Super-sharp resonances in chaotic wave scattering

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by  Marcel Novaes

Super-sharp resonances in chaotic wave scattering
Marcel Novaes
Departamento de F´ısica, Universidade Federal de S˜ao Carlos, S˜ao Carlos, SP, 13565-905, Brazil
Wave scattering in chaotic systems can be characterized by its spectrum of resonances, zn =
En−iΓ2n, where En is related to the energy and Γn is the decay rate or width of the resonance.
Ifthecorresponding ray dynamicsis chaotic, a gap is believed todevelop in thelarge-energy limit:
almost all Γn become larger than some γ. However, rare cases with Γ < γ may be present and
actually dominate scattering events. We consider the statistical properties of these super-sharp
resonances. We find that their number does not follow the fractal Weyl law conjectured for the
bulkof thespectrum. Wealso test,for a simple model, theuniversal predictionsof random matrix
theory for density of states inside the gap and the hereby derived probability distribution of gap
2 size.
1
0 PACSnumbers: 03.65.Sq,05.45.Mt,05.60.Gg
2

n
Scattering of waves in complex media is a vast area to converge to γ. More natural variables are their expo-
a
of research, from oceanography and seismology through nentials, and we find that e Γ0 e γ decreases with E
J − −
−
acoustics and optics, all the way to the probability am- accordingtoapowerlaw,whoseexponentiswellapprox-
6
1 plitude wavesofquantummechanics[1–4]. We focus our imated by d α.
−
attention on the important class of systems for which Averyfruitful approachtochaoticscatteringofwaves

] the complexity is not due to the presence of random- is random matrix theory (RMT) [9], in which the sys-
D
ness or impurities, but rather because the correspond- tem’spropagator(theGreen’sfunctionofthewaveequa-
C
ing ray dynamics is chaotic. The presence of multiple tion) is replaced by a random matrix, whose spectral
.
n scattering leads to very complicated cross-sections, with properties are studied statistically. The RMT prediction
i stronglyoverlappingresonancesthat,althoughdetermin- for the density of resonance states inside the gap was
l
n istic,haveapparentlyrandompositionsandwidths. Itis derived in [10]. In this work we derive the RMT predic-
[ notuncommonthatinagivensituationonlythesharpest tion for the probability distribution of e Γ0 e γ, and
− −
−
1 resonancesarerelevant,withthe othersprovidinganap- compare both these predictions to numerical results in a
v proximately uniform background. specific system.
6 For concreteness of terminology,we consider quantum Waves in chaotic systems can be modeled by the so-
2
mechanical systems, but our results are general. We de- called‘quantum maps’. These are N N unitary matri-
33 note resonances by E iΓ and call Γ the width or the ceswhereN 1/~,andthelargeene×rgylimitE is
− 2 ∼ →∞
. decayrate. Inthelarge-energylimitagapdevelopsinthe replaced by the limit of large dimension, N . This
1 → ∞
resonance spectrum: most resonances have their widths approachhas been used to study transport properties of
0
2 largerthan γ,which is the averagedecayrate ofthe cor- semiconductorquantum dots [11], entanglementproduc-
1 responding classical (ray) dynamics. This was noticed tion[12],thefractalWeyllaw[13],fractalwavefunctions
v: longago[5],andmorerecentlytherehavebeenattempts [14, 15], proximity effects due to superconductors [16],
i to prove it rigorously [6]. We are interested in the rare among other phenomena. Scattering can be introduced
X
case of states inside the gap, i.e. with Γ < γ, which we by means of projectors. The propagator becomes a sub-
r call super-sharp resonances. unitary matrix of dimension M < N, and its spectrum
a
comprises N M zero eigenvalues and M resonances of
The distribution of typical resonances in chaotic sys-
−
the form z =e i(En iΓn/2).
tems is conjectured to follow the so-called fractal Weyl n − −
Asourdynamicalmodelweusethe kickedrotator. Its
law[7]: theirnumbergrowswithE asapowerlaw,whose
classical dynamics is determined by canonical equations
exponentisrelatedtothefractaldimensiondoftheclas-
of motion in discrete time,
sical repeller, the set of rays which remain trapped in
the scattering region for infinite times, both in the fu-
K
ture and in the past [8]. In a numericalexperiment with q =q +p + sin(2πq ) (mod1) (1)
t+1 t t t
4π
a simple model, we find that the number of super-sharp
K
resonances, denote it by , does not follow this law. p =p + [sin(2πq )+sin(2πq )] (mod1). (2)
NSSR t+1 t 4π t t+1
It does grow with E according to a power law, but the
exponent seems to be insensitive to γ or the dimension This system is known to be fully chaotic for K > 7,
of the repeller. with Lyapunov exponent λ ln(K/2). Its quantization
≈
We also investigatethe dependence with energyof the [16, 18] yields a N dimensional unitary matrix U. We
widthofthesharpest(andusuallymostimportant)reso- setequalto zeroafractionof1 µ ofits columns,corre-
−
nance. LetthisbedenotedΓ . AsEgrows,itisexpected sponding to a ‘hole’ in phase space. Here µ corresponds
0
2
1,0
1,0
0,8
d
0,8 a
N=500
b 0,6 N=1500
P N=5000
0,6 1 N=10000
0,4
0,4
0,2
0,2
0,0
0,00 0,02 0,04 0,06 0,08 0,10
0,0 x
0,7 0,8 0,9 1,0
m
1,0
FIG. 1: Scaling exponents as functions of µ for the open
kickedrotator. d,αandβarerelatedtothefractalWeyllaw, 0,8
the number of super-sharp resonances and the width of the
N=500
gap, respectively. We see that α is approximately indepen- 0,6 N=1500
dentof thesize of theopening. Thesolid lineisa bestlinear P N=5000
2 N=10000
fit to d, while the dashed line is d−hαi, where hαi≈0.66 is 0,4
theaverage valueof α.
0,2
to the fraction of rays which escape the scattering re- 0,0
gion per unit time, so µ=e γ. We do not consider very 0,0 0,1 0,2 0,3 0,4 0,5
− x
smallvaluesofµ,whichwouldcorrespondtowidelyopen
systems.
We find numerically that Nα, while e Γ0 FIG. 2: Top panel: P1 =Pr(r2−µ<x) for the open kicked
SSR −
N ∼ − rotator,atseveraldimensions,forµ=0.8. Lowerpanel: same
e−γ ∼N−β. The exponentsareplottedasfunctions ofµ for P2 = Pr(ηe(r2−µ) < x), where ηe is defined in the text.
in Figure 1,together with d, the numericallydetermined
Solid line is a rescaled RMT prediction.
exponent in the Weyl law. We see that, somewhat sur-
prisingly, in this range α is approximately constant, i.e.
insensitive to the dimension of the repeller. The super- This calculation can be found in [10]; we sketch it here
sharp resonances do not follow the fractal Weyl law. On for completeness. The probability density for r = e Γ/2
−
the other hand, the exponent β has approximately the is known exactly [17], and when M =µN and N 1 it
≫
same slope as d. We find the relation d β = α to be can be approximated to
−
approximately fulfilled, which is to be expected since it
says that the number of super-sharp resonances, Nα, P (r) (µ r2)
∼ P(r) ∞ 1+erf η√µ − , (4)
is proportionalto the totalnumber ofresonances, Nd, ≈ 2 r
∼ (cid:20) (cid:18) (cid:19)(cid:21)
times the width of the gap, N β.
−
∼ where erf is the error function and
We now turn to an RMT treatment of the problem.
RMT for quantum maps amounts to taking matrices
N
uniformly distributed in the unitary group. In [17], an η = (5)
ensemble of truncated unitary matrices was introduced s2µ(1 µ)
−
as a model for scattering, and it was shown that, as
is now the large number. Clearly, (4) will be different
N,M with µ = M/N held fixed, the probability
density→of∞r =e−Γ/2 converges to fsreotmtinPg∞r2(r=) µon+lyǫ/ifηµ, t−heνp0riosboafbitlhiteyodridsterribouft1io/nη.foArfttheer
1 µ 2r variable ǫ becomes
P (r)= − , (3)
∞ µ (1 r2)2
− P(ǫ)=√π(1 erf(ǫ)). (6)
if r2 < µ and to 0 otherwise. The expression (3) was −
tested numerically for a chaotic quantum map in [18] This is the densiety of states inside the gap. It is only
and found to be an accurate description of the bulk of non-zero in a small region that shrinks as N 1/2 in the
−
the spectrum, provided a rescaling was performed to in- asymptotic limit. Its integral provides the probability
corporate the fractal Weyl law. that r2 µ be less than some value x. This we denote
−
Letusstartwiththedensityofsuper-sharpresonances. by Pr(r2 µ<x).
−
3
InFigure2aweseePr(r2 µ<x)for the openkicked
−
rotator, at various dimensions for µ = 0.8. We see that 1,0
asN growsthevaluesofrbecomemorelocalizedaround
µ. In Figure 2bwe introduce a scaledvariableη(r2 µ), 0,8
−
but with η different from η in that it involves the actual
exponent β instead of the RMT prediction 1/2: 0,6 N=1000
e P N=2500
1 N=5000
e 0,4 N=7500
N β N=10000
η = . (7)
2µ(1 µ) 0,2
(cid:18) − (cid:19)
All curves fall on teop of each other, indicating that this 0,0
is the correct scaling. Moreover, the shape of the curve 0,000 0,025 0,050 0,075 0,100 0,125
x
agrees with (6).
Letus nowconsiderthe probabilitydistributionofthe
largest eigenvalue of the propagator, which corresponds 1,0
to the sharpest resonance and whose modulus we de-
note R. Largest eigenvalue distributions are important 0,8
in different areas of mathematics [19] and physics [20],
N=1000
0,6
and have even been measured [21]. A similar calcula- P N=2500
2 N=5000
tion to the one below can be found in [22]. Let ( z ) N=7500
P { } 0,4 N=10000
denotethejointprobabilitydensityfunctionforalleigen-
values. Whenintegratedoverallvariablesfrom0tox,it 0,2
givestheprobabilitythatalleigenvaluesaresmallerthan
x. Therefore, if Q(x) is the probability that the largest 0,0
0,0 0,1 0,2 0,3 0,4 0,5
eigenvalue be smaller than x, then
x
M
Pr(R<x)= ( z ) d2z . (8) FIG. 3: Top panel: P1 = Pr(R2 −µ < x), where R is the
i
Z[0,x]M P { } i=1 modulusofthelargesteigenvalue,fortheopenkickedrotator
Y at several dimensions, for µ = 0.8. Lower panel: same for
The jpdf of the eigenvalues is [17]: P2 = Pr(ηe2(R2 −µ)2 < x). Solid line is a rescaled RMT
prediction.
1..M M
( z ) z z 2 (1 z 2)N M 1. (9)
i j i − −
P { } ∝ | − | −| | where we have defined the function
i<j i=1
Y Y
1+erf(z)
It is a usual trick to write z z 2 in terms of the (z)=ln . (12)
i<j| i− j| L 2
Vandermonde determinantQ|detA|2, where Aij = zji−1. (cid:18) (cid:19)
This can be shown to be equal to M!detB, where B = One interesting question that can be answered at this
ij
zij−1(zi∗)i−1. Each element of the matrix B depends on pointis,howlikelyisitthatatruegapwillexistatµfor
a single variable,and the integrationdecouples. The an- a finite value of N? The probability that all eigenvalues
gularpartoftheintegralsdiagonalizesthe matrixand,if are smaller than µ is simply given by the exponential in
M =µN and N 1, the result becomes (11) with x = 0. It is thus proportional to e−c√N for
≫
some constant c.
M x Notice that (11) has some similarity with the Tracy-
Pr(R<x) (1 y)N(1−µ)yjdy. (10) Widomdistribution[23]ofthelargesteigenvalueofGaus-
∝ −
jY=0Z0 sian ensembles of RMT, in the sense that it involves the
exponential of the integral of a function that satisfies
Thisresultisexact,butabitcumbersome. Approximat-
a non-linear differential equation, (z) = 2z (z)
ing the integrand by a Gaussian function we arrive at L′′ − L′ −
( (z))2. It is not a Painlev´e transcendent, however.
Pr(R < x) ∝ Mℓ=0 12 + 21erf η x2−µ+ ℓ(1N−µ) . LR′ eturning to the calculation, let us change variable
This result canQbe funrther simphlifi(cid:16)ed by exponentia(cid:17)tiiong to δ = ηx2 and obtain exp 2µη δ∞L(z)dz . Clearly,
the product into a sum, setting the scale as ℓ = 2µηξ this function does not converge as η . Assuming
and approximating the sum by an integral. We get δ to be large, we use (z)(cid:8) Re z2→/2√∞πz(cid:9)and inte-
−
L ≈ −
grate by parts to get exp 2µηe δ2/4√πδ2 . In order
−
Pr(R2−µ<x)∝exp(cid:26)2µηZ0∞dξL ηx2+ξ (cid:27), (11) to have a finite limit, wenm−ust have ηe−δ2/oδ2 = O(1).
(cid:0) (cid:1)
4
This implies that δ2 = y+W(η), where W is the Lam- This work was supported by CNPq and FAPESP.
bert function, which for large η can be approximated as
W(η) logη loglogη. Therefore, if instead of R we
≈ −
consider the variable
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to appear.
whether they are universal or system specific.