Quantization of a two-dimensional map with a strange attractor

Quantization of a two-dimensional map with a strange attractor

Volume 99A, number 4 PHYSICS LETTERS 28 November 1983 QUANTIZATION OF A TWO-DIMENSIONAL MAP WITH A STRANGE ATTRACTOR Robert GRAHAM Fachbereich Phys...

305KB Sizes 0 Downloads 66 Views

Volume 99A, number 4

PHYSICS LETTERS

28 November 1983

QUANTIZATION OF A TWO-DIMENSIONAL MAP WITH A STRANGE ATTRACTOR Robert GRAHAM Fachbereich Physik, Universita't Essen, W. Germany Received 23 August 1983

A two-dimensional map with a strange attractor is quantized. The quantum density matrix and the Wigner function is obtained, and its relation to the classical strange attractor is analyzed.

Classical dynamical systems with sensitive dependence on initial conditions have been the subject of intense study in recent years. Both, non-integrable conservative systems, and systems with strange attractors have been investigated. A particularly simple description of such systems is provided by discrete maps, which arise naturally as return maps of continuous phase flows. It is an interesting question, how such maps behave after quantization. For instance, this question comes up when one enquires, how chaos in conservative classical systems is reflected in their quantum properties, and the question has received considerable attention in this context [1,2]. However, one may also pose the equally important question, how systems exhibiting strange attractors behave after quantization. To the best of my knowledge, this question has not yet been considered, despite the fact that in quantum optics, e.g., there are many well-known examples of quantum systems exhibiting strange attractors in the classical limit. Another class of such systems is based on Josephson junctions, where again quantum effects, dissipation, and chaotic behaviour occur simultaneously. It is the purpose of this letter to explore some typical features of such systems. To this end we consider a special class of discrete maps, which can be treated analytically, both, in their quantized form, and also in their classical limit, where they exhibit a strange attractor and sensitive dependence on initial conditions. The comparison of both solutions allows us to discuss the fate of the classical strange attractor in the quantized map. The classical models to be quantized satisfy the equations of motion _1 g, qn+l = 2qn (rood 1), Pn+l - ~Pn (gn), (1,2) with g(qn + 1/2) = g ( q n ) ,

0 <~qn < 1,

- ~ ' < Pn < + oo.

(3,4)

The jacobian of the map (1), (2) equals one, and the map is locally area preserving. However, the two stripes 0 <~qn < 1/2 and 1[2 ~
in phase space are attracted to a f~miteregion of phase space under the iteration of the map, and the dynamics on the attractor in the region has sensitive dependence on initial conditions. The map (1), (2) is therefore locally conservative, but globally dissipative, and it has a strange attractor. We note that eqs. (1), (2) form the area preserving special case of a class of models introduced by Kaplan and Yorke [3] for numerical study. These classical models have since also been solved analytically [4,5], also for the case when external classical noise is added on the right-hand side [6,7]. The map (1), (2) is a physically interesting model, since it provides an illuminating qualitative approximation [8] for the return map of the Lorenz model [9], which occurs naturally in a realistic dissipative quantum system like the single mode laser [10]. 0.031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland

131

Volume 99A, number 4

PHYSICS LETTERS

28 November 1983

We are here interested in a quantized version of eqs. (1), (2). Due to the preservation of volume in (p, q) space, it is natural to consider qn, Pn as a canonically conjugated pair of variables and to cast eqs. (1), (2) into the hamiltonian form

qn+l = qn + ~H(qn, Pn+l)/~Pn+l ,

Pn+l = Pn - ~n(qn, Pn+l)/~qn •

(5)

We note that (5) implies preservation of area dPn+ldqn+ 1 = dPndqn and of the Poisson bracket {Pn+l, qn+J ) = (Pn, qn } = 1. It does not imply conservation of H, however, contrary to the more familiar case, when H is the generator of infinitesimal time steps. The hamiltonian, which reduces eq. (5) to eqs. (1), (2) is given by H = [2qn (rood 1) - q n ] Pn+l + 2g(qn)"

(6)

We note that it is not possible to eliminate qn from H in favour of qn+l" Hence the hamiltonian (6) only generates forward time-steps, in agreement with the fact that the map (1), (2) is not invertible. We now use the hamiltonian (6) to define the quantized map we wish to consider. The variable 2rrq we interprete as an angle, its canonically conjugate p[2n as the associated angular momentum. In the coordinate representation we have quantum mechanicaUy p[2n = (h[i)(a[aq), ond angular momentum is quantized p/2rc = hl if we impose the condition of the uniqueness of the wavefunction ~kl(q+ 1) = fit(q). We now quantize the general map (5) by adapting the quantiza. tion method via path integrals [11]. We restrict ourselves to hamiltonians which are at most quadratic in the momentum variable p. The Schr6dinger equation of the map (5) then takes the form 1

~n+l(q) = f dq' K(qlq')t~n(q'), 0 where the transition amplitude K(q [q') is obtained from the classical hamiltonian H(qn, Pn+ 1) by

(7)

+¢o

K(qlq') = f

2~hexp{(i/fi)[P(q - q') - H ( q ' , p ) ] ) [1 + a2H(q',p)/aq'ap] 1/2 .

(8)

_ o o

The kernel K(q Iq') is unitary if the hamiltonian map (5) is invertible. A unitary time evolution cannot be associated with the non-invertible map (1), (2), represented by the hamiltonian (6), since it generates only steps forward in time, which is connected with the fact that the map is globally dissipative, as discussed above. However, we can incorporate these properties in the quantum map, by applying the quantization principle only locally and separately for the intervals 0 ~ 1/2 is determined. According to the principles of quantum theory the determination of this alternative implies the projection of the wavefunction ~bn(q) onto the two orthogonal subspaces (Iqn); 0 ~
1

dq' K(q[q')~n(q' ) + exp(i6n)

f

dq'K(q~q').qJn(q')

(9)

112

with -I-oo

6(x - m) .

K(q Iq') = ~/26 (m°d 1)(q _ 2q') exp [-2(i[h)q(q ')1 , 6 (m°d 1 )(x) = m=_oo

132

(10)

Volume 99A, number 4

PHYSICS LETTERS

28 November 1983

We note that the Schr6dinger equation (9) is compatible with the boundary condition qJn(q+ 1) = ~kn(q),which we shall impose on its solutions. In order to eliminate the random phases 6n it is useful to introduce the density matrix

(q IPn Iq') = t~n(q)~kn (q') ,

(11)

where the bar denotes averaging over the random phase factors. We normalize the density matrix on the unit interval 0 ~
1

1

(q [Pn+l Iq ) = ~ [<½q(mod 1)IPn I~q'(mod 1)) + (½[q(mod 1) + 1] [On [3 [q'(mod 1) + 1])] X exp {-2(i/h) [g(½q) --g(½ q ')] },

(12)

where we made use of the property (3) ofg(q). Eq. (12) defines the quantum map which we wish to consider. We now turn to the solution of eq. (12), which we obtain by direct iteration. It is most conveniently stated in terms of the Wigner function [12] +oo

Wn(q,p)= f -~---~edX-"

(q +~ttXlPnlq_½hx)

(13)

for 0 ~< q < 1, - ~ < p ~< ~, which takes the form 2n-1 +0.

Wn+k(q,p)=_2_ n ~

f

dx exp(_ipx/l~)((q+½x)(m°dl) +m 2n

m =0

IPlc

Xexp{,~_l_~[g((q+~x)(m°dl)+m))_g((q-½x)(m°dl)+m)] 2t 2t

(q-½x)(modl)+m } 2n

}

For n --*oo and k ~ ~ this expression can be simplified considerably by using 2-n(q (qlPn [q) ~ 1 (cf. below), and we obtain for the Wigner function the exact result 2n-1

W.~(q,p)=

lim 2 - n n---~ oo

~) m=O

(14)

• +x) (mode

1) ~ 0 and

+~dx ( . 2i~[ ((q+½hx)(modl)+m) ((q-½hx)(modl)+m)]} f "~n exp - k o x - ~ g - g t= 1 2l 2t

(15) We now discuss some aspects of this solution. First of all we note that the probability density of q, given by the diagonal elements ( q IPn Iq) of the density matrix or the integral over p of the Wigner function, evolves independently from the offdiagonal elements and satisfies the classical equation of motion. It approaches 1 for n --* ~. All moments and correlation functions involving q, and not p, are therefore given by their classical expressions and show mixing and sensitive dependence on initial conditions as in the classical case. Quantum effects appear only in expressions containing the momentum variable p. Next we take the classical limit of the Wigner function in the steady state, W~(q, p), and obtain, after carrying out the integral over x lim ~0

W**(q,p) =

2n - 1 lim 2 - n ~ /5(p n--*~, m=O

Fm(q)),

(16)

with 133

Volume 99A, number 4

28 November 1983

PHYSICS LETTERS

e~

F m (q) = - ~ 2- lg,((q + m)/21+ 1).

(17)

I=0

This result agrees with the known exact probability density of the classical model (1), (2) in the steady state [5,7]. It exhibits the classical strange attractor as the closure of the set of curves p = Fm(q), which forms the support of the classical phase space probability density, to which the Wigner function is reduced in the classical limit. Next we determine the leading quantum corrections to the classical result (18), by expanding the potential g in eq. (15) to third order in h (the second order terms in ~ cancel). The integral over x can then still be done explicitly and gives an Airy function in place of the classical f-function in eq. (16). We therefore have the semi-classical correspondence rule

8 09 - F m (q))~ [A m (q)[- 1/3 Ai(sign(Am(q)) [p _ F m (q)]/1Am(q)[1/3),

(18)

with

Am(g ) ~l2 C 2 - 3k- 2g,,,((q =

+

m)12 k)

(19)

k=l

which associates the semi.classical Wigner distribution to the classical probability density. Otherwise the (16) remains unchanged to this order in ~I. Thus, quantum interference effects between trajectories with the same value of m lead to a delocalization of the branches of the strange attractor in p-direction on a scale I Am(q)[ 1/3. The binary representation of the integer m, by construction, encodes the prehistory of the trajectories; it contains a 1 in the Inlth place ifqn >1 1/2 and a 0 ifq n < 1/2 for - o o < n ~< - 1 [8]. Trajectories with different values of m are found not to interfere quantum mechanically; only transition amplitudes along trajectories with the same prehistory interfer with each other. These are just all the trajectories which, classically, would end up on a given branch of the attractor at n - 0. Their mutual interference delocalizes that branch by associating with it an Airy function of local width IAm (q)l 1/3. The Airy functions are real but oscillate on the negative side of their argument between positive and negative values. The superposition therefore considerably modifies the classical measure, described by 6-distributions on the attractor. Finally, we turn to the exact quantum result (15). The integral over x gives 8-functions of p at integer multiples of lr~, i.e. we have the exact correspondence rule

8(p - Fm(q) ) ~

~

8(t9 - 7riTi)Wi],m(q),

(20)

T~- _ oo

with 1

W'[.m(q) = f dx exp' -2zrix~T '

0

2i ~

g

(q + x)(mod 1) +m

~/k = 1

2k

-g '

(q - x) (mod 1) +m 2k

(21) '

which replaces the semi-classical result (18). Integrating eq. (15) over q we obtain, after some rearrangements, the exact probability distribution of p as +oo

W**(p) = ~ l=

8(p

-

2#hl)Pl,

(22)

- - oo

with 2n-1

pl= l i m 2 - n

D

n---~

m=O

134

1

n

(23)

Volume 99A, number 4

PHYSICS LETTERS

28 November 1983

Thus, the strange attractor has disappeared in the full quantum ensemble and the momentum variable is instead quantized in integer units o f 2rrfi. The probability to find the state with quantum number 1 in the statistical ensemble of the steady state is P I . We note that P1 is positive and normalized, as it must be. We conclude that the novel procedure for quantizing a dissipative system presented here leads back to a probability measure concentrated on a strange attractor in the classical limit, but demonstrates that the strange attractor is not preserved in the full quantum result. The reasons are (i) the delocalization of the individual branches of the attractor due to quantum interference between trajectories with the same prehistories and (ii) the superposition of the broadened quasi-probability densities (which are positive and negative) associated with different branches. However, at least in the model we have considered here sensitive dependence on initial conditions and mixing are still present in correlation functions of the quantized map even though the strange attractor has disappeared. This work was begun during the author's stay at the Institute for Theoretical Physics in Santa Barbara. The hospitality and superb working conditions provided by this institute, as well as financial support by the Deutsche Forschungsgemeinschaft are gratefully acknowledged.

References [1] G. Casati, B.V. Chirikov, F.M. Izraelev and J. Ford, in: Stochastic behavior in classical and quantum hamiltonian systems, eds., G. Casati and J. Ford, Lecture Notes in Physics, Vol. 93 (Springer, Berlin, 1979) p. 334. [2] M.V. Berry, N.L. Balazs, M. Tabor and A. Voros, Ann. Phys. 122 (1979) 26; J.H. Hannay and M.V. Berry, Physica 1D (1980) 267. [3] J.L. Kaplan and J.A. Yorke, Lecture Notes Math. 730 (1979) 228. [4] R.V. Jensen and C.R. Oberman, Phys. Rev. Lett. 46 (1981) 1547. [5] D. Mayer and G. Roepstorff, Strange attractors and asymptotic measures of discrete time dissipative systems, Preprint, Aachen (1982). [6] E. Ott and J.D. Hanson, Phys. Lett. 85A (1981) 20. [7] R. Graham, Exact solution of some discrete stochastic models with chaos, preprint, Essen (1982). [8] M. H6non and Y. Pomoau, Lecture Notes Math. 565 (1976) 29. [9] E.N. Lorenz, J. Atmos. Sci. 20 (1963) 130. [10] H. Haken, Phys. Lett. 53A (1975) 77; R. Graham, Phys. Lett. 58A (1976) 440. [11] R.P. Feynman and A.R. Hibbs, Quantum mechanics and path integrals (McGraw-Hill, New York, 1965). [12] E. Wigner, Phys. Rev. 40 (1932) 749.

135