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?