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MODEL OF AN EXCITON-PHOTON SYSTEM
INTERACTING WITH THE LATTICE

By

Mikhail Valeryevich Katkov

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Physics and Astronomy

2006

ABSTRACT

MODEL OF AN EXCITON-PHOTON SYSTEM
INT ERACTIN G WITH THE LATTICE

By

Mikhail Valeryevich Katkov

The purpose of my thesis is to provide a model and theoretical analysis of a
matter-light system in which the exciton-phonon and the exciton-photon interac-
tions are treated on the same ground. The system considered is a semiconductor
polymer chain coupled to a single electromagnetic mode in a cavity. The excitations
of the chain have a mixed exciton-photon character and are described as polaritons.
Polaritons are coupled to the lattice by the deformation potential interaction and
can propagate in the chain. I find that the presence of optical excitation in the poly-
mer induces strain on the lattice. I use a BCS variational wave function to calculate
the chemical potential of the polaritons as a function of their density. The validity
of the variational approach is checked be analyzing the case of a short chain with
only two unit cells. In the case of a long chain and for a strong coupling with the
lattice, the system undergoes a phase transition corresponding to the self-trapping
of polaritons. The role of the exciton spontaneous emission and cavity damping is

discussed in the case of homogeneous optical lattice strain.

ACKNOWLEDGMENTS

I would like to thank my advisor Prof. Carlo Piermarocchi for his full support in

both the professional and personal aspects.

iii

Contents

Introduction 1

1 Excitons 6

1.1 Frenkel excitons ............................... 7

1.2 Wannier excitons .............................. 9

1.3 Excitons as composite particles ...................... 10

2 Exciton-lattice interaction 13

2.1 Deformation Potential ........................... 14

2.2 Heuristic analysis .............................. 17

2.3 SSH Hamiltonian .............................. 19

3 Ab initio electronic structure calculations 23

3.1 Density Functional Theory ......................... 24

3.2 Calculation methods ............................ 27

3.3 Calculation results ............................. 29

4 Light-exciton interaction 32

4.1 Electromagnetic field quantization .................... 33

4.2 Two level system coupled to a single cavity mode: dressed states . . 34

4.3 N two level systems coupled to a cavity mode: Dicke model ..... 38

4.4 Coherent states ............................... 39

4.5 Variational Polariton wave function and energy minimization . . . . 41

5 Exciton-phonon interaction in a microcavity 45

5.1 The model .................................. 46

5.2 Energy minimization ............................ 47

5.3 Two sites model ............................... 50

6 Results 52
6.1 Numerical procedure: Steepest descent method of functional mini-

mization ................................... 53

6.2 Low polariton density ........................... 55

6.3 Intermediate density: self—trapping .................... 56

6.4 Saturation .................................. 62

6.5 Homogeneous deformation and role of damping ............ 64

iv

Conclusions

Bibliography

70

73

1.1

1.2
2.1

3.1

3.2

3.3

4.1
4.2

4.3

4.4

6.1

List of Figures

A physical picture of exciton hopping: 1) an electron is excited from
a valence shell in the first unit cell; 2) it moves into unoccupied ex-
cited state of a nearby cell; 3) the Coulomb repulsion on the valence
electrons in the second cell pushes a valence electron towards the

hole left in the valence shell of the first cell. ...............
Pair excitation spectrum versus momentum K .............

The dependence on the site index of (a) the exciton density, (b) the
force due to the exciton density gradient, and (c) the displacement

due to the force ...............................

Polydiacetylene electron charge density distribution. Darker regions

correspond to a greater charge density. .................

Structural model of a polydiacetylene chain with H atoms as side

groups. ....................................

PDA band structure along the PDA symmetry axis in k-space. The
dashed line is the zero level in the middle of the conductive and

valence bands .................................

One dimensional cavity used to quantize electromagnetic field.

Elements of quantum optics / Pierre Meystre, Murray Sargent III. Berlin;
New York: Springer-Verlag, 1990. Dressed atom energy level diagram.
The dashed lines are energy iegenvalues for atom-field system with
no interaction energy. Solid lines include atom-field interaction. . .
Polariton chemical potential as a function of the polariton density
for (5 / g = 0 solid line, 6 / g = 1 (dashed line), and (5 / g = 10 ((dot-
dashed line). At low density )1 corresponds to the lower polariton
energy, at large density the chemical potential become equal to the

10

3O

31
33

38

energy of the cavity photon since the excitonic component is saturated. 43

Exciton density times 2 as a function of excitation density for 6 / g =
0 (solid line), 6 / g = 1 (dashed line), and (5 / g = 10 (dot-dashed
line). In the limit of a large polariton pooulation the exciton density

saturates at half filling. ........................... 44

T. Hjorteland,The Action Variational Principle in Cosmology , Institute
of Theoretical Astrophysics University of Oslo (1999). The method
of Steepest Descent approaches the minimum in a zig-zag manner,

where the new search direction is orthogonal to the previous .....

vi

54

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

6.10

Polarization w for (5/t0 = 0.01, g/to = 0.01, x/to = 6, pp = 3 x
10-3 (dotted line), pp = 6 x 10-3 (dashed line), pp = 6 x 10-2 (dot-

dashed line), and pp = 10 (solid line). ..................

Polariton chemical potential as a function of the polariton density
for 5/1‘0 = 0.01, g/to = 0.01, X/to = 0 (solid line), X/to = 4.5
(dashed line), and x/ to = 6 (dot-dashed line). Circuses show the

analytical solution regions ..........................

Photon density as a function of polariton density for x/ to = 0 (solid
line), x/ to = 4.5 (dashed line), and x/ to = 6 (dot-dashed line).

The three basic Jacobian elliptic functions, for the parameter m=0.99.
The solid line is sn, the dotted line is cn, and the dashed line is dn.
Note that the period of dn is half that of the other two, and that dn
is nodeless. All Iacobian elliptic functions may be constructed from
these three. Of the 12 possible functions, these three shapes are the
only normalizable ones which differ from each other by more than a
translation along the horizontal axis or a renormalization along the

vertical axis [37]. ..............................

Polarization 1p for J/to = 0.01, g/to = 0.01, X/to = 6, pp 2 4 x 10‘3

near the symmetry-breaking point .....................

Exciton density times 2 as a function of the polariton density for
X/to = 0 (solid line), X/to = 4.5 (dashed line), and X/to = 6 (dot-
dashed line).

Force as a function of the site index for x/to = 6, pa = 6 x 10‘3

(dashed line), per = 6 X 10“2 (dot-dashed line) ..............

Displacement due to the force as a function of the site index for
(S/to = 0.01, g/to = 0.01, x/to = 6, pp = 6 x 10-2 and the cor-
responding scheme of the chain distortions showing the position of

atoms before and after the system excitation. ..............

Effective lattice potential as a function of displacement. The figure
has been obtained for the following set of parameters: 158 = wx —
2t0 = 2.282 eV, D = 27a = 6.1 eV, 5 = 2.5-103m/s, a = 1.5 nm,
M = 2.110-21 g, wo = 2.282 eV, and c = SzM/az = 82eV/a2.
The excitation corresponds to pex ~ 0.1. ac = g’ = 1 meV, and
we 2 2.242 eV. T = 400 er and I‘ = 700 er for the upper and

lower curve, respectively. The curves are displaced for clarity. . . . .

vii

56

57

59

60

62

63

64

65

3.1

5.1

6.1

List of Tables

Ab initio calculated PDA properties in comparison to the experimen-
tal data: energy gap; exciton mass; exciton deformation potential. . . 31

Ground state energies calculated exactly and using the trial wave
functions. The calculations were made using the following set of pa-
rameters: (0,; = 0.9Eg, t1; = 0.5153, g = 0.253. .............. 51

Limits of Jacobian Elliptic functions and integrals [40] ......... 60

viii

Introduction

Recent advances in nanospectroscopy have demonstrated the possibility of ad-
dressing a single conjugated polymer chain in a solid matrix [1, 2]. These optical
studies of single chains overcome limitations due to ensemble averaging and in-
homogeneous broadening and give a clear picture of the exciton dynamics. Most
interestingly, a one-dimensional (1D) singularity in the optical density of states,
a T“2 temperature dependence of the the exciton lifetime, [3] and a macroscopic
quantum spatial coherence [4] have been observed in isolated red polydiacety-
lene (PDA) chains. These features suggest that the behavior of optical excitations
in polymer chains can be very close to that of an ideal semiconductor quantum
wire system.

In this thesis, we study excitons in a polymer chain coupled to a single elec-
tromagnetic mode in a cavity. The resulting mixed states of excitons and cavity
photons can be described in terms of polariton quasiparticles. We will focus on
the properties of polaritons in the presence of a deformation potential interaction
with the lattice. The main contribution of this thesis consists in proposing and an-
alyzing a new theoretical model which includes the essential physical ingredients
in a simple and elegant manner. These essential ingredients are: the 1D nature of
the optical excitations, the strong exciton-lattice coupling (polaronic effects) and
the strong exciton-photon coupling (polaritonic effect). Polaronic and polaritonic
effects in 1D systems have been separately studied in the past [5], [6], [7]. How-

ever, the interplay of these two strong interactions has not been studied before.

Excitons in polymer chains provide an example of a system where this interplay
gives rise to new physical effects. The presence of an optical cavity enhances such
effects because it enhances the light-matter coupling. We will consider a modi-
fied Su—Schrieffer—Heeger [8](SSH) model, describing the propagation of excitons
in the polymer [7], with an additional term that takes into account the coupling
with a single cavity mode. Excitons are modeled as excitation of two level systems
localized within the unit cells of the polymer chain, and a variational mean-field
approach is used to calculate the properties of the ground state of the systems at
zero temperature as a function of the density of polaritons. A similar approach was
used to study the transition between a polariton Bose Einstein condensate (BBC)
to a laserlike behavior for polaritons in a cavity [9]. The coupling to the lattice
by the deformation potential adds new features to the ground-state properties of
polaritons. One of the main results of this thesis consists in showing that a self-
trapping [10] of polaritons occurs at a threshold value for the polariton density.
This means that even if the polymer chain is homogeneously excited, the polariza-
tion of the medium will be spatially inhomogeneous, showing bright spots coming
from the luminescence of self-trapped polaritons. This mechanism could give rise
to a BBC of polaritons which localize spontaneously without the need of external
traps or strain fields.

Exciton-polaritons are one of the most promising candidates for the realiza-
tion of BECs in condensed matter systems [11]. Exciton-polaritons lasing [12] and
matter-based parametric amplifiers [13] are additional important applications of
these quasiparticles. Exciton-polaritons in organic systems are particularly inter-
esting due to the large excitonic oscillator strength and to their strong coupling to
phonons, which gives rise to strong optical nonlinearities [14]. Evidence of polari-
tonic effects in a single polydiacetylene chains has been recently reported [15].

We will focus on semiconductor polymer chains with a non-degenerate ground

state. In principle many different materials have the exciton-phonon coupling strong

enough for this optical self-trapping effect. [16] A well known example in this
class of materials is polydiacetylene (PDA) [17], which we will use throughout the
thesis. Polydiacetylene [18], has been studied for its strong nonresonant optical
nonlinearities and exciton-phonon effects. Phonon-mediated optical nonlinearities
have been observed in this material [14]. However, we will keep our theory gen-
eral in such a way that it can be extended to polymers with similar properties.

The geometry of the system consists of a single polymer chain embedded in
a 30 optical cavity. The total length of a polymer chain is typically of the same
order or smaller than the wavelength of light at the exciton resonance. We con-
sider only a single discrete cavity mode coupled to the excitons. This is justified
by the fact that in organic systems the exciton bandwidth is of the order of several
tens of meV, much smaller than the energy separation between the fundamental
and first excited 0D cavity modes (typically around 2 eV). As a consequence the
photon confinement, in contrast to other systems, the linear momentum is not a
good quantum number for polaritons in this case. Different geometries for organic
systems in optical cavity have been considered in the literature. For instance, an
interesting interplay of the exciton-cavity and exciton-LO-phonon dynamics has
been predicted in the case of planar cavity structures [19]. We chose the cavity
fundamental mode energy below the exciton creation energy. Thus, there is no ab-
sorption of energy since the cavity photons are in the transparency region.

This thesis is organised as follows. First we give a review of the exciton concept
in Chapter 1 as a composite particle of an electron and a hole held together by
Coulomb attraction. We describe the two classes of excitons: Frankel and Wannier,
giving commutation rules for the former and an effective hydrogenic model for the
later, where we pay attention to the statistical properties of excitons.

In Chapter 2 we introduce the deformation potential, which is responsible
for interaction of the excitons with acoustic phonons. Following Toyazawa [20],

we show stability of a bound state due to exciton-phonon coupling in ID. We

introduce the Su Schrieffer Heeger (SSH) Hamiltonian [6] which describes an
exciton-lattice interacting system via the deformation potential. Using variational
approach we show how to get a non-linear Schrodinger equation, describing the
exciton localization. Then, we discuss an analytical solution to the equation in the
case of an infinite polymer chain.

To obtain some parameters for a polymer chain such as the atomic positions,
exciton mass, energy gap and deformation potential, we use ab initio quantum-
mechanical calculations in Chapter 3. For these purposes we employ a set of
programs for electronic structure calculations within the density functional the-
ory (DFT). Also, we give a brief description of DFT in this Chapter. We find a good
agreement of the results obtained from ab initio calculations with the experimental
data.

In Chapter 4 we introduce the concept of polaritons as mixed states of pho-
tons and optical excitations. First we consider the problem of a two level system
coupled to a single optical mode using the dressed-atom picture. Then we extend
the model to a system of N two-level oscillators. After reviewing the concept of
coherent states, we build a polariton wave function. To find the ground state we
minimize the energy of the exciton-photon interacting system by the use of the
variational approach. We analyze the chemical potential of the polariton system
and a mixed exciton-photon nature of the system depending on the polariton den-
sity.

Our ultimate purpose is to combine these two types of interactions by consider-
ing an exciton-phonon system in a microcavity. In Chapter 5 we give our model of
such a 1D system of excitons coupled to lattice deformations as well as to a single
cavity mode of the electromagnetic field. The Hamiltonian of our model consists
of two terms corresponding to the Dicke model [5] and the SSH model [21]. Also,
we compare the variational results to an exact calculation with two sites in order to

establish the validity of our approach. Since the system is strongly nonlinear, it is

hard to solve it analytically except for some limiting cases. We provide the details
of the numerical energy minimization procedure (the steepest descent method of
functional minimization) used to find the distribution of polaritons in the chain
and the total energy of the systems.

The results on the self-trapping phase transition as well as analytical results
that can be obtained is some limits are described in Chapter 6. We present there
our results of the optical polarization function for different regimes of the self-
trapping. The optical polarization localization induces local forces on the lattice,
which create the lattice distortions. The chemical potential of the system and pho-
ton / exciton density shows a discontinuity at the points of the self-trapping phase
transition because of the polariton number redistribution between the excitonic
and photonic parts . Also, in the last part of this Chapter we analyze the role of the
excitonic spontaneous emission and of the finite Q factor of the cavity. Conclusions
are in the last Chapter, which also contains a discussion of possible experimental

observations of the predicted effect.

Chapter 1

Excitons

In this chapter we will review the exciton concept, which will be used through-
out this thesis. Simply put, an exciton is an electron and a hole held together by
Coulomb attraction. An exciton localized within a single unit cell is called a Frenkel
exciton. Frenkel excitons appear mainly in molecular crystals, polymers, and bio-
logical molecules. An exciton in the opposite limit is called a Wannier exciton, that
corresponds to an electron hole pair spread over many unit cells. Wannier exci-
tons are typical for inorganic semiconductors. In between these two limits, there
are intermediate excitons. We will see that excitons in polydiacetylene (PDA) be-
long to this intermediate class and are therefore particularly difficult to describe
theoretically.

In Section 1.1 we describe Frenkel excitons, and we introduce the paulion com-
mutation rules for the exciton creation and annihilation operators. These Paulions
commutator rules are a consequence of the composite nature of excitons. In Section
1.2 we introduce Wannier excitons, which are quite different from that of Frenkel
excitons, and can be described using an effective hydrogenic model. Finally, in Sec-
tion 1.3 we give a brief discussion of the exciton lifetime and the exciton statistical

properties. In particular, we discuss the limitations of the bosonic approach.

 

 

 

 

 

 

3

Figure 1.1: A physical picture of exciton hopping: 1) an electron is excited from a
valence shell in the first unit cell; 2) it moves into unoccupied excited state of a
nearby cell; 3) the Coulomb repulsion on the valence electrons in the second cell
pushes a valence electron towards the hole left in the valence shell of the first cell.

1.1 Frenkel excitons

Excitons can move through a solid. In the case of Frenkel excitons, this motion
is viewed as hopping of both the electron and the hole from one site to another.
A physical picture of this looks as the following. An electron is excited from a
valence shell. Then it may move into unoccupied excited state of a nearby cell. The
Coulomb repulsion on the valence electrons in this excited cell pushes one valence
electron towards the hole left in the valence shell of the original cell. This effectively
makes the hole to follow the excited electron. The schematic of the process is shown
in Fig. 1.1. Alternatively, it may be seen as an excited electron that attracts the hole
in the valence shell of the new cell.

In general the Frenkel exciton Hamiltonian has the form [22]
H... = 213,133,, + E],,,,,B,’:Bm, (1.1)
n run

where 8;, B" are operators of creation and annihilation of an excitation at lattice

site n, respectively. E g is the energy gap, and In," is the interaction parameter for a
transfer between two molecules at site 71 and m. The first part of the Hamiltonian
represents on—site excitation, while the second one describes the exciton transfer
process. '

The B2], Bn are Pauli operators [22], which obey Fermi anticommutation rela-

tions at the same site

3,3; + 3:3,, = ,

3,3,, = 8:3; = 0, (1.2)
and Bose commutation relations between different sites

[3,13+ — [Ban] = [3:33;] = 0,m 7e n, (1.3)

"l —

The anticommutation relations forbid two excitons from being localized in the

same unit sell. Applying a Fourier transformation to

B" ~iknabk I (14)

1
2 —— Ze
V N k
where N is the number of lattice sites, k is the wave vector, a is the site separation

in a periodical chain, we determine bk as the annihilation operator of an exciton

with a wave vector k. For bk and b; commutation relations are
[27,, bki] = o, [b,’;, 19;] = o

and

1.
2;B,,B,, N3
N I

+ _ , _
[bk’rbkl - 5k,k N

= 5,9,, — 2 (1.5)

where Nex is the number of excitons. Thus, at low-exciton concentration, the Pauli

operator are replaced by Bose ones.

1.2 Wannier excitons

The picture of Wannier excitons is quite different [23]. In this picture, electrons and
holes orbit each other as free particles in the background of the underlying lattice.
The Wannier exciton can be seen as the analogous of a hydrogen atom. The under-
lying lattice is taken into account by the electron and hole mass renormalization, by
the screening of the Coulomb interaction, and by considering the scattering with
phonons and impurities.

The Wannier exciton wave function consists of an atomic-orbital part 4),,(x) ,
which describes the relative electron-hole motion, and by a plane-wave factor eiKR
describing the electron-hole center of mass motion. In this phase term, R and K

are the center of mass coordinate and wave vector, which corresponds to a crystal

momentum. The total Wannier exciton energy is given by the expression:

 

—e2 #18
E = E , 1.6
ex 8 + 2aexeon2 + 2(me + m,,) ( )
where E8 is the electron-hole energy gap, aex is the excitonic Bohr radius,
hze
“ex : “2—‘0" r (1-7)
e mex

mm, is the reduced excitonic mass mex = mam}, / (me + mh). The energy spectrum is
shown in Fig. 1.2. The Coulomb attraction is reduced by the static dielectric con-
stant, 60. If the electron and hole masses are close to the free electron mass, the
binding exciton energy is ~ Ry/ 65, where Ry is the Rydberg energy. For semicon-
ductors, 60 ’-_‘=’ 10, so the exciton binding energies are in the range of 100meV, and

the separation between electron and hole is in the range of 50A.

E. Hammum and H. Hang. Condensation effects of excitons

(nKl

Eu

 

 

 

 

 

O K
Figure 1.2: Pair excitation spectrum versus momentum K

1.3 Excitons as composite particles

Since excitons are made of particles and antiparticles, they intrinsically have a fi-
nite lifetime. Both Frenkel and Wannier excitons have a probability for the excited
electron and the hole to recombine by photon emission. The recombination rate is
proportional to the square of the electron-hole relative wave function at zero dis-
tance, which for a hydrogenic Wannier excitons is 422(0) = 1 / 7mg}. That implies
that smaller excitons decay faster. In particular, Frenkel excitons are expected to
have a shorter lifetime than Wannier excitons.

An exciton is composed of two fermions with spin %. 50 following the rules of
angular momentum composition we have %®% = 0 ® 1. This suggests that that

excitons may behave as bosons. However, since they are composite bosons, this is

10

true only to a certain level of approximation. The classical review of Hanamura
and Haug [24] derived the exciton commutation relationships in the general case,
which we briefly present here.

The most general electron-hole pair state can be written in the form

lex) = Z ckkiazbg, |0) , (1.8)
kk’

where a; and b; are operators of electron and hole creation, and [0) is the vacuum
state for pair generation.

If we determine the eigenfunctions of the Hamiltonian, which takes into ac-
count the crystal—band structure and includes the Coulomb interaction, and write

lex) 2 CL," IO), it can be obtained that

C2,. = 26K,k+kl¢n(l)atb;~ (1.9)
k,k’

with l = (mh/M)k — (me/M)k’, where M = me + mh, and 4),,(1) = f d3¢n(x)e'”"‘
is the Fourier transform of the orbital wave function of the relative motion, which
is the solution to the standard hydrogenic equation. The commutation relations of

exciton operators are
[c c,,]=o [c’r ct 1:0
K,m Kn I Km! K’n’
and

- (m__,K m
[CK’N"CI+(,nl = 6K,K’on1nl_ Z¢n’(1w I K_I)¢"( 41K_l)aK—IaK’—l

-Z¢;(I I<’)I—4>n( %Klbx-1bK’—I

= 5K,K'5n,n’ + 0(na3x) , (1.10)

11

where n = N/ V is the exciton density. We use the completeness of the orbital
wave functions in the derivation. Thus, the excitons will act as bosons as long as
the density of electronic excitation is small compared to a3”, or in other words the

interparticle spacing is large compared with the correlation length aex.

12

Chapter 2

Exciton-lattice interaction

In this chapter we consider the interaction of electrons and excitons with the lattice
in a one-dimensional system. We will see that this interaction results in a lattice de—
formation due to the coupling of the excitons or electrons with acoustic phonons
via the deformation potential. In Section 2.1 we will first discuss the concept of
deformation potential in a general way. Optical phonon modes are also present
in polymer chains. However, we are not going to discuss their role since the op-
tical excitation can be easily tuned far away from the vibronic resonances associ-
ated with the optical phonon modes. On the contrary, the acoustic phonons always
present in the system, since there is no energy threshold for their creation. In Sec—
tion 2.2 we discuss the simple heuristic version of Toyozawa’s theory [20] show-
ing that in 1D there is always a stable exciton bound state for a long chain due to
the exciton-phonon coupling. In Section 2.3 we introduce the Su Schrieffer Heeger
(SSH) Hamiltonian [6], which was developed to describe electron transport prop-
erties in polymer chains. Also in this chapter we derive a non-linear Schrbdinger
equation describing the exciton localization by deformation potential, and we dis-

cuss a simple analytical solution to that equation.

13

2.1 Deformation Potential

The concept of a linear deformation potential was first justified within the effective
mass approximation by Bardeen and Shockley [25]. Within this theory, the energy
of an electron in one band is expanded in powers of a quantity characterizing the
strength of the lattice strain. The expansion is kept to the first power of this pa-
rameter, which makes the theory linear. Neglecting terms of the second order is
equivalent to considering the effective masses unchanged by the strain.

Under applied forces a crystal is strained resulting in a change of volume and

shape. The displacement of an ion in the crystal is characterized by

u=r'—r, (2.1)

where the position of the ion is r before the strain is applied and 1" after it. When the
crystal is strained, the distances between ions is affected. If the distance between

two neighboring ions is initially

 

dl = \/21x§ + ax; + dx§ , (2.2)

then after the strain is applied, it will be equal to

 

dz’ = (/dx'12 + dx’22 + dx’32, (2.3)
where dx; = dx; + d 11,-. The distance dl’2 can be expressed as
d1’2 = c112 + Zeikdxidxk, (2.4)

where a tensor of rank two has been introduced as

1 am 811;, au) Bu,
. _ _ _ + _ + __ . 2.
E'k 2 (ax, ax, Bx; axk) ( 5)

14

The second rank tensor eik is called the strain tensor. As it can be seen from the
definition (2.5), this tensor is symmetric and can be therefore reduced by finding
the three principal axes. Along the i-th principal axis the expression for the com-

pressed length is

dx; 1' V1 + Zeiidx; . (2.6)

In the case of a weak strain, we keep only the lowest order in the displacement.

Then the components eik of the strain tensor are determined by the expression

_ 1 Bu,- Bu),
‘ 5 (a7; + :77) ' ‘2”

and the relative elongation along the i-th axis is given by

d.’.—d-
#:‘/1+2£fi—1%£fi. (28)
t

In general the energy shift for the k-th non-degenerate band extremum is expressed

as

AEk = EDijEij . (2.9)
if

The coefficients of this expansion form a second rank tensor called the deformation
potential tensor. Note that e ,1- is a dimensionless quantity. In isotropic crystals, the

deformation potential tensor is just a single coefficient, and
Zen-(r) = div u(r) (2.10)
it

represents a local elastic deformation in the continuum limit.
In this thesis we are going to consider semiconductor systems, and we will fo-

cus on the change in energy for the electronic states at the bottom of the conduction

15

band and at the top of valence band. These energy shifts are then written as
AEC = acdiv u(r) (2.11)

and

A130 = avdiv u(r) , (2.12)

where ac and av are constants representing the deformation potentials for the con-
duction and valence band, respectively. The values of ac and av can be obtained by
measuring the carrier mobilities, or by a quantum-mechanical calculation of the
band energy as a function of the lattice constant, which will be shown in Chap-
ter 3. Deformation potentials are usually of the order of several electron volts [3]
[7]-

The displacement u(r) can be expanded in terms of the acoustic modes of the

lattice vibrations as

i + —'
22., “Uq(bvquer1|S|qqr+baqe 'q'), (2.13)

where p and S are the density and the longitudinal sound velocity in the crys-
tal, naq is a unit vector in the direction of displacement of the normal mode, with
a = 1, 2, 3 representing the three branches of the acoustic mode, and V is the quan-

tization volume. The relative compression of the volume is given by

- 1* e—iqr
d1vu(r :Zq 2——pVhS|q| i-q naq(baqe'qr —-qu ). (2.14)

This expression shows that only longitudinal phonon modes are involved in the
deformation potential interaction in the linear approximation. Also, since the inter-
action coefficient in Eq. (2.14) is proportional to q i , we notice that short-wavelength

phonons play a dominant role.

16

In this thesis we will deal with excitons, composed by one electron and one

hole. The deformation potential for an electron-hole pair is then written in the form
Ll(re, rh) ._—_. aediv u(re) — ahdiv u(rh) . (2.15)

Thus, the exciton-phonon interaction results from the sum of phonon-electron and
phonon-hole interaction. For a Frenkel exciton localized within a unit cell the ex-

pression (2.15) can be written as
”(rx) 2 axCllV u(rx) , (2.16)

where ax = a, — a), and rx = re = rh.

2.2 Heuristic analysis

If the deformation potential interaction is strong enough, it may happen that the
presence of an electron or an exciton causes a deformation of the lattice which
in turn traps the carriers [26]. This self-trapping mechanism is referred to as a
polaronic effect. In order to analyze the conditions for the self-trapping, we start
by assuming that the electron is trapped in a region of length L. Thus, the electron
kinetic energy due to the localization is h2/ 2m L2, where m is either the exciton or
electron mass. Also, we assume that the lattice is deformed locally , i e. only where
the electron is. Assuming a constant expansion A in the length L, the elastic energy
is C AZL / 2 (energy per length L, where C is an elastic constant). The electro-phonon
interaction contributes to the total energy with an additional —DA, where D is the
deformation potential. Thus, the total energy is given in the form:

r12 CAZL __

E=— ——
2mz.2+ 2

DA . (2.17)

17

In order to understand the physical origin of the self trapping we first minimize

the energy with respect to A, setting 37’:- = 0. This leads to

D
A _ a. (2.18)
For this value of A the energy is
r? 02
— m3 ’ a' (2'19)
which has a minimum for any value of D at
hz
L, ,~ 2 — . 2.20
fl )1 mD3 ( )
This shows that the minimum energy is
D4
E ,- = -———-— , 2.21
m n ZCZIIZ ( )

which implies that a stable bound state always exists in 1D case if the chain length
is greater than the localization length Lmin.

In the consideration above we used the adiabatic approximation assuming the
excitons move faster than the lattice can respond. Since the vibrational energy
quantum is much smaller than the exciton kinetic energy, the lattice motion can
be treated classically. For the validity of using classical description in the adiabatic
approximation

h

m—S‘ >> Lmin r (222)

where S is the sound velocity. h/ 1115 is the analogy of the Compton wavelength,

which is about 10“7 m for the typical semiconductor materials. Substituting the

18

expression (2.21) into the condition (2.22), we get

— >> 1 , (2.23)

which is satisfied for the most of semiconductive materials we consider.

2.3 SSH Hamiltonian

We will use the Su Schrieffer Heeger (SSH) Hamiltonian [6] to describe the inter-
action of excitons with the lattice displacement (phonons). The Hamiltonian de-
scribes the kinetic energy of the ions, the elastic energy of the lattice in the har-
monic approximation, and the transfer energy which depends linearly on the rela-

tive displacement of the ions. It has the following form

 

2 C
HSSH : 2 2px; + Z 5(un-H _ “11)2 _ 2 Ltn+l,n(B;+1Bn + BZBn+1) ° (2-24)
n n n

M, C, u", and p" are the mass, spring constant, total displacement, and momentum
of the n-th site of the tight-binding chain. B: , B" are operators of creation, annihi-
lation of excitons in a singlet spin state. The hopping term is infill," 2: to — 7(un+1 —
u n), where 7 is proportional to the exciton-phonon deformation potential interac-
tion as D = 27a. If the lattice is given a uniform dilation d, then the exciton energy
would increase by 27nd. The value of to can be determined by the exciton effective
mass m as to = hz/Zmaz.

In case of Frenkel excitons, they are localized within single unit cells. Thus the

exciton wave-function can be written in the form

l4>> = ZBnBllmn , (2.25)

where IO)" means a state with no exciton at site n. Consider the adiabatic approxi-

l9

mation defined by
(IHI) = <4>|Hssnl4>> = HI. + Hex, (2.26)

which contains a free lattice and an excitonic term. The classical equations of mo-

tion for the chain in the adiabatic approximation [7] can be written as

EN”)
811,,

 

Mil" = — = C(u,,+1— u,,) — C(un — u,,_1) + F". (2.27)
In continuum limit, the strain force on the site n, F", is related to the gradient of the

exciton density as

(2.28)

 

For a stationary solution, we set Eq. (2.27) to zero, which gives the local displace-

ment due to the strain force as
u, = —C“1 / / 13,, dzn, (2.29)

where n is a continuous variable now. Then from Eq. (2.28) we find

6“it
an

 

= -—-21C|8..|2 +aA, (2.30)

where A is a dimensionless constant of integration that gives finite-size effects. The
choice of A = 0 implies that the total length of the chain is not fixed, which does
not affect the results in the limit of a wire much longer than the extension of the
self-trapping region. The force constant C can be expressed in terms of the sound
velocity 8 as C = SZM/az.

Using a variational approach, we can minimize the energy (IH I) (Eq. 2.26)with

respect to the parameter 8", which is subject to the constraint
Zlfilz =1. (2.31)
I!

20

In continuum limit, that leads to the equation

828,, D 811,,

—nfi2fin+~5;7)-+7;flnig—==yfin, (282)

where y is a Lagrange multiplier. Then from Eqs. (2.30) and (2.32) we get the

non-linear Schrodinger equation

325:1 Dz 2
—t0(25n + an2 )‘l' mfirllfinl = ”5!! - (2-33)

 

The only solution describing the excitonic wave function to Eg. (2.33) which van-

ishes at infinity is
)3 = g sech[Y(n — n0)] , (2.34)
where
2 2
‘Y==.32—331§, c235)
2M52h

Integrating Eq. 2.29, we get the lattice deformation as

Ba
11,, 2 _MS—2 tanh[Y(n — n0)] + naA (2.36)

For A equal to zero a chain must shorten on a polaron creation according to Eq.
2.36. In principle A can be choosen in such a way that the total length will be
unchanged. The electronic energy is

mD“a2

-——————— :237
8A4284h2 ( )

Eex = “2t0 —

The first term is just the energy of a free exciton with k = 0 (an electron at the
bottom of the conduction band and a hole at the top of the valence band). The
second term is the exciton binding energy. The case of a finite chain, in which the

localisation length is comparable to the chain length, is considered in Chapter 6.

21

 

2
a
0 20 40 60 80 100W” ()

J Fn (b)
0 20 40 V5 80 100

I

“ u C
0 20 40‘Co 80100n()

 

 

 

Figure 2.1: The dependence on the site index of (a) the exciton density, (b) the force
due to the exciton density gradient, and (c) the displacement due to the force

22

Chapter 3

Ab initio electronic structure

calculations

In order to obtain some parameters for our numerical calculations of the optical
polarization function we used ab initio quantum-mechanical calculations. We have
employed PWscf (Plane-Wave Self-Consistent Field), which is a set of programs
for electronic structure calculations based on the density functional theory and
density functional perturbation theory. This package uses plane wave basis sets
and pseudopotentials. PWscf [27] can currently perform the following kinds of

calculations:
0 ground-state energy and one-electron (Kohn-Sham) orbitals
0 atomic forces, stresses, and structural optimization

0 molecular dynamics on the ground-state Born-Oppenheimer surface, also

with variable-cell

phonon frequencies and eigenvectors at a generic wave vector, using Density-

Functional Perturbation Theory

effective charges and dielectric tensors

23

o electron—phonon interaction coefficients for metals

0 interatomic force constants in real space

In Section 3.1 we summarize the main results of the density functional theory
(DFT), and in Section 3.2 we give a description of the methods used for the cal-
culation. In Section 3.3 we present out numerical results for the band gap, elec-
tron/ hole masses, and deformation potential for PDA (polydiacetylene). The re-
sults of these calculations are going to be used throughout the thesis as input pa-

rameters for our theoretical modeling.

3.1 Density Functional Theory

Traditional methods in electronic structure calculations are based on the many-
electron wave function, for instance in the Hartree-Fock theory, which have a lim-
ited applicability because of the difficulties of the related formal equations. How-
ever, if the primary interest is in the ground-state properties rather than in the elec-
tronic excitations, there is an alternative and workable approach based on the den-
sity functional theory [30]. In this method the many-body electronic wave func-
tion is replaced by the electronic density as the basic quantity, and the calculation of
ground-state properties is reduced exactly to a problem of noninteracting electrons
in an effective potential. The effective potential includes the external potential and
the effects of the Coulomb interactions between the electrons such as the exchange
and correlation interactions. DFT is based on the Hohenberg-Kohn [28] variational
principle, which relates the ground-state energy to the charge density. From this a
set of one-electron-like self-consistent equations, the Kohn-Sham equations [29],
can be derived, which give an exact description of ground-state properties at least
in principle.

We will discuss below the derivation and some formalism of DFT. [29] As usual,

we assume that the Born-Oppenheimer approximation is valid, meaning that elec-

24

trons move in an external potential V created by the nuclei of the considered mole-
cules. An electron state is described by a wave function ‘I’( F1, . . . , ?N), which satis-

fies the many-electron Schrbdinger equation

HT 2 [T + V + U] ‘i’ = iii—5V? +iV(?,-) + Ema-ii) ‘1’ = E‘I’ , (3.1)
where H is the electronic molecular Hamiltonian, N is the number of electrons and
U is the electron-electron interaction. The operators T and U are so-called universal
operators as they are the same for any system, while V is system dependent or non-
universal. The difference between a single-particle problem and the much more
complicated many-particle problem just arises from the interaction term LI.

DFT provides a way to map the many-body problem with U onto a singe-body
problem without U. In DFI‘ the key variable is the density n(?) given by

n(?) = N/d3r2/d3r3- --fder‘lJ*(?,72,...,7N)“Y(7,q2,...,?N) . (3.2)

Hohenberg and Kohn proved that this relation between the wave function and
the density can be inverted for the ground state density 710(7), i.e. ‘I’g is a unique

functional of no,

‘1’0 = ‘Yolnol (3.3)

Consequently, all other ground state observables F are also functionals of n0
(1:) ["0] = (‘I’olnol lFl ‘I’olnoli - (34)
Thus the ground state energy is also a functional of no

E0 = E[n0] '2 (Y0[n0] IT-i- V + U] Yoh‘lol) , (3.5)

25

where the external potential can be written explicitly in terms of the density
V[n] = / V(?)n(”r’)d3r. (3.6)
When V is known for a specific system the functional
E[n] = T[n] + U[n] + / V(?)n(?)d3r (3.7)

has to be minimized with respect to n, assuming reliable expressions for T[n] and
U [n]. The energy minimization will give the ground state density no and all other
ground state observables. By solving the so-called Kohn—Sham equations of the

non-interacting system

[‘2—V2+V(?)r12¢1(H):57i¢(?)r (3-8)

 

we can find the orbitals 4),- that reproduce the density n(r ) of the original many-

body system
N
n(?) 5 n5(?) = Z|¢,(?)|2 . (3.9)

V5 is an external effective potential in which the particles are moving and it can be

written as

 

Vs— .. v+ /|_.__82715(:)d3r’+ ch[ns(?)], (3.10)

where the second term denotes the so-called Hartree term describing the electron-
electron Coulomb repulsion, while the last term ch is called the exchange corre-

lation potential, which includes all the many-particle interactions.

26

3.2 Calculation methods

The common procedure of all the DFI‘ methods is based on solving the one-electron
eigenvalue problem 3.8, where self-consistent iterations of the equation are re-
quired between the crystal potential and the electronic density [30]. The main
difference in the different computational methods consists in the kind of functions
used to expand the crystal eigenfunctions. On one hand the expansion in a plane-
wave basis set appears naturally to describe electron wave functions of poorly lo-
calized electrons in the crystal, e.i conduction states as well as valence states close
in energy to the Fermi level. On the other hand, an exceedingly high number of
plane waves is needed to reproduce correctly the crystal core states. This difficulty
can be overcome either by simulating the potential with a pseudo-potential which
reproduces correctly the energies of the weakly bound states, or by orthogonal-
izing the plane waves to the core states. The first approach is referred to as the
pseudopotential methods and the second one as the orthogonalized plane-wave
methods (OPW). In order to review the essential aspects of the OPW methods, we

consider the plane waves [30]

W" (k, r) =

 

1
ex ik+h, -r, 3.11
where k is a vector in the Brillouin zone and h" is a vector of the reciprocal lat-

tice. Knowing the crystal core states |¢c(k, r)), the projector PM" (k) on the space

spanned by them can be defined as

P“"’(k) = E I¢.(k.r)> <4>.(k.r)l . (3.12)

Thus the orthogonalized plane wave X" (k, r) can be defined by

X,,(k, r) 2 [1 — P‘O"’(k)]W,,(k, r) . (3.13)

27

Using this basis, the secular equation no longer gives the core states, and the con-
duction and valence states can be reproduced by a small number of X, (k, r). The

crystal potential is represented by a sum of atomic-like potentials

V(r) = Z V(r — d). — rm) . (3.14)

d;1,rm

centered at the site dy of each elementary cell denoted by the lattice vector rm.
Then the matrix elements can be expressed in terms of the Fourier transform of the

atomic-like potentials as

V1.61) = 317 / V(r)e><pliq'rdrl: (3.15)

and the orthogonality terms are (W,, (k + h", r) [4), (k, r) ). Thus the secular equation

in this wave set reads

(Xn(k+hn,r)|H ~— EIXn(k+h,,,r)) =

: ilk + hmlz _ El5mn 'l‘ Eexp[—i(hm — hr!) ‘ditlvplhml _ hn) +
di‘

+ 203 - EC)(W,,(k,r)|¢C(k,r))|(¢C(k,r)|Wn(k,r)) - (3.16)

C

The basic idea of the pseudopotential method is, as mentioned above, to eliminate
the core and try to reproduce its effect on the valence states by means of a fictitious
potential. A pseudopotential can be generated in a variety of ways either through
scattering theory, or by fitting atomic properties or crystal levels. Pseudopotentials
arise naturally in the OPW methods, where Eq. 3.16 can be seen as a secular equa-

tion between plane waves with a nonlocal energy-dependent pseudopotential

Vps(k ‘l' hrmk + hm E) = Zexpl—i(hn1 _ hit) ° d}l]‘/}l(h"l) "' hn) +
‘0‘

+ BE — Ec)<w..(1<. 0112.3. r)>l<4>c(k, r)|w..(k. 1)) . (3.17)

C

28

The pseudopotential method can be seen as an OPW method without the core or-
thogonalization part, which adds a smooth function in the vicinity of the core. In
a typical approach pseudopotentials are given in the form of few Fourier compo-

nents [30]., thus the secular equation reads

(Wm(k,r,r)|H — Ean(k,r,r)) =

= [(k + 11m)2 - Elémn + ZeXp[—i(hm — h") - d,.]Vps,,.(hm — h"). (3.18)
did
In such approach the pseudopotential reproduces the effect of the total potential

on the valence electrons.

3.3 Calculation results

In order to obtain some realistic parameters for the further numerical calculations
of the structure properties, we have used PWscf to perform ab initio calculations
on PDA. First we made a self-consistent calculations within the density functional
theory for the atomic position and electron charge density. Performing molecu-
lar dynamics calculations on the ground-state with variable-cell, we determined
the C, H atomic positions. The electron charge density is shown in Fig. 3.1, where
darker regions correspond to a greater charge density. The corresponding structure
of PDA is shown in Fig. 3.2. There are single bonds between C1(2) and H1(2) and
also C2 and C3, double bonds between C1 and C2, and triple bonds between C3 and
C4, which makes the distance between them shorter compared to that between the
other C atoms. Then, we performed PDA band structure calculations. The bands
calculated, using the plane wave basis sets, are shown in Fig. 3.3. Based on these
result we obtained the following PDA parameters involved in the numerical cal-
culation of the optical polarization function. The energy gap between the valence

and conduction band measured at k = 0 is equal to 2.0eV. The effective electron

29

 

 

 

Figure 3.1: Polydiacetylene electron charge density distribution. Darker regions
correspond to a greater charge density.

Ii

[.1

Figure 3.2: Structural model of a polydiacetylene chain with H atoms as side
groups.

and hole masses calculated from the second derivatives of the dispersion curves
of the conduct and valence bands at k = 0 are 0.25m, and 0.36me, correspondingly,
where m, is the electron mass. By changing a PDA unit cell length from the equi-
librium, we get the PDA electron and hole deformation potentials: De = 2.3eV and
D), = —7.3eV. The corresponding PDA exciton deformation potential determined

as D, — D), is 9.6eV. Table 3.1 gives the calculated properties of PDA in comparison

30

 

 

 

9.2 .\ A /

. 0 V

 

El
1.

 

 

 

 

 

 

 

 

 

 

0 1t/a 21t/a

Figure 3.3: PDA band structure along the PDA symmetry axis in k-space. The
dashed line is the zero level in the middle of the conductive and valence bands.

to those obtained from the experimental data [2], [7], [3].

! PWscf [ experimental

 

E8 (6V) 2.0 2.3
(1n; + 111;) /me 0.6 0.3
D, - D), (eV) 9.6 6.1

 

 

Table 3.1: Ab initio calculated PDA properties in comparison to the experimental
data: energy gap; exciton mass; exciton deformation potential.

31

Chapter 4

Light-exciton interaction

In this chapter we will introduce the concept of polaritons as mixed states of pho-
ton and optical excitations. The simplest model for an optical excitation is given
by a two-level oscillator, which will be considered here. The electromagnetic field
will be described using a second quantization approach. This approach is impor—
tant to describe many physical features as the spontaneous emission, the Lamb
shift, resonance fluorescence, and ”non-classical” states of light, such as squeezed
states. In many problems, the state of the electromagnetic field can be described
by coherent states. This approach is equivalent to a semiclassical description of
the light-matter interaction. After reviewing the concept of coherent states, we will
show how to build a polariton wave function using coherent states. This polariton
wave function will be used in a variational approach throughout this thesis.

In Section 4.1 we review the second quantization for a single-mode electromag-
netic field. In Section 4.2 we consider the problem of a two level system coupled
to a single quantized mode, and we introduce the dressed-atom picture. In Section
4.3 we extend the model to N two-level oscillators coupled to a single cavity mode
(the Dicke model). In Section 4.4 we define the coherent state of the electromag-
netic field, which will be used to model a singe-mode electromagnetic field in an

optical cavity. In Section 4.5 we introduce the variational wave function and find

32

Mirror Mirror

I.\

//////// /
><
lN
//////// '\

 

 

Figure 4.1: One dimensional cavity used to quantize electromagnetic field.

the ground state of the interacting excitons and photons.

4.1 Electromagnetic field quantization

We consider a cavity of volume V, enclosed by mirrors as shown in Fig. 4.1. Clas-

sically the electromagnetic field is described by the Hamiltonian
H = / (e0152 + BZ/HOWV, (4.1)

where E and B are the magnitudes of the electric and magnetic fields, 60 and no are
the permittivity constants, respectively. The integral is taken over the volume V.
In the second quantization form, the Hamiltonian for the field can be rewritten in

terms of the photon annihilation and creation operators c and c'r as
H = hwcwc + 1/2) . (4.2)
The corresponding eigenstates of the Hamiltonian satisfy the equation

Hln) = hwc(n + 1/2) , (4.3)

33

where It may be interpreted as the number of photons in the state In). The electric

field operator can be expressed in terms of creation and annihilation operators as
E = £(c + c"), (4.4)
where the electric field per photon is given by

_ )th

and the action of creation, annihilation operators on In) is

CI") = filn - 1)
clln) = \/n+1|n+1), (4.6)

respectively.

4.2 Two level system coupled to a single cavity mode:
dressed states

In describing the matter-field interaction we start with the simple problem of a
two-level atom coupled to a single quantized mode of the field, and we introduce
the so-called ”dressed-atom” picture. [31]. The interaction Hamiltonian between

an atom and the electric field in the dipole approximation has the form
V = --d - E , (4.7)

where E is the electric field and d is the dipole moment. For quantized fields, E

and d become the dipole moment and electric field operators, respectively. Using

34

photon operators, the interaction Hamiltonian becomes

v : (c+c*)(g8*+g*8), (4.8)

NIH

where 8* and B are operators of creation and annihilation of an optical excitation.
The full Hamiltonian for the two level system and the electromagnetic field then
reads

HAL = hwcclc + %(c + 1:")(g8+ + g*B) + EgBlB, (4.9)

where E g is the energy difference between the ground and the first excited states.
The coupling constant g can be chosen to be real without loss of generality.

A basic concept we will use in this thesis is the rotating-frame approximation.
Considering the terms in the interaction energy Eq.(4.9), we note that cB’r corre-
sponds to the absorbtion of a photon and the excitation of the atom. Conversely,
clB corresponds to the emission of a photon and the deexcitation of the atom.
These two combinations are kept in the rotating-frame approximation, and the re-
maining two pairs, clBI and cB, are dropped in this approximation. This can be
clarified by writing the time evolution of the operators in the Heisenberg picture

as

C(t)B+(t) = C(O)B+(O)e—i(hwc-Eg)t

C(t)+8(i) = C(O)+B(O)ei(hwc_53)t. (4.10)

Notice that these ”resonant” pairs vary very slowly when the energies of the pho-

ton and the excitation are close. On the contrary, the antiresonant terms

C(t)B(t) ___ C(O)B(O)e—i(hwc+Eg)t

C(t)+B(t)+ —_- c(0)*8(0)*ei<"wc+58)‘, (4.11)

35

oscillates very fast and tend to average to zero. Dropping the combinations (4.11),

we obtain the Hamiltonian

1
HAL = hwccic + EgBlB + §g(cB+ + clB) .

(4.12)

Notice that the light matter interaction couples only the atom-field states In, 1)

and In + 1, O) for each value of n. We can break then the Hilbert space of the cavity

photon system in many decoupled manifolds and we consider each manifold in-

dependently. Solving the secular equation on a given manifold, we get the energy

eigenvalues

1

1
Eh = h(n+1)hwc+§(7€n+6),

where the detuning

§=Eg_hwc

and the quantized Rabi frequency

 

72,. = \/.52 +g2(n + 1) .
The energy eigenvectors are

II) = cos0n|n,1)-sin9nln+1,0)

Ih) = sin6,,In,1)+coanIn+1,0),

36

(4.13)

(4.14)

(4.15)

(4.16)

where

 

 

c059 R" _5
" (A72, —5)2 +g2(n + 1) '
sin 8,. g V n + 1 (4.17)

 

 

\/(Rn -5)2+82(" + 1) '

The state given by Eq. (4.16) are called dressed states describing the two-level atom
interacting with a single-mode field in a given fixed number state for the photon
field. The unperturbed states of noninteracting Hamiltonian are called bare states.

The energy separation between two dressed states within the same manifold
is R", which reaches its minimum at resonance when E3 = we. In this case, the

dressed states Eq. (4.16) reduce to

1
II) = 72(I"11)-|"+1r0>)
HI) = 1(In,1)+|n+1,0)) (4.18)

3|

2

with eigenvalues

E, = h(n+1)wc—%hg\/n+1

1
E), = h(n+1)wc+§hg\/n+1. (4.19)

Dressed atom energy level diagram is shown in Fig. 4.2. We note that the rotating
frame approximation will hold as long as |5I remains much smaller than wc and

E8.

37

 

h(n+l)fl

 

 

 

 

n(n- l )0 0

Figure 4.2: Elements of quantum optics / Pierre Meystre, Murray Sargent III. Berlin; New
York: Springer—Verlag, 1990. Dressed atom energy level diagram. The dashed lines
are energy iegenvalues for atom-field system with no interaction energy. Solid lines
include atom-field interaction.

4.3 N two level systems coupled to a cavity mode: Dicke
model

In order to introduce the polariton concept which we will use later for a chain,
we consider the Dicke model [5] which describes localized, physically separated,
saturable excitations coupled to a single mode of the electromagnetic field by the
dipole interaction. We extend the Hamiltonian for a single two level system in
Eq. (4.9) to a set of N two-level atoms. In the rotation-wave approximation dis-

cussed above, the Hamiltonian has the form

HD = hwcctc + E §(B;c + c*B,.) + 2: £38,113, . (4.20)
n n

38

If the two-level atoms are near the ground states )2" (BZBH) << N, the operator

1
8* = —— 8;, (4.21)
m {3

which creates the photon-exciton coupling, is approximately bosonic, and the Hamil-
tonian (4.20) is that of two coupled bosons. (Here we rescale the coupling constant
3” = g\/N) Polaritons are defined as eigenstates of such a model. To extend the po-
lariton concept beyond the low-excitation limit, following Estham and Littlewood
[32], we generalize the concept of a polariton by defining it as a quantum of exci-
tation of the coupled light-matter system. The polariton number is then the total

number of photons and excitons,
Np = (etc + E BIB") , (4.22)
n

which is a conserved quantity for the model.

4.4 Coherent states

We will describe the state of the field in the cavity in terms of coherent states, rather
than with photon number states. In fact, it can be proved that the electromagnetic
field in an optical cavity can be well represented by a coherent state [33]. A co-
herent state distributes its quantum-mechanical uncertainty equally, which means
that the phase and amplitude uncertainty are approximately equal. Conversely, in
a Fock state with a well-defined number of particles the phase is completely un-
certain.

There are several ways to introduce coherent states. To emphasize their nearly
classical character, we define the coherent states by the constraint that the classical

energy of the field is equal to the quantum-mechanical energy for a coherent state

39

lg). That leads to the factorization condition

(CICICIC) = <C|C+ICI<CIC|€>- (4-23)

It can be shown that the definition (4.23) implies that the coherent states are eigen-

states of the annihilation operator

CIC>1 = AMI/1: (4.24)

which can be used as an equivalent definition of the coherent states. The coherent

state lg) can be expressed in terms of the number states In) as
_ -|/\|2/2 A: 4 2
— e n . . 5
This gives the probability of finding n photons in the coherent state as
— , (4.26)

which is called a Poisson distribution, where (n) is the average photon number.
For a coherent state we have (n) = IAIZ.

A coherent state can also be expressed in terms of the vacuum state IO) as
14>. = (AH-“10> = Dune). (4.27)

where D(_>_) is called the displacement operator. The displacement operator can
be seen as a shifting operator for the vacuum, and the coherent states IC) are often

called displaced states of the vacuum.

40

4.5 Variational Polariton wave function and energy min-
infizafion

We choose a trial wave function for the system of two level atoms as a superpo—
sition of a coherent state of photons and a BCS state of the fermions [32] in the

form

14.4.6 42> -—- loin (4.10),, + W810.) . (4.28)

where IQ), is the coherent state of the cavity mode with the order parameter A,
and I0)" and |1),, denote the number of excitons in the site 11. A, an, 8", and 4),, are

variational parameters, a,, and 8,, are subject to the single-occupancy constraint
|o.,.|2 + |)3,,|2 = 1. (4.29)

We fix the overall phase in such a way to have A real and introduce the phase (pn
to make an and 8,, real.

By introducing the optical polarization function
wn = ”(8; + Bn) = 2‘Xnfin I (430)

the total energy reads

_ 2 84 E8
E—hwcA +—2—;ip,,cosq)ndn——2-;(/1—tpfi, (4.31)

in the assumption IanI > IBnI.

To find the ground state of the system at a fixed (Np) we minimize the func-
tional (H — uNp) with respect to the function 1])" and with respect to the A and 41,.
parameters. By minimizing the energy with respect to (p, we can set 4),, = 7r as-

suming g > O and anti" > 0. Finally, the excitation parameters 11),, and A are given

41

by the solutions of

<th — 14M — £24.. = o. (4.33)

where u is a Lagrange multiplier constraining the average polariton number, which
plays a role of the chemical potential of the polariton system. Since the system is
composed of N identical and noninteracting two level atoms, the variational coef—
ficients, and therefore the polarization function 1pn do not depend on the index n.
We will see later in this thesis that the interaction with the lattice will break this
symmetry.

After rescaling A’ = A / x/N and g’ = g\/N and eliminating ([1,, in Eqs. (4.32)

and (4.33), we get the equation for the chemical potential

~ 1 8/2

wC = I
4 (E82 +g’2/l’2

where thc = hwc — u and hEg = hEg — 14. Equation (4.34) is analogous to BCS gap

 

(4.34)

equation, with an order parameter A’.

At a very low polariton density p can be obtained by expanding the Eq. (4.34)

(158 + free, — (m + 3'2) , (4.35)

where 6 = E g —- have is the optical detuning. Notice that in this low density limit the

for small A as

NIH

’1:

chemical potential corresponds to the energy of the lowest dressed cavity-exciton
state, i.e. the lower polariton. At finite polariton number we can calculate it numer-
ically. The results of our numerical calculations for the p dependence on the polari-
ton density p p = N p / N are given in Fig. 4.3 for several values of the detuning (5. At
low densities the p is given by the conventional linear-response polariton energy

( 4.35). When the density increases the excitonic component of the polariton pop-

42

 

 

 

 

 

Figure 4.3: Polariton chemical potential as a function of the polariton density for
o' / g = 0 solid line, (5 / g = 1 (dashed line), and (5 / g = 10 ((dot-dashed line). At low
density it corresponds to the lower polariton energy, at large density the chemi-
cal potential become equal to the energy of the cavity photon since the excitonic
component is saturated.

ulation saturates, and the polaritons become more photon-like. As a consequence
we obtain that at high densities the chemical potential approaches hwc, which is
the energy of the cavity photon.

Fig. 4.4 shows the exciton density ( f I 8,, Izdn) / N times 2 for the same parame-
ters as in Fig. 4.3 as a function of pp. Notice that this quantity reaches 1 in the limit
of large polariton density, corresponding to a half filling of the excited electronic
states. This maximizes the polarization and hence minimizes the dipolar interac-

tion energy between the excitons and the cavity.

43

 

Exciton density times 2

 

 

 

 

Figure 4.4: Exciton density times 2 as a function of excitation density for (5 / g 2 0
(solid line), (5 / g = 1 (dashed line), and 5 / g = 10 (dot-dashed line). In the limit of
a large polariton pooulation the exciton density saturates at half filling.

44

Chapter 5

Exciton-phonon interaction in a

microcavity

In Chapter 2 we studied the interaction of excitons with the lattice (phonons) in
a one-dimensional system. In that case, the exciton-photon interaction just repre-
sented a pumping mechanism to generate the exciton population. On the other
hand, in Chapter 4 we describe the model of N two-level oscillators coupled to a
single cavity mode. Of course, it would be of interest to treat the exciton-phonon
and the exciton-photon interactions on the same ground. By considering the two
interactions coherently, we can expect the possibility of controlling the system by
some field parameters such as the polariton number and the cavity-exciton detun—
ing. This Chapter deals with the exciton-photon and exciton-phonon models and
is organized as follows: in Sect. 5.1 we introduce the model. In Sec. 5.2 we illustrate
the variational approach used to calculate the ground state properties of the sys-
tem. In Sec. 5.3 we compare the variational results to an exact calculation with two
sites in order to establish the validity of our approach. In this Chapter the model

and its limitations are discussed. The main results will be presented in Chapter 6.

45

5.1 The model

The model consists of the 1D system of excitons coupled to lattice deformations as
well as to a single cavity mode of the electromagnetic field. The Hamiltonian can
be written as

H = HD + HSSH . (5.1)

The first term corresponds to the Dicke model [5] of an ensemble of two level

systems coupled to a single electromagnetic mode
HD = w,c*c+;§(8;c+c*8,) +wx;B;Bn, (5.2)

where cox is the exciton energy. We will use it = 1 throughout the Chapter. 3; and
B" are operators of creation and annihilation of excitons in a singlet spin state, and
cl and c are creation and annihilation operators for the cavity photons of energy we.
Each site n of the model represents a single monomer of the polymer chain. The
parameter g indicates the exciton-cavity dipolar coupling constant. In the Dicke
model, the atoms do not have translational degrees of freedom and there is no
direct transfer of optical excitation from one atom to the other. Here, we need to
add two additional features to the Dicke model: (i) the excitations can hop from site
to site and move along the backbone of the polymer chain, and, (ii), the hopping
of the excitation depends on the relative position of the sites in the lattice, which
can move around their equilibrium position. As discussed in Chapter 2, this can

be represented by the excitation transfer in the SSH form as

H Li + 2 C (u u )2
n 2M n 2 n "
_ ZtflflrnwkflBn + B;Bn+1) , (5.3)
n

46

where t,,+1,,, = to — 7(u,,+1 - un). The SSH model was used to describe the elec-
tronic transport in polyacetylene chains [8], and was extended to the case of exci-
tonic transport in polydiacetylene [7]. We use the same definitions for the parame-
ters as in Chapter 2.

We consider a 3D optical cavity giving a discrete spectrum of cavity modes.
The size of the cavity can be chosen so that the fundamental mode at energy we is
resonant or nearly resonant with the exciton energy. Typically, the total length of
the polymer chain is smaller than the wavelength of the light, therefore we can as-
sume that the strength of the light-matter interaction g is constant in the site index
n. Also, in the case of organic materials the excitonic bandwidth, given by 2t0, is
of the order of several tens of meV, while the energy separation between different
3D cavity modes is typically around 2 eV. Therefore, the coupling of excitons with

higher cavity modes can be neglected.

5.2 Energy minimization

We consider a polariton trial wave function which is a product of a coherent state

for photons and a BCS state for excitons as [32]

14.4.4.4» = M1 (“410). +8‘“’"finl1>n) , (54)

n

where A, an, 8,,, and (p,, are variational parameters, and IA) represents a coherent
state for the cavity. The coefficients an and 8,, are subject to the single-occupancy

constraint

(4,12 + [8,8 = 1. (5.5)

IO) is the vacuum state of the cavity mode, and I0) ,, and I1)" denote the ground and
the excited state of the two level system at the site n. We assume A real and we fix

the overall phase to make an and 8,, real. In contract to the discussion in Chapter

47

4, due to the hopping, the variational coefficients will depend on the index n in the
general case.

We can interpolate continuously the wave function by transforming the discrete
sum 25:1 in a continuous integral ION dv. In this way, we can define the optical po-
larization 1p(u) = 2a( u)8(1/), where a and 8 are continuous functions of u. We can
express the total energy of the system as the expectation value of the Hamiltonian
in Eq. (5.1) with the trial wave function in Eq. (5.4). In continuum form, the total

energy can be expressed as
5:6442 5m

1 N P2 72 72 2 72 2 I 2 2
+2/0 IM+Cu +t0(1,b +1P(P I—IOIP +7114) +84‘PCOS‘P‘9’XV1—‘I’Id"!

where rp(1/), p(v) and u(v) are also continuous functions. The prime indicates the
derivative with respect to the variable 11. In deriving Eq. (5.6) we have assumed that
Ia(v)I > I8(1/)I along the chain, which corresponds to the condition of negative
detuning between the exciton resonance and the cavity mode. Notice that the total
polariton number
2 2 2 N 1 2

bb=A-e/MMW=A—H§—§/ 1—¢dv 6%

is a conserved quantity. In order to find the ground state of the system at a fixed po-

lariton number we perform a variational minimization of (H — uN p) with respect

to the functions 1}), (p, u and with respect to the constant A. From the condition

48

5 (E — pr) = 0 we obtain the system of equations

41(4)”le + (fit/22') + $254.56., = 0. (5.8a)
Cu” + ill/’2’ = 0, (5.8b)
(“’X _ ”)‘P (5.8c)

2t ”+2t 1— ’2 —2 u’— Acos —————=0,
01/) Dipl (P) 71/) g (P M

(w, — u)A + g / wcos (pdu = 0. (5.8d)

From Eqs. (5.8a) and (5.6) we can see immediately that, in the assumption a8 > 0,
and g > 0 we can take the solution (0 = 7:, since this variable has no constraint.
That makes the global phase of the optical polarization constant along the chain
and out of phase with respect to the cavity field. In order to have a closed equation
for 1p we can integrate the equation for u in Eq. (5.8b) and substitute in Eq. (5.8c).

By direct integration of Eq. (5.8b) we can write
I _ __'I’_ 2
u —-—- 2CIIIJI +aA, (5.9)

where A is a dimensionless constant of integration. We choose A = O, which im—
plies that the total length of the polymer is not fixed. Then again, the force constant
C can be expressed in terms of the sound velocity 8 as C = 52M / a2. Finally, elim-
inating the displacement u, the system of equation in Eq. (5.8) can be rewritten in

the form

(wx - VIII) _

—2l’olp — toll)” — XIPB — 3A + W — 0 , (5.103)
(w, — p) — i / tpdv = 0, (5.10b)

where the coefficient of the cubic term x = D2/4MSZ. This system of coupled

equation will be solved numerically in Chapter. 6

49

5.3 Two sites model

In order to check the validity of the BCS trial wave function in Eq. (5.4) and the
accuracy of the variational approach, we compare in this section the exact solution
of a 2-sites problem with the result obtained with the variational approach. The

exact solution is obtained by writing the wave function in the form
W = UOIOONP) + ’71I01NP — 1) + '72I10NP - 1) + ’73I11NP — 2) ,

where the first term in the right-hand side corresponds to the state with zero exci-
tons and Np photons, the second and third terms describe the states with one exci-
ton and Np — 1 photons, and the last term is a state with two excitons and Np — 2
photons. Here, we assume that the number of polaritons in the system Np > 1. In
this section we will also consider the hopping tu as a fixed parameter. Using this
form for the wave function in the Schrbdinger equation with the Hamiltonian in

Eq. (5.1) we obtain a set of equations for 17,:

1.170 = chP’io +§\/1—V;(’71 + '72) ,

i271: 51771 + §(\/I_\I_I:'Io + WIT—1’73) - t12772 ,

1.'72 = 51’72 + glVN—P’io + W43) — t12’71 ,

1773 = (wc(NP — 2) + 260x) 773 +gx/W—1M1 + 172) ,

where 81 = wc(N p — 1) + tax. The energy spectrum and, in particular, the ground
state energy are found by direct diagonalization.

Using the trial wave function approach described in the previous section, we

50

Np Eo, exact Eo, trial WF
4 2.993 3.089
20 16.928 16.972
100 87.837 87.857
200 177.012 177.027
1000 893.528 893.521

 

 

 

Table 5.1: Ground state energies calculated exactly and using the trial wave func-
tions. The calculations were made using the following set of parameters: w, =
0.9Eg, t1; = 0.515 , g = 0.253.

find that

(4.4.13. CPIHIa,A,l3, (P) = «M2 + wxwi + 6%) -

2
—2t12a181a282 cos((p1 — (p2) +gA 241,8,- cos q),- . (5.11)
i=1

Without a loss of generality, the following substitutions are made in Eq. (5.11):

 

(p,- = 7r, A = \/ Np - 8% - 8%, a, = 1 — 8?. The resulting expression depends only
on 8,. The minimum value of (a, A, 8, goIH Inc, A, 8, (p) is considered as the ground
state energy. Table 5.1 shows the comparison between the exact and the variational
results. The ground state energy was calculated for different values of the photon
number in the cavity. The ground state energies calculated with the variational
approach are in good agreement with the exact calculation. As expected, the exact
ground state energy is slightly smaller than the energy calculated using the trial
wave function. This difference can be related to correlation effects not included in
the trial wave function. Also notice that the difference between the ground state
energy calculated with the two approaches decreases at a larger number of the
photons. In fact, the correlation effects between the excitons and the cavity photons
are expected to disappear for a photon number much larger that the excitation

number in the system.

51

Chapter 6

Results

In this chapter we will make an analysis of the exciton-lattice interaction in a micro-
cavity by solving the system of equations for the polarization function and the cav-
ity field order parameter. The system of Eqs. (5.10) describes the coherent exciton-
phonon and exciton-photon interactions, meaning that the phase between the the
polarization and the cavity field, which is given in terms of the coherent state,
remains constant. This phase is fixed by the dipole exciton-phonon interaction.
We keep the cavity field energy below the exciton energy, implying hat we deal
with virtual excitons. Thus, dissipation processes are not considered in the system.
Also, we assume that the polariton number, which is the total number of photons
and excitons, is a conserved quantity for our system. The coupling of polaritons to
the lattice brings self-attraction to the system, i.e. exitons appear to be trapped in
the self-created matter field. This nonlinear attraction depends both on the exciton
density and the deformation potential. Starting from no polaritons and increasing
their number, the system goes through several phase transitions due to the change
in the excitation regimes.

The system of Eqs. (5.10) is strongly non-linear. It is practically impossible
to solve it analytically except for some limiting cases, which we will discuss in

Sec. 6.3. Thus, we have to make a numerical analysis. In Sec. 6.1 we describe the

52

method of functional minimization and how it can be applied to our problem of
solving the non-linear system of equations. In the other sections of this Chapter we
present analytical results for some limiting cases as well as numerical calculations
for the model introduced in Chapter 5. By calculating the chemical potential, we
found self-trapping phase transitions, which are also reflected in a sharp change
of the exciton/ photon number as a function of the polariton number. In Sec. 6.2
we discus the limit of low polariton density when the cubic term in Eq. (5.10a) and
correspondingly the self-trapping effect can be neglected. In Sec.6.3 we discuss the
self-trapping and analyze the system when the fermionic nature of the excitons
can be taking into account as a perturbation. With further increase of the polariton
density the internal fermionic structure gives rise to a hard-core repulsive term,
leading to the saturation of the exciton state, which we describe in Sec. 6.4. In the
last Section we analyze the system in the assumption that the lattice deformation
is homogeneous, which allows us to obtain an analytical expression for the total
energy by a direct solution of the equation of motion for the excitons and cavity

photons.

6.1 Numerical proCedure: Steepest descent method of
functional minimization

Having established the validity of the variational approach, we can study the gen-
eral case of a long chain. In order to study on the same ground the inter-cell hop-
ping and the presence of the cavity field we solve the system of Eqs. (5.10) nu-
merically. There are several methods can be used for solving the problem, e.g the
Runge-Kutta method, Shooting method, Relaxation method [35]. For the purpose
of solving the system, we use the steepest descent method of functional minimiza—
tion [35]. Several methods have been proposed to improve the method [36]; how-

ever, the steepest descent method of functional minimization is efficient enough for

53

r

START

ll

(«A _

/ "

 

Figure 6.1: T. Hjorteland,The Action Variational Principle in Cosmology , Institute of
Theoretical Astrophysics University of Oslo (1999). The method of Steepest De-
scent approaches the minimum in a zig-zag manner, where the new search direc-
tion is orthogonal to the previous

solving the system. This method is efficient in solving numerically Gross-Pitaevskii
equations for BECs, which are similar to our nonlinear equations [34]. The method
consists of projecting an initial trial state onto the minimum of an effective energy
H by propagating the state in an imaginary time. The graphical illustration of the
steepest descent method is given in Fig. 6.1. We start from an initial field parameter
A(T = O) and a trial function tp(r = 0), then A(T)and 112(7) are evaluated in terms

of the equations

 

 

 

aip(r) __ (5H

8T — 3WAT) (6'1)
and

8A(T) _ _ 3H

61 _ EMT)’ (6'2)

where 3 indicates a constrained derivative that preserves normalization. Eqs. (6.1)
and (6.2) define a trajectory in the parameter space for the optical polarization and
the field parameter A. At each step we move a little bit down the gradient — 53.3% and

—— 2%. The end product of the iteration corresponds to the self-consistent minimiza-

54

tion of the energy. The time dependence is just a label for different configurations.

In practice we chose a step Ar and iterate the equations

 

 

3H
42(1' + Ar) ~ 1P(T) — ATt—SWT) , (6.3)
A(T + Ar) z A(T) — AT 3H (6.4)

EMT)
by normalizing 1p and A to the total number of polaritons Np at each iteration. The
time step Ar controls the rate of convergence. The system is described using 100
points and periodic boundary conditions and one parameter for the cavity photon
field. As a test we compared our numerical calculations with some analytical limit
cases. For the trial initial 1p we used a random values on each site and also a form
corresponding to an analytical limit that will be discussed in the next section. The
number of iteration depends on the convergence rate and the choice of the initial

trial function. Typically, we used 105 — 106 iterations.

6.2 Low polariton density

The results of the numerical solution for the function 1]) are shown in Fig. 6.2 for
the case x = 6t0. At very low excitation densities pp = Np/N, where N is the
number of sites, polaritons behave as simple bosons. In that limit the self-trapping
effect is absent since the nonlinear attractive potential is proportional to the local
polariton density and can be neglected in this low density limit. Mathematically,
this limit can be described by approximating tp/ W z 1]: in the last term of
Eq. (5.10a). The cubic term 423 is not strong enough to give rise to the self-trapping
effect in this limit [37]. In other words, a nonhomogeneous solution to Eq. 5.10a
would have an overlap much bigger than the periodicity of the system, imposed
by the boundary conditions. Note that symmetry-breaking always exists for box

boundary conditions, which we do not consider here.

55

 

 

 

 

 

l T I f T T
QOBV z
:5 , _, ....... . ______ \
ch] "’ til l
50.6" : l|| || _.
c? ~ .' ll '1
O I il| i
.5 0.4H ' l | '. z
3 : . ' '.

'C‘. ’ . ,‘ l a
5302— 5 ' '. '-. _
o . ' ll ‘ I
pd : I \ I
I ............. I ....... r ............. \_ ....................
O 1— .. L-’ . 2’ . \\_L n \._J_ z
o 20 4.0 . 6O 80 100
Sltelndex

Figure 6.2: Polarization 4) for 6/t0 = 0.01, g/to = 0.01, X/to = 6, pp 2 3 x 10‘3
(dotted line), pp = 6 x 10-3 (dashed line), pp = 6 x 10-2 (dot-dashed line), and
pp = 10 (solid line).

In the absence of the self-attractive term the distribution of the optical polariza-

tion is homogeneous. In this low excitation limit the chemical potential it is simply

I
y=§(w3(+wc—(/62+g’2), (6.5)

where cog, = wx — 2t0 (energy at the bottom of the excitonic band), 6 = (as, — we is

given by

the optical detuning, and g’ = gm. Notice that in this limit the chemical potential
does not depend on the coefficient of the cubic term in Eq. (5.10). Moreover in this

limit the chemical potential corresponds to the energy of the lowest polariton at

k=O.

6.3 Intermediate density: self-trapping

By increasing the number of polaritons we observe a critical polariton number at
which a polarization symmetry-breaking transition occurs. A similar symmetry-
breaking transition was found in the case of BBC with an attractive nonlinear in-

teraction [37]. Our numerical calculations for the dependence of it on the polariton

56

constant solutions
conventional

. elliptic solutions
linear-response /
o .. 3:.“ ‘ " 9
. : ‘ . 1-. :-_ tr" 1 ; .‘ . '-

self—tmpping '
\

-0.2

 

*‘G
0
§
.4.
D
3
_Q-
g
a.
3
IA

3
I
I
I
I
I
I
I
g
° — ""§\?

l- | 4
. . . i
sollton-likei solutions / ii’
-0.6 - 39 -

,“flat-top"solqtions . ' "jkineticlterm dJ'sappeor'ance

io“ 10'“ 10’ 10’

 

 

 

Pp

Figure 6.3: Polariton chemical potential as a function of the polariton density for
6/t0 = 0.01, g/to = 0.01, X/lo = 0 (solid line), x/to = 4.5 (dashed line), and
x/ to = 6 (dot-dashed line). Circuses show the analytical solution regions.

density p p = N p/ N are shown in Fig. 6.3 for several values of the cubic term x,
where the circled regions are those we are able to find analytical solutions .

When the exciton number increases following the polariton number, a self-
trapping occurs, caused by the polariton-lattice coupling (dashed line in Fig. 6.2).
There is a drop in the chemical potential at the point of symmetry-breaking (Fig. 6.3)
due to the self-trapping. The breaking of the spatial homogeneity gives a negative
contribution to the energy of the system. Also, this makes exciton-like states more
energetically favorable than photon-like states, which results in a sharp decrease
of the photonic component for nonzero x, as shown in Fig. 6.4. The self-trapping
effect brings a discontinuity both in the chemical potential 8, and in the photon (ex-
citon) density as a function of the polariton density. Since the effective attractive
potential depends on both ltpl2 and x, the point of symmetry breaking for higher
x corresponds to smaller values of the polariton density pp = (Np) / (N).

Starting from the small excitation limit, we can expand tp/ \ /1 — 412 z o + %¢3.

57

 

 

 

 

l l l l
100* 4 _
III/I.
1’."
Z04 10-2_ ’4” in J
N a’I‘l // i
(< . : LII [WI
10“L I r
-6 1 1 1 1
10 _ - - -
104 103 102 10‘ 100 101
Pp
Figure 6.4: Photon density as a function of polariton density for x/to =
0 (solid line), X/to = 4.5 (dashed line), and x/to = 6 (dot-dashed

line).

The second term introduces an effective repulsion related to the intrinsic fermionic
nature of the excitons and decreases the cubic term x by (cox — fl) / 2. In this small
excitation limit some analytical forms for the solution to Eq. (5.10a) for the system
with a fixed exciton number can be written in terms of Jacobian elliptic functions.
Using the identity 4)” 2 ip’ (dip’ /d1p), we write down Eg. (5.10a) in this limit in the
form

Id I _
F13.)— = ‘(X — wLZ—EMP ——gA+ (wx — t0 _ ”)4’ I (6'6)

After integrating both the sites of Eq. 6.6, we get

tot}!

cox—y

(7C—2)
2

 

imp/2 = — (P4 — Zg/IIP + (wx — to — y)l[12 + d , (6.7)

where d is a constant of integration. using the separation of variables and integrat-

ing again, we get the expression

 

 

to fay: / \/_(X_ “P , (6.8)

WWW — 23M + (wx — to — W2 + d

58

 

sn(u l m) etc.

 

l
O

 

 

 

 

Figure 6.5: The three basic Iacobian elliptic functions, for the parameter m=0.99.
The solid line is sn, the dotted line is cn, and the dashed line is dn. Note that
the period of dn is half that of the other two, and that dn is nodeless. All Iacobian
elliptic functions may be constructed from these three. Of the 12 possible functions,
these three shapes are the only normalizable ones which differ from each other by
more than a translation along the horizontal axis or a renormalization along the
vertical axis [37].

which is an elliptic integral. It is always possible to express an elliptic integral as a
sum of normal elliptic integrals of the first, second, and the third kinds [38]. Then
by the use of Cayley transformation [39] Eq. (6.8) can be inverted, and ip(v), which
is the solution to the polarization function, can be given in terms of the Iacobian
elliptic functions. There is a total of 12 such functions. Only three of them (sn, cn

and dn) have distinct physical forms. The function sn(uIm) may be written in the

integral form,
d t

o \/1 — t2\/1 — th’

 

(6.9)

 

where u = sn“l(x) so that x : sn(uIm). The functions cn(u|m) and dn(uIm) may

then be defined by the equations

 

(u)=Im \/1— sn2( (|),m
n()=uIm \/1 —— msn2( u(Im). (6.10)

 

59

 

0.098 1 i . I a i . 1 fl

0.096

Polarization, \II=20LB
E
A

0.092

 

 

 

 

1 l 4 l 1 1 l 1
o 20 49 . 60 80 100
Site index

 

Figure 6.6: Polarization II) for 25/10 = 0.01, g/to = 0.01, x/to : 6, pp = 4 x 10“3
near the symmetry-breaking point.

We plot them in Fig. 6.5. The limits of the sn, cn and dn, along with complete
elliptic integrals K (m) and E(m), are listed in Table 6.1. The period of the sn and
en functions is 4K(m), while that of the dn function is 2K(m).

Particularly, if we neglect the second term in the right hand side of Eq. (6.7), the
solution to the nonlinear Schrodinger equation (NLSE) is given by one of the Jaco-
bian elliptic functions. The cn function solves the NLSE for repulsive nonlinearity,
while only the cm and dn functions solve the NLSE for attractive nonlinearity. Only
the dn function satisfies the periodic boundary conditions in the assumption that
the polarization function does not vanish (42,, > 0). Thus, it should be a solution
for the optical polarization to the nonlinear equation (5.10a) when the excitonic
number is greater the photon one. If the cubic term of the NLSE is weak, the para—
m=0 m=1
sn(uIm) sin(u) tanh(u)
cn(uIm) cos(u) sech(u)
dn(u|m) 1 sech(u)

K(m) 7T/2 oo
E(m) 7I/2 1

 

 

 

Table 6.1: Limits of Jacobian Elliptic functions and integrals [40]

60

meter m —-> 0+ and dn —1 1. In the opposite case of the strong nonlinear attraction,
m —+ 1‘ and dn —> sech, that leads to the function collapse in a sharp peak. The
numerical solution for the polarization 1]) near the symmetry breaking is shown
in Fig. 6.6. The shape of the function plot corresponds to dn, which looks like an
asymmetric wave function on top of the constant solution. This solution is unstable
and has a very slow convergence.

With a small increase of the polariton number p X the excitonic component starts
to significantly exceed the photonic one because the self-trapping makes it energet-
ically favorable. The polarization function collapses in a sharp peak because of the
strong nonlinear attraction. In case when the photon number is much smaller than
the exciton number, we can neglect the fourth term in Eq. (5.10a), since as seen
from Eq. (5.10b), A is inversely proportional to co, — y. In this limit, when the local-

ization length is much smaller than that of the chain, the solution can be written in

 

the form
1]) = Asech[Y(v — v0)I , (6.11)
where
_. m N
Y: (X 2 ) P, (6.12)
41‘0

A: 0%, (6.13)

and the chemical potential is determined by the equation

(x - “ii—LEVNIZ’
16m . (6.14)

 

y=wx—2t0—

Alternatively, we can say that when the cubic term in the Eq. (5.10a) is quite strong

m ——> 1 and dn(uIm) —> sech(u)

61

 

Exciton density times 2

 

 

 

10 . 1 1 1 1 1 1 r
-4 -2 O 2 4

10
Pp

 

Figure 6.7: Exciton density times 2 as a function of the polariton density for
x/to = 0 (solid line), X/to = 4.5 (dashed line), and X/lo = 6 (dot-dashed
line).

6.4 Saturation

By further increasing the polariton density, the internal fermionic structure of the
excitons gives rise to a hard-core repulsion term, which leads to the saturation of
the exciton states, making the polarization distribution broader and flatter at the
top (dot-dashed line in Fig. 6.2). When the saturation spreads over the whole chain
length, the polarization distribution (solid line in Fig. 6.2) becomes homogeneous
again. We expect a discontinuity of )1 again at this point as a consequence of the
disappearance of the gradient of the polarization function (kinetic term), which
gives a positive contribution to the energy. With the further increase of the exci-
ton density the hopping effect is reduced due to the blocking. Fig. 6.7 shows the
exciton density f I8I2dv/ N multiplied by 2 as a function of pp, which reaches 1 in
the limit of large excitation density regardless the value of x, corresponding to half
filling of the exciton band. The half filling maximizes the polarization and hence
minimizes the dipole interaction energy between the excitons and the cavity pho-
tons. In this saturation regime the polaritons become photon-like, since an added

excitation mainly contributes to the cavity mode, and thus the chemical potential

62

 

 

 

 

 

0.2 l fl l ' l ' l ‘ l
- .1
. -- I
ll '1
Ii 'I
01* li ll _
N l I ' I
‘> ’ H N
n . I
Q) 0 —-/——;h.—L. ._ _ __.,..______
8 1'1 If" ‘i -’ T
O 1 l i -'
LII-I l I l l
on H i? —
H i.‘
H 'l
1' I,“
I I
_02 1 1 1 1 1 1' 1 L 1 1
' 0 20 4.0 . 60 80 100
Slte index

Figure 6.8: Force as a function of the site index for x/ to = 6, p6,, 2 6 x 10’3‘ (dashed
line), per 2 6 x 10‘2 (dot-dashed line).

finally approaches wc (see also Fig. 6.3). In this saturation regime the expressions
for the polarization function 42 and the chemical potential y are given by the fol-

lowing expressions

0
= ——-——-—— 6.15
1P ,———-§2 + ()2 ( )
and
I2
j, = w, _ ____3 (6.16)

4W ’

where O = g\/_N_,g’ = g/N, and 6 = cox — we.

A gradient in the density of polarization produces a force on the ions according
to Eq. (5.8b). The force is stronger at the edges of the saturation region of the Iipl2
distribution as seen in Fig. 6.8, and is positive to the left from the center of the
symmetry-breaking point and negative to the right. This reduces of the total length
of the chain due to the interaction with the electromagnetic cavity mode.

The classical displacements due to such a force, which can be found from Eqn. 2.29,
with the corresponding scheme of the chain distortions, are shown in Fig. 6.9 for
6/to = 0.01, g/to = 0.01, X/to = 6, pp = 6 x 10‘2. It shows a significant change in

the region of the exciton localization.

63

 

 

Displacement, *y/ZC

.L.
O
l
1

 

1 l 1 l 1 l 1

20 4o . 6O 80 100
Slte Index

®—+€§>—+ €9~+ 9+ 0' O 40 +0 G—Q 4—@ <—®

Without light

63’ 9 ®® 9.0063 ‘3 @
Withlight

 

 

 

C

Figure 6.9: Displacement due to the force as a function of the site index for 6/ to =
0.01, g/to = 0.01, x/to = 6, pp = 6 x 10’2 and the corresponding scheme of
the chain distortions showing the position of atoms before and after the system
excitation.

6.5 Homogeneous deformation and role of damping

The lattice deformation in the presence of optical excitations can be understood by
analyzing the total energy of the system in the assumption that the lattice defor-
mation is homogeneous [41], [42]. As seen in the previous section, this assumption
is justified for a systems in the saturation regime or for a polariton density below
the critical value for self-trapping. The total energy of the lattice is modified by
the presence of polaritons. The homogeneous deformation allows us to obtain an
analytical expression for the total energy of the system, and to analyze the effect of

the finite linewidth of polaritons. We will include both the spontaneous emission

64

 

 

 

 

 

100 ' t ' I ' I ' I ' I
75 e -
50 - -
F=O.4n1cV
A
:q>) 25 ~ \ , ‘
E 0 >— \ ‘ s ‘ ‘ ’ ’ I I I .1
b . ‘ “I ------
I
-25 - '1 a
l
_50 _ F=0.7meV q
.I
'75 n 1 1 1 n 1 1 1 1 1 1 T
-0.03 -0.02 -0.0l 0.00 0.01 0.02 0.03
u/a

Figure 6.10: Effective lattice potential as a function of displacement. The figure has
been obtained for the following set of parameters: E8 = wx — 2t0 = 2.282 eV,
D = 2712 = 6.1eV, s = 2.5 - 103m/s, a = 1.5 nm, M = 2.110-21 g, wo = 2.282 eV,
and C = SZM/a2 = 82eV/a2. The excitation corresponds to p1.Jr ~ 0.1. a, = g’ = 1
meV, and we 2 2.242 eV. F = 400 peV and F = 700 er for the upper and lower
curve, respectively. The curves are displaced for clarity.

65

of the excitons and the finite Q factor of the cavity mode. Instead of using the vari-
ational approach of Chapter 5, we will solve directly the equations of motion for
the excitons and cavity photons.

We start by using the Fourier transform

1 .
3,, = — Ee’km’bk, (6.17)
W 1

where N is the number of lattice sites, k is the wave vector, and a is the lattice

separation, to define the operator bk as the annihilation operator of an exciton with

a wave vector k. The Hamiltonian in Eq. (5.1) can then be rewritten as

H: E;— (un+1- un)2I +

+£wkbgbzk+Efl (,k k’) b 6,,
k,k’

+wcc+c + g§c+b0 + 33’ch , (6.18)

where wk = cox —- 2t0 cos(ka) and

f(,k k’)= Till] un11—un)
n

X Ie—ikaeina(k—k’)+eik’aeina(k’— k)Ix ' (6.19)

We consider a finite-length chain with periodic boundary conditions. The equilib-
rium lattice displacement is homogeneous and will be indicated by u : un+1 — u".

Notice that in this limit f (k, k’) is diagonal
f (k, k’ ) = 27cos(ka)u§k1k1 . (6.20)

Eqs. (6.20) and (6.18) show that in this homogeneous case there is no mixing of

polariton with different k vector. Since the cavity couples to the k = 0 exciton

66

mode only, the polariton modes at k 76 0 are completely decoupled and do not
enter in the dynamics. Taking into account Eq. (6.20), the excitonic part in H re-
duces to wo(u)bgbg, where w0(u) = cox — 2t0 + 2711 is the energy of the exciton at
k = 0 renormalized by the lattice deformation potential. We also add a term that
describes the pumping of the cavity mode by an external field and represent the

full system by

 

2 2
_ mfi
H _ Zn: I2M + 2 I +
I
+w0(u)b5bo + 32—(clbo + bgc) + wcc+c +

@0361.th + C‘l'e-iwct) I (6.21)

where E0 represents the electric field of the external pump and 1c is the coupling
between the external pump and the cavity mode. The operators Bn are fermionic on
the same site, but commute for different sites. Therefore, the commutation relation

for b0 reads
2 X: B; 8,,

[60,115] = 1 — —"——

N (6.22)

Since we will assume here that there are no excitations of phonon modes at k 75 0,

we have that )2 BIB" = )3 bzbk ~ bgbo. The equations for the expectation values of
N k

the polarization (b0) = p, exciton density (bgbo) = nx and cavity photon operator

(c) = A can be obtained using the standard factorization scheme [43] and read

I .
A = —i IwcA + %p + 55—08%de — acA , (6.23)
2; £3 2 £2_
p — zIw0(1 N )p+ 2 (1 N )I Pp, (6.24)
. .8, 271x :1 :1:
nx=—13(1—W)(p A—pA )—2Fn, (6.25)

where we have introduced the spontaneous emission rate of the excitons I' and

67

the damping of the cavity mode ac, due to the finite Q-factor of the cavity. The
equation for A can be explicitly integrated and gives

1 _8_’2 53 -1...
m _. 2 “c + Zace . (6.26)

This expression can be used in Eqs. (6.24) and (6.25) which give for the steady state

-— _-8___ __ _ _Z"_x - e’i 110
p—( 122C(1 N))/(wc w0(1 N)+zI‘+140‘C(1 N ,

(6.27)

where we have defined p = fie‘iw“.
The u-dependent total potential energy U(u) of the system can be obtained
by substituting the solution for p, A, and "x which depend on 11, into the initial

Hamiltonian(6.21). At weak excitation we obtain

U(u) ~ él-z—Cl‘u2 + w0(11)nx(u)+

I 2
(g ) — -2—I:I nx(u) +wcl (6.28)

w _
+C 2a ac

 

where I = (KEG/Zac)2 and

l(if/2)2
15 + 27102 + r2 '

 

nx(u) ~ Ipl2 = ( (629)

Notice that nx has a resonant behavior as a function of 11, which is a consequence
of the deformation potential shift in the excitonic energy.

In the absence of light the potential energy LI depends quadratically on u with
a minimum at u = 0. When the laser is switched on, this dependence changes due
to the presence of optical excitations in the system. Due to the resonant behavior
in Eq. (6.29), it is energetically favorable to have u yé 0 if this reduces the total
energy of the system. Figure 6.10 shows the potential energy for some values of

parameters. The parameters used in these calculations are typical of polydiacety-

68

lene chains. We observe that the potential II has a parabolic dependence with a
superposed Lorentian contribution due to the optical excitation. This Lorentian
contribution can be either positive or negative depending on the relative strength
of the exciton-cavity coupling, spontaneous emission rate and cavity damping. The
actual behavior can be explicitly calculated using Eq. (6.28). In the figure we con-
sider two particular sets of parameters providing a minimum of the energy that

corresponds to contraction and expansion of the lattice.

69

Conclusions

The main part of this thesis has been devoted to the analysis of the exciton-photon
system interacting with the lattice. We have investigated the optically-induced lat-
tice strain in a single polymer chain in a cavity. We have extended the excitonic
SSH model, describing the exciton propagation in the chain, to include the effect
of the cavity electromagnetic field. Using a polariton picture, we have obtained
a system of integro-differential nonlinear equations for the spatial distribution of
the optical excitations in the chain. At low excitation regime we have been able to
get analytical solutions to this system of equations in terms of the Jacobian elliptic
functions. Numerical calculations have been performed by the use of the steepest
descent method of functional minimization. We find solutions describing a self-
trapping of the polaritons which saturate when the excitation density is increased.
This saturation induced by the hard-core repulsion is related to the internal fermi-
onic structure of excitons.

The chemical potential of the polaritons shows a discontinuity at a threshold
value of the polariton density. This critical density corresponds to the onset of the
self-trapping. The self-trapping causes a sharp increase in the excitonic component
and decrease in the photonic component of the polariton wave function. With the
polariton density increase the system becomes more photon-like because of the
excitonic part saturation. That makes the chemical potential approach the cavity
photon energy in the limit of the high excitation.

We have also considered the role of the finite radiative recombination rate of the

70

excitons and the finite Q factor of the cavity. These can be studied in a direct way
in the case of a homogeneous strain field. We have found that both a contraction
and expansion of the lattice are possible, depending of the relative strength of the
exciton-cavity coupling, radiative recombination, and cavity Q factor.

We have neglected exciton-exciton Coulomb effects, which may affect the pic-
ture, especially in the high excitation regime. This deserves further investigations.
However, it is known that the BCS variational ansatz in Eq. (5.4) provides a good
description of the ground state of an electron-hole system both in the low excita-
tion regime (excitonic BBC) and in the high excitation regime (BCS pairing) [44].
The fact that the electron-hole system here interact with the cavity electromag-
netic field gives an additional justification to this ansatz since light-dressing can
reduce Coulomb features beyond mean field like screening and broadening [45].
The exciton-exciton correlations may change the magnitude of the effective para-
meters such as the electrom / hole dispersion and the coupling constant [46].

The results suggest that local lattice deformations can be finely controlled by
the polariton density and detuning. This control scheme could have interesting
applications not only to polymers but also to some other organic materials where
a strong exciton-phonon interaction exists, like e.g. in J-aggregates, or DNA [47].
Recently-developed experimental techniques can detect light-induced lattice dis-
placement in molecules and semiconductor systems. The quantity that is measured
in electron or x-ray diffraction experiments would be Bun / an, i.e. the local defor-
mation due to the light-induced changes in the lattice. Typically this quantity is in
the range 10 — 100 "1A, which is reasonable for an experimental observation of the
effect. As an example, we can propose to use ultrafast electron diffraction (UED) to
detect light-induced lattice displacement [48]. A typical second-generation UED
apparatus is composed of a pulse laser, a picosecond electron gun, a free-jet expan-
sion source, and a singe-electron detection system. Diffraction images are recorded

with and without the system excitation by a charge-coupled device camera at the

71

end of a phosphor scintillator-fiber optic-image intensifier chain in the detector
chamber. This technique has a spatial resolution of about a milliangstrom. In these
experiments the laser pulse is typically used to thermally excite a system. The elec-
tron diffraction is then used for the detection of structural changes due to heating.
The structural changes discussed in this thesis have a completely different nature,
since they originate from the strong exciton-light and exciton-lattice coupling. In
contrast to heating, the coherent nature of the exciting field is important in the
structural deformations we have investigated. The results of this thesis could open
to novel investigations on this coherent melting, based on laser and optical control

in condensed matter systems.

72

Bibliography

[1] J. G. Miiller, U. Lemmer, G. Raschke, M. Anni, U. Scherf, J. M. Lupton, and J.
Feldmann, Linewidth-Limited Energy Transfer in Single Conjugated Polymer Mole-
cules, Phys. Rev. Lett. 91, 267403 (2003)

[2] T. Guillet, I. Berréar, R. Grousson, J. Kovensky, C. Lapersonne-Meyer, M. Shott,
and V. Voliotis, Emission of a Single Conjugated Polymer Chain Isolated in Its Single
Crystal Monomer Matrix, Phys. Rev. Lett. 87 087401 (2001).

[3] F. Dubin, J. Berréar, R. Grousson, T. Guillet, C. Lapersonne-Meyer, M. Shott,
and V. Voliotis, Optical evidence of a purely one-dimensional exciton density of states
in a single conjugated polymer chain, Phys. Rev. B 66 113202 (2002).

[4] F. Dubin, R. Melet, T. Barisien, R. Grousson, L. Legrand, M. Shott, and V. Volio-
tis, Macroscopic coherence of a single exciton state in an organic quantum wire, Nature
Physics 2 32 (2006).

[5] R. H. Dicke, Coherence in Spontaneous Radiation Processes, Phys. Rev. 93, 99
(1954).

[6] W. P. Su, I. R. Schrieffer and A. ]. Heeger, Soliton excitations in polyacetylene,
Phys. Rev. B, 22 2099 (1980).

[7] E. G. Wilson, A new theory of acoustic solitary-wave polaron motion, Polarons and
exciton-polarons in one dimension: the case of polydiacetylene, J. Phys. C 16 6739
(1983); ibidem 16 1039 (1983).

[8] A. J. Heeger, S. Kivelson, J. R. Schrieffer and W. P. Su, Solitons in conducting
polymers, Rev. Mod. Phys. 60, 781 (1988).

[9] M. H. Szymanska and P. B. Littlewood, The crossover between lasing and polariton
condensation in optical microcavities, Solid State Commun. 124, 103 (2002).

[10] Y. Toyozawa, Optical Processes in Solids, Cambridge University Press, Cam-
bridge (2003).

[1 1] H. Deng, G. Weihs, C. Santori, I. Bloch, and Y. Yamamoto, Condensation of Semi-
conductor Microcavity Exciton Polaritons, Science 298 199 (2002).

[12] H. Deng, G. Weihs, D. Snoke, I. Bloch, and Y. Yamamoto, Polariton Lasing
Vs. Photon Lasing in a Semiconductor Microcavity, Proc. Nat. Acad. Sci. 100 15318
(2003).

73

[13] M. Saba et al., High-temperature ultrafast polariton parametric amplification in
semiconductor microcavities, Nature 414, 731 (2001).

[14] B. I. Greene, J. Orenstein, and S. Schmitt—Rink, All-optical nonlinearities in organ-
ics, Science 247 679 (1990); B. I. Greene et al., Phonon-Mediated Optical Nonlinearity
in Polydiacetylen,Phys. Rev. Lett. 61 325 (1988).

[15] F. Dubin, J. Berréar, R. Grousson, M. Shott, and V. Voliotis, Evidence of polariton-
induced transparency in a single organic quantum wire, Phys. Rev. B 73 121302(R)
(2006)

[16] K. 5. Song and R. T. Williams, Self-trapped excitons, Springer, Berlin (1996).

[17] Polydiacetylenes, edited by D. Bloor and R. R. Chance, NATO Advanced Study
Institute, Series B: Physics (N ijhoff, Dordrecht, 1985).

[18] Polydiacetylenes, edited by D. Bloor and R. R. Chance, NATO Advanced Study
Institute, Series B: Physics (Nijhoff, Dordrecht, 1985).

[19] V. M. Agranovich, M. Litinskaia, and D. G. Lidzey, Cavity polaritons in micro-
cavities containing disordered organic semiconductors, Phys. Rev. B 67, 085311 (2003).

[20] Y. Toyozawa, Self-Trapping of an Electron by the Acoustical Mode of Lattice Vibra-
tion, Prog. Theor Phys. 26 29 (1961).

[21] A. J. Heeger, S. Kivelson, J. R. Schrieffer and W. P. Su, Solitons in conducting
polymers, Rev. Mod. Phys. 60, 781 (1988).

[22] H. Zoubi and G. C. La Rocca, Phys. Rev. B 72, Exciton-polariton kinematic inter-
actions in organic microcavities, 125306 (2005)

[23] S. A. Moskalenko and D.W. Snoke,Bose-Einstein Condensation of Excitons and
Biexcitons (Cambridge University Press, Cambridge, 2000).

[24] E. Hanamura and H. Haug, Condensation eflects of excitons, Phys. Rep. 33, 209
(1977)

[25] J. Bardeen and W. Shochley, Deformation Potentials and Mobilities in Non-Polar
Crystals, Phys. Rev. 80, 72 (1950).

[26] G. Whitfield and P. B. Shaw, Interaction of electrons with acoustic phonons via the
deformation potenth in one dimension, Phys. Rev. B 14,3346(1976).

[27] www.pwscf.org

[28] P. Hohenberg and W. Kohn, Inhomogeneous Electron Gas, Phys. Rev. 136, B864
(1964).

[29] W. Kohn, L. J. Sham, Self-Consistent Equations Including Exchange and Correla-
tion Effects, Phys. Rev. 140, A1133 (1965).

74

[30] P. Giannozi, G. Grosso and G. Pastory Parravichi, Theory of Electron States in
lattice and Superlattices, Rivista Del Nuova Ciment026 N .3 (1990).

[31] P. Meystre, M. Sargent HI, Elements of Quantum Optics, Berlin; New York:
Springer-Verlag (1990).

[32] P. R. Eastham and P. B. Littlewood, Bose condensation of cavity polaritons beyond
the linear regime:The thermal equilibrium of a model microcavity, Phys. Rev. B 64
235101 (2001).

[33] RJ. Glauber, Coherent and Incoherent States of the Radiation Field, Phys. Rev. 131
2766 (1963).

[34] F. Dalfovo and S. Stringari, Bosons in anisotropic traps: Ground state and vortices,
Phys. Rev. A 53, 2477 (1996).

[35] W.H. Press, Saul A. Teukolsky, W.T. Vetterling, B.P. Flannery Numerical Recipes
in Fortran 77, Cambrige University.

[36] I. Stich, R. Car, M. Parrinello, and S. Baroni, Conjugate gradient minimization of
the energy functional: A new method for electronic structure calculation, Phys. Rev. B
39 4997 (1989).

[37] L. D. Carr, C. W. Clark, and W. P. Reinhardt, Stationary solutions of the one-
dimensional nonlinear Schrodinger equation. 1. Case of repulsive nonlinearity, Station-
ary solutions of the one-dimensional nonlinear Schrodinger equation. II. Case of attrac-
tive nonlinearity Phys. Rev. A 62, 063611 (2000); ibidem 62,063610(2000).

[38] Nandbook of Elliptic Integrals for Engineers and Scientists, Springer-Velag, New
York, Heidenberg, Berlin, (1971).

[39] Introduction to elliptic functions, John Wiley and Songs, Inc, New York (1953).

[40] Handbook of Mathematical Functions, edited by M. Abramowitz and LA. Stegun,
National Bureau of Standards, Washington, DC (1964).

[41] M. V. Katkov, Y. V. Pershin, and C. Piermarocchi, Theory of cavity-polariton self-
trapping and optical strain in polymer chains, Phys. Rev. B 74, 1 (2006)

[42] Part of the results in this we obtained in cooperation with Dr.Y. Pershin.

[43] V. M. Axt and A. Stahl, A dynamics-controlled truncation scheme for the hierarchy
of density matrices in semiconductor optics, Z. Phys. B 93, 195 (1994).

[44] C. Comte and P. Nozieres, Exciton bose condensation - the ground-state of an
electron-hole gas .1. Mean field description of a simplest model, I. Physique 43 1069
(1982)

[45] C. Ciuti et al., Strongly Driven Exciton Resonances in Quantum Wells: Light-
Induced Dressing versus Coulomb Scattering, Phys. Rev. Lett. 84, 1752 (2000).

75

[46] I. E.Perakis and TV. Shahbazyan, Canonical transformation approach to the ultra-
fast nonlinear optical dynamics of semiconductors, Int]. Mid. Phys. B 13, 869 (1999).

[47] E. M. Conwell and S. V. Rakhmanova, Polarons in DNA, Proc. Natl. Acad. Sci.
USA 97 4556 (2000).

[48] H. Ihee et al., Direct Imaging of Transient Molecular Structures with Ultrafast Dif-
fraction, Science 291 458 (2001).

76

 

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lIIIIIIIIIIIIIIIIIIIIIIIIHIIIIIIl