Classical and Quantum Transitions to Chaos for a Family of Periodically Driven Hamiltonians

Annals of Physics  PH5504 annals of physics 246, 369380 (1996) article no. 0031

Classical and Quantum Transitions to Chaos for a Family of Periodically Driven Hamiltonians Cesar R. de Oliveira* Universidade Federal de Sa~o Carlos, Departamento de Matematica, C.P. 676, Sa~o Carlos, SP, 13560-970 Brazil Received May 22, 1995

The classical and quantum behaviors of a family of periodically driven hamiltonians are compared. By the resonance overlap criterion the perturbation field strength = c for the classical chaotic transition is estimated. The quantum spectral transition is analyzed in terms of the photonic approximation. From the validity of the photonic approximation and quantum delocalization condition the functional dependence of = c is recovered in a purely quantum way.  1996 Academic Press, Inc.

1. Introduction Among the most interesting problems in quantum theory posed nowadays are the perturbation of self-adjoint operators with dense pure point spectrum and the understanding of the quantum signatures of classical chaos. In reference to the latter it is expected that one could, somehow, anticipate the classical chaotic border from quantum arguments. The classical chaos is particularly interesting in periodic driven systems duo to the possibility of unperturbed energy diffusion with time. A general formalism to predict the classical chaotic border for periodic driven systems is the so-called overlapping criterion [1, 2]. Although it is based on rough arguments and approximations, it has proven helpful and generally gives reasonable estimates when compared to numerical experiments. The quantum mechanics of such classical systems can be investigated by studying the properties of the quasi-energy operator (also called Floquet hamiltonian) [3, 4] K=&i

 +H(t) t

where H(t) is the time dependent (periodic) hamiltonian of the perturbed system. In general K has dense spectrum even for the unperturbed case; one can thus expect that the studies of the quantum counterparts, of some classical chaotic driven systems, will be characterized by perturbations of self-adjoint quasi-energy operators with dense spectra, and so may deserve special considerations. * E-mail address: Oliveirapower.ufscar.br.

369 0003-491696 18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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A formalism to study spectral properties of the quasi-energy operator is the so-called photonic approximation [5, 6, 7]. Although it employs rather crude approximations it has been a useful tool in the study of localization properties of quantum systems. It also consists of a continuous counterpart of the Maryland construction [2], i.e., a map from nonkicked periodic driven systems to tight-binding models with pseudo-random potentials. In Ref. [5] the quantum mechanics of the model H=

p2 ++x+=x cos(|t), 2

x0,

with +, =>0, was considered. By applying the photonic approximation it was possible to get the value of = q for the quantum localization-delocalization transition. Notice that = q was previously obtained in [8] from very difference arguments, which were based on know facts for the kicked rotator and a Siberian argument. From the validity of the photonic approximation and the delocalization condition the classical border for chaos was recovered in a purely quantum way. In this paper we extend the above quoted results to the family of hamiltonians H: =

p2 ++x : +=x cos(|t), 2

x0

(1.1)

where 0<:1. The classical threshold for chaos is obtained through the overlapping criterion as discussed in Section 2. In Section 3a we present the photonic approximation and in Section 3b it is applied to hamiltonians 1.1. The validity and consistency conditions for the photonic approximation are discussed in Section 4. In Section 5 we show the results of some numerical calculations. Our results indicate that the recovery of classical chaos border presented in [5] for :=1 was not accidental. Recall that for :=1 the classical border does not depend on | and it was also obtained from quite different arguments [8]. Since the chaotic transition for the classical system 1.1 and the possibility for a quantum localization-delocalization transition do not seem to have appeared anywhere, such study has its own interest. In fact, it is found that a classical transition to chaos occurs for 0<:1; see relation 2.5. Also a quantal transition from power-law localized eigenstates to extended ones is expected to occur according to relation 3.9 and Section 4, for any 0<:1. Another motivation to consider system 1.1 is the possible influence, on localization properties, of different laws governing successive level spacing of the unperturbed autonomous hamiltonian system (see 3.5), in contrast to the uniform spacing of the harmonic oscillator and the kicked rotator linear law. I would like to stress that the recovery of the classical chaotic border from quantal arguments presented here is to be regarded just as a first step in such direction. The possible generalizations of the methods considered to other situations are not clear yet.

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TRANSITIONS TO CHAOS

2. Classical Chaotic Threshold The hamiltonians 1.1 model a particle elastically bouncing on a fixed wall at x=0 subjected to nonuniform potential (:{1) and a monochromatic perturbation. This family of hamiltonians is integrable for ==0 or in case the wall is absent. To begin with we introduce action-angle variables (we shall usually drop the explicitly : dependence) (J, %)=(J : , % : ) for the unperturbed family of hamiltonians 1.1, J=J(E )=

1 2?

? sgn( p) %(x, J )= k2

|

- 2 k1

 - 2(E&+&x )= ?+ :

xxmax 0

1 - 1&y :

1:

E (2+:)2: (2.1)

dy.

E denotes the energy and x max =(E+) 1:. k 1 and k 2 are : dependent parameters. Notice that we have imposed %(0)=0 and %(x max )=\?. Figure 1 shows the general behavior of W=W : #xx max . A major problem for the analytical treatment of system 1.1 is the lack of an explicit inversion formula x=x(%, J) for most : values. According to the resonance overlapping criterion [1, 2] the m th and (m+1) th classical resonance of system 1.1 will overlap for =>= m , with = m given by =m =

|2 d0 4 16m |V m | dJ m

\ +

&1

(2.2)

where 0(J )=dEdJ is the frequency of the classical motion, J m is the action at the mth resonance m0(J m )=|, and V m is the m th Fourier coefficient of x(%, J m ).

Fig. 1. Typical curve W(%)=x(J, %)x max . The jump discontinuity of its first derivative at %=0 is apparent.

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From now on we indicate all relations of proportionality by a generic parameter c=c(:). The eventually different values of c should arise no confusion. For instance, we shall also indicate both k 1 and k 2 that appeared above by c(:). We have d0 =c(:)(E(J ) +) &2:. dJ

(2.3)

Due to the fixed wall at x=0, W is a continuous function of % with a jump discontinuity of its first derivative with respect to % at %=0, then it is found that |V m | rc(:)

x max 1 Em =c(:) 2 2 m m +

\ +

1:

(2.4)

where E m =E(J m ). By inserting 2.3 and 2.4 into 2.2, and taking into account that 0=|m we get = m =c(:) + 1(2&:)

| m

\+

2(1&:)(2&:)

.

(2.5)

Since our approach to the corresponding quantum system will hold only for 0<:1 (see Section 4), we shall restrict our analysis to this range of : values. Notice that by 2.5 it is seen that unbounded classical chaotic diffusion is possible for this range of :, and that only for :=1, = m does not depend on m and |. For 1<:<2 the critical value = m increases with m and no chaotic diffusion takes place.

3. Quantum Approach In this section we discuss the photonic approximation and then apply it to the family of hamiltonians 1.1. Our main goal is to provide a purely quantum approach for the localization-delocalization transition, i.e., the photonic approximation is used to indicate spectral transitions for the quasi-energy operators associated with 1.1. 3.1. The Photonic Approximation It is well known the important role played by quasi-resonant single-photon transitions for the description of quantum systems under monochromatic perturbations. Under suitable conditions an approximate description of the quantum system H=H 0 +=x cos(|t)

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373

can be obtained by restricting the quantum states to the subspace spanned by the quasi-resonant (or photonic) states. The latter are specified in terms of the quasienergy operator K=&i

 +H 0 +=x cos(|t). t

It is assumed that H 0 has a complete set of eigenvectors H 0  n(x)=u n  n(x), so that the complete set of eigenfunctions and eigenvalues of the corresponding quasi-energy operator is given by f r, n(x, t)= n(x) exp(&ir|),

Kf r, n =(u n &r|) f r, n .

The matrix elements of K in this basis are = ( f r, n | K | f r$, n$ ) =(u n &r|) $ n, n$ $ r, r$ + (  n | x | n$ )($ r, r$+1 +$ r, r$&1 ) 2

(3.1)

Although exact, equation 3.1 is tractable only in special cases; here we approximate 3.1 by considering only the photonic states associated with the initial unperturbed eigenfunction f n0 , 0 with corresponding eigenvalue u n0 . I.e., the unperturbed eigenstates  n(r) whose energy levels u n(r) are closest to (u n0 +r|). We shall denote the rth photonic state by |r) and by 2 r the corresponding detuning from exact resonance: 2 r =u n(r) &u n0 &r|. Now we project 3.1 onto the subspace spanned by the photonic states and thus obtain the photonic hamiltonian (, which has matrix elements = (r| ( | j)=(u n0 +2 r ) $ r, j + (  n(r) | x | n( j) )($ r, j+1 +$ r, j&1 ) 2

(3.2)

If we neglect the eventual eigenvalue correction for small = and write the solution of the eigenvalue equation (,=u n0 , in the form ,= r , r |r) we get 2 ( n(r) | x | n(r+1) ) , r+1 +(  n(r) | x | n(r&1) ) , r&1 + 2 r , r =0. = Neglecting this eigenvalue correction can be justified in terms of the expected dense spectrum of quasi-energy operators, so that in case u n0 happens not to be an eigenvalue of ( there would be arbitrary near such eigenvalues. This point may present some subtleties; in order to assure dense spectrum of K it may be necessary to exclude some particular values of |, which should hardly manifest, for instance, in numerical calculations. We shall leave such considerations to the more mathematically inclined reader. Since for large r we have Z r #(  n(r) | x | n(r+1) ) r(  n(r) | x

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| n(r&1) ), we find that the amplitudes , r of the eigenfunctions on the photonic states satisfy 2 2r , r =0. (3.3) , r+1 +, r&1 + =Z r This equation describes the photonic approximation. An obvious limit of validity of 3.3 is that | has to be larger than the local level spacing. Another condition requires the level width for single-photon transition to be smaller than the energy spacing between quasi-resonant levels; if the level width is identified with the Rabi width, this condition can be written as |>= |Z r |

(3.4)

In principle one could calculate both Z r and 2 r and study numerically the solution of 3.3, which resembles a discrete one-dimensional Schrodinger equation. In case the corresponding solutions are square summable we have localization, otherwise we have extended states. In the next subsection we apply the photonic equation for hamiltonians 1.1 and study the behavior of their solutions through an analogy with a specific tight-binding model. 3.2. Quantum Transition for H : Now we estimate the function dependence of Z r =Z r(:) and 2 r =2 r(:) for the family 1.1 and study the localization-delocalization transition through 3.3. Since the main role is played by large values of r we are justified in using semi-classical expressions. From the BohrSommerfeld semi-classical quantization condition applied to 1.1 we get the following set of eigenvalues, for ==0, u n =c(:)( + 1:(n&14)) 2:(2+:),

n=1, 2, 3, ...

(3.5)

Recall that c(:) denotes a parameter that depends only on :. By its very definition 2 r cannot be larger than one half of the unperturbed level spacing / r #u n(r)+1 &u n(r) (or u n(r) &u n(r)&1 ). It is convenient to normalize the detunings by defining _ r #2 r / r , which will be distributed over the interval [&0.5, 0.5]. For large r we have u n(r) rr|

and

/rr

u n(r) n

so that 2 r =/ r _ r rc(:) + 1:(r|) (:&2)(2:) _ r . A semiclassical expression [9] for the matrix elements Z r is Zr r

1 2?

|

?

x(%, J r ) exp(iN r %) d% &?

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(3.6)

TRANSITIONS TO CHAOS

375

where N r is the number of unperturbed levels between u n(r) and u n(r+1) . Notice that Z r rV Nr rc(:)

x max . N 2r

Since N r r|/ r and for large r we have x max r(r|+) 1: it follows that Z r rc(:) + 1: | &(1+:): r (:&1):. It is worth noting that this expression for Z r recovers the expression obtained, in the case :=1, from direct application of exact matrix elements [5]. We have then | 32 2r _r . rc(:) Zr -r Inserting this expression into 3.3 one finds the photonic equation for 1.1 , r+1 +, r&1 +c(:)

| 32 =-r

_ r , r =0.

(3.7)

Notice that this equation does not depend on + and the remarkable decaying of the ``potential'' as r &12 for any allowed value of :. On account of the photonic equation we discuss the localization-delocalization transition for 1.1 and left the discussion of the validity of 3.7 to the next section. Our numerical computations indicate that _ r are uniformly distributed over [&0.5, 0.5], and that the autocorrelations are rapid decaying. Should _ r be strictly random, we could think of 3.7 describing a Schrodinger operator with decaying randomness as r &12. The true random case was rigorously considered by Delyon, Simon and Souillard [10, 11]. On the other hand, it is known that pseudo-randomness is often sufficient for the validity of localization properties that are rigorously proved for genuine random sequences [12]. Thus, although the sequence _ r in 3.7 is not a strictly random one, we still expect the DSS localization results to apply. Now we recall, in a convenient way, the above quoted DSS localization results and refer to [10, 11] for details. They considered the model . r+1 +. r&1 +*

V(r) -r

. r =E. r ,

for a large class of random potentials V(r) and proved that there exists * 1 such that if *>* 1 the spectrum is almost surely pure point with power localized eigenfunctions, and for any small closed neighborhood of the origin, i.e., E=0, there exists * 2 such that for *<* 2 the spectrum in this neighborhood is almost surely pure singular continuous. It was also found that with probability one the corresponding eigenfunctions . r satisfy 2 (3.8) |. r | 2 rA |r| &B* for some positive parameters A and B depending on E.

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In our case such results indicate that 3.7 has a transition at === q from power localized eigenfunctions for =<= q to extended states for =>= q . Moreover, the exponent ruling the algebraic decay of eigenfunctions has the form c(:) | 3= 2. Now the quantum threshold for delocalization = q can be found by the condition that the eigenfunctions are not square summable, which translated to 3.7 implies that the above exponent is equal to 12, so that = q =c(:) | 32

(3.9)

Notice that = q does not depend on + and has the same functional dependence for all allowed : values (see next section). By comparing the expressions for = q with = m one sees room for the occurrence of dynamical localization for certain parameter values. Indeed, the dynamical localization phenomenon has been observed, from direct numerical integration of the equations of motion, in the case :=1 [8]. Summing up, on account of the photonic approximation and the DSS results, we have strong indications for a quantal transition for (1.1) from localized eigenfunctions to extended ones. These indications are supported by numerical calculations in the case :=12, as described in Section 5.

4. Consistency Conditions In this section we first address the validity of the application of the photonic approximation to (1.1), i.e., Eq. (3.7). Then we recover the functional dependence of the classical chaotic border (2.5) from quantal consistency conditions. By the relation (3.5) it follows that the level spacings are decreasing only for :<2, and this puts the first restriction on the applicability of the photonic approximation to (1.1). The condition (3.4) for (1.1) can now be written out |>c(:) =| &(1+:): + 1: r (:&1):.

(4.1)

We see that for :>1 condition (4.1) is not satisfied for every r, so that we cannot apply the photonic approximation to these cases. We are then left with 0<:1. Except for ``quantum resonances'' [3, 13] and certain symmetries [14], it is commonly assumed that a necessary condition for quantum delocalization is the presence of classical unbounded energy growth. Thus, if besides (4.1) we require quantum delocalization, it follows that both conditions are consistent only if the interval = q <=c(:) + 1:(|r) (:&1):, so that =>c(:) + 1:(|r) (:&1):.

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(4.2)

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TRANSITIONS TO CHAOS

Now we make use of classical relations to change variables. Since r|ru n(r) r E(J m ), where J m is the action at the closest classical resonance, i.e., E(J m )= c(:) + 2(:&2)0(J m ) 2:(:&2) and m0(J m )=|, we can write (|r) (1&:): =c(:) (+ 1:(|m)) 2(1&:)(:&2). After inserting this into (4.2) we find =>c(:) + 1(2&:)

| m

\+

2(:&1)(:&2)

(4.3)

The threshold in (4.3) has the exact functional dependence of the classical border for chaos (2.5). In summary, for every 0<:1 we have recovered the classical chaos border from the validity of the photonic approximation and the quantum delocalization condition, i.e., from pure quantum arguments.

5. Numerical Results To support the prediction that a spectral transition should occur at === q we carried out some numerical calculations. Starting with a pair (, 0 , , 1 ) and inserting the numerically calculated detunings 2 r and matrix elements Z r into 3.3, we use the transfer matrix method to iteratively compute (, r , , r+1 ). Based on the results of DSS, with a generic choice of (, 0 , , 1 ) one should obtain, after a large number of iterates, a norm &, r&#(|, r | 2 + |, r+1 | 2 ) 12 increasing as r ;, with ; being the exponent ruling the algebraic decay of eigenfunctions of 3.7. Therefore, we can use the following equation 1 N 1 N : log &, r&=const+; : log r N r=1 N r=1

(5.1)

to numerically computing the exponent ;. In general we do not have explicit expressions for x(%, J ), which imposes some complications for calculating Z r . In the case :=1 exact matrix elements are known and it served as a suitable test to the photonic approximation approach [5]. However, :=1 is a rather peculiar value: it is on the border for unbounded classical chaotic diffusion; the photonic approximation does not hold for :>1; only for :=1 the classical border for chaos 2.5 does not depend on | and m, and is also a linear function of +. Thus, we have been carried out the above outlined numerical procedure also for the case :=12. By 2.1 we have found E(J ) +

2

,+2? 3

\ +_ \ +& 4% % ,=arccos 1& +2 _ ? \ ?+ &

x 12(%, J )=

1+2 cos

2

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2

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CESAR R. DE OLIVEIRA

Fig. 2. The exponent ; for :=12 as a function of =| 32 for two values of | (see the inset). For |=0.1 we have the initial condition n 0 =15 and +=1.0 in Eq. (3.3). For |=0.04 we have n 0 =100 and +=0.5. The full line is the best-fitting equation ln(;)=&1.94&2.05 ln(=| 32 ).

for 0%?, and extended to an even function to &?<%<0. In this way, we can numerically calculate Z r by 3.6. As in the case :=1 the computed values of ; from 5.1 do not depend on the initial choice (, 0 , , 1 ) and, for large =, very small ``potential values'' come into play in 3.7, so we used an average over small intervals of | to improve the numerical stability. This corresponds to disorder averaging in the DSS model. From the numerically computed values of ;, see Fig. 2, we checked the functional dependence ;(=)=c= &2| 3 suggested by the results of DSS. From numerical fittings we obtained = q by noting that extended states are not square-summable; so the transition occurs at ;=12, which leads to = q r0.54 | 32 and we have numerically estimated the constant c(12) in 3.9.

6. Conclusions Although we have considered a specific family of hamiltonians, in this paper we have done a step towards the quantal derivation of classical chaos border. An important ingredient in such derivation was the photonic approximation; despite its apparent crudeness it has successfully been applied to the case :=1 and now also to 0<:<1. The consistency of our results supports its usefulness.

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We made use of the overlapping criterion and showed that a classical transition to chaos occurs for model (1.1), provided 0<:1. For this range of : values we have predicted a quantum transition from power-localized to extended states. Next, we stress the main points in such approach: v by applying the photonic approximation to (1.1), we have got tight-binding hamiltonians with pseudo-random potentials decaying according to an inverse square-root law for all allowed values of :. Those models have a counterpart in the rigorous theory of truly random potentials. v the photonic approximation does not apply if the classical system (1.1) does not present a transition to chaos, i.e., :>1. v the pseudo-random potentials in (3.7) do not depend explicitly on +. v the quantum transition threshold = q has the same functional dependence for any allowed value of :. v the classical chaotic border was recovered from the validity of the photonic approximation and the condition for quantum delocalization. Of course, we had to use some classical expressions; however, just as a good dictionary. v since = m and = q have very different functional dependence on + and |, we have a concrete possibility of dynamical localization also for 0<:<1. v the presence of the cos (|t) term indicating the monochromatic perturbation in H : was crucial. It was due to this term we have got a tight-binding model with nearest-neighbor coupling, and then permitted the use of the DSS results to clue in the behavior of the solutions of (3.3) in case of model (1.1). It is well known that singular continuous spectra are quite unstable under operator perturbations and approximations. Thus, at the moment we can not say if the singular continuous spectrum, rigorously proven for the corresponding strictly random potential case, also characterizes the delocalized regime of (1.1). We would like to add that it is not clear how to extend the arguments here presented to an acceptable degree of generality, including other models as well as different kinds of perturbations; some of these difficulties rest on the lack of rigorous results. To finish we accentuate that the photonic approximation is a way to deal with perturbations of dense point spectra, and it is not at all obvious how far is its range of applicability.

Acknowledgments I acknowledge the helpful collaboration of G. Casati and I. Guarneri in an early stage of this work. I also thank the financial support of Fundaca~o de Amparo a Pesquisa do Estado de Sa~o Paulo (FAPESP, Brazil).

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References 1. B. V. Chirikov, Phys. Rep. 52 (1979), 263. 2. G. Casati and L. Molinari, Progr. Theor. Phys. Suppl. 98 (1989), 287. 3. J. Belissard, in ``Trends and Developments in the Eighties'' (S. Albeverio and Ph. Blanchard, Eds.), World Scientific, Singapore, 1985. 4. H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, ``Schrodinger Operators,'' Springer, Berlin, 1987. 5. C. R. de Oliveira, I. Guarneri, and G. Casati, Europhys. Lett. 27 (1994), 187. 6. R. V. Jensen, S. M. Susskind, and M. M. Sanders, Phys. Rep. 201 (1991), 1. 7. G. Casati, I. Guarneri, and D. L. Shepelyansky, ``Classical Chaos, Quantum Localization, and Fluctuations: A Unified View,'' Preprint NASF-ITP-89-56, Univ. California, Santa Barbara, 1989. 8. F. Benvenuto, G. Casati, I. Guarneri, and D. L. Shepelyansky, Z. Phys. B 84 (1991), 159. 9. L. Landau and M. Lifshits, ``Mecanique Quantique. Theorie non Relativiste,'' Mir, Moscow, 1966. 10. F. Delyon, B. Simon, and B. Souillard, Ann. Inst. H. Poincare 42 (1985), 283. 11. F. Delyon, J. Stat. Phys. 40 (1985), 621. 12. N. Brenner and S. Fishman, Nonlinearity 4 (1992), 211. 13. F. Izrailev and D. L. Shepelyansky, Theor. Mat. Fiz. 43 (1980), 417. 14. I. Guarneri and F. Borgonovi, J. Phys. A 26 (1993), 119.

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