Effective Noise Theory for the Nonlinear Schroedinger

Transcription

Eective Noise Theory for the
Nonlinear Schrödinger
Equation
Erez Michaely
Eective Noise Theory for the
Nonlinear Schrödinger
Equation
Research thesis
Submitted in Partial Fulllment of the
Requirement for the degree of
Master of Science in Physics
Erez Michaely
Submitted to Senate of
the Technion - Israel Institute of Technology
Haifa, Israel, JUNE 2012, Sivan 5772
THE RESEARCH THESIS WAS DONE UNDER THE SUPERVISION OF PROF. SHMUEL
FISHMAN AT THE DEPARTMENT OF PHYSICS.
THE GENEROUS FINANCIAL HELP OF THE TECHNION IS GRATEFULLY
ACKNOWLEDGED.
Publications from this thesis:
•
E. Michaely and S. Fishman. Eective noise for the nonlinear Schrödinger equation with
disorder. Phys. Rev. E, 85:046218, 2012.
•
E. Michaely and S. Fishman.
Statistical properties of the one dimensional Anderson
model relevant for the nonlinear Schrödinger equation in a random potential. Accepted
for publication in EPJ
This research was supported in part by the US-Israel Bi-national Science Foundation (BSF)
and by the Israel Science Foundation (ISF).
5
This thesis work is dedicated to my beloved parents Osnat and Aharon Michaely, who were
supportive in every aspect possible during my grad school and my life. Thank you
6
Contents
1 Introduction
1.1
1.2
5
The Nonlinear Schrödinger Equation with a Random Potential
. . . . . . . . . .
5
1.1.1
Motivation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.1.2
Spreading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
Theoretical Analysis
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Eective Noise Theory and its Numerical Tests
13
2.1
Eective Noise Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Statistical Properties of
2.3
Fn (t)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1
Fn (t)
Exhibits the Power Spectrum of Noise
2.2.2
Fn (t)
Has Rapidly Decaying Auto-correlation Function
2.2.3
Stationarity of
2.2.4
Averages of
Fn (t)
Fn (t)
and
11
. . . . . . . . . . . . . . . .
13
19
20
. . . . . . . . . .
20
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
Cn (τ )
The Scaling of the Second Moment
. . . . . . . . . . . . . . . . . . . . . . . . .
M2
with
ξ
. . . . . . . . . . . . . . . . . . . .
22
25
3 Statistical Properties of the Anderson Model Relevant to the NLSE With
Random Potential
27
3.1
3.2
Estimate of Scaling of the Overlap Sums
Vnm1 ,m2 ,m3
with
ξ
in the Regime of Weak
Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
Vmm12 ,m3 ,m4
30
Distributions of the Overlap Sums
in the Regime of Weak Disorder
.
3.3
3.4
m1 = m2 = m3 = m4 = 0
3.2.1
The Case
3.2.2
The Case of
m1 = m2 = 0; m3 = m4 = 1 .
. . . . . . . . . . . . . . . . . .
32
3.2.3
The Case of
m1 = m2 = m3 = 0; m4 = 1 .
. . . . . . . . . . . . . . . . . .
33
3.2.4
Distribution of
Vmm12 ,m3 ,m4
. . . . . . . . . . . . . . . . . . . . .
When 3 or 4 Dierent
mi
30
are Involved . . . . .
33
. . . . . . . . . . . . . . .
33
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
Φnm1 ,m2 ,m3
for Weak Disorder
3.3.1
Distribution of
En
3.3.2
Distribution of
Φ+ ≡ E n + E m
. . . . . . . . . . . . . . . . . . . . . . . .
34
3.3.3
Distribution of
Φ− ≡ En − Em
. . . . . . . . . . . . . . . . . . . . . . . .
37
3.3.4
Distribution of
2 ,m3 ,m4
Φm
m1
. . . . . . . . . . . . . . . . . . . . . . . . . .
40
Strong Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4 Toy model
45
4.1
Denition of the Toy Model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Results of the Numerical Calculations
. . . . . . . . . . . . . . . . . . . . . . . .
5 Summary and Discussion
45
47
52
5.1
Validity of the Eective Noise Theory
. . . . . . . . . . . . . . . . . . . . . . . .
5.2
Possibility for the Breakdown of the Eective Noise Theory
5.3
Statistical Properties of the Anderson Model Relevant for the NLSE with a Ran-
. . . . . . . . . . . .
52
53
dom Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
5.4
Toy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
5.5
5.6
5.7
5.8
Some Details of the Numerical Calculations - Split Step Method
Fn (t)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
M2
Vnm1 ,m2 ,m3
with
ξ
with
ξ
56
. . . . . . . . . . . .
62
. . . . . . . . . . . . . . . . . . .
63
. . . . . . . . .
66
List of Figures
|ψ|
2
The probability distribution of
2.2
Secont moment of
2.3
Fn (t)
2.4
Distribution of
2.5
|C (τ )|
2.6
Dependency of
3.1
Dierent averages of
3.2
Average inverse participation number as a function of
3.3
Distribution of
3.4
and
Cn (τ )
. . . . . . . . . . . . . . . .
18
. . . . . . . . . . . . . . . . . . . . . . . . .
19
for some disorder . . . . . . . . . . . . . . . . . . . . . . . . . .
23
ψ
as function of
over lattice size
x
2.1
t.
Fn (t)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
for dierent
t0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
M2
on
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
ξ
Vmm12 ,m3 ,m4
as a function of
ξ
. . . . . . . . . . . . . . . . . .
ξ
29
. . . . . . . . . . . . . .
34
V00,0,0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
Distribution of
V00,1,1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.5
Distribution of
V00,0,1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.6
Averages of
3.7
Distribution of
En
for weak disorder
3.8
Distribution of
3.9
Distribution of
3.10 Comparing
V00,1,2 and V01,2,3
Φ+
V
37
. . . . . . . . . . . . . . . . . . . . . . . . .
38
Φ+
for weak disorder . . . . . . . . . . . . . . . . . . . . . . . . .
38
Φ+
for weak disorder . . . . . . . . . . . . . . . . . . . . . . . . .
39
Φ−
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
for the strong disorder regime . . . . . . . . . . . . . . . . . . . . .
43
3.11 Distribution of
3.12 Averages of
ξ
. . . . . . . . . . . . . . . . . . . . .
with
as function of
Φ
En
for strong disorder
. . . . . . . . . . . . . . . . . . . . . . . .
44
Φ+
and
ψ
Φ−
for strong disorder . . . . . . . . . . . . . . . . . . . .
t
4.1
Second moment of
(toy model) . . . . . . . . . . . . . . . . .
49
4.2
Dierent modes excitation (toy model) . . . . . . . . . . . . . . . . . . . . . . . .
50
4.3
Second moment of
ψ
4.4
tm as
β
5.1
Distribution of
5.2
Average auto-correlation function decay
5.3
Scaling
M2
with
ξ
for
β=1
5.4
Scaling
M2
with
ξ
for
β = 3.5
a function of
as a function of
44
as a function of
. . .
51
. . . . . . . . . . . . . . . . . . . . . . . . . . .
51
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
(toy model)
Fn t = 103
t
for extremly short time (toy model)
. . . . . . . . . . . . . . . . . . . . . . .
59
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
List of Tables
5.1
Numerical values for the distribution of
F0
. . . . . . . . . . . . . . . . . . . . . .
58
5.2
F3
. . . . . . . . . . . . . . . . . . . . . .
58
5.3
F10
5.4
|C0 (τ )|
5.5
hVnm1 ,m2 ,m3 i
2
. . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
hVnm1 ,m2 ,m3 i
and
E
D
2
(Vnm1 ,m2 ,m3 ) −
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6
5.7
Numerical values of
ν
as a function of
β
D
2
(Vnm1 ,m2 ,m3 )
E
58
60
63
. . . . . . . . . . . . . .
64
. . . . . . . . . . . . . . . . . . . . . . .
65
Abstract
The Nonlinear Schrödinger Equation (NLSE) with a random potential is a paradigm for
competition between randomness and nonlinearity. It is also of experimental relevance for experiments in optics and in atom optics. In spite of extensive exploration for the last two decades
the elementary properties of its dynamics are still not understood. Dynamical localization is
found for the linear Anderson model. In particular in one dimension in the presence of a random
potential typically a wave packet that is initially localized will remain localized for arbitrary long
time. The elementary open question is whether this holds also in the presence of nonlinearity.
In numerical calculations it is found that spreading by subdiusion takes place for a wide range
of parameters and for a long time. Analytical and rigorous arguments indicate that this subdiffusion cannot be asymptotic in time. The basic justication that was given for the subdiusion
is that the nonlinear term acts as noise. In this thesis this assumption was reformulated and
tested numerically. It was found that in the relevant regime the nonlinear term behaves as noise
with a rapidly decaying correlation function, required for subdiusion. It was also found that
this eective noise is stationary. A scenario for the failure of the eective theory in the long time
limit is outlined. Several statistical properties of the linear Anderson model that are relevant for
the NLSE with a random potential were calculated. A toy model for the NLSE was proposed.
1
List of Symbols and Abbreviations
• ψ
- wavefunction
• H0
• β
- linear Hamiltonian (Anderson model)
- nonlinearity strength
• εx
- on site potential
• n, m1 , m2 , m3 , m4
• Z
- lattice index
- natural number
• W
- disorder strength
• N
- wavefunction norm
• H
- nonlinear Hamiltonian
• un
- eigenfunction of the Anderson model
• xn
- localization center
• ξ
- maximal localization length
• ξm
• σ
- localization length of mode
m
- power of nonlinear term
2
• n ψ 2 - index of refraction
• Ql
- nonlinear classical chain eigenfunction
• ωl
- nonlinear classical chain eigenvalue
• cm -
expansion coecient
• Em
- eigenenergy of the Anderson model
• Φnm1 ,m2 ,m3 • Vnm1 ,m2 ,m3
• Fn (t)
• Eψβ
• ρ
total phase
- overlap sum
- eective noise
- energy of
ψ
- norm density
(1)
(1)
(2)
(1)
• C, C0 , C0 , C0 , C1 , C0 , C̄
• η, η1 , η2 • P
exponent of
- numerical constant
ξ
- number of resonant modes
• A0 , A1 , A2 , A3 , A4
- numerical constant
• γ
- exponent of
• T
- equilibration time
• D
- diusion constant
β
• M1
- rst moment
• M2
- second moment
• F̂n (ω)
- Fourier Transform of
Fn (t)
3
• Sn (ω)
• F˜n (t)
• ω
- power spectrum of
- shifted
Fn (t)
Fn (t)
- frequency
• Cn (τ )• y1 , y2
auto-correlation function of
- scaling variable
• Φ+
- sum of two eigenenergies
• Φ−
- dierence of two eigenenergies
• σ (ξ)
• ũn
• ψ̃
Fn (t)
- width of the tted gaussian
- toy model eigenfunction
- toy model wavefunction
•
NLSE - Nonlinear Schrödinger Equation
•
i.i.d - independent identically distributed
•
GPE - Gross-Pitaevskii Equation
•
FPU - Fermi-Pasta-Ulam
•
RHS - right hand side
•
PDF - probability distribution function
4
Chapter 1
Introduction
1.1
The Nonlinear Schrödinger Equation with a Random
Potential
The Nonlinear Schrödinger Equation (NLSE) with a random potential is a fundamental problem. In spite of extensive mathematically rigorous, analytical and numerical explorations, the
elementary properties of its dynamics are not known. The problem is relevant for experiments
and its resolution will shed light on many problems in chaos and nonlinear physics. It may also
stimulate novel experiments. On a one dimensional lattice, the NLSE with a random potential
is given by,
2
i∂t ψ (x, t) = H0 ψ (x, t) + β |ψ (x, t)| ψ (x, t) ,
(1.1)
H0 ψ (x, t) = − [ψ (x + 1, t) + ψ (x − 1, t)] + εx ψ (x, t) ,
(1.2)
where
while
x ∈ Z;
and
{εx }
is a collection of independent and identical distributes (i.i.d.) random
variables uniformly distributed in the interval
Hamiltonian
H0
W W
−2, 2
where
W
is the disorder strength. The
is the Anderson model in one-dimension [1, 2, 3, 4, 5, 6]. It is important to
5
note that for the dynamics generated by (1.1) there are two constant of motions.
•
The
`2
norm,
N =
•
X
x
2
|ψ (x, t)| .
(1.3)
The energy given by the Hamiltonian [7]
H=
X
x
"
2
− (ψ (x + 1) ψ ∗ (x) + ψ ∗ (x + 1) ψ (x)) + εx |ψ (x)| +
Therefore the NLSE is regarded chaotic. The classical variables are
role of coordinate and
ψ ∗ (x)
is the conjugate momentum.
β
4
|ψ (x)| .
2
ψ (x)
(1.4)
that plays the
The NLSE is a Hamilton
equation of this Hamilotnian.
The interesting question about the dynamics of (1.1) is: will an initial wave function
ψ (x, t = 0),
which is localized in space, spread indenitely for large times, and in particular in the asymptotic
limit,
t → ∞.
Surprisingly, the answer to this elementary question is not known in spite of
extensive research in the last two decades [8, 9, 10, 11, 12, 13].
The NLSE (1.1) is regarded
as a representative of many nonlinear problems. Therefore, its understanding may shed light
on dynamics generated by other nonlinear equations, e.g, nonlinear Klein-Gordon and FPU
(Fermi-Pasta-Ulam) equations.
The dynamics of (1.1) is completely understood in the two
limiting cases. In the absence of the random potential (εx
wavepacket will spread indenitely, for all values of
= 0,
for all
x)
an initially localized
β , unless solitons are formed.
In the discrete
case, unlike the continuous case, the formation of the solitons cannot be established rigorously
[14]. The continuous version of this model is in fact an integrable problem [7]. For attractive
nonlinearity,
β < 0, solitons are found while for repulsive nonlinearity, β > 0, complete spreading
takes place. In the presence of randomness
(W > 0),
but for
Anderson model (1.2) where it is rigorously established that
β =0
Eq. (1.1) reduces to the
all the eigenstates are exponentially
localized in one-dimension with probability one [1, 2, 4, 5, 6]. At long scales the eigenfunctions
behave as
un (x) ∼ e−|x−xn |/ξ ,
6
(1.5)
where
xn
is the localization center and
ξ
is the localization length. Consequently, diusion is
suppressed and in particular a wavepacket that is initially localized will not spread to innity.
This is the phenomenon of Anderson localization. In two-dimensions it is known heuristically
from the scaling theory of localization, that all the states are localized, while in higher dimensions
there is a mobility edge that separates localized and extended states [3, 4].
The behavior of the dynamics generated by (1.1) is very dierent in the two extreme limits
(W = 0, β 6= 0)
and
(W 6= 0, β = 0).
Therefore, it is a paradigm for the exploration of the
competition between randomness and nonlinearity.
The nonlinear term of the form
2
β |ψ| ψ
used in (1.1) is just one representative possibility,
which is used in this thesis for the sake of clarity. In several theoretical studies it is replaced by
[8, 15, 10, 16]
2σ
Hσ = β |ψ|
where
σ>0
ψ,
(1.6)
is arbitrary. Other types of nonlinear terms appear in experimental realizations.
1.1.1 Motivation
The NLSE was derived for a variety of physical systems under some approximations. In the
present subsection two major examples are outlined (namely Nonlinear optics and Bose-Einstein
Condensates) and the classical example of the FPU system, that is a related classical problem,
is presented.
Nonlinear Optics
It was derived in classical optics, where
ψ
is the electric eld, by expanding the index of
refraction in powers of the electric eld, keeping only the leading nonlinear term [17]. Let the
the form,
2
|ψ|
, then for weak elds it takes
2
2
4
n |ψ| = n0 + n1 |ψ| + O |ψ| .
(1.7)
index of refraction depend on the intensity of the electric eld
7
The nonlinear term in (1.1) corresponds to a weak eld so that the quartic correction is negligible.
In several important cases
2
2
n |ψ| saturates, namely, lim|ψ|2 →∞ n |ψ| = const.
For example
in the induction technique [18, 19] the index of refraction takes the form,
2
n |ψ| =
n0
1 + |ψ|
2.
(1.8)
In optics, Eq. (1.1) corresponds to the paraxial approximation where the propagation direction plays the role of time.
In this approximation the variation in the index of refraction
in space is weak, and therefore there is only a small change in the propagation direction, and
back-scattering is negligible.
Interacting Bose-Einstein Condensates (BEC)
For Bose-Einstein Condensates (BEC), the NLSE is a mean eld approximation, where the
term proportional to the density
β|ψ|2
approximates the interaction between the atoms.
In
this eld the NLSE is known as the Gross-Pitaevskii Equation (GPE) [20, 21, 22, 23, 24]. It
was rigorously established, for a large variety of interactions and of physical conditions, that
the NLSE (or the GPE) is exact in the thermodynamic limit (for systems with number of
particles going to innity and with bounded energy per particle). Repulsive interactions obeying
−σ
V (x) = V (−x) falling fast enough as 0 ≤ V (x) ≤ C ∗ hxi
1/2
hxi ≡ 1 + x2
are assumed. In this case the wavefunction
product of
N
with
C∗ > 0
and
σ > 5,
where
of the system turns to be a tensor
single particle wavefunctions, each satisfying the GPE [25, 26]. Experiments on
spreading of wavepackets of cold atoms in a random optical potential were recently performed
[27, 28, 29, 30]. In those experiments as in experiments in optics, the random potential exhibits
correlations and therefore deviates from the model presented in (1.1).
Nonlinear Classical Problems
Equations similar to (1.1) are found also in classical mechanics. For example, a vibrating string
could be approximated as a chain of nonlinear oscillators. This is the celebrated Fermi-PastaUlam (FPU) problem [31]. The equation of motion of one oscillator in the chain is given by,
mẍn
= k [(xn+1 − xn ) − (xn − xn−1 )] +
8
(1.9)
h
i
2
2
α (xn+1 − xn ) − (xn − xn−1 )
which is known as the
α-model,
mẍn
which is known as the
β -model.
boundary condition and
N
or
k [(xn+1 − xn ) − (xn − xn−1 )] +
h
i
3
3
β (xn+1 − xn ) − (xn − xn−1 )
=
(1.10)
The eigenmodes and eigenvalues of the linear part for Direchlet
oscillators are given by,
Ql (n)
r
=
2
πl
sin n
N
N
(1.11)
with frequencies
ωl = 2ω0 sin
where
ω0 =
p
k/m and l = 1, 2, . . . N .
Expanding
xn (t) =
N
X
πl
,
2N
(1.12)
xn using the eigenmodes of the linear problem,
cl (t) Ql (n)
(1.13)
l=1
and using the orthogonality of the modes, gives
c̈n = −ωn2 cn + α
for the
α-model
Vnm1 m2 cm1 cm2
(1.14)
Vnm1 m2 m3 cm1 cm2 cm3
(1.15)
m1 m2
and,
c̈n = −ωn2 cn + β
for the
X
β -model,
where
Vnm1 m2
and
X
m1 m2
Vnm1 m2 m3
are some complex expressions involving overlap
sums of three and four eigenmodes correspondingly.
Those equations could be brought to a
similar form as (1.1) written in linear problem eigenmodes.
1.1.2 Spreading
For linear problems, all aspects of the dynamics are determined by the spectral properties,
namely the eigenvalues and the eigenfunctions. This is not correct for nonlinear problems. For
9
example, for small
β
in (1.1) there are stationary and quasi-periodic states which are exponen-
tially localized [32, 33, 34]. This however does
not
imply that an initially localized wavepacket
will not spread, contrary to the case of a linear system with a bounded localization length (but
here some caution is in place [35] .
The important question of the eld is whether a wavepacket, that is initially localized in
space, will indenitely spread for dynamics controlled by (1.1).
A simple heuristic argument
indicates that spreading will be suppressed by randomness. If unlimited spreading takes place,
the amplitude of the wave function will decay since the
`2
norm,
N , is conserved.
Consequently,
the nonlinear term will eventually become negligible, and Anderson localization will take place
as a result of the randomness, as was conjectured by Fröhlich
et al
[36]. However, in numerical
calculations performed by Shepelyansky [37] for the kicked rotor with a cubic nonlinear term,
Anderson localization (that takes place in the absence of the nonlinear term) was destroyed and
sub-diusion takes place. Similar spreading was found numerically also by Shepelyansky and
Pikovsky [8] and by Flach and coworkers [11, 10]. Therefore, the naive argument for localization
of (1.1) has to be reconsidered and a proper theory should be developed. A natural question is
what can we conclude from the numerical simulations ? The main problem is that dynamics of
(1.1) are chaotic. The dynamics are generated by the Hamiltonian (1.4), where the NLSE (1.1)
is the corresponding Hamilton's equation with the conjugate variables
the nonlinearity, the motion in the
ψ (x)
and
ψ ∗ (x).
ψ (x),ψ ∗ (x) phase-space will be typically chaotic.
Due to
Therefore,
the numerical solutions of (1.1) are not the actual solutions. In order to draw conclusions it is
assumed that they are statistically similar to the correct solutions. Since it is a system of an
innite number of degrees of freedom there is no real theoretical support for this assumption.
If we use the fact that only a nite number of the
ψ (x)
variables are involved, there is a
competition between two eects. Chaos is enhanced by increase in eective number of degrees
of freedom, and suppressed by the decreasing amplitude of the spreading wavepacket. A scaling
theory indicates in the asymptotic limit localization takes place [38]. Theoretical [39, 40, 12, 13]
and rigorous [9] analysis indicated that the spread is at most logarithmic in time.
10
1.2
Theoretical Analysis
In this section I present a convenient way to analyze the various regimes starting from the short
time regime and up to the asymptotic long time regime. Various authors use an expansion of
the wavefunction in terms of the eigenstates,
ψ (x, t) =
X
um (x),
and eigenvalues,
Em ,
H0
of
as,
cm (t) e−iEm t um (x) .
(1.16)
m
For the nonlinear equation the dependence of the expansion coecients,
cn (t) ,
is found by in-
serting this expansion into (1.1), resulting in the equation of motion of the expansion coecients
cn (t)
X
i∂t cn = β
m1 ,m2 ,m3
Vnm1 m2 m3 c∗m1 cm2 cm3 eitΦn
m1 ,m2 ,m3
where
Φnm1 ,m2 ,m3
≡ Fn (t)
(1.17)
is the total phase
1 ,m2 ,m3
= En + Em1 − Em2 − Em3 .
Φm
n
and
Vnm1 m2 m3
(1.18)
is an overlap sum
Vnm1 m2 m3 =
X
un (x) um1 (x) um2 (x) um3 (x) .
(1.19)
x
This sum is negligibly small if the various eigenfunctions are not localized in the same region of
the order of the localization length,
The eigenvalues
En ,
ξ.
the eigenfunctions
un (x) ,
overlap sums depend on the random potentials,
Consequently,
potentials,
En , un (x)
and
Vnm1 m2 m3
the expansion coecients,
{εx }
and therefore they are
cn (t)
and the
random variables.
take dierent values for the various realizations of the
{εx }.
The outline of the thesis is as follows. In Chapter 2 an eective noise theory is outlined and
its assumptions are tested numerically. In particular the statistical properties of the nonlinear
term in (1.1) are computed. This is the main result of this thesis. The results of Chapter 2 are
published in [41]. In Chapter 3, we present statistical properties of the Anderson model which
are relevant for the NLSE with random potential [42]. In Chapter 4 a toy model for the NLSE
11
a with a random potential problem is presented. Chapter 5 is the summary of this thesis. Also
some open problems are discussed there.
12
Chapter 2
Eective Noise Theory and its
Numerical Tests
In this Chapter an eective noise theory [10, 11, 41] is presented and its assumptions are outlined
in Sec.
2.1.
The assumptions for the eective noise theory, on
numerically in Sec. 2.2. Averages of
Fn (t)
Fn (t)
of (1.17) is checked
over dierent realizations are presented in Appendix
A. The results of this section are published [41].
2.1
Eective Noise Theory
In this Section the phenomenological theory [10, 11] is presented for the spreading that is found
numerically. It will be presented in form that we found reasonable [41]. First we note that it
is clear that not all the content of the initial wavepacket spreads for all values of
rigorously shown, that for suciently large
β,
β.
It was
the initial wavepacket cannot spread so that its
amplitude everywhere vanishes at innite time [43]. The proof makes use of the conservation of
the
`2
norm of the wavefunction and the conservation of the energy
13
Eψβ ≡ H,
where
of
H0 ,
H
is given by (1.4). The expectation value of (1.2),
is bounded by the maximal eigenvalue
E (max) .
(2.1)
If we assume
Eψβ=0 ,
E (max) = maxm Em ≤ 2 + W/2,
2
limt→∞ supx |ψ (x)| = 0,
then
Eψβ = Eψβ=0 < E (max) ,
term in (1.4) vanishes. If the initial wavefunction is localized, for
then
Eψβ > E (max) .
that is the expectation value
β
namely
since the
Eψβ=0 ≤
4
|ψ (x)|
suciently large and positive
Since energy is conserved this leads to a contradiction with the assumption
that in the innite time limit the function
|ψ (x)|
spreading of a fraction of the wavepacket.
site or started at one linear eigenstate of
[37, 44, 43, 10, 11].
2
vanishes for every site
x.
It does not contradict
For a wavepacket initially localized on one lattice
H0 ,
sub-diusion was found in numerical experiments
The purpose of the theory presented in what follows, is to explain the
spreading that takes place after some time.
It was found numerically that after some initial
time the shape of the wavepacket is similar to the one presented in Fig 2.1. Note that Fig 2.1 is
plotted in log scale so that the area should not be conserved. It consists of a relatively at region
at the center and exponentially decaying tails. The theory of [10, 11] assumes spreading from
the relatively at region of the wavepacket to the region where the amplitude of the wavepacket
is small. Let
m1 , m2
and
m3
designate eigenstates of
within the at region, and let
n
H0
with the centers of localization found
designate a state with a center of localization found in the tail
of the wavepacket, but in the vicinity of the at region. Therefore, spreading will take place to
the region where the
n-th
state is localized. In particular
|cm1 |2 ≈ |cm2 |2 ≈ |cm3 |2 ≈ ρ
here
ρ
(2.2)
is the average density and,
|cn |2 ρ.
It is assumed that the RHS of (1.17) is a random function denoted by
(2.3)
Fn (t).
We turn to estimate
its typical behavior. First we note that the overlap sums (1.19) are random functions. Within
the scaling theory pf localization one expects that for suciently weak disorder their various moments are determined by the localization length. For the case where all indices
14
(n, m1 , m2 , m3 )
are identical the average is just the inverse participation ratio which is proportional to 1/ξ . For
the general case the scaling theory suggests it is a function only of
theories leads us to assume it is a power of
ξ.
ξ.
Experience with scaling
Therefore we try the approximate forms,
(1)
hVnm1 ,m2 ,m3 i = C0 ξ −η1 ,
(2.4)
and for the second moment we try to t to,
(2)
< |Vnm1 ,m2 ,m3 |2 >= C0 ξ −2η2 .
Here
the
(1)
C0
mi
and
and
(2)
C0
are constants and
ξ ∼ W −2
ξ
We should note that
is actually energy dependent. For weak disorder in the center of the
[45, 46], this relation holds for most energies in the energy band [45]. In what
follows we will estimate the values of
sites
is an average over realizations. We note that when
n are all dierent the average of the overlap integrals vanishes.
the localization length
band,
< .. >
(2.5)
(xn , xm1 , xm2 , xm3 ),
η1
and
η2
for various disorder strengths and for various
which are within the localization length. Otherwise the sum (1.19) is
negligible. It is demonstrated in Chapter 3 and that issue is discussed there in great details.
that this is indeed the case and there is a typical magnitude of the value of the of the overlap
sum (1.19) and it scales as,
V = C1 ξ −η
where
by
ξ
C1
is a constant at least for the dominant
Vnm1 ,m2 ,m3 .
(2.6)
Here and in what follows we denote
the localization length in the center of the band.
Making the assumption that
the order of
ξ3
Fn
is random because the sum on the RHS of (1.17) consists of
terms, at least for weak disorder. These are rapidly oscillating in time, and it is
a nonlinear function of the
cmi (t).
This assumption will be tested in detail in the next Section.
The RHS of (1.17) is assumed to take the form [11]
Fn = V Pβρ3/2 fn (t) =
where
C1
C1
Pβρ3/2 fn (t)
ξη
(2.7)
is a constant and
P = A0 β γ ξ α ρ
15
(2.8)
is proportional to the number of "resonant modes", namely ones that strongly aect the dynamics of the state
n.
Although it is reasonable to assume that the number of resonant modes
is proportional to the density
ρ
a strong argument for it is missing, nevertheless it is consistent
with all numerical results [10, 11]. We assume here the form (2.8) where
dependent of
β
and
ξ.
A0
is a constant in-
In the end of this section we argue that within these assumption
in agreement with the assumption of [11, 10]. The value of
α
γ=1
is estimated numerically (see Sec.
2.3). Under these assumptions (1.17) reduces to:
i∂t cn (t) = Fn (t)
We assume
Fn (t)
(2.9)
can be considered random with rapidly decaying correlations and that the
distribution function of
fn (t)
is stationary.
Consequently the integral of correlation function
C (t0 ) = hf (0) f (t0 )i, where h..i is the average over the random potential, converges.
results in
C1
cn (t) = −i η Pβρ3/2
ξ
ˆ
Integration
t
dt0 fn (t0 )
(2.10)
0
leading to
< |cn (t)|2 >=
where
A1
is a constant.
A1 2 2 3
P β ρ t = A1 A20 β 2(γ+1) ρ5 ξ 2α−2η t
ξ 2η
The value of
achieved when it takes the value
ρ.
< |cn (t)|2 >
(2.11)
increases with time and equilibrium is
Transitions between states of the type of
n
(states with
small amplitude) are ignored in this model. The required time for equilibration is
T =
1
(2.12)
Bξ −2 ρ4
where we dene
B = A1 A20 β 2(1+γ) ξ 2α−2η+2
The equilibration time
T
varies slowly compared to
words there is a separation of time scales.
t
(2.13)
(see discussion after (2.21)).
On the time scale
T
In other
the system seems to reach
equilibrium by a diusion process and the density becomes constant in a region that includes
the site
n.
Hence on this time scale it seems to equilibrate. On a longer time scales, there is an
16
even longer equilibration time scale, and the resulting diusion is even weaker. The consistency
dT
dt
of the argument results of the fact that
variations of
ρ
and
T
→0
t
are slow on the scale of
for
t → ∞.
Therefore it is assumed that the
. This assumption is checked in the end of this
section. The resulting diusion coecient is
D=C
where
C
is a constant.
ξ2
= CBρ4
T
(2.14)
The assumption is that the nonlinear term generates a random walk
with the characteristic steps
T
and
ξ
in time and space. At time scales
t T,
there is diusion
and
M2 = Dt,
(2.15)
where
M1 =
X
x
and the variance
M2 =
X
x
2
x |ψ (x, t)|
2
(2.16)
(x − M1 ) |ψ (x, t)|
are the rst and second moments. Since the second moment
2
M2
(2.17)
is inversely proportional to
ρ2
one nds
where
A2
1
= A2 CBρ4 t
ρ2
(2.18)
1
1/3
= (A2 CBt) .
ρ2
(2.19)
is a constant. Therefore
The second moment satises:
M2 =
1
2/3
A2
(CBt)1/3
(2.20)
in agreement with the numerical results presented in Fig. 2.2, and
T =
1
Bξ −2 ρ4
2/3
=
C 2/3 A2 ξ 2 t2/3
Cξ 2
=
t.
1/3
M2
B
17
(2.21)
Figure 2.1: (color online) Probability distribution
8
β = 1, t = 10
(top blue/solid curve);
5
t = 10
|ψ|
2
x for W = 4 and for
β = 0, t = 105 (bottom
over lattice sites
(middle red/gray curve);
black curve). (Fig. 2 of [8]).
The density
dρ
dt
∼t
and
− 67
dρ
dt
ρ and the equilibration time T
and
dT
dt
∼t
− 31
t.
T
and
D
Since in the NLSE
β
1
ρ2 ∼ t− 3
. First note that in the long time limit
dT
dt . Ergo for the derivation of the equilibration time
long scales of spreading
for large
change with time as
t→∞
and
2
T ∼ t3 .
Hence for
both derivatives vanish
ρ can considered constant and on
can be considered constant. Therefore the theory is consistent
appears only via the combination
(2.13) and (2.14) only in the power
4
(that is in the combination
2
β |ψ (x)|
β 4 ρ4 )
The crucial assumption of the theory presented in this section is that
, it can appear in
therefore
γ = 1.
Fn (t) behaves as noise
with a rapidly decaying correlation function (in order for (2.11) to be correct). This assumption
was explicitly tested [41] and prsented in Sec. 2.2. The reason for
Fn (t)
to behave as a random
variable is that the sum (1.17) consists of many terms with random phases, and the dynamics
of the
cn (t)
are chaotic, since these are generated by the nonlinear Hamiltonian,
H.
A crossover to the regime (2.20) from the regime where a dierent power law is found is
presented in [47].
18
Figure 2.2:
M2
versus time in log-log plots. For
W =4
and
β = 0, 0.1, 1, 4.5
((o)range, (b)lue,
(g)reen, (r)ed). The disorder realization is kept unchanged. The dashed straight line guides
the eye for exponent 1/3 (Fig. 2 of [11]).
2.2
In this Section the statistical distribution of
Fn (t)
Fn (t)
is explored.
dependent NLSE (1.1) was solved numerically for a nite lattice of
of the random potential
For this purpose the time
N
sites, for
ε (x) and for W = 4 (which is the disorder strength) .
NR
realizations
The wavefunction
ψ (x, t) at time t was calculated for a single site excitation namely the initial condition ψ (x, 0) =
δx,0
using the split step method. The details of the numerical calculation are presented in the
appendix A. The expansion (1.16) of
ψ
in terms of eigenfunctions of the linear problem (1.2)
yields,
i∂t cn (t) =
X
x
2
β |ψ (x, t)| ψ (x, t) un (x) eitEn ≡ Fn (t) .
19
(2.22)
This equation was used to calculate
whether
Fn (t)
numerically for a lattice of
N
sites. In order to check
Fn (t) can be considered as noise we calculated its power spectrum and auto-correlation
function. First we present results obtained for times up to
N = 1024
for a single site excitation at
realizations.
M2 ∝ t1/3
t = 0.
t = 105
for
β = 1, W = 4 (ξ ≈ 6.4),
The calculation was preformed for
NR = 50
For nearly all these realizations it was found that the second moment grows as
in agreement with the results of [8, 10, 11] as presented in Sec. 2.1. We focus rst
on such realizations and present the results for a specic realization in Fig. 2.3
2.2.1
Fn (t)
Exhibits the Power Spectrum of Noise
The power spectrum is
2
Sn (ω) = F̂n (ω) ,
where
1
F̂n (ω) = lim √
t̃→∞
t̃
ˆt̃
0
(2.23)
Fn (t) · e(−iωt) dt.
(2.24)
is the nite time Fourier transorm. It is plotted for some realization in Fig. 2.3.a for n=0. It
exhibits a peak around
|ω0 | ≈ 1.72
and its width is
4ω ≈ 0.1.
The nite width is characteristic
of noise. Also the Fourier transform of
F̃n (t) = Fn (t) · e−iω0 t
will exhibit a wide power spectrum near
ω = 0,
with the width of
(2.25)
4ω
that is characteristic of
noise.
2.2.2
Fn (t)
Has Rapidly Decaying Auto-correlation Function
The auto-correlation function of
Fn (t)
is
Cn (τ ) = Fn (t) · Fn∗ (t + τ )
20
(2.26)
where bar denotes time average
For
F̃n (t)
placed by
g (t) ≡ limt̃→∞
´t
1 e
g (t) dt.
e
t 0
we dene the auto-correlation function
F̃n (t)
. In Fig. 2.3 .b we plot
zoomed version is plotted.
(R)
Cn
C˜n (τ )
that is just (2.26) with
= Re (Cn (τ ))
for
n=0
Fn (t)
re-
while in Fig.2.3 c the
|ω0 | ≈ 1.72 that is
= Re C̃n (τ ) , presented in
Note an oscillation of frequency of the order
superimposed on the function. In the corresponding plots of
(R)
C̃n
Fig.2.3.d and Fig.2.3.e, one does not nd this oscillation. Behavior of the imaginary part of the
auto-correlation function
in Fig.2.3 are for
n = 0.
en(I) = Im C
en (τ )
C
is similar (see Fig.2.3.f ).
Similar results were found also for
n=3
and
All results presented
n = 15.
the auto-correlation function decays by 2 orders of magnitude on the scale of
order of
2π/4ω ∼ 65).
Therefore the correlation of
We see that
4τ ≈ 140
(of the
F̃n (t) behaves as the one of noise with short
time correlations. For realizations where the growth of the second moment
M2 ∼ t1/3
was not
found, the power spectrum was found to be substantially narrower by 2 orders of magnitude.
The calculations were repeated for
β=2
where similar results were found, and for
β = 0.5.
For
the latter case the number of realizations where it was found that the second moment grows
like
t1/3
is substantially smaller than for
β =1
or
β = 2.
In all cases where the width of the
power spectrum was small the typical growth of the second moment
and vice versa.
M2 ∼ t1/3
was not found
This demonstrates the strong relation between the eective noise behavior and
the diusive growth of the second moment. It also demonstrates the dierent behavior of various
realizations of the randomness.
We turn now to test the distribution of
sequence of points separated by
ta > 4τ
F̃n (t).
F̃n (t)
, that is for points where the values of
uncorrelated, and compute the distribution of
presented in Fig. 2.4 for
For this purpose we sample
F̃n (k · ta )
for
k = (1, 2, ..K).
for a
F̃n (t)
are
The results are
t = 105 , ta = 200, K = 500.
2.2.3 Stationarity of Fn (t)
In Fig. 2.4 and in Fig. 2.5 we demonstrate that the distribution of
Fn (t) is stationary.
t0 .
in Fig. 2.5 we show that the auto-correlation function is independent of
21
Namely
2.2.4 Averages of Fn (t) and Cn (τ )
We calculated the averages of
Fn (t)
and
Cn (τ ),
A.
22
the numerical data is presented in Appendix
−4
x 10
0.04
(a)
(b)
3
0.03
0.02
0.01
0
S0(ω)
C(R) (τ)
2
0
−0.01
1
−0.02
−0.03
0
−2
−1.5
ω
−1
−0.5
0
0
0.03
0.5
1
1.5
2
τ
2.5
4
x 10
0.03
(c)
(d)
0.025
0.02
0.02
0.01
g
(R)
C0 (τ)
C(R)
(τ)
0
0.015
0
0.01
−0.01
0.005
0
−0.02
−0.005
−0.03
0
20
40
60
τ
80
100
120
0
0.5
1
τ
1.5
2
2.5
4
x 10
−3
x 10
12
(e)
(f)
0.03
10
0.025
8
g
(I
)
|C0 (τ)|
g
(R)
|C0 (τ)|
0.02
6
0.015
4
0.01
2
0.005
0
0
0
20
40
60
80
100
τ
120
140
160
180
0
50
τ
100
150
Figure 2.3: The correlation Cn (t) and power spectrum Sn (ω) of Fn (t) for W = 4 , β = 1,
N = 1024, t = 105 ,n = 0. (a) The Power Spectrum S0 (ω), (b) The auto-correlation function
˜(R)
(R)
(R)
C0 (τ ), (c) The zoomed C0 (τ ), (d) The auto-correlation function C0 (τ ), (e) the
˜(R)
˜(I)
zoomed C0
(τ ), (f ) the zoomed C0 (τ )[see text].
23
160
0.18
(a)
(b)
0.16
140
0.14
120
0.12
100
P(Y)
P(Y)
0.1
80
0.08
60
0.06
40
0.04
20
0
−0.5
0.02
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0
−1
0.5
−0.8
−0.6
−0.4
Y
and
0
Y
0.2
0.4
0.6
0.8
Y = F̃n (k · ta ) where k = (1, 2, ..K) , K = 500, ta = 200
the bin size 0.0596 . (a) For the same realization used in Fig. 1 . (b) The
distribution of values found for all NR = 50 realizations.
Figure 2.4: The distribution of
t = 105
(R)
−0.2
1
,
Figure 2.5: The absolute value of the auto-correlation function for the same realization as Fig.
2.3 with dierent
t = t0
24
dened in (2.26)
2.3
In this Section we will estimate the exponent
α dened in (2.8).
M2
with
ξ
For this purpose we write (2.20)
in the form
1
M2 = At 3
(2.27)
A = A4 ξ ν
(2.28)
with
where
ν=
2
3
(α − η + 1)
step method to obtain
(see (2.13)) while
A4
is a constant independent of
1
M2 ∼ t 3
account. This was the case for nearly all the
plots like Fig. 2.6. For
β.
ν
1 < β < 3.5
The exponent
NR
realizations for
and
α.
using the fact that
α
until
t = 106
at some stage of the calculation were taken into
ξ>7
and
regimes it was not satised for a signicant number of realizations. Fixing
strong uncertainty of
We used the split
ψ (x, t) for dierent realizations (NR = 30) and computed ψ
. Only realizations which satised
for various values of
ξ.
η ≈1
of (2.8) takes the values
These results indicate that
of magnitude but not a verication of this power law.
For further numerical data see appendix B.
25
β
β < 4.
we estimate
we nd that
ν
from
1.235 < ν < 1.71
1.85 < α < 2.56
A∼ ξ ν .
In the other
. We note the
It is an estimate of the order
6
5
4
y
3
2
1
0
−1
0
0.5
1
1.5
2
2.5
3
3.5
4
x=ln(ξ)
A dened by (2.27) and (2.28) for β = 1 (blue circles) and for
ξ . We denote y = ln (A) and x = ln(ξ) . From the least square t we
ν = 1.684 for β = 1 (blue) and ν = 1.395 for β = 3 (red).
Figure 2.6: The dependence of
β=3
(red squares) on
nd
26
Chapter 3
Statistical Properties of the
Anderson Model Relevant to the
NLSE With Random Potential
In this Chapter the statistical properties of overlap sums of groups of four eigenfunctions of
the Anderson model for localization as well as combinations of four eigenenergies are computed.
Some of the distributions are found to be scaling functions, as expected from the scaling theory
for localization. These enable to compute the distributions in regimes that are otherwise beyond
the computational resources. These distributions are of great importance for the exploration of
the NLSE in a random potential since in some explorations the terms we study are considered as
noise and the present work describes its statistical properties. Some of the results were published
[41] and some were submitted for publication [42].
27
3.1
ξ
Vnm1 ,m2 ,m3
with
in the Regime of Weak Disorder
The overlap sum
Vnm1 ,m2 ,m3
is a random function. In this subsection the scaling of its typical
values with the maximal localization length [45]
ξ≈
96
W2
(3.1)
is evaluated. This relation holds in the limit of weak disorder. In the numerical calculations
presented in this chapter
W
is varied as the control parameter and the localization length is
calculated from (3.1). The estimate (3.1) is a reasonable approximation for
as was checked explicitly (and used) in this Section. We note that the
W < 5.5 or ξ > 3.15
Vnm1 ,m2 ,m3
of substantial magnitude when all the centers of localization of the states
are within a distance
ξ.
Only such overlap sums are considered.
sums over realizations vanishes unless
,
n = m1
and
are considered.
over
NR = 5000
are varied.
m2 = m3
and all permutations.
We calculated
D
E
2
|Vnm1 ,m2 ,m3 |
un , um1 , um2 , um3
The average of the overlap
consists of two pairs of identical values
In this Section only such matrix elements
and
hVnm1 ,m2 ,m3 i
(where
h·i
denotes average
xn , xm1 , xm2 , xm3 are xed fractions of ξ , while ξ (and W )
D
E
2
hVnm1 ,m2 ,m3 i ∼ ξ −η1 and |Vnm1 ,m2 ,m3 |
∼ ξ −2η2 while the variance
realizations) while
Assuming
E
D
2
2
(Vnm1 ,m2 ,m3 ) − hVnm1 ,m2 ,m3 i
Fig. 3.1 . We conclude that
variable
(n, m1 , m2 , m3 )
take values
Vnm1 ,m2 ,m3
scales as
ξ −2η3 ,
we estimate these exponents from Figures like
η1 ≈ η2 ≈ η3 ≈ 1
. Therefore the typical magnitude of the random
η = 1.
Although this result is expected from the scaling
scales as (2.6) with
theory of localization, it is not obvious appriory. In particular it is not clear what is the eect
of cancellations of various terms resulting of opposite signs.
In summary for a crude evaluation one can assume (2.6) holds with
28
η = 1.
−3
−4
−5
(b)
−6
y
−7
−8
−9
−10
(r)
−11
−12
(g)
−13
2
2.5
3
3.5
4
4.5
5
5.5
x=ln(ξ)
Figure 3.1: Average of the overlap sums with pairs of identical indices. A log-log plot of (b)
y = ln
0, ξ , ξ
V0 3 3
a function of
, (r)
y = ln
*
0, ξ , ξ
V0 3 3
2 +
and (g)
y = ln
*
0, ξ , ξ
V0 3 3
2 +
−
0, ξ , ξ
V0 3 3
2 !
as
x = ln(ξ) , for the parameters N = 512 , NR = 5000. The localization length
11 < ξ < 103. The least square t leads to η1 = 1.039 , η2 = 0.958 and
η3 = 0.853 respectively .
varies in the interval
29
3.2
Distributions of the Overlap Sums
Vmm12 ,m3 ,m4
in the Regime
of Weak Disorder
We explore the values of
Vmm12 ,m3 ,m4
in the regime of weak disorder (which corresponds to
ξ ).
Most of the explorations are numerical. The lattice size
relatively long localization length
is xed at
N = 500.
For each realization of the
ε (x),
we computed the eigenfunctions
and ordered them in space by the center of norm coordinate, dened by
We choose
um1
with
m1 = 0
x
un
x · u2n (x)
.
to be the eigenfunction centered in the lattice. We studied only
0,0,0
,
the following quantities V0
Vmm12 ,m3 ,m4
xn =
P
V00,1,1
,
V00,0,1 ,V00,1,2
are large. Combinations with
mi & ξ
and
(taking
V01,2,3
representative of values where
m1 = 0) have negligible values because
the overlap sum is a sum of exponentially decaying functions in space of the form of (1.5). We
calculated these values for
NR = 2 · 104
strengths in the weak disorder regime
the values of
25 . ξ . 103.
realizations, and repeated this calculation for 7 disorder
1 ≤ W ≤ 2 where the maximal localization length ξ
We computed the distributions of the
Vmm12 ,m3 ,m4
takes
as follows. We
2 · 104 values of Vmm12 ,m3 ,m4 , one for each realization. We know that Vmm12 ,m3 ,m4 m ,m ,m P 2
must satisfy 0 < Vm 2 3 4 < 1 because the eigenfunctions are all normalized
x un (x) = 1.
1
m ,m ,m We made a histogram of the values Vm 2 3 4 in number of bins Nbins = 500 in the interval
1
calculated
[0, 1],
the resulting bin size is
δx = 0.002.
In order to get the distribution we normalized the
values of the histogram, dividing them by the number of realizations,
In the calculation of the statistical properties of the
according to the number of dierent indices
Vmm12 ,m3 ,m4
NR .
we distinguish dierent groups
mi .
3.2.1 The Case m1 = m2 = m3 = m4 = 0
In the case where all indices are equal we have chosen them to be zero. In this case
V00,0,0 =
X
x
u40 (x) ≡ V0 .
30
(3.2)
It is just the inverse participation ratio. Its distribution was calculated analytically by Fyodorov
and Mirlin [48] and was found to satisfy scaling, that is if
V0
and the localization length is
ξ
P (V0 , ξ)
is the probability density of
then, if one denes a scaling variable
y0 = V0 ξ
(3.3)
its probability density is
P (y0 ) =
In [48] this scaling was found to hold in a
1
P (V0 , ξ) .
ξ
(3.4)
narrow range of energy.
In the present work we
demonstrate numerically that it is an excellent approximation also when the maximal localization length,
ξ
(3.1) is used. The scaling function is dierent from the one of [48].
V0
First we verify that the average of
satises
hV0 i =
where C is a constant independent of
ξ,
and
scaling relation (in agreement with [41]).
ξ
C
ξ
(3.5)
is given by (3.1), as maybe expected from the
This is clear from Fig.
3.2, and it is found that
C = 1.296 . . ..
The probability density function (PDF) as a function of
V0
is presented in Fig. 3.3a . A
typical function tted to the numerical data is shown in Fig. 3.3b for
W =1
and it takes the
form
2
P (V0 ) = c1 e−(c2 +c3 ·ln(V0 ))
with
c1 = 127.7 . . ., c2 = 9.097 . . .
and
c3 = 1.865 . . ..
(3.6)
We found that the scaling (3.3) and (3.4)
holds for all weak disorder strengths studied as shown in Fig.3.3c. The resulting scaling function
is
2
P (y0 ) = a1 e−(a2 +a3 ln(y0 )) .
with
a1 = 1.21 . . ., a2 = 0.539 . . .
and
(3.7)
a3 = 1.71 . . ..
What is the reason for the scaling? From (1.19) and (1.5) it is clear that the magnitude of
each of the
umi
1
is of order √
ξ0
while the number of terms in the sum that contribute substantially
31
is of order
ξ0 .
Therefore
V0 ,
although random, it is typically proportional to
1
ξ0 . Note that all
the contribution to the sum (1.19) are positive.
If the calculation is conned to a narrow energy,
ξ0
is practically constant and
function found in [48]. In the case we study the energy of the site
P (y0 )
is the
m1 = 0 (middle of the lattice)
varies as the realizations change and an eective average over the realizations is performed.
Since the density of states (see Fig. 3.7) and the localization length as a function of energy are
at at the center of the band, where the localization length is maximal and takes the value close
to (3.1), terms with this value of the localization length dominate the overlap sum (1.19). It is
worthwhile to note that the scaling function (3.7) we found is dierent from the one found in
[48]. It is practically the average of the function found in [48] over energy.
Now we consider the cases where the
mi
take two dierent values say
V00,1,1
of
V00,0,1 .
3.2.2 The Case of m1 = m2 = 0; m3 = m4 = 1
Also here an argument similar to the one presented in the previous section holds, but the
localization lengths of the two wave functions involved are dierent,the overlap sum is of the
order
1
ξ0 +ξ1 , therefore
denoted here by
ln (V1 )
ln (V1 ).
V1 ,
rather than
V00,1,1
behaves as
1
ξ . In order to investigate the distribution of
V00,1,1 ,
which consists of many near zero values, we generated the histogram of
V1
. In Fig. 3.4 we present the distribution of
P (ln (V1 ))
as a function of
The best t for the scaling function, in terms of the scaling variable
y1 = V1 ξ
is shown there as well. As expected
(3.8)
hV1 i satises a relation similar to (3.5) but with C = 0.429 . . .
(in agreement with the results of Sec. 3.1 where
32
η=1
was found [41]).
3.2.3 The Case of m1 = m2 = m3 = 0; m4 = 1
Let us denote
V00,0,1 ≡ V2 .
estimate the typical value of
It is of order
V22
=
*
From the denition (1.19) it is clear that
V2
we study
X
u30
2
V2
(x) u1 (x)
x
C̄ = 0.566 . . . independent of ξ .
hV2 i = 0.
Therefore to
. It can be estimated by
!2 +
≈
*
1
. Therefore it is reasonable that
ξ02 (3ξ1 +ξ0 )
q
with
X
u60
(x) u21
(x)
x
2
V2 ∼
!+
.
(3.9)
1
ξ 3 . Indeed one nds
hV22 i = C̄ξ −1.5
(3.10)
The scaling is dierent from the one found in Subsec. 3.2.2
where only the case was two pairs of identical indices and permutation was calculated.
This
motivates us to introduce the scaling variable
y2 = V2 ξ 1.5 .
In Fig.
P ln V22
3.5 we show the distribution of
scaling variable
y2 .
(3.11)
as a function of
ln V22
in terms of the
3.2.4 Distribution of Vmm ,m ,m When 3 or 4 Dierent mi are Involved
2
1
3
4
In this case we could not nd any simple scaling relation. The averages are found to be exponential in
3.3
ξ,
as one can see from Fig. 3.6 (dierent from the case of Sec. 3.1).
1 ,m2 ,m3
Φm
n
In this Section we explore the statistical properties of
distribution of the eigenenergies
En .
Φ
for Weak Disorder
dened in Chapter 1 (1.18) and the
For the weak disorder regime we xed the lattice size
33
−3
−3.2
−3.4
−3.6
z
−3.8
−4
−4.2
−4.4
−4.6
−4.8
−5
3
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
5
x=ln(ξ)
Figure 3.2: a log-log scale of
z = ln (hV0 i)
as a function of
x = ln (ξ) . The numerical results are
z = −a0 x + b0 for a0 = 0.95 . . .
represented by circles while the line is the best t. The t is
and
b0 = 0.26 . . ..
N = 500 and computed the eigenenergies for NR = 103 realizations.
for 7 dierent values of the disorder strength ,1
<W <2
We repeated this calculation
which correspond to
25 . ξ . 103.
3.3.1 Distribution of En
The distribution of the eigenenergies for the weak disorder regime, as plotted in Fig.
symmetric around
E = 0
3.7 is
and characterized by convex function in the middle and sharply
decaying function at the boundaries.
3.3.2 Distribution of Φ+ ≡ En + Em
We calculated the distribution of the sums of two eigenenergies obtained for the same realization.
The motivation for calculation of these sums is from the terms where
are arbitrary in (1.17). In Fig. 3.8 we plot distributions of
34
Φ+
m1 = m2 = 0
and
m3 , m4
with various disorder strengths.
120
120
(b)
(a)
80
80
0
P(V0)
100
P(V )
100
60
60
40
40
20
20
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0
0.1
V0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
V0
1.2
(c)
1
0
P(y )
0.8
0.6
0.4
0.2
0
0.5
1
1.5
2
2.5
3
3.5
y0
Figure 3.3: The PDF for various values of weak disorder strength. (a) The PDF as a function
V0 for: blue circles W = 1 (ξ ≈ 103), green squares W = 1 16 (ξ ≈ 75) , red crosses W = 1 31
(ξ ≈ 58), turquoise dots W = 1 12 (ξ ≈ 46), purple pluses W = 1 23 (ξ ≈ 37), green stars W = 1 56
(ξ ≈ 30), black rhombus W = 2 (ξ ≈ 25). (b) The PDF for W = 1, the results of the
simulation are presented by blue circles, while the solid line is given by (3.6). (c) P (y0 ) as a
function of y0 . The data collapse indicates the scaling of the PDF with the localization length
ξ . The colored lines correspond to various values of the disorder strength. The black thick line
of
is the best t for the log normal distribution (3.7).
35
(a)
(b)
0.08
0.035
0.07
0.03
0.06
P(ln(y1))
P(ln(V1))
0.04
0.025
0.02
0.05
0.04
0.015
0.03
0.01
0.02
0.005
0.01
0
0
−18
−16
−14
−12
−10
−8
−6
−4
−2
−14
−12
−10
−8
ln(V )
−6
−4
−2
0
2
ln(y )
1
1
Figure 3.4: The PDF of
ln (V1 )
for various values of weak disorder strength presented in Fig
3.3. (a) The PDF as a function of
ln (V1 )
(b) The scaled PDF as a function of
ln (y1 )
the black
thick line represents the best t. The data collapse indicates the scaling of the PDF with the
localization length
ξ.
The colored lines correspond to various strengths of disorder.
(a)
0.08
(b)
0.07
0.02
P(ln(y22))
P(ln((V2)2))
0.06
0.015
0.05
0.04
0.01
0.03
0.02
0.005
0.01
0
−40
−35
−30
−25
−20
−15
−10
−5
0
−25
0
ln((V2)2)
Figure 3.5: As in Fig. 3.4 the PDF of
−15
−10
−5
0
ln(y22)
ln V22
for various values of weak disorder strength
ln V22 . (b) The scaled PDF as a function of
best t. The number of bins here is 200.
presented there. (a) The PDF as a function of
ln y22
−20
. The black thick line is the
36
−3
−3.5
(a)
(b)
−3.5
−4
−4
−4.5
−4.5
−5
z
z
−5
−5.5
−5.5
−6
−6
−6.5
−6.5
−7
−7
−7.5
20
30
40
50
60
70
ξ
80
90
100
110
−7.5
20
120
30
40
50
60
70
80
ξ
90
100
110
120
s
2 V00,1,2
z = ln
as a function of ξ . The t is z = −0.037 · ξ − 2.5
s
2 V01,2,3
z = ln
as a function of ξ . The t is z = −0.036 · ξ − 3.1
Figure 3.6: (a)
Note the maximal value of the distribution decreases with
W.
(b)
We found the following relation
between the maximal value of the distribution which we will denote
Φm
and
ξ,
Φm (ξ) = 0.13ξ 0.15
(3.12)
3.3.3 Distribution of Φ− ≡ En − Em
We calculated the distribution of the dierences between two eigenenergies obtained for the
same realization. Despite the symmetric nature of the distribution of
from
Φ+
Φ− = 0 ,
Ei
the value of
Φ−
diers
because of level repulsion [49, 50]. Therefore we anticipate a relatively large peak at
as shown if Fig. 3.9. A comparison between
Φ−
and
One can see that, the distributions dier substantially only near
of level repulsion.
37
Φ+
is presented in Fig. 3.10.
Φ− = 0
and
Φ+ = 0
because
−3
1.2
x 10
1
P(E)
0.8
0.6
0.4
0.2
0
−3
−2
−1
0
1
2
3
E
En for various weak disorder strength. blue circles W = 1 (ξ ≈ 103),
W = 1 16 (ξ ≈ 75) , red crosses W = 1 13 (ξ ≈ 58), turquoise dots W = 1 12
2
5
pluses W = 1 (ξ ≈ 37), green stars W = 1 (ξ ≈ 30), black rhombus W = 2
3
6
(ξ ≈ 25).
green squares
(ξ ≈ 46),
purple
0.25
+
P(Φ )
0.2
0.15
0.1
0.05
−4
−3
−2
−1
0
1
2
3
4
5
Φ+
Figure 3.8: The distribution of
Φ+
for strengths of disorder as in Fig. 3.7 using the same
symbols.
38
0.25
−
P(Φ )
0.2
0.15
0.1
0.05
−4
−3
−2
−1
0
1
2
3
4
Φ−
Figure 3.9: (a) The distribution of
Φ−
for strengths of disorder as in Fig. 3.7. Number of bins
used is
100.
0.3
0.45
(a)
(b)
0.25
0.2
0.35
P(Φ+) , P(Φ−)
−
P(Φ ) , P(Φ )
0.4
+
0.15
0.1
0.3
0.25
0.05
0.2
0
−4
−3
−2
−1
0
+
1
2
3
0.15
4
−0.1
−
Φ , Φ
number of bins used is
10
0
+
Φ
Figure 3.10: (a) The PDF of
2
−0.05
Φ+
used is
39
0.1
0.15
Φ− (green circles) for W = 1, total
Φ = 0 and Φ− = 0, total number of bins
(blue squares) and
. (b) zoom of (a) around
0.05
−
,Φ
103 .
+
3.3.4 Distribution of Φmm ,m ,m
2
1
3
4
2 ,m3 ,m4
Φm
= Φ is a combination of 4 eigenenrgies.
m1
needs to compute
N4
of these combinations. To avoid lengthy computations we are presenting
a much smaller number of realizations and use a smaller lattice size.
will present these distribution for
disorder strengths in the range
NR = 10
1 ≤ W ≤ 4,
distribution is found for all values of
W,
A
which correspond to
W =1
6.5 < ξ < 103.
N = 128
A Gaussian like
with the values of
σ
−Φ2
σ2
(3.13)
is the width of the gaussian. A t is presented in
A = 0.1335 . . .
[12]. Next we calculated the width of each Gaussian
σ
and
σ = 4.278 . . .
as a function of
ξ
in agreement with
and found
σ (ξ) = b1 · ξ −b2 + b3
with
b1 = 5.924 . . ., b2 = 0.865 . . .
the case of very weak disorder,
and
b3 = 4.17 . . .,
ξ 1,
for 7
as shown in Fig. 3.11a. The form of the distribution is
is the normalization constant and
Fig. 3.11b for
In this subsection we
realizations on a lattice with size
P (Φ) = Ae
where
Φ one
In order to calculate the distribution of
(3.14)
this function is presented in Fig. 3.11c. For
we see the value of
σ
approaches
σ → 4.17 . . .,
in agreement with the value found for the distribution plotted in Fig. 3.11d where
40
which is
W = 0.
0.14
0.15
(a)
(b)
0.12
0.1
P(Φ)
P(Φ)
0.1
0.08
0.06
0.05
0.04
0.02
0
−15
−10
−5
0
Φ
5
10
0
−10
15
5.6
−4
−2
0
Φ
2
4
6
8
10
(d)
(c)
0.14
5.2
0.12
5
0.1
P(Φ)
σ
−6
0.16
5.4
4.8
0.08
4.6
0.06
4.4
0.04
4.2
0.02
4
−8
0
20
40
60
ξ
80
100
0
−8
120
−6
−4
−2
0
Φ
2
4
6
8
Φ for various values of weak disorder strength. (a) blue circles W = 1
(ξ ≈ 103), green squares W = 1 21 (ξ ≈ 46) , red crosses W = 2 (ξ ≈ 25), turquoise dots
W = 2 21 (ξ ≈ 16.5), purple pluses W = 3 (ξ ≈ 11.4), green stars W = 3 12 (ξ ≈ 8.4), black
rhombus W = 4 (ξ ≈ 6.5). Number of bins = 1000. (b) the PDF as a function of Φ with
W = 1, the red dashed line is the t (3.13). (c) σ as a function of ξ , the red dashed line it the
t 3.14. (d) The PDF as a function of Φ for W = 0, the red dashed line it the t (3.13) with
values A = 0.1335 and σ = 4.27.
41
3.4
Strong Disorder
In the case of strong disorder one does not expect scaling to work [51]. Indeed we could not
nd a scaling distribution for
V0
and
V1
dened in Sec.
Their averages scale with dierent powers of
the
En
exhibits a maximum near
E=0
ξ
2 for the regime of strong disorder.
as is clear from Fig. 3.12. The distribution of
as one can see from Fig. 3.13, while for weak disorder
a minimum is found there (compare Fig.
3.7 to Fig.
3.13).
The distributions of
presented in Fig 3.14 exhibit a linear dependence on the values of
The distribution of
Φ
Φ+
and
Φ−
Φ+ and Φ−
respectively.
is similar to the one found for weak disorder, Fig. 3.11b ts even better
gaussian distribution.
42
0.125
−0.6
(b)
(a)
−0.7
0.12
−0.8
0.115
Z
z
−0.9
−1
0.11
−1.1
0.105
−1.2
−1.3
0
0.2
0.4
0.6
0.8
1
1.2
0.1
1.4
ln(ξ)
1
1.2
1.4
1.6
1.8
2
ξ
2.2
2.4
2.6
2.8
3
−2.15
(c)
−2.2
−2.25
−2.3
z
−2.35
−2.4
−2.45
−2.5
−2.55
−2.6
−2.65
1
1.2
1.4
1.6
Figure 3.12: (a)
(b)
z = hV1 i
1.8
2
ξ
2.2
2.4
2.6
2.8
3
z = ln (hV0 i) as a function of ln (ξ) with a linear t z = −0.39 · ln (ξ) − 0.63.
ξ with the t z = −0.01 · ξ + 0.13. (c) z = ln (hV2 i) as a function
of ξ with a linear t z = −0.21 · ξ − 2
as a function of
43
−3
1.5
x 10
P(E)
1
0.5
0
−8
−6
−4
−2
0
2
4
6
8
E
Figure 3.13: The PDF of En for various strong disorder strength. blue circles W = 6
(ξ ≈ 2.85), green squares W = 6 32 (ξ ≈ 2.3) , red crosses W = 7 13 (ξ ≈ 1.9), turquoise dots
W = 8 (ξ ≈ 1.6), purple pluses W = 8 23 (ξ ≈ 1.36), green stars W = 9 13 (ξ ≈ 1.17), black
rhombus W = 10 (ξ ≈ 1.02).
0.15
(b)
(a)
0.12
0.1
P(Φ−)
+
P(Φ )
0.1
0.08
0.06
0.05
0.04
0.02
0
−10
−5
0
5
0
10
Φ+
−10
−5
0
5
10
Φ−
Figure 3.14: Distribution function in the regime of strong disorder. Symbols are as in Fig.
3.13. (a) The PDF of
Φ+ .
44
(b) The PDF of
Φ− .
Chapter 4
Toy model
In this chapter a simple toy model that exhibits some properties of the Nonlinear Schrödinger
equation with a random potential was introduced and explored numerically. Its main feature is
extremely slow spreading around the initial state.
4.1
Denition of the Toy Model
The dynamics of (1.1) are conveniently described in terms of the eigenstates of the corresponding
linear model (1.2) by (1.17). These dynamics are characterized by :
1. Conservation of the norm :
Nx (t) =
X
x
2
|ψ (x, t)| = 1,
(4.1)
and for the orthonormal basis (1.5)
Nc (t) =
X
n
45
2
|cn (t)| = 1
(4.2)
2. The
Vnm1 m2 m3
are short ranged, namely the values decay in the distance between the
localization centers of the states
3. The
Vnm1 m2 m3
4. The
Emi
um1 , um2 , um3 , un .
are random variables.
are the eigenvalues of the linear problem (1.2).
Therefore they are correlated
random variables.
5. The
Emi
are correlated with
Vnm1 m2 m3
.
In the present work we introduce a toy model that satises some properties of (1.1) as presented
by (1.17). The hope is that it will enable to develop a model that is much simpler than the NLSE
with a random potential that will shed light on its behavior. For this model the eigenvectors of
H0
are replaced by,
1
ūn (x) = √ (δx,n + δx−1,n )
2
(4.3)
with a periodic boundary condition :
where
N
1 ū −N (x) = √ δx, −N + δx, N
2
2
2
2
is the size of the lattice.
Since (4.3) are localized
fore the
(4.4)
Vnm1 m2 m3
Vnm1 m2 m3
becomes non zero only for
m1 ,m2 , m3 ∈ (n, n ± 1).
There-
satisfy property (2) but not (3). The model satises property (4.2) of the
norm conservation,
The wave function is
∂
∂ X
2
Nc (t) =
|cn (t)| = 0
∂t
∂t n
ψ̄ (x, t) =
X
cn (t) e−iEn t ū (x) .
(4.5)
(4.6)
n
The norm
Nx
satises
Nx =
X
X
X
1X
2
2
ψ̄ (x, t)2 =
|cn | +
cn c∗n−1 + cn−1 c∗n ≤ 2
|cn | .
2
x
n
n
n
46
(4.7)
Two versions of the model are considered:
1. The
En
are independent random variables uniformly distributed in the interval
2. The
En
are the eigenvalues (in a random order) of
variables uniformly distributed in the interval
H0
where
εx
are independent random
W
[− W
2 , 2 ].
Version (2) satises property (4), but both versions do not satisfy property (5) since
is not random. The choice of
W̄
and
W
un
i∂t cn (t) = β
Vnm1 m2 m3
will be specied in the next section.
In order to dene the model it is useful to write (1.17) in terms of
replacing the
W̄
[− W̄
2 , 2 ].
ūn
dened by (4.3) by
given by (1.2) resulting
XX
ūn (x) ūm1 (x) ūm2 (x) ūm3 (x) c∗m1 cm2 cm3 eit(En +Em1 −Em2 −Em3 )
(4.8)
x {mi }
using (4.6) it leads to:
∂t cn (t) = −iβ
4.2
X
x
2
ei·t·En ūn (x) ψ̄ (x, t) ψ̄ (x, t) .
(4.9)
Results of the Numerical Calculations
The behavior of (4.9) for various times was computed for various values of
β.
The time is
measured in the natural (dimensionless) units of (1.1). Typically averages of various realizations
are presented.
The initial condition chosen is the following:
cn (t = 0) = δn,0
47
(4.10)
that means ,
1
ψ̄ (x, t = 0) = √ (δ0,x + δ−1,x )
2
The dierential equation (4.9) was solved by using the Runge-Kutta algorithm.
gorithm does not conserve the norm
Nc (t)
Therefore we have limited ourselves to
The Hamiltonian
εx .
H0
This al-
for long times which results of numerical errors.
t ≤ 104
such that
of (1.2) was diagonalized for
|Nc − 1| ≤ 0.06
W =2
for
R = 1000
realizations of the
The resulting eigenvalues were used in the calculations for version 2 of the toy model. One
would like the
En
of version 1 to correspond in some way to the eigenvalues of version 2. For
this purpose the eigenvalues from each realization were ordered in an ascending order, namely
such that the eigenvalues of the
r-th
realization satisfy
(r)
E1
The value of
W̄
(r)
≤ E2
≤ . . . ≤ En(r) .
(4.11)
was chosen as the average extension, namely:
1 X (r)
(r)
EN − E1 .
R r=1
R
W̄ =
(4.12)
In the calculations we present in what follows we use a lattice of size
various numbers
and
ψ̄ (x, t)
R
of realization and various values of
β.
The
cn (t)
N = 20, t ≤ 104 ,
were calculated from (4.9)
was calculated from (4.6). The second moment was obtained from (2.17).
The second moment was calculated for the 2 versions of the model and the results are
presented in Fig. 4.1. An obvious growth is found for
β ≥ 2.
The time we could run it was,
unfortunately, too short to identify the power of the growth of
D
E
2
|cn |
m2 .
In Fig.
4.2 the various
are plotted as a function of time for version (1) of the model. The behavior for version
2 is similar. We note that the long time behavior is extremely slow. Moreover
X
|n|≥5
2
|cn | ≤ 0.02
that is extremely small, hence the growth of the second moment results mainly of spreading
near the initial site.
48
(b)
(a)
50
50
45
β=3
45
40
40
β=3
35
30
25
β=2
20
<m2(t)>
<m2(t)>
35
25
20
β=2
15
15
β=1
10
5
0
30
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
5
β=0.5
0
10000
t
Figure 4.1: Average over
β=1
10
β=0.5
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
t
R = 100
realizations,
hm2 (t)i
for the the 2 versions for dierent
β:
(a) version 1, (b) version 2.
From Fig.4.1 and Fig. 4.2 we note that there is a big change for short time. Therefore we
studied in great detail the short time behavior. The results for the second moment are presented
in Fig. 4.3. We note a maximum found for some value of
t
followed by oscillations. We checked
that these oscillations involve mainly the initial site and its nearest neighbors. We denote by tm
the time where the maximum in Fig. 4.3 is found. In Fig. 4.4
49
tm
is plotted as a function of
β.
(a)
(b)
1
0.25
0.9
β=0.5
0.8
β=1
0.2
0.5
β=3
0.4
0.15
2
β=2
0.6
<|c |2(t)>
2
<|c0| (t)>
0.7
0.1
β=3
0.3
0.2
β=2
0.05
β=1
0.1
0
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0
10000
β=0.5
0
1000
2000
3000
4000
t
(c)
7000
8000
9000
0.09
0.045
0.08
0.04
0.035
<|c |2(t)>
β=3
0.06
4
0.05
0.04
β=2
0.03
0.02
0.03
β=3
0.025
0.02
0.015
β=2
0.01
0.01
β=1
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
0.005
0
β=1
0
1000
2000
3000
t
10000
(d)
0.05
0.07
<|c3|2(t)>
6000
t
0.1
0
5000
4000
5000
6000
7000
8000
t
D
E
2
R = 100 realizations of |cn | (t) of version
n = 0, (b) n = 2, (c) n = 3, (d) n = 4.
50
1 for dierent
β:
(a)
9000
10000
(a)
30
(b)
35
β=5
β=5
30
25
β=3
25
β=3
<m2(t)>
<m2(t)>
20
15
20
15
10
10
β=2
5
0
0
1
2
3
4
5
6
7
β=2
5
β=0.5
β=0.5
8
9
0
10
0
1
2
3
4
5
t
6
7
8
9
10
t
R = 1000
hm2 (t)i
realizations for
for various values of
β:
(a) version
1, (b) version 2
(b)
9
9
8
8
7
7
6
6
m
10
5
t
tm
(a)
10
5
4
4
3
3
2
2
1
1
0
0
0.5
1
1.5
2
2.5
3
β
3.5
4
4.5
0
5
0
0.5
1
1.5
2
2.5
2
1.5
1.5
ln(tm)
2
1
0.5
0
0
−1.5
−1
−0.5
4
4.5
5
1
0.5
−2
3.5
(d)
2.5
m
ln(t )
(c)
2.5
−0.5
−2.5
3
β
0
0.5
1
1.5
−0.5
−2.5
2
ln(β)
Figure 4.4: The values of
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
ln(β)
tm (β)
for the 2 versions: (a,c) version 1, (b,d) version 2.
51
Chapter 5
Summary and Discussion
5.1
Validity of the Eective Noise Theory
The eective noise theory was discussed in detail in Chapter 2. It was introduced in [37] and
was further developed in [10, 8, 11]. It was found to be consistent with the numerical results
in some regimes. In Chapter 2 a version of this theory was presented and tested numerically.
In particular the distribution of the eective driving
Fn
dened in (1.17) was studied .
The
correlation function was calculated as well and was found to be characterized by a wide power
spectrum and rapid decay with time. These were found only for realizations where subdiusion
with the second moment growing as
and the approximation of
Fn
t1/3
is found, indicating the relation between this spreading
as eective noise.
The distribution of the
Fn (t)is
found to be
stationary. These results are purely numerical and support the eective noise theory. An obvious
challenge is to obtain these results analytically. We determined that the behavior
(2.28)), with
1.235 < ν < 1.71
dependence of
P
on
ξ
A ≈ ξν
(see
is a reasonable approximation. From this we conclude that the
(2.8) is controlled by the exponent
1.85 < α < 2.56.
Although
ξ
varied
over one decade and the evaluation of the exponent is crude we believe it may give the correct
order of magnitude.
52
5.2
Possibility for the Breakdown of the Eective Noise
Theory
For the eective noise theory it is essential that
Fn (t)
the number of terms in the sum (1.17) that resonate with
not be too small. The density
ρ
and therefore
P
can be considered random. For this
n
should be large,namely
decrease with time. If
P
P
should
is very small there
may be a situation that as a result of uctuations, the sum (1.17) is dominated just by one term
and therefore it is eectively quasi periodic. If spreading is a result of the randomness of
will stop then. Let us rst estimate the time scale required to spread so that
P ≈ 1.
Fn ,
it
For this
purpose let us write (2.8) in the form
P ≈ Aξ α ρ
where
small.
when
A = A0 β .
ρ
Since
t
decreases with time
(5.1)
there is a time scale when
P
will become very
Assuming the constants are of the order of unity, using (2.13) and (2.18) the time
P≈1
t∗
satises
1
ξ 2α · 1 ≈ 1
ξ 2(α−η+1) t∗ 3
or
(5.2)
1
ξ ( 6 α+ 3 (η−1)) ≈ t∗ 3
(5.3)
t∗ ≈ ξ ( 6 α+2(η−1))
(5.4)
t∗ ≈ ξ δ
(5.5)
7
2
resulting in
21
for
1.85 < α < 2.56
where
and
6.54 < δ < 9
The time required for,
η=1
η=1
P 1,
when the eective noise theory may fail is even larger. We used
since this is the scaling of the
overlap sums for localization length
Vnm1 ,m2 ,m3
with nonvanishing average. These dominate the
ξ.
53
Such a scenario may enable to reconcile the numerical results where subdiusion is found
[52, 8, 43, 44, 10, 11] with the analytical results predicting asymptotic spreading that is at most
logarithmic [52, 9, 53, 12]. These points should be subject of future research.
5.3
Statistical Properties of the Anderson Model Relevant
for the NLSE with a Random Potential
In some cases of weak disorder it was demonstrated that scaling holds. In particular it was
shown that for weak disorder (or large localization length
V00,1,1
,
V00,0,1
y1 = ξV00,1,1
are functions of these variables and of
and
y2 = ξ 3/2 V00,0,1
where
ξ
via one scaling variable,
V00,1,2
and
V01,2,3 .
y0
was calculated numerically (3.7). We
In addition to the fundamental interest,
the scaling function can be extremely useful for the case when
obtain the distribution of
Vmm12 ,m3 ,m4
y0 = ξV00,0,0 ;
is the maximal localization length given by (3.1) (see
(3.3), (3.8) and (3.11)). The distribution function of
could not nd a scaling function for
ξ
ξ) the distribution functions of V00,0,0 ,
ξ
is very large.
It enables to
for regimes where numerical calculations require a basis
of size that is beyond the available computer resources. Also the averages and variances of the
Vmm12 ,m3 ,m4
were computed for some
scale simply with
ξ
{mi } .
In most cases of weak disorder we analyzed, these
as one could guess from the distribution functions of the scaling variable.
In some cases the averages are exponential in
ξ
(see Fig. 3.6).
The distribution functions of combinations of energies of the Anderson model were studied
as well. A dierence between the distribution of
near
Φ+ = 0, Φ− = 0.
Φ+ = En + Em
It is a signature of level repulsion.
Em1 + Em2 − Em3 − Em4
and
Φ− = En − Em
The distribution of
was found
m2 ,m3 ,m4
Φm
=
1
was found to be gaussian and the dependence of the variance on the
localization length was computed see (3.14).
5.4
Toy Model
In Chapter 4 a toy model with an equation of the evolution of the expansion coecients of
the wave function (1.17) or (4.8) that is similar to the one of the NLSE (1.1) is presented and
54
explored numerically. It is found to exhibit spreading similar in some aspects to the one found
for the NLSE. In particular the spreading is slow and involves a small number of sites near the
initial site, as is the case for the original NLSE reported in [8] and [11]. On the basis of this
observation one may speculate that slow spreading over a small region in the vicinity of the
initial position is characteristic of a family of nonlinear equations with random potentials. For
short times oscillation around the initial site was found. We hope that for the model presented
here it will be much easier to obtain analytical results than for the original NLSE. So far such
results were not found.
55
Appendix A
5.5
Fn (t)
Fn (t)
In this appendix the statistical distribution of
is explored.
dependent NLSE (1.1) was solved numerically for a nite lattice of
of the random potential
linear problem (
??).
β
were used. The wavefunction
ψ (x, 0) = δx,0
sites, for
NR
ψ (x, t)
at time
t
realizations
Various values
was calculated for
using the expansion (1.16) in terms of eigenfunctions of the
In particular (1.17) can be written in the form
i∂t cn (t) =
X
x
2
β |ψ (x, t)| ψ (x, t) un (x) eitEn ≡ Fn (t)
This equation was used to calculate
Fn (t)
numerically for a lattice of
subsection the values of the parameters used are
and
N
ε (x) for various values of the randomness parameter W .
of the nonlinear parameter
the initial condition
For this purpose the time
N
(5.6)
sites. In the present
NR = 104 , N = 160, t = 103 , W = 4, 5, 7, 8, 10,
β = 0.1, 0.5, 1.
The complex function
Fn (t)
can be written as
To test its distribution, histograms of the values of
the range between the extremal values of
Fn ).
(R)
Fn
(R)
Fn
(I)
+ iFn
and
where
(I)
Fn
(R)
Fn
and
(I)
Fn
are real.
were prepared (100 bins in
Fig.5.1 is prepared for
W =4
and
β = 1.
The
distribution is found to take the form
ζ
P (y) = P0 e−a|y|
was tted and the
P0, , a, ζ
(5.7)
constants are obtained from the t. The quality of the t is rated by
56
the parameter
R2
[54], where
R2 = 1
represents the best possible t. The results summarized
in Table 5.1 are obtained from Figures like Fig.(5.1) for various
distributions of the real and imaginary parts are similar.
functional form takes the value
reasonable approximations is
n=3
and
n = 10
ζ ≈ 0.5.
P0 ≈ 0.15
W
and
β.
The parameter
ζ
We note that the
that controls the
The variation of the other parameters is larger but a
and
a ≈ 10.
Results of this nature were found also for
. The results are presented in Table 5.2 and 5.3. We do not understand the
signicance of the various parameters. These were calculated mainly to verify the consistency
of the analysis.
0.16
0.16
Fitted Values
(a)
(b)
0.14
P0=0.3382 ±0.0418
0.12
a=17.2±1.48
ζ=0.4132±0.029
R2=0.9962
P0=0.322±0.0377
a=18.37±1.69
ζ=0.4326±0.03
R2=0.9959
0.12
0.1
P(y)
0.1
P(y)
Fitted Values
0.14
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0
−0.1
0.1
−0.08
−0.06
−0.04
−0.02
y=Re[F0(t=103)]
0
0.02
0.04
y=Im[F0(t=103)]
(R)
P (y) of (a) y = F0
t = 103 and (b)
W = 4 , β = 0.1, N = 160, NR = 104 . The Red curve is the best
Figure 5.1: The normalized distribution
(I)
y = F0
t = 103
for
t to
(5.7) (with the parameters presented in the inset).
In order to explore the correlations of
Fn (t)
we dene
fn ≡ Fn − hFn i
where
h·i
denotes the
ensemble average. The auto-correlation function is dened as
D
E
Cn (τ ) = fn (t) · fn∗ (t + τ )
where bar denotes time average
g (t) ≡ limt̃→∞
57
´t
1 e
g (t) dt.
e
t 0
(5.8)
We denote
(R)
Cn (τ ) = Cn
+
0.06
0.08
0.1
(R)
t = 103
ζ
F0
parameters
t = 103
a
ζ
P0
0.4372
0.9986
0.2272
16.87
0.4382
0.9978
0.4502
0.9965
0.2199
15.56
0.4401
0.9956
16.86
0.5017
0.9964
0.1536
17
0.5053
0.9966
0.1233
17.72
0.546
0.9958
0.1338
15.85
0.5087
0.9927
8
0.0868
13.34
0.5013
0.9902
0.09151
12.96
0.4856
0.9931
4
0.1438
7.081
0.4886
0.9917
0.1395
8.813
0.5546
0.9977
5
0.2044
6.317
0.3668
0.9823
0.1947
6.545
0.3863
0.9804
6
0.094
7.232
0.5665
0.988
0.09789
7.123
0.5437
0.9912
8
0.0077
6.12
0.5325
0.9939
0.0754
6.125
0.5437
0.987
4
0.1378
9.416
0.569
0.9934
0.137
10.38
0.5993
0.998
5
0.1313
6.909
0.4632
0.9943
0.1154
8.004
0.5399
0.9963
6
0.1119
6.261
0.463
0.9921
0.1086
6.508
0.4857
0.9871
8
0.08121
5.67
04805
0.9834
0.07183
6.341
0.5544
0.9887
W
P0
a
0.1
4
0.2207
16.84
5
0.2105
15.92
6
0.1554
7
1
(I)
F0
R2
β
0.5
Table 5.1: The values of the tted coecients
(R)
parameters
F3
a
t = 103
ζ
P0 , a, ζ
of (5.7) for dierent values of
(I)
R2
R2
t = 103
ζ
W
P0
0.1
4
0.1606
16.08
0.4591
0.9841
0.138
17.42
0.4978
0.9777
5
0.1438
23.16
0.5597
0.9875
0.1381
24.63
0.5792
0.9895
4
0.1572
7.796
0.4681
0.9584
0.152
7.897
0.4792
0.9623
5
0.12
10.29
0.625
0.9961
0.1269
9.559
0.5865
0.9958
,
(R)
parameters
F10
a
of (5.7) for dierent dierent values of
(I)
F10
a
t = 103
ζ
R2
P0
0.4635
0.9992
0.4558
41.34
0.4765
0.999
0.2389
0.9992
5.784
35.7
0.3092
0.9981
15.07
0.3331
0.9994
1.412
14.97
0.3283
0.9991
23.43
0.4193
0.9991
1.553
21.38
0.3913
0.9992
β
W
P0
0.1
4
0.4637
38.65
5
17.8
27.99
4
1.347
5
1.369
0.5
t = 103
ζ
P0 , a, ζ
W .
,
P0 , a, ζ
W .
58
.
R2
β
0.5
P0
F3
a
β,W
β
R2
of (5.7) for dierent dierent values of
β
10
9
8
7
4
6
2
5
C0 (τ)
6
0
4
−2
3
−4
2
−6
1
−8
0
100
200
Fit
|C0 (τ)|
(b)
(a)
8
300
400
500
τ
600
700
800
900
0
1000
Fitted Values
P1=7.959±0.018
τ0=−19.61±1.25
σ=286.9±1.6
R2=0.9973
0
100
200
300
400
500
τ
600
700
800
900
1000
W = 4, β = 0.1, NR = 104 ,
(R)
N = 160. The red solid curve is |C0 (τ )| the blue dashed curve is C0 (τ ) and the green point
(I)
dashed curve is C0 (τ ) . (b) The auto-correlation function |C0 (τ )| for W = 4, β = 0.1,
NR = 104 , N = 160. The red solid curve represents the numerical data of |C0 (τ )| and the blue
Figure 5.2: The auto-correlation function of
Fn
at
n = 0,
for
dashed curve is the tted function (the tted parameters are in the inset).
(I)
iC
n
r
(R)
Cn
2
(R)
(R)
(I)
(I)
Cn and Cn are real. In Fig. 2 we plotted Cn (τ ) , Cn (τ ) and |Cn (τ )| =
2
(I)
4
+ Cn
for W = 4, β = 0.1, NR = 10 , N = 160 and n = 0. In Fig. 5.2b a t to
where
Cn (τ ) = P1 e−(
is presented and the parameters were tted for
τ0 −τ
σ
W =4
2
)
and
(5.9)
β = 0.1.
The t works also for other
values of the parameters and the results are presented in Table 5.4 . We see that the t is good
for various values of the parameters. We found the Gaussian decay presented here superior to
other ts we tried including power-laws and exponential. Results of similar nature were found
also for
n = 10
and
n = 3.
We conclude that the decay of the correlation function is very rapid.
59
parameters
2
)
R2
-19.61
286.9
0.9973
-18.62
290.5
0.998
8.879
-44.12
331.5
0.997
9.119
-63.16
357.3
1
10
9.804
-105.8
370.5
0.997
4
199.8
-5498
60.83
0.9936
5
224.5
-13.63
66.41
0.987
6
211.6
0.8531
54.4
0.9938
8
213.8
3.565
54.08
0.9956
4
199.8
-5.498
60.83
0.9936
5
215.4
-7.776
62.8
0.9899
6
207.8
-0.864
55.69
0.9926
8
216.5
1.993
57.11
0.9944
W
P1
0.1
4
7.959
5
8.369
7
8
1
τ0 −τ
σ
σ
β
0.5
P1 e−(
τ0
Table 5.4: The values of the tted parameters
P1 , τ0 , σ
60
of (5.9)for dierent parameters
β , W.
61
Appendix B
5.6
Vnm1 ,m2 ,m3
with
ξ
The overlap sum
Vnm1 ,m2 ,m3
is a random variable. In this subsection the scaling of its typical
values with the maximal localization length [45]
ξ≈
96
W2
(5.10)
is evaluated. This relation holds in the limit of weak disorder. In the numerical calculations presented in this paper we vary
proximation for
W
as the control parameter. The estimate (5.10) is a reasonable ap-
W < 5.5 or ξ > 3.15 as was checked explicitly (and used) in this subsection.
note that the
Vnm1 ,m2 ,m3
of the states
un , um1 , um2 , um3
We
takes value of substantial magnitude when all the centers of localization
are within a distance
ξ.
Only such overlap sums are considered.
The average of the overlap integral over realizations vanishes unless
(n, m1 , m2 , m3 )
consists
n = m1 and m2 = m3 and all permutations. We calculated
D
E
2
|Vnm1 ,m2 ,m3 | and hVnm1 ,m2 ,m3 i (where h·i denotes average over NR = 5000 realizations) while
of two pairs of identical values ,
n, m1 , m2 , m3 are xed fractions of ξ , while ξ (and W ) are varied. Assuming hVnm1 ,m2 ,m3 i ∼ ξ −η1
E
D
E
D
2
2
m ,m ,m 2
∼ ξ −2η2 while the variance (Vnm1 ,m2 ,m3 ) − hVnm1 ,m2 ,m3 i scales as ξ −2η3 ,
and |Vn 1 2 3 |
we estimate these exponents and the results are presented in Table 5.5 and Table 5.6 for various
values of the parameters (in Table 5.6 only cases where
the order of magnitude
smaller
that
Om
Om
hVnm1 ,m2 ,m3 i = 0
is presented (the size of a term is
10Om ).
are presented). Also
It is understood that the
is, the less it aects the result of this work. From Table 5.5 and Table 5.6 we note
η1 ≈ η2 ≈ η3 ≈ 1
scales as (2.6) with
. Therefore the typical magnitude of the random variable
η = 1.
Another estimate of the overlap sum is by
Vabs =
62
DP
n,m1 ,m2 ,m3
|Vnm1 ,m2 ,m3 |
E
Vnm1 ,m2 ,m3
as a function of
hVnm1 ,m2 ,m3 i
n = 0, m1 = 0, m2 = 0, m3 = 0
n = 0, m1 = 0, m2 = ξ/2, m3 = ξ/2
n = 0, m1 = 0, m2 = ξ/3, m3 = ξ/3
n = 0, m1 = 0, m2 = ξ/6, m3 = ξ/6
n = ξ/2, m1 = ξ/2, m2 = ξ/6, m3 = ξ/6
n = ξ/2, m1 = ξ/2, m2 = ξ/3, m3 = ξ/3
D
E
2
2
(Vnm1 ,m2 ,m3 ) − hVnm1 ,m2 ,m3 i
η1
0.759
0.959
1.001
1.01
1.003
1
η3
n = 0, m1 = 0, m2 = 0, m3 = 0
n = 0, m1 = 0, m2 = ξ/6, m3 = ξ/6
n = 0, m1 = 0, m2 = ξ/6, m3 = ξ/6
n = 0, m1 = 0, m2 = ξ/6, m3 = ξ/6
n = ξ/6, m1 = ξ/6, m2 = ξ/6, m3 = ξ/6
n = ξ/6, m1 = ξ/6, m2 = ξ/6, m3 = ξ/6
Table 5.5: The exponents
size
N = 512
η1
and
η3
Om
0.435
0.9
0.881
0.831
0.892
0.832
(−2) − (−3)
(−4) − (−6)
(−4) − (−6)
(−3) − (−5)
(−3) − (−5)
(−3) − (−5)
for the average and the variance of
NR = 5000. We varied the
11 < ξ < 150 (0.8 < W < 2.9).
, number of realizations
interval
W.
Om
(−1) − (−2)
(−2) − (−3)
(−3) − (−5)
(−2) − (−3)
(−2) − (−3)
(−2) − (−3)
Vnm1 ,m2 ,m3
It was claimed [55] on the basis of numerical and analytical estimates that
hence
Vabs v ξ −2 ln ξ
Vabs v ξ −1.7
and
−1/2
η = 1.7
. Numerically it was found that
Vabs v W 3.4
(we checked this results independently).
purposes of our work the other estimates are relevant and we assume
For
ξ 11
we couldn't nd smooth curves of
equal to the integer part of ξ/a where
are signicant, since
ξ
of the overlap sums as
a is xed and ξ
Vabs v W 4 ln (W )
which corresponds to
We believe that for the
η = 1.
The reason is that the
varies. For small
ξ
the jumps in
mi
are
Vnm1 ,m2 ,m3
does not cover many integers. The results obtained indicate that scaling
ξ −1
assume (2.6) holds with
5.7
Vnm1 ,m2 ,m3 .
for lattice
localization length in the
holds for values
ξ < 11.
In summary for a crude evaluation one can
η = 1.
The numerical values of the exponent
ν
M2
with
ξ
dened in (2.28) obtained from plot like Fig. 5.3 and
Fig. 5.4 are shown in Table 5.7.
63
D
(Vnm1 ,m2 ,m3 )
2
E
n = 0, m1 = 0, m2 = 0, m3 = 0
n = 0, m1 = 0, m2 = 0, m3 = ξ/2
n = 0, m1 = 0, m2 = 0, m3 = ξ/3
n = 0, m1 = 0, m2 = 0, m3 = ξ/6
n = 0, m1 = 0, m2 = ξ/2, m3 = ξ/2
n = 0, m1 = 0, m2 = ξ/3, m3 = ξ/3
n = 0, m1 = 0, m2 = ξ/6, m3 = ξ/6
n = ξ/2, m1 = ξ/2, m2 = ξ/6, m3 = ξ/6
n = ξ/2, m1 = ξ/2, m2 = ξ/3, m3 = ξ/3
η2
Om
0.6145
(−2) − (−3)
(−4) − (−6)
(−4) − (−6)
(−3) − (−6)
(−3) − (−6)
(−2) − (−3)
(−2) − (−5)
(−3) − (−5)
(−3) − (−5)
1.168
1.1805
1.03
0.934
0.9525
0.9315
0.959
0.928
η2 for the second moment of the overlap integral V0m1 ,m2 ,m3 where the
N = 512 , Number of realizations NR = 5000. We varied the localization length
in the interval 11 < ξ < 150 (0.8 < W < 2.9).
Table 5.6: Exponent
lattice size is
β=1
5
(a)
4
y
3
2
1
0
−1
0
0.5
1
1.5
2
2.5
3
3.5
x=ln(ξ)
A dened by (2.20) and (2.21) on ξ , for β = 1. The localization
1.5 < ξ < 24 . y = ln (A) and x = ln(ξ) . The red solid line is
y = 1.684 · x − 1.1, resulting in ν = 1.684.
length is varied in the interval
64
β=3.5
5
(b)
4.5
4
3.5
y
3
2.5
2
1.5
1
0.5
0.5
1
1.5
2
x
2.5
3.5
ξ , for β = 3.5. The
2.6 < ξ < 25.3. y = ln (A) and x = ln(ξ) . The
y = 1.235 · x + 0.329 , resulting in ν = 1.235.
A
3
dened by (2.20) and (2.21) on
localization length is varied in the interval
solid line is
β
ν
1
1.684
1.3
1.677
1.8
1.779
2
1.712
3
1.4
3.5
1.235
ξ
1.9 < ξ < 25.5
2.6 < ξ < 25.5
2.6 < ξ < 25.5
1.9 < ξ < 25.5
2.6 < ξ < 25.3
2.6 < ξ < 25.3
red
A4
0.33
0.414
0.408
0.505
1.1
1.389
Table 5.7: The value of the exponent
ν
for dierent values of the constant
realizations is
65
NR = 30.
β.
The number of
Appendix C
5.8
Some Details of the Numerical Calculations - Split Step
Method
We used the split step method to obtain the time evolution starting from the initial wavefunction.
The lattice size
N
used is
512 or 1024.
The reason we used the relativity large lattice is because
we wanted to avoid boundary eects, namely we required the wavefunction amplitude to be
smaller than
10−12
on the boundary. The time step used in the split step method is
dt = 0.1.
We used this time step because it is small enough relative to the time scales in the system at
hand and large enough in order to complete the numerical calculation in reasonable time. It is
the smallest time step used in [10, 11]. The initial condition used is a single site excitation in
the middle of the lattice denoted by
xn = 0
namely,
66
ψ (x, t = 0) = δx,0 .
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72
‫תיאורית רעש אפקטיבי עבור משוואת‬
‫שרדינגר הלא לינארית‬
‫ארז מיכאלי‬
‫‪i‬‬
ii
‫תיאורית רעש אפקטיבי עבור משוואת‬
‫שרדינגר הלא לינארית‬
‫חיבור על מחקר‬
‫לשם מילוי חלקי של הדרישות לקבלת התואר‬
‫מגיסטר למדעים בפיזיקה‬
‫ארז מיכאלי‬
‫הוגש לסנט הטכניון ־מכון טכנולוגי לישראל‬
‫סיוון התשע''ב חיפה יוני ‪2012‬‬
‫‪iii‬‬
‫תודות‬
‫מחקר זה נעשה בהנחייתו של פרופסור שמואל פישמן בפקולטה לפיזיקה בטכניון‪.‬‬
‫אני מודה לטכניון על התמיכה הכלכלית הנדיבה בהשתלמותי‪.‬‬
‫‪iv‬‬
‫הקדשה‬
‫מחקר זה מוקדש לאסנת ואהרון מיכאלי הורי היקרים שתמכו בי‪ ,‬בכל רובד אפשרי‪ ,‬במהלך לימודי התואר‬
‫ובמהלך כל חיי‪.‬‬
‫תודה‬
‫‪v‬‬
‫תקציר‬
‫משוואת שרדינגר הלא לינארית עם פוטנציאל אקראי הינה בעיה בסיסית שמקורה בניסויים באופטיקה‬
‫ובאופטיקה של אטומים‪ .‬הבעיה הינה הבעיה הייצוגית של תחרות בין האקראיות והלא לינאריות‪ .‬דינמיקה‬
‫לינארית בנוכחות פוטנציאל אקראי מראה תופעה מאוד מיוחדת בשם "לוקליזציית אנדרסון"‪ .‬אנדרסון בשנת‬
‫‪ 1958‬פרסם מאמר מהפכני שניבא העדר דיפוזיה בגבישים מסוימים בעלי פוטנציאל אקראי‪ .‬תופעת לוקליזציית‬
‫אנדרסון באה לידי ביטוי בתחומים רבים כגון הולכה חשמלית‪ ,‬התפשטות גלי אור‪ ,‬עיבוי בוזה־איינשטיין‬
‫ואקוסטיקה‪ .‬במהלך השנים תופעת הלוקליזציה הפכה מובנת יותר‪ .‬במימד אחד ובשני מימדים לכל ערך של‬
‫פוטנציאל אקראי נמצא את תופעת הלוקליזציה‪ ,‬בעוד שלמקרה התלת מימדי קיים ערך סף של האנרגיה של‬
‫המערכת המפריד בין מצבים ממוקמים ולא ממוקמים‪ .‬במסגרת עבודה זו נתמקד במקרה החד מימדי בו ידוע‬
‫כי כל הפונקציות העצמיות של מודל אנדרסון‪ ,‬כלומר מודל שבו יש פוטנציאל אקראי הקבוע בזמן‪ ,‬הן ממוקמות‬
‫אקספוננציאלית‪ .‬כמו כן‪ ,‬ידוע כי במודל אנדרסון חבילת גל לא תתפשט ללא הגבלה‪ ,‬כלומר פונקציית גל‬
‫ממוקמת‪ ,‬תתחיל להתפשט אבל לאחר זמן מסוים ההתפשטות תיעצר‪ .‬במקרה של משוואת שרדינגר הלא‬
‫לינארית עם פוטנציאל אקראי‪ ,‬התשובה לשאלה היסודית ‪ :‬מה יקרה לפונקציית גל )או חבילת גל(‪ ,‬הממוקמת‬
‫בהתחלה‪ ,‬בזמנים ארוכים אסימפטוטית? לא ידוע‪ .‬האם האיבר הלא לינארי האחראי להרחבת פונקציית הגל‬
‫על ידי אינטראקציות בין האופנים העצמיים השונים ינצח ותתקיים התפשטות או האקראיות של הפוטנציאל‬
‫תנצח ובסוף תתקיים לוקליזציה‪ ,‬לפי מודל אנדרסון? במטרה לענות על השאלה הזאת בוצעו מחקרים אנליטיים‪,‬‬
‫מתמטיים ונומריים נרחבים במהלך עשרים השנים האחרונות‪ .‬ישנן מספר גישות סותרות שבאות ליישב מחלוקת‬
‫זאת‪ .‬ישנם מספר טיעונים היוריסטיים המצביעים על קיום לוקליזציה בגלל שימור הנורמה של פונקציית הגל‬
‫ובסופו של דבר הזנחת האיבר הלא לינארי המותיר את הדינמיקה להיות שקולה לזו של מודל אנדרסון‪ .‬חישובים‬
‫נומריים שבצעו בשנים האחרונות מראים את ההפך הם מראים־ תת־דיפוזיה‪ .‬כחלק מהניסיון להסביר את‬
‫התוצאות הנומריות פותחו מספר תיאוריות המסבירות את ההתפשטות התת־דיפיוזית הזאת‪ .‬במסגרת חיבור‬
‫זה‪ ,‬נציג את הפירוש שלנו במסגרת אחת מהתיאוריות האלו ־ תיאורית הרעש האפקטיבי ונבדוק את ההנחות‬
‫של התאוריה הזאת בצורה נומרית‪.‬‬
‫בפרק הראשון של החיבור‪ ,‬נציג מבוא לשטח המדובר‪ .‬נציג את המושגים העיקריים של מרחק לוקליזציה‪,‬‬
‫חוזק האקראיות וחוזק האי־לינאריות‪ .‬כמו כן נציג עובדות ידועות על משוואת שרדינגר הלא לינארית ונציג‬
‫שני קבועי תנועה‪ ,‬הנורמה של פונקציית הגל והאנרגיה )ההמילטוניאן הקלאסי היוצר את משוואת שרדינגר‬
‫הלא לינארית עם הפוטנציאל האקראי( המעידה על אופיה הכאוטי של הבעיה‪ .‬כמו כן נציג תיאור של מערכות‬
‫‪vi‬‬
‫פיזיקליות הרלוונטיות למשוואת שרדינגר הלא לינארית‪) :‬א( מערכת אופטית לא לינארית שבה השדה החשמלי‬
‫משפיע על מקדם השבירה ונוצרת תגובה לא לינארית להתקדמות חבילת הגל‪) .‬ב( מערכת של עיבוי בוזה־‬
‫איינשטיין בקירוב של השדה הממוצע‪ .‬בהמשך הפרק נציג שיטות אנליזה מקובלות בשטח‪ ,‬כמו פיתוח פונקציית‬
‫הגל במצבים העצמיים של מודל אנדרסון‪ ,‬כלומר מצבים עצמיים ממוקמים אקספוננציאלית במרחב‪ .‬פיתוח זה‬
‫מתגלה כמאוד נוח לעבודה תאורטית ונומרית בתחום‪.‬‬
‫הפרק השני‪ ,‬שהוא העיקרי בחיבור‪ ,‬מתאר את התאוריה של הרעש האפקטיבי ואת ההנחות הבסיסיות‬
‫שלה‪ ,‬תיאוריה זו מתבססת על עבודתם של פלאך )‪ ,(Flach‬שפליינסקי )‪ (Shepelyansky‬ושותפיהם‪ .‬התאוריה‬
‫מתבססת על עבודות נומריות שבוצעו במהלך המחקר האינטנסיבי בשנים האחרונות‪ .‬העובדות העיקריות הן‪:‬‬
‫פונקציית הגל מתפשטת בצורה של תת־דיפוזיה‪ ,‬בחלק גדול מתחום הפרמטרים של הבעיה‪ ,‬עם אקספוננט‬
‫דיפוזיה של ‪) 1/3‬להבדיל מ ־ ‪ 1‬בדיפוזיה רגילה(‪ .‬כמו כן נמצא שלאחר זמן מספיק ארוך פונקציית הגל היא‬
‫כמעט אחידה על תחום רחב סביב האזור משם התחילה ההתפשטות‪ ,‬הן במרחב האמיתי והן במרחב של‬
‫המצבים העצמיים של הבעיה הלינארית‪ .‬בתיאוריה זו אנחנו מניחים שהתפשטות של חבילת הגל נובעת מרעש‬
‫אקראי הנובע מאינטראקציות בין מספר גדול של האופנים השונים‪ .‬כאשר פונקציית הגל מתפשטת אל המצב‬
‫העצמי מחוץ לאזור השטוח‪ ,‬מצב זה הופך להיות חלק מהמצבים המחוללים רעש והתהליך הזה חוזר על עצמו‬
‫וגורם להמשך ההתפשטות‪ .‬במסגרת התאוריה ניתן למצוא את קצב ההתפשטות של פונקציית הגל המתאים‬
‫לתוצאות החשבונות הנומריים שנעשו בתחום‪ .‬תוצאות אלו מתאימות לתת־דיפוזיה עם אקספוננט שווה ל‬
‫‪ .1/3‬עובדה זו הופכת את התאוריה למעניינת ורלוונטית‪ .‬בפרק זה אנחנו מציגים את התוצאות של בדיקת‬
‫ההנחות‪ ,‬קרי קיום תכונות של רעש אקראי עם אוטו־קורלציה שדועכת מהר בזמן‪ .‬אנחנו מראים כי קיים קשר‬
‫חזק בין תת־דיפוזיה ורעש‪ .‬בסיום החיבור אנחנו מציגים תרחיש אפשרי שבו התאוריה הופכת ללא רלוונטית‬
‫משום שההנחות שלה נשברות‪ .‬תרחיש זה קורה בזמנים שקשה להגיע אליהם בחישובים נומריים‪ .‬החשבונות‬
‫הנומריים המוצגים בפרק זה מתבססים על אלגוריתם ה"הצעד המפוצל"‬
‫)‪step‬‬
‫‪ (split‬שבעזרתו נהוג לפתור‬
‫את משוואת שרדינגר הלא לינארית ובעיות לא לינאריות נוספות‪ .‬בגלל אופייה הכאוטי של הבעיה החשבונות‬
‫נומריים לזמנים ארוכים אינם מתכנסים לפתרון האמיתי של הבעיה ואנחנו נאלצים להסתמך על התכנסות של‬
‫גדלים סטטיסטיים כמו ממוצעים‪.‬‬
‫הפרק השלישי‪ ,‬מתאר ממוצעים והתפלגויות של גדלים של מודל אנדרסון שחשובים לבעיה של משוואת‬
‫שרדינגר הלא לינארית עם פוטנציאל אקראי בעיקר בתחום שבו הפוטנציאל האקראי חלש‪ .‬בדקנו שני גדלים‬
‫חשובים להתפשטות של פונקציית הגל‪ .‬הגדלים הם "סכום החפיפה" של ארבע פונקציות עצמיות של הבעיה‬
‫הלינארית ו"הפאזה הכללית" בדינמיקה המורכבת מקומבינציה של ארבע אנרגיות עצמיות של הבעיה הלינארית‪.‬‬
‫‪vii‬‬
‫אנחנו מבחינים‪ ,‬בין שתי קבוצות של "סכומי חפיפה" הקשורות למספר האינדקסים השונים בהם‪ .‬במקרה של‬
‫הפאזה הכללית אנחנו מציגים התאמות נומריות שונות של פונקציית ההתפלגות של קומבינציות שונות של‬
‫האנרגיות העצמיות‪ .‬אנחנו מאמינים שבעזרת ידע על ההתפלגויות ויתר התכונות הסטטיסטיות של "סכומי‬
‫החפיפה" ו"הפאזה הכללית" אנחנו נוכל למצוא "מודל צעצוע" שיתאר בצורה פשוטה את הבעיה הסבוכה של‬
‫משוואת שרדינגר הלא לינארית עם פוטנציאל אקראי‪ .‬הצגת מודל כזה עשויה לשפוך אור על הדינמיקה של‬
‫הבעיה המקורית שכרגע הפתרון שלה נראה רחוק‪.‬‬
‫בפרק הרביעי‪ ,‬אנחנו מציעים "מודל צעצוע"‪ ,‬שדומה בחלק מהתכונות שלו לבעיה המקורית ופותרים אותו‬
‫בצורה נומרית‪ .‬המודל מראה התפשטות מאוד איטית מסביב למצב ההתחלתי‪ .‬אנחנו סבורים שמודלים אחרים‬
‫עשויים להיות מוצלחים יותר מבחינת תיאור הבעיה של משוואת שרדינגר הלא לינארית עם פוטנציאל אקראי‪.‬‬
‫הפרק החמישי‪ ,‬הינו פרק הסיכום של התיזה‪ .‬אנחנו מציגים בו את סיכום התוצאות ומדגישים את הנקודות‬
‫החשובות‪ ,‬לדעתנו‪ ,‬שהתגלו במהלך המחקר הזה וכמו כן את השאלות הפתוחות‪ .‬חקר משוואת שרדינגר הלא‬
‫לינארית עם פוטנציאל אקראי רחוק מלהסתיים‪ ,‬ויש עוד הרבה כיווני מחקר מעניינים‪ ,‬שלא מוצו‪.‬‬
‫‪viii‬‬

Effective Noise Theory for the Nonlinear Schroedinger

Transcription

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