anderson localization for the nonlinear schrödinger equation (nlse): results and puzzles yevgeny...
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Anderson Localization for the Nonlinear Schrödinger Equation (NLSE):
Results and Puzzles
Yevgeny Krivolapov, Avy Soffer, and SF
The Nonlinear Schroedinger (NLS) Equation
2
oit
H
1 1n n n n nV 0H
V random Anderson Model0H
1D lattice version
2
2
1( ) ( ) ( ) ( )
2x x V x x
x
0H
1D continuum version
Does Localization Survive the Nonlinearity???
0 localization
Does Localization Survive the Nonlinearity???
• Yes, if there is spreading the magnitude of the nonlinear term decreases and localization takes over.
• No, assume wave-packet width is then the relevant energy spacing is , the perturbation because of the nonlinear term is
and all depends on• No, but does not depend on • No, but it depends on realizations • Yes, because time dependent localized
perturbation does not destroy localization
n1/ n
2/ n
Does Localization Survive the Nonlinearity???
• Yes, wings remain bounded
• No, the NLSE is a chaotic dynamical system.
Experimental Relevance
• Nonlinear Optics
• Bose Einstein Condensates (BECs)
Numerical Simulations
• In regimes relevant for experiments looks that localization takes place
• Spreading for long time (Shepelansky, Pikovsky, Molina, Kopidakis, Komineas Flach, Aubry)
????
Pikovsky, Sheplyansky
S.Flach, D.Krimer and S.Skokos Pikovsky, Shepelyansky
2log x 2log x
t
st
Perturbation Theory
The nonlinear Schroedinger Equation on a Lattice in 1D
2
oit
H
1 1n n n n n 0H
n random Anderson Model0HEigenstates E0
m mn m nH u u
( ) ( ) miE t mn m n
m
t c t e u
The states are indexed by their centers of localization
mu
Is a state localized near m
This is possible since nearly each state has a Localization center and in box size M approximately M states
21 2 3 1 3
1 2 3
1 2 3
( ), , *
, ,tm m m mi E E E E tm m m
m m m m mm m m
i c V c c c e
Overlap 1 2 3 31 2, ,m m m mm mm
m n n n nn
V u u u u
of the range of the localization length
perturbation expansion
( ) (1) 2 (2) 1 ( 1)( ) ...... ....o N Nn n n n nc t c c c c
Iterative calculation of ( )lncstart at
(0)0( 0)n n nc c t
start at (0)
0n nc step 1
0( )(1) 000
tni E E t
n nc iV e
0n0( )
(1) 000
0
1 ni E E t
n nn
ec V
E E
0n (1) 0000t 0c iV
Secular term to be removed by ' (1) ....n n n nE E E E
here (1) 0000nE V
Secular terms can be removed in all orders by
' (1) 2 (2) ......n n n n nE E E E E
0n0( )
(1) 000
0
1 ni E E t
n nn
ec V
E E
The problem of small denominators
The Aizenman-Molchanov approach
'
( ) ( ) miE t mn m n
m
t c t e u New ansatz
Leading order result
• Starting from a state localized at some site the wave function is exponentially small at distances larger than the localization length for time of the order
0 2
1t
Strategy for calculation of leading order
Equation for remainder term
AverageSmall denominators
Probabilistic bound
Bootstrap
Equation for remainder term
We need to show
or just
We will limit ourselves to the expansion
remainder
'
( ) ( ) niE t nx n x
n
t c t e u Expansion
Where the equation for is
Where the linear part is
removed secular term
To simplify the calculations we will denote
such that
which leads to
Small denominators
Where Si are small divisor sums, defined bellow.
Small denominators
We split the integration to boxes with constant sign of the Jacobian
Using the following inequality
The integral can be bounded
The integral could be further bounded
where
We show this in what follows
Use Feynman-Hellman theorem
and take j significantly far.
Sub-exponential
Minami estimate
We exclude intervals
Exclude realizations of a small measure
the difference between is not small
The effect of renormalization
Using
(leading order in )
One finds
where are the minimal and maximal Lyapunov exponents
Therefore is not effected for
A bound on the overlap integral is
Bound on the growth of the linear term
Using
Therefore
Chebychev inequality
same for
Therefore
similar for
Where the equation for is
Nonlinear terms
linear
The Bootstrap Argument
some algebra gives
or by renaming the constants
Summary
• Starting from a state localized at some site the wave function is exponentially small at distances larger than the localization length for time of the order
0 2
1t
Higher ordersiterate
21 2 3 1 3
1 2 3
1 2 3
1 1 1( ), , ( )* ( ) ( )
, , 0 0 0
t
m m m m
(N)m
N N r N r si E E E E tm m m r s l
m m m mm m m r s l
i c
V c c c e
NR number of products of 'V s
1N 'm s in 3 3N places
3 3
1N
NR
N
subtract forbidden combinations
ln 27lim ln 2
4N
N
R
N
Number of terms grows only exponentially, not factorially!!No entropy problem
typical term in the expansion
2 NNN N NR T e T
The only problem NT
it contains the information on the position dependence n
complicated because of small denominators
Can be treated producing a probabilistic bound
21 2( ) /( )Pr b n n Na N a NN
nc e e e e
1 3a 2 1a
at order
N exponential localization for
*n n* 2n N
1a
b
What is the effect of the remainder??
Questions and Puzzles
• What is the long time asymptotic behavior??
• How can one understand the numerical results??
• What is the time scale for the crossover to the long time asymptotics??
• Are there parameters where this scale is short?