Physics@Technion

Anderson localization
in some nonlinear
systems
Yevgeny Bar-Lev (né Krivolapov),
Shmuel Fishman and Avy Soffer,
Technion, May 2010

Outline
� Anderson localization
� Some Nonlinear phenomena
� Nonlinear optics
� Nonlinearity in many-body physics
� Nonlinearity and Disorder: Motivation
� Heuristics and numerics
� Perturbation theory
� Results
� Summary

Anderson localization
� Anderson model (1958)
where are independent random variables
� All eigenstates are exponentially localized
° = » ¡1
un (x) » exp ¡° (x ¡ xn)
were is the inverse localization length
� Dynamical localization: Transport is
exponentially suppressed

Some nonlinear phenomena
BEC
He 4
Many-Body
Nonlinearity due to
approximations
Dynamics
Gross-Pitaevskii or
NL Schrodinger eq.
Optics/Hydrodynamics
Nonlinearity due to
media
Waves
NL materials
Solitons

Nonlinear Schrödinger Equation
(NLSE)
� Transverse modes of light propagating in
nonlinear media
i @A
@z
= ¡ 1
2k0
@2A k0n2
¡
@x2 n0
jAj 2 A
� Bose-Einstein Condensate in a mean field
approximation
i~@tà = ¡ ~2 @
2m
2Ã @x2 + ¯ jÃj2 Ã

Properties of NLSE
� Without external potential and in the
continuum the equation is integrable
Nonlinearity
Attractive
Repulsive
Bright solitons
Dark solitons,
Spreading
wavepacket

Nonlinearity and Disorder: Motivation
Disorder
Localization of
all eigenfunctions.
transport is
suppressed
?
Nonlinearity
Enhancement of
transport
by soliton
formation

Does dynamical localization
survives the nonlinearity?
The model: motivated by experimental relevance
NLSE with a random potential
i@tà = ¡J [Ã(x +1) +Ã(x ¡1)] +"xà +¯ jÃj 2 Ã

Simple heuristic arguments
� Yes, if there is spreading the magnitude of the nonlinear term decreases
and localization takes over.
� Depends, for a localization length of the relevant energy spacing is ,
and the perturbation because of the nonlinear term is
therefore for small β there is no spreading. (Shepelyansky)
� Depends, there will be no spreading for large enough β (theorem), but there
is subdiffusion even for very small β (Flach)
� Yes, because quasiperiodic localized perturbation does not destroy
localization (Soffer, Wang-Bourgain)
»
» ¡1
¯ jÃj 2 ¼ ¯» ¡1

Numerics – second moment
Pikovsky, Shepelyansky
YK
Ð
2
log10 x ® Ð x 2®

Numerics - wavefunction
Pikovsky, Shepelyansky

Problem of all numerics
� “Asymptotics curse”: It is impossible to decide
whether there is a saturation in the expansion of
the wavefunction. In any case it looks like the
expansion is very slow, at most sub-diffusional.
� Convergence: All long time numerics are done
without convergence to true solution.
� Time scale: The time scale of the problem is not
clear

Perturbation theory: Change basis
and expand in
� Expand in eigenstates of the linear Anderson model
� The equation for the expansion coefficients is
where
� Expand cn
in powers of

Example: 1 st order
(0)
n n0
c �
�
� �1 � 000 i( En �E0
) t
c0 ��iVn
e
�t n � 0
n �
n �
0
0
i( E E ) t
(1) 000 �1�e� n � n � �
En�E0 c V
� �
The first problem: Secular terms
c � �iV �t
1 000
0 0
� �
The second problem: Small denominator problem

Elimination of secular terms
Change
Expand
For example:
à (x; t) = X
à (x; t) = X
i@tcn = ¡ E (0)
n ¡ E0 ¢
n cn + ¯ X
cn (t) e
n
¡iE0 nt
cn (t) e un (x)
n
¡iEnt
un (x)
m1m2m3
V m1m2m3
n c ¤ m1cm2cm3e i(E0 n+E0 m ¡E
1 0 m ¡E
2 0 m3)t i@tc (1)
n = ¡E (1)
n ±n0 + V 000
n e i(E 0 n ¡E0 0)t
We set
E (1)
0 = V 000
0

Bounding the general term
The general term is a product of terms of the form
³ m1m2m3
n ´
Using Cauchy-Shwartz inequality
hj³ m1m2m3
n j s i ·
We find
*
¯
1
V m1m2m3
n
E 0 n ¡ fE0 g mi
¯E0 n ¡ fE0g ¯
mi
2s
¯
+ 1=2
Ð m1m2m3 jVn j 2s®1=2 ;
hj³ m1m2m3
n j s ¡1
i · D" 0e 3 (°¡"0 )s ijxn¡xm ij

Structure of general term

Bounding the general term
Finally, we can show using generalized Hölder inequality
* ¯ ¯¯¯¯ X
³ m1m2m3
n ³ m4m5m6
m1
fmig
or for the expansion coefficient
D¯¯c ¯
(k) ¯
n
sE
±
¢ ¢ ¢ ³ 000
mN¡1
¯
¯s+
¯
±
· F (k)
± e¡(°¡"¡"0 )sjxnj
· e 2k F (k)
± e¡(°¡"¡"0 )sjxnj
All the expansion coefficients are exponentially localized uniformly in time !
Does it prove dynamical localization ?

The remainder
Equation of motion
i@tQn = Wn (t) + X
Mnm (t) Qm + X
Qn (t = 0) = 0
m
m
¹Mnm (t) Q ¤ m + F (Q)
For t · t one can neglect the nonlinear term (Bootstrap argument)
¤
jQn (t)j · A¯ N+1 t ¢ e ¡°jnj
t ¤ » O ¡ ¯ ¡1¢
?
t ¤ » O ¡ ¯ ¡N¢

Main result – Logarithmic front
X
jQnj 2 < " 2
If we require that it is satisfied for some
n>¹n
µ ¶ N A¯ t
¹n (t) = » ln
"
The front after which the solution is exponentially bounded and quasi-
periodic in time propagates at most logarithmically !*
* Similar to a conjecture of W.-M. Wang
¹n

¡¹n (t)
µ ¶ N A¯ t
¹n (t) = » ln
"
jª (n; t)j
?
¹n (t) n
Exponential tail Exponential tail

Numerical implementation

lim
jQnj
= 0
¯N Is ?
¯!0

A new time scale
If the series is asymptotic it has an optimal order
Therefore
N! ¢ ¯ N » 1
N = ¯ ¡1
t¤ = O ¡ ¯ ¡N¢ = O
³ 1
¡
¯ ¯
Is this time scale connected to the Lyapunov exponent of the system ?
´

Summary
� Renormalized perturbation theory was developed, including secular
terms removal in all orders, bounding of the general term and the
remainder (for finite time).
� Symbolic and numerical implementation of the theory was done.
Good agreement with numerical solutions and fast calculation.
� The wavepacket spreads at most logarithmically up to large
times.
� The asymptotic nature of the series was supported numerically.
� A new time scale was proposed.

Open questions
� Does the logarithmic front persists to
arbitrarily long time ?
� Are the series asymptotic ?
� Is the new time proposed time scale
connected to the Lyapunov exponent of
the system ?
Thank you !