On the classical-quantum correspondence for periodically time dependent systems

Chaos, Solitom

& Fracti

Vol. 5, No. 7, pp. 1143-1167, 1995 Copyright 0 1995 Elswier Science Ltd Printed in Great Britain. AU rights reserved 09604779/95$9.50 + .oo

On the Classical-Quantum Correspondencefor Periodically Time Dependent Systems MARTIN

HOLTHAUS*

Department of Physics, Center for Nonlinear Sciences, and Center for Free-Electron Laser Studies, University of California, Santa Barbara, CA 93106, USA

Abstract-A simple, but instructive approach to quantum dynamics close to classical nonlinear resonancesis discussed.It is shown that a quantum system can be fairly insensitive to a transition to chaos in the corresponding classical system; the same approximation can describe states on both sides of a ‘chaos border’. Tunneling between quantized invariant manifolds in periodically driven systems, as well as ‘scarring’ of PIoquet states, is described analytically.

1. INTRODUCTION

What is the effect of classical Hamiltonian chaos on the corresponding quantum system? This question can be traced back even to the days of the old quantum theory. If there are n constants of motion in a Hamiltonian system with n degrees of freedom, the classical phase space is completely stratified into invariant n-tori [l], and the semiclassical Einstein-Brillouin-Keller (EBK) quantization rules [2-41 can be employed to approximately calculate quantum mechanical energy eigenvalues and eigenfunctions. But how to proceed if there are less than the required n constants of motion, if the tori are not there at all? This problem, which used to be a rather severe conceptual threat to the old quantum theory [2], lost its sharpnesswith the advent of wave mechanics. Schriidinger’s equation has solutions, no matter whether the corresponding classical system is integrable or chaotic. Nevertheless, it has become more and more obvious recently [5,6] that these solutions behave somehow ‘different’ if there is classical chaos, in a sense that is sometimes hard to pin down precisely. Since the occurrence of chaotic dynamics in classical mechanics is a rule, rather than an exception, a thorough study of the classical-quantum correspondence in fully or partially chaotic systemsis of importance for a deeper understanding of quantum mechanics itself. The correspondence can also be investigated in systems under the influence of an external time dependent force; a periodically forced quantum system, the highly excited hydrogen atom in a strong microwave field, is one of the experimental cornerstones [7] in the present discussion of ‘quantum chaos’. Of course, a system with a periodically time dependent Hamiltonian operator H(t) = H(t + T) does not have energy eigenvalues or energy eigenfunctions. But there is a substitute: according to the Floquet theorem, periodic time dependence gives rise to quasienergies E and Floquet functions u(t), defined as solutions of the eigenvalue equation [8,9] [H(t) - i&]u(t) = w(t) *Present address: Universittlt Marburg, Fachbereich Physik, Renthof 6, 35032Marburg, Germany. 1143

(1)

1144

M. HOLTHAUS

with periodic boundary conditions in time, u(t) = u(t + T). The wave functions W(t) = u(t)e-‘”

(2)

then solve the time dependent Schrodinger equation. (Throughout this article, a system of units with A = 1 will be used.) If u,(t) is an eigensolution with quasienergy E,, then the product u,(t) exp (knot) is an eigensolution with quasienergy E, + mo, provided that m is an integer, and w = 27r/T. Therefore, the quasienergy spectrum can be organized into Brillouin zones, with the ‘fundamental zone’ -w/2 < E < w/2. There is an elegant way to formulate the Floquet theorem. If H(t) = H(t + T), then the time evolution operator U( t, 0) has the representation [lo, 111 U(t, 0) = P(t)exp(-iGt),

(3)

with a unitary, periodic operator P(t) = P(t f T), and a self-adjoint, time independent operator G. The spectrum of G is the quasienergy spectrum, modulo w. If the Schrodinger wave function v(t) is transformed according to the new function G(t) satisfies the equation i&$(t)

= G+(t).

(5) Thus, the unitary transformation (4) can be interpreted as a transformation to a new frame of reference, seen from which the dynamics is described by the Hamiltonian G. The Floquet theorem guarantees that this transformation always exists, but its explicit construction, as well as that of G, is a formidable problem in most cases. If the classical system is integrable, Floquet states are associated with invariant manifolds in a way that is very similar to the association of energy eigenstates with invariant tori [12]. If we restrict ourselves, for simplicity, to periodically driven systems with one degree of freedom, the classical dynamics can be described in the odd-dimensional extended phase space {(p, x, t)} spanned by momentum, position, and time; and T-periodic vortex tubes [l] replace the usual tori. The semiclassical quantization rule can then be written as [12,131 (p dx - H dt) = 2n(k + u/4), (6) I Yl where the quantization path y1 winds once around a T-periodic vortex tube, u is its Maslov index, and the integer k is the semiclassical quantum number. After the ‘quantized’ tubes have been determined, the quasienergies are given by E= -4

r (pdx - Hdt)

mod 2n/T,

where y2 is a T-periodic path along a quantized tube, and the range of integration is one period. Note that in these rules the Poincare-Cartan form p dx - H dt replaces the familiar ‘p dx’ in the usual EBK quantization conditions. Typically, a periodically driven system is neither completely regular nor completely chaotic, but both types of motion can coexist [14], sometimes with a very sharp boundary separating an almost regular from a chaotic region of phase space [15]. A particularly important situation occurs if a strong external periodic perturbation introduces a resonance, large enough to support several quantum states: then inside the mainly regular resonant island there are invariant periodic vortex tubes, and one may try to apply the semiclassical quantization rules. But what about the surrounding stochastic sea? Does the character of a Floquet state change dramatically when its vortex tube is destroyed?

1145

Periodically time dependent systems

These questions will be investigated in detail in the following sections. First, a simple analysis of quantum dynamics close to ‘large’ classical resonances will be given in Section 2. The following Sections 3 and 4 deal with the study of specific examples. In Section 3 it will be demonstrated that a quantum system can be remarkably insensitive to the classical transition to chaos; the same approximate formula can describe both states that are associated with invariant vortex tubes and states whose tubes have been destroyed. Section 4 discusses the fact that the incompatibility of the T-periodic boundary conditions imposed on the Floquet states and the NT-periodic tubes of a primary N:l resonance gives rise to a genuine quantum effect, which can be interpreted as coherent tunneling between quantized vortex tubes. The analysis also shows how ‘scarring’, i.e. the enhancement of quantum mechanical probability density along unstable periodic orbits, can occur in periodically driven systems. As discussed in the concluding Section 5, this finding may be of some importance for the interpretation of scars that have been identified in microwave experiments on highly excited hydrogen. Finally, an Appendix justifies the naive approach adopted in Section 2.

2. CLASSICAL RESONANCES IN QUANTUM MECHANICS

Let us consider a classical, one-dimensional, nonlinear oscillator described by a Hamiltonian H,(p, x), and let us assume that this system is driven by an external monochromatic force of strength L and frequency o. The total Hamiltonian is then given by H(p, x, t) = H&l,

x) + Axcos(ot).

(8)

After a canonical transformation (p, x) + (I, 8) to action-angle variables of the undriven oscillator, the unperturbed Hamiltonian depends only on the action variable 1. For simplicity, the symbols Ho and H are kept to denote the transformed Hamiltonian functions by H,(Z) and H(Z, 6, t), respectively. If the origin of the angle, 6 = 0 = 27r, is chosen such that the unperturbed trajectory is an even function of I?, then x = x(Z, 8) can be expanded in a cosine series,

so that the Hamiltonian can be expressed as H(Z,

6,

t)

=

H,(Z)

+

AxJco(z)cos(ot)

+

~~*.(z){c0s(ni3 n

+

ot)

+

cos(n6

-

wt>l.

1 (10)

If 8 = w/N, so that the time required for one cycle of unperturbed motion coincides roughly with N cycles of the external drive, then the Nth primary resonance occurs [14]: one of the arguments of the cosine functions becomes stationary. Since we have

in the unperturbed system, let us define the resonant action IN by the condition Q(Z,) = f.

M. HOLTHAUS

1146

In order to describe the dynamics close to the resonance, the usual approximations [16] are made: Ho is expanded quadratically around IN, Hot0 = E,, + Q(ZNW

PO2 + -7 2M

(13)

with AZ= I-

IN

(14)

Em = Ho(ZN)

(15)

M-1 = d2Hot0 (16) dZ I,’ In addition, only the resonant term in (10) is kept, and the coefficient xN(Z) is approximated by the constant TV. Then one is left with + ycos(N6-

H(Z, 6, t) = E,, + !2(ZN)AZ + E

cot),

(17)

where Y = ~XNVN).

(18)

Since (I, 8) is a canonically conjugate pair, so is (AZ, 19). Therefore, one can naively ‘quantize’,

and arrive at the Schriidinger equation Q(Z,) Eres+--------i

a 319

1

a2

2M

at?

+ ycos(N6-

wt) q(s, t).

(20)

Since, in general, canonical transformations and quantization do not commute, equation (20) can give an approximate description of the near-resonant dynamics in the quantum counterpart of the driven oscillator (8) only in a semiclassical regime where this noncommutativity does not matter too much, i.e. under the assumption that the classical resonance covers enough of phase space to support several quantum states. Rather than discussing the validity of the present procedure in detail now, it will be justified a posteriori in the Appendix by showing that a purely quantum mechanical analysis leads, up to minor corrections, to the same result. Substituting Nt9 - ot = 2.2 and defining $z, t) = ~((s, t), terms linear in a/(az) cancel because of the resonance condition (12), and one obtains N2 ?I2 + ycos(22) $
(21)

The ansatz +(z, t) = x(z>exp(-iWt)

(22)

then immediately leads to a Mathieu equation in standard form [17], x”(z) + [a - 2q co.5(2z)]x(z)

= 0,

(23)

Periodically time dependent systems

1147

with parameters a = F(

W - E,,)

2M~iv(Z,) 4My (25) N2 N2 ’ According to Floquet’s theorem, the Mathieu equation has solutions of the form x(z) = PV(.z)eivZ,where Y is a characteristic exponent, and P,,(z) = P,(z + ‘rr) is a r-periodic function. The corresponding solutions of (20) read

q=-=

w(6, t) = P,([N6 - wt]/2)exp(i[vNt‘l/2

- v0t/2 - Wt]).

(26)

Since 6 is an angle variable, defined modulo 2n, the wave functions q(S, t) must be 2nperiodic in 6. This requirement is compatible with (26) only if the characteristic exponent is of the form y =
j integer.

(27)

If the Mathieu parameter q is specified, the allowed values of the other parameter a are thereby restricted to certain values ak(v, q) (for which v = y(j) actually occurs as a characteristic exponent), which will be labeled by an integer k. Hence, the ‘energies’ W can take on only discrete values Wj,k. The solutions (26) can then be identified with Floquet wave functions Uj,k(~, t) exp (-iEj,kt), and their quasienergies are found to be &j,k =

Wj,k

+ ~

mod o

N2 ak(v 9 q) + @! mod CD, = &,, + 8M N where j = 0, 1, . . ., N - 1. The present solution is an example of an approximate construction of the transformation (4). The new Schradinger equation (5) with a time independent Hamiltonian is given by (21), which describes a fictitious particle with a ‘renormalized’ mass, moving in a cosine lattice. Since A6 = 277corresponds to AZ = Nn, the periodicity interval covers N wells. The operator G is the Mathieu operator, augmented by the proper boundary conditions. There are two special cases of particular interest. For N = 1 and N = 2 the quantities ak(Y, q) coincide with the characteristic values [17] of the Mathieu equation, which have been tabulated in some detail [18] and can be determined numerically with only minor effort. These two cases will be studied in the next sections, with the help of specific model systems. 3. THE DRIVEN PARTICLE IN A BOX

A fairly simple, but highly instructive model is the periodically driven ‘particle in a box’. In a system of units where the particle’s mass, the half-width of the box, and Planck’s quantum h are unity, its Hamiltonian takes the form H(p, x, t) = $p2 + V,(X) + J.xsin(ot), with

(29)

M. HOLTHAUS

1148

After a transformation to action-angle variables (with 6 = 0 corresponding to the right turning point), the Hamiltonian (10) reads H(Z, 6, t) = !fC. + $ =C ,, J-jsin(n6 8 n 1,3,5, . ?I2

+ 0X) - sin(n6 - ot)}.

(31)

Primary resonances occur for odd N only. They are located at (32)

and, within the pendulum approximation [16], their half-widths are given by (33)

Figures 1 and 2 show Poincare sections for o = 5$, so that I, = 20. The driving amplitude chosen for Fig. 1 is il/& = l/64. Above a stochastic sea, the island chain of the primary N = 3 resonance is visible, still well separated from the large resonance with N = 1. The numerically determined half-width of this resonance agrees very favorably with the theoretical value AZ1 = 5. When the perturbation strength is increased further, the stochastic sea grows. In addition, the homoclinic tangle originating from the hyperbolic periodic orbit of the N = 1 resonance gives rise to a stochastic layer around the resonant island. Thus, the area of the island of regular motion shrinks. For h/c.? = 0.057784375 (see Fig. 2) the layer is already connected to the stochastic sea; the N = 3 island chain has been completely destroyed. Equation (33) now yields AZ, = 9.6 and therefore overestimates the width of the actual island. However, AZ, agrees well with the sum of the layer width and the half width of the island. The situation considered in these two figures is ideally suited to investigate the influence of the transition to chaotic dynamics in a classical system on its quantum mechanical

30

0

.2

.4

.6

.6

1

angle 19/ 2~ Fig. 1. PoincarC section for the classical periodically driven particle in a box (29) with w = 56 A/d = 0.015625,

and

Periodically time dependent systems

0

.6 .4 / 2n angle 6

.2

.8

1149

1

Fig. 2. Poincar6 section for the classical periodically driven particle in a box (29) with w = 56 and A/& = 0.057784375.In contrast to Fig. 1, the N = 1 resonance is now surrounded by a broad stochastic layer.

counterpart. The closed contours seen inside the resonant N = 1 island are sections of vortex tubes with a plane t = const. Since these tubes are T-periodic, the semiclassical quantization rules (6), (7) apply. In the case of Fig. 2, about 8 Floquet states can be associated with invariant vortex tubes ‘inside’ the resonant island, but for other states that originate from low-lying eigenstates of the unperturbed system such an association seems no longer possible. What, then, happens on the border to the stochastic sea? The approximate analysis of the preceding section becomes fairly simple for N = 1, since the characteristic exponent (27) is restricted to Y = 0 only, and the index j becomes redundant. The allowed values of ak(O, q) are identical with the characteristic values that give rise to r-periodic Mathieu functions. Those values associated with even Mathieu functions are usually referred to [17] as ao, u2, u4, . . . ; those associated with odd functions as bz, b4, b6 . . . If one defines k = 0, 2, 4, . . . ak(q)

=

k = 1, 3, 5, . . . ’

ak(q) bk+l(q)

(34)

the approximate formula (28) for the near-resonant quasienergies can be written as 1 Ek

=

Em

+

-‘+k(q)

8M

mod o,

(35)

with

M-1 = 5

(36)

and the Mathieu parameter

q2!E. IT4

(37)

As has been emphasized by Reich1 and Lin [19,20], the particle in the box is a particularly

1150

M. HOLTHAUS

convenient model to study the role of classical resonances in quantum mechanics. The unperturbed Hamiltonian H,(Z) = $Z2/8 depends only quadratically on the action, so that a quadratic expansion around IN introduces no error, and the Fourier coefficients x,(Z) = 8/(&z*) do not depend on the action at all. Moreover, the Bohr-Sommerfeld quantization of the unperturbed system, Z + n, with n = 1, 2, 3, . . . (note that there is no Maslov correction associated with the two ‘hard wall’-reflections) yields the exact energy eigenvalues En ; the parameter M-’ is identical to the parameter Ey introduced in the quantum mechanical resonance analysis (see Appendix). The same treatment also yields (cf. (A16)) 4’

4Wxlr+l) -(= 11 E’: ?r4( (2r + 1>*1 Since the quantum number of the resonant state, r, has to be large for the analysis to be meaningful, this result is compatible with (37). Figure 3 now shows a part of the numerically determined quasienergy spectrum of the driven particle in the box for the same frequency o = 5* as employed in Figs 1 and 2, so that the resonant state is I = 20. The quasienergies displayed are those that originate from the lowest 30 energy eigenvalues of the undriven system. Figure 4, on the other hand, shows approximate quasienergies according to (39, for k = 0, 1, . . ., 14. The approximation is remarkably good for all 15 states, even up to ~/CL?= 0.06. What makes this result interesting is the fact that the approximation refers to two seemingly different types of Floquet states. The quantum number k appearing in (35) can be identified with the integer k in the semiclassical rule (6): for N = 1, the cosine potential in the effective Schrijdinger equation (21) has only a single well within the periodicity interval, and the state most tightly bound in this well corresponds to the innermost quantized vortex tube with k = 0, and so forth. For ;1/02 = 0.057784375 (see Fig. 2) the regular island of the fundamental resonance can carry no more than 8 quantized tubes (even disregarding the occurrence of higher order island chains close to the chaos border), and one might expect the semiclassical quantum number k to become meaningless for

.

3 w

Fig. 3. Part of the quasienergy spectrum of the quantum mechanical particle in a box (w = 5~9). The quasienergies displayed originate from the lowest 30 energy eigenvalues of the unperturbed system.

Periodically time dependent systems

1151

x/w2 Fig. 4. Approximate quasienergies according to (39, for the same parameters as those employed in Fig. 3. The ‘ground state’ k = 0 (originating from the resonant unperturbed state r = 20) is indicated by the arrow.

k 3 8, since the corresponding invariant tubes do no longer exist. But it is not that simple. There is a regime where the destruction of classical vortex tubes has hardly any influence on the quantum mechanical spectrum, and the same analytical approximation can describe eigenvalues both of states that are semiclassically associated with invariant manifolds and states for which such a simple association does not hold. Thus, the present example shows that a quantum system can ignore a transition to ‘soft chaos’ [5] in its classical counterpart to a large extent. It is also instructive to investigate the eigenfunctions, i.e. the Floquet states. To begin with, Fig. 5 shows a contour plot of the probability density ]u(x, t)]’ of the Floquet state that originates from the unperturbed state with n = 10, for the same parameters as in Fig. 2. Again, this state is much more ‘regular’ than the Poincare section seems to suggest: the only effect of the external perturbation is to induce a slight periodic oscillation of the 10 lobes of the unperturbed state. The Figs 6-9 show the densities of near-resonant Floquet states with k = 0, 4, 9, and 12. If a Floquet state can be associated with an invariant tube, its probability density is mainly determined [12] by the projection of that tube from the extended phase space {(p, x, c)} to the ‘configuration space’ {(x, t)}, in complete analogy to the time independent case [21]. Thus, the density of the state with k = 0 is determined by the projection of the ‘smallest’ (in diameter), innermost quantized tube, the density of states with higher k by the projection of correspondingly ‘larger’ tubes. Since all invariant vortex tubes encircle the stable, elliptic periodic orbit of the N = 1 resonance, the flow of probability density is mainly determined by this orbit. In particular, the density of the state with k = 0 (see Fig. 6) is strongly concentrated in its immediate vicinity. (It is easy to show that the elliptic periodic N = 1 orbit for the Hamiltonian (29) follows the external force and hits the left wall (x = -1) at times t = T/4 modulo T = 27r/o, and the right wall (X = 1) at t = 3T/4 mod T. In contrast, the hyperbolic orbit hits the right wall at t = T/4 mod T, and the left one at t = 3 T/4 mod T .) The state with k = 12 (Fig. 9), however, which can certainly not be associated with an invariant tube, behaves differently: it is a ‘scar’ [22], since its density peaks along the

M. HOLTHAUS

1152

.8

.6 h .

.2

Fig. 5. Contour plot of the probability density \u(r, t)12 of the Ploquet state that originates from the unperturbed eigenstate n = 10, for o = 5trr and @0r = 0.057784375(cf. Fig. 2).

.8

.6 h .

-1

-.5

0

.5

1

X

Fig. 6. Probability density of the Floquet state k = 0, for the same parameters as in Fig. 2. This state, which originates from the unperturbed state n = r = 20, is the resonance-induced ground state; its density is concentrated along the elliptic periodic orbit.

unstable, hyperbolic periodic N = 1 orbit. Note that this orbit moves in phase opposition to the elliptic one. The good agreement of the exact and the approximate quasienergy spectrum up to k = 14 suggests that the simple analysis of Section 2 can also describe this state. Now the required solutions ~~(2) of the Mathieu equation (23) are simply the n-periodic Mathieu functions. The parameters o = 5r? and A/G?= 0.057784375 employed in Figs 5-9 correspond to a Mathieu parameter q = 92.455, and the characteristic value al,(q)

Periodically time dependent systems

1153

.8

Fig. 7. Probability density of the Floquet state k = 4, for the same parameters as in Fig. 2.

.8 .6 h .u .4 .2

Fig. 8. Probability density of the Floquet state k = 9, for the same parameters as in Fig. 2.

(see (34)) is 184.9105784, very close to 2q. (The characteristic values azr(q) corresponding to even Mathieu functions can efficiently be computed as eigenvalues of the Mathieu operator -d2/dz2 + 2qcos(2z) in the space spanned by {l/v(n), ~(2/n)cos(2mz); m a 1); those corresponding to odd functions, bzl(q), as eigenvalues in the space spanned by {d(2/7r) sin (2mz); m a 1)). This means that the effective Schriidinger equation (21) admits 12 states with an energy lower than the top of the cosine well, and the energy of the state with k = 12 almost coincides with the top of the well. A fictitious classical particle, moving with that energy in the cosine well, would spend most of its time in the immediate

1154

M. HOLTHAUS

.8

96 h \u .4

.2

0 X

Fig. 9. Probability density of the Floquet state k = 12, for the same parameters as in Fig. 2. This wave function shows signs of scarring; its density is strongly enhanced along the hyperbolic periodic orbit.

vicinity of z = 0 = rr, so that the quantum mechanical probability density should be strongly enhanced at those points. This is contirmed in Fig. 10, which shows the square of the Mathieu function x12(z). Since z = r/2 corresponds to the elliptic periodic orbit of the original system, and z = 0 = 1~ to the hyperbolic one, this line of reasoning gives an intuitively clear understanding of the ‘scar’ in Fig. 9. Simultaneously, one obtains a criterion for the occurrence of scarring in periodically driven systems: the enhancement of density along the hyperbolic orbit of an N = 1 resonance should be most pronounced when

.8

Fig. 10. Square of the Mathieu function x&z), corresponding to the characteristic value a12(q) for q = 92.455. The function is normalized such that [adz&(z) = 1.

1155

Periodically time dependent systems

a characteristic value a,,(q) coincides with 2q. (Odd Mathieu functions corresponding to a value b,,(q) vanish at z = 0 = 7~,but become large in a neighborhood of this point.) The quantum mechanical treatment (see Appendix) yields the approximate near-resonant Floquet states [23]

U&X, t) = Cx(km)lr + m)emim”‘, m

where &m) _ - -1 Rdzxk(z)e-2’mz ll I 0

(40)

are the Fourier coefficients of the a-periodic Mathieu functions. Even though this simple formula appears to have no direct ‘knowledge’ of classical mechanics, it can describe wave functions with a classical background. Figure 11 shows the result of the evaluation of (39) for k = 0, Fig. 12 the result for k = 12. Both the projected vortex tube and the scar are approximated fairly well. (For both figures, terms from m = - 19 up to m = +19 have been taken into account, and f has been replaced by t - T/4.) 4. THE DRIVEN PARTICLE IN A TRIANGULAR WELL

The approximate quantization scheme of Section 2, or the quantum mechanical analysis in the Appendix, holds for arbitrary N. But there is a problem with the simple EBK-like vortex tube quantization for any resonance other than N = 1: the central periodic orbit, as well as the surrounding tubes, will be NT-periodic, rather than simply T-periodic. If a ‘quantized’ tube is selected according to (6), and the associated semiclassical wave function is constructed, then the density of this wave function will again be determined by the projection of the tube to the {(x, t)} space, and will, therefore, also be NT-periodic. But

1 .8

.6 h . .4

0 -1

-.5

0

.5

1

X

Fig. 11. Approximate probability density of the Floquet state k = 0 according to (39), for the same parameters as in Fig. 2 (cf. Fig. 6).

1156

M. HOLTHAUS

1

.8 .6 .h .4 .2 0

Fig. 12. Approximate probability density of the Floquet state k = 12 according to (39), for the same parameters as in Fig. 2 (cf. Fig. 9).

that is incompatible with the T-periodic boundary conditions imposed on the Floquet states. To express the same thing differently: the second semiclassical rule (7) can only be applied if N = 1. What happens in the other cases will be studied in this section exemplarily for N = 2, using the model of the driven particle in a triangular well: H(p, x, t) = tp” + V(x) + Axsin(

(41)

where V(x) =

Im

: -&

i

(42)

This model has also been employed by Benvenuto et al. [24]; a detailed investigation of its ‘kicked’ version has been presented by Shimshoni and Smilansky [25]. In action-angle variables, the unperturbed Hamiltonian reads

f&(Z) = 0213,

(43)

2 and ATZ2J3 H(Z, 6, t) = H,(Z) + ___ sin (Wt) - - 33Lz2’3C1-{sin(rzt9 (37r)“3

7T(3s)“3

n=l

+ wt) - sin(n6 - wt)}.

n2

(44) Primary resonances are found at zN2fE; 303 their half-widths are given by

(45)

1157

Periodically time dependent systems

Whereas the resonances accumulate at Z = 0 for the particle in the box, with a width that decreases as l/N, the triangular well has primary resonances at arbitrarily large actions, with widths that grow proportional to N ‘. Thus, with increasing N the resonances will cover more and more phase space, and support more and more vortex tubes. A Poincare section showing the N = 1 resonance and the two islands of the iV = 2 resonance, as well as several higher order island chains, is displayed in Fig. 13, for A.= 0.4 and o = 0.95468 (chosen such that the energies of the unperturbed states it = 30 and n = 31 in the quantum system differ by o/2). The Bohr-Sommerfeld quantization of the unperturbed well, Z --, n - l/4 with n = 1, 2, 3 . . (where the Maslov correction -l/4 accounts for the phase loss at the ‘soft’, right turning point) does not yield the exact quantum mechanical energy eigenvalues. These are given by

En=%, where {-z,}

(47)

2l/3 are the zeros of the Airy funcion Ai(

Since [17] (48)

with 5 z-4 + Ez-6 + . . .), 48 f(z) - z */3 1+1z-*--36 ( the semiclassical result is recovered if f(z) is approximated by the leading term only. The parameters (16) and (25), M-r and 4, are found to be

M-1 = -04

(50)

N41? 4N*$A 4= 7’

(51)

60

0

.2

.4

.6

.a

1

angle 6 / 2n Fig. 13. PoincarC section for the driven particle in a triangular well (41) with o = 0.95468 and A = 0.4.

M. HOLTHAUS

1158

Obviously, the parameter E’: appearing in the quantum analysis agrees with M-’ in the approximation f(z) = z 2b if the Maslov correction is neglected. The dipole matrix elements for the triangular well system are given by [26] It=m

22rjz,/3 (nl4m)

=

-22p,(

Zn - znJ2

(52)

nZm

and an expansion of (11x( r f N) shows that (51) agrees with (A16) to leading order in N/r, again neglecting the Maslov correction. But although the numerical values of the parameters entering the approximate formulae (28) and (A22) differ slightly, their structures are identical. For N = 2 there is j = 0 and j = 1 only, so that the near-resonant quasienergies (A22) (with r = 30 and 6 = 0) can be expressed as [27] mod o, &o,k= E, + b%dd (53) where (Ye is one of the characteristic values {ao, b2, u2, bq, . . ,} belonging to Ir-periodic Mathieu functions, and qk = E, + @a,(q)

+ ;

mod w,

(54)

where now CY~is one of the characteristic values {br , a3, b3, us, . . .} that are associated with 2n-periodic functions. The asymptotic behavior 1171of these characteristic values, b f+l - 4 5-

241+5

l!

1p+3’4exp (-4j/(q)),

q-,

00,

(55)

shows that for large q the near-resonant spectrum consists of two almost identical groups, separated by w/2. This prediction is confirmed by a numerical calculation. Figure 14 shows a part of the

Fig. 14. Part of the quasienergy spectrum for the particle in a triangular weu (o = 0.95468). The quasienergies displayed originate from the unperturbed states n = 10-55. The arrows indicate the ‘ground state doublet’ k = 0.

Periodically time dependent systems

1159

spectrum for w = 2-(&r - EN) = 0.95468 (cf. Fig. 13); the ‘ground states’ k = 0 in the two sequences (53) and (54) are indicated by arrows. For clarity, only quasienergies originating from unperturbed states n = 10 to n = 55 have been plotted, so that the N = 1 resonance, which influences states around it = 5, is not visible. Again, there is a mismatch between the size of the resonant islands and the number of quantum states described by the approximation: each of the N = 2 islands in Fig. 13 can carry about 6 states, but the corresponding two quasienergy ‘fans’ in Fig. 14 consist of several more states. The appearance of the two almost, but not exactly identical groups (53) and (54) has a very transparent interpretation [27]. The central elliptic 2T-periodic orbit of the N = 2 resonance has two realizations, displaced from each other in time by T. Hence, there are two families of 2T-periodic vortex tubes, again displaced by T. If, instead of associating two Floquet states separately with two individual 2T-periodic quantized tubes, one associates each of the two states with both tubes, so that the Floquet states appear in pairs of almost equal density, then the problem of the incompatibility of the 2T-periodic vortex tubes and the required T-periodicity of the Floquet states can be resolved. Figure 15, which shows the density of one of the two k = 0 states for a time interval of two periods, confirms that this construction is correct: the density is concentrated along both realizations of the periodic orbit. As expected, the density of the other state with k = 0 is found to be almost the same. The difference of w/2 between (53) and (54) can be regarded as the manifestation of a classical effect. Since the vortex tubes are 2T-periodic, the fundamental periodicity length for the near-resonant states is 2T, rather than simply T. Hence, the width of the corresponding Brillouin zone is o/2, rather than cu. This type of ‘dimerization’ can be seen in exact analogy to its counterpart in solid state physics: if the periodicity interval in a spatial lattice is doubled from a to 2a, then the width of the Brillouin zone shrinks from 2x/u to r/a. In contrast, the slight difference between quasienergies with the same k in the two groups i = 0 and i = 1, as expressed by the difference (55) of the characteristic values, reflects a geniune quantum effect. The projection of a tube describing a classical particle 2

1..5

h . u

1

.5

0

Fig. 15. Contour plot of the probability density of one of the two states with k = 0 in the driven triangular well, with parameters as in Fig. 13. The density follows both realizations of the 2T-periodic elliptic orbit, with a strong interference at the crossing points of their projections to the {(x, t)} plane.

1160

M. HOLTHAUS

moving (with positive momentum) towards the slope of the triangular well crosses, at certain moments in time, the projection of a tube describing a particle moving (with negative momentum) away from the slope, although the tubes themselves remain well separated in the extended phase space {(p, X, t)}. But it is the projection to the {(x, t)} plane, rather than the tubes themselves, that determines the probability density ]u(x, t)12 of the Floquet states [12], and there has to be an interference of both ‘possibilities’ in those regions of space-time where different projected tubes intersect. The contour plot in Fig. 15 shows this effect rather clearly, and it is even more impressive in the 3D-plots of the density of states with k = 0 (Fig. 16) and k = 1 (Fig. 17). In this way, the two Floquet states associated with a pair of quantized vortex tubes are coupled by quantum tunneling, and the two sequences (53) and (54) can be interpreted as one sequence of tunneling-split doublets. For example, the ground state splitting, i.e. the splitting between the two members of the pair with k = 0, is given by he0 = ;~‘:(m)

- so(q)) (56)

e l&5’:

-i? 43i4exp (-42/(d), 4 7T) and the splittings between the higher pairs follow similarly from (55). The exponential suppression of the tunneling splitting with j/(q), i.e. with d(n), reflects the growth of the resonant islands with the square root of the driving amplitude, see (46). Of course, the

Fig. 16. Probability density of one of the two states with k = 0 in the driven triangular well. The parameters are as in Fig. 13, the displayed intervals in space and time are the same as in Fig. 15.

Fig. 17. Probability density of one of the two states with k = 1 in the driven triangular well.

Periodically time dependent systems

1161

validity of the approximation (56) is restricted to moderate values of A.which lead to the formation, but not destruction, of large resonances. The interpretation of the near-resonant spectrum (53), (54) as a sequence of tunnelingsplit pairs becomes even more transparent in the light of the transformed Schriidinger equation (5). As already noted, the effective Hamiltonian G describes a fictitious particle in a cosine lattice with N wells in the periodicity interval (see (21)), which reduces to a simple double well for N = 2. The two minima of this double well potential correspond to the two realizations of the elliptic periodic orbit, and the finding that the Floquet states are ‘delocalized’ over two vortex tubes reflects the fact that the eigenfunctions of the double well are delocalized over both wells. It should be noted, however, that the effective mass of the fictitious particle in the double well potential is negative. To finish this section, Figs 18 and 19 show the density of two other Floquet states in the driven triangular well. The complexity of these states is significantly greater than that of states with low k, but they show clear signatures of classical phase space structures. 5. DISCUSSION

After these clear and seemingly clean numerical results it is somewhat embarrassing to confess that a lot of dirt has tacitly been swept under the carpet. Actually, periodically driven quantum systems are much more complicated than the present ‘hands-on’ approach seems to suggest. For example, the question whether the quasienergy spectrum is singular or absolutely continuous is usually not trivial at all [28,29]. The problem can be illustrated by the following ‘Gedankenexperiment’, which mimics the numerical procedure: to determine the quasienergy spectrum for a system like the driven particle in a box or in a triangular well, one can diagonalize the ‘one-cycle-evolution operator’ U(T, 0). Of course, for all numerical purposes one has to work in a finite basis set consisting of, say, 1 functions; and the Schrodinger equation will be approximated by a system of 1 ordinary differential equations. The approximate evolution operator U[(T, 0) is then an 1 x I

2

1.5

.5

-0

5

10

15

20

25

Fig. 18. A Floquet state for the driven particle in a triangular well (parameters as in Fig. 13).

1162

M. HOLTHAUS

Fig. 19. Another example of a Floquet state for the driven particle in a triangular well (parameters as in Fig. 13).

matrix, and its eigenvalues should approximate some of the eigenvalues of the real system. When 1 is enlarged, the number of quasienergies in each Brillouin zone grows. The ‘new’ eigenvalues will inevitably approach the ‘old’ ones at certain values of the parameter A, which will result in the formation of (possibly extremely narrow) avoided quasienergy crossings [30]. (It is assumed that there is no symmetry which allows a crossing. The Hamiltonian of a driven particle in a box, for instance, remains invariant under the combined operation x + -x and t + t + T/2, and the Floquet functions have even or odd parity under this operation. The argument can then be applied within each of these two symmetry classes.) When the basis size 1 becomes very large, these avoided crossings will, in a naive sense, be ‘dense’ in a plot of quasienergies versus A. What, then, happens for I + m? Will a certain quasienergy level E(A) that appeared to be smooth in a low-I computation be completely ‘destroyed’, or will the size of the avoided crossings decrease fast enough so that there is ‘something’ left? To phrase the problem in more mathematical terms: even for A = 0 the quasienergy spectrum of the particle in a box or in a triangular well is a dense pure point spectrum, assuming a generic choice of the frequency o. What happens if an operator with such a spectrum is perturbed, i.e. for nonvanishing A? Fortunately, this question can be answered on a rigorous level [31,32] for the driven particle in a box. The unperturbed system has increasing energy gaps and the perturbation is bounded. Hence, it follows [31] that the perturbed system has a dense pure point quasienergy spectrum as well. However, the situation is entirely different for the triangular well problem. In this case, the unperturbed system has decreasing energy gaps and the periodic perturbation is unbounded; and the question of the nature of the quasienergy spectrum seems to be unresolved at present. Benvenuto ef al. [24] have suggested that the driven triangular well system exhibits a transition from a point spectrum to a continuous spectrum at a certain, finite value of the perturbation strength A. But there is also support for a different hypothesis. There are arbitrarily large primary resonances for arbitrarily small values of A (see (46)), and there should be some form of quantum tunneling between them. Since these resonances extend up to infinite action, it appears possible that the spectrum is actually continuous for arbitrarily small, but nonzero, A. If this were true, a

Periodically time dependent systems

1163

quantum mechanical particle in a triangular well could escape to infinity under the influence of a driving force with a small amplitude, whereas the corresponding classical particle would remain bounded by KAM barriers. However, since this breaking of the classical-quantum correspondence would be due to a tunneling effect, it would manifest itself only after very long times. On a shorter time scale the assumption of a point spectrum, although still a fiction, would at least be a good one; a quasienergy spectrum like that displayed in Fig. 14 (computed in a large, but finite basis) can describe the quantum dynamics at least for a reasonably long, finite time. In any case, further ‘hard’ mathematical results on periodically forced quantum systems are desirable. Leaving the dirt under the carpet for the moment, the examples treated in the present paper provide, nevertheless, quite useful insights into quantum dynamics. One of the most important results is the finding that a highly excited, strongly driven quantum system is capable of forming new ground state-like states. The quantum number IZ, which labels the eigenstates of the unperturbed system H,, loses its dynamical significance in the presence of a strong driving force. In contrast, the quantum number k introduced in the analysis of near-resonant states refers to the effective Hamiltonian G in (5); it labels states even in the presence of the force in a consistent way. It makes, therefore, a lot of sense to interpret the wave function shown in Fig. 6 no longer as the excited state n = r = 20 of the driven particle in the box, but as the ground state of a new Floquet-Mathieu excitation, which, for convenience, will be abbreviated as ‘Floton’ in the following discussion. Thus, the wave functions shown in Figs 7 and 8, and even the ‘scar’ in Fig. 9, may be regarded as certain excited states of a Floton brought into being by an N = 1 resonance; the approximate equation (35) describes the Floton spectrum. This interpretation of near-resonant quantum dynamics becomes particularly interesting when applied to experimental results obtained on highly excited hydrogen atoms interacting with microwave radiation [7]. It is known that states close to classical resonances are relatively harder to ionize than their neighbors, a finding that is usually explained as ‘trapping in nonlinear resonances’. Although such an explanation doubtlessly contains a good deal of truth, it has a serious deficiency, since classical language is used to describe a quantum mechanical phenomenon. The semiclassical vortex tube quantization provides a bridge between the classical and the quantum world, and shows that the Floquet wave functions of states ‘trapped inside a resonance’ actually follow the stable periodic orbit [33]. The (strictly quantum mechanical) ‘Floton’ formulation even has a certain charm: for N = 1, for example, the unperturbed resonant state rz = r turns into the Floton ground state k = 0, the neighbors IZ = r f 1 into k = 1 and k = 2, and so forth, and-of course!-a ground state is more stable than the excited states! In a striking paper, Jensen et al. [34] suggested that the experimentally observed anomalous stability of states for which there is no trapping island, like n = 62 for a microwave frequency W/~P= 36.02 GHz, could be caused by a ‘scarred’ wave function. Considering the fact that for this frequency the center of the N = 1 resonance lies close to the action of the unperturbed state with principal quantum number n = r = 57, and that for a microwave amplitude L = 17.4 V/cm the size of the resonant island in a one-dimensional model is of the order of a few 27rh [34], and, finally, that the scar originates from the state rz = 62 = (r + 5), it seems more than likely that this scar is precisely of the type shown in Fig. 9, i.e. that it can be described in terms of a fictitious particle with an energy close to the top of the effective cosine potential of an N = 1 resonance. But if that is true, if the scar is just another excited state of the Floton, why should it be more stable than its neighbors, as found in the experiments [35]? This question deserves to be studied in greater detail. As a starting point, it is interesting to note that the expectation value (uk(&]uk) of the unperturbed Hamiltonian operator,

1164

M. HOLTHAUS

taken for the approximate near-resonant Floquet states (39), is time independent:

(57) Since ( ukl - ia,luk) vanishes, it is easy to show that the time-averaged expectation values of H,, + hx cos (ot) are equal to the quasienergies, i.e. the energies of the fictitious particle in the cosine potential. These energies, of course, vary monotonically with the Floton quantum number k. But if one plots the expectation value (57) for the particle in the box with parameters as in Figs 5-9, one obtains Fig. 20. The expectation value increases monotonically with k only up to the classical chaos border (k = 8) and then starts to fluctuate, with a very pronounced minimum at the scar, k = 12. If a similar result could be established for the hydrogen atom, one might have a clue for a rigorously quantum mechanical explanation of the stability of states associated with scarred wave functions. Further experimental and theoretical efforts are needed. Finally, it is worthwhile to point out that the investigation of near-resonant quantum dynamics can be extended to the case where the driving amplitude )L varies in time. With the help of the adiabatic principle it is then possible to define ‘generalized n-pulses’ [36] which lead to selective excitation of a prescribed target state within a weakly anharmonic multilevel system. Such pulses are of great interest in chemical physics, in the area of laser-assisted control of molecular dynamics [37]. This last remark may underline the fact that the study of quantum dynamics close to a classical chaos border is not an esoteric subject of interest only to a few dedicated individuals. On the contrary, it might well create a significant impact on a broad field of science. Acknowledgements-This paperhas its roots in a joint work [12] with H. P. Breuer,whom I thank for a long and pleasant collaboration. Part of the material was developed during the course Physics 252, held at UCSB in spring 1992. I thank M. Sherwin for giving me the opportunity to teach a part of this course, and for many discussions. I am also grateful for discussions with B. Birnir, M. E. Flattt, B. Galdrikian, D. W. Hone and W. Kohn. Last, but not least, it is a pleasure to thank G. Ahlers and S. J. Allen for kind hospitality. This work was supported by the Office of Naval Research (NOOO14-92-J-1452)and by a Feodor Lynen grant from the Alexander von HumboldtStiftung.

580 560

480

I

0

I

I

I I

I

I I

5

I

I

10

I I

I

I I

15

k

Fig. 20. Expectation value (u~]Hs(nk) for the driven particle in a box, with parameters as in Figs 5-9. Note the local minimum at the scar, k = 12.

Periodically time dependent systems

1165

REFERENCES 1. V. I. Arnold, Mathematical Methods of Classical Mechanics. Springer, New York (1978). A. Einstein, Verh. Dtsch. Phys. Ges. (Berlin) 19, 82 (1917). L. Brillouin, J. Phys. Radium 7, 353 (1926). J. B. Keller, Ann. Phys. (N.Y.) 4, 180 (1958). M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics. Springer, New York (1990). K. Nakamura, Quantum Chaos-A New Paradigm of Nonlinear Dynamics. Cambridge University Press,

2. 3. 4. 5. 6.

Cambridge (1993). 7. For a review, see: P. M. Koch, in Chaos and Quamum Chaos, Lecture Notes in Physics Vol. 411. Springer, Berlin (1992). 8. Ya. B. Zel’dovich, Zh. Eksp. Theor. Fiz. 51, 1492 (1966); [Sov. Phys. JETP 24, 1006 (1967)]. 9. V. I. Rims, Zh. Eksp. Theor. Fiz. 51,1544 (1966); [Sov. Phys. JETP 24, 1041 (1967)]. 10. W. R. Salzman, Phys. Rev. A 10, 461 (1974). 11. F. Gesztesy and H. Mitter, J. Phys. A 14, L79 (1981). 12. H. P. Breuer and M. Holthaus, Ann. Phys. (N. Y.) 211, 249 (1991). 13. F. Bensch, H. J. Korsch, B. Mirbach and N. Ben-Tal, J. Phys. A 25, 6761 (1992). 14. A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion. Springer, New York (1983). 15. H. P. Breuer, K. Dietz and M. Holthaus, Physica D 46, 317 (1990). 16. B. V. Chirikov, Phys. Rep. 52,263 (1979). 17. M. Abramowitz and I. A: Stegun (Ebs.),‘Handbook of Mathematical Functions. Dover, New York (1972). 18. Tables Relatina to Mathieu Functions. Columbia Universitv Press. New York (1951). \ , 19. L. E. Reich1 &d W. A. Lin, Phys. Rev. A 33, 3598 (1986). 20. W. A. Lin and L. E. Reichl, Phys. Rev. A 37, 3972 (1988). 21. V. P. Maslov and M. V. Fedoriuk, Semi-classical Approximation in Quantum Mechanics. Reidel, Dordrecht (1981). 22. E. J. Heller, Phys. Rev. Lett. 53, 1515 (1984). 23. M. Holthaus, in Proceedings of the Yukawa International Seminar “Quantum and Chaos: How Zncompatible?“, Kyoto, August 24-28 (1993); Prog. Theor. Phys. Suppl., 116, 417 (1994). 24. F. Benvenuto, G. Casati, I. Guameri and D. L. Shepelyanski, Z. Phys. R 84, 159 (1991). 25. E. Shimshoni and U. Smilansky, Nonlinearity 1, 435 (1988). 26. R. G. Gordon, J. Chem. Phys. 51, 14 (1969). 27. M. Holthaus and M. E. Flatte, Phys. Lett. A 187, 151 (1994). 28. J. Bellissard, in Trends and Developments in the Eighties, edited bv S. Albeverio and Ph. Blanchard. World Scientific, Singapore (1985). 29. J. S. Howland. Ann. Inst. Henri Poincare 49. 309 (1989). 30. J. von Neumann and E. Wigner, Phys, Z. 36, 467‘(1929). 31. J. S. Howland, Ann. Inst. Henri Poincare 49, 325 (1989). 32. J. S. Howland, J. Phys. A 25, 5177 (1992). 33. J. Henkel and M. Holthaus, Phys. Rev. A 45, 1978 (1992). 34. R. V. Jensen, M. M. Sanders, M. Saraceno and B. Sundaram, Phys. Rev. Len. 63,277l (1989). 35. B. E. Sauer. M. R. W. Bellermann and P. M. Koch. Phvs. Rev. Lett. 68., 1633 (1992). \ I 36. M. Holthaus and B. Just, Phys. Rev. A 49, 1950 (1994)’ 37. For a review, see: W. S. Warren, H. Rabitz and M. Dahleh, Science 259, 1581 (1993). 38. G. P. Berman and G. M. Zaslavsky, Phys. Lett. A 61,295 (1977).

APPENDIX The approximate formula (28) for near-resonant quasienergies had been found by naively quantizing the Hamiltonian (17), disregarding the fact that canonical transformations and quantization do not commute. In this appendix, the approximate result will be justified by a strictly quantum mechanical analysis [38]. In addition, an approximate expression for near-resonant Floquet states will be derived. Let Ho be the Hamiltonian operator of an unperturbed nonlinear oscillator, and E, and In) its energy eigenvalues and eigenfunctions: HoIn) = EnIn).

(4

Now consider the driven system H(t)

= Ho + hxcos(wt)

(3

and assume the existence of an N: 1 resonance, i.e. assume that close to a ‘resonant’ state lr) the (slowly varying) level spacing is approximately given by w/N:

1166

M. HOLTHAUS

where the prime indicates differentiation with respect to n. A reasonable ansatz for a wave function that consists mainly of a superposition of near-resonant states is

tin, t) = Cc,(t)ln) exp [ -i( E, + (n - r)z)t), n

(A4)

with coefficients c.(t) that obey the equation E, - E, - (n - r);

c,(t)

+ ~~cos(wt)el(n-m)wf~N(nlxlm)c,(t). m

Expanding the energy eigenvalues quadratically around E,, keeping only the resonant terms m = n ?I N, and assuming that the matrix elements (n]x]n + N) can be approximated by a common value, e.g. (rlxlr + N), one finds

if"(f) = fE:(n - r)%,(t) + 6(Tl - r)C,(t) + V(C,+,(t) + C,-,,,(t)),

646)

V = ;(r(xjr + N)

(A7)

where

and

The coefficients now separate into N groups {c,(t)]n = I + j + mN; j = 0, . . ., N - l}. Redefining the indices within each of these groups, n-r-j

mz

N

’

j = 0, . . ., N - 1,

NJ)

the N equations for the coefficients become iCm(f) =

$N2E:(m + j/N)%,,,(t) + N6(m + j/N)c,,,(f) + V(c,+,(t) + c,-l(t)).

(AlO)

Assuming that the coefficients decay sufficiently fast around m = 0, the c,(t) of each group are represented as Fourier coefficients of a 2rr-periodic function fj(q) [38]:

c,(t) = &- zrdplfi(q$)e-i~~e-iW~ I0 =-

1

2Nn

2Na I ,,

(All)

i(m+j/N)qe-iWr,

dVggj(cp)e-

with gj(p))

=

G4W

fj(cp)eijqlN.

Hence, one obtains Wgj(cP)

= -~N~E;-$

+ T$

+ 2vcos(cp)

(

gj(&

6413)

1

The substitution 9, = 22, gj(cp) = exp (-2iSZ/(NEy))xj(Z) 5

+ U-

leads to a standard Mathieu equation [17] 2qCOS(2Z)

xj(Z)

= 0

6414)

with

2s 2 aAE-+ c-1 N2E’:

qL!!l= N2E:

NE’:

4A(r\xlr + N) N2E” ’

6415) (Am

A F’loquet solution xi(Z) = P,(z)eivZ gives gj(P)) = pdd-4 exp (iid2 - &d(NC)I), (-417) so that a comparison with the required form (A12) now restricts the characteristic exponents to the discrete values V=y(j)=Zi+28NE; ’ N and the functions fj(q) can be identified as the periodic parts of the corresponding Mathieu functions, f,(V)

= pvlj)(d2).

(Al@ (A19

Periodically time dependent systems

1167

The necessary restriction of v also restricts the allowed values of the parameter a to certain values uk(v, q). which, by (A15), leads to the quantization of the energies W. Again, the integer k is used to label different solutions. A particular Mathieu function Pflj),k gives rise to the coefficients = ~~zudaP,j),~(~~)e-im~e-iw,.~r

c,(t)

(-42’3)

0 3

X(m)e-iW,,kr, 1.k

and the associated Schrijdinger wave functions are w(X, t) = ~$$lr

+ j +

+

~N)e-imw'eXp{-i(E,

Wj,k

+ jw/N)t}

(-421)

m -

uj,k(x,

t)exp(-i&j,kt).

Since the functions uj,k(x, t) defined here are manifestly 2a/w-periodic, they are the approximate near-resonant Floquet states; their quasienergies are Ej,k

=

E, +

Wj,k

+ ~

mod UJ

(-422) = E, - &

+

+E;Uk(V,

q) + 5

mod w.

, If there is a resonant state jr) which satisfies the relation (A3) exactly, then 6 = 0, and the characteristic exponents (AH) coincide with the exponents (27) that appear in the previous result (28). If the level spacing varies sufficiently slowly, S will be small, and setting 6 = 0 is still a good approximation. Compared to (28), the mass parameter M-* (see (16)) has now been replaced by E:, and the Fourier coefficient x~(IN) in the definition of the Mathieu parameter q (25) has been replaced by the matrix element 2( rlnlr + N). The example of the particle in a triangular well explicitly demonstrates that the values of the parameters entering (28) and (A22) are not necessarily identical, but the difference is small for large quantum numbers. Most importantly, although the numerical values of the parameters can differ slightly, both equations have exactly the same structure. Thus, a classical primary N:l resonance leads, in the semiclassical limit, to N disjoint groups of Floquet states in the corresponding quantum system.