Effects of a non-linear coupling term in both classical

Effects of a non-linear coupling term in both
classical and quantum discrete non-linear
Schrödinger equation
———
Etude des conséquences de l’ajout d’un terme de
couplage non linéaire dans les équations de
Schrödinger discrète non linéaire, classiques et
quantiques
Lionel Jimenez
Supervised by J.C. Eilbeck
Heriot Watt University
April-July 2006
Contents
1 Introduction
3
2 Classical approach
2.1 Classical dimer . . . . . . . . . . . .
2.1.1 Stationary solutions . . . . .
2.1.2 Stability . . . . . . . . . . . .
2.1.3 Bifurcation diagram . . . . .
2.1.4 Non stationary solutions . . .
2.2 Trimer . . . . . . . . . . . . . . . . .
2.2.1 A new parametrization . . . .
2.2.2 Interpretation . . . . . . . . .
2.3 Conclusion on the classical approach
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3 Quantum approach
3.1 Operators and the Hamiltonian . . . . . . . .
3.2 Quantum dimer . . . . . . . . . . . . . . . . .
3.2.1 The Hamiltonian . . . . . . . . . . . .
3.2.2 Energies of the |0, ni and |n, 0i states
3.2.3 Interpretation . . . . . . . . . . . . . .
3.3 Quantum lattice . . . . . . . . . . . . . . . .
3.3.1 The Hamiltonian . . . . . . . . . . . .
3.3.2 Behaviour when 1 = 0 . . . . . . . . .
3.3.3 Resonance behaviour . . . . . . . . . .
4 Conclusion
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Bibliography
22
A Details
24
A.1 Eigenvalues of the stability matrix . . . . . . . . . . . . . . . 24
A.2 Calculation details of the trimer parametrization . . . . . . . 24
1
B Figures
25
B.1 Behavour near the bifurcation point . . . . . . . . . . . . . . 25
B.2 Trimer bifurcation diagram when 1 = 0 . . . . . . . . . . . . 26
B.3 Energies when 1 = 0 . . . . . . . . . . . . . . . . . . . . . . . 27
2
Chapter 1
Introduction
The discrete nonlinear Schrödinger equation(DNLS) has been studied a lot
over the last 20 years. The main reason is that it provides a good explanation for phenomenon such as localisation of energy(Self-Trapping) in
certain molecules, that is why it is sometimes called Discrete Self Trapping(DST) equation. For example the study of the C − H stretching in
the dihalomethanes such as CCl2 H2 or CBr2 H2 can be described using a
discrete nonlinear Schrödinger equation(DNLS) with a linear coupling. A
non linear coupling could refine the model and reveal new behaviours for
particular values of the parameters that can be obtain arranging or creating
molecules in different configurations. But this is a very general equation
that can be also found in optics for example, so the parameter we will use
will be dimensionless, and if taken a dimension that will be cm−1 ,as it would
be for the C − H stretching study.
The purpose will be to study the effects of this new term theoretically.
We will try to get analytical results and to compare it to the usual results
obtained using a linear coupling. So we will first try to add such a term
in a classical model, as classical approach usually provides interesting general results before considering a quantum case(QDNLS) that is supposed to
better describe the problems as we are working at a quantum scale. For
both approach we will begin with only a few degrees of freedom f , that is
consistent with the fact that for example the study of the C − H stretching
in the dihalomethanes involves only 2 degrees of freedom. After that, when
possible, we will try to generalise the results to more degrees of freedom.
3
Chapter 2
Classical approach
The theoretical study of the DST equation finds applications in many cases
in the quantum world(e.g. crystals with special properties), it seems therefore natural to study the quantum equivalent of the DST equation. But as
usual in physics we will start with the classical approach that provides interesting general results and we will later discuss the quantum case in section
3.
Although physical system such as nano-crystals include more than 2 or 3
degrees of freedom, you can model some molecules[5] using simple 2 or 3
degrees of freedom models known as dimers and trimers. The advantage
is clearly that f degrees of freedom models are not possible to be solved
analytically whereas the dimer with its two conserved quantities(energy and
number) is fully integrable. Previous studies were mainly about the nonlinear dimer with on-site nonlinearity [6], [7] i.e. with a classical Hamiltonian
of the form:
H=
f
X
i=1
(ω0 |Ai |2 −
X
γ
A∗i Aj
|Ai |4 ) − 0
2
i,j i6=j
The object of this work is to study the effect of adding a nonlinear coupling
or nonlocal nonlinearity as in [4]. The new classical Hamiltonian will now
be:
f
X
X
1
γ
0 A∗i Aj − |Ai Aj |2
(2.1)
(ω0 |Ai |2 − |Ai |4 ) −
H=
2
2
i=1
2.1
i,j i6=j
Classical dimer
We start with the simple case of f = 2 degrees of freedom. Using the usual
dynamic equations
dAj
∂H
=
i
dt
∂A∗j
4
The Hamiltonian 2.1 leads to the dynamic equations:
dA1
= ω0 A1 − γ|A1 |2 A1 − 0 A2 − 1 |A2 |2 A1
dt
dA2
i
= ω0 A2 − γ|A2 |2 A2 − 0 A1 − 1 |A1 |2 A2
dt
i
(2.2)
Defining the norm or number N by
N=
f
X
i=1
|Ai |2
it is straightforward to check that Ṅ = 0 and Ḣ = 0, thereby giving the two
conserved quantities needed for this new dimer to remain integrable.
2.1.1
Stationary solutions
Let us first discuss stationary solutions of the form:
A1 = φ1 e−iωt
and
A2 = φ2 e−iωt
Using 2.2 and the definition of N leads to this set of 3 coupled equations
with 3 unknown quantities(φ1 , φ2 and ω)
φ2
+ 1 |φ2 |2
φ1
φ1
ω0 − ω = γ|φ2 |2 + 0
+ 1 |φ1 |2
φ2
N = |φ1 |2 + |φ2 |2
ω0 − ω = γ|φ1 |2 + 0
(2.3)
Subtracting the first two equations and assuming that φ1 and φ2 are real,
one finds:
0
(γ − 1 ) −
(φ22 − φ21 ) = 0
(2.4)
φ1 φ2
Combining this with the definition of N obviously gives 3 different solutions:
The symmetric solution(↑↑)
r
r
N
N
and φ2 =
φ1 =
2
2
with ω↑↑ = ω0 −
N
(γ + 1 ) − 0
2
The antisymmetric solution(↑↓)
r
r
N
N
N
φ1 =
and φ2 = −
with ω↑↓ = ω0 − (γ + 1 ) + 0
2
2
2
5
The so-called soliton solution(↑ .)
φ1 =
"
N
2
φ2 =
"
N
2
s
!# 12
420
1+ 1− 2
N (γ − 1 )2
s
!# 12
420
1− 1− 2
N (γ − 1 )2
ω↑. = ω0 − γN
Interpretations
The first two solutions are the usual results for coupled oscillators. However
the so-called soliton solution seems really different. First of all it exists only
0
for γ > 2
N +1 , that is for a on-site nonlinearity strong enough. Then, if the
nonlinearity increases until γ − 1 0 then φ1 → N and φ2 → 0 and all the
energy is located on just one oscillator(this explains the designation ’↑ .’,
the name soliton coming from the fact that is a typical localised nonlinear
effect.) All these solutions are very similar to the case when there is only a
linear coupling and we can point out that replacing γ − 1 by a new γ 0 in
the expression of φ1 and φ2 leads to the exact same results. This does not
work with the frequencies though, but we will show after 2.1.4 that this γ0
remains important.
2.1.2
Stability
To study the stability of such a problem we have now to introduce small
variations in the amplitudes as:
φ01 = φ1 + δu1
φ02 = φ2 + δu2
δu1,2 are both complex and depend on time and φ1,2 being solution of 2.3.
The frequency is unchanged. This leads(keeping only first order terms) to
an evolution equation for the variations:
˙
<(δu)
0 B
<(δu)
=
=(δu)
A 0
=(δu)
δu being a vector and A and B being 2x2 matrices defined below:
δu1
W = ω0 − ω
δu =
δu2
A=
−W + γφ21 + 1 φ22
0
0
−W + γφ22 + 1 φ21
6
B=
−0 − 21 φ1 φ2
W − 3γφ21 − 1 φ22
−0 − 21 φ1 φ2
W − 3γφ22 − 1 φ21
For the solutions to be stables we want the eigenvalues of the matrix AB to
be purely real and negatives([1], [7]). The calculation(A.1) of the eigenvalues
gives the following results:
0
The ↑↓ solution is stable for γ > 1 − 2
N
20
The ↑↑ solution is stable for 0 < γ < N + 1
0
The ↑ . solution is stable for γ > 2
N + 1
These results are close to the linear coupling solution, and again replacing
γ by γ 0 leads to the same result
2.1.3
Bifurcation diagram
Figure 2.1 shows the bifurcation diagram(i.e. γ vs W = ω0 − ω) of the
modified dimer in red and the usual dimer in blue(Unstable solution are in
doted line).
We have the same global behaviour except that there’s a shift in the
epsilon1=0
400
Gamma
300
200
100
0
0
100
0
100
200
300
W
epsilon1=2*epsilon0
400
400
Gamma
300
200
100
0
200
W
300
400
The highest branch is the ↑↓ branch.
The one that becomes dashed is the ↑↑ branch.
The one with the different steep is the ↑ . branch.
Figure 2.1: Bifurcation diagram for different values of 1 (0 = 30cm−1 )
bifurcation point. This shift is due to the non local nonlinearity that tends
7
to reduce the effects of the local nonlinearity by a −1 term, so replacing γ
by γ 0 we have the same coordinates for the bifurcation point.
2.1.4
Non stationary solutions
As the dimer is integrable, it is interesting to search for non-stationary
solutions. It will inform us about the behaviour of the 2 oscillators and will
highlight the bifurcation showing how we move from one solution to another.
For the usual dimer this has been done in [6], we basically follow the same
method using formula on elliptic integrals([8])
Major steps of the calculation
Using 2.2 and introducing p = |A2 |2 − |A1 |2 one can find:
d2 p
= 20 p [(γ − 1 )(A∗2 A1 + A∗1 A2 ) − 20 ]
dt2
(2.5)
We have now to find out an expression for A∗2 A1 + A∗1 A2 . Equations 2.2 and
their conjugates give us a straightforward to solve differential equation that
leads to
A∗2 A1 + A∗1 A2 =
γ − 1 2
(pt=0 − p2 ) + (A∗2 A1 + A∗1 A2 )t=0
40
(2.6)
Then combining 2.5 and 2.6 one finds a Duffing’s equation
d2 p
dt2
= Ap − Bp3
A = 20 (γ − 1 )(A∗2 A1 + A∗1 A2 )t=0 + 12 (γ − 1 )2 p2t=0 − 20
B = 21 (γ − 1 )2
For simple initial conditions(e.g. A2 (t = 0) = N ) this can be solved in terms
of elliptic functions depending on the value of the different parameters:
0
If 2A < N 2 B, that is γ < 2
N + 1
γ − 1
p = N cn −0 t, k =
20
0
If 2A > N 2 B, that is γ > 2
N + 1
20
γ − 1
N t, k =
p = N dn −
2
γ − 1
dn and cn being the Jacobi’s elliptic functions of parameter k. The difference
between the two behaviours is obvious on figure 2.2, while the green plot is
always strictly positive, the red one oscillates between 1 and -1. This means
that in the first case the energy is localised mainly on the first oscillator,
whereas in the second case the energy oscillates between the two oscillators.
This confirms the conclusion of the bifurcation diagram. If we check what is
8
Figure 2.2: behaviour of the dimmer for γ <
1 (green)
20
N
+ 1 (red) and γ >
20
N
+
happening at the bifurcation(B.1), we see that the green plot tends to have
a longer period and a higher amplitude of oscillation and stay mainly near
0 and that the red plot is radically changing to be mainly near 0 except
for some peaks. One must also notice that when γ = 1 i.e. γ 0 = 0 the
behaviour is strictly sinusoidal, just as if there was no non-linearity at all.
2.2
Trimer
We propose a first step into many degrees of freedom by treating the case of
f = 3 which is called the trimer. We will only study the stationary solutions,
as the time dependent solutions seem much more complicated to find. We
now have a new set of equations for the amplitudes and the frequency:
φ2 φ3
ω0 − ω = γ|φ1 |2 + 0
(2.7)
+ 1 |φ2 |2 |φ3 |2
+
φ1 φ1
φ3 φ1
2
ω0 − ω = γ|φ2 | + 0
+ 1 |φ1 |2 |φ3 |2
+
φ2 φ2
φ1 φ2
+
+ 1 |φ1 |2 |φ2 |2
ω0 − ω = γ|φ3 |2 + 0
φ3 φ3
N
= |φ1 |2 + |φ2 |2 + |φ3 |2
There are 2 trivial cases to start with. First if one of the amplitude is 0, we
are back to the already solved dimer case, then if the amplitudes are all equal,
obviously we have φ1 = φ2 = φ3 = √N3 and W = ω0 − ω = (γ − 21 ) N3 + 20 .
Just as in the usual trimer we have a single complex solution([1]). For the
9
other solutions a parametric solution has been found in the linear coupling
case([1]). As this can not be used(or adapted easily) for the modified trimer,
we propose a new parametrization that can be use with 1 6= 0 as well as
with 1 = 0(usual case).
2.2.1
A new parametrization
We first assume that 2 amplitudes are equal, that is φ1 = a, φ2 = φ3 = b.
After some algebra(A.2) starting from 2.7 we find that
b−
a 2 a2
W + γN − 0
=
−
2
4
γ − 1
(2.8)
Therefore giving us a parametrization in terms of hyperbolic sine and cosine
instead of the sine and cosine obtained in the usual case([1]). x will be
−0
our new parameter, and let W +γN
be written as A. We can’t make any
γ−1
assumption on the sign of A so we will discuss both signs separately.
A>0
Using the fact that ch(x)2 − sh(x)2 = 1 we have:
√
√
a = 2 Ash(x) and b = Aex
(2.9)
And we have to keep in mind that A depends on the parameter as well. To
be able to use this parametrization we have to find which range of values
correspond to which behaviour and to plot the bifurcation diagram including
all the branches, that means being able to find W (x) and γ(x). From 2.7
first and last equation we find two non linear coupled equations:
γN − 0 − W
ex
γN − 0 − W 2x
sh(x)2 + 0
+ 21
e
γ − 1
sh(x)
γ − 1
1
1 N
γN 1 −
= 0 + W −
4sh(x)2 + 2e2x
4sh(x)2 + 2e2x
W = 4γ
(2.10)
(2.11)
given the fact that combining 2.10 and 2.11 leads to a second order equation
on γ we are able to find(using Maple if necessary) two different expressions
for γ(x) and then W (x), expressions we can substitute in 2.9 to have a(x)
and b(x). We will not give the expressions here as they might not fit in a
page and they are easy to find using Maple.
A<0
We proceed exactly the same way but the results are modified. Let us just
give the equations, A0 = −A:
√
√
a = 2 A0 ch(x) and b = A0 ex
10
W = 4γ
2.2.2
ex
−γN + 0 + W 2x
−γN + 0 + W
ch(x)2 + 0
+ 21
e (2.12)
γ − 1
ch(x)
γ − 1
1
1 N
γN 1 +
(2.13)
= 0 + W +
2
2x
4ch(x) + 2e
4ch(x)2 + 2e2x
Interpretation
After verifying that putting 1 = 0 in the previous equations gives the same
behaviours and branches as in [1], we are now able to plot everything we
need to understand the behaviour of the modified trimer. Figure 2.3 shows
the amplitude of a(red) and b(blue) while the parameter x scales from -4 to
4.
First we didn’t use the full range(theoretically ] − ∞; ∞[ of the parameter as it appears that we can see some asymptotic behaviour already in
this range. We can guess that the range between -4 and -1 on figure 2.3
correspond to the (↑ ..) as |a| |b|, a closer analysis of the data shows that
on (a) this corresponds to a branch with negative values of γ, whereas (b)
correspond to the desired (↑ ..) branch. Between -1 and 0 in (a) we have
opposite sign for a and b but rather the same amplitude, this corresponds
to the (↑↑↓) branch. Between 0 and 1 in (a) we have both a and b positive
but this time b a, so that this is the (↑↑ .) branch. In both figure, we can
see for x > 1 that we have a = b which should represent the (↑↑↑) branch,
but again a closer analysis shows that it corresponds only to one or 2 points
on the bifurcation diagram, which is good because we assumed that a 6= b,
excluding this branch in the parametrization. If we now plot the bifurcation
diagram(Figure 2.4) we are able to identify the branches, and tell that there
is no major difference with the usual case(B.2), except again for a shift.
2.3
Conclusion on the classical approach
After the study of the f = 2 and 3 cases, we are now able to draw some
conclusion about the general behaviour and the influence of the parameter
1 .
We have shown that there are solutions with localised energy in both cases
and that the influence of 1 is to weaken the local nonlinearity. This new
term does not sharply modify the behaviour which makes it simple to solve
analytically, but does not show some interesting new behaviour. We must
then generalise the results we have about the f = n case that is mainly the
existence of a localised solution of the type (↑ .....etc) even though the 1
parameter tends to weaken the local nonlinearity.
11
(a)
1
0.5
0
−0.5
−1
−4
−2
0
x
(b)
2
4
−2
0
x
2
4
1
0.5
0
−4
Figure 2.3: Amplitudes of a(red) and b(blue) function of the parameter x in
the case A > 0(a) and A < 0(b)
600
540
480
gamma
420
360
300
240
180
120
60
0
0
60
120
180
240
300
W
Figure 2.4: Trimer branches(no stability has been considered) 1 = 0 =
30cm−1
The yellow curve is the ↑↑↑ branch
The highest red curve is the ↑↑↓ branch
The other red one is the ↑↑ . branch
The green red is the ↑ .. branch
12
Chapter 3
Quantum approach
The classical approach gave us some interesting results about this new nonlinear coupling term, but the behaviours we predicted are very similar to
the usual linear coupling case. The study of the usual quantum case leads
to the quantum breathers phenomenon([5],[9]), that is a localised state in
terms of number of excitation that latter progress through the lattice while
delocalising. A recent study([4]) has tried to look at non linear coupled
Hamiltonian and finds out a new resonance-like behaviour. This behaviour
is also theoretically studied in the case of low number of quanta and big
lattice, but in a certain approximation. The object of this chapter will be to
get some exact result on the quantum dimer and compare the predictions to
the classical ones, and after that to look at the big lattices case but without
the approximation made in [4].
3.1
Operators and the Hamiltonian
We have to quantise the problem, we will proceed as in [5]. That is:
A∗i ↔ b†i
Ai ↔ bi
bi and b†i respectively being the usual lowering and raising operators defined
using Dirac’s notation as:
√
bj |n1 , n2 , ..., nj , ..., nf i =
nj |n1 , n2 , ..., nj − 1, ..., nf i
p
†
nj + 1|n1 , n2 , ..., nj + 1, ..., nf i
bj |n1 , n2 , ..., nj , ..., nf i =
We now quantise N
N̂ =
f
X
N̂j
N̂j = b†j bj
j=1
nj being the eigenvalues of N̂j and describing the number level on site(oscillator)
P
j, and n = fj=1 nj being the total excitation number. And finally we quan13
tise H according to [5]
Ĥ = −
γ
− ω0 N̂j + b†j bj b†j bj + 0 (b†j bj+1 + b†j bj−1 ) + 1 b†j bj b†j+1 bj+1
2
2
f X
γ
j=1
(3.1)
3.2
Quantum dimer
3.2.1
The Hamiltonian
The dimer’s Hamiltonian is now:
i
γ
γh † 2
Ĥ = ω0 −
N̂ −
(b1 b1 ) + (b†2 b2 )2 − 0 (b†1 b2 + b†2 b1 ) − 1 b†1 b1 b†2 b2
2
2
As N̂ and Ĥ obviously commute, an eigenfunction of N̂ is an eigenfunction
of Ĥ. If we take a fixed value of number of excitation n, the most general
eigenfunction of N̂ will be
|ψi = c0 |n, 0i + c1 |n − 1, 1i + ... + cn |0, ni
where the cj are complex constants we will determine using the fact that
|ψi has to be an eigenfunction of Ĥ as well. This is
H~c = E~c
where ~c is the column vector(c0 , c1 , ..., cn ) and H = H0 − 0 V is the matrix
associated to Ĥ in the |n − j, ji base. Let us now calculate the elements of
the matrices H0 and V . It is well known([2],[5]) that:
γ
Ĥ0 |n − j, ji =
ω0 n − (n + n2 − 2nj + 2j 2 ) − 1 b†1 b1 b†2 b2 |n − j, ji
2
γ 2
(3.2)
=
ω0 n − (n + n) + (γ − 1 )(n − j)j |n − j, ji
2
Thereby showing that H0 is diagonal of values given by 3.2. We can also
find the matrix V :
p
b†1 b2 |n − j, ji = (n − j + 1)j|n − j + 1, j − 1i
p
b†2 b1 |n − j, ji = (n − j)(j + 1)|n − j − 1, j + 1i
Which gives a tridiagonal matrix:

√
n
0
...
0
0
p 0
 √n
2(n − 1) p 0
...
0

p 0

2(n − 1)
0
3(n − 2)
...
0
 0

...
.
.
.
.
.
.
.
.
.
.
..
V =  ...
p
p

 0
...
3(n − 2) p 0
2(n − 1) 0

√
 0
2(n − 1)
0
n
...
0
√
0
...
0
0
n
0
14











To find the quantum analogy of the classical soliton solutions we are interested in finding the energies of states involving |0, ni and |n, 0i.
3.2.2
Energies of the |0, ni and |n, 0i states
Since the (1, n) and (n, 1) positions in the matrix V are 0, one could think
these two states are uncoupled, but in fact they are through the coupling
between each adjacent state. To find out how, we will use the perturbation
theory just as in [2].Using this theory supposes that γ − 1 0 . We have
some data(3.3) about γ and 0 , but 1 has still to be measured, so we will
0
). As we can see on 3.3 we have
assume that it is small(typically 1 ∼ 10
γ − 1 ∼ 40 , that we will taken as enough to make the approximation. It
is important to see that at orders lower than n both states have the same
energy, so the calculation will require a nth order perturbation theory.
Molecule
0
γ
CCl2 H2
29.54
127.44
CBr2 H2
32.80
125.45
CI2 H2
33.69
124.25
V alues1 of γ and 0 (in cm−1 , taken in [5])
(3.3)
We will use the following notations for the approximated energies and
corresponding eigenvectors:
E = E0 + 0 E1 + . . . + k0 Ek + . . .
~c = c~0 + 0 c~1 + . . . + k0 c~k + . . .
Now that we know we are talking about vectors for c1 . . . ck we will omit the
symbol ~. Defining the operator L = H0 − E0 I and using the perturbation
theory we obtain the set of equations:
Lc0 = 0
Lc1 = (E1 − V )c0
Lc2 = (E1 − V )c1 + E2 c0
..
.
(3.4)
Lck = (E1 − V )ck−1 + E2 ck−2 + . . . + Ek c0
As we are interested in the |0, ni and |n, 0i states, 3.2 gives us E0 = ω0 n −
γ
2
0
2 (n + n). And one can remark that the operator L now depends on γ =
γ − 1 :
L = diag(α0 , α1 , . . . , αn ) αi = (γ − 1 )
1
multiply it by
hc
e
to have it in Joules
15
So formally, it is just like the usual case, but with γ 0 instead of γ. We will
not develop the entire detailed calculation as it would take pages and it is
all done in [2]. The important point is the result:
n0
2nno
(3.5)
+o
∆E =
(n − 1)!(γ − 1 )n−1
(γ − 1 )n−1
3.2.3
Interpretation
It as been show([5],[2]) that this energy splitting correspond to eigefunctions
of the form:
1
|ψi± = √ (|n, 0i ± |0, ni)
2
Comparing the results with the classical one, something seems surprising at
first glance: nothing tells us about a localisation of the energy on one of
the two oscillators as the probability of finding energy on one or another
oscillator is equal(because of the symmetry). But there are experimental
evidence that a localisation of energy can happen, even in the quantum
case. Let us explain it by taking a solution that would be a combination of
|ψi±
1
1
+
−
|ψ(t)i = √ |ψi+ e−iE t + √ |ψi− e−iE t
2
2
Then at t = 0 all the energy is located on the first oscillator, and most will
remain there until the phase difference changes to π that is in a time τ :
τ∼
π~
∆E
where ∆E = |E + − E − |. Giving us an estimation of the oscillation time:
τ∼
π~(n − 1)!(γ − 1 )n−1
2nn0
This time grows rapidly2 with n , showing that the energy remains located
in one oscillator in agreement with the classical case(2.1.1). Once again
the term −1 tends to weaken the local non linearity and could even give
a null time for γ = 1 , radically changing the behaviour of the oscillators,
the energy not being localised anymore. Just like in the classical approach,
we have a behaviour that is closer to a fully linear Hamiltonian for this
particular value of 1 .
2
Numerically, this time becomes rapidly longer than any measurable time
16
3.3
Quantum lattice
3.3.1
The Hamiltonian
On lattices we generally use a slightly different Hamiltonian coming from a
quantisation done after a change of variables.
Ĥ = −
f
X
γ
j=1
2
b†j b†j bj bj + 0 (b†j bj+1 + b†j bj−1 ) + 1 b†j bj b†j+1 bj+1
(3.6)
We can notice that Ĥ is translationally invariant, so k is a good quantum
number and defining the translation operator T̂ by:
T̂ |n1 , n2 , . . . , nk−1 , nk , nk+1 , . . . , nf i = |nf , n1 , . . . , nk−2 , nk−1 , nk , . . . , nf −1 i
Ĥ and T̂ commute. This Hamiltonian still commutes with N̂ , so as usual
we will use eigenfunction of both T̂ and N̂ so that we can block diagonalise
Ĥ and only study a block for fixed n and k. We will here only study the
case n = 2 which is far less complex, we will also only consider the case f
odd, since f even can be obtained with the same method. Let us introduce
an eigenstate of both T̂ and N̂ :
|ψτ i = c1
f
X
(T̂ /τ )
j−1
j=1
+c3
f
X
j=1
j=1
(T̂ /τ )j−1 |1, 1, 0, . . . , 0i
(T̂ /τ )j−1 |1, 0, 1, . . . , 0i + . . .
+c(f +1)/2
f
X
j=1
T̂ |ψτ i = τ |ψτ i
|2, 0, 0, . . . , 0i + c2
f
X
(T̂ /τ )j−1 |1, 0, 0, . . . , 0, 1, 0, . . . , 0i
τ = eik
If we are interested in a particular block of Ĥ for a fixed τ (and n = 2) the
energies can be determined by finding the eigenvalues of the matrix H2,τk
that satisfies
H2,τk ~c = E~c
17
Calculating the elements of H2,τk is straightforward and leads to the following (f + 1)/2 × (f + 1)/2 trigonal matrix:
√ ∗


γ
2q
0
0
0 ... 0
√
 2q
1
q∗ 0
0 ... 0 



.. 
∗
 0
q
0 q
0 ... . 



.. 
∗
q = 0 (1 + τ ) (3.7)
H2,τk = −  0
0
q
0 q
... . 



 ..
.. .. ..
 .
.
.
. 0 


 0
0
... 0
q
0 q∗ 
0
0
0 ... 0
q p
3.3.2
Behaviour when 1 = 0
This case is well known([5],[9]) and correspond to
pthe quantum breathers.
This means there is a new energy band E(k) = − γ 2 + 1620 cos2 (k/2) corresponding to a localised energy that remains here on a significant time scale
before slowly delocalising. In the limit of f → ∞ there is also a continuum
band located between ±20 cos(k/2) corresponding to non-localised states.
The eigenvalues can be seen in B.3.
3.3.3
Resonance behaviour
Resonance behaviour shown by [4]
For arbitrary values of 1 , [4] showed numerically that we have the same
global behaviour as in 3.3.2. But it highlights the fact that for 1 = γ,
the propagation along the lattice is much faster and then the energy does
not remain localised as we see on 3.1. This is this special behaviour for
this particular value of 1 which we call resonance behaviour. [4] gives an
expression for the energies involved in this resonance by considering only
the localised states that are the one of the form(and their translations):
|2, 0, . . . , 0i
and
|1, 1, 0, . . . , 0i
Then the matrix 3.7 is only a 2 × 2 square matrix whose eigenvalues are easy
to get:
s
γ − 1
γ − 1 2
E± = −γ −
±
+ 1620 cos2 (k/2)
(3.8)
2
2
Without the approximation
First we will assume that the matrix is semi-infinite that is f → ∞. Some
corrections might be found in the large but finite case([10] for the usual
18
Figure 3.1: Numerical simulation of the evolution of nj . φt is for the time
and n is for the lattice sites(taken from [4] thanks to C.Falvo)
case) but I hadn’t enough time to prospect this. So let λ be an eigenvalue of
matrix M = −H2,τk /|q|(to get the energy we will only have to multiply λ by
−|q|) and let us introduce ~v = [v1 , v2 , . . . , vk , . . .] an associated eigenvector.
We will also use g = γ/|q| and d = |q|/1 . We have then (M − λI)~v = 0,
which can be written in term of coordinates:
√ q∗
(g − λ)v1 + 2 v2 = 0
|q|
√ q
1
q∗
2 v1 + ( − λ)v2 + v3 = 0
(3.9)
|q|
d
|q|
q∗
q
vk−1 − λvk + vk+1 = 0 k ≥ 3
|q|
|q|
The last equation is a two-step difference equation which general solution
is:
vk = Aµk+ + Bµk−
∗
q
− λµ + |q|
µ2 = 0. Then
√
λ ± λ2 − 4
µ± = |q|
2q ∗
µ+ and µ− being the roots of
q
|q|
We will now use a change of variables(J. Dorignac, private communication):
λ = X + X1 so that:
µ± = Xeiθ (+),
1 iθ
e (−) using q = |q|eiθ
X
19
As we want the solutions to be bounded(that is vanishing at infinity), let us
choose vk = X k−2 ei(k−2)θ , k ≥ 2 with X between 0 and 1. The eigenvalues
will be then determined using the first two equations of 3.9:
X 3 + (d − g)X 2 + (dg + 1)X − d = 0
(3.10)
Any solution of 3.10 with X between 0 and 1 will correspond to an energy
of a bounded state.
Even though this equation is not easy to solve analitically we can find some
results. First, depending on the values of parameters d and g there is either
1 or 2 bounded solution.
In the usual case when γ → 0 the breather band tends to go just at the
border of the continuum, and with γ increasing the soliton band goes away
of the continuum([5]). That gave us the idea to look for which values of the
parameters we have a solution at the border of the soliton band. That is,
changes in variables taken in account, when X = 1. This condition leads to
the relation:
2
2|q|1
g=
⇔ γ=
(3.11)
1−d
1 − |q|
When this is verified we have 3 solutions: one is for X > 1, one is for
0 < X < 1 and of course X = 1. A numerical inquiry shows that for value
of g greater than this limit we still have the first two solutions and the one
that was X = 1 decrease to become a bounded solution(J. Dorignac, private
communication). For values lower than the limit the last solution increases
becoming a non-bounded solution. So we will take 3.11 as a limit between
1 and 2 bounded states.
We now have to find an expression for these energies. We will use a low
coupling approximation: |q| ∼ 0 γ ∼ 1 that is with the new parameters:
g 1 and d 1. Then we can say that d − g ≈ −d − g, and d is an obvious
root of 3.10 and we find the following roots (for g ≥ 2):
p
g ± g2 − 4
d,
2
Of course exactly two of them are between 0 and 1, as g is above the limit
defined in 3.11. And if we go back to the energies we find for the bounded
states:
2 + |q|2
∼ −1
E1 = −γ E2 = − 1
1
So we have 2 breather bands located around −γ and −1. When γ = 1 , the
energies are rather equal, which is important as the resonance behaviour
is explained in [4] as a result of the hybridation of the 2 bands. This is
another way to explain this hybridation qualitatively, and of course in the
limit |q| = 0 these are exact results.
20
Chapter 4
Conclusion
The classical part provided us with usual results, that is in fact we have
the same behaviours except that γ becomes γ 0 . Such results are not very
important since no new behaviour is found, but there is still one relevant
thing: the fact that we can have γ 0 = 0 changes radically the behaviour for
this value of the parameter.
Concerning the quantum approach, in the dimer case we can fairly draw the
same conclusion: there is no radical change of behaviour except for γ 0 = 0
that changes from a localised energy to a delocalised energy. About the
lattice, we have one major change that is the resonance behaviour.
But when looking closer, one could say that it is the same kind of conclusion
as before, the fast delocalising energy being typically a linear effect and the
localisation being typically nonlinear. So what is common to all the results
is that for certain values of both the local and nonlocal nonlinearity, we see
a surprisingly linear-like behaviour.
It would be interesting to further study the lattice case to, for example, get
better approximations(or exact) energies for the two breather bands. Other
further works should also include 2 and 3 dimensional case as well as the
study of non-translational invariant lattices.
21
Bibliography
[1] J.C.Eilbeck,P.S. Lomdahl and A.C.Scott. The discrete self-trapping
equation. Physica D, 16:318-338, 1985.
[2] L.Bernstein, J.C.Eilbeck and A.C.Scott. The quantum theory of local
modes in a coupled system of nonlinear oscillators. Nonlinearity 3
(1990) 293-323.
[3] J.C. Eilbeck and M.Johansson. The discrete nonlinear Schrödinger
equation-20 years on. Proceedings of the third conference:
Localization & Energy Transfer in Nonlinear Systems (2002)
44-67
[4] C.Falvo, V.Pouthier and J.C. Eilbeck. Fast energy transfer mediated by multi-quanta bound states in a nonlinear quantum lattice.
Submitted 2006
[5] A.C.Scott. Nonlinear Science (second edition) Oxford texts in
applied and engineering mathematics, Oxford University Press (2003)
[6] V.M.Kenkre and D.K.Campbell Self-traping on a dimer: Timedependent solutions of a discrete nonlinear Schrödinger equation
Physical Review B, 34-7(1986)
[7] J.C.Eilbeck,P.S. Lomdahl and A.C.Scott. Stationary solitons on finite
lattices. Proceedings of the III International Symposium on
Selected Topics in Statistical Mechanics, Dubna, USSR,
22-26 August 1984, 2, 328-339 (1985).
[8] Derek F. Lawden.
Springer-Verlag (1989)
Elliptic functions and applications
[9] J.C.Eilbeck Some exact results for quantum lattice problems. Proceedings of the Third Conference: Localization &
Energy Transfer in Nonlinear Systems (2003) 177-186.
[10] J.C.Eilbeck R.L.Pego On the eigenvalues and eigenvectors of some
large trigonal matrices To be published
22
Thanks
I would like to thank Chris Eilbeck for accepting to be my tutor for 4 months,
for his disponibility, advice and his general kindness. I would also like to
thank him for all the extra advice he gave me concerning living in Scotland
and climbing. I also thank Jerome Dorignac for his precious contribution
to the last part. I would like to thank Emily and Siobhan for helping me
during the first days I was here, and I finally thank Nicol Craig and Scott
Massie for beeing my flatmates and providing me interesting vues on the
8 = D problem.
23
Appendix A
Details
A.1
Eigenvalues of the stability matrix
For the (↑↑) solution:
λ=0
and
− 40 − 20 1 N + 20 γN
For the (↑ .) solution:
λ=0
and
− N 2 γ 2 + 2N 2 γ1 − N 2 21 + 420
For the (↑↓) solution:
λ=0
A.2
and
− 40 + 20 1 N − 20 γN
Calculation details of the trimer parametrization
Subtracting the first two equation of 2.7 leads to:
W (a − b) = γ(a3 − b3 ) + 0 (b − a) + 1 (−b3 − a2 b + 2b2 a)
assuming1 that a 6= b and using that −b3 − a2 b + 2b2 a = (a − b)(b2 − ab) we
find that:
W = γ(a2 + b2 + ab) − 0 + 1 (b2 − ab)
If 1 = 0 you just have to ’complete the square’ in the a2 + b2 + ab expression
to find a parametrization in term of sine and cosine. But here we introduce
N to substitute a2 by N − 2b2 and find:
b2 − ab =
W + γN − 0
γ − 1
It is then straightforward to find 2.8
1
That explains why the parametrization does not take in account the ↑↑↑ branch
24
Appendix B
Figures
B.1
Behavour near the bifurcation point
1
= 0.9995(red) and
Figure B.1: Behavour around the bifurcation( γ−
20
1.0001(green))
25
B.2
Trimer bifurcation diagram when 1 = 0
600
540
480
Gamma
420
360
300
240
180
120
60
0
0
60
120
180
240
300
W
Figure B.2: Trimer branches(no stability has been considered) 1 = 0, 0 =
30cm−1
The yellow curve is the ↑↑↑ branch
The highest red curve is the ↑↑↓ branch
The other red one is the ↑↑ . branch
The green red is the ↑ .. branch
26
B.3
Energies when 1 = 0
Figure B.3: Eigenvalues for n=2, in grey the continuum and lower the
breather band(taken from [9])
27