Frequency Dependence of Quantum Localization in a Periodically Driven System Manabu Machida, Keiji...

Frequency Dependence of Quantum Localization

in a Periodically Driven System

Manabu Machida, Keiji Saito, and Seiji Miyashita

Department of Applied Physics, The University of Tokyo

Matrices of Gaussian Orthogonal Ensemble (GOE) are real symmetric, and each element of them is a Gaussian distributed random number. 2

,2 1,0 jiijij HH

GOE Random Matrix

E.P. Wigner introduced random matrices to Physics. Wigner, F.J. Dyson, and many other physicists developed random matrix theory.

VtHtH )()( 0

0H V

)sin()( tAt

and are independently created GOE random matrices.

is fixed at 0.5.

Hamiltonian

A varies.

Typical Hamiltonian for complexly interacting systems

under an external field.

/2

0

)(i

expT dttHF

iexpF

Floquet Theory

0 nn F

Energy after nth period:

We define,

Energy fluctuates around satE

0

0

HnTH

nT

Comparing

/

Saturated !

Solid line

Esat is normalized so that the ground state energy is 0 and the energy at the center of the spectrum is 1.

nn H satEwith

0.02

0.1

0.2

0.41.0

as a function of satE /

How to understand the localization?

(i) Independent Landau-Zener Transitions

Wilkinson considered the energy change of a random matrix system when the parameter is swept.

M. Wilkinson, J.Phys.A 21 (1988) 4021

M. Wilkinson, Phys.Rev.A 41 (1990) 4645

We assume transitions of states occur at avoided crossings by the Landau-Zener formula, and each transition takesplace independently.

How to understand the localization?

Transition probability

Probability of finding the state on the lth level

Diffusion equation:

X

The integral on the exponential diverges.

Therefore,

1sat E

Quantum interference effect is essential!

for any

How to understand the localization?

The global transition cannot be understood only by the Landau-Zener transition.

The random matrix system The Anderson localization

In each time interval T, the system evolves by the Floquet operator F.

The Hamiltonian which brings about the Anderson localization evolves in the interval T,

THU A

iexp

F

AH

How to understand the localization?

(ii) Analogy to the Anderson Localization

ApQp

'',

A iitiivHiii

i

: random potential distributed uniformly in the width W

:Hamiltonian for the Anderson localization

m

m

W

tpmp

AA

iv

mpmp QA

N

mm

m EpE1

)1(Qsat

How to understand the localization?

mPmP QQ

Let us introduce in order to study -dependence of the quantum localization.

F. Haake, M. Kus, and R. Scharf, Z.Phys.B 65 (1987) 381

K. Zyczkowski, J.Phys.A 23 (1990) 4427

min

1

2

0

2

0

2

0minmin 1;:min1 N

rNN

rN

We count the number of relevant Floquet states in the initial state.

minN

99.0r

One important aspect of the quantum localization

/2

0 e

-dependence of Nmin

ba

NN1/

11

*min

*min

Phenomenologically,

Parameters in the phenomenological function of minN

/1Q eP

m/QQ emPmP

minN

minN (numerical)

/2

0 e

/Q e hP

h

h : unknown amplitude

This fact suggests the local transition probability originates in the Landau-Zener transition.

The quantum localization occurs in this random matrix due to the quantum interference effect. On the other hand, the Landau-Zener mechanism still works in the local transitions.

To be appeared in J.Phys.Soc.Jpn. 71(2002)

Conclusion