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arXiv:cond-mat/0011141v2 [cond-mat.dis-nn] 30 Jan 2001

Reducing nonideal to ideal coupling in random matrix description of

chaotic scattering: Application to the time-delay problem

Dmitry V. Savin1,2, Yan V. Fyodorov3, and Hans-J¨ urgen Sommers1

1Fachbereich Physik, Universit¨ at-GH Essen, 45117 Essen, Germany

2Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia

3Department of Mathematical Sciences, Brunel University, Uxbridge, UB8 3PH, United Kingdom

(Received 8 November 2000; published in Phys. Rev. E 63 (March 2001), Rapid Communication)

We write explicitly a transformation of the scattering

phases reducing the problem of quantum chaotic scattering

for systems with M statistically equivalent channels at non-

ideal coupling to that for ideal coupling. Unfolding the phases

by their local density leads to universality of their local fluctu-

ations for large M. A relation between the partial time delays

and diagonal matrix elements of the Wigner-Smith matrix is

revealed for ideal coupling. This helped us in deriving the

joint probability distribution of partial time delays and the

distribution of the Wigner time delay.

PACS numbers: 05.45.-a, 24.60.-k, 73.23.-b

The random matrix theory (RMT) is generally ac-

cepted to be an adequate tool for describing various

universal statistical properties of quantum systems with

chaotic intrinsic dynamics, see Ref. [1] and references

therein. In particular, one can distinguish two variants

of the RMT approach allowing one to address the chaotic

nature of quantum scattering. The first one [2] considers

the scattering matrix S as the prime object without any

reference to the system Hamiltonian. The probability

distribution P(S) of S at the fixed energy E of incident

particles is chosen to satisfy a maximum entropy principle

and natural constraints which follow from the unitarity

and causality of S, and the presence (or absence) of the

time-reversal(TRS) and spin-rotation (SRS) symmetries,

P(S) ∝

????

det(1 −¯S†¯S)

det(1 −¯S†S)2

????

(βM+2−β)/2

.(1)

Such a distribution is known as the Poisson kernel [3]

and uses the phenomenological average (or optical) S-

matrix S(E) as the set of input parameters. Without

loss of generality¯S can be considered as diagonal [4].

P(S) depends also on the number of scattering channels

M and the symmetry index β [β=2 for a system with

broken TRS, and β=1(4) if the TRS is preserved and the

SRS is present (absent)].

The approach proved to be a success for extracting

many characteristics important in the theory of meso-

scopic transport [5]. However, correlation properties of

the S-matrix at close values of energy E as well as spec-

tral characteristics of an open system related to the so-

called resonances turn out to be inaccessible in the frame-

work of such an approach, essentially because of the one-

energy nature of the latter. To address such quantities

one needs to consider the HamiltonianˆH of the quantum

chaotic system as the prime building block of the the-

ory. It amounts to treatingˆH as a large N × N random

matrix of appropriate symmetry and relating S to the

Hamiltonian by means of standard tools of the scatter-

ing theory [6,7]. This idea supplemented with the super-

symmetry technique of ensemble averaging [8] resulted

in advance in calculating S-matrix correlation functions

[6,9] and many other related characteristics such as, e.g.,

time delays [10–12], see Refs. [11,1] for a review.

In the limit N→∞ one can prove [13] the equivalence

of both mentioned approaches by deriving the Poisson

kernel (1) from the Hamiltonian approach (see also Ref.

[11]), with the average S-matrix being

S(E)ab=1 − γa[iE/2 + πν(E)]

1 + γa[iE/2 + πν(E)]δab,(2)

independent of β. Here, the average density of states

ν(E)=π−1?1−(E/2)2determines the mean level spac-

ing ∆=(νN)−1of the closed system, and phenomenolog-

ical constants γc>0 characterize the coupling strength to

continuum in different scattering channels (c=1,...,M).

The particular case of ideal coupling,¯S=0 [when the

transmission coefficients equal unity for all channels, see

Eq. (6) below], plays an especiallly important role for the

S-matrix approach [14]. Equation (1) simplifies then to

P0(S)=const, which is invariant under the transforma-

tions of S, leaving the measure invariant. Such a situ-

ation corresponds to the so-called Dyson’s circular en-

semble (CE) of unitary matrices and is much simpler to

handle analytically.

A general situation of nonideal coupling,¯S ?=0, turns

out to be much more complicated. It is natural to ex-

pect, however, that results obtained for the case of non-

ideal coupling could be related to those at ideal coupling.

Although many useful ideas around such a relation were

discussed in the literature [2,13,11,15], we are not aware

of explicit relations, to the best of our knowledge.

In this Rapid Communication we consider the most

simple but physically important case of statistically

equivalent channels.We demonstrate the validity of

the following simple statement (and discuss several ap-

plications of it): Let S(E) = Uˆ s(E)U†, where ˆ s(E) =

diag(e2iδ1(E),...,e2iδM(E)), be the random S-matrix at

the energy E, the distribution of which is given by the

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Poisson kernel (1) with an explicit parameterization of

S(E) from Eq. (2) (γc=γ for all c). Then for every E the

transformation of the eigenphases δc(E)

?

πν(E)

φc= arctan

1

?1

γtanδc(E) +E

2

??

(3)

maps them to the eigenphases φcof the random scatter-

ing matrix, the distribution of which is given by the CE of

the same symmetry. In particular, the joint probability

density function (JPDF) of φcis [1]

?

a<b

p0

?{φc}?∝

??e2iφa− e2iφb??β.(4)

The matrix U of energy dependent eigenvectors uni-

formly distributed in the orthogonal, unitary, or symplec-

tic group (for β=1,2, or 4, respectively) is not affected by

map (3). This is a consequence of statistical equivalence

of the scattering channels.

The suggested transformation was first noticed and

verified in Ref. [11] for the case of broken TRS (β=2)

and further exploited in Ref. [12]. It can be easily gener-

alized to the other symmetry classes as follows. We cal-

culate first the Jacobian of the transformation (3). After

a simple algebra it can be represented as

∂φc

∂δc

=

cos2φc

πνγ cos2δc

=

T

∗e2iδc|2.

|1 − S

(5)

where

T(E) = 1 − |S(E)|2=

4γ πν(E)

1 + γ2+ 2γ πν(E). (6)

is the energy dependent transmission coefficient [6]. We

note that exactly the same factors as Eq. (5) appear in

Eq. (1). Employing the identity

??e2iφa− e2iφb??=

we substitute Jacobians (5) into Eq. (4) and, making use

of the identity?M

p?{δc}?∝

a<b

??e2iδa− e2iδb??

????

cosφacosφb

πνγ cosδacosδb

????, (7)

a<bfafb= (?

??e2iδa− e2iδb??β

cfc)M−1, arrive at

?

M

?

c=1

????

∂φc

∂δc

????

β(M−1)/2+1

. (8)

With Eq. (5) taken into account, we immediately rec-

ognize in this expression the JPDF of the eigenphases

corresponding to Poisson kernel (1). Due to the scalar

nature of transformation (3) it does not change the ma-

trix U of eigenvectors.

Let us start with considering the mean density ρ(δ) of

scattering (eigen)phases at arbitrary coupling. It is self-

evident that the phases in the CE (i.e. for the case S=0)

are uniformly distributed on the unit circle, the average

density being merely ρ0(φ) =(1/M)?

cδ(φ−φc)=1/π.

The corresponding density for S?=0 is not constant. In-

deed, using the identity ρ(δ)dδ=ρ0(φ)dφ, we see that

????

Although simple, this relation is an important one and es-

tablishes the physical meaning of the Jacobians of trans-

formation (3) relating them to the corresponding densi-

ties of the scattering phases. Density (9), being expressed

in terms of S only, does not depend on the particular

choice of S used in the derivation as long as the average

S-matrix is proportional to the unit matrix.

It is instructive to look at Eq. (8) in the limit of large

number of channels when the typical difference δa−δb∼

1/M ≪ 1.Then one can expand δc= δ0+˜δc (˜δc≪1)

around, say, δ0. The leading contribution is given by

p?{˜δc}?∝?

further goes to p0

distribution (4) of the CE upon the proper rescaling of

the phases,

ρ(δ) =1

π

∂φ

∂δ

????=

T

|1 − S

∗e2iδ|2.(9)

a<b|˜δa−˜δb|β?

c|∂φc/∂δc|β(M−1)/2+1

δ0

?{˜φc}?∝?

, which

a<b|˜φa−˜φb|βand agrees with

˜φc= |∂φc/∂δc|δ0˜δc= πρ(δ0)˜δc. (10)

We see that in the limit M ≫ 1 the local fluctuations

of the phases unfolded by their local density turn out to

be uniformly described by the CE at arbitrary coupling

strength. Such a universality in statistics of phases of

random unitary (scattering) matrices has much in com-

mon with that typical for eigenvalues of random Hamil-

tonian matrices [1] and is in agreement with results of

realistic numerical simulations for M = 23 [16].

Let us now consider an application of the same ideas to

the time-delay problem, where such a universality reveals

itself explicitly. Following the original wave-packet anal-

ysis by Eisenbud, Wigner and Smith [17] it is natural to

define [11] the partial time delays via the energy deriva-

tive of the scattering phases, τc= 2¯ h∂δc/∂E. Their sta-

tistical properties have been studied in much detail in

the framework of the Hamiltonian approach for the case

of broken [11] and preserved TRS as well as in the whole

crossover region of gradually broken TRS [12]. Recently,

some of these predictions were successfully verified on the

model of a quantum Bloch particle chaotically moving in

a superposition of ac and dc fields [18].

In particular, the mean density of partial time delays

P(τ)=(1/M)?

ple at ideal coupling, T=1, when it reads as

cδ(τ−τc) turns out to be especially sim-

P0(t=τ/tH) =(β/2)βM/2

Γ(βM/2)

e−β/2t

tβM/2+2,(11)

with tH=2π¯ h/∆ being the Heisenberg time.

Eq. (10), the partial time delays at ideal and nonideal

coupling (τ(0)

c

and τc, respectively) are simply related as

Due to

τ(0)

c

= 2¯ h∂φc/∂E = πρ(δc)τc.(12)

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Here, we have neglected the smooth nonresonant depen-

dence of ρ(δ) on E. Since the phase and its derivative

(the partial time delay) are uncorrelated quantities in the

CE [19], their joint distribution factorizes: ? p0(φ,τ(0))=

S ?=0. The relation ? p0(φ,τ(0))dφdτ(0)= ? p(δ,τ)dδ dτ be-

of partial time delays at nonideal coupling as

?π

0

∂(δ,τ)

?π

0

(1/π)P0(τ(0)).This is not the case for ? p(δ,τ), when

tween them allows us, however, to represent the density

P(τ) =

dδ

????

∂(φ,τ(0))

????? p0

?φ(δ),τ(0)(δ,τ)?

=

dδ

π[πρ(δ)]2P0(πρ(δ)τ).(13)

One can easily convince oneself [20] that such a formula

reproduces in every detail the expression obtained in Ref.

[11] by means of supersymmetry calculations. It is worth

mentioning that the density of phases (9) is independent

of the underlying symmetry and therefore Eq. (13) is also

valid for the crossoverregime of partly broken TRS. [Note

that in the crossover regime P0(t) is a slightly more com-

plicated function, see Ref. [12]].

Expression (13) is the proper one for generalization to

the JPDF of the partial time delays, w?{τc}?. Before

doing this, we first establish a useful relation between

τc and the matrix elements of the Wigner-Smith time-

delay matrix Q = −i¯ h(∂S/∂E)S†[17]. Writing S in the

eigenbasis representation as S = Uˆ sU†, one obtains

U†QU = −i¯ h∂ˆ s

∂Eˆ s†+ i¯ h

?

ˆ s,U†∂U

∂E

?

ˆ s†, (14)

where [,] denotes the commutator. The matrix ˆ s being

diagonal, the diagonal elements of the second term in

Eq. (14) are zero, whereas the first term is exactly the

diagonal matrix of the partial time delays. Thus, the

partial time delays coincide with the diagonal elements

of the time-delay matrix taken in the eigenbasis of the

scattering matrix,

τc= [U†QU]cc. (15)

The physical meaning of the diagonal elements of the

time-delay matrix is well known: they describe the time

delay of a wave-packet incident in a given channel [17,21].

Thus, relation (14) sheds more light on the physical

meaning of the somewhat formally defined partial time

delays. In particular, one expects that for the case of

ideal coupling the inherent rotational invariance of the

problem makes all the basises statistically equivalent and

thus the JPDF of diagonal elements of the Q-matrix

should coincide with that of partial time delays.

The latter claim can be substantiated as follows. Fol-

lowing the insightful paper [19], it is convenient to con-

sider the “symmetrized” time-delay matrix Qs

Qs= S−1/2QS1/2= −i¯ hS−1/2∂S

∂ES−1/2.(16)

This similarity transformation unveils the symmetry

which is hidden in Q: Qs is already a real symmetric

(hermitian, or quaternion self-dual) matrix for β = 1, (2,

or 4). In the eigenbasis of S the diagonal elements of Qs

and those of Q coincide. Moreover, in the case of chaotic

scattering with ideal coupling, the matrix Qsturns out to

be statistically independent of S, their joint probability

density being?P0(S,Qs)=P0(S)W0(Qs), where

W0(Qs) ∝ θ(Qs)det(Qs)−3βM/2−2+βe−(β/2)tHtrQ−1

s

(17)

is the probability density of the time-delay matrix [19].

The latter is manifestly invariant under the choice of the

basis for Qsproving the above statement on the relation

between statistics of partial time delays and diagonal el-

ements of the Wigner-Smith matrix.

To find the corresponding JPDF w0

integrate out all off-diagonal elements of Qs which is a

hard problem in general. For the case of unitary sym-

metry, β = 2, one can perform the job by splitting the

integration into that over the matrix ˆ q = diag(q1,...,qM)

of eigenvalues of Qsand that of the eigenvectors, V ,

?

with ∆(ˆ q) =?

stands for the remaining integral over the unitary group

which can be done, following Ref. [22], by means of the

famous Itzykson-Zuber formula [23]. Finally, we find it

more convenient to define the generating function of par-

tial time delays rather than the JPDF itself and obtain

?{τc}?one has to

wu

0

?{τc}?∝d[ˆ q]θ(ˆ q)∆2(ˆ q)

det(ˆ q)3Me−tHtrˆ q−1Q?{τc}?,(18)

a<b(qa− qb) being the Vandermonde de-

terminant. Here Q?{τc}?=?d[V ]?M

c=1δ?τc−(V ˆ qV†)cc

?

?

e−i(k1τ1+...+kMτM)?

τ∝

det[ψj(kl)]

?

a<b(ka− kb),(19)

where ψj(kl) =

spans the values l = 1,...,M and j = 0,1,...,M − 1.

Such an expression allows us to calculate all the mo-

ments and correlation functions of partial time delays by

a simple differentiation. Moreover, setting in the preced-

ing equation k1=...=kM=k and calculating the corre-

sponding limit in the right-hand side, we come to a con-

venient representation for the distribution Pu

Wigner time delay, tw=(τ1+...+τM)/MtH, for a system

with broken TRS and ideal coupling to continuum,

?∞

−∞

?∞

0dq qj−3Me−iklq−tH/q, the index l

M(tw) of the

Pu

M(tw) ∝dkeiMktwdet?ψ(n)

j

(k)?, (20)

where ψ(n)

The distribution of the Wigner time delay was earlier

calculated explicitly only for the case of M=1 [11,12,24],

when it follows from Eq. (11). Compact expression (20)

is valid for β=2 and arbitrary M [25]. For M=2, Eq. (20)

can be integrated further to yield

j

(k)≡dnψj(k)/dkn, and j,n=0,...,M−1.

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P2(tw) ∝ t−3(β+1)

w

e−β/twU?β+1

0dy ya−1(1+y)b−a−1e−zybe-

2,2β + 2,

β

tw

?,(21)

with U(a,b,z) = [1/Γ(a)]?∞

ing the confluent hypergeometrical function.

represented the above distribution (21) in a form covering

all β = 1,2,4 which will be verified below. In particular,

the asymptotic behavior at tw ≫ 1 is P2(tw) ∝ t−β−2

in agreement with the known universal tail t−βM/2−2,

which is typical for the time-delay distributions in open

chaotic systems [11,12,19,18].

To verify Eq. (21) for β=1,4, it is convenient to con-

sider a general problem of finding the distribution?

of Qs. This distribution is found to be

Here we

w

W0(?Q)

of the n×n submatrix?Q standing on the main diagonal

W0(?Q)∝θ(?Q)det(?Q)−β(M/2+n−1)−2e−βtHtr?

The particular case n=1 reproduces result (11) of the

Hamiltonian approach. Equation (22) for n=2 helps in

calculating the joint distribution ? w0(t1,t2) of two partial

U?β

?

Q−1/2. (22)

time delays t1,2=τ1,2/tHfor arbitrary M. One obtains

? w0(t1,t2)

P0(t1)P0(t2)∝

2,βM

2+β+2,

(t1t2)β/2

β

2t1+

β

2t2

?

.(23)

The knowledge of ? w0(t1,t2) allows us to find further the

prove the formula (21) for any β. As follows also from

Eq.(23), there exist nonvanishing correlations between

the partial time delays. They are, however, of differ-

ent nature as compared to the correlations between the

proper time delays (the eigenvalues of Q) which show re-

pulsion [19].

For¯S ?= 0 the matrices S and Qs cease to be statis-

tically independent variables and do correlate. There-

fore statistical properties of diagonal elements of Q in

arbitrary basis (save the eigenbasis of S) are different

from that of partial time delays, unless coupling is ideal.

Still, the JPDF w?{τc}?

nonideal coupling can be found by repeating basically

the same steps which lead to Eq. (13).

? p?{δc},{τc}?d[δ]d[τ] = ? p0

[which follows from Eq. (17)], allows us to relate w?{τc}?

and w0

?????

=d[δ]

distribution of the Wigner time delay for M=2 and thus

of the partial time delays at

The identity

?{φc},{τ(0)

c }?d[φ]d[τ(0)], to-

gether with the statistical independence of φc and τ(0)

c

?{τc}?as follows:

?

?

w({τc}) =d[δ]

∂?{φc},{τ(0)

c }?

∂?{δc},{τc}?

M

?

c=1

?????? p0

?{φc},{τ(0)

?{πρ(δc)τc}?, (24)

c }?

[πρ(δc)]p?{δc}?w0

where d[δ] means the product of differentials.

In conclusion, we suggest the transformation of the

scattering phases, allowing one to reduce the problem of

quantum chaotic scattering with statistically equivalent

channels at arbitrary coupling to that for ideal coupling.

Applications of this transformation to statistical proper-

ties of phases and those of time delays are discussed.

We are grateful to V.V. Sokolov for critical com-

ments. The financial support by SFB 237 “Unordnung

and grosse Fluktuationen” (D.V.S. and H.J.S.), RFBR

Grant No. 99–02-16726 (D.V.S.), and EPSRC Grant No.

GR/R13838/01 (Y.V.F.) is acknowledged with thanks.

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?2π

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can be also derived directly from Eq. (17).

0dδ f(pcosδ+q sinδ)=2?π

0dδ f(?

g2−1cosδ]−1, g=2/T−1.

p2+q2cosδ) allows

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4