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Faculty of Fundamental Problems of Technology

PRACA DYPLOMOWA

Collective spontaneous emission from asystem of quantum dot

Wildan Abdussalam

sªowa kluczowe:

superradiance, short-range and long-range couplings, weak and

strong excitation regimes, regular and random arrangements,

homogeneous and inhomogeneous QD ensembles

krótkie streszczenie:

This thesis describes the enhanced emission in the small ensemble of QDs

in random and regular arrangement under weak and strong excitation

regimes

Wrocªaw 2012

Promotor: Prof. dr. hab. in» Paweª Machnikowski

"I would like to thank to my supervisor, Professor Pawel

Machnikowski, for priceless help upon my master study in Wroclaw

University of Technology"

Contents

1 Quantum dots 9

1.1 Quantum dots ensemble . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.1 CdSe/ZnSe system . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Self-Assembled Quantum Dots . . . . . . . . . . . . . . . . . . . . 10

1.2.1 Basic concept in epitaxial growth . . . . . . . . . . . . . . 10

1.2.2 Stranski-Krastanow QDs . . . . . . . . . . . . . . . . . . . 11

2 Model 13

2.1 The systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Evolution 27

3.1 Single QD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Quantum dots array . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Runge-Kutta ODE . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Results 39

4.1 Double quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Quantum dots array . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3 Mismatch energy inuence . . . . . . . . . . . . . . . . . . . . . . 49

5 Conclusions 53

5

Introduction

Double quantum dots (DQD) and quantum dot arrays have attracted enor-

mous attention in recent years particularly associated with the superradiance

phenomenon which is capable of being applied into laser control of spontaneous

collective radiation [1]. Dicke phenomenon is associated with the description of a

spontaneously radiating gas where all atoms or molecules interact with a common

radiation eld [2]. In this regard, the question arises of whether quantum dots

(QDs) may also exhibit signatures cooperative radiation and hence have to be

considered as coupled quantum systems.

The experiment progress shows that the QDs do not behave like individual inde-

pendent objects as long as they form an ensemble of QDs [3]. In this experiment

QDs are placed within mesas square and arranged randomly in the lateral en-

semble. The size of mesas is subsequently reduced to observe the eect of the

number of the dots on the decay rate. Those mesas are then irradiated with

photoluminescence under weak excitation regime and the decay rate is reduced

upon the mesa size reduction that yield the reduction of coupling.

Our model is attempted to describe the physical phenomena above. The inter-

play between the interference terms of the emission rates, mismatch energies,

long- and short-range couplings on the decay rate of excitons is described for the

case of double dots placed in the lateral array. This work was published in the

ref. [17]. Further, the role of random and regular arrays under weak and strong

excitation regimes is also described. This work has also been submitted to the

7

8 CONTENTS

Acta Polonica A and currently is being reviewed. Last but not least, The open

question whether the superradiance eect occurs in the long- or short-range cou-

plings is also described in the last section of chapter 4. This work is aimed to

shed some light on the interpretation of the experiment [3] in which enhanced

emission was observed in a quantum dot ensemble in which the dipole coupling

energies on the typical inter-dot distances were much smaller than the average

transition energy mismatch between the dots.

This thesis is organized as follow. Chapter one contains the overview of basic

properties of QDs and the description of most popular method of their fabrica-

tion. Second chapter contains The model describing the system of single QD,

double QDs and quantum dots array. The third chapter contains the evolution

and numerical model in the QDs. The fourth-chapter describes the decay rate

of exciton recombination and photon emission in either two-dot or quantum dots

array cases. This work was supported by the Foundation for Polish Science un-

der the TEAM programme, co-nanced by the European Regional Development

Fund.

Chapter 1

Quantum dots

This chapter contains overview of basic properties of QDs and description of

the most popular fabrication method of their fabrication based on the ref. [4].

1.1 Quantum dots ensemble

QDs are nanometer-sized three-dimensional structures which conne electrons

and holes in dimensions associated with De Broglie wavelength. As a result, the

energy levels are quantized and for that reason they are also often referred as ar-

ticial atoms. Many researchers are fascinated from the physics of semiconductor

quantum dots (QDs) and their high potential for applications in photonics and

quantum information technology. One of the most investigated type of QDs is The

II-VI system CdSe/ZnSe which is the material of choice for the yellow/green/blue

spectral range. The use of CdSe QDs instead of ZnCd(S)Se quantum wells as ac-

tive region in laser diodes is well-promissing candidate to overcome the stability

obstacles in ZnSe based optoelectronic devices [5].

1.1.1 CdSe/ZnSe system

This fabrication of CdSe/ZnSe system is based on the references [3] and [6].

The II-VI system CdSe/ZnSe self-assembled of quantum dots formed upon epitax-

9

10 CHAPTER 1. QUANTUM DOTS

ial growth is related to the lattice-missmatched heteroepitaxy Stranski-Krastanow

(SK) growth mode. In a simplied picture, the strain is released by the forma-

tion of islands without introducing dislocations after the two-dimensional (2D)

growth of a few-monolayers (ML)-thick layer. The sample modeled here [6] was

grown by molecular beam epitaxy on a GaAs substrate and a 200-nm-thick GaAs

buer layer. Nominally, 1.3 monolayers of CdSe were deposited on a 50-nm-thick

layer of ZnSe. QDs form by self-assembly. Finally, the QDs were capped with

a second 25-nm-thick layer of ZnSe. Upon the self-assembled The CdSe layers

were deposited by migration enhanced epitaxy at 0.029 ML/s and are embedded

in a 40-50-nm-thick ZnSe buer layer and a 20-25-nm-thick ZnSe cap layer. The

growth temperature was 280 0C for the II-VI layers [6]. On one part of the sample,

square-shaped mesas were formed by electron beam lithography and dry etching.

Single mesas were used to prove the existence of zero-dimensional states. For the

purpose of statistical averaging, another part of this sample was patterned with

such mesas arranged in arrays of size 100 µm × 100 µm and separations of 1 µm,

2 µm, 5 µm and 10 µm. The mesas were designed with edge lengths ranging from

60 nm to 1,000 nm, but the actual lengths were about 50 nm larger. For all of

the experiments, the sample was mounted to the cold-nger of a liquid-helium

bath cryostat and kept at a temperature of 2 K [3].

1.2 Self-Assembled Quantum Dots

In this section, the basic information of epitaxial growth concept is presented,

based on the ref. [4].

1.2.1 Basic concept in epitaxial growth

Epitaxial deposition takes place when a single-crystalline material A is de-

posited on a clean surface of a single crystal B (substrate) under proper condition.

When A and B is the same (dierent), the process is referred to a homoepitaxy

1.2. SELF-ASSEMBLED QUANTUM DOTS 11

Figure 1.1: The scheme of the three modes for heteroepitaxial growth of a material A on asubstrate B: layer-by-layer or FM, island growth or VW, or layer-plus-island orSK. In the gure, it is assumed that the lattice constant of material A is largerthan the lattice constant of B (strained heteroepitaxy) [4].

(heteroepitaxy). The material A can be provided, e.g., by thermal evaporation of

material from hot crucibles (molecular beam epitaxy or through molecular gas)

grows at the substrate surface (metal organic vapour phase epitaxy). The crystal

growth must clearly be a nonequilibrium process since the surface processes such

as condensation and evaporation must counterbalance.

According to Bauer [7], the lm growth may be classied in either of the follow-

ing three modes: layer-by-layer or Frank-van der Merwee (FM), island growth

or Volmer-Weber, or layer-plus island or Stranski-Krastanow (SK). These modes

are shown in g. 1.1.

1.2.2 Stranski-Krastanow QDs

In this section, the overview of Stranski-Krastanow growth InAs/GaAs is de-

scribed based on ref. [4]. The lm growth in Stanski-Krastanow QDs is started

with the preparation of three-dimensional island tetragonal formation for InAs

on GaAs (001) undergone by at lm to match the substrate lattice. It yields

mist dislocation, when the thickness of the deposited lm exceeds the certain

critical value consequently yielding deep traps for charge carriers. The island is

subsequently overgrown in order to provide a well-dened connement potential.

Even if the morphology is well characterized before capping, strong changes can

occur during the subsequent capping. Therefore, one must pay attention not to

use the shape of uncapped QDs to argue the morphology of buried QDs.

12 CHAPTER 1. QUANTUM DOTS

Figure 1.2: Schematic illustration of the fabrication of unstrained GaAs/AlGaAs QDs start-ing from InAs QDs (a-d). RT-STM image of an AlGaAs nanohole (D = 7 nm)with depth of about 4 nm (e). Micro-PL spectra of a single QD (D = 7 nm, d =2 nm) at increasing excitation intensity (f). Partially adapted from [9].

Once InAs QDs are overgrown, the GaAs surface above them can be used as

the "substrate" for the deposition of a new layer of QDs called vertical stacks of

InAs/GaAs QDs. This is useful to increase the volume density of QDs in a sample

(which is particularly important for application such as laser based on QDs). If

the spacer between two subsequent layer is thin enough, QDs in the upper layer

tend to form right on top of the buried QDs because of the tensile strain present

on the GaAs regions above buried QDs.

The GaAs surface with nanoholes is also able to be used as a template for

the fabrication of groups of closely spaced QDs, which may act as lateral QD

molecules [8]. The growth process is shown in g. 1.2. Nanoholes are obtained

by using a nominal etching depth of 5 nm. Bow-tie shaped nanoholes with the

depth of 5-6 nm are subsequenlty overgrown with InAs at 500 0C. Once an InAs

coverage of 0.8-1.4 ML the surface is planarized, as schematically shown in the

g. 1.2(b), X-ray diraction data show that holes are lled by a diluted InGaAs

alloy [9]. By increasing the amount of deposited InAs, pairs of small QDs begin

to be formed on top of the original nanoholes and are invariably aligned along

the [110] crystal direction. With further growth such QDMs grow in size and a

single QDs start appearing.

Chapter 2

Model

This chapter contains the overview of the ensemble of QD model. Since QD

is an articial atom we use an atomic model based of ref. [10] with some modied

derivations. Let us consider the ensemble of QDs which consists of N identical

two-level system coupled to a continuum of quantized EM modes, and possibly

to an external driving eld near resonance. The N QDs conned to a region

small compared to the spontaneous pulselength and the frequency shift due to

EM coupling is also taken into account.

2.1 The systems

Let us consider a collection of N identical non-overlapping QDs, at positions

~r1,...,~rN , coupled to a quantized multimode electromagnetic eld. Each QD Aα

is assumed to have only two states |±〉α, separated by energy εα = ε = ε+ − ε−.

Using the dipole approximation, the hamiltonian is able to be written with,

H =∑

α εασ†ασα + 1

2

∫d3rD

2(~r)ε0εr

+ ε0c2[∇× ~A(~r)

]2

− 1ε0εr

∑α

(~dασα + ~d∗ασ

†)· ~D(~rα)

(2.1)

13

14 CHAPTER 2. MODEL

Where ~rα is the position of the dot α, εα is the transition energy at the dot α, ~D(~r)

is electric displacement, ~A(~r) is electromagnetic potential, dα is the interband ma-

trix element of the electric dipole at the the dot α and εr is electric constant of the

semiconductor. Let us label the rst-part of hamiltonian as HQD(dots), second-

part is He−m(electromagnetic eld) and third-part is HI(QD-eld interaction) Let

us now dene ~A(~r) and ~D(~r) in the creation and annihilation operators:

~A(~r) =∑

~k,λ

√~

2ω~kε0εrVe~k,λe

i~k·~rb~k,λ + h.c.,

~D(~r) = i∑

~k,λ

√~ω~kε0εr

2Ve~k,λe

i~k·~rb~k,λ + h.c..

(2.2)

Where V is the volume of system, b~k,λ is the annihilation operator of photon, e~k,λ

is the polarization of the e-m wave with wave vector ~k and polarization λ and ω~k

is frequency of the e-m wave with wave vector ~k. By substituting Eq. (2.2) into

Eq. (2.1) we obtain

He−m =∑

~kλ ~ω~kb†~k,λb~k,λ,

HI = −i∑

α

∑~kλ

√~ω~k

2ε0εrV

(e~k,λe

i~k·~rαb~k,λ − e∗~k,λe−i~k·~rαb†~k,λ

)·(~dασα + ~d∗ασ

†α

).

Hence in the interacting picture with respect to H0 = HQD + He−m

H = −i∑

α

∑~kλ

√~ω~k

2ε0εrV

(e~k,λb~k,λe

i~k·~rαe−iω~kt − e∗~k,λb†~k,λe−iω~k·~rαeiω~kt

)

×(~dασαe

−iεαt/~ + ~d∗ασ†αe

iεαt/~).

(2.3)

Let us now calculate the average values of physical quantities at any time t.

For instance, the average number of excitons is 〈∑

α σ†ασα〉. The calculation is

solved in the Heisenberg picture, by calculating the evolution operators, generated

by Hamiltonian H. For any operator Q associated with the electron (exciton)

2.1. THE SYSTEMS 15

subsystem then we have

Q =i

~[H,Q]t ≡

i

~[H(t), Q(t)] (2.4)

Since[b~k,λ, Q

]= 0 we obtain

Q = 1~∑

α

∑~kλ

√~ω~k

2ε0εrV[e~k,λe

i~k·~rαe−iω~kt(~dα [σα, Q] e−iεαt/~ + ~d∗α

[σ†α, Q

]eiεαt/~

)b~k,λ

−e~k,λei~k·~rαeiω~kt

(~dα [σα, Q] e−iεαt/~ + ~d∗α

[σ†α, Q

]eiεαt/~

)b†~k,λ].

(2.5)

Here all operators are at the time t which is not written explicitly. Since the

carrier-photon coupling, the evolution of the exciton associated with the operator

Q involves the photon operators b~k,λ (t). In order to nd these operators we write

down the evolution equation for them:

b~kλ = i~

[H, b~kλ

]= 1

~∑

β~qλ′

√~ω~q

2ε0εrV

[b~q,λ′† , b~q,λ

]e†~k,λe

−i~k·~rαeiω~qt×

×(~dβσβe

−iεβt/~ + ~d∗βσ†βe

iεβt/~),

= 1~∑

β~qλ′

√~ω~q

2ε0εrVe†~k,λe

−i~k·~rαeiω~kt(~dβσβe

−iεβt/~ + ~d∗βσ†βe

iεβt/~).

This is able to be formally integrated over time to give

b~kλ(t) = b~kλ(t0) +∑

β

√~ω~q

2ε0εrV

∫ t0dτ e†~k,λe

−i~k·~rβeiω~kτ×

×(~dβσβ(τ)e−iεβτ/~ + ~d∗βσ

†β(τ)eiεβτ/~

).

(2.6)

16 CHAPTER 2. MODEL

This is subsequently substituted into the equation of motion for Q (Eq. 2.5) and

we obtain

Q = 1~∑

α~kλ

√~ω~k

2ε0εrVe~k,λei

~k·~rαe−iω~kt×

×(~dα [σα(t), Q(t)] e−iεαt/~ + ~d∗α

[σ†α(t), Q(t)

]eiεαt/~

)b~kλ(0)

−e∗~k,λe−i~k·~rαeiω~ktb†~kλ(0)

(~dα [σα(t), Q(t)] e−iεαt/~ + ~d∗α

[σ†α(t), Q(t)

]eiεαt/~

)+

+ 1~2∑

αβ~kλ

~ω~k2ε0εrV

e~k,λei~k·~rαe−iω~kt×

×(~dα [σα(t), Q(t)] e−iεαt/~ + ~d∗α

[σ†α(t), Q(t)

]eiεαt/~

)×

×∫ t

0dτ e∗~k,λe

−i~k·~rβeiω~kτ(~dβσβ(τ)e−iεβτ/~ + ~d∗βσ

†β(τ)eiεβτ/~

)+

−e−i~k·~rαeiω~kt∫ t

0dτ e~k,λe

−i~k·~rβeiω~kτ(~dβσβ(τ)e−iεβτ/~ + ~d∗βσ

†β(τ)eiεβτ/~

)×

×e∗~kλ(~dα [σα, Q] e−iεαt/~ + ~d∗α

[σ†α, Q

]eiεαt/~

).

(2.7)

We shall be interested in the average values of operators, that is, 〈Q〉 is of interest.

The terms of Eq. (2.7) start from the part 1/~2 untill the end is able to be written

2.1. THE SYSTEMS 17

in the following way

1~2∑

ijαβ

(~d

(i)α [σα(t), Q(t)] e−iεαt/~ + ~d

(i)∗α

[σ†α(t), Q(t)

]eiεαt/~

)

×∫ t

0dτ(~d

(j)β σβ(τ)e−iεβτ/~ + ~d

(j)∗β σ†β(τ)eiεβτ/~

)×

×∑

~kλ

~ω~k2ε0εrV

e(i)~k,λe

(j)∗~k,λ

ei[~k·(~rα−~rβ)−ω~k(t−τ)]+

−∑

ij1~2∑

αβ

∫ t0dτ(~d

(j)β σβ(τ)e−iεβτ/~ + ~d

(j)∗β σ†β(τ)eiεβτ/~

)×

×∑

~kλ

~ω~k2ε0εrV

e(i)~k,λe

(j)∗~k,λ

+ e−i[~k·(~rα−~rβ)−ω~k(−τ)]×

×(~d

(i)α [σα, Q] e−iεαt/~ + ~d

(i)∗α

[σ†α, Q

]eiεαt/~

).

(2.8)

where the third and fth term is called the memory function denoted R(ij)αβ (t− τ).

It contains the phase factor e−i[~k·(~rα−~rβ)−ω~k(−τ)] which oscillates very quickly as

a function of ~k unless t− τ ∼~k·(~rα−~rβ)

ω~k∼ n|~rα−~rβ|

C. We may say more denitely,

the memory depth is on the order of τmem ∼ nck0∼ 1

ω0, where ω0 is the frequency

cut-o, ω0 ∼ CD, where D is the size of QD. This time scale is extremely short

compared to the time scale of the secular evolution of σα(t) and hence we may

replace σβ(τ) → σβ(t) under the integral (since σ cannot change much between

18 CHAPTER 2. MODEL

the times t− τmem and t). We thus may obtain Eq. (2.8) in this way

1~2∑

ij

∑αβ~d

(i)α [σα(t), Q(t)] e−iεαt/~~d

(j)β σβ(t)

∫ t0dτe−iεβτ/~R

(ij)αβ (t− τ)

+~d(i)∗α

[σ†α(t), Q(t)

]eiεαt/~~d

(j)β (τ)σ†β(t)

∫ t0dτeiεβτ/~R

(ij)αβ (t− τ)+

+~d(i)α [σα(t), Q(t)] e−iεαt/~~d

∗(j)β σ†β(t)

∫ t0dτeiεβτ/~R

(ij)αβ (t− τ)

+~d(i)∗α

[σ†α(t), Q(t)

]eiεαt/~~d

(j)β (τ)σ†β(t)

∫ t0dτe−iεβτ/~R

(ij)αβ (t− τ)+

−∫ t

0dτeiεβτ/~R

∗(ij)αβ (t− τ)~d

∗(j)β σ†β(t)~d

(i)α [σα(t), Q(t)] e−iεαt/~+

−∫ t

0dτe−iεβτ/~R


(j)β σβ(t)~d

∗(i)α

[σ†α(t), Q(t)

]eiεατ/~+

−∫ t

0dτeiεβτ/~R


∗(j)β σ†β(t)~d

∗(i)α

[σ†α(t), Q(t)

]eiεατ/~+

−∫ t

0dτe−iεβτ/~R


(j)β σβ(t)~d

(i)α [σα(t), Q(t)] e−iεατ/~.

(2.9)

Further, let us substitute τ = t − s. Let us also neglect all the terms which

consist of the factors e±i(εα+εβ) because they oscillate very quickly and do not

have contribution to the dynamics. As a result we obtain

1~2∑

ij

∑αβ~d

(i)α~d∗(j)β [σα(t), Q(t)]σ†β(t)e−i(εα−εβ)t/~

∫ t0dse−iεβs/~R

(ij)αβ (s)+

−~d(i)∗α

~d(j)β σβ(t)

[σ†α(t), Q(t)

]e−i(εβ−εα)t/~ ∫ t

0dseiεβs/~R

∗(ij)αβ (s)+

+~d∗(i)α

~d(j)β

[σ†α(t), Q(t)

]σβ(t)e−i(εβ−εα)t/~ ∫ t

0dseiεβs/~R

(ij)αβ (s)+

−~d(i)α~d∗(j)β σ†β(t) [σα(t), Q(t)] ei(εβ−εα)t/~ ∫ t

0dτe−iεβs/~R

∗(ij)αβ (s).

(2.10)

2.1. THE SYSTEMS 19

Note that the second term in each line is the hermitian conjugate of the rst term.

Let us now attempt to calculate the expression of the form∑∫ t

0dseiωsR

(ij)αβ (s)~d

(i)α~d∗(j)β .

Let us asume that t τmem so that the upper integration limit is able to be

extended to ∞ (R(ij)αβ (t) = 0 for t τmem, anyway). First, let us perform

the summation over polarizations. The polarization vectors are transverse so

that the three vectors e~k1 , e~k2 , and k form a complete orthogonal set. Further,

the completeness relation may be written∑

λ e∗(i)~kλe

(j)~kλ

+ k(i)k(j) = δij and hence∑λ e∗(i)~kλe

(j)~kλ

= δij − k(i)k(j).

Next, let us integrate over the orientation of the vector ~k and then we obtain

R(ij)αβ (s) =

∑~kλ

~ω~k2ε0εrV

e∗(i)~kλe

(j)~kλei[

~k·~rαβ−ω~ks] →∑~kλ

~ω~k2ε0εrV

(δij − k(i)k(j)

)ei[

~k·~rαβ−ω~ks].

Where ~rαβ = ~rα − ~rβ. By changing the summation into integration we nd

R(ij)αβ (s) = V

(2π)3

∫d3~k

~ω~k2ε0εrV

(δij − k(i)k(j)

)ei[

~k·~rαβ−ω~ks],

∑ij R

(ij)αβ (s)~d

(i)α~d∗(j)α = 1

(2π)3

∫d3~k

~ω~k2ε0εr

[~dα~d

∗β −

(dαk)(

d∗βk)ei[

~k·~rαβ−ω~ks]].

The integration over ~k shall be performed in the spherical coordinates relative

to the orientation of the ~rαβ vector (which is, ~rαβ is along the "z" axis). Let us

write

~k = k

sin θ cosϕ

sin θ sinϕ

cos θ.

and obtain

∑ij R

(ij)αβ (s)~d

(i)α~d∗(j)α = 1

(2π)3

∫ π0

sin θdθ∫ 2π

0dϕ∫ k0

0k2dk

~ω~k2ε0εr×

×[~dα~d

∗β −

(dαk)(

d∗βk)]ei[k·rαβ cos θ−ω~ks].

(2.11)

20 CHAPTER 2. MODEL

where

∫ π

0

sin θdθ

∫ 2π

0

dϕek·rαβ cos θ = 2π1

ik · rαβ2i sin(k · rαβ) = 4π

1

k · rαβsin(k · rαβ).

We now can write

∑ij R

(ij)αβ (s)~d

(i)α~d∗(j)α = 1

(2π)3

[~dα~d

∗β∇α∇β − (dα∇α)

(d∗β∇β

)]×

×∫∞

0d~k

~ω~k2ε0εr

∫ π0

sin θdθ∫ 2π

0dϕei[k·rαβ cos θ−ω~ks],

= ~4π2ε0εr

[~dα~d


(d∗β∇β

)] ∫∞0d~kω~k

sin(k·rαβ)k·rαβ

e−iω~ks.

Using the identity∫∞

0dse−i(ω−ω~k)s = πδ

(ω~k − ω

)− iP 1

ω~k−ωand substituting

ω~k = cknwe nd

∫∞0dseiωs

∑ij R

(ij)αβ (s)~d

(i)α~d∗(j)α

= ~4π2ε0εr

[~dα~d


(d∗β∇β

)]1~rαβ×

×π∫∞

0dk c

nsin (k · rαβ) δ

(cnk − ω

)− iP

∫∞0dk c

n

sin(k·rαβ)ckn−ω

,

= ~4π2ε0εr

[~dα~d


(d∗β∇β

)]1~rαβ×

×π sin (k · rαβ) Θ(ω)− iP

∫∞0dk

sin(k·rαβ)k−k0

.

(2.12)

Where k0 = nωcand we may denote D(αβ)

⊥ =[~dα~d


(d∗β∇β

)]. Let

2.1. THE SYSTEMS 21

us now dene

∇α∇βsin(k·rαβ)

rαβ= ∇α

[k cos(k·rαβ)

rαβ− sin(k·rαβ)

r2αβ

](−~rαβrαβ

),

=

[−k2 sin(k·rαβ)

rαβ− k cos(rαβ)

r2αβ− k cos(rαβ)

r2αβ+ 2

sin(k·rαβ)r3αβ

](~rαβrαβ

)·(−~rαβrαβ

)+

+

[k cos(k·rαβ)


r2αβ

](~rαβr2αβ· ~rαβrαβ− 3

rαβ

)

, =k2 sin(k·rαβ)

rαβ.

(dα∇α)(d∗β∇β

) sin(k·rαβ)rαβ

= (dα∇α)

[k cos(k·rαβ)


r2αβ

](−

~d∗β~rαβ

rαβ

),

=

[−k2 sin(k·rαβ)

rαβ− k cos(rαβ)

r2αβ− k cos(rαβ)

r2αβ+ 2

sin(k·rαβ)r3αβ

]×

(−

~d∗β ·~rαβrαβ

)·(~dα~rαβrαβ

)+

[k cos(k·rαβ)


r2αβ

]×

×(~d∗β~rαβ

r2αβ·~dα~rαβrαβ−

~dα ~d∗βrαβ

)

, =k2 sin(k·rαβ)

rαβ

(~dαrαβ

)(~d∗β rαβ

)+

+

[3kcos(k·rαβ)

r2αβ− 3

sin(k·rαβ)r3αβ

](~dαrαβ

)(~d∗β rαβ

)+

−[k cos(k·rαβ)


r2αβ

]~dα ~d∗βrαβ

.

22 CHAPTER 2. MODEL

Thus,

[~dα~d


(d∗β∇β

)] sin(k·rαβ)rαβ

=

= k3[~dα ~d∗β −

(~dαrαβ

)(~d∗β rαβ

)]sin(k·rαβ)k·rαβ

+[(~dα~d

∗β

)− 3

(~dαrαβ

)(~d∗β rαβ

)]×

×[

cos(k·rαβ)(k·rαβ)

2 −sin(k·rαβ)(k·rαβ)

3

]

= k3 23Fαβ (krαβ)

∣∣∣~dα∣∣∣ ∣∣∣~dβ∣∣∣Where

Fαβ (krαβ) = 32[~dα ~d∗β −

(~dαrαβ

)(~d∗β rαβ

)]sin(k·rαβ)k·rαβ

+

+[(~dα~d

∗β

)− 3

(~dαrαβ

)(~d∗β rαβ

)]×

×[

cos(k·rαβ)(k·rαβ)

2 −sin(k·rαβ)(k·rαβ)

3

]

The rst (real) term in Eq. (2.12) contributes only for ω > 0, that is, only in the

2.1. THE SYSTEMS 23

rst two terms in Eq. (2.10). Hence, Eq. (2.10) can be written in the form

14πε0εr~

∑αβσα(t)†Q(t)σβ(t)e−i[εβ−εα]t/~D

(αβ)⊥ −Q(t)σ†α(t)σβ(t)e−i[εβ−εα]t/~Dαβ

⊥

−σ†β(t)σα(t)Q(t)ei[εβ−εα]t/~D∗αβ⊥ + σ†β(t)Q(t)σα(t)ei[εβ−εα]t/~D∗αβ⊥ sin(kβrαβ)

rαβ+

− i4π2ε0εr~

∑αβ[σα(t)Q(t)σ†β(t)eiωβαt − σβ(t)Q(t)σ†α(t)eiωαβt

]D

(αβ)⊥ P

∫∞0dk

sin(k·rαβ)k+kβ

+

+[σ†α(t)Q(t)σβ(t)eiωαβt − σ†β(t)Q(t)σα(t)eiωβαt

]D

(αβ)⊥ P

∫∞0dk

sin(k·rαβ)k−kβ

+

− i4π2ε0εr~

∑αβ[−Q(t)σα(t)σ†β(t)eiωβαt + σβ(t)σ†α(t)Q(t)eiωαβt

]D

(αβ)⊥ P

∫∞0dk

sin(k·rαβ)k+kβ

+

+[−Q(t)σ†α(t)σβ(t)eiωαβt + σ†β(t)σα(t)Q(t)eiωβαt

]D

(αβ)⊥ P

∫∞0dk

sin(k·rαβ)k−kβ

,

(2.13)

where kβ =nεβ~c , and ωαβ =

εα−εβ~ . In the second line of Eq. (2.13), the indices

are changed α ↔ β and let us assume∣∣∣kβ − ~k0

∣∣∣ π, hence we may replace

sin (kβrαβ) → sin(~k0rαβ

), where k0 = n~ε

~c and ~ε is the average transition energy

in the ensemble. In this way, we obtain for the rst and second line

k303πε0εr~

∑αβ Fαβ (krαβ)

[σ†α(t)Q(t)σβ − 1

2

σ†α(t)σβ(t), Q(t)

]eiωαβt (2.14)

In the third and fourth line from Eq. (2.13), approximating kβ ' k0, we change

α ↔ β and all the terms cancel each other. In the fth and sixth line for α 6= β

from Eq. (2.13) we have[σα, σ

†β

]= 0. Let us perform the change of indices and

24 CHAPTER 2. MODEL

approximate kβ ' k0, we obtain

− i4π2ε0εr~

[σ†ασβ, Q

]eiωαβtD

(αβ)⊥

1rαβ

[∫∞0dk

sin(k·rαβ)k+k0

+∫∞

0dk

sin(k·rαβ)k−k0

]

= − i4π2ε0εr~

[σ†ασβ, Q

]eiωαβtD

(αβ)⊥

∫∞∞ dk

sin(k·rαβ)k−k0

= − i4π2ε0εr~

[σ†ασβ, Q

]eiωαβtD

(αβ)⊥

1rαβπ

cos (k0rαβ) ,

= − i4π2ε0εr~

[σ†ασβ, Q

]eiωαβtD

(αβ)⊥

cos(k0rαβ)rαβ

.

(2.15)

Let us now dene

∇α∇βcos(k·rαβ)

rαβ= ∇α

[−k sin(k·rαβ)

rαβ− cos(k·rαβ)

r2αβ

](−~rαβrαβ

),

= [−k2 cos(k·rαβ)

rαβ+

k sin(krαβ)r2αβ

+k sin(krαβ)

r2αβ+

+2cos(k·rαβ)

r3αβ](~rαβrαβ

)·(−~rαβrαβ

)+

+

[−k sin(k·rαβ)


r2αβ

](~rαβr2αβ· ~rαβrαβ− 3

rαβ

),

=k2 cos(k·rαβ)

rαβ.

2.1. THE SYSTEMS 25

(dα∇α)(d∗β∇β

) cos(k·rαβ)rαβ

= (dα∇α)

[−k sin(k·rαβ)


r2αβ

](−

~d∗β~rαβ

rαβ

),

=

[−k2 cos(k·rαβ)

rαβ+ 2

k sin(rαβ)r2αβ

+ 2cos(k·rαβ)

r3αβ

]×

(−

~d∗β ·~rαβrαβ

)·(~dα~rαβrαβ

)+

[−k sin(k·rαβ)


r2αβ

]×

×(~d∗β~rαβ

r2αβ·~dα~rαβrαβ−

~dα ~d∗βrαβ

),

=

(k2 cos(k·rαβ)

rαβ− 3

k sin(k·rαβ)r2αβ

− 3cos(k·rαβ)

r3αβ

)(~dαrαβ

)(~d∗β rαβ

)+

+(~dα~d

∗β1)[

k sin(k·rαβ)r2αβ

+cos(k·rαβ)

r3αβ

].

Hence

Dαβ⊥

cos(k·rαβ)rαβ

=

=[~dα ~d∗β −

(~dαrαβ

)(~d∗β rαβ

)]k20cos(k0·rαβ)

k·rαβ−[(~dα~d

∗β

)− 3

(~dαrαβ

)(~dβ rαβ

)]×

×[k0 sin(k0·rαβ)

r2αβ+

cos(k0·rαβ)r3αβ

]

= −43k3

0Gαβ (k0rαβ)∣∣∣~dα∣∣∣ ∣∣∣~dβ∣∣∣

(2.16)

Where

Gαβ (krαβ) = 34−[~dα ~d∗β −

(~dαrαβ

)(~d∗β rαβ

)]cos(k·rαβ)k·rαβ

+

+[(~dα~d

∗β

)− 3

(~dαrαβ

)(~d∗β rαβ

)]×

×[

sin(k·rαβ)r2αβ

+cos(k·rαβ)

r3αβ

]

(2.17)

26 CHAPTER 2. MODEL

If the displacement eld is 0 initially (no external eld present) then the rst and

second line from Eq. (2.7) vanishes upon averaging and the equation of motion

for the observable Q reads

Q =∑

α εα[σ†ασα, Q

]+∑

α 6=β iGαβ(k0rαβ)|~dα||~dβ|k30

3πε0εr~

[σ†ασβ, Q

]ei(εα−εβ)t/~

+∑

α,β

Fαβ(k0rαβ)|~dα||~dβ|k303πε0εr~

[σ†αρσβ − 1

2

σ†ασβ, Q

].

(2.18)

Where Ωαβ =Gαβ(k0rαβ)|~dα||~dβ|k30

3πε0εr~ and Γαβ =Fαβ(k0rαβ)|~dα||~dβ|k30

3πε0εr~ . We thus may

simplify Eq. (2.18) to become

Q =∑

α εα[σ†ασα, Q

]+∑

α 6=β iΩαβ

[σ†ασβ, Q

]ei(εα−εβ)t/~

+∑

α,β Γαβ[σ†αρσβ − 1

2

σ†ασβ, Q

].

(2.19)

This expression will be the fundamental equation of motion in the theory of

spontaneous emission from an N-atom system. The properties of Eq. (2.19) can

be found in the appendix.

Chapter 3

Evolution

In this chapter the evolution part of single QD is described based on the

third of right part of Eq. (2.19). In the second part, quantum dot array is also

described based on the Eq. (2.19).

3.1 Single QD

Now let us consider only evolution part of the single QD model taken from

the third of right part of Eq. (2.19),

ρ =∑α

Γα

[σαρσ

†α −

1

2σ†ασα, ρ

]. (3.1)

σ†σ, ρ is anticommutator and Γ is associated with the average transition either

from the ground state to the excited state (Γ+) or from the excited state to the

ground state (Γ−). In this case, α is represented by (+) and (−) hence equation

(3.1) turns out to be,

ρ = Γ+

[σ+ρσ− − 1

2σ−σ+, ρ

]+ Γ−

[σ−ρσ+ − 1

2σ+σ−, ρ

], (3.2)

in which σ+ = |1〉〈0| represents transition from ground state |0〉 to higher state

|1〉 and σ−=|0〉〈1| which represents emission from higher state |1〉 to ground state

27

28 CHAPTER 3. EVOLUTION

|0〉. Let us substitute those operators to the equation (3.2) and hence the equation

(3.2) turns out to be:

ρ = Γ+[|1〉〈0|ρ|0〉〈1| − 12(|0〉〈1|1〉〈0|ρ+ ρ|0〉〈1||1〉〈0|)]

+ Γ−[|0〉〈1|ρ|1〉〈0| − 12(|1〉〈0|0〉〈1|ρ+ ρ|1〉〈0|0〉〈1|)]

(3.3)

this master equation will subsequently be written in the matrix form

ρ =

〈0|ρ|0〉〈0|ρ|1〉〈1|ρ|0〉〈1|ρ|1〉

(3.4)

in this case the density matrix is parametrized in this form

ρ =

1+z2

x−iy2

x+iy2

1−z2

(3.5)

whereby x, y, and z are the coordinates on the Bloch sphere. Let us consider only

〈0|ρ|0〉 as an example by taking the average of equation (3.3) in the state |0〉

〈0|ρ|0〉 = −Γ+〈0|ρ|0〉+ Γ−〈1|ρ|1〉, (3.6)

Hence,

z2

= Γ+(1+z2

) + Γ−(1−z2

),

z = (Γ− + Γ+)(z − Γ−−Γ+

Γ−+Γ+),

(3.7)

Now, let us dene:

u = z − Γ− − Γ+

Γ− + Γ+

, (3.8)

3.1. SINGLE QD 29

Thus,

u = −(Γ− + Γ+)u,

∫duu

= −(Γ− + Γ+)∫dt,

lnu = −(Γ− + Γ+)t,

u = Ce−(Γ−+Γ+)t.

(3.9)

Hence,

z = Ce−(Γ−+Γ+)t + Γ−−Γ+

Γ−+Γ+,

ρ00 = 12

[1 + Ce−(Γ−+Γ+)t + Γ−−Γ+

Γ−+Γ+

].

(3.10)

After simplifying, the equation becomes

ρ00 =Γ−

Γ− + Γ+

+ (1− φ0 −Γ−

Γ− + Γ+

)e−(Γ−+Γ+)t. (3.11)

Since we know the value of z thus we can also nd ρ11

ρ11 =Γ+

Γ+ + Γ−+ (1− e−(Γ−+Γ+)t) + φ0e

−(Γ−+Γ+)t. (3.12)

Now let us solve the other part which is ρ10 and ρ01 in order to obtain the value

of x and y. Both parts describe exciton coherence between the ground state and


excited state. One nds

〈1|ρ|0〉 = Γ+[−12〈1|ρ|0〉] + Γ−[−1

2〈1|ρ|0〉],

〈1|ρ|0〉 = −12〈1|ρ|0〉[Γ+ + Γ−],

x+iy2

= −12[x+iy

2][Γ+ + Γ−],

x+ iy = x2[Γ+ + Γ−]− iy

2[Γ+ + Γ−].

(3.13)

It consequenlty yield the real part,

x = −x2[Γ+ + Γ−],

x = Ce−12

[Γ++Γ−]t,

(3.14)

and the imaginary part,

iy = − iy2

[Γ+ + Γ−],

y = Ce−12

[Γ++Γ−]t.

(3.15)

Let us now dene,

x+iy2

(0) = µ∗,

C [1 + i] = µ∗,

C = µ∗

1+i.

(3.16)

3.1. SINGLE QD 31

and for the other part,

x−iy2

(0) = µ,

C [1− i] = µ,

C = µ1−i .

(3.17)

The C constant in the Eq. 3.16 and Eq. 3.17 is then able to be substituted into

ρ01 and ρ10. We have just found ρ00, ρ01, ρ10 and ρ11 as

ρ00 = Γ−Γ++Γ−

+ (1− φ0 − Γ−Γ++Γ−

)e−[Γ++Γ−]t

ρ01 = 12µe−

12

(Γ++Γ−)t

ρ10 = 12(µ∗)e−

12

(Γ++Γ−)t

ρ11 = Γ+

Γ++Γ−(1− e−[Γ++Γ−]t) + φ0e

−[Γ++Γ−]t

(3.18)

Those results are presented back in the matrix form.

ρ =

ρ00 ρ01

ρ10 ρ11

(3.19)

ρ00 is the probability of having excitons in the system (ground state of QD), ρ11 is

the probability of having one excitonin the QD (excited state), ρ01 and ρ10 are the

quantum coherences between the ground and excited state. The diagonal parts

describe the exciton recombination in the single QD system while the o-diagonal

parts describe the dephasing of the system when the QD is in superposition of

the ground and excited states.


3.2 Quantum dots array

Let us consider N QDs placed in laterally aligned structures and separated

by distances ~rαβ. Each QD is modeled as a two-level system and contains zero

or one exciton. The Hilbert space of the many-dot system in our model is then

spanned by the empty dot state |0..0..0〉, all the states ß1, ..., iα, ...iN , in which iα

= 0, 1 within 2N states. The transition energies for the interband transition in

the system are

Eα = E + εα (3.20)

The dots are characterized by the identical interband matrix elements of the

dipole moment operator d . We assume that the excitons in the dots are heavy

hole excitons. The dots are coupled by an interaction Vαβ which can be either of

dipole-dipole character (long-range dispersion force) or result from carrier tunnel-

ing (short-range, exponentially decaying interaction). The coupling is represented

by a Hamiltonian corresponding to the unitary part of eq.(2.19).

H =N∑α=1

εασ†ασα +

N∑α 6=β=1

Vαβσ†ασβ, (3.21)

nα = σ†ασα, α = 1,2,...6 (simulation is taken up to 6 dots) denotes the exciton

number operators. Now let consider the operator σ†α and σα, in which α = 1,

2, ....,6, creating and annihilating an exciton in the dot α. In this many-dot

approximation, for instance in two dots case, |n2n1〉 denotes a state with n1

excitons in the 1st dot and n2 excitons in the 2nd dot in which (ni can be 0 or 1)

then,

σ2|11〉 = |01〉, σ†1|10〉 = |11〉, σ1|10〉 = 0, σ†2|10〉 = 0, etc.,(3.22)

where the third-part from the left of Eq. (3.22) means that we may not annihilate

a non-existing exciton and the fourth-part means that we may not create an

3.2. QUANTUM DOTS ARRAY 33

exciton in a dot which is already occupied. Shortly written,

σ1|n2n1〉 = n1|n2, n1 − 1〉, σ†1|n2n1〉 = (1− n1)|n2, n1 + 1〉 (3.23)

The coupling Vαβ is composed of two contributions: short-range V(sr)αβ and long-

range V(lr)αβ (dipole). The long-range dipole coupling written in the eq.(3.21)is

described by

V(lr)αβ = −~Γ0G(k0rαβ), (3.24)

where

Γ0 =|d0|2k3

0

3πε0εr(3.25)

is the spontaneous emission (radiative recombination) rate for a single dot, ε0 is

the vacuum permittivity, εr is the relative dielectric constant of the semiconduc-

tor, and

k0 =nE

~c, (3.26)

where c is the speed of the light n =√εr is the refractive index of the semicon-

ductor, and

G(x) =3

4

[−(

1− |d · rαβ|2) cosx

x+(

1− 3|d · rαβ|2)(sinx

x2+

cosx

x3

)].(3.27)

where rαβ = rαβ/rαβ and d = d/d, where d is the interband matrix element

of the dipole moment operator which is assumed identical for both dots. For a

heavy hole exciton, d =(d0/√

2)

[1,±i, 0]T , so that for a vector rαβ in the xy

plane one has |d · rαβ|2 = 1/2 and for rαβ in the z direction one has |d · rαβ|2 = 0.

The short-range coupling is described by

V(sr)αβ = V0e

−rαβ/r0 . (3.28)


0

2

4

6

0 20 40 60 80 100

V/h-

(ns-1

)

r12 (nm)

(a)

Vlr/h-

Vsr/h-

0

1

2

0 200 400 600 800

V/h-

, Γ

12 (

ns-1

)

r12 (nm)

(b)

Vlr/h-

Γ12

Vsr/h-

Figure 3.1: Interference term of the decay rate Γ12 and the short and long-range couplingamplitudes Vlr, Vsr as a function of the lateraly placed inter-dot distance. In(a), the small distance section is shown, while in (b) the oscillating tail at largerdistances is visible.

The eect of the coupling to the radiation eld is accounted by including the dis-

sipative term in the evolution equations, which describes radiative recombination

of excitons. The equation of evolution of the density matrix is then given by an

equation 2.19.

ρ = − i~

[H0, ρ] +2∑

α,β=1

Γαβ

[σαρσ

†β −

1

2

σ†ασβ, ρ

+

], (3.29)

where

Γαα = Γββ = Γ0, Γαβ = Γβα = Γ0F (k0rαβ) , (3.30)

with

F (x) =3

2

[(1− |d · rαβ|2

) sinx

x+(

1− 3|d · rαβ|2)(cosx

x2− sinx

x3

)]. (3.31)

and .......+ denotes the anticommutator. The dipole (long-range) coupling

between the interband dipole moments associated with the excitons in the dots

exhibits a long-range nature, with the typical1

R3behaviour at short distance

while the short-range interaction vanishes exponentially at distances on the or-

3.3. RUNGE-KUTTA ODE 35

der of few nanometers [17]. Fig. 3.1 shows both dipole (eq.3.24) and tunnel

coupling (eq.3.28) between the dots and the interference terms of the emission

rates (eq.3.30), with CdSe/ZnSe parameters, which appear due to constructive

and destructive interference of amplitude for radiative recombination of excitons

from pairs of QDs.

For the simulation, eq. (3.29) is then rewritten using general parametrization

of the density matrix in the basis |0..0..0〉, |0..0..1〉, |0..1..0〉, ... , |1..1..1〉 whose

the form as follow,

ρ =

x0 + ix1 x2 + ix3 .... .... .....

... x10 + ix11 .... .... ....

.... .... ..... .... ....

.... .... x2(Na+b) + ix2(Na+b)+1 ....

..... .... .... .... x2N2−1) + ix2N2

,(3.32)

Whereby N is the number of states and N = 2n, whereby N is the number of QDs.

Thus, the matrix element in the row a and column b is parametrized as x2(Na+b) +

ix2(Na+b)+1. Equation of motion for the vector (x1, ..., x2N+1) is subsequently

solved numerically by using GSL library and the radiative recombination of QD

array is obtained from the occupations (diagonal part of the matrix). The method

of solving the equation of motion is presented in detail in the next section in this

chapter. The diagonal elements of matrix represent the radiative recombination

of excitons while the o diagonal elemets represent the phase decoherence of

system. Since this thesis focus on the collective emission of QDs it thus only

consider the diagonal part as radiative recombination of excitons.

3.3 Runge-Kutta ODE

The Runge-Kutta ODE procedure is utilized to solve the dierential equation

of motion for the vector yielded from the expansion of the Hamiltonian and


evolution part spanned in the matrix form. In this section, the short review of

Runge-Kutta method is described [11]. The fourth-order Runge-Kutta method

(RKT4) with adaptive step size has proved to be robust and capable of industrial-

strength work. To understand this important tool let us derive the simpler second-

order method. The fourth order is just more derivation which shall not presented

in detail here.

A Runge-Kutta algorithm for integrating a dierential equation is based upon

the formal integral of dierential equation,

dy

dt(t) = f(t, y)→ y(t) =

∫f(t, y)dt→ yn+1 = yn +

∫ tn+1

tn

f(t, y)dt. (3.33)

The approximation enters by expanding f (w tn, y w yn) in a Taylor series about

the midpoint of the integration interval

f(t, y) w f(tn+1/2, yn+1/2

)+(tn+1/2

) dfdt

(tn+1/2

)+ ξ(h2). (3.34)

When Eq. 3.34 is substituted into Eq. 3.33, the integrals of(t− tn+1/2

)vanishes

and a higher-order algorithm than Euler's is obtained even though the use of the

same number of terms,

∫f(t, y)dt w f

(tn+1/2, yn+1/2

)h,→ yn+1 w yn + hf(tn+1/2, yn+1/2). (3.35)

The Euller's method thus can be used to express yn+1/2 which is not given by the

initial condition,

yn+1 w yn +dy

dt

h

2= yn +

1

2hf(tn, yn). (3.36)

The second order Runge-Kutta algorithm is then obtained,

yn+1 w yn + k2, k2 = hf

(tn +

h

2, yn +

k1

2

),k1 = hf (tn, yn) . (3.37)

3.3. RUNGE-KUTTA ODE 37

It is easily seen that the derivative function f is evaluated at the ends and mid-

point of the interval, so that only the initial value of the unknown y is required.

In RKT4, there are four gradient (ki) terms to provide a better approximation to

f(t, y) near the midpoint, and they can be determined with just four subroutine

calls,

yn+1 = yn + 16

(k1 + 2k2 + 2k3 + k4) ,

k1 = hf (tn, yn) ,

k2 = hf(tn + h

2, yn + k1

2

),

k3 = hf(tn + h

2, yn + k2

2

),

k4 = hf (tn + h, yn + k3) .

(3.38)

Chapter 4

Results

In this chapter, the coupling inuence on the radiative decay of both double-

dot and QD array is described. It is necessary to know the inuence of the

coupling and arrangement (regular/random) because of the experiment fact that

the coupling was reduced by reducing the number of QDs [3]. This coupling

should depend on the number of QDs [12] and should decrease with the inverse

distance between QDs [13].

In these simulations, the used parameters for the QDs ensemble is a typical

CdSe/ZnSe on QD system: Γ0 = 2.56 ns−1, n = 2.6, E=2.59 eV. For the short

coupling the amplitude is V0 = 5 meV and the range is r0 = 15 nm. However, for

the double dots system, in order to only study the interplay between coupling (V),

mismacth energies ε and interference term Γ, the used parameters corresponds

to typical InAs/GaAs system with Γ0 = 1.0 ns−1, n = 3.3, E = 1.3 eV which

produces the coupling and interference parameter as depicted in g. 4.1. The

values of the two couplings as well as the interference term of the decay rate Γαβ

are plotted as a function of the distance between the dots. In this gure the

distance values are marked for which the decay as the function of distances shall

be discussed in the following section.

39

40 CHAPTER 4. RESULTS

0

1

2

0 5 10 15 20

V/h-

(ps-1

)

r12 (nm)

(a)

A

Vlr/h-

Vsr/h-

0

1

2

0 200 400 600 800

V/h-

, Γ

12 (

ns-1

)

r12 (nm)

(b)

B

C D E

Vlr/h-

Γ12

Vsr/h-

Figure 4.1: Interference term of the decay rate Γ12 and the short and long-range couplingamplitudes Vlr, Vsr as a function of the lateraly placed inter-dot distance withInAs/GaAs parameter. In (a), the small distance section is shown, while in (b)the oscillating tail at larger distances is visible. Note the dierent scales in (a)and (b).

Figure 4.2: The alignment of lateral double quantum dots (DQD)

4.1 Double quantum dots

In this section the interplay between interference term, short- and long range

couplings, and mismatch energies is described. This result is mainly based on

the ref. [17]. Let us now consider the case of coupling according to Fig. 4.1 in a

laterally aligned structure of DQD which induces the decay rate of excitons with

the initial state chosen to be(|01〉+|10〉√

2

), whereby the two-digit kets denote the

occupation of the respective dots.

In Fig. 4.3 the result shows the numerical simulations based on Eq (3.29). On

each plot, the average number of excitons in the system is shown as a function of

time for identical dots (ε = 0) and for slightly non-identical dots with ε = 0.01

meV. We study the decay of exciton population for various distances between the

dots and compare the evolution for the two type of couplings.

For identical dots, the exciton decay time for superradiant initial state strongly

4.1. DOUBLE QUANTUM DOTS 41

0

0.2

0.4

0.6

0.8

1

(a)Exci

ton o

ccupat

ion

identical dots, LR coupling

ABCDE

(b)

identical dots, SR coupling

ABCDE

0

0.2

0.4

0.6

0.8

1

0 1 2 3

(c)Ex

cito

n o

ccupat

ion

t (ns)

LR (ε = 0.01 meV), non-identical dots

ABCDE

0 1 2 3

(d)

t (ns)

SR (ε = 0.01 meV), non-identical dots

ABCDE

Figure 4.3: The exciton occupation (the average number of excitons in the system) for aninitial single-exciton state corresponding to a superradiant delocalized superpo-sition. (a) and (b) show the evolution for a pair of identical dots coupled bylong-range dipole forces and by short-range tunnel coupling, respectively. (c) and(d) show the evolution for a pair of non-identical dots, for the two kinds of cou-plings as previously. The labels A,...,E refer to the values of the inter-dot distancemarked in gure 4.1.

depends on the distance between the dots. This is due to the oscillations and

decay of the interference term Γ12 . For the dots placed at a short distance

(case A), Γ12 ∼ Γ0 and the decay has a strongly collective character, which is

manifested by the faster emission visible in g. 4.3 (a,b) [24]. The collective

eect gets weaker as the distance between the dots grows and Γ12 decreases (B).

For some values of the distance, Γ12 < 0 (C) superradiant initial state behaves as

subradiant initial state. Then, the amplitudes for photon emission from the two

dots interfere destructively and the decay gets slower than the usual exponential

decay with the rate Γ0 (the initial state becomes subradiant). Whenever Γ12 = 0,

the decay rate is the same as for an individual dot (D). Comparison of g. 4.3 (a)

and (b) shows that for identical dots, these eects do not depend on the coupling

and are therefore the same, irrespective of the presence and physical nature of

the interaction between the dots.

For dots that dier by the relatively small transition energy mismatch of 2ε

= 0.02 meV, almost all this non- monotonic dependence of the emission rate on


0

0.2

0.4

0.6

0.8

1

(a)Exci

ton o

ccupat

ion


ABCDE

(b)


ABCDE

0

0.2

0.4

0.6

0.8

1

0 1 2 3

(c)Ex

cito

n o

ccupat

ion

t (ns)

LR (ε = 0.01 meV), non-identical dots

ABCDE

0 1 2 3

(d)

t (ns)


ABCDE

Figure 4.4: The exciton occupation (the average number of excitons in the system) for an ini-tial single-exciton state corresponding to a subradiant delocalized superposition.(a) and (b) show the evolution for a pair of identical dots coupled by long-rangedipole forces and by short-range tunnel coupling, respectively. (c) and (d) showthe evolution for a pair of non-identical dots, for the two kinds of couplings aspreviously. The labels A,...,E refer to the values of the inter-dot distance markedin gure 4.1.

the distance disappears. The reason is that the oscillations of the interference

term take place in the distance range where the coupling between the dots is

very weak and is dominated already by a small energy mismatch assumed here,

which destroys collectivity of the emission process [24]. The only exception is

the smallest distance shown in this plot, where the coupling is suciently strong.

By comparing g. 4.3(c) and g. 4.3(d) one can see that also in this case, the

evolution of the exciton occupation is nearly the same for both systems except

for the short-range coupling in the same short distance as the long coupling, in

this case the short-range coupling is not suciently strong to be compared with

mismatch energy and thus the oscillation remains visible. At larger distances

both couplings are negligible compared to the energy mismatch.

Let us now consider another initial state of DQD in which the case of sub-

radiant initial state in the form(|01〉−|10〉√

2

). For identical dots, as depicted in

g. 4.4(a,b), the exciton decay time for the sub-radiant initial state also strongly

depends on the distance between the dots. This is due to the oscillation and decay


of interference term Γ12. However, for the dots placed at a short distance (case

A) Γ12 ∼ Γ0 and either Vlr or Vsr Γ12, the sub-radiant initial state exhibits

stable emission while the superradiant state decays exponentially with a twice

larger rate. In the intermediate distance range (case B) either Vlr or Vsr = 0,

the decay is not exponential. Further, for some values of the distance, Γ12 < 0

(C) subradiant initial state behaves as superradiant intial state. Then, the am-

plitudes for photon emission from the two dots interfere constructively and the

decay gets faster than the usual exponential decay with the rate Γ0 (the initial

state becomes superradiant). Whenever Γ12 = 0, the decay rate is the same as for

an individual dot (D). Comparison of g. 4.4 (a) and (b) shows that for identical

dots, these eects do not also depend on the coupling and are therefore the same,

irrespective of the presence and physical nature of the interaction between the

dots.

For dots that dier by the relatively small transition energy mismatch of 2ε

= 0.02 meV, as depicted in the g. 4.4(c,d), similar to the previous initial state,

almost all this non-monotonic dependence of the emission rate on the distance

also disappears and the only exception is the smallest distance shown in this plot

whereby the coupling is suciently stronger than mismatch energy (g. 4.4 c).

At larger distances both couplings are also negligible compared to the energy

mismatch. However, since the mismatch energy is suciently larger than either

coupling or interference term it dominate the decay process producing oscillation

upon the decay process with the frequency similar to the mismatch energy.

4.2 Quantum dots array

In this section the comparison between random and regular arrangement un-

der weak and strong excitation regimes is described. Let us consider a quantum

dot ensemble placed in the xy plane. Each QD is modeled as a two-level system

(empty dot and one exciton). We consider ordered and random 2D arrays. The or-


dered arrays are double dots, triangular aligned structure of the dots, rectangular

aligned structure of the dots, penta and hexagonal structures. Those ensembles

are arranged in laterally aligned array as depicted in the Fig. 4.5a. Here, the value

of D is 26 nm and radius of QD (R) is 10 nm with the QDs density of n ≤ 1011

cm−2. For the case of random arrays, the ensemble is randomly distributed over

square mesas with the restriction that the inter-dot distance cannot be smaller

than 10 nm as depicted in g. 4.5b.

The spectral properties of the dots are modeled by a Gaussian distribution of

their transition energies with the standard deviation σ. On each plot, the photon

emission in the system is shown as a function of time for identical dots (σ = 0)

and non-identical dots with the realistic value of fundamental transition energy

standard deviation σ = 18.4 meV. Further, the ensemble is irradiated by weak

(superradiant single-exciton state) or strong (fully inverted state) excitations in

order to distinguish the excitons and photon probability occupation properties

under the inuence of either weak and strong exctitation conditions.

To assess whether in an ensemble with the density as given in the previous

paragraph the QDs can interact collectively couple to the radiation eld, it must

be shown that changes of the PL decay rate are a function of the number of

interacting QDs and their respective separation. Therefore, The QD arrays shown

in Fig. 4.5a are placed onto square mesas with 83 nm × 83 nm for 6 dots, 71

nm × 71 nm for 5 dots, 63 nm × 63 nm for 4 dots, 55 nm × 55 nm for 3 dots

and 45 nm × 45 nm for 2 dots depicted in g 4.5b. As depicted in Fig. 4.6

a and b, for identical QDs under weak excitation regime1, it is shown that the

exponential decay gets slower with reducing number of the dots which implies

that the coupling is reduced by reducing number of the dots. It is clear that

QDs do not behave as independent particles as long as they form an ensemble of

QDs. Again, there is very slight dierence between the role of short-range and

long-range coupling on the decay of excition occupation. However, for the case

1weak excitation regime is the other term for superradiant inital state


(a) The regularly lateral ensemble of QD. (b) The randomly lateral ensemble of QD

Figure 4.5

0

0.2

0.4

0.6

0.8

1

(a)Ex

cito

n o

ccu

pat

ion


2 dots3 dots4 dots5 dots6 dots

(b)



0

0.2

0.4

0.6

0.8

1

0 1

(c)Exci

ton

occ

upat

ion

t (ns)



0 1

(d)

t (ns)



Figure 4.6: The decay of excitons of superradiant initial state |0....1〉+....+|1....0〉√N

(weak excita-

tion condition) in the regularly lateral structure of QDs array.

non-identical dots shown in g. 4.6 c and d, if σ is larger than the values of

short-range coupling (Vsr), long-range coupling (Vlr), and the interference term

(Γ), the enhanced emission eect completely disappear from the system.

In order to nd out whether a randomly distributed ensemble of QDs behaves

similar to a regular array of QDs let us now study the exciton recombination

from a random QD ensemble in the same weak excitation regime. In each case,

let us calculate an average of 100 realizations of the evolution in systems with a

given number of dots but diering in their positions within the sample plane as

shown in Fig. 4.5b. For the random array case under weak excitation regime, the

excitation lifetime as a function of the mesa size is shown in Fig. 4.7. The lifetime


0

100

200

300

400

500

10-3

10-2

10-1

Mesa size (µm2)

Lif

etim

e (p

s)

Lr identical dots

Sr identical dots

Lr σ=18.4 meV

Sr σ=18.4 meV

Figure 4.7: The lifetime of excitons within randomly lateral ensemble under weak excitationregime.

for the mesas with a greater number of the dots is shorter than for the mesas

with a smaller number of the dots. This occurs due to constructive coupling and

interference among greater number of the dots inducing the decay ensemble. The

exciton lifetime is similar to the regular array case, which means that there is

very little dierence in arranging the dots either randomly or regularly under the

weak excitation regime.

The fact that QDs do not necessarily behave as independent particles proved

by the increase of exciton emission rate at low excitation needs to be assured by

the signature of photon emission rate under both weak and strong excitation2.

In this case, the increase of emission rate at low excitation could yield in a de-

layed outburst of radiation under strongly inverted initial condition [14]. Let us

thus rstly determine the photon emission both under strong and weak excitation

regime within the regularly lateral ensemble of the dots which is obtained by nu-

merical rst order dierentiation Γ = ∆P/∆t from the excitation recombination

shown in the g. 4.6. Let us consider the y-axis in the g. 4.6 as P whereby P

is the probability of exciton occupied the state in the time t. The ensemble of ei-

ther regularly or randomly ordered QDs is then irradiated under weak excitation

2Strong excitation regime is the other term for the inverted initial state


0

4

8

(a)

Γ [

ns-1

]


2 dots4 dots6 dots

(b)


2 dots4 dots6 dots

0

1

2

3

0 0.2 0.4 0.6 0.8 1

(c)

Γ [

ns-1

]

t (ns)

LR (σ = 18.4 meV), non-identical dots

2 dots4 dots6 dots

0 0.2 0.4 0.6 0.8 1

(d)

t (ns)

SR (σ = 18.4 meV), non-identical dots

2 dots4 dots6 dots

Figure 4.8: The decay of photon emission of the superradiant single-exciton initial state|0....1〉+....+|1....0〉√

Nin the regularly lateral structure of QDs array.

regime.

Fig. 4.8 a and b show that the photon emission rate under weak excitation

condition for the regular identical dots ensemble experiences faster decay as in-

creasing number of dots in the ensemble. Moreover, the weaker emission with

slower decay rate appears in the certain delay time by reducing the number of

the dots in the small ensemble. It implies that reduced coupling in short and

long range coupling slows down the emission rate of photons. However, for non-

identical dots ensemble g. 4.8 c and d, if the σ is greater than both couplings

and interferences, it diminishes the enhanced emission eect.

Let us now determine the photon emission under strong excitation regime

whereby the ensemble is irradiated with strong excitation so that each QD is

initially occupied by an exciton. Fig. 4.9 a and b show that the photon emis-

sion under strong excitation condition for the regular identical dots ensemble

experiences outburst of emission in certain delay time which gets stronger with

increasing number of the dots in the small ensemble. It implies that enhanced

emission eect appears within the regular ensemble of QDs. It is also clear that

QDs systems exhibit signatures of cooperative radiation and hence ought to be

considered as coupled quantum systems. However, for non-identical dots ensem-


0

10

20

(a)

Γ [

ns-1

]



(b)



0

4

8

12

16

0 0.2 0.4 0.6 0.8 1

(c)

Γ [

ns-1

]

t (ns)



0 0.2 0.4 0.6 0.8 1

(d)

t (ns)



Figure 4.9: The decay of photon emission for the inverted initial state |11...1〉 in the regularlylateral structure of QDs array.

ble g. 4.9 c and d, for realistic system in which σ is greater than both couplings

and interferences, it diminishes the enhanced emission eect.

In order to nd out whether a randomly distributed ensemble of QDs behaves

similar to a regular array of QDs let us now also study the photon emission from

a random QD ensemble in the same weak and strong excitation regimes. In each

case, let us calculate an average of 100 realizations of the evolution in systems

with a given number of dots but diering in their positions within the sample

plane as can be seen in Fig. 4.5b. Then, the emission of photon is subsequently

investigated under both weak and strong excitation regimes.

As depicted in Fig. 4.10 that the decay rate of photon gets slower when re-

ducing the number of QDs within the ensemble. This is similar to the photon

emission from a regular QD array in which the decay rate of photon also gets

slower by reducing the number of QD. It therefore can be said that there is very

slight dierence whether the dots are placed either randomly or regularly. Never-

theless, the random order is more relevant to the experiment in which long-range

coupling inuence on the decay rate of photon depicted at g. 4.10 (a) shows

dierent decay from the short-range coupling. This is due to stronger coupling

of short-range than long-range coupling in the range of r > 50 nm in which the

4.3. MISMATCH ENERGY INFLUENCE 49

0

2

4

6

8

10

12

(a)

Γ [

ns-1

]


2 dots4 dots6 dots

(b)


2 dots4 dots6 dots

0

1

2

3

0 0.2 0.4 0.6 0.8 1

(c)

Γ [

ns-1

]

t (ns)


2 dots4 dots6 dots

0 1

(d)

t (ns)


2 dots4 dots6 dots

Figure 4.10: The decay of photon emission for the superradiant single-exciton initial state|0....1〉+....+|1....0〉√

Nin the randomly lateral structure of QDs array.

overall distances among QDs are around these values. One can therefore conclude

that in the real system the enhanced emission of photon occurs due to the short-

range coupling, which may result from tunneling eect of bound-electron within

QD. However, again, if σ is greater than the dissipative part (Γ) and coupling

(Ω) the enhanced emission eect shall disappear from the QD ensemble.

Again, the random array under strong excitation regime is also studied. Fig.4.11

shows that enhanced emission eect appears within identical dot ensemble in

which the delayed outburst of photon emission is visible. This also implies that

there is very slight dierence arranging the dots either randomly or regularly un-

der both weak and strong excitation regimes. However, if the σ is much stronger

than either dissipative and coupling parts, again, it diminishes the enhanced

emission within the ensemble of QD.

4.3 Mismatch energy inuence

In an experiment, it is dicult to realize a system with nearly identical dots.

The inhomogeneity of the QD ensemble leads to the destruction of enhanced

emission. Therefore, in order to ensure whether the superradiance eect really


0

5

10

15

20

(a)

Γ [

ns-1

]



(b)



0

4

8

12

16

0 0.2 0.4 0.6 0.8 1

(c)

Γ [

ns-1

]

t (ns)



0 0.2 0.4 0.6 0.8 1

(d)

t (ns)



Figure 4.11: The decay of photon emission for the inverted initial state |11...1〉 in the randomlateral QDs ensemble.

appear within the inhomogeneous QDs system and whether higher value of inho-

mogeneity than either coupling and interference terms yields the destruction of

enhanced emission eect, let us determine the role of the standard deviation of

fundamental transition energy. Let us consider a random array of 4 and 6 dots

as an example for this case, the photon emission is then observed as a function

of decay time with the σ as a control variable.

Fig. 4.12 shows the inuence of the QD inhomogeneity on the photon emission

of 4 and 6 dots system under strong excitation regime which retains to produce

enhanced emission eect. The eect of enhanced emission gradually disappears

with the increasing of inhomogeneity of the system. This implies that in the

distance between dots within size of mesa 4 × 10−3µm2 the emission occurs for

both long- and short-range coupling. However, for the case 6 dots, whereby the

distances between the dots become larger than for 4 dots case due to larger size of

mesa, The enhanced emission occurs in the short-range coupling in a wider range

of σ since at the same range of distance, as depicted on g. 3.1 the long-range

coupling is rather weak than short-range coupling.

It ought to be emphasized, nevertheles, that the collective nature of the in-

teraction between QDs and the electromagnetic eld is remain essential dot the

4.3. MISMATCH ENERGY INFLUENCE 51

0

5

10

(a)

Γ [

ns-1

]

4dots, long-range coupling

σ = 0 σ = 10 µeVσ = 20 µeV

σ = 18.4 meV

(b)

4dots, short-range coupling

σ = 0σ = 10 µeVσ = 20 µeV

σ = 18.4 meV

0

4

8

12

16

20

0 0.2 0.4 0.6 0.8 1

(c)

Γ [

ns-1

]

t (ns)

6dots, long-range coupling

σ = 0σ = 2 µeV

σ = 10 µeVσ = 18.4 meV

0 0.2 0.4 0.6 0.8 1

(d)

t (ns)

6dots, short-range coupling

σ = 0σ = 2 µeV

σ = 10 µeVσ = 18.4 meV

Figure 4.12: The decay of photon emission of 4 dots and 6 dots under strong excitation regime(initial state |11...1〉) in the randomly ordered QD ensemble induced by variousmismatch energies.

observed eect. According to Eq. 2.19, the collective electromagnetic coupling

induces the system dynamics in two ways: It is not only as a medium for the

dipole interactions described by the coupling constant Vαβ but also has a role

as the triger for the appearance of interference terms Γαβ whereby α 6= β in

the dissipative part. These terms are not present in the hypothetical case of

QDs emitting to separate reservoirs yet still coupled by interactions [28]. Both

random and regular order of QDs ensemble does not show any enhancement of

spontaneous emission under the long-range coupling condition.

In view of relatively large inhomogeneity of QD transition energies, suciently

strong coupling is required to stabilize the collective nature of emission. For com-

mon interdot separations, fundamental dipole interactions are too weak to have

a role in the kinetic emissions. However, the presence of short-range interaction

due to tunnel coupling between the dots leads to the enhanced emission in the

quantitative agreement with experimental results [3].

Chapter 5

Conclusions

In this work the collective spontaneous emission from a system of quantum

dots have been studied. We have shown that the radiative decay of exciton

occupation in a pair of coupled quantum dots depends on the distance between

the dots as a result of the spatial dependence of the interference term governing

the interaction with the quantum electromagnetic eld. For non-identical dots,

the emission rate depends on the interplay of the energy mismatch between the

dots and the coupling between them. Although the two kinds of couplings that

are present in the system (short-range and long-range coupling) have essentially

dierent physical nature and properties, they may lead to the same dynamics of

the observed collective emission.

We have shown that the delayed outburst of radiation, typical for the super-

radiant emission develops in the luminescence from regular arrays or randomly

distributed ensembles of quantum dots in the strong excitation regime. The way

the dots are distributed in the sample plane (regular vs. random) makes very lit-

tle dierence on the photon emission. This means that the system response is not

dominated by accidental clustering that might appear in the random distribution

case and lead to strongly enhanced contribution to the overall emission from pairs

of accidentally very closely spaced dots. Moreover, in both cases, the superradi-

ant maximum is completely washed out if the realistic degree of inhomogeneity

53

54 CHAPTER 5. CONCLUSIONS

of the fundamental transition energies is taken into account. Thus, enhanced

emission observed experimentally under weak excitation does not imply that true

superradiance will be manifested for a fully inverted system.

Last, but not least in the relevant small ensemble of the dots, the enhanced

emission eect in the inhomogeneous ensemble of QD does not occur under long-

range coupling regime but only in the short-range coupling regime since only

the short-coupling is suciently strong to stabilize the inhomogeneity within the

small-range of dot ensemble.

55

Appendix

Superradiance-evolution equations

The detection probability at the time t at the point ~r is proportional to the

radiation intensity [20].

I (~r, t) = 〈E(−) (~r, t)E(+) (~r, t)〉

For compact system, (rαβ ct), one can express the electric eld operators by

the electronic operators, which leads to

W (~r, t) =3Γ

8π

[1− |r · p|2

]∑α,β

ei~kr·~rαβ〈σ†α(t)σβ(t)〉

whereby σα = |0〉α〈1|α and k = ωcwe assume that ∆~kr · ~rαβ 1, whereby

∆k is the dispersion of emitted wavelengths. Let thus determine the quantities

〈σ†α(t)σβ(t)〉 ≡ xαβ at any time t. The evolution of electronic quantities is given

by the equation

xαβ = i1~∑

γ εγ〈[σ†γσγ, σ

†ασβ]〉+ i

∑γ 6=δ Ωγδ〈

[σ†γσγ, σ

†ασβ]〉+∑

γ,δ Γγδ〈σ†γσ†ασβσδ − 12

(σ†γσδσ

†ασβ + σ†ασβσ

†γσδ)〉.

hence it is obtained,

[σ†γσγ, σ

†ασβ]

= σ†γ[σγ, σ

†ασβ]

+[σ†γ, σ

†ασβ]σγ

=∑†

α

[σγ, σ

†α

]σβ +

∑†α

[σ†γ, σβ

]σγ + σ†γσ

†α [σγ, σβ] +

[σ†γ, σ

†α

]σβσγ

=∑†

α

[σδ, σ

†α

]σβ +

∑†α

[σ†γ, σβ

]σδ

Since

[σα, σ

†α

]= −|+〉〈−|−〉〈+| + |−〉〈+|+〉〈−| = −|+〉〈+| + |−〉〈−| = 1− 2σ†ασα

56 CHAPTER 5. CONCLUSIONS

hence

[σ†γσγ, σασβ

]= σ†γ

(1− 2σ†γσγ

)σβδαδ − σ†α

(1− 2σ†βσβ

)σγδγβ

However, σ†γσ†γ = 0; σβσβ = 0, thus only −σγσβδαγ + σ†ασγδγβ hence,

∑γ

εγ〈[σ†γσγ, σ

†ασβ]〉 = εασ

†ασβ − εβ〈σ†ασβ〉 = (εα − εβ)σ†ασβ

In the second term, the following commutator appears

[σ†γσδ, σ

†ασβ]

= σ†γ[σδ, σ

†α

]σβ + σ†α

[σ†γ, σβ

]σδ − σ†γσ†α [σγ, σβ] +

[σ†γ, σ

†α

]σβσδ

= σ†γ(1− 2σ†γσδ

)δαδσβ − σ†α

(1− 2σ†βσβ

)δγβσδ

Hence,

[σ†γσδ, σ

†ασβ]

=∑

γ 6=δ Ωγδ

[σ†γ(1− 2σ†ασα

)δαδσβ − σ†α

(1− 2σ†βσβ

)δγβσδ

]=

∑γ 6=δ Ωγδσ

†γσβ

∑γ 6=α Ωβδσ

†ασδ − 2

∑γ 6=α Ωγασ

†γσ†ασασβ

+2∑

δ 6=β Ωγασ†ασ†βσβσδ

After averaging

∑γ 6=δ Ωγδ〈

[σ†γσδ, σ

†ασβ]〉 =

∑γ 6=α Ωγασγβ − 2

∑δ 6=β Ωβδσ

†ασδ∑

γ 6=α Ωγαxγααβ + 2∑

δ 6=β Ωγαxαββδ,

where xαβγδ = 〈σ†ασ†βσγσδ〉; the averages of this kind vanish if there was initially

only one excitation in the system (since there are 2 annihilation operators in

sequence). If we restricted the discussion to merely one excitation then the system

57

(without dissipation) closes. The dissipative term is

−12

(σ†γσδσ

†ασβ + σασβσ

†γσδ)

= −12

[σ†γ(δαδ + σ†ασδ

)σβ + σ†α

[δγβ + σ†γσβ

)σδ]

= −σ†γσ†ασβσδ(−1

2δαδσ

†γσβ − 1

2δγβσ

†ασδ + σ†γσ

†ασασβδαδ + σ†ασ

†βσβσδβ

)= xγαβδ − 1

2(δαδxγβ + δγβxαδ) + σ†γσ

†ασασβ + σ†ασ

†βσβσδ.

Altogether

−12

∑γ,δ Γγδ (δαδxγβ + δγβxγβ + δγβxαδ) = −1

2

∑γ Γγαxγβ − 1

2

∑δ Γβδxαδ∑

γ Γγασ†γσ†ασασβ +

∑γ Γβωσ

†ασ†βσβσγ.

Hence, in the case of just one excitation,

xαβ =i

~(εα − εβ)xαβ + i

∑γ 6=α

Ωγαxγβ − i∑α 6=β

Ωβγxαγ −1

2

∑γ

(Γγαxγβ + Γβγxαγ)

with the denitions

Ωαβ(x) =3

4Γ

−[1− |d · rαβ|2

] cosx

x+[1− 3|d · rαβ|2

] [sinx

x2+

cosx

x3

],

Γαβ(x) =3

2Γ

[(1− |d · rαβ|2

) sinx

x+(

1− 3|d · rαβ|2)(cosx

x2− sinx

x3

)].

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