Frequency Dependence of Quantum Localization in a Periodically Driven System

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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
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GOE Random Matrix
Matrices of Gaussian Orthogonal Ensemble (GOE)
are real symmetric, and each element of them is
a Gaussian distributed random number.
E.P. Wigner introduced random matrices to
Physics. Wigner, F.J. Dyson, and many other
physicists developed random matrix theory.
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Hamiltonian
and
are independently created GOE random matrices.
is fixed at 0.5.
varies.
Typical Hamiltonian for complexly interacting
systems under an external field.
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Floquet Theory
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Energy after nth period
We define,
Energy fluctuates around
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Comparing
with
1.0
Saturated !
0.4
0.2
0.1
Esat is normalized so that the ground state
energy is 0 and the energy at the center of the
spectrum is 1.
0.02
Solid line
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as a function of
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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 takes place independently.
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How to understand the localization?
Probability of finding the state on the lth level
Transition probability
Diffusion equation
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How to understand the localization?
The integral on the exponential diverges.
The global transition cannot be understood only
by the Landau-Zener transition.
Therefore,
for any w
Quantum interference effect is essential!
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How to understand the localization?
(ii) Analogy to the Anderson Localization
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,
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How to understand the localization?
Hamiltonian for the Anderson localization
random potential distributed uniformly in the
width W
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Let us introduce in order to study
w-dependence of the quantum localization.
We count the number of relevant Floquet states in
the initial state.
F. Haake, M. Kus, and R. Scharf, Z.Phys.B 65
(1987) 381
K. Zyczkowski, J.Phys.A 23 (1990) 4427
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One important aspect of the quantum localization
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w-dependence of Nmin
Phenomenologically,
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Parameters in the phenomenological function of
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(numerical)
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unknown amplitude
This fact suggests the local transition
probability originates in the Landau-Zener
transition.
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Conclusion
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)