Quantum Chaos and 1=f Noise
A. Relan
˜
o, J. M. G. Go
´
mez, R. A. Molina, and J. Retamosa
Departamento de Fı
´
sica Ato
´
mica, Molecular y Nuclear, Universidad Complutense de Madrid, E-28040 Madrid, Spain
E. Faleiro
Departamento de Fı
´
sica Aplicada, E.U.I.T. Industrial, Universidad Polite
´
cnica de Madrid, E-28012 Madrid, Spain
(Received 25 February 2002; published 22 November 2002)
It is shown that the energy spectrum fluctuations of quantum systems can be formally considered as a
discrete time series. The power spectrum behavior of such a signal for different systems suggests the
following conjecture: The energy spectra of chaotic quantum systems are characterized by 1=f noise.
DOI: 10.1103/PhysRevLett.89.244102 PACS numbers: 05.45.Mt, 05.40.–a, 05.45.Pq, 05.45.Tp
The understanding of quantum chaos has greatly ad-
vanced during the past two decades. It is well known that
there is a clear relationship between the energy level
fluctuation properties of a quantum system and the large
time scale behavior of its classical analogue. The pioneer-
ing work of Berry and Tabor [1] showed that the spectral
fluctuations of a quantum system whose classical ana-
logue is fully integrable are well described by Poisson
statistics; i.e., the successive energy levels are not corre-
lated. In a seminal paper, Bohigas et al. [2] conjectured
that the fluctuation properties of generic quantum sys-
tems, which in the classical limit are fully chaotic, co-
incide with those of random matrix theory (RMT). This
conjecture is strongly supported by experimental data,
many numerical calculations, and analytical work based
on semiclassical arguments. A review of later develop-
ments can be found in [3,4].
We propose in this Letter a different approach to quan-
tum chaos based on traditional methods of time series
analysis. The essential feature of chaotic energy spectra in
quantum systems is the existence of level repulsion and
correlations. To study these correlations, we can consider
the energy spectrum as a discrete signal, and the sequence
of energy levels as a time series. For example, the se-
quence of nearest level spacings has similarities with the
diffusion process of a particle. But generally we do not
need to specify the nature of the analogue time series. As
we shall see, examination of the power spectrum of
energy level fluctuations reveals very accurate power
laws for completely regular or completely chaotic
Hamiltonian quantum systems. It turns out that chaotic
systems have 1=f noise, in contrast to the Brown noise of
regular systems.
The first step, previous to any statistical analysis of the
spectral fluctuations, is the unfolding of the energy spec-
trum. Level fluctuation amplitudes are modulated by the
value of the mean level density E, and therefore, to
compare the fluctuations of different systems, the level
density smooth behavior must be removed. The unfolding
consists in locally mapping the real spectrum into an-
other with mean level density equal to one. The actual
energy levels E
i
are mapped into new dimensionless
levels
i
,
E
i
!
i
NNE
i
;i 1; ...;N; (1)
where N is the dimension of the spectrum and
NNE is
given by
NNE
Z
E
1
dE
0
E
0
: (2)
This function is a smooth approximation to the step
function NE that gives the true number of levels up to
energy E. The form of the function E can be deter-
mined by a best fit of
NNE to NE.
The nearest neighbor spacing sequence is defined by
s
i
i1
i
;i 1; ...;N 1: (3)
For the unfolded levels, the mean level density is equal to
1andhsi1. In practical cases, the unfolding procedure
can be a difficult task for systems where there is no
analytical expression for the mean level density [5].
Generally, two suitable statistics are used to study the
fluctuation properties of the unfolded spectrum. The
nearest neighbor spacing distribution Ps gives informa-
tion on the short range correlations among the energy
levels. The
3
L statistic makes it possible to study
correlations of length L: As we change the L value, we
obtain information on the level correlations at all scales.
By contrast, in this paper we characterize the spectral
fluctuations by the statistic
n
[6] defined by
n
X
n
i1
s
i
hsi
X
n
i1
w
i
; (4)
where the index n runs from 1 to N 1. The quantity w
i
gives the fluctuation of the ith spacing from its mean
value hsi1. The function h
2
n
i is closely related to the
covariance matrix and thus provides important informa-
tion on level correlations. Recently, it has been shown [7]
that, under certain assumptions, h
2
n
i is a logarithmic
function of n for the RMT ensembles.
VOLUME 89, NUMBER 24 PHYSICAL REVIEW LETTERS 9DECEMBER 2002
244102-1 0031-9007=02=89(24)=244102(4)$20.00 2002 The American Physical Society 244102-1
The subject of the present work is the function
n
,
instead of h
2
n
i. It represents the deviation of the unfolded
excitation energy from its mean value n. From our point
of view, the function
n
has a formal similarity with a
time series. For example, we may compare the energy
level spectrum with the diffusion process of a particle.
The analogy is clear if the index i of the nearest level
spacings is considered as a discrete time, and the spacing
fluctuation w
i
as the analogue of the particle displace-
ment d
i
from the collision at time i to the next collision.
Certainly, there are some differences. For instance, there
is no limitation for a single particle displacement d
i
,
whereas w
i
> 1 for a nearest level spacing sequence
because there cannot exist negative spacings. We also note
that the amplitude and sign of the displacements are given
by a certain probability function that may be very com-
plex and depend on the previous particle trajectory, i.e.,
on the values d
j
, j<i. On the other hand, in a spectrum
the position of each level, and consequently w
i
, depends
not only on the lower energy levels, but also on those with
higher energy. Considered as a time series, there is a
strong dependence on the past as well as on the future
history. Nevertheless, in spite of these peculiarities, the
analogy exists, and the function
n
is the analogue of the
particle total displacement at time n.
Our aim is to study the
n
signal of chaotic quantum
systems. We can analyze their spectral statistics with
numerical techniques normally used in the study of com-
plex systems and try to relate the emerging properties
with some universal features that appear in many other
branches of physics. One of those techniques is the cal-
culation of the power spectrum Sk of a discrete and
finite series
n
given by
Skj
^
k
j
2
; (5)
where
^
k
is the Fourier transform of
n
,
^
k
1
N
p
X
n
n
exp
2ikn
N
; (6)
and N is the size of the series.
As an example of a very chaotic system, we take the
atomic nucleus at high excitation energy, where the level
density is very large. To obtain the energy spectrum,
shell-model calculations for selected nuclei are per-
formed, using realistic interactions that reproduce well
experimental data of nuclei in a mass region. The
Hamiltonian matrices for different angular momenta,
parity, and isospin are fully diagonalized, and careful
global unfolding is performed. Then, sets of 256 consecu-
tive levels of the same J
T, from the high level density
region, are selected. To characterize the statistical proper-
ties of the
n
signal, we calculate an ensemble average of
its power spectrum, in order to reduce statistical fluctua-
tions and clarify its main trend. The average hSki is
calculated with 25 sets.
Figure 1 shows the results for a typical stable sd shell
nucleus,
24
Mg, with matrix dimensionalities up to about
2000; and for a very exotic nucleus,
34
Na, with dimen-
sions up to about 5000, in the sd proton and pf neutron
shells. Clearly, the power spectrum of
n
follows closely a
power law. We may assume the simple functional form
hSki
1
k
: (7)
A least squares fit to the data of Fig. 1 gives 1:11
0:03 for
34
Na,and 1:06 0:05 for
24
Mg. These re-
sults raise the question of whether there is a general
relationship between quantum chaos and the power spec-
trum of the
n
fluctuations of the system.
Probably, the simplest and most reliable way to clarify
this issue is to compare
n
and hSki for Poisson energy
levels and random matrix spectra. Random matrix theory
plays a predominant role in the description of chaotic
quantum systems [3,4]. It deals with three basic
Hamiltonian matrix ensembles: the Gaussian orthogonal
ensemble (GOE) of N-dimensional matrices, applicable
for time-reversal invariant chaotic systems with rota-
tional symmetry or integer spin and broken rotational
symmetry; the Gaussian unitary ensemble (GUE), appli-
cable for chaotic systems in which the time-reversal in-
variance is violated; and the Gaussian symplectic
ensemble (GSE), applicable for time-reversal invariant
chaotic systems with half-integer spin and broken rota-
tional symmetry. There are, of course, other more com-
plex ensembles such as deformed ensembles, band matrix
ensembles, etc., but they will not be considered in this
work. Instead, we include an ensemble of diagonal ma-
trices whose elements are random Gaussian variables. We
call it the Gaussian diagonal ensemble (GDE).
Figure 2 shows the signal
n
for a GDE (Poisson) and
a GOE spectrum of dimension 1000. Clearly, the two
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5
log 〈S(k)〉
log k
34
Na
24
Mg
FIG. 1. Average power spectrum of the
n
function for
24
Mg
and
34
Na, using 25 sets of 256 levels from the high level density
region. The plots are displaced to avoid overlapping.
VOLUME 89, NUMBER 24 PHYSICAL REVIEW LETTERS 9DECEMBER 2002
244102-2 244102-2
signals are very different. It is also illustrative to compare
those signals with a discrete time series xt, with 1=k and
1=k
2
power laws, generated with the random-phase ap-
proximation procedure used in [8]. The similarity of the
2 time series with the Poisson spectrum and the
1 time series with the GOE spectrum is obvious.
To compute the average hSki, we generate 30 different
matrices of dimension 1000 for each type of random
matrix ensemble. Figure 3 shows the results of these
calculations in a decimal log-log scale. In all the cases,
the main trend is essentially linear, except for very high
frequencies, where some deviation is observed, probably
due to finite size effects.
Ignoring frequencies greater than logk 2:2, the fit to
(7) gives
GDE
1:99 with an uncertainty near 2%.The
spectrum of any matrix pertaining to GDE consists of N
uncorrelated levels. This is due to the diagonal character
of the matrix and to the fact that its matrix elements are
independent random variables. Consequently, the nearest
level spacings are also uncorrelated, and
n
is just a sum
of N 1 independent random variables. The power spec-
trum of such a signal is well known to present 1=k
2
behavior, and that is in full agreement with our numerical
value for in the Poisson spectrum. Furthermore, Berry
and Tabor [1] showed that, in a semiclassical integrable
system, the spacings s
i
are random independent variables
for i 1. As a consequence, their
n
power spectrum
behaves as 1=k
2
. However, this behavior may be modified
by the levels of the ground state region.
By contrast, the spectrum of any GOE member of large
dimension is generally considered the paradigm of cha-
otic quantum spectra. It presents level correlations at all
scales. The same applies to GUE and GSE, in increasing
order of level repulsion. As is well known, the nearest
neighbor spacing distribution for these three ensembles
behaves as Pss
for small s, where is known as the
level repulsion parameter. For our diagonal ensemble with
Poisson statistics, 0,while 1, 2, and 4 for GOE,
GUE, and GSE, respectively [3,4].
The power spectrum of
n
for these three ensembles is
displayed in Fig. 3. The fit of hSki to the power law (7) is
excellent. For the exponents, we obtain
GOE
1:08,
GUE
1:02,and
GSE
1:00. In all the cases, the error
of the linear regression is about 2%. The three ensembles
yield the same power law, with ’ 1. This is in agree-
ment with preliminary results [9] showing that a loga-
rithmic behavior of h
2
n
i, as for the GOE [7], is related to a
1=k
behavior of the
n
power spectrum, with 1.
The small deviations from 1 observed in our numeri-
cal results are probably simple finite size effects of the
matrix dimensions, which introduce additional inaccura-
cies through statistical fluctuations and the unfolding
procedure. As Table I shows,
GOE
approaches 1 as the
matrix dimensionality increases.
Clearly, the power spectrum hSki behaves as 1=k
both in regular and chaotic energy spectra, but level
correlations decrease the exponent from the 2 limit
for uncorrelated spectra to apparently a minimum value
1 for chaotic quantum systems.
The concept of quantum chaos has no precise definition
as yet. Quantum systems with classical analogues are
considered chaotic when their classical analogues are
chaotic. Quantum systems without classical analogues
may be called chaotic if they show the same kind of
fluctuations as chaotic quantum systems with classical
analogues. In practice, the Bohigas-Gianoni-Schmit con-
jecture is generally used as a criterion.
-10
0
10
20
30
40
0 200 400 600 800 1000
x(t)
t
-30
-20
-10
0
10
20
0 200 400
600 800 1000
δ
n
n
FIG. 2. Comparison of the
n
function for Poisson (dashed
line) and GOE spectra (solid line), with a standard time series
xt with 1=k
power spectrum, for 2 (dashed line) and
1 (solid line).
-4
-3
-2
-1
0
1
2
3
4
5
0 0.5 1 1.5 2 2.5 3
log 〈S(k)〉
log k
GDE
GOE
GUE
GSE
FIG. 3. Power spectrum of the
n
function for GDE (Poisson)
energy levels, compared to GOE, GUE, and GSE. The plots are
displaced to avoid overlapping.
TABLE I. Dependence of the power spectrum exponent
GOE
on the ensemble matrix dimensionality N.
N 128 512 2048
GOE
1.10 1.09 1.06
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244102-3 244102-3
The results obtained above for the power spectrum of
the
n
statistic suggest the following conjecture.
The energy spectra of chaotic quantum systems are
characterized by 1=f noise.
This conjecture has several appealing features. It is a
property characterizing the chaotic spectrum by itself,
without any reference to the properties of other systems
such as GOE. It is universal for all kinds of chaotic
quantum systems, either time-reversal invariant or not,
either of integer or half-integer spin. Furthermore, the
1=f noise characterization of quantum chaos includes
these physical systems into a widely spread kind of sys-
tems appearing in many fields of science, which display
1=f fluctuations. Thus, the energy spectrum of chaotic
quantum systems exhibits the same kind of fluctuations
as many other complex systems. However, there is no
indication that 1=f spectral fluctuations in a quantum
system imply 1=f noise in its classical analogue.
Neither have we found any relationship with 1=f noise
in classical chaotic phenomena such as intermittency [10].
Chaotic quantum energy spectra are characterized by
strong level repulsion and strong spectral rigidity. In
terms of the conventional
3
L statistic, strong rigidity
corresponds to small values of
3
L and slow, logarith-
mic dependence on L. We can try to interpret what
spectral rigidity means in terms of the
n
function.
Rigidity of the energy spectrum means that the devia-
tions of the energy spacings s
i
from their mean value
hsi1 are generally small, and that the spectrum is
organized in such a way that a deviation of a spacing
from the mean tends to be balanced by neighboring spac-
ings. Therefore it is unlikely to find a long series of
consecutive spacings all above or below the mean spacing.
In a time series, antipersistence means that an increas-
ing or decreasing trend in the past makes the opposite
trend in the future probable. In the present approach,
where a quantum energy spectrum is considered as a
time series, the spectral rigidity is analogous to antiper-
sistence. We have seen that a rigid energy spectrum gives
rise to a
n
power spectrum of 1=k
type with 1.On
the other hand, as is well known, a time series with a 1=k
power spectrum where ’ 1 is very antipersistent.
Therefore, the interpretation of spectral rigidity as the
analogue of antipersistence is consistent with the behav-
ior of the power spectrum.
In summary, we have seen that for quantum systems
the
n
function can be considered as a time series, where
the level order index n plays the role of a discrete time.
The power spectrum hSki of
n
has been studied for
representative energy spectra of regular and chaotic quan-
tum systems. Neat power laws hSki 1=k
have been
found in all cases. For Poisson spectra, we get 2,as
expected for independent random variables. For spectra
of atomic nuclei at higher energies, in regions of high
level density, and for the GOE, GUE, and GSE ensembles,
we obtain 1.
These results suggest the conjecture that chaotic quan-
tum systems are characterized by 1=f noise in the energy
spectrum fluctuations. This property is not a mere statis-
tic to measure the chaoticity of the system. It provides an
intrinsic characterization of quantum chaotic systems
without any reference to the properties of RMT en-
sembles. As is well known, 1=f noise is quite ubiquitous.
It characterizes sunspot activity, the flow of the Nile river,
music, and chronic illness [11]. And we believe that it
characterizes quantum chaos as well.
This work is supported in part by Spanish Government
Grants No. BFM2000-0600 and No. FTN2000-0963-
C02.
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