Journal of the Physical Society of Japan
Vol. 71, No. 10, October, 2002, pp. 2427–2433#2002 The Physical Society of Japan

Frequency Dependence of Quantum Localization in a Periodically Driven System
Manabu M ACHIDA!, Keiji SAITO and Seiji M IYASHITA
Department of Applied Physics, School of Engineering, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656(Received June 5, 2002)

We study the quantum localization phenomena for a random matrix model belonging to the Gaussianorthogonal ensemble (GOE). An oscillating external field is applied on the system. After the transient
time evolution, energy is saturated to various values depending on the frequencies. We investigate thefrequency dependence of the saturated energy. This dependence cannot be explained by a naive pictureof successive independent Landau–Zener transitions at avoided level crossing points. The effect ofquantum interference is essential. We define the number of Floquet states which have large overlap with
the initial state, and calculate its frequency dependence. The number of Floquet states shows
approximately linear dependence on the frequency, when the frequency is small. Comparing the
localization length in Floquet states and that in energy states from the viewpoint of the Anderson
localization, we conclude that the Landau–Zener picture works for the local transition processes between
levels.
KEYWORDS: quantum dynamics, quantum localization, random matrix, quantum chaos
DOI: 10.1143/JPSJ.71.2427

1.

Introduction

The diffusive phenomena have been actively studied inquantum systems with a complex level structure in thepresence of a periodic external field. Under a periodic
perturbation, when a system shows so-called quantum
chaotic nature, it is widely believed that energy diffusionof the system eventually ceases and the energy saturates at a
finite value. So far, only limited number of these systems
have been actually studied. The kicked rotator model1) andthe kicked top model2) are typical quantum models whichshow the quantum chaotic nature under a periodic external
field. The difference between classical and quantum energy
diffusion has been discussed by investigating these models.
In the quantum kicked rotator model, starting initially fromthe ground state, the system evolves diffusively only for a
finite time. After then the diffusive time-evolution ceases
and the average energy is saturated to a finite value. That is,
the system cannot absorb energy after that time. This is aremarkable difference from the classical diffusion, where theenergy increases unlimitedly. The mechanism of thisquantum localization in the model was discussed in thecontext of the Anderson localization.3,4) Recently the kickedrotator model is experimentally realized using Hydrogen andSodium atoms, and the quantum localization is observed.5–9)
It is known that random matrices well describe characteristics of the level statistics of the systems which show
chaotic nature in the corresponding classical models.10,11)
Therefore we study the quantum localization process ofrandom matrices to extract universal features. In this paper,
we investigate the time evolution of a system whose
Hamiltonian is taken from the Gaussian orthogonal ensemble (GOE) with a periodic perturbation, and try tocapture the universal features on the relation between thesaturated energy and the frequency of the perturbation. So
far, several types of time-dependent random matrices are
adopted to study the quantum diffusion process.12–15)!

E-mail: machida@spin.t.u-tokyo.ac.jp

Wilkinson calculated the energy diffusion constant using
the Landau–Zener transition formula16–18) when the perturbation acts on the system linearly in time.13) He clarified thedifference of diffusion constants between GOE and theGaussian unitary ensemble (GUE). Wilkinson neglected theeffect of interference of quantum phase, which plays acrucial role in the quantum localization in the presence of
periodic perturbation.14) Cohen and Kottos considered theenergy diffusion before the saturation in the Wigner’s
banded random matrix (WBRM) model with an oscillating
perturbation, and calculated the diffusion coefficient as afunction of the amplitude and frequency of the perturbation.12) Taking these previous studies into account, we study
how the Landau–Zener picture is related to the quantum
localization by investigating the dependence of the saturated
energy on the frequency of the perturbation.
We first numerically demonstrate that the quantum
localization occurs in the system, which cannot be explainedby a naive application of an independent Landau–Zener
picture to the system. In order to understand the quantum
localization intuitively, we consider the overlap between
Floquet eigenstates and the initial ground state, and find thatthe overlap exponentially decays in the Hilbert spacespanned by Flouqet eigenstates, which directly means thatthe ground state is localized in this representation. We define
the relevant number of Floquet eigenstates which has a large
overlap with the initial ground state. This number was
originally introduced as a quantity which corresponds to the
Lyapunov exponent in the quantum kicked top model by
Haake et al.2) The dependence of the relevant number on thefrequency is numerically investigated. We numerically findthat this number linearly depends on the frequency of theperturbation in small frequency regime. Employing aplausible proportional relation between the localization
length in the energy space and that in the Floquet space,which is also adopted in the case of the kicked rotator model,
and taking account of this linear dependence, we find theLandau–Zener mechanism works in the view of theAnderson localization.

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J. Phys. Soc. Jpn., Vol. 71, No. 10, October, 2002

M. MACHIDA et al.

Fig. 1. Energy spectrum of H for N ¼ 500 as a function of ! around (a) the ground state and (b) the center.

This paper is organized in the following way. In §2, we
explain the random matrix model and the numerical methodof time evolution. In §3, we study the frequency dependenceof the saturated average energy. In §4, we discuss thequantum localization by introducing the number of relevant
Floquet states. Finally in §5, we give summary anddiscussion.

crossings are seen in Fig. 1. There are no degeneracies oflevels although some energy levels are seen to be crossing
due to the line width.
Now we consider the time evolution of a state j i of thesystem;@ð3Þ
i j i ¼ HðtÞj i:@t2.

Here we take h! ¼ 1 and the initial state j 0 i is taken to be
the ground state. The state j 1 i after a period T ¼ 2$=!
from t ¼ 0 is expressed using the Floquet operator20) F;" Z 2$=!
#
ð4Þ
j 1 i ¼ Fj 0 i; F ' T exp &iHðtÞdt ;Model and Method

We consider time-reversal invariant systems in an
oscillating external field. The random matrix ensemble
appropriate for describing the spectral statistics of thesesystems is the Gaussian orthogonal ensemble (GOE).
Random matrices well describe characteristics of energy
spectra of complex quantum systems which have noconserved quantities.
We shall consider the total Hamiltonian given byHðtÞ ¼ H0 þ !ðtÞV;ð1Þ

where H0 denotes the non-perturbed part of the Hamiltonian,
and !ðtÞV is the perturbation part with a time-dependent
parameter !ðtÞ. We take H0 and V from the ensemble of theGOE random matrices with the dimension N. This model
corresponds to the realistic systems such as billiard systems
with a time-dependent boundary, or complex spin systems
with a time-dependent external field. Matrix elements H0ij
and Vij are taken from independent Gaussian randomvariables with mean zero and with the variance: varðH0ij Þ ¼
1 þ "ij and varðVij Þ ¼ 1 þ "ij . Note that both of the nonperturbed and perturbed terms have the same statisticalproperties. The density of states #ðEÞ at the energy E is given
by Wigner’s semicircle law for large N,19)
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffiffi
ð2Þ
4N & E :#ðEÞ ¼
2$N

In this paper, we confine !ðtÞ to the sinusoidal form,!ðtÞ ¼ A sinð!tÞ with A ¼ 0:5. Eigenvalues of a GOE
random matrix show a structure with level repulsion as afunction of !. Energy spectrum of a system of N ¼ 500
around the ground state, and around the center (E ¼ 0) aredepicted in Figs. 1(a) and 1(b), respectively. Many avoided

0where T means the time ordered product. Therefore the state
after the nth period j n i is written asnni ¼ F j

j

0 i:

ð5Þ

Floquet eigenvectors fj%ig form a complete set in the Hilbert
space, and Floquet eigenvalues are aligned on a unit circle ofthe complex plane;
Fj%i ¼ ei&% j%i:

ð6Þ

We calculate F numerically by integrating the Schrödinger
equation (3) for a period using the fourth order decomposition of time-evolution operator.21) The time evolution ofenergy is expressed in terms of Floquet eigenvalues andeigenstates. The energy after the nth period is expressedusing the Floquet operator;
h

n jHj

ni ¼ h¼
3.

y nn
0 jðF Þ HF j

Xeinð&% &&%0 Þ

%0 ;%

h

0i

0
0 j% ih%j

00 ih% jHj%i:

Esat as a Function of !

3.1 Numerical results
We define gð%; %0 Þ and Esat as,
gð%; %0 Þ ' h

00
0 j%ih%jH0 j% ih% j

0 i;

ð7Þ

J. Phys. Soc. Jpn., Vol. 71, No. 10, October, 2002

Esat '

X

gð%; %Þ;%and rewrite eq. (7) as,X
h n jH0 j n i ¼
ei nð&% &&% Þ gð%; %0 Þ

M. M ACHIDA et al.

ð8Þ

0

%0 ;%

¼ Esat þ

X$
%<%0

%
ei nð&% &&% Þ gð%; %0 Þ þ c.c. :0ð9ÞIn the second term of the right-hand side of eq. (9), theexpectation values of terms for f% 6¼ %0 g oscillates with theperiod 2$=ð&%0 & &% Þ. Thus, we see h n jH0 j n i fluctuatesaround Esat . Therefore, Esat is regarded as the saturated valueof the energy.
Figure 2 shows the time evolution of h n jH0 j n i (zigzag
curve), and the corresponding Esat (horizontal line) withrespect to !=$ ¼ 0:02, 0.1, 0.2, 0.4, and 1.0 for N ¼ 500.
Figure 3 shows Esat as a function of ! obtained from fivesamples for N ¼ 256, 500, and 700. We should note that the
maximal saturated energy is zero because the energy
spectrum is symmetric about zero energy (E ¼ 0), and thismaximal energy corresponds to the high temperature limit.
Here we normalize Esat such that the distance between E ¼ 0
and the ground state is 1. Indeed in classical systems, theenergy diffusion continues forever, which means that Esat
converges to 0.
On the other hand, when ! is very small (! ¼ 0:02$), the2429

time evolution of the system is almost adiabatic, and theaverage energy changes little.22) When ! becomes slightly
larger (! ¼ 0:1$ and 0.2$), transitions between levels begin
to occur at avoided crossings (that is, nonadiabatic
transitions). Thus the system absorbs energy and EðtÞincreases. However EðtÞ saturates before it reaches to 0.
The value of Esat gradually approaches to the center of theenergy spectrum (Esat ¼ 0) as ! grows larger (! ¼ 0:4$ and1.0$). These finite saturations of Esat are caused by theFig. 2. Zigzag curves represent the time evolution of the average energy
EðtÞ. Horizontal lines represent Esat . A pair of EðtÞ and Esat denotes, fromthe bottom, data for ! ¼ 0:02$, 0:1$, 0:2$, 0:4$, and 1:0$.

Fig. 3. Esat as a function of ! for (a) N ¼ 256, (b) N ¼ 500, and (c) N ¼ 700. Errorbars show the variance among the data obtained
from five samples.

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J. Phys. Soc. Jpn., Vol. 71, No. 10, October, 2002

M. MACHIDA et al.

quantum effects, and this feature is called the quantum
localization.1,10) The quantum localization was first discovered in the kicked rotator model, which shows chaotic
motion in the classical limit.1)
3.2

Failure of a picture of independent Landau–Zenertransitions
As shown in the previous section, the energy is saturated
to a finite value. We here show that this saturation cannot berealized without considering the quantum interference effect.
Let us consider the small frequency regime where transitions
between energy levels take place only between the twolevels at an avoided level crossing point. In such a region,
we can introduce the well-known Landau–Zener picture foreach transition. Wilkinson proposed a theory of theevolution of the energy in a random matrix model with atime-dependent perturbation.13) His theory assumes thattransitions take place only at avoided crossings and thetransition probability is determined by the Landau–Zener
formula.17) This implies that the sweeping speed of theparameter is sufficiently slow, and that multiple level
scatterings are not relevant. In the case that the field
increases linearly in time, he found that this approximationof independent Landau–Zener scattering gives a goodresult.14)
If we apply a simple independent Landau–Zener picture to
our case, we will find,Esat ¼ 0:

ð10Þ

In what follows, we derive eq. (10). We assume that eachtransition occurs at an avoided level crossing point and theprobability is given by the Landau–Zener formula. Let us
consider the transition rate R which is the probability perunit time that the system makes a transition to one of the twoneighboring states. This quantity R was already given for thecase where the external field increases linearly in time.13)
We apply the known expression to the present oscillatingcase. The transition rate R is determined by statistical
distributions of the energy gaps and the asymptotic slopes at
avoided crossing points. The average speed of the parameter!ðtÞ is denoted by j!!_j. For the present oscillating field, j!!_j isroughly estimated as j!!_j ’ A!. In addition, let # denote thedensity of states and '! denote the typical difference between
the asymptotic slopes of an avoided crossing. Under theconditions of slow speed, h! '! j!!_j#2 ( 1, which corresponds
to the situation that transitions of states only occur at
avoided crossings to the adjacent levels, R is expressed as,
ð11Þ
R / h! 1=2 '! 1=2 j!!_j3=2 #2 / h! 1=2 j!!_j3=2 #3=2 :We consider the time evolution of the occupation probability
for the eigenstate with energy E at time t, f ðE; tÞ. We canderive the diffusion equation for the occupation probabilityf ðE; tÞ of the state by extending Wilkinson’s theory to thecase that #ðEÞ is not constant;
&
'
"
#
@#f
@ dE
@
@f
!
_
þ j !j
#f ¼
D#
;
ð12Þ
@t
@E d!
@E
@E
where D ¼ R=#2 . Using the equation of continuity:
&
'
@#
@ dE
!
_
þ j !j# ¼ 0;
@t
@E d!

ð13Þ

eq. (12) is rewritten as,
"
#
@f
dE @f
@
@f
!
_
#
þ
D#
:
# ¼ &j!j@t
d! @E @E@E

ð14ÞLet us consider the steady state f ðEÞ ¼ f ðE; 1Þ. Then theleft-hand side of eq. (14) gives zero. Note
pffiffiffiffiffithat
ffiffiffiffiffiffiffiffi#
ffiffi is given
pffibyffiffiffi
1
the semicircle law (2), #ðEÞ ¼ 2$N
L2 & E2 ðL ¼ 2 N Þ
when N is very large. Therefore, f ðEÞ is solved as"Z E 0#ZE
ð15Þ
c1 expkð(Þd( dE0 þ c0 ;f ðEÞ ¼
&L

&L

where c0 and c1 are constants of integration, and&'
1 ! dE&1 @ðD#Þ
_
j !j
&#
:
kðEÞ '
D
d!
@E

ð16Þ

When we integrate eq. (15), the second term of kð(Þdiverges, because
Z E0
Z E0
1 @D#
d ln #
ð17Þ
&d( / &d( ! &1;
D# @(
&L
&L d(
where we used D ¼ R=#2 and eq. (11). Therefore,f ðEÞ ¼ c0 :

ð18ÞAs a result, the average energy always becomes 0 (the centerof the energy spectrum);Z
Esat ¼ E#f ðEÞdE ¼ 0:
ð19Þ
Thus we find that the idea of independent Landau–Zener
scattering is not applicable at least to a periodic system.14,23)
We cannot ignore the quantum mechanical interference
effect on the quantum localization.
3.3

Phenomenological interpretation of the quantum
localization in analogy to the Anderson localization
We here phenomenologically interpret the quantum
localization. The quantum localization in the present system
reminds us the Anderson localization where a particle in thepresence of a random potential is spatially localized.24) The
system for the Anderson localization is described by the
Hamiltonian:XX
HA ¼
vi jiihij þ t
jiihi0 j:ð20Þ
0i

hi;i iHere, jii means the state of the particle at the ith site, vi is thevalue of a random potential at the site i which is uniformly
distributed with a width W, and t represents hopping to oneof the nearest neighbor sites. In this particle system, theaverage hopping probability pA is expressed as pA ) jtj=W.
The probability pA ðmÞ that the particle exists on the mth sitefrom the initial (0th) site is estimated as pA ðmÞ )25)
pm
That is, the particle is localizedA ) expð&m logðW=jtjÞÞ.
in the space exponentially. This localization is caused by thequantum interference effects.
Let us consider the analogy to the Anderson localizationin the quantum localization of the present model. We
associate the state jii in eq. (20) with the ith adiabatic stateof H0 . Furthermore we focus on the frequency regime wheretransitions between levels take place only at avoided
crossings. We numerically confirmed that this situation is
actually realized for small !. In this case, we define a

J. Phys. Soc. Jpn., Vol. 71, No. 10, October, 2002

M. M ACHIDA et al.

Fig. 4. The probability PQ ðmÞ that the state is on the mth level of thesystem (1) for N ¼ 500 and ! ¼ 0:1$. It is obtained after 600 periods.

transition probability pQ between adjacent levels, whichcorresponds to pA in the Anderson localization, and we findthe common characteristics between the present situation in
the random matrices and the Anderson localization. That is,
the localization occurs due to the quantum inference effectamong many transitions between states. We may write theoccupation probability pQ ðmÞ that a state is on the mth levelas pQ ðmÞ ) pm
Q , where pQ is a transition probability between
the states. Actually fast relaxation is observed numerically inpQ ðmÞ of a typical example, which is depicted in Fig. 4.
If we assume an exponential form in the analogy to theAnderson localization, the saturation energy Esat is written
as,Esat ¼

N
X
ðm&1Þ
pQ
Em =Z;

ð21Þ

m¼1

of H0 and Z is thewhere Em is the mth eigenvalueP
normalization factor, Z ¼ Nm¼1 pm&1
Q . Equation (21) gives afinite saturation value of Esat .
4.

The Relevant Number of Floquet Eigenstates

In previous sections, we showed the quantum localizationin the random matrix model and considered an intuitive
interpretation. In this section, we consider the quantum
localization from the viewpoint of the Floquet theory. As
seen in eq. (8), the quantum dynamics is determined by theoverlaps between the initial state and Floquet eigenstates,i.e., jh%j 0 ij2 , and Esat is obtained by Floquet eigenstates.
Thus we here discuss this overlap.
We found numerically that the distribution of jh%j 0 ij2
decays approximately following an exponential function
when we arrange the states in order of the magnitude of theoverlap as is shown in Fig. 5. This property is one of thecharacteristics of the quantum localization. We study how
many Floquet states are involved in the ground state, anddefine the minimal number Nmin of the Floquet states bywhich the initial state j 0 i is covered within a ratio r:
(
N
min
X
1
jh 0 j%ij2 > r; jh 0 j%ij2
Nmin ¼ min N min :
N
%¼1
)
ð22Þ
* jh

0 j% þ 1ij

2

:

2431

Fig. 5. The dependence of the overlap jh%j 0 ij2 on the number %.
jh%j 0 ij2 is arranged in order of the magnitude. The overlap decays
exponentially. Here ! ¼ 0:1$ and N ¼ 500.

This quantity is the same as the quantity used in the different
context by Haake, Kus, and Scharf.2) Here, we taker ¼ 0:99. Figure 6 shows Nmin as a function of !=$ obtained
from five samples for N ¼ 256, 500, and 700.
We introduce the fitting function for Nmin ð!Þ in the form:()
1!
Nmin ð!Þ ¼ Nmin 1 & (
ð23Þ
)b :
ð!=!! Þa þ 1!
This function fits the data quite well. In eq. (23), Nmin
, a, b,!
and ! are determined from the least-squares method. The
results are shown in Table I. From eq. (23), Nmin ð!Þ isextremely small when ! ( !! . This is partially because theenergy gaps in lower energy region are so large comparing
with ! that transition to higher levels from the ground state
seldom occurs [see Fig. 1(a)], and therefore the systembehaves almost adiabatically.22) On the other hand, Nmin ð!Þ
is well fitted by a linear function (dashed line) for the regionof ! ) !! .
So far, we observed localization in the energy space andalso in the space of Floquet states. Now we discuss on
localization length of the two localizations. In particular, we
focus on the linear dependence of Nmin ð!Þ on ! in the small! region. We employ a phenomenological argument using
the analogy to the Anderson localization as explained in§3.3. Let us express the transition probability pQ using somefunction "ð!Þ in the form:pQ ) exp½&"&1 ð!Þ,:ð24Þ

The occupation probability pQ ðmÞ that a stationary state is on
the mth level is expressed as pQ ðmÞ ) pm
Q )exp½&"&1 ð!Þm,. Thus we can regard "ð!Þ as the localization length in the energy space. Now let l denote theTable I. Values of the parameters in eq. (23) obtained by the least-squares
method.
Variables

Dim (256)

Dim (500)

Dim (700)

!
Nmin

0.79

0.80

0.80

a

2.3

2.6

3.0

b

0.67

0.47

0.54

!! =$

0.15

0.10

0.11

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J. Phys. Soc. Jpn., Vol. 71, No. 10, October, 2002

M. MACHIDA et al.

Fig. 6. Nmin as a function of !=$ for (a) N ¼ 256, (b) N ¼ 500, and (c) N ¼ 700. Errorbars show the variance among the data obtained
from five samples. The fitted line and also linear fitting in the intermediate region are shown in each graph.

localization length in the Floquet space, i.e.,
jh%j 0 ij2 ’ ce&%=l , where c is the normalization constant.
From the definition (22), Nmin is expressed using the length l
as
l
ð25Þ
Nmin ’ & lnð1 & rÞ / l:N
Since we now focus on the linear dependence of Nmin on !,we write,
Nmin ¼ )! þ *;

ð26Þ

where ) and * are constants. We note that * is negligibly
small. Then we havel / !:

ð27Þ

As has been assumed in the the kicked rotator model todiscuss the diffusion constant,26) we assume the proportionality between l and "ð!Þ,l / "ð!Þ:ð28Þ

Then, we obtain from eqs. (27) and (28),"ð!Þ ¼ !=h:ð29Þ

Here h is an unknown amplitude which may depend on N, A,
the typical size (! of gaps at avoided crossings, and thedifference '! of the two asymptotic slopes at an avoided
crossing. This ! dependence is consistent with that of the

Landau–Zener formula, because eqs. (24) and (29) give
"
#
h
ð30Þ
pQ ) exp & :!
Actually, we numerically confirmed that transitions between
levels take place only at avoided crossings in the region
where Nmin linearly depends on !. Thus we suppose that theamplitude of local transition probability is originated in theLandau–Zener transition, although the phase interference
has an important effect on the global phenomena.
5.

Summary and Discussion

We have studied numerically the time evolution of theaverage energy of the GOE random matrix model under aperturbation of a sinusoidal function of time. In §3, we
showed that the quantum localization occurs in this model.
We have also studied the frequency dependence of thesaturated energy Esat , and we showed that we can not rely on
the picture of independent occurrence of Landau–Zenertransitions in the present case because the suppression of thequantum diffusion is essentially due to the phase interference
effect of quantum mechanics. We discuss this quantum
localization in analogy to the Anderson localization. In §4,
we introduced the relevant number of Floquet states Nmin ,and proposed a form of Nmin as a function of !. Nmin isalmost zero when ! is nearly zero, and it saturates when ! issufficiently large. In the intermediate region, Nmin grows

J. Phys. Soc. Jpn., Vol. 71, No. 10, October, 2002

linearly as ! increases. This linear dependence of Nmin on !
implies that the localization length l in the Floquet
eigenstates also depends linearly on ! [eq. (25)]. On
condition that " / l, the localization length " of theeigenstates of the Hamiltonian is also proportional to !.
This dependence of " on ! implies the Landau–Zenermechanism governs the local transitions.
Acknowledgments
This study is partially supported by Grant-in-Aid from the
Ministry of Education, Culture, Sports, Science andTechnology of Japan. The computer calculation waspartially carried out at the computer center of the ISSP,
which is gratefully acknowledged.
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