonian system
aa r X i v : . [ n li n . C D ] J un Preprint Number : IITM/PH/TH/2007/6
Classical bifurcations and entanglement in smooth Hamiltonian system
M. S. Santhanam , V. B. Sheorey and Arul Lakshminarayan ∗ Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India. Max Planck Institute for the Physics of Complex Systems,N¨othnitzer Strasse 38., Dresden 01187, Germany (Dated: October 23, 2018)We study entanglement in two coupled quartic oscillators. It is shown that the entanglement,as measured by the von Neumann entropy, increases with the classical chaos parameter for genericchaotic eigenstates. We consider certain isolated periodic orbits whose bifurcation sequence affects aclass of quantum eigenstates, called the channel localized states. For these states, the entanglementis a local minima in the vicinity of a pitchfork bifurcation but is a local maxima near a anti-pitchforkbifurcation. We place these results in the context of the close connections that may exist betweenentanglement measures and conventional measures of localization that have been much studied inquantum chaos and elsewhere. We also point to an interesting near-degeneracy that arises in thespectrum of reduced density matrices of certain states as an interplay of localization and symmetry.
PACS numbers: 05.45.-a, 03.67.Mn, 05.45.Mt
I. INTRODUCTION
The study of entanglement is currently an active areaof research in view of it being a physical resource forquantum information theory, quantum computing, quan-tum cryptography and teleportation [1]. At a classicallevel, entanglement does not have a corresponding coun-terpart. However, increasingly it is being realized thatthe nature of classical dynamics, whether it is regular orchaotic, affects entanglement in the quantized version ofthe system [2]. In general, larger chaos in the systemleads to larger entanglement production. This has beenestablished by considering kicked top models [3], bakersmap [4], Dicke model [5], billiard in a magnetic field [6],kicked Bose-Einstein condensates [7] and N -atom Jaynes-Cummings model [8]. In contrast to these studies, therole of classical bifurcations in entanglement of chaoticsystems has not received much attention. Even thoughentanglement is a purely quantum attribute, it is never-theless affected by the qualitative nature of the dynamicsin phase space. The results to this effect are obtainedprimarily in the context of quantum phase transitions inthe ground state of infinite systems in which the entan-glement is maximal at critical parameter values [12]. Forinstance, for case of ions driven by laser fields and cou-pled to a heat bath, i.e, a form of Dicke model was shownto exhibit maximal entanglement of its ground state atthe parameter value at which classical system bifurcates.Similar result for the ground state was reported from thestudy of coupled tops, a generalization of the two dimen-sional transverse field quantum Ising model [9] as wellfrom Jahn-Teller models [10]. The ground state entangle-ment of mono-mode Dicke model is shown to be related ∗ Permanent address: Department of Physics, Indian Institute ofTechnology Madras, Chennai, 600036, India. to Hopf bifurcation [11]. Qualitatively similar results fortwo component Bose-Einstein condensate are also known[13]. In all these cases, the treatment is confined mostlyto the ground state of the system that exhibits criticalityand involves one single classical bifurcation.Do these results hold good for chaotic, smooth Hamil-tonian systems that do not exhibit criticality in the senseof phase transitions ? As opposed to a single bifurcation,what happens in bifurcation sequences where stabilityloss and stability gain interleave one another ? Boththese questions explore the connection between chaos andentanglement in a physical setting that is different fromthe earlier studies. In the context of this work, we ex-amine a Hamiltonian system whose classical dynamics iscontrolled by a single tunable parameter. The changesin the parameter leads to changes in the phase spacestructure; for instance regularity to chaos transition andbifurcation sequences of fixed points. Typically, chaoticsystems display a sequence of bifurcations. Consider, forinstance, the coupled oscillator systems, a paradigm ofchaos for smooth Hamiltonian systems and is related toatoms in strong magnetic fields, the quadratic Zeeman ef-fect problems [14]. In these cases, one particular sequenceof bifurcation is a series of pitchfork and anti-pitchforkbifurcations [15]. The pitchfork corresponds to a peri-odic orbit losing stability and in the Poincar`e section thisappears as a elliptic fixed point giving way to a hyper-bolic fixed point. The anti-pitchfork is when the periodicorbit gains stability. In this work, we consider coupledquartic oscillators and show that the entanglement in thehighly excited states of the system is modulated by clas-sical bifurcations. We could place this in the context ofworks that lend support to the notion that for genericone-particle states there is a strong correlation betweenentanglement and measures of localization [16, 17, 18].
II. ENTANGLEMENT IN A BIPARTITESYSTEM
A pure quantum state | Ψ i composed of many subsys-tems | φ i i is said to be entangled if it cannot be writtendown as a direct product of states corresponding to eachof the subsystem. | Ψ i entangled = | φ i ⊗ | φ i ⊗ | φ i ........ ⊗ | φ n i (1)Thus, entanglement implies stronger than classical cor-relations. If ρ = | ψ ih ψ | is the density matrix representa-tion for a pure state | ψ i , then the reduced density matrix(RDM) can be obtained by applying the trace operationto one of the degrees of freedom. Thus, ρ = Tr | ψ ih ψ | ρ = Tr | ψ ih ψ | (2)are two RDMs, whose one of the degrees of freedom istraced out. The notation Tr i denotes that the trace oper-ation is applied on the i th degree of freedom. Schmidt de-composition [1] provides a representation for | ψ i in termsof product of basis states, | ψ i = N X i =1 p λ i | φ i i (1) | φ i i (2) (3)where | φ i i (1) and | φ i i (2) are the eigenvectors of the RDMs ρ and ρ respectively, and λ i are the eigenvalues of eitherof the RDMs. The von Neumann or the entanglemententropy of pure state is given by, S = − N X i =1 λ i log λ i (4)Thus, when S = 0, the subsystems are not entangled andwhen S >
0, they are entangled. The Schmidt decompo-sition provides a compact and unique representation forthe given eigenstate (unique in the generic case when thenon-zero spectrum of the RDM is nondegenerate).
III. HAMILTONIAN MODEL ANDBIFURCATION SEQUENCEA. Quartic oscillator
We consider the Hamiltonian system given by, H = p x + p y + x + y + αx y (5)with α being the tunable chaos parameter. For α =0 , ,
6, the system is classically integrable and becomespredominantly chaotic as α → ∞ . This has been exten-sively studied as a model for classical and quantum chaosin smooth Hamiltonian systems [19] and exhibits qualita-tively similar dynamics as the host of problems involvingatoms in strong external fields. In the limit α → ∞ , it is also of relavance as model of classical Yang-Mills field[20]. To study the quantum analogue of this system, wequantize it in a symmetrized basis set given by, ψ n ,n ( x, y ) = N ( n , n ) [ φ n ( x ) φ n ( y ) + φ n ( x ) φ n ( y )](6)where N ( n , n ) is the normalization constant and φ ( x ) φ ( y ) is the eigenstate of unperturbed quartic oscil-lator with α = 0. The choice of this form of basis set isdictated by the fact that the quartic oscillator has C v point group symmetry, i.e., all the invariant transforma-tions of a square. Hence we have chosen the symmetryadapted basis sets as in Eq. 6 from A representation of C v point group.Thus, the n th eigenstate is,Ψ n ( x, y ) = X j ( n ,n )=1 a n,j ( n ,n ) ψ n ,n ( x, y ) (7)where a n,j ( n ,n ) = h ψ ( x, y ) | Ψ n ( x, y ) i are the expansioncoefficients in the unperturbed basis space. Note that n , n are even integers and a n,j ( n ,n ) = a n,j ( n ,n ) in A representation of C v point group. The eigenvalueequation is solved numerically by setting up Hamiltonianmatrices of order 12880 using 160 even one-dimensionalbasis states. B. Bifurcation sequence in quartic oscillator
In a general chaotic system many bifurcation sequencesare possible. However, a two dimensional Hamiltoniansystem can exhibit only five types of bifurcations [15].One such prominent sequence is a series of pitchfork andanti-pitchfork bifurcation shown schematically in Fig 1.To reiterate, a pitchfork bifurcation takes place when astable orbit loses stability and gives rise to two stableorbits. Anti-pitchfork bifurcations happen when a sta-ble orbit is spontaneously born due to the merger of twounstable orbits. We will focus on a particular periodicorbit, referred to as the channel orbit in the literature[21], given by the initial conditions { x, y = 0 , p x , p y = 0 } ,which displays such a bifurcation sequence. The Poincar´esection in the vicinity of the channel orbit has interestingscaling properties and the orbit itself has profound influ-ence on a series of quantum eigenstates, called localizedstates, in the form of density enhancements or scars [22].Such density enhancements due to channel orbits havealso been noted in atoms in strong magnetic fields or thediamagnetic Kepler problem [23] as well.The stability of the channel orbit in the quartic oscil-lator in Eq (5) is indicated by the trace of monodromymatrix J ( α ) obtained from linear stability analysis. Itcan be analytically obtained for the channel orbits [24]as, Tr J ( α ) = 2 √ (cid:16) π √ α (cid:17) . (8)The channel orbit is stable as long as | Tr J ( α ) | < J ( α ) = ±
2. From
Chaos parameter
FIG. 1: The schematic of a typical bifurcation sequenceinvolving a series of pitchfork (circles) and anti-pitchfork(square) bifurcations as a function of chaos parameter. Thesolid lines indicate that the orbit is stable and dashed lineindicate instability.FIG. 2: The Poincare section for the quartic oscillator in Eq.5 in shown for (a) α = 90 and (b) α = 90 .
5. Note that at α = 90 the periodic orbit undergoes a pitchfork bifurcation. this condition, it is clear that the bifurcations take placeat α n = n ( n + 1), ( n = 1 , , .... ). Thus the channel or-bit undergoes an infinite sequence of pitchfork and anti-pitchfork bifurcations at α = α n . Note that for n = 9, wehave α = 90 as one of the pitchfork bifurcation points.The Poincar´e sections displayed in Fig 2 shows that thestable channel orbit at α = 90 (Fig 2(a)) bifurcates andgives birth to two new stable orbits (Fig 2(b)) while thechannel orbit itself becomes unstable. Thus, pitchforkbifurcations take place at α n = 2 , , , , , ..... andanti-pitchfork at α n = 6 , , , , .... . This can be ob-served in the plot of Tr J ( α ) as a function of α shown inFig 3. IV. QUARTIC OSCILLATOR STATES ANDREDUCED DENSITY MATRIXA. Quartic oscillator spectra
The quantum spectrum of the quartic oscillator is ex-tensively studied and reported [19, 22, 28]. For the pur-poses of this study, we note that two classes of eigen-states can be identified. The first one is what we call ageneric state whose probability density | Ψ n ( x, y ) | cov-ers the entire accessible configuration space. Most of theeigenstates fall in this class and they are instances of α -3-2-10123 Tr J( α ) FIG. 3: The linear stability of the channel orbit as a functionof α . The orbit is stable for | Tr J ( α ) | <
2. The pitchforkbifurcation points are indicated by circles and anti-pitchforkbifurcations are indicated by squares.
Berry’s hypothesis that the Wigner function of a typicalchaotic state condenses on the energy shell [25]. In Fig4(a), we show the expansion coefficients for the 1973rdeigenstate of the quartic oscillator counted sequentiallyfrom the ground state for α = 90. Notice that the stateis delocalized over a large set of basis states. These classof states are well described by random matrix theory.The second class of states is the localized states, whichhas enhanced probability density in the vicinity of theunderlying classical periodic orbits. Theoretical supportfor this class of states based on semiclassical argumentsis obtained from the works of Heller [26], Bogomolny andBerry [27]. As a typical case, Fig 4(b) shows the expan-sion coefficients for the 1972nd state which is localizedover very few basis states in contrast to the one in Fig4(a). In this work, we concentrate on a subset of sucheigenstates whose probability density is concentrated inthe vicinity of the channel periodic orbit. This set ofstates are nearly separable and can be approximately la-belled by a doublet of quantum numbers ( N,
0) using theframework of adiabatic theory [22, 28]. Note that such alabeling is not possible for the generic states since theyare spread over a large number of basis states.
B. Reduced density matrix
In this section, we compute the eigenvalues of the RDMand the entanglement entropy of the quartic oscillatoreigenstates as a function of the chaos parameter α . Interms of the expansion coefficients in Eq. (7), the ele-ments of RDM, R x , can be written down as, h n | ρ ( x ) | n ′ i = M X n =1 K n ,n a n ,n a n ,n ′ , (9)where the normalization constant K n ,n = 1 if n = n and = 1 / n = n . In this case, the y -subsystemhas been traced out. Similarly another RDM, R y ,with elements h n | ρ ( y ) | n ′ i can be obtained by tracingover x variables. Let A represent the eigenvector ma-trix of order ( M + 2) / a n ,n , where i -20-15-10-50 0 20-100 | a n,j(n ,n ) | j(n ,n ) L og λ i (a)(b) (c)(d) FIG. 4: Quartic oscillator eigenstates for α = 90 in the unper-turbed basis. (a) 1973rd state (delocalized), (b) 1972nd state(localized). The inset is the magnification of the dominantpeak. The eigenvalues of the RDMs for (c) 1973rd state and(d) 1972nd state. The inset in (d) is the magnification of thedominant eigenvalues that display degeracy. n , n = 0 , , ....M labels the rows and columns respec-tively. Then, in matrix language, the RDM R x = A T A is matrix of order ( M + 2) / M = 318 and we numerically diagonal-ize the RDM of order 160. The eigenvalues of RDM fora typical delocalized state and a localized state is plot-ted in Fig 4(c,d). In general, the dominant eigenvaluesfall exponentially, though with different rates, for boththe generic and typical localized state indicating thatthe Schmidt decomposition provides a compact represen-tation for the given eigenstate. Earlier such a behaviorwas noted for coupled standard maps [2]. The first fewdominant eigenvalues of RDM for localized states display(near-)degeneracy (see Fig 4(d)). This arises as a conse-quence of ( i ) C v symmetry of the potential due to whichthe eigenvector matrix is symmetric, i.e, a n ,n = a n ,n and ( ii ) the localization is exponential in the directionperpendicular to that in which the quanta of excitationis larger [28], i.e, a N,n ∝ exp( − ωn ), where ω > N .The origin of near-degeneracy can be understood by byconsidering a simple model of 4 × n is suppressed such that a n,j ( n ,n ) = a n ,n ), P = a , a , a N, a N +2 , a , a , a N, a N +2 , a N, a N, a N,N a N +2 ,N a N +2 , a N +2 , a N +2 ,N a N +2 ,N +2 . (10)Here we have only used the one-dimensional quartic os-cillator quantum numbers (0 , , N, N + 2) because the lo-calized states can be approximately well represented byall possible doublets arising from these quantum num-bers. For instance, an adiabatic separation with the( N,
0) manifold gives a good estimate for the energy ofits localized states [28]. The representation gets betteras we add more 1D quantum numbers to the list above. The exponential localization implies that a n ,n ≈ n ∼ n . Further, a n ,n ≈ n , n << N . Thus,elements a N,N ∼ a N +2 ,N ∼ a N +2 ,N +2 ∼ a , ≈
0. Then,we can identify a block matrix B with non-zero elementsas, B = (cid:18) a N, a N +2 , a N, a N +2 , (cid:19) (11)Then, the eigenvector matrix P can be approximated as, P ≈ (cid:18) T (cid:19) . (12)Under the conditions assumed above, the RDM separatesinto two blocks which are transpose of one another. Thus,the reduced density matrix will have the form, R = P T P = (cid:18) BB T
00 B T B (cid:19) (13)Since the eigenvalues remain invariant under transpo-sition of a matrix, i.e , the eigenvalues of BB T and B T B are identical and hence we obtain the degeneracy.Though we use a 4 × N = 264and the dominant eigenvalue of RDM using the approx-imate scheme in Eqns (10-13), is λ = 0 . C. Entanglement entropy
Entanglement entropy for each eigenstate is computedfrom the eigenvalues of the RDM using Eq (4). In Fig5, we show the entanglement entropy of the quartic os-cillator at α = 30 for one thousand eigenstates startingfrom the ground state. The localized states have valuesof entanglement entropy much lower than the local aver-age as seen from the dips in the figure. Most of them aremuch closer to zero and substantiate the fact that theyare nearly separable states. In the next section we willshow that the entanglement entropy of localized state ismodulated by the bifurcation in the underlying channelperiodic orbit.The generic delocalized states, on the other hand, formthe background envelope seen in Fig 5. These chaoticstates are not affected by the bifurcations in the iso-lated orbits. It is known that such delocalized states can D S FIG. 5: (Color Online) Entanglement entropy for the quarticoscillator at α = 30 from ground state to 1000th state. Thelocalized states have lower value of entanglement entropy asseen from the dips in the curve. The solid red curve is S RMT ,the RMT average of entanglement entropy. be modeled using random matrix theory and hence thedistribution of their eigenvectors follows Porter-Thomasdistribution [29]. The entanglement entropy can also becalculated based on RMT assumptions and it is knownto be [30], S RMT = ln( γM ) where γ ≈ / √ e and M isthe dimensionality of the reduced density matrix. In thecase of quartic oscillator, the Hilbert space is infinite indimension and we take M to be the effective dimension M eff of the RDM. One indicator of the effective dimen-sion of the state is the inverse participation ratio of theeigenstates. Based on this measure and due to C v sym-metry of the quartic oscillator, we have M eff = D where D is the state number. Thus, the effective dimension ofRDM is, M eff = √ D . Finally, we get for the entangle-ment entropy, S RMT = ln( γM eff ) ∼ ln( γ √ D ) . (14)In Fig 5, S RMT is shown as solid red curve and it correctlyreproduces the envelope formed by the delocalized stateswhile the localized states stand out as deviations fromRMT based result, namely, S RMT . V. ENTANGLEMENT ENTROPY ANDBIFURCATIONS
In this section, we show the central result of the pa-per that the entanglement entropy is a minimum at thepoints at which the underlying periodic orbit undergoesa pitchfork bifurcation. As pointed out before, the lo-calized states of the quartic oscillator are characterizedby the doublet ( N,
0) and are influenced by the chan-nel periodic orbit. We choose a given localized state, say,with N = 200 and compute the entanglement of the samestate, i.e, (200 ,
0) state as a function of α . The state thatcan be characterized by the doublet (200 ,
0) will be a lo-calized state at every value of α . The result is shown inFig 6 as the curve plotted with open circles. The values α S FIG. 6: (Color Online) Entanglement entropy as a functionof α . The three curves correspond to different ( N,
0) typelocalized states; solid circles (240,0), open circles (200,0) andsquares (210,0). The positions of pitchfork bifurcations (tri-angle up) and anti-pitchfork bifurcations (triangle down) aremarked on both the x -axes. of α at which the pitchfork and anti-pitchfork bifurca-tion takes is marked in both the horizontal axes of thefigure as triangle-up and triangle-down respectively. Forthe purpose of easier visualization, they are connectedby vertical lines. Notice that the entanglement entropyattains a local minima in the vicinity of every classicalpitchfork bifurcation and it attains a local maxima nearevery anti-pitchfork bifurcation. As Fig 6 shows, similarresult is obtained for two different localized states with( N = 210 ,
0) and ( N = 240 , V ( x, y ) = x + y + βx y ,where β is the chaos parameter.At a pitchfork bifurcation, as shown in Fig. 2, thefixed point corresponding to the channel periodic orbitloses stability and becomes a hyperbolic point. The cen-tral elliptic island seen in Fig 2(a), breaks up into twoislands. The localized state that mainly derives its sup-port from the classical structures surrounding the stablefixed point suffers some amount of delocalization, butis largely supported by the stable regions. At an anti-pitchfork bifurcation, the hyperbolic point becomes anelliptic fixed point and the orbit has gained stability anda small elliptic island just comes into existence. Hence,the eigenstate is still largely delocalized since the smallelliptic island is insufficient to support it. This heuristicpicture which is quite sufficient to explain oscillations inlocalization measures is seen to be surprisingly valid evenfor the somewhat less intuitive measure of entanglement.It has been noted earlier that when the correspondingclassical system undergoes a pitchfork bifurcation, theentanglement entropy defined by Eq (4) attains a maxi-mum [9], and it has been conjectured to be a generic prop-erty. We note that this, apparently contradictory result,is however in the context of an equilibrium point under-going a bifurcation and the relevant state is the groundstate, whereas in the case we are studying here the or-bit that is bifurcating is a periodic orbit and the statesare all highly excited. In this situation there is a muchtighter correlation between more conventional measuresof localization (Shannon entropy, participation ratio etc.)and entanglement.As the parameter α is increased, the quartic oscillatorgets to be predominantly chaotic and this should implyincrease in entanglement. However, this is true only forthe generic delocalized states as seen in Fig 5. The local-ized states are influenced not so much by the increasingvolume of chaotic sea but by the specific periodic orbitsthat underlie them. Hence, for these states, it is onlyto be expected that the qualitative changes in the phasespace in the vicinity of the corresponding periodic orbitsaffect quantum eigenstate and hence its entanglement aswell. This can be expected to be a generic feature ofentanglement in quantum eigenstates of mixed systems. VI. CONCLUSIONS
In summary, we considered a smooth Hamiltonian,namely the two-dimensional, coupled quartic oscillator as a bipartite system. We study the effect of classicalbifurcations on the entanglement of its quantum eigen-states. The quartic oscillator is a classically chaotic sys-tem. One particular class of eigenstates of the quarticoscillator, the localized states are scarred by the chan-nel periodic orbits. We have shown that the entangle-ment entropy of these localized states is modulated bythe bifurcations in the underlying channel periodic orbit.When this orbit undergoes a pitchfork bifurcation, theentanglement attains a local minimum and iwhen it un-dergoes an anti-pitchfork bifurcation the entanglement isa local maximum. Physically, this is related to the pres-ence or the absence of elliptic islands in the phase spacein the vicinity of the channel orbit. We expect this tobe a general feature of bipartite quantum systems whoseclassical analogue display bifurcation features.
Acknowledgments
We thank J. N. Bandyopadhyay for discussions andcomments. [1] M. A. Nielsen and I. L. Chuang,
Quantum Computationand Quantum Information (Cambridge University Press,Cambridge, 2000).[2] A. Lakshminarayan, Phys. Rev. E , 036207 (2001); K.Furuya1, M. C. Nemes, and G. Q. Pellegrino, Phys. Rev.Lett. , 016217 (2004);C. Mejia-Monasterio, G. Benenti, G. G. Carlo, G. Casati,Phys. Rev. A , 066201 (2003); J. N. Bandyopadhyay and A. Laksh-minarayan, Phys. Rev. E , 016201 (2004).[4] A. J. Scott and C. M. Caves, J. Phys. A , 9553 (2003).[5] Xi-Wen Hou and Bambi Hu, Phys. Rev. A , 042110(2004).[6] Marcel Novaes and Marcus A. M. de Aguiar, Phys. Rev.E , 045201(R) (2004).[7] Q. Xie and W. Hai, Eur. Phys. J. , 265 (2005).[8] R. M. Angelo, K. Furuya, M. C. Nemes, and G. Q. Pel-legrino, Phys. Rev. A , 043801 (2001);[9] A. P. Hines, R. H. McKenzie and G. J. Milburn, Phys.Rev. A , 042303 (2005).[10] A. P. Hines, C. M. Dawson, R. H. McKenzie and G. J.Milburn, Phys. Rev. A , 022303 (2004).[11] M. C. Nemes et al. , Phys. Lett. A , 60 (2006). [12] T. Osborne and M. Nielsen, Phys. Rev. A , 032110(2002); A. Osterloh et al., Nature , 608 (2002).[13] Q. Xie and W. Hai, Eur. Phys. J. , 277 (2006).[14] H. Friedrich and D. Wintgen, Phys. Rep. , 37 (1989).[15] J.-M. Mao and J. B. Delos, Phys. Rev. A , 1746 (1992).[16] A. Lakshminarayan, V. Subrahmanyam, Phys. Rev. A , 052304 (2003).[17] H. Li, X. Wang, and B. Hu, J. Phys. A:Math. Gen. ,10665 (2004).[18] H. Li and X. Wang, Mod. Phys. Lett. B , 517 (2005).[19] O. Bohigas, S. Tomsovic and D. Ullmo, Phys. Rep. ,45 (1993).[20] W.-H. Steeb, J. A. Louw and C. M. Villet, Phys. Rev. D , 1174 (1986); A. Carnegie and I. C. Percival, J. Phys.A , 801 (1984).[21] K. M. Atkins and G. S. Ezra, Phys. Rev. A , 93 (1994).[22] B. Eckhardt, Phys. Rev. A , 3776 (1989); J. Za-krzewski and R. Marcinek, Phys. Rev. A , 7172 (1990).[23] K. Muller and D. Wintgen, J. Phys. B , 2693 (1994);D. Delande and J. C. Gay, Phys. Rev. Lett. , 1809(1987).[24] H. Yoshida, Cell. Mech. , 363 (1983); Physica ,128 (1987).[25] M. V. Berry[26] E. J. Heller, Phys. Rev. Lett. , 1515 (1984); in Les
Houches LII, Chaos and Quantum Physics, edited byM.-J. Giannoni, A. Voros, and J. Zinn-Justin, (North-Holland, Amsterdam, 1991).[27] M. V. Berry, Proc. R. Soc. Lond. A , 219 (1989); E.B. Bogomolny, Physica D , 169 (1988).[28] M. S. Santhanam, V. B. Sheorey and A. Lakshminarayan, Phys. Rev. E , 345 (1998).[29] F. Haake, Quantum Signatures of Chaos , (Springer-Verlag, Berlin, 2000).[30] J. N. Bandyopadhyay and A. Lakshminarayan, Phys.Rev. Lett.89