anderson localization for the nonlinear schrödinger equation (nlse): results and puzzles
DESCRIPTION
Anderson Localization for the Nonlinear Schrödinger Equation (NLSE): Results and Puzzles. Experimental Relevance. Yevgeny Krivolapov, Hagar Veksler, Avy Soffer, and SF. Nonlinear Optics Bose Einstein Condensates (BECs). Competition between randomness and nonlinearity. - PowerPoint PPT PresentationTRANSCRIPT
Anderson Localization for the Nonlinear Schrödinger Equation (NLSE):
Results and Puzzles
Yevgeny Krivolapov, Hagar Veksler, Avy Soffer, and SF
Experimental Relevance
Nonlinear Optics
Bose Einstein Condensates (BECs)
Competition between randomness and nonlinearity
The Nonlinear Schroedinger (NLS) Equation
2
oit
H
( ) ( 1) ( 1) ( ) ( )x x x V x x 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
Work in progress, several open questions
Aim: find a clear estimate, at least for afinite (long) time
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 some time dependent quasi-
periodic localized perturbation does not destroy localization
n1/ n
2/ n
Does Localization Survive the Nonlinearity?
• No, the NLSE is a chaotic dynamical system.
• No, but localization asymptotically preserved beyond some front that is logarithmic in time
Numerical Simulations
• In regimes relevant for experiments looks that localization takes place
• Spreading for long time (Shepelyansky, Pikovsky, Molina, Kopidakis, Komineas, Flach, Aubry)
• We do not know the relevant space and time scales• All results in Split-Step• No control (but may be correct in some range)• Supported by various heuristic arguments
Pikovsky, Sheplyansky
S.Flach, D.Krimer and S.Skokos Pikovsky, Shepelyansky
2log x 2log x
t
stb, g, r0.1, 1, 5
p
oit
H
Modified NLSE
22m x t
Shepelyansky-Pikovsky arguments –No spreading for 2p Flach, Krimer and Sokos
1
1p
1 for 0p
H. Veksler, Y. Krivolapov, SF
0 200 400 600 800 10000
10
20
30
40
50
60
t
<m
2>
0 5 10 150
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
p
ba
0 5 10 150
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
p
2m
t p
p1 2( ) ( )c c
100
102
104
106
108
10-2
10-1
100
101
102
103
104
t
m2
(b)
(g)(r)
(t)
(p)
(y)
2m
tp= 1.5, 2, 2.5, 4, 8, 0 (top to bottom)
Also M. Mulansky
Heuristic Theories that Involve Noise Assumptions Clearly Contradict Numerical Results:
1.Nothing happens at 2. approach to localization at
2p 0p
These theories may have a range of validity
Perturbation Theory
The nonlinear Schroedinger Equation on a Lattice in 1D
2
oit
H
( ) ( 1) ( 1) ( ) ( )x x x x x 0H
n random Anderson Model0HEigenstates ( ) ( )m m mu x E u x0H
, miE tm m
m
x t c t e u x
The states are indexed by their centers of localization
mu Is a state localized near mx
This is possible since nearly each state has a Localization center and in a box size M there are approximately M states
,m mx x x
mu x D e e
, ,D D
realization
for realizations of measure 1
Indexing by “center of mass” more stable
21 2 3 1 3
1 2 3
1 2 3
( ), , *
, ,tn m m mi E E E E tm m m
n n m m mm m m
i c V c c c e
Overlap 1 2 3
1 2 3
, , ( ) ( ) ( ) ( )m m mn n m m m
x
V u x u x u x u x
of the range of the localization length
perturbation expansion( ) (1) 2 (2) 1 ( 1)( ) ......o N N N
n n n n n nc t c c c c Q
Iterative calculation of ( )lnc
start at (0)
0( 0)n n nc c t
1 2 31 2 31 2 3
1, 3 n m n m n mn m m m
x x x x x xx x x xm m mnV V e e
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 (fractional moments)
'
( ) ( ) miE t mn m n
m
t c t e u New ansatz
Step 2
1 2 31 2 3
1 2 3
1 2 3
1 2 31 2 3
1 2 3
1 2 3
2 1 0 0
, ,
0 1 0
, ,
2
n m m m
n m m m
i E E E E tm m mt n n m m m
m m m
i E E E E tm m mn m m m
m m m
i c V c c c e
V c c c e
1 002 0nn nE V n
0 0
0
0 0
00 0000
0 0
00 000
02
00 000
0, 0
00 0002
0 0
2 0
2
2
3
m m
n
n m
n m n
mi E E t i E E tm
m m
ni E E tn n
nt n m
i E E tn m
m n m
mi E E E t i E E tn m
m m
V Ve e n
E E
V Ve
E Ei c
V Ve
E E
V Ve e
E E
0n
Secular term removed
1
2 n n
n n
c t c td t
c t c t
Second order
To different realization
0 50 100 150 200 250 300 350 400-1
-0.5
0
0.5
1
t
im(c
n)
= 0.1 site = 64
0 50 100 150 200 250 300 350 400-1
-0.5
0
0.5
1
t
re(c
n)
= 0.1 site = 64
0 50 100 150 200 250 300 350 4000
0.2
0.4
0.6
0.8
t
d1(n
)
= 0.1 site = 64
0 50 100 150 200 250 300 350 400-6
-4
-2
0
2
4
6x 10
-3
t
im(c
n)
= 0.1 site = 65
0 50 100 150 200 250 300 350 400-6
-4
-2
0
2
4
6x 10
-3
t
re(c
n)
= 0.1 site = 65
0 50 100 150 200 250 300 350 4000
0.5
1
1.5
2
t
d1(n
)
= 0.1 site = 65
1
2 n n
n n
c t c td t
c t c t
Second order
Red=perturbative, blue=exact
1
2 n n
n n
c t c td t
c t c t
0 50 100 150 200 250 300 350 400-6
-4
-2
0
2
4
6x 10
-3
t
im(c
n)
= 0.1 site = 65
0 50 100 150 200 250 300 350 400-6
-4
-2
0
2
4
6x 10
-3
t
re(c
n)
= 0.1 site = 65
0 50 100 150 200 250 300 350 4000
0.5
1
1.5
2
t
d1(n
) = 0.1 site = 65
0 50 100 150 200 250 300 350 400-1
-0.5
0
0.5
1
t
im(c
n)
= 0.1 site = 64
0 50 100 150 200 250 300 350 400-1
-0.5
0
0.5
1
t
re(c
n)
= 0.1 site = 64
0 50 100 150 200 250 300 350 4000
0.05
0.1
0.15
0.2
t
d1(n
)
= 0.1 site = 64
Third order
0 50 100 150 200 250 300 350 4000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Qex
and Qlin
t
||Q|| 2
0 50 100 150 200 250 300 350 4000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Qex
and Qlin
t
||Q|| 2
Second order Third order
Blue=exact, red=linear
2
2 2 0.1n n
n
nn
c t c td t
c t
Second order
Aim: to use perturbation theoryto obtain a numerical solution that is controlled aposteriory
General removal of secular terms
1 2 31 2 3
1 2 3
1 2 3
0
n m m m
t n n n n
i E E E E tm m mn m m m
m m m
i c E E c
V c c c e
000
0 1 1 10 0
0
1k k k k nn n n n n n n n
n
VE c E c E E
E E
1 2 31 2 3
1 2 3
1 2 3
1
0
1 1 1
0 0 0
n m m m
kk k s s
t n n ns
k k r k r si E E E E tr s lm m m
n m m mm m m r s l
i c E c
V c c c e
Remove secular termsIn order k with
Number of terms in higher orders
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
iterate
NR number of products of 'V s
1N 'm s in 3 3N places
3 3
1N
NR
N
subtract forbidden combinations
2 3 11
1 2 1
0 0 000 000
1 2 3 1,0,0, , ,0, ,0, ,0,0,0,0
k
k k
m m mmn m m m
k
V V V V
m m m m
The solid line denotes the exact numerical solution of the recurrence relation for and dashed line is the asymptotic upper bound on this solution
kR 2 .ke
lnlim
27ln 2
4
N
N
R
N
kR
Bounding the General Term
1 2 31 2 3
1 2 3
1 2 3
1
0
1 1 1
0 0 0
n m m m
kk k s s
t n n ns
k k r k r si E E E E tr s lm m m
n m m mm m m r s l
i c E c
V c c c e
1 2 3 4 5 6
1 5 6
terms
000 000
k
m m m m m mn m m m
results in products of the form
1 2 3
1 2 3
i
m m mm m m nn
n m
V
E E
1 2 3 1 2 3
1/2
1/22
2
1
i
s sm m m m m mn ns
n m
VE E
1where 0
2s
Using Cauchy-Schwartz inequality
Small denominatorsIf the shape of the squares of the eigenfunctions sufficiently different,using a recent result of Aizenman and Warzel we propose
Assuming sufficient independence between states localized far away
Chebychev inequality
Bound on products
1/
1 1
k k kk
i ii i
x x
and with the help of the generalized Hoelder inequality
Average over realizations of measure 1
continue to longer products for example
Bound on general term
The Bound deteriorates with order
The remainder
( ) (1) 2 (2) 1 ( 1)( ) ......o N Nn n n n
Nn
nc t
Q
c c c c
Bound by a bootstrap argument
26· 2 ·n nx xcNnQ t M t e t C e e
One can show that for strong disorder C and the constant are multiplied by
( ) 0f
The Bound on the remainder
It is probably proportional to exp( )
Front logarithmic in time1
lnx t
Bound on error
• Solve linear equation for the remainder of order
• If bounded to time perturbation theory accurate to that time (numerics)
• Order of magnitude estimate if asymptotic hence for optimal order (up to constants).
0t
0 1N t ! 1NN 0 !t N
N
Perturbation theory steps
• Expansion in nonlinearity
• Removal of secular terms
• Control of denominators
• Probabilistic bound on general term
• Control of remainder
Summary
1. A perturbation expansion in was developed2. Secular terms were removed3. A bound on the general term was derived 4. Results were tested numerically5. A bound on the remainder was obtained, indicating
that the series is asymptotic.6. For limited time tending to infinity for small nonlinearity,
front logarithmic in time or 7. Improved for strong disorder8. Earlier result: exponential localization for times shorter
than
lnx t
2
2ln t
Open problems
1. Can the logarithmic front be found for arbitrary long time?
2. Is the series asymptotic or convergent? Under what conditions?
3. Can the series be resummed?
4. Can the bound on the general term be improved?
5. Rigorous proof of the various conjectures on the linear, Anderson model
6. How to use to produce an aposteriory bound?