anderson localization for the nonlinear schrödinger equation (nlse): results and puzzles yevgeny...
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![Page 1: Anderson Localization for the Nonlinear Schrödinger Equation (NLSE): Results and Puzzles Yevgeny Krivolapov, Avy Soffer, and SF](https://reader038.vdocuments.us/reader038/viewer/2022103122/56649cce5503460f9499990e/html5/thumbnails/1.jpg)
Anderson Localization for the Nonlinear Schrödinger Equation (NLSE):
Results and Puzzles
Yevgeny Krivolapov, Avy Soffer, and SF
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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
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Does Localization Survive the Nonlinearity???
0 localization
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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
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Does Localization Survive the Nonlinearity???
• Yes, wings remain bounded
• No, the NLSE is a chaotic dynamical system.
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Experimental Relevance
• Nonlinear Optics
• Bose Einstein Condensates (BECs)
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Numerical Simulations
• In regimes relevant for experiments looks that localization takes place
• Spreading for long time (Shepelansky, Pikovsky, Molina, Kopidakis, Komineas Flach, Aubry)
????
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Pikovsky, Sheplyansky
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S.Flach, D.Krimer and S.Skokos Pikovsky, Shepelyansky
2log x 2log x
t
st
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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
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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
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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
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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
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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
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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
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Strategy for calculation of leading order
Equation for remainder term
AverageSmall denominators
Probabilistic bound
Bootstrap
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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
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Where the equation for is
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Where the linear part is
removed secular term
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To simplify the calculations we will denote
such that
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which leads to
Small denominators
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Where Si are small divisor sums, defined bellow.
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Small denominators
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We split the integration to boxes with constant sign of the Jacobian
Using the following inequality
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The integral can be bounded
The integral could be further bounded
where
We show this in what follows
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Use Feynman-Hellman theorem
and take j significantly far.
Sub-exponential
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Minami estimate
We exclude intervals
Exclude realizations of a small measure
the difference between is not small
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The effect of renormalization
Using
(leading order in )
One finds
where are the minimal and maximal Lyapunov exponents
Therefore is not effected for
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A bound on the overlap integral is
Bound on the growth of the linear term
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Using
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Therefore
Chebychev inequality
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same for
Therefore
similar for
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Where the equation for is
Nonlinear terms
linear
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The Bootstrap Argument
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some algebra gives
or by renaming the constants
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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
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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
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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
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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
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What is the effect of the remainder??
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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?