Coherent photons
from a solid-state artificial atom
Clemens Matthiesen
St John’s College
Department of Physics
University of Cambridge
This dissertation is submitted for the degree of
Doctor of Philosophy
May 2013

This dissertation is the result of my own work and includes nothing
which is the outcome of work done in collaboration except where
specifically indicated in the text.
This dissertation does not exceed the word limit of 60000 words.
Meinen Eltern.
Acknowledgements
I have left the acknowledgements to be written last, after finishing
the ‘serious’ parts of this thesis. After all, it should be the easiest bit.
So I thought, but it turns out it is quite difficult to summarise four
years of work (and life) on one page. I had the pleasure of working
with many fantastic colleagues and they have contributed greatly to
the results presented later on, in teaching, training, discussing (more
passionately since the arrival of Claire) and working together.
First, I should and want to thank my supervisor Mete Atatu¨re for his
academic guidance throughout the years, while giving me the freedom
to pursue interesting (at least initially) side projects. He has been in-
creasingly busy, but his door is still always open (quite literally) and
he somehow always manages to find the right words to motivate.
I am very grateful to the super-postdoc Nick (now super-group leader
I expect) who taught me optics, research practices and the lab tech-
niques when I was in my first year. I hope Eleni is doing well! I have
fond memories of working with Chaoyang in my first year (I had real
difficulties following your labwork in the first few months!). Visiting
him, now that he is a senior academic, and his group (Yu-Ming, Yu
He and Yu-Jia) in Hefei was a real pleasure.
Tina was around when I pressed first buttons in the lab (x up and
down, y up and down, NOT z) and her presence has been a constant in
the group. Seeing some parts of California with her, on the pretext of
going to ‘the-world’s-leading-something-optics-something-conference’
was really fun (I mean Photonic West.). I also want to thank Pe-
ter Humphreys for his valuable work on the Michelson interferometer
during his Part III project.
The last two years have seen a renewed QD team: one-by-one Claire,
Jack, Megan, Carsten and Rob joined and brought with them fresh
ideas and a solid theoretical background. The working atmosphere
in both the lab and the Room 935A (coincidence? I think not) has
always been relaxed and driven at the same time, and I feel very much
at home there. I am also very grateful to Carsten, Jack and Rob for
proof-reading this thesis, and picking up on the occasional factual er-
ror as well.
The friendly, outgoing and fun nature of the whole AMOP-MESS
(QINM now?) group has made the past years into a very enjoyable
experience and I thank members here if I haven’t mentioned them
earlier: Yong, Yuri, Josh (who should get a master in labview with
his PhD), Ben, the NV team(s) of Helena, Dhiren, and recently Lina
and David. Paul filled the role of experienced post-doc very well, and
I was able to learn a lot from him, when I was meant to teach our lab
techniques. The one year he spent with us was a fruitful one and I
hope we keep up the communication. Zhenyong Li brought a differ-
ent set of skills to the lab during his stay and I hope his research is
benefiting from the new experiences.
Jeroen Elzerman and Andrew Ferguson kindly agreed to examine my
thesis work and conduct the viva, and I am grateful for their thorough
reading and insightful questions and comments.
I am very much looking forward to get back to the lab and work with
my teammates on the exciting projects planned for the future. Thanks
to Prof. Richard Phillips’s continued support of my JRF applications,
Prof. Skolnick’s multiple reference letters and Mete’s willingness to
keep me, I will be able to spend another few years in Cambridge. I
have also enjoyed enlightening physics discussions with Prof. Phillips.
Finally, I cannot be grateful enough to Kelly. After nearly three
years of being in Cambridge together I still don’t know she manages
to balances the long working hours with everything else so well.
Summary
Single spins confined in semiconductor quantum dots - artificial atoms in the
solid-state - are attractive candidates for quantum mechanical bits, the funda-
mental units and building blocks of a quantum computer. The ability to address
quantum dot spins optically allows us to initialise and manipulate the state of
the quantum bit. Gaining information on the qubit, for example by reading out
its state, not only requires state-selective optical excitation, but also access to
the single photons scattered in response by the quantum dot. Further, for a dis-
tributed computer architecture where nodes of few quantum bits are interlinked
via optical communication channels photonic quantum bits are required to faith-
fully transmit the quantum information.
In this thesis we advocate resonant excitation of quantum dot transitions and
collection of the resonance fluorescence to address two outstanding challenges:
generating dephasing-free single photons for use as flying quantum bits and single-
shot spin readout. To this end we investigate the spectral and first-order coher-
ence properties of quantum dot resonance fluorescence. In particular, we directly
observe highly coherent scattering in the low Rabi frequency limit which has
remained unexplored for solid-state single photon emitters so far. At the same
time, interactions with the semiconductor environment are revealed and quan-
tified through their optical signatures: exciton-phonon coupling, nuclear spin
dynamics and local electric field fluctuations signal a departure from the ideal
atom-like behaviour.
Taking advantage of the laser-like coherence of single phase-locked quantum dot
photons in the Heitler regime, we demonstrate near-ideal two-photon quantum
interference. This benchmark measurement is a precursor for the photonic en-
tanglement of distant quantum dot spins in a quantum optical network, and the
results here predict a high fidelity operation.
Finally, moving to tunnel-coupled quantum dot molecules we show that the over-
lap of carrier wave functions in two closely spaced quantum dots forms new spin-
selective optical transitions not available in single quantum dots. Then, the pres-
ence or absence of scattered photons reveals the electron spin. Intermittency in
the quantum dot resonance fluorescence allowed us, for the first time, to observe
spin quantum jumps in real-time.
Both achievements - highly coherent photons and spin readout - provide the
missing link to attempt creation of a small-scale quantum network now.
Contents
Contents vii
1 Introduction 1
1.1 Quantum Information Processing . . . . . . . . . . . . . . . . . . 3
1.1.1 DiVincenzo’s 5 + 2 and quantum optical networks . . . . . 4
1.2 Semiconductor quantum dots . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Quantum information, metrology and mesoscopic physics . 7
1.3 QD level structures . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 Exciton fine structure . . . . . . . . . . . . . . . . . . . . . 9
1.3.2 The nuclear Overhauser field . . . . . . . . . . . . . . . . . 10
1.3.3 Trion levels under magnetic and electric fields . . . . . . . 11
1.4 Coherent control of single spins: state of research . . . . . . . . . 13
2 Light-matter interaction of two levels: resonance fluorescence 18
2.1 Optical Bloch equations and the density matrix . . . . . . . . . . 19
2.2 First-order coherence of resonance fluorescence . . . . . . . . . . . 25
2.3 Second-order coherence of resonance fluorescence . . . . . . . . . . 27
3 Experimental setup and techniques 30
3.1 Confocal microscopy . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 QD Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.1 General considerations for sample design . . . . . . . . . . 34
3.2.2 The ‘Chef 2’ sample . . . . . . . . . . . . . . . . . . . . . 35
3.2.3 QD device processing . . . . . . . . . . . . . . . . . . . . . 37
3.2.4 Solid immersion lenses . . . . . . . . . . . . . . . . . . . . 43
vii
CONTENTS
3.3 Single QD spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3.1 Gated photoluminescence . . . . . . . . . . . . . . . . . . 49
3.3.2 Differential transmission and reflection . . . . . . . . . . . 50
3.3.3 Resonance fluorescence . . . . . . . . . . . . . . . . . . . . 52
3.4 Quantum optics in the lab: interferometric measurement techniques 53
3.4.1 Spectral measurements . . . . . . . . . . . . . . . . . . . . 53
3.4.2 First-order coherence . . . . . . . . . . . . . . . . . . . . . 54
3.4.3 Second-order coherence . . . . . . . . . . . . . . . . . . . . 57
4 A two-level system in a solid-state environment: resonant QD
spectroscopy 60
4.1 Subnatural linewidth single photons . . . . . . . . . . . . . . . . . 61
4.1.1 Performance of the measurement system . . . . . . . . . . 61
4.1.2 Saturation and power broadening . . . . . . . . . . . . . . 64
4.1.3 Spectral and statistical properties at low powers . . . . . . 66
4.1.4 Wave-particle duality in the Heitler regime . . . . . . . . . 73
4.2 Effects of the solid-state environment . . . . . . . . . . . . . . . . 76
4.2.1 Spectral diffusion . . . . . . . . . . . . . . . . . . . . . . . 76
4.2.2 Phonon coupling . . . . . . . . . . . . . . . . . . . . . . . 87
4.2.3 Nuclear Overhauser field . . . . . . . . . . . . . . . . . . . 97
4.3 Detuned excitation: quantum to classical Rayleigh scattering . . . 101
5 Optical manipulation of electric fields and quantum dot elec-
trometry 107
5.1 The observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.1.1 One-colour steady-state measurements . . . . . . . . . . . 108
5.1.2 Modulated shifts: Two-colour steady-state . . . . . . . . . 109
5.2 The origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.2.1 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.2.2 Electric field . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3 Time-resolved electric field dynamics and QD electrometry . . . . 117
5.3.1 Considerations for QD electrometry . . . . . . . . . . . . . 117
5.3.2 Dynamics of laser-induced electric field . . . . . . . . . . . 119
viii
CONTENTS
6 Coherent single QD photons for quantum optical networks 125
6.1 Considerations for two-photon interference . . . . . . . . . . . . . 126
6.2 Mutual coherence: Phase-locking of QD fluorescence to laser . . . 127
6.3 Photon shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.4 Photon indistinguishability . . . . . . . . . . . . . . . . . . . . . . 132
6.5 Chapter discussion and outlook . . . . . . . . . . . . . . . . . . . 137
7 Single-shot readout of the electron spin 139
7.1 Why quantum dot molecules? . . . . . . . . . . . . . . . . . . . . 140
7.2 Quantum dot molecule sample . . . . . . . . . . . . . . . . . . . . 141
7.3 Quantum dot molecules . . . . . . . . . . . . . . . . . . . . . . . 142
7.4 Quantum dot molecule spectroscopy . . . . . . . . . . . . . . . . . 144
7.5 Steady-state spin dynamics . . . . . . . . . . . . . . . . . . . . . . 153
7.6 Time-resolved spin dynamics . . . . . . . . . . . . . . . . . . . . . 155
7.6.1 State mixing and spin pumping in quantum dot molecules 158
7.7 Spin readout fidelity . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.8 Improving readout fidelities . . . . . . . . . . . . . . . . . . . . . 163
7.9 Chapter outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
8 Conclusions and outlook: towards a small-scale quantum net-
work 168
References 172
ix
Chapter 1
Introduction
The last two decades have seen a tremendous interest in the emerging field of
quantum information technology. The promise of quantum information process-
ing is an exponential speed-up of a range of computational problems whose calcu-
lation is not feasible with current technology. In the pursuit of control of materials
at the fundamental - quantum - level, this research has already developed a rich
toolbox of techniques that enabled advances in metrology and sensing applica-
tions, and is driving progress in understanding and designing materials at the
nanoscale.
Binary logic, the basis of classical computers, despite its obvious successes, is
not very good at dealing with complex system problems, e.g. many-body sim-
ulations. In these problems the number of computational steps scales exponen-
tially with the number of elements, such that any simulation becomes rapidly
intractable. Protein folding provides an example of such a complex system where
understanding the spatial configurations is also key to understanding diseases
like Alzheimer’s. As another example, quantum mechanical processes themselves
cannot be efficiently modelled with classical computers as there is no classical
equivalent for most quantum effects. The solution is a computer that does not
operate on classical physics, but on the basis of quantum mechanics, where en-
tanglement of different quantum bits is exploited for massive parallel computing.
At this stage research evolves from obtaining information about physical systems
to an approach where “information is physical”[1] and constitutes a part of a
1
complex system.
Today’s paradigm of a quantum computer, dubbed the “Quantum Internet” [2],
is a network consisting of nodes of one to a few stationary quantum bits, such
as single atoms, solid-state impurities, electron or hole spins connected via flying
quantum bits, such as single photons. Quantum information processing based on
solid-state spins was kick-started by a proposal by Loss and DiVincenzo less than
15 years ago [3]. The requirements for a realisation are summarised in what are
known as the 7 DiVincenzo criteria [4]. Starting with a physical represen-
tation of a quantum bit, control over individual quantum bits, pairs of quantum
bits and the exchange of information between separate quantum bits form the
ingredients necessary to perform arbitrary operations and algorithms.
Getting to the stage of a working prototype of a quantum computer is a huge
technical challenge, and at the present time many research groups, academic,
governmental and commercial alike, are exploring a number of physical systems
that can form the building blocks.
This thesis is concerned with small parts of this huge challenge for one partic-
ular physical system: optically active semiconductor quantum dots. We loosely
summarise the aims of this particular work as follows. The generation of clean
photons from a semiconductor quantum dot. Of course, clean is not a proper
scientific term, but purposely chosen, as the requirements on photons change
with the respective goals. For the greatest part of this thesis we are concerned
with generating photons of high coherence, with as little dephasing as possible.
However, in applications such as spin readout, scattering rates and the need for
recycling transitions take precedence.
Phrased differently, in this thesis we discuss resonant generation of single pho-
tons from single quantum dots and tunnel-coupled quantum dot molecules in the
context of quantum information processing.
In the introduction we cover the basics of what quantum dots (QDs) are and give
a brief overview on different types of QDs and their properties. We will put this
work in a wider context by presenting the main motivation, quantum information
processing, and review the standard criteria required for a physical realisation of a
2
quantum bit and the transfer of quantum mechanical information via flying quan-
tum bits. Chapter 2 provides the theoretical background for the thesis. It covers
light-matter interaction for the simplest case of a two-level system. This forms the
theoretical basis to calculate the properties of resonance fluorescence and links to
the standard techniques of quantum optical interferometry, namely the first- and
second-order correlation functions. Chapter 3 introduces the experimental system
and lab techniques, from sample processing to resonant spectroscopy. Applying
these techniques, Chapter 4 translates the resonance fluorescence theory from
Chapter 2 into experiments, highlighting the near-ideal behaviour of QD optical
transitions in the low power limit of resonant excitation. This includes the direct
observation of highly coherent scattering, which has remained largely unexplored
for solid-state single photon emitters. Deviations from this ideal behaviour are
explored in the second half of the chapter, where we reveal signatures of exciton-
phonon coupling, effects of nuclear spin dynamics, and spectral wandering in the
statistical properties of resonance fluorescence. Chapter 5 investigates proper-
ties of optically induced electric fields in our QD device structure. Chapter 6
demonstrates the potential of shaped single coherent photons for quantum inter-
ference applications, exemplified in two-photon interference measurements in a
Hong-Ou-Mandel interferometer. Chapter 7 illustrates single-shot spin readout
in a vertically stacked tunnel-coupled quantum dot molecule. Finally, conclusions
and a very brief outlook on further experimental work are given in Chapter 8.
1.1 Quantum Information Processing
Quantum computing proposes to make use of intrinsically quantum mechanical
features such as superposition of states and entanglement. Whereas classical bits
take only the values 0 or 1, the state of a quantum bit, or short qubit, is given
by a linear superposition of its two basis states |ψ〉 = c0|0〉 + c1|1〉, where the
coefficients are in general complex. It has been shown that some problems can be
solved more ‘efficiently’ using qubits, most notably Shor’s algorithm to factoring
high digit numbers [5]. In addition to these improvements on classical computing,
new, entirely quantum mechanical concepts can be realised, with quantum cryp-
tography and quantum key distribution as the high profile applications which are
3
already commercially available
1.1.1 DiVincenzo’s 5 + 2 and quantum optical networks
A set of requirements a physical qubit needs to fulfill has been formulated by
DiVincenzo and co-workers in 2000 [4] which quickly became the Commandments
of quantum computing. We quote them here directly:
1. A scalable physical system with well characterised qubits. What are the
qubit basis states and how are they coupled? What other states or inter-
actions may play a role? The scalability requirement is more involved; we
need not only consider qubit-qubit coupling, but also the physical resources
involved.
2. The ability to initialise the state of the qubits to a simple fiducial state, such
as |000 . . .〉. We need to set the exact state (phase and amplitude) of the
system before any operations are performed.
3. Long relevant decoherence times, much longer than the gate operation time.
In short, a quantum state loses memory of its phase (dephasing) and am-
plitude (relaxation) after some time and any operation on a state that is no
longer well defined is meaningless. Mechanisms for error correction exist,
see for instance Ref. [6] and the standard estimate is that the decoherence
time needs to be larger than the gate operation by a factor of 104 − 105.
4. A set of universal quantum gates. Any operation can then be broken down
into a number of gate operations. For a single qubit this translates to having
full control over amplitude and phase of the qubit. Conceptually, the qubit
state is visualised as a point on the surface of a sphere (Bloch sphere) and
experimentally we need to be able to reach any point on the surface.
5. A qubit-specific measurement capability to read out the results of a set of
operations.
4
The starting point to any physical implementation is to identify a two-level system
that serves as qubit. Two-level systems or quasi-two-level systems can be found
in many areas of physics, so there is a wealth of qubit candidates. A number of
these are presented in Ref. [7], or more recently in Ref. [8]. Electron or hole
spins in quantum dots are attractive candidates as they can be controllably in-
troduced, spin states can be manipulated optically, and/or using external electric
and magnetic fields. Equally important, research suggests very long decoherence
times. We will elaborate on the performance of spins in QDs in the context of
these requirements in section 1.4.
The last two requirements are slightly separate from the previous ones; they are
concerned with communicating quantum information between separate qubits
and over macroscopic distances.
6. The ability to interconvert stationary and flying qubits. This translates to
creating entanglement between the stationary qubit, e.g. the QD spin, and
a photon.
7. The ability faithfully to transmit flying qubits between specified locations,
which requires a communication channel.
Requirements 6 and 7 lead to the vision of a quantum optical network we men-
tioned earlier - the Quantum Internet. Figure 1.1 from Ref. [2] shows a graphical
representation of the concept. Theoretical proposals based on this idea are al-
ready well-developed.
1.2 Semiconductor quantum dots
In general, QDs (sometimes called quantum boxes) refer to semiconductor struc-
tures that confine electrons (or holes) in nanoscale potentials in all three spatial
dimensions. As such they show many properties known from atoms: The three-
dimensional quantum confinement gives rise to a δ-function-like density of states
(i.e. discrete energy levels) and a shell structure is observed when electrons are
added one by one [9; 10; 11]. Single quantum dots have been termed ‘artificial
atoms’ for these reasons. Semiconductor quantum dots have been the subject of
5
Kimble, H. J. The quantum internet.
Nature 453, 1023-1030 (2008)
Figure 1.1: Illustration of the quantum internet as a network of nodes connected by
non-classical communication channels (top) and the transfer of quantum information
between two qubits in cavities. Reproduced from Ref. [2].
much research during the past 20 years. First proposed by Arakawa and Sakaki in
1982 as highly efficient and temperature-independent emitters in the active areas
of semiconductor diode lasers [12], they are now used in a number of research
areas.
Realisation of these structures was made possible following the advances in fab-
rication techniques such as molecular beam epitaxy or metalorganic chemical
vapour deposition [13] that allow layer by layer growth of semiconductors while
controlling the composition of each layer.
There are two main techniques for creating quantum dots: self-assembly and elec-
trical definition. Both types of dots are commonly used in experiments aiming
to control quantum properties like the electron spin and will be briefly introduced:
• Electrically defined QDs: metallic surface gates above a two-dimensional
electron gas at an interface, typically GaAs/AlGaAs, can be used to deplete
the electron gas below them, isolating small islands of electrons from the
electron gas. The small size of the islands leads to quantisation in the
remaining two dimensions. These electrically defined dots are also called
lateral QDs and the strength and shape of their confining potential can be
6
conveniently modified by changing the gate voltage, allowing an all-electric
approach to spin control. In such a way tunnel-coupled double and triple
QD structures have been realised [14].
• Self-assembled dots: the quantum dots used in this study were grown in
Stranski-Krastanov mode1 [17]. Layered semiconductor structures are cre-
ated using molecular beam epitaxy. A higher bandgap substrate layer, typi-
cally GaAs, is grown, followed by ∼ two monolayers of a lower bandgap ma-
terial, InAs in our case. This is called the wetting layer. As a consequence
of local strain fields due to the mismatch of lattice constants between two
layers islands form in the wetting layer at a critical thickness. The growth
conditions determine size and shape of these islands, hence the confinement
energy. Self-assembled QDs are usually lens-shaped with a diameter of a
couple of tens of nanometres and a stronger confinement in growth (z-) di-
rection (few nm). The wetting layer containing the QDs is overgrown with
GaAs, in some cases only partially, such that the QD height can be con-
trolled in an additional annealing step. When another InAs wetting layer
is added 10-20 nm above the previous, QDs nucleate preferentially on top
of QDs in the first layer, thus vertically coupled QD molecules are created.
Gating the sample allows control over the number of charges resident in the
QD. For an introduction and a review of QD growth see for instance Ref.
[18].
1.2.1 Quantum information, metrology and mesoscopic
physics
QDs provide optically addressable two-level systems in the the solid state with
good optical properties. Excited state lifetimes on the order of one nanosec-
ond promise high single photon emission rates. This makes QDs attractive for
quantum key distribution. Electrically driven single photon sources have been
1There are other important self-assembly mechanisms that lead to creation of quantum dots,
such as precipitation of colloidal QDs (e.g. CdTe and CdSe) in solution[15], or QDs formed as
interface fluctuations in a quantum well [16]. For the purpose of spin control the electrically
defined QDs and Stranski-Krastanov QDs offer the most desirable properties and are used in
current research.
7
developed [19] and tested in laboratories [20].
The confinement of single spin carriers provides partial protection from the solid-
state environmental effects that rapidly dephase and relax any unprotected ini-
tially well-defined states. Spin manipulation, taking advantage of the optically
excited states, allows gate operation times of picoseconds [21; 22] which is con-
siderably faster than using electric fields or microwave excitation.
Spins and charges in quantum dots are not completely isolated, however, and the
question of how atom-like these artificial atoms really are is an important one to
answer. What role does the semiconductor environment play? Bound electron-
hole pairs in QDs, excitons, still interact with the lattice phonons, electron and
exciton spins experience the dynamics of the nuclear spin bath via the hyperfine
interaction. QD transition energies are sensitive to electric field fluctuations in
the environment, dynamics of nearby charge traps or neighbouring QDs. These
effects pose challenges to quantum information applications, where unwanted in-
teraction should be kept to a minimum, but open up other opportunities.
The sensitivity to environmental effects can be exploited for metrology appli-
cations. This has been spectacularly demonstrated for magnetometry with the
nitrogen vacancy centre in diamond [23]. For QDs, electrometry has been pro-
posed and demonstrated [24]. At the same time, optical thermometers based on
colloidal QDs are advocated in biological sciences (see for example Ref. [25]).
The environmental effects on the QD optical properties are topic of section 4.2
in Chapter 4, and Chapter 5 contains an application of QD electrometry.// Next
we look at the QD energy level structure in more detail.
1.3 QD level structures
This section contains a brief discussion of the level structure for the neutral and
the negatively charged QD ground states. The effects of external electric and
magnetic fields of the levels are touched upon, and we introduce some interactions
with the solid-state environment. These will be picked up again at the appropriate
places in the experimental chapters.
8
1.3.1 Exciton fine structure
We are working with QDs which are either empty or single electron charged in
their ground state. The excited states, accessible via absorption of a photon, are
then the neutral exciton (a single electron-hole pair) and the negatively charged
exciton (called trion), respectively. The heavy- and light-hole bands are strongly
separated in quantum confinement, so that holes in the QD are to a good ap-
proximation heavy-holes with total angular momentum Jhh = 3/2
1. The energy
and fine structure of the optical transitions are governed by Coulomb interactions
which we separate into direct Coulomb interaction and exchange interaction.
Direct Coulomb interactions for two particles a and b are, in general, given by
VCoulomb =
∫ ∫
drdr′ |r− r′|−1 φa (r)∗ φb (r′)∗ φa (r)φb (r′) , (1.1)
where the φi (x) are the orthonormal wavefunctions of particles i (electron e or
hole h) at location x. Exchange interaction
Vexchange =
∫ ∫
drdr′ |r− r′|−1 φa (r)∗ φb (r′)∗ φa (r′)φb (r) (1.2)
has consequences for the fine structure of the neutral exciton. Neglecting ex-
change, there are four degenerate neutral exciton states at zero magnetic field:
↑⇓, ↓⇑, ↑⇑ and ↓⇓, where single-lined arrows denote electron spin projection and
double-lined arrows the heavy hole projections. Creation and recombination of
↑⇓, ↓⇑ excitons is allowed by electric dipole selection rules and hence they are
‘bright’, while the ↑⇑ and ↓⇓ excitons are forbidden and ‘dark’. Electron-hole
exchange gives rise to three terms [28]: The isotropic exchange splits dark and
bright excitons by an energy δ0. The anisotropic exchange δ1 hybridises the two
bright excitons in the absence of magnetic field into orthogonal, linear superposi-
tions ↑⇓ ± ↓⇑ [29]. Their splitting is known as XY -splitting. The dark excitons
are split by δ2. Anisotropic electron-hole exchange vanishes for QDs with cir-
1The approximation of holes in QDs as pure heavy-holes is a valid approximation in many
cases, but QDs do exhibit a small degree of mixing with the light-holes states ⇑+1/2 and ⇓−1/2.
For single QDs a spread of admixtures has been observed; for our single QD samples the
admixture is typically of a few percent [26]. In coupled QDs heavy-light hole mixing can be
enhanced through an offset of the vertically stacked QDs relative to each other [27].
9
cular symmetry, but we observe it for all QDs in our sample. The exchange
Hamiltonian for basis states ↑⇓, ↓⇑, ↑⇑, ↓⇓ is [28]
Hexchange =
1
2

δ0 δ1 0 0
δ1 δ0 0 0
0 0 −δ0 δ2
0 0 δ2 −δ0
 . (1.3)
1.3.2 The nuclear Overhauser field
We have to keep in mind that, while QDs are termed artificial atoms and show
some atomic-like properties, charges are confined in a mesoscopic environment
and interactions with the solid-state matrix can give rise to very ‘un-atomic’
effects.
A self-assembled In/GaAs quantum dot consists of N ∼ 105 atoms and each one
of them has a nuclear spin (spin 3/2 for Ga and As and 9/2 for In). A single
electron or hole spin confined in the QD interacts with the nuclear spins through
the hyperfine interaction. For electrons the dominant term of this interaction [30]
is the Fermi contact interaction1
HHF =
∑
k
AkSˆ · Iˆkδ (r−Rk) , (1.4)
where Rk gives the position of the kth nucleus with spin Iˆk, r the position of the
electron with spin Sˆ, and the hyperfine coupling constant A reflects the spatial
dependence of the electron wave function in the QD. The effect of nuclear spins
is commonly described as an effective magnetic field BN acting on the electron.
The magnitude of this field, named Overhauser field, is then given as
BN =
∑
k AkIˆk
gµB
. (1.5)
The Overhauser field effect can be understood as the electron spin precessing in a
time-varying magnetic field, where the field strength and direction at any time is
1We note that other terms, such as dipole and quadrupole interaction play a role too, see
for instance Refs. [31; 32].
10
given by the sum of all nuclear spins vectors. If there is no preferential alignment
of the nuclear spins, for instance in the absence of an external magnetic field,
then the individual spins are randomly oriented. This gives rise to a spherical
symmetry of the Overhauser field distribution. While the mean of the Overhauser
field is zero, the most likely configuration has a finite field strength.
The Overhauser field evolves on timescales of microseconds [33] while remaining
frozen on shorter timescales. Its root-mean-square magnitude can be estimated
from the number of nuclear spins and is typically on the order of a few tens of
millitesla [34].
Eq. (1.4) includes electron-nuclear spin flip-flop terms, and hyperfine interaction
is the dominant source of spin relaxation at low magnetic fields. This relax-
ation is due to direct coupling between the ground states. At finite fields the
incommensurate splitting of the QD ground state compared to the nuclear spin
splitting renders spin relaxation due to the flip-flop terms inefficient. However,
electric-dipole forbidden transitions can be enabled by concomitant nuclear spin
flips. Research into this link between electron spin and nuclear polarisation has
revealed interesting physics [35; 36] and partial control of the nuclear spins has
been demonstrated [37; 38; 39; 40]. A controllable nuclear field can be employed
as resource in quantum control [41].
The influence of the nuclear spin bath on QD emission spectra will be discussed
in Chapter 4, section 4.2.3.
1.3.3 Trion levels under magnetic and electric fields
Zeeman shifts. So far research into electron or hole spins in optically active
QDs has focussed mostly on single spins as qubits. Experimental studies into
other approaches, such as using two spins, have only emerged recently [42; 43;
44]. Our approach also concentrates on a single spin. Under magnetic field the
degeneracies of the single electron ground states and the excited trion states are
lifted according to the electron (ge) and hole (gh) g-factors:
HZeeman = −µB
(
ge,zSe,z − gh,z
3
Je,z
)
Bz,
11
Figure 1.2: Selection rules for Faraday and Voigt geometry under magnetic
field. The spin and charge states for each level are given: arrows denote spin projection
for electrons (filled circles) and holes (empty circles). In the excited states the two
electrons form a spin singlet, so that the total spin is determined by the hole spin
projection.
where µB is the Bohr magneton and we have chosen the magnetic field to be
in the z-direction and neglected the diamagnetic shift, which adds a quadratic
dependence on the magnetic field [45]. Two geometries are being explored in
experiments: in Faraday geometry the magnetic field is parallel to the sample
growth axis and coincides with the direction of optical axis. In Voigt geometry
the magnetic field is perpendicular to the growth and optical axis. Selection rules
differ for these two geometries. We show the level structure and selection rules
for the trion transitions in Fig. 1.2.
Faraday geometry is used in Chapter 5. The solid-state environment (see pre-
vious section) relaxes the selection rules, so that there is a small probability of
relaxation into the opposite ground state when cycling one of the allowed tran-
sitions (branching ration ∼ 200:1). The ground states are very weakly coupled,
leading to a finite spin-flip rate (see 1.3).
Stark shifts Excitons in QDs have a large permanent electron dipole moment,
as the electron and hole are physically separated by a ∼ nanometre distance [46].
Due to the local confinement in the low bandgap QD, the exciton energy can be
tuned over many times its radiative linewidth by external electric fields without
ionising. The energetic shift of QD photoluminescence as a function of applied
12
Figure 1.3: Relaxation of selection rules in the trion transition as a result of interac-
tions with the environment.
vertical field F is dominated by the change of the exciton energy and contains
both linear and quadratic terms:
∆Eexciton = −µFapplied + αF 2applied , (1.6)
where µ is the permanent dipole and α the polarisability. In our structures the
Stark shift is entirely linear to a good approximation; the quadratic term has
been observed for larger applied fields in different QD device structures [46; 47].
1.4 Coherent control of single spins: state of
research
Here, we summarise the current state of research into coherent control of spins
in QDs. It will become obvious that not all DiVincenzo criteria can be realised
to equal extents in single QDs and how coupled QDs, quantum dots molecules
(QDMs), offer a natural extension that allows spin state readout.
A complete quantum operation on a single qubit can be separated into three
parts: state preparation/initialisation, coherent manipulation and state readout
at the end. From the DiVincenzo criteria we pick the three requirements relating
to spin control and the requirement on coherence times. We give an overview
13
of recent progress in these areas, focussing mostly on optically active QDs. The
numbering below follows the DiVincenzo criteria introduced earlier, so, having
chosen spin up and down as basis states, we start with number two:
2. State initialisation was first demonstrated in 2006 [48]. It relies on the
state mixing in Faraday geometry described above (see Fig. 1.3). A spin-
flip Raman transition occurs after cycling a trion transition for a few µs,
and provided that spin relaxation is much slower than this optical ‘spin
pumping’, the spin state is initialised. Transferring this protocol into Voigt
geometry, optical spin pumping initialises the spin within a few ns [49] .
This is possible as the excited trion state is coupled equally to both single
electron spin ground states. However, initialisation fidelities can suffer from
detuned pumping via the other excited state (cf. Fig. 1.2). Similar optical
pumping has also been observed for single holes [50].
3. Much effort has also been put into determining spin coherence times (the
dephasing time T2) and prolonging it. We distinguish here between the
inhomogeneous dephasing time T ?2 which is derived from a time-averaged
measurement and equivalent to an ensemble dephasing time, and the homo-
geneous dephasing time T2 which would be measured for a single spin in a
single measurement. The inhomogenous dephasing time T ?2 of the electron
spin is consistently found to be on the order of 1 ns, limited by averaging
over the different Overhauser field configurations for individual measure-
ments [39; 51; 52; 53]. However, the Overhauser field is fluctuating slowly
(∼ µs or slower [33]), so we should expect homogeneous T2 times in the
microsecond range as well. Optical spin-echo of a single spin [52] and ma-
nipulation of a spin ensemble subset [54] do indeed recover T2 ∼ few µs.
Use of dynamic decoupling techniques has resulted in a tremendous im-
provement of these numbers for electrically defined QDs, reaching dephas-
ing times in excess of 100 µs [55]. In parallel, research efforts directed at
influencing the nuclear spin bath and quieting down the fluctuations have
shown some success [37], and coherent population trapping measurements
suggest an improved inhomogeneous dephasing time of a few microseconds.
Further, qubits encoded in the two-electron singlet-triplet basis can be en-
14
gineered to be insensitive to environmental fluctuations and the inhomoge-
neous dephasing time is in on the order of 100 ns, or longer [44].
Finally, much attention has also been paid to hole spin qubits due to their re-
duced Fermi contact hyperfine interaction. Inhomogeneous dephasing times
in the range of 2 to 20 ns have been reported [43; 56; 57], and T2 ∼ 1 µs
with a a Hahn spin-echo [56].
With ps manipulation capabilities, these coherence times would be sufficient
for 104 − 105 operations, i.e. error correction. We note, that each optical
operation may still induce some decoherence, however, so it remains to be
seen what the performance in actual applications will be like.
4. The requirement of a universal set of quantum gates is experimentally
equivalent to having full control over the Bloch sphere for a single qubit. In
order to reach any point on the Bloch sphere rotations about two axes are
necessary. Typically one rotation axis can be associated with controlling
the amplitude of the qubit (rotations in zx− or zy−plane) at a fixed phase,
the other rotation sets the phase of the qubit (xy-plane). Much progress
has been made in coherent optical control of electron spins. Complete con-
trol has been achieved for single spins [22; 58] and an ensemble subset [59].
Press et al. [22] and Kim et al. [58] operate in Voigt geometry, where the
electron ground states can be coupled using two-photon Raman transitions,
cf. Fig. 1.2. Rabi oscillations between ↑ and ↓ are observed, demonstrating
control over the qubit amplitude. Fast Larmor precession in a high mag-
netic field provides the second rotation for Press et al. as is evidenced by
Ramsey interference. In [58] the free Larmor precession is complimented by
geometric phases that can be imparted to one spin component of a coherent
superposition of ↑ and ↓. This adds control to the otherwise free precession
about the magnetic field.
The same level of control has been demonstrated for single hole spins in a
QD [43; 56; 57]. Further, research conducted in the United States’s Naval
Research Laboratory could extend the control scheme to two independent
spins (both electrons and holes) in tunnel-coupled QD molecules. Em-
ploying exchange interaction between the spins and geometric phase gates
15
allowed the implementation of two-qubit rotations - the first demonstration
of both single- and two-qubit control for QDs [42; 43].
With regard to electrically defined QDs, rotations of 2-electron qubits in
electrically defined double QDs have been achieved via control of the ex-
change interaction [51] and more recently by creating a magnetic field gra-
dient across the QDM [38]. The combination of the rotations enable full
control over the Bloch sphere.
In Faraday geometry spin rotations have been performed using microwaves
[60]. While operation in Voigt geometry allows completion of a gate oper-
ation in a few tens of ps [22], the need to rely on microwave rotations have
limited spin manipulation in Faraday geometry to ∼ 100 ns.
5. Spin readout has mostly relied on averaging the results of many (> 104)
measurements to obtain reliable results. Attempting to directly measure the
population of a spin state in Voigt geometry leads to spin shelving within
a few optical cycles, so that only 1-2 photons are scattered in each readout
attempt. Standard photon collection and detection efficiencies are around
0.1-1% [53; 61], requiring ∼ 104 attempts to distinguish between spin up
and down states. The collection and detection rates are mostly limited by
the photon extraction from the high-index semiconductor into free space.
Research to remedy this problem is progressing well though, with reports
of 70 % sample outcoupling efficiencies in specialised sample designs [62].
Single-shot readout in Faraday geometry is less hopeless; the reduced back-
action of the readout laser causes spin shelving in ∼ 1 µs [26] here and
state-of-the-art collection efficiencies should allow single-shot readout now.
Non-destructive readout techniques, such as Faraday rotation [63] and Kerr
rotation [64] have been successfully applied to determine the spin state of
a single electron, but require integration over many milliseconds.
The first successful attempt at single-shot spin readout is detailed in Chap-
ter 7 of this thesis. We use a QD molecule consisting of two vertically cou-
pled QDs where a transition to a molecular excited state is spin-sensitive
and recycling.
6. Spin-photon entanglement: The sixth of the Divincenzo criteria was
16
demonstrated independently by three groups recently [53; 65; 66]. They
employ a single electron spin in a single QD in Voigt geometry. As each
excited state forms a Λ-system with the two ground states the polarisation
(and frequency) of a single scattered photon is entangled with the state
of the QD electron spin. Measurements of spin-photon correlations in two
independent bases were then used to prove the entanglement.
17
Chapter 2
Light-matter interaction of two
levels: resonance fluorescence
In this chapter we recap the near-resonant light-matter interaction of a simple
two-level system. This is a standard topic in quantum optics textbooks and as
such it would be pointless to provide a comprehensive treatment here. However,
Chapter 4 and, to some extent, Chapter 6 are concerned with determining the
extent to which optical QD transitions confirm with the predictions for ideal two-
level systems. So we give here a summary of theoretical background and sketch
out derivations of resonance fluorescence properties. Important reference for this
chapter are Loudon’s The Quantum Theory of Light [67], chapters 2, 7.9 and
8 in particular, Quantum Optics by Scully and Zubairy [68], chapters 5 and 10,
and Elements of Quantum Optics by Meystre and Sargent [69], chapters 4 and 16.
The aim of this chapter is to sketch out how we can derive the properties of
the emission from a resonantly driven two-level system. The resonance fluores-
cence is defined by its spectral and statistical properties, which can be calculated
- and measured - through field- and intensity-correlation functions. The reso-
nance fluorescence spectrum, first derived by Mollow in 1969 [70], can be found
with a fully quantum mechanical treatment, i.e. with both quantised fields and
a quantised emitter, see Ref. [71] for example. A semi-classical approach, where
the optical field is classical, gives the same results though, and for simplicity this
18
ГωL
ω
L ω L
ω
QD
ω
L ω L
Г
ω L ωQD
ω
‹
Г
ωL ω
| ›1
| ›2
Figure 2.1: Sketch of two-level system and optical field. The ground (|1〉) and excited
(|2〉) states are coupled by a monochromatic field of energy ~ωL which is close to the
energy difference for the two levels. Spontaneous emission broadens state |2〉 and its
population relaxes with rate Γ.
approach will be followed. Details of the calculation can also be found in Ref.
[72], a subsequent paper by Mollow. The optical Bloch equations obtained this
way not only reveal the steady-state behaviour of populations and coherences of
the two-level-system for near-resonant driving frequencies, but solving the density
matrix equation of motion is also key to obtaining the first-order correlation func-
tion, and hence the spectrum of the scattered light (the resonance fluorescence).
The intensity correlation function can be derived in a similar fashion.
2.1 Optical Bloch equations and the density ma-
trix
The premise. First we introduce the parameters of the two-level system and
the light field. Figure 2.1 shows a sketch of the energy levels. Ground and
excited states, denoted |1〉 and |2〉, are separated by an energy ~ω. For a QD
neutral exciton transition the ground state would correspond to the crystal ground
state, while the excited state is a single exciton. In the sketch the excited state
is broadened due to spontaneous radiative relaxation to the ground state, and
the width Γ indicates the decay rate. In the absence of other interactions, this
19
decay rate is given by the inverse excited state lifetime which we denote T1.
This timescale governs the decay of excited state population, but also the decay
of the QD polarisation. To account for additional dephasing processes of the
polarisation we introduce the dephasing time T2 here. We distinguish between
radiative dephasing and any additional pure dephasing, quantified by γp:
1
T2
=
1
2T1
+ γp. (2.1)
The optical field is a single monochromatic mode of frequency ωL = ω + ∆,
where ∆ is the detuning from the QD resonance. The QD-field interaction is
conveniently described using the QD density matrix:
ρ = |Ψ〉 〈Ψ|
= (c1 (t) |1〉+ c2 (t) |2〉) (c?1 (t) 〈1|+ c?2 (t) 〈2|)
= |c1|2 |1〉 〈1|+ c1c?2 |1〉 〈2|+ c2c?1 |2〉 〈1|+ |c2|2 |2〉 〈2| (2.2)
where the QD state |Ψ〉 is a linear superposition of its two basis states, with
time-dependent coefficients ci (t). The density matrix elements ρij are obtained
as
ρij = 〈i| ρ |j〉 . (2.3)
The diagonal elements give the populations and satisfy ρ11 + ρ22 = 1. The off-
diagonal elements ρ12 = ρ
?
21 are related to the QD polarisation and called the
‘coherences’.
Optical Bloch equations. The evolution of the density matrix elements for
the QD-light interaction is described by the equation of motion
ρ˙ = − i
~
[Hint, ρ]− 1
2
(γρ+ ργ) . (2.4)
Here the first term describes the QD-field interaction and the second term adds
population relaxation. The interaction Hamiltonian contains the electric dipole
20
interaction with the radiation field and we give it in the interaction picture here:
Hint =− ~ΩU †0 (t)
(
e−iφ |2〉 〈1|+ eiφ |1〉 〈2|)U0 (t) cosωLt . (2.5)
The radiation field drives transitions between the QD ground and excited states
and the Rabi frequency Ω describes the rate. It is linked to the amplitude of the
radiation field E and the dipole matrix element D12 = 〈1|x |2〉 (here for a field
polarised along x) as
Ω =
|D12|E
~
.
With interaction operator U0 (t) = e
−iω2t |2〉 〈2|+ e−iω1t |1〉 〈1|, where ~ω1,2 corre-
spond to the energies of ground and excited states, and ω = ω2−ω1 the interaction
Hamiltonian becomes
Hint =− ~Ω
2
(
eiωt−iφ |2〉 〈1|+ e−iωt+iφ |1〉 〈2|) (eiωLt + e−iωLt) . (2.6)
Multiplication of the two brackets on the right-hand side yields four terms, two
of which oscillate at the difference frequency ± (ω − ωL), and two oscillating at
the sum frequency ± (ω + ωL). Discarding of the latter two in the rotating wave
approximation and writing the difference frequency as ∆ we arrive at the final
interaction picture Hamiltonian:
Hint =− ~Ω
2
(
e−iφ |2〉 〈1| ei∆t + eiφ |1〉 〈2| e−i∆t) . (2.7)
For the relaxation terms in Eq. (2.4) we assume for now just the excited state
lifetime T1. The matrix γρ+ ργ then becomes
γρ+ ργ =
(
0 ρ12
T1
ρ21
T1
2ρ22
T1
)
. (2.8)
21
Using Eq. 2.7 and Eq. 2.8 in Eq. 2.4 we evaluate the density matrix elements’
equation of motion:
˙ρ12 = i
Ω
2
e−i∆t (ρ22 − ρ11)− ρ12
2T1
(2.9)
˙ρ22 = i
Ω
2
(
ei∆tρ12 − e−i∆tρ21
)− ρ22
T1
(2.10)
With the substitution ρ˜12 = e
i∆tρ12 we can take out the oscillating factors. Fur-
ther, we include pure dephasing processes phenomenologically by replacing 2T1
in Eq. (2.9) with T2. Then the equations of motion are equivalent to the Bloch
equations derived for nuclear magnetic resonance of spin 1/2 systems [73] and
hence called the optical Bloch equations :
˙˜ρ12 = i
Ω
2
(ρ˜22 − ρ˜11)− ρ˜12
(
1
T2
− i∆
)
(2.11)
˙˜ρ22 = i
Ω
2
(ρ˜12 − ρ˜21)− ρ˜22
T1
. (2.12)
Steady-state solutions. To relate the Bloch equations to measurable quan-
tities we have to link the electric field operators for QD emission to the density
matrix elements. To this end we introduce the transition operators first.
σ− = |1〉 〈2| (2.13)
σ+ = |2〉 〈1| (2.14)
The time dependence of their expectation value follows
〈σ− (t)〉 = ρ˜21e−iωLt . (2.15)
The positive frequency part of the electric field operator is proportional to the
lowering operator and defined as
E+
(
r, tˆ
)
=
eω2esc ·D12
4pi0c2r
σ−
(
tˆ
)
. (2.16)
22
tˆ is the retarded time and esc the polarisation vector of the scattered field. Mea-
suring the intensity of the scattered light is proportional to the single-time first-
order correlation time of the field:
I ∝
∫ T
0
dt
〈
E−
(
tˆ
)
E+
(
tˆ
)〉 ∝ ∫ T
0
dt
〈
σ+
(
tˆ
)
σ−
(
tˆ
)〉
. (2.17)
With
σ+
(
tˆ
)
σ−
(
tˆ
)
= |2〉 〈2| ,〈
σ+
(
tˆ
)
σ−
(
tˆ
)〉
= ρ˜22 (2.18)
we find that the intensity is proportional to the steady-state excited state popu-
lation ρ˜22 (t→∞) = ˜ρ22,ss. Taking ˙˜ρ22 = 0, ˙˜ρ12 = 0 in Eq. (2.11) and (2.12) we
find
I ∝ ˜ρ22,ss = 1
2
Ω2
∆2 T2
T1
+ 1
T1T2
+ Ω2
. (2.19)
The scattered intensity is actually composed of two parts1: I = Icoh + Iinc. The
coherent term maps out the magnitude of off-diagonal density matrix elements
as
Icoh ∝ | ˜ρ12,ss|2 = 1
4
Ω2
(
∆2 + T−22
)(
∆2 + 1
T 22
+ T1
T2
Ω2
)2 . (2.20)
Figure 2.2 (left-hand side) provides a visualisation of the scattering intensity
according to (2.19) for T2 = 2T1 as a function of detuning (in units of 1/ (2piT1))
and square of the Rabi frequency Ω which scales as the power of the exciting
field. The power scale is in units of the saturation power Ω2sat = 1/ (2T
2
1 ). We see
a saturation of the scattering intensity towards a steady-state population of 1/2,
and at the same time a broadening of the scattering response (power broadening)
1This can be seen in the quantum mechanical Langevin-Bloch equations which include a
noise term. The noise term separates the spontaneous, incoherent emission originating from
the influence of vacuum fluctuation on the QD dipole motion [69] from the coherent response
of the dipole to the radiation field.
23
Figure 2.2: Plots of the total scattering intensity (left) and coherent fraction of the
radiation (right). The linewidth is given by 1/ (2piT1) and the saturation power is
proportional to Ω2sat = 1/
(
2T 21
)
.
as a function of excitation power. The plot on the right side shows the coherent
fraction Fcoh = Icoh/I with the same parameters. Towards zero power, or strong
detuning, the coherent part dominates the scattering. We assumed T2 = 2T1 for
the two plots, which corresponds to an absence of pure dephasing. We illustrate
the effect of pure dephasing on the coherent and incoherence intensity parts in Fig.
2.3 for the resonance condition (∆ = 0). Panel a shows the power dependence of
the coherent fraction for T2 = 2T1 (no pure dephasing) as blue curve, the green
curve is for T2 = T1, while T2 = 0.5T1 for the red curve. In the right panel we
show the total scattering intensity (black) together with the incoherent (blue)
and the coherent (green) parts for a dephasing time T2 = T1. The coherent part
reaches a maximum at the saturation point, where Fcoh = T2/4T1. It approaches
zero as the power is increased. The sensitivity of the coherent fraction to the
pure dephasing rate makes it a good benchmark for measurements of T2. We
experimentally study the saturation behaviour for a single QD in Chapter 4.
24
0 . 1 1 1 0 1 0 00 . 0
0 . 5
 
 
Flu
ore
sce
nce
 Int
ens
ity
E x c i t a t i o n  p o w e r  ( 1 / T 1 T 2 )
0 . 1 1 1 0 1 0 00 . 0
0 . 5
1 . 0
 
 
Coh
ere
nt f
rac
tion
E x c i t a t i o n  p o w e r  ( 1 / T 1 T 2 )
ba
Figure 2.3: Influence of dephasing time on coherent and incoherent contributions to
the fluorescence. a, coherent fraction for T2 = 2T1 (blue curve), T2 = T1 (green) and
T2 = 0.5T1 (red).b, total scattering intensity (black), incoherent contribution (blue)
and coherent contribution (green) for T2 = T1. The laser detuning is kept to zero
detuning in both panels.
2.2 First-order coherence of resonance fluores-
cence
The first-order coherence of resonance fluorescence is evaluated through the cor-
relation function
g(1) (τ) =
〈
E−
(
tˆ
)
E+
(
tˆ+ τ
)〉〈
E−
(
tˆ
)
E+
(
tˆ
)〉 (2.21)
=
〈σ+ (t)σ− (t+ τ)〉
〈σ+ (t)σ− (t)〉 .
The challenge of evaluating g(1) (τ) lies in calculating a two-time expectation
value. The trivial case of τ = 0 gives us
g(1) (0) = 1 . (2.22)
Calculating Eq. (2.22) is made possible with the quantum regression theorem. It
states that if the density matrices of a system (the two-level atom/QD) coupled
to a reservoir (the radiation field modes) are uncorrelated at time t = 0, that
is ρsystem−reservoir (0) = ρsystem (0) ρreservoir (0), then the equation of motion for the
two-time operator expectation values obeys that for the single-time operator ex-
pectation values. In other words, knowing the complete time dependence of the
transition operators 〈σ+ (t)〉 and 〈σ− (t)〉 is sufficient to calculate the first-order
25
coherence.
In particular, the equations of motion for the single- and two-time transition
operator expectation values have the general solution [72]
〈σ− (t+ τ)〉 eiωLt+τ = α1 (τ) + α2 (τ) 〈σ− (t)〉 eiωLt
+ α3 (τ) 〈σ+ (t)〉 e−iωLt
+ α4 (τ) 〈σ+ (t)σ− (t)〉 (2.23)
〈σ+ (t)σ− (t+ τ)〉 eiωLt = α1 (τ) 〈σ+ (t)〉+ α2 (τ) 〈σ+ (t)σ− (t)〉 eiωLt (2.24)
+ α3 (τ) 〈σ+ (t)σ+ (t)〉 e−iωLt
+ α4 (τ) 〈σ+ (t)σ+ (t)σ− (t)〉 (2.25)
Similar expressions hold for 〈σ+ (t+ τ)σ− (t+ τ)〉 and 〈σ+ (t)σ+ (t+ τ)σ− (t+ τ)σ− (t)〉.
In the two-time correlation function above the expectation values of the α3 and
α4 vanish. Noting the relation between transition operator expectation values
and the density matrix elements from (2.18) and that
〈σ− (t)〉 =ρ˜21e−iωLt , (2.26)
we conclude that a knowledge of the time evolution of the density matrix elements
ρ˜12 (t), ρ˜22 (t) allows us to evaluate g
(1) (τ). Both derivation and the final solution
for ρ˜12 (t), ρ˜22 (t), including finite detuning ∆ and pure dephasing γp, are lengthy.
For reference, a solution using Laplace transform was published by Torrey1 [76]
and we will not recreate it here. If we are interested in the stationary limit, that
is, the steady-state first-order correlation function, we can take t → ∞ and use
the steady-state solutions to the optical Bloch equations in Eq. (2.25). From the
first-order coherence g(1) (τ) the emission spectrum follows directly as
S (∆ν) ∝
∫ ∞
0
dτ
〈
E− (τ)E+ (0)
〉
e−i∆ντ + c.c (2.27)
Explicit expressions for g(1) (τ) and S (∆ν) are given in the Chapter 4 (Eq. (4.4)
and (4.5)) and compared to the experimental measurements of QD resonance
1The original manuscript contains a number of typographic errors, for a corrected version
see, for instance, Ref. [74], which contains only one mistake [75].
26
fluorescence.
2.3 Second-order coherence of resonance fluo-
rescence
The second-order coherence of resonance fluorescence is defined by the second-
order correlation function
g(2) (τ) =
〈E− (t)E− (t+ τ)E+ (t+ τ)E+ (t)〉
〈E− (t)E+ (t)〉2 (2.28)
=
〈σ+ (t)σ+ (t+ τ)σ− (t+ τ)σ+ (t)〉
〈σ+ (t)σ− (t)〉2
.
Without the quantum regression theorem we can again only find the τ = 0 value:
g(2) (0) = 0 , (2.29)
since σ+ (t)σ+ (t) = |2〉 〈1| |2〉 〈1| = 0 (and similarly for σ− (t)σ− (t)).
The formal derivation of the intensity correlation function follows the one outlined
above for the field correlation function. However, we can arrive at the same
result with a bit of physical insight. The experimental configuration relating
to the intensity correlation function we are usually interested in consists of the
time-resolved detection of a second photon conditioned on the detection of a first
photon. A sketch of the experimental arrangement is provided in Fig. 2.4. The
detection of the first photon projects the QD two-level system into the ground
state. Measuring the averaged probability of photon detection at a subsequent
time just maps out the excited state population
g(2) (τ) ∝ 〈σ+ (τ)σ− (τ)〉 = ρ˜22 (τ) (2.30)
for the condition that ρ˜22 (τ = 0) = ρ˜12 (τ = 0) = 0. The normalisation is given
by the steady state value of the excited state population. Using Torrey’s solution
27
to obtain the time evolution of ρ˜22 we find
g(2) (τ) = 1− e−η|τ | ×
(
cos (µ |τ |) + η
µ
× sin (µ |τ |)
)
, (2.31)
where
η =
1
2
×
(
1
T1
+
1
T2
)
,
µ =
√
Ω− 1
4
×
(
1
T1
− 1
T2
)2
.
Figure 2.4 illustrates the three common interferometer types (used also in this
thesis) to measure first- and second-order coherence, together with examplary
curves for a strongly resonantly (∆ = 0) driven two-level system. Panel a shows
a cavity with a tunable mirror: a Fabry-Perot interferometer to record spectra.
Panel b displays a Michelson interferometer for first-order coherence measure-
ments, whereas panel c shows a Hanbury-Brown and Twiss interferometer for
intensity correlation measurements. The setups will be discussed in more detail
in section 3.4. The oscillations in the g(1) and g(2) curves in panels b and c are due
to Rabi oscillations of the two-level system. The modulation of the fluorescence
intensity due to the coherent oscillations gives rise to a three-peaked spectrum,
the Mollow triplet - see panel a, where the sidebands are separated from the cen-
tral peak by the Rabi frequency. Note that in the spectrum we have omitted the
elastic contribution for simplicity.
28
QD RF
LO 
+80 MHz
Spectrum
analyser
tΔt
λ/2
τ
-10 -5 0 5 10
Time delay (Γτ)
-10 -5 0 5 10
Time delay (Γτ)
-6 -4 -2 0 2 4 6
RF spectrum 
-1
(2πΓ t)
a b c
Figure 2.4: Schematics of optical interferometers to measure first- and second-order
coherence (top) with examples of corresponding spectra/coherences (bottom). The
bottom curves are simulations for a coherent two-level system and the abscissa is made
dimensionless by normalising to Γ = 1/T1. a, Fabry-Perot interferometer. b, Michelson
interferometer. c, Hanbury-Brown and Twiss interferometer.
29
Chapter 3
Experimental setup and
techniques
In this chapter we introduce the experimental setup and experimental techniques
used in the following chapters. In the flow-chart 3.1 we attempt to give an
overview and a quick summary. The diagram represents the techniques (and
equipment) available in the lab. The confocal microscope with the QD sample
forms the central part of the setup and will be described first. We briefly discuss
the basics of confocal microscopy and microscope design and then follow the de-
velopment from QD wafer to a Schottky diode device where the QD charge can
be controlled using external electric field. Optical characterisation using above
bandgap optical excitation as well as resonant spectroscopy are introduced in this
context and we discuss the use of numerical aperture enhancing solid immersion
lenses (SILs) to improve photon extraction. The properties and working prin-
ciples of the optical interferometry techniques employed in the detection of QD
fluorescence are covered at the end of the chapter.
3.1 Confocal microscopy
The central piece of the experimental apparatus consists of a confocal microscope
that allows optical addressing and collecting the fluorescence from single QDs,
which are held at, or close to, the liquid helium temperature of 4.2 K. Two
30
Figure 3.1: Summary of experimental setup and techniques. The optical excitation
column on the left displays which laser wavelengths are available and what they are used
for. The detection column summarises detection instruments and techniques, grouped
according to the nature of the detection. cw: continuous wave.
confocal microscope setups were used in this thesis, one in conjunction with a
liquid helium flow cryostat (Oxford Instruments MicrostatHires) and the other
one attached to a liquid helium bath cryostat (similar to the attocube attoliquid
system). The flow cryostat was used for QD wafer and sample characterisation,
while long-term experiments were all performed with the bath cryostat system.
However, the working principle is identical in both cases, so we will start with
the general features.
Figure 3.2 provides a sketch of a confocal microscope setup. First, the microscope
head and the sample need independent motional degrees of freedom. In the case
of the flow cryostat setup, the sample is kept stationary, while the microscope sits
on three stacked linear direct drive stages (PI 405.DG for lateral displacement, PI
451 for lifting). For the bath cryostat, the microscope is attached to the cryostat
and the sample is positioned on piezoelectric slip-stick xyz-positioners (attocube
ANPxyz101). The microscope provides four degrees of freedom in each arm to
allow for accurate alignment of the excitation beam and efficient fibre coupling
in the collection arm. As the microscopes are used for resonant spectroscopy, we
31
SMF
SMF
90:10 BS
Sample
indendendent
positioning
Polarisation
optics
Objective
lens
Figure 3.2: Confocal microscope with sample. Sample and microscope head can be
moved independently from each other so that different positions on the sample surface
are in focus. The confocal points in excitation (horizontal) and collection (vertical)
arm are given by single-mode optical fibres (SMF). A beamsplitter (BS) with a high
transmission to reflection ratio combines and separates the paths
cannot use dichroic mirrors to combine/separate excitation and collection paths.
In order to minimise photon loss in the collection path (vertical arm in Fig.
3.2) the beamsplitter is strongly unbalanced, typically with a transmission to
reflection ratio of ∼10:1. Single-mode fibres (SMF) are used for input and output
to provide single mode beams and rejection of background light. Polarisers and
waveplates can be inserted into each arm. In the case of the flow cryostat setup,
the objective lens (Mitutoyo 50x M Plan APO NIR) is kept at room temperature
outside the cryostat housing and at liquid helium temperatures inside the bath
cryostat (Thorlabs C110TME-B or C240TME-B). The bath cryostat system is
illustrated in greater detail in Fig. 3.3. In panel a we identify the confocal
microscope elements discussed before (see sketch in Fig. 3.2) and highlight a few
additional features. Panel b shows the part of the microscope that is kept at
cryogenic temperatures during experiments, namely objective lens, sample, and
positioners. To avoid mechanical failure of the objective lens during cooldown and
at cold, we use single-piece aspheric lenses. The numerical aperture (NA) of 0.4
and 0.5 is chosen to match the light extraction with a zirconia super-hemispherical
solid immersion lens (discussed later this chapter). Progressive close-ups of the
sample in panels c and d show both sets of gates and a zirconia solid immersion
lens glued to the sample surface.
32
Figure 3.3: a, Photograph of helium bath cryostat insert and microscope when pulled
out of cryostat. In comparison to the sketch in Fig. 3.2 an imaging camera (Watech
120N+) is included here. Rotational and translational elements are highlighted in both
microscope arms. A photodiode (Thorlabs SM1PD1A) measures transmitted power of
the excitation laser, enabling power calibration and stabilisation. The cryostat insert
contains the objective lens, sample and sample positioners, held together in a 30mm
cage system (see b). It is sealed off from the outside and provides optical access through
a BK-7 glass window at the top. b, Photograph of bottom end of the microscope cage
system when pulled out of the insert. The location of objective lens, sample and posi-
tioners are indicated by the arrows. Teflon strips (white) keep electric connections in
place, out of the way of the optical beam and the outside edges of the cage. In order to
cool down the insert (with sample etc.) is first vacuum pumped to ∼ 10−5 mbar and
then filled with 20-25 mbar of helium gas. This helium gas provides heat exchange with
the outside and cools sample and positioners to liquid helium temperatures when the
insert is half-way immersed in helium in the bath cryostat. c, Close-up of a contacted
QD sample with zirconia SIL. Gold wirebonds connect to the Schottky contacts on the
top-right side. On the bottom-left, the bigger Ohmic contact are visible. d, Close-up
of the Schottky contacts and bondpads. The light blue structure stems from the semi-
transparent titanium gate that forms the Schottky contact. At the top, an additional
layer of titanium and gold provides pads for wirebonding.
33
3.2 QD Sample
3.2.1 General considerations for sample design
The QD electronic level structure offers transitions to excited states that can be
optically addressed and scatter single photons in response. It is via these sin-
gle photons that information is obtained, e.g. about the spin state, and acting
as flying quantum bits, entanglement can be generated or transferred to equiv-
alent systems. As such the QD sample design is driven by efforts to maximise
photon extraction. This is particularly crucial as the QDs are buried in a high
refractive index semiconductor (GaAs, n∼3.5) and total internal reflection at
the semiconductor-air interface limits outcoupling of photons optics to ∼ 2%.
Approaches to ameliorate this problem are nicely illustrated in Ref. [77]. One
approach is to manipulate the local density of optical states, such that light is
preferentially emitted into spatial modes that allow for good collection. This can
be achieved by embedding the quantum dots into microcavities during semicon-
ductor growth and post-processing. An alternative approach uses geometric op-
tics, mainly relying on solid immersion lenses (SILs). Here the emitter properties
are not modified; the strategy is to minimise total internal reflection by employ-
ing spherical interfaces. Cavity solutions can be processing-intensive, while the
second approach is very simple and offers broadband improvements at the same
time. The SIL approach is used throughout this thesis and will be explored in
more detail in section 3.2.4 of this chapter.
To go from a QD sample to a useful QD device we have to add an element of con-
trol. Attempting to control individual spin qubits requires the ability to control
the charge state of individual QDs first, making this a very desirable feature. We
also note that using QDs as single-photon sources under resonant excitation may
not be straightforward with samples without charge-control [78]. Charge-state
control can be partly achieved by appropriate doping [52], i.e. by incorporating
additional charges closeby and hoping these will be trapped in QDs. This is a
probabilistic process, which does not guarantee QD charging stability. A more
flexible solution was pioneered by Warburton et al. [11]. Here, the QD is placed
between two electronic gates, one of which acts as electron reservoir and the po-
34
Si doped GaAs
GaAs SSIL
~mm diameter
Ti top gate
~nm thickness
coupled quantum
dots
air gap, sacrificial layer
~ micron thickness
Distributed Bragg
Reflector
Ohmic Au contact
to back gate
AlGaAs
AlAs
...
GaAs 
AlGaAs
back gate
InAs
spacer
GaAs
AlGaAs
AlAs
InGaAs
Si doped GaAs
T
Capping (90 nm)
Tunnel barrier (50 nm)
Capping (10 nm)
QDs (2 monolayers)
Tunnel barrier (35 nm)
Back gate (40 nm)
Spacer (130 nm, 73.5 nm)
DBR (20x 82.4 nm,73.5 nm)
Buffer and substrate
Si doped GaAs
Zr SIL
Top gate (6nm)
Capping (90nm)
Barrier (50nm)
QDs (2 monolayers)
Tunnel barrier (35nm)
Back gate (40nm)
Spacer (130nm)
DBR (20x 82.4nm,73.5nm)
Ohmic AuGeNi 
contact to back gate
AlGaAs
AlAs
x20
GaAs 
AlGaAs
InAs
Buffer and substrate
Zr SIL
Ohmic AuGeNi 
contact to back gate
Si doped GaAs
Top gate (6nm)
Capping (90nm)
Barrier (50nm)
QDs (2 monolayers)
Tunnel barrier (35nm)
Back gate (40nm)
Spacer (130nm)
DBR (20x 82.4nm,73.5nm)
Buffer and substrate
GaAs
AlGaAs
AlAs
InGaAs
Zr SIL
Ohmic AuGeNi 
contact to back gate
Ti Top gate (6nm)
Capping (90nm)
Barrier (50nm)
QDs (2 monolayers)
Tunnel barrier (35nm)
Back gate (40nm)
Spacer (130nm)
DBR (20x 82.4nm,73.5nm)
Schottky diode
Figure 3.4: QD sample structure. Left: Colour-coded structure of the QD wafer ‘Chef
2’. Layer thicknesses are indicated to the right of the sketch (not to scale). The Si
doping concentration in the back gate is nominally 4 · 10−18 cm−3. DBR: distributed
bragg reflector. Right: Schottky diode structure with zirconia solid-immersion lens
(Zr SIL). An ohmic contact is formed with the back gate through GaAs etching and
evaporating and annealing gold-germanium-nickel (AuGeNi) on the sample. The top
gate is a Schottky contact made of a thin titanium film.
tential difference between them determines the potential of QD electronic levels
with the respect to the Fermi level.
3.2.2 The ‘Chef 2’ sample
We will illustrate the concepts introduced above on the example of the sample
used throughout most of this thesis. The wafer was grown by Maxime Hugues
at the National Centre for III-IV Technologies in Sheffield and dubbed ‘Chef 2’.
A schematic of sample structure including layer thicknesses is shown on the left-
hand side of Fig. 3.4. The right-hand side displays a sketch of the device after
processing, highlighting the Schottky diode section. A zirconia solid immersion
lens glued on to the sample surface enhances focussing and photon collection.
Controlling the potential difference between top and bottom gate allows us to
control the charge state of the QDs inbetween. Figure 3.5 illustrates the princi-
ple of this charge tuning (for electrons) by looking at the electronic band structure
of the Schottky diode under different bias voltages. z denotes the growth direc-
35
35 nm 13 nm 200 nm
0V
Ef
n+
1e
2e
Schottky
contact
z
GaAs
AlGaAs
InGaAs
V
Ef
n+
1e
2e
Schottky
contact
Figure 3.5: QD Charge control in Schottky diodes. Left: Electronic band structure
of the QD diode heterostructure under zero applied bias. The Fermi level (Ef) is below
the first QD electronic state, so the QD is uncharged. Right: Under forward bias the
first QD level shifts below the Fermi energy and an electron tunnels in from the back
contact.
tion. The layer materials are indicated in the same colour code as before. The
back contact is GaAs heavily n-type doped with electron-donating silicon atoms
replacing gallium in GaAs. The Fermi level Ef is then pinned to just below
the GaAs conduction band edge in the n-doped layer and, at zero applied bias,
halfway in between the conduction and valence band at the Schottky contact.
The electronic levels for having the QD occupied with 1 or 2 electrons are indi-
cated. Applying a gate voltage effectively shifts the QD energy levels with respect
to the Fermi energy. Conduction band electrons from the Fermi sea in the back
contact can only tunnel into the quantum dot if unoccupied states are below or
within kT of the Fermi energy Ef . Due to its small size, and consequently a small
capacitance, and its discrete energy levels a charging energy ∆Echarging is needed
to place another electron into the QD [79]. For liquid helium temperatures (∼
4 K) ∆Echarging  kT , such that the number of electrons in the dot is well de-
fined as long as the Fermi energy is not aligned with the QD level (within kT
of the energy level). This effect is known as quantum Coulomb blockade. Apply-
ing a forward bias, electrons are added one by one. These effects can be clearly
seen in photoluminescence studies, where the energy of emitted photons changes
abruptly as the number of resident electrons is changed [11; 80]. The electric field
36
across the diode also shifts the QD exciton energy levels via the DC Stark effect,
which was introduced earlier (section 1.3.3). A bias-dependent map of the QD
photoluminescence from a ‘Chef 2’ device is discussed later in this chapter.
From the band structure diagrams in Fig. 3.5 we can also understand the function
of the other layers in the diode. Going up in z-direction from the back contact,
the following GaAs layer provides the tunnel barrier between the Fermi sea in
the back contact and the QD layer (cf. labels in Fig. 3.4). The barrier thick-
ness determines the tunneling time for an electron in and out of the QD. Given
the exponential tunneling time dependence on thickness this can stretch from
sub-nanosecond to second timescales. For a different sample with a nominally
identical barrier thickness of 35 nm a tunnelling time ≤ 0.1 s was deduced from
spin relaxation measurements [26]. For the ‘Chef 2’ sample, a similar measure-
ment suggests microsecond tunneling times. This difference may be due to the
dopant diffusion in the back contact which varies with concentration. Next, the
InAs QD layer is covered with 10 nm of GaAs 1. Following the GaAs cap, 50
nm of AlGaAs form a barrier (due to the larger bandgap) for carriers that could
tunnel from the Schottky contact into the QD layer. Finally, the top GaAs layer
provides a cap that prevents the AlGaAs from oxidising.
The distributed Bragg reflector (DBR, see Fig. 3.4) consists of 20 layers of
GaAs/AlGaAs with thicknesses such that reflection in the spectral region around
970-980 nm is enhanced. The relative position of the QD layer with respect to
the DBR places the QDs in an antinode of the reflected field.
3.2.3 QD device processing
Once a sample structure is designed, growth is carried out externally. For the
duration of this PhD work samples were purchased from the EPSRC National
Centre for III-V Technologies in Sheffield. We are grateful to Maxime Hugues
and Edmund Clarke who performed the growth there.
1The influence of this capping layer thickness has been subject of a recent study by Houel
et al. [81]. The authors observe strong (several linewidths) fluctuations of the QD resonance
frequency for 30 nm capping thickness, while this feature is strongly reduced for a capping
thickness of 150 nm. The spectral jumps are attributed to charges trapped at the interface of
capping layer and the following AlGaAs barrier. The capping layer thickness of 10 nm in our
devices is chosen to avoid bound states at the QD energy levels at the GaAs/AlGaAs interface.
37
Figure 3.6: Schematic of QD photoluminescence through above bandgap excitation.
When the energy of the exciting laser is greater than the GaAs bandgap, electrons are
promoted from the valence to conductance band, leaving behind holes. These electron-
hole pairs quickly diffuse into the lowest available energy configuration, e.g. the s-shell
of a quantum dot, and recombine radiatively.
The QD density required for different applications (e.g. QD lasers vs. single QD
spectroscopy) varies widely, and achieving a specific uniform density in growth
requires careful calibration. For the low density samples we need for our research
the rotation stop method is employed to provide some flexibility on the QD
density. With this method the standard rotation of the wafer during growth
is interrupted at the start of the InAs deposition, such that the InAs density
and consequently the QD nucleation density varies across the two-inch wafer.
Typically it varies from zero QDs in several mm2 at one edge to many tens per
µm2 or higher at the opposite end. In between there is a strip where the density
is appropriate for single QD research. Hence the first step in producing a good
QD device is to identify the region of appropriate density.
3.2.3.1 QD photoluminescence
Characterisation of new wafers and sample processing relies on the ability to
generate and detect QD fluorescence in a straightforward manner. Fortunately
there is a very convenient method: photoluminescence. Excitons can be gener-
ated in abundance in the direct bandgap material GaAs by excitation at above
bandgap energies. The electron-hole pairs relax quickly into the lowest available
energy configuration and recombine radiatively. If quantum dots happen to be in
the spatial vicinty, excitons can fall into these potential traps, giving rise to QD
38
in
cr
ea
si
ng
 d
en
si
ty
A
B
C
D
E
A
B
CDE
5
0
m
m
100 μm
Figure 3.7: Investigation of QD density for a wafer grown with the rotation stop
method. The top left image is taken from the wafer delivery note from the EPSRC
National Centre for III-V Technologies in Sheffield. It is annotated to link the spatially
resolved photoluminescence maps shown to the right and below to locations on the
wafer.
photoluminescence on recombination (see Fig. 3.6). In our case, we employ a
diode laser at 785 nm. Laser scattering and luminescence from the GaAs and the
wetting layer quantum well is filtered out with a longpass filter (e.g. Thorlabs
FEL0900)1.
3.2.3.2 Sample density check
Investigating the QD density starts by cleaving a ∼ 6 mm wide strip off the
wafer, either along the major or the minor wafer axis, roughly following the
expected density gradient. After cooling down the strip in the flow cryostat
1Alternatively, exciting the InAs wetting layer provides an efficient venue for QD photolu-
minescence generation.
39
we employ above bandgap excitation and deliberately decollimate the 785 nm
laser before the microscope objective, such that a large sample area (>100 x
100 µm2) is illuminated. Imaging the sample surface through a 900 nm longpass
filter reveals spatially localised fluorescence from individual QDs. Figure 3.7
displays on the top left a room temperature photoluminescence map supplied by
EPSRC National Centre for III-V Technologies. The wavelength of the maximum
intensity peak of the photoluminescence spectrum is colour-coded as function
of position on the wafer. The map is annotated to illustrate the QD density
measurements performed in our lab: The direction of the QD density gradient
is indicated by the white arrow, the darker area shows the sample strip that
was cleaved for analysis and the lettered dots show the approximate position of
selected spatial luminescence maps. Points A and E are approximately 20 mm
apart. For location E the QD density is in the region of 10 QDs/µm2. At this
density, individual QDs can still be easily addressed individually using resonant
excitation, as they emit at different wavelengths. However, experiments showed
that the QD optical properties suffered. Spectral jumps and strong spectral
diffusion (evident in broad absorption lineshapes) were typically observed for
the QDs investigated (∼ a dozen). For densities of one to a few QDs per µm2
(comparable to location D in Fig. 3.7) these problems are less severe, here about
half of the QDs exhibit jumps and diffusion. QD samples with densities below
1/µm2 are typically clean, with absorption linewidths close to the radiative limit
at low excitation power and hence preferred for measurements going beyond above
bandgap photoluminescence.
3.2.3.3 Cleanroom processing
Once the location of suitable QD density on the wafer is established, pieces of
approximately 6 x 5 mm2 size are cleaved and prepared for processing in the
cleanroom of the Semiconductor Physics group at the Cavendish Laboratory.
The sample size is determined by the size of mask features. A margin of about 1
mm is necessary to avoid problems with photoresist edge beads. The processing
steps mostly follow standard procedures for high electron mobility Hall-bars. We
give a brief outline of the procedure here.
40
1. Ohmic contacts. The Ohmic contacts require annealing, so they have to
be done before other metal gates. After cleaning, the positive photoresist
Shipley 1813 is deposited on the sample. Spinning the sample (5500 RPM
for 30 seconds) distributes the photoresist evenly across the sample (save
the edge beads), with a thickness of ∼ 1.3 µm. After baking for 60 seconds
at 115◦ C to harden the resist, the sample is exposed under the appropriate
mask section (the one containing the ohmic contact shape). Developing
uses MF-319 for about 2.5 minutes. To get a contact to the buried n-doped
layer, we etch the sample (i.e. the now developed area) roughly 3/4 the
distance to the n-doped layer using GaAs etch. The sample is cleaned in
a 10 % HCl solution before thermal evaporation of about 200 nm of gold-
germanium-nickel (AuGeNi). Following the lift-off, annealing is achieved in
a rapid thermal annealer at a temperature of 370 ◦C.
2. Schottky gates. Preparations for the photolithography are similar to be-
fore, but the photoresist is baked at 90◦ C. After exposure, the sample is
soaked in chlorobenzene for 2 minutes; this produces sharper edges in the
development.
Photon collection happens through the Schottky gate, so this needs to be
as thin as possible. The evaporation of nominally 5 nm 1 titanium is per-
formed by cleanroom staff (thanks are due to Graham Winiecki and Melanie
Tribble of the Semiconductor Physics group) using electron beam physical
vapour deposition. Due to the smaller feature size the lift-off process usu-
ally requires sonicating the sample very briefly (and carefully). The shape
of the Schottky pads can be seen in Fig. 3.3 d.
3. Schottky contacts. Wirebonding to a 5 nm thin layer of titanium is more
than challenging. To enable gold ball wirebonding, another deposition of
titanium and gold is needed at the location of the bondpad. This area is
shaded golden in the photograph in Fig. 3.3 d. The photolithography steps
are identical to the ones for the Schottky gates. The evaporation is done
in a thermal evaporator, typically 20 nm of titanium are deposited first,
followed by 40 nm of gold. The additional titanium improves the adhesion
1Usually ends up being ∼ 6 nm thick, occasionally also 30 nm . . .
41
Figure 3.8: I-V characteristics of two QD diodes at room temperature. The left
photograph represents a ‘good’ gate, the one on the right is a ‘bad’ gate.
of the gold layer.
4. Diode behaviour check. Before proceeding to the final steps of mounting
the sample on the chip carrier and wirebonding, the I-V characteristics of
each gate are checked at room temperature with a curve tracer. Figure 3.8
shows two photographs of the curve-tracer in the cleanroom. The horizontal
scale is set to 0.5 V per division, the vertical scale is 50 µA per division. The
photograph on the left displays a ‘good’ I-V curve. In forward bias, current
starts passing at around 0.3 V. The photograph on the right (example of a
bad gate) shows some Ohmic character, indicating leakage current between
the gates.
5. Packaging. Sample processing is completed by mounting the sample on
a leadless ceramic chip carrier (using silver paste as glue) and wirebonding
the working gates to the metalised castellations on the chip carrier using
42
a goldball bonder1. The wirebonds for Schottky and Ohmic contacts are
visible in Fig. 3.3.
3.2.3.4 Gate check at 4K
It is advisable to check the QD response to biasing the gates in the flow cryostat
before placing it (semi-permanently) in the bath cryostat. This should show that
devices with ‘good’ I-V curves (see Fig. 3.8 left) influence radiative recombination
rates in accordance with the band diagram in Fig. 3.5. Figure 3.9 displays six
spatial photoluminescence maps under increasing bias. Starting in reverse bias,
we can make out two clearly separated areas on the sample surface (red lines show
borders as guide to the eye). In reverse bias excitons ionise (electrons tunneling
into the back gate) faster than radiative recombination in QDs can take place,
so we do not expect luminescence. The dark area in the map for -150 mV bias
is covered by the titanium gate, whereas QD emission is visible from regions not
covered by the gate. Under increasing forward bias, QD emission is visible in the
gated area indicating that radiative recombination occurs faster than tunneling
timescales, whereas the ungated area looks unchanged. For this sample, emission
is most intense around 450 mV bias; here most QDs are occupied by one or two
electrons. At high forward bias, current starts passing between the n-doped layer
and the Schottky gate and emission quenches.
3.2.4 Solid immersion lenses
The problem of efficient light extraction from QDs was already mentioned at
the start of this section, together with the approach employed: solid immersion
lenses. The solid immersion microscope was introduced by Mansfield and Kino
in 1990 [82]. In analogy to liquid immersion, e.g. in oil, the idea is to fill the
objective space with a material of high refractive index nSIL, and thus increase
the numerical aperture of the microscope to NA = nSILsin(θ), where θ is the
angular semi-aperture in the objective space. For optical studies of QDs there
1The gold wire can easily rip off the bondpad (and part of the sample surface) instead of
sticking to it, rendering a week or two (and a precious piece of sample) useless. Patience and
careful calibration of the bonder parameters are recommended.
43
-150mV 150mV 350mV
450mV550mV750mV
100 μm
Figure 3.9: Influence of bias across the QD diode on the photoluminescence intensity
under above bandgap excitation. The applied bias is displayed in the top left corner
of each image, with positive voltages corresponding to forward bias. The edges of a
Schottky gate are indicated by red lines. Only areas under the Schottky gate react to
the bias.
44
Figure 3.10: Illustration of solid immersion lens geometries, reproduced from The
solid immersion lens by Kino [83]. The supersphere geometry is also known as Weier-
strass geometry.
are two beneficial effect: improved focusing and light extraction.
In order to fill the objective space fully, spherical surfaces separating the immer-
sion material from objective lens are desirable. Focal points form at two locations
in spheres of high refractive index, so SILs come in two standard geometries which
are illustrated in Fig. 3.10. For features directly underneath the planar surface of
a hemispherical SIL the lateral magnification increases by nSIL due to the wave-
length being λSIL = λ0/nSIL, as does the NA, while the size of the focal spot is
reduced by a factor 1/nSIL compared to the vacuum case. Super-hemispherical
SILs (SSILs) ideally increase the magnification and the NA by n2SIL (with nSIL
being the upper limit of the NA) while the spot size should decrease by 1/n2SIL.
Beating the (vacuum) diffraction limit is possible with SILs and this has been
demonstrated in a number of works [84], see Ref. [85] for a review.
The important advantage of using SILs for QDs is a reduction in total internal
reflection, where an order of magnitude increase in photon extraction is predicted
[77; 86], e.g. going from 1-2 percent without SIL to 30 percent with a GaAs SSIL.
Successful implementations of SILs for QD spectroscopy and other nanophotonic
applications have been reported in a number of works [87; 88; 89]. Using a high-
index glass SIL and a GaA SSIL, respectively, a resolution of 350 nm (for λ=950
nm light) [90] and an increase in transmission contrast by a factor of seven [91]
45
were reported.
In practice, the achievable reduction in focal size and the NA enhancement de-
pend on the optical quality of the SIL and the contact to the object. Simulations
suggest that the size of any air gap between the SIL and the target material
strongly influences focusing abilities [83; 88], as the evanescent part of the fo-
cused spot does not contribute to the intensity at the object location anymore.
For less than 10 % deviation from the ideal case, the air gap should be on the
order of λ/5. For some materials it has been possible to integrate the SIL directly
into the device (most notably for diamond colour centres, see Refs. [92; 93] for
example), thus eliminating the challenge of creating good optical contact, but
this has not been achieved yet for GaAs.
However, even for imperfect contact with the sample SILs, and SSILs in partic-
ular, funnel QD emission that made it out of the semiconductor and reduce the
NA necessary to collect it. It should also be pointed out that SSILs are highly
achromatic [94] while SILs do not suffer from this.
In this thesis we employ 2 mm diameter SSILs made from cubic Zirconia. The
refractive index of n ∼ 2.17 is higher than that of most common ‘optics’ materi-
als. In terms of refractive index, GaAs SSILs would be preferable, but these are
not readily available. The Zirconia SSILs are glued on the sample surface using a
transparent mounting compound 1. The contact to the sample surface is checked
by focusing a laser through the SIL and observing the reflection from the focal
spot at the sample/SIL interface. Two distinct reflections are visible for air gaps
on the order of 1 µm and bigger. Airgaps of a few hundred nm and smaller cannot
be measured easily. Typically we see an enhancement in collection efficiency by
a factor 5-10 in the flow cryostat system with an objective NA of 0.42.
Figure 3.11 illustrates the concepts of increased magnification and imaging with
SILs with experimental (and hence flawed) data. A Zirconia SSIL and a GaAs
SSIL are mounted on the same test sample and we image QD fluorescence in the
flow cryostat setup with a 0.42 NA objective lens (Mitutoyo M Plan Apo NIR).
The QDs can be approximated as point-sources, so imaging them is equivalent
to measuring the point-spread-function (PSF) of the microscope. Fig. 3.11a dis-
1Obtained from the glass workshop in the Cavendish Laboratory. The exact product is not
known, but it is similar (or identical) to the ‘KLEER-MOUNT’ acrylic resin from MetPrep.
46
plays this effective PSF for a QD not covered by either SIL. The linecuts through
the image reveal a full-width-at-half-maximum (FWHM) of 8 pixels, which cor-
responds to a physical width of ∼ 1.4 µm. For a diffraction limited microscope
and a point-source emitting at 950nm we would expect a PSF of ∼ 1.15 µm.
The discrepancy is probably due to the fact that the microscope objective is not
corrected for focussing through the glass cover of the flow cryostat (0.5 mm thick-
ness). Figs. 3.11b and 3.11c show images for QDs under a Zirconia and a GaAs
SSIL, respectively. Using SSILs the image should be magnified by n2SIL, so we
divide the measured FWHM in pixels by n2SIL to extract the PSF. The results
are summarised in Fig. 3.11d and compared to the ideal case. We are quite a
bit off the ideal case (about a factor of three). Reasons for this are likely to be
the optical contact at the SIL/sample interface where an air gap of a few µm
was measured, and chromatic dispersion1. However, in both cases we still reach
a PSF equivalent to that of an ideal system (without SIL, but) with an NA of 1.
We note that not reaching the diffraction limit also implies that the microscope
alignment is less sensitive to movement of the sample with respect to the micro-
scope - an advantage in liquid helium cryostats which need refilling or continous
flow.
The addition of a SSIL with good optical contact to the gated sample con-
cludes the device processing and preparation part. Finished samples can then be
cooled down in the bath cryostat for resonant studies on single QDs. The relevant
techniques are described in the following section.
3.3 Single QD spectroscopy
Three optical techniques are used for spectroscopy and time-resolved measure-
ments: photoluminescence (PL), differential transmission/reflection (DT/DR)
and resonance fluorescence. Characterisation of the optical transitions of indi-
vidual QDs and QDMs employs all three techniques, where each plays a different
role. The main results of this thesis rely on photon coherence properties and high
measurement bandwidth and are obtained with resonance fluorescence.
1Sampling over a large number of QDs would give a more representative picture, as we
would sample over a range of emission wavelengths at the same time.
47
Figure 3.11: Point-spread function (PSF) of confocal microscope with and without
SIL. The PSF is obtained by imaging the fluorescence from a single QD onto a camera
and fitting a Gaussian to the intensity profile in two orthogonal directions. a, PSF for
imaging through air with a 0.42 NA objective. b, PSF for imaging through a zirconia
SIL in Weierstrass geometry (SSIL) with a 0.42 NA objective. a, PSF for imaging
through a GaAs SIL in Weierstrass geometry with a 0.42 NA objective. d, Plot of the
PSF versus the refractive index of the immersing medium. The black points show the
data from a-c, the solid red line displays the theoretical PSF for a diffraction-limited
setup (0.42 NA). The black dashed line shows the theoretical curve for a NA=1. We
assume an emission wavelength of 950 nm here.
48
3.3.1 Gated photoluminescence
The process of photogeneration by above bandgap excitation was covered in
3.2.3.1. With the sample processed into a Schottky diode optical transitions
of different charge states can be identified. This is illustrated in Fig. 3.12 where
we present the photoluminescence/gate voltage map of a single QD. The pho-
toluminescence is spectrally dispersed in a 0.75 m monochromator (Princeton
Instruments SP2750) using a 1200 groves/mm grating and detected with a liquid
nitrogen-cooled charge coupled detector. The electric field across the QD corre-
sponding to the applied bias is given by the top abscissa. Operating far below
saturation three distinct transitions are identified: the neutral exciton transition
X0 and the positively and negatively charged trion transitions X1+ and X1−.
The charges involved in the recombination are indicated next to the data. This
map can be likened to an optical fingerprint of a QD as charging points, emis-
sion energies and splittings between transitions vary for each QD. However, some
features are consistent for all QDs in our samples. Most prominently, the X1−
transition is redshifted typically by 4-4.5nm (∼ 1.3 THz or 5.5 meV) with respect
to the X0 transition. The change of emission wavelength as function of the gate
voltage is due to the DC Stark shift. It is worth noting that here we do not
see a clear transition in gate voltage between different charge ground states, and
the optical transitions overlap considerably in applied bias. The reason for this
overlap is the slow tunneling time for charges in the QD: this may take place
on a many µs timescale, much slower than radiative recombination. Hence the
QD can be optically charged, and we see e.g. X1− emission at a bias where the
QD is empty in its ground state. PL spectroscopy is great in its simplicity, but
has several drawbacks: The resolution is limited to the spectrometer resolution
(∼ 30 µeV for a 1800 groves/mm grating) while the transition linewidth of QD
transitions is typically a few 100 MHz (∼ 1µeV).
49
electron
hole
vacuum
0 250 500
942
944
946
948
950
952
X
0
X
1+
Gate voltage (mV) 
W
a
v
e
le
n
g
th
 (
n
m
)
X
1-
-10 0 10 20
1316
1314
1312
1310
1308
1306
1304
E
n
e
rg
y
 (
m
e
V
)
Electric field (kV/cm)
Figure 3.12: Photoluminescence map of a single QD. Sequential charging of the QD
with gate voltage is visible. The linear shift of the emission energy with gate voltage
is due to the Stark effect.
3.3.2 Differential transmission and reflection
Differential transmission and reflection (DT/DR) in the context of QD spec-
troscopy refers to a homodyne detection technique of resonant QD scattering
which offers very high spectral resolution while being experimentally robust at
the same time. Here we will only outline the method and its range of application;
for a comprehensive description see Ref. [95]. The simplifying assumptions made
there do apply in our case. DT and DR use interference of the laser radiation
with the resonantly scattered field from the QD (homodyning) to pick up the
weak QD scattering. If we consider a laser interacting resonantly with a single
dipole transition, such as the X0 transition in the QD, we have the electric field
of the laser EL, the coherent (Rayleigh) scattering from the dipole Ecoh and the
incoherent scattering Einc. Putting the fields on a photodetector in transmission
geometry we measure the intensity of the fields
Idetected = |EL + Ecoh + Einc|2
=IL + Icoh + Iinc + 2< (EL · Ecoh)
+ 2< (EL · Einc) + 2< (Ecoh · Einc) . (3.1)
50
In a typical QD experimental system (in the absence of cavities) coupling the
laser field to the QD transition is inefficient, i.e. EL  Ecoh,inc. In this case
the second-order terms Icoh and Iinc are smaller than fluctuations in the laser
intensity and can be neglected. For the cross-terms (the interference terms) we
have to take phase relations and the detector bandwidth into account. Fields
with random phase fluctuations faster than the detector bandwidth will average
to zero and the interference terms will not contribute to measured intensity. This
applies to the cross-term (EL · Einc). However, the interference of laser and co-
herent scattering should be stable due to the fixed phase between them. Still
this term is quite small compared to the laser intensity. To obtain a clean signal
we modulate the cross-term by fast modulation of the gate voltage (few KHz)
such that the QD transition is modulated in and out of resonance using the Stark
shift. The laser intensity term is not affected. Phase-sensitive detection of the
photodetector signal removes the laser intensity term and allows us to measure
the absorption profile of the transition by scanning the laser frequency (or gate
voltage) across the QD resonance.
Experimentally, the photodetector for DT (Thorlabs FDS-1010) is placed directly
underneath the QD sample in the cryostat. For DR we use a fibre-coupled pho-
todetector at room temperature (Femto photoreceiver EO-200). In both cases the
photocurrent is converted to voltage and amplified by typically 108V/A before
demodulation in the Lock-In amplifier (Stanford Research 830 DSP). The spec-
tral resolution is only limited by the spectral stability of the resonant laser (<2
MHz in our case) which is much smaller than the QD linewidth. DT and DR are
homodyne spectral measurements. The measurement bandwidth is determined
by the gate modulation speed which sets a lower bound to the integration time
constant of the Lock-In amplifier and the signal-to-noise given amplifier and lock-
in amplifier specifications. In our setup the modulation frequency is typically ∼
2 KHz with integration times of 10-100 ms.
First QD spectroscopy measurements using this technique were reported in Refs.
[96; 97]. Since then it has become a well-used and important technique in the
quantum dot community, enabling progress in the coherent control of single and
coupled QDs [42] and metrology [24] by providing a resonant (and hence high
resolution) observation tool. Given all the advantages it remains a purely spec-
51
troscopic technique. QIP applications require access to resonantly generated pho-
tons, i.e. collection of the QD resonance fluorescence only, where the excitation
laser is suppressed.
3.3.3 Resonance fluorescence
Applications in quantum communication and computation rely on high coherence
indistinguishable photons [98], and the ability to create entanglement between
spin qubits and photonic qubits [2]. This can only be achieved with resonant
excitation and collection of the resonance fluorescence.
Following the first calculations of the resonance fluorescence spectrum by Mollow
[70] in 1969 this direct photon generation method was quickly demonstrated in
atomic beams [99; 100; 101]. Resonance fluorescence is now routinely employed
in optical investigations of atomic gases, single trapped ions and atoms [102]. In
these systems an orthogonal geometry for excitation and collection of the fluo-
rescence can be employed, such that the excitation laser does not make it to the
detector. This is crucial as the fluorescence signal is orders of magnitude weaker
than the laser and both are essentially at the same frequency.
For quantum dots the first attempts replicated the orthogonality of excitation and
detection in a semiconductor environment using waveguide structures to confine
the excitation light to one plane [103; 104] or a one-dimensional waveguide [105].
In our experimental setup we excite and collect both QD fluorescence and laser
scattering along the same path (see Fig. 3.3). The QD transition dipole has a
preferential in-plane orientation, independent of excitation orientation and po-
larisation, therefore we excite in a linear polarisation at ∼ 45 degrees to the
transition polarisation and collect in the orthogonal polarisation. This cross-
polarisation ideally removes all laser at the cost of filtering half of the scattered
QD photons. Since the first demonstration of QD resonance fluorescence with
this method [106] the laser suppression has been improved and is on par with
trapped atom systems [107]. The experimental performance of our setup will be
discussed at the beginning of the following Chapter.
The realisation of techniques to access the resonance fluorescence from quantum
dots has enabled advances in the quantum dot field: the strictly resonant excita-
52
tion was shown to reduce photonic decoherence when compared to nonresonant
excitation [108]. Further, taking advantage of the spin-selective QD transitions,
required for spin-based QIP applications, QD resonance fluorescence allowed de-
tailed studies of spin dynamics [26; 109], excitation-induced sideband broadening
[110] and the realization of single-shot spin readout [111], subject of Chapter 7.
The recent demonstration of QD spin-photon entanglement [53; 65] is a further
key achievement, enabled by the ability to access QD resonance fluorescence.
3.4 Quantum optics in the lab: interferometric
measurement techniques
This section explains the basics and the setups used for interferometric mea-
surements. In analogy to the theoretical chapter we take a look at first-order
coherence (time-resolved and spectrally) and second-order coherence. In Fig. 3.1
we already indicated the experimental capabilities and Fig. 2.4 of Chapter 2
illustrated the three common interferometer types used in this thesis. Below we
discuss our particular setups in more detail.
3.4.1 Spectral measurements
Figure 3.13 shows the setup of the Fabry-Perot interferometer (FPI) on the left
side. The interferometer (Exfo TL Laser Spectrum Analyzer) has a free spectral
range of 30 GHz and can be aligned to a finesse of better than 1000 with a trans-
mission of ∼ 25%. Two single mode optical paths are aligned trough the cavity,
one with maximum finesse for the QD fluorescence and one with moderate finesse
(∼ 150) for a reference laser. The job of the reference laser is to stabilise and scan
the cavity across the QD fluorescence spectrum. To this end the reference laser is
frequency-stabilised before entering the interferometer (linewidth < 2 MHz) and
the transmission through the FPI is detected on a photodiode. The measured
transmission feeds back to control the distance between the cavity mirrors, hence
fixing the cavity resonance relative to the reference laser frequency. By scanning
the reference laser frequency we scan the cavity resonance at the same time. For
improved stability during scans the reference laser is further power stabilised in
53
Figure 3.13: Setup of Fabry-Perot interferometer (left) and measurement of its resolu-
tion (right). RF: resonance fluorescence, APD: avalanche photodiode, PD: photodiode,
PID: Proportional-Integral-Differential gain controller, SMF: single-mode fibre, MMF:
multi-mode fibre.
the FPI setup. Both stabilisation procedures indicated in Fig. 3.13 employ com-
mercial PID controllers (Standford Research Systems SIM960). The right side
of Fig. 3.13 shows a measurement of the FPI linewidth. Here the signal (QD
RF in the setup) is given by a single mode laser at fixed frequency (linewidth of
a few MHz) and the reference laser scans the cavity resonance across the signal
frequency in steps of 8 MHz. The signal is recorded by a single photon count-
ing avalanche photodiode (APD). The Lorentzian fit to the data, displayed as
red curve, yields a full width at half maximum or equivalently, a spectral resolu-
tion, of 25 MHz. The alignment of the cavity is controlled by three independent
piezoelectric transducers attached to one cavity mirror and as such suffers from
piezo creep. In our setup a finesse greater than 1000 can be sustained for ∼
1 hour (sometimes considerably shorter), then realignment of the piezoelectric
transducers is necessary.
3.4.2 First-order coherence
The first-order coherence of light is related to its spectrum by a Fourier trans-
form, so the information obtained in either measurement should be equivalent.
However, experimentally there are advantages and disadvantages associated with
the specific setups used and both measurement techniques can complement each
54
Figure 3.14: Setup and calibration of the Michelson interferometer. Left: The input
signal (QD resonance fluorescence here) is sent through a single-mode fibre (SMF), split
into stationary and moving paths, recombined on the same beamsplitter and coupled
into a SMF connected to an avalanche photodiode (APD). Two shutters (S1, S2) allow
measurements of the intensity in each path. An additional delay is indicated by dashed
lines. Right: Measurement of the visibility of the first-order coherence for a single-mode
laser. The inset displays part of a fringe measurement and the calibration procedure
used to obtain the data points in the main figure.
other.
The setup for the Michelson interferometer is shown on the left in Fig. 3.14. The
signal (QD RF) is divided into two arms by a beamsplitter (nominally 50:50).
One arm is stationary: the signal is reflected back onto itself and coupled into
a single-mode fibre connected to an APD. The mirror on the other arm sits on
a linear stage (Physik Instrumente DG505.6DG) with 17 nm design resolution
and 150 mm travel such that a delay of up to 1 ns can be reached relative to
the stationary arm. An additional delay can be incorporated, indicated by the
dashed path and mirrors, which covers delay times around 2.5-3 ns.
Measuring the first-order coherence of the signal field requires careful alignment:
the reflected/transmitted paths from stationary and moving arm need to be
collinear and spatially overlapping. Non-collinearity results in a phase varia-
tion across the beam (averaging out interference when integrating over the beam
area), while imperfect beam overlap reduces the amplitude of the interference.
A good method of aligning is to use a high coherence laser as signal and project
the recombined beam onto a surface a long way away and observe the fringe pat-
tern that arises through the spatially varying phase difference. Minimising the
55
number of interference fringes while keeping the beam overlap one should arrive
at the point where the beam can be completely extinguished by slowly scanning
the moving arm. Coupling the beam into the single mode fibre and recording
the APD signal as the moving arm is scanned then shows the expected sinusoidal
intensity dependence. Assuming an (infinitely) coherent input field the visibility
of the interference signal, defined as
V =
Imax − Imin
Imax + Imin
, (3.2)
where Imax(Imin) corresponds to the maximum (minimum) of the intensity, quan-
tifies the instrument response function. A visibility of 1 characterises the ideal
situation.
In our case the linear stage is not completely flat such that scanning the moving
arm across the full range shows a nonlinear variation in beam overlap. The first-
order coherence of the input field is obtained from measuring the fringe visibility
for a number of set delays between the arms. To calibrate the interference signal
at each delay, two mechanical shutters (S1 and S2 in Fig. 3.14) are employed.
First we slowly move the stage and measure the fringe visibility for a number of
periods (5-10), then close shutter S1 and record the intensity in the moving arm,
close shutter S2 to record any background signal (APD dark counts mostly) and
open S1 to record the intensity in the stationary arm. Exemplary data for this
procedure is displayed in the inset on the right of Fig. 3.14
When measuring the first-order coherence of QD resonance fluorescence a se-
quence is added in the calibration where the QD transition is shifted off resonance
via gate control and laser leakage is recorded. Given different intensities I1, I2 in
each arm, the expected visibility for a monochromatic source is
V =
Imax − Imin
Imax + Imin
=
2
√
I1I2
I1 + I2
(3.3)
and the measured visibility is compared to this ideal case. After the calibration
measurement, the stage moves to the next set delay (e.g. a distance of 1 mm)
and the procedure repeats.
The plot on the right in Fig. 3.14 displays the results for a laser (several MHz
56
0 1 0 0 0 2 0 0 0 3 0 0 00 . 7 0
0 . 7 5
0 . 8 0
0 . 8 5
0 . 9 0
0 . 9 5
1 . 0 0
 
 
Vis
ibili
ty o
f g(1
) (τ)
D e l a y  τ ( p s )
A = 0 . 9 9 3 8 ( 5 )
w = 6 6 7 0 ( 4 0 )  p s
21 ( / )2 wA e 
−
Figure 3.15: Coherence meaurement for a free-running external cavity laser (Toptica
DL pro 940). The data at short delays is the same as in Fig. 3.14, the long delay data
is obtained using the additional delay line.
linewidth) as input field. The visibility across the travel range is consistently
around 99% (average 99.3%, standard deviation <0.5%), very close to the ideal
case1. At longer delays, for example when the additional delay stage is incor-
porated, the finite laser coherence becomes apparent. Figure 3.15 displays the
g(1) (τ) measurement for a free-running external cavity diode laser. It is reason-
able to expect a Gaussian decay of laser coherence in this kind of measurement:
The laser linewidth is narrow at short timescales (< 100 KHz for ∼ µs acquisi-
tion), however mechanical instabilities broaden the linewidth on slower timescales.
Roughly speaking, the laser central frequency may wander monotonously in one
direction on sub-second timescales, and randomly when measured longer. The
parameter for a Gaussian fit (blue curve) are shown in the lower left of the figure.
3.4.3 Second-order coherence
The second-order coherence or intensity-correlation function g(2)(τ) describes the
light’s intensity statistics in time. This is equivalent to the arrival time statistics
of photons at a detector. Hence with a perfect detector we only have to record
a sufficiently long timetrace of detector clicks and then evaluate the correlation
1For positive delays > 300 ps the noise is higher, due to decreasing beam overlap at the end
of the travel range of the stage.
57
- 2 - 1 0 1 20 . 0
0 . 5
1 . 0
 
 
Inte
nsit
y (a
.u.)
T i m e  ( n s )
2 x  P e r k i n - E l m e rS P C M - A Q R H - 1 6O r t e c  T A C / M C A
S t a r t  &  S t o p :3  p s  p u l s e ,  9 5 7 n mF W H M :  6 2 0  p s
b
- 1 0 1 20 . 0
0 . 5
1 . 0
 
 
Inte
nsit
y (a
.u.)
T i m e  ( n s )
1  x  P e r k i n - E l m e rS P C M - A Q R H - 1 6O r t e c  T A C / M C A
S t a r t :  e l e c t r i c  p u l s eS t o p :  3  p s  p u l s e ,  8 2 5 n mF W H M :  4 8 0  p s
a
Figure 3.16: Timing resolution of APD single photon counters. a, Response of single
APD to a pulses of a few ps width at 825 nm recorded in a start-stop configuration
with the Ortec TCA/MCA. A modelocked Ti:Sa laser provides optical pulses at ∼ 80
MHz repetition rate. The start trigger is obtained by sending a fraction of the pulsed
laser light to a photodiode with 1.4 GHz bandwidth and amplifying the output. After
heavy attenuation the laser pulsetrain is coupled into the APD whose electrical output
gives the stop signal. The jitter of the electric start pulse is negligible compared to
the APD jitter. b, Response function of two APDs in a Hanbury-Brown and Twiss
configuration. The red line is least-squares fit to the data using a Voigt function. The
Lorentzian contribution to the linewidth is about 1/3.
function, i.e. the detection time differences for all pairs of clicks. For real detec-
tors we run into problems when the features of interest, typically the intensity
correlation function at small delays, occur on timescales smaller than the detec-
tor deadtime. The way around this is to measure g(2)(τ) in a Hanbury-Brown
and Twiss setup [112], a sketch of which was shown in Fig. 2.4. The signal is
split into two parts of (ideally) equal intensity and sent to two detectors. Each
detector may have a large deadtime, but the time difference between two closely
spaced photons can still be resolved if they leave different output ports of the
beamsplitter.
The experimental (optics) setup for g(2)(τ) measurements is less involved than
that for g(1)(τ) or the spectrum S(ν) as no moving parts are involved. The signal
from a single-mode fibre is sent to a free-space beamsplitter and the two output
ports are coupled into single photon detectors via coupling into single-mode fi-
bres. Alternatively, the signal is coupled directly into a fibre-based beamsplitter
(FONT Canada) with the two output fibres plugged into the detector fibre ports.
58
Two systems are available in the lab that convert the detector coincidences into
correlation data. The first one is the time-to-amplitude converter from Ortec
(ORTEC Model 567). It converts pulse detection time differences between the
two input ports (start and stop are fixed) into the amplitude of an ouput pulse.
The pulse height is digitised and stored with a multi-channel analyser (MCA
926M) before being sent to a computer. The timing resolution of the instrument
amounts to ∼ 20 ps for coincidence time delays of 100-200 ns. The second system
is the 8-channel time-to-digital converter ‘quTau’ from quTools. In addition to
recording time-differences between any two of the input channels it also allows
time-tagging events from all channels in the same run (no loss of information).
The timing resolution is specified as 81 ps.
In almost all measurements presented in this thesis we use fibre-coupled Perkin-
Elmer single photon counting modules (SPCM-AQRH16). At 950 nm wavelength
they feature a detection efficiency about 20-25 % and dark count rates ∼ 20 Hz.
The timing jitter of individual modules varies between 450-500 ps, depending on
module, detection wavelength and detection rate [113]. In the Hanbury-Brown
and Twiss configuration this corresponds to a measurement uncertainty of 600-700
ps. These numbers are important as they, together with the timing resolution of
the correlation electronics, represent the system response function and any mea-
sured signal is a convolution of the actual signal with the response function of the
instrument. Two measurements for the system response function are presented
in Fig. 3.16, for single photon detection on the left and two-photon detection on
the right.
59
Chapter 4
A two-level system in a
solid-state environment: resonant
QD spectroscopy
Having covered some theoretical background on the near-resonant light-matter
interaction of two-level systems in Chapter 2, and described our experimental
capabilities in Chapter 3, we now turn to the experimental study of a real system.
Subject of study are the resonantly driven neutral exciton (X0) and the negatively
charged trion (X1−) QD transitions at zero external magnetic field. With a fine-
structure splitting of several GHz for our samples both X0 transitions individually
should behave like a good two-level system. As for the X1− transitions at zero
external magnetic field, we would na¨ıvely expect the two sets of transitions to be
degenerate and hence also follow the predictions for a two-level system.
In this chapter we explore the properties of QD resonance fluorescence. We look
at the standard features expected for two-level systems: saturation behaviour,
power broadening and the first- and second-order correlation functions. In the
low and moderate excitation power regime (around saturation and below) we
find that the X0 transition approximates an ideal two-level system well. In the
Heitler regime we directly observe highly coherent scattering from a single QD;
a first for solid-state single photon emitters. A closer look at the spectral and
statistical properties of the fluorescence reveals deviations from the ideal two-level
60
behaviour. We identify and quantify influences on the QD emission properties:
Spectral diffusion modifies scattering rates, exciton-phonon coupling provides a
dephasing mechanism and finally, the nuclear Overhauser field invalidates the
two-level approximation for the X1− transition.
Some of the material in section 4.1 was published in [114]. Measurements of the
acoustic phonon sidebands (Fig. 4.11) were taken and analysed together with
Claire Le Gall.
4.1 Subnatural linewidth single photons
4.1.1 Performance of the measurement system
The technical challenge to observe the QD resonance fluorescence is to suppress
sufficiently the laser background at the same frequency. Quantifying the perfor-
mance of our experimental system in terms of the laser background suppression
versus the collection and detection efficiency of QD photons has to be the first
step in our investigation.
Our experimental arrangement, discussed in detail in the previous chapter, re-
lies on a confocal microscope where we use the combination of cross-polarisation
and confocal rejection of the collection fiber to suppress laser reflections in the
detection path. The two polarisers are aligned to the dark-field configuration
by sending a strong laser at the QD resonance frequency through the confocal
microscope and minimising the transmission. Starting from the dark-field config-
uration and rotating the collection polariser to maximise transmission we measure
a factor of 107-108 difference in power at the collection fibre end. Conversely, com-
paring the photoluminescence intensity from a saturated X1− transition in the
dark-field configuration to the intensity measured when the collection polariser is
absent, we find about a factor two difference. Translating this suppression of laser
reflection into a QD signal-to-laser background ratio depends on the excitation
Rabi frequency ΩR, on how efficiently we excite the transition and how well we
collect the QD fluorescence.
Figure 4.1 a-d presents the excitation power dependence of the resonance fluores-
cence from QD ‘Claude’ in order to quantify both the collection efficiency of our
61
0
2 0 0 k
4 0 0 k
6 0 0 k
1 0 0
1 k
1 0 k
 
 
RF 
Sig
nal
 (s-1
)
 
Bac
kgr
oun
d (s
-1 )
A P D  d a r k  c o u n t s
0 . 1 1 1 0 0
5 0 0
1 0 0 0
1 5 0 0
 
 SB
R
N o r m .  P o w e r  ( W s a t 2 )
- 1 2 - 8 - 4 0 40
1 0 0 k
2 0 0 k
3 0 0 k
d
c
b
 RF
 Sig
nal
 (s-1
)
 
 
D e t u n i n g  ( G H z )
5 3 0  M H z
a
- 3 6 . 4 - 3 5 . 7 - 3 5 . 00
2 5 0 
 
 
 
Figure 4.1: Analysis of collection efficiency and background contribution of resonance
fluorescence on the example of QD ‘Claude’. a, Integrated photon counts for resonance
fluorescence at the saturation point as a function of laser detuning. The red curve
is a Lorentzian fit to the data (open circles). Inset: Total off-resonance background
counts (blue bars) compared to the APD dark counts alone (yellow bars). b, Integrated
photon detection events when the laser is on resonance with the QD transition (red
triangles) and c, off resonance (black triangles). d, resonance fluorescence signal-to-
background ratio (SBR). At saturation (Ω2sat = 1/ (T1T2)) a value of 1050 is achieved.
The excitation power for b-d is scaled in units of the saturation power.
62
system and the fraction of residual laser background in the detected photons.
In panel a the laser frequency is fixed and photodetection events from a single-
photon counting avalanche photodiode (APD) are recorded (open blue circles) as
a function of gate bias. The bias Stark-shifts the QD transition and is expressed
as frequency detuning here. The full-width at half-maximum of the Lorentzian
fit (red curve) to the data is 530 MHz in linear frequency. Here, the excitation
power corresponds to the saturation point. The inset displays the residual back-
ground for large detunings (blue bars) which includes the contribution of APD
dark counts, measured separately and shown in yellow. In panels b and c the
red and the black triangles display the total photodetection events per second
on- and off-resonance, respectively, as a function of excitation power normalized
to the saturation power. The solid curves represent the theoretically expected
behaviour for the data in each panel and the dashed blue line indicates the mean
dark count level.
For an excitation power at the saturation point (Ω2sat = 1/ (T1T2), where Ω de-
notes the Rabi frequency and T1, T2 lifetime and pure dephasing time) we collect
photons at a rate of 1.2-1.5 MHz into our single-mode optical fiber, while the
laser and detector background contribute <0.01% of the total signal. With a
detector quantum efficiency in the range of 20-25% we register 3 · 105 detector
clicks per second. The blue circles in panel d show the obtained ratio of the res-
onance fluorescence signal to total background (SBR) for each laser power along
with the expected power dependence (blue curve). At a tenth of saturation, the
residual laser contribution to the background falls below the dark count level (30
Hz) limiting the SBR.
Turning to the combined photon collection and detection efficiency, we can esti-
mate this number from the saturation curve and the QD excited state lifetime.
For a lifetime of 600-700 ps, which is typical for our sample we estimate a modest
0.1% total collection and detection efficiency for QD Claude. Starting from the
detector and working towards the QD sample we estimate the losses in the collec-
tion path as follows: detector efficency ∼ 0.2, coupling into the collection fibre ∼
0.5, polarisation suppression ∼ 0.5, polariser transmittance ∼ 0.85, beamsplitter
transmittance ∼ 0.92, transmittance through glass plate/objective lens/SSIL ∼
0.8. Altogether this give 0.0275 ≈ 1/35. The remaining factor then sets the
63
outcoupling efficiency to 3.5 %1.
4.1.2 Saturation and power broadening
4.1.2.1 Saturation of a two-level system
The data in Fig. 4.1 b follows the saturation behaviour we expect for the two-level
system and was already discussed in the theoretical background. As a reminder,
the emission rate ΓRF is given by the excited state population and its lifetime:
ΓRF = ρ22 × 1
T1
=
1
T1
× 1
2
Ω2
(T1T2)
−1 + Ω2
, (4.1)
where we assume resonant excitation at zero detuning. The square of the Rabi
frequency (Ω2) is proportional to laser power, which is the experimentally mea-
surable quantity. The link between excitation power and Rabi frequency is made
through the saturation point; where the emission rate is half of its maximum.
This point can be determined very reproducibly and from Eq. (4.1) we can see
that the saturation power corresponds to Ω2sat = 1/ (T1T2). Using this to rear-
range we have
ΓRF =
1
2T1
× s× (T1T2)
−1
(T1T2)
−1 + s× (T1T2)−1
=
1
2T1
× s
1 + s
, (4.2)
where s is the saturation parameter2.
4.1.2.2 Power broadening
Power broadening of a two-level system is expected to follow
ΓFWHM =
2~
T2
×
√
1 + Ω2T1T2 = ΓFWHM (0)×
√
1 + s, (4.3)
1We note that the DBR stopband for this sample is centred around 980 nm (designed to
be 960 nm) and reflectivity measurements show a intensity difference of factor 2-3 between the
centre of the stopband and the wavelength of QDs we usually work with (950-960 nm). Count
rates above 1 MHz have been measured at longer wavelengths
2One of the assumptions of the two-level model is that lifetime and dephasing time do
not change with excitation power. The former is a solid assumption while the latter is not
necessarily true [110; 115; 116]. However, this effect becomes only apparent for s 1.
64
0 . 0 1 0 . 1 1 1 0 1 0 0
1
3 0 0  M H z
 
 
Line
wid
th (
GH
z)
P o w e r  ( Ω2s a t )
2 5 0  M H z 0
2
4
6
8
1 0
Me
asu
rem
ent
 ba
ndw
idth
 (Hz
*)
b
0 . 0 1 0 . 1 1 1 0 1 0 0
1
Line
wid
th (
GH
z)
P o w e r  ( Ω2s a t )
3 0 0  M H z
2 5 0  M H z 0
2
4
Me
asu
rem
ent
 ba
ndw
idth
 (Hz
*)
a
Figure 4.2: Power broadening of the X0 (a) and X1− (b) transitions of QD ‘Claude’.
Experimental data is shown as black dots. The grey shaded area displays the theoretical
behaviour for zero-power linewidths of 250 MHz (lower boundary) and 300 MHz (upper
boundary). An effective measurement bandwidth is indicated by the blue curve.
where the saturation parameter is used again to simplify the expression. Ideally,
the linewidth in the limit of small Rabi frequencies reflects the dephasing time T2
and with the knowledge of the lifetime T1 we can extract a pure dephasing time. In
the absence of pure dephasing (T2 = 2T1), the radiative linewidth is recovered. It
is important to note that the linewidth here is the absorption linewidth, obtained,
for example, by scanning a narrow-band laser across the transition and recording
the total emission intensity. This should be distinguished from the emission
linewidth, which can be considerably different and is treated in the next section.
Figure 4.2 a and b display the linewidth of the X0 and X1− transitions of QD
‘Claude’, respectively. The data, shown as black dots, are extracted by fitting
Lorentzians to absorption linescans such as the one in Fig. 4.1 a. The linewidth
uncertainty for each fit is about the size of the dots in the figure. The grey
band indicates the line broadening expected from 4.3 for radiative linewidths
between 250 and 300 MHz and no pure dephasing. 250 MHz radiative linewidth
is expected from lifetime measurements for this sample. Above the saturation
point the experimental data are mostly within this band, while the mismatch
with theory is clear below saturation, in particular for the X1− data set. Also
shown is the effective measurement bandwidth (blue line): It is the inverse of
the time it takes to scan across a 250 MHz window (one radiative linewidth).
This is in the range of 1-10 Hz, i.e. measuring the FWHM of the transition takes
65
about 1 s for the measurements presented. The deviations from the ideal case are
attributed to spectral diffusion, rather than fast dephasing and will be explored in
the second half of this chapter (see section 4.2.1). Further, the difference between
the X0 and X1− is due to the nuclear Overhauser field (see section 4.2.3).
4.1.3 Spectral and statistical properties at low powers
The spectrum of resonance fluorescence. Resonant excitation of a two-
level system drives the coherences and population in the Bloch sphere according to
the optical Bloch equations presented in the theoretical background. The emitted
photons directly reflect the dynamics of the system, both of the population (in
the emission rate) and the coherences (in the spectrum). In order to capture the
full picture through measurements we need to look at the spectral and statistical
properties of the photons using the correlation functions introduced theoretically
and experimentally earlier.
The spectrum of resonance fluorescence for a laser exactly on resonance is given
by1 the sum of incoherent and coherent components:
S (∆ν) = Iinc S˜inc (∆ν) + Icoh S˜coh (∆ν)
= ρ22
(
1− 1
2
T˜2
T˜1 + Ω2T˜ 21 T˜2
)
S˜inc (∆ν) + ρ22
(
1
2
T˜2
T˜1 + Ω2T˜ 21 T˜2
)
S˜coh (∆ν)
= ρ22
{(
1− 1
2
T˜2
T˜1 + Ω2T˜ 21 T˜2
)
1
2piρ22
((
Ω2T˜ 21
)−1
+ 2
) × [ 1/T˜2
∆ν2 + 1/T˜2
2
+
2ρ22
Ω2
×
(
Aη/2−B (∆ν − µ) / (8µ)
(∆ν − µ)2 + η2 +
Aη/2 +B (∆ν + µ) / (8µ)
(∆ν + µ)2 + η2
)]
+
(
1
2
T˜2
T˜1 + Ω2T˜ 21 T˜2
)
δ (∆ν)
}
(4.4)
1It is surprisingly difficult to write down this equation without typographic mistakes and
wrong signs - a good proportion of QD resonance fluorescence papers suffer from this issue.
66
where ∆ν is the emission detuning in linear frequency, T˜1,2 = 2piT1,2, S˜ refers to
the normalised (area of unity) spectrum and we use shorthands
A =Ω2 +
1
T˜1
(
1/T˜1 − 1/T˜2
) ,
B =2Ω2 +
(
3
T˜1
− 1
T˜2
)
− 2
T˜1
(
1
T˜1
− 1
T˜2
)
,
µ =
√
Ω2 − 1
4
(
1
T˜1
− 1
T˜2
)
,
η =
1
2
(
1
T˜1
+
1
T˜2
)
.
The normalisation of Eq. 4.4 is then such that the integrated intensity corre-
sponds to the excited state population ρ22, linking it to the saturation curve we
are familiar with.
At high excitation power the incoherent component reveals the Mollow triplet,
while at low excitation power the spectrum is single-peaked. The emergence of
sidebands in the spectrum of resonance fluorescence at Rabi frequencies Ω Ωsat
arises as a consequence of Rabi oscillations between ground and excited states
and is considered the hallmark of resonance fluorescence. It was among the first
features sought and observed experimentally, first for atoms [99], but also for
QDs [103; 104; 105; 106]. In this limit of Rabi frequency, the emission is sponta-
neous and incoherent with respect to the excitation laser. Remarkably, resonance
fluorescence is expected to show a crossover from incoherent to coherent pho-
ton scattering, accompanied by a drastic change in the emission spectrum as a
function of Rabi frequency. Perhaps surprisingly, the limit of vanishing Rabi fre-
quency in the resonant interaction has remained relatively unexplored for atoms,
ions and any single solid-state emitter [117]. Here, we will investigate this limit
for QD resonance fluorescence and report on the first direct observation of highly
coherent light scattering from a solid-state emitter. We employ a shorthand,
the ‘Heitler regime’, to denote the situation where the Rabi frequency is much
smaller than the spontaneous emission rate and pure dephasing is negligible, after
W. Heitler who described the response of a two-level system to monochromatic
67
excitation in the early days of quantum optics.
Figure 4.3 displays a series of resonance fluorescence spectra for QD ‘Claude’.
For each row of the figure the excitation laser power is fixed and its frequency is
set on resonance with the QD transition. The resonance fluorescence spectrum
is recorded with the scanning Fabry-Perot cavity (see section 3.4.1 for instru-
ment details and response function). In the left panels we plot the experimental
data (as filled red circles) and a calculated spectrum based on Eq. (4.4) and the
Fabry-Perot response function (green continuous line) in linear scale. The middle
panels show the data and the convolved theoretical spectrum (green dashed line)
in a semi-logarithmic scale, together with the coherent (blue continuous line) and
incoherent (red continuous line) components. Both components of the spectrum
are convolved with the FP response function and offset for clarity. The panels
on the right give the theoretical spectra without convolution. These spectra in-
tegrate to give the value of ρ22. To the right of the plot we note the experimental
saturation parameter s and the parameters for the theoretical spectra.
The excitation laser bandwidth is very narrow on small timescales (< 100 KHz
for ∼ µs acquisition), however slow spectral wandering broadens the linewidth on
the timescale of the data acquisition (∼ seconds) to a few MHz. In the simulation
we take a top-hat function of 3 MHz width to account for this laser linewidth.
This is the narrow peak on top of the incoherent component in the right panels.
This spectrum in the right panels is convolved with a Lorentzian of 25-35 MHz
width and scaled to the data to arrive at the spectra (green lines) in the left and
middle panel.
Starting from the top, at a laser power corresponding to s = 2.35 both com-
ponents are clearly visible in a linear scale (left panel). The incoherent part
dominates the deconvolved spectrum (right panel); here the coherent fraction
is 0.3. As we decrease the laser power, going down the panels, the incoherent
component is increasingly masked by the coherent part. Above saturation, some
structure is visible on a semi-logarithmic scale in wings of the incoherent part,
whereas for s ≤ 1 the incoherent spectral component is reduced to a single peak.
At s = 0.1 where theory predicts a coherent fraction of 0.9 the entire measured
spectrum collapses to a Fabry-Perot cavity resolution-limited Lorentzian. Any
residual incoherent component of resonance fluorescence in this regime is less
68
s = 2 . 3 5
Ω = 0 . 2 7 1  G H z
T 1 = ( 2 pi 2 5 0 M H z ) - 1T 2 = 2 T 1
s = 1 . 8 0
Ω = 0 . 2 3 7  G H z
T 1 = ( 2 pi 2 5 0 M H z ) - 1T 2 = 2 T 1
s = 1 . 1 9
Ω = 0 . 1 9 3  G H z
T 1 = ( 2 pi 2 5 0 M H z ) - 1T 2 = 2 T 1
s = 0 . 6 9
Ω = 0 . 1 4 6  G H z
T 1 = ( 2 pi 2 5 0 M H z ) - 1T 2 = 2 T 1
s = 0 . 2 4
Ω = 0 . 0 8 6  G H z
T 1 = ( 2 pi 2 5 0 M H z ) - 1T 2 = 2 T 1
- 0 . 5 0 . 0 0 . 5
 
Spe
ctru
m (
a. u
.)
- 0 . 5 0 . 0 0 . 5F r e q u e n c y  ( G H z ) - 0 . 5 0 . 0 0 . 5
0 . 0
0 . 2
0 . 4
2 0
4 0
- 0 . 5 0 . 0 0 . 5
 
 
Spe
ctru
m (
a. u
.)
- 0 . 5 0 . 0 0 . 5 - 0 . 5 0 . 0 0 . 5 0 . 0
0 . 2
0 . 4
2 0
4 0
- 0 . 5 0 . 0 0 . 5
 
 
Spe
ctru
m (
a. u
.)
- 0 . 5 0 . 0 0 . 5 - 0 . 5 0 . 0 0 . 5 0 . 0
0 . 2
0 . 4
2 0
4 0
- 0 . 5 0 . 0 0 . 5
 
 
Spe
ctru
m (
a. u
.)
- 0 . 5 0 . 0 0 . 5 - 0 . 5 0 . 0 0 . 5 0 . 0
0 . 2
0 . 4
2 0
4 0
- 0 . 5 0 . 0 0 . 5
 
 
Spe
ctru
m (
a. u
.)
- 0 . 5 0 . 0 0 . 5 - 0 . 5 0 . 0 0 . 5 0 . 0
0 . 2
0 . 4
2 0
4 0
- 0 . 5 0 . 0 0 . 5
 
 
Spe
ctru
m (
a. u
.)
- 0 . 5 0 . 0 0 . 5 - 0 . 5 0 . 0 0 . 5 0 . 0
0 . 2
0 . 4
2 0
4 0
s = 0 . 1
Ω = 0 . 0 5 6  G H z
T 1 = ( 2 pi 2 5 0 M H z ) - 1T 2 = 2 T 1
Figure 4.3: Resonance fluorescence spectra of QD ‘Claude’ as function of exitation
power. The data (red circles) are recorded with a scanning Fabry-Perot cavity of 30
MHz resolution. Continuous and dashed lines show calculated spectra for the respective
experimental parameters. With the exception of the right panels, the calculated spectra
are further convolved with the Fabry-Perot response function and scaled to the data.
Left panels: data and convolved calculated spectrum. Middle panel: as before, but in
a semi-logarithmic scale. Coherent (blue) and incoherent (red) components are shown
individually for clarity. Right panels: Calculated spectra without convolution.
69
than the detectable level within the spectral resolution and the signal-to-noise
ratio of our measurements. We clearly observe subnatural linewidth fluorescence
here, consistent with predictions for a simple two-level system in the absence of
any pure dephasing. In fact, the theory curves for a purely radiatively broadened
system, i.e. T2 = 2T1, fit the measured spectra for all power levels presented
in Fig. 4.3. This is strong evidence that QDs provide clean atomic-like optical
transitions with little to no dephasing at timescales comparable to their lifetime.
The spectral resolution of the Fabry-Perot cavity of ∼ 30 MHz limits us at this
point to perform more quantitative analysis of the QD resonance fluorescence
spectrum. We switch to Fourier transform spectroscopy and measure the first-
order correlation function in a Michelson interferometer (see section 3.4.2 for
details on the setup).
The first-order correlation of resonance fluorescence. The derivation of
the first-order correlation for a resonantly driven two-level system was sketched
out in the theoretical background. Separating the spectrum into incoherent and
coherent components and neglecting the oscillations at the optical wavelength we
have:
g(1) (τ) = Iinc g
(1)
inc (τ) + Icoh g
(1)
coh (τ)
= ρ22
(
1
2
exp−|τ |/T2 + exp−|τ |/2
(
1
T1
+ 1
T2
) (
N cos (µ |τ |) +M sin (µ |τ |)
))
+ ρ22
1
1 + Ω2T1T2
T2
2T1
, (4.5)
where τ is the time delay between the fields interfered in the interferometer and
we have taken the detuning to be zero. The coherent part of eq. (4.5) has to be
modified to account for the finite laser coherence and laser spectral wandering
encountered in experiments. From the instrument calibration (cf. Fig. 3.15) a
Gaussian decay is reasonable, such that
Icoh g
(1)
coh (τ) =
(
ρ22
1
1 + Ω2T1T2
T2
2T1
)
exp
− 1
2
( |τ |
τL
)2
, (4.6)
70
where τL is around 5-7 ns. The amplitude of the g
(1) (τ) expression given above
scales as the excited state population ρ22. In order to compare it to measured
interference visibilities we divide by ρ22 and obtained an expression with unity
amplitude for all Rabi frequencies.
Experimental results on first-order coherence measurements for the X0 transition
of QD ‘Claude’ are presented in Fig. 4.4. Here we drive the QD on resonance with
a free-running external cavity diode laser; the measurement protocol is detailed
in section 3.4.2. Figure 4.4 a shows experimental data as open symbols and fits
according to Eqs. (4.5), (4.6) as solid lines for four excitation powers, indicated
on the right-hand side in terms of the saturation parameter s.
We make two immediate observations. First, in agreement with the subnatural
linewidth emission spectra from Fig. 4.3, the coherence of the QD resonance
fluorescence is sustained on timescales considerably longer than the spontaneous
lifetime of ∼ 640 ps (250 MHz radiative linewidth). Second, we would expect the
visibility measured around zero time delay to be equal to that of the laser (∼
0.99, see Fig. 3.14 for reference). For QD resonance fluorescence the maximum
amplitude is ∼ 0.88. This discrepancy suggests the presence of a very fast coher-
ence decay channel for part of the fluorescence. For a stepsize of about 10 ps in
Fig. 4.4 a, the spectral width of the fast decay channel is expected to be in the
THz frequency range, which is consistent with phonon-assisted transitions as we
will see later in this chapter.
Panel b presents a power series of interference visibilities obtained at long delays
(filled symbols), together with the measurement for the excitation laser (open
squares). The laser coherence was measured at intensities and scanning speeds
similar to the resonance fluorescence data and allows us to account for the finite
laser coherence. In panel c, we plot the coherent fraction as extracted from the
fits in panel a (blue circles) and the power series in b (black squares) as a func-
tion of excitation power. To compare to the predictions for a simple two-level
system we consider only the zero-phonon line (ZPL) here1 and correct for the
1that is, we correct for the finite visibility at zero time delay
71
2 4 0 0 2 6 0 0 2 8 0 00 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0
 
 
|g(1)
(t)|
T i m e  d e l a y  τ ( p s )
 s  =  0 . 0 6 s  =  0 . 1 0 s  =  0 . 2 2 s  =  0 . 7 4 s  =  1 . 2 3 s  =  2 . 4 0 l a s e r
cb
0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0 3 0 0 00 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0
s  =  0 . 2 2
s  =  0 . 7 4
s  =  1 . 2 3
s  =  2 . 4 0
 
 
|g(1)
(t)|
T i m e  d e l a y  τ  ( p s )
0 . 1 10 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0
 
 
Coh
ere
nt f
rac
tion
 of 
ZPL
E x c i t a t i o n  p o w e r  ( u n i t s  o f  s )
a
Figure 4.4: First-order coherence measurements of QD resonance fluorescence and
evaluation of the coherent fraction. a, g(1) measurements at short and long delays
(data as open symbols) with fits according to Eqs. (4.5) and (4.6). b, Long delay
measurements for a series of excitation powers compared to the laser coherence. c,
Coherent fraction of QD resonance fluorescence as extracted from panels a and b. The
grey band gives the theoretical prediction for 1.8T1 < T2 < 2T1.
72
laser coherence. The grey band displays the coherent fraction according to
Fcoh =
1
1 + Ω2T1T2
T2
2T1
=
1
1 + s
T2
2T1
, (4.7)
where the upper limit is set by T2 = 2T1 and the lower limit by T2 = 1.8T1.
4.1.4 Wave-particle duality in the Heitler regime
The concept that two-level systems emit photons one at a time is well-known
in quantum optics. First proposed theoretically by Kimble and Mandel [118]
in 1976, the first measurement of photon antibunching in the intensity correla-
tion g(2) (τ) was shown by Kimble, Mandel and Dagenais a year later [101; 119].
Since the potential of single photon sources in quantum communication [98] and
computation [120] was establised, research into this field has exploded. First an-
tibunching measurements using above bandgap excitation on quantum dots stem
from 2000 [121; 122], and great experimental efforts have led to efficient opti-
cally pumped QD single photon sources [123], and also electrically controlled QD
single photon diodes in laboratories [19]. Under resonant excitation the photon
statistics, captured through the second-order correlation function g(2) (τ), reflect
the dynamics of the driven system, and allowed the observation of QD exciton
Rabi oscillations [104].
Where g(1) (τ), or equivalently the emission spectrum, measures the wave proper-
ties of the photons, the intensity autocorrelation g(2) (τ) represents the photon’s
particle nature. These are, of course, complementary features. For incoherent
pumping, i.e. above band excitation the intensity autocorrelation g(2) (τ) is linked
to the excited state lifetime T1, whereas the field correlation function g
(1) (τ) links
to its coherence time T2. In other words, the coherence τc,photon of the single pho-
tons is constraint to the emitter lifetime: τc,photon = T2 ≤ 2T1.
Under resonant excitation in the Heitler regime, however, photons scatter elas-
tically: When the Rabi frequency is much smaller than the spontaneous emission
rate and pure dephasing is negligible, QD emission is predominantly coherent.
Then the coherence properties of resonance fluorescence photons (and thus their
73
- 0 . 5 0 . 0 0 . 5 F r e q u e n c y  ( G H z )
Spe
ctru
m (
a.u
.)
 
- 5 0 5
1
 g(2)
(τ) 
 
T i m e  d e l a y  ( n s )
Figure 4.5: Wave-particle duality in the Heitler regime. Left: The emission spectrum
is Fabry-Perot resolution limited, and much narrower than the radiative linewidth (in-
dicated by the shaded area). The subnatural linewidth indicates a coherence T2  2T1.
Right: The intensity autocorrelation at the same measurement setting demonstrates
antibunching on the timescale of the excited state lifetime.
spectrum) are liberated from the QD transition lifetime. Instead, the coherence
is derived from the excitation laser properties. The subnatural linewidth emission
spectrum from the previous section is displayed on the left-hand side in Fig. 4.5
together with the radiative linewidth as grey shaded area. On the right-hand side
we display the g(2) (τ) intensity autocorrelation measurement under identical ex-
citation conditions. The data (dark red curve) demonstrates clear antibunching
on the timescale of the excited state lifetime. Here, the recorded curve is limited
by the detector response, such that the dip close to the zero time delay cannot be
fully resolved. The dotted black line shows a deconvolved fit to the data, based
on equation (2.31)1.
These results reveal the counter-intuitive nature of resonance fluorescence in the
Heitler regime: The QD operates as a quantum converter of a weak coherent
state into a coherent single photon stream. The photons arrive one at a time to
our detection system, but are phase-correlated with each other and with the laser
1Spectral diffusion of the QD transition during the long g(2) (τ) measurement needs to be
taken into account to produce the blue fit curve. This will be discussed in more detail in section
4.2.1.
74
field from which they originated.
At this point we can draw some intermediate conclusions from the combination
of spectral, g(1) (τ) and g(2) (τ) measurements:
• Fits to spectra and g(1) (τ), in particular the evaluation of the coherent
fraction in Fig. 4.4, allow us to put a number on the QD transition coherence
time of T2 = (0.95± 0.05) · 2T1. The results show that QD transitions are
intrinsically mostly free of pure dephasing, contrarily to earlier reports [104].
• In the limit of weak excitation and negligible pure dephasing (the Heitler
regime), photons are predominantly coherently scattered. Their coherence
is no longer bound by the transition lifetime; it can be significantly longer.
Our measurements point to a time-averaged linewidth of several MHz. At
the same time, antibunching persists on the timescale of the transition
lifetime.
• Within the measurement uncertainty the coherently scattered photons fol-
low the coherence of the laser which is limited by spectral wandering to
several MHz. In other words, a quantum dot generates single photons with
laser-like coherence free from any dephasing processes affecting the QD light
emission.
The observation of highly coherent scattering is relevant and exciting for a num-
ber of applications in quantum information processing with QDs. First, issues
thought to be inherent in solid-state systems, such as spectral diffusion, modify
the probability to scatter a photon, but do not change its spectral properties.
This, in combination with the fact that the absolute photon frequency is pinned
to the laser frequency can be of crucial advantage in quantum interference ap-
plications, where indistinguishable photons are needed at different times or from
separate sources. Second, not only the frequency, but also the photon phase is
expected to be determined by the excitation laser, which should allow us to shape
phase and spectrum of single photons deterministically. Some of these ideas will
be explored experimentally in chapter 6.
We note that Nguyen et al. report similar measurements, see Ref. [124].
75
4.2 Effects of the solid-state environment
The first part of this chapter explored experimental evidence of the near-ideal
behaviour of QD emission under low power resonant excitation. However, even
in these basic quantum optical measurements, deviations from the theoretical
predictions were present: Absorption linewidths were broader than expected,
particularly for the X1− transition (cf. Fig. 4.2), the contrast of the first-order
coherence was limited (cf. Fig. 4.4) and the antibunching dip in the second-order
coherence was narrower than predicted (Fig. 4.5).
To explain these effects we have to consider the solid-state environment of the QD.
Here we focus on three effects in particular: spectral diffusion, exciton-phonon
coupling and the nuclear Overhauser field.
4.2.1 Spectral diffusion
Spectral diffusion describes the effect of transition resonance shifts over time. It is
usually attributed to the uncontrolled electric fields arising from the dynamics of
defects (metastable trapped charges) or other QDs nearby. The time-dependent
electric fields shift the QD resonance via the Stark effect (see intro section). Par-
ticular sample circumstances and models of defect dynamics have been subject
of a number of studies recently [78; 81; 125; 126]. Timescales and magnitudes
vary greatly between different structures and samples in these reports. Here we
want to quantify the severity of spectral diffusion in our gated sample (Chef 2
sample), and we employ resonance fluorescence to obtain precise information on
bandwidth and amplitude of spectral diffusion1. Information is extracted from
time-resolved resonance fluorescence detection: We record a series of 10-s long
resonance fluorescence detection time traces with a resolution of 10 µs and treat
the data using statistics on photon arrival probabilities and time and compare
the result to a simple model.
1Note added: High quality measurements, also using QD resonance fluorescence and yielding
similar results, were obtained in the group of R. Warburton very recently [127].
76
4.2.1.1 Spectral diffusion amplitudes
First we take the 10-s long timetraces and compile histograms of the probability
to detect a set number of counts per detection bin. Given the overall small
collection and detection efficiency (< 0.2%) of the setup we expect the detections
to obey Poissonian statistics. Then, for a mean number of m photons per bin,
the probability distribution of kbin counts per bin is
P (kbin) =
mkbin exp−m
k!
. (4.8)
The mean m is linked to the QD fluorescence through the collection and detection
efficiency ηdetection and the excited state population (see Eq. (4.2)):
m (s,∆) = ηdetection × tbin × 1
2T1
s
1 + s+ 2 (2pi∆)2 T1T2
, (4.9)
where tbin is the duration of a detection bin and ∆ is the detuning of the laser
from resonance. For numerical data treatment it is convenient to work with
m (s,∆) = a× s+ 1
1 + s+ 2 (2pi∆)2 T1T2
, (4.10)
where the fraction is normalised to unity for zero detuning (∆ = 0) and the
prefactor a can be conveniently linked to the experimental mean for a given bin
size.
The effect of spectral diffusion is to make the laser detuning into a time-varying
quantity ∆ (t). When measuring for sufficiently long we expect to cover the full
extent of the spectral diffusion, such that ∆ (t) can be described by a probability
distribution. In terms of the probability distribution of kbin counts per bin, the
Poisson distribution (4.8) is modified to
P (kbin) =
∑
∆
W (∆)× m (s,∆)
kbin exp−m(s,∆)
k!
. (4.11)
The weighting function W (∆) effectively describes how much time is spent at each
detuning, and we can reasonably assume that it follows a Gaussian distribution
77
for long measurement times:
W (∆) = e
− 1
2
(
∆−δ
∆FWHM
√
8 ln 2
)2
. (4.12)
To allows us to describe the effect of spectral diffusion when setting (or attempt-
ing to set) the laser at a finite detuning δ from the QD resonance the Gaussian
is centred on δ. The amplitude of the spectral diffusion is given by the width of
the Gaussian (∆FWHM) and the running variable is ∆.
Predictions from the model (Eqs. (4.11), (4.12)) are compared to the experimen-
tal data in Fig. 4.6. Here the laser is set on resonance with the X1− transition of
QD ‘Isidor II’ and resonance fluorescence counts are recorded continuously for 10
s with 10 µs resolution. The measurement is repeated a number of times and for
different excitation powers. The data are rebinned to move the mean counts per
bin to around 30 or greater to allow accurate fitting. The resulting histograms
are normalised to unity and displayed as circles in Fig. 4.6 a-f. The blue curve is
a fit of the model to the data; the main parameters are displayed in the top right
corner of each plot. The red curve shows the histogram expected for a transition
without spectral diffusion. In panels a,c and e the data are well fitted1 assuming
the laser (or equivalently the weighting function W (∆)) is set on resonance, and
the diffusion coefficient ∆FWHM is in the range of 100-150 MHz. This is consistent
with most of the 10-s long pieces of data of which we only display a few here. In
some cases however, histograms such as in panels b, d and f are obtained. The
data agree very well with the model assuming that the QD resonance is centred
on a fixed detuning from the laser for the duration of the measurement. At the
same time, the diffusion coefficients for these data is similar to the zero detuning
cases.
We note that the histogram shape (and amplitude) is a sensitive function of both
the diffusion width and the detuning, and a particular shape is uniquely described
by both parameters.
In panel g, we summarise the diffusion coefficients for a few data sets (including
data with non-zero detuning). For this particular QD, the diffusion coefficients
1An additional peak is visible in the histogram data, mainly for higher excitation power.
This is attributed to a distinct spectral jump of the QD resonance, a feature commonly observed
in high-density samples.
78
range between 100-200 MHz on a timescale of 10 s, with the data in panels b, d,
f implying that an average of the full extent of spectral diffusion is not reached in
10 s. Panel h displays the effect spectral diffusion is expected to have on (slow)
absorption linewidth measurements: We take the measured diffusion amplitudes
∆FWHM from panel g and calculate the resulting absorption lineshape. Roughly
speaking, the spectral diffusion broadens the absorption lineshapes by 50 MHz
compared to the transform limit (250 MHz radiative linewidth) which is given
by the lower limit of the grey band. This compares reasonably well with the
experimental data presented earlier (cf. Fig. 4.2).
4.2.1.2 Spectral diffusion timescales
To access information on the diffusion timescale we could divide the 10-s long
data into smaller chunks and repeat the histogram treatment, but the analysis is
fairly cumbersome and becomes less accurate as the amount of data is reduced.
At the same time, the assumption of a Gaussian detuning distribution is a pri-
ori not valid anymore. Two more suitable methods for analysing the spectral
diffusion dynamics are decomposition into Fourier components (via Fast Fourier
Transforms) in the frequency domain and calculating the intensity autocorrela-
tion in the time domain. Both of them are very powerful tools for this purpose.
Here, we concentrate on the autocorrelation1. The intensity autocorrelation func-
tion g(2) (τ) was introduced earlier as a way to quantify the photon statistics at
timescales comparable to transition lifetimes. As a reminder, the intensity auto-
correlation is given by
g(2) (τ) =
〈I (t) I (t+ τ)〉
I2av
. (4.13)
Calculating the correlations between events separated in time is a means to pick
up on dynamics in the data. For a source with entirely Poissonian character,
such as an ideal laser, the intensity output is entirely uncorrelated for any τ ,
such that the autocorrelation is alway unity. Fluctuations manifest themselves as
bunching (i.e. g(2) (τ) > 1) in the correlation function, with their characteristic
1Note: A detailed analysis of noise in resonance fluorescence using FFTs was recently put
on the arXiv by the group of R. Warburton [127].
79
0 . 0 0 . 5 1 . 0 1 . 5 2 . 0
1 0 0
1 2 5
1 5 0
1 7 5
2 0 0  
 
Diff
usio
n w
idth
 ∆ FW
HM
(MH
z)
E x c i t a t i o n  p o w e r  ( u n i t s  o f  s )
0 2 0 4 0 6 0 8 0 1 0 0 1 2 00 . 0 0
0 . 0 2
0 . 0 4
0 . 0 6  
 
P(k
bin)
C o u n t s  p e r  b i n  k b i n
p o w e r   =   0 . 6  s
∆ d i i f f u s i o n  =  1 3 0  M H z
∆ d e t u n i n g  =      0  M H z
0 2 0 4 0 6 0 8 0 1 0 0 1 2 00 . 0 0
0 . 0 2
0 . 0 4
0 . 0 6  
 
P(k
bin)
C o u n t s  p e r  b i n  k b i n
p o w e r   =   0 . 6  s
∆ d i i f f u s i o n  =  1 1 5  M H z
∆ d e t u n i n g  =    6 9  M H z
e
h
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0
0 . 0 0
0 . 0 2
0 . 0 4
 
 
P(k
bin)
C o u n t s  p e r  b i n  k b i n
p o w e r   =   0 . 3  s
∆ d i i f f u s i o n  =  1 3 4  M H z
∆ d e t u n i n g  =      0  M H z
0 2 0 4 0 6 0 8 0 1 0 00 . 0 0
0 . 0 2
0 . 0 4
0 . 0 6
 
 
P(k
bin)
C o u n t s  p e r  b i n  k b i n
p o w e r   =   0 . 3  s
∆ d i i f f u s i o n  =  1 5 0  M H z
∆ d e t u n i n g  =  1 3 5  M H z
0 1 0 2 0 3 0 4 0 5 0 6 0
0 . 0 0
0 . 0 2
0 . 0 4
0 . 0 6
0 . 0 8
 
 
P(k
bin)
C o u n t s  p e r  b i n  k b i n
p o w e r   =   0 . 1  s
∆ d i i f f u s i o n  =  1 0 8  M H z
∆ d e t u n i n g  =    6 5  M H z
0 1 0 2 0 3 0 4 0 5 0 6 0 7 0
0 . 0 0
0 . 0 2
0 . 0 4
0 . 0 6
0 . 0 8  
 
P(k
bin)
C o u n t s  p e r  b i n  k b i n
p o w e r   =   0 . 1  s
∆ d i i f f u s i o n  =  1 0 7  M H z
∆ d e t u n i n g  =      0  M H z
a
g
f
dc
b
0 . 1 1
2 5 0
3 0 0
3 5 0
4 0 0
4 5 0
 
 
Est
ima
ted
 ab
sor
ptio
n lin
ewi
dth
 (M
Hz)
E x c i t a t i o n  p o w e r  ( u n i t s  o f  s )
Figure 4.6: Histogram analysis of spectral diffusion effects. a-f, The QD X1− transi-
tion is driven on resonance (nominally) and detection events are recorded for 10 s each
time. Intensity statistics for rebinned data are shown as circles. Fits using the spectral
diffusion model are shown as blue curves and the histograms expected in the absence
of spectral diffusion are shown as red curves.g, Summary of diffusion amplitudes. h,
Extrapolated effects on the absorption linewidth.
80
10
-3
10
-2
10
-1
10
0
1.70
1.75
1.80
1.85
1.90
1.95
2.00
In
te
n
s
it
y
 c
o
rr
e
la
ti
o
n
 g
(2
) (
) 
 (
a
.u
.)
Time delay  (s)
τ
τ
I II III
Figure 4.7: Intensity autocorrelation functions for 19 resonance fluorescence time
traces. The data are binned to 1 ms resolution. Three regions of different behaviour
are identified and colour-coded.
timescales and their amplitudes directly mapped to the amplitude and decay time
(the correlation time) of the bunching. We apply this method to QD resonance
fluorescence.
To get a first impression of the situation we have a look at a set of measurements,
binned to a low resolution. Figure 4.7 presents a semi-logarithmic plot of the
unnormalised1 second-order correlation functions for 19 resonance fluorescence
time traces, each of 10-s length and binned to 1 ms resolution. The antibunching
effect at delays τ comparable to the emitter lifetime does not play a role here as
the time resolution tbin  T1.
We identify three regions of different behaviour: All correlations show clear
bunching and a strong similarity with each other for time delays up to ∼ 100
ms (lightly shaded region marked ‘I’). For the intermediate time delays indicated
by region ‘II’ in the plot (∼ 100 ms to ∼ 500 ms) the correlation functions for
1to keep the curves separated
81
individual time traces follow a mostly monotone trend, but differ from each other.
Region ‘III’, with delays greater than 0.5 s looks messy. There are no obvious
trends, the g(2) (τ) amplitude varies widely (wildly), and each individual time
trace shows unique behaviour.
Let us start with the analysis of the behaviour in region ‘III’. We first note that
the curves are inherently more noisy here as τ approaches the total recording
time of 10 s. On top if that, if spectral diffusion-induced fluctuations in the res-
onance fluorescence persisted only on timescales shorter than the measurement
time, we would expect to reach a steady-state in the correlation function. There
would be no correlation between detection events separated by large delays τ :
g(2) (∞) −→ 1 (for a normalised distribution). No such steady-state is reached
in our data which covers correlation delays up to 5 s. We conclude that the cor-
relation time for slow fluctuation is  5 s and these fluctuations contribute the
largest amplitude to the total noise.
Region ‘II’ in Fig. 4.7 shows the transition from the fast to the slow fluctuation
regime and does not yield much additional insight, so we move to the fast dynam-
ics in region ‘I’. To maximise the resolution, the data is kept in the original bin
size of 10 µs here. Figure 4.8 displays four exemplary autocorrelations (a-d) as
green continuous curves in a semi-logarithmic scale, two on resonance (a, c) and
two for a moderate detuning (b, d). The bunching signature already observed
in Fig. 4.7 is resolved in good detail. It is dominated by a decay on a ∼ 1 ms
timescale. A weaker decay feature can be made out at fast timescales (∼ 20 µs),
in particular for the resonant cases. A third decay is present around 10 ms time
delay; this one is more obvious for detuned excitation. Using a sum of three
exponential decays we can generate good fits to the data, which are shown as
dashed red lines.
Revealingly, there is additional structure in the detuned g(2) (τ): a series of alter-
nating peaks and troughs at delays times which are multiples of 10 ms. This is
the manifestation of 50 Hz noise and points to electric field noise picked up via
the sample gates. Panels e-h summarise the decay times and amplitudes found
through fitting. The three correlation times (shape- and colour-coded) are consis-
tently in the same range for different data sets and powers. The picture is similar
for detuned excitation (panel f), but it is not always possible to identify the fast
82
Figure 4.8: Spectral diffusion timescales and amplitudes. a-d, Intensity correlations
for QD resonance fluorescence with a timing resolution of 10 µs. Three decay timescales
are identified and summarised in panels e-h.
83
2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0
0 . 0 0
0 . 0 2
0 . 0 4
0 . 0 6
0 . 0 8
0 . 1 0
0 . 1 2  
 
Me
an 
diffe
ren
ce f
rom
 un
ity
T i m e  t r a c e  n u m b e r  ( c h r o n o l o g i c a l  o r d e r )
Figure 4.9: Analysis of slow diffusion amplitudes. Using data from region ‘III’ in Fig.
4.7 absolute deviations from the steady-state (g2(∞) = 1) are evaluated. The black
squares show the mean of the deviations for each time trace, the grey rectangles display
the range of the deviations.
decay timescale and reasonable fits can be generated with two exponential decays
only. The plots of the correlation amplitudes (panels g and h) use the same shape
and colour code as before and confirm the 1 ms correlation to be the dominant
feature.
Having identified the noise amplitudes and timescales in region ‘I’, can we com-
plete the picture and also quantify the amplitude of the very slow fluctuations?
Going back to Fig. 4.7 we calculate the absolute deviation from the ideal steady-
state value for g(2) (τ > 1 s). The mean (black squares) and the extent of the de-
viations (grey shaded rectangles) for 20 consecutive resonance fluorescence time
traces are presented in Fig. 4.9. The amplitude (∼ maximum of the grey area)
of the deviation is greater or comparable to the 1 ms noise contribution for most
of the individual 10-s long time traces. Over the course of the measurement for
the 20 traces, the total deviations exceed the magnitude of the fast contributions
(cf. Fig. 4.8 g, h) significantly.
The origin of spectral diffusion. The intensity autocorrelation is a powerful
tool to obtain quantitative information about spectral diffusion timescales and
amplitudes. Can we link the results to their origin, i.e. identify which diffusion
timescale stems from which physical mechanism? The two diffusion mechanisms
we have to consider are electric and magnetic field noise. Electric fields shift the
84
QD transition resonance via the Stark effect and give rise to an effective detuning
of the excitation laser from resonance. Part of the electric field noise is of exter-
nal origin, such as the 50 Hz noise we have seen earlier. Some will be due to the
dynamics of charges close to the QD. The magnetic field noise comes from the
randomly varying cumulative Overhauser field from the nuclear spin environment
of the QD. The data shown in this section are from the negatively charged trion
transition X1−, so any magnetic field splits the ground and excited state with the
respective splitting determined by the direction and magnitude of the field. Then
we go from a simple two-level system to four levels that, again depending on the
orientation of the B-field, support two or four optical transitions (and mixtures of
these). In any case, however, there is a splitting between the optical transitions.
The effects on absorption lineshapes are sketched out in Fig. 4.10, in a for electric
fields, and b for magnetic fields. The top panels of a and b show two lineshapes
each which represent the absorption profile at two different times. In the sketch
for electric field fluctuations (panel a) the two Lorentzians are shifted by 100 MHz,
in b we change the splitting between a sum of two Lorentzians from 100 to 200
MHz. The bottom panels display the absolute difference between two lineshapes.
The amplitude of the absolute difference tells us how sensitive the fluorescence
intensity is to E- and B-field fluctuations as a function of laser detuning from
the unperturbed QD transition resonance. The sensitivity to electric field noise
is greatest at the slope of the absorption lineshape. In contrast, the sensitivity
to magnetic fields is greatest on resonance.
Relating this information to the correlation times and amplitudes in Fig. 4.8, we
tentatively assign the 10 µs noise to the nuclear Overhauser field dynamics. It is
stronger on resonance, and the timescale is about the same order of magnitude
as expected for the nuclear correlation time [33]. The 10 ms noise is quite clearly
due to electric field fluctuations: its amplitude is dramatically enhanced at finite
detuning. With regards to the 1 ms dynamics, the situation is less clear. The
correlation amplitudes on- and off-resonance do not differ much. At the same
time, the amplitude shows a dependence on the excitation laser power and possi-
bly a saturation behaviour. With the currently presented data we cannot clearly
link it to laser-induced fluctuations of the environment and this will be further
investigated. As for the very slow noise, this can be compensated by manually
85
0 . 0
0 . 5
1 . 0
 
 
RF 
sign
al (
a.u
.)
- 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 00 . 0
0 . 1
|∆ dif
fusi
on(B
-fie
ld)|
E x c i t a t i o n  l a s e r  d e t u n i n g  ( G H z )
0 . 0
0 . 5
1 . 0
 
 
RF 
sign
al (
a.u
.)
- 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0
0 . 0
0 . 2
|∆ dif
fusi
on(E
-fie
ld)|
E x c i t a t i o n  l a s e r  d e t u n i n g  ( G H z )
a b
Figure 4.10: Effects of electric (a) and magnetic (b) field changes on the QD trion
transition. a, A change in the electric field at the QD Stark shifts the transition
frequency. The magnitude of the change is greatest at the slope of the resonance. b, A
change in the magnetic field changes the splitting between the optical transitions and
it most strongly felt on resonance.
changing the applied gate voltage and hence is due to changes of electric-field.
Consequences for long measurements: high-resolution intensity corre-
lations. Experiments based on two-photon detection with sub-nanosecond tim-
ing resolution, such as measuring antibunching in a Hanbury-Brown and Twiss
setup, or cross-correlations in a Hong-Ou-Mandel interferometer may require ac-
quisition times of many hours, depending on the excitation power and detuning.
As such, these measurements can suffer significantly from spectral diffusion. Care
must be taken in the data treatment then to include the diffusion effects. We
give an appropriate expression for the intensity autocorrelation g(2) (τ):
g(2) (τ,∆,∆FWHM, δ) =
∑
∆
g(2) (τ,∆)×W (∆FWHM,∆, δ)× L2 (∆)∑
∆
(. . . , τ →∞) . (4.14)
Here, we calculate the sum of g(2) curves for a range of excitation detunings ∆,
weighted by a product with the Gaussian diffusion function W (∆FWHM,∆, δ)
(centred on δ with a FWHM of ∆FWHM) and the square of the power broadened
Lorentzian linewidth L = s+1
1+s+2(2pi∆)2T1T2
(decribing the amplitude of the two-
photon signal). The expression is normalised by division with its long-time limit.
86
We will use this formalism in section 4.3 of this chapter for g(2) measurements at
finite detuning.
4.2.2 Phonon coupling
The electron and hole wavefunctions of excitons in QDs are given by an envelope
function depending on QD shape and composition and the fundamental Bloch
functions of the solid-state lattice potential. As such the excitons couple to the
lattice vibrations of the embedding matrix and this coupling provides a dissipation
mechanism. While the confinement in quantum dots gives rise to discrete energy
levels and inhibits phonon-driven population relaxation (the ‘phonon bottleneck’
[128; 129]), dephasing of the optical transition can still be phonon assisted.
4.2.2.1 Acoustic phonon sideband
Experimentally, phonon coupling in QDs was first observed in the photolumines-
cence spectra of II-IV QDs [130]. At cryogenic temperatures the spectrum was
composed of a narrow peak sitting on top of a broad asymmetric background.
The narrow peak is known as the zero-phonon line, as the emission energy is
determined by the optical transition only, while the broadband is due to acoustic
phonon-assisted fluorescence and referred to as the phonon sideband. Similar re-
sults were obtained for InAs/GaAs QDs using four-wave mixing [131], and later
also in photoluminescence [132; 133].
To investigate the effects of exciton-phonon coupling in our samples, we have to
record the resonance fluorescence emission spectrum over a much wider frequency
range than covered earlier in the scanning Fabry-Perot experiments. Measure-
ments of the QD resonance fluorescence spectrally dispersed in the spectrometer
on a 1200 grooves/mm grating are presented in Fig. 4.11 using a logarithmic
intensity scale. The broad phonon sideband is very clearly visible. Before going
into details of the data we will briefly review the theoretical background.
A comprehensive treatment of linear exciton-phonon coupling based on the in-
dependent boson model was published by Krummheuer et. al in 2002 [134] and
accounts for the phonon sidebands. Dephasing of the zero-phonon line can be
included with quadratic coupling to acoustic phonons [135; 136] but will be ne-
87
glected here. Following Ref. [134], the exciton-phonon coupling Hamiltonian is
given by
Hexc−phon = ~
∑
q
(
geqbqc
†c− ghqbqd†d+ H.c.
)
, (4.15)
where the sum is over the phonon wavevectors q and there is a summation for each
phonon branch (e.g. the longitudinal acoustic, longitudinal optical and transverse
acoustic phonons). The phonon coupling matrix elements g
e/h
q quantify the cou-
pling for the specific branch, phonon wavevector and the charge carrier (electron
e or hole h) involved. c and c† (d and d†) are the annihilation and creation oper-
ators for electrons (holes) and vice versa for the phonons (bq, b
†
q). The coupling
matrix elements are specified by a form factor, given by the wavefunction, and a
factor linked to the specific coupling mechanism. The coupling mechanisms are
known from bulk electron-phonon interaction as the deformation potential (for
acoustic phonons), the piezoelectric electron-acoustic phonon coupling and the
electron-optical phonon Fro¨hlich interaction [137].
For excitons in QDs the deformation potential is the dominant coupling mecha-
nism by a wide margin [134; 138]. A simple physical picture of the exciton-phonon
interaction and how it leads to phonon sidebands is obtained when we consider
the exciton creation and annihilation. The exciton is localised in the small volume
of the QD and its presence affects the lattice equilibrium position. Upon photon
absorption the lattice deforms (polaron formation) to reach the new equilibrium
position, and the process repeats when the exciton recombines. The deformation
process is fast (high bandwidth), meaning that the interaction with phonons is
strong on this first very short timescale and weak thereafter.
The phonon dispersion of longitudinal acoustic phonons is linear for small wavevec-
tors and the density of states is proportional to the square of the phonon energy.
The form factor determines which wavevectors are coupled via the deformation
potential. The range of phonon energies involved is given by the spatial extent
of the electron and hole wavefunctions and thus linked to the shape of the QD:
smaller QDs probe a wider range of frequencies than larger QDs and should there-
fore couple more strongly to acoustic phonons. Intuitively we can understand this
as an averaging of the carrier-phonon interaction over the size of the QD: when
88
the phonon wavelength is considerably shorter than the smallest extent of the
QD, its effect will not be felt. In the emission spectra of Fig. 4.11 the size of
phonon sideband directly reflects the range of acoustic phonon frequencies that
contribute to the interaction. Further, the sideband is very strongly asymmetric
with almost its entire weight on the lower energy side of the zero-phonon line.
This can be linked to the phonon occupation probability which is governed by
the Bose-Einstein distribution:
n (q) =
1
e~ω(q)/kbT − 1 . (4.16)
At 4.2 K kbT (kb being the Boltzmann constant) corresponds to 0.36 meV. The
probability of absorption of a phonon with energy ~ωq is proportional to the
occupation number n (q), while the emission of a phonon goes as n (q) + 1. At
low energies the emission process is overwhelmingly more likely than absorption.
Hence the QD fluorescence is shifted to the red as some energy is taken away by
phonon emission.
Figure 4.11 a and c show the emission spectra (normalised to the peak intensity)
for the neutral exciton transition X0 (a) and for the charged exciton transition
X1− (c) driven at about twice their saturation powers. The similarities of their
spectra is confirmed in Fig. 4.11 b, where both spectra are displayed on the same
scale by subtracting the energy of the zero-phonon line peak. The similarity ex-
tends over the entire spectrum, even the small hump at 1 meV is consistent for
both transitions. We can conclude that the interaction with acoustic phonons
takes place exclusively in the excited state, and the presence of an additional
charge carrier does not affect the interaction.
The weight of the phonon sideband relative to the total emission intensity is con-
sistently around 12% for different QDs. The existence of a strong phonon side-
band provides an explanation for the limited constrast in first-order coherence
measurements of the QD resonance fluorescence (cf. Fig. 4.4): The coherence
time of the sideband photons is inversely proportional to the sideband width and
can only be detected in a range of a few ps around zero time delay.
Figure 4.11 d shows a close-up of the X0 phonon sideband in linear scale for
an excitation power significantly above saturation. An additional weak peak is
89
dc
9 5 2 9 5 4 9 5 6 9 5 8 9 6 0 9 6 2
1 E - 4
1 E - 3
0 . 0 1
0 . 1
1 R e s o n a n t l y  d r i v e nX 1 -  t r a n s i t i o n
p o w e r :  2  s
Nor
ma
lize
d e
mis
sion
 spe
ctru
m
W a v e l e n g t h  ( n m ) 9 5 0 9 5 2 9 5 4 9 5 6
0
2 x 1 0 - 3
4 x 1 0 - 3
6 x 1 0 - 3
8 x 1 0 - 3
Nor
ma
lize
d e
mis
sion
 spe
ctru
m
W a v e l e n g t h  ( n m )
R e s o n a n t l y  d r i v e nX 0  t r a n s i t i o n
p o w e r :  6 0  s
b i e x c i t o n  e m i s s i o n
- 1 0 - 5 0 5
1 E - 4
1 E - 3
0 . 0 1
0 . 1
1  X 0  s p e c t r u m
 X 1 -  s p e c t r u m
Nor
ma
lize
d e
mis
sion
 spe
ctru
m
E n e r g y  ( m e V )9 4 8 9 5 0 9 5 2 9 5 4 9 5 6 9 5 8
1 E - 4
1 E - 3
0 . 0 1
0 . 1
1
Nor
ma
lize
d e
mis
sion
 spe
ctru
m
W a v e l e n g t h  ( n m )
R e s o n a n t l y  d r i v e nX 0  t r a n s i t i o n
p o w e r :  2  s
a b
Figure 4.11: Acoustic phonon sidebands in QD resonance fluorescence. a, spectrum
for X0 transition. c, spectrum for the X1− transition. b, Comparison of the two. d,
Phonon-assisted two-photon processes appear for strong excitation powers.
90
present in this spectrum and its energy corresponds to the biexciton line of the
QD. It is only detected for high excitation powers (> 10 × s) and the intensity
grows as the square of the excitation power. This points to a two-photon absorp-
tion process, where the emission of an acoustic phonon bridges the energy gap
between the exciton and the biexciton state.
The details of the phonon sidebands will not be discussed further here. Consider-
ing the implications for QIP with QD resonance fluorescence we have to consider
that sideband photons have very short decoherence times (∼ ps) and are not
suitable for interference applications (such as distant entanglement through in-
terference). In these cases the QD fluorescence has to be filtered to exclude the
phonon sideband. Where only the fluorescence intensity matters, as it is the case
for state readout there are no negative effects.
We also note that the contribution of the phonon sideband to the fluorescence
can vary considerably between different samples. For instance Konthasinghe et
al. report a sideband contribution of 5% [125]. Finally, given the very good
signal-to-noise ratio with which we resolve the sideband, resonance fluorscence is
a suitable technique to study exciton-phonon effects in QDs.
4.2.2.2 Phonon-broadening of the zero-phonon line
In the previous discussion of phonon coupling we focussed on the acoustic side-
bands. These are satisfactorily explained considering a Hamiltonian linear in
the phonon displacement [134], and good quantitative agreement to experimen-
tal data can be found for realistic (i.e. non-Gaussian) wavefunctions [139]. The
model does not predict a broadening of the zero-phonon line itself, however, while
this was observed in the first experiments already [130; 131]. The broadening as
a function of lattice temperature can be recovered in a theoretical treatment that
accounts for a quadratic coupling to acoustic phonons [136].
Recently, an additional excitation power-dependent dephasing has been found in
several experiments under resonant excitation [110; 115; 116]. The importance of
this excitation-induced dephasing is significant: ultrafast optical control of spin
qubits relies on the optically excited state to achieve fast spin rotations via virtual
transitions [22; 42]. For excitonic qubits (see Ref. [140] for example) it is the
91
coherence between ground and excited state that has to be addressed optically
and it is therefore strongly sensitive to such a mechanism.
Excitation-induced dephasing has been treated theoretically [115; 116; 141; 142]
and explains published experimental data well. The additional dephasing is a
consequence of Rabi oscillations under high power resonant excitation excitation.
Rabi oscillations dress the two-level system such that three optical transitions
are created. The size of the splitting of ground and excited state is given by
the Rabi frequency Ω and achievable Rabi frequencies are in the energy range
where acoustic phonons couple to the QD exciton. The presence of the addi-
tional levels enhances phonon transitions at the Rabi frequency. For a range of
Rabi frequencies ∼ few GHz < Ω < ωc, where ωc is the cutoff frequency for QD
exciton-phonon coupling, the dephasing is predicted to behave as the square of
the Rabi frequency.
Measuring the excitonic coherence (through the spectrum or the first-order corre-
lation function) for Rabi frequencies in this regime is experimentally challenging
as the signal, given by the QD fluorescence, is limited to its saturation value,
while the laser background increases (without bound) as the square root of the
Rabi frequency. To extract information about dephasing times with less strict
conditions on signal-to-background ratios one can chose to measure part of the
spectrum only, e.g. the Mollow triplet sidebands [110]. These are spectrally re-
moved from the excitation laser background and their width is related to the
dephasing time.
Here we present measurements of the full first-order correlation signal from a
single QD driven in continuous-wave mode. Rabi oscillations can be recorded in
a time-resolved fashion via the first-order correlation and the decay of the oscilla-
tions is directly linked to the dephasing time T2. Performing these measurements
is possible in our case due to the good suppression of laser background (cf. Fig.
4.1). Additionally, at laser powers where s ≥ 200 we modulate the QD resonance
between on- and off-resonance with respect to the excitation laser using the DC
Stark effect. Phase-sensitive detection is then employed to pick up the QD reso-
nance fluorescence contribution. The modulation speed is on the order of 1 KHz
and serves to eliminate fluctuations in the laser background during the course of
an interference fringe measurement (see section 3.4.2 for the measurement pro-
92
tocol). Using this technique we recover the original resonance fluorescence signal
provided there is no interference between laser leakage and the QD signal, i.e.
the coherent fraction Fcoh  1. This condition is well satisfied for excitation at
zero detuning and s > 100.
Figure 4.12 a-j presents a series of first-order correlation measurements on the
X1− transition of QD ‘Claude’. Blue squares represent experimental data and
the red line is a fit according to Eq. (4.5). The values for the main fit parameters,
the Rabi frequency Ω and the dephasing time T2, are indicated in each panel. To
relate the Rabi frequency to the saturation parameter used in previous sections,
Fig. 4.12 a is for an excitation power of s ∼ 10, and Fig. 4.12 j for s ∼ 4000.
The oscillations in the visibility of g(1) are the signature of Rabi oscillations:
Starting with the maximum visibility at zero time delay1 a pi rotation of the two-
level system is obtained when the visibility reaches the first minimum. A revival
of visibility follows until we reach a time delay corresponding to a 2pi rotation,
where the fringe amplitude is maximum again.
The agreement between data and theory is remarkable for intermediate Rabi fre-
quencies (Ω ≤ 4GHz) and a trend of decreasing T2 times with increasing power
is clear. At higher Rabi frequencies differences to the fit function appear, most
visible at longer delays and highest powers. Figure 4.12 i contains data for longer
delays and is a particularly clear example: Here the amplitude of the coherent
oscillations goes through a minimum, while maintaining an intermediate level of
coherence. At longer delays the oscillations pick up again, but the phase is flipped
by pi. The presence of this additional beat note in the data points to more fre-
quency components in the spectrum than accounted for in the fit function, which
is for an ideal two-level system. Since the data is based on the fluorescence from
a singly charged QD (the X1− transition), a likely reason for the beating is a
nuclear Overhauser-split ground state. We will confirm this suspicion in the next
section and ignore it for now. To be able to extract meaningful numbers for the
T2 time the fits to the data are made to fit the amplitude of the Rabi oscillations
after the beating. Still, this is a caveat for the data interpretation. A proper
treatment would account for the effect of the nuclear field, but this is beyond the
1The amplitude of the first-order correlation around zero time delay is limited by the phonon
sideband contribution, as discussed earlier.
93
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 00 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0 W  =  2 1 . 5 2  G H zT 2  =  0 . 4 3  n s
Vis
ibili
ty o
f g(1
) (τ)
D e l a y  τ ( n s )
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 00 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0 W  =  1 4 . 6 3  G H zT 2  =  0 . 6 0  n s
Vis
ibili
ty o
f g(1
) (τ)
D e l a y  τ ( n s )
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 00 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0 W  =  1 0 . 3 1  G H zT 2  =  0 . 7 6  n s
Vis
ibili
ty o
f g(1
) (τ)
D e l a y  τ ( n s )
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 00 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0 W  =  8 . 1 2  G H zT 2  =  0 . 8 1  n s
Vis
ibili
ty o
f g(1
) (τ)
D e l a y  τ ( n s )
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 00 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0 W  =  4 . 4 7  G H zT 2  =  0 . 9 0  n s
Vis
ibili
ty o
f g(1
) (τ)
D e l a y  τ ( n s )
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 00 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0 W  =  2 . 9 2  G H zT 2  =  0 . 9 4  n s
Vis
ibili
ty o
f g(1
) (τ)
D e l a y  τ ( n s )
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 00 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0 W  =  2 . 3 6  G H zT 2  =  1 . 0  n s
Vis
ibili
ty o
f g(1
) (τ)
D e l a y  τ ( n s )
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 00 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0 W  =  1 . 7 3  G H zT 2  =  1 . 0 1  n s
Vis
ibili
ty o
f g(1
) (τ)
D e l a y  τ ( n s )
dc
ji
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 00 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0 W  =  1 . 1 2  G H zT 2  =  1 . 0 7  n s
Vis
ibili
ty o
f g(1
) (τ)
D e l a y  τ ( n s )
g h
b
fe
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 00 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0
 
 
Vis
ibili
ty o
f g(1
) (τ)
D e l a y  τ ( n s )
W  =  0 . 8 5  G H zT 2  =  1 . 0 8  n s
a
Figure 4.12: First-order coherence measurements for theX1− transition of QD Claude
under strong resonant excitation. Data is shown as blue squared, error bars are indi-
cated by vertical lines. Fits according to Eq. (4.5) are shown as continuous red lines.
94
scope of this section (and this thesis1).
In Fig. 4.13 we summarise the findings from the first-order correlation measure-
ments. Panel a shows the power dependence of the pure dephasing rate
γp =
1
T2
− 1
2T1
. (4.17)
Data are shown as open triangles. The upwards pointing red triangles were ob-
tained from g(1) measurements taken in the space of a few days (in between two
helium refills of the cryostat) such that the optical alignment was very similar.
The downwards pointing blue triangles originate from different data sets. First,
before going into details on the exact dephasing behaviour, we note there is a
marked dependence of the dephasing rate on the Rabi frequency: In the range
studied here, the dephasing rate increases (within the scatter of independent mea-
surements) monotonously with Rabi frequency over one order of magnitude, from
γ (Ω ≈ 1 GHz) ∼ 25 MHz to γ (Ω ≈ 20 GHz) ∼ 260 MHz.
Two black lines, one dashed and one dotted show fits to the data: The dashed
line assumes a parabolic dependence on the Rabi frequency with a constant offset
(no linear term), while the dotted line is based on a linear dependence with a con-
stant offset. For Rabi frequencies below 10 GHz the linear function matches the
data well, but strongly deviates at higher frequencies. In contrast, the quadratic
function (dashed) yields good agreement only at higher Rabi frequencies (Ω >
few GHz). The constant term in the fit overestimates the dephasing significantly
as we go towards small Rabi frequencies. In the inset the same data is shown, but
with the abscissa scaled as the square of the Rabi frequency. Here we can see,
perhaps more clearly, that the dephasing rate is approximately linear in Ω2 at
higher powers and deviates from this behaviour at lower powers. This deviation is
at odds with the expected Ω2-dependence predicted by acoustic phonon-coupling
and currently there is no intuitive explanation. We note that the effect of the
Overhauser field at different excitation powers is not clear: The beating is obvious
at high Rabi frequencies, but the beat note shifts to lower frequencies at lower
powers. Further, the regime where the dephasing rate fits a linear Ω dependence
is not covered in the published experimental investigations [110; 115; 116].
1at this point
95
0 5 1 0 1 5 2 0 2 50
2
4
6
8
1 0
T 2 .
 Ω
R a b i  f r e q u e n c y  Ω ( G H z )
1 1 00 . 0
0 . 1
0 . 2
0 . 3
Pur
e d
eph
asin
g ra
te γ
 (G
Hz)
R a b i  f r e q u e n c y  Ω ( G H z )
0 2 0 0 4 0 0
0 . 1
0 . 2
γ (G
Hz)
Ω2 ( G H z 2 )
a b
Figure 4.13: Power-dependence of pure dephasing rates from the data in Fig. 4.12.
a, Pure dephasing rate as a function of Rabi frequency (inset: as function of squared
Rabi frequency). Dashed line: fit according to a + bΩ2. Dotted line: fit according to
e + dΩ. b, Number of 2pi oscillations within the dephasing time as a function of Rabi
frequency.
Finally, panel b of Fig. 4.13 illustrates the consequences of excitation-induced
dephasing for coherent control schemes that involve QD excitonic states. Here
we show the product of the dephasing time T2 and the Rabi frequency Ω. This
quantifies the number of full (2pi) Rabi oscillations performed within the dephas-
ing time. In the absence of excitation-induced dephasing the data should follow
a straight line. In contrast, the product saturates within the accessible range
of Rabi frequencies, reaching a maximum rotation of 9 × (2pi). Extrapolating
this trend to even larger Ω, the product should decrease until the Rabi frequency
exceeds the cut-off frequency of the exciton-phonon coupling. The cut-off can be
estimated from the peak of the phonon sideband in Fig. 4.11 as ωc ∼ 1 meV ≈
250 GHz. For QIP applications it is of course desirable to operate at Ω  ωc,
however reaching these powers is not feasible in cw with current sample struc-
tures. Excitation with ps pulses may allow us to reach this goal in the foreseeable
future.
Overcoming the issue of dephasing due to phonon coupling with very strong ex-
citation is one option. Another interesting approach would be to engineer the
density of phonons states, for instance through microfabrication [143].
96
b- 0 . 5 0 . 0 0 . 50 . 0
0 . 5
1 . 0 X 0  t r a n s i t i o ns  ~  0 . 1
 
 
Spe
ctru
m (
a. u
.)
F r e q u e n c y  ( G H z )
- 0 . 5 0 . 0 0 . 50 . 0
0 . 5
1 . 0
 
 
Spe
ctru
m (
a. u
.)
F r e q u e n c y  ( G H z )
X 1 -  t r a n s i t i o ns  ~  0 . 0 5
a
Figure 4.14: Emission spectra of resonantly driven X0 (a) and X1− (b) transitions.
The excitation power is such that ≥ 90 % of the scattering should be coherent.
4.2.3 Nuclear Overhauser field
The effects of the nuclear spin bath on properties of a single resident electron spin
in a self-assembled QD were discussed briefly in the introductory chapter. Here we
present the experimentally observable signatures of the nuclear Overhauser field
in the resonance fluorescence emission spectra of a single QD. The effect is most
accessible when we excite weakly, far below the saturation power, but present at
any excitation power. We start the discussion by comparing the emission spectra
in Fig. 4.14 of the X0 and X1− transitions from the same QD (‘Claude’) at simi-
lar excitation conditions. The X0 spectrum shows the familiar resolution-limited
coherent Rayleigh peak. The X1− spectrum sports two sidebands in addition,
spaced roughly symmetric about the Rayleigh peak and centred at ± 200 MHz.
To understand the origin of the sidebands we have to consider the optical level
structure of the trion transitions in the presence of the Overhauser field.
For any orientation the Overhauser field may be decomposed into two compo-
nents, see the left-hand side of Fig. 4.15: a Faraday-like field parallel to the
optical axis (out-of-plane), and a Voigt-like field which is orthogonal to the op-
tical axis and can point in any in-plane direction. We note that the comparison
to Faraday and Voigt configurations is not entirely accurate. The ground states
split in the same way, but the analogy is not valid for the excited state, where
any splitting is due to the heavy-hole response to the field. Given the vanish-
ing Fermi contact hyperfine interaction for hole spins the excited state splitting
97
Faraday geometry
Voigt geometry
B
B
optical axis
in-plane
out-of-plane
B
nuc
out-of-plane
+
-
L
1
L
1
iL
1
+ 
-
B (t)
nuc
in-plane
L
2
L
2
iL
2
B (t)
nuc
ω
laser
ω
laser
Figure 4.15: Influence of the Overhauser field on the transition selection rules. The
out-of-plane component gives rise to Faraday-configuration selection rules, while the
in-plane components give rise to Voigt configuration selection rules. For a Faraday
field we expect a single emission frequency in the Heitler regime, while three emission
frequencies are available in Voigt configuration.
is negligible. For convenience we continue the analogy with Voigt and Faraday
components, however. The Faraday component splits the X1− into two circularly
polarised transitions with little cross-talk (branching ratio ∼ 200:1). Changes
in the Faraday component of the Overhauser field change the splitting between
the two transitions. For instance, if the Faraday field were to switch direction,
the transition energies would switch (see sketch on the top right-hand-side of
Fig. 4.15). The exciting laser is linearly polarised and its energy corresponds
to the mean of the two transitions, i.e. the transition energy in the absence of
the Overhauser field. Considering the weak excitation limit where coherent scat-
tering dominates we expect to scatter Rayleigh photons from either of the two
transitions with equal weight. Raman scattering is not allowed by the selection
rules and only occurs rarely. Hence the spectrum should be similar to the X0
spectrum: we expect a single peak at the laser frequency.
The electron g-factor is isotropic so a Voigt field splits the spin ground state as
before (cf. sketch at the bottom right-hand side of Fig. 4.15). For holes in our
sample the valence band mixing is small [26; 28] and hence the in-plane g-factor
is small. At the same time the hyperfine interaction with the nuclei is weak as
the hole orbitals have p-symmetry. With the resulting negligible excited state
98
splitting the level structure can be approximated by a Λ-system: both ground
states couple equally to the excited state. The transition energies vary with the
magnitude of the Voigt Overhauser field and the in-plane direction determines
the linear selection rules with respect to the laser polarisation: the two transitions
are orthogonally polarised (indicated by L and L¯ in the sketch) for any in-plane
direction, but the exact polarisation state varies with the azimuthal angle. For
weak excitation both transitions still scatter Rayleigh photons, but at the same
time coherent Raman scattering into either ground state is allowed, such that
three optical frequencies exist. While the energy of the Rayleigh scattering is
fixed by the laser, the Raman photon energy is determined by the ground state
splitting. For a given Voigt field we expect Raman photons blue and red detuned
from the Rayleigh peak by half the ground state splitting. A Voigt field with a
fixed magnitude then gives rise to three narrow (subnatural linewidth) peaks. As
the Voigt Overhauser field evolves it changes the ground state splitting and the
energy difference between Raman and Rayleigh scattering follows. The resulting
spectrum should feature two symmetric and extended Raman sidebands1.
The X1− spectrum from Fig. 4.14 b can be qualitatively explained when both
out-of-plane and in-plane nuclear Overhauser fields are considered. To verify this
picture and gain quantitative insight on the nuclear field amplitude Jack Hansom
and Claire Le Gall have developed a numerical model to calculate the emission
spectra of the four-level trion system in the presence of a Gaussian nuclear field.
The model shall not be explained here in detail as it is not my work (see Ref.
[144]). Here it is used to compare simulation results to a few experimental spectra.
Figure 4.16 shows X1− spectra obtained for QD ‘Claude’ at different excitation
powers (experimental data as green squares). Emission spectra are calculated
for a range of Overhauser field variances. In the calculation the field variance is
varied in steps of 10 MHz (corresponding to ∼ 2.5 mT magnetic field) up to 200
MHz (∼ 50 mT) while experimental parameters (e.g. excitation parameter s and
lifetime T1) are fixed. We display three theoretical curves for each experimental
spectrum: a best-fit is shown as continuous red curve, a dashed curve for a vari-
ance about 100 MHz and a dotted curve for a variance of 200 MHz.
The main features of the measured spectra are well reproduced for a nuclear field
1this holds for detuned excitation we well
99
- 1 0 1
0 . 0
0 . 5
1 . 0
Spe
ctru
m (
a. u
.)
F r e q u e n c y  ( G H z )
s  =  9B n  v a r i a n c e 1 0 0  M H z 1 5 0  M H z 2 0 0  M H z
- 0 . 5 0 . 0 0 . 5
0 . 0
0 . 5
1 . 0 s  =  1B n  v a r i a n c e 1 0 0  M H z 1 5 0  M H z 2 0 0  M H z
Spe
ctru
m (
a. u
.)
F r e q u e n c y  ( G H z )
hg
fe
dc
b
- 1 0 1
0 . 0
0 . 5
1 . 0
Spe
ctru
m (
a. u
.)
F r e q u e n c y  ( G H z )
s  =  1 4B n  v a r i a n c e 1 0 0  M H z 1 5 0  M H z 2 0 0  M H z
- 1 0 1
0 . 0
0 . 5
1 . 0
Spe
ctru
m (
a. u
.)
F r e q u e n c y  ( G H z )
s  =  7 . 9B n  v a r i a n c e 1 0 0  M H z 1 4 0  M H z 2 0 0  M H z
- 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0
0 . 0
0 . 5
1 . 0 s  =  3 . 7B n  v a r i a n c e 1 0 0  M H z 1 4 0  M H z 2 0 0  M H z
Spe
ctru
m (
a. u
.)
F r e q u e n c y  ( G H z )
- 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0
0 . 0
0 . 5
1 . 0 s  =  1 . 8 5B n  v a r i a n c e 1 0 0  M H z 1 7 0  M H z 2 0 0  M H z
Spe
ctru
m (
a. u
.)
F r e q u e n c y  ( G H z )
- 0 . 5 0 . 0 0 . 5
0 . 0
0 . 5
1 . 0
 
 
Spe
ctru
m (
a. u
.)
F r e q u e n c y  ( G H z )
s  =  0 . 4B n  v a r i a n c e 1 0 0  M H z 1 6 0  M H z 2 0 0  M H z
- 0 . 5 0 . 0 0 . 5
0 . 0
0 . 5
1 . 0
 
 
Spe
ctru
m (
a. u
.)
F r e q u e n c y  ( G H z )
s  =  0 . 1 6B n  v a r i a n c e 1 0 0  M H z 1 5 0  M H z 2 0 0  M H z
- 0 . 5 0 . 0 0 . 5
0 . 0
0 . 5
1 . 0 s  =  0 . 0 2B n  v a r i a n c e 1 1 0  M H z 1 6 0  M H z 2 0 0  M H z
 
 
Spe
ctru
m (
a. u
.)
F r e q u e n c y  ( G H z )
a
Figure 4.16: Emission spectra for resonantly excited X1− transition at zero external
magnetic field. Data (green squares) are measured with a scanning Fabry-Pero cavity
of ∼ 30 MHz resolution. The continuous curves (red, black dashed, black dotted) are
results from simulations for a four-level system and Gaussian nuclear field variance.
100
variance of ∼ 140-160 MHz corresponding to ∼ 25 mT. The emission spectra are
most sensitive to the Overhauser field for weak excitation and the pronounced
sidebands allow accurate determination of the variance. At higher excitation
powers the effect of the nuclear field on the emission spectra is visually less strik-
ing than in the low power limit. Here, the central peak of the Mollow triplet is
broadened and the sidebands assume an asymmetric shape in addition to being
broadened. We note that in panels g and h, the sidebands appear to split into
two peaks, while the simulated spectra are always single-peaked and smooth.
This points to a non-Gaussian field distribution, possibly due to a laser-driven
dynamic spin polarisation [35]. An actual splitting of the sidebands would be
consistent with the beating observed in the first-order coherence measurements
at very high excitation powers (cf. Fig. 4.12).
In conclusion, the emission spectra of the negatively charged trion transition can
be well explained by taking a fluctuating nuclear Overhauser field into account.
Conversely, the resonance fluorescence spectra serve as a direct and sensitive
probe for interactions with the nuclear spin bath. Along these lines it may be
interesting to measure the positively charged trion transition to investigate hole-
nuclear spin dynamics [109; 145]. Finally, we note that the Overhauser field could
be used as a resource for coherent control of spins in the absence of an external
field [144].
4.3 Detuned excitation: quantum to classical
Rayleigh scattering
The correspondence principle states that in some limit, usually for large particle
number or high temperature quantum effects are masked and classical physics
can describe a system adequately. When considering resonant light scattering,
the limit of strong excitation gives rise to Rabi oscillations and the Mollow triplet,
manifestations of quantum coherence between the optically coupled levels. At low
driving powers however, the coherent part of resonance fluorescence dominates
(resonant Rayleigh scattering) and the spectrum corresponds to that of a classi-
101
cal driven harmonic oscillator. The statistical properties of the fluorescence still
remain profoundly quantum mechanical though: antibunching is observed at all
excitation powers. This interesting property was discussed earlier. Rayleigh scat-
tering is usually thought of as a classical effect and antibunching is not expected.
A trivial reason not to observe antibunching is that Rayleigh scattering is rarely
measured on the single-particle level, but as we shall see there is more subtle
physics going on.
Where antibunching signifies the quantum-ness of single two-level systems, a
second-order correlation of unity is reserved for coherent fields, i.e. lasers. Clas-
sical fields are thermal and this is characterised by a bunching behaviour in the
second-order correlation: g
(2)
thermal (τ → 0) > 1. Rayleigh scattering is usually as-
sociated with a strong detuning (many linewidths) of the polarising field from
the atomic resonance. Large excitation detunings provide an alternative way (to
low power resonant excitation) to generate elastic scattering. More precisely, for
detunings 2piδ > 1
T2
√
1 + s, i.e. detunings greater than the half-width at half
maximum of the absorption linewidth, elastic scattering becomes dominant, even
at high driving powers. We will use a QD to investigate off-resonant elastic scat-
tering in the single-emitter limit.
Here we use the QD X0 transition (of QD ‘Isidor II’) and set the excitation
power to the saturation value (s=1) to have a a decent signal strength while
avoiding to dress the two-level system. Figure 4.17 a shows the QD fluorescence
spectrum in a semi-logarithmic scale for a set of detuning values between ∼ ±1
GHz. Two of the experimental spectra are highlighted to illustrate the effect of
detuning. Close to resonance (dark blue shaded spectrum, δ ∼ 50 MHz) coherent
and incoherent parts exist with roughly equal strength. As detuning is increased
the broad incoherent component dies away, and at δ ∼ 750 MHz the coherent
peak alone remains visible. Figure 4.17 b displays the theoretical prediction for
the dependence of the coherent fraction on detuning (for s=1, T2 = 1.9T1) in the
absence (red dashed line) and in the presence of spectral diffusion (blue line).
The coherent fraction of the total fluorescence Fc behaves as
102
- 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5
δ ~   7 5 0  M H z
δ ~   5 0  M H z
 
 
RF 
spe
ctru
m (
a.u
.)
F r e q u e n c y  ( G H z ) - 1 0 1
0 . 4
0 . 6
0 . 8
1 . 0
∆F W H M  ~       0  M H z
∆F W H M  ~   2 0 0  M H z
 
 
RF 
Coh
ere
nt f
rac
tion
E x c i t a t i o n  d e t u n i n g  δ ( G H z )
a b
Figure 4.17: Dependence of emission spectra and coherent fraction on detuning of
the excitation laser. a, Emission spectra for X0 transition for s = 1 on a logarithmic
intensity scale. The coherent fraction increases strongly with laser detuning. b, The-
oretically expected detuning dependence of coherent fraction in the presence (dashed
red) and absence (blue) of spectral diffusion for s = 1 and T2 = 1.9T1.
Fc =
(2piδ)2 T2 (2T1)
−1 + (2T1T2)
−1
(2piδ)2 + (1 + s)T−22
(4.18)
as a function of detuning. In analogy to the spectral diffusion model used before,
the expression is modified to
Fc,diffusion (δ) =
∑
∆
Fc (∆)× L (∆)×W (∆FWHM,∆, δ)∑
∆
L (∆)×W (∆FWHM,∆, δ) . (4.19)
In general, the coherent fraction is enhanced at small detuning δ ∼ 0, and de-
creased at larger detuning.
The intensity autocorrelation g(2) is recorded for each detuning |δ| in Fig. 4.17
a. Figure 4.18 displays four representative measurements with detailed fit func-
tions. The top panels contain the data as dark red curve with grey shading. The
blue continuous curve is a convolution of the theoretical expression based on Eq.
(2.31) and including spectral diffusion (see Eq. 4.14) and the system response
function (cf. Fig. 3.16). The deconvolved fit is plotted as dashed black curve.
For clarity the convolved (middle panels) and deconvolved (bottom panel) curves
are displayed again without the data in the lower panels. Here, the blue curves
are based on the fits to the data, while the red curves neglect spectral diffusion,
103
but use the same parameters otherwise.
The detuning modifies the effective Rabi frequency according to Ω˜ =
√
Ω2 + (2piδ)2,
where Ω = s/ (T1T2) is the bare Rabi frequency. In the limit of large detuning,
i.e. large effective Rabi frequency, the amplitude of the coherent oscillations in
the intensity autocorrelation is not centered around unity, unlike in the case for
large Rabi frequencies at zero detuning. In the latter case the oscillations go from
zero at τ = 0 to two after a pi rotation and decay symmetrically around unity
within the coherence time T2. With a large excitation detuning the intensity
autocorrelation can exceed the value of 2, which is the τ = 0 limit for chaotic
light. The deconvolved fit to the data in Fig. 4.18 crosses this limit. As the
detector resolution is finite, the contrast of the oscillations washes out and we
average between peaks and troughs. For the largest detuning presented here, δ ∼
810 MHz, the antibunching signature has almost completely vanished and the
autocorrelation is dominated by the first peaks on either side. The data might
be described more accurately as bunching, instead of antibunching !
This discussion may seem a bit artificial here; of course the fluorescence is still
antibunched - as it always is for a single two-level system. It is the timing resolu-
tion that prevents us from measuring this part. Extending the scheme from Fig.
4.18 further in experiments is challenging, since the signal is strongly reduced for
large detuning and the data acquisition becomes unreasonably long. To complete
the picture we keep the theoretical prediction only and set the detuning to twenty
times the natural linewidth. The result is presented in Fig. 4.19. The amplitude
of the oscillation approaches four around τ = 0. Averaging over an oscillation
period gives a simple curve
g(2) (τ)averaged = 1 + exp
−|τ |/T2 (4.20)
which is consistent with the behaviour of a classical light source.
The apparent bunching in the fluorescence can be explained intuitively1 with
higher-order corrections to Rayleigh scattering [69]. We note that off-resonant
QD excitation and sideband correlations were explored in the strong excitation
regime by Ulhaq et al. [146].
1i.e. no quantitative analysis will be given
104
12 ∆         ~   3 3 0  M H z∆F W H M  ~   2 2 0  M H z
1
2
g(2) (
τ)
- 5 0 5
1
2
T i m e  d e l a y  τ ( n s )
dc
1
2 ∆         ~   8 1 0  M H z∆F W H M  ~   3 8 0  M H z
1
2
g(2) (
τ)
- 5 0 5
1
2
T i m e  d e l a y  τ  ( n s )
1
2 ∆         ~   5 5 0  M H z∆F W H M  ~   1 5 0  M H z
1
2
g(2) (
τ)
- 5 0 5
1
2
T i m e  d e l a y  τ  ( n s )
1
2 ∆         ~       0  M H z∆F W H M  ~   2 8 0  M H z
1
2
g(2) (
τ)
- 5 0 5
1
2
T i m e  d e l a y  τ  ( n s )
a b
Figure 4.18: Second-order correlation functions for finite detunings. Top panels:
data (noisy red, grey shading) with a fit (blue curve) taking into account spectral
diffusion and the system resolution and the deconcolved fit (dashed black). Middle
panels: Convolved fit with (blue) and without (red) spectral diffusion. Bottom panels:
deconvolved fit with (blue) and without (red) spectral diffusion.
105
- 5 0 50
1
2
3
4
∆   =   5 0 0 0  M H z
 
 
g(2) (
τ)
T i m e  d e l a y  τ  ( n s )
Figure 4.19: Calculated intensity correlation function for s = 1, ∆ =5000 MHz (red
curve). The blue curve shows the effect of finite timing resolution for this calculation.
106
Chapter 5
Optical manipulation of electric
fields and quantum dot
electrometry
It has been observed by a number of groups working on resonant QD spectroscopy
[58; 81] that the QD transition shifts as a function of the resonant excitation
power. This effect has not been explored in detail, to my knowledge, but is im-
portant in the context of coherent optical spin manipulation [22], as resonance
shifts in the optical transitions lead to phase shifts in the spin evolution. If not
properly accounted for, this is a source of dephasing.
Here, we study the power dependence of resonance shifts in the steady-state, i.e.
under continuous-wave illumination, and time-resolved shifts under pulsed excita-
tion. Ruling out laser-induced heating we identify charging of the heterostructure
via optical absorption in the QD gates as the cause. The effect may be exploited
for fast spectroscopy, or optical switches.
Data acquisition and analysis for this chapter has benefited from collaboration
and discussion with the members of the QD team. In particular, I am grateful
to Martin Geller for fruitful discussions on thermometry and how to measure
of the Fermi-Dirac distributions. Rob Stockill contributed to the thermometry
and electometry measurements. I am grateful for the patience of other QD team
members while these measurements were taken.
107
b1 0 - 4 1 0 - 3 1 0 - 2 1 0 - 1 1 0 0
- 3
- 2
- 1
0
 
 
Res
ona
nce
 fre
que
ncy
 (G
Hz)
E x c i t a t i o n  p o w e r  ( u n i t s  o f  s )
( ) ( )( ) 1 0 . 2 7 G H z  l n ( )r e s r e sv v p s p s= = −
- 4 - 2 0 2
0 . 0
0 . 5
1 . 0
 
 
Nor
ma
lise
d in
ten
sity
D e t u n i n g  ( G H z )
 0 . 0 0 0 3 5  s 0 . 0 0 3      s 0 . 0 4 4      s 0 . 1 8        s
a
Figure 5.1: Shifts of QD resonance as a function of resonant excitation power. a,
Series of absorption lineshapes obtained by scanning the gate voltage across the X0
transitions for different laser powers.b, Summary of resonance shifts as function of
power. For the red data points the power is decreased in subsequent measurements,
while it is increased for black data points. The dashed line is a fit according to Eq.
(5.1).
5.1 The observation
5.1.1 One-colour steady-state measurements
In this section we describe the observation of laser-induced resonance shifts and
quantify the magnitude of the shifts. Figure 5.1 a shows a series of absorption
lineshapes of the neutral exciton transitions (QD ‘Isidor II’) for different exci-
tation powers. We keep the excitation laser frequency fixed for all powers and
scan the gate voltage across the two resonances. Knowing the Stark shift (∼ 410
MHz/mV) the gate voltage is translated into frequency again. For the range of
powers displayed the QD resonance shifts by almost 2 GHz, which exceeds the
transition linewidth by far. Panel b shows the frequency shifts over five orders
of magnitude in excitation power on a semi-logarithmic scale. For reference, the
saturation point (s=1) corresponds to about 10 nW power incident on the QD
sample. Data from two sets of measurements are presented. For the red data
points the excitation power of successive line scans was approximately halved
(i.e. starting at higher powers), while we doubled the power with each scan for
the black data points. If the shift were to originate from a QD-intrinsic property
we would expect to see a saturation behaviour as the QD is strongly driven. This
108
is clearly not the case. On a semi-logarithmic scale, the resonance shifts follow a
straight line. A fit to the data using the empirical model
νres = νp=s + a× ln (p (s)) (5.1)
is shown as dashed black line. The constant in Eq. 5.1 gives a reference frequency
and the constant of proportionality is a ≈ −270 MHz. This corresponds to a
redshift of the QD transition of approximately one radiative linewidth every time
the power is increased by a factor 2.5. Over the range of powers studied there is
no significant deviation from the logarithmic dependence.
5.1.2 Modulated shifts: Two-colour steady-state
We can repeat the measurement, but this time keeping the resonant laser power
fixed while adding a second laser. This second laser, denoted pump laser, is
strongly red-detuned from the QD excitonic resonance (by about 15 nm). Its
energy is then also far below the bandgap of GaAs and the wetting layer. We use
a grating to remove the pump laser from the detection path. Figure 5.2 a shows
a two-dimensional plot of the X0 lineshapes as function of the pump laser power.
The first data column in the plot shows the case where only the resonant laser,
now called probe laser, is incident on the sample. Then the pump laser is added
and its power is doubled with each column. The QD transition frequency shifts
in a similar fashion to the single colour experiments of Fig. 5.1. We conclude that
the resonance shift depends on total (below bandgap) laser power incident on the
sample, regardless of the detuning to the QD transition: it is not a resonance
phenomenon.
Next, we investigate the bandwidth of the resonance shifts. To this end, the probe
laser is kept operating in continuous-wave mode while the pump laser amplitude
is modulated (modulated depth of 1) by a square wave of variable frequency. The
corresponding data is shown in 5.2 b. Here, the peak power of the pump laser
is 12 times higher than the probe laser power. At low modulation frequencies
four transitions are distinctly visible: the fine-structure split X0 transitions jump
between their steady-state resonances at the two power levels. Going to faster
modulations first broadens the individual transitions and then smears out the
109
- 4
- 2
0
2
4
P p r o b e :  0 . 2  sc w
P p u m p :  1 2  x  P p r o b e  
m o d u l a t e d
∆p u m p  ~  1 5  n m
X 0  p r o b e  +  p u m p
1 0 01 0P u m p  m o d u l a t i o n  f r e q u e n c y  ( H z )
Rel
ativ
e fr
equ
enc
y (G
Hz)
1
X 0  p r o b e
- 2
0
2
4 P p r o b e :  0 . 2  s ,  c wP p u m p :  0 - 2 5 0  x  P p r o b e ,  c w
∆p u m p  ~  1 5  n m
1 2 83 2820 . 50 P u m p  p o w e r  r e l a t i v e  t o  p r o b e
Rel
ativ
e fr
equ
enc
y (G
Hz)
0 . 1 3
a b
Figure 5.2: Two-colour shifts of QD resonance. a, the resonant laser (probe laser)
is kept at constant power and the X0 lineshapes are measured as a function of the
strongly detuned pump laser power. b, The probe power is constant and the pump
laser amplitude is modulated on and off for a range of modulation frequencies.
transition between the steady states. At a modulation frequency of about 1 KHz
only two transition frequencies remain and a new steady-state is reached: The
modulation exceeds the bandwidth of the laser-induced shifts, and the response
corresponds to that for a time-averaged incident power.
5.2 The origin
We consider two mechanisms that could lead to resonance shifts as a function of
incident power: build-up of an electric field and local heating of the sample due
to residual absorption. First we discuss sample heating and QD thermometry.
5.2.1 Temperature
It is well known that increasing the temperature of a semiconductor leads to
shrinkage of the band gap, due to thermal expansion of the lattice and increased
electron-phonon scattering. This is observed as a red-shift in the optical emission
from semiconductors, both for bulk and confined states. For a wide tempera-
ture range the famous empirical Varshni model [147] describes the band gap-
temperature relation well. Early measurements on QD ensembles seemed to con-
firm the validity of the model [148]. At very low temperatures deviations from
110
the Varshni model are apparent, however, and photoluminescent spectroscopy of
single QDs enabled accurate measurements here [149]. Recently, the temperature
dependence of the resonance frequency of single QDs has been measured in res-
onant transmission experiments by Kroner et al. [150] with very high resolution
(∼ µeV). They found a saturation of the resonance frequency for temperatures <
10 K. QD resonance fluorescence measurements as a function of temperature on
a different sample agree qualitatively while still resolving resonance shifts below
10 K temperature [151]. Close to the liquid helium temperature resonance shifts
remain small, however, in this case too. Based on these reference we can esti-
mate the change of temperature corresponding to our measured resonance shifts
to be on the order of several K (∼ 5 K). To confirm or refute local heating as
source of the resonance shifts we should attempt to measure the local temperature
independently.
QD Thermometry A possible indicator of the QD temperature was intro-
duced in the previous chapter: phonon sidebands in the resonance fluorescence.
The strength, and more sensitively, the shape of the sideband depends on tem-
perature as the thermal occupation of phonon modes changes [130; 131]. Power-
dependent sideband measurements do not show such a change in our measure-
ments however, despite the good quality of the data (cf. Fig. 4.11).
There is an alternative way to measuring temperature in our gated devices. The
back contact is made conductive by Si-doping and the chemical potential µF is
pinned at the dopant energy level. The broadening of the chemical potential is
given by the Fermi-Dirac distribution
nFD =
1
e(E−µF)/(kBT ) + 1
. (5.2)
If this broadening can be measured we have access to a primary thermometer. We
wish to probe the broadening with the QD ground state. The idea relies on carrier
thermalisation between the back contact and the QD ground state and is sketched
in Fig. 5.3: The occupation of the QD electronic ground state is linked to the
occupation in the back contact at the same energy. For instance, when the energy
of the QD one-electron state is more than kBT below the chemical potential the
111
occupation
energy
V
gate
μ
F
Figure 5.3: Sketch of the conduction band for part of the QD device. On the left is the
Si-doped back contact. The chemical potential µF is determined by the dopant concen-
tration. The occupation of electronic states in the back contact obeys the Fermi-Dirac
distribution which is shown as blue curve (occupation vs. energy) and additionally
indicated by the shading. The QD ground state (e.g. the single electron ground state)
is tunnel-coupled to the reservoir such that the occupation is in equilibrium with the
occupation in the back contact. Probing the QD ground state occupation as a function
of the gate voltage provides a way to map out the Fermi-Dirac distribution.
single electron is stable in the QD. Tunneling to and from the back contact does
not occur because no free states are available at this energy in the back contact.
When the QD one-electron energy level is raised to the chemical potential plenty
of free states are available in the back contact and the QD occupation of 0.5
reflects the occupation of the back contact. Increasing the reverse bias further
empties the QD as states of the same energy are not occupied in the back contact.
Continuing to use the example of the single electron ground state the occupation
of the ground state can be measured through optical absorption at the trion
transition energy. Sweeping the QD one-electron ground state energy through
the chemical potential via the applied bias and measuring the trion absorption
probability should then map out the Fermi-Dirac distribution.
Figure 5.4 a shows an exemplary two-dimensional voltage-frequency map of the
X1− resonance fluorescence intensity. The gate voltage range is chosen such that
the QD moves from the zero to the one-electron ground state. In Fig. 5.4 b
we plot the peak intensity for each data row from panel by fitting Lorentzians
to the gate voltage sweeps at fixed frequency. The abscissa scale is obtained by
translating the gate voltage into energy relative to the chemical potential using
112
- 2 0 20
2
4  
 
Inte
nsit
y (a
.u.)
E n e r g y  r e l a t i v e  t o  
 F ( m e V )
2 8 0 3 0 0 3 2 03 1 3 8 8 0
3 1 3 8 8 5
3 1 3 8 9 0
G a t e  v o l t a g e  ( m V )  
Fre
que
ncy
 (G
Hz)
0 . 0 0 0
8 0 . 0 0
1 6 0 . 0
2 4 0 . 0
3 2 0 . 0
4 0 0 . 0
a b
Figure 5.4: QD thermometry with resonance fluorescence. a, Gate voltage-frequency
map of the X1− transition. For low gate voltages the QD is empty and there is no signal.
As the forward bias is increased the QD ground state occupation moves smoothly from
0 to 1 electron and the resonance fluorescence signal is recovered - proportional to the
single electron ground state occupation. b, Plot of the resonance fluorescence intensity
as function of energy detuning between QD ground state and chemical potential. The
red line is a fit to the Fermi-Dirac distribution.
the lever arm method:
∆E1e, gs = e
dbarrier
ddiode
×∆Vgate, (5.3)
where the change in energy of the one-electron ground state E1e, gs depends on
the change in applied bias ∆Vgate and the ratio of distance between the back gate
and the QD layer (dbarrier) to the total length of the Schottky diode (ddiode). A
fit to the Fermi-Dirac distribution (Eq. (5.2)) is shown as red curve and gives
the temperature as 5.6 ± 0.2 K. It should be pointed out here that the lever
arm method is a first-order approximation and prone to systematic errors. In
particular, the distance dbarrier varies from the value specified for the growth as
charges from the doped back contact spill into the tunnel barrier and shorten the
distance. This leads to an overestimate of the temperature. Further, the laser
power should be kept constant throughout the measurement, also while scanning
the frequency. Stabilisation on the optical table is possible to considerably better
than 1 %, but the microscope beamsplitter transmission/reflection shows a fre-
quency dependence, which, depending on the particular frequencies, can give rise
to a 10 % change in power at the QD over ∼ 10 GHz. These systematic errors
113
should not keep us from testing the laser-induced heating hypothesis though, as
long as conditions can be kept comparable in subsequent measurements.
In Fig. 5.5 we present an analysis of QD resonance maps with a resonant probe
and an off-resonant pump laser. The raw data is shown in panels a, b and c.
Here, the resonant laser is fixed at 0.05 of the saturation power and the pump is,
in order of the panels, absent, ten times the power of the probe and 100 times
the probe power (panels a-c respectively). The analysis follows the procedure
outlined above, and panel d displays the peak fluorescence intensity as function
of QD ground state detuning from the chemical potential. Data for the three
measurements and fits according to Eq. (5.2) are colour-coded: red corresponds
to the data in panel a, green to panel b and black to panel c. The similarity
between the data sets is striking and the temperatures given by the fits agree
with each other within their errors1.
To summarise: so far, we have not seen any evidence for laser-induced heating.
We would expect a temperature change of several K, but this is ruled out by
measurements of the Fermi distribution. However, the QD resonance maps of
Fig. 5.5 do provide us with evidence of laser-induced electric-fields that shift the
QD transition and this is subject of the next section.
As an aside, we note that Fermi distribution measurements with optically ac-
tive QDs may be used in thermometry applications. Here we obtain a precision
close to 100 mK. The accuracy is limited by systematic errors in the lever arm
calculation to about 2-3 K. A big improvement here could be a more accurate
determination of the effective tunnel barrier thickness dbarrier (see Eq. (5.3)),
for instance through cross-sectional scanning tunneling microscopy of the QD
heterodiode structure. For practical purposes, integrating a Schottky diode for
thermometry may not be convenient in some applications, and the required elec-
tric field can additionally affect the semiconductor environment. By nature the
QD thermometer is local, so to achieve a spatial temperature map we need to
measure a set of QDs at reasonably well-resolved positions. The spatial temper-
1We note the temperatures are different from what was obtained for the exemplary data
in Fig. 5.4. The current data set was recorded at a different time (time difference of some
weeks) and as such the alignment of the microscope was different, mainly in the excitation
polarisation. Consequently the frequency dependence of microscope beamsplitter reflectance
and transmittance differs for both measurements.
114
Figure 5.5: QD thermometry in the presence of a pump laser. a-c, QD resonance
fluorescence maps in gate voltage-frequency space for pump powers of zero (a), ten
times the probe power (b), and 100 times the probe power (c). d, Peak intensities
versus detuning from the chemical potential, fitted with the Fermi-Dirac distribution.
Red data points and fit correspond to panel a data, green to panel b and black to panel
c. e, Position of QD resonance in gate voltage-frequency space for panels a-c. Positions
where the intensity exceeds 5, 10 and 20 % of the full intensity are highlighted.
115
ature resolution is then given by the ability to map the QD locations. As for the
accessible temperature range, QDs are efficient emitters up to ∼ 50 K and this
limits the use to cryogenic temperatures. The main attraction of this method is
that we have a primary thermometer, that is, the thermometer does not require
calibration. We note that other methods, for example temperature-dependent
emission energy can be exploited for thermometry purposes (see Ref. [152] for a
review) and this principle has been successfully applied for colloidal II-VI QDs
[25].
5.2.2 Electric field
QD resonance shifts through temperature and electric-field changes can be told
apart by their dependence on the bias applied to the Schottky diode. If we
consider the 2D QD resonance maps in gate voltage and transition frequency
space (see Fig. 5.5 a-c) a temperature increase will push the entire transition to
smaller frequencies. The gate voltage range should remain unchanged, as it is
determined by the QD charging properties, which are tied to the applied bias.
In contrast, electric field changes (independent from the applied bias) should not
alter the transition frequencies, but shift it along the gate voltage axis. Going
back to the analysis in Fig. 5.5, we move to panel e. Here, we plot the QD
resonance positions from the 2D maps as lines. The colour code is indicated in
the legend and was already used in panel d. The resonance position is obtained
through Lorentzian fits to each data row of panels a-c. For ease of comparison,
three data points are highlighted for each measurement. These mark the positions
(in gate voltage/frequency) where the QD transition reaches ≥ 5, 10 and 20 %
of the full signal strength. Within the noise these positions occur at constant
frequency, but shifted gate voltage. Hence the observed resonance shifts can be
firmly attributed to a laser-induced change in the electric field felt by the QD.
When the pump laser shifts the QD resonance, the gate voltage needs to be
increased to recover the QD resonance. In our case the laser-induced electric field
counteracts the applied field, it acts to reduce it. Further, the reversibility and
‘smoothness’1 of the shifts suggests that it is not due to dynamics of local defects,
1and the reproducibility on different QDs
116
but rather due to free carrier absorption in the gates of the heterostructure.
In the following section we will investigate details of the electric-field shift from an
electrometry point-of-view, with particular regard to the dynamics of the electric
field.
5.3 Time-resolved electric field dynamics and
QD electrometry
Here we turn to time-resolved measurements to obtain more information on the
bandwidth of the resonance shifts, and at the same time investigate the perfor-
mance of resonance fluorescence-based electrometry.
5.3.1 Considerations for QD electrometry
First, we discuss our results in the context of QD electrometry. Using the QD
resonance shifts as means to infer details about the QD environment was already
used in the investigation of spectral diffusion in section 4.2.1. To recap: changes in
the electric field around the QD act on the permanent dipole of the electron-hole
pair1 of excited QD states and as such translate to change in the QD transition
frequency. This Stark shift is linear in the range of electric field changes studied
here. The electric field sensitivity then depends on the change in optical signal
versus the noise in the detection.
The ability to probe electric fields using the optical response from QDs under
resonant excitation was explored in Ref. [24]. There, the use of homodyne detec-
tion in differential transmission and reflection schemes was advocated to pick up
on both DC and AC fields with a Lockin-Amplifier. Houel et al. used differential
transmission to locate the position of defects in the tunnel barrier close to a QD
[81]. The recent work by Kuhlmann et al. exploits resonance fluorescence to
measure electric (and nuclear) field noise timescales [127].
One strategy for using QD resonance fluorescence for electrometry is to set the
probe laser frequency at a detuning where the response to electric-field changes
1in particular in the growth direction
117
is greatest. This detuning is given by the maximum of the first derivative of the
lorentzian QD absorption lineshape and corresponds to
δmax =
1
2pi
√
1 + s
6T1T2
. (5.4)
Electric fields can be sensed if the signal change due to the QD resonance shift
exceeds the shot noise of the signal at the set detuning. If the QD photon de-
tection rate as a function of detuning ∆ and detection/collection efficiency η is
given by I(∆) = η × ρ22
T1
and the Stark shift coefficient by α, then electric field
changes δE for which
δE ≥
√
I (δmax)
∂
∂∆
I (δmax)× α
(5.5)
can be measured. To give an example using parameters from our system and
sample, for a typical QD with a natural linewidth of 250 MHz, a Stark shift
coefficient of 80 KHz (V/m)−1 and a combined collection/detection efficiency
giving 500 kcounts/s on resonance at saturation we obtain a theoretical electric
field sensitivity of
δE & 50V
m
√
Hz
−1
(5.6)
In practice, some of this sensitivity is taken away by constant ‘sensing’ of spectral
diffusion. While this is in principle already electric field sensing, it reduces our
ability to detect electric field changes on top of this background.
A different strategy to electric field sensing is to scan across the QD resonance
while detecting photons and fit the absorption lineshape to obtain the exact QD
resonance. This approach, employed in the next sections yields experimental
sensitivities around 50 (V/m) /
√
Hz or better, on top of the uncontrolled spectral
diffusion and without operating under optimum conditions.
For the remainder of this chapter we will shift the focus back to investigating
the dynamics of the laser-induced electric field. For simplicity we will continue
working in units of gate voltages, but we keep in mind that the following results
are a practical application of electrometry: to convert the changes in gate voltage
118
into electric field changes we just have to divide by the length of the QD diode
(185 nm).
5.3.2 Dynamics of laser-induced electric field
Measurement technique Measurements with a modulated pump laser were
introduced in Fig. 5.2. Now, the duty cycle of the modulated pump laser is
reduced ( 1) to allow the transition to relax back into its steady-state after the
pump pulse. We use an electro-optic modulator with high bandwidth (5 GHz)
modulated by a digital delay/pulse generator (SRS DG645) with a transition time
< 2 ns to create fast laser pulses and record time-resolved fluorescence detection
events with a time-to-amplitude converter (quTools quTAU). Figure 5.6 shows a
time-resolved map of the QD fluorescence for a pump pulse of 20 µs duration and
a repetition rate of 50 Hz. The probe laser frequency is fixed and time-resolved
data is recorded for 60 seconds at each gate voltage. Data acquisition is triggered
by the same pulse generator that drives the EOM and lasts in this case for 20
ms with a resolution of 200 ns. The top left panel shows the first 100 µs of the
raw data, where the pump pulse is incident after about 20 µs, while the top right
panel shows the remainder of the data, here binned to 2 µs resolution. In the
bottom panels we trace the QD resonance position, obtained through Lorentzian
fits for each time delay (200 ns resolution in both bottom panels).
The QD response to the pump pulse is characterised by a fast and monotonous
resonance shift during the pulse. Two timescales are present in this response
risetime; this will be discussed for higher timing resolution data presented later
on. Following the pump pulse, the transition drops rapidly to slightly lower gate
voltages and then slowly relaxes to its equilibrium position. The rapid drop is
instantaneous within the resolution of this data, while the slow relaxation hap-
pens on the order of milliseconds. Comparing these timescales to the bandwidth
measurement with a square-wave modulated pump in Fig. 5.2 we identify the
slow millisecond relaxation time as the dominant part: When the repetition rate
exceeds roughly 1 KHz, the transition does not relax back to its ‘unperturbed’
resonance and a new steady-state exists.
119
Figure 5.6: Time-resolved response of QD transition to a pump pulse of 20 µs dura-
tion. Top panels show the raw data, bottom panels the QD resonance (in gate voltage)
which is obtained by fitting Lorentzians to constant-time cuts in the top panels. Top
left: pixel timing resolution of 200 ns, 0.6 ms integration per pixel. Top right: pixel
timing resolution of 2 µs, 6 ms integration. Bottom: Timing resolution of 200 ns in
both panels. The pump pulse peak power 400 times the probe power.
120
Detailed dynamics Having introduced the experimental technique to mea-
sure the laser-induced electric field changes we take a closer look at the dynamics
now. The timing resolution in Fig. 5.6 was too low to resolve the the response
to the onset of the pump pulse, so here we use faster pump pulses and higher
timing resolution. In Fig. 5.7 we present data for pulses of nominally 2 and 10
ns duration. The plots on the left (a and c) show the system response functions
for the two pulses. These are measured by sending the pump laser only to the
QD and detecting part of its reflection into the collection fibre, using the same
equipment as for the time-resolved pump-probe experiments. For a rectangular
pulse of nominally 2 ns duration we obtain am approximately Gaussian shaped
pulse with a FWHM of 1.4 ns. The shape is determined almost entirely by the
bandwidth of the pulse generator. Panel b shows the dynamics of the QD res-
onance in a pump-probe measurement using the pump pulse from panel a. The
data is derived from a time-detuning map as before in Fig. 5.6. The response of
the QD resonance suggests that the pump-induced electric field follows the pulse
intensity closely, but the width of the feature (∼ 3 ns) is broader than the pump
pulse width. The fall time of the response seems to be slightly slower than the
rise time. Taking into account the original width of the pump pulse, the response
of the electric field felt by the QD is about 2.5 ns, corresponding to a bandwidth
of 400 MHz. Panels c and d of Fig. 5.7 display the pulse shape and the QD
resonance response for a pump pulse of nominally 10 ns duration.
Interestingly, the QD resonance displays an ‘overshoot’ behaviour: once the pump
pulse has passed, the QD resonance shifts slightly past its original resonance
into the opposite direction, before relaxing back to the original resonance setting
within 10-100 ns. This phenomenon may be related to a backaction of the charge
distribution in the QD gates after the quick build-up and discharge of the pump-
induced field, but this is not resolved at the moment.
To complete the picture we employ QD electrometry to look at laser-induced
fields for pump pulses with durations ranging from 100 ns to 200 µs in Fig. 5.8.
In each panel we show the dynamics of the QD resonance over the full time range
of the measurement in the main plot. The immediate response to the pump pulse
is shown in more detail in the insets of each panel. Here we highlight the timing
of the pulse and its duration by shading the respective area in the inset in green.
121
dc
b
2 0 0 3 0 0 4 0 0
0 . 0 0
0 . 0 5
0 . 1 0
 
 
Cha
nge
 in r
eso
nan
t ga
te v
olta
ge 
(mV
)
T i m e  ( n s )
2 6 0 2 8 0
 
 
F W H M  ~  3  n s
2 6 0 2 7 0 2 8 0 2 9 0
Inte
nsit
y (a
. u.
)
T i m e  ( n s )
P u m p  p u l s eF W H M  ~  9 . 6  n s
1 0 0 0 1 5 0 0
0 . 0
0 . 2
0 . 4
0 . 6
0 . 8
 
 
Cha
nge
 in r
eso
nan
t ga
te v
olta
ge 
(mV
)
T i m e  ( n s )
1 1 5 0 1 2 0 0
 
 
F W H M  ~  9 . 7  n s
2 5 5 2 6 0 2 6 5 2 7 0 2 7 5
Inte
nsit
y (a
. u.
)
T i m e  ( n s )
P u m p  p u l s eF W H M  ~  1 . 4  n s
a
Figure 5.7: Time-resolved QD electrometry for pump pulses of 1.4 ns (a, b) and 10 ns
(c, d) duration. a, c, Measurements of the pump pulse intensity profile. b, d, Response
of the QD resonance. The timing resolution in panels a-c is about 0.5 ns, limited by
the APD jitter. The timing resolution of panel d data is 1 ns. The integration time
per pixel for the original data for b is 25 ms and 5 ms for d.
122
For a pump pulse of 100 ns (see Fig. 5.8 a) the response follows the rectangular
shape of the pump initially. Similarly to the data for shorter pulses presented
in Fig. 5.7 there is an overshoot immediately following the pump pulse. How-
ever, the response curve crosses the steady-state (0 mV on the ordinate axis) a
second time, reaches a maximum and only then decays monotonically back to
the equilibrium gate voltage. Increasing the pump pulse duration to 500 ns in
panel b we find clear differences between the intensity profile of the pump pulse
and the electric field response felt by the QD: After the initial fast rise of the
response curve (which follows the rise time of the optical pulse closely) the QD
resonance continues to shift in the same direction while the pump laser power
is constant, but at a much slower rate. In panel b this shift is approximately
linear in time. The sharp drop of the response curve follows the fall time of the
optical pulse again, but no overshoot is visible this time. Further increasing the
pump pulse duration to 2 and 200 µs in panels c and d, respectively, it becomes
obvious that the QD resonance approaches a new steady-state position during the
constant intensity part of the pump pulse. Here the behaviour is approximated
by a (1− exp(−t/τc)) dependence. The characteristic time scale τc is some tens
of microseconds (cf. panel d).
The slow final decay back to the unperturbed resonance shows a non-trivial depen-
dence on pump pulse power and duration. It varies from few tens of microseconds
to milliseconds for the data presented here. The decay is further influenced by
the finite repetition rate of the measurement scheme. For a detailed study of the
relaxation rate the repetition rate should be significantly reduced.
In conclusion, we have presented a detailed study of laser-induced QD resonance
shifts, quantifying both amplitudes and the timescales. Performing QD thermom-
etry and electrometry we have identified the origin of the resonance shifts as laser
absorption in the QD sample gates leading to the build-up of an electric field.
While this conclusion is perhaps less exciting it allows us to disentangle this un-
wanted effect from other experimental data. Further, the investigation confirms
the potential of optical electrometry, where resonance fluorescence enables high
sensitivity and high bandwidth operation. Finally, having identified the sample
gates as problematic, we can look into sample structures which avoid the issue.
123
Figure 5.8: Tracing the QD resonance for pump pulses from 100 ns to 200 µs duration.
The inset displays the dynamics around the pump pulse in greater detail. a, 30 ms
integration time per time bin of 10 ns. b, as a. c, 15 ms integration time per time bin
of 50 ns. d, 60 µs integration time per 400 ns time bin.
124
Chapter 6
Coherent single QD photons for
quantum optical networks
The ‘Quantum Internet’ envisions remote stationary qubits connected via fly-
ing qubits, where local operations and entanglement between the distant qubits
enables quantum computation. Quantum dots offer a natural solution where a
trapped spin encodes the stationary qubits, while resonantly scattered photons
can carry the spin information over macroscopic distances. The latter was re-
cently exploited to demonstrate spin-photon entanglement for QDs [53; 65; 66].
The missing link so far is an experimental demonstration of distant QD spin en-
tanglement. Distant entanglement has been successfully demonstrated for other
physical systems, most notably for trapped atoms [153; 154] and recently also for
the nitrogen-vacancy defect centre in diamond [155]. Entanglement is created by
interference and detection of two photons - each entangled with the host qubit
they originate from. This non-local entanglement based on photon quantum in-
terference has been identified as a promising and robust approach [156; 157]. The
robustness stems from the probabilistic nature of entanglement creation, placing
fewer constraints on detection efficiencies, but relies crucially and fundamentally
on indistinguishable photons from separate qubits: In the presence of pure de-
phasing or spectral diffusion photons become distinguishable and do not interfere.
Photon detection then erroneously heralds qubit entanglement. Coherent scatter-
ing from a QD provides an attractive way of generating dephasing-free photons,
125
protected to some degree from spectral diffusion, and we investigate their poten-
tial for quantum interference here.
This chapter builds on the observation of subnatural linewidth QD photons de-
tailed in Chapter 5. Ref. [158] contains the publication of material from this
chapter. We first demonstrate that the coherently generated single photons from
a single self-assembled InAs quantum dot display mutual coherence with the ex-
citation laser on a timescale exceeding three seconds. Exploiting this degree of
mutual coherence we tailor the coherent photon waveforms by shaping the ex-
citation laser field. In contrast to post-emission filtering, this technique avoids
both photon loss and degradation of the single photon nature. By engineering
pulsed waveforms of single photons, we then demonstrate that separate photons
generated coherently by the same laser field are fundamentally indistinguishable,
lending themselves to creation of distant entanglement through quantum inter-
ference.
Results presented in this chapter stem from a collaborative effort of the QD team.
In particular, Carsten H. H. Schulte built the heterodyning setup and performed
and analysed the measurement on QD fluorescence - laser mutual coherence (see
Fig. 6.1). Martin Geller and Claire Le Gall contributed greatly to the long and
tiring process of getting to the stage of collecting good two-photon interference
data1.
6.1 Considerations for two-photon interference
QD photon coherence is affected by the generation process. Above bandgap exci-
tation, while technically easy to implement, relies on the incoherent relaxation of
excitons in the QD ground state and emitted photons suffer from pure dephasing.
This was confirmed by interference of two photons emitted by the same QD at
different times. Keeping this time separation to a few nanoseconds avoids spectral
diffusion affecting interference of the photon pair and interference contrasts of ∼
70% were reported [159; 160]. However, two separate QDs experience indepen-
dent environmental dynamics: the environmental noise is uncorrelated. In this
1This includes two-photon interference measurements with two independent quantum dots
during the autumn and winter of 2011-2012 which did not make it into this thesis.
126
case two-photon interference from separate QDs using above bandgap excitation
showed only a modest contrast of 18 % [161].
Resonance fluorescence avoids optically exciting the host material and therefore
suppresses these dynamics. However, the ambient charge fluctuations, for ex-
ample, can still broaden a transition via spectral wandering as we discussed in
section 4.2.1. While not causing significant pure dephasing, the ‘wobble’ of the
QD resonance renders the use of QDs in a quantum optical network problem-
atic. Understanding the origin of these dynamics and eliminating them is a key
challenge. Alternatively, approaches to suppress their effects should be pursued.
6.2 Mutual coherence: Phase-locking of QD flu-
orescence to laser
Elastic scattering from an optical transition represents an alternative approach to
high quality photon generation, which avoids significant population in dephasing-
prone excited states. Unlike the spontaneously emitted photons, the elastically
scattered photons in the Heitler regime are expected to be phase-locked to the
excitation laser. This phase-locking translates to an ideally infinite mutual coher-
ence and accentuates their potential for quantum interference applications [162].
In well-isolated experimental systems, such as trapped atoms and ions, this mu-
tual coherence was observed via an optical heterodyning technique [163], which
allowed studying the broadening of elastic scattering due to motional and vibra-
tional dynamics [164; 165], when the species are driven by a single frequency laser.
The effect of environment dynamics, apparent through spectral wandering for in-
stance, can still limit the mutual coherence between the excitation laser and the
photons generated in the Heitler regime: QD resonance shifts translate to phase
shifts between the elastically scattered single photons and the excitation laser.
Therefore, phase-sensitive heterodyning measurements to determine the extent
to which phase-locking to the excitation laser occurs are necessary in order to
determine the suitability of these photons as flying qubits.
Figure 6.1 a shows the experimental arrangement used to perform optical hetero-
dyning measurements. QD resonance fluorescence at frequency ν superimposed
127
+
AOM
Spectrum Analyser
Laser
Local
Oscillator
PD 2PD 1
QD
Signal
-100 0 100 200 300 400
0
5
10
N
oi
se
 p
ow
er
 (µ
V²
)
Relative frequency (Hz)
-0.6 0 0.6
299
mHz
a
ν
νδ
b
ν
Figure 6.1: Mutual coherence between QD resonance fluoresence and the excitation
laser. a, Illustration of the experimental arrangement used for optical heterodyning
of QD resonance fluorescence. Acousto-optical modulation (AOM) provides the shift
of the local oscillator frequency by δν 210 kHz. The outputs of the two photodiodes
(PD1,2) are subtracted electrically and sent to a spectrum analyser. b, Typical power
spectrum of the beating signal as a function of frequency relative to δν. Inset: high-
resolution spectrum for 5 s of continuous data acquisition. Spectra with sub-hertz res-
olution reveal phase coherence between QD photons and excitation laser on a timescale
of seconds.
128
with a strong local oscillator originating from the excitation laser but shifted in
frequency by δν ∼ 210 kHz. This frequency difference results in a beat note
in the difference signal of two balanced photodiodes (PD1,2). A spectrum anal-
yser calculates the power spectrum via a fast Fourier transform (FFT) algorithm.
The linewidth and the lineshape of the power spectrum reflect directly the phase
stability, i.e. the mutual coherence, between the two fields during the course of
the measurement. In particular, periodic dynamics of the relative phase appear
as sidebands and aperiodic fluctuations appear as broadening of the spectrum.
Figure 6.1 b displays a representative raw spectrum with a strong peak at the
beating frequency. The inset shows that a Gaussian lineshape with a FWHM of
299 mHz fits the data of 5-second continuous acquisition. We note here that the
spectral bandwidth is partly due to the the system response resolution of 200 mHz
for the measurement settings and any external noise sources in the setup. Longer
acquisition time yields higher spectral resolution, hence narrower linewidths (e.g.
172 mHz for 10-second acquisition time), but phase fluctuations induced by the
mechanical instability of the setup (and any spectral diffusion) reduce the signal
quality and limit the obtainable linewidth.
Nevertheless, even these raw linewidths indicate that every photon is phase-locked
to the excitation laser with mutual-coherence time exceeding 3 seconds, which cor-
responds to a mutual-coherence length of one million kilometres. These results
confirm the prediction for an ideal two-level system that each photon inherits
fully the coherence properties of the excitation laser in this alternative photon
generation process. The absence of significant linewidth broadening in this mea-
surement confirms that the main contribution to spectral diffusion for our sample
(and this QD) occurs on timescales of seconds or slower.
6.3 Photon shaping
Methods currently available to control single photon wavepackets include direct
spectral filtering of the photons [166] and phase [167] or amplitude [168; 169] ma-
nipulation via photon transmission through electro-optic elements. Wavepacket
control during the photon generation process has only been achieved for trapped
atoms inside optical cavities using multi-pulse sequences [170; 171].
129
0 2 4 6 8 1 0 d
b 
Inte
nsit
y (a
.u.)
a e
fc - 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4L a s e r L a s e r  s p e c t r u m
L a s e r
 
Inte
nsit
y (a
.u.)
L a s e r  s p e c t r u m
Q D  s p e c t r u m
Q D  s p e c t r u m
- 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4
 
Inte
nsit
y (a
.u.)
0 2 4 6 8 1 0
 
Inte
nsit
y (a
.u.)
T i m e  ( n s )
- 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4
 
Inte
nsit
y (a
.u.)
F r e q u e n c y  ( G H z )
- 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4
 
Inte
nsit
y (a
.u.)
F r e q u e n c y  ( G H z )
Figure 6.2: Tailoring waveforms of single photons. a, The intensity of the excitation
laser is modulated in time using an electro-optic modulator driven with a 200-MHz sine
wave. b, The spectrum of the modulated laser, measured through a FabryPerot cavity
(resolution ∼ 20 MHz), shows sidebands at the modulation frequency. c, d, A different
waveform modulation with the corresponding spectrum for the laser. This waveform
is designed to suppress the spectral component at the original carrier frequency. e, f,
Measured spectra of the QD photons generated from the two synthesized laser wave-
forms given in a, c, respectively. The elastically scattered photons replicate the laser
spectrum within the linewidth of the transition, which is indicated by the shaded grey
area.
130
The coherent nature of elastic scattering provides a means for coherent synthesis
of single photon waveforms directly in the photon generation process without the
need for spectral filtering or optical cavities. In order to demonstrate this ability,
we use an electro-optic modulator (Jenoptik AM980) driven by a sine wave signal
generator (Rohde & Schwarz SMF100A) to encode the excitation laser field. The
throughput I of the EOM as a function of applied bias Vbias follows
I (Vbias) ∝ 1
2
cos (kVbias + φoffset) +
1
2
, (6.1)
where constant k relates the bias voltage to a phase shift of the cosine function
and φoffset determines the working point of the EOM. Consider driving the EOM
with a sine wave of frequency νmod and amplitude A. The intensity modulation
of the laser is then given by
I (t) ∝ 1
2
cos (kA sin (νmodt) + φoffset) +
1
2
(6.2)
Complex intensity patterns can be generated with when the amplitude of the
modulation exceeds the pi-Voltage of the modulator: kA > pi. The resulting
spectrum is given by the Fourier transform of the first-order correlation function
of the output field of the EOM.
Figure 6.2 presents two examples of coherent control of single photon waveforms
by this technique. The first example of a synthesized laser waveform and the cor-
responding spectrum are displayed in panels a and b, respectively. The temporal
measurement in panel a is performed using a photodiode with 8-GHz bandwidth,
while the spectrum in panel b is measured using the scanning Fabry-Perot inter-
ferometer. The amplitude modulation of the laser in time gives rise to sidebands
in the spectrum at the modulation frequency of 200 MHz. Panels c and d display
another example for encoding the laser field with a more complicated pattern.
The amplitude of the applied modulation is greater than the pi-Voltage and in
the resulting spectrum the component at the carrier frequency (zero detuning) is
strongly suppressed.
The single photon QD emission spectra for the two examples are presented in
panels e and f. The 200-MHz spectral spacing and the relative strength of the
131
frequency components are dictated by the laser spectrum for each example, in ad-
dition to being weighted by the QD transition linewidth. The radiative linewidth
(250 MHz) is indicated as gray shaded areas in panels e and f for comparison1. In
both cases (Fig. 6.2 e,f), antibunching in the intensity-correlation measurements
is fully sustained. We note that while these examples are based on amplitude
modulation of the laser field, encoding phase to each spectral component is pos-
sible in this scheme.
These measurements demonstrate that, as a consequence of their mutual coher-
ence with the laser field, the single photon waveforms can be synthesized deter-
ministically without post-generation filtering which can lead to photon loss and
degradation of the single photon quality. The accessible bandwidth for waveform
synthesis is restricted only by the mutual coherence for low frequencies, and the
QD transition linewidth for high frequencies.
6.4 Photon indistinguishability
A key requirement of a photonic link in a quantum network is the ability to
create time-synchronised indistinguishable single photons from separate sources.
However, the uncorrelated environment fluctuations experienced by separate QDs
under nonresonant excitation lead to variations in temporal and spectral overlap
of photon wavepackets [161; 172; 173]. Imparting mutual coherence on indepen-
dent photons via a common excitation laser can therefore be a crucial advantage
in achieving indistinguishable wavepackets. Therefore, we perform Hong-Ou-
Mandel style two-photon interference [174] experiments of two QD photons scat-
tered from the same QD at different times to quantify the indistinguishability of
synthesised photon waveforms with complex spectra. We note that in all interef-
erence measurements that follow, the QD fluorescence is spectrally filtered via a
1600 grooves per millimeter grating to remove the phonon sideband.
A schematic of the experimental setup is shown in Fig. 6.3 a. Figure 6.3 b
displays the laser pulse train (top panel) used to generate coherent QD photons
in the Heitler regime (bottom). The laser pulses are derived from a continuous
1We note that the measured QD spectrum is broadened by spectral diffusion beyond the
radiative limit.
132
QD
In
te
ns
ity
(a
.u
.)
Laser
0 10
Time (ns)
-10 0 10 20
0.5
1.0
d e
g(
2)
(
)
Delay  (ns)
0.5
1.0
┴
g H
O
M
(2
) (
)
-10 0 10 20
0
5
10
c
g H
O
M
(2
) (
)/g
(2
) (
)-
1
Delay  (ns)
b
0.5
1.0
||
g H
O
M
(2
) (
)
-10 0 10 20
0
5
10
g H
O
M
(2
) (
)/g
(2
) (
)-
1
Delay  (ns)
/2
QD
photons
Polari-
sation
controllers
50:50
APDs
3 ns
Delay
a
τ
λ
ττ
τ
τ
τ
τ
ττ
τ
Figure 6.3: Indistinguishability of waveform-synthesised photons. a, Illustration
of the experimental setup for two-photon interference from a single QD. b, Time-
synchronised single QD photons (bottom panel) are generated by exciting the QD
transition with laser pulses of 500-ps width and 300-MHz repetition rate (top panel).
c, The intensity autocorrelation function for QD photons is recorded in a Hanbury-
Brown and Twiss setup. A two-photon probability per pulse below 5% demonstrates
the quantum nature of the emitter. d,e Two-photon interference measurements for
QD photons of orthogonal (d) and parallel (e) polarisation. The bottom panels dis-
play the normalised differences to the intensity autocorrelation in c, respectively. The
lack of two-photon interference for orthogonally polarized photons in d gives rise to
coincidences at zero time delay, yielding a peak in the bottom panel. For photons of
parallel polarisation, the absence of coincidences around zero time delay reflects suc-
cessful two-photon interference in e. The ratio of the areas in the bottom panels yields
a raw visibility of 0.926. Experimental data in ce are shown as continuous dark red
curves, while a simulation for each case based on system parameters is shaded in light
red. The data are recorded in 162-ps time bins with 600-ps timing resolution.
133
Figure 6.4: First-order interference in the Hong-Ou-Mandel interferometer. We moni-
tor single photon detection events during the Hong-Ou-Mandel experiment. Top panel:
First-order interference occurs for coherent QD photons as a consequence of varying
phase difference between both interferometer arms. The contrast of first-order interfer-
ence quantifies the fraction of coherently scattered QD emission. Bottom panel: The
time traces of detection events for photons of orthogonal polarisation stay nearly con-
stant, as photons are distinguishable in their polarisation. The QD transition is shifted
out of resonance periodically via DC Stark effect to monitor the background counts.
This is visible as abrupt drops in intensity.
wave laser by modulating the EOM transmission with independent control on
the duration and the repetition rate. Here, the laser pulses are 500-ps long with
a repetition rate of 300 MHz. For these settings the temporal response of the
QD transition is visible as an exponential tail in the coherent conversion of laser
pulses into single photons1. Before we describe the two-photon interference mea-
surements in detail we want to confirm that the the pulsed single photon stream
predominantly consists of coherent scattering.
First-order interference in the Hong-Ou-Mandel interferometer. In ad-
dition to the coincidence events that manifest the two-photon interference we also
record the individual detection events on both avalanche photodiodes (APDs)
(see Fig. 6.3 a). Figure 6.4 presents two 1200-s long measurement of the single-
photon detections of pulsed QD fluorescence on the two APDs. The top panel
1As an aside, it may be interesting to create photons with the Gaussian or double-
exponential waveforms, which can improve the excitation probability [175].
134
corresponds to the case where the photons in both interferometer arms have the
same polarisation, while the bottom panel shows a detection time trace for pho-
tons of orthogonal polarisation. In the former case, the single-photon count rate
shows signatures of first-order interference. This is due to the phase fluctuations
between both interferometer arms as the path lengths vary slowly over time due
to mechanical wandering and slight changes in ambient temperature1. This first-
order interference is strongly suppressed, as it should be, for photons of nominally
orthogonal polarisation in the bottom panel. The contrast of first-order interfer-
ence quantifies the fraction of coherently scattered QD emission Fcoh which is
Fcoh ≈ 0.92 for the data in Fig. 6.4.
Two-photon interference Figure 6.3 c shows the measured intensity-correlation
function g(2) (τ) for these photons. The missing peak at zero time delay in co-
incidence detection evidences strong antibunching. For pure photonic states, a
Hong-Ou-Mandel-style two-photon interference measurement in the vicinity of
zero time delay is expected to follow
g
(2)
HOM (τ) =
(
t21 + r
2
1
)
g(2) (τ) + 2r1t1
(
1− η + ηg(2) (τ)) , (6.3)
where t1 and r1 are the transmission and reflection coefficients of the first beam-
splitter used to form the two input arms of the second beam splitter (cf. Fig. 6.3
a). The variable η quantifies the photon indistinguishability including any imper-
fections of the measurement apparatus. For fully indistinguishable (η = 1) pho-
tons g
(2)
HOM (τ) reduces to g
(2) (τ), while for fully distinguishable (η = 1) photons
Eq. (6.3) takes the form g
(2)
HOM (τ) = (t
2
1 + r
2
1) g
(2) (τ) + 2r1t1. We use the polari-
sation of the input photons as means to vary η , such that parallel (orthogonal)
polarisation leads to a maximum (minimum) value for η. These measurements,
performed in conditions identical to those of Fig. 6.3 c , are shown in panels 6.3
d and 6.3 e for orthogonal and parallel input polarisation states, respectively.
In order to quantify the two-photon interference fidelity we compare the data
1Further, for the purpose of data calibration the QD transition was periodically shifted out
of resonance with respect to the exciting laser, so that the background counts due to detector
dark counts and laser leakage were recorded. This is visible as sharp drops in intensity in the
data
135
directly with the ideal case of g
(2)
HOM (τ) = g
(2) (τ): the bottom panels of Fig. 6.3
d and e show
g
(2)
HOM (τ)− g(2) (τ)
g(2) (τ)
for their respective interference measurement. Integrating the normalised dif-
ference curves for both polarisation cases over one repetition period (∼ 3.3 ns)
centred around zero time delay, we can extract a raw contrast of the two-photon
interference of CHOM = 1−
(
A‖/A⊥
)
= 0.926± 0.016.
The raw contrast of the two-photon interference constitutes a lower bound to
photon wavepacket indistinguishability, as the measurement is affected by imper-
fections of the setup. In our case these include polarisation degradation in the
single mode fibre beamsplitter and an imbalance of beamsplitter coefficients. The
conventional treatment of these imperfections [159; 176] assumes that the pho-
ton coherence time τc is much shorter than the time delay in the interferometer,
τc  ∆tdelay. The expected interference contrast is then
CHOM = 1− A‖
A⊥
= 1− 2r2t2p‖ (D − 1) + r
2
2 + t
2
2
2r2t2p⊥ (D − 1) + r22 + t22
(6.4)
where D is the distinguishability of photon pairs. Experimentally we measure the
fraction A‖/A⊥ then determine 1−D, knowing the values and error estimates for
all parameters involved in Eq. (6.4). For the data in Fig. 6.3 we obtain a cor-
rected contrast of unity within the experimental uncertainty (Ccor = 1.03±0.05).
For coherently scattered photons the correction for the interferometer imperfec-
tions is not straightforward as their coherence time exceeds the interferometer
time delay, therefore we choose to correct only for the polarisation mismatch
which affects coherent and incoherent photons in the same way. The expected
interference contrast then yields
CHOM = 1− A‖
A⊥
= 1− 1− p‖ +Dp‖
1− p⊥ +Dp⊥ . (6.5)
This lower bound on photon wavepacket indistinguishability evaluates to 0.96±
0.04 and is still higher than all indistinguishabilities reported for non-resonantly
generated QD photons. He et al. obtained similar interference visibilities for
136
resonantly generated incoherent photons recently [177]. We stress that the lack
of correlation between the environment dynamics of multiple QDs will inevitably
lead to the degradation of photon indistinguishability for the incoherently gen-
erated photons, while the coherent generation process reported here offers some
protection from such effects.
6.5 Chapter discussion and outlook
We have demonstrated the generation of fundamentally indistinguishable, coher-
ent single photons from a solid-state quantum emitter with a high degree of con-
trol and flexibility in waveform synthesis. These phase-locked photons are ideally
suited for quantum interference applications, which form the basis of quantum
communication [162], linear optics quantum computation [98; 120], and distant
entanglement schemes [156; 157]. The price to pay for this unprecedented pho-
ton quality is excitation efficiency. To obtain a coherent fraction of about 0.9 in
pulsed excitation, photons are probabilistically generated with 5-10% efficiency
(1.5-3 · 107 Hz for the two-photon interference measurements presented here).
While generation rates are lower than those achievable in deterministic excitation
schemes, the trade-off between photon rates and photon quality is well justified
for two main reasons: first, quantum interference applications can work well
with probabilistic gates [157; 178], while the low photon quality reported so far
[161; 173] rules out the use of QD photons in quantum information applications.
Further, non-unity photon collection and detector efficiencies evidently render any
measurement-based schemes probabilistic. Second, an entanglement rate scaling
with the two-photon detection efficiency [157] is replaced advantageously by one-
photon detection [156; 162] utilising the first-order coherence of the coherently
scattered photons.
The technique presented in the work is not limited to QDs and can be extended
to other quantum systems in the optical domain as well as to superconducting
circuits in the microwave domain.
One next immediate step in the direction of a quantum network is to extend the
two-photon interference to two independent quantum dots. We have attempted
137
this measurements, in fact before measuring the two-photon interference pre-
sented in this chapter. Raw visibilities in the pulsed experiment around 40%
were obtained, however, without filtering the phonon sideband and limited po-
larisation control. The main reason the experiment was not pursued further was
that the second QD sample showed strong spectral diffusion and blinking (due to
the high QD density). Correcting for the experimental imperfections gave a lower
bound on photon indistinguishability of > 70%, but was not reported due to the
experimental imperfections. We note that Konthasinghe et al. have reported on
continuous-wave two-photon interference from two independent QDs under res-
onant excitation recently [179]. In the presence of strong spectral diffusion they
obtain a two-photon visibility around 40%.
138
Chapter 7
Single-shot readout of the
electron spin
In this chapter we discuss the dynamics of few spins confined to a pair of vertically
stacked QDs. The electronic levels of the two QDs can be tuned into resonance
so that tunnel coupling leads to the development of molecular states. We demon-
strate how transitions into these molecular states can be used to monitor the spin
projection of a single electron confined to one of the QDs. This chapter is based
on [111].
In the first part of this chapter we explain how spectroscopy of QD molecules
(QDMs) can be linked to electron tunnel coupling in various spin and charge
configurations. In this context the detailed energy level structure for the case
of a single electron confined to one QD and a neutral exciton transition in the
other QD is derived and compared to measurements. The concept of spin readout
by recycling a molecular transition is picked up: We investigate if there are any
obvious shortcomings to using the transition to the molecular T+1/2 state as read-
out channel. In the second part time-averaged and time-resolved measurements
demonstrate the conditionality of the molecular transitions on the single electron
spin state and we show that spin dynamics can be monitored in real-time. The
fidelity of the spin readout is discussed.
Despite being the last experimental chapter of this thesis, the work was actually
done first (2009-2010). As such, both the photon collection efficiency and the
139
suppression of laser background for resonance fluorescence for the measurements
presented here are somewhat worse than the performance described in earlier
chapters. In the final part of this chapter we extrapolate from the measured spin
readout fidelities to fidelities potentially afforded by the improved collection effi-
ciency and background suppression. I also note that measurements were devised,
performed and analysed in a strong collaboration with Chao-Yang Lu and Dr
Nick Vamivakas who share equal credit for the results. I also acknowledge the
contribution of Yong Zhao whose calculations on QDM energy level served as a
model for those I present in this chapter. Antonio Badolato and Stefan Fa¨lt grew
the sample.
7.1 Why quantum dot molecules?
We may remember from the QD level structure discussion and the summary on
coherent QD spin control schemes in Chapter 1 that spin state readout is a bot-
tleneck for single spin qubits in optically active QDs: Fast optical initialisation
and manipulation on the one hand and reliable readout on the other hand cannot
be realised at the same time in this system. Our approach is to decouple the
two parts using a tunnel-coupled quantum dot molecule. With a single electron
resident in one QD of the pair while the other QD is empty we expect optical
transitions into four different molecular states (see Fig. 7.1) when an exciton
is added in the empty QD. The nomenclature for ground and excited states is
explained in the figure caption. Transitions of the single electron charged QD,
denoted ‘host’ QD, can be used for state initialisation via optical pumping as
discussed before, whereas the transitions to molecular states via the second QD,
denoted ‘readout’ QD, are reserved for readout. Under magnetic field the four
molecular transitions are separated in energy, and backaction effects on the host
electron spin should be strongly reduced compared to the single QD case. Previ-
ous research by Kim et al. [180] has shown in time-averaged measurements that
the transition to the T+1/2 state is robust against spin pumping and conditional
on the host spin state.
Our aim is to perform real-time spin readout by observing intermittent resonance
fluorescence from the 1→ T+1/2 transition. For ions, intermittent resonance fluo-
140
Figure 7.1: Cartoon of the ground and excited states of the QDM. The
left (right) half of each square shows charges in the host (readout) QD where top (bot-
tom) squares contain electrons (holes). Double-ended arrows denote optical transitions,
single-ended arrows show weak decay channels. The single electron ground states are
labeled 0 and 1, the ‘atomic’ trion states in the host QD are S+3/2, S−3/2 (electrons
in singlet state) and the molecular states are denoted Tx. Subscripts give the total
angular momentum in each case.
rescence has revealed quantum jumps back in 1986 already [181; 182; 183] and it
was later observed for other systems [184]. The 1→ T+1/2 transition is expected
to be a robust recycling transition and the suitability of this transition as readout
channel for the host electron spin will be investigated.
7.2 Quantum dot molecule sample
The sample is a Schottky diode heterostructure (see Fig. 7.2) with two InAs
QD layers separated by 13 nm (separation of bottom of the first wetting layer
to bottom of the second wetting layer). The distance to the ohmic contact is 30
nm, and the length between the ohmic and Schottky gate is 200 nm. Growth
conditions are such that the energy levels in the readout QD (marked red in the
sketch) are redshifted in comparison to levels in the host QD (blue). Electron
141
30 nm d=13 nm
L=200 nm
n+
Figure 7.2: Dimensions of the sample.
tunneling in the configuration of a single electron in the host QD and an exciton
in the readout QD is possible by tuning the gate voltage. Hole energy levels in
the two QDs are far detuned so that hole tunneling does not occur.
7.3 Quantum dot molecules
Tunnel-coupling between QDs shows characteristic signatures in photolumines-
cence spectra, most prominently avoided crossings, or anticrossings [185]. These
avoided crossings are a result of hybridisation of the two energy levels that are
brought into resonance. Detailed (magneto-)photoluminescence studies have been
published, identifying which charge and pin configurations in the QDM are in-
volved in tunneling at each of the observed anticrossings [27; 185; 186; 187].
Fig. 7.3 shows a PL-gate voltage map of the host (a) and the readout QD
(b). The magnitude of the anticrossings give information about the strength of
the tunnel coupling for the charge states involved. While we extract values for
exchange splittings and tunnel coupling from these PL maps we rely on the reso-
nant high-resolution differential transmission (DT) to identify optical transitions
and their respective ground states unambiguously.
Before moving on to the DT measurements and theoretical modelling we briefly
discuss the simplest case that can give rise to an anticrossing, a single electron
142
Figure 7.3: PL map of the the host QD (a) and readout QD (b). Neutral exciton and
trion emission lines have been indicated.
in the QDM, to obtain some intuition for the more complex cases. We define
the basis states of the system as the electron being in the host QD (|1〉) and the
electron being in the readout QD (|2〉):
|1〉 = e†host |0〉
|2〉 = e†readout |0〉 ,
where e†QD is the fermionic creation operator for an electron in the QD defined
in the subscript and |0〉 denotes the vacuum state. The energy levels for the
two states can be tuned into resonance by changing the gate voltage. We relate
the energy difference δE between the two levels to the gate voltage and sample
dimensions:
δE =
e (V0 − Vg)
L
d+ δE0, (7.1)
where V0 is the offset of the conduction band edge and the Schottky contact at
zero bias, L is the distance between the gates, d is the distance of charges in the
host QD to charges in the readout QD (cf. Fig. 7.2). δE0 provides a constant
offset. We designate state |1〉 to be the energy reference: as we are only interested
in the relative energies we subtract the single particle energy and only state |2〉
depends linearly on the applied gate voltage. The matrix tunneling element
143
- 1 0
- 5
0
5
1 0
 
 
Ene
rgy
 (in
 un
its 
of t
e)
G a t e  v o l t a g e  ( a .  u . )
Figure 7.4: Energy levels for a single electron in a tunnel-coupled QDM.
is given by te = 〈φhost|h|φreadout〉, where φhost and φreadout are the orthonormal
wavefunctions of the electron in state |1〉 and |2〉, respectively, and h is the single-
particle Hamiltonian. The Hamiltonian for the coupled system is then
H1e =
(
0 te
te δE
)
. (7.2)
Solving for the eigenvalues we obtain energies E1,2
E1,2 =
δE
2
± 1
2
√
δE2 + 4t2e (7.3)
The energies (7.3) are plotted in Fig.7.4 The anticrossing is 2te in magnitude.
7.4 Quantum dot molecule spectroscopy
We are interested in the gate voltage regime where the QDM ground state is given
by a single electron in the host QD, and the QDM excited state has an exciton
added in the readout QD.
Here, we present a calculation of the QDM’s electronic states and optical tran-
144
sition energies relevant for our readout scheme. We compute the dependence of
the energies on gate voltage and external magnetic field and compare the simu-
lation to our measured DT maps. In the model we choose basis states that are
eigenstates in the limit of weak electron-hole exchange and no tunnel coupling
between the host and readout QD. As we will see later this choice of basis comes
rather close to the actual set of eigenstates under high magnetic fields. The nu-
merical model allows us to make a statement about the ‘purity’ of the readout
state T+1/2, and hence possible errors in the readout scheme. The effects of state
mixing, e.g. due to heavy-light hole mixing, can be discussed in this context as
well.
The QDM ground state consists of a single electron in the host QD whereas
the excited states consist of two electrons and one hole within the two QDs. In
the excited state the two electrons occupy the lowest orbital levels and can both
reside in the same QD or one electron can be in each dot. For our model (and
sample) the heavy hole is localized in the readout QD. From the DT spectroscopy
(see Fig. 7.5) it is obvious that tunnel coupling takes place in the excited states,
so we consider the following 12 excited states of two electrons and one hole:
|1〉 = e†host↑e†host↓h†readout⇑ |0〉 |7〉 = e†host↑e†host↓h†readout⇓ |0〉
|2〉 = e†readout↑e†readout↓h†readout⇑ |0〉 |8〉 = e†readout↑e†readout↓h†readout⇓ |0〉
|3〉 = e†host↑e†readout↓h†readout⇑ |0〉 |9〉 = e†host↑e†readout↓h†readout⇓ |0〉
|4〉 = e†host↓e†readout↑h†readout⇑ |0〉 |10〉 = e†host↓e†readout↑h†readout⇓ |0〉
|5〉 = e†host↑e†readout↑h†readout⇓ |0〉 |11〉 = e†host↑e†readout↑h†readout⇑ |0〉
|6〉 = e†host↓e†readout↓h†readout⇓ |0〉 |12〉 = e†host↓e†readout↓h†readout⇑ |0〉
Here (e, h)†dot,σz is the fermionic creation operator for an electron (e) or a hole (h)
in host or readout dot, with spin projection σz. States |1〉, |2〉, |7〉, |8〉 are the
only configurations with two electrons in the same dot. The electrons are in a
spin-singlet state in these cases; triplet states are far removed in energy and do
not take part in any interactions. The states |4〉, |6〉, |9〉, |11〉 are ‘dark’, whereas
the states |3〉, |5〉, |10〉, |12〉 are ‘bright’.
The Hamiltonian we consider is:
145
H = Hparticle +HCoulomb +Hexchange +Htunnel (7.4)
For the purpose of a numerical solution we note
1. Hparticle is the sum of single particle energies. The absolute energies of
two electrons and one hole in the potential of the quantum dot molecule in
different spin configurations are very similar, so we choose the configuration
with all charges in the readout dot (states |2〉 and |8〉) as reference energy
and calculate energies relative to the reference. This reference is set in the
absence of Coulomb and exchange interactions and does not change with
applied gate voltage.
2. The Coulomb terms for two particles a and b are defined as before (Eq.
(1.1)). We reduce the Coulomb interactions to four terms. The subscripts
denote which charges are considered and if the interaction is direct (both
particles in the same dot) or indirect (in separate dots). Values can be
estimated from photoluminescence (PL) maps and found in the literature
[185; 187; 188; 189]:
Vee,direct ≈ 20meV
Vee,indirect ≈ 10meV
Veh,direct ≈ −24meV
Veh,indirect ≈ −10meV
3. Exchange interaction is defined in Eq. (1.3). Two of the three terms
for electron-hole exchange can be deduced from spectroscopic data. The
isotropic exchange splitting between dark and bright excitons can be ex-
tracted from the PL map in Fig. 7.3 to be δ0 ≈ 220 µeV. The anisotropic
exchange δ1 can be measured from the DT spectrum in Fig. 7.5a when the
host QD is charged with two electrons and hence in a spin singlet configu-
146
ration. Then the host QD spins do not take part in exchange interactions
and we measure δ1 ≈ 17.6 µeV. The dark exciton splitting δ2 is not visible
in photoluminescence measurements, but expected to be of the same order
as δ1. The interdot electron-electron exchange Jee is very small, from [185]
we estimate it to be ∼ 1 µeV. It couples states |3〉 to |4〉 and |9〉 to |10〉.
4. Tunnel coupling takes place between states |1〉, |2〉 and |3〉, |4〉 and between
|7〉, |8〉 and |9〉, |10〉. The magnitude can be extracted from the PL maps
and is te ≈ 330 µeV for this QDM.
Lastly, changing the gate voltage shifts the detuning δE between electron
energy levels in the host dot and readout dot, enabling electron tunneling.
Expressing the Hamiltonian of Eq. (7.4) in the basis |1〉-|12〉 block diago-
nalizes the Hamiltonian into 2 blocks. One block couples states |1〉-|6〉 and
the other couples states |7〉-|12〉 1. Eq. (7.5) is the Hamiltonian matrix for
states |1〉-|6〉 (the block for states |7〉-|12〉 is essentially the same).
H|1〉−|6〉 =
−2δE + V1 0 te −te 0 0
0 V2 te −te 0 0
te te −δE + V3 + δ0 Jee δ1 0
−te −te Jee −δE + V4 − δ0 0 δ2
0 0 δ1 0 −δE + V5 + δ0 0
0 0 0 δ2 0 −δE + V6 − δ0

(7.5)
Due to space constraints placeholders were used instead of the full Coulomb
1D. Kim et al. [180] observe another anticrossing in the excited state level structure where
coupling between states |5〉 and |9〉 occurs. We see the same effect in other dot pairs, but
for this particular one it happens at magnetic fields outside the range of data presented. A
phenomenological term (2 µeV) has been included in the final Hamiltonian.
147
matrix elements:
V1 = 2Veh,indirect + Vee,direct
V2 = 1Vee,direct + 2Veh,direct
V3 = Veh,direct + Veh,indirect + Vee,indirect
V4 = Veh,indirect + Veh,direct + Vee,indirect
V5 = Veh,direct + Veh,indirect + Vee,indirect
V6 = Veh,direct + Veh,indirect + Vee,indirect
Diagonalising the Hamiltonian we determine both the eigenenergies and eigen-
states as a function of gate voltage and magnetic field which we then compare
with the measured data.
Fig. 7.5 presents differential transmission (DT) spectra of the host and readout
QD optical transitions as a function of gate voltage at zero external magnetic
field. The top panel displays the trion transition of the host QD where the labels
0e/1e/2e denote the ground state charge configurations and the vertical dashed
lines indicate the 1-electron ground state range. The bottom panel gives the
transition map for the readout QD in the same gate voltage range. Throughout
the relevant gate voltage range the readout QD remains uncharged. The voltage
dependence of the splitting between the two optical transitions indicates tunnel
coupling between host and readout QD. Signs of the avoided crossing are only
visible in the readout QD spectrum, so the tunnel coupling takes place between
the exciton electron and the host dot electron, as assumed in the calculation. In
Fig. 7.5 b we present the calculated level structure by diagonalising Eq. (7.5)
near the anticrossing. The linear Stark shift common to all lines in the measured
spectrum has been neglected in the simulation. The five lines show the evolu-
tion of the doubly degenerate eigenenergies with applied gate voltage (the two
eigenenergies not shown in Fig. 7.5 b are higher in energy and take no part in the
interactions in this gate voltage range). We can relate the transitions observed
experimentally (Fig. 7.5 a) to the green and blue lines in the simulation. We
148
Figure 7.5: Transmission spectroscopy on the quantum dot molecule and
calculated level structure. a, Transmission spectra for the QDM at zero magnetic
field. The laser frequency is scanned for fixed gate voltages. The top panel shows the
transmission spectrum for the host QD trion transition. The 0e/1e/2e labels denote the
ground state charge configuration of the host. The bottom panel shows the transmission
spectrum of the readout QD transitions. In the gate voltage range bounded by the
vertical white lines the readout QD ground state is uncharged. b, top panel Calculated
level structure for the QDM at zero magnetic field around the avoided crossing. The red
lines correspond to the singlet-like spin configuration and anticross. The blue doublet
passes through the avoided crossing unaffected and can be associated with the bright
basis states with parallel electron spins (T±-like), while the dashed grey line originates
from the parallel electron spin dark states. The green line goes from dark to bright
as it wiggles through the avoided crossing. It consists of basis states with antiparallel
electron spins and has T0-character. b, bottom panel Comparison of data from a
to the simulation. The arrow indicates the gate voltage region where time-resolved
measurements are performed later.
149
subtract the Stark shift from the data and compare simulation and experiment
directly (Fig. 7.5 c). None of the other transitions show up in the spectrum as
they are far detuned (red lines) and/or optically dark (grey lines).
A closer look at the evolution of the eigenvector reveals how we can understand
the effect of tunnel coupling between two electrons in the presence of a hole. Go-
ing back to Fig. 7.5 b, the red line is the only one to show an anticrossing, which
would be expected from a singlet state for a 2-electron system. The eigenvector at
the lower gate voltage end is composed mostly of basis states |3〉 and |10〉, which
are antiparallel spin combinations of one electron in each dot with opposite hole
spins. Moving closer to the anticrossing the basis states with both electrons in
the host dot (|1〉 and |7〉) are gaining importance. At the anticrossing the states
|1〉,|7〉 and |3〉,|10〉 are roughly of equal strength and weakly admixed with the
dark states |4〉 and |9〉. At even higher gate voltages (cut off in the figure) basis
states |1〉 and |7〉 dominate. The lower red line shows reverse character. We
associate this behaviour with a molecular singlet state modified by electron-hole
exchange. Next, the blue line shows almost no dependence on gate voltage as
it moves through the anticrossing, characteristic of a molecular triplet state. Its
eigenvector is a combination of initially all bright basis states (|3〉,|5〉,|10〉,|12〉).
|5〉 and |12〉 contribute the most, and around the anticrossing the eigenvector is
almost entirely made up of the parallel electron spin combinations |5〉 and |12〉.
The green line starts off as an admixture of all dark basis states (|4〉,|6〉,|9〉,|11〉).
As it wiggles through the anticrossing region its composition changes to include
the bright antiparallel spin basis states (|3〉,|10〉) at the expense of the dark par-
allel spin ones (|6〉,|11〉). On the higher gate voltage side of the avoided crossing
the bright states |3〉,|10〉 dominate, reminiscent of a T0 triplet in the 2-electron
case. The lower grey dashed lines are very similar in nature to the green line
at low gate voltages: they are combinations of all dark states. For gate voltages
above the avoided crossing only the contributions from the dark parallel spin
states (|6〉,|11〉) remain. In the comparison to the two-electron case the green
and grey lines represent T± states and do not take part in the tunnel coupling.
We move on to discuss the magnetic field dependence in Faraday geometry of
the bright QDM transitions relevant for the readout scheme (green and blue line
150
Figure 7.6: Energy levels and QDM optical transitions as a function of
magnetic field. a, Evolution of eigenenergies with magnetic field at the gate voltage
marked by the arrow in Fig. 7.5 c. The degeneracy of different angular momentum
states is lifted for all lines. From top to bottom we have the bright molecular triplet-like
states, the dark (optically forbidden) triplets and the molecular singlet-like ones. The
black pair at the bottom shows the splitting of the single electron ground state. b, The
transition energies are obtained by subtracting the respective ground state energies
from each excited state. c, DT spectra for the readout QD transitions as a function of
magnetic field in the Faraday configuration. The first avoided crossing between T+3/2
and T−1/2 states occurs at 2 T as indicated by the red rectangle. At larger magnetic
fields the two highest frequency transitions are T+3/2 and T+1/2, respectively.
151
in Fig. 7.6b and 7.6c). Figure 7.6 presents simulation (Fig. 7.6a and 7.6b) and
experiment (7.6c) for DT spectra as a function of magnetic field. In the simu-
lation we only consider Zeeman shifts, neglecting the diamagnetic shift which is
also present in the data and common to all transitions. Figure 7.6a shows the
QDM eigenenergy evolution as a function of magnetic field. Electron and hole
g-factors are found from the data in Fig. 7.6c. The top 4 lines correspond to
the 4 bright triplet-like eigenstates and at the bottom of the figure the single
electron ground states are shown in black. For convenience we label these four
eigenstates according to the basis state that contributes most in each case. This
is the notation used when we introduced the readout scheme in section 7.1. At
low magnetic fields going from higher to lower energies we have the T+1/2 state
(corresponding basis state |12〉 in Eq. (7.4)), the T−1/2 state (basis state |5〉) and
then the T+3/2 (basis state |3〉) and T−3/2 states (basis state |10〉). Around 2 T
magnetic field the T+3/2 state and the T−1/2 state anticross so that the T+3/2
state is second highest in energy for higher magnetic fields. Fig. 7.6b displays
the transition energies expected from the energy level scheme in Fig. 7.6a using
these labels. The experimental raw data is given in Fig. 7.6c.
In order for our readout to work efficiently we need the T+1/2 readout state to
have as little admixture of basis states other than |12〉 as possible. We use the
numerical model to evaluate the composition of the T+1/2 state in terms of the
basis states at 2.3 T, the magnetic field value which is later used for readout.
The four biggest contributions to the eigenvector are:
T+1/2 (B = 2.3T) =
√
0.9813 |12〉+
√
0.0181 |10〉+
√
3.3 · 10−4 |9〉+
√
4 · 10−5 |5〉 .
All remaining contributions are ≤ √10−5. The admixture of |10〉 means that the
polarisation of emitted photons is slightly elliptical, but it does not introduce
errors in the readout, as the host spin state is 1. Reduced readout fidelities are
caused by admixtures of the basis states with host spin 0: |9〉 and |5〉. Con-
sidering that radiative recombination of |9〉 is dipole-forbidden the admixture of
|5〉 is expected to be the dominant contribution to readout error. In this case,
measurement of a photon from the T+1/2 excitation has a fidelity error of < 10
−4.
This is expected to be much smaller than other experimental errors. As such we
152
consider the T+1/2 state to be satisfactorily ‘pure’ for the purpose of spin readout.
Having discussed the purity of the readout state, we briefly turn to the other tran-
sitions. Due to the anticrossing at 2 T the intermediate transitions are roughly
equal superpositions of basis states |3〉 and |5〉 at 2.3 T. A second anticrossing is
expected at higher fields between the two lower energy levels (pair of green lines
in Fig. 7.6 a) [180]. This anticrossing mixes basis states of different host spin
orientations. The optical transition to the T+1/2 state remains the only transition
that does not hybridise with the other transitions.
The analysis confirms that the 1 → T+1/2 transition is suitable as a recycling
transition to deduce the host QD spin state.
7.5 Steady-state spin dynamics
In order to show that there is a direct link between the transitions to specific
molecular states and the host spin orientation we perform time-averaged two-
colour resonance fluorescence measurements. The host spin is deterministically
set to either 0 or 1 by continuous optical pumping [48] and we monitor resonance
fluorescence counts while driving the 1 → T+1/2 transition. Unless specified
otherwise, all following experiments are performed at 2.3 T and the laser is linearly
polarised. A 1 nm bandwidth filter is used to reduce laser leakage from the
pumping laser.
In the left panel of Fig. 7.7a the pumping laser is scanned across the 0→ S+3/2
transition (detuning δpump) while the readout laser is fixed and detuned from
the 1→ T+1/2 transition by δreadout (see sketch at the bottom of Fig. 7.7 for the
transitions involved). For finite pump detuning δpump, i.e. inefficient pumping into
the 1 state, we observe a constant resonance fluorescence signal when the readout
laser is resonant with the 1 → T+1/2 transition. In the double resonance case
spin pumping into state 1 is efficient and the resonance fluorescence signal from
the readout transition is enhanced. In the right panel of Fig. 7.7a the pumping
laser initialises state 0 continuously when its detuning δp from the 1 → S−3/2
transition is small. Here the resonance fluorescence signal is suppressed in the
double resonance case.
153
Figure 7.7: Steady-state spin dynamics monitored via molecular transitions.
Sketches at the bottom of the figure indicate which transitions are being driven. The
red (blue) arrow denoted the readout (pump) laser.
154
A third test of the conditionality of the molecular transitions on the host spin
state is shown in Fig. 7.7b. The pumping laser is set to initialise the host spin in
state 0. The right part of Fig. 7.7 b is identical in conditions to the right panel
of a, but as the readout laser is red detuned further, it hits resonance with the
molecular transitions 0→ T+3/2 first and then 0→ T−1/2. These transitions light
up as the host spin is pumped into the 0 state, consistent with the reduction of
signal in the 1→ T+1/2 channel.
Another interesting feature is observed in Fig. 7.7 b: when the pumping laser
is off-resonant, i.e. not preparing the host spin state, no resonance fluorescence
signal is visible from the 0 → T+3/2,T−1/2 transitions, while the 1 → T+1/2 is
continuously very bright. This points to a spin pumping process where the host
spin state is flipped by driving the molecular 0→ T+3/2,T−1/2 transitions.
7.6 Time-resolved spin dynamics
Having confirmed the purity of the T+1/2 readout state in simulations and veri-
fying that
1. resonance fluorescence from the readout transition shows conditionality on
the host spin state and
2. it does not exhibit optical spin pumping unlike the two spectrally closest
transitions,
we move on to real-time resonance fluorescence monitoring from the readout tran-
sition. Host spin dynamics are expected to be reflected in intermittent resonance
fluorescence.
Fig. 7.8a shows a 300-ms long time trace of the resonance fluorescence (2 ms
time bins) when the readout laser is driving the 1→ T+1/2 transition below sat-
uration. Clear jumps in the fluorescence between two levels are observed. We
denote the bright fluorescence level as ‘on’ and the dark levels as ‘off’. The his-
togram for the frequency of fluorescence counts per time bin for 1200 seconds of
data is displayed in Fig. 7.8b. It shows two modes which we associate with ‘on’
and ‘off’. In 7.8c the frequency histogram for ‘off’ durations, i.e. the duration
155
Figure 7.8: Time-resolved resonance fluorescence from the readout transi-
tion. a 300 ms of intermittent resonance fluorescence. b, Histogram of photocounts.
The distribution is considerably broader than would be expected from a, this is due to
spectral diffusion during the 1200 seconds of recording data. c, Statistics ‘on’ the off
durations reveal single exponential decay.
of a dark period, is plotted. It follows a single exponential decay with a decay
constant Toff=14.9 ms.
Next, we monitor resonance fluorescence while pumping the host spin into state
1 or 0, in analogy to the time-averaged measurements. Fig. 7.9a shows counts
in 250 µs time bins, where for negative times the readout laser only is active.
The time trace exhibits switching between on and off levels as before. At positive
times a pumping laser resonant with the 1 → S−3/2 transition forces the host
spin into state 0 and the readout fluorescence is quenched. In the top right panel
of Fig. 7.9b we show a histogram of fluorescence counts when only the readout
laser is driving the 1 → T+1/2 transition on resonance (time bin of 4 ms). The
left panel shows a zoom-in of the lower tail of the histogram, which is due to ‘off’
periods in the fluorescence stream. The same arrangement of plots is used in the
bottom panel of b where a weak pumping laser drives the 0 → S+3/2 transition.
Counts in the lower histogram tail are reduced by a factor of 3.
These time-resolved measurements demonstrate the same conditionality of flu-
orescence on the host electron spin as the time-averaged measurements: Dark
periods in the photonstream can be associated with spin state 0 and bright peri-
ods with spin state 1.
156
Figure 7.9: Time-resolved resonance fluorescence from two-colour experi-
ments. a, Quenching of fluoresence when the pumping laser initialises the spin state
to 0 at positive times. b, Comparison of count histograms when only the readout laser
is active (top panels) to simultaneously pumping into spin state 1. The left panels show
the lower tails of the full histograms in more detail.
157
7.6.1 State mixing and spin pumping in quantum dot
molecules
Before quantifying the fidelity of the readout technique we need to discuss a
puzzling feature of the resonance fluorescence counts histograms: In Fig. 7.8b,
and even more obviously in Fig. 7.9 b the modes corresponding to ‘on’ and
‘off’ are highly imbalanced. For thermal equilibrium we expect the population
of the Zeeman sublevels to obey Boltzmann statistics. In our case this corre-
sponds very closely to a balanced distribution, so that the histogram areas for
‘on’ and ‘off’ should be equal. We associate the imbalance of populations with
the optical pumping observed in Fig. 7.7 b when the host spin state is 0. We
expect that measurement-induced spin flips originate from these transition be-
ing driven nonresonantly when the 1 → T+1/2 transition is probed on resonance
by the readout laser. In order to quantify the optical spin pumping we use a
previously reported protocol [26] to measure the spin pumping rate directly for
the transition spectrally closest to the readout transition. Fig. 7.10 a explains
the protocol in a sketch. Panel b exhibits the timescale for optical pumping as a
function of spectral detuning. The shortest resonance fluorescence decay constant
is (0.324±0.020) µs obtained on resonance when the transition is driven strongly
into saturation.
At the magnetic field used throughout these measurements (2.3 T) the main mech-
anism relaxing the optical selection rules for single QDs is the heavy-light hole
mixing, and the induced spin-flip rate was shown to be independent of the exter-
nal magnetic field (beyond 1T) [26]. For QDMs the hole-mixing induced spin-flip
rate can be further enhanced by a lateral offset between the two QDs [187]. We
define an admixed heavy-hole state ⇑˜ =⇑ (+3/2) + + ⇑ (+1/2) + − ⇑ (−1/2).
Optical spin pumping can be mediated through the following mechanism then(
↑ 0
0 0
)
~ω→
(
↑ ↓
0 ⇑˜
)
te→
(
↑↓ 0
0 ⇑˜
)
te→
(
↓ ↑
0 ⇑˜
)
~ω→
(
↓ 0
0 0
)
,
where we have used the illustration of QDM states from Fig. 7.1 . We expect
to observe efficient spin pumping from transitions that have contributions from
158
cycle
t
APDt
spin
pumping
GV
1e
0e
on
off
APD
E
F
E
F
E
F
E
F
E
F
E
F
on
off
AOM
Figure 7.10: Measurement of optically induced spin-flip transition timescale
from the 0 → T+3/2 transition. Left panel, Protocol for spin pumping measure-
ments. The 0→ T+3/2 transition is driven when the AOM is on. The gate voltage is set
to recycle the host QD electron at the beginning of each cycle. The APD ‘on’ window
moves in consecutive cycles across the region shaded in red. The Zeeman sublevels of
the host QD electron relative to the Fermi level and their occupation at several stages
of the pulse sequence are indicated schematically below. For spin relaxation processes
much slower then spin pumping times, T1  Tsp, and tcycle ≥ Tsp, an electron initially
in the 1 state will stay in the 1 state until it is recycled. Right panel, laser detuning
dependence of the time-resolved RF at 2.3 T magnetic field obtained for strong excita-
tion laser power. The shortest exponential decay constant occurs on resonance (0.324
± 0.020 µs) and is prolonged for finite detunings.
159
O f f  d u r a t i o n  ( m s )
 
2 0 4 0 6 0 8 0
1
1 0
1 0 0
 
 
 
Fre
que
ncy
 of 
off
dur
atio
n
a b
0 . 1 0 . 2 0 . 3 0 . 4 0 . 5
1 0
1 0 0
 
 
T off 
(ms
)
R a b i  f r e q u e n c y  ( u n i t s  o f  γ )
Figure 7.11: Statistics on off durations for different laser powers. a, Fre-
quency of off duration for laser power Ω ∼ 0.45T−11 (s = 0.4), light grey (Ω ∼
0.23T−11 (s = 0.1), dark grey) yields a decay constant Toff =6.5 ms (Toff =14.9 ms).
b, Toff as a function of laser power.
the antiparallel electron spin basis states |3〉 and |10〉 (specifically the the 0 →
T+3/2 and 1 → T−3/2 transitions). At 2.3 T both the 0 → T+3/2 and 0 →
T−1/2 transitions are admixtures containing basis state |3〉, consistent with the
observation of optical spin pumping.
From the optical pumping rate obtained on resonance we can estimate that the
readout laser, detuned from the 0→ T+3/2 transition by 2pi×8 GHz, causes spin
flips in a few to tens of ms when the host electron is in state 0, depending on
its driving power. In order to provide evidence for this off-resonant pumping we
vary the readout laser power and compare the duration of dark periods. Fig.
7.11a shows two ‘off’ duration histograms for laser powers of s = 0.4(light grey)
and s = 0.1 (dark grey), giving decay constants Toff of 6.5 ms and 14.9 ms,
respectively. In Fig. 7.11b we plot the extracted Toff values for a range of laser
powers. The expected dependence of Toff on laser and external parameters is
1
Toff
=
1
T0→1
+
γsfΩ
2
4∆2 + T−21 + 2Ω2
, (7.6)
where Ω, T1, ∆, γsf and T0→1 denote the Rabi frequency, excited state lifetime,
spectral detuning between the readout laser and the 0 → T+3/2 transition, rate
of optically induced spin flips from state 0 to 1 and the natural spin relaxation
time from state 0 to 1. We set ∆ = 2pi×8 GHz, 1/T1 = 2pi×250 MHz and
160
γsf = 2pi × 0.49 MHz from our measurements. The solid black line is obtained
for T0→1 =140 ms, while the lower and upper boundaries of the region shaded in
grey are for T0→1 =50 ms and T0→1 =1 s, respectively.
7.7 Spin readout fidelity
The fact that we can observe abrupt jumps in the fluorescence level, corresponding
to spin flips of the host electron, means that we can determine the spin state for
each measurement time bin. We can think of a time trace of the readout transition
fluorescence as a continuous stream of single-shot readout attempts.
In order to estimate the fidelity of our readout technique we analyse a set of
continuous wave (cw) photodetection time traces, like the one shown in Fig. 7.8
a. We consider each time bin of the time trace to be a single-shot readout attempt.
Every time bin below a set threshold is labeled as ‘off’, and we assign spin state
0 to the host QD electron, and vice versa for time bins above the threshold. The
fidelity is evaluated as 1 − p1e1 − p0e0, where p1 and p0 are the probabilities of
finding the electron in state 1 or 0, and e1 and e0 are the errors in assigning state
1 and 0. A readout error is made when the fluoresence count of a time bin is
above (below) the threshold, but the electron spin was actually in state 0 (1) at
the start of the measurement.
We identify two sources of error when deciding if a particular time bin is ‘on’
or ‘off’. The first one (en0/e
n
1) is due to photon shot noise: as the photons obey
Poisson statistics we observe a spread of photocounts per time bin. We can
associate one distribution with each of the two host electron spin states. The
distribution for state 0 is given by the shot noise of the laser leakage through the
crossed polarisers, while the distribution for state 1 is given by both the laser
leakage and the shot noise of photodetection from cycling the readout transition
1→ T+1/2. Both distributions can be obtained individually from the time traces.
The fraction of each distribution on the other (wrong) side of the threshold value
constitutes the error en0/e
n
1 in each case. There is one more step involved in
obtaining the error value. When operating in cw mode, the spin flips due to
off-resonant pumping of the 0→ T+3/2 transition skew the histogram in favour of
the 1 state. The imbalance of state 0 and 1 populations is a consequence of the
161
cw readout, so we rescale the distributions obtained for states 0 and 1 to equal
areas, corresponding to the case of equal populations, i.e. no spin information
prior to a single measurement attempt: p1 = p0 = 0.5. For the data in Fig. 7.8
the overlap is below 0.1%. The best threshold value is the one that minimises
the overlap of the rescaled distributions in this case.
The second, more severe source of error (ed0/e
d
1) is spin flips during the course of a
time bin/readout attempt. We note that a spin flip during the measurement does
not necessarily lead to a readout error: spin flips that happen towards the end of
a time bin do not affect the correct measurement outcome, since the photocounts
remain on the correct side of the threshold. Spin flips early in the time bin,
however, do result in erroneous assignment of the spin state. Where exactly we
cross the line from correct to incorrect readout depends on where the threshold
value between ‘on’ and ‘off’ is. For a threshold that is approximately centred
between the ‘on’ and ‘off’ distributions we can say that only spin flips during the
first half of the time bin lead to wrong readout. The spin flip time from 0 to 1
(Toff) has been measured for various laser powers and is presented in Fig. 7.11.
For the data set of Fig. 7.8 this value is Toff=14.9 ms and the time bin is 2 ms.
From this, the probability of a spin flip in the first half of the time bin, i.e. in
1 ms is ≈ 1/14.9 ≈ 0.07. For spin flips from 1 to 0 we assume a timescale of ∼
100 ms which is a conservative estimate of the natural T1-time from Fig. 7.11b.
The chance of a spin flip in 1 ms is then ≈ 1/100 = 0.01. Summing up the errors
yields a fidelity of
1− p0e0 − p1e1 = 1− 0.5(0.07 + 0.01 + O(10−3)) = 0.96.
The magnitude of the ed0/e
d
1 error is determined by the length of the time bin
compared to the spin-flip timescale: longer time bins increase the signal-to-noise
ratio as they separate the shot noise distributions for states 0 and 1 and hence
reduce the en0/e
n
1 error. At the same time longer time bins increase the chance that
a spin flips during a readout attempt. We analyse the fidelity of the high laser
power data set (Ω ∼ 0.49T−11 in Fig. 7.11b). The time bin for this measurement
is 300 µs, while Toff has been determined as 5.2 ms, so that the e
d
0 error is reduced
(≈ 0.15/5.2 ≈ 0.03). This is at the expense of the ed0/ed1 error, which is of similar
162
magnitude. We sum the errors and obtain
1− p0e0 − p1e1 = 1− 0.5(0.03 + 0.002 + 0.035 + 0.055) = 0.94
in this case. We conduct a systematic analysis of error dependencies on integra-
tion times and driving powers in the next section.
7.8 Improving readout fidelities
Compared to the QDM sample used for spin readout, the devices based on the
‘Chef 2’ wafer used in the earlier chapters represent an improvement of factor ∼
4 in photon extraction and the background suppression is improved by a factor of
∼ 20. With these parameters we can project what readout fidelities are possible
with a QD molecule sample with optimised structure and a DBR layer.
We quickly summarise the strategy for spin readout and an optimisation of the
readout fidelity first: State-readout using resonance fluorescence is achieved by
probing a spin-selective transition for a set duration and comparing the number
of photodetection events to a suitably chosen threshold. Essentially, distinguish-
ing two spin state translates to telling if a particular transition is active/on or
off: If the number of photodetection events is above the threshold the transition
is assumed to be on (and the spin state is deduced), if the number of detec-
tions is below the threshold the transition is off. An erroneous assignment of the
spin state is made when the photodetection counts are above the threshold even
though the transition is off and vice versa. The readout error is given by the
weighted probabilities of these events; denoting the individual errors eon and eoff
we have a total error of e = 1/2 (eoff + eon) in the case of equal spin populations.
In general, two sources contribute to readout error, both dependent on the du-
ration of the readout attempt, the binsize. On the one hand, the photodetection
obeys poissonian statistics and two distributions can be associated with the on
and off cases. Any overlap of the two distribution gives rise to an error pro-
portional to the overlap. Minimising this error, i.e. separating the distributions
requires increasing the binsize. On the other hand, finite spin lifetimes mean
that during some measurement attempts a spin flip occurs that can obscure the
163
readout. Spin flips contribute to the area between the on/off distributions.
Any measurement should be short compared to the spin lifetime and smaller bin-
sizes decrease this type of error. Additionally, processes that shorten the natural
spin lifetime such as optical pumping can limit the fidelity further. Once the
statistics for spin dynamics and photon detection are known the threshold for
readout is chosen such that the overall error is minimised.
We mimic spin readout by switching the X1− transition of QD ‘Claude’ (sin-
gle QD, Chef 2 sample, zero external magnetic field) on and off resonance with
respect to the exciting laser and comparing photon statistics for the two cases.
This gives us the first type of error discussed above. Figure 7.12 a displays the
statistics for an excitation laser power at saturation. APD counts are recorded
on and off resonance with 10 µs resolution for a cumulative duration of 2 seconds.
The inset shows an exemplary timetrace where the trion transition is driven for
100 ms on resonance first and then 100 ms off resonance. The timetraces are
rebinned to 30 µs resolution for the histogram. APD dark counts have been
measured separately and are shown as well. The dashed curves are distributions
expected from poissonian statistics. In Fig. 7.12 b we plot the overlap of on/off
histograms vs. binsize for a range of powers from a tenth of saturation to close
to 200 times saturation. For a given bin size, we see the overlap decrease first as
the driving power and hence the QD emission rate is increased until it reaches
a minimum and increases for higher powers as the background rises while QD
emission saturates. Due to the finite sample size all on/off histograms separate
completely for larger bin sizes. Kinks and non-monotonous decay with bin size
are due to the discrete nature of photon counting.
Next, we include lifetime and spin pumping effects: From Eq. (7.6) and Fig.
7.11 we have a natural spin lifetime of T0→1 ∼ 140 ms. No spin pumping effects
are observed for spin down (state 1), but spin up (state 0) is pumped into the
down state when the readout laser is active. This is due to off-resonant (detuning
∆ = 2pi×8 GHz) driving of a transition that pumps the spin efficiently from up to
down at a rate of γsf = 2pi×0.49 MHz when on resonance with the spin-pumping
transition.
The results, see Fig. 7.13, are given for two excitation powers. Here, red corre-
164
0 5 1 0 1 5 2 0
1 0
1 0 0
1 0 0 0
1 0 0 0 0
1 0 0 0 0 0
 o n  r e s o n a n c e o f f  r e s o n a n c e d e t e c t o r  d a r k  c o u n t s
 A P D  c o u n t s  p e r  3 0  µs  b i n
 
 
num
ber
 of 
eve
nts 0 5 0 1 0 0 1 5 0 2 0 0
0
5
1 0
APD
 cou
nts
per
 10
µs b
in
t i m e  ( m s )
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 01 E - 3
0 . 0 1
0 . 1
1
1 0 n o r m a l i s e dp o w e r : 0 . 1 0 . 2 0 . 3 5 0 . 6 1 . 0 2 . 0 5 3 . 6 5 . 3 8 . 5 2 0 . 7 5 7 . 4 1 8 9
 
 
Ove
rlap
 of 
on/
off 
dist
ribu
tion
s (%
)
B i n s i z e  ( µs )
Figure 7.12: Signal-to-background and noise analysis of resonance fluorescence for
spin readout. Photodetection events are recorded with 10 µs resolution when the QD
transition is driven on and off resonance (see inset of a for exemplary time traces). a,
Histogram of photocounts per time of 30 µs for resonant driving (blue), off-resonant
driving (red) and no driving (black). b, Overlap of on-resonance and off-resonance
photodetection histograms as function of detection binsize for a series of excitation
powers.
165
sponds to the readout error at saturation whereas green is for a fifth of saturation
power. We include the individual errors for spin up (up-pointing triangles) and
spin down (down-pointing triangles) and the average (filled circles). Fidelities
higher than 99 % can be reached within 20 µs at saturation and a peak fidelity
of 99.59 % is achieved in 40 µs. Slightly higher fidelities are possible at smaller
excitation powers, but come at a cost of significantly larger binsizes: we predict
99.73 % fidelity for a fifth of saturation and a binsize of 120 µs (see arrows in Fig.
7.13).
0 5 0 1 0 0 1 5 0 2 0 00 . 0 1
0 . 1
1
1 0
R e a d o u t  f i d e l i t y  -  s p i n  u p  a n d  d o w n  c o m b i n e d
 
 
Rea
dou
t er
ror 
(%)
B i n s i z e  ( µs )
0 . 4 1 0 . 2 7
Figure 7.13: Estimated spin readout error for a QDM with improved photon collec-
tion efficiency (corresponding to the efficiency obtained with the Chef 2 sample). The
QDM parameters, such as off-resonant spin pumping rate and spin lifetime, are from
measurements on the actual QDM sample. Spin pumping, spin lifetime and the his-
togram overlap from Fig. 7.12 contribute to the calculation. Red data is for a readout
power at saturation (s = 1), while the green data is for s = 0.2. Up-pointing triangles
give the error for spin up readout, down-pointing triangles for spin down readout, and
the solid circles represent the average error.
A promising approach to improve the readout fidelities further might be to opti-
mise parameters limiting QDM readout. In our case this is spin pumping of the
up spin on the order of ms. Increasing the parameter ∆ in Eq. (7.6) by a fac-
tor two (which can be achieved by increasing the tunneling interaction slightly)
would bring the maximum fidelity up to 99.9 %.
We note that all results here rely on a simple threshold method; more sophisti-
cated methods, such as maximum likelihood have been shown to greatly improve
readout fidelities [107]. Quantum error correction schemes typically assume read-
out errors on the order of 10−3, which can be satisfied even with the threshold
166
methods.
7.9 Chapter outlook
In conclusion, we have observed spin-dependent quantum jumps in a QDM via
resonance fluorescence. The spin state of a single electron confined to one QD
of the vertically stacked QD pair is monitored by virtue of a molecular recycling
transition that is conditional on the electron spin state. The use of a QDM
eliminates the need to address the same transition for spin state manipulation and
readout, overcoming a main limitation of single QD qubits. While QDM molecules
have been investigated before, also with respect to spin readout, the combination
of this physical system with the high bandwidth of resonance fluorescence has
made single-shot readout possible. Our measurements with the QDM indicate
single-shot fidelities of ∼ 95% for measurement times of 250 µs, but with the
improved collection efficiency and signal-to-background ratio we extrapolate a
readout fidelity > 99.5% for a measurement time of 30 µs. Equally exciting,
the prospect of using single-shot readout in a verification scheme for distant spin
entanglement would decrease the measurement time by the inverse of the squared
combined collection/detection efficiency when compared to the use of single QDs.
For current collection efficiencies this time gain would be at least a factor of 1000.
167
Chapter 8
Conclusions and outlook:
towards a small-scale quantum
network
Eight chapters on - how does this thesis fit together as a whole and where do we
go from here? In the introduction we outlined the vision of the quantum internet:
A quantum optical network of individual quantum dots, or better quantum dot
molecules, entangled through quantum interference of their single photons. The
smallest possible network would be just two remote spin qubits which can be
manipulated individually and entangled via two-photon interference.
Removing obstacles in the way of achieving this next step of a small-scale network
has provided the motivation throughout the thesis. We were concerned with two
issues mainly:
1. Photon quality. This was driven by the following question: Can we entan-
gle two QDs - with high fidelity - using photon interference? Splitting this
question into smaller parts we aimed to answer: How can we generate high
coherence single photons from a QD? What are the limits on coherence?
How do environmental dynamics affect the photon quality?
2. Spin readout. Single-shot readout was, out of the first five, the only
missing DiVincenzo criterium for spins in QDs at the start of this thesis
work. Can we overcome the limitation of laser backaction flipping the spin
168
before its state is determined? What are the readout fidelities and how can
they be optimised?
The first point was investigated in Chapters 4,5 and 6 of this thesis. We set out to
systematically study the first- and second-order coherence of QD resonance fluo-
rescence in the previously unexplored regime of small Rabi frequencies, Ω . 1/T1.
We could report the direct observation of highly coherent single photon emission
- a somewhat novel feature in the solid-state community that dominates QD re-
search, despite being a standard topic in quantum optics textbooks. The emission
of subnatural linewidth single photons was detailed in the first part of Chapter 4
and is one of the main results of this thesis.
While this observation is a sign of negligible pure dephasing we still found marked
deviations from the properties predicted for an isolated two-level system. These
influences of the solid-state environment - spectral diffusion, exciton-phonon cou-
pling and Overhauser field dynamics - were explored in the second half of Chapter
4. One other influence on the QD optical transition, this time not a ‘natural’ but
laser-induced effect, was identified as an optically induced electric field in Chap-
ter 5.
The sensitivity of the QD transition to these additional interactions spell out a
serious challenge for the realisation of entanglement:
• Uncorrelated spectral diffusion for independent QDs reduces their spectral
overlap and hence limits photon interference probabilities.
• Uncorrelated Overhauser fields dephase independent spin qubits within the
inhomogeneous dephasing time of a few nanoseconds.
Characterising and quantifying the extent of these challenges is a first step. The
next important step is to minimise or to avoid them.
Chapter 6 detailed one approach to the photon interference part: relying on
coherent QD scattering locks the scattered photon’s frequency to that of the ex-
citing laser and avoids any pure dephasing. We used two-photon interference of
single QD emission to confirm that photons created this way are fundamentally
indistinguishable and envision that for entanglement purposes separate QDs are
addressed by the same laser. Of course, there are downsides to this scheme, most
169
importantly the low excitation efficiency. A better solution would be to eradicate
spectral diffusion in the first place. . . Future work will pursue the two approaches.
A starting point for the reduction of spectral diffusion will be an active stabilisa-
tion of the QD transition.
Going to the second main issue of single-shot spin readout: For the simple optical
level structure of a singly charged QD there are limited resources to satisfy all
of the first five DiVincenzo criteria at once. In particular, spin initialisation and
optical spin rotations rely on excited states coupled strongly to both qubits basis
states. This translates to a balanced branching ratio of excited to ground state
transitions. Spin readout on the other hand requires, in the face of non-unity
probability of photon collection and detection, a recycling transition.
Since these opposing requirements cannot be satisfied for single QD we turned to
quantum dot molecules in Chapter 7. There, the level structure is far more com-
plex than the simple two-level system we considered all the way up to Chapter
8, but it allows for a spin-sensitive recycling transition.
The more complex level structures of QDMs gives rise to further options for co-
herent spin control and it is likely that QDMs will replace single QDs as the main
workhorse for QIP applications. This is evidenced by the ongoing research which
has enabled 2-qubits gates [42] and protected qubits states [44] already.
Distant spin entanglement remains the motivation for further work. This the-
sis provides solutions to some of the challenges we face along the road, but some
remain to be solved.
We finish this chapter and the thesis by listing a few of the challenges which will
be taken on next.
• Improving the collection efficiency. Total internal reflection at the semiconductor-
air interface limits outcoupling of photons into the collection optics to 1-
2%.We have employed solid immersion lenses to ameliorate this issue and
increased outcoupling efficiency to 6-7%. However, theoretical considera-
tions suggest that optimal index matching alone should allow an increase
in extraction to 30% [77] and with improved sample design the extraction
efficiency should approach 99% [190]. We will explore new sample designs
170
following these proposals.
This technical improvement is particularly crucial if we want to verify spin-spin
entanglement. In the protocol of Duan et al. [153; 157] two-photon interference
and detection projects the qubits into entanglement. Hence the probability of
an entanglement event scales as the square of the collection/detection efficiency.
Then the spin states need to be measured to verify the entanglement. Using a
single QD as qubit host we only scatter 1-2 photons in the readout attempt so the
readout success rate for both qubit scales as the square of the collection/detection
efficiency again. Verified entanglement rates then go as the fourth power in total.
With our current system this translates to roughly one event every ten minutes.
If single-shot readout is available, however, the rate scales as the square only,
which would be a significant improvement. Entanglement rates could be further
increased using Cabrillo’s entanglement protocol [156] for example, but technical
requirements increase.
• Spin coherence and the Overhauser field. Long spin coherences have been
reported for frequency-domain measurements in combination with optical
feedback on the nuclear field [39]. The use of feedback with time-domain
spin rotations is still outstanding though. Further, dynamical decoupling
schemes, which have been applied very successfully for electrically defined
QDs [55] and NV centres [191], are an outstanding challenge as well.
• Efficient coherent scattering. Coherent scattering rates are 10-20 times
lower than for deterministic excitation schemes. It may be possible to com-
bine deterministic photon generation with coherent scattering: A very short
(picoseconds) optical pulse moves the population almost entirely into the
excited state and then a long weak pulse creates the two-level system co-
herences necessary for coherent scattering. This should keep the generation
process deterministic while still allowing coherent scattering to exist along-
side spontaneous emission.
171
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