![The number of nodal domains ℵ N (full circles) for the chaotic half-circular microwave rough billiard. Full line shows a least squares fit ℵ N = a 1 N + b 1 √ N to the experimental data (see text), where a 1 = 0.058 ± 0.006, b 1 = 1.075 ± 0.088. The prediction of the theory of Bogomolny and Schmit [2] a 1 = 0.062.](https://www.researchgate.net/profile/Leszek-Sirko/publication/8128675/figure/fig3/AS:667172997181457@1536077846936/The-number-of-nodal-domains-N-full-circles-for-the-chaotic-half-circular-microwave_Q320.jpg)
![The normalized signed area distribution N Σ A for the chaotic half-circular microwave rough billiard. Full line shows predicted by the theory asymptotic limit N Σ A ≃ 0.0386, Blum et al. [1].](https://www.researchgate.net/profile/Leszek-Sirko/publication/8128675/figure/fig4/AS:667172997173264@1536077846951/The-normalized-signed-area-distribution-N-S-A-for-the-chaotic-half-circular-microwave_Q320.jpg)
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arXiv:0903.1965v1 [nlin.CD] 11 Mar 2009
Experimental investigation of nodal domains in the chaotic
microwave rough billiard
Nazar Savytskyy, Oleh Hul and Leszek Sirko
Institute of Physics, Polish Academy of Sciences,
Aleja Lotnik´ow 32/46, 02-668 Warszawa, Poland
(Dated: August 26, 2004)
Abstract
We present the results of experimental study of nodal domains of wave functions (electric field
distributions) lying in the regime of Shnirelman ergodicity in the chaotic half-circular microwave
rough billiard. Nodal domains are regions where a wave function has a definite sign. The wave
functions ΨNof the rough billiard were measured up to the level number N= 435. In this way the
dependence of the number of nodal domains ℵNon the level number Nwas found. We show that
in the limit N→ ∞ a least squares fit of the experimental data reveals the asymptotic number of
nodal domains ℵN/N ≃0.058 ±0.006 that is close to the theoretical prediction ℵN/N ≃0.062.
We also found that the distributions of the areas sof nodal domains and their perimeters lhave
power behaviors ns∝s−τand nl∝l−τ′, where scaling exponents are equal to τ= 1.99 ±0.14
and τ′= 2.13 ±0.23, respectively. These results are in a good agreement with the predictions
of percolation theory. Finally, we demonstrate that for higher level numbers N≃220 −435 the
signed area distribution oscillates around the theoretical limit ΣA≃0.0386N−1.
1
In recent papers Blum et al. [1] and Bogomolny and Schmit [2] have considered the
distribution of the nodal domains of real wave functions Ψ(x, y) in 2D quantum systems
(billiards). The condition Ψ(x, y) = 0 determines a set of nodal lines which separate regions
(nodal domains) where a wave function Ψ(x, y) has opposite signs. Blum et al. [1] have
shown that the distributions of the number of nodal domains can be used to distinguish
between systems with integrable and chaotic underlying classical dynamics. In this way
they provided a new criterion of quantum chaos, which is not directly related to spectral
statistics. Bogomolny and Schmit [2] have shown that the distribution of nodal domains
for quantum wave functions of chaotic systems is universal. In order to prove it they have
proposed a very fruitful, percolationlike, model for description of properties of the nodal
domains of generic chaotic system. In particular, the model predicts that the distribution of
the areas sof nodal domains should have power behavior ns∝s−τ, where τ= 187/91 [3].
In this paper we present the first experimental investigation of nodal domains of wave
functions of the chaotic microwave rough billiard. We tested experimentally some of impor-
tant findings of papers by Blum et al. [1] and Bogomolny and Schmit [2] such as the signed
area distribution ΣAor the dependence of the number of nodal domains ℵNon the level
number N. Additionally, we checked the power dependence of nodal domain perimeters l,
nl∝l−τ′, where according to percolation theory the scaling exponent τ′= 15/7 [3], which
was not considered in the above papers.
In the experiment we used the thin (height h= 8 mm) aluminium cavity in the shape of
a rough half-circle (Fig. 1). The microwave cavity simulates the rough quantum billiard due
to the equivalence between the Schr¨odinger equation and the Helmholtz equation [4, 5]. This
equivalence remains valid for frequencies less than the cut-off frequency νc=c/2h≃18.7
GHz, where c is the speed of light. The cavity sidewalls are made of 2 segments. The rough
segment 1 is described by the radius function R(θ) = R0+PM
m=2 amsin(mθ +φm), where
the mean radius R0=20.0 cm, M= 20, amand φmare uniformly distributed on [0.269,0.297]
cm and [0,2π], respectively, and 0 ≤θ < π. It is worth noting that following our earlier
experience [6, 7] we decided to use a rough half-circular cavity instead of a rough circular
cavity because in this way we avoided nearly degenerate low-level eigenvalues, which could
not be possible distinguished in the measurements. As we will see below, a half-circular
geometry of the cavity was also very suitable in the accurate measurements of the electric
field distributions inside the billiard.
2
FIG. 1: Sketch of the chaotic half-circular microwave rough billiard in the xy plane. Dimensions
are given in cm. The cavity sidewalls are marked by 1 and 2 (see text). Squared wave functions
|ΨN(Rc, θ)|2were evaluated on a half-circle of fixed radius Rc= 17.5 cm. Billiard’s rough boundary
Γ is marked with the bold line.
The surface roughness of a billiard is characterized by the function k(θ) = (dR/dθ)/R0.
Thus for our billiard we have the angle average ˜
k= (hk2(θ)iθ)1/2≃0.488. In such a
billiard the dynamics is diffusive in orbital momentum due to collisions with the rough
boundary because ˜
kis much above the chaos border kc=M−5/2= 0.00056 [8]. The
roughness parameter ˜
kdetermines also other properties of the billiard [9]. The eigenstates
are localized for the level number N < Ne= 1/128˜
k4. Because of a large value of the
roughness parameter ˜
kthe localization border lies very low, Ne≃1. The border of Breit-
Wigner regime is NW=M2/48˜
k2≃35. It means that between Ne< N < NWWigner
ergodicity [9] ought to be observed and for N > NWShnirelman ergodicity should emerge.
In 1974 Shnirelman [10] proved that quantum states in chaotic billiards become ergodic for
sufficiently high level numbers. This means that for high level numbers wave functions have
to be uniformly spread out in the billiards. Frahm and Shepelyansky [9] showed that in the
rough billiards the transition from the exponentially localized states to the ergodic ones is
more complicated and can pass through an intermediate regime of Wigner ergodicity. In
this regime the wave functions are nonergodic and compose of rare strong peaks distributed
3
over the whole energy surface. In the regime of Shnirelman ergodicity the wave functions
should be distributed homogeneously on the energy surface.
In this paper we focus our attention on Shnirelman ergodicity regime.
One should mention that rough billiards and related systems are of considerable interest
elsewhere, e.g. in the context of dynamic localization [11], localization in discontinuous quan-
tum systems [12], microdisc lasers [13, 14] and ballistic electron transport in microstructures
[15].
In order to investigate properties of nodal domains knowledge of wave functions (elec-
tric field distributions inside the microwave billiard) is indispensable. To measure the wave
functions we used a new, very effective method described in [16]. It is based on the pertur-
bation technique and preparation of the “trial functions”. Below we will describe shortly
this method.
The wave functions ΨN(r, θ) (electric field distribution EN(r, θ) inside the cavity) can be
determined from the form of electric field EN(Rc, θ) evaluated on a half-circle of fixed radius
Rc(see Fig. 1). The first step in evaluation of EN(Rc, θ) is measurement of |EN(Rc, θ)|2.
The perturbation technique developed in [17] and used successfully in [17, 18, 19, 20] was
implemented for this purpose. In this method a small perturber is introduced inside the
cavity to alter its resonant frequency according to
ν−νN=νN(aB2
N−bE2
N),(1)
where νNis the Nth resonant frequency of the unperturbed cavity, aand bare geometrical
factors. Equation (1) shows that the formula can not be used to evaluate E2
Nuntil the term
containing magnetic field BNvanishes. To minimize the influence of BNon the frequency
shift ν−νNa small piece of a metallic pin (3.0 mm in length and 0.25 mm in diameter) was
used as a perturber. The perturber was moved by the stepper motor via the Kevlar line
hidden in the groove (0.4 mm wide, 1.0 mm deep) made in the cavity’s bottom wall along the
half-circle Rc. Using such a perturber we had no positive frequency shifts that would exceed
the uncertainty of frequency shift measurements (15 kHz). We checked that the presence
of the narrow groove in the bottom wall of the cavity caused only very small changes δνN
of the eigenfrequencies νNof the cavity |δνN|/νN≤10−4. Therefore, its influence into the
structure of the cavity’s wave functions was also negligible. A big advantage of using hidden
in the groove line was connected with the fact that the attached to the line perturber was
4
FIG. 2: Panel (a): Squared wave function |Ψ435(Rc, θ)|2(in arbitrary units) measured on a half-
circle with radius Rc= 17.5 cm (ν435 ≃14.44 GHz). Panel (b): The “trial wave function”
Ψ435(Rc, θ) (in arbitrary units) with the correctly assigned signs, which was used in the recon-
struction of the wave function Ψ435(r, θ) of the billiard (see Fig. 3).
always vertically positioned what is crucial in the measurements of the square of electric field
EN. To eliminate the variation of resonant frequency connected with the thermal expansion
of the aluminium cavity the temperature of the cavity was stabilized with the accuracy of
0.05 deg.
The regime of Shnirelman ergodicity for the experimental rough billiard is defined for N >
35. Using a field perturbation technique we measured squared wave functions |ΨN(Rc, θ)|2
for 156 modes within the region 80 ≤N≤435. The range of corresponding eigenfrequencies
was from ν80 ≃6.44 GHz to ν435 ≃14.44 GHz. The measurements were performed at 0.36
mm steps along a half-circle with fixed radius Rc= 17.5 cm. This step was small enough to
reveal in details the space structure of high-lying levels. In Fig. 2 (a) we show the example
of the squared wave function |ΨN(Rc, θ)|2evaluated for the level number N= 435. The
5
perturbation method used in our measurements allows us to extract information about the
wave function amplitude |ΨN(Rc, θ)|at any given point of the cavity but it doesn’t allow
to determine the sign of ΨN(Rc, θ) [21]. Our results presented in [16] suggest the following
sign-assignment strategy: We begin with the identification of all close to zero minima of
|ΨN(Rc, θ)|. Then the sign “minus” maybe arbitrarily assigned to the region between the
first and the second minimum, “plus” to the region between the second minimum and the
third one, the next “minus” to the next region between consecutive minima and so on.
In this way we construct our “trial wave function” ΨN(Rc, θ). If the assignment of the
signs is correct we should reconstruct the wave function ΨN(r, θ) inside the billiard with the
boundary condition ΨN(rΓ, θΓ) = 0.
The wave functions of a rough half-circular billiard may be expanded in terms of circular
waves (here only odd states in expansion are considered)
ΨN(r, θ) =
L
X
s=1
asCsJs(kNr) sin(sθ),(2)
where Cs= (π
2Rrmax
0|Js(kNr)|2rdr)−1/2and kN= 2πνN/c.
In Eq. (2) the number of basis functions is limited to L=kNrmax =lmax
N, where
rmax = 21.4 cm is the maximum radius of the cavity. lmax
N=kNrmax is a semiclassical
estimate for the maximum possible angular momentum for a given kN. Circular waves with
angular momentum s > L correspond to evanescent waves and can be neglected. Coefficients
asmay be extracted from the “trial wave function” ΨN(Rc, θ) via
as= [π
2CsJs(kNRc)]−1Zπ
0ΨN(Rc, θ) sin(sθ)dθ. (3)
Since our “trial wave function” ΨN(Rc, θ) is only defined on a half-circle of fixed radius Rc
and is not normalized we imposed normalization of the coefficients as:PL
s=1 |as|2= 1. Now,
the coefficients asand Eq. (2) can be used to reconstruct the wave function ΨN(r, θ) of the
billiard. Due to experimental uncertainties and the finite step size in the measurements of
|ΨN(Rc, θ)|2the wave functions ΨN(r, θ) are not exactly zero at the boundary Γ. As the
quantitative measure of the sign assignment quality we chose the integral γRΓ|ΨN(r, θ)|2dl
calculated along the billiard’s rough boundary Γ, where γis length of Γ. In Fig. 2 (b)
we show the “trial wave function” Ψ435(Rc, θ) with the correctly assigned signs, which was
used in the reconstruction of the wave function Ψ435(r, θ) of the billiard (see Fig. 3). Using
the method of the “trial wave function” we were able to reconstruct 138 experimental wave
6
FIG. 3: The reconstructed wave function Ψ435(r, θ) of the chaotic half-circular microwave rough
billiard. The amplitudes have been converted into a grey scale with white corresponding to large
positive and black corresponding to large negative values, respectively. Dimensions of the billiard
are given in cm.
functions of the rough half-circular billiard with the level number Nbetween 80 and 248
and 18 wave functions with Nbetween 250 and 435. The wave functions were reconstructed
on points of a square grid of side 4.3·10−4m. The remaining wave functions from the
range N= 80 −435 were not reconstructed because of the accidental near-degeneration
of the neighboring states or due to the problems with the measurements of |ΨN(Rc, θ)|2
along a half-circle coinciding for its significant part with one of the nodal lines of ΨN(r, θ).
These problems are getting much more severe for N > 250. Furthermore, the computation
time trrequired for reconstruction of the ”trial wave function” scales like tr∝2nz−2, where
nzis the number of identified zeros in the measured function |ΨN(Rc, θ)|. For higher N,
the computation time tron a standard personal computer with the processor AMD Athlon
XP 1800+ often exceeds several hours, what significantly slows down the reconstruction
procedure.
Ergodicity of the billiard’s wave functions can be checked by finding the structure of the
energy surface [8]. For this reason we extracted wave function amplitudes C(N)
nl =hn, l|Niin
the basis n, l of a half-circular billiard with radius rmax , where n= 1,2,3...enumerates the
7
FIG. 4: Structure of the energy surface in the regime of Shnirelman ergodicity. Here we show the
moduli of amplitudes |C(N)
nl |for the wave functions: (a) N= 86, (b) N= 435. The wave functions
are delocalized in the n, l basis. Full lines show the semiclassical estimation of the energy surface
(see text).
zeros of the Bessel functions and l= 1,2,3... is the angular quantum number. The moduli
of amplitudes |C(N)
nl |and their projections into the energy surface for the representative
experimental wave functions N= 86 and N= 435 are shown in Fig. 4. As expected, in the
regime of Shnirelman ergodicity the wave functions are extended homogeneously over the
whole energy surface [6]. The full lines on the projection planes in Fig. 4(a) and Fig. 4(b)
mark the energy surface of a half-circular billiard H(n, l) = EN=k2
Nestimated from the
semiclassical formula [7]: q(lmax
N)2−l2−larctan(l−1q(lmax
N)2−l2) + π/4 = πn. The peaks
|C(N)
nl |are spread almost perfectly along the lines marking the energy surface.
An additional confirmation of ergodic behavior of the measured wave functions can be also
8
FIG. 5: Amplitude distribution P(ΨA1/2) for the eigenstates: (a) N= 86 and (b) N= 435 con-
structed as histograms with bin equal to 0.2. The width of the distribution P(Ψ) was rescaled
to unity by multiplying normalized to unity wave function by the factor A1/2, where Ade-
notes billiard’s area. Full line shows standard normalized Gaussian prediction P0(ΨA1/2) =
(1/√2π)e−Ψ2A/2.
sought in the form of the amplitude distribution P(Ψ) [22, 23]. For irregular, chaotic states
the probability of finding the value Ψ at any point inside the billiard, without knowledge
of the surrounding values, should be distributed as a Gaussian, P(Ψ) ∼e−βΨ2. It is worth
noting that in the above case the spatial intensity should be distributed according to Porter-
Thomas statistics [5]. The amplitude distributions P(ΨA1/2) for the wave functions N= 86
and N= 435 are shown in Fig. 5. They were constructed as normalized to unity histograms
with the bin equal to 0.2. The width of the amplitude distributions P(Ψ) was rescaled to
unity by multiplying normalized to unity wave functions by the factor A1/2, where Adenotes
billiard’s area (see formula (23) in [22]). For all measured wave functions in the regime of
Shnirelman ergodicity there is a good agreement with the standard normalized Gaussian
9
FIG. 6: The number of nodal domains ℵN(full circles) for the chaotic half-circular microwave
rough billiard. Full line shows a least squares fit ℵN=a1N+b1√Nto the experimental data (see
text), where a1= 0.058 ±0.006, b1= 1.075 ±0.088. The prediction of the theory of Bogomolny
and Schmit [2] a1= 0.062.
prediction P0(ΨA1/2) = (1/√2π)e−Ψ2A/2.
The number of nodal domains ℵNvs. the level number Nin the chaotic microwave rough
billiard is plotted in Fig. 6. The full line in Fig. 6 shows a least squares fit ℵN=a1N+b1√N
of the experimental data, where a1= 0.058 ±0.006, b1= 1.075 ±0.088. The coefficient
a1= 0.058 ±0.006 coincides with the prediction of the percolation model of Bogomolny and
Schmit [2] ℵN/N ≃0.062 within the error limits. The second term in a least squares fit
corresponds to a contribution of boundary domains, i.e. domains, which include the billiard
boundary. Numerical calculations of Blum et al. [1] performed for the Sinai and stadium
billiards showed that the number of boundary domains scales as the number of the boundary
intersections, that is as √N. Our results clearly suggest that in the rough billiard, at low
level number N, the boundary domains also significantly influence the scaling of the number
of nodal domains ℵN, leading to the departure from the predicted scaling ℵN∼N.
The bond percolation model [2] at the critical point pc= 1/2 allows us to apply other
results of percolation theory to the description of nodal domains of chaotic billiards. In par-
ticular, percolation theory predicts that the distributions of the areas sand the perimeters
10
FIG. 7: Distribution of nodal domain areas. Full line shows the prediction of percolation theory
log10(hns/ni) = −187
91 log10(hs/smin i). A least squares fit log10(hns/ni) = a2−τlog10(hs/smini) of
the experimental results lying within the vertical lines yields the scaling exponent τ= 1.99 ±0.14
and a2=−0.05 ±0.04. The result of the fit is shown by the dashed line.
lof nodal clusters should obey the scaling behaviors: ns∝s−τand nl∝l−τ′, respec-
tively. The scaling exponents [3] are found to be τ= 187/91 and τ′= 15/7. In Fig. 7
we present in logarithmic scales nodal domain areas distribution hns/nivs. hs/sminiob-
tained for the microwave rough billiard. The distribution hns/niwas constructed as nor-
malized to unity histogram with the bin equal to 1. The areas sof nodal domains were
calculated by summing up the areas of the nearest neighboring grid sites having the same
sign of the wave function. In Fig. 7 the vertical axis hns/ni=1
NTPNT
i=1 n(N)
s/n(N)repre-
sents the number of nodal domains n(N)
sof size sdivided by the total number of domains
n(N)averaged over NT= 18 wave functions measured in the range 250 ≤N≤435. In
these calculations we used only the highest measured wave functions in order to minimize
the influence of boundary domains on nodal domain areas distribution. Following Bogo-
molny and Schmit [2], the horizontal axis is expressed in the units of the smallest possi-
ble area s(N)
min,hs/smini=1
NTPNT
i=1 s/s(N)
min, where s(N)
min =π(j01/kN)2and j01 ≃2.4048 is
the first zero of the Bessel function J0(j01) = 0. The full line in Fig. 7 shows the pre-
diction of percolation theory log10(hns/ni) = −187
91 log10(hs/smin i). In a broad range of
11
FIG. 8: Distribution of nodal domain perimeters. Full line shows the prediction of percolation
theory log10(hnl/ni) = −15
7log10(hl/lmini). A least squares fit log10(hnl/ni) = a3−τ′log10 (hl/lmin i)
of the experimental results lying within the range marked by the vertical lines yields τ′= 2.13±0.23
and a3= 0.04 ±0.21. The result of the fit is shown by the dashed line.
log10(hs/smin i), approximately from 0.2 to 1.3, which is marked by the two vertical lines
the experimental results follow closely the theoretical prediction. Indeed, a least squares fit
log10(hns/ni) = a2−τlog10(hs/smini) of the experimental results lying within the vertical
lines yields the scaling exponent τ= 1.99 ±0.14 and a2=−0.05 ±0.04, which is in a good
agreement with the predicted τ= 187/91 ≃2.05. The dashed line in Fig. 7 shows the
results of the fit. In the vicinity of log10(hs/smini)≃1 and 1.2 small excesses of large areas
are visible. A similar situation, but for larger log10(s/smin)>4, can be also observed in the
nodal domain areas distribution presented in Fig. 5 in Ref. [2] for the random wave model.
The exact cause of this behavior is not known but we can possible link it with the limited
number of wave functions used for the preparation of the distribution.
Nodal domain perimeters distribution hnl/nivs. hl/lminiis shown in logarithmic scales
in Fig. 8. The distribution hnl/niwas constructed as normalized to unity histogram
with the bin equal to 1 . The perimeters of nodal domains lwere calculated by identi-
fying the continues paths of grid sites at the domains boundaries. The averaged values
hnl/niand hl/lminiare defined similarly as previously defined hns/niand hs/smini, e.g.
12
FIG. 9: The normalized signed area distribution NΣAfor the chaotic half-circular microwave rough
billiard. Full line shows predicted by the theory asymptotic limit NΣA≃0.0386, Blum et al. [1].
hl/lmini=1
NTPNT
i=1 l/l(N)
min, where l(N)
min = 2πqs(N)
min/π = 2π(j01/kN) is the perimeter of the
circle with the smallest possible area s(N)
min. The full line in Fig. 8 shows the prediction
of percolation theory log10 (hnl/ni) = −15
7log10(hl/lmini). Also in this case the agreement
between the experimental results and the theory is good what is well seen in the range
0.2<log10(hl/lmini)<1.2, which is marked by the two vertical lines. A least squares fit
log10(hnl/ni) = a3−τ′log10 (hl/lmini) of the experimental results lying within the marked
range yields τ′= 2.13 ±0.23 and a3= 0.04 ±0.21. The result of the fit is shown in Fig. 8
by the dashed line. As we see the scaling exponent τ′= 2.13 ±0.23 is close to the exponent
predicted by percolation theory τ′= 15/7≃2.14. The above results clearly demonstrate
that percolation theory is very useful in description of the properties of wave functions of
chaotic billiards.
Another important characteristic of the chaotic billiard is the signed area distribution
ΣAintroduced by Blum et al. [1]. The signed area distribution is defined as a variance:
ΣA=h(A+−A−)2i/A2, where A±is the total area where the wave function is positive
(negative) and Ais the billiard area. It is predicted [1] that the signed area distribution
should converge in the asymptotic limit to ΣA≃0.0386N−1. In Fig. 9 the normalized
signed area distribution NΣAis shown for the microwave rough billiard. For lower states
13
80 ≤N≤250 the points in Fig. 9 were obtained by averaging over 20 consecutive eigenstates
while for higher states N > 250 the averaging over 5 consecutive eigenstates was applied. For
low level numbers N < 220 the normalized distribution NΣAis much above the predicted
asymptotic limit, however, for 220 < N ≤435 it more closely approaches the asymptotic
limit. This provides the evidence that the signed area distribution ΣAcan be used as a
useful criterion of quantum chaos. A slow convergence of NΣAat low level numbers N
was also observed for the Sinai and stadium billiards [1]. In the case of the Sinai billiard
this phenomenon was attributed to the presence of corners with sharp angles. According
to Blum et al. [1] the effect of corners on the wave functions is mainly accentuated at low
energies. The half-circular microwave rough billiard also possesses two sharp corners and
they can be responsible for a similar behavior.
In summary, we measured the wave functions of the chaotic rough microwave billiard
up to the level number N= 435. Following the results of percolationlike model proposed
by [2] we confirmed that the distributions of the areas sand the perimeters lof nodal
domains have power behaviors ns∝s−τand nl∝l−τ′, where scaling exponents are equal
to τ= 1.99 ±0.14 and τ′= 2.13 ±0.23, respectively. These results are in a good agreement
with the predictions of percolation theory [3], which predicts τ= 187/91 ≃2.05 and τ′=
15/7≃2.14, respectively. We also showed that in the limit N→ ∞ a least squares fit of the
experimental data yields the asymptotic number of nodal domains ℵN/N ≃0.058 ±0.006
that is close to the theoretical prediction ℵN/N ≃0.062 [2]. Finally, we found out that
the signed area distribution ΣAapproaches for high level number Ntheoretically predicted
asymptotic limit 0.0386N−1[1].
Acknowledgments. This work was partially supported by KBN grant No. 2 P03B 047
24. We would like to thank Szymon Bauch for valuable discussions.
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