Table Of ContentQuantum signatures of classical multifractal measures
Moritz Sch¨onwetter1 and Eduardo G. Altmann1
1Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
A clear signature of classical chaoticity in the quantum regime is the fractal Weyl law, which
connectsthedensityofeigenstatestothedimensionD oftheclassicalinvariantsetofopensystems.
0
Quantumsystemsofinterestareoftenpartiallyopen(e.g.,cavitiesinwhichtrajectoriesarepartially
reflected/absorbed). In the corresponding classical systems D is trivial (equal to the phase-space
0
dimension), and the fractality is manifested in the (multifractal) spectrum of R´enyi dimensions
D . In this paper we investigate the effect of such multifractality on the Weyl law. Our numerical
q
simulationsinarea-preservingmapsshowforawiderangeofconfigurationsandsystemsizesM that
(i) the Weyl law is governed by a dimension different from D =2 and (ii) the observed dimension
0
oscillates as a function of M and other relevant parameters. We propose a classical model which
considers an undersampled measure of the chaotic invariant set, explains our two observations,
5
and predicts that the Weyl law is governed by a non-trivial dimension D < D in the
1 asymptotic 0
semi-classical limit M →∞.
0
2
PACSnumbers: 05.45.Mt,05.45.-a,05.45.Df
n
a
J I. INTRODUCTION While a straightforward application of the fractal Weyl
4 law to partially open systems predicts a trivial scaling
1 Fractality is a well-known signature of chaotic dynam- (N M),asalsoarguedin[14],non-trivialscalingshave
ics [1–3]. In open quantum systems the fractality of the been∼reported in numerical investigations [11]. In order
]
D classical dynamics appears in the distribution of eigen- toinvestigatetherobustnessoffractalityintheWeyllaw
states[4]andintheWeyllaw[5]. Recentstudiesconsid- we study it at the transition between monofractal and
C
eredtheeffectofperiodicorbits[6]andweakchaos[7–9] multifractal. ThisisdonebyvaryingthereflectivityR
. ≥
n and were conducted on 4-dimensional Hamiltonian sys- 0ofanabsorbingregionlocalizedinthephase-space. For
i tems [7] and different area-preserving maps [6, 8–10]. In all nonzero reflectivities we obtain non-trivial Weyl laws
l
n these systems the fractal Weyl law can be written as witheffectivedimensionsthatshowstrongoscillationsas
[ a function of system size and other parameters.
2 N(νi > ν cutoff) MD0(S)/2, (1) Wearguethattheseobservationsareduetoaneffective
v | | | | ∼ undersampling of the classical measure in the quantum
9 (S) regime and that the non-trivial scaling remains valid in
where M is the dimension of the Hilbert space, D is
8 0 the semiclassical limit M .
the fractal dimension of the chaotic saddle, and ν
8 | |cutoff →∞
is a cutoff separating long-lived from short-lived states
0
0 with eigenvalues νi.
. Optical cavities [11] and other relevant physical sys- II. FRACTALS IN CLASSICAL DYNAMICS
1
tems [12] are not completely open but instead show par-
0
5 tial absorption. A simple physical picture is a billiard As numerically convenient examples of Hamiltonian
1 in which trajectories hitting the boundary are partially chaoticsystemsweconsiderherefullychaoticsymplectic
: reflectedandpartiallyabsorbed. Incontrasttoopensys- maps of the form f :(cid:126)x (cid:126)x , defined in a bounded
iv tems(fullabsorption),inpartiallyabsorbingsystems,the phase-space (cid:126)x Ω. Tonin(cid:55)→corpn+or1ate the effect of partial
X box-countingdimensionD0 isanintegernumberequalto absorption we∈associate to each trajectory an intensity
r the phase-space dimension [13]. Fractality in the classi- I , with I = 1, I = R((cid:126)x )I , and R((cid:126)x) [0,1] is a
n 0 n+1 n n
a cal dynamics of such systems is still present [11] and can realvaluedfunctiondescribingthereflectivity∈ofdifferent
bequantifiedusingthespectrumofR´enyidimensions[1] regions in the phase-space.
Inordertoinvestigatethedifferencesbetweenpartially
D = 1 lim ln(cid:80)iN=(1ε)µqi, (2) and fully open systems it is instructive to consider the
q
1 q ε→0 ln1/ε special case of R((cid:126)x)=R<1 in a small absorbing region
−
of the phase-space (i.e. R((cid:126)x)=1 everywhere else). If
where µ((cid:126)x) is the relevant probability measure R=0 in the absorbing region the system corresponds to
(cid:82)
( dµ((cid:126)x) 1) in the phase-space Ω and the sum is over usualopenmapsandthesetofinitialconditions(cid:126)x with
Ω ≡ (cid:82) 0
ε-sized boxes Bi (with 0 < µi := Biµ((cid:126)x)d(cid:126)x) used to limn→∞In =1 form the forward trapped set of the map
partition the phase-space. For q =0, D is recovered. A (the stable manifold of the chaotic saddle [2, 15, 16]). In
0
distributionµ((cid:126)x)withanon-trivialdependenceofD on thiscasethemeasureisuniformlydistributedonafractal
q
q is called multifractal. support and the R´enyi spectrum is flat (D = D ). In
q 0
In this paper we investigate the quantum mechanical analogy, for a partially absorbing system 0 < R < 1 the
signatures of multifractality in the classical phase-space. measure in a phase-space region B (e.g., the i-th box of
i
2
a) baker map b) cat map
1 1 1 1
1 1
n R<1 + n R<1 +
y n y n
y y
0 0 0 0
x x x x
0 n 1 0 n+1 1 0 n 1 0 n+1 1
FIG.1: (Coloronline)Classicaldynamicsofarea-preservingmaps(x ,y )(cid:55)→(x ,y )withabsorbingregions. Thepairs
n n n+1 n+1
of plots a) and b) show the action of one application of the baker’s map (x ,y )=(3x−(cid:98)3x (cid:99),y /3+(cid:98)3y (cid:99)/3) and the
n+1 n+1 n n n
cat map (x ,y )=(2x +y mod 1,3x +2y mod 1), respectively. The areas surrounded by dotted lines in the left
n+1 n+1 n n n n
plots indicate the absorbing regions (R < 1) of each map. The dark regions in the right plots show the first images of the
absorbing region.
a partition by a regular grid) should be proportional to bakermap catmap
1 1 Imax
the average intensity I for n of randomly drawn
n
→ ∞
(cid:126)x B . The finite time normalised measure of B after
0 i i
∈
n iterations can thus be written as
I y y
n,i
µn,i = (cid:80)(cid:104) (cid:105) , (3)
I
i(cid:104) (cid:105)n,i
where the average intensity ... is computed over ini-
(cid:104) (cid:105)n,i 0 0 0
tialconditions(cid:126)x0 Bi. Thisdefinitionisconsistentwith 0 x 1 0 x 1
∈
the usual definition of the stable manifold measure [1]
considering the intensities I [0,1] of trajectories as 2.0 2.0
∈ numerical
weights (see also Ref. [12]).
q q
Asrepresentativecasesoffullychaoticarea-preserving D analytical D
numerical
maps we investigate in detail the baker [1] and the cat
1.5 1.5
map [17] as defined in Fig. 1. In the case of the baker 0 2 4 6 8 10 0 2 4 6 8 10
map the absorbing region was chosen in order to allow q q
for comparison with analytical calculations. In the case
of the cat map we chose an arbitrary asymmetric region. FIG. 2: (Color online) Multifractal properties of the maps
The essential point for our investigation of the role of defined in Fig. 1. Top row: finite time approximation of a
typical phase-space distribution of the measure µ defined in
partial opening is that both choices lead to a non-trivial i
Eq. (3). Bottom row: R´enyi spectrum D for R=0.3 (see
D for R=0. In Fig. 2 we report the distribution of the q
0 AppendixAfortheanalyticalcalculationsonthebakermap).
measure µ in the phase-space for a particular reflectivity
and the spectrum D . For the baker map the calcu-
q
lations can be done analytically (see Appendix A) and
confirm the validity of our numerical methods of esti-
where ˆ is the quantised closed system and ˆ is a pro-
mating µi and Dq. In both maps D0 = 2 and Dq is U P
jectorthatcontainsalltheinformationaboutabsorption.
non-trivial, confirming the result of Ref. [13] that the
In the case of full absorption, ˆ removes the part of the
apparent self-similar distribution of µ observed in the P
quantum wave-function ψ(x) that overlaps with the ab-
phase-space is properly quantified only through the full
sorbingregion, but doesnot alterthe remainingpart. In
multifractal spectrum D .
q thecaseofpartialabsorption, ˆ reducestheintensityby
(cid:12) (cid:12)2 P
a factor R(x) as (cid:12) x ˆ ψ (cid:12) =R(x) ψ(x)2. See App. B
(cid:12) (cid:12)
(cid:104) |P| (cid:105) | |
III. FRACTALS IN QUANTUM SYSTEMS
for the matrix representations of the maps used in this
paper.
In this section we introduce the quantised version of
InanM-dimensionalHilbertspace(effective(cid:126)= 1 )
the maps of Sec. II and compute Weyl’s law (1). By 2πM
open quantum map we mean a propagator [18] maps ˆ have have M left (and right) eigenstates with
M
complex energies (cid:15) = E +iΓ /2. ˆ acts like a time
i i i
ˆ = ˆˆ, (4) evolution operator so its eigenvaluesMν are related to (cid:15)
i i
M UP
3
for an improved analysis of the scaling, which according
to the Weyl law prediction (1) should be d 1, where
102 a)quantumbakermap b)quantumcatmap d =D 1isthepartialfractaldimensionof0t−heforward
0 0
101 R=0.0 R=0.0 trapped−setalongtheunstabledirection. D0 denotesthe
100 full fractal dimension of this set. It is connected to the
10 1 dimension of the chaotic saddle D(S) by [2]
− 0
10 2
− (S)
101 R=0.001 R=0.0001 D(S) =2D 2 d = D0 . (6)
100 0 0− ⇔ 0 2
10 1 The usual fractal Weyl law is confirmed in Fig. 4ad for
−
10 2 thecaseR=0. ForsmallR, thecurvesfollowtheR=0
−
101 R=0.01 R=0.001 case for small M before deviating from it. With increas-
)
ν|100 ing R, the deviation is observed already for smaller M.
(ρ|10−1 Interestingly, even for large R and M the numerical re-
10 2 sults do not approach a constant value as predicted by
−
101 R=0.1 R=0.01 the classical dimension d0 = 1. Instead, a non-trivial
100 scaling with an effective dimension between 0 and 1 is
10 1 observed. Evenifthenumericalresultsdonotallowfora
−
10 2 conclusionregardingthescalinginthelimitM ,the
− →∞
101 R=0.2 R=0.1 roughlyconstantscalingforabroadrangeofvaluesinM
100 shows the importance of the effective dimension (similar
10 1 effective dimensions have been employed to investigate
−
classical [20] and quantum [8] Hamiltonian systems with
10 2
−
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 mixed phase-space).
In order to more carefully analyse the scaling we de-
ν ν
| | | | fine the local dimension d as the slope of the N/M
loc
vs. M curves between two nearby matrix sizes M. Fig-
ures 4be show the dependency of d on the choice of
FIG. 3: Distribution of eigenvalues |ν|, see Eq. (5))in the loc
the cutoff ν . For small cutoffs ν < ν
quantisedbaker(a)andcat(b)maps. Twomatrixsizeswere | |cutoff | |cutoff | |typ
used: M =37 (gray line) and M =39 (black line) [19]. The we observe dloc = 1, in agreement with the observa-
verticaldashedlinemarksthecutoffvalueusedforthefractal tion in Fig. 3 that the distribution of eigenvalues be-
Weyl law calculations in Fig. 4ad. comesconcentratedaround|ν|typ(almostallstatescount
as long-lived). A more interesting behavior appears for
ν > ν , where at least two regimes governed
by b|y|cnutoonff-trivi|al|tdyp can be identified before d starts to
loc loc
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) fluctuate for ν 1 due to a lack of statistics: (I1)
νi =exp −(cid:126)i(cid:15)i =exp −(cid:126)iEi exp −21(cid:126)Γi . (5) an regime of|in|ctuetromffe→diate |ν|cutoff with strong (almost
periodic) oscillations of d ; and (I2) a regime in which
loc
Since Γi ∈ R+0 and Ei ∈ R it follows that |νi| ∈ [0,1]. dloc ≈ d(0R=0). These regimes are particularly visible for
eH.ger.elaofntegr-lwiveedusseta|νtei|sthoaqvueanνitif(cid:47)yt1he(Γliifetim1)e.ofthestate, sfomrallalrRge(rbRec.auSseur|pνr|itsyipng→ly,0)F,ibgsu.t4ccafnsbheowidethntaitfierdegaimlsoe
| | (cid:28)
Wecalculatedtheeigenvaluesof ˆ forboththebaker I1 increases for increasing M (strong oscillations cover a
M
andcatmapfordifferentreflectivitiesRandmatrixsizes widerrangeofcutoffs). Thissuggeststhattheoscillating
M =3n withn N[19]. Thenormaliseddistributionsof (local) scaling is not an artifact of small M but instead
∈
ν are shown in Fig. 3. The states concentrate around that it will prevail in the large M limit.
i
| |
a typical value ν , which is ν 0 for R = 0, In summary, our numerical observations of the quan-
| |typ | |typ ≈
increaseswithR,andcanbeapproximatedasthemedian tum maps show surprising features. In particular: (i)
of ρ(ν ). For increasing matrix-sizes M the distribution the classical prediction d0 =1 was not observed; (ii) the
| |
becomesmoreconcentratedaround ν . Toinvestigate effective scaling varies smoothly with R; and (iii) strong
| |typ
thistendencyinfurtherdetail,andinlinewiththefractal oscillations we observed in dloc vs. |ν|cutoff.
Weyl law, we concentrate on the number of eigenstates
with ν ν .
Fig|urie|s≥4|ad|custhoffow the relative number of eigenstates IV. SAMPLING MULTI-FRACTAL MEASURES
N(ν : ν ν )/M as a function of the matrix-
i | i|≥| |cutoff
size M. For this plot, ν was chosen in such a way In this section we propose an explanation for the scal-
| |cutoff
that ν > ν but still small enough to allow the ing of the number of long-lived states with M observed
| |cutoff | |typ
calculation of the scaling for a wide range of M, see ver- in the previous section based on the classical dynamics
ticaldashedlinesinFig.3[27]. ThedivisionbyM allows of the system.
4
quantumbakermap quantumcatmap
32 33 34 35 36 37 38 39 32 33 34 35 36 37 38 39
100 100
a) d)
M
/
)
off 10−1
ut
c
ν|10−1
|
>
10 2
| −
ν
(|
N R=0.0 R=0.01 R=0.2 R=0.0 R=0.001 R=0.01
R=0.001 R=0.1 R=0.4 R=0.0001 R=0.005 R=0.1
10 2 10 3
− −
101 102 103 104 101 102 103 104
M M
1.2 1.2
b) e)
1.0 1.0
0.8 0.8
c 0.6 0.6
o
dl
0.4 0.4
0.2 R=0.0 R=0.1 0.2 R=0.0 R=0.005
R=0.001 R=0.2 R=0.0001 R=0.01
0.0 R=0.01 R=0.4 0.0 R=0.001 R=0.1
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8
ν ν
| |cutoff | |cutoff
1.2 1.2
c) f)
1.0 1.0
0.8 increasing M 0.8
c 0.6 0.6 increasing M
o
dl
0.4 0.4
0.2 0.2
0.0 R=0.01 0.0 R=0.0001
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8
ν ν
| |cutoff | |cutoff
FIG. 4: (Color online) Scaling behaviour of the quantum states for the baker (a-c) and cat (d-f) maps. Top row: Fraction
of eigenvalues above the cutoff value |ν| as a function of the system size M at different reflectivities. The dotted lines
cutoff
mark the expected scaling for R = 0 (d(R=0) = ln(2)/ln(3) for the baker map and d(R=0) ≈ 0.42 for the cat map). Middle
0 0
row: Localscalingexponentd (M):=(lnN(M)−lnN(M/3))/ln3,whereN(M)denotesN(ν :|ν |≥|ν| ),atM =39
loc i i cutoff
for different values of R as a function of |ν| (colors and symbols as in top row). We show results up to a cutoff where
cutoff
N(M)<0.005M afterwhichpointthelackofresonancesleadstostrongoscillationsofd . Theverticaldashedlinesindicate
loc
the cutoff values used in the top row. The small ticks on the upper axis mark the positions of the medians of the eigenvalue
distributions (≈|ν| ). Bottom row: d (M) as a function of the cutoff for M ={37,38,39} (lighter shades correspond to
typ loc
smaller M).
5
A. General Argument p =1 (1 µ )S, where (1 µ )S is the proba-
i i i
− − −
bility that the box is rejected S times.
We first consider how the usual fractal Weyl law is
For one realization of this process the available volume
obtained from the classical phase-space of fully absorb-
V(M) is proportional to the number N of counted seg-
ing maps [5, 9]. Lu et al. [5] argue that the states can b
ments. Therefore, point 3. is generalised to 3’. as
be considered as non-overlapping and that each of them
covers an area of h = 1/M. This localisation happens M M
(cid:88) (cid:88)
in the forward trapped set (for the left eigenstates [28]) N(M) E[N ]= p = 1 (1 µ )S, (7)
b i i
∝ − −
[4, 21], which is smooth. Therefore, the number of dif- i=1 i=1
ferent long-lived states depends on the distribution of
where E[...] is the expected value computed over mul-
the forward trapped set in the perpendicular direction
tiple sampling realisations and the number of trials S
(along the backward trapped set or, equivalently, along
is associated to the total number of states S = S M.
the unstable manifold of the chaotic saddle). The rea- 0
In the case of a fully absorbing region, µ = 1/
soning above leads to the following constructive classical i
N
for all boxes intersecting the forward trapped set
procedure:
N
and µ = 0 everywhere else. Eq. (7) simplifies to
i
1. Cover an intersection of the forward trapped set E[N ]= (cid:0)1 (1 1/ )S(cid:1) S→∞ and we recover
b
with segments of size h=ε=1/M. N − − N −−−−→ N
the fractal Weyl law. Analogously, in the fully reflecting
S→∞
2. Count the number Nb of boxes (segments) needed case µi = 1/M and E[Nb] M and we recover the
−−−−→
to cover the forward trapped set, which estimates traditional Weyl law.
the phase-space volume available to the long-lived At the heart of the reasoning above is a crucial dif-
eigenstates. ference between the computation of d0 and of the Weyl
scaling. d is obtained by first taking the limit of sam-
0
3. The number of long-lived states N(M) is propor-
pled trajectories S to infinity S (construction of
tional to Nb. the measure) and afterwards com→pu∞ting the scaling of
Since Nb Md0 we obtain N(M) Md0, the fractal filled boxes in the limit of small box sizes ε → 0. In
∼ ∼ the Weyl law the analogues of these two limits (S
Weyl’s law (1). The scaling exponent d0 is the par- → ∞
and ε 0) happen simultaneously when M be-
tial fractal dimension of the forward trapped set along → → ∞
causethemaximumnumberofeigenstatesgrowswithM
theunstabledirection(6). SchomerusandTworzydlo[9]
(S M) and the phase-space resolution increases with
provide an alternative explanation based on the phase- ∝
M (h = ε = 1/M). Therefore, the scaling in the Weyl
spaceareasthatdonotescapeuptoagiventime(which
law corresponds to the estimation of d when sampling
grows with M). Both explanations are equivalent be- 0
the measure with a sample size
cause asymptotically the h-sized boxes intersecting the
invariantsetcorrespondtothenon-escapingphase-space S =S M =S ε−1 (8)
0 0
areas.
We now generalize the ideas above to the case of par- in (7). The proportionality constant S effectively con-
0
tial absorption. In this case we expect for large M the trolshowmanysegments(long-livedstates)willbefound
long-lived states to be distributed according to the clas- for the different M values and therefore plays a similar
sical measure µ of the forward trapped set. Therefore, role as the cutoff value ν in the quantum case.
it is essential to consider a procedure that accounts for | |cutoff
the non-uniformity of µ. In line with the usual interpre-
tation of µ as a probability measure, we modify point 2. B. Application to Baker Map
above and attribute to each segment i a probability p
i
of being counted as long-lived (i.e., a probability that a
We now apply the general sampling method proposed
long-livedstateislocalizedinit). Thisprobabilityhasto
in the previous section to the baker map (defined in
belarge(small)wherethemeasureµ ofthesegmentiis
i Fig. 1). The expected number of long-lived boxes E[N ]
b
large (small) because states localise on high measure ar- for ε=3−n can be computed analytically (see Appendix
easofthephase-space. Atthesametime,forlargeM the
C)
number of possibly long-lived states grows and p should
i
grow because of the multiple chances of one state being (cid:88)n n! (cid:18) Rl (cid:19)S
localized in segment i. The simplest stochastic process E[N ] =3n 2n−l 1 . (9)
b n
− l!(n l)! − (2+R)
agreeing with these two intuitions corresponds to per- l=0 −
forming S independent trials with a success-probability
In Fig. 5 we plot Eq. (9) and its local derivatives as
µ . This can be incorporated in the procedure described
i
a function of its three parameters ε, S , and R. The
above by replacing point 2. by 0
similarities to the quantum observations shown in Fig. 4
2’. Count each segment i as part of the ac- are remarkable (the comparison is based on the identi-
cessible phase-space volume with a probability fications ε = h = 1/M, N(ν : ν ν ) E[N ],
i | i|≥| |cutoff ∝ b
6
31 32 33 34 35 36 37 38 39 31 32 33 34 35 36 37 38 39 310311312
100 100
a)
M
/
)
off
]Nb 10 1 νcut|10−1 R=0.4
[E − >| R=0.2
ε R=0.1
|
R=0.4 R=0.01 (ν| R=0.01
N R=0.001
R=0.2 R=0.001
R=0.0
R=0.1 R=0.0 10 2
10−2 −100 101 102 103 104 105 106
100 101 102 103 104
M
1/ε
1.1
FIG. 6: (Color online) Comparison of the quantum data to
b)
1.0 the classical model. The symbols show the number of long-
0.9 lived states. The solid lines are the classical values of aE[Nb]
obtained by choosing S and a for each R so that the right-
0
0.8
mosttwopointsofthequantumdataarematchedascloseas
oc 0.7 possible.
dl
0.6
0.5 R=0.4 R=0.001
and the similar role played by S and ν . In partic-
0.4 R=0.1 R=0.0001 ular, we reproduce the three ma0in obs|er|vcauttoioffns reported
R=0.01 R=0.0
0.3 in Sec. III: (i) the scaling of the curves is different from
10−10 10−8 10−6 10−4 10−2 100 102 d0 = 1; (ii) for increasing R, the effective scaling devi-
1/S0 ates more and more from d(R=0) and approaches a non-
0
trivial dimension (Fig. 5a); (iii) the local slope d show
loc
1.1 strong oscillations with S (Fig. 5b) in a range of S
0 0
1.0 c) values which increases with M (Fig. 5c). One quantum
feature which is not immediately apparent in Fig. 5b is
0.9
(R=0)
the plateau of d d , the regime I2 mentioned at
0.8 increasing M loc ≈ 0
the end of Sec. III. To understand this we need to take a
dloc 0.7 closerlookatthemappingfromS0 tothecutoff|ν|cutoff.
0.6 The limiting case S0 can be safely identified with
→ ∞
ν 0. Inthislimitourmodelcorrectlyreproduces
0.5 | |cutoff →
the quantum observations (d 1). The identification
0.4 loc →
0.3 R=0.001 pofostshibeleopbpuotsilnesgslmimeiatnSin0gf→ul,0sinwciethin|νt|hcuistolffim→it b1oitshatlhsoe
10 10 10 8 10 6 10 4 10 2 100 102
− − − − − numberofresonances,andN becomeverysmall,andas
b
1/S0 a result the determination of dloc becomes prone to fluc-
tuations. Betweentheselimits,wecanassumethat1/S
0
grows monotonically with ν . However, the func-
| |cutoff
FIG. 5: (Color online) Scaling behaviour of the classical tional form relating 1/S and ν is unknown (e.g.,
0 | |cutoff
expectedvalues(9)forthebakermap. Theblackdottedline it depends on the distribution of eigenstates) which does
denotesd(R=0). a)Rescaledexpectedvalueofacceptedboxes notallowforacarefulquantitativecomparisonofthetwo
0
at an arbitrary sample size of S(M) = S0M with S0 = 0.3. plots (e.g., a non-linear transformation of the x-axis in
E[Nb] grows at fixed ε monotonically with R. The shaded Fig. 5b could generate the plateaus around d(R=0) seen
areas cover the range between E[N ]±σ[N ], where σ is the 0
b b in Figs. 4bd.
standard deviation. b) Scaling exponent d (M = 39) as a
loc
We now search for a quantitative comparison between
function of 1/S . The lower the value of R, the lower the
0
curve intersects the dashed vertical line (the S choice used our classical sampling model and the quantum observa-
0
in a). c) dloc(M = 3n) with n ∈ {7,8,9}. Lighter shades tions. The only parameter we have to fit is S0 in the
correspond to smaller M. sampling model, which according to the arguments of
Sec . IVA should reproduce the scaling of the long-lived
states with large M. Accordingly, for each value of R
we choose the value of S which reproduces the local
0
7
slope between the last two N/M points in the quantum
100
data (Figs. 4ad) and we shift the E[N ] curve obtained
b
to match the rightmost point of each quantum data-
set (this last shift fixes the proportionality factor a in
N(M) = aE[N ]). We restrict S to the range (cid:2)3−1,3(cid:3),
b 0
M
sboestthaatgrSeeims eonftthweitohrdtehreowf Mholseinrcaengteheosfeqvuaalunetsumgivdeatthae. ]/Nb 10−3 R=0.4
Because of the oscillations in d there may be more [E R=0.2
loc
R=0.1
than one choice of S [29]. The results for the different
0 R=0.01
R shown in Fig. 6 confirm that our model describes the R=0.001
a)
data also for values of M much smaller than the region R=0.0
used in fixing S0,a. 10−6100 105 1010 1015 1020
The quantitative success of our sampling model mo- M
tivates us to consider its behavior in the semiclassical
limit M . In Fig. 7a we plot Eq. (9) in an increased 1.0
range of→M∞=1/ε values using for the different R curves d0
d0.3
thevalueofS0 usedinFig.6b(estimatedfromthequan- 0.9 d1 d∞
tcuilmlatdioantsa)(.seWeeRaegfa.i[n22s]eefotrheaprreevsieenwceonoflloogg--ppeerriiooddiiccitoiess- ptotic 1.0
iinn fdractaanlsd).foWr hthilee rtahnegsee oosfcMillataioccnessspiblalye ianmthaejoqruraonle- dasym 0.8 00..89 dd00..12
loc d0.3
tum computations, they do not hinder the estimation of 0.7 0.7
d0.4
an asymptotic dimension dasymptotic as the general trend b) 0.60.001 0.01 0.1 1.0
of the curves. For R = 0 the results confirm that for 0.6
d = d(R=0) and that our sampling correctly ac- 0.0 0.2 0.4 0.6 0.8 1.0
asymptotic 0
countsfortheusualfractalWeyllaw. Moreinterestingly, R
d < 1 for all R, strongly suggesting that the
asymptotic
non-trivialscalingd observedinthequantumdataare
loc
FIG. 7: (Color online) Asymptotic scaling of the
not finite M effects but persist in the asymptotic limit.
classical prediction. a) Extrapolation of E[N ] of the
This implies that the fractal Weyl law of partially open b
curves in Fig. 6a without the shift (i.e. a=1). R
systems does not follow the classical prediction d = 1.
0 increases from the black (bottom) to the brown (top)
Comparing the R dependence of d with the an-
asymptotic curve. The dashed lines follow the asymptotic scaling
baleytuisceadl ctaolceustlaimteadtedqthiinsdsiccaaltiensg.that roughly q ≈ 0.3 can tMhedasaymveprtoatgice−1iswictohmdpaustyemdptootivcer=a(cid:104)lllnNplo(nMsMs)i−−blllennNMc((cid:48)Mom(cid:48))b(cid:105)i,nawthioenres
(cid:8)(M,M(cid:48)):M(cid:48) <M ∧(M,M(cid:48))∈{310,311,...,340}2(cid:9). In b)
we show these dimensions together with their standard de-
V. DISCUSSIONS viations. In grey are analytically obtained dq curves, see
Eq. (A13). Note, that for fixed R and q∈R+, d ≥d ≥d
0 q ∞
[1].
In summary, we have investigated the statistical prop-
erties of long-lived eigenstates of chaotic systems with
partial absorption. Contrary to the naive prediction ob-
tained combining the classical fractal dimension d and rangeofcutoffsin ν . Inanypracticalsituationthelimit
0 | |
the fractal Weyl law, we observe numerically that for M isnotachievableandtherelevanceofourmodel
→∞
a broad range of parameters the number of long-lived isthatitallowstounderstandtheeffectivedimensionob-
states grows sub-linearly with the dimensionality M of served quantum mechanically. We have also considered
the Hilbert space. We explain these observations by alternative models in which we assume that the eigen-
considering a model which is consistent with the quan- statesconcentrateexclusivelyonthephase-spaceregions
tumtoclassicalcorrespondenceinthesemiclassicallimit with highest measure µ, or on boxes with a µ-dependent
M but that accounts for an undersampling of the probability that we compute without including a sam-
clas→sica∞l measure at finite M. This undersampling com- pling process. None of these models is able to reproduce
bined with the strong spatial fluctuations of the multi- the quantum observations, see Appendix D.
fractal classical measure is responsible for the effective Our results are not limited to area-preserving maps
reduction of the available phase-space volume for long- or to systems with localized absorbing regions. Instead,
lived eigenstates. they should be visible in systems of any dimension for
Our most surprising finding is that even the asymp- which a multifractal spectrum of a classical measure µ
totic(M )scalingofthenumberoflong-livedstates appears. In n-dimensional systems the long-lived reso-
→∞
differ from the d =1 prediction. This result is compat- nances are expected to concentrate in n-cuboids of vol-
0
ible with our numerical simulations at least for a broad ume h and the sampling has to be carried out on a hy-
8
perplaneintersectingthesevolumes. Ingeneralpartially- the measure is constant in the horizontal direction, we
absorbing systems there is no analogous fully open sys- can restrict ourselves to a vertical line and calculate the
tem (R 0 for the cases above). In these cases, the partial dimensions d along this line. Assume this line
q
scaling o→f Weyl’s law is expected to follow the dimen- is covered by disjoint segments of length ε = 3−n. The
sion of the regions of highest probability[30] for small M dimensions of the full distributions are then simply ob-
(analogoustod oftheR=0caseintheexamplesabove) tained from
0
and show a transition towards the dimension d
asymptotic
D =1+d (A4)
oftheundersampledµ(forthelargeM case). Themulti- q q
fractalityinthelocalizationofwavefunctionscanappear with
alsowithoutaclearclassicalcounterpart,e.g. inthecase
1 ln(cid:80)3n µq
of phase transitions in disordered systems [23, 24]. It d = lim i=1 i. (A5)
q
would be interesting to investigate whether the number 1 q n→∞ nln3
−
of such states show similar dependencies on system size
Here µ is the intensity on the i-th segment normalised
i
as the ones we observed.
by I , so that
tot,n
3n
(cid:88)
Acknowledgments µ =1 (A6)
i
i=1
We thank A. B¨acker, R. Ketzmerick, M. K¨orber, and Or, again using multinomial factors:
T. T´el for insightful discussions and suggestions.
n n−k(cid:18) (cid:19)
(cid:88)(cid:88) n
µ =1 (A7)
n;k,l
k,l
Appendix A: Analytical D in the baker map k=0 l=0
q
with
Assume a ternary baker map as defined in Fig.1 with RkRl Rn−k−l
reflectivities RL in the left, RM in the middle and RR in µn;k,l = L IM R (A8)
the right vertical strip. In this system it is possible to tot,n
calculatethenumberofnecessaryiterationstogetanex- With that we can write the sum in (A5) as
actresultuptosomeminimumε . Thesizeoffeatures
min 3n
(stripes)intheternarybakermapafterniterationsisat (cid:88)
least 1/3n. If ε is smaller than that value then the ln µqi = (A9)
min
box-counting will pick up the features as areas and D i=1
will approach 2. 0 (cid:88)n n(cid:88)−k (cid:0)kn,l(cid:1)(cid:0)RLkRMl RRn−k−l(cid:1)q
ln = (A10)
sitLyeitnRtLh,eM,tRop∈h[0o,r1iz]o.nAtaftletrhtihrde fiisrsRt it,etrhateiocnentthrealinstternip- k=0 l=0 (RL+RM+RR)nq
L
hasanintensityofR ,andthebottomthirdcarriesR . (Rq +Rq +Rq)n
M R ln L M R = (A11)
The total intensity is (R +R +R )nq
L M R
(Rq +Rq +Rq)
Itot,1 =RL+RM+RR. (A1) nln L M Rq (A12)
(R +R +R )
L M R
The next iteration gives the following total intensity:
Thus we obtain for d
q
Itot,2 = RL2 +RLRM+RLRR + (Rq+Rq +Rq)
RMRL+RM2 +RMRR + (A2) dq = 1 lim nln(RLL+RMM+RRR)q . (A13)
RRRL+RRRM+RR2 1−q n→∞ nln3
Thisisindependentofnsowecandropthelimitandget
and so forth. At the n-th step we obtain the total inten-
sity ln (RLq+RMq +RRq)
d = (RL+RM+RR)q . (A14)
q
n n−k(cid:18) (cid:19) (1 q)ln3
I = (cid:88)(cid:88) n RkRl Rn−k−l (A3) −
tot,n k,l L M R The limit q 1 can be obtained using L’Hˆopital’s rule.
k=0 l=0 →
= (RL+RM+RR)n d :=lim 1 ln(cid:80)3i=n1µqi
1
q→11 q nln3
Here we used the fact that the number of strips with −
a certain combination of reflectivities RkRl Rn−k−l is ln(RL+RM+RR)
given by the multinomial coefficient (cid:0)n(cid:1):L=M Rn! . ⇒d1 = ln3 (A15)
k,l k!l!(n−k−l)!
R lnR +R lnR +R lnR
Now we focus on the multifractal dimension spectrum L L M M R R.
(2)ofthedistributiongeneratedbythismap. First,since − (RL+RM+RR)ln(3)
9
Asymptotically d goes to Plugging in the expression for µ we get
q n;k,l
lim d = ln(RL+RM+RR)−ln(Rmax), (A16) E[Nb]= (C2)
q→∞ q ln3 (cid:88)n n(cid:88)−k(cid:18) n (cid:19) (cid:32) RkRl Rn−k−l (cid:33)S
where Rmax :=max{RL,RM,RR}. k=0 l=0 k,l 1− 1− (RLL+RMMR+RR)n .
Equation (9) follows from setting R = 1, R = R,
Appendix B: Quantised maps L,R M
carrying out the inner sum, and renaming the indices.
Appendix D: Discussion of alternatives to sampling
The matrix representation of the propagator of the
quantised version of the baker map defined in Fig. 1 is
Here we want to show that taking into account only
1 0 0 the regions of phase-space with the highest measures by
M/3
baker = baker 0 √R1M/3 0 (B1) a cutoff in the measure can not reproduce the scaling
M U · 0 0 1 observed in the quantum maps.
M/3
First, let us consider a fixed cutoff. In this case it is
with easy to see that for any value of µcutoff = const. there
exists a n∗ with
F 0 0
M/3
Ubaker =F−M1· 00 FM0/3 F0 (B2) µn;k,l <µcutoff, ∀n>n∗.
M/3
n→∞
Therefore d 0.
where FM;m,n = √1M exp(cid:2)−2Mπi(n+1/2)(m+1/2)(cid:3) are The seco0n−d−−c−→ase is an n-dependent cutoff value
FreosuernitearttiorannasfnodrmFa−Mt1iodnesnforotems tphoesiitniovnertsoemtroamnsefnotrummatrioepn-. aµlclutRoffL,,nM,=R <IIctuotto,1nff twheitrhe caolsnosteaxnitstIscuatoffn.∗ Iwnitthhiµsnc;ka,sle<if
Foranexplanationofaquantisationschemeseee.g.[25]. µ , n > n∗. In other words: d n→∞ 0. An ex-
cutoff,n 0
Analogously, the quantised cat map has the following ∀ −−−−→
ception here arises if R = 1 and R [0,1). Then,
L,R M
form: for all 0 < I < 1 after a number o∈f iterations n∗
cutoff
1 0 0 all middle thirds fall below the cutoff d0 n→∞ ln2/ln3
M/6 −−−−→
from above.
Mcat =[αM;m,n]M · 0 √R1M/2 0 (B3) So,byassociatingthenumberoflong-livedeigenstates
0 0 1
M/3
ofboxesaboveagiventhresholdweobtainaneffectivedi-
mension contradicting our numerical observations. This
where M is even and the entries are [26]
indicatesthatthe(large)regionswithlowmeasureµstill
α = 1 exp(cid:20)2πi(cid:0)m2 mn+n2(cid:1)(cid:21). (B4) contain substantial portions of the eigenstates.
M;m,n The third alternative we want to discuss is the case
√M M −
whereeachboxcontributestothelong-livingphase-space
volume proportionally to its intensity
Appendix C: Derivation of E[N ]
b
p I =µ I . (D1)
n;k,l n;k,l n;k,l tot,n
∝
Startingfromthegeneralequation(7)fortheexpected
AsbeforeweidentifyM =3n. Inthiscontextweneedto
number of accepted boxes here we show how to arrive at
limitthecontributionofeachboxp =min 1,αI
n;k,l n;k,l
the formula for the baker map (9). andcomputeE[N ]=(cid:80)n (cid:80)n−k(cid:0)n(cid:1)p .{Thefacto}r
The i-th box is found with a probability pi =1 (1 b k=0 l=0 k,l n:k,l
µ )S. Analogous to (A8) we can restrict oursel−ves t−o α plays a similar role as the cutoff in the sense that for
i
different boxesanddefinep :=1 (1 µ )S. The α (α 0) we count all (zero) boxes. This method
expected value of the numbne:rk,olf foun−d bo−xesni;ks,l rep→ro∞duces→the correct scalings in the cases RL,M,R = 1,
andR =1,R =0. However,itfailstoreproducethe
L,R M
3n n n−k(cid:18) (cid:19) increaseoftheeffectivedimensionwithRandM andthe
(cid:88) (cid:88)(cid:88) n
E[N ]= p = p . (C1) oscillating behaviour of d with ν .
b i n;k,l loc cutoff
k,l | |
i=1 k=0 l=0
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