Environmental Coupling in a Quantum
Dot as a Resource for Quantum Optics
and Spin Control
Jack Hansom
Department of Physics
University of Cambridge
This dissertation is submitted for the degree of
Doctor of Philosophy
Churchill College June 2015

Declaration
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.
It is not substantially the same as any that I have submitted, or, is being concurrently
submitted for a degree or diploma or other qualification at the University of Cambridge or
any other University or similar institution except as declared in the Preface and specified in
the text. I further state that no substantial part of my dissertation has already been submitted,
or, is being concurrently submitted for any such degree, diploma or other qualification at the
University of Cambridge or any other University of similar institution except as declared in
the Preface and specified in the text.
This dissertation does not exceed the word limit of 60000 words.
Jack Hansom
June 2015

This thesis is dedicated to my parents.

Acknowledgements
A bit more than four years ago, I was first shown around the lab. Clemens showed me his most
recent exciton Rabi oscillations, Tina tried to explain the Jahn-Teller effect (which mostly
went over my head), and Josh showed me images of bow-tie antennae. I was impressed with
their enthusiasm in showing yet another prospective student around their respective setups. I
knew from then on that this group would provide an enjoyable and cooperative environment
for my PhD.
First, I’d like to thank Mete Atatüre for taking me on, and for his continued guidance and
support. His enthusiasm and excitement are an essential part of what makes the group both
dynamic and fun. His occasional lectures on the art of milk frothing also provide a useful
distraction.
I’ve been fortunate to work with an extremely talented bunch of people within the QD team,
which has made for an enjoyable and intellectually challenging environment. Following
the theme of the thesis, this environment has been a tremendous resource in terms of
technical expertise and enlightening discussions, and all the members of the QD team deserve
acknowledgement: Alex, Carsten, Claire, Clemens, Lukas, Megan, Paul and Rob. I would
also like to thank Jake Taylor, whose insightful comments and suggestions have contributed
greatly to the outcome of this thesis.
The group has grown quite a bit since I first joined, and every new arrival has brought with
them new insight and energy to the group. As well as contributing to the atmosphere of
collaborative enthusiasm in the group, I’d like to thank the rest of the AMOP-MESS team for
helpful discussions and technical assistance from which this thesis has benefited significantly:
Ben, Carmen, Christian, David-Dominik, Dhiren, Helena, Jan, Josh, Lina, Tina, and Yury.
I would like to thank the administrative staff, and particularly Pam, whose efficiency and
organisation will never cease to amaze.
My writing is sometimes in need of a good deconvolution, and I’m grateful to Rob, Claire,
Mete and my father for proof-reading my thesis and giving me their suggestions and correc-
tions.
I would also like to thank Prof. Jonathan Finley and Dr. Andrew Ferguson who agreed to
examine this thesis and provided many insightful questions and comments during the viva.
viii
Beyond the physics, I will always have fond memories of going to conferences and other
outings with the group: from teaching kids and (not-so-) grown-ups about soap computers at
the Green Man festival to hiking among the redwoods of California.
Finally, I would like to thank my friends and family for their love and support. In particular,
I’d like to thank LeAnn for her continued support throughout my PhD, even with an ocean
between us.
Abstract
A single spin confined to a semiconductor quantum dot is a system of significant interest for
quantum information science, as a potential optically-addressable qubit. In many respects, a
quantum dot behaves like a single atom with high quality single photon emission. By control-
ling the light-matter coupling in such a system, it is possible to generate highly non-classical
states of light and coherently control a single spin confined to the quantum dot.
A departure from the ideal atomic picture appears once we consider the mesoscopic environ-
ment with which the quantum dot interacts. Charge fluctuations in the surroundings of the
quantum dot affect the photon emission frequency leading to inhomogeneous broadening.
Further broadening of the emission is caused by coupling to phonon modes of the host
semiconductor material. Finally, coupling between the spin of a confined electron and a
large bath of nuclear spins residing in the quantum dot leads to fast dephasing of the electron
spin. All of these effects are typically considered detrimental to the potential use of quantum
dots for quantum technologies. In this thesis, we develop the environmental coupling of a
negatively charged quantum dot as a resource for quantum optics and spin control.
First, the phonon-assisted fluorescence is shown to be a useful independent channel for feed-
back stabilisation of the quantum dot emission frequency, without requiring a measurement
of the indistinguishable zero-phonon line. With stabilisation, the corresponding frequency
broadening is drastically improved, and the sub-Hz frequency fluctuations are no longer
resolved.
Next, we show low-power resonance fluorescence emission spectra of the negatively charged
trion transition. In the low power regime of resonance fluorescence, the excited state is not
populated and most of the emission is coherent. In addition to elastic Rayleigh scattering,
we observe coherent Raman sidebands, linked to an effective magnetic field created by the
hyperfine interaction, the Overhauser field. This fluctuating effective field lifts the electron
spin degeneracy in the absence of a magnetic field, and dictates the optical selection rules
of the trion system. These spectra therefore allow for a measurement of the time-averaged
distributions of in-plane and out-of-plane Overhauser field components.
In the final part of the thesis, we use this hyperfine-generated Λ-scheme to optically create
electron spin superpositions through two-colour excitation and coherent population trapping.
We then show that rapid shifts in the relative phase of the lasers lead to initialisation of the
electron spin into a rotated dark state.

Table of contents
1 Introduction 1
1.1 Quantum Dots as Artificial Atoms . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Quantum Dot Growth . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 From Confined Exciton to Light . . . . . . . . . . . . . . . . . . . 2
1.1.3 Sample Structure and External Tuning Parameters . . . . . . . . . 4
1.2 Quantum Information Processing with Quantum Dots . . . . . . . . . . . . 7
1.2.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Environmental Coupling in Quantum Dots . . . . . . . . . . . . . 9
2 Theory of Resonance Fluorescence from a Two-Level System 13
2.1 Semiclassical Model of Light-Matter Interaction . . . . . . . . . . . . . . . 14
2.2 Resonance Fluorescence Correlation Functions . . . . . . . . . . . . . . . 16
2.2.1 First Order Correlation Function . . . . . . . . . . . . . . . . . . . 18
2.2.2 Second Order Correlation Function . . . . . . . . . . . . . . . . . 21
2.3 Phase Space Distribution of Resonance Fluorescence . . . . . . . . . . . . 22
2.3.1 Wigner Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.2 Squeezing in Resonance Fluorescence . . . . . . . . . . . . . . . . 25
3 Experimental Methods for Quantum Dot Resonance Fluorescence 29
3.1 Optics with Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.1 Confocal Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.2 Resonant Excitation . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Resonance Fluorescence Spectroscopy and Photon Statistics . . . . . . . . 35
3.2.1 Absorption Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.2 Emission Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.3 Intensity Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . 41
xii Table of contents
4 Phonon-assisted Resonance Fluorescence Frequency Stabilisation 47
4.1 Electric Field Noise in Quantum Dots . . . . . . . . . . . . . . . . . . . . 48
4.2 Phonon Sideband Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 Feedback Stabilisation Technique . . . . . . . . . . . . . . . . . . . . . . 56
4.4 Stabilisation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4.1 ZPL Intensity Fluctuations . . . . . . . . . . . . . . . . . . . . . . 58
4.4.2 Stabilization Bandwidth . . . . . . . . . . . . . . . . . . . . . . . 62
4.4.3 Inhomogeneous Broadening . . . . . . . . . . . . . . . . . . . . . 64
5 Hyperfine Effects in Resonance Fluorescence 65
5.1 Mean-field Approach to the Hyperfine Interaction . . . . . . . . . . . . . . 66
5.2 Optically Induced Spin Dynamics . . . . . . . . . . . . . . . . . . . . . . 71
5.2.1 Finite Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2.2 Zero Applied B-field . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3 Coherent Raman Spectroscopy under Zero Magnetic Field . . . . . . . . . 77
5.3.1 Deviation from Two-Level System . . . . . . . . . . . . . . . . . . 77
5.3.2 Calculation of Four-Level Resonance Fluorescence Spectrum . . . 79
5.3.3 High Resolution Measurement of the Nuclear Spin Distribution . . 81
5.4 Nuclear Spin Polarisation under Low Magnetic Fields . . . . . . . . . . . . 86
6 Hyperfine-Assisted Coherent Population Trapping of an Electron Spin 91
6.1 Theory of Coherent Population Trapping . . . . . . . . . . . . . . . . . . . 92
6.1.1 Two-colour Excitation of an Ideal Three-Level System . . . . . . . 92
6.1.2 Extension to the Hyperfine-split Trion System . . . . . . . . . . . . 96
6.2 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.3 Hyperfine-assisted Coherent Population Trapping . . . . . . . . . . . . . . 99
6.4 Spin Control via Phase Modulation . . . . . . . . . . . . . . . . . . . . . . 108
7 Conclusion 113
7.1 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.2 Towards a Photonic Cluster-state Machine Gun . . . . . . . . . . . . . . . 117
References 119
Chapter 1
Introduction
In this introduction, we aim to clarify the terms used in the title of the thesis. First the system
under study, consisting of self-assembled semiconductor quantum dots (QD), is introduced.
We then briefly recap the state of the art of QD research against the background of quantum
information science. Finally, we introduce the environment mentioned in the title, and its
problematic aspects in terms of QD-based quantum optics and spin control.
1.1 Quantum Dots as Artificial Atoms
1.1.1 Quantum Dot Growth
Progress in the field of quantum information science with QDs has been made possible
by the developments in heteroepitaxial growth techniques. The QDs studied here were
grown by molecular beam epitaxy (MBE) in the Stranski-Krastanov (SK) growth mode
[1]. MBE allows for layer-by-layer crystal growth of high-quality semiconductor materials
controlled on the monolayer scale allowing abrupt changes in materials. Complex layered
heterostructures can be grown in this way, which has been crucial to the development of
high-quality optically-active QDs. For our samples, InAs is grown on top of a GaAs substrate.
These two materials have similar crystal structure but a lattice constant mismatch of ≈ 7%.
At first, the InAs layer adapts to the substrate lattice and strain accumulates. After a critical
layer thickness, the strain relaxes and islands of InAs form spontaneously.
The random nature of the SK growth process means that QDs are formed at random locations
and have randomly distributed sizes and morphologies. This isn’t critical for the single-QD
studies discussed in this thesis as we can afford to spend some time looking for the right QD.
However, this is a major challenge for scaling QD-based technologies and has motivated
research into the deterministic positioning of optically-active QDs [2].
2 Introduction
By carefully choosing appropriate growth conditions, islands of dimensions comparable to
the exciton Bohr radius in the bulk material can be grown. The InAs direct band gap (0.43
eV) is much smaller than that of GaAs (1.52 eV), which leads to a creation of a confining
potential for electrons and holes within the QD. This 3 dimensional confinement leads to a
discrete energy spectrum of excitons confined to the QD.
The exact energy spectrum depends on the QD size and composition. This can be tuned by
varying the growth parameters, or alloying induced by post-growth annealing. Specifically,
the height of the dot can be tuned through partial capping of the QD layer with GaAs and
subsequent annealing. The typical QDs grown in this fashion are lens shaped with a diameter
of order 10 nm and a height of around 3 nm, and the confinement is therefore strongest in the
z-direction. This technique blue-shifts the energy spectrum to ≈ 1.3 eV such that common
Si detectors can be used to study these QDs.
1.1.2 From Confined Exciton to Light
We have established that charge carriers, i.e. conduction band electrons and valence band
holes, can be trapped by the confining potential created by the QD. Light emission is then
linked to the recombination of excitons, bound electron-hole pairs, confined to the QD. This
process is schematically depicted in Fig. 1.1. Light emission from the QD can be induced
by optical creation of excitons within the host matrix, with above bandgap excitation. The
excitons rapidly decay to the lowest available energy state, i.e. into the QD. The subsequent
recombination on a timescale of hundreds of ps leads to the emission of a photon. The entire
process is called photoluminescence. An example spectrum obtained via this process is
shown in panel B of Fig. 1.1. Discrete peaks can be seen in the spectrum, confirming the
atom-like nature of the QD. The peaks which are not labeled are linked to emission from
other QDs within the collected area. For the three most intense peaks, generated by the same
QD, each peak is associated to a different charge complex within the QD. In addition to
electron-hole pairs, single charges can be trapped by the QD during the above-band excitation
of the sample, and these will occupy the ground state of the QD. The charge state of the QD
therefore fluctuates in time, and three peaks are observed in a time-averaged measurement.
The labels indicate the charge of the ground state, X0 being a neutral exciton (electron-hole
pair), and X- being a charged trion (two electron singlet and one hole). Coulomb interactions
will lead to different energies for exciton, trion, and biexciton states, leading to different
emission wavelengths.
GaAs and InAs have similar band structures, and both have a direct band-gap between a
p-type valence band and an s-type conduction band. The latter has S = 1/2 and therefore
two-fold degeneracy. In the presence of spin-orbit coupling, the valence band is composed of
1.1 Quantum Dots as Artificial Atoms 3
Fig. 1.1 A. Schematic diagram depicting photoluminescence from a QD. The filled circles
represent conduction band electrons and the hollow circles depict valence band holes. The
green arrow represents optical above bandgap excitation, and the red arrow represents
exciton recombination along with the emission of a photon. B. Example photoluminescence
measurement. The labels represent the charge complexes linked to each emission peak.
4 Introduction
six [J||Jz] states. Spin-orbit coupling leads a large splitting (0.41 eV in InAs) between the
[1/2||±1/2] doublet and the [3/2||±3/2,±1/2] quadruplet. This doublet is referred to as
the split-off band. The degeneracy of the quadruplet is further lifted at k ̸= 0 in bulk due
to the influence of other bands (described by the Luttinger Hamiltonian [3]), forming two
bands with ±3/2 and ±1/2 angular momentum projection at the Γ-point. These are called
heavy and light hole bands respectively, due to their differing curvatures in dispersion. In
self-assembled QDs, the degeneracy between these two bands at k = 0 is lifted by strain,
leading to a large energy splitting of tens of meV between the heavy and light hole bands [4].
The heavy-hole band then constitutes the single-particle hole ground state. A combination
of heavy-hole and s-type electron has total angular momentum J =±1,±2. As photons are
spin 1 particles, only the first type of exciton recombination is allowed by an electric-dipole
transition. Hence the J =±1 excitons are called bright and the J =±2 dark.
In the case of the neutral exciton, additional complexity arises from the lowered symmetry
of the QD due to the growth process. The anisotropic electron-hole exchange interaction
admixes the bright exciton states, giving two orthogonal linearly polarised transitions [4].
This is often called the X-Y splitting (≈ 10 µeV ) for this reason.
1.1.3 Sample Structure and External Tuning Parameters
Electric field Tuning
Several problems arise from what we have discussed so far. Firstly, self-assembled QD
emission energies are tied to their size and morphology which is randomly distributed from
dot-to-dot. Secondly, the data shown in Fig. 1.1 shows that multiple charge configurations
can exist within the QD. In order to mitigate these effects, we use QDs embedded within a
Schottky diode structure shown in Fig. 1.2. An n-doped GaAs layer is grown 35 nm below
the QD layer providing an electron reservoir with Fermi energy EF. A semi-transparent
Titanium top-contact provides the Schottky barrier. A voltage can be applied between this
top-gate and an Ohmic contact to the n-doped layer, thereby tuning the charging energy of
the QD relative to the Fermi level of the electron reservoir. Crucially, Coulomb interactions
between electrons confined to the QD raise the two-electron charging energy compared with
the single electron energy. For temperatures lower than this splitting, the QD can therefore
be deterministically charged with a single electron. Working at 4K satisfies this condition.
Despite energy-conservation prohibiting further electron tunneling into the QD, a second
order process involving simultaneous tunneling to and from the Fermi sea is allowed, called
cotunneling. This process gives rise to electron spin relaxation and its rate can be tuned
1.1 Quantum Dots as Artificial Atoms 5
Fig. 1.2 Schematic diagram representing the band structure in the gated heterostructure. The
dashed blue line represents the Fermi energy, and the dashed gray lines represent the charging
energies within the QD, split by the Coulomb blockade. A voltage can be applied between
the Ti Schottky contact and the n-doped layer to tune the charging energy levels relative to
the Fermi level.
through the tunnel barrier thickness. The AlGaAs layer between the QD layer and Schottky
contact prevents tunneling between these two, due to its larger band gap.
A further benefit of using a gated sample is that it offers us a tuning knob for emission
energy through the quantum confined Stark effect. The electron and hole wavefunctions of a
confined exciton are spatially separated, to some extent. This electric dipole is affected by
external electric fields [5], thereby modifying the electron and hole energy levels and the
emitted photon energy after recombination. Thanks to the confinement provided by the QD,
the frequency tunability of the emission can be orders of magnitude larger than the linewidth
without ionising the exciton, for a given charge configuration.
Charged Exciton Level Structure
Another important experimental tuning knob is given by applying a magnetic field of con-
trolled magnitude and direction. Here, we briefly discuss the impact of such an external
magnetic field on the level structure of the QD. As it is of interest for the work shown in
this thesis, we will focus on the negatively charged QD level structure. Both the orientation
and magnitude of the applied field will significantly modify the emission spectrum through
the Zeeman effect. In the experiments shown here, we only use a magnetic field aligned
along the growth direction (called Faraday geometry), but it is instructive to also consider the
6 Introduction
Fig. 1.3 A. Trion level structure under an applied magnetic field in the growth direction:
the Faraday geometry. The red arrows indicate dipole allowed optical transition. The filled
blue arrow represents the ground state electron spin and the hollow arrow represents the
heavy-hole pseudo-spin. B. Trion level structure under an applied magnetic field orthogonal
to the growth direction: the Voigt geometry.
orthogonal (Voigt) geometry, particularly when we come to discuss the effective magnetic
field of the nuclei in chapters 5 and 6.
The ground state in a negatively charged QD is given by the electron spin, which can have
the following spin projections along the growth direction { |↑⟩= |1/2⟩ , |↓⟩= |−1/2⟩ }. The
lowest excited state is composed of two electrons in a singlet state and a heavy-hole spin. The
paired electron spins play no role, and the excited states therefore have angular momentum
projection { |⇑⟩= |3/2⟩ , |⇓⟩= |−3/2⟩ }. The electron triplet states have significantly higher
energies and can safely be neglected here.
In the Faraday configuration, both excited and ground states split according to the hole
and electron out-of-plane g-factors, as shown in Fig. 1.3 A. Of the four optical transitions,
only two have angular momentum difference ∆M = 1 and are optically allowed. The Zeeman
splitting seen experimentally is then given by the difference of electron and hole g-factors.
In this geometry we have two separate cycling transitions with ideally no diagonal relaxation.
In practice, due to light-heavy hole mixing, the excited states have a finite component with
angular momentum |±1/2⟩. The diagonal transitions are then dipole-allowed, which allows
for optically-induced electron spin pumping [6].
1.2 Quantum Information Processing with Quantum Dots 7
In the Voigt configuration, shown in panel B, the Zeeman Hamiltonian admixes the
z-projections of angular momentum [4]:
|⇑⟩x =
1√
2
(|⇑⟩z+ |⇓⟩z)
|⇓⟩x =
1√
2
(|⇑⟩z−|⇓⟩z)
|↑⟩x =
1√
2
(|↑⟩z+ |↓⟩z)
|↓⟩x =
1√
2
(|↑⟩z−|↓⟩z)
All four transitions between ground and excited states are then optically allowed. In this case
ground and excited state splittings are determined by the in-plane electron and hole g-factors.
To first approximation, all transitions have equal oscillator strengths and are linearly polarised
as indicated in panel B. The presence of spin-flip diagonal transitions allows for ultrafast
optical electron spin control [7].
1.2 Quantum Information Processing with Quantum Dots
1.2.1 State of the Art
One of the main motivations for the recent research into optics with quantum dots has been
finding a good physical system with which to process quantum information. A spin confined
to a quantum dot could provide such a system, and a great deal of research has been done to
see whether this system fulfills the DiVicenzo criteria [8] for quantum computation. These
criteria give a list of conditions a quantum system must fulfill in order to be useful for
quantum information processing. Here, we will briefly summarise the state of the art in QD
research compared with these criteria, with a particular focus on negatively charged QDs.
The first criterion is having a well defined qubit. As we have mentioned earlier, QDs
allow the trapping of a single electron, for which the spin states can be considered as qubit
states. Depending on the magnetic field used, electron spin relaxation between these states
is limited by cotunneling processes or phonon-induced decay. This relaxation can be made
longer than ms [9, 10], with an appropriate tunnel barrier and at moderate magnetic fields.
The second condition is to be able to initialise the qubit into a pure state. Considering
the level structures of Fig. 1.3, this can be done by optical pumping. The rate at which
one can initialise this state depends on the branching ratio, and therefore on the magnetic
8 Introduction
field geometry. This has been shown both in the Faraday [6] and Voigt geometries [11], and
initialisation fidelity higher than 99% can be obtained.
A further criterion states that a complete set of unitary operations is needed, and can
be separated into two requirements of full single-qubit control and a two-qubit entangling
gate. For single-qubit gates, various forms of coherent spin control have been shown in
QDs. The Voigt geometry is the most useful in this instance, as a single electron spin can be
manipulated optically through the spin-flip Raman transitions [7]. These optical transitions
allow spin control on a timescale of ps through ultrafast pulses. These pulses dress the
level structure through the AC Stark effect allowing coherent rotations about one axis of the
Bloch sphere, while rotations about an orthogonal axis are achieved either through Larmor
precession in the external magnetic field, or ultrafast geometric phase gates [12].
Another criterion concerns the interaction of the qubit with the environment, and how
quickly the quantum information stored in the qubit is lost. This is quantified through the
coherence time T2 of the qubit and will be discussed in more detail later in this section. How
long the coherence time needs to be depends on the time required for qubit manipulation. This
ratio is a standard benchmark for the comparison of different quantum systems. With quantum
error-correcting codes [13], decoherence times 104 times longer than qubit manipulation
times would be required. The latter is where QDs figuratively shine, as the qubit manipulation
can be done optically on a timescale of ps, much faster than systems requiring microwave
spin manipulation.
High fidelity and fast readout of the spin is also required. Again, this can be done optically
with QDs. Single-shot readout is only possible in the Faraday geometry [14], due to the
quasi-cycling transition. However the transitions have an appreciable spin-flip branching
ratio (1:200) which limits the readout fidelity. High fidelity single-shot readout has been
achieved with a qubit encoded in a QD molecule [15]. Reaching such a stage in single
quantum dots will require further development of non-destructive readout techniques, such
as Faraday rotation [6] or Kerr rotation [16] of a probe beam, which could be enhanced to
single-shot level by quantum dot spin coupling to a quantum well [17].
Two further criteria concern the interconversion of stationary and flying qubits. Optically-
active QDs naturally provide such a link due to its spin-correlated optical transitions. The
presence of entanglement between spin and emitted photon required for such a link was
shown recently with QDs [18–20]. This brings us back to the two-qubit entangling gate
requirement that we mentioned earlier. One feasible architecture with QDs is based on the
idea of a ’quantum internet’ [21] composed of a set of quantum dot nodes connected by
optical links. Entanglement between distant QD spins can then be created probabilistically
through, for instance, a Bell state measurement of two photons emitted from each QD. This
1.2 Quantum Information Processing with Quantum Dots 9
effectively transfers the spin-photon entanglement to spin-spin entanglement. Combined with
the aforementioned single qubit manipulation, this would satisfy the criterion of a universal
set of quantum gates. However, this method relies on photons emitted by different QDs to
be quantum mechanically indistinguishable in order to erase which-path information in the
entanglement transfer. In this context, the solid-state environment once again comes into
play.
1.2.2 Environmental Coupling in Quantum Dots
Although they are sometimes called artificial atoms due to their discrete optical transitions,
charges confined to quantum dots exhibit a qualitatively different behaviour. Departures from
the atomic picture are caused by the presence of a mesoscopic environment of nuclear spins
and charge carriers, as well as coupling to phonon modes of the host lattice.
Hyperfine Effects in Quantum Dots
The SK-grown InGaAs quantum dots used in this thesis have very advantageous optical
properties, in terms of quantum efficiency and single-photon quality. However, every naturally
occurring isotope of all the atomic elements forming the QD have a nuclear spin: 9/2, 3/2, and
3/2 for the most common isotopes of In, Ga, and As respectively. Charge carriers confined
to the QD therefore interact with a large bath of nuclear spins. In fact, the confinement
of the carrier wavefunction within the QD exacerbates this interaction compared with a
delocalised carrier. Indeed, the reduced size of the sampled nuclear spin bath means that
random fluctuations no longer average to zero, leading to appreciable effects even in the
absence of nuclear spin polarisation. This interaction can, to a good approximation, be
treated as a mean field with a given statistical distribution and correlation time. This effective
nuclear hyperfine magnetic field is called the Overhauser (OH) field, after the prediction by
Albert Overhauser [22] that such an effective nuclear field can be generated by transfer of
electron spin polarisation through the hyperfine interaction. Later, it was demonstrated that
optical spin pumping of the electron offers an efficient means of generating significant OH
fields [23], and this technique has since been a mainstay of optical studies of nuclear spin
effects.
In terms of quantum information processing, random fluctuations of the OH field are
problematic for the use of charges confined to QDs as local qubits. Indeed, a finite OH field
distribution leads to uncertainty in the instantaneous Larmor frequency of the electron and
consequently to spin dephasing. While the electron spin feels an effective magnetic field due
to the nuclear spins, the opposite is also true and the nuclear spins precess in the effective
10 Introduction
field produced by the electron spin. The two systems feed back onto each other and the
dynamics of electron spin relaxation and dephasing are non-trivial. The Merkulov-Efros-
Rosen model [24] describes the dynamics of electron spin relaxation in the presence of a
random OH field and predicts multiple characteristic timescales. The fastest timescale is
linked to the random orientation of the OH filed with respect to the initialised electron spin.
The OH field modifies the precession axis shot-to-shot and a time-averaged measurement
will exhibit dephasing on a timescale inversely proportional to the OH field dispersion [24].
The second fastest timescale is then linked to precession of the individual nuclear spins about
the effective electron magnetic field, or Knight field. This model has been recently validated
experimentally for InGaAs QDs, with the first and second timescales of order 1 ns and 300
ns respectively [14]. These two timescales are spin conserving within the QD, and further
decay on longer timescales correspond to spin diffusion outside of the QD.
In terms of using the electron spin as a qubit, the hyperfine interaction therefore leads to
a short inhomogeneous dephasing time T ∗2 . Fortunately, as the nuclear spin diffusion is
relatively slow compared to the electron spin manipulation times, much of the fast dephasing
can be counteracted by using a spin-echo pulse scheme on the electron spin [25].
The growth process affects the nuclear spin dynamics, and hyperfine effects in the SK-
grown QDs under study exhibit significant differences compared with unstrained QDs. The
strain field is highly position-dependent within the QD and is maximum at the boundary with
the GaAs matrix [26]. This gives rise to inhomogeneous electric fields which interact with
the quadrupole moment of the nuclei, giving inhomogeneous spin splittings within the QD
[27]. Nuclear spin diffusion through energy-conserving flip-flop processes is therefore sup-
pressed which greatly prolongs the OH field correlation time [28], in the absence of a Knight
field. Consequently, nuclear spin polarisation decays slowly compared with unstrained QDs
[28, 29] and large degrees of nuclear spin polarisation can be achieved, corresponding to
an effective magnetic field of several Tesla [30]. These QDs therefore offer an interesting
testbed of a non-Markovian central-spin system.
The Overhauser effect isn’t the only method allowing for control of the nuclear spin bath.
Resonant optical excitation has also lead to striking observations of dynamic nuclear spin
polarisation. In contrast to the Overhauser effect, this form of nuclear spin polarisation is
independent on the polarisation of light used, is bidirectional, and is seen in both Faraday
and Voigt geometries [26, 31]. This manifests itself experimentally as top-hat absorption
spectra with a width up to an order of magnitude larger than the zero-field linewidth, with
a hysteresis in laser frequency scan direction. These effects arise due to an effective non-
collinear hyperfine term (Sˆz.Iˆx), giving qualitatively different behaviour from the well-know
flip-flop terms (Sˆ+.Iˆ−+ Sˆ−.Iˆ+) [27, 32, 33].
1.2 Quantum Information Processing with Quantum Dots 11
Finally, despite strain-field induced inhomogeneous broadening of nuclear spin transitions,
radio-frequency control has also recently been achieved with self-assembled QDs [34, 35].
All these techniques show viable routes to OH field control and even perhaps a reduction in
its fluctuations. Achieving such a narrowing would extend the coherence time of the electron
spin without requiring complex dynamical decoupling schemes.
The hyperfine interaction in QDs will be discussed in chapters 5 and 6, particularly in
the regime of low-magnetic field and low degree of nuclear spin polarisation. We show that
the nuclear spin bath provides a quasi-static quantisation axis for the electron spin, giving a
level structure intermediate between the Faraday and Voigt geometries. Optical spin control
is then shown to be possible within this level structure.
Optical Linewidth Broadening via Charge Fluctuations and Phonon Coupling
The proposed ’quantum internet’ architecture mentioned in the previous section requires
reliable spin-photon interfaces, for which the quality of the emitted photon is key. One of
the figures of merit of interest here is the indistinguishability of photons emitted at different
times or by different QDs, which would ideally be unity for transform-limited emission
with identical radiative lifetimes. The aforementioned nuclear spin dynamics will again
be important, but other environmental couplings will contribute to a broadening of the
emission beyond the transform limit: charge fluctuations within the semiconductor sample
and coupling to bulk acoustic phonon modes. Depending on the sample quality, the former
can lead to spectral diffusion or blinking of the emission in time. The state of the art in terms
of indistinguishability of incoherently generated (postselected) photons from independent
QDs is 82%, achieved in resonant excitation, and is limited by such spectral diffusion of both
emitters [36].
Different approaches are possible to mitigate the effect of spectral wandering. By moving
to the low power limit of excitation, where the emission spectrum follows that of the laser,
near-unity indistinguishability can be achieved [37], at the expense of lowered emission
probability. However, intensity fluctuations due to spectral wandering can still negatively
impact the visibility of two-photon interference measurements [38] even in the coherent
emission regime. Embedding the QDs within an optical cavity [39] could lead to a Purcell
enhancement of the radiative decay rate thereby broadening the transform limit beyond the
inhomogeneous contribution. Alternatively, external controls, such as electric and magnetic
fields, can be used as a control knobs for feedback stabilisation of the emission frequency
[40–42].
12 Introduction
Due to the discrete nature of the energy levels within the QD, phonon coupling does not
lead to direct energy relaxation of the confined ground state excitons. However, phonon-
assisted radiative recombination occurs, leading to the creation of a broad component of
the spectrum [43, 44]. This component is a further source of photon distinguishability.
In practice, a spectral filter can be used to reject this part of the spectrum, in which case
phonon-assisted processes can be thought of as an added optical loss in the detection path.
In self-assembled QDs, this mechanism is a significant contributor to the total photon loss
as the phonon-assisted emission typically corresponds to ≈ 10−15% of the total emission
intensity. Nonetheless, this value compares favourably with other solid-state emitters, such
as nitrogen-vacancy or silicon-vacancy centers in diamond where the emission fraction into
the phonon-sideband accounts for ≈ 96% [45] and ≈ 20% [46] respectively.
These two environmental couplings, and their impact on the QD spectrum, are the topic
of chapter 4. Instead of treating the phonon-assisted emission as a loss, we show that it can
be used to provide feedback stabilisation of the zero-phonon emission, thereby mitigating
the effect spectral diffusion arising from charge fluctuations.
Chapter 2
Theory of Resonance Fluorescence from
a Two-Level System
The problem of a single two-level atom interacting with a near-resonant electromagnetic field
- resonance fluorescence - is a very simple one to state, but has been one of the most studied
systems in quantum optics since its inception. It is of interest from a fundamental point
of view, as the system is central to the understanding of light-matter coupling. Resonance
fluorescence is also of interest as a method of generation of single photons for quantum
information processing [47–49], quantum relays [50], and quantum networks [21]. Finally,
resonance fluorescence is of particular interest for this thesis as an investigative tool to
uncover the solid-state physics affecting the QD. Indeed, compared with incoherent excitation
methods such as photoluminescence which creates a large number of interacting free carriers
in the environment, resonant excitation below the band gap is relatively non-invasive. In
addition, the coherent nature of the light-matter interaction allows emission spectroscopy for
which the resolution is no longer limited by the radiative lifetime of the excited state.
The aim of this chapter is to give an overview of the predictions of resonance fluorescence
and the theoretical tools used throughout the thesis to calculate observable correlation
functions. In the first two sections of the chapter, we will review the textbook theory of
resonance fluorescence and state its main predictions about the quantum state of the emitted
light, particularly its first and second order coherence. In the third section we will deviate
from textbook knowledge and discuss the phase-space properties of resonance fluorescence,
particularly in the low power regime. Quadrature squeezing is the predicted outcome of the
marriage of wave-like and particle-like characteristics in this regime. The content of this
final section is adapted from a manuscript in preparation at the time of writing [51].
14 Theory of Resonance Fluorescence from a Two-Level System
Fig. 2.1 Illustration of a two-level system interacting with a near-resonant electric field.
Ω is a parameter related to the amplitude of the driving field and describes the rate of
coherent energy transfer between field and atom. Γ is an incoherent decay rate describing the
spontaneous emission.
2.1 Semiclassical Model of Light-Matter Interaction
The theory of resonance fluorescence describes the interaction between a two-level system
and a near-resonant oscillating electric field and is a standard feature of most quantum optics
textbooks [52–55]. Although most of the thesis involves more energy levels due to the
presence of a hyperfine interaction, the two-level results illustrate the features of resonance
fluorescence useful for solid-state spectroscopy and can be generalised to a system containing
both spin and exciton states. The basic picture, along with the relevant physical parameters,
is shown in Fig 2.1. Under the dipole approximation, an electric dipole in an electric field is
described by the following interaction Hamiltonian:
Hint =−D.E(R, t) (2.1)
2.1 Semiclassical Model of Light-Matter Interaction 15
where D = q.r is the electric dipole moment for charge q and position operator r. E(R, t)
is the electric field at the location of the atom. The interaction between the atom and an
intense oscillating electric field can be treated semi-classically, with the field being treated as
a classical (linearly polarised) oscillating electric field: E(t) = εcos(ωl t) xˆ. After applying
the rotating-wave approximation, the interaction Hamiltonian can be written:
Hint =
h¯Ω
2
[
σ+e−iωlt +σ−eiωlt
]
(2.2)
Ω=
eε
h¯
⟨2|x |1⟩
where e is the electronic charge, and we have assumed that the field is polarised along xˆ
and both the field amplitude and dipole matrix element have been absorbed into the Rabi
frequency Ω. x is the position operator along xˆ, σ± are the atomic raising/lowering operators
and ωl is the laser frequency. Using the notation of Fig. 2.1, we define the basis as:
|1⟩=
(
1
0
)
; |2⟩=
(
0
1
)
(2.3)
The full Hamiltonian, after neglecting the approximately constant electric field energy term,
can be written conveniently in the atomic basis :
H = h¯
(
0 Ω2 e
iωlt
Ω∗
2 e
−iωlt ωX
)
(2.4)
where h¯ωX is the excited state energy. The dynamics of this system are then computed with
the Liouville equation:
dρ
dt
=− i
h¯
[H,ρ]+L [ρ] (2.5)
whereL [ρ] is the Lindblad superoperator describing the irreversible dynamics of the system,
namely spontaneous emission and any pure dephasing processes. In the absence of the latter,
this superoperator takes the form [52]:
L [ρ] =
Γ
2
(2σ+.ρ.σ−−σ+.σ−.ρ−ρ.σ+.σ−) (2.6)
where Γ is the radiative decay rate of the excited state. Equation 2.5 gives rise to a set of
coupled differential equations, called the Bloch equations. By flattening the density matrix
into vector form and using the fact that its trace is 1, we can rewrite the Bloch equations for
16 Theory of Resonance Fluorescence from a Two-Level System
the slowly varying density matrix elements in a simple matrix form[54]:
dS
dt
=−M.S+Λ (2.7)
S =
e−iωltρ12eiωltρ21
ρ22
 ; Λ=
 iΩ/2−iΩ/2
0
 ; M =
Γ/2+ iδ 0 iΩ0 Γ/2− iδ −iΩ
iΩ/2 −iΩ/2 Γ

where δ = ωl−ωX is the laser detuning from resonance and ρi j = ⟨i|ρ | j⟩. Pure dephasing
terms can be added phenomenologically by making the replacement Γ/2→ Γ/2+ γ in the
diagonal terms of M corresponding to the atomic coherences. Solving this equation in the
steady-state limit dSdt = 0 gives the steady state atomic populations and coherences. Analytical
solutions exist for this simple system [53] and for exactly resonant excitation, the steady-state
density matrix is given by:
ρatom =
 Γ2+|Ω|2Γ2+2|Ω|2 iΩ∗ΓΓ2+2|Ω|2
−iΩΓ
Γ2+2|Ω|2
|Ω|2
Γ2+2|Ω|2
 (2.8)
This simple expression is particularly useful as it gives the dependence of the QD populations
on the input laser power. This allows for experimental calibration of the laser power without
requiring full knowledge of the field amplitude at the QD or its dipole matrix element. For
this we use another definition of the Rabi frequency :
Ω= Γ
√
s
2
(2.9)
where s is called the saturation parameter and is proportional to the input laser power. From
equations 2.8 and 2.9, it follows that the s = 1 condition is obtained when the excited
state population is half the high-power limit. So far these equations describe the state of
the atom only, but in our experiments we always measure the emitted field. In the far-
field limit, we can use the relation between source-field operator and atomic operator [53]
E+sf (r, t)∝σ−(t−r/c). Thus we can relate the time dependence of the measured field directly
to the time dependence of the atomic operators by making the appropriate substitution.
2.2 Resonance Fluorescence Correlation Functions
Experimental techniques in optics are tailored for the measurement of correlation functions
and the experiments presented here are concerned with those of resonance fluorescence
2.2 Resonance Fluorescence Correlation Functions 17
Fig. 2.2 A. Power dependence of the different components of the resonance fluorescence
intensity. B. Incoherent and coherent fractions of resonance fluorescence demonstrating the
existence of two power regimes.
from a QD. Analytical expressions exist for the most commonly used correlation functions,
but we will not quote them here and instead concentrate on the interpretation of the main
results. As we are interested in applying these results to a more complex four-level system,
we will present a general method for the calculation of correlation functions which was used
throughout the thesis in order to simulate the data. As an example, one of the most often
studied correlation functions of resonance fluorescence is the intensity autocorrelation which
is a useful tool for the study of the photon statistics of the emission. It is written as follows:
G(2)(t1, t2) =
〈
E−(t)E−(t+ τ)E+(t+ τ)E+(t)
〉
∝
〈
σ+(t)σ+(t+ τ)σ−(t+ τ)σ−(t)
〉
(2.10)
where we have used the relation between field and atomic operators in the far field limit and
neglected the time retardation. Under the Born-Markov approximation, the initial states of
the atom and reservoir are separable:
ρ(0) = ρatom(0)⊗ρR(0) (2.11)
18 Theory of Resonance Fluorescence from a Two-Level System
Defining U(τ) as the unitary time-evolution operator for the total system of atom and
reservoir, the second order correlation can be written:
G(2)(τ) =Tr
[
σ+U†(τ)σ+σ−U(τ)σ−ρ
]
=Tr
[
σ+σ−U(τ)σ−ρσ+U†(τ)
]
=Tratom
[
σ+σ−TrR
[
ρ
′
atom(τ)⊗ρR(τ)
]]
(2.12)
where we have used the cyclic property of the trace, and we have switched from the Heisen-
berg picture to the Schrodinger picture. Here, ρ ′atom(τ) = [σ−ρatomσ+](τ) is the result
of time-propagation of the projected steady state density matrix ρatom. Equation 2.12
is equivalent to the single time correlation function ⟨σ+(τ)σ−(τ)⟩ with the replacement
ρatom(τ)→ [σ−ρatomσ+](τ). Equation 2.12 is a specific instance of the quantum regression
theorem allowing the calculation of a two-time correlation from single-time expectation
values, assuming a Markovian bath. From here, all we need is a method for propagating a
general density matrix in time. For this, we effectively trace over the reservoir by calculating
the non-unitary evolution given by the Liouville equation. Starting from the Bloch equations
written in matrix form given by equation 2.7, the atomic state at time τ takes the following
form [54]:
S
′
(τ) = e−MτS
′
(0)+Tr[ρ
′
atom(0)]
(
1− e−Mτ)M−1Λ (2.13)
where the term Tr[ρ ′atom(0)] has been included to preserve the trace of ρ
′
atom(τ). S
′
(τ) is a
vector of elements of ρ ′atom(τ) defined similarly to equation 2.7.
2.2.1 First Order Correlation Function
The first order correlation function of an optical field describes its phase coherence and is
therefore measured with an interference experiment, typically either a Michelson interferom-
eter or a Fabry-Perot etalon. In the Heisenberg picture, this can be written as:
G(1)(τ) =
〈
E−(τ)E+(0)
〉
(2.14)
This expression gives the correlation between the field at time t and time t + τ , and is
therefore naturally linked to the coherence time of the field. The zero delay value relates to
2.2 Resonance Fluorescence Correlation Functions 19
Fig. 2.3 Calculated spectra of resonance fluorescence for different excitation powers of 0.1, 1
and 10. The laser detuning from the bare resonance is zero in all cases. The dashed grey line
corresponds to the spectrum of spontaneous emission corresponding to incoherent pumping.
A finite spectral resolution is included, which limits the width of the low power spectrum.
the experimentally measured photon count, for integration time T , as follows:
I ∝
∫ T
0
dt
〈
E−(t)E+(t)
〉
(2.15)
∝
∫ T
0
dt
〈
σ+(t)σ−(t)
〉
= T ρ22
To gain some insight into the different contributions to resonance fluorescence, we can
decompose the atomic operators as follows:
σ±(t) = ⟨σ±(t)⟩+δσ±(t) (2.16)
where ⟨σ±(t)⟩ is given by the steady state solution to the Bloch equations, and δσ±(t) is
a correction due to the effect of noise operators representing the interaction with vacuum
fluctuations in the bosonic reservoir [55]. Inserting 2.16 into 2.15, and using ⟨σ+(t)⟩ =
⟨σ−(t)⟩∗ we obtain:
I ∝
∫ T
0
dt |⟨σ+(t)⟩|2+
∫ T
0
dt
〈
δσ−(t)δσ+(t)
〉
≡Icoh+ Iinc (2.17)
20 Theory of Resonance Fluorescence from a Two-Level System
Here we have separated the resonance fluorescence into two components: Icoh which is
linked to a coherent energy transfer from the driving field to the atomic dipole and into the
resonance fluorescence field, and Iinc which is due to spontaneous emission from the atom.
These two components have very different first order coherence properties and their relative
contributions vary with driving power. We can relate the coherent component of resonance
fluorescence to the atomic coherences:
Icoh ∝
∫ T
0
dt |⟨σ+(t)⟩|2
∝T |Tr [σ+ρ]|2 = T |ρ12|2 (2.18)
Figure 2.2 shows how the different components of the total emitted fluorescence vary with
driving power. Two regimes are apparent: the first is the low power regime (s < 1) for
which the intensity is approximately linear in driving power and the coherent component
dominates, and the high power regime (s > 1) where the intensity increases sub-linearly with
power while the incoherent component dominates. This can be understood as bleaching of
the optical dipole when the Rabi frequency far exceeds the excited state decay rate. The
emission tends towards a rate of Γ/2, showing the phenomenon of saturation characteristic
of systems with a finite number of excited states. In this regime, the atomic coherences are
averaged out and the steady-state is a statistical mixture of ground and excited states. The
main difference between the coherent and incoherent components, as their names suggest,
lies in their spectral properties. For a stationary process, the Wiener-Khinchin theorem states
that the spectrumS is given by the Fourier transform of the first order correlation function
of resonance fluorescence [55]:
S (ν) ∝
∫ ∞
0
dτ
〈
E−(τ)E+(0)
〉
e−iντ + c.c. (2.19)
By separating the slowly varying atomic operators S±(t) = e∓iωltσ± in the same way as
equation 2.16, we can distinguish the coherent and incoherent components:
S (ν) ∝
∫ ∞
0
dτ
〈
S+(τ)S−(0)
〉
e−i(ν−ωl)τ + c.c.
S (ν) ∝
∣∣〈S+(t)〉∣∣2 ∫ ∞
0
dτ e−i(ν−ωl)τ + c.c.
+
∫ ∞
0
dτ
〈
δS+(τ)δS−(0)
〉
e−i(ν−ωl)τ + c.c. (2.20)
The first term, related to the steady-state atomic coherence, gives a delta function centered
at the laser frequency. The second component gives rise to other characteristic features
2.2 Resonance Fluorescence Correlation Functions 21
of resonance fluorescence such as spontaneous emission and the Mollow triplet [56]. The
Mollow triplet is linked to Rabi oscillations which modulate the excited state population, and
their features have a spectral width limited by the excited state lifetime. The full derivation
of the resonance fluorescence spectrum is beyond the scope of this chapter, but can be
done by numerically solving the equations of motion in the Heisenberg picture and taking
their Laplace transform. We refer the interested reader to section 5.3.2 for the calculation
of the spectrum of the four-level system. Simulations of spectra for two-level resonance
fluorescence are shown in Fig 2.3 and show the transition between coherent emission in
the low power regime to incoherent emission characterised by a multi-peak spectrum with
widths of order Γ and a splitting corresponding to the Rabi frequency. In this thesis, we are
mainly concerned with using resonance fluorescence as a tool to uncover the underlying
solid-state physics so we will be focusing on the low power regime which gives us access to
sub-linewidth resolution in emission spectroscopy.
2.2.2 Second Order Correlation Function
The second order correlation of the resonance fluorescence field is a widely studied function
in quantum optics as it describes the photon statistics of the field. This function can be
written as follows:
G(2)(τ) =
〈
E−(0)E−(τ)E+(τ)E+(0)
〉
(2.21)
From this expression and the fact that E−E+ is proportional to the photon number operator,
it is clear that G(2)(τ) gives the correlation between photon number at t and t+τ . In practice,
this is measured by correlating arrival times of clicks on single photon detectors. This
function is useful from a fundamental quantum optics perspective as it describes the quantum
nature of photon emission from an atom. From a more technical perspective, the second
order autocorrelation also holds an imprint of the magnitudes and timescales of intensity
noise and is therefore useful as a tool for characterisation of the solid-state environment. In
terms of atomic operators, this becomes:
G(2)(τ) =
〈
S+(0)S+(τ)S−(τ)S−(0)
〉
(2.22)
It is clear from this expression that the two level nature of the emitter will lead to a zero-time
correlation of zero, as σ+(0)σ+(0) = 0, whereas the long-time value is ⟨S+(τ)S−(τ)⟩2 =
ρ222. This condition of a negative value of G(2)(0)−G(2)(∞), referred to as antibunching, is
a frequently used test of a quantum emitter as this effect has no classical counterpart. The
22 Theory of Resonance Fluorescence from a Two-Level System
Fig. 2.4 Calculated intensity autocorrelation functions of resonance fluorescence for excitation
powers for different driving powers and laser detuning from the bare resonance. T1 is the
excited state lifetime.
intensity autocorrelation then maps out the probability of the atom being in the excited state
a time τ after being projected into the ground state. Figure 2.4 shows that the antibunching
feature at zero delay survives for a wide range of driving powers and detunings. With
perfect detectors the antibunching feature would be visible for all powers and detunings,
but for realistic timing resolutions the parameter space where this effect is visible becomes
finite. The Rabi oscillations seen in the three peaked spectrum of Fig 2.3 can be seen in the
time domain in the intensity autocorrelation as oscillations in the excited state probability.
Equivalent oscillations are obtained by detuning the laser from resonance leading to the
notion of a generalised Rabi frequency Ωeff =
√
Ω2+δ 2.
2.3 Phase Space Distribution of Resonance Fluorescence
In the previous section, we have discussed textbook examples of the first- and second-order
coherence properties of resonance fluorescence. Notably we have shown that in the low power
regime, the resonance fluorescence field inherits the coherence properties of the excitation
laser while showing single photon emission. That is, the emission is phase coherent but the
emitted field is not a coherent state. This may seem like a surprising result at first, considering
that the phase of a single photon state is undefined [53]. In this section, we will deviate from
textbook knowledge and will present an illustrative resolution of this wave-particle duality
by looking at the properties of resonance fluorescence in phase space. We will describe a
phase-space quasi-probability distribution often used in the quantum optics community: the
2.3 Phase Space Distribution of Resonance Fluorescence 23
Wigner function. With the help of this distribution, we will then provide an intuitive picture
of resonance fluorescence accounting for its first- and second-order coherence properties.
2.3.1 Wigner Functions
Phase space distributions are very useful tools in classical mechanics. These distributions
describe the joint probability distribution of two variables representing degrees of freedom of
the system. When these variables are related, an initial point in phase space will travel along
a predictable trajectory. For a harmonic oscillator this trajectory is elliptical for a phase space
defined by position and momentum. In quantum mechanics, it is in general not possible to
define a joint probability distribution function for conjugate variables such as position and
momentum due to the uncertainty principle. However, it is possible to construct distributions
which have many characteristics of phase space distributions, while violating one or more of
the axioms of probability. These are therefore called quasiprobability distributions. Several
such distribution functions are commonly used in quantum optics [57, 58] and we choose to
concentrate on the Wigner function [59] which provides a quasiprobability distribution of
position and momentum which is well defined for all states, while giving negative values
for non-classical states. It is formally defined as the Weyl transform of the density matrix
[60, 61]:
W (x, p) =
1
h
∫
e−ipy/h¯ψ(x+ y/2)ψ∗(x− y/2)dy (2.23)
It can be seen that this definition has the useful property that the probability distribution in x
can be recovered by integration over p, using
∫
eipx/h¯d p = hδ (x):∫
W (x, p)d p = ψ∗(x)ψ(x) (2.24)
Here, we will consider a single mode of the electromagnetic field which is described
identically to a harmonic oscillator, with the quadrature operators Xˆ1 = 12(aˆ+ aˆ
†) and
Xˆ2 = 12i(aˆ− aˆ†) taking the role of position and momentum operators. The nth eigenstate of a
harmonic oscillator in the position representation is:
Ψn(x) =
1
π
1
4
e−
x2
2 Hn(x) (2.25)
Where Hn(x) is the nth Hermite polynomial. By inserting 2.25 into 2.23, we obtain the
Wigner function W|n⟩⟨n| for the n-photon Fock state:
W|n⟩⟨n| =
(−1)n
π
e−p
2−x2Ln
[
2(p2+ x2)
]
(2.26)
24 Theory of Resonance Fluorescence from a Two-Level System
Fig. 2.5 Wigner function and corresponding quadrature distributions for the vacuum state (A)
and the single photon Fock state (B). The dashed yellow line represents the intersection of
the Wigner function with the zero plane.
Here, Ln [x] is the n-th order Laguerre polynomial. The n = 0 and n = 1 Wigner functions
are shown in Fig 2.5 along with the marginal distributions describing the single quadrature
probability distributions obtained by integrating over the other quadrature. The vacuum
state minimises the Heisenberg uncertainty relation for quadrature operators ∆Xˆ21∆Xˆ
2
2 ≥ 116 ,
and both marginal distributions are Gaussian with a mean of zero and a finite variance
corresponding to ∆Xˆ21,2 =
1
4 . Coherent states have the same distribution albeit displaced
from zero, and any classical state can be described as a sum of displaced vacuum Wigner
functions[61] with all features thereby being limited in width by the uncertainty principle.
The Wigner function of a single photon state shows straightforwardly why we cannot interpret
the Wigner function as a probability distribution. Indeed there is a region around zero for
which the distribution is negative which is a direct indication of the nonclassicality of the
state. However, for any state, the negative regions are balanced by positive regions such that
the observable, the marginal distribution, only contains positive probabilities. The Wigner
functions of Fock states have been reconstructed experimentally [62, 63] by measuring the
marginal distributions and performing inverse Radon transforms. We also note that the
distribution is circularly symmetric about zero, meaning that the single photon state, and
indeed any Fock state, has no defined phase.
2.3 Phase Space Distribution of Resonance Fluorescence 25
2.3.2 Squeezing in Resonance Fluorescence
With the Wigner function formalism in our toolkit, we can move on to discuss the state of
light in resonance fluorescence. We have already described how to obtain the steady state of
the atom under continuous illumination and will use these results here to analyse the emitted
state of light. The key point is that the coherent interaction between the driving field and the
atomic dipole creates steady state atomic coherences in the rotating frame which will map
to a coherent superposition of vacuum and single photon Fock state in the output field. In
general, a linear superposition of eigenstates does not lead to a linear superposition of the
corresponding Wigner function, and the state of light will exhibit non-trivial properties such
as phase coherence with the driving field and quadrature squeezing [64]. We use the relation
between field and atomic operators in the far field limit (aˆ ∝ σˆ−) to map the steady state
atomic density operator onto the { |0⟩ , |1⟩ } subspace of the outgoing field. For illustrative
purposes, we furthermore rewrite the density matrix as a mixture of vacuum and a pure
superposition state:
ρRF = A |Ψ⟩⟨Ψ|+B |0⟩⟨0| (2.27)
where:
|Ψ⟩=α |0⟩+β |1⟩
A =
Γ2+ |Ω|2
Γ2+2 |Ω|2 ; B =
|Ω|2
Γ2+2 |Ω|2
α =
Γ
Γ+ i |Ω| ; β =
−iΩ
Γ+ i |Ω|
(2.28)
The Wigner function of the full state is then:
WρRF(X1,X2) = AW|Ψ⟩⟨Ψ|(X1,X2)+BW|0⟩⟨0|(X1,X2) (2.29)
Combining equations 2.23, 2.25, 2.28, and 2.29 we obtain:
WρRF(X1,X2) =
(
B+A |α|2
)
W|0⟩⟨0|(X1,X2)+A |β |2W|1⟩⟨1|(X1,X2)
+
4A
π
ℜ(α∗β (X1− iX2))e−X21−X22
(2.30)
26 Theory of Resonance Fluorescence from a Two-Level System
Wigner functions are plotted for finite driving powers of s = 0.355 and s = 10 (Fig 2.6) and
in the limit of high driving power (Fig 2.7). The low power Wigner function no longer shows
the phase invariance of the Fock states. In fact, by inspection of equation 2.30, the chosen
phase is determined by the atomic coherence ρ21 = α∗β which is in turn imparted by the
driving field through the complex Rabi frequency. This way resonance fluorescence exhibits
wave-like properties despite showing anti-bunching. However, despite appearing similar to a
classical field in a first order coherence measurement, the resonance fluorescence state cannot
be a coherent state as it displays antibunching [65] and sub-Poissonian statistics [66]. The
state therefore must deviate from a displaced vacuum state and becomes distorted as shown in
Fig 2.6. In the linear regime (s << 1), the resonance fluorescence Wigner function remains
approximately Gaussian with a small displacement, similar to a coherent state. As the
driving power increases, so does the displacement until the excited state population becomes
non-negligible at which point the Wigner function loses its Gaussian character, due to the
lack of available multi-photon states. This distortion of the state leads to a redistribution
of phase space density from one quadrature to the other. There is therefore a range of
driving powers for which the variance in one quadrature is lower than the vacuum state at
the expense of increased variance in the other quadrature. This quadrature squeezing can
be seen in the individual quadrature distribution of Fig 2.6 A which are compared to the
vacuum distributions. Here, the quadrature variance reaches ∆X21 ≈ 0.219 corresponding to
87.5% of the vacuum variance. Figure 2.6 B and Figure 2.7 show the behaviour when the
driving power increases. In this regime, the atomic coherence decreases with power, leading
to a loss of phase coherence with the driving field. In the limit of high driving power, the
state is a statistical mixture of the n = 0 and n = 1 Fock states and the circular symmetry
is restored. The quadrature distributions become non-Gaussian and exhibit equal variances
twice as large as the vacuum variance. For certain powers, resonance fluorescence therefore
exhibits classical first order coherence inherited from the driving field while keeping the
quantum nature imparted by the atomic transition. Combining both of these effects naturally
leads to the conclusion that quadrature squeezing must also occur. This effect was predicted
in 1981 by Walls and Zoller [67] for which the direct experimental demonstration has only
recently been achieved [51], validating the full quantum picture of resonance fluorescence.
2.3 Phase Space Distribution of Resonance Fluorescence 27
Fig. 2.6 Contour plot showing the Wigner function and corresponding quadrature distributions
for resonance fluorescence driving at s = 0.355 (A) and s = 10 (B). The dashed yellow line
shows the intersection with the z=0 plane. The orange and blue curves are the two quadrature
distributions, and the dashed grey line is the calculated quadrature variance for the vacuum
state.
Fig. 2.7 Wigner function and corresponding quadrature distributions for resonance fluores-
cence driving in the limit of high power

Chapter 3
Experimental Methods for Quantum Dot
Resonance Fluorescence
In this chapter, we give an overview of the experimental techniques used throughout this
thesis, with a particular focus on resonance fluorescence as the technique of most interest.
Resonant excitation of a quantum emitter offers a highly sensitive probe of environmental
coupling in QDs and allows for control of a local spin qubit through coherent light-matter
interaction [7, 68]. In this context resonance fluorescence is the central technique for
the measurements shown in the following chapters, and we will discuss the experimental
methods used for measurement of its emission spectrum and photon statistics. We will
show measurements taken with a neutral exciton transition alongside the corresponding
experimental setup in order to compare with the predictions of chapter 2 and discuss the
interpretation of the neutral exciton as a two-level system before moving on to a more
complicated level scheme in later chapters. The data shown in this chapter were taken over
many years in collaboration with the members of the quantum dot subgroup. In particular,
the Fabry-Perot spectral measurement of the neutral exciton transition was taken by Clemens
Matthiesen and is presented with his permission.
3.1 Optics with Quantum Dots
3.1.1 Confocal Microscopy
For all of the measurements presented in this thesis, we use a confocal microscope mounted
on top of a sample insert. A diagram of the microscope and cryostat is shown in Fig 3.1.
The excitation fibre is a polarisation-maintaining single-mode fibre, and is followed by a
18.4 mm focal length aspheric lens for collimation of the beam within the microscope. A
30 Experimental Methods for Quantum Dot Resonance Fluorescence
Fig. 3.1 A. Diagram of microscope head above the helium bath cryostat. The beam splitters
used are designed to give 90% transmission. The translation and rotation stages give four
degrees of freedom for excitation and detection arms. HWP: half-wave plate; LP: linear
polariser. B. Diagram of the sample position within the bath cryostat. The sample is glued to
a three-axis nanopositioner stack below the objective lens. Superconducting coils within the
cryostat allow magnetic fields up to 7 T at the sample aligned along the optical axis. The
electrical connection to the sample gate is not shown.
half-wave plate in the excitation path allows for excitation with arbitrary linear polarisation
at the sample. A beam-splitter with 90% transmission reflects part of beam towards the
sample, and the transmitted beam is incident on a photodiode to monitor the power. The
imbalance in transmission and reflection coefficients minimises the loss of QD photons being
transmitted through two beam-splitters. The reading on the photodiode is fed back onto
an acousto-optic modulator to stabilise the laser power at the microscope. A single-piece
aspheric lens with NA= 0.5 is mounted above the sample and acts as the objective lens. Both
the laser reflection and the QD emission are collected by the same objective and directed to
the detection fibre. A combination of a waveplate (either half-wave or quarter-wave) and
polariser can be placed just before the detection fibre in order to reject the laser reflection
from the sample. Figure 3.1 shows a typical arrangement of the polarisation optics, but the
mounts allow for their removal, and the exact setup can be changed to suit the experimental
requirements. The initial collimation of the beam is done with the insert outside the cryostat.
Rotation and translation mounts in the microscope arms allow for fine adjustment of the
alignment, with four degrees of freedom for both excitation and detection paths. The QD
sample is mounted on a set of three-axis nano positioners (Attocube ANC 150/3) within an
3.1 Optics with Quantum Dots 31
Fig. 3.2 Gate voltage dependence of single QD Photoluminescence taken with an excitation
power well below the saturation point of the transitions. The labels indicate the nature of the
level system corresponding to each peak. The width of each peak is limited to the resolution
of the spectrometer (≈ 0.02 nm).
insert placed in a helium bath cryostat. The sample insert is first pumped to ≈ 10−5 mbar,
and then filled with ≈ 20 mbar of helium acting as an exchange gas for thermalisation of
the sample. The cryostat contains a superconducting coil allowing for the application of
an external magnetic field aligned along the optical axis (Faraday geometry). The sample
under study is grown with 40 alternating layers of GaAs and AlGaAs below the n-doped back
contact. This structure forms a distributed Bragg reflector (DBR) optimised for broadband
reflection in the near-IR in order to direct more of the QD emission into the objective lens.
The high refractive index of GaAs is problematic for extraction of light into vacuum due to
the shallow angle of total internal reflection with respect to the optical axis. To mitigate this
effect, a super-hemispherical cubic Zirconia solid immersion lens [69] is mounted on top
of the sample. We are working in the Weierstrass configuration [70, 71] which theoretically
allows for an improvement of n2 in the effective NA. This is only true for a SIL refractive
index matching that of the sample, so, while they are more readily available than GaAs SILs,
using a cubic Zirconia SIL (n = 2.15 compared with n≈ 3.54 at 950 nm for GaAs) is not
ideal for extraction efficiency.
Once the optics are aligned, we can start characterising QDs, for which the first step is
usually measuring the spectrum of photoluminescence. For this, we send 785 nm diode laser
emission through the excitation fibre and couple the detection fibre into a 0.75 m diffraction
grating spectrometer (Princeton Instruments Spectra Pro SP-2760) containing a 900 nm
32 Experimental Methods for Quantum Dot Resonance Fluorescence
longpass filter to reject the driving laser and luminescence from the GaAs matrix and InAs
wetting layer. The dispersed beam is then read out on a liquid nitrogen-cooled CCD array
(Acton Spec-10). The spectrometer is first calibrated by measuring the spectrum of a tunable
single-mode laser while measuring its frequency on a high resolution wavemeter (High
Finesse WS Ultimate). The excitation energy is above the GaAs band gap and therefore
generates a large number of excitons which quickly relax to the lowest energy configuration.
Some of the excitons decay into the quantum dot to occupy the discrete confined exciton
energy levels. Fig 3.2 shows a spectrum of photoluminescence from a single quantum dot.
Three discrete emission energies can be seen corresponding to the different charge states
in the QD. Photon emission in all three cases is due to the recombination of an electron-
hole pair, but the presence of an additional charge in the quantum dot leads to a Coulomb
renormalisation of the emission energy leading to three different peaks. The labels beside the
peaks describe the charge in the QD ground state. Of course, the data shown in Fig. 3.2 are
not sufficient to justify this labelling, but the different charge configurations can be identified
with additional polarisation-resolved and magnetic field-dependent photoluminescence [4].
We find that, although the X0 transition energy varies considerably from dot to dot, the
splitting between exciton and trion energies is fairly consistently ≈ 4 nm in our sample.
The relative intensities of these three peaks depend on the gate voltage. The electric field
tunes the QD charging energies across the Fermi energy given by the n-doped back contact.
Therefore certain charge states are allowed in certain gate voltage regions. The gate voltage
applied to the sample controls the charge state of the QD to some extent, but one can see
significant overlap between the voltage regions showing emission corresponding to different
charge states. The same processes describing exciton relaxation into QD excited states also
lead to single charge carriers (conduction band electron or valence band hole) relaxing into
the QD ground state, and multiple charge complexes are observed at a single gate voltage. A
competition between the exciton radiative lifetime and the carrier tunneling rates determines
the extent to which the peaks overlap in gate voltage. The emission wavelength is also seen
to change linearly with gate voltage, revealing the quantum-confined Stark effect described
in chapter 1.
3.1.2 Resonant Excitation
Photoluminescence gives a relatively simple first step to QD characterisation but the amount
of information that can be extracted is limited. By its very nature, photoluminescence is a
fairly invasive method of probing the QD, as a large number of excitons is generated in its
surroundings and we have little control over the charge state of the QD at any given time.
We are interested in probing the intrinsic fluctuations in the environment which affect the
3.1 Optics with Quantum Dots 33
QD transition on the order of its linewidth, and therefore a ’cleaner’ excitation method is
desirable. Resonant excitation provides a more direct method of addressing the transition
and allows for resolutions much lower than the radiative linewidth (≈ 240 MHz), whereas
the diffraction grating spectrometer has a resolution limited to ≈ 7 GHz. The resolution in
resonant excitation is typically limited either by the driving laser coherence or the spectral
response of the detection setup. We use frequency-tunable external cavity diode lasers
(Toptica DL pro, Toptica TA pro, New-Focus Velocity 6319) for which the emission is
single-mode to a good approximation. In practice, their long-time linewidths are limited
by our frequency stabilisation protocol. We send part of the laser emission to a wavelength
meter , and we feed back onto the laser frequency control. Typically, the remaining frequency
fluctuations are ≈ 2 MHz giving a resolution several orders of magnitude lower than the
radiative linewidth.
The main practical difficulty with resonant excitation in a confocal setup is separating
the reflected driving laser from the QD emission. In principle these two fields do not have
identical spectra, but in practice they cannot be separated spectrally by using common filters.
Fortunately, there are other degrees of freedom in which the two fields are separable. As
described earlier, changes in electric field applied to the QD lead to linear shifts in transition
energy. This fact can be used to isolate the QD component of the detected field in a technique
known as differential reflection (DR) [72]. Typically, the reflected laser field is several orders
of magnitude more intense than the QD field, and the laser background would dominate in
a simple absorption spectrum. However, we can use the fact that, in the low laser power
regime, part of the QD emission is coherent and will interfere with the laser to perform a
homodyne measurement:
Idetected = |Elaser+EQD|2 ≈ |Elaser|2+2 |Elaser.EQD|
≈Ilaser+2η cos(φ)
√
Ilaser.Icoh (3.1)
where Icoh is the coherent fraction of the QD resonance fluorescence intensity, φ is the
relative phase between the QD emission and laser, η is a factor accounting for the different
polarisations of reflected laser and resonance fluorescence, and we have assumed |Ecoh|2 ≪
|Elaser|2. The first term of equation 3.1 is the dominant term but we can subtract it by using
the implicit dependence of the second term on gate voltage. For this, we modulate the gate
voltage with an amplitude larger than the exciton radiative linewidth and demodulate the
intensity measured on a photodiode at the same frequency with a lock-in amplifier. The
constant term in equation 3.1 is thereby eliminated and we are left with a signal of amplitude
proportional to
√
Ilaser.Icoh ∝
√
s |ρ12|. DR is a useful method for probing the steady-state
34 Experimental Methods for Quantum Dot Resonance Fluorescence
atomic coherence, but doesn’t give access to the QD photons, making measurements such
as the intensity autocorrelation of resonance fluorescence impossible. In addition, the
measurement relies on integrating the signal over a few modulation periods, meaning that the
measurement bandwidth is limited by the gate RC constant.
Another degree of freedom we can use to separate QD emission from reflected laser is
polarisation. Typically, we drive the QD with a linearly polarised laser, which can therefore
be suppressed with another linear polariser in the collection arm. The polarisation of the QD
resonance fluorescence is determined by the transition selection rules, which in general don’t
coincide with the driving laser polarisation and the QD emission will be partly transmitted by
the detection polariser. In practice, we minimise the laser throughput for a given excitation
polarisation by rotating the collection polariser and then iterate for different excitation
polarisations in order to find a global maximum in the extinction ratio. Typically this ratio
reaches values of order 107−108 and is highly frequency dependent. The use of a SIL is
helpful in this context, as it reduces the size of the focal spot. This means that less laser power
is required to saturate the QD and the signal-to-background ratio increases. In addition, it is
sometimes necessary to add a diffraction grating between the laser and the excitation fibre to
improve the laser spectrum when it shows broadband emission due to enhanced spontaneous
emission as well as a single-mode peak. This can improve the polarisation extinction ratio
by several orders of magnitude depending on the laser used. With this extinction ratio, we
routinely achieve signal-to-laser background ratios above 100 at s = 1 measured with single
photon counting avalanche photodiodes. The half-wave plate following the polariser in the
detection arm is used to match the axis of the polarisation-maintaining fibre in order to avoid
intensity fluctuations at the output due to motion of the fibre.
These two methods are central to the measurements shown in this thesis and are common
techniques within the QD community, but both have their own advantages and disadvantages.
DR is a simple technique to implement and allows for arbitrary excitation polarisation
but leaves no access to the QD photons and suffers from a bandwidth limited by the gate.
Polarisation suppression on the other hand gives access to the QD single photons and allows
measurements of the first and second order correlation functions described in chapter 2
at the expense of freedom in excitation frequency and polarisation. A third measurement
technique, developed for the measurements shown in this thesis, allows for both access to
resonance fluorescence photons and excitation with multiple frequencies and polarisations.
This method relies on spectral separation of phonon-assisted resonance fluorescence and is
further described in chapter 4 when the exciton-phonon coupling is discussed in more detail.
3.2 Resonance Fluorescence Spectroscopy and Photon Statistics 35
3.2 Resonance Fluorescence Spectroscopy and Photon Statis-
tics
3.2.1 Absorption Spectroscopy
The first step in resonant characterisation of a QD typically consists of a measurement of the
DR contrast as a function of laser frequency and gate voltage. An example two-dimensional
DR map is shown in Fig. 3.3 and gives a far more precise value for the transition frequency
and DC Stark shift compared with non-resonant excitation. In most measurements, it is
more convenient to change the gate voltage rather than the laser frequency and an accurate
measurement of the Stark shift is therefore needed to calibrate the laser detuning in frequency
units. Figure 3.3 B shows a typical low-power DR measurement as a function of laser
detuning from resonance for a given laser frequency. As discussed in Chapter 1, the X0
transition consists of the transition between the crystal ground state and a heavy-hole exciton.
The degeneracy between the two heavy-hole excitons is lifted by the strain field in the
quantum dot giving an anisotropic electron-hole exchange term [4]. Two peaks are apparent
with a splitting (commonly called the X-Y splitting) determined by the anisotropic exchange
term. Both transitions have similar oscillator strengths, and the relative size of the two peaks
is mainly determined by the excitation polarisation. By varying the angle of the excitation
polarisation, one can see that the relative contributions are anti-correlated and each peak
can be suppressed at angles differing by 90 degrees. Hence, the two transitions are linearly
cross-polarised with each other. In Figure 3.3, the suppressed peak is still visible due to finite
ellipticity of the excitation polarisation at the sample location. What we are dealing with then
is a three level system. However, if the X-Y splitting is large compared with the linewidth,
spectral and polarisation selectivity allow us to isolate a two-level system. This QD shows a
small X-Y splitting of 1.8 GHz compared to the typical value of ≈ 6 GHz observed with this
sample.
As discussed in chapter 2, two-level effects appear in a wide variety of measurements.
One of the simplest measurements, and one which is typically used to calibrate the Rabi
frequency, is power-dependent absorption spectroscopy. These reveal saturation effects in
different forms: the total fluorescence intensity reaches a maximum as the driving power
is increased while the absorption linewidth begins increasing linearly with Rabi frequency
[53]. Furthermore, the transition between the two power regimes is reflected in a decay of
the coherent fraction of the fluorescence intensity. The latter effect can be probed straight-
forwardly with DR measurements, as these rely on the interference between the resonance
fluorescence field and the reflected laser. When the laser intensity dominates over the QD
36 Experimental Methods for Quantum Dot Resonance Fluorescence
Fig. 3.3 A. 2D map of DR as a function of gate voltage and laser frequency. The linear
shift in resonance frequency is due to the DC Stark effect. Measuring the gradient allows a
conversion between laser frequency detuning and gate voltage detuning. B. DR measurement
as a function of laser detuning from resonance for a fixed laser frequency. The excitation
polarisation is linear at an arbitrary angle with respect to the neutral exciton transitions. C.
DR measurements with excitation polarisations matching one of the two transitions. All
curves are normalised to the peak of Voigt fits.
3.2 Resonance Fluorescence Spectroscopy and Photon Statistics 37
intensity, the measured contrast as a fraction of the total intensity on the photodiode is
proportional to
√
Icoh
Ilaser
∝ |ρ12||Ω| . Figure 3.4 shows a power dependence of the DR contrast
and resonance fluorescence count rate. In the low power regime, the fluorescence intensity
is linearly dependent on driving power and is mainly due to Icoh, which explains why the
DR contrast is constant while the RF counts increase linearly. The subsequent drop in DR
contrast reflects the transition to incoherent fluorescence, while the RF count rate starts to
show sub-linear behaviour. These two effects can be used to calibrate the saturation power.
Assuming the reflected laser power is the main contribution to the total intensity (i.e. the
contrast is small), the point of half maximum contrast directly gives s = 1. For the RF
measurement, s = 1 corresponds simply to the point of half maximum count rate. For this
QD, we find maximum resonance fluorescence count rates of 1.6 MHz. From the two level
theory, we expect the emission rate to saturate at Γ2 which, considering the measured lifetime
of T1 = 0.58 ns, would give a maximum emission rate of 862 MHz. The discrepancy arises
from a large number of factors, which can be separated into a product of collection efficiency
and detection efficiency. We use Si APDs which typically have detection efficiencies of
≈ 20% at a wavelength of 950 nm meaning the efficiency of collection into the detection
fibre is around 0.9 %. One of the main factors limiting this value is the fraction of dipole
radiation which is directed to the 0.5 NA objective lens, even with the aid of a DBR and SIL.
Further effects such as finite transmission of the optics, fibre coupling efficiency and partial
polarisation rejection of the RF account for the rest of the collection efficiency.
Saturation of the two-level absorption also leads to an increase in absorption linewidth. In
an ideal system, the low-power spectrum is Lorentzian with a linewidth given by the excited
state lifetime ∆ν = 12πT1 . In this sample, the spectrum is typically a composite of Lorentzian
and Gaussian distributions and exhibits a linewidth≈ 2∆ν . This is the first indication that the
two-level approximation is incomplete for such a solid-state emitter, where interactions with
a fluctuating environment lead to inhomogeneous broadening. This broadening is caused
by charge fluctuations in the sample and hyperfine coupling to an uncontrolled nuclear spin
environment. The former will be the main topic of interest in Chapter 4, while the latter will
be studied in further detail in Chapters 5 and 6. Reference [73] shows a dependence of the
absorption linewidth on laser frequency sweep rate, showing that the increased width is not
caused by pure dephasing of the transition.
3.2.2 Emission Spectroscopy
Absorption spectroscopy allows us to probe the steady-state atomic population ρ22 and
coherence |ρ12| but misses key features of resonance fluorescence such as Rabi flopping.
Additionally, Fig. 3.4 A. shows the expected drop in coherent fraction as the driving power
38 Experimental Methods for Quantum Dot Resonance Fluorescence
Fig. 3.4 A. Peak DR contrast (orange circles) and peak resonance fluorescence count rate
(green triangles) as a function of driving laser power . The dashed orange curve is a fit
to the functional form of
√
Icoh
s and the green curve is a fit to the functional form of Itotal.
B. Resonance FWHM extracted from individual Voigt fits to the lineshape as a function
of driving power. The dashed curve is a fit of the form
√
A+Bs. The dashed grey curve
represents the expected low power FWHM given by the transform limit 1/(2πT1) for an
ideal two-level system.
3.2 Resonance Fluorescence Spectroscopy and Photon Statistics 39
Fig. 3.5 Diagram of the scanning Fabry-Perot (FP) interferometer setup. The reference laser
(blue beam) transmission is recorded on a photodiode (PD), the output of which is used as
an input to a PID controller. The output of the PID is fed back to the controller to control
the cavity length via a piezoelectric transducer. The resonance fluorescence collected from
the microscope (red beam) is simultaneously sent through the cavity and the transmission is
recorded on an avalanche photodiode (APD).
increases, but only gives us a decay relative to the low-power value and therefore limited
information on the first order coherence. To measure the first order coherence of resonance
fluorescence, we use a scanning Fabry-Perot interferometer setup shown in Fig. 3.5. A
frequency-stabilised reference laser is sent through the FP and the throughput is measured
on a photodiode. The cavity is composed of two confocal mirrors, for which the relative
alignment is controlled by three piezoelectric transducers and one piezoelectric transducer
controls the distance between the mirrors. The initial cavity alignment is done by ramping
one of the voltages and minimising the width of the transmission peak with the other
piezo actuators. Once the cavity is aligned, scanning the reference laser frequency gives a
Lorentzian transmission peak of typically 20 MHz FWHM.
The cavity length is initially set to give half-maximum transmission of the reference
beam, and this transmission is fed back via PID control to the controller to stabilise to this
setpoint. Changing the reference laser frequency thereby tunes the cavity resonance by the
same frequency difference. The signal from the detection fibre is simultaneously sent through
the cavity and recorded on an APD. The peak transmission is typically on the order of 10%
and the integration times used are therefore much longer than for an absorption spectrum. A
comparable spectrum is taken by tuning the QD off resonance via gate voltage in order to
measure the spectrum of laser leakage. This reference spectrum is then subtracted from the
raw data to obtain the resonance fluorescence emission spectrum.
40 Experimental Methods for Quantum Dot Resonance Fluorescence
Fig. 3.6 A. X0 emission spectrum taken around s = 1 with a Scanning Fabry-Perot measure-
ment. The width of the spectrum is limited by the Fabry-Perot linewidth of ≈ 16 MHz. The
dashed line is a two-level simulation including the experimental parameters of driving power
and spectral resolution. B. X0 emission spectrum taken at a similar driving power with a
grating spectrometer. The dashed curves are guides to the eye. ZPL stands for zero-phonon
line and PSB stands for phonon side-band. A background measurement with the QD off
resonance was subtracted for both measurements.
3.2 Resonance Fluorescence Spectroscopy and Photon Statistics 41
Fig. 3.7 Diagram depicting a Hanbury Brown and Twiss setup for measurement of resonance
fluorescence intensity autocorrelation.
Figure 3.6 shows emission spectra measured with two methods: the first obtained with the
scanning FP with a spectral resolution of ≈ 16MHz and a free spectral range on the order of
a few GHz, and the second with a grating spectrometer with a resolution of order 10 GHz and
a much larger detuning range. The FP measurements show a low power emission spectrum
consistent with a two-level picture: a sub-linewidth peak due to coherent emission is present
on top of a broad peak due to incoherent emission. The presence of a large proportion of
coherent emission in the low power regime suggests that the neutral exciton transition suffers
little from pure dephasing [37, 74] and is consistent with a simple two-level picture. The
second measurement, in contrast, shows a broad sideband in addition to the narrow peak
centered at the laser frequency. This sideband stems from exciton coupling to bulk acoustic
phonon modes [43, 44] leading to phonon-assisted fluorescence even in the low power regime.
The phonon sideband typically accounts for 10− 15% of the total emission intensity and
leads to a ps timescale decay of the first order coherence [75]. This phonon coupling, and its
impact on spectral indistinguishability, is covered in more detail in chapter 4.
3.2.3 Intensity Autocorrelation
As discussed in chapter 2, the intensity autocorrelation function gives useful information both
on the quantum nature of photon emission in resonance fluorescence and on the classical
noise sources affecting the emission. With an ideal detector and no photon loss, one would
simply need to record arrival times of all photons and then compute the distribution of time
delays between photons. However, technical limitations in realistic detectors have led to
the adoption in the quantum optics community of a different correlation method called the
42 Experimental Methods for Quantum Dot Resonance Fluorescence
Hanbury Brown and Twiss setup shown in figure 3.7. Single photon detectors, such as
the avalanche photodiodes used in this work, suffer from a recovery time after detecting
a photon during which no detection can be made. For the detectors used here this dead
time ranges from 30 ns to 50 ns. Antibunching, with features on the order of the radiative
lifetime, therefore cannot be seen with a single detector setup. Instead, a 50/50 beam splitter
is used to split the resonance fluorescence field which is then measured on two separate
detectors. One detector is labeled ’start’ and the other ’stop’. The electronic correlator (quTau
time-to-digital converter) then builds a histogram of arrival times of ’stop’ pulses conditioned
on the arrival of a ’start’ pulse. The exact algorithm used to build up the histogram depends
on the correlator used. In the case of the quTau, once a ’start’ pulse is detected, a correlation
count is added for every subsequent ’stop’ received until a new ’start’ pulse is detected. If no
’stop’ pulse is detected within a set time window, no correlations are added to the histogram
and the correlator waits for another ’start’ pulse. In the limit of low count rates, the histogram
generated is equivalent to the intensity autocorrelation function multiplied by a constant
factor determined by the collection and detection efficiencies. At high count rates, such
that the probability of detecting two ’start’ pulses within the time window is significant, the
long-delay coincidences are underestimated. In fact, the histogram decays exponentially
with a timescale corresponding to the mean ’start’-’start’ delay. One way of correcting for
this effect is to introduce loss in the resonance fluorescence (e.g. by insertion of a neutral
density filter) such that the ’start’-’start’ mean delay is long compared to the time window,
or one can fit the exponential decay e−t/τ and correct by dividing the measured histogram
by (1− e−t/τ). The first option leads to a longer histogram acquisition time, so we usually
opt for the second option. An electronic delay between the ’start’ and ’stop’ signals offsets
the zero-delay point to a finite delay in the electronics, and the correction described above
therefore does not accidentally correct for classical noise sources which would also appear
as exponential decays.
Figure 3.8 shows intensity autocorrelation measurements of resonance fluorescence from
the X0 transition for two driving powers. The zero delay value is below 0.5 for both of these
demonstrating that we indeed have a two-level system emitting single photons. However,
there is significant deviation from the ideal two-level theory, particularly for the high power
measurement where g(2)(0)≈ 0.3. This is mostly due to the finite timing resolution of the
HBT setup determined mainly by the timing jitter of the APDs used. The timing resolution
of the HBT setup is obtained independently by measuring the width of pulses emitted from
a mode-locked Titanium-sapphire laser at a similar mean count rate. The width of these
pulses is less than 2 ps, whereas the width obtained via HBT measurement is typically
≈ 600 ps. Further deviation from the simple two-level theory is introduced because of the
3.2 Resonance Fluorescence Spectroscopy and Photon Statistics 43
inhomogeneous broadening caused by the environment. Therefore, Rabi flopping is not tied
uniquely to laser power even when the mean detuning is zero, and the Rabi frequency tends
to be higher than one would expect for a given driving power. In most simulations in this
thesis, this broadening is included by averaging over a Gaussian distribution of detunings
which is QD-dependent. On timescales far longer than the radiative decay, we observe a flat
autocorrelation. This suggests that there is no significant decay into a dark state on a longer
timescale.
We also use intensity autocorrelation measurements as a probe of the impact of environ-
mental fluctuations on resonance fluorescence. These fluctuation timescales are typically far
greater than the dead time of our detectors, therefore we can use a single detector in order to
reduce the integration time needed. In classical optics, the intensity autocorrelation can be
written [76]:
G(2)(τ) = lim
t→∞⟨I(t+ τ)I(t)⟩ (3.2)
where I(t) is the field intensity at time t. Practically, this can be measured by taking a
timetrace of resonance fluorescence for a given time T and time bin tbin. To obtain an
autocorrelation from a single timetrace, we do the following. We take two subsets of length
T ′ from the initial timetrace, one starting at t = 0 and the other at t = τ . We then multiply
these two traces, element-by-element, and take the mean of the resulting vector. In the limit
of large T ′, this value is equivalent to the right hand side of equation 3.2. By first subtracting
the mean count rate squared, and then dividing by the same value, we obtain a normalised
g(2)(τ)−1. An example intensity autocorrelation of X0 resonance fluorescence obtained with
a single detector is shown in figure 3.9. This autocorrelation is the average autocorrelation of
10 timetraces taken under the same experimental conditions, with T = 10 s and tbin = 200µs.
The simple two-level model predicts that correlations in the resonance fluorescence intensity
should decay on a timescale of order Γ. However correlations still persist at timescales
of ms and even seconds, betraying relatively slow environmental fluctuations. One has to
be cautious in assigning these fluctuations to the solid-state environment of the QD, as the
correlations could also arise from more trivial technical reasons, e.g. electronic noise on
the gate or intensity fluctuations caused by motion of the detection fibre. However, the
fluctuations on the timescales shown are highly detuning- and power-dependent suggesting
that the intensity noise is in fact related to frequency fluctuations of the transition [73, 77].
The measurements presented in this chapter show that, in many ways, the neutral exciton
transition of a quantum dot behaves as a two-level system. Indeed, the short-time photon
statistics show that it is a good single photon emitter and that it doesn’t suffer significantly
from decay into dark states. Furthermore, the presence of coherent emission at low powers
signifies that the transition doesn’t suffer from pure dephasing. However, the presence of the
44 Experimental Methods for Quantum Dot Resonance Fluorescence
Fig. 3.8 Measured intensity autocorrelation of resonance fluorescence from the neutral
exciton transition at driving powers of s = 0.1 and s = 4.5 shown as orange and green
symbols respectively. The solid curve is a simulation with the parameters shown in the
legend, where ∆diff is the FWHM of the detuning distribution representing spectral diffusion
and ∆t is the time FWHM of the instrument response function. The dashed curves are
simulations without spectral diffusion and without convolution with the instrument response
function.
3.2 Resonance Fluorescence Spectroscopy and Photon Statistics 45
Fig. 3.9 Intensity autocorrelation of resonance fluorescence from the neutral exciton transition
at a driving power of s = 1 and long delays. Correlations in the RF intensity can be seen as
g(2)(τ)−1 > 0.
environment is still clear in most of these measurements: charge and nuclear spin fluctuations
lead to inhomogeneous broadening while phonon coupling gives a homogeneous broadening
even at cryogenic temperatures. These effects are generally considered detrimental for the use
of quantum dots as either single photon emitters or spin-photon interfaces. In the following
chapters, we will add ground state spin into the mix and study these diverse environmental
effects in more detail, with a particular focus on using this environment as a resource.

Chapter 4
Phonon-assisted Resonance Fluorescence
Frequency Stabilisation
There exist multiple measures to quantify the quality of a single-photon emitter. In many
respects, QDs show promise due to their high quantum efficiency, large oscillator strength,
and low g(2)(0). Additionally, a lot of progress has been made in the integration of QDs
within semiconductor nanostructures [78–80], leading to increased collection efficiencies
and the prospect of on-chip architectures. From the perspective of the use of a single
quantum emitter for quantum information, one of the most important figures of merit is the
indistinguishability of independently emitted photons. Indeed, most quantum information
processing protocols rely on interference of two single photons generated independently
and a poor indistinguishability is a severe roadblock, e.g., imperfect spectral overlap would
limit the entanglement fidelity of two distant QD spins in a distributed quantum network
architecture [21]. As we have already briefly touched upon, the solid-state environment leads
to inhomogeneous broadening of the exciton transition, thereby reducing the spectral overlap
of independently emitted photons. Measurements of QD photon indistinguishability remain
limited by environmental noise, even under resonant excitation [36, 81, 82]. Near-unity
indistinguishability with QDs can be achieved despite the noisy environment in the regime
of coherent single photon emission [37], as the spectrum is dictated by the excitation laser.
However, this technique suffers from a low photon emission rate due to the low excitation
power required. Furthermore, this method still suffers from the effect of inhomogeneous
broadening, in the form of a reduction of emission intensity. If the intensity noise is large
enough, this can in turn lead to a lowered two-photon interference visibility [38].
In this chapter, we concentrate on the effects of electric field fluctuations and phonon coupling
on the QD resonance fluorescence. We further demonstrate a technique which makes use
of the phonon-assisted emission to feedback on the zero-phonon line (ZPL) frequency,
48 Phonon-assisted Resonance Fluorescence Frequency Stabilisation
thereby reducing the inhomogeneous broadening without sacrificing ZPL emission. Similar
stabilisation protocols have been demonstrated with solid-state quantum emitters, but rely
either on periodically scanning across the resonance which limits the bandwidth [41], or
require the detection of a significant fraction of the zero-phonon emission to obtain an error
signal [40, 42].
The measurements shown in this chapter, published in Ref [83], were taken in collabora-
tion with the rest of the QD team and Carsten H. H. Schulte in particular. This chapter builds
up on work done by Megan J. Stanley and Clemens Matthiesen, published in Refs [77, 84],
quantifying the timescales and amplitudes of different noise sources in QD RF. In particular,
the histogram analysis technique of Ref [84] was used to extract the detuning distribution.
4.1 Electric Field Noise in Quantum Dots
In this chapter we examine the negatively charged trion transition in the absence of an
externally applied magnetic field. An example absorption linescan is shown in Fig 4.1 taken
over a few seconds. The typical lifetime of the trion excited state in this sample is ≈ 0.7
ns, while the linewidth of the low-power absorption spectrum is 620 MHz, corresponding
to almost three times the transform limit. As examined in Refs [73, 77], two physical
mechanisms contribute to this inhomogeneous broadening. Firstly, the fluctuating OH field
lifts the degeneracy between the ground states, and to a lesser extent the excited states. This
random distribution of spin splittings leads to a broadening of the time-averaged absorption
linewidth. The coupling to the nuclear spin bath is discussed in more detail in chapter 5 and
6. Secondly, fluctuations in the charge environment of the QD affect the permanent dipole
moment of the excited state and consequently alter the transition frequency. Refs [73, 77]
show that OH and electric field noise sources occur at different timescales. The OH field
fluctuations typically occur on timescales between 10 µs and 100 µs whereas electric field
fluctuations show timescales anywhere between 100 µs and ≈ 1 s. These fluctuations can
be seen in the time-domain by using a frequency- and power-stabilised single-mode laser,
initially tuned on resonance with the X- transition. Panel B of Fig. 4.1 shows a typical
timetrace with an integration time of 100ms. The effect of slow electric field fluctuations
on the instantaneous transition frequency can be seen as large RF intensity fluctuations.
The amplitude of these fluctuations is large relative to the corresponding shot noise for the
same mean count rate, showing that we are far from having an ideal quantum emitter. The
motivation behind the work presented in this chapter is then clear: one would like to minimise
the effect of inhomogeneous broadening, both to reduce the photon distinguishability as well
as improving photon emission rates under resonant excitation.
4.1 Electric Field Noise in Quantum Dots 49
Fig. 4.1 A. Low-power differential reflection absorption spectrum (orange diamonds) along
with a Voigt fit (brown curve). The best fit FWHM is 620± 10 MHz. The dashed curve
shows the transform-limited spectrum for a transition with an excited state lifetime of 0.7
ns. B. Example resonance fluorescence timetrace (blue curve) for zero initial detuning and
an excitation power of s = 1. The integration time in this measurement is 100ms. The
red rectangle represents ±1σ for a Poisson distribution with the same mean count rate and
integration time.
To see what impact a realistic feedback stabilisation protocol would have, we turn to
a more quantitative measurement of the noise timescales in this particular QD. For this,
we measure 30 s long timetraces with an integration time of 200µs at an excitation power
corresponding to s = 1. There exist noise contributions below 200µs, but we know that the
gate on our sample has a bandwidth of ≈ 3 kHz, so feedback stabilisation of faster noise
sources would be impossible for this sample. Figure 4.2 shows the average noise power
spectrum and intensity autocorrelation g(2)(τ)−1 for 100 such measurements. In this chapter,
we use a normalisation such that the integrated power spectral density gives the average
intensity variance over 30 s, as follows [73, 85]:
NQD( f ) =
(tbin)2
T
∣∣∣∣DFT [ N(i)< N(i)>
]∣∣∣∣2 (4.1)
where N(i) is the number of counts in the ith bin of width tbin, T is the total measurement
time, and we take the Discrete Fourier Transform (DFT). The measurement shows a noise
spectral density decreasing with increasing frequency. A general 1/ f -like trend accounts
for most of the noise spectral density. The additional frequency dependence is due to the
presence of an ensemble of fluctuators [73, 86] representing individual charge traps in the
environment, each with its own characteristic timescale describing its charging dynamics.
This typically gives a roll-off type spectrum in addition to the 1/ f -like dependence. A similar
50 Phonon-assisted Resonance Fluorescence Frequency Stabilisation
Fig. 4.2 A. Average noise spectral density of resonance fluorescence at a driving power of
s = 1 (green curve). A reference measurement is obtained by un-suppressing the reflected
laser (grey curve) such that the mean count rate is comparable with the green curve. The high
frequency value corresponds to the noise spectral density due to shot noise. The dashed red
curve is a fit to A f−α , and the orange shaded area is a similar fit with an added Lorentzian
decay of 15Hz FWHM. B. Intensity autocorrelation function of resonance fluorescence. The
dashed red curve is a multi-exponential fit with decay timescales as shown.
measurement is taken by detuning the QD and unsuppressing the laser reflection such that
the mean count rate is the same as the RF measurement. This reference spectrum is flat
at frequencies above 1 Hz, and gives an estimate of the contribution of shot noise to the
measurement. There is an increase of noise power at low frequencies due to instabilities
in the setup, which could be linked to polarisation fluctuations at the output of the PM
detection fibre, which are mapped to intensity fluctuations due to the polarisation dependence
of the optics used before detection (see next section). At frequencies above ≈ 1 kHz, the RF
noise spectral density is of the same order of magnitude as shot noise. This tells us that a
frequency stabilisation scheme operating with a bandwidth of 1 kHz would be sufficient to
get the RF intensity noise down to the shot-noise level. The presence of individual fluctuators
can be seen more clearly in the intensity autocorrelation [77, 87], which is well fit by a
multi-exponential decay with four characteristic timescales ranging from 1ms to 1s. Each
exponential decay timescale corresponds to a correlation time describing an environmental
noise source, e.g. charge traps created by nearby defects [86].
The intensity noise shown in Fig. 4.2 betrays an underlying ZPL frequency fluctuation.
There are various ways that this time-averaged detuning distribution can be measured. One
possibility is to record RF intensity while scanning the laser frequency across the resonance
and building statistics on the central frequency distribution. However, this method is limited
both by the experimentally achievable sweep rates and by finite count rates which reduce the
4.1 Electric Field Noise in Quantum Dots 51
Fig. 4.3 Two example histograms of X- resonance fluorescence (blue columns), taken for 1s
long timetraces at s = 1 with an integration time of 200µs. The blue dashed curve is a best
fit assuming a Gaussian detuning distribution of shot-noise limited histograms. The dashed
orange curve is the Poisson distribution corresponding to the same mean count rate.
quality of the peak fitting routine. Another less demanding method, developed by Clemens
Matthiesen and Megan J. Stanley [84], is based on fitting the histograms of RF timetraces to
a Gaussian detuning distribution of shot-noise limited histograms. For this, we assume that
the time-averaged charge state of the environment leads to a Gaussian distribution of central
frequencies, and that the measurement time is long enough to sample the entire distribution.
From each timetrace, the mean detuning and Gaussian diffusion width can be extracted.
Fluctuations faster than the measurement time will appear in the diffusion width, whereas
slower fluctuations will lead to a different mean detuning in every subsequent timetrace. Two
example histograms of 1 s long timetraces are shown in Fig. 4.3, along with the best fit and
the corresponding fit parameters. By performing a χ2 analysis, we find that the uncertainty
on the best fit mean detuning is≈ 20 MHz. For this QD, we find an average diffusion FWHM
of 230 MHz which is on the order of the transform-limited linewidth, which explains the
absorption linewidth measured in Fig. 4.1. This inhomogeneous broadening will have a
detrimental effect on experiments relying on photonic coalescence from independent QDs
[21, 48–50]. A calculation of the distribution of spectral overlaps for two emitter suffering
from inhomogeneous broadening is shown in Fig. 4.4. For diffusion widths on the order
of the radiative linewidth, the average spectral overlap reduces significantly, which is why
Hong-Ou-Mandel type experiments with incoherently generated photons have shown photon
indistinguishabilities significantly lower than 1 [36, 81]. Developing a method for reducing
52 Phonon-assisted Resonance Fluorescence Frequency Stabilisation
Fig. 4.4 Simulation of the distribution of spectral overlaps for photons emitted by two
uncorrelated emitters. A. Example of spectral overlap for two emitter with normalised
Lorentzian homogeneous spectra (red and green peaks) detuned with respect to the other
emitter. The blue shaded region shows the result of multiplication of the spectra of the
two emitters, for which the integral is proportional to the spectral overlap. B. Calculated
distributions of spectral overlaps for 2500 samples of a Gaussian detuning distribution. ∆diff
represents the FWHM of the Gaussian distribution. Each spectral overlap is normalised to
the integral of the squared homogeneous lineshape. The dashed lines represent the average
spectral overlap for each diffusion width.
4.2 Phonon Sideband Filtering 53
the diffusion width by an order of magnitude would therefore lead to significant improvement
to photon indistinguishability.
This calculation of course assumes that spectral diffusion is the only source of indistin-
guishability affecting QDs. In the following section, we will discuss a second source of
photonic distinguishability: exciton-phonon coupling.
4.2 Phonon Sideband Filtering
The effect of phonon coupling in QDs, due to their low-dimensionality, is qualitatively
different from the case of bulk semiconductors. The discrete energy spectrum in QDs
prevents direct phonon-mediated energy relaxation, and creates a so-called phonon bottleneck
[88]. However, virtual transitions involving phonon absorption or emission still occur and
contribute to broadening of the optical transition in QDs [89]. Historically, phonon-induced
dephasing in QDs has been observed as a fast decay of the exciton coherence in four-wave
mixing [90] and as a temperature-dependent broadening of the emission spectrum [43, 44].
The spectral shape is well reproduced by a model considering a coupling of the exciton
to longitudinal-acoustic (LA) phonon modes through the deformation potential [89]. This
model predicts three components to the spectrum: a peak at the exciton transition energy
associated with the emission of a single photon (zero-phonon line), and blue (red) detuned
sidebands associated with absorption (emission) of a single LA phonon. The exact spectral
shape is determined by the temperature and the shape of the dot. The size of the dot, and
therefore the extent of the exciton wavefunction, determines the lower limit of the wavelength
of the coupled phonon modes. High temperatures, meaning high mean phonon numbers in
the LA modes, cause an increase in the rate of phonon-assisted processes compared with the
zero-phonon radiative decay and a subsequent increase in the spectral linewidth. Fortunately,
the coupling rate to low-energy LA phonons tends to zero, and the ZPL itself doesn’t suffer
from significant dephasing. Several experiments have shown that, after filtering out the PSB,
the ZPL shows photon emission with high indistinguishability [36, 37, 81]. Figure 4.5 B
shows a resonance fluorescence spectrum at 4K. We find that typically ≈ 90% of emission is
in the ZPL (86% for this QD) and the phonon-assisted emission is highly asymmetric about
the ZPL. The latter effect reflects the low occupation number of the LA phonon modes, and
phonon emission is consequently far more likely than absorption.
By working with such a solid-state emitter, we therefore find that approximately a tenth of
the emission intensity has to be thrown away in order to obtain near-unity indistinguishability.
The idea behind the stabilisation scheme presented in this chapter is that, instead of discarding
the PSB emission, we use it as an secondary channel to measure the instantaneous frequency
54 Phonon-assisted Resonance Fluorescence Frequency Stabilisation
Fig. 4.5 A. Schematic diagram of the optical filtering setup used to measure the zero-
phonon line (ZPL) and phonon sideband (PSB) of QD resonance fluorescence simultaneously.
f describes the focal length of the lens. B. Resonance fluorescence spectrum of the X-
transition under an excitation power corresponding to s = 1 taken with the flip-mirror stage.
C. Resonance fluorescence spectra taken through the ZPL (blue curve) and PSB (red curve)
paths for the same excitation power and integration time as B.
of the ZPL. This way we can feedback on the transition frequency without needing to sacrifice
any of the useful ZPL photons. In order to simultaneously measure both spectral components,
we use the setup shown in Fig. 4.5. The phonon sideband peak is only ≈ 1 nm detuned
from the ZPL, so we cannot simply use a dichroic mirror or a bandpass filter. Instead, in
order to obtain a broadband collection of the PSB we use a 4-f-like setup. QD Resonance
fluorescence is dispersed on a 1600 grooves/mm diffraction grating and a lens focuses the
different spectral components at different points in the Fourier plane. By using a sharp-edged
mirror, we selectively reflect part of the spectrum back onto the grating, where the beam is
re-collimated and coupled into a single-mode fibre. The component which is not reflected is
coupled into a different single-mode fibre. A flip-mirror can be inserted before the grating
in order to measure a reference spectrum. Figure 4.5 C shows spectra of the ZPL and PSB
modes under the same excitation condition. The two spectral components are well separated
into two spatial modes, although there is some leakage of the PSB emission into the ZPL
4.2 Phonon Sideband Filtering 55
Fig. 4.6 A. Photon count rates with the ZPL path coupled to an APD, with the laser on
resonance (red circles) and off resonance (blue triangles). The Signal-to-Background Ratio
(SBR) is plotted as gray squares. The red curve is a fit to the standard two-level saturation
behaviour, the blue curve is a linear fit, and the gray curve is the ratio of the two fits. B.
Similar measurement to A, but with the PSB path coupled to the same APD. The blue line is
flat, and the background is solely limited to the dark count rate of the APD of ≈ 50 Hz.
mode, probably due to scattering at the mirror edge. The peak count rate of the ZPL mode
is ≈ 85% that of the reference spectrum taken through the flip-mirror path. This loss is
mainly due to the efficiency of the grating (≈ 90% for linearly polarised light) as well as
finite transmission of the other optics. The peak PSB count rate is ≈ 60% compared with
the reference spectrum count rate at the same wavelength. This efficiency is lower due to
the PSB beam double-passing the grating. Figure 4.6 shows count rates for the two modes,
measured with the same avalanche photodiode. We measure around 4% of the ZPL count
rate with the PSB mode, giving 17 kHz at a driving power of s = 1. For a wide range of
powers, the signal to background ratio (SBR) is significantly higher than 100, as there is very
little cross-talk between the two modes and therefore no laser background in the PSB mode.
For all the powers shown here, the SBR is limited by the dark count rate of the APD. For the
56 Phonon-assisted Resonance Fluorescence Frequency Stabilisation
ZPL mode the background is limited by imperfect polarisation suppression of the reflected
laser.
Is this count rate high enough to stabilise against the noise described in the previous
section? A rough estimate of the stabilisation bandwidth that the PSB mode would give
can be obtained by considering the time needed to differentiate between the QD being on
resonance and being detuned by a certain amount. From Fig. 4.6, driving at s= 1 we obtain a
PSB count rate of 17 kHz on resonance. For the detuned PSB count rate, we assume a mean
detuning equal to that of Fig. 4.1, where Imax = 416 kHz and Imean = 343 kHz. The detuned
PSB count rate would then be ImeanImax .17 kHz. The integration time needed for the difference
between these two signals to be equal to shot-noise is around 2 ms. In practice the bandwidth
will depend on the experimental implementation, but in principle the PSB count rates shown
here should be sufficient to deal with most electric field noise sources for this QD.
4.3 Feedback Stabilisation Technique
Figure 4.7 shows the experimental setup used to convert the PSB channel into an error signal
for feedback stabilisation of the ZPL frequency. A single mode laser, which is frequency-
stabilised to ±5 MHz and power-stabilised to ±1%, is used to resonantly excite the trion
transition at a driving power corresponding to s = 1. The resonance fluorescence is sent
through the filtering setup and both the ZPL and PSB paths are coupled to separate APDs.
The ZPL channel is used for characterisation and plays no further part in the stabilisation
protocol. A simple measurement of the count rate from the PSB APD can give us information
about the mean detuning from resonance but cannot distinguish between positive and negative
detunings. For this we use a function generator to apply a small-amplitude modulation of
the gate voltage. This modulation will be mapped onto the PSB signal, the phase of which
gives the sign of the detuning. We demodulate the PSB signal with a lock-in amplifier to
obtain a detuning-dependent error signal (shown in Fig. 4.8 B for a lock-in gain of 80 dB
a modulation of 1.5 kHz and time constant of 1 ms) which can be used as input of a PID
controller. The output u(t) of a PID controller is related to the error signal ε(t), defined as
the difference between the measured value and the setpoint, as follows:
u(t) = P.ε(t)+ I
∫ t
0
ε(t)dτ+D
dε(t)
dt
(4.2)
Here D is not used as it can sometimes lead to amplification of high frequency noise, P and
I are optimised by minimising the intensity variance after stabilisation. The output of the
PI controller is fed back as a DC correction to the gate voltage. The modulation used here
4.3 Feedback Stabilisation Technique 57
Fig. 4.7 Diagram of the feedback stabilisation scheme. We use a confocal microscope with
polarisation rejection of the excitation laser to collect QD resonance fluorescence. The
optical filtering stage is the one described in Fig. 4.5. LP: linear polariser, APD: avalanche
photodiode, and FG: function generator.
58 Phonon-assisted Resonance Fluorescence Frequency Stabilisation
is ≈ 0.5 mV corresponding to a modulation of the ZPL central frequency of ≈ 150 MHz in
linear frequency. The absorption spectrum is consequently broadened from 620±10 MHz
to 760± 10 MHz. This unfortunately leads to a drop in peak count rate, but corresponds
to a decrease of only 3% for this modulation amplitude. Figure 4.8 B shows that the error
signal goes as the derivative of the lineshape, as the modulation amplitude is small compared
with the linewidth. From panels C and D, we see that stabilisation beyond 1kHz will not
be possible, both because the signal-to-noise ratio (SNR) drops below unity for integration
times lower than 1 ms and because we are limited by the bandwidth of the Schottky diode.
Higher stabilisation bandwidths may be obtained by increasing the modulation amplitude,
at the price of lowering the peak ZPL count rate, but would also necessitate a device with a
faster gate.
4.4 Stabilisation Results
We now turn to a characterisation of the stabilisation protocol, by comparing the intensity
and frequency noise characteristics of the ZPL channel with and without stabilisation. For an
appropriate ’unstabilised’ comparison, a reference measurement is taken with no feedback or
modulation applied to the gate voltage. All measurements shown in this section are taken at
a driving power of s = 1 for a QD initially set to be on resonance.
4.4.1 ZPL Intensity Fluctuations
As a first characterisation step, we look at the difference in ZPL intensity noise between
stabilised and unstabilised measurements. Figure 4.9 shows a relatively long (2000 s)
measurement with an integration time of 100 ms. This experiment is thus sensitive only to
the 1/ f -like component of noise, which appears as slow and high amplitude fluctuations in
intensity. These are apparent in the unstabilised case (green curve) where the intensity drops
almost by half at certain points throughout the measurement, and the stabilisation protocol
(orange curve) shows suppression of this slow noise. Due to the gate voltage modulation, the
stabilised curve shows a peak count rate ≈ 3% lower than the unstabilised case. However,
for long measurements, suppression of 1/ f -like noise leads to an increase of mean count
rates by 7.3%.
For a comparison of the faster noise characteristics, we measure 100 ZPL resonance
fluorescence time traces of 30 s length and 200 µs integration time. The noise spectral
densities for stabilised and unstabilised cases are shown in Fig. 4.10, after subtraction of a
reference measurement taken with laser reflection. A significant improvement is seen with
4.4 Stabilisation Results 59
Fig. 4.8 A. Normalised absorption spectra with (red circles) and without (green diamonds)
gate modulation and a driving power of s = 1. The curves are fits to Voigt profiles. The gray
bar represents the amplitude of the modulation in linear frequency. B. Mean error signal
obtained at the output of the lock-in amplifier (blue circles) with ±1σ shown as a shaded
region. The blue curve shows a fit to the derivative of a Lorentzian profile. C. Signal-to-noise
ratio (purple triangles) as a function of lock-in time constant for a modulation at 1.5 kHz.
The dashed curve is a fit to a power law with best fit exponent of 0.27±0.04. D. Amplitude
of the error signal (purple squares) as a function of modulation frequency for a fixed time
constant of 1 ms and a fixed modulation amplitude. The gray curve is a fit to a Gaussian
decay and serves as a guide to the eye.
60 Phonon-assisted Resonance Fluorescence Frequency Stabilisation
Fig. 4.9 Timetraces of resonance fluorescence for an initial condition of zero detuning. The
integration time is 100 ms and the driving power corresponds to s = 1. The green (orange)
trace is the measurement without (with) modulation and stabilisation. The right panel is
the corresponding histogram over the entire measurement time. The dashed lines represent
the mean count rate which is 370 kHz for the unmodulated measurement and 397 kHz with
stabilisation.
stabilisation, particularly in the low frequency region where the noise power is reduced by
more than 2 orders of magnitude. The additional peaks in the stabilised spectrum arise from
the gate modulation at 1.5 kHz and extra noise due to the PI electronics. From the perspective
of using this stabilisation scheme to improve photonic coalescence between independent QDs,
the added noise at the modulation frequency is not necessarily limiting, as the same function
generator can be used to drive both QDs in phase. With the normalisation of equation 4.1, the
integrated power spectral density gives the average variance in ZPL intensity over 30s. For
the unstabilised case this gives a variance 44 times higher than the corresponding shot-noise
(estimated with the laser reference measurement), whereas with stabilisation it is only twice
shot-noise. Of course, this comparison with shot noise is slightly arbitrary in the sense that
the result depends on experimental parameters such as collection and detection efficiency.
However, we can see that the noise contribution due to electric field fluctuations is reduced
by a factor of 22. A more complete characterisation of the stabilisation would require an
estimate of the bandwidth of stabilisation which can be compared with the electric field noise
characteristics. Fig. 4.10 shows an improvement in intensity noise up to ≈ 1 kHz. However,
the interpretation of this improvement at high frequencies is not trivial, due to the detuning
dependence of electric field noise. As demonstrated in [73, 77], the sensitivity to electric
field noise varies as the derivative of the homogeneous lineshape and therefore has a local
minimum at zero detuning. Thus, by suppressing the slow frequency fluctuations, the mean
4.4 Stabilisation Results 61
Fig. 4.10 Average noise spectral density of 30 s long resonance fluorescence time traces with
an integration time of 200 µs where the measurement with (without) stabilisation is shown
in orange (green). The broad peak in the stabilised spectrum corresponds to the applied
gate modulation frequency of 1.5 kHz, and other peaks are caused by the introduction of
additional noisy electronics. A reference spectrum, taken with unsuppressed laser with a
similar mean count rate, is subtracted to obtain both curves. The gray dashed curve shows
the shot-noise level for a measurement at comparable count rate.
62 Phonon-assisted Resonance Fluorescence Frequency Stabilisation
detuning is reduced along with the sensitivity to electric field noise. This explains why the
noise spectral power is lower in the stabilised case even for frequencies above the modulation
frequency, which cannot correspond to a stabilisation bandwidth.
4.4.2 Stabilization Bandwidth
Obtaining a stabilisation bandwidth from a noise spectrum is not simple due to the con-
founding factor of detuning dependent noise sensitivity. As explained earlier, the intensity
autocorrelation gives us information about the distinct fluctuation timescales corresponding
to individual fluctuators. With no stabilisation we find that a multi-exponential decay with
four characteristic timescales (1.2, 6.8, 44, and 752 ms) fits the data well. Our stabilisation
scheme corrects the emission frequency of the QD, but as the corrections to the electric
field required are small, we assume that our stabilisation scheme does not affect the inherent
charge dynamics in the environment. Therefore, we can use the knowledge of the charge
dynamics to see the effect of the stabilisation at each timescale. Figure 4.11 A shows the
intensity autocorrelation with and without stabilisation. We fit the stabilised curve with a
multi-exponential decay with the same timescales as the unstabilised case. An extra decaying
oscillation at 1.5 kHz is included to model the gate modulation. The ratio of best fit amplitude
with and without stabilisation at the same timescale (’Noise contribution ratio’) is plotted
in panel C and shows a complete suppression of the two slowest timescales. The fact that
each timescale is reduced by a different factor is a first indication that the improvement
is not solely due to a lower sensitivity. For a more precise estimate of the bandwidth we
can look at the decay timescale of the oscillations in the autocorrelation. This decay is
linked to the acquisition of an error signal and subsequent compensation, and its timescale
therefore quantifies the bandwidth of the stabilisation. To motivate this interpretation, a
similar measurement for higher value of P is shown in panel B. A revival of the oscillations
is seen along with a phase flip, due to the feedback overshooting the setpoint because P is
too high. This is a common problem with PID control systems, and the data shown here are
with optimised values of P and I. The best fit to the decay of oscillations in the orange curve
gives a stabilisation bandwidth of 191±4 Hz. This is comparable to the fastest electric field
fluctuations in the sample and should lead to a significant reduction in the inhomogeneous
width of the ZPL. A modest improvement of the collection or detection efficiency would
allow the complete suppression of electric field noise for this QD. This could be achieved by
embedding the QD in an optimised dielectric structure [91–93], combined with a structure
allowing faster gate control[94, 95]. In principle, with near-unity collection and detection
efficiencies, PSB count rates could allow for partial feedback correction of nuclear spin
fluctuations. However, reaching these timescales would require vast technical improvements.
4.4 Stabilisation Results 63
Fig. 4.11 A. Intensity autocorrelation of 30 s long resonance fluorescence timetraces with
an integration time of 200µs. The green circles are the same data as Fig.4.3 taken with no
stabilisation. The orange circles show the stabilised measurement with optimised values of
P and I. The oscillations at 1.5 kHz are caused by the gate voltage modulation. The solid
curves are multi-exponential fits with the decay timescales shown. The fit to the orange data
further includes a decaying sinusoid component to fit the oscillations. B. Two stabilised
autocorrelation measurements for two values of P. A revival of oscillations can be seen for
higher P along with a phase flip. C. Ratio of the best fit decay amplitudes extracted from
A at the four timescales after and before stabilisation. A value of zero indicates that noise
at that timescale is completely suppressed. The dashed curve is a Gaussian fit serving as a
guide to the eye.
64 Phonon-assisted Resonance Fluorescence Frequency Stabilisation
Fig. 4.12 Distribution of central frequencies with (without) stabilisation shown as an orange
(green) histogram. The central frequency is extracted from histogram analyses of 781
measurements of 1 s long timetraces. The curves are Gaussian fits with 23 MHz and 154
MHz FWHM for the stabilised and unstabilised measurements respectively.
4.4.3 Inhomogeneous Broadening
The decrease in intensity noise shown above is encouraging, but a more useful figure of
merit, in terms of the use of QDs as sources of on-demand indistinguishable single photons,
would be the corresponding time-averaged frequency distributions. As described in section
4.1, we record a large number of 1 s long timetraces with an integration time of 200µs with
and without stabilisation and perform a histogram analysis to extract the mean detunings for
each histogram. The distribution of this fit parameter over 781 measurements is shown in Fig.
4.12. In the stabilised case, we fit to a sum of two Gaussian detuning distributions separated
by the modulation amplitude (150 MHz in QD frequency) in order to model the square-wave
gate modulation. The distribution for the stabilised data is centered at 70 MHz corresponding
to half the modulation amplitude, and it is ≈ 150 MHz for the unstabilised case as the QD
drifted off resonance throughout the measurement. From the Gaussian fits, we find that the
distribution FWHM decreases from 154 MHz to 23 MHz with the stabilisation scheme. By
performing a χ2 analysis on the fitting routine, we estimate that the standard error on the
best fit is 23 MHz. Therefore, the measured stabilised frequency distribution is limited by
the fitting routine and is likely to be significantly narrower. Sub-Hz frequency fluctuations
are therefore suppressed to the extent that they can no longer be resolved.
Chapter 5
Hyperfine Effects in Resonance
Fluorescence
An electron spin confined to a QD shows promise as an efficient spin-photon interface, and as
such is a strong candidate for an optically-addressable stationary qubit forming the basis of an
optical quantum information processor. However, the coupling of this spin degree of freedom
to a mesoscopic nuclear spin bath is an important consideration. Random fluctuations of the
nuclear spin bath lead to uncertainty in the electron Zeeman splitting and consequently to
spin dephasing. This hyperfine coupling in QDs has drawn a large part of the community’s
focus in recent years [27, 34], both as a central challenge to future QD application and as a
fundamental realisation of the central spin problem. We concentrate here on the low-magnetic
field regime, where hyperfine effects are obscured by the broad radiative linewidth of QDs.
In this chapter, we present measurements of X- resonance fluorescence emission spectra
and intensity autocorrelations. First, we will show that optically-induced electron spin
pumping persists down to zero applied field showing that the hyperfine interaction lifts
the spin degeneracy and provides a quantisation axis for the electron spin. This zero-field
quantisation affects both the spin-splitting as well as the optical selection rules, changing
from Voigt-like to Faraday-like depending on the orientation of the effective magnetic field.
Both of these effects are seen in low-power emission spectra, revealing coherent Raman
transitions in the Lambda-system created by the Overhauser field. The low-power regime
of resonance fluorescence is a key tool, enabling spectroscopy with a resolution below the
radiative linewidth. The measured spectra contain information on the nuclear spin bath, and
provide a direct measurement of the time-averaged distribution of magnitudes and orientations
of the Overhauser field. In the final part of the chapter, we explore the effect of a small
external magnetic field on the emission spectra and provide high-resolution measurements of
subsequent nuclear spin polarisation and anisotropic nuclear spin distributions.
66 Hyperfine Effects in Resonance Fluorescence
The intensity autocorrelation data shown here and their subsequent analysis are based on
data published in reference [96]. The results shown here were obtained in collaboration with
the rest of the QD team, in particular Carsten Schulte and Claire Le Gall who helped coax
the temperamental FP cavity into shape in order to take these measurements. The data of Fig.
5.12, 5.10 and the neutral exciton data in Fig. 5.7 were taken by Clemens Matthiesen and are
shown with his permission.
5.1 Mean-field Approach to the Hyperfine Interaction
We are working here with the X- trion transition between a single conduction band electron in
the ground state and the lowest energy trion as the excited state. The level structure assumed
for the theoretical modeling is shown in Fig. 5.1. At zero external magnetic field, there is
no Zeeman splitting of the states, but the spin eigenstates are still in general non-degenerate
due to the hyperfine interaction with the nuclear spin bath. The ground state manifold of the
charge complex is composed of two orthogonal non-degenerate electron spin states and the
excited state is composed of an electron spin singlet and a heavy hole. The electron spin
singlet does not contribute to the excited state splitting, which is therefore determined by
the heavy hole hyperfine interaction. In general, the hyperfine interaction between spin S
and nuclear spin I can be decomposed into three contributions: the Fermi contact hyperfine
interaction (δ (r)S.I form), the dipole-dipole interaction ([(r.S)(r.I)]−S.I form) and the
coupling to the orbital angular momentum (L.I form) [97, 98]. The symmetry of the carrier
wavefunction determines which of these contributes most to the strength of the interaction.
For s-type wavefunctions, such as the conduction band electron in the ground state, the
Fermi contact interaction dominates and leads to an isotropic interaction with a strength
proportional to the electron wavefunction at the nucleus position. This interaction can be
written as follows:
HˆFC = ν0∑
j
A j
∣∣ψ(r j)∣∣2(Iˆ jz Sˆz+ Iˆ j+Sˆ−+ Iˆ j−Sˆ+2
)
(5.1)
where ν0 is the two atom cell volume, A j is the hyperfine constant of nucleus j with r j its
position, ψ(r) is the electron wavefunction. The first term within the brackets has an effect
similar to that of an external magnetic field which splits the electron spin states. Here, z
represents the quantisation axis of the electron spin, which is determined by both the applied
field and the instantaneous nuclear spin state. Considering the O(105) nuclei within the
electronic wavefunction, we take the mean field approach. We define the resulting effective
5.1 Mean-field Approach to the Hyperfine Interaction 67
Fig. 5.1 X- Level structure in the absence of an external magnetic field, for different orienta-
tion of the OH field (orange arrow). The straight green arrow represents the linearly polarised
driving field, and the curved arrows represent dipole allowed elastic and inelastic transitions.
Filled black arrows represent a conduction band electron spin, and hollow arrows represent a
valence band hole pseudo-spin.
magnetic field affecting the electron, the Overhauser (OH) field [27], as follows:
BN =
ν0
µBge∑j
A j
∣∣ψ(r j)∣∣2 〈Iˆ j〉 (5.2)
where µB is the Bohr magneton, ge is the isotropic electron g-factor, and BN is the effective
nuclear magnetic field. The mean field picture then leads to a frozen Zeeman-like splitting.
The dynamics of the system are given by the second term in the brackets of equation 5.1,
which leads to the exchange of spin angular momentum between the confined electron and
the individual nuclei. This term results in electron spin relaxation, as well as the possibility
of polarising the spin ensemble by exciting the dot circularly and transferring the resulting
electron polarisation to the nuclei via flip-flop [99].
The excited state hyperfine interaction is determined by the heavy-hole spin. The valence
band has mainly p-type symmetry [100] and the hole has no overlap with the nuclei, therefore
suppressing the Fermi contact interaction. The dipole-dipole form of the hyperfine interaction
dominates, but its strength has been measured to be an order of magnitude smaller than the
electron hyperfine coupling [101, 102]. For this reason, we neglect the hyperfine interaction
in the excited states in this chapter. The confinement is strongest in the z-direction, leading
68 Hyperfine Effects in Resonance Fluorescence
to a splitting of heavy and light hole bands, which we take to provide a hole spin quantisation
axis along the z-direction in the absence of hyperfine splitting.
The strong confinement of the electron wavefunction means that the electron spin samples
a relatively small ensemble of nuclear spins. Therefore, a random distribution of nuclear
spin orientations in the QD is not averaged to zero, and the OH field generally has a finite
dispersion proportional to 1/
√
N, where N is the number of nuclei contained within the
electron wavefunction. We treat fluctuations in the nuclear spin bath as a random walk of the
magnitude and orientation of the OH field. As the electron hyperfine interaction is isotropic,
we treat the x, y and z components of the OH field as independent. In the absence of nuclear
spin polarisation, each OH field component follows a Gaussian distribution centered at
zero with equal dispersions. The magnitude of the OH field determines the electron spin
splitting. Additionally, The electron spin quantisation axis follows the OH field, and the
selection rules fluctuate accordingly as shown in Fig. 5.1. On the one hand, for an OH
field aligned along the optical axis the selection rules are the same as those obtained in
the Faraday geometry, giving two circularly polarised transitions with a small branching
ratio for the spin-flip transitions. On the other hand, an in-plane OH field gives Voigt-like
selection rules with equal contribution of spin-conserving and spin-flipping transitions. In a
time-averaged measurement, with the OH field free to evolve in three dimensions, we expect
selection rules somewhere in between these two limiting cases, with an additional broadening
of the ground state splitting. Under low power resonant excitation, this should lead to the
observation of coherent elastic (Rayleigh) as well as inelastic (Raman) contributions to the
fluorescence emission spectrum. Both coherent Raman and Rayleigh transitions occur via a
virtual level, but the linewidth of the former will be broadened by the dispersion of the OH
field distribution, whereas the latter will have coherence properties determined by the driving
laser.
The basis and notation used in the calculation is the following: |1⟩= |↓⟩ , |2⟩= |↑⟩ , |3⟩=
|↓↑⇓⟩ , |4⟩ = |↑↓⇑⟩, where single arrows represent electron spin states and double arrows
represent heavy hole pseudo-spin states.The component of the Hamiltonian representing the
Fermi coupling, HˆFC, can be represented as follows in the { |0⟩ , |1⟩ } basis for the ground
state:
HˆFC =
µBge
2
BN.σ
=
µBge
2
(BX .σˆX +BY .σˆY +BZ.σˆZ)
=
µBge
2
|BN |
(
cos(θ)σˆZ + sin(θ)
(
eiφ σˆ++ e−iφ σˆ−
))
(5.3)
5.1 Mean-field Approach to the Hyperfine Interaction 69
where the σˆµ operators are the usual Pauli spin operators for the ground states. θ and φ are
the polar and azimuthal angles relative to the optical axis, respectively. The quantum dot is
additionally coupled to a classical oscillating linearly polarised electric field describing the
excitation laser, which is approximately resonant with both dipole allowed trion transitions
(1→ 3, and 2→ 4). In the dipole and rotating wave approximations, this component of the
Hamiltonian, Hˆrad, can be written as follows:
Hˆrad =− (D1,3+D2,4) .E
=Ω1,3
(
σ31e−iωlt + c.c
)
+Ω2,4
(
σ42e−iωlt + c.c
)
(5.4)
where Di, j = µi, jσi j + µ j,iσ ji is the dipole operator of the transition i → j, σi j = |i⟩⟨ j| in
our previously defined basis, Ωi, j is the Rabi frequency (taken to be real) and µi, j is the
dipole matrix element describing transition i→ j. ωl is the laser angular frequency. The total
Hamiltonian of the system can then be written:
Hˆ = HˆFC+ Hˆrad+
4
∑
i=3
h¯ωTσii+ HˆZeeman (5.5)
where the third term describes the degenerate excited state energy, and the fourth term
describes the Zeeman splitting due to an external field in the Faraday geometry. In matrix
form this gives:
Hˆ =

µBge
2 (|BN |cos(θ)+BZ) µBge2 |BN |sin(θ)e−iφ h¯2Ω1,3eiωlt 0
µBge
2 |BN |sin(θ)eiφ −µBge2 (|BN |cos(θ)+BZ) 0 h¯2Ω2,4eiωlt
h¯
2Ω1,3e
−iωlt 0 h¯ωT + µBgh2 BZ 0
0 h¯2Ω2,4e
−iωlt 0 h¯ωT − µBgh2 BZ

where gh is the hole pseudo-spin g-factor in the zˆ direction, and BZ is the magnitude of the
applied magnetic field. We can then use the methods described in chapter 2 to calculate the
second order correlation function for the trion system. The methods used for the calculation
of the emission spectrum are shown in the section 5.3.2.
Most of the data shown in this chapter were obtained with a single QD (nicknamed
’Felix’). In order to simulate the data, a few parameters are needed, namely: the radiative
decay rate, the distribution describing the spectral wandering, and the electron and hole
g-factors. The first is obtained with a pulsed measurement. The excitation laser is passed
through a high-bandwidth electro-optic amplitude modulator. Voltage pulses longer than
the radiative lifetime are sent to the EOM, and the subsequent time-resolved RF is recorded.
By subtraction of a reference time-resolved measurement with unsuppressed laser at a
70 Hyperfine Effects in Resonance Fluorescence
Fig. 5.2 Magnetic field-dependent absorption spectroscopy with a negatively charged QD
(A) and a neutral QD (B). The QD is driven below saturation with a linearly polarised
laser. We fit all curves to a sum a two Voigt peaks and extract the frequency difference to
obtain the Zeeman splitting. The extracted g-factors for X0 and X- are shown inset. The X0
measurement shows a dispersive component due to interference of the QD RF with the laser
background.
comparable count rate, we are able to obtain a decay in RF once the driving laser is switched
off, giving a 1/e time of 740 ps. The measurement of an absorption spectrum, combined with
the knowledge of the radiative lifetime, gives the extent of the inhomogeneous broadening.
The electron and hole g-factors can be measured separately by comparing Zeeman shifts of
bright and dark excitons in the Voigt geometry [103]. Having access only to the Faraday
geometry in this experiment, it is not feasible to obtain an independent measurement of both
g-factors. Instead, we measure the Zeeman splitting of the exciton and trion transitions, and
make an educated guess of the electron g-factor based on measurements on similar QDs,
to obtain the hole g-factor. Figure 5.2 shows B-field dependent absorption of X- and X0
transitions. In a first-order approximation, we would expect these to show the same g-factors
corresponding to gX = |ge−gh|. This is indeed what is seen, both for this dot and two others
in the sample, despite there being a large range of gX (from 0.8 to 1.9). For similar InAs QDs,
typical values of the out-of-plane electron g-factor are typically around ge =−0.6 [4] and do
not vary significantly with exact QD shape, as opposed to the hole g-factor [104]. For the
simulations presented here, we therefore assume an isotropic electron g-factor of ge =−0.6
and an out-of-plane hole g-factor of gh = 0.3. Figure 5.2 additionally shows the difficulty of
performing measurements of hyperfine effects in the low B-field regime and in the absence
of nuclear spin polarisation: even under 23 mT, no splitting can be resolved and the expected
OH field is commensurate with both the radiative linewidth and the spectral diffusion.
5.2 Optically Induced Spin Dynamics 71
5.2 Optically Induced Spin Dynamics
We turn to autocorrelation measurements of the X- transition. For the ideal two-level
system discussed in chapter 2, besides the antibunching and Rabi oscillations decaying on a
timescale of the radiative lifetime, we expect a complete loss of correlations on all longer
timescales. Intensity autocorrelations, and their detuning dependence, have been used to show
fluctuations of the OH field on timescales ranging from tens to hundreds of µs [73, 77]. Here
we focus on shorter timescales, from zero delay up to tens of ns. This time window, shorter
than OH field dynamics and longer than the radiative lifetime, allows for unobstructed access
to the optically-induced electron spin dynamics, revealing a sub-linewidth level structure
determined by the hyperfine interaction.
5.2.1 Finite Magnetic Field
First, we look at the photon statistics with an applied field large enough to split the two
transitions. Figure 5.3 shows intensity autocorrelation measurements taken at 76 mT in the
Faraday geometry, driving the QD below saturation at s = 0.1. Using such a low driving
power ensures that no bunching arises due to a combination of Rabi oscillations with a finite
timing resolution. Measurements at six different detunings are taken. When the laser is
set at equal detuning from both transitions (measurement 4), a small degree of bunching
is seen (g(2)−1 = 0.23). Additionally, The antibunching dip almost disappears, due to the
increased effective Rabi frequency which cannot be resolved with our correlation setup. As
the laser is detuned from the point of symmetry, strong bunching appears and continues
to increase even when red-detuned from both transitions. The bunching arises due to the
spin-flip Raman transition. An optical transition is associated with each of the spin states,
and when the laser is further detuned from one transition than from the other, the electron
spin is shuffled from an optically bright state to a slightly darker one. This effect leads to
photon correlations on a timescale of a few optical cycles and creates bunching in g(2). The
clear detuning dependence indicate that the bunching arises from competing processes of
optically-induced spin pumping, repumping, and spin relaxation. From exponential fits to
the decay of the bunching, we extract similar timescales of ≈ 40 ns for all measurements.
This timescale corresponds to a ≈ 2.5 optical cycles at this power, and already shows that the
hyperfine interaction modifies the time-averaged branching ratio in the low magnetic field
regime.
72 Hyperfine Effects in Resonance Fluorescence
Fig. 5.3 A. Detuning-dependent resonance fluorescence (blue circles) from the X- transition
at a driving power below saturation (s ≈ 0.1) under a finite magnetic field in the Faraday
configuration with 1 s integration time. The dashed lines show the lineshapes of the two
transitions, derived from a fit to the data. The green points show the detuning settings
and average RF intensity for the measurements shown in panel B. B. Normalised intensity
autocorrelations taken at the detunings shown in A for s= 0.1. Each curve is offset for clarity,
and the dashed line represents the g(2) = 1 condition. C. Extracted bunching amplitudes
from exponential fits to the data shown in B. The dashed line is a quadratic fit and serves as a
guide to the eye.
5.2 Optically Induced Spin Dynamics 73
Fig. 5.4 A. Low power (s = 0.1) intensity autocorrelation for zero mean detuning. The blue
curve shows the measured data, normalised to the long term value (at τ = 10µs). The orange
curve is a simulation using a four level master equation including a diffusion width equal to
Γ and averaged over an isotropic OH field distribution with a dispersion of 22 mT. A finite
timing resolution of 0.6 ns is also included in the calculation. B. Simulations of intensity
autocorrelations for different driving powers indicated in the legend. The long-time bunching
amplitude is determined by the absorption linewidth and is most clearly observed in the low
power regime.
5.2.2 Zero Applied B-field
We perform a similar measurement in the absence of an external magnetic field, shown
in panel A of figure 5.4. Again, we drive the QD at s = 0.1, initially on resonance. A
significant degree of bunching (g(2)−1≈ 0.25) reveals the spin-flip transitions enabled by
the in-plane component of the OH field. The decay timescale of 38± 3 ns is consistent
with that of Fig. 5.3 for the same driving power, confirming that the same mechanism -
optically-induced electron spin flips - is at play in both cases. Panel B shows intensity
autocorrelation calculations at different driving powers and shows bunching in all cases.
However, there is a clear dependence of the bunching amplitude and timescale on power,
marking the transition between two sources of bunching. In the low-power limit, where the
mean spin splitting is commensurate with the homogeneous linewidth, the bunching occurs
on a long timescale and is related to electron spin dynamics. In contrast, in the high power
regime both transitions are homogeneously power-broadened to such an extent that the spin
splitting becomes negligible; the bunching amplitude is then entirely due to Rabi oscillations,
and no bunching is expected beyond a timescale on the order of the radiative lifetime.
Bunching on a timescale of tens of ns could arise from blinking of the transition on
a fast timescale [87]. However, this mechanism should not depend on laser-detuning on
the scale of the linewidth, therefore a detuning-dependence should unambiguously point
74 Hyperfine Effects in Resonance Fluorescence
to optically-induced electron spin flips as the underlying mechanism. Looking back at the
finite field measurement of Fig. 5.3, we see that the bunching amplitude shows a local
minimum when the laser is symmetrically detuned from both transitions. In this condition,
spin pumping and repumping rates are equal and neither spin state is ’darker’ than the other.
We should observe a similar behaviour in the zero-field case, due to the finite electron spin
splitting generated by the OH field.
However, a detuning dependence is not as straightforward to obtain as in Fig. 5.3, due to
spectral wandering on the order of the transform-limited linewidth. The largest contribution
to spectral wandering is the slow 1/ f -like noise (c.f. chapter 4) and sub-linewidth control
of detuning is rendered difficult for long measurements, such as intensity autocorrelation.
Instead, we initially set the laser on resonance and save correlation histograms every 50
s while monitoring the mean count rate within each time window. We then postselect the
histograms according to their mean count rate, and obtain separate histograms for each
intensity bin. The post-selection technique is depicted in Fig. 5.5. By comparison of the
intensity bins with an absorption spectrum measured on a shorter timescale (≈ 2 s), we can
convert the intensity bins to detuning bins. The bunching amplitude for each detuning bin
is extracted by fitting each post-selected histogram with an exponential decay (panel C).
The bunching amplitudes are summarised in panel D, showing a sub-linewidth detuning
dependence. This confirms that the degree of bunching depends on optically-induced spin
pumping and repumping rates.
To fit the data in panel D, we develop a simple model based on solving the 4-level Bloch
equations described by the Hamiltonian of equation 5.5. With a finite ground state splitting
and a laser detuned from the bare resonance, a steady state ground state population imbalance
can be created through preferential excitation of one of the transitions. The calculated laser
detuning dependence of the created populations is shown in Fig. 5.6. This imbalance in
turn leads to correlations in the RF signal from the QD on the timescale of optical pumping.
In order to extract a bunching amplitude from the calculated steady-state spin polarisation,
we compare the expected autocorrelation in two regimes: for timescales much smaller than
the optical spin pumping rate (but longer than the radiative lifetime) g(2)(τ < TSP), and the
long timescale autocorrelation g(2)(τ → ∞). Ignoring further correlations due to electric
field and OH field fluctuations and working in the low power regime (ρSS11 +ρ
SS
22 ≈ 1), the
unnormalised long timescale autocorrelation value can be approximated as:
G(2)(τ → ∞) ∝ (ρSS11L [∆1]+ρSS22L [∆2])2 (5.6)
where ρSSii is the steady state population of ground state i, andL [∆i] is the photon detection
rate from the transition associated with spin i with a Lorentzian dependence on laser detuning
5.2 Optically Induced Spin Dynamics 75
Fig. 5.5 a. Mean RF count rate (green circles) for an integration time of 50 s, driving at s = 1
initially on resonance. The dashed lines represent the postselection bins used to measure
a detuning-dependent g(2)(τ). The blue and red regions correspond to the post-selected
histograms shown in panel c. b. Absorption linescan taken over ≈ 2s used to convert the
data from intensity bins to detuning bins. c. Two representative post-selected histograms
(blue and red curves) corresponding to the intensity post-selection of a. These curves are
normalised to the long term correlation value. We fit exponential decays to the bunching
component of the correlation and extract the bunching amplitude (represented by the arrows).
d. Extracted bunching amplitudes (blue circles) as a function of post-selected detuning. The
error bars represent the standard error in the exponential fits to each post-selected histogram.
The red curve shows the calculated bunching amplitude from a 4-level rate-equation model
assuming an isotropic OH field distribution with a dispersion of 18±1mT
76 Hyperfine Effects in Resonance Fluorescence
- 1 0 - 8 - 6 - 4 - 2 0 2 4 6 8 1 00 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0
 
 
Gro
und
 sta
te p
opu
latio
n
 ρ11 ρ22
- 1 0 - 8 - 6 - 4 - 2 0 2 4 6 8 1 0 0 . 0 0 0
0 . 0 0 5
0 . 0 1 0
0 . 0 1 5
 ρ33+ρ44
Exc
ited
 sta
te p
opu
latio
n
D e t u n i n g  ( u n i t s  o f  Γ )
O H  s p l i t t i n g
Fig. 5.6 Detuning dependence of the calculated steady-state atomic populations for resonant
excitation of the X- transition at s = 0.1. The red and blue curves show the ground state
populations and the dashed grey line shows the total population in the excited state manifold.
The OH field is fixed with a magnitude corresponding to the splitting in the dashed curve.
∆i. For timescales shorter than the optical pumping rate, as the electron spends a fraction of
time ρSSii in spin state i, we have:
G(2)(τ < TSP) ∝ ρSS11L [∆1]
2+ρSS22L [∆2]
2 (5.7)
The bunching amplitude BA is then given by:
BA =
G(2)(τ < TSP)
G(2)(τ → ∞) −1 (5.8)
We are then left with two free parameters: the OH field dispersion (assuming a Gaussian
isotropic distribution) and Gaussian broadening of the line due to electric field noise on a 50s
timescale. By fitting the data of Fig. 5.5, we perform a χ2 analysis to find 18±1 mT and
80±10 MHz respectively. This model is simplistic, in that it ignores both the antibunching
at small timescales and the spin pumping rate, but it gives a good quantitative agreement
with the data. Although the model depends strongly on the time-averaged magnitude of
the OH field, it has no dependence on its orientation. The decay timescale of the bunching
corresponds to ≈ 2.5 optical cycles. This is inconsistent with the branching ratio in the
Faraday geometry (≈ 0.5%) indicating that the OH field has a significant in-plane component,
but this measurement alone doesn’t give a quantitative estimate of the in-plane OH field
variance. A different measurement method is needed to obtain a good estimate of the full
OH field distribution.
5.3 Coherent Raman Spectroscopy under Zero Magnetic Field 77
5.3 Coherent Raman Spectroscopy under Zero Magnetic
Field
We move to measurements of the first order coherence of the trion system, again in the
low magnetic field regime. As discussed in chapter 2, the two-level resonance fluorescence
is composed of coherent and incoherent fractions, which display different spectral proper-
ties. In the low power regime, the coherent fraction dominates giving a spectral linewidth
limited either by the excitation laser or the measurement resolution. This regime therefore
offers a significant improvement in resolution over absorption spectroscopy and allows for
measurement of sub-linewidth features due to the hyperfine interaction.
5.3.1 Deviation from Two-Level System
Panel A of Fig. 5.7 shows a comparison of emission spectra obtained with a driving power
of s = 0.1 for the neutral exciton and the negatively charged trion transitions. For both
measurements, the FP transmission function is determined with a measurement of reflected
laser and displays a Lorentzian lineshape with a FWHM of ≈ 20MHz. The X0 spectrum
shows the expected two-level behaviour, i.e. a single line centered at the laser frequency with
a width given by the FP response. This sharp feature is also observed in the case of the X-
spectrum, indicating Rayleigh scattering. However, in sharp contrast with the X0 transition,
two broad sidebands are also seen either side of the peak of elastic scattering. These are
typically detuned from the laser frequency by ≈ 200 MHz and have a linewidth of ≈ 250
MHz. These reveal a deviation from two-level behaviour due to inelastic scattering.
Panel B shows a similar comparison between low power X- emission spectra with and
without an applied magnetic field. The application of 100 mT completely suppresses the
sidebands, and a behaviour similar to that of a two-level system is recovered. At first glance, it
seems that several mechanisms can plausibly explain the observation. Perhaps the sidebands
stem from the incoherent contribution to the fluorescence, and by applying the magnetic field,
we are simply detuning the QD from the laser and thereby increasing the coherent fraction of
the scattering. The sidebands in this case would signify the onset of the Mollow triplet [105],
and the splitting between the sidebands would be highly power- and detuning-dependent.
Figure 5.8 shows measurements of the trion RF emission spectra as a function of laser
detuning from resonance. We find that the integrated intensity drops, following the absorption
lineshape of the transition. However, the shape of the spectrum remains almost unaffected:
the detuning between sidebands is constant, and the fraction of emission in the sidebands
does not decrease with detuning. These observations point to the fact that these sidebands
78 Hyperfine Effects in Resonance Fluorescence
Fig. 5.7 A. Typical RF emission spectra measured at low driving power s= 0.1 for the X- (red
circles) and X0 (blue circles) transitions normalised to the peak count rate. B. Low-power RF
emission spectra measured with (green circles) and without (red circles) an external applied
magnetic field in the Faraday configuration.
Fig. 5.8 Detuning dependence of the X- RF emission spectra measured at low driving power
s ≈ 0.1. The laser frequency was kept constant, and the detuning was controlled with the
gate voltage. The emission is centered at the laser frequency, but the curves are offset by
an amount corresponding to the applied gate voltage for clarity. The dashed grey curve
represents the absorption lineshape typically measured for this QD.
5.3 Coherent Raman Spectroscopy under Zero Magnetic Field 79
arise from a coherent Raman transition. The scattering occurs via a virtual level and the
difference in energy is accounted for with an electron spin flip in the basis defined by the OH
field.
5.3.2 Calculation of Four-Level Resonance Fluorescence Spectrum
In this section, we present the theoretical methods used for the calculation of the emission
spectrum of resonance fluorescence, with a negatively charged QD. The hyperfine coupling
to the nuclear spin bath is treated in a mean-field approach, as described in section 5.1. For
the coupling to the driving laser field, we will borrow the methods discussed in chapter 2,
in relation to the two-level system, which we will generalise to the four-level system here.
Once again, we can describe the dynamics in the system with a Liouville equation, using the
Hamiltonian of equation 5.5:
dρ
dt
=− i
h¯
[H,ρ]+L [ρ] (5.9)
where,L [ρ] is the Lindblad superoperator describing the irreversible dynamics of the system,
namely spontaneous radiative decay and other pure dephasing processes. This takes a slightly
different form to 2.6, as we are dealing with two distinct optical transitions. Working in the
basis defined in section 5.1, we have two dipole allowed transitions: |1⟩ → |3⟩ and |2⟩ → |4⟩.
The Lindblad superoperator, excluding for the moment any pure dephasing terms, becomes:
L [ρ] =
Γ13
2
(2σ13ρσ31−σ33ρ−ρσ33)
+
Γ24
2
(2σ24ρσ42−σ44ρ−ρσ44) (5.10)
where Γ13 and Γ24 are the radiative decay rates corresponding to transitions |3⟩ → |1⟩ and
|4⟩ → |2⟩ respectively. For a stationary stochastic process, according to the Wiener-Khinchin
theorem, the spectrum is simply given by the Fourier transform of the first order correlation
function of the electric field operator describing the amplitude of the emitted field. In the
Heisenberg picture this is given by equation 2.19. The relation between electric field and
atomic operators once again differs from the two level case:
E−(t) ∝ µ13σ13(t)+µ24σ24(t) (5.11)
where µ13 and µ24 are the dipole moments associated with both transitions. So far, we have
ignored the vectorial nature of the dipole operators. In contrast with the two-level system,
we will have to take the detection polarisation into account which will introduce cross-terms
80 Hyperfine Effects in Resonance Fluorescence
in the first order correlation function. Assuming a left-hand circularly polarised |3⟩ → |1⟩
transition, a right-hand circularly polarised |4⟩→ |2⟩ transition, a linearly polarised excitation
along xˆ and a linearly polarised detection along yˆ, the detected field can be written:
E−det(t) ∝ iµ13σ13− iµ24σ24 (5.12)
In order to calculate the emission spectrum, we operate in the Heisenberg picture and solve
the Langevin equations of motion. These equations of motion have a form similar to the
Liouville equation for the density matrix in the Schrodinger picture but also consider a noise
operator. We will operate in the Markovian regime where the noise is delta correlated in
time so its stationary two-time contribution will be zero. Therefore, the Langevin equation
of motion will closely follow equation 5.9, with σ ji(t) replacing ρi j. In a similar way to
equation 2.7, we can write the resulting equations of motion for the slowly varying atomic
operators in matrix form:
∂
∂ t
Ψ(t) =
∂
∂ t

S12
S13
...
S43
S44
= LΨ(t)+Λ+F(t) (5.13)
where Si j = Ai jσi j, and Ai j is a time dependent coefficient used to cancel out terms oscillating
at the driving frequency. The condition of unit trace has been used to eliminate S11, giving
rise to vector Λ, F(t) is a vector describing the noise operators, and L is a time-independent
15x15 matrix omitted here for lack of space. The RF spectrum can then be obtained by
inserting the detected field of equation 5.12 into equation 2.19. Assuming that |µ13|= |µ24|,
we obtain:
S (ν0) ∝
∫ ∞
0
dτ ⟨S31(τ)S13(0)⟩e−i(ν0−ωl)τ + c.c
−
∫ ∞
0
dτ ⟨S31(τ)S24(0)⟩e−i(ν0−ωl)τ − c.c
−
∫ ∞
0
dτ ⟨S42(τ)S13(0)⟩e−i(ν0−ωl)τ − c.c
+
∫ ∞
0
dτ ⟨S42(τ)S24(0)⟩e−i(ν0−ωl)τ + c.c (5.14)
Each term is closely related to the Laplace transform of the correlation functions
〈
Ψi(τ)Ψ j(0)
〉
with p =−i(ν0−ωl) as the variable of the kernel. Therefore, all we need to calculate the
5.3 Coherent Raman Spectroscopy under Zero Magnetic Field 81
spectrum is to find the values of the Laplace transforms corresponding to the four terms in
equation 5.14, which we now write Φi, j(p) =
∫ ∞
0
〈
Ψi(τ)Ψ j(0)
〉
epτ .
We first differentiate the correlation function using equation 5.13, and take zero contribu-
tion from the noise operators as they have a delta-function correlation in time:
d
dτ
〈
Ψi(τ)Ψ j(0)
〉
=
〈
(LΨ(τ)+Λ)iΨ j(0)
〉
(5.15)
Then, using the property of the Laplace transform of a differential (L[ f ′(t)] = pL[ f (t)]−
f (0)) and of a constant (L[A] = Ap ), we take the Laplace transform of the previous expression:
pΦi, j(p)−
〈
Ψi(0)Ψ j(0)
〉
=
15
∑
k=1
LikΦk, j(p)+
Λi
p
〈
Ψ j(0)
〉
(5.16)
After defining ϕ j(p) = (Φ1, j(p),Φ2, j(p), · · · ,Φ15, j), this can be conveniently rewritten in
matrix form and the problem is reduced to simple algebra:
(pI15−L)ϕ j(p) = ζ j(p) (5.17)
where I15 is the 15x15 identity matrix, and the ith component of ζ j is
〈
Ψi(0)Ψ j(0)
〉−
Λi
p
〈
Ψ j(0)
〉
, which can be found by calculating the stationary solutions of the Langevin
equations of motion. Being a problem with 15 simultaneous equations, the calculation of
ϕi, j is rather cumbersome to perform analytically so it is done numerically in this thesis. In
order to obtain spectra with finite linewidths, a real part λ is added to the Laplace variable p,
such that p =−i(ν0−ωl)+λ . This parameter gives rise to a Lorentzian spectral resolution
with a FWHM of 2λ , corresponding to the Fabry-Perot interferometer cavity linewidth. By
comparison with equation 5.14, the full spectrum is then given by:
S (p) ∝ℜ
{
ϕ2,8(p)+ϕ7,13(p)−ϕ7,8(p)−ϕ2,13(p)
}
(5.18)
5.3.3 High Resolution Measurement of the Nuclear Spin Distribution
Combining the mean-field model discussed earlier in this chapter with the theoretical methods
of the previous section, we can calculate the expected trion emission spectra for a given
nuclear spin distribution. Figure 5.9 shows a low power (s = 0.4) emission spectrum along
with an absorption spectrum for reference. The spectral features of the emission vary on a
scale smaller than the absorption linewidth, highlighting the benefit of using the low-power
regime to resolve hyperfine features. The coloured spectra are simulations for different fixed
OH fields with a very narrow spectral resolution. The individual spectra are composed of
82 Hyperfine Effects in Resonance Fluorescence
- 0 . 6 - 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4 0 . 6 0 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0
Abs
orp
tion
 spe
ctru
m (
a.u
)
D e t u n i n g  ( G H z )
0 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0
 
Em
issi
on 
spe
ctru
m (
a.u
)
B n u c
Fig. 5.9 Measured absorption (grey diamonds) and emission (green circles) spectra at s= 0.4.
For the absorption spectrum, the detuning refers to driving laser detuning from the trion
resonance. For the emission spectrum, the detuning refers to the FP detuning from the laser
frequency. Different coloured lines show calculated spectra for different fixed OH field
magnitudes varying from 12 mT to 47.5 mT, weighted by the nuclear spin distribution. The
solid green curve is the calculated time-averaged spectrum with a finite resolution and an
OH dispersion of 22mT.
5.3 Coherent Raman Spectroscopy under Zero Magnetic Field 83
Fig. 5.10 X- RF emission spectrum (white circles) for a driving power of s = 0.016. The
curves are simulations for different ground state dephasing rates as shown. The inset shows
the reduced χ2 for different dephasing rates for a best fit analysis. The value for χ2 reaches a
minimum for T 2ground ≈ 70T 1excited. The data are therefore consistent with long electron
spin coherence. The dashed curve shows the χ2min+1 and gives an indication of the lower
limit of best fit spin coherence time.
three narrow peaks: the elastic Rayleigh peak with a width limited to the laser coherence,
and two Raman peaks either side of the Rayleigh peak. The Raman peaks are broadened
by laser-induced spin relaxation and ground state decoherence. Their detuning from the
Rayleigh peak is directly given by the instantaneous ground state splitting. Additionally, the
integrated intensity in the Raman sidebands relative to the Rayleigh peak is determined by
the branching ratio for the diagonal transitions. The exact shape of the emission spectrum
is therefore a complex function of OH field magnitude and orientation, as well as electron
spin coherence. The full spectrum is obtained by averaging over a Gaussian distribution of
OH field components in three dimensions. These spectra are insensitive on in-plane angle
φ so it is convenient to work with a cylindrical coordinate system, where the OH field is
decomposed into its in-plane and out-of-plane components. On the one hand, the shape of
the sideband relates to the time-averaged distribution of OH field magnitudes. On the other
hand the integrated intensity in the sideband relative to that of the Rayleigh peak relates to
the time-averaged optical selection rules, and therefore the OH field orientation. Therefore, a
single spectrum simultaneously contains information on both the in-plane and out-of-plane
OH field distributions. For this measurement we find that an isotropic distribution with a
dispersion of 22 mT fits the data well.
84 Hyperfine Effects in Resonance Fluorescence
Fig. 5.11 A. Differential reflection map of the X- transition. The green and red labels represent
the measurement conditions used in panel B and C. The cotunneling region corresponds to the
gate voltage at which the X- emission begins to broaden and loses intensity. B. X- emission
spectrum taken at low power (s= 0.1) far away from the cotunneling region showing coherent
Rayleigh and Raman peaks. The solid curve shows a simulation for an OH field dispersion of
18 mT with negligible pure dephasing. C. Emission spectrum taken in the cotunneling region,
showing a relative decrease of coherent emission and a broad incoherent peak. The solid
curve is a simulation with an OH field dispersion of 18 mT, a ground state pure dephasing
rate equal to the radiative decay rate, and a detuning of 30 MHz.
Figure 5.10 shows the dependence of the simulated spectra on pure dephasing of the
ground states. The dataset used as a comparison is taken at a very low power s = 0.016
in order to minimise broadening of the Raman peaks through laser-induced electron spin
relaxation. The presence of pure dephasing of the ground states is expected to significantly
broaden the sidebands, and we find that the data is consistent with little pure dephasing of
the electron spin. A χ2 analysis suggests a lower bound on the pure dephasing rate at least
an order of magnitude longer than the radiative decay rate. The simulations shown here
assume a static level structure, but fast fluctuations of the OH field would appear as a pure
dephasing in the best fit to the data. Therefore, the data of Fig. 5.10 suggest that the model of
the electron experiencing a ’frozen’ OH field on the timescale of radiative decay seems valid.
The nature of the sample structure readily offers a method of tuning the electron spin
5.3 Coherent Raman Spectroscopy under Zero Magnetic Field 85
Fig. 5.12 A. (a) Emission spectra (hollow circles) measured for different driving laser Rabi
frequencies as shown. The solid curves are the best fits with peak height and OH field
dispersion as fit parameters. (b) Summary of OH field dispersion values which give the best
fits at different Rabi frequencies, where the error bars arise from a χ(2) analysis. The line and
shaded region show the mean best fit OH field dispersion and the standard error respectively.
relaxation rate through cotunneling with the back contact [106]. Figure 5.11 shows emission
spectra measured for the same driving power (s = 0.1) with two different applied gate
voltages. Panel B shows a measurement which is taken far away from the edge of the
charging plateau (panel A), where cotunneling rates are slow [106], showing a behaviour
similar to Fig. 5.10. Panel C, in contrast, shows a broad incoherent spectrum well fit with the
addition of a pure dephasing rate comparable to the radiative decay rate. This verifies that
the coherent Raman transitions are highly dependent on the electron spin coherence.
As we have seen from Figs. 5.3 and 5.4, the electron spin dynamics are drastically
modified in the presence of the driving laser. In order to see what impact the resonant
excitation has on the OH field distribution, we turn to a power dependence of the trion
emission spectrum. Figure 5.12 shows emission spectra under different driving powers below
(s ≈ 0.1), above (s ≈ 38), and close to the saturation point (s ≈ 3). The sidebands, which
are clearly separated from the Rayleigh peak in the low power measurement, are no longer
resolved at intermediate power. Increasing the power further generates the well know Mollow
triplet, and the system behaves similarly to a two-level system. The solid curves are best fits
to simulations using the model described earlier, with the OH field dispersion as the fitting
parameter. Panel B shows a summary of best fit OH field dispersions for different driving
86 Hyperfine Effects in Resonance Fluorescence
powers. We find no significant variation in best fit value over three orders of magnitude
in driving power, indicating that, although the driving power may change the timescale of
OH field dynamics [77], the time-averaged distribution is unaffected by the measurement.
Furthermore, no significant nuclear spin polarisation is created, as expected for a linearly
polarised driving field.
The data shown in this section support the conclusion that, in the absence of an external
magnetic field, fluctuations in the nuclear spin bath of the QD lead to a fluctuating electron
spin splitting and quantisation axis. On the timescale of electron spin dynamics, the ’frozen’
nuclear spin model fits the data well, and the hyperfine generated Λ-system can be described
as quasi-static. In the absence of an external magnetic field, the time-averaged OH field
distribution is isotropic with a Gaussian distribution. For the three QDs measured here,
we obtain best fit dispersions of 22± 1 mT, 19.5± 0.5 mT, and 18± 1 mT. The OH field
provides a spin splitting and optical selection rules propitious for optical spin control. We
have already shown this, to some extent, with single-laser electron spin pumping (c.f. Fig.
5.4). The principle of OH-enabled optical spin control will be extended to two-colour
excitation schemes in chapter 6.
5.4 Nuclear Spin Polarisation under Low Magnetic Fields
Already in Fig. 5.7, we have seen that the hyperfine effects in the emission spectrum disappear
with the application of a modest out-of-plane magnetic field. The B-field dictates the electron
spin quantisation axis and the associated optical selection rules. The branching ratio in the
Faraday geometry, determined by light-heavy hole mixing, is small, therefore the emission
spectrum is similar to that of a single two-level system. In this section, we take a closer
look at the transition between the low- and high-magnetic field regimes. Figure 5.13 shows
emission spectra measured at s = 1 for different out-of-plane magnetic fields, commensurate
with the OH field dispersion. The solid curves are simulations assuming a fixed isotropic
OH field distribution, with no mean nuclear spin polarisation. This assumption fits the data
well in the case of low (2.6 mT) and high (100 mT) fields, but a striking deviation is seen
at intermediate fields. Here the simulations predict increased detuning between the Raman
sidebands due to the magnetic field giving an additional contribution to the electron spin
splitting, and a decrease in Raman contribution to the total spectrum. However, the data do
not show the predicted increase in spin splitting. We fit the spectra at each magnetic field
with a sum of two Gaussian peaks (representing the Raman sidebands) and one Lorentzian
(for the elastic scattering) peak. Panel B shows the summary of the ratio of best fit Gaussian
area to Lorentzian area, as a function of magnetic field. As expected, this decays on a scale
5.4 Nuclear Spin Polarisation under Low Magnetic Fields 87
Fig. 5.13 A. Low-power X- emission spectra (squares) for different values of applied magnetic
field in the Faraday geometry. The solid curves are simulations assuming an isotropic OH
field distribution with a constant dispersion of 22 mT. B. Ratio of the sideband area to the
Rayleigh peak area extracted from fits to three peaked spectra. The error bars arise from
standard errors in the area fit parameters. The solid curve is a Lorentzian fit to the data and
serves as a guide to the eye. C. Ground state Zeeman splitting (grey circles) and their mean
(red circles) extracted from three peaked fits to data. The solid blue curve shows the expected
behaviour assuming a constant isotropic OH field distribution with a dispersion of 22 mT.
The error bars show the standard deviation of measured Zeeman splittings. The dashed line
shows a linear best fit to the data, and its gradient is consistent with zero.
commensurate with the zero-field OH field dispersion. However, the best fit Raman detuning
(panel C) is constant up to 100 mT, above which the Raman sideband area decreases such
that it cannot be fit. If the OH field distribution were unaffected by the B-field, we would
expect a dependence of the form
√
B2N+B
2
ext, which is shown as a solid blue curve.
Clearly, the data are inconsistent with a fixed OH field distribution. In this case, the
nuclear spins are polarised so as to minimise the electron spin splitting. Nuclear spin
polarisation in the low magnetic field regime has been observed [107], but was induced by
circularly polarised optical excitation. Here the excitation polarisation is linear, and the laser
does not selectively address either spin state due to the sub-linewidth splitting.
Figure 5.14 shows an emission spectrum under an applied magnetic field of 29 mT. The
solid curves are simulations with different OH field distributions. The red curve shows
88 Hyperfine Effects in Resonance Fluorescence
- 0 . 5 0 . 0 0 . 50 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0
 
 
3 8  m T
0
RF 
spe
ctru
m (
a.u
)
D e t u n i n g  ( G H z )
B n u c x
B n u c z
- B e x t0
2 2  m T
Fig. 5.14 Low power X- emission spectrum (hollow circles) measured with an applied field
of 29 mT in the Faraday configuration. The red curve is a simulation assuming an isotropic
OH field with a dispersion of 22 mT, equivalent to the best fit value at zero field. The green
curve is a simulation for a mean OH field equal and opposite to the applied field, with a
dispersion of 22 mT. The blue curve shows a simulation assuming an anisotropic OH field
distribution with a mean equal and opposite to the applied field, an in-plane dispersion of
22 mT and an out-of-plane dispersion of 38 mT. The inset shows the OH field distributions
assumed in the corresponding simulations.
5.4 Nuclear Spin Polarisation under Low Magnetic Fields 89
the calculated spectrum for an isotropic distribution, with the same dispersion as the best
fit value for zero-field and no nuclear polarisation. Here, the increase in electron Zeeman
splitting can be seen with an increased splitting of the Raman sidebands. The green curve
shows the calculated spectrum for an isotropic distribution with the same dispersion, but
with a mean polarisation equal and opposite to the applied B-field, in agreement with Fig.
5.13 C. In this case, the Raman detuning matches the data, but the relative Raman sideband
intensity is over-estimated. This indicates that the time-averaged Zeeman splitting fits the
data, but the fraction of time that the OH field spends in-plane is overestimated, pointing to
an increase in fluctuations in the out-of-plane direction. The blue curve is obtained by lifting
the requirement of isotropy, and assuming an out-of-plane dispersion larger than in-plane.
We find an in-plane dispersion of 22 mT, unchanged from the zero-field value, and an out-of-
plane dispersion of 38 mT fits the data best. The increased out-of-plane fluctuations, possibly
linked to the nuclear spin polarisation, cause the time-averaged branching ratio to decrease
and the Raman contribution decreases accordingly. The data are obtained by averaging over
a few single FP scans, and the signal-to-noise ratio obtained clearly distinguishes between
the different models. Typically, we obtain standard errors of 1-3 mT for the dispersion in
both orientations, and the model with an isotropic distribution is clearly inconsistent with the
data.
The Raman transitions are linked to the flip-flop term of the Hamiltonian and are therefore
accompanied by a nuclear spin flip. In principle, this effect does provide a mechanism for
creation of nuclear spin polarisation by pumping the diagonal transition [108]. However,
we operate in a regime where the electron spin states are not resolved in absorption and we
therefore should have approximately equal spin pumping rates in both directions through
this mechanism. This would lead to laser power-dependent dynamics but shouldn’t lead to a
mean nuclear spin polarisation.
A qualitatively different form of nuclear spin polarisation under resonant excitation has
previously been observed, dubbed the ’dragging’ effect [26, 31]. The name arises from
the striking lineshapes observed for neutral and charged excitons in the Faraday and Voigt
geometries. Under an applied B-field larger than 1 T, scanning the laser frequency across the
blue Zeeman transition leads to a flat-top spectrum with a width up to an order of magnitude
larger than the zero-field value. Additionally, this effect is bi-directional indicating it is not
driven by electron spin pumping. The nuclear spin polarisation is mediated by the electron
spin without flipping it. The non-collinear interaction, arising either from the hole hyperfine
interaction [33] or from the quadrupolar field in highly strained QDs [32], leads to a laser
detuning-dependent polarisation rate which provides a feedback mechanism locking the QD
transition on resonance. It is not clear whether the same mechanism underlies the nuclear
90 Hyperfine Effects in Resonance Fluorescence
spin polarisation observed here, but the fact that both observations occur under resonant
excitation with linearly polarised light indicates a possible link. However, the absorption
scans of Fig. 5.2, measured under the same magnetic field as Fig. 5.14, show no such
’dragging’ feature.
In summary, we note that the measurement of low-power emission spectra presented here
offers a high-resolution probe of nuclear spin polarisation processes in the low magnetic
field regime, allowing access to in-plane and out-of-plane dispersions simultaneously. This
method could allow for the study of nuclear spin polarisation processes extended into the
low B-field regime.
Chapter 6
Hyperfine-Assisted Coherent Population
Trapping of an Electron Spin
One of the most important considerations for the experimental realisation of quantum
information processing is the interaction between the qubit and its environment. Coupling
to an uncontrolled and fluctuating environment leads to loss of information stored within
the qubit. However, we also need to control and measure the qubit, which requires a finite
coupling to the outside world, in our case through a bosonic reservoir (the field collected with
our microscope). In practice it is hard, but feasible, to isolate the qubit from selected parts of
its environment. One can directly engineer a qubit such that its coupling to a subset of its
environment is minimised [109], or one can use pulse schemes which isolate the qubit from a
given spectral region of environmental noise [25, 110]. Alternatively, the environment can be
addressed and controlled directly, in order to either limit its fluctuations [31, 111] or change
its spectral properties such that it no longer affects the qubit [112, 113]. Much research effort
has, in parallel, focused on developing the environment of a qubit as a resource, which can
potentially allow for dissipation-driven quantum computation [114–117] and coupling of
distant qubits [118, 119]
In the previous chapter, we have demonstrated that, in the limit of low magnetic field, the
hyperfine coupling with the mesoscopic nuclear spin bath provides a quasi-static quantisation
axis. In addition, the random orientation of the effective magnetic field provides access
to spin-flip Raman transitions. In this chapter, we use this hyperfine-generated Λ-scheme
to optically create electron spin superpositions via two-colour resonant excitation, through
coherent population trapping (CPT). We further show that control of the optical phase, along
with laser powers, allows for arbitrary coherent spin state preparation in the basis dictated by
the Overhauser field.
92 Hyperfine-Assisted Coherent Population Trapping of an Electron Spin
The results shown in this chapter are adapted from Ref. [96], and were taken in collab-
oration with other members of the QD team, particularly Carsten Schulte, Claire Le Gall,
and Clemens Matthiesen. Jake M Taylor performed the theoretical analysis of two-colour
excitation of the trion system, and his input was central in devising the experiments and
understanding its results. Other members of the lab also deserve acknowledgement for
tolerating our use of all the lab wavemeters for sustained periods of time.
6.1 Theory of Coherent Population Trapping
In the first part of this section, we will outline the basic concept of CPT with a three-level
system and how this effect is typically measured. Of course, the trion system under study
here is a four level system with random selection rules, and the theoretical methods involved
for two-colour excitation of such a system will be discussed briefly in the second part of the
section.
6.1.1 Two-colour Excitation of an Ideal Three-Level System
The original and simplest system studied for CPT [120] is a three-level system, arranged in a
Λ-configuration, as shown in Fig. 6.1. Two relatively long-lived ground states are connected
to a single excited state through two electric dipole transitions. Two lasers each excite a
separate optical transition near-resonantly, and are assumed not to address the other transition.
Experimentally, this can easily be achieved by using transitions with orthogonal polarisations,
or with a large enough ground state splitting such that the detuned excitation can safely be
ignored.
Here we work in the following basis:
|1⟩=
 10
0
 ; |2⟩=
 01
0
 ; |3⟩=
 00
1
 (6.1)
The Hamiltonian describing the level scheme shown in 6.1 corresponds to the following:
Hˆ = h¯

ωGS
2 0
Ωa
2 e
iωat
0 −ωGS2 Ωb2 ei(ωbt+φ)
Ωa
2 e
−iωat Ωb
2 e
−i(ωbt+φ) ωX
 (6.2)
where h¯ωGS is the ground state energy splitting, h¯ωX is the excited state energy, φ is a relative
phase between the fields driving transitions a and b of frequencies ωa and ωb respectively. In
6.1 Theory of Coherent Population Trapping 93
Fig. 6.1 Illustration of a three-level system interacting with two near-resonant electric fields,
each addressing a dipole-allowed optical transition. Ωa,b is the Rabi frequency of transition
a,b. Γa,b are incoherent decay rates describing the spontaneous emission from the excited
state to states |1⟩ and |2⟩. The total radiative decay rate from state |3⟩ is then Γ= Γa+Γb.
ΓGS is the incoherent relaxation rate between ground states. In this approximation of the
trion system, the ground state splitting is given by the OH field magnitude BN .
94 Hyperfine-Assisted Coherent Population Trapping of an Electron Spin
order to see that such a two-colour excitation scheme leads to the existence of states which
do not couple to the excited state, we will first ignore all relaxation mechanisms. A pure state
of the system, written in the basis of bare states is then:
|Ψ⟩=
 AB
C
 (6.3)
Its dynamics, neglecting irreversible dynamics caused by coupling to the environment,
are given by the time-dependent Schrodinger equation leading to three coupled differ-
ential equations for each component of the state. Working in the interaction picture
(A = A˜ei
ωGS
2 t ,B = B˜e−i
ωGS
2 t ,C = C˜e−iωX t), we obtain:
˙˜A =− iΩa
2
eiδatC˜
˙˜B =− iΩb
2
eiφeiδbtC˜
˙˜C =− i
2
(
Ωae−iδat A˜+Ωbe−iδbt B˜e−iφ
)
(6.4)
where δa,b is the detuning of laser a,b from the transition |1⟩ → |3⟩ , |2⟩ → |3⟩. Considering
the third equation at the two-photon resonance condition (TPR) δa− δb = 0, it is clear
that for a certain combination of A˜ and B˜, the excited state population will no longer be
time dependent. That is, for an initial ground state superposition fulfilling the condition
ΩaA˜+ΩbB˜e−iφ = 0, there will be no optical excitation. This state is called the dark state in
such a system and can be written:
|Ψ⟩= Ωb |1⟩−Ωae
iφ |2⟩√
Ω2a+Ω2b
(6.5)
Spontaneous emission and ground state decay, which we have neglected in the above,
will change the situation slightly. However, in general there will still be a ground state
superposition which will be much longer lived than its orthogonal state. In other words, we
can rewrite the system such that an effective driving field only addresses one ground state,
in which case we have fast optical pumping into the dark state. In the dressed ground state
manifold, we effectively create an imbalance between spin decay rates: in one direction it is
linked to the optical pumping rate and in the other to the ground state decoherence. Thus,
for a system with long coherence times, we can initialise the state on a timescale of order
6.1 Theory of Coherent Population Trapping 95
0 . 0 2 0 . 0 4
 
 
γ G S  =  1  Γ          0 . 1  Γ          0 . 0 0 1  Γ   
| ρ3 3 |
- 2 0 2
- 2
0
2
 
 
δa  ( Γ )
δ b (
Γ)
Fig. 6.2 Calculated excited state population for two-colour excitation of the three-level
system, as a function of the laser detunings from resonance. The lasers each have a driving
power of s = 0.1, and negligible ground state relaxation and dephasing is assumed. The right
panel shows a linecut of the left panel along the red dashed line, for different ground state
pure dephasing rates, as shown in the legend.
Γ with high fidelity. The lifetime of the created dark state depends critically on the ground
state decoherence, as it is a coherent superposition state.
Typically, the presence of CPT is observed experimentally as a sharp spectral feature,
namely a drop in absorption when the lasers satisfy the TPR condition. To model a realistic
three-level system, with excited and ground state relaxation and pure dephasing, we numeri-
cally solve the Bloch equations in the steady-state. The calculated excited state population
as a function of pump and probe detunings is shown in Fig. 6.2, in the case of low-power
excitation of s= 0.1 for both lasers. The creation of a dark state can clearly be seen as a sharp
drop in absorption along the diagonal, where the TPR condition is fulfilled. Furthermore,
the visibility of the dip and its width depend sensitively on the ground state coherence,
and for a pure dephasing rate equal to the radiative decay rate, the dip disappears. In the
dressed ground state manifold, the effective spin relaxation rates in both directions become
commensurate, and no net spin polarisation is observed.
However, the predicted two-colour spectrum depends critically on driving power. The
situation becomes more complex when the pump Rabi frequency is larger than the radiative
decay rate, in which case the two-level system addressed by the pump is dressed. The
absorption spectrum of the probe will then exhibit a splitting proportional to the pump
Rabi frequency. This effect, known as the Autler-Townes splitting, has been observed
96 Hyperfine-Assisted Coherent Population Trapping of an Electron Spin
- 2 - 1 0 1 2
- 2
- 1
0
1
2
 
 
δa  ( Γ )
δ b (
Γ)
0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8
 
 
| ρ3 3 |
Ωb  =  0 . 2 3  Γ         0 . 7 1  Γ         2 . 2 3  Γ   
Fig. 6.3 Calculated excited state population for two-colour excitation of the three-level
system, as a function of the laser detunings from resonance. The laser addressing transition a
has a driving power of s = 10, and the laser addressing transition b has a driving power of
s = 0.1. Negligible ground state relaxation and dephasing is assumed. The right panel shows
a linecut of the left panel along the red dashed line, for different Rabi frequencies describing
transition b, as shown in the legend.
previously for the neutral exciton [121, 122] and the charged trion [68] transitions of a QD.
The calculated absorption spectrum for a pump power of s = 10 is shown in Fig. 6.3. When
the pump is on resonance with transition a, the probe absorption spectrum shows a splitting,
with still zero absorption at zero detuning. The zero-detuning dip in this case is caused both
by CPT and the onset of Autler-Townes splitting. An unambiguous demonstration of CPT
therefore requires driving powers well below saturation, where the Autler-Townes splitting is
smaller than the radiative linewidth.
6.1.2 Extension to the Hyperfine-split Trion System
The trion system under consideration is more complex than the simple three-level model
of the previous section. Here we will follow the four-level model discussed in chapter 5
and briefly outline the added complications to the theoretical modeling of CPT in such a
system. Firstly, there is a heavy-hole spin in the excited state, leading to two near-degenerate
states. Secondly, the spin splittings are determined by a randomly evolving OH field and
will generally be smaller than the excited state linewidth. Thirdly, fluctuations in the OH
field orientation lead to random optical selection rules. We therefore have a double Λ-
6.1 Theory of Coherent Population Trapping 97
system where each excited state is in general coupled to both ground states. We also cannot
selectively excite any one branch of the trion system as we do not have spectral or polarisation
selectivity in a time-averaged measurement. All these effects mean the theoretical treatment
of two-colour excitation in this system is not as simple as the previous section suggests. It
is not even clear, a priori, that CPT can occur in this system considering the presence of a
second excited state [123].
Adding a second linearly polarised excitation laser into the system is not as trivial as the
aforementioned three level model. As the OH field evolves randomly, the optical selection
rules in general do not coincide with the laser polarisations, therefore we obtain terms in the
Hamiltonian which oscillate with multiple frequencies. Thus, one of the main difficulties
in analysing this system is the fact that we do not have a rotating frame which allows for
the cancellation of these terms. The theoretical analysis of CPT in the trion system was
performed by Jake M Taylor, and we will therefore only briefly discuss the main results
here. The full analysis is given in the supplementary information of Ref. [96]. By treating
the driving fields as perturbations, the eigenvalues of the ground state effective Hamiltonian
can be calculated. Their imaginary parts correspond to decay rates of the ground state spin
eigenstates, i.e. bright and dark state decay rates. For equal and low laser powers, in the
absence of an external field, these can be written as follows, after removing an overall real
term to both eigenvalues:
ε± ≈−iγ/2±
√
(B−δ )2/4− γ2 sin2(θ)/4 (6.6)
where δ is the angular frequency difference between the two lasers, θ is the angle the OH
field makes with the optical axis, B is the ground state splitting in angular frequency due to
the OH field magnitude, and γ is related to the optically-induced spin flip rate:
γ =
Ω2aΓ
δ 2a +Γ2
(6.7)
where δa is the detuning of laser a from transition |1⟩→ |3⟩. A long-lived dark state therefore
exists for γ2 sin2(θ)> (B−δ )2 as one eigenstate consequently has a longer lifetime than the
other. Furthermore, this dark state is a superposition state in the basis defined by the OH field
orientation.
In order to compare the measurements with theory, the detuning dependent absorption
is calculated by integrating the time-dependent Liouville equation numerically. In general,
the solutions will oscillate at the difference frequency of the lasers, so the mean absorption
98 Hyperfine-Assisted Coherent Population Trapping of an Electron Spin
over several periods is taken as the pseudo steady state. An average over 150 nuclear spin
configurations is then taken to simulate random fluctuations of the OH field.
6.2 Experimental Methods
All the measurements discussed in this chapter deal with two-colour excitation of a Λ-scheme.
This entails some practical complications over the standard resonance fluorescence schemes
used in previous chapters. Specifically, we require two narrow-band resonant lasers to excite
the two branches of the Λ-system and we simultaneously need to detect the QD fluorescence
with a high signal-to-noise ratio. In addition, we want the two lasers to be linearly orthogonal
to each other, such that they address different optical transitions and to avoid interference
effects. For this, we combine the lasers at a polarising beam-splitter before coupling into the
polarisation-maintaining (PM) excitation fibre. The fibre has two stress members within the
cladding in order to break the circular symmetry and induce birefringence within the fibre.
Consequently, there are two preferential input polarisations between which the cross-talk
is small allowing for high polarisation extinction ratio. A half-wave plate is used before
the fibre in order to match the axis of the PM fibre. We measure the polarisation extinction
ratio at the microscope head to be typically ≈ 1000. We use two separate external-cavity
diode lasers, each frequency- and power-stabilised with PID feedback control to ±2MHz
and ±1% respectively. The long-time mutual coherence between the two lasers is limited by
the frequency stabilisation scheme and is around 5 MHz for long measurements [124]. This
value is our frequency resolution in two-colour absorption spectroscopy.
The detection of resonance fluorescence using cross-polarisation of the detection path
is unfeasible with this excitation scheme, as the laser leakage will dominate over the QD
signal. Additionally, the detuning between the lasers will be typically on the order of the
electron spin splitting, which, as seen in the previous chapter, is around 200 MHz. Separating
the two lasers spectrally would therefore require a complex setup, such as a high finesse
cavity or stabilised interferometer. Instead, we detect the QD fluorescence with two different
methods: a slightly modified differential reflection technique, and the phonon-sideband
detection discussed in chapter 4.
For differential reflection with two-colour excitation, we again modulate the gate voltage at a
frequency of order 1 kHz and demodulate the signal obtained on a photodiode. However,
one of the two laser inputs is suppressed by using a linear polariser in the detection path. The
degree of polarisation suppression is typically much worse than with single-laser excitation,
as we do not have a linear polariser in the excitation path and therefore rely on the polarisation
extinction ratio of the PM fibre. This detection method is useful as it allows us to measure
6.3 Hyperfine-assisted Coherent Population Trapping 99
the response to one laser independently of the other. We can therefore perform ’pump-
probe’ experiments which are a standard technique used to demonstrate CPT in systems
with optically resolved ground states [68, 125, 126]. Considering the Λ-system shown in Fig.
6.1, this would give a DR contrast which is proportional to one of the optical coherences,
i.e. ∝ |ρ13|√sprobe if the laser addressing transition |2⟩ → |3⟩ is suppressed. However, if the
pump signal is not completely suppressed, this technique projects both signals onto the
same polarisation and can lead to interference effects. Such effects can lead to spectrally
sharp drops in signal which is also the telltale sign of CPT. Using differential reflection can
therefore lead to ambiguous results, as well as being a relatively slow measurement.
The second technique used is the phonon-sideband detection described in chapter 4. This
method allows us to measure the excited state population with arbitrary flexibility in excitation
frequency and polarisation and is therefore ideally suited to the measurement of CPT with
sub-linewidth spin splitting. Another advantage of this technique over differential reflection
is that it gives access to the QD photons with no laser background and therefore allows for
high-bandwidth correlation measurements.
6.3 Hyperfine-assisted Coherent Population Trapping
In this section, we show two-colour absorption spectroscopy measurements at low magnetic
fields. As discussed previously, two different mechanisms will lead to a similar effect,
namely a drop in absorption in a certain spectral region fulfilling the TPR condition. Figure
6.4 shows two-colour differential reflection measurements taken at combined laser power
approximately equal to the saturation point (panels A and B) and well above it (panels C and
D). The high power data shows that the pump laser dresses part of the trion system, which the
probe sees as a splitting larger than the radiative linewidth in the absorption spectrum. This
is the Autler-Townes splitting, arising from the coherent light-matter interaction. Observant
readers may also note a sharp peak at equal laser frequencies, absent from the theoretical
prediction. This is caused by imperfect laser suppression, and the pump laser leakage is
projected onto the probe polarisation. The sharp feature is thus a heterodyne signal when the
detuning corresponds to the gate modulation frequency. Lowering the laser powers reduces
the Autler-Townes splitting until it is sub-linewidth, and a dip is still observed at the TPR.
This dip is a result of CPT, combined with the onset of an Autler-Townes splitting. This
measurement highlights the importance of working in the low-power regime, to obtain a
signal unambiguously related to CPT alone.
Figure 6.5 shows two-colour measurements for equal laser powers corresponding each
to s = 0.1, detected with the phonon sideband. In this case we expect the Autler-Townes
100 Hyperfine-Assisted Coherent Population Trapping of an Electron Spin
Fig. 6.4 A. Measurement of QD absorption with two-colour resonant excitation detected in
differential reflection. Laser 2 (pump) is suppressed by a cross-polarisation technique. Laser
1 and 2 drive the QD below saturation, at s ≈ 0.3 and s ≈ 0.9. B. Simulation of |ρ23| as a
function of laser detuning for parameters corresponding to A assuming a three level system
with fixed spin splitting. C. Similar measurement to A, but with laser 1 and 2 driving at
higher powers of s≈ 15 and s≈ 0.6. D. Simulation of |ρ23| for parameters corresponding to
panel C, assuming a three level system with fixed spin splitting.
6.3 Hyperfine-assisted Coherent Population Trapping 101
splitting due to either laser to be ≈ 0.22Γ meaning that its effect on the absorption at the
TPR will be smaller than the CPT contribution. At zero-magnetic field, we observe an
increase of QD fluorescence when both lasers are close to resonance, with a spectrally
sharp drop in signal around the TPR. The noise in such measurements is mainly caused by
slow spectral wandering of the central frequency of the transition. This can be confirmed
by comparing the noise properties for symmetrically detuned lasers and equally detuned
lasers. The noise in the former region (upper left and lower right edge of the central peak)
is reduced compared with the latter region (upper right and lower left), consistent with a
noise source leading to a random fluctuations in the central frequency. Panel B shows a four
level simulation for experimental parameters corresponding to panel A. A small degree of
nuclear spin polarisation is added empirically as an added contribution to the electron spin
splitting. A four-level numerical simulation assuming a mean Zeeman splitting of 400 MHz
gives a best fit to the data. In both the data and the simulation, a decrease in fluorescence
along the diagonal corresponding to the TPR indicates preparation of the electron spin in a
coherent dark state. When an external magnetic field is applied (panels C and D) the situation
changes considerably. Four peaks are observed, corresponding to the different permutations
of two lasers addressing two optical transitions. Notably, the upper-left and lower-right peaks,
corresponding to the two lasers addressing different transitions, are more intense than the
other two, corresponding to both lasers addressing the same transition. This is due to optical
pumping of the spin in the latter, which is prevented by optical repumping when the lasers
address different transitions. Furthermore, no sharp decrease of fluorescence is seen on these
peaks, showing that the Raman transitions are suppressed and the fraction of the OH field
distribution giving rise to CPT is reduced.
Figure 6.6 shows the measured QD fluorescence as a function of two-photon detuning
(δ1− δ2)/2 for the condition δ1 + δ2 = 0. This corresponds to a diagonal linecut of the
measurements shown in Fig. 6.5, along the dashed green line. Panels A and B show the
measured absorption at 0 mT and 18.4 mT respectively. The 0 mT curve is an average of two
separate linecuts. A narrow CPT dip is clearly visible in the absence of an applied field, at
the TPR. Two simulations are shown as a solid blue curve and a dashed green curve, obtained
with the four-level and three-level models respectively. In the case of the four-level model,
the measured OH dispersion is used along with a pure dephasing rate of 2π×1 MHz, and the
best fit is obtained for a mean spin splitting of 400 MHz. For the three level model, a fixed
ground state splitting is assumed and a dephasing rate of 2π×2.6 MHz gives a best fit. There
is good qualitative agreement between the models, indicating that the interpretation of the
trion as a Λ-system is valid. The low visibility of the CPT feature is a result of its intermittent
nature: only a fraction of the OH field distribution provides the necessary Λ-scheme for CPT.
102 Hyperfine-Assisted Coherent Population Trapping of an Electron Spin
3 1 4 3 6 0 3 1 4 3 6 13 1 4 3 5 9
3 1 4 3 6 0
3 1 4 3 6 1
 
 
L a s e r  1  f r e q u e n c y  ( G H z )
Las
er 2
 fre
que
ncy
 (G
Hz)
DC
B
3 1 4 3 6 0 3 1 4 3 6 1
3 1 4 3 6 0
3 1 4 3 6 1
 
 
L a s e r  1  f r e q u e n c y  ( G H z )
Las
er 2
 fre
que
ncy
 (G
Hz)
- 0 . 6 - 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4 0 . 6
- 0 . 6
- 0 . 4
- 0 . 2
0 . 0
0 . 2
0 . 4
0 . 6
δ 2
 (G
Hz)
δ1  ( G H z )
- 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4
- 0 . 4
- 0 . 2
0 . 0
0 . 2
0 . 4
δ 2
 (G
Hz)
δ1  ( G H z )
A
Fig. 6.5 A. Measurement of QD absorption with two-colour resonant excitation with phonon
sideband detection. Laser 1 and 2 drive the QD below saturation, both at s ≈ 0.1. B.
Simulation of QD fluorescence as a function of laser detunings for parameters corresponding
to A for a four-level model with a mean ground-state splitting of 400 MHz. C. Similar
measurement to A, but with an applied magnetic field of 52 mT. D. Simulation of QD
fluorescence for parameters corresponding to panel C, assuming a four-level model with a
mean ground-state splitting of 800 MHz.
6.3 Hyperfine-assisted Coherent Population Trapping 103
Fig. 6.6 A. Measured QD fluorescence (red diamonds) as a function of two-photon detuning
for the condition δ1 + δ2 = 0. The blue solid curve shows a simulation for the four level
model with a mean ground state splitting of 400 MHz and a pure spin dephasing rate of
γGS = 2π×1MHz. The effect of electric field is added as a Gaussian distribution of central
frequencies of ≈ 150MHz. The green dashed curve is a simulation for a three level system
with a pure spin dephasing rate of γGS = 2π × 26MHz, and a similar Gaussian central
frequency distribution. B. QD fluorescence as a function of two-photon detuning with the
application of a Faraday-geometry magnetic field of approximately the same magnitude as
the OH field. The dashed curve is a fit to a Voigt lineshape.
Therefore, stable coherent dark states are generated intermittently and sustained for a finite
period of time, dictated by the slow dynamics of the nuclei. This is in contrast to systems
showing low-visibility CPT due to fast pure dephasing of the ground state on the same order
of magnitude as the radiative decay rate [125]. With an applied field comparable to the OH
field dispersion, the CPT dip vanishes as we recover a two-level response. The absorption is
additionally broadened as the magnetic field increases the mean Zeeman splitting. However,
a single-peak Voigt profile fits the data well, suggesting that coherent effects such as CPT no
longer play an important role.
In Fig. 6.6, the effect of CPT is clear from the lineshape when scanning across the TPR.
Similarly to the data shown in chapter 5, we expect the CPT visibility to depend on the
applied magnetic field, due to its effect on the optical selection rules. However, one has to be
careful in the method used to measure the CPT visibility in measurements such as Fig. 6.6.
We measure the CPT visibility from the loss of QD absorption at the TPR when the second
laser is switched on, compared with a purely linear power dependence. For this, we compare
the count rate at the center of the dip at the TPR (corresponding to ω1 ≈ ω2) with the count
rate measured with one of the lasers far detuned. The raw data are shown in panel A of Fig.
104 Hyperfine-Assisted Coherent Population Trapping of an Electron Spin
6.7. A non-monotonic behaviour is seen, with a minimum around 15mT. This loss of signal
arises from three separate mechanisms: the onset of saturation of the transitions even at low
powers, incoherent spin pumping, and CPT. We therefore subtract the first two contributions
in order to extract the corrected CPT visibility (shown in panel B of Fig. 6.7). At 0 T, the
detuning from either transition is comparable to the linewidth, and spin pumping does not
contribute to the absorption sub-linearity, as the optically induced pumping/repumping rates
are far greater than the spin relaxation rate. When a field is applied which starts to split
the lines (Bext > 20 mT), the lasers will preferentially address one spin state when ω1 ≈ ω2
and δa ≈ 0. Therefore, an imbalance is created between optically-induced pumping and
repumping rates. The latter becomes comparable to the spin relaxation rate when the Zeeman
splitting is larger than the radiative linewith, while the pumping rate is still ≈ 26 MHz (see
Fig. 5.3 and 5.4). An increase in power therefore forces the electron to spend more time in
the spin state for which the laser is detuned, leading to an added sub-linearity in the power
dependence. We model this with rate equations to describe the electron spin populations.
This model therefore does not take into account any coherent effect such as CPT. We quantify
the sub-linearity through the loss defined as:
Loss = 1− P1 · I(P2)
P2 · I(P1) (6.8)
where P1 and P2 are the single- and two-laser combined powers, I(P2) is the photon detection
rate at an excitation power of P2. In our measurement we use equal laser powers, hence
P2 = 2 ·P1. A loss of 0 indicates a perfectly linear power dependence, and a loss of 0.5 means
that doubling the combined laser power does not increase the QD absorption. The average
signal for a given power and detuning from both transitions is given by:
I(P) =
T1(P) ·S(1,∆1,P)+T2(P) ·S(2,∆2,P)
T1(P)+T2(P)
(6.9)
where T1(P) is the average time spent in spin state i, and S(i,∆i,P) is the photon detection
rate when the electron is in spin state i (before the spin is pumped into the other state) with
the lasers equally detuned from the corresponding vertical transition (green arrows in Fig.
5.1) by ∆i. Defining Γi(∆i) as the rate at which the spin in state i is pumped (both optically
and through spin relaxation) to the other spin state, the average fraction of time spent in state
i is then:
Ti(∆i) =
1
Γi(∆i)
=
1
L [∆i]ΓOP+ΓGS
(6.10)
6.3 Hyperfine-assisted Coherent Population Trapping 105
HereL [∆] is a Lorentzian profile normalised toL [0] = 1 , ΓOP is the optical spin pumping
rate at zero detuning, and ΓGS is the non-optical spin relaxation rate which is assumed to be
detuning independent and magnetic field independent over the range measured.
Here, both the photon emission rates and the time spent in either spin state depend on the
applied laser power. We assume that both transitions saturate in the same way as a simple
two-level system, i.e. when doubling the laser power, with s being the saturation parameter:
S(1,∆i,P = 2s) =
2s
2s+1
s
s+1
S(1,∆i,P = s) (6.11)
We also assume that the optically induced pumping rates, as they are proportional to the
excited state population, follow the same power dependence and do not depend on B-field on
the scale of tens of mT. We justify this assumption by noting that the decay timescales for 0
T and 76 mT are the same for equal laser powers (c.f. Figs. 5.4 and 5.3).
After integrating over the OH field distribution, and using the parameter for ΓOP obtained
from Fig. 5.4, we are left with one free parameter: ΓGS. Fig. 6.7 shows simulations for
different spin relaxation rates alongside the raw data. These values are consistent with the
spin relaxation rate observed for a different QD on the same sample. The best fit for this
dot gives a ground state relaxation rate of 4.7 kHz. We then subtract the corresponding
simulation from the raw data to obtain the corrected CPT visibility shown in panel B of
Fig. 6.7. This shows an exponential decay on a scale smaller than the OH field dispersion,
qualitatively consistent with the suppression of the Raman transitions through the application
of a Faraday-geometry magnetic field seen in Fig. 5.13. However, both decays are highly
sensitive on the degree of dynamic nuclear spin polarisation, which could vary significantly
between a single-laser measurement with fixed detuning and a two-colour measurement
with scanning lasers. Therefore, we do not necessarily expect the same decay scale for both
measurements.
Finally, the dependence of this feature on electron spin coherence is probed straight-
forwardly by comparing measurements taken at the center of the charging plateau and at
at the plateau edge, where cotunneling rates are enhanced, thereby significantly reducing
spin coherence. Figure 6.8 shows two-colour differential reflection measurements, where
we use low-power pump and probe laser excitation (s = 0.05 for each laser). The probe is
kept on resonance and the suppressed pump is scanned across the resonance. The contrast
measured is proportional to |ρ13|√sprobe and a drop in contrast indicates a sub-linearity of the
response (c.f Fig. 3.4). In the center of the charging plateau, the contrast drops to ≈ 83%
of the single-laser value at the TPR condition, whereas in the cotunneling regime it drops
to ≈ 96%, corresponding to the expected saturation behaviour for these powers. For the
106 Hyperfine-Assisted Coherent Population Trapping of an Electron Spin
B
0 1 0 2 0 3 0
0 . 0
0 . 1
0 . 2
 
 
CP
T v
isib
ility
B - f i e l d  ( m T )
∆ΒN U C
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0
0 . 1
0 . 2
0 . 3
 
 
Los
s in
 sig
nal
B - f i e l d  ( m T )
1 0  k H z
4 . 7  k H z
1  k H z
s a t u r a t i o n
A
Fig. 6.7 A. Sub-linearity in QD fluorescence (grey circles) measured at equal laser frequencies.
The blue, light green and dark green curves are simulations taking into account incoherent
spin pumping and saturation for different values of spin relaxation rate shown as labelled. The
dashed grey line shows the B-field-independent contribution from saturation alone. The red
area is an exponential fit and represents the loss in signal not accounted for by saturation and
spin pumping, corresponding to CPT. B. Decay of the CPT visibility obtained by subtracting
the light green curve in panel A from the data. The blue curve shows an exponential fit, and
the measured OH field dispersion to within ±1σ is shown as an orange area. The error bars
are calculated by propagation of the min-max values of count rates obtain at the TPR, and
the standard error in best fit amplitudes of fits to single-laser lineshapes.
6.3 Hyperfine-assisted Coherent Population Trapping 107
- 0 . 6 5 - 0 . 6 0 - 0 . 5 5
3 1 4 3 4 5
3 1 4 3 5 0
3 1 4 3 5 5
- 0 . 6 5 - 0 . 6 0 - 0 . 5 5
 
∆ν  =  0 . 5  G H z ∆ν  =  1 . 1  G H z
Las
er f
req
uen
cy (
GH
z) 3 . 1 4 4 E 5
G a t e  v o l t a g e  ( V )
3 . 1 4 3 E 5
0 . 0 0 0
0 . 2 0 8 0
0 . 4 1 6 0
0 . 0
0 . 2
0 . 4
 Co
ntra
st (
%)
 Co
ntra
st (
%)
- 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 1 . 50 . 8
0 . 9
1 . 0
S a t u r a t i o n
 
 
B e x t  =  2 . 6 m TDR
 sig
nal 
(a.u
.)
P u m p  l a s e r  d e t u n i n g  ( G H z )
C P Tc o n t r i b u t i o n
BA
Fig. 6.8 A. Single-laser differential reflection measurement for different applied gate voltages
near the cotunneling region. The left panel shows two scans taken far from (blue circles)
and close to (red circles) the cotunneling region, at laser frequencies corresponding to the
blue and red lines in the bottom panel. The solid curves are Voigt fits with best fit FWHM as
displayed. B. Two-colour differential reflection measurement as a function of pump laser
detuning for the two gate voltage positions shown in panel A. Both pump and probe laser
are set to s = 0.05, and the pump reflection is suppressed. The curves are normalised to the
mean value far away from pump resonance. The upper dashed grey line shows the expected
loss in signal arising from saturation alone at these driving powers.
108 Hyperfine-Assisted Coherent Population Trapping of an Electron Spin
magnetic field used, we wouldn’t expect optical spin pumping to contribute significantly
to the sub-linearity. Therefore, this measurement indicates that the extra term leading to a
sub-linearity is linked to the electron spin coherence, consistent with hyperfine-enabled CPT.
6.4 Spin Control via Phase Modulation
In the previous section, we have shown that the hyperfine interaction allows for the steady-
state preparation of electron spin superpositions of the form:
|ψ⟩ ∝ α |↑, nˆ(t)⟩−βeiφ |↓, nˆ(t)⟩ (6.12)
where the unit vector nˆ(t) is defined by the instantaneous orientation of the adiabatically
evolving OH field, ↑ and ↓ are electron spin projections along nˆ(t), α and β are effective
Rabi frequencies and φ depends on the relative phase between the lasers. This state and its
orthogonal ground state span the dressed basis. It is well known that the polar angle of the
dark state within the ground state Bloch sphere is given by the ratio of Rabi frequencies. This
fact forms the basis of stimulated Raman adiabatic passage [127]. However, the created dark
state also depends on the relative phase between the two driving fields, which determines the
azimuthal angle within the Bloch sphere. Adiabatically changing the phase then coherently
rotates the electron spin in the OH field basis. However, we do not have access to high
fidelity spin readout with such low magnetic fields and therefore cannot demonstrate such
coherent rotations. Instead, to expose the optical phase dependence of the dark state, we
impose a non-adiabatic phase jump after preparing a dark state and measure its effect on the
time-resolved QD fluorescence. This rapid phase manipulation leads to a new rotated dressed
basis in which the electron spin gains a finite bright state component, thereby generating
fluorescence. Subsequent photon emission pumps the electron spin into the new dark state.
Figure 6.9 shows the experimental setup used to evidence the phase dependence of
the dark state. In order to maximise the mutual coherence of the driving fields, we use a
single laser which we separate into two beams. One beam is sent through an acousto-optic
modulator (AOM). We use the first order diffraction, which is frequency shifted by the AOM
driving frequency (80MHz). The second beam is sent through an electro-optic modulator
(EOM) which provides a phase offset proportional to the applied voltage. The EOM used has
a bandwidth of ≈ 20 GHz, meaning that in practice our experimental bandwidth is limited
by the voltage supply. The two beams are then combined on a polarising beam-splitter and
sent to the cryostat. Voltage pulses with a rise-time of ≈ 2 ns are sent to the EOM in order
to impart a rapid phase jump. In the absence of an external magnetic field, the selection
6.4 Spin Control via Phase Modulation 109
Fig. 6.9 Experimental setup used to measure time-resolved correlations between fast phase
shifts and QD absorption. A single laser is split into two paths, one frequency shifted with an
acousto-optic modulator (AOM), and the other phase-shifted with an electro-optic modulator
(EOM). Single photon detection events are correlated with voltage pulses sent to the EOM.
PBS: polarising beam-splitter; APD: avalanche photodiode; λ/2: half-wave plate.
rules evolve according to the OH field, therefore the lasers do not address a single transition
selectively and a relative phase of the lasers is not necessarily mapped directly to a relative
phase of the Rabi frequencies. We therefore apply a small (8.4 mT) magnetic field to split the
excited states and ensure that the applied phase shift affects only one of the Rabi frequencies.
The added contribution to the electron spin splitting then roughly matches the detuning
between lasers, and the TPR condition is approximately met.
Figure 6.10 shows an example time-resolved fluorescence measurement, correlated with
a rapid phase shift for laser powers of sEOM = 0.2 and sAOM = 0.05. In this case most of
the response is due to the phase-controlled laser, and we can therefore take a comparable
background measurement by blocking the frequency-shifted laser. However, the imbalance
in powers leads to dark states with a finite z-projection (in the OH dictated basis), which will
consequently limit the maximum bright state overlap induced by the phase pulse.
Two features can be observed in the measurement: a sharp dip correlated with the falling
edge of the phase pulse, and a small increase in fluorescence thereafter. The former effect is
due to the finite fall-time of the voltage pulses sent to the EOM, leading to a frequency shift
(middle panel) which tunes the laser off-resonance for a finite time. After the phase pulse,
we see an increase in fluorescence which decays on a timescale of order 10 ns, corresponding
110 Hyperfine-Assisted Coherent Population Trapping of an Electron Spin
N
or
m
al
is
ed
 c
oi
nc
id
en
ce
s
0.94
0.96
0.98
1.00
1.02
0 20 40
Delay (ns)
back-
ground %
signal %
-1
0
1
2
Ap
pl
ie
d
vo
lta
ge
 (V
)
-1
0
∂φ
/∂
t (
t)
(a
.u
.) 
rapid
phase
change
phase-controlled laser only
both lasers
Fig. 6.10 Example time-resolved QD fluorescence measurement (lower panel) with the AOM
path blocked (unblocked) shown as blue (red) columns. Both measurements are normalised
to their long-time mean, but the red data are offset for clarity. The upper panel shows the
measured voltage pulse sent to the EOM showing a fall time of ≈ 2 ns. The middle panel
shows the derivative of the upper panel and corresponds to an instantaneous frequency shift
of the driving laser, which correlates with a decrease in QD fluorescence. After the ≈ 2 ns
long dip, the data in the bottom panel are fit with an exponential decay, for which the best fit
amplitude quantifies the intermittent fluorescence intensity.
6.4 Spin Control via Phase Modulation 111
to a few optical cycles at this power. An increase in fluorescence on this timescale is not
seen in the single-laser reference measurement, indicating that the feature is indeed related
to the creation of a bright state component. A smaller increase in fluorescence decaying
on a significantly longer timescale is observed in the reference measurement, related to the
EOM response to the voltage pulses. We fit both curves, starting ≈ 2 ns after the dip, with
exponential decays and record the best fit amplitude. We subtract the single-laser value from
the two-laser measurement to quantify the intermittent fluorescence amplitude linked to the
creation of a bright state component.
Figure 6.11 shows a summary of extracted best fit amplitudes for the two-laser and single-
laser measurements. The single-laser measurements show a quadratic dependence on voltage
pulse amplitude, suggesting an effect linearly dependent on driving power. In contrast, the
two-laser measurement shows a consistently higher signal with a voltage dependence which is
not a simple quadratic. Subtracting the quadratic fit to the reference measurements we obtain
the background-subtracted data shown in panel C. The non-monotonic dependence on pulse
amplitude is well fit with a sinusoidal behaviour, which is consistent with phase-induced
rotations of the dressed basis. The maximum signal of ≈ 0.75% results from a combination
of low CPT visibility in a time-averaged measurement (≈ 17%) which is further reduced
with the application of a small magnetic field (≈ 3−5% at 8.4 mT, c.f. Fig. 6.7) and the
imbalanced power ratio reducing the maximum phase-induced overlap with the bright state.
Furthermore, the sharp dip caused by the finite fall-time of the EOM obscures the peak in
fluorescence leading to a further decrease in signal. Panel A shows a sketch of the effect of
phase pulses in this system. First the spin is initialised into the dark state, and the phase pulse
leads to a rotation of the bright/dark basis (blue line) about the z-axis in the OH dictated
quantisation axis. The electron spin does not follow the rotation adiabatically and therefore
has a finite overlap with the new bright state. Thereafter, the QD fluoresces with an intensity
proportional to this bright state component and is pumped into the new dark state within a
few optical cycles. The protocol therefore allows for incoherent rotation of the prepared spin
state, within the basis given by the OH field. Along with control of relative laser powers, this
gives access to arbitrary spin state preparation in the nuclear basis. We note that, although
arbitrary preparation of spin states can be done in the Bloch sphere given by nˆ(t), we do not
know the instantaneous orientation of the OH field. However, this lack of knowledge does
not lead to dephasing in a time-averaged protocol, in contrast to working in a fixed lab frame,
as the OH field evolves quasi-statically.
112 Hyperfine-Assisted Coherent Population Trapping of an Electron Spin
CB
0 1 2
0 . 0
0 . 5
1 . 0
Inte
rmi
tten
t flu
ore
sce
nce
 am
plitu
de 
(%)
E O M  p u l s e  a m p l i t u d e  ( V / V pi)
0 1 2 3 4 5
0
1
 
Inte
rmi
tten
t flu
ore
sce
nce
 am
plitu
de 
(%)
P u l s e  a m p l i t u d e  ( V )
 t w o - l a s e r s i n g l e - l a s e r
A
Fig. 6.11 A. Representation of the effect of a laser phase shift on the electron spin state (red
arrow) in the Bloch sphere representing the ground state manifold. Before the phase shift, the
electron spin is prepared in the dark state. The phase shift rotates the dark/bright basis (blue
line) in the bare basis. The electron is then projected onto the new dark/bright basis, and the
fluorescence intensity is proportional to its overlap with the new bright state. B. Summary
of intermittent fluorescence amplitude for different Voltage pulse amplitudes. The orange
circles show the data with both paths unblocked, and the blue triangles show the data with the
AOM path blocked. The dashed curve is a quadratic fit to the data. C. Background-corrected
intermittent fluorescence amplitude, obtained by subtracting the dashed curve from the
orange circles in panel B. The x-axis is rescaled such that the sine fit has a period of 2π . The
error bars represent the standard deviation in correlation rates and are limited by shot noise.
Chapter 7
Conclusion
7.1 Summary and Outlook
In this thesis, we have discussed the impact of the environment on the spin and optical
properties of a solid-state atom-like system. The results presented in chapter 3 show that, in
many respects, the neutral exciton transition of a QD behaves much like an ideal two-level
atomic transition. Many of the predictions of resonance fluorescence from a well-isolated
quantum emitter are verified in this mesoscopic solid-state emitter, namely the transition
from coherent to incoherent scattering regimes along with coherent Rabi oscillations and
robust photon antibunching.
Two inconsistencies with the two-level picture emerge due to interactions with the solid-
state environment. Firstly, charge fluctuations in the host semiconductor matrix lead to
a time-averaged random distribution of transition frequencies. Secondly, exciton-phonon
coupling leads to phonon-assisted fluorescence. These effects lead to broadening of the
emission spectrum and consequently to photonic distinguishability, posing a challenge in
view of using QDs as spin-photon interfaces. In chapter 4 we demonstrated a technique which
makes use of the phonon-assisted emission as a probe of frequency fluctuations caused by
charge dynamics in the environment. This probe is then used to narrow the zero-phonon line
by feedback stabilisation on the gate voltage. The inhomogeneous broadening on a sub-Hz
timescale is reduced by at least a factor of 7, which should allow a significant improvement in
fidelities of operations requiring photonic interference from different emitters in uncorrelated
environments. Furthermore, the intensity noise characteristics in resonance fluorescence are
improved by a factor of 22 with a stabilisation bandwidth of 190 Hz.
However, there are several noise sources affecting the QD transition at faster timescales
[73, 77, 87]. In order to obtain transform-limited emission, a stabilisation bandwidth of
≈ 50 kHz [73] would be required. The stabilisation bandwidth of the protocol shown here
114 Conclusion
is limited both by the collection efficiency and by the gate bandwidth. Electrical gates
with bandwidths up to GHz have been demonstrated [94] in samples with similar optically
active QDs. The limiting factor then is photon extraction efficiency from the sample. In the
sample used here, with a cubic Zirconia SIL and a DBR below the QD layer, the fraction
of emission collected by the objective lens is around 6%. Fortunately, the MBE process
required for the growth of optically-active QDs is readily compatible with the creation of
complex heterostructures and lithography techniques. Various cavity QED structures have
been demonstrated with QD emitters [78, 79, 128, 129], which can improve photon extraction
efficiencies while being compatible with a gated structure [130]. Improvements in collection
efficiency in a broadband structure are also possible [93], with collection efficiencies up to
≈ 99% predicted with a dielectric antenna structure [91, 92].
Such an improvement in collection efficiency would certainly allow for the complete suppres-
sion of charge noise affecting the QD emission. We can speculate on whether or not nuclear
spin fluctuations can also be fully suppressed with this protocol, assuming certain technical
advancements. Firstly, moving from the Si APDs used here with detection efficiencies of
≈ 20% to superconducting nanowire single photon detectors with detection efficiencies reach-
ing 90% [131] would provide a four-fold improvement in stabilisation bandwidth. We can
further speculate that an order of magnitude improvement in collection of the QD emission
into the objective may be feasible with a dielectric antenna structure. Consequently, phonon-
assisted stabilisation bandwidths of order 10 kHz may be possible. Further improvement
could be obtained through a higher modulation amplitude or by sampling part of the ZPL,
although both of these would lead to a loss of indistinguishable QD emission. It therefore
seems that such a stabilisation scheme may be able to correct for the slowest components of
nuclear spin noise, arising from nuclear spin diffusion outside the QD [28] on a timescale of
ms, but the fastest timescales linked to nuclear spin precession in the Knight field (< 100 µs)
may be out of reach.
In chapter 5, we showed first and second order coherence measurements of resonance
fluorescence from a negatively charged QD, in the low magnetic field regime. The second-
order correlation shows bunching on a timescale of tens of ns, with a sub-linewidth detuning
dependence. These observations point towards a hyperfine electron spin splitting in the
absence of a magnetic field. Furthermore, the optical spin pumping timescale of tens of
ns shows that the Faraday-like selection rules are broken by the hyperfine interaction, and
spin-flip Raman transitions are dipole allowed. Emission spectra in the low power limit show
deviations from a simple two-level picture. In addition to a resolution-limited peak at the
laser frequency, we observe symmetrically detuned sidebands arising from coherent Raman
7.1 Summary and Outlook 115
transitions in the hyperfine-split four level system. The shape and size of these sidebands
relative to the elastic peak provide a direct measurement of the time-averaged OH field
distribution. In the absence of a magnetic field, this distribution is isotropic and Gaussian
with a dispersion of ≈ 20 mT depending on the QD. The signal-to-noise ratio obtained in
these measurements gives a standard error of ±1 mT using the OH field dispersion as a
fitting parameter. Under applied magnetic fields commensurate with the OH field dispersion,
we further observe indications of nuclear spin polarisation counteracting the effect of the
magnetic field on the electron spin splitting, in combination with increased nuclear spin
fluctuations in the direction of the applied field. Using coherent Raman spectroscopy allows
the measurement of fine features such as small degrees of anisotropy as well as sub-linewidth
nuclear spin polarisation. This technique therefore has applications in the further study of
nuclear spin physics in the low-magnetic field regime.
On a more speculative note, a high-resolution measurement of the coherent Raman sideband
could serve as a form of nuclear spin narrowing. By conditioning on the detection of a photon
at a given Raman detuning, a subsequent experiment relying on the electron spin coherence
can be significantly improved. Indeed, the electron spin inhomogeneous dephasing time T∗2
is inversely related to the time-averaged dispersion of the OH field distribution. The question
then is: how much can we narrow the OH field distribution by conditioning on a photon
detection for a given FP width? In a ’frozen’ nuclear spin configuration, the width of the
Raman sideband is given by a combination of the FP linewidth and the optically-induced spin
relaxation. Using low driving powers is then beneficial, but also reduces the detection rate of
conditioning photons. The best FP alignment obtained here gives a transmission FWHM of
10 MHz, compared to an optically-induced relaxation rate of ≈ 2π×4 MHz for a driving
power of s = 0.1 (corresponding to the bunching decay rate of Fig 5.4). Conditioning on a
Raman photon therefore reduces the OH field FWHM from ≈ 200 MHz to ≈ 10 MHz, as
shown in Fig. 7.1 A. Such a conditioning measurement would also prepare an anisotropic
distribution with an orientation predominantly in-plane, therefore providing a Lambda system
with well-known ground state splitting. The OH field correlation time is of order 10 µs,
much longer than the dark state initialisation time of order 10 ns (c.f. decay timescale of Fig.
6.10). Therefore, such a conditioning measurement should therefore significantly increase
the visibility of subsequent coherent population trapping of the electron spin. Panel B shows
a comparison between three-level CPT calculations for two different ground state dephasing
rates. The green curve corresponds to the best fit dephasing rate fitting the coherent popu-
lation trapping data of Fig. 6.6, corresponding to an empirical dephasing rate of 2π × 26
MHz, and the orange curve is the calculation with a 20-fold reduction in dephasing rate, rep-
resenting the 20-fold reduction in OH field dispersion due to the conditioning measurement.
116 Conclusion
B
- 1 0 10 . 0 0
0 . 0 1
0 . 0 2
0 . 0 3
0 . 0 4  
 ρ 33
( δ1 −δ2 )/2 ( Γ )
 F i l t e r e d U n f i l t e r e d
0 2 0 0 4 0 00 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0
 
 
Unn
orm
alis
ed 
pro
bab
ility
E l e c t r o n  s p i n  s p l i t t i n g  ( M H z )
 F i l t e r e d U n f i l t e r e d
A
Fig. 7.1 A. The green curve shows the un-normalised ground state splitting distribution
corresponding to the measured OH field dispersion of chapter 5. The orange curve shows the
expected distribution after conditioning on the detection of a Raman photon for a FP cavity
detuned by 220 MHz from the driving laser and a FP FWHM of 10 MHz. B. Simulated
two-colour absorption spectrum as a function of two-photon detuning. The green curve is
obtained with a ground state pure dephasing rate of 2.6 MHz, corresponding to the best fit of
the data in chapter 6. The orange curve shows a simulation for a dephasing rate 20 times
smaller, corresponding to the extent of the narrowing of the distribution in A.
Here we have assumed that the best fit dephasing rate of the three-level model is proportional
to the OH field dispersion. The CPT visibility, and consequently the preparation fidelity,
is predicted to increase significantly. As discussed in chapter 6, the three level model is
incomplete for the trion system and we stress that the validity of the prediction of Fig. 7.1 is
therefore limited. Indeed, this prediction ignores possible nuclear spin polarisation as well as
the effect of the conditioning on the selection rules. Nevertheless, testing such a prediction
would provide an interesting extension to the work shown in this thesis.
In chapter 6, we have shown that the hyperfine interaction provides a quantisation axis
for the electron spin and adequate selection rules for optical spin control. Through two-
colour excitation of the trion transition, we observe a sharp decrease in absorption at the
two-photon resonance which is the spectroscopic signature of coherent population trapping.
Furthermore, we have shown that control of the relative phase of the two driving fields allows
the initialisation of the electron spin in an arbitrary superposition state in the basis dictated
by the OH field. For an example of a possible extension of the work shown in chapter 6, in
the following section we describe the proposal of Ref. [132] for generating long strings of
entangled photons using the trion system in the low magnetic field regime.
7.2 Towards a Photonic Cluster-state Machine Gun 117
7.2 Towards a Photonic Cluster-state Machine Gun
The DiVicenzo criteria mentioned in chapter 1 describe the necessary conditions a qubit must
satisfy in order to provide an adequate building block for a quantum computer. These criteria
explicitly follow the circuit model of quantum computation. Another model of quantum
computation has been developed over the past decade based on using strings of photons
entangled in so-called cluster states [133], or more generally graph states [48]. In this
protocol, quantum computation can be achieved by measuring single photons from the cluster
state, the result of which is fed forward to choose the basis for the following measurement.
This protocol is called one-way quantum computation as the initial resource is destroyed
throughout the process. One-way computation, including classical feed-forward, has been
demonstrated experimentally [134], providing a proof-of-principle for the use of graph states.
However, so far the initial states have been generated with parametric down-conversion. This
technique is fundamentally probabilistic and the generated state inherits the Poisson statistics
of the pump laser, therefore requiring low driving powers in order to minimise multi-photon
events. Therefore, a more scalable method for photonic cluster-state generation would be
highly desirable.
One such proposal [132] involves using a negatively charged QD, and its spin-photon
entanglement properties [18–20] to generate large cluster states. By periodic excitation of
the trion transition, multiple subsequently emitted photons can be entangled via the electron
spin. Two circularly polarised transitions are allowed between |↑⟩ and |⇑⟩, or |↓⟩ and |⇓⟩
(c.f Fig. 1.3). If the electron spin is initially prepared in a superposition state, say |↑⟩+ |↓⟩,
and is subsequently excited with a linearly polarised optical pulse (broadband compared
with the spin splittings), subsequent recombination leads to an entangled spin-photon state:
|↑,σ+⟩+ |↓,σ−⟩. Further photon scattering from this state would lead to a GHZ-type state. In
order to obtain a cluster state, a unitary operation must be applied to the electron spin between
scattering events. The Larmor precession of the spin in the magnetic field oriented along the
y-direction can provide such an operation. For instance, if the time between exciting pulses
corresponds to a quarter of the precession period, the state before the second scattering event
is: (|↑⟩+ |↓⟩) |σ+⟩+(|↓⟩− |↑⟩) |σ−⟩. After a second scattering event and another quarter
period rotation, the full state becomes:
|↑,σ+,σ+⟩+ |↓,σ+,σ+⟩+ |↓,σ−,σ+⟩− |↑,σ−,σ+⟩ (7.1)
+ |↓,σ−,σ−⟩− |↑,σ−,σ−⟩− |↑,σ+,σ−⟩− |↑,σ+,σ−⟩
which is a 3-qubit cluster state for a logical encoding of |↑⟩ ≡ 1, |↓⟩ ≡ 0, |σ+⟩ ≡ 0, and
−|σ−⟩ ≡ 1. This procedure can ideally be repeated for a time limited by the coherence time
118 Conclusion
of the electron spin, and long strings of photons can in principle be generated. Furthermore,
this protocol is relatively insensitive to inhomogeneous broadening of the transition compared
with a ’quantum internet’ architecture composed of different QDs. The electron spin can
then be disentangled from the photonic string with a measurement in the computational
basis. Relying on Larmor precession for the intermediate operations leads to a complication:
the finite lifetime of the excited state (in which the Larmor precession is different from the
ground states) leads to errors in the created cluster state. Therefore, a Larmor precession
period slower than the excited state lifetime is required, or equivalently, the electron spin
splitting has to be sub-linewidth. This is where the work shown in this thesis relates to this
proposal. In chapter 6, we have demonstrated a method for the preparation of an electron spin
superposition state in the low-magnetic field regime, thus satisfying the requirement for a
sub-linewidth spin splitting. As discussed in the previous section, the fidelity of the dark state
initialisation may be significantly improved by conditioning on the detection of a Raman
photon. Furthermore, the uncertainty in Larmor precession is no longer determined by the
full extent of the OH field distribution but by the FP resolution. This leaves us with the final
electron spin readout. To achieve this, a high-power σ+-polarised detuned laser pulse could
be used to modify the hyperfine level structure to recover Faraday-like selection rules through
the AC Stark effect. The effective Faraday geometry would then allow a measurement of the
electron spin in the computational basis. The electron spin splitting due to the OH field is
of order 100 MHz, compared with AC Stark shifts of several tens of GHz which have been
used for ultrafast spin control [135]. Therefore, the created level structure can be made to
resemble the Faraday geometry to a good degree.
Demonstrating such a protocol will not be trivial, and we expect some rather daunting
technical difficulties to arise. First and foremost, an alternative to the cross-polarisation
technique used throughout the thesis will be needed, as the qubits are encoded in the photon
polarisation. Using a time-window may allow the suppression of the laser background
to some extent, but further technical improvements in excitation efficiency will probably
be needed to achieve a reasonable signal-to-background ratio. Secondly, the conditioning
scheme requires sampling part of the emission and is therefore a significant source of photon
loss while the limited transmission efficiency of the FP will lead to long waiting times for the
conditioning photon.
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