"Quantum Chaos" with Time-Periodic Hamiltonians ...

Progress of Theoretical Physics Supplement No.

98, 1989

"Quantum Chaos" with Time-Periodic Hamiltonians Giulio CASATI and Luca MOLINARI

Dipartimento di Fisica dell'Universita di Milano and L N. F. N., Sezione di Milano, via Celona 16, 20133 Milano (Received December

19, 1988)

Time-periodic Hamiltonian systems are an easily accessible area for comparing quantum and classical chaos. This pedagogical paper reviews the two important models of the kicked rotator and the Hydrogen atom in a microwave, both characterized by the quantum phenomenon of dynamical localization, which strongly affects the classically diffusive behavior.

Contents § O. Introduction § 1. The Floquet operator 1.1. An autonomous formulation for time-dependent Hamiltonians 1.2. The periodic case 1.3. Spectral properties of the Floquet operator 1.4. The A. C. Stark-Lo Surdo effect 1.5. Exact examples § 2_ Kicked dynamics 2.1. General discussion 2.2. The Maryland construction § 3. Tight binding models § 4. The kicked rotator 4.1. The classical model 4.2. The quantum description 4.3. Anderson's dynamical localization 4.4. Some comments § 5. The hydrogen atom in a microwave field 5.1. Introduction 5.2. Kepler's map 5.3. Quantum numerical experiments § 6. Conclusions Appendices A The Devil's staircase B Liouville and Diophantine numbers C Chirikov's criterion of overlapping resonances D The Sturm basis for the hydrogen atom

287

288

G. Casati and L. Molinari

§ O.

Introduction

"Quantum chaos" is not a theory: It is the definition of a rapidly expanding area of research whose aim is to explore the modifications introduced by quantum mechanics into the dynamics of classical systems which manifest deterministic chaotic behavior. The problem is wide and difficult, but of great importance for several reasons, a deeper understanding of the correspondence between classical and quantum descriptions and the justification of quantum statistical mechanics. On one side the stability criteria for the trajectories of points in phase space, on the other the spectral features and pattern of wave packets: We face again a wave-particle duality. There is evidence that the classical regular or chaotic motions have their quantum counterpart in the statistics of level spacings, respectively Poissonian or Wigner-Dyson. This has been tested in many cases where the two regimes are accessible by changing a parameter, like billiards, nonlinear oscillators. Bounded and conservative systems with a finite number of particles, however, are characterized by a pure point energy spectrum and a recurrent dynamics: This leaves no room for any kind of chaotic behavior in the time evolution of the wave function. To circumvent this limitation, one may consider periodically driven Hamiltonian systems. Chaotic behavior in the classical dynamics of such systems is almost a rule, even in one configurational dimension. The relevant quantum numbers are the eigenvalues of the Floquet operator, giving the evolution of the state over one period. The allowance for a continuous spectrum is a chance which may favor analogies with the classical evolution. Among the simple models that may be devised for a fair understanding of both quantum and classical dynamics, much research has been devoted to the "periodically kicked rotator", introduced in 1977 at the Como conference. Many surprising facts originated from it. The classical dynamics is described by the famous Standard Map: a prototype of two dimensional measure-preserving chaotic maps. The quantum dynamics has shown the very interesting phenomenon of "dynamical Anderson localization", which turns on as soon as the discreteness of the Floquet spectrum is being resolved, and limits the wave packet spreading over the rotator's basis. This behavior may be strictly compared to that occurring in tight binding models of solid state, by means of a mapping (the Marylandconstruction). Localization is expected to occur also in the much more realistic and relevant case of the ionization of Hydrogen atoms interacting with a microwave field, in a range of frequencies that is presently under laboratory control. The quantum theory so far developed (1984) which provides a gross description of the phenomenon, and numerical computations seem to be in good agreement with the most recent experiments. This may support the hypothesis that dynamical localization is likely to constitute a general feature of quantum chaos. The present understanding in quantum chaos often originates more from qualitative than quantitative considerations, analogies, and indications coming from numerical experiments. Mathematical rigour, though desirable, is a rare occurrence: Only few are the landmarks in this "terra incognita".

"Quantum Chaos" with Time-Periodic Hamiltonians

289

The aim of this paper is to illustrate the main achievements in the study of some simple time-periodic models of quantum chaos. It is addressed to the non-specialist, and therefore some complementary sections have been included. § 1.

The Floquet operator

The beautiful Floquet theory,9) dating 1883, is concerned with the solutions of differential equations with periodic coefficients, like ji +cv(t)y=O,

cv(t+ T)=cv(t).

If Yi(t) are two independent solutions, also Yi(t + T) are: Therefore Yi(t + T) = ~jar:;yj(t). For the general solution F(t)= ~iCiYi(t) one then has F(t + T) =~r:;ar:;ciYit). The occurrence of F(t+ T)=aF(t) is implied by det(a-aI)=O. In this situation, defining the characteristic exponent jJ. by a=exp(jJ. T) one, may write the general solution as F(t)=exp(jJ.t)¢(t), with ¢(t) aT-periodic function.

In quantum mechanics, a main tool for investigating time periodic systems is the Floquet operator, giving the one period evolution of the wave function. Its eigenvectors are solutions of Schrodinger's equation with the property ¢(t)=ex p ( -

~ ct )¢(t) ,

¢(t + T)= ¢(t) .

For the physicist c is a "quasienergy" value, and the quasienergy spectrum takes the place of the spectrum of the time-dependent Hamiltonian in the description of the dynamical properties of the system. In this paragraph we introduce these important concepts. 1.1.

An autonomous formulation for time-dependent Hamiltonians

In the usual formulation of quantum mechanics, given a time-dependent Hamiltonian fl(t) acting on a Hilbert space .JC, the solutions of Schrodinger's equation fl(t)¢(t)=in

~~

(1·1)

evolve according to a Unitary Propagator,7) a two-parameter family of unitary operators connecting the wave function at different tirnpc> ¢(t)= D(t, s)¢(s) .

(1· 2)

The iterative solution of the formal integral equation that may be derived from (1·1) (1·3)

is given by Dyson's expansion with time ordering, since Hamiltonians at different times do not in general commute

(1·4)

290


The propagator has the following properties : (a)

O(t, s) O(s, r)= O(t, r),

(b)

I, O(t, s)t = O(t, S)-I= O(s, t).

(c)

O(t, t)=

There is however a different and elegant way to describe time-dependent systems, in which time is no more a distinguished variable and the theory is more symmetric. In classical mechanics, starting with a time-dependent Hamiltonian H(P, q, t), it is always possible to produce an equivalent Hamiltonian description which is conservative with respect to a new, fictitious time variable TJ. The phase space is extended to include the ordinary time t as a canonical configurational variable, with the "energy" as conjugate momentum. The new Hamiltonian is K(p, q; E, t)=H(p, q, t)+ E .

(1°5)

The equations of motions for t and E are dt = aK =1 dTJ

aE

'

showing that the energy term works as a Lagrangian multiplier, restricting TJ to be t up to an additive constant, thereby giving the ordinary Hamilton's equations for P and q. Moreover, the variations of E are minus the variations of H, and this justifies the identification of E with the energy of an external field responsible for the variations of ~e energy H(t) of the system. The picture will be pushed further with quantization. This formal procedure allows the straightforward exploitation of the many nice properties of autonomous Hamiltonian systems. A similar construction was applied by Howland6 ),IO) in quantum mechanics with a time-dependent Hamiltonian ii(t), acting on a Hilbert space..9C with norm 11011.9f. Let us define the extended Hilbert space V(5R.,..9C) of functions f/J: t-+f/J(t)E..9C such that

The classical phase variables t and E are quantized into a multiplication operator (Tf/J)(t)= tf/J(t) and a time derivative E= - ina/at respectively, satisfying a canonical commutation relation [E, f]=-in. The quantum operator corresponding to K is

- = H(t) - -

K

a

inTi'

(1°6)

A solution of the extended Schr6dinger's equation in the auxiliary time TJ

(107) is formally given by the action of a one parameter group on the initial condition f/J (TJ=O)=f/JoELZ(5R.,..9C) : (loS)


291

A direct computation shows that the function O(t, t - r; )¢o(t - r;) is a solution of (1° 7)

with the same initial condition ¢o(t); this establishes the connection between the two descriptions. For any function ¢EV(!R, .9C) (1°9)

It is interesting to remark that the operator K, like the energy operator E, satisfies the canonical commutation, relation [K, f] = - iii. Then, by von Neumann's theorem it is unitarily equivalent to E, generator of time translations

f= vtfv.

K=VtEV ,

(1°10)

The latter relation impiies that V acts as a multiplication operator with respect to the variable t: (V¢)(t)= V(t)¢(t). Being unitarily equivalent to E, the operator K shares with it an absolutely continuous spectrum. In terms of the corresponding unitary groups e- iqKn,= V t e-iqElh V

or (e-iqKlh¢)(t)= Vt(t) V(t- r;)¢(t- r;).

Comparing with formula (1°9), Oct, s)= vt(t) yes) . 1.2.

(1°11)

The periodic case

Let us suppose hereafter that the Hamiltonian is time periodic: ii(t+ T)=ii(t), implying the further property (d)

O(t+ T,

s+ T)= Oct, s),

which encloses the full knowledge of the time evolution into the set O(s+ r, s), r:::;;: T. The propagator over one period O(s+ T, s) defines the Floquet operator Fs, whose spectral properties do not actually depend on the initial time s, since they are unitarily equivalent: Fs=O(s, t)FtO(s, t)t. We shall therefore refer to F=O(T,O). The relation O(nT,O)=F n shows that the Floquet operator describes the long term behavior of the quantum evolution, and is therefore of primary importance in the study of time periodic quantum systems. In Howland's description one notices that K now commutes with the one-period time translation operator (P¢)(t)=¢(t + T) [K, P]=O,

-

(z-)

P=exp tiTE .

(1°12)

In a way similar to Bloch's construction for periodic crystals, one removes the degeneracy arising from periodicity in the time configurational variable by restricting the operator K to the subspace of periodic functions, or the space V([O, T],.9C) with boundary conditions ¢(T)=¢(O). The restriction KT is known as the quasi-energy operator. For time periodic systems the quasienergy operator KT plays essentially the same role as the Hamiltonian for autonomous systems. Its application to atomic systems in the field of a monochromatic radiation goes back to the works of ZeI' dovich2 ) and Ritus,3) in 1966. Let us sketch the construction. The eigenvalue equation for P is solved by

292


functions of the type PA(t)=eiAt¢.(t)

with

¢A(t+ T)=¢A(t).

Any function in the extended Hilbert space is a superposition of such states parameterized by ilE[O, 2n/T]

f

¢(t)= dilc(iI)PA(t) .

In other words, one has a continuous decomposition of V(IR,.9C) into P-invariant subspaces .9C Aof functions which are the product of exp(iilt )times a periodic function. The eigenvalue equation of the operator K, restricted to any such subspace, is (fi(t)-in

1t )eiAt¢.(t)=ceiAt¢.(t)

or K¢A=(C-nil)¢A.

Neglecting the translation by nil, the action of the operator K on each subspace is described by the quasi-energy operator KT, previously defined. By definition, the quasienergy point spectrum is found by solving the eigenvalue equation in V([O, T],.9C) (1·13) One may however consider other equivalent equations, that will be given after the following two remarks about the spectrum: (1) If ¢ is one solution of (1·13), also exp( - i2nnt/n T)¢(t) is a solution, with quasienergy c+2nn/T. Every value in a(KT) is therefore replicated an infinite number of times and possible proper eigenvalues of KT may turn to be immersed in a shifted replica of the continuum part of the quasi energy spectrum. (2) In the particular case of a Hamiltonian not depending on time, the spectrum of the quasienergy operator is given by

-----r-

{ c±n( 2nn ) ,cEa(H), - nEN} . a(KT)=

(1·14)

Both remarks suggest the interpretation of the system as a coupled matter-photon system. In place of Eq. (1·13), one may solve SchrOdinger's equation (1·1) with the requirement that its solution is of the form ¢(t)=exp(-ict/n)¢(t), with ¢(t+T) =¢(t). Equation (1·9), written for 'fJ= T and taking into account the periodicity of functions, shows that the Floquet operator F is unitarily equivalent to exp (- iTKT/n) ; the quasienergy spectrum (modulo 2n/T) may thus also be found solving in .9C the eigenvalue equation (1·15) For an outlook on the applications and approximations of the Floquet operator, the reader may consult Chu's paper. 22 ) 1.3.

Spectral properties of the Floquet operator

We briefly review some of the definitions and spectral properties that will be often


293

recalled in the next paragraphs, with particular reference to Floquet operators. According to the spectral theorem for a unitary operator F on a Hilbert space..9C, one has the following orthogonal decomposition of the space into F-invariant subspaces: ..9C =..9C pptB..9C actB..9Csing ,

(1'16)

respectively called the pure point, the absolutely continuous and the singular continuous subspace. They are defined in terms of spectral measures, and their relation to Lebesgue measure. For any vector if; in ..9C there exists a positive measure (the spectral measure) J.I.~ such that (1'17) (a) The pure point subspace ..9C pp is spanned by the eigenvectors of F, solutions of (1'15) with the quasi energy values forming a denumerable set: the pure point spectrum (Jpp(KT ). The spectral measure for a vector in this subspace is concentrated on this point set, and zero elsewhere. The eigenvectors (Floquet states) corresponding to different eigenvalues are orthogonal. Not surprisingly, initial states belonging to ..9Cpp give rise to a recurrent dynamics, as discussed by Hogg and Huberman. 17l This means that the mean value for any observable is an "almost periodic" function of time (Vs>03 Te and {Te} such that 1/(t+Te)-/(t)l. Fln,O>=exp 2 If qi(t) are periodic, then

(1·30)

In, T>=ln, 0> and the quasienergy spectrum is discrete, with

eigenvalues

( 1)

cn=nQ n+ 2

with

1

(T dr Q=yJo x2(r)'

§ 2.

Kicked dynamics

2.1. General discussion To provide simple models that Classically show a chaotic behavior and that at the same time allow a reasonable understanding of the corresponding quantum picture, is not a simple task. A strikingly simple model that fulfills this requirement is the "periodically kicked rotator", first introduced by Casati, Chirikov, Izrailev and Ford, in 1977, which has since then become the source of many fruitful investigations. Before entering into a detailed discussion of the model (§ 4), let us put forward some general informations about kicked models, and their relation with tight binding models of solid state physics. One may in fact consider a generalization of the kicked rotator, in which an autonomous integrable Hamiltonian is periodically perturbed by impulsive terms of infinitesimal duration (kicks). The classical equations of motion are studied in the form of a map connecting phase space points after subsequent kicks. Maps are very well suited for computer experiments and allow precise information on the long time behavior and the character of the dynamics, in a way similar to Poincare surface of section technique. For -simplicity we restrict to the case of one configurational dimension. The general form of the Hamiltonian is the following:

298


H(P, q, t)=Ho(p, q)+ V(q)

f:

11=-00

8(t-nT).

(2·1)

The evolution between two kicks is given by the free dynamics, ruled by Ho, and is supposed to be known. Let To(p, q) be the image under the stationary free Hamiltonian flow of the point (p, q) after one period. Taking T(p, q) as the image over one period in the full kicked dynamics, a kick and a free evolution, one has the following canonical (area preserving) map: (p', q')= T(P, q)= To(P- V'(q), q).

-

(2·2)

Going to the quantum description, in the q representation, the wave function solves

-

~ (Hor/J)(q, t)+ V(q)r/J(q, t)n~ ...8(t-nT)=i1iTt.

(2·3)

The delta term should not be considered with too much mathematical concern: its meaning is given by the following statement. Being nT the instant of a kick, the wave function gains a phase change according to r/J(q, nT+E)=e-tv(ql/"r/J(q, nT-E).

(2·4)

Between two kicks the evolution is given by the free Hamiltonian. The main features of the quantum dynamics are contained in the spectrum of the Floquet operator, propagating the state vector from time zero to time T: (2·5) As an illustrative example, let us consider the Hamiltonian (2·6)

H(p, 0)=p+kcosO'};.8(t-nT), n

leading to the very simple map: 0'=0+ T, p'=p+ksinO. The map may be iterated to give

PN=Po+kNsin( 00+ N21 T)

with

The kinetic energy , averaged over initial phases 80 in (0,27[-), grows quadratically in N when T is a rational multiple of 27r. For the typical irrational case the energy remains bounded, while it grows more slowly than quadratically if T/27r is very close to a rational. Though obviously not showing a transition to chaos, this example shares the same two-fold behavior, with a resonant and a nonresonant regime, of the much more complex case of the kicked rotator (where the momentum is squared). The eigenvalue equation with quasienergy E for the Floquet operator is

which, for T/27r irrational, easily provides the explicit value of r/J£(O) given r/J£(O). The time behavior of the energy is the same as in classical mechanics, and it can be


299

shown that the different regimes correspond to different characters of the Floquet spectrum. This linear case, but with a general periodic potential V(O), has been investigated by Grempel et al. 2) and Berry.S) In another paper,I) Berry and others analyze the dynamics of the kicked Hamiltonian H(p, q, t)=#/2+(A/2k)q2/J"£,a(t-nT) n

(2°7)

with a particular emphasis on the evolution of Wigner's distribution. The harmonic potential (k=l), leading to a linear phase map, is solved exactly, giving a discrete quasienergy spectrum. Another model, with a finite number of states which suits well for numerical calculations, is the "kicked top".5) If ji (i=1, 2, 3) are angular momentum matrices in representation j, the j2-preserving Hamiltonian is

r

- (np)H= lz+ (nk)-2 2j Is "£,a(t-nr).

(2°8)

The classical limit is obtained for j-+oo, after rescaling Xi=j;jj. The case P=7C/2 is investigated, finding classical chaos at k::::::3. The Floquet operator O=exp( -

~; js2)exp( -iPj2)

shows a Poissonian spacing distribution of quasienergies at low values of k, and Wigner-Dyson distribution, typical of random matrices, for k large. In general, though the Floquet operator for kicked dynamics may be written explicitly, the investigation of its spectral properties is a difficult problem. Very few are the theorems; some insight has been provided by numerical computations and the possibility of establishing an equivalence of these kicked models with "tight binding models" through the Maryland construction, the subject of the next paragraph. Some rigorous results on one-dimensional tight binding models will be reviewed in § 3. 2.2.

The Maryland construction

Starting from the Hamiltonian with angular potential (2°9)

Grempel, Fishman and Prange2) have shown through an ingenuous formal construction that the eigenvalue equation for the Floquet operator e-(il1t)V(S)(e-(i/21t) TP n r/J)( 0)

= e-(il1t)Ter/J( 0)

is equivalent to an eigenvalue problem for a "tight binding model" of a peculiar type. The eigenvalue equation is transformed according to the following steps: [e-(il1t)T(p"/2-e) e-(iI21t) V(S) - e(i/21t)V(S)] e(i/21t) V(S) r/J( 0) =0 , [e-(il1t)T(p"/2-e>(

l-i

tan ~~) )-(1+ i tan ~~))Jcos( 2~)e(i/21t)V(S)r/J(0)=0,

300


which, after setting 1(B)=1/2[1 +exp(i/n) V(B)]¢(B) (the mean oithe wave function before and after a kick) becomes [i(l + e-(i/II)T(pn/2 - e »-1(1 -

e-(i/II)T(pn/ 2 -

e»

-tan( ~~») ]¢( B) =0 .

Expanding 1(B) in the angular momentum basis 1(B)="2.":=_",cre irfi , the final result is obtained: (2·10) eT Tm=tan ( 2n

n(n-l)T 4

m

n)

,

(2·11)

Equation (2·10) is the eigenvalue equation for an electron in crystal with sites m, site energies Tm and hopping matrix elements Ws. The usefulness of the construction requires the numbers Tm to be a random sequence. Moreover, the potential must satisfy the bound Ivi < ;rn, for otherwise Wr is divergent. This difficulty may be removed by a slightly different construction by Shepelyansky.7) The linear case n=l in (2·9) with the particular choice V(B)=2narctan(kcosB - E) leads to an equation with only nearest-neighbor interaction (2·12) that will be considered in the next section, where some results on tight binding models are reviewed. § 3.

Tight binding models

In this section we give some useful results about lattice discretizations of Schr6dinger's equation, known to solid state physicists as "tight binding models". These models play an important role in the investigation of transport properties of solids at low temperature, where the electron wave function becomes very sensitive to local impurities and imperfections of the crystal lattice. In a paper dating 1958, Anderson1) showed that in three dimensions a random potential may produce the localization of the wave function: ¢(x)~exp-Ix-xol/';:, with';: known as the "localization length". This localization property was shown by Mott and Twose3 ) to occur in d = 1 no matter how small the disorder is; it is the result of a subtle interplay between tunneling and interference and in principle is out of the reach of perturbation theory. For d >2 one may get extended states from localized ones above a critical energy (Mott threshold), showing up in a transition insulator-conductor. Some exact mathematical results are available, mainly for one dimensional models. They often rely on properties from number theory: A feature that is shared by kicked dynamics and relates to the problem of small denominators, familiar to the KAM theory. On the Hilbert space 12(Z) of sequences, with norm 11¢11=("2.nEzl¢nI 2)1/2, one defines the discretized laplacian


301 (3·1)

Contrary to the continuum case, it is customary to associate to the kinetic energy the laplacian with positive sign; moreover, its diagonal term is usually included into the potential, which is specified by a sequence of numbers {Vn}nez. In this way, the Hamiltonian H is a sum of a nearest-neighbor interaction term (HotP)n = tPn+l + tPn-l and a local term describing the interaction energy of the electron with the crystal site (VtP)n= VntPn. The Hamiltonian is thus tridiagonal, or a Jacobi matrix. A tight binding model is described by the eigenvalue equation

(3·2) which is conveniently given a matrix form

(Pn = Hn(E) (Pn-l with the definitions

(P =(tPn+l) n

¢n'

The solution is found iteratively starting from an initial condition (Po

(Pn=Hn(E)···H1(E)(Po = Kn(E)(Po , (P-n = H_n+l(E)-l ... Ho(E)-l (Po = K-n(E) (Po. An important notion is that of spectral density of the Hamiltonian H. To define it, one first restricts to a finite interval - L~ n~L, with boundary condition tP±IL+lI=O. HL is then a symmetric matrix of order 2L+1. For any value of E in (3·2), once tP-(L+ll is fixed equal to zero, one may calculate tPL+l by means of the transfer matrix, with generally a nonzero result. The eigenvalues of HL are then given by the zeros of tPL+l(E). For example, in the free case the eigenvalues of the matrix HOL are easily calculated to be E no= 2cos(mr/(2L + 2)), (n=l, 2, ... , 2L+ 1). The eigenvalue distribution p(E) defined as the number of eigenvalues less than E divided by (2L+ 1), in the limit L-HXJ admits a spectral density k(E) whose support is a(H), such that p(E)=fE'"!l:.Efik(E'). For the free case: dko(E)=7r- 1(4-E 2)-1/2dE, the spectrum is absolutely continuous and is given by the interval [-2,2]. The asymptotic properties of the transfer matrices Kn are relevant for the behavior of eigenfunctions and hence in the investigation of the spectral properties of the Hamiltonian. In this regard, two important parameters are the forward and backward Lyapounov's exponents, which measure the rate of exponential growth of the matrix norm (3·3) Positiveness of a Lyapounov's exponent may imply both exponentially growing and decaying solutions in one direction, depending on the initial condition. A theorem by Furstenberg and Kesten2l stating equality and positiveness of these exponents for

302


products of random matrices enabled Matsuda and Ishii to extend the result to tight binding models with random potential. For the free Hamiltonian: ro(E)=arcosh(IEI/2) if lEI >2, and is zero for lEI ~2. The Lyapounov exponent is related to the spectral properties of H through Thouless' formula: r(E)= fioglE- E'ldk(E').

(3·4)

In fact, for the finite interval we stated that the eigenvalues are solutions of rpL+l(E) =0, then rpL+l(E)~II(E-Ei) and loglrpL+l(E)I~~ilogIE-Eil which goes in (3'4) in the continuum limit. The theorem of Ishii-Pastur-Kotani states that the absolutely continuous part of the spectrum of H is the closure of the set {Elr(E)=o}ess. We now mention three very interesting cases: 1) Anderson models The potential {Vn} is chosen as a collection of identically distributed random variables. The relevant theorem is the following (Kunz and Souillard8 »): If the density is bounded and has compact support, then H has a pure point spectrum with exponentially localized eigenfunctions. Lloyd's model 4 ) corresponds to the case of Lorentzian distribution P(V)

(3'5)

the localization length ~(E) may be calculated exactly: (3·6)

2) Incommensurate periodic potentials An example is the "almost Mathieu equation" (or Harper's Eq.), with potential Vn=Acos27r(an+ 8) ,

(3'7)

8 is a free parameter and a is irrational, for otherwise the Bloch construction applies. Andre and Aubry7) have proven that the Lyapounov exponent satisfies the inequality r(E)~log(IAI/2) and therefore for IAI~2 the Hamiltonian H(8) has no a.c., spectrum. Positivity of r does not necessarily imply a point spectrum. In fact, when a is a Liouville number (that is the period of V(8) is almost commensurate to the lattice spacing), the following Gordon's inequality holds: [rp~+l + rp~-l] ~ [rp12 + rp02]/ 4 precluding a point spectrum and leaving us with a singular continuum spectrum (Avron and Simon). Simon and Bellissard succeeded in showing that aSIng has a Cantor set structure for a dense set of couples (A, 8). 3) "The Maryland model" This example, by Fishman Grempel and Prange 10) (see also Ref. 13)), is of particular interest for our purpose (2 '12). The potential is not bounded Vn=A tan(7ran+ 8) .

(3'8)


303

Again a is required to be irrational, and 8 different from those values that may give singulari ties. The Lyapounov exponent is given by the convolution

(3'9) and turns to be always positive, implying the absence of the a.c. part of the spectrum. It may be proven that for a a Liouville number the Hamiltonian has neither a pure point spectrum. However, suppose that a is a Diophantine irrational [Appendix B], then H has a pure point spectrum for all A> 0 and phases 8. The eigenvalues Em are solutions of

f

A 7i

dE' , ( 1 8) (Em - E')2 + ,12 dko(E ) =frac ma +Z-7i

.

(3·10)

The eigenfunctions display an exponential decay. § 4.

4.1.

The kicked rotator

The classical model

This is the most popular model of kicked dynamics, and perhaps the simplest. It may describe a pendulum in an intermittent gravitational field, or the dynamics of a rotating electric dipole which is periodically kicked by a photon. The Hamiltonian is (4 '1)

where 8 is the angular position of the rotator, p the ang~lar momentum, k the perturbation strength. In the present case, integration of Hamilton's equations over one period yields the well-known Standard Map for the canonical variables. It was first introduced by Chirikov,2) while considering the motion of a charged particle in a magnetic bottle. Most important, it appears as an approximation of the dynamics for several periodic systems close to the separatrix. 30 ) Let us quote a few facts about it. The map looks very simple

p'=p+ksin8, 8'=8+p'T,

(4 '2)

but nevertheless hides a very complicated dynamical pattern. It may be viewed as a canonical transformation with generator

F(p', 8)= p' 8 +-f /'" ,.:.;.....,

.

/ o~.~~~-W~~~~~~~~~~~~~-u

'b.o

1.

2.

3.

4.

5.

6.

e

Fig. 1. K=0.97 The phase space plot shows the iterations of 37 initial points randomly chosen in the square 21r X21r. Several islands are visible and two invariant curves, which prevent diffusion in action P.

O~~~~~~LLLill~UU~LL~~~~~~

"b.O

1.

2.

3.

4.

5.

6.

e

Fig. 2. K =3.0 Again the iterations of 37 initial points. The chaotic region has invaded most of the avail· able space. A large island surrounding the fixed point e=1r, P=O still survives.

2OCO.

4000.

6000.

8000.

N

10000.

Figs. 3 and 4. K=3.0 The diffusive behavior of the classical mean energy, compared with the quantum behavior, which shows localization. The mean value