Table Of ContentSignature of Dynamical Localization in the Lifetime Distribution of Wave-Chaotic
Dielectric Resonators
Oleg A. Starykh1, Philippe R. J. Jacquod1, Evgenii E. Narimanov2 and A. Douglas Stone1
1 Department of Applied Physics, P.O.Box 208284, Yale University, New Haven, CT 06520-8284
2 Bell Laboratories– Lucent Technologies, 700 Mountain Ave., Murray Hill NJ 07974
(February 1, 2008)
0
0
0
on the angle of incidence of the ray. In particular, rays
2
Weconsidertheeffectofdynamicallocalizationonthelife-
bouncing at the cavity’s boundary with an angle of inci-
n times of the resonances in open wave-chaotic dielectric cav- dence χ smaller than the critical angle, χ =sin−1(1/n)
a ities. We show that dynamical localization leads to a log- c
J (angles of incidence are defined from the normal to the
normal distribution of the resonance lifetimes which scales
3 with the localization length in excellent agreement with the boundary),aretransmittedbyrefractionwithhighprob-
resultsofnumericalcalculationsforopenroughmicrocavities. ability, while those with χ > χc are trapped by total
] internal reflection, and can only escape with low proba-
n PACS numbers: 42.55.Sa, 05.45.Mt, 42.25.-p
bility by tunneling (evanescent leakage). Semiclassically
n
- the (dimensionless) angular momentum of the ray in a
s The study of lifetime distributions of finite quantum
circular cavity is m=nkR sinχ, where k =2π/λ is the
i systems weakly coupled to a continuum is a subject of 0
d wavevector(invacuum)andR istheradiusofthecavity.
. active experimental and theoretical investigation. The 0
at natureofthespectrumofresonancesdependsstronglyon Hence a ray with angular momentum m > kR0 will be
stronglytrappedwhereasone withm<kR will rapidly
m thenatureofthestatesofthefinitesystem“inisolation”. 0
escape. Correspondingly,resonantstates with mean val-
- For example if those states are ergodically extended and
d structureless over the system then the resonances will ues hmi > kR0 will have long lifetimes, whereas those
n withmeanvalues less thankR willhaveshortlifetimes,
show the behavior expected from random matrix the- 0
o i.e. thereisathresholdvaluem =kR forstrongescape
c ory, the famous Porter-Thomas distribution in the case c 0
in angular momentum space. In an undeformed (circu-
[ of a single channel [1]. A close relative of this resonance
lar)cavitymisanintegralofmotionandtherearemany
distribution has been measured in quantum dots in the
1 exponentially long-lived“whispering gallery” resonances
v Coulombblockaderegime[2,3]. Morerecentlyithasbeen
with m>kR .
7 pointed out that optical cavities with partially or fully 0
For a generically deformed cavity angular momentum
1 chaotic ray dynamics would have interesting resonance
0 is not conserved, nor is there any other second constant
properties and efforts have been made to characterize
1 of motion beyond the energy [8]. Hence the angular mo-
their distribution in various limits [4–6]. In a geometry
0 mentum can fluctuate. The scale of those fluctuations
0 which is approximately translationally invariant in one
depends on the existence of KAM tori in phase space
0 direction the wave equation becomes a scalar equation
which limit the diffusion in angle of incidence. Beyond
/ with a close formal analogy to the Schr¨odinger equation
at andthephysicsoftheresonancespectrumbecomesessen- some critical value of the deformation these barriers are
m destroyedandclassicalrayswithinitialangularmomenta
tiallythesamefortheopticalandquantumsystems. We
m much larger than m can now diffuse to arbitrarily
- will henceforth consider cylindrical dielectric resonators c
d low angular momentum and escape by refraction [4]. As
n which are translationally invariant along their axis, but
a result, even for kR 1 the lifetime of rays start-
o can be deformed in their cross-section. The analogue of 0 ≫
ing with m > m is not exponentially long; the corre-
c theclassicallimitoftheSchr¨odingerequationisthelimit c
v: ofrayopticswhenthewavelengthofthe electromagnetic sponding level width Γm can be estimated from the dis-
tance to the critical value in angular momentum space:
i field is much shorter than the typical radius of the cav-
X Γ = D/(m m )2. (Here D is the effective diffusion
r ity λ ≪ R0. We will regularly use the term ”quantum” comefficient in p−hasce space, which in principle can depend
a to describe properties of the wave solutions which differ
on m.) One might then guess that a cavity with such
fromthebehaviorofraysinthesamegeometry. Themo-
chaotic ray dynamics will no longer support any high-Q
tionofalightraywithinthecavityisidenticaltothatofa
resonances. Howeverthis is not necessarilythe case,due
pointmassina classicalbilliardandthe resultingbound
to the phenomenon of ”dynamical localization” [9]. It
states are the analogue of the eigenstates of ”quantum
is nowwell-knownthatjust asarandomsystemexhibits
billiards” [7]. However,unless the index of refraction, n,
exponentiallocalizationinreal-spaceduetoAndersonlo-
is taken infinite, none of these states are truly bound,
calization, the same kind of destructive interference can
there always being some non-zero probability of escape
occur in a chaotic dynamicalsystem,and suppress diffu-
from the cavity. Moreover in the case of a simple dielec-
sion in the relevant phase space [10].
tric cavity the escape probability is strongly dependent
Theconditionfortheonsetofdynamicallocalizationis
1
that the diffusion time across the system be longer than time distribution with scaling properties directly related
the Heisenberg time defined by the inverse level spac- to the system’s localization length ξ.
ing of the cavity: t ¯h∆−1. For longer times than First, to illustrate the effect of dynamical localization
H
∼
t a wavepacket starts to “resolve” the discreteness of on the physical properties of the modes in the open cav-
H
thespectrumandthespreadinginangularmomentumis ity, we show in Fig.1 the real-space structure of two
suppressed. Basedonananalogywiththekickedrotator modes of a deformed cylindrical microcavity defined ac-
[11], the localization length ξ is determined by the clas- cording to the model described immediately below. The
sical diffusion rate D, ξ D. Consider a state centered two modes correspond to exactly the same shape of
∼
aroundanangularmomentumm suchthatm m ξ. the cavity, corresponding to fully ergodic classical ray
0 0 c
− ≤
Inthiscasewavepacketscanescapebeforetheirdiffusion dynamics, have the same average angular momentum
ceases and the classical picture is adequate. Two differ- m2 1/2 0.5nkR , but differ in their wavevector and
0
h i ≈
ent statistical behaviors are possible in this regime. If as a consequence (see below) differ in their localization
the escape is determined by slow diffusion to a bound- lengths. Asa resultoneresonanceis inthe quantumdif-
ary where escape can occur the survival probability has fusive regime and the other in the localized regime. The
beenstudiedrecentlybyseveralauthors[12–14]andthey qualitative difference is immediately apparent; the dif-
find characteristicpower-law distributions. Here the dif- fusive mode emits much more strongly and has a more
fusionconstantsatisfiesL D L2,andittakesmany dense spatial structure due to the large angular momen-
≪ ≪
collisions to cross the available phase space (for the op- tumspreadinthestate. Thelocalizedmodeontheother
tical cavities the role of the system size L is played by hand emits weakly andappears to have a caustic similar
2nkR , the number of angular momentum states avail- to a regular whispering gallery mode, but a closer look
0
able). When the diffusion constant is larger, D L2, at its spatial structure shows the pattern of nodes has
∼
the motion is ballistic in the sense that the phase space an irregular character entirely different from the usual
is crossed in a few collisions; this situation leads to the whisperinggallerymodesofcircularresonatorsascanbe
Porter-Thomasdistribution of resonance widths and the seen on Fig. 2.
related distributions mentioned above [1,2,15,16]. How- We now define the model corresponding to Figure
everwhendynamicallocalizationdominatesthenthelife- 1 and 2. We consider an optically inactive, cylindri-
time of a localized state centered around angular mo- cal microcavity with an index of refraction n > 1.
mentum m such that m m ξ becomes exponen- The cross-section perpendicular to the cylinder’s axis
0 0 c
− ≫
tially longer than the corresponding classical diffusion is given by a circle perturbed by M harmonics of ran-
time to the classical emission threshold. Thus one has dom amplitude 1/l a 1/l, R(φ) = R [1 +
ℓ 0
− ≤ ≤
the possibility of high-Q resonances of completely non- M a cos(ℓφ)]. The average roughness of the surface
ℓ=2 ℓ
classical, pseudo-random character, something not con- iPs defined as κ¯ = <κ2(φ)> , κ(φ) = (dR/dφ)/R .
φ 0
sidered in the optics literature to our knowledge (except This model was inptroduced by Frahm and Shepelyan-
inaveryrecentexperimentinthemicrowaveregime[17]). sky [11] with the condition of perfectly reflecting walls,
Ittherefore becomesofinterestto understandthe statis- and they referred to it as the rough billiard to contrast
tical distribution of resonance lifetimes in such a situa- with the smoother quadrupolar deformations considered
tion. byNo¨ckelandStone[4]. Howeverthe spatialwavelength
In the localized regime ξ/L 1, the angular momen- of the roughness is still assumed to be large compared
≪
tum components of wavefunctions decay exponentially to the wavelength of the resonance. The advantage of a
away from their centers and one naturally expects ex- rough boundary is that the transition to classical chaos
ponentially long average lifetimes for states centered far is achieved with much smaller amplitude of deformation
above the classicalemission threshold m0 mc >ξ. Re- making it easierto explore the parameter regimeof fully
−
cently No¨ckelandStone [4]comparedthe exactlifetimes chaoticclassicalmotionanddynamicallylocalized”quan-
ofresonancesofquadrupole-deformedmicrocavitieswith tum” behavior. As we shall see below, the open rough
the mean classical diffusion time and found the lifetimes billiard has scaling and statistical properties essentially
to be significantly longer in certain cases; they conjec- identical to a quasi-one-dimensional disordered system,
tured that these discrepancies arose from incipient dy- whereas the quadrupole billiard does not. For the rough
namical localization. Indeed, dynamical localization has billiard the classicaldynamics can be well approximated
been shown to occur in certain closed cavities [18,11] by a discrete map for which Chirikov’s overlap criterion
andaveryrecentexperimentalpaperconfirmedthisphe- [19] gives an estimate of the critical roughness κ¯ above
c
nomenon in microwavecavities of similar shape to those which the classical dynamics becomes fully chaotic as
studied below [17]. However no detailed study has been κ¯ M−5/2. The two deformation parameters M and
c
made of the statistical and scaling properties of the life- κ¯ a∼llow to reach a classically fully ergodic regime char-
time in this regime. These are the main topics of the acterized by a diffusion constant (averaged over angular
current work. Below we will show that this localized momentum) D = 4(κ¯nkR )2 for κ¯ κ¯ and quantum
regimeischaracterizedbyaverybroad(log-normal)life- 3 0 ≫ c
2
FIG.2. Details of the wave intensity corresponding to the
topresonanceshowninFig. 1. Theintricatestructureofthe
wave intensity is due to classically chaotic boundary scatter-
ingandmakestheresonanceclearlydifferentfromastandard
whispering-gallery like high-Q resonance.
c−2∂2E(r,t) = n2(r,φ) 2E(r,t), where the refraction
indext satisfies n(r) = n∇inrside the cavity, and n(r) = 1
outside.
We use the approach in which the resonances widths
in wavevector are given by the imaginary part of the
wavevector of the quasibound states defined by the fol-
lowingmatchingconditions. Firstweexpandthe electric
field in the angular momentum basis (r r)
≡| |
∞
E(r,t)=e−ickt imA (kr)eimφ, (1)
m
FIG.1. Top: IntensityplotofaresonancewithnkR0 =50 mX=−∞
in therough cavity with κ=0.08, M =15 and n=2.5. Red
where
color corresponds to the maximum of the intensity, and blue
totheminimum. Bottom: Intensityplotofaresonancewith
α H+(nkr)+β H−(nkr) if r R(φ),
nkR0 =150 for thesame set of parameters as above. Am =(cid:26) γmmHm+m(kr) m m othe≤rwise. (2)
This expansion corresponds to the so-called Siegert
mechanically, one gets a localization length ξ D [9] so
∼ boundary conditions [20] in which the states have only
that the dynamically localized regime is determined by
an outgoing component at infinity (H±(x) are Hankel
κ¯2nkR 1[11]. Keepingparametersofthecavityfixed m
0
≪ functions of first and second type, respectively). Such
andvaryingkR one is able to accessstates with greatly
0
boundary conditions cannot be satisfied for real k and
different localization lengths.
areonlysatisfiedfordiscretecomplexk. Itcanbeshown
We restrict ourselves to the simplest case of TM-
polarized electric field E(r) parallel to the cylinder axis thatthe imaginarypartofthek values whichsatisfy Eq.
(2)arethepolesofthetrueunitary(on-shell)S-matrixof
forwhichboththe fieldandits derivativearecontinuous
the scattering problem. From the expansion coefficients
at the cavity’s boundary. This restriction is primarily
inEq. (2)wedefine vectorsα~,β~ and~γ. Thefieldsinside
for convenience, the TE modes obey a slightly different
andoutsidethecavityarerelatedbythecontinuityofthe
scalar equation which can be treated in a similar man-
field and its derivative on the boundary and (after inte-
ner. Itshouldbe mentionedhoweverthatsemiconductor
grationaroundtheboundary)oneoftheseequationscan
quantumcascademicro-cylinderlasersstudiedin[5]emit
be used to eliminate ~γ leaving a linear relation between
solelyintheTMmodeduetoaselectionrule. Maxwell’s
α~ and β~. The matrix expressing this relation we call ˜.
equation reduce then to a single scalar wave equation, S
Moreover the regularity of the field at the origin r = 0
3
implies α~ = β~, thus a secular equation for the resonant from the expectation value of an antihermitian operator
values of k is obtained of the form: V taken over eigenstates α(0) of the matrix , which
| i M
describes some effective “closed cavity”
˜α~ =α~. (3)
S
We use the notation ˜-matrix because in the case of Γ=b0hα(0)|V|α(0)i=b0 α(m0)∗Vmm′α(m0)′ (4)
S mX,m′
a closed billiard the matrix so-defined is actually the
unitary S-matrix of the scattering problem of a wave Explicitely,
incident outside the impenetrable billiard [24,22]. In
our case the matrix so-defined is non-unitary for any =(J′J′) (1/n)(J′H+′) H−H+ −1 H−J , (5)
M −
n< andforrealk, the eigenvaluesof ˜havethe form (cid:0) (cid:1) (cid:0) (cid:1)
λ =∞exp(i(ϕ + iδ )), where both ϕ Sand δ > 0 are V is the antihermitian part of , and the matrix ele-
reralfunctionsrof mormentum k. The surbscript rr numbers ments (Z¯Z) are defined as M
states in a deformed cavity where angular momentum is
im−ℓ
notconserved. Exactquantizationofthecavity- solving Z¯Z = dφ Z¯ (k)Z (k) ei(m−ℓ)φ, (6)
Eq. (3) exactly - implies λr =1 so that the exact imple- (cid:0) (cid:1)ℓm 2π Z ℓ m
mentation of this procedure requires finding a complex
Z (k) and Z¯ (k) stand for either H±(kR(φ)), the
k = q iγ such that ϕ (q iγ) = δ (q iγ) = 0. The m m m
corresp−ondinginverselifretim−eΓ isthreng−ivenbyctimes Bessel function Jm(nkR(φ)), or their derivative. The
r coefficient b in (4) depends only on the hermitian
γ via the dispersion relation, Γ =cγ. Howeverapproxi- 0
r part of , i.e. it is determined by the properties
matelifetimescanbefoundbyamuchmoreefficientpro- of theH0“closMed system”, b−1 = α(0) ∂ /∂k α(0)
cedure which is extremely helpful if one wishes to study 0 h | H0 | i ≈
full distributions as we do here. First, it is straightfor- mα(m0)∗(J′J′′)mm′α(m0)′, and can be regarded as a nor-
ward to show [23] that when γ is small, simply finding Pmalization factor. An eigenstate, localized at angular
momentum m m kR , will have exponentially
the complex eigenphases λr =exp(i(ϕr+iδr)) for real k 0 ≫ c ≈ 0
small width Γ. Therefore, for such a resonance in the
determines the imaginary part of k on resonance by the
relation: γ = δ/ϕ′ whereϕ′ isthe derivativeofthereal semiclassical limit Γ ≪ ∆, and Eq. (4) is appropriate.
− More details on this formalism can be found in [6].
part of the phase with respect to momentum for real k.
For an exponentially localized state one generally has
Moreover,sincethisderivativecanbeshowntobeslowly-
varying on the scale of the level spacing ∆k, it is not
m m
0
necessaryeventoquantizetherealpartofthephase(i.e. αm exp | − | (7)
| |∼ (cid:18)− 2ξ (cid:19)
tofindthe realk whichmakesϕ(k)=2π integer). The
×
functionϕ′ canbeeasilycalculatedforthecircularcylin- We consider the regime of large localization length
derandthisrelationandtheassumptionofslowvariation ξ κ¯2k2R2 1. Since the rough billiard is classically
ofthederivativecanbeconfirmedexplicitly(forthecase ch∼aotic, the0p≫hases of the coefficients α change rapidly
m
where γ is small). Therefore we can generate large life- withtheangularmomentum, anditisnaturaltoassume
time ensembles simply by diagonalizingthe matrix ˜for that its correlation function satisfies
S
real k and extracting the imaginary phase δ, by which
means we generate ∼2nkR0 inverse lifetimes per diago- hα∗ℓ+mαℓiℓ =δm,0h|αℓ|2iℓ (8)
nalization. This procedure is motivated by the work of
DoronandFrischatwhonotedthatthestatisticalproper- where the average is performed over an angular mo-
ties ofclosedbilliardschangedlittle awayfromthe exact mentum interval l [m0 δℓ/2,m0+δℓ/2] such that
∈ −
quantization condition (in their case it was the distri- 1 δℓ < ξ. This behavior is illustrated in the inset to
≪
bution of splittings of semiclassically degenerate states) Fig. 3 for one typical set of cavity parameters, corrobo-
[24]. The linear relation between δ and γ has been in- rating the validity of the assumption (8).
r r
dependently proposed earlier by Hackenbroich [25] and As follows from Eq. (6) and the definition of V, the
demonstrated for the case of the circle. matrixelements Vmm′ inangularmomentumrepresenta-
To best relate the localization properties of the eigen- tionvaryonascaleofδm κ¯kR0 1,andthereforewe
∼ ≫
states,whichapplytoaclosedcavity,to thedistribution canreplacethe productα∗mαm′ inEq. (4)by its average
of lifetimes in an open cavity, we employ a perturbative value hα∗mαm′i over the interval |m−m′| ∼ κ¯kR0. To-
formalism which was recently developed specifically to gether with (8) this leads to the diagonalapproximation
treat open optical resonators[6] (it is similar in spirit to
well-known quantum perturbative scattering approaches nkR0
α V α α 2V (9)
such as R-matrix theory in the single-level approxima- h 0| | 0i≈ | m| mm
X
tion). Accordingtothattheorynarrowresonancewidths |m|=kR0
Γ ∆ (∆ is the resonance spacing) can be computed
The matrix element
≪
4
1
Vmm =−2n[(J′H+′)mℓ(H−H+)−ℓℓ1′(H−J)ℓ′m−c.c.]
2 >
(10) 2|m
*α 0.5
includes both the refractive (classical) escape from the <|
>/
resonator (for |ℓ|,|ℓ′| < mc = kR0) and the “tunnel- αm
ing escape ”(corresponding to evanescent leakage [4], for ρ) m+l0.0
|ℓ|,|ℓ′| > mc). To evaluate the sum over angular mo- P(1 *α<
menta we use the stationary phase-based technique, de- 0 5 10 15
l
veloped in [26] in the context of the calculation of level
splittingsinroughbilliard. The“classical”refractioncon-
tribution is found to be
k2R m kR
Γclass 0 exp 0− 0 . (11) 0
m0>mc ≈ nξ (cid:18)− ξ (cid:19) 0 1 ρ/<ρ> 2 3
Thisresultshowsthatanexponentially-longlifetime can
FIG.3. Distributionoftheinverseparticipationratioρfor
be due to the exponentially-small wavefunction com-
n = 3, M = 15, and (nkR0,κ¯) = (150,0.02) (solid line),
ponent leaking outside the classically totally-internally-
(100,0.03) (dashed line) and (50,0.06) (long-dashed line).
reflected region. To see if this process controls the life- κ¯nkR has been kept constant, which result in a stable av-
timeweneedtocomparethisresultwiththedirect“tun- erage IPR hρi = 0.1±0.01. Inset: Normalized correlation
neling escape”contribution to the lifetime. The latter function (8) for nkR0 =100, and κ¯=0.03.
process involves angular momenta only above emission
threshold m . For an estimate, it is then sufficient to
c average 1/ξ [27]. Moreover Frahm and Shepelyansky
evaluatethelinewidthofthestateabovem inthecircu- 0
c have explicitly shown [11] that the problem of the rough
lar cavity, which can be thought of as a state with zero
billiard maps onto a variant of the kicked rotor problem
localizationlength. We find thatthe tunneling contribu-
and hence to an ensemble of band randommatrices [29],
tion is also exponentially small (n 1),
≫ which also describe quasi-1d disordered systems. Here
Γtunn = (1/n)Im H+ (kR )/H+ (kR ) the angular momentum index plays the role of the site
m0>mc − m0−1 0 m0 0
(cid:2) (cid:3) coordinatesinthe disorderedsystems,with anideallead
≈exp(cid:16)2qm20−m2c +2m0ln(mc) (the continuum) accessibleatmc =kR0. Collisions with
the rough boundary correspond to random hopping be-
−2m0ln(m0+qm20−m2c)(cid:17). (12) tween sites, which are at most κ¯nkR0 lattice spacings
apart [11]. Once the critical angular momentum, corre-
Competition between the classical escape [Eq.(11)] and
sponding to the classicalemissionborder m , is reached,
c
the tunneling [Eq.(12)]is strongestfor (m m )/m
0 c c the wavepacket escapes the system and never returns.
− ≪
1,i.e. whenthewidthofthetunnelingbarrierissmallest.
Hencewemayassumethattheinverselocalizationlength
Comparing the two contributions in this region we find
inourproblemhasanapproximatelynormaldistribution
that Γclass Γtunn for ξ √kR , and Γtunn Γclass in
0 around its mean (our ensemble here is of boundary real-
≫ ≫ ≫
the opposite limit. Restriction on the range of ξ weak-
izations)
ens as one moves to higher angular momentum, and for
(m m )/m 1 the classical escape mechanism al- (1/ξ 1/ξ )2
0 − c c ∼ P(ξ) exp[ − 0 ], (13)
waysdominatesoverthetunnelingone. Thetunnelinges- ∼ − 2σ2
capethereforeisrelevantonlyforverysmalldeformations
Therefore,asfollowsfrom(11)and(13),theresonance
κ (kR )−3/4, which produce short localizationlength.
≪ 0 lifetime is distributed log-normally,
Thus, lifetime of the states with very short (essentially
zero) localization length is determined by the tunneling ln2(t/t )
0
P(t)d(ln(t)) exp[ ]d(ln(t)) (14)
escape, whereas that of the more extended states with ∼ −2σ2(m kR )2
0 0
−
ξ 1 by the classical emission from their exponentially
≫ where
weak tails at m =kR .
c 0
Havingestablishedarelationbetweenthe lifetime and
ln(t )=(m kR )/ξ (15)
0 0 0 0
the localization length of a corresponding closed cavity, −
the distribution of lifetimes then follows from the distri- andthederivationisdoneinaleadinglogarithmapprox-
bution of localization lengths. In one-dimensional and imation, so that the pre-exponential factor in (11) is ne-
quasi-one-dimensionaldisordered systems it is now well- glected. Thisresultisthenverynatural: thedistribution
established that the distribution of the inverse localiza- of lifetimes in open dynamically-localized cavities is log-
tion length is typically normally distributed around an normalfor the resonanceslocalizedfar fromthe classical
5
emission threshold m kR . This is entirely analo- 0.4
0 0
≫
gous to the conductance distribution of localized chains,
−2
which will be log-normal for a fixed distance from the )
2
|m
ends [27] (see [28] for log-normal distribution of delay 0.3 ψ −4
times/resonance widths). We note that the relationship n(|
between dynamical localization and Anderson localiza- L −6
)
tion was first placed on firm footing in a seminal paper (t 0.2 −8
by Fishman, Grempel and Prange [10]. P 0 50 100
m
Equation (14) essentially relies on the two assump-
tions: first , Eq. (8) that the phase of the wavefunc-
0.1
tion components are randomly distributed with no long-
range correlations and second, that the eigenstates are
exponentially localized with a normal distribution of lo-
0.0
calization lengths. We now test the validity of these two 5 10 15 20
Ln(t)
assumptions for our roughmicrocavities. The validity of
Eq. (8) is confirmedbythe sharpdropofthe correlation FIG. 4. Lifetime distribution for n = 3.5, nkR ≈ 110,
function α∗ α / α 2 for m > 0 which is clearly κ¯ = 0.018 and M = 15 for states with average angular mo-
h ℓ+m ℓiℓ h| ℓ| iℓ
seen from the numerical results presented in the inset to mentum m0/(nkR0) = 0.5 (circles), 0.6 (squares), 0.7 (dia-
Fig. 3. The localization properties are investigated by monds) and 0.8 (triangles). This last distribution is fitted
according to Eq. (14) with σ ≈ 0.04 (dashed line). The
computing the distribution of the Inverse Participation
Ratio (IPR) defined as ρ = α 4/ α 2. The emission border is at mc ≈ 0.29nkR. Each distribution is
m| m| m| m| constructed from 8000 to13000 lifetimes obtained from 1013
IPRmeasuresthe inversenumPberofeffecPtive eigenstates
boundaryrealizations of therough cavity. Inset : Typical lo-
components and thus allows to distinguish between lo-
calizedeigenstateforthesameparameterset. Thedashedline
calized and delocalized states. Generally, in the local-
indicatesanexponentialdecaycorrespondingtoalocalization
ized phase ρ is independent of the system size since at length of ξ=16.
h i
most ξ L sites contribute to the sum and ρ ξ−1.
≪ h i ∼
In the other limit of ergodic states, all sites contribute
equally and ρ L−1 in this case. Between these two movesawayfromtheclassicalemissionbordermc =kR0.
limits, a varhietiy∼of behaviors may occur depending on Furthermore, the agreement with Eq. (14) is quantita-
the inner structure of the eigenstates. In Fig. 3 we tively confirmed by a direct fit of the broadest of these
showIPRdistributionsforthreedifferentparameterssets distributions (see dashed line on Fig.4).
corresponding to the same average localization length We expect by analogy to the scaling theory of local-
ξ (κ¯nkR )2 9. The three distributions are indeed ization that the logarithmic average of the lifetime will
0
sta∼bleunderpar≡ametervariationskeepingξconstantand exhibit a universal scaling behavior. This expectation is
this shows that not only the average IPR/localization confirmed by the data shown in Fig. 5 where we present
length [11], but also the full IPR distribution obeys a numerical results for the scaling obeyed by lnt0. Log-
one-parameterscalingwithκ¯nkR. Thesituationsisvery averagedlifetimes for different parameter sets have been
similar to the one studied in Ref. [29] for BRM with a computed for at least 2000 lifetimes in narrow energy
band width 1 b √2ξ for which the IPR distribution windows δm/(nkR0) = 0.2 around given angular mo-
was analytical≪ly co≈mputed. Similar deviations from [29] mentum m0 > mc for different values of indices of re-
as seen on Fig. 3 are also present for BRM with not too fraction 1.5 n 4, wavelengths 75 nkR0 180
≤ ≤ ≤ ≤
large band widths, so that these numerical results con- androughnessesκ¯. Allpresentedresultsareinthe local-
firmtheuniversalityofthedynamicallylocalizedregime, ized regimes ξ < m0 mc and the corresponding curves
−
quite analogoustoquasi-one-dimensionaldisorderedsys- have been put on top of each other by a one-parameter
tems. We also illustrate this exponential localization by scaling. Fig. 5 demonstrates the validity of the linear
showing one typical state in the inset to Fig. 4. relation (15) as is indicated by the straight line. The
Having tested the validity of the main assumptions exact parameter dependence of the scaling can be de-
on which (14) relies, we present in Fig. 4 distribution duced from the analogy to the scaling theory of localiza-
of lifetimes for the classically-chaotic,dynamically local- tion. In this case the Thouless conductance g = Γ/∆ǫ
izedregimeoftheopenroughmicrocavity. The distribu- is the scaling quantity (or its logarithm in the localized
tions shown correspond to resonances centered in inter- regime); here Γ is the resonance width, and ∆ǫ is the
valsofwidthδm/(nkR )=0.1aroundangularmomenta mean level spacing. In our case Γ = cγ (where γ is
0
m /(nkR ) = 0.5, 0.6, 0.7 and 0.8, well above the clas- the width in momentum space), but ∆ǫ differs from the
0 0
sical threshold m /(nkR ) 0.29. Clearly, the distri- corresponding Schr¨odinger equation, due to the differ-
c 0
butions are log-normal and≈their widths increase as one ent dispersion relation for the wave equation. Taking
this into account one finds that the analogue of the di-
6
mensionless conductance is g n2kcγR2. and it is the 5
∼ 0
logarithm of this dimensionless quantity which we plot
against(m kR)/ξ∗. Fig. 5 allows to identify the scal-
0− 0
ingparameterξ∗ withthelocalizationlengthuptoafree
>
parameter. That this scaling holds for the rough cavity )
2 0 (κnkR)2/M
demonstrates that localization length is independent on R −5 0
the angular momentum as is expected for an homoge- γc 0.0 0.5 1.0
k
neously diffusive system. The situation is fundamentally
2n −10
differentforaquadrupolarcavity(M =2)ascanbeseen ( 0.2
onFig.5(seeblackdiamonds). Obviouslyonescalingpa- Ln ξ∗/M
rameterisnotsufficienttobringthecorrespondingcurve < −15 0.1
ontopoftheotheronessatisfying(15). Thisindicatesan
angular momentum dependent diffusion constant which 0.0
−20
directly follows from the effective local map derived for 0 5(m −kR)/ξ∗10 15
this particularcase[31]. Furthermore,inthe regimecor-
0
respondingtothepresenteddataforthequadrupolarde- FIG.5. Scaling of the average logarithm of lifetime hlnt0i
formation, small invariant torii and islands of stability vs. (m0 − kR)/ξ∗ for parameters 95 < nkR < 195,
stillsurvive,resultinginstronglylocalizedwavefunctions 2.5 < n < 5, 0.008 < κ¯ < 0.03 and 10 < M < 30. Open
with short localization length determined essentially by symbols correspond to the rough deformation for which a
thesizeofremainingclassicalstructures. Becauseofthis, one-parameter scaling exists. The full diamonds correspond
and unlike the situation in the rough cavity, lifetime of to a quadrupolar billiard (M = 2) for which the diffusion is
affectedbyclassicalinvariantstructuresinphasespace. Inset
such states is determined by the tunneling escape (see
: Localization length as extracted from the scaling shown on
discussion after Eq.(12)). Therefore a clean demonstra-
themain pannel.
tion of exponential dynamical localization is difficult in
the quadrupolar billiard.
Furtherconfirmationthattheextractedscalingparam- are nonetheless smooth on the scale of the wavelength)
eter is indeed related to the system’s localization prop- shouldbe ofinterestin opticalstudies ofscattering from
erties is given in the inset to Fig. 5 where ξ∗ is plotted small particles, however their random nature seem to
against the diffusion constant D = (κ¯nkR0)2 as derived make such resonances unsuitable for applications.
fromtheeffectiveroughmap[11]. Thisinsetgivesanun- We have benefitted from interesting discussions with
ambiguousconfirmationoftheabovederivedrelationbe- F. Borgonovi,C.Texier andY. Fyodorovand wouldlike
tweenlocalizationlengthandlog-averagedlifetime. Note to thank G. Maspero for sending us his thesis and A.
thatξ∗ hasalineardependenceonthediffusionconstant D. Mirlin for communicating us several interesting ref-
D even at small D where the relation ξ D does not erences. We acknowledge the support of the Swiss NSF
∼
hold. For large D however,the relation ξ∗ ξ holds. and the NSF grant No. PHY9612200.
∼
To summarize, we have presented the first study of
the lifetime distributions ofaquantum-chaoticopensys-
temin the dynamically localizedregime. This study was
greatlyexpeditedbythelinearrelationbetweenthecom-
plex phases of the eigenvalues of the nonunitary scatter-
ing matrix away from exact quantization and the imag-
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8
