Chaotic scattering and transmission fluctuations

Physica D 50 (1991) 367-390 North-Holland

Chaotic scattering and transmission fluctuations E. Doron, U. Smilansky Department of Nuclear Physics, the Weizmann Institute, Rehovot 76100, lsrael

and A. Frenkel Electromagnetics Department, RAFAEL, Haifa 31021, Israel Received 20 August 1990 Revised manuscript received 14 January 1991 Accepted 15 January 1991 Communicated by R.S. MacKay

We discuss the phenomenon of chaotic scattering and its application in the study of transmission of electrons in mesoscopic devices as well as the transmission of microwaves through junctions. We show that the fact that the ray optics (classical dynamics) is chaotic, implies fluctuations in the observed transmission coefficients, whose statistics is determined by the theory of random matrices. We also show how the classical distribution functions which reflect the chaotic nature of the classical dynamics, determine the dependence of the correlations observed in the fluctuating transmission coefficients on external parameters. The time domain properties of chaotic scattering systems are also examined, and are shown to depend on the chaotic nature of the classical dynamics, together with a wave mechanical enhancement in time reversal invariant systems. Finally, we study the role of absorption and discuss its effects on the transmission fluctuations and their statistics.

I. Introduction I n t e r e s t in c h a o t i c s c a t t e r i n g is s t e a d i l y increasing, d u e to t h e fact t h a t this p h e n o m e n o n is m a n i f e s t e d in a large variety o f physical systems. This is t r u e n o t only for p r o b l e m s in classical dynamics, b u t also for systems w h e r e waves a r e s c a t t e r e d f r o m objects w h o s e extension is large relative to t h e w a v e l e n g t h o f the s c a t t e r e d waves. A m o n g t h e l a t t e r systems, t h e p r o p a g a t i o n o f e l e c t r o n s in m e s o s c o p i c systems [1-8], t h e scattering o f light a n d o t h e r e l e c t r o m a g n e t i c r a d i a t i o n by reflecting o b s t a c l e s [9-12], or m o l e c u l a r [13, 14] a n d n u c l e a r [15-18] r e a c t i o n s a r e t h e most o u t s t a n d i n g e x a m p l e s (a review o f t h e field, a n d a c o m p r e h e n s i v e l i t e r a t u r e list can b e f o u n d in review p a p e r s by E c k h a r d t [19] a n d Smilansky [20]).

T h e classical c h a o t i c s c a t t e r i n g is b e s t o b s e r v e d by m e a n s o f t h e ' r e a c t i o n function', which shows t h e d e p e n d e n c e o f t h e r e a c t i o n o u t c o m e on the c o n d i t i o n s which define t h e i n c o m i n g channels. C h a o t i c s c a t t e r i n g results in fractal r e a c t i o n functions which oscillate on all scales a n d which a r e c o n t i n u o u s only on t h e c o m p l e m e n t o f a C a n t o r set. This r e m a r k a b l e b e h a v i o u r is a t t r i b u t e d to t h e p r e s e n c e o f a ' s t r a n g e r e p e l l e r ' which is comp o s e d o f u n s t a b l e orbits t h a t a r e t r a p p e d f o r e v e r by the s c a t t e r e r . T h e t r a p p e d p e r i o d i c orbits f o r m the skeleton of the 'strange repeller', and their u n s t a b l e a n d stable m a n i f o l d s r e a c h o u t to t h e a s y m p t o t i c d o m a i n . A s c a t t e r i n g t r a j e c t o r y app r o a c h e s t h e r e p e l l e r by following a stable direction a n d leaving in an u n s t a b l e direction. W h i l e s p e n d i n g t i m e in t h e p r o x i m i t y o f t h e h e t e r o c l i n i c tangle, t h e t r a j e c t o r y ' l o s e s its m e m o r y ' , which

0167-2789/91/$03.50 © 1991 -Elsevier Science Publishers B.V. (North-Holland)

E. Doron et al. / Chaotic" scattering and transmission fluctuations

368

leads to the complicated structures observed in the reaction function. The intersection of the stable (or unstable) manifold with any line which defines the incoming channel in the asymptotic phase space is a Cantor set, whose Hausdorff dimension d is one of the parameters which characterize the scattering problem. A feature which is common to a large class of chaotic scattering problems is the fact that typically the probability that a trajectory stays within the interaction region for at least the time t is given by

/

c~ e

~'

(l)

for long enough times. (The probability for short times is nongeneric and may contain contributions from trajectories which do not approach the heteroclinic tangle.) It can be shown [21] that, for hyperbolic systems, the p a r a m e t e r y is related to the Lyapunov exponent, which characterizes the set of trapped orbits ,~, and to the Hausdorff dimension d by y = (1 - d ) A .

(2)

These universal classical features have also been found to induce a typical behaviour in systems which involve scattering of waves whose short wave limit corresponds to chaotic scattering. (That is, e.g., quantum problems whose Newtonian counterpart is chaotic, or 'physical' light scattering where the 'geometrical' optics is chaotic.) The S matrix which contains the entire information on the scattering process was shown to fluctuate in a typical way, namely, the fluctuations about its m e a n obey the statistics of one of the random ensembles of unitary matrices [22]. These ensembles were discussed in detail by Dyson, and, since we consider here only problems with time reversal symmetry, we shall always refer to the 'orthogonal ensemble', known also as the 'circular orthogonal ensemble' o r C O E for short. The relation between classical chaotic scattering and the applicability of random matrix theory in wave scattering was motivated by semiclassical

\ \

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/ //

/ \

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Fig. 1. The junction model. The asymmetrized version used in the calculations is drawn in heavy lines.

considerations (see below) and amply supported by numerical solutions of model problems. In the present p a p e r we would like to continue the investigation of chaotic wave scattering, and report some new results which allow us to interpret the fluctuating cross sections and their correlations, in terms of properties which we derive from the chaotic classical dynamics. The general expressions and results will be discussed in the framework of the semiclassical approximation. We shall demonstrate the general results and show their usefulness in a particular set of problems: the transmission of waves through waveguide junctions in the plane. These junctions are characterized by an 'interaction region' which is fed by 'leads' or waveguides (see fig. 1). The motivation for studying such systems comes from the extensive theoretical and experimental work done on the transport of electrons through wire junctions in the ballistic regime and the quenching of the quantum Hall effect [2-7]. Recently, a description of such a system was given in terms of classical transition probabilities to go from one lead to another, calculated using a "billiard" model [23]. However, this method cannot be used

E. Doron et al. / Chaotic scattering and transmission fluctuations

to calculate fluctuation properties of the S matrix, as it is completely classical. In the systems described above, the asymptotic incoming and outgoing waves are transmission modes. Each mode corresponds to a standing wave in the transverse direction and a traveling wave in the longitudinal direction. The longitudinal motion is characterized by a longitudinal wave number k I. In 'billiard' problems, where the interaction area and the channels are defined in terms of reflecting walls, the transversal wave number k t and the longitudinal wave number k l satisfy

k2t + k Z = k

2,

(3)

where k 2 is proportional to the total energy, k t is quantized and can take only a finite number of values k t = ('rr/d)n, where d is the width of the lead and 1 <_n < [dk/w]. The channels are therefore labeled by an index a which specifies the lead, and also by n for the particular mode. The S matrix is of finite dimension due to energy conservation. IS.... ;t3.m]2 gives the transmission probability, or the probability of reflection back to the same lead if a =/3. Strictly speaking, the S matrix should be dec o m p o s e d into its "direct p a r t " Sdi r and "fluctuating part" Sf. The former is due to "fast" processes, while the latter is due to "slow" processes, in which the underlying classical chaos manifests itself. It is only Sf which obeys the statistics of the COE, and S obeys them only in the limit Sdi r << Sf. In the systems which we will discuss here this condition is indeed satisfied. The definition of the asymptotic wave modes has a simple classical analogue: The ray propagates in the wave guide in a zig-zag fashion with a constant angle relative to the wave guide axis, whose value is arctan(kt/k~). The transverse mom e n t u m Pt, measured in units of h, takes the values + k t. The magnitude of the transverse m o m e n t u m can be considered as an 'action variable' I: d I = Ip, l ~ .

(4)

369

The conjugate angle variable 0 is then given in terms of the position Yt along the transverse direction by Yt

0 = "rr~- s g n ( P t ) .

(5)

The quantization condition is now simply

I=n,

n=1,2 .... ,

(6)

where n defines the wave mechanical mode or channel number. The reaction function in the present case gives the dependence of the transverse action (or propagation angle relative to the lead axis, see above) in the channel /3 as a function of the angle variable in channel a. A typical deflection function for the junction shown in fig. 1 is plotted in fig. 2. The corresponding staying time distribution is plotted in fig. 3, and it agrees very well with the expression (1). The structure of the S matrix reflects the symmetries in the corresponding junction. In the statistical analysis of the S matrix, it is advantageous to reduce the S matrix as much as possible, since submatrices corresponding to different irreducible symmetries are statistically independent. This reduction is most naturally achieved by removing from the junction all parts which can be incorporated by symmetry operations. Thus, the four-lead junction which is shown in fig. 1 is replaced by its irreducible eighth (drawn with heavy lines), where only one half of a lead survives, and the transmission problem reduces to a reflection problem. It should be emphasized that the results obtained for the reduced problem have the same physical content as the full problem, or of any of its reducible parts. In order to obtain the complete S matrix, one should also solve the wave equation subject to N e u m a n n boundary conditions on one or more of the symmetry lines. One would then construct the full S matrix from linear combinations of the various symmetry-reduced S matrices. Since the wave-

E. Doron et al. / Chaotic scattering and transmission fluctuations

370

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functions corresponding to different symmetry classes do not interact, the matrix elements of S matrices corresponding to different symmetries are statistically independent and add together in the usual way for independent random variables. Thus, the correlation length of fluctuations in the full S matrix elements would be the same as in the symmetry-reduced S matrices, and the vari-

ance would increase linearly with the number of symmetry classes. The p a p e r is organized in the following way. The next section will give a summary of the results already known to hold for the S matrix in the semiclassical regime. The new relations, which are mainly concerned with various autocorrelation functions, will be presented later. The semiclassical theory will be illustrated by results which were obtained numerically by solving the wave equation in the interaction region. The junction system has the advantage that it possesses two length scales: the characteristic size of the interaction area, and the transversal dimension of the lead. The former determines the interference pattern in the interaction area, as well as the density of the resonances. The latter sets (for a given energy) the dimension of the S matrix. In the extreme semiclassical approximation, both length scales should be much longer than the wavelength. By varying the two length p a r a m e t e r s independently we may approach the semiclassical limit in different ways, which we shall exploit when comparing the theoretical predictions with the numerical results. This is illustrated in section 3, where the time domain properties of the S matrix are examined. The time domain is a natural place in which to examine quantum-classical

E. Doron et al. / Chaotic scattering and transmission fluctuations

correspondence, as the classical dynamics for billiards is independent of the energy. In section 4 we discuss the effects of weak absorption on chaotic wave scattering, and in particular we investigate the extent to which energy loss affects

371

the predictions made for lossless systems. We shall summarize the results in section 5. The numerical method which we used, as well as some of the checks on its accuracy are described in the appendix.

2. The semiclassics of the S matrix

In this and the following sections we shall focus our attention on the system shown in fig. 1, drawn with heavy lines. This system consists of two parts: the semi-infinite waveguide, and the scattering cavity (which replaces the symmetric junction because of the symmetry considerations mentioned above). The indices of the S matrix are determined by the quantized value of the transverse momentum, as explained in the previous section. The basis for the semiclassical treatment of scattering problems is the expression, derived by Miller [24] (see also ref. [25]),

1 /l/2,$--.~10/'

Snn'=()---~wl ~lO0(s,

exp[iqO(s'(I'I')-iaru(~'/2] "

(7)

Here I and I' are the classical action variables con,~sponding to channels n and n', respectively, 0 is the initial angle, V(s) is the Maslov index, and q~(s)(I, I') is the reduced action along the path s [26], measured in units of h. The summation is carried out over all paths s which are incident onto the cavity and emerge from it with actions I and I', respectively. The pre-exponential factor is interpreted as the square root of the contribution of the s trajectory to the total classical transition probability, and we denote

10I'-'=P(~)(I,I').

2"n" ~ s ~

(8)

It must be noted that semiclassical orbit sums such as (7) are valid only as long as the difference in action between close orbits is smaller than the pre-exponential factor. However, in the case of chaotic scattering both the mean distance between neighbouring orbits and the pre-exponential factor decrease exponentially with the length of the orbit with approximately the same exponent, and therefore (7) remains valid even for long orbits. The instances in which orbits do coalesce (caustics) are rare, and, since in this paper we are interested mainly in averaged quantities, can be ignored. In ref. [26] it was conjectured that if the underlying classical dynamics is chaotic, the resulting S matrices of the quantum problem show statistical behaviour corresponding to that of one of Dyson's circular ensembles. The choice of the ensemble depends on the symmetries inherent in the dynamics. For time reversal invariant systems the behaviour is expected to be that of the circular orthogonal ensemble. This conjecture was substantiated [20] by using (7) to calculate the correlations between different elements of the S matrix. For S matrices of dimension L ( = [kd/~r]) one can define the correlation functions

372

E. Doron et aL / Chaotic scattering and transmission fluctuations

where the average is taken over a small square around n, n'. Then, it was shown semiclassically that for large enough matrices the underlying chaotic dynamics implies

C.,(n) = a(n),

G,,.

=0.

(9)

In other words, the real and imaginary parts of the S matrix are independent random variables. Further substantiation of the random matrix hypothesis was obtained in ref. [22], where the statistics of the S matrix eigenphases were studied. We shall not go into details here, as this paper will not deal with eigenvalue statistics. We shall now restrict ourselves to 'billiard' systems of the kind described in section 1. Here the velocity of the particles is constant. Thus, the path length inside the interaction region is a much more natural variable than the delay time. Relation (1) can be rewritten as P(I)

= YI e-v't,

(10)

for l much larger than some minimal path length. We shall need in the following an expression for the probability of a path to reflect n times off a specific segment of the wails. Assuming a 'mean free' path length lfr~e between reflections one can write

(ll)

p ( n ) = e,,,1,,7,,(1 - e ~,,) e-,,~,,

for n > nmin, with %, = y t l f .... and n.,~,, the minimal number of reflections possible, n.,~. is generally of the order of 1 or 2 in most problems. In wave mechanical calculations the important parameter will not be the energy but the wavenumber k. This is because the Schr6dinger equation, in the absence of a magnetic field, reduces to the Helmholtz equation (V 2 + k2)~/y = 1),

(12)

in which k is the only relevant parameter. An important property of the S matrix is its auto-correlation with respect to changing the energy, or m our case changing the wavenumber k. This correlation function is defined as

c..,(K)

=
(13)

+

where the averaging is done over a classically small k interval Ak. Using (7) in (13) one gets to lowest order in K [26]

C,,~,( K ) = L

PC~)(I, I') expl IK~j~-(14) 3' 4- .',"

k "

In the short wave limit, A@>> 1, and the nondiagonal term in (14) averages to zero. The partial derivative in the diagonal term is just acI)/ak = l, the path length traversed inside the interaction area,

E. Doron et al. / Chaotic scattering and transmission fluctuations

373

and so we can write

C~,(I<) = f dl(

(15)

PH'(k,l))k ei~',

where Pit(k, l) is the classical probability that a I ~ I' transition will have a trajectory whose path length within the interaction area is in the interval [l, l + d/]. This is just given by (10), and is independent of I, I'. So, assuming that the variation of 3' in the interval [k, k + Ak] is sufficiently small, we get for the correlation function

1 1 - iK/Aj

=( s*..,(k - ½K) s..,(k + " . ) L

1

1 + t~/

1 + (K/yt) 2

(16)

Together with the assumed COE statistics (9) we can show that the auto-correlation function for the fluctuating part of the cross sections for large S matrices should be given by

< O'nm(k - }K) ~rn,rn,(k -f- IK)> k


1

"~nn'~mm'1 +

(K/yz) 2 '

(17)

where d = e - ( e ) is the fluctuating part of the cross section. Expressions of this type were derived by Ericson [27, 28] for the correlations of nuclear cross sections, and so chaotic scattering provides a dynamical mechanism for generating Ericson fluctuations. However, relation (16) holds even if the poles of the S matrix do not overlap, and is valid for small matrices also. To test these statistical predictions, the S matrix was computed for the junction model at various values of R and k, giving various S matrix dimensions L. The method of calculation is described in the appendix. Random matrix theory (RMT) predicts an asymptotically (as the S matrix dimension L increases) Poissonian distribution of the absolute value squared of the S matrix elements, with a mean of 2/L and 1,/L for diagonal and nondiagonal elements, respectively. Figs. 4a and 4b show the distribution function of the absolute value squared of the diagonal and nondiagonal elements of the S matrix, respectively, for matrices of size 6 x 6 and up. The elements were normalized by a factor of L + 1, in order to get rid of the matrix size dependence. The dashed lines in the graphs represent the theoretical Poisson distribution with a mean of 2 and 1 for the diagonal and nondiagonal elements, respectively, and the agreement with the computed element distribution is good. The correlation between the real and imaginary parts of the S matrix elements was found to be ~ 0.001, so the real and imaginary parts are essentially uncorrelated. This also compares well with the COE predictions.

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4 iN,finalized Sal,~l~

6

0

"~'

'

i

'

r-I

2 3 INorn}ahzed S~. ~,=~]2

I

~

4

Fig. 4. Distribution of the absolute value squared of the diagonal S matrix elements for small dimensions: (a) diagonal elements; (b) off-diagonal elements. The dashed lines denote the theoretical Poisson distribution.

374

Chaotic scattering and transmission fluctuations

E. D o r o n et al. /

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Is~l2 Fig. 5. Distribution of the absolute value squared of the S matrix elements for small dimensions. Graphs (a) to (f) show distributions for matrices of dimension 2 to 7, respectively. The solid line is the prediction of (18).

For smaller matrices the distribution is no longer expected to be Poisson. In ref. [29] various statistics of small matrices belonging to the C O E were derived. In particular, the probability distribution of the absolute value squared of the diagonal elements (the diagonal cross sections) was shown to be P(JS..[ 2) = ½( L - 1)(1 - J S . . J 2 ) (L-3)/2.

(18)

Fig. 5 shows the numerical distributions versus theory for matrices of dimension 2 to 7. One can see that there is good agreement. In particular, (18) predicts that for 2 x 2 matrices the diagonal cross sections tend to get large values, for 3 x 3 matrices they are uniformly distributed, and for large matrices small values are more probable. This fact makes the 2 x 2 distribution easily distinguishable from all the others.

E. Doron et al. / Chaotic scattering and transmission fluctuations 1.0

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Fig. 6. The real part of the transition amplitude $23 as function of k, for R = 0.748.

I . . . . -I

2

Fig. 7. Absolute value squared of the auto-correlation function of the transition amplitudes S.m as function of k, for R = 0.748, averaged over all matrix elements.

The wavenumber correlation function was computed for R = 0.748. Fig. 6 shows the real part of $23 as a function of k. We notice that the amplitude fluctuates widely, with an apparently constant correlation length. The auto-correlation of ]Snm]2, averaged over the whole matrix, is shown in fig. 7. The average correlation length is ~ 0.5, which compares with the value 0.4 predicted by (16). So far we have shown that the present system displays the same features predicted to be universal for systems whose classical analogue is chaotic. We shall now show that these considerations can be extended to other directly observable quantities. In particular we address the problem of finding the auto-correlation function of the S matrix under changes of a general parameter A of the model. A could parametrize a change in the geometry of the scattering junction, the strength of an external field (e.g. a magnetic field), or any other perturbation of the system. Repeating the derivation of (15) one gets

G"m(A)

=

(S*m(A

/,-,

+

½A)Snm(A-2' A ))A= \l ~T P ~ f ' ) e x p /li A O~(s)]\

(19)

The partial derivative of the reduced action is

OA -

aA f dt(~-ao)= f dt

Od~"~ Oqi OPi O'~POqiOA oqi + °JfaPiaTiOA + ~ X - P i - ~ - q i - ~ ] •

(20)

Using Hamilton's equations of motion a~/ap i = qi and aW/aq i = - / ~ / w e get

0¢I) OA

[ =

-

ap i ] final qi-~-A-Jinitia,

+F,

(21)

where

F=

fdt OJU a--X

(22)

E. Doron et al. / Chaotic scatterb~g and transmission fluctuations

376

F can be interpreted as the integrated impulse imparted to the trajectory by the 'force' a#Z/~/OA.The term in the square brackets in (21) depends only on the initial and final points of the path s. If the parameter A does not affect the asymptotics of the system this term vanishes, and we are left with F only. However, for some choices of A the asymptotics are affected by the change in A. Then, this term can be written as a sum of two terms: a (possibly large) path-independent term ~'], and a path-dependent term ~ * ~ , which will typically be much smaller than the average value of F. For example, consider the correlation function (19) for the junction model with R, the circular arc radius, as the parameter. F will be the sum of the radial momenta imparted at reflections off the circular arc, and so will be of the order of nk, where n is the number of collisions with the wall. A change of R induces a change in the width of the channels, and so affects the asymptotic region as well. The asymptotic term in (21) for a transition n -~ m will be

q~-

-[-

initial =

c3d ] q'- Yfinal Od

Yinitialad

= .5"~ +

The maximum change in the endpoint coordinates will be Y~n,~l- Yiniti~l < d, and k~~=

,~2 = (Tr/dZ)( mYfina,-

(23)

nw/d,

and so

(24)

nyi,itia.) < k.

So we see that the distribution of F - ,~2 and that of F will be essentially the same, if in most cases n>l. Inserting (21) into (19) one is left with

G,,m(A)

~ f ~ dF( Ptr(A,F) ), e iAF.

(25)

PIr(A, F)

is the classical probability density of F, averaged over a range of A where it does not significantly change. Thus, S matrix fluctuations, which involve complicated interference of waves, can be expressed as a Fourier transform of the classical probability density function of F. The classical paths may be broken up into segments, where each segment is defined as the part of the trajectory between two successive reflections from one of the walls, for example the circular segment in the junction model. The integral over the classical path in (22) can be written as a sum of integrals over path segments. We denote by ~ the contribution of the jth segment. If the motion is chaotic one can take the Fj to be independent random variables with the same distribution, with an average F h and variance ~,. Then the distribution of the total F, for paths containing n segments, can by the central limit theorem be approximated by the Gaussian distribution

p(F[n)_

1

((F-nFh)2)

exp -

2n~t '

.

(26)

The probability density function for having a path composed of n segments in a chaotic problem was

E. Doron et aL / Chaotic scattering and transmission fluctuations

377

given in (11). So, the total probability density function for F is

P(F)

= e'~mi"~'"(1 e -v")

~

l

(

~/2-rrncrh- exp

-

( F - nFh) 2 2norh

) y~n .

(27)

n ~nmi n

Inserting (27) into (25) gives us a sum over Fourier transforms of Gaussians,

L

d x ~

exp

ixt

2~r

=exp(iXt-

~' r t 2 ),

(28)

which gives finally

IG.,~(A)

I

~(1

E

- e-~'") e ~m''~'"

exp [ .(iF~

- ':~,,A ~ - ~.)1

~/ ~ / ~ rain

~) 7.)]"

= G. m (0) ¢ ! : e - ~',) exp (_--_ ~ ,, m__~n~ l1 -- exp(iFhA

-

½o-hA 2 --

(29)

Eq. (29) shows that the S matrix correlation function can be expressed in terms of a few parameters 7,, F h and ~rh, which characterize the classical chaotic system in a very general way. Note that only the average and variance of F are required, and not more detailed information about its distribution function. In order to test the above result the S matrix was calculated for k = 70 and R varying between 0.5 and 0.85. The relevant quantity F here is the total radial force exerted by the circular segment on the cavity. This force acts when a particle is reflected off the circle, and so is indeed a sum of discrete impulses. Fig. 8 shows the distribution function P(F] n = 10). One can see that the distribution is well approximated by a Gaussian, with F h = 0.5k and crh -- 0.25k 2. At these p a r a m e t e r values the expected correlation length is approximately proportional to 7,. y~ for the range of R used varies between 0.9 and 0.2, and so the correlation length is not expected to be constant. To overcome this problem we normalize the R axis by the expected correlation length, by defining r = R / I ( R ) , where l(R) is the width of the correlation function (29). A plot of the real part of $33 versus R is shown in fig. 9a. One can see that the frequency of oscillation varies quite a lot over the R range. The value of the real part of $33 versus the normalized radius r is shown in fig. 9b. Here, the fluctuations seem similar over the entire r range, implying that the functional dependence of the correlation length on R is correct. Fig. 10 shows the average auto-correlation function of IS(r)12. The average correlation length of this normalized sequence is calculated to be 1.2, in good agreement with theory. Another interesting quantity we can evaluate is the correlation length of transmission fluctuations induced by a change in the strength B of an external magnetic field perpendicular to the plane of the junction. Fluctuations of this kind were observed experimentally [4, 30]. We will try to use (29) in order to predict the correlation length for the junction model, for small values of B. In the presence of magnetic field pointing in the 2 direction the Lagrangian is

~=

½m~2 + B( e / 2 c )( ," x '~)z - v ( , - ) ,

(30)

E. Doron et al. / Chaotic scattering and transmission fluctuations

378

~

]-~

T"~ [ • , ,

~

~-,

/ ~ : --,

o,25:~

i 0

7P

L o.15~r O.

, "~,t

j

o. 6:

'x',"~,,:t

/ /

i0 i

J'I

o8i

0,20~

/

0.4~

,~l<

-

/

/

/

/

/

\, \

/

\\\

• \ \..

o.

0.2

~

o,oo

~L

0

2

I ~ ,

. . . .

4

8

6

o.oi,

10

~

.

i

~ I

-

. . . .

h .....

4

I

2

~_.

0

I" (~hailge or R)

2

4

~r

Fig. 8. Distribution of F when changing the circle radius, for R = 0.91 and n = 10. The solid line is a best fit of a Gaussian distribution.

Fig. 10. Auto-correlation of the transition probabilities IS,,,,,I 2 as function of normalized circle radius r, averaged over all matrix elements.

where V(r) defines the junction boundaries. F is therefore

F =

(e/2c)f dt ( r X i')~.

(31)

The boundary conditions in the leads are independent of B, and so ~ and ~ 2 I~) vanish in this case. We arbitrarily define the origin to be at the center of the circle which defines the circular arc wall. We also suppose that the magnetic field is small, so that the motion in the cavity between collisions follows approximately straight lines. The angular m o m e n t u m r x # is constant between collisions, and so

= (e/c)Aj,

(32)

where Aj is the area of the triangle formed by the origin and the two endpoints of the line. This area must be taken with its sign, which is + and - for anti-clockwise and clockwise motion, respectively.

0 t>

r,

J

C0 l Dr;

:a}

i c

•

0.5

, ~

] .......

J

0.6

0.7

I 0.8

......

'6b

o~

(/'lli/II' i

',b}

~fu

i,j

~'

L

j 0.9

40

60

80

R

Fig. 9. The real part of the transition amplitude $33 as function of: (a) circle radius R, and (b) normalized circle radius r.

E. Doronet al. / Chaoticscatteringand transmissionfluctuations

379

2.C

1.5

1.0

0.5

0.0 ~ -

0.5

0 F (change of B)

Fig. 11. Distribution of F when changing the magnetic field strength, for R = 0.91 and n = 10. The solid line is a best fit of a Gaussian distribution.

0.5

W e can r e d e f i n e F i to be the value o f t h e integral t a k e n b e t w e e n successive reflections off the curved section of the cavity. T h e a v e r a g e value of Fj is F h = 0, b e c a u s e t h e m o t i o n inside the j u n c t i o n is b o u n d e d , a n d t h e v a r i a n c e is s o m e value (e/c)2o-n, w h e r e ~rn is d e p e n d e n t on g e o m e t r y a n d field s t r e n g t h only (the field s t r e n g t h d e p e n d e n c e is not very strong, as can b e seen by inspection). Fig. 11 shows an e x a m p l e o f the c a l c u l a t e d d i s t r i b u t i o n function of F , f r o m which crh can easily be extracted. W e insert t h e s e values into (19) to get

1 2 (1--e-'Yn) exp[---~nmin(e/c) o'hB 2 ] IGnm(B)[=Gnm(O) 1 -exp[-½(e/c)2o-hB2-yn ]

F o r nmi n

=

1 a n d y , << 1 o n e can show that

be= ( c / e ) ~ .

IG,,m(B)I

(33)

falls to half its B = 0 value w h e n B = b c, w h e r e

(34)

F l u c t u a t i o n s in the m a g n e t o r e s i s t a n c e w e r e s e e n e x p e r i m e n t a l l y for four-wire j u n c t i o n s in ref. [4]. B e c a u s e we d o not know t h e p r e c i s e g e o m e t r y of t h e j u n c t i o n s u s e d we c a n n o t d e r i v e h e r e p r e c i s e values for the c o r r e l a t i o n length o f t h e s e fluctuations. H o w e v e r , we can m a k e an o r d e r of m a g n i t u d e e s t i m a t e . T a k i n g t h e size o f the j u n c t i o n to be ~ 700 n m we get, a c c o r d i n g to (34), a c o r r e l a t i o n length b c ~ 10 mT, which c o m p a r e s well with t h e m e a s u r e d c o r r e l a t i o n lengths o f 5 - 3 0 m T ~1.

3. The time domain U n t i l now we have e x a m i n e d p r o p e r t i e s of the S m a t r i x as a f u n c t i o n o f the e n e r g y (or the w a v e n u m b e r ) . This a p p r o a c h c o n s i d e r s t h e s c a t t e r i n g p r o c e s s as a s t e a d y state, w h e r e the i n c o m i n g a n d #1During the final stages of preparation of this work we received a preprint [31] where bc was independently derived for mesoscopic systems.

E. Doron et al. / Chaotic scattering and transmission fluctuations

380

scattered wave intensities depend on the energy only. A dual approach is to consider a narrow wave packet (a pulse) which impinges on the scatterer, and measure the distribution of the scattering products in time. The two approaches are intimately related: the S matrix in the time domain is the Fourier transform of S(E). An equivalent duality holds in "billiard" problems between the m o m e n t u m (wavenumber) and the path length representations. In the present section we shall investigate chaotic scattering from the time domain (path length domain) point of view, and consider the Fourier spectrum of S ( k ) as the response of the cavity to a localized burst of energy. As such we would expect the spectrum to follow approximately the classical path length distribution function. To put this idea in a semiclassical framework, we consider

2/j

[Sd(l) I =

dkSn.(k)e

-~k'

2/

(35) tt '

[ Snd(/

=

d k S n m ( k ) e ~k,

,

(36)

In~-m

where we have explicitly separated averaging over the diagonal and nondiagonal elements of S. Inserting the semiclassical expression (7) and carrying out the Fourier transform we get the double sum

f d k Sn,.,( k ) e

ik'2 ~ ~ ( P(Hsl)P(l)i')) '/2(~.( l - l (s)) (~( l - l (C)) e ir:O',--'', )/2

= P ( l ) + E (P¢]')P~"))'/26~.( l - l(~)) 6..(l - l (¢)) e i~('' ~,,)/2,

(37)

S 4- S'

where the delta functions are broadened by the fact that we take only a finite range of k values into consideration. As before when we average over many matrix elements we neglect the contribution of pairs of different paths. This leaves us with the diagonal term P(I). However, if the system exhibits time reversal symmetry then in the case n = m for each path s there exists a distinct time reversed path s' with the same boundary conditions, action, Maslov index and probability. Such pairs of conjugate paths contribute t o ]Sd(/)l 2 a term which equals the diagonal term, so that ISd(l)l 2 = 2P(l). This is not the case for nondiagonal matrix elements n 4=m, since time-reversed paths do not obey the same boundary conditions. Hence, time reversal symmetry implies [Sd(l) ]z ~ 2 e - ~ " , ^

(38)

2

]Snd(/)l ~ e -z'#.

(39)

The average Fourier spectrum of S ( k ) is shown in fig. 12, for a cavity with R = 0.748. The upper curve is the average over diagonal elements of the S matrix and the lower curve over the off-diagonal terms. The behaviour is clearly exponential over some three decades, which reflects the expected precision of the quantum calculations. The slope for the two curves is similar and equal to 0.45, in good agreement with the value 7z = 0.4 predicted by the classical theory. Note that this is an averaged quantity, and Fourier spectra of individual S matrix elements may deviate from this form. Deviations are expected in the low l regime, where the probability density P(l) is still affected by the contributions of nongeneric short trajectories.

E. Doron et al. / Chaotic scattering and transmission fluctuations

381

101

10 0

10-1

10 2

10 3

0

5

i0

20

15 I

Fig. 12. Fourier spectrum of S,,m(k), averaged over all diagonal elements (top graph) and off-diagonal elements (bottom graph), for R = 0.748. The value of 39 for this system is 39 = 0.4. The dashed lines are a best fit of an exponential to the data, with slopes of 0.479 and 0.478 for diagonal and off-diagonal elements, respectively.

A n o t h e r interesting time domain quantity is the time delay (or the path length inside the cavity) as a function of k. This quantity is defined as [32-34]

l(k)

=

~

1

T r ([S ,OS) ~-

,

1

= ;-L ~ S . m

OS,,.,

1 ( d e t ( S ) ) * ~0- d e t ( S ) . Ok - i L

(40)

nm

l ( k ) fluctuates widely with k as we pass close to the various poles of the S matrix. However, its average value is the wave mechanical analogue of the m e a n path length of a particle in the classical system, and so the effects of the classical chaotic dynamics should be visible here. W e calculate the average value of l ( k ) semiclassically by inserting (7) into (40). Using the relation 0q~(s))/0k = l (s) one gets

. )

OSnm\

- IS,,,. ~

I k = E ( P~]')P~]"))'/2 l(S)( exp[ i( q)("' - q~(*') - i w( v~. - v" ) / 2 ] )k"

(41)

SS '

H e r e ( )k denotes averaging over an interval of k where the n u m b e r of propagating modes L is constant. As before this double sum can be separated into a diagonal s = s' sum and a nondiagonal s 4= s' sum. Also, if the system is invariant u n d e r time reversal, there is a contribution from conjugate pairs of paths in the case m = n. The diagonal and conjugate pair contributions are i n d e p e n d e n t of k, while the nondiagonal term fluctuates with k, and so averages out. So, neglecting the contribution of the nondiagonal term in (41) and taking time reversal conjugation into account one gets

( - iS*.,,,, O S n m / O k ) k = 2 ( l ) / L = (1)/L

for n = m , for n 4~ m,

(42)

where ( l ) is the average length of the classical paths inside the interaction area, which in chaotic systems

E. Doron et al. / Chaotic scattering and transmission fluctuations

382

is approximately 3'71. Inserting (42) into (40) gives

[(L +

(l(k))k =

1)~LIT{'.

(43)

The significance of (43) can be seen by looking at extreme cases. If L >> 1 the wave mechanical mean time delay (or path length) approaches the classical value, as expected. However, in the junction model it is possible to choose d and k such that k is much larger than unity. In this case the behaviour inside the cavity can be described by means of semiclassical relations, but L = [kd/rr] is small, so that only a few channels are propagating in the waveguide. Then a wave mechanical enhancement of the time delay appears. In the extreme case of kd/w < 2, only one waveguide channel is open, and the wave mechanical time delay is double the classical value. This result should be contrasted with the corresponding result for elastic scattering. In both cases, the S matrices reduce to unimodular numbers. However, for elastic scattering, the staying time (or length), given by the derivative of the phase with respect to E (or k), approaches the classical staying time for h ~ 0. In the case of a single open channel, the quantal staying time is double the classical one. We shall discuss the origin of this p h e n o m e n o n in the last section. In order to test the validity of (43), (l(k)>k was computed for S matrix dimensions L between 1 and 7. It was not possible to use the same system for all L, for computational reasons. Instead, several systems were used, with values of yt ranging from 0.1 to 0.4. In fig. 13 the ratio l(L)/l(L + 1) is plotted for all the systems used. The solid line represents the prediction of (43). We can see that within the numerical accuracy there is a reasonable agreement for all systems used, and in particular the enhancement due to time reversal symmetry is clearly seen. The value of (l> for the one-channel and multi-channel cases can be seen in fig. 14, with R = 0.91 and R = 0.748, respectively, y71 for R = 0.91 is 5, while (l> is 9.2, which is not far from 2 y i 1. The wave mechanical enhancement of the average path length is due to the fact that the dynamics is invariant under time reversal. This enhancement should therefore disappear in the presence of a magnetic field. However, this transition will not be abrupt but gradual. Switching on the magnetic field B

1.4~

,

,

'

[

"

I

. . . .

P

/

0\

[ , , , , ] 50

ill

55

60 k

65

70

201

\

i

15( i0( 5(

1.0 L 0

i

I

2

4

6

L

Fig. 13. Ratio of l ( i ) to I(L + 1) for five different systems, having values of Yl between 0.1 and 0.4. The solid line is the theoretical result given by (43).

c

50

55

60 k

65

¼

7O

Fig. 14. The scattered path length versus k. (a) R = 0.91, only one channel open; (b) R = 0.748, 4-5 channels open.

E. Doronet al. / Chaoticscatteringand transmissionfluctuations will cause the actions of the time reversed pair s and s' to change by so their contribution to the sum in (41) is Al(k) =

383

(b@/OB)B each, to first order, and

~P~l]'l{S)exp(2i~BB) = f o d' L ~ dFP(FL')P(l)lexp(2iFB),

(44)

where F is the same quantity used in the calculation of the magnetic field correlation function. P(l) is given by (10), and P(FII) is given by substituting n = I/lfree into (26). We also define trl = trh/lfree. Then the integral in (44) can be evaluated analytically, giving

AI=

1

)2,

1 + (2B/b~) 2

(45)

with b c being exactly the width of the magnetic field auto-correlation function (34). Thus, the time reversal enhancement of the staying time is switched off once the magnetic field reaches a value which is of the order of the magnetic field correlation length be.

4. Chaotic scattering in the presence of absorption In most problems of practical relevance, absorption of energy and flux due to interactions with the environment cannot be avoided. Absorption introduces a new time scale, which is inversely proportional to the rate of energy loss. It can also be thought of in terms of a width (the "absorption width") which is added to the "escape width" resulting from the coupling of the system to the continuum. We shall be concerned here mainly with situations where the absorption width is but a small fraction of the escape width. We shall show that even under such circumstances, absorption may affect the results in an appreciable way. A direct consequence of absorption is that the S matrix ceases to be unitary, with Idet(S)l < 1. In such cases the S matrix fluctuations can no longer be described in terms of the theory of the unitary matrix ensembles. To the best of our knowledge, a theory for random, sub-unitary matrices is not yet available, and the connection between chaotic scattering in the presence of absorption and the corresponding random matrix theory is not known. The main purpose of the

present section is to provide some information on the effects of absorption on the S matrix statistics, and to offer an intuitive argument which could be used to interprei the numerical data. We hope that this will be of help for further systematic studies of this problem. A natural way to introduce absorption to billiard problems of the kind considered here, is to add an imaginary part i a to the wavenumber k. For microwave cavities this corresponds to introducing absorption through a complex index of refraction, and for electrons in a solid state device, the imaginary part comes from an effective (local) optical potential. The advantage of this approach is that the numerical method which we use to calculate the S matrix (see the appendix) can be easily generalized to complex wavenumbers. Moreover, the addition of an imaginary part to the wavenumber can be also considered as a convolution of S(k) with a Lorentzian with the appropriate width. Hence, if one calculates S(k) for real k values over a sufficiently wide and dense grid, one can study the effect of various absorption widths by convolving the same data with different Lorentzians. This method has the advantage that the computational load is very small.

384

E. Doron et al.

/ Chaotic" scattering and transmission fluctuations

A convenient measure of the overall effect of absorption is given by the value of ]det(S)[. If we include absorption by adding a small imaginary part to the wavenumber, we can write

d e t [ S ( k + ice)]

=det[S(k)]exp(-ce~-k) = det[S(k)] exp[-cel(k)L], (46)

where det(S(k)) = exp(iqffk)). ¢ ( k ) is real on the real axis, because of unitarity. Here use was made of the analyticity of S(k) (and of ¢ ( k ) ) in a strip around the real axis [12]. The meaning of (46) is that the effective path length along which the absorption takes place is the Wigner length l(k) which was defined in section 3 in (40). However, l(k) is a quantum quantity, which fluctuates strongly as k is changed, and whose mean {l(k)} may differ substantially from the mean classical path length, due to interference between time reversed trajectories. For systems where the leads transmit only a small number of modes, this interference effect becomes significant, reaching a factor of 2 in the extreme case of a single mode. In fig. 15 the value of Idet(S)l 2 and of e x p [ - 2 c e / ( k ) L] are compared, for a 3 × 3 matrix with ce = 0.01. The agreement is very good, and in

o.~#~

~

~

~d'~

,.

J

most places it is difficult to separate the two graphs. As noted above, S(k + i a ) can be derived from S(k) by convoluting it with a Lorentzian of width ce. This, by the convolution theorem, is equivalent to multiplying the Fourier transform of S(k), S(1), with the Fourier transform of the Lorentzian, which is exp(-ce[ll). However, as we noted bef o r e , [,~ij(l)]2 decreases exponentially, with slope Y. Thus, if c~ <
o.4 V I

t .......

70

I

72

....

_

IS[il 2

ISiil 2

{iS, i2 ) ,

(47)

I 74

76

78

80

k

Fig. 15. [det[S(k)][2 (solid line) and e x p [ - 2 a l ( k ) L ] (dashed line) as functions of k, for S matrices of dimension L = 3 and absorption coefficient a = 0.01.

where the S matrix includes the effects of absorption. By considering the normalized quantities, we delete the obvious effect of the size of ce on the mean of ISiil 2. The numerical distributions of

/

E. Doron et al.

Chaotic scattering and transmission fluctuations

the normalized diagonal elements are shown in fig. 16 for S matrix dimension L ranging between 1 and 6, with a = 0.01. The surprising result is that the distribution of the diagonals of an S matrix of dimension L which includes absorption, is very similar to the predictions of COE for unitary S matrices (no absorption) of dimension L + 1! Moreover, further simulations show that this result is not sensitive to the value of a over a wide range. This numerical result is consistent

~o-

I

I

with the intuitive argument that the absorption effect can sometimes be mimicked by adding an unobserved channel to the S matrix. The ( L + 1) x ( L + 1) matrix is unitary, but its L x L minor is subunitary, and the absorption in this picture corresponds to the flux which is diverted to the extra channel. If the flux in the unobserved channel would be of the order of the mean flux per channel, we could justify the use of COE statistics (for ( L + 1 ) x ( L + 1) unitary matrices) to

I

_

2.5

(~)

8-

385

2,0

F

_

l

T

6-v

i

4 --

~~"

~.5 t.o 0.5

o

o.o

I 0

0.2

I

0,4

0.6

0.8

1

0

I

t

I

2.0

I

I

1.5 E"

t.O

- :=

1_,0

C~

0.5

0.5

0

I

I

I

[

q

0.2

0.4

0.6

0.8

1

0.0

o.z5

Is;,I~ I

2.5

0.4

0.6

I

I

I

0.8

3_

ls;,I~

1.5

0.0

0.2

IS:J

~-

I--

I

[

I

I

o.s 0.75 Is;,Ia

--

t

I

I

I

I

I -~---

2.0 4-"

1_.5

~"

1.o

E-"

0.5

% I - - I- . ~I

0.0 0

0.2

0.4

0.6

Is;,I~

I 0.8

1--1 o o

0.2

0.4

0.6

0.8

I-L

Fig. 16. Distribution of the normalized absolute value squared of the diagonal matrix elements of S matrices which include absorption. The absorption coefficient is a = 0.01. Graphs (a) to (f) show distributions for matrices of dimension L = 1 to 6, respectively. The solid line is the prediction of (18) for matrices of dimension L + 1.

386

E. Doron et al. / Chaotic scattering and transmission fluctuations

account for the properties of L x L subunitary matrices. Our numerical results were obtained for weak absorption, and yet this naive picture applies surprisingly well, as can be seen from fig. 16.

5. Summary and conclusions The main purpose of this article was to point out the consequences of the classical chaotic dynamics on the propagation of waves in simple devices. We have shown that the S matrix, and hence, the transmission and reflection coefficient fluctuate when the wavelength or any other parameter of the system is varied. The statistics and the correlations of these fluctuations were investigated in detail. It was shown that the statistics of the fluctuations follow the predictions of the theory of random matrices (COE). The autocorrelation functions were shown to be expressible as Fourier transforms of classical functions, which determine the distribution of the variables which are canonically conjugate to the variable whose variation induces the S matrix fluctuations. That is, e.g., the auto-correlations in the energy (wavenumber) are determined by the classical distribution of the delay time (path length in the interaction area), the fluctuations due to changes of the shape of the cavity are determined by the distribution of the corresponding momenta, and the classical distribution of areas determines the quantal fluctuations as the magnetic field varies. Due to the chaotic nature of the classical dynamics, the classical distributions assume very simple forms, which depend on only a few parameters. Of prime importance is the p a r a m e t e r y, which is directly related through eq. (2) to the intrinsic p a r a m e t e r s which characterize the strange repeller of the classical system. The results which were discussed in the present work emphasize the generic properties of the classical dynamics, and their universal counterparts in the wave dynamics. However, many clas-

sical systems show their generic behavior only for trajectories which have spent sufficiently long time in the interaction area. Those which dwell there for a short time only do not acquire the stochastic aspects of the chaotic dynamics. The contributions of the latter trajectories to the S matrix form its nongeneric part, and since it corresponds to short interaction times it varies slowly with energy. This is the smooth part Sd~r of the S matrix, which is also referred to as the 'direct' part. The chaotic dynamics which corresponds to long dwell times generate the fluctuating part S t of the S matrix. The statistics of S t. is not independent of Sj~r, since the total S matrix has to be unitary, and not any of its constituents. The unitarity constraints and the modified C O E statistics were discussed in ref. [29]. An intriguing result was obtained when we discussed the mean quantum delay time (length of path in the interaction region). We showed that due to time reversal symmetry, the resulting mean time depends on whether one uses the diagonal or nondiagonal S matrix elements as input. In the extreme situation where only a single travelling mode exists, the S matrix reduces to a unimodular complex number, and the delay time is twice the classical result. If we consider, in contrast, elastic scattering, where the S matrix is a unimodular complex number as well, the same definition of a delay time coincides with the classical delay time! The difference between the two examples is that the transverse degree of freedom, although being ' p o p u l a t e d ' only in its lowest eigenfunction, is not degenerate. Rather, it corresponds to a superposition of two travelling waves, with Pt = -+ kt. A trajectory which satisfies the boundary conditions Pt[i,f = ± k t e n t e r s and leaves the interaction region at transverse positions Y~,t, which are seldom the same. This is why time-reversed trajectories are distinct phase space paths whose contributions to the S matrix add coherently, and hence the factor of two, In proper elastic scattering, there is no additional degree of freedom to distinguish timereversed trajectories. Thus, the trajectory coin-

E. Doron et al. / Chaotic scattering and transmission fluctuations

cides with its time reversed counterpart, and the corresponding interference effect is absent. Finally, we would like to mention that this expression of time reversal symmetry, and the resulting elastic enhancement has the same physical origin as the corresponding enhancement (also called ' w e a k localization') which is o b s e r v e d in backscattered waves from random media [36, 37]. The effects of absorption on chaotic scattering were also studied. It was shown that in the case of weak absorption the total energy loss is the same as the loss over a path of length l ( k ) . The distribution function of the diagonal elements of the S matrix was also shown numerically to be affected by absorption, in a fashion similar to the effect of the addition of an unobserved channel to the system. The above theoretical results are relevant to experiments, and a preliminary, albeit convincing example was reported recently [35]. A more detailed experiment is now under way. The study of transmission fluctuations in mesoscopic junctions is another domain where our results are of relevance. Typical leads have transverse dimensions of the order of ~ 100-200 rim, and for carriers with Fermi wavelength of AF ~-50 nm there are only a few open modes [4]. An interesting experiment in the present context would be to measure the reflectivity of a device, and observe the disappearance of the ' w e a k localization' enhancement as a function of the magnetic field. The same experiment could also measure the transmission fluctuations and thus provide the range of field values where the effects of time reversal symmetry are not noticeable. Mesoscopic devices produced by advanced lithographic methods are still rather crude as far as the smoothness of the lines which delineate the interaction area is concerned. A problem which was not addressed by us in the present work is the influence of details whose scale is intermediate between the overall shape parameters and the wavelength. Classically speaking, such details will probably reduce the importance of direct processes, will increase the mean dwell

387

time and will enhance the stochasticity of the classical description. The effect on the quantum fluctuations should depend on the ratio between the roughness scale and the wavelength. To the best of our knowledge this subject has not yet been addressed.

Acknowledgements We would like to thank Dr. E. Levin for his help during the first stages of the experimental work. We profited from discussions with Professor H.A. Weidenmiiller, and from valuable suggestions for which we are indebted. The research was supported in part by research grants from the U S - I s r a e l binational science foundation (BSF), and the G e r m a n - I s r a e l i Foundation for Scientific Research and Development (GIF).

Appendix. The wave mechanical calculation method This appendix will describe the method used to calculate the S matrix of a scattering problem of the type discussed in this paper. The method is quite general and should be applicable, with variations, to many other systems. The time-independent Schr6dinger equation reduces in our case to the Helmholtz equation (12) with the appropriate boundary conditions, which are the vanishing of the wavefunction at the edges of the conductors. We may also impose conditions of vanishing derivative on the wavefunction on some segments of the cavity, if we wish to study other symmetry classes. Inside the infinitely long straight waveguide, the solutions of eq. (12) can be written as the superposition of a finite number of propagating and an infinite number of exponentially decaying

E. Doron et a L / Chaotic scattering and transmission fluctuations

388

modes:

linear equations,

q t ( x > 1)

Y'. [J4,,( krx=,)sin(4nOx=,)lce,, n=l

L

= ~ (ite-ik,(x

1)+

L

R, e i k t ~ - , , ) s i n ( ~ y )

/=1

/=1 zc

~c

+

R,

A1,

/=L+I

where It, R t denote incident and reflected amplitudes, respectively. Inside the junction the solutions of the Helmholtz equation, imposing vanishing amplitude on the diagonal, can be written in polar coordinates as

for matching the wavefunction at the interface, and

Y'~ [J4,,( kt')sin(4nO)]ee,, n=l

k= L

zc

qt(r.O) = ~ c~.J4~(kr)sin(4nO ). n

(A.5) /=L+I

= Eik,(R,-l,)s'nlTY (A.2)

,.{Iv

]

/=1

1

(A.6) The problem of finding the solution to (12), with the appropriate boundary conditions, is now reduced to that of finding suitable conditions for a=(C~l,C~ 2 . . . . ) so that qr will vanish on the one-eighth of a circle that comprises the only part of the boundary where the wavefunction is not identically zero, by virtue of (A.2). This is accomplished by demanding that (A.2) vanish on a set of Nzero appropriately chosen points {ri, 0 i} on the eighth-circle. This gives us a set of linear conditions on a

for matching the normal derivative. Performing the derivative and taking the sine transform with respect to y of the two sides of the equations gives a discrete set of linear conditions, two for each waveguide channel. Written in matrix form these conditions are l ' l a = R + 1,

(A.7)

i F2a = R - I

(A.8)

for matching the internal wavefunction and derivative to the open channels in the waveguide, and

Y'. [ J4n( kri)sin(4nOi)]cen = O, n-1

i = 1,2 . . . . . N z.....

I=L+I

(A.3) .;Tza = 0,

or in matrix form ~/1 a = 0.

(A.4)

Second, the two representations of the wavefunction, (A.2) inside the junction and (A.1) inside the conductor, must be made to match at the intersection of the two. This gives two sets of

(A.9)

signifying that there is no energy coming in through closed channels. One may also write an expression matching the internal wavefunction to the reflected waves in the closed channels. However, our detector is taken to be at the end of an infinite waveguide, and so we are not interested in the closed channels. The (real) matrices F~. 2 have L rows, as we have a condition for each

E. Doron et al. / Chaotic scattering and transmission fluctuations

open channel. The matrix ~ 2 has in principle infinitely many rows. However, in practice one limits the number of closed channels taken into consideration to some number Nctosed, as channels further up are only weakly coupled to the internal wavefunction. Eqs. (A.4) and (A.9) can be united into the expression ~¢¢'¢~= 0,

(A.10)

where ~ " now contains N A = N~ero + Ndos~d rows. We now have at our disposal all the equations relevant to the problem: (A.10), (A.7) and (A.8). We now set all but the first (Nzero + L + Nc~o~d) elements of ~ to zero, in order to get a uniquely solvable set of equations. Note that if we succeed in getting a result that satisfies the boundary conditions it does not matter that we truncated the Bessel series. The approximation lies not with this truncation but with the approximate boundary conditions that we impose. In order to derive the S matrix, connecting R and I, we need to insert conditions (A.10) into (A.7), (A.8). The difficulty in doing this stems from the fact that the conditions implied in (A.10) are not orthogonal conditions. Nearby points on the circle tend to give very similar conditions for vanishing of the wavefunction, and so the matrix is, to numerical precision, not of full rank. Thus, methods such as Gaussian elimination will not work. What we need to do is to separate the information containing part of ~¢" from the linear dependencies, to get orthogonal conditions. This can be done by writing the singular value decomposition (SVD) [38] of :~/:

~'=U.X.V

t,

(A.11)

where U and V are unitary matrices, and E is a (nonsquare) matrix with positive (or zero) values along the diagonal, and zero elsewhere. If we insert (A.11) into (A.10), drop U and d e f i n e / 3 ' = v t a , (A.10) can be written as /3[ = 0,

i = 1,2 . . . . . r a n k ( ~ ' ) .

(A.12)

389

The first r a n k ( d ) rows of V t now contain all the information, to numerical precision, contained in ~¢', while the remaining rows of V t contain nothing but numerical noise. However, it is convenient to take into account all the rows, as this prevents having to change the length of a and, now that it has been separated out, the numerical noise does no harm. This procedure can be checked by taking an arbitrary vector/3 fulfilling (A.12), multiplying it by V t to get a vector a , and multiplying it by d . The result is zero up to machine precision, even when the matrix ~e" contains a few identical rows. We can now easily insert the zero conditions into (A.7), (A.8). We multiply F 1 and F 2 on the right by V, and take the last L columns, to get square matrices F1.2- Then we define S2 = Fzf~ - 1, to get R = (1 - ia)

1(1 q'- i ~ 2 ) I

=

Sl.

(A.13)

The unitarity of the S matrix is ensured if g2 is symmetric. The method of calculation described above includes the two parameters Nzero and NcJosed. Nzero must be chosen large enough so that zeroing the chosen points will imply a very low value for the wavefunction on the segments connecting the points (this also implies that the zero conditions for adjacent points are very similar). The choice of Nzero, as well as of the zeroed points themselves, can be checked by calculating the value of qr on the boundary, where it is supposed to be zero. The values must be much smaller than the typical value for points inside the cavity, and the amplitude along the boundary must be as constant as possible, to ensure an efficient distribution of zeroed points { R i , Oi}. The value of NclosecI must be chosen large enough so very little flux is coupled to the cavity via the untreated closed channels. However, choosing values which are too high leads to differences of many tens of orders of magnitude between different elements of the matrices, and so to numerical difficulties. For those reasons this method is limited in the maxi-

390

E. Doron et al. / Chaotic scattering and transmission fluctuations

mum k for which meaningful results can be obtained. A good check on the results is the asymmetry present in the matrix X2,

~ i j -- ~ j i = max , ij ~'~ij "~ ~ j i

(A.14)

which controls the unitarity of S. The magnitude of the deviation from symmetry in ,(2 gives an idea of the precision of the calculation of S. When 3 << 1 the results are reasonable, and otherwise the values of Nzero, the choice of the points {R i, 0 i} or Nciosed must be changed.

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