290

Progress of Theoretical Physics Supplement No. 145, 2002

Frequency Dependence of Quantum Localization
in a Periodically Driven SystemManabu Machida,∗) Keiji Saito and Seiji Miyashita
Department of Applied Physics, School of Engineering,
The University of Tokyo, Tokyo 113-8656, Japan(Received April 1, 2002)
We study the quantum localization of a GOE random matrix model driven by a periodic
external ﬁeld. After the diﬀusion is suppressed, the average energy oscillates around someﬁnite value Esat . We study how Esat depends on the frequency of the external ﬁeld. In order
to investigate this phenomenon, we also study the frequency dependence of the relevant
number of Floquet states Nmin , and discuss its dependence in the view of the Anderson
localization in the region where the frequency is not large.

Recently active research has been done on the dynamical nature of quantum
systems whose energy levels have complex structures. Especially, the quantum localization is known to take place in these systems under a periodic external ﬁeld. The
quantum localization was ﬁrst predicted theoretically 1) and also realized experimentally. 2) - 6) Although the quantum localization has mainly studied for simple models
such as the kicked rotator model 1) and the kicked top model, 7) some studies haverecently done on random matrix models which take account of many-body interaction. 8), 9) In this paper, we investigate the frequency dependence of the quantum
localization of the system of a random matrix model.
We consider time-reversal invariant systems under an oscillating external ﬁeld.
The random matrix ensemble appropriate for describing the spectral statistics ofthese systems is Gaussian orthogonal ensemble (GOE). 10) Eigenvalues of a GOE
random matrix repulse each other, and they never degenerate. Points where energy
levels nearly degenerate are called avoided crossings. We take H0 and V to be
GOE random matrices. Elements of these matrices are taken from the Gaussiandistribution. The mean value of each element is zero, and the variance is 1 fordiagonal element and 2 for oﬀ-diagonal element. Using H0 and V , the Hamiltonianis given as follows;
H(t) = H0 + λ(t)V(1)with λ(t) = A sin(ωt) (A = 0.5).
We take the ground state as the initial state |ψ0  at time t = 0. State |ψ1  after
a period is expressed by operating the time-evolution operator for a period F (theFloquet operator 11) ); |ψ1  = F |ψ0 . Eigenfunctions of F form the complete set of
the Hilbert space and eigenvalues of F are aligned on a unit circle in the complex
plane;
F |ν = eiφν |ν.
∗)

E-mail: machida@spin.t.u-tokyo.ac.jp(2)

Frequency Dependence of Quantum Localization

291

In the same way, the state after the nth period is expressed as, |ψn  = F n |ψ0 .
Therefore, the average energy after the nth period is written as,ψn |H0 |ψn  =ein(φν −φν  ) ψ0 |ν  ν|ψ0 ν  |H0 |ν.

(3)

ν  ,ν

Note that H(t = 0) = H0 .
In this equation, the energy oscillates aroundEsat ≡ν|ν|ψ0 |2 ν|H0 |ν,

(4)

after the diﬀusion is suppressed. Hence, we regard Esat as the saturated energy.
In the present study, we normalize Esat such that the distance between E = 0
and the ground state is 1. Figure 1 shows Esat as a function of ω for systemsof several sizes. When ω is very small, the average energy hardly changes dueto the adiabatic theorem: 12) when ω gets slightly larger, transfers of occupation
probabilities of energy levels begin to occur at avoided crossings. We see that Esat
approaches to the center of the energy spectrum (Esat = 0) asymptotically as ωgrows. Note that the energy spectrum is symmetric about its center.
As is seen in Eq. (4), most of the ω-dependence of Esat is determined by |ν|ψ0 |2
because Eq. (4) is approximately rewritten as Esat  ν |ν|ψ0 |2 Eν , where Eν iseigenvalue of H0 whose eigenvector is the closest to |ν. Therefore we focus on thisoverlap and employ the minimal number Nmin of Floquet eigenvectors necessary tocover the initial state |ψ0  to within a ratio r. 7) That is, we deﬁne Nmin as follows;
Nmin =




Nmin

1min Nmin :|ψ0 |ν|2 > r; |ψ0 |ν|2 ≥ |ψ0 |ν + 1|2 ,Nν=1(5)where N is the size of the Hamiltonian. We take r = 0.99. Figure 2 shows Nmin asa function of ω for N = 256, 500 and 700. In Fig. 2, when ω is nearly zero, Nmin

Esat

0-0.2
-0.4
-0.6
-0.8
-1.0

0

0.2

0-0.2
-0.4
-0.6
-0.8
-1.0
0 1
0.4 0.6

dim(256)dim(500)
dim(700)
2 3 4
0.8
15 6
1.2

ω/π

Fig. 1. Esat as a function of ω. The Esat for 256, 500 and 700 dimension are shown respectively.
The inset also shows Esat as a function of ω for a wide range of ω.

292

M. Machida, K. Saito and S. Miyashita

Nmin

0.8
0.6
0.8
0.6

0.4

dim(256)dim(500)
dim(700)
tangential line

0.4
0.2

0.2
0000.2

0.4

01
0.6

2 3 4
0.8
1ω/π

5 6
1.2

Fig. 2. Nmin as a function of ω. The Nmin for 256, 500 and 700 dimension are shown respectively.
The inset also shows Nmin as a function of ω for a wide range of ω.

is close to zero because almost no transitions of states occur due to the adiabatictheorem. 12) On the other hand, Nmin is approximately constant when ω is suﬃciently
large because the occupation probability is spread over almost the whole states before
the quantum diﬀusion is suppressed.
In Fig. 2, we also ﬁnd empirically that the form of Nmin (ω) is ﬁtted by∗
1−Nmin (ω) = Nmin

1[(ω/ω ∗ )a + 1]b.

(6)

The values of a, b and ω ∗ obtained from
Table I. Values of the parameters in Eq. (6).
the least-squares method are shown in
They are obtained by the least-squares
Table I. In Eq. (6), Nmin (ω) has a linear
method.
shape in the intermediate region aroundparameter dim(256) dim(500) dim(700)
the inﬂection point (ω ∼ 0.1π). Hence,
∗Nmin
0.79
0.80
0.80
we regard the tangential line at the ina
2.3
2.6
3.0
ﬂection point as Nmin (ω) in the intermeb
0.67
0.47
0.54
diate region. The tangential line at theω∗
0.15
0.10
0.11
inﬂection point of Nmin in Eq. (6) in thecase N = 700 is also depicted in Fig. 2.
Now, we focus on the intermediate region, where Nmin grows linearly. We ﬁndnumerically that transitions of states take place only at avoided level crossings whenω ∼ 0.1π. We investigate the quantum localization in the following way, since a
direct application of the Landau-Zener transition 13) does not explain the observed
ω-dependence of Esat and Nmin . 14), 15) For the localization, the eﬀect of quantum
interference takes important role. In the following, we associate the quantum localization of the system with the Anderson localization. 16) Here we study the case withsmall ω, where the Landau-Zener transition gives main mechanism of level transfer.
The corresponding system, which shows the Anderson localization, has random
potential U with the width W and constant hopping t. Since transitions occur at

Frequency Dependence of Quantum Localization

293

avoided crossings, we believe that the Landau-Zener transition takes place and the2transition probability at each avoided crossing is proportional to exp − 2σAωπ cos ωt ,where  is the size of gaps of avoided crossings, and σ is the size of the diﬀerence ofslopes of two asymptotic lines at avoided crossings. We assume that the corresponding hopping constant t after a period of the external ﬁeld has the form¯2t ∼ exp −h(N, A) ω −1 ,σ̄(7)where h(N, A) is some function of N and A, ¯ is the typical size of , and σ̄ is thetypical size of σ. The probability pm that the state occupies the mth site from theinitial 0th site is given in the study of the Anderson localization by 17)pm ∼|t|W

m



¯2
∼ exp −h(N, A) ω −1 m .
σ̄

(8)On the other hand, the probability to ﬁnd the state at the mth level of H(t) is foundto be proportional to exp (−Γ m), where Γ −1 is the localization length. Comparing
2them, we have, Γ ∼ h(N, A) ¯σ̄ ω −1 . Using the fact Nmin ∼ Γ −1 , we obtainNmin ∼σ̄ω.
¯2(9)Equation (9) suggests linear dependence of Nmin on ω. In Fig. 2, we ﬁnd this relation
except for very small value of ω.
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