Proc. Waseda Int. Sympo. on Fundamental Physics —New Perspectives in Quantum Physics—
J. Phys. Soc. Jpn. Vol. 72 (2003) Suppl. C pp. 109–110c
2003
The Physical Society of Japan

Frequency Dependence of Quantum Localization in a Periodically Driven System
Manabu Machida,∗ Keiji Saito and Seiji Miyashita
Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan(Received January 30, 2003)

Under a periodically driven external ﬁeld or a temporally oscillating boundary, quantum fewbody systems whose corresponding classical systems show chaos are modeled by the randommatrices with an oscillating parameter. In such systems, time evolution of the system fromthe ground state shows the quantum diﬀusion. However, due to periodic oscillation of theparameter, quantum interference eﬀect prevents the energy of the system from diverging and theenergy saturates after some transient time. We show an example where the energy eventually
ﬂuctuates around a constant saturated energy. Since the value of this saturated energy depends
on the frequency of the driving, we also investigate the dependence of the saturated energy on
the frequency in the small frequency region.
KEYWORDS: quantum dynamics, quantum localization, random matrix

is |ν, with the eigenvalue eiφν ;
i 2π/ω
F = T exp −H(t) dt , 0Under periodic driving force, the initial state which is
the ground state eventually settles down in a state corresponding to a ﬁnite temperature. This saturation of
the energy is a well known phenomenon, but its dependence on the frequency of the periodic force is still anopen problem. We study such evolution in a quantum
few-body system, and uncover the dependence of the saturation energy on the frequency. This driving could bean oscillating external magnetic ﬁeld acting on a quantum spin system, an oscillating boundary of quantum
billiards, or so. Under such driving forces, saturationof the energy occurs after some transient time. If interactions of the system are strong or the boundary of abilliard has less symmetry, dynamics of the corresponding classical systems becomes chaos, and Hamiltoniansof the systems can be represented by random matrices.1)
Hence, the random matrix theory gives a powerful tool
to study quantum systems which have chaotic nature
not only for their static characters, such as the spectral
statistics, but also for their dynamics,2–4) although thecorrespondence between random matrices and Hamiltonians of realistic systems is not trivial.5) Using a randommatrix model with a sinusoidally changing parameter,
we present a model where the energy eventually saturates and ﬂuctuates around a constant saturated energy.
We investigate how the saturated energy depends on thefrequency of the driving ﬁeld.
The random matrix model we shall study is written as
H(t) = H0 + λ(t)V.

F |ν = eiφν |ν(ν = 1, . . . , N ),

(2a)
(2b)

where T denotes the time-ordered product and we arrange ν so that |ν|ψ0 |2 > |ν  |ψ0 |2 for ν > ν  . Then,the state |ψn  after the n-th period is written as|ψn  = F n |ψ0 .

(3)Using this fact, the expectation value E(n) of energy
after the n-th period can be expanded by the Floquet
eigenstates;
E(n) = ψn |H0 |ψn 
=ein(φν  −φν ) ψ0 |νν|H0 |ν  ν  |ψ0 .

(4)

ν,ν 

Now, we shall show E(n) ﬂuctuates around a constantenergy Esat , which depends on the frequency ω, when nis suﬃciently large. The time average of E(n) gives thesaturated energy Esat .
m−1

1 E(n)m→∞ m
n=0|ν|ψ0 |2 ν|H0 |ν=E(n) ≡ lim

(5)

ν(1)

≡ Esat ,Here H0 and V are random matrices of dimension N independently drawn from Gaussian Orthogonal Ensemble(GOE). The elements satisfy (H0 )ij  = 0, (H0 )2ij  =1 + δij , (V )ij  = 0, (V )2ij  = 1 + δij . The parameter λ(t) denotes a periodic driving with the frequency ω:λ(t) = A sin(ωt), A = 0.5.
Let the initial state |ψ0  at t = 0 be the ground state of
H0 . We deﬁne the Floquet operator F whose eigenstate

where we use the relation ein(φν  −φν ) = δν,ν  . The variance can be estimated as22var[E(n)] = E(n) − E(n) = E 2 (n) − Esat(6) |,≤ ||H0 ||2 , ||H0 || ≡ max|(H)
0jj

j,j

where we assume the nonresonance condition that all
the diﬀerences of the eigenvalues are diﬀerent, that is,
if φν1 − φν2 = φν4 − φν3 = 0 such that |νi |ψ0 |2 =∗ E-mail address: machida@spin.t.u-tokyo.ac.jp

109

J. Phys. Soc. Jpn. Vol. 72 (2003) Supplement CE(n) [energy]

110

1M. Machida et al.

ω=0.4π0.8

ω=0.2π0.6Here we estimate the eﬀective pumping rate p(k)from the ground state to an excited state |k [Esat =k p(k)Ek ]. Here, Ek is the k-th eigenvalue of H0 andcorresponding to Ek . Gener|k is the eigenstate of H0 
ally, this rate is given by6) ν |ν|ψ0 |2 |ν|k|2 . Let  be
the localization length in the Floquet space:0.4|ν|ψ0 |2 ∝ e−ν/ .

ω=0.1π0.2

In addition, it has been suggested,7)00200

400600800

10.8p(k) =|ν|ψ0 |2 |ν|k|2 ∼ e−k/ .

0.4
0.2
00.4

0.6

0.8

1

1.2

ω/π
Fig. 2. Esat as a function of ω/π. N = 700. Each point is obtained
from being averaged over five samples. Error bars show thevariance.

0, i = 1, 2, 3 and 4, then ν1 = ν4 , ν2 = ν3 . Thiscondition is plausible because H0 is a random matrix
and eiφν distributes randomly on a unit circle. Equation(6) implies that E(n) does not deviate far from Esat .
Figure 1 shows time evolution of the average energy,E(n), for three values of ω in the case of N = 700. Indeed, we see that E(n) ﬂuctuates around Esat . The energy level distribution has a symmetric shape about thecenter of the spectrum. We remind that an energy unit
is so determined that the ground state energy is zero andthe center of the spectrum is one.
In Fig. 1, Esat clearly depends on the frequency ω. We
shall focus on the saturated energy Esat . In order to see
its dependence on the frequency, we study ﬁve samples,and plot the average of Esat as a function of ω in Fig. 2.
There, N = 700 and error bars show variances of Esat .

(9)

νWe have pointed out that  is proportional to ω 6) in theregion where the scattering process consists of successive transitions of the state to adjacent levels at avoided
level crossing points (the Landau–Zener region). Thus
we conclude that in this region (we numerically checkedω  0.1π is in the Landau–Zener region) p(k) is expressed asp(k) ∼ e−αk/ω ,0.60.2(8)Thus, we haveFig. 1. Time evolution of E(n) is shown with the corresponding
Esat (horizontal line). N = 700 and ω = 0.1π, 0.2π and 0.4π,
respectively.

Esat|ν|k|2 ∼ e−|ν−k|/ .

1000

n [period]

0(7)

(10)

where α is a constant.
This conclusion means that, in the quantum localization, the pumping rate in the saturated state depends on
ω exponentially, which is not compatible to the pictureof independent Landau–Zener transition8, 9) at avoided
level crossing points.6) This picture ignores the quantum
phase interference and gives the result Esat → 1 regardless of ω. The ω dependence of p(k) in eq. (10) leads that
Esat → 0 at very small ω. In addition, eq. (10) implies
that the saturated value Esat is always below 1. This isqualitatively consistent with the ω dependence of Esat inFig. 2 at small ω. Quantitative comparison between Esat
calculated using eq. (10) and the numerical data is a future issue, since it needs more data points in Fig. 2 andfor example, in eq. (10), α is an unknown amplitude. In
short, to have the form as shown in Fig. 2, the quantumphase interference is essential.

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