Third-Order Frequency-Resolved Photon Correlations from a Single Quantum Dot's Resonance Fluorescence

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Graduate Theses and Dissertations Graduate School

March 2021

Yamil A. Nieves González University of South Florida

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Scholar Commons Citation Nieves González, Yamil A., "Third-Order Frequency-Resolved Photon Correlations from a Single Quantum Dot's Resonance Fluorescence" (2021). Graduate Theses and Dissertations. https://scholarcommons.usf.edu/etd/8834

This Dissertation is brought to you for free and open access by the Graduate School at Scholar Commons. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. Third-Order Frequency-Resolved Photon Correlations from a Single Quantum Dot’s

Resonance Fluorescence

Yamil A. Nieves González

A dissertation submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy Department of Physics College of Art and Sciences University of South Florida

Major Professor: Andreas Muller, Ph.D. Myung Kim, Ph.D. Dmitry Voronine, Ph.D. Theresa Evans-Nguyen, Ph.D.

Date of Approval: February 11, 2021

Keywords: Semiconductor, Photon statistics, Quantum optics, Quantum emitter

Para mi padre y madre, Anibal y Zoraida, quienes no saben ingles y, por tanto, no comprederan nada en esta disertación y aún así se enorgullecerán igual

para mi abuelo, Luis Camargo, quién, de niño, me empujó a una carrera en medicina, pero mi rebelde ser optó por estudiar Física.

Ellos fueron quienes me criaron para alcanzar este nivel con todo su amor y apoyo. Acknowledgements

I would ﬁrst like to express my sincere appreciation and gratitude to my advisor, Dr. Andreas Muller, who remains my best role model as a scientist & mentor. Without your insightful feedback, guidance and support this dissertation would not be possible. I am also grateful to my committee members, Dr. Myung Kim, Dr. Dmitry Voronine, and Dr. Theresa Evans-Nguyen, for their valuable time throughout this process. Additionally, I would like to thank Dr. Garrett Matthews and the APS Bridge Program, for their continuous help and advice outside of research throughout my grad school career. I would like to extend my appreciation for the USF Physics department staﬀ, Daisy, Jimmy and James, for their hard-work and dedication (and James’ hilarious emails). You provided me with the tools that I needed to choose the right direction and successfully complete my graduate-level education. Finally, I would like to thank my friends and colleagues, Sebastian, Jaspreet, Olga, Corisa, Greg, Brian and Danish, for all the stimulating discussions and fun we have had in these past years. Table of Contents

List of Figures ...... iii

List of Tables ...... vi

List of Abbreviations ...... vii

Abstract ...... viii

Chapter 1: Introduction ...... 1 1.1 The Concept of the Photon ...... 1 1.2 Types of Electromagnetic Radiation ...... 4 1.3 Applications ...... 5 1.4 N-photon Sources ...... 6 1.5 Quantum Dots ...... 7 1.6 Overview ...... 9

Chapter 2: Light-Matter Interaction ...... 12 2.1 The Driven Two-Level Model ...... 12 2.1.1 Non-Interacting Hamiltonian ...... 13 2.1.2 Coupled Hamiltonian and the Dressed State Picture . . . . . 15 2.1.3 Optical Bloch Equations ...... 17

Chapter 3: First-Order Photon Correlations ...... 20 3.1 The Mollow Triplet ...... 20 3.2 Experimental One-photon Spectrum ...... 23

Chapter 4: Second-Order Photon Correlations ...... 26 4.1 Sensing Method ...... 30 4.2 The Two-photon Spectrum ...... 32

Chapter 5: Third-Order Photon Correlations of Resonance Fluorescence from a Quantum Dot ...... 37 5.1 Three-photon Auto-correlations Measurements ...... 39 5.1.1 Experimental Setup ...... 39 5.1.2 Auto-correlation Measurements ...... 42

i 5.1.3 Discussion ...... 45 5.1.4 Conclusion ...... 48 5.2 Three-photon Cross-correlations Measurements ...... 50 5.2.1 Experimental Setup ...... 50 5.2.2 Cross-correlation Measurements ...... 52 5.2.3 Discussion ...... 54 5.2.4 Conclusion ...... 58

Chapter 6: Concluding Remarks ...... 60

References ...... 61

Appendix A: Copyright Permissions ...... 72

Appendix B: Author’s Work ...... 74 B.1 List of Publications ...... 74 B.1.1 Conference Proceedings ...... 74

ii List of Figures

Fig. 1.1: A quantum particle is represented by a wave packet...... 3

Fig. 1.2: Simpliﬁed schematic illustration of the electronic density of states in semiconductors with 3, 2, 1, and 0 degrees of freedom...... 7

Fig. 1.3: Simpliﬁed Jablonski diagram illustrating diﬀerent types of photoluminescence processes...... 8

Fig. 1.4: Schematic illustration of the epitaxial growth of self-assembled QDs.9

Fig. 2.1: Schematic representation of the superposition of a two-level system driven at near-resonant condition by a monochromatic light ﬁeld... 14

Fig. 2.2: Dressed state picture as a consequence the interaction between the two-level system and monochromatic laser ﬁeld where the two states of each doublet in Figure 2.1 become “coupled” by the Rabi frequency of the electric ﬁeld...... 16

Fig. 3.1: Power spectrum for a two-level system driven exactly on resonance (∆ = 0), with decay rate κ = 0.2 GHz, for diﬀerent Rabi frequencies as given by Benjamin Mollow [86] through Equation (3.2)...... 21

Fig. 3.2: Dressed state picture of a resonantly driven two-level system generating resonance ﬂuorescence with a decay rate κ...... 22

Fig. 3.3: Schematic representation for the experimental setup for the measurement of the power spectrum from a resonantly driven two-level system system...... 23

Fig. 3.4: Theoretical and experimental one-photon spectrum of the resonance ﬂuorescence from a single QD...... 25

Fig. 4.1: Schematic representation of a Hanbury Brown & Twiss setup for the measurement of the time-delay between photons, τ = t2 − t1...... 26

iii Fig. 4.2: Second-order photon correlations for incoherent (square), coherent (triangle), and single-photon (circle) sources...... 27

Fig. 4.3: Schematic representation of the sensors theory of frequency-resolved photon correlations...... 30

Fig. 4.4: Schematic representation of the modiﬁed Hanbury Brown & Twiss setup for the measurement of correlations between two ﬁltered photons...... 32

Fig. 4.5: Two-photon resonance ﬂuorescence spectrum, for photons ﬁltered at frequencies ω1 and ω2 respectively, arriving simultaneously (τ = t2 − t1 = 0) at the detectors...... 33

Fig. 4.6: Theoretical and experimental measurements between the one-photon spectrum and the two-photon auto-correlation spectrum...... 34

Fig. 4.7: Dressed-state picture formalism for the quantization of energy states by two-photons...... 35

Fig. 5.1: Three-photon resonance ﬂuorescence spectrum for photons ﬁltered at frequencies ω1, ω2 and ω3 respectively, arriving simultaneously (τ1 = t2 − t1 = 0 and τ2 = t3 − t1 = 0) at the detectors...... 38

Fig. 5.2: Photograph of the inside of the cryostation illustrating the orthogonal excitation/detection geometry of the InAs quantum dots samples in a planar microcavity containing alternating layers of GaAs/AlAs Bragg mirrors...... 40

Fig. 5.3: Fabry-Perot etalon with adjustable resonance frequency between two thermoelectric coolers (TEC) for optimal heat distribution...... 40

Fig. 5.4: Schematic representation of the experimental setup for the measurements of third-order photons auto-correlations...... 41

Fig. 5.5: First (a), second (b), and third (c) order ﬁltered resonance ﬂuorescence spectrum at time-delays τ1 = τ2 = 0...... 43

iv Fig. 5.6: Third-order auto-correlation maps are shown for experimental (a) and theoretical (b) data for a ﬁltering frequency ωf − ωL ≈ Ω/2, and for experimental (c) and theoretical (d) data for a ﬁltering frequency ωf − ωL ≈ Ω...... 44

Fig. 5.7: Simulation for the normalized one-photon spectrum (a) the auto- correlated three-photon spectrum at zero time-delay (b) for which all parameters are identical to those in Figure 5.5 except for and increased Rabi frequency, Ω/2π = 15 GHz...... 46

Fig. 5.8: Dressed state picture formalism for the quantization of energy states by three-photons...... 48

Fig. 5.9: Simulation for the third-order auto-correlation maps for the ﬁlter frequencies (a) ωf − ωL = Ω/2, and (b) ωf − ωL = Ω/3 corresponding to the pair of superbunching peaks in Figure 5.7b...... 49

Fig. 5.10: Schematic (top) and actual (bottom) experimental setup for the measurement of three-photon cross-correlations...... 51

Fig. 5.11: Experimental three-photon correlation maps as a function of time delays τ1 = t2 − t1 and τ2 = t3 − t1 with fixed Rabi frequency of Ω/2π ≈ 1.88 GHz for different filtering frequency configurations with filter bandwidths of Γ1/2π = 0.5 GHz, Γ2/2π = 0.75 GHz, and Γ3/2π = 0.67 GHz ...... 52

Fig. 5.12: Dressed-state picture formalism for (a) three-photon correlations corresponding to the origin of central feature of all correlation maps in Fig- ure 5.11, and (b) two-photon correlations corresponding to the origin of the vertical, horizontal, and diagonal features of each correlation map...... 55

Fig. 5.13: Experimental and theoretical Mollow triplet lineshape for ﬁlter bandwidths Γf /2π = 0.33 GHz and Γf /2π = 0.75 GHz...... 58

v List of Tables

Table 5.1: Raw data for the indicated correlation maps showing the normalized 3rd and 2nd-order correlations at τ1 = τ2 = 0...... 53

vi List of Abbreviations

FWHM...... Full Width at Half Maximum

HBT...... Hanbury-Brown and Twiss

QD...... Quantum Dot

SPCM...... Single-Photon Counting Module

SD...... Spectral Diﬀusion

TLS...... Two-Level System

vii Abstract

The ability to control the photon emission from light sources has sparked an increasing interest in the field of quantum optics due to potential applications in the development of quantum technologies. Motivated by such applications, photon statistics have been a powerful investigative tool used to characterize the properties of light sources capable of emitting a bundle of N photons. As such, the first-order and second-order photon correlations continue to be researched extensively providing invaluable information about the light and the mechanism underlying its generation. Meanwhile, third-order photon correlations have remained majorly unexplored due to the high degree of experimental challenges involving such measurements. This work presents experimental progress towards constructing the complete spectrally filtered three-photon spectrum of the resonance fluorescence generated from a single semiconductor quantum dot under strong monochromatic light excitation, and explores the effects imposed by the limited resolution of the filter bandwidth and environmental influ- ences due to spectral diffusion. We investigated third-order auto-correlation measurements, in which photons were identically filtered, resulting in correlation maps that are functions of two relative time-delays. These measurements revealed accentuated nonclassical charac- teristics amongst the distinguishing features of the three-photon spectra following faithful agreement with theory. Additionally, we investigated third-order correlations where photons originate from opposite sidebands of the power spectrum. These three-photon cross- correlation measurements resulted in correlation maps with “anti-bunching” features as a consequence of correlations arising from the same sideband, and “bunching ridges” due to opposite sideband correlated photons.

viii Chapter 1 Introduction

Information is all around us—on books, pictures, internet, DNA, etc.—and the way we process this information is constantly changing. From Pascal’s arithmetic machine to small computers that can ﬁt in your pocket (a.k.a. smartphones) we seek smaller, faster and more powerful devices that are able to store and process this information. With the coali- tion of quantum physics and computing theory we are on the verge of a new technological revolution in quantum computation & information processing. By harnessing the laws of quantum mechanics, quantum computers are expected to enhance considerably the information processing power to the point that computation currently out of reach, even for classical super-computers, would become feasible. Before achieving such technology, a ﬁne control over the quantum particle used as the information carrier is required. Quantum states of light are expected to play a crucial role in transmitting quantum information over distances. Therefore, a fascinating subject of research in quantum optics is the science of correlations between photons.

1.1 The Concept of the Photon

The study of the properties of light is one of the most fundamental aspects of the natural sciences that has puzzled scientific minds for centuries. The foundations of the field of optics (such as reflection, refraction, diffusion, and vision) were established by philosophers from ancient Greece during Antiquity (between 8th century B.C. and the 2nd A.D.) from Plato, to Aristotle, to Euclid and Ptolemy, followed by scholars of the “House of Wisdom” during the

1 Islamic Golden Age (between 8th − 14th century) from Al-Kindi, to Ibn-Sahl, all the way to “the father of modern optics” al-Haytham. These concepts on light rendered a large effect in the development of optics up until the appearance of Newton in the late 17th century. Sir Isaac Newton is without a doubt one of the defining figures in the history of science. Newton is mostly known by his Principia (which states his universal law of gravitation and his laws of motion that formed the foundations of classical mechanics) and by co-discovering mathematical methods now included in the field of calculus. It is, however, important to note that one of Newton’s major contributions was in the field of optics. He was the first to show that color is an intrinsic property of light, and advocated for its corpuscular theory claiming that “light is comprised of colored particles”, despite evidence that supported the wave behavior (e.g., Grimaldi’s observation of the diffraction of light). After a long debate among prestigious names that spanned for decades, it seemed that Huygens’s wave theory, Young’s double-slit experiments, Fresnel’s work on diffraction and polarization, and Maxwell’s electromagnetic field theory were able to explain most of the phenomena of light, electricity, and magnetism. The electromagnetic wave theory had tri- umphed and the particle interpretation was believed to be definitely buried. No one suspected that the dawn of the twentieth century would bring a revolution in our understanding of the nature of light yet again. The beginning of the 20th century carried two uncertainties involving light that brought forth the birth of two revolutionary theories that cause major transformations in the field of physics. One was the unexpected result by the Michelson-Morley experiment [1] that showed the speed of light is the same regardless of the motion of the Earth. The other was the uncertainty surrounding the nature of blackbody radiation at high frequencies, commonly known as the Rayleigh-Jeans ultraviolet catastrophe. The former lead to Einstein’s famous Theory of Special Relativity [2], while the latter ultimately led Max Planck to postulate that radiation energy can be divided into a number of discrete “energy elements”, , that must be proportional to the frequency ν, that is = hν, where h is a constant that eventually

2 carried Planck’s name [3]. Planck’s seminal work is gen- Particle? erally regarded today as the birth of Quantum Theory. . . The arrival of quantum mechanics brought with it the revival of the particle versus wave nature of light controversy. In 1905, Einstein used Planck’s “energy elements” Wave? . to explain the photoelectric effect (emission of electrons . from a material after absorbing electromagnetic radiation). Einstein proposed that, not just the energy but the radiation itself is quantized in discrete energy packets which he called “light quantum” [4]. This theory marked the beginning of the modern concept of the photon. Wave-particle duality Both! A photon is generally created via a process known . . as spontaneous emission, which occurs when a quantum mechanical system—such as a molecule, an atom, or an artificial atom—decays from an excited state to a lower Figure 1.1: A quantum particle is represented by a wave packet. energy state by releasing a quantized amount of energy. In 1956, Hanbury-Brown and Twiss (HBT) discovered another surprising property of light as they were trying to measure the angular diameter of the stars by studying the correlation between intensity fluctuations [5]. The HBT experiment consisted on splitting a low-intensity beam of light (simulating that of a star) into two components using a half-silvered mirror, each component detected separately through two photo-multipliers. In their experiment, HBT claimed that two photons had been positively correlated (i.e., detected at the same time) when it had been expected that only single photons were to be detected due to the low intensity of the source. This effect brought in a lot of controversy into the world of quantum theory [6]. At the time, this result seemed to go against the foundations of quantum theory since, assuming a photon as a small localized indivisible particle, there would be no way to detect

3 two photons at the same time. Although initially rejected, the HBT effect was soon under- stood to be the manifestation of the wave-particle duality of light [7]. By simply treating photons as a system of bosons (wave-like objects) which can be accumulated in the same energy state, the quantum effect is then a result of the “bunching” of monochromatic photons (photons with a single optical wavelength or frequency). Along with the HBT development, another famous optical effect of similar quantum nature was discovered which provided further insight on the behaviour between two identical photons. In 1987, physicists Chung Ki Hong, Zhe Yu Ou and Leonard Mandel demonstrated a technique based on the interference of two identical single-photons as they enter through the two input modes of a 50:50 beamsplitter, where both photons have an equal chance (50%) of either being reflected or transmitted [8]. Due to destructive interference, the probability that both photons are reflected (or transmitted) decreases the more indistinguishable the photons are. Therefore, two photons with identical properties will always exit the beamsplitter together as a pair, this is commonly known as the Hung-Ou-Mandel (HOM) effect. The discovery of the HBT and HOM effects, along with the advent of the laser, led to the development of modern quantum optics as it was shown that all kinds of light sources can be differentiated in terms of their photon statistics [9].

1.2 Types of Electromagnetic Radiation

For a broad range of quantum processes, including those involving multiparticle interactions [10], photon statistics have been a powerful investigative tool used to characterize the coherence properties of an electromagnetic ﬁeld. Instead of simply being linked to monochromatic wave sources and a constant phase diﬀerence, in its essence, the degree of coherence is the measure of the temporal correlations between photons. The correlations between the detection time of subsequently emitted photons serve as the primary tool for characterizing the type of radiation emitted through light-matter interactions at the few or single quantum level. For example, for photons originating from

4 blackbody radiation (i.e., incoherent light sources such as the sun or an incandescence light bulb) coincidences are systematic since photons arrive together in groups of two or more at the detector, exhibiting strong positive correlations as seen in the bunching reported by the HBT experiment. Meanwhile, for stimulated radiation (i.e., coherent light sources such as a laser), no mutual correlations are shown since photons arrive at the detectors arbitrarily indicating no time relationship between them. In addition to the thermal and stimulated light sources, there is a third type of radiation that cannot be explained through classical methods, displaying negative correlations: the single-photon sources. A single-photon source emits light as well spaced photons within its emission lifetime after being optically, or electrically, excited causing them to arrive at the detector one at a time and therefore producing the characteristic of photon antibunching. Ideally, a single- photon source: (1) emits deterministic (i.e., on-demand) single photons at any arbitrary time deﬁned by the user, (2) have a 0 % probability of emitting multiple-photons, (3) emits identical (indistinguishable) photons, and (4) allows for a fast repetition rate [11–13]. These non-classical states of light serve a wide range of applications in the so called “quantum information processing” where the data processing is done using a quantum system as the information carrier.

1.3 Applications

The ability to control the photon emission from light sources is a key component in the development of quantum technologies [14–17]. These systems use the quantum-mechanical states of a two-level quantum system (a system with only excited and ground energy states) as quantum bits (also know as ‘qubits’). These qubits replace the traditional binary bits (0 or 1) used in classical computers in order to enhance considerably the processing power. While the traditional binary bits can only be in one state or another, qubits harness the laws of quantum mechanics where coherent superposition allows for both states to exists simultaneously.

5 Through massive parallel processing, quantum computers are expected to perform a wide variety of applications like the ability to: crack popular encryption schemes with quantum cryptography [18–21], extend the range of distribution of quantum resources through quantum repeaters [22–24], optical imaging using light that has never interacted physically with the object to be imaged through ghost imaging [25–27], complete transfer of information between particles through quantum teleportation [28–31], or making highly sensitive measurements with high-resolution through quantum metrology [32–35]. The pursuit of such quantum applications makes it increasingly important to expand the available quantum sources in order to be capable of emitting a bundle of N photons.

1.4 N-photon Sources

One useful approach to produce N-photons is by generating a correlated bundle of photons, where one photon may be used to ‘herald’ the existence of the other photons. A common example of such approach is via a nonlinear process in bulk such as spontaneous parametric down-conversion in crystals [36–39] and waveguides [40], or four wave mixing in optical fibers [41–43] and cavities [44]. While heralded single-photon sources are very practical for many of the applications mentioned on Section 1.3, these methods suffer from significant drawbacks as the repetition rate must be significantly reduced in order to avoid producing more than the desired amount of photons. This has led researchers into developing truly deterministic single-photon sources. A potential approach to produce deterministic single-photons (or entangled photons through cascade emission [45]) is from two-level quantum systems such as atoms [46–49], molecules [50–52], ions [53, 54], mesoscopic quantum wells [55], or a nitrogen-vacancy colored center in diamond [56–59]. Another promising N-photon emitter with excellent optical stability, compared to other solid state systems, is the semiconductor quantum dot [60].

6 1.5 Quantum Dots

Quantum dots (QD) are semiconductor crystals which size ranges within only a few nanometers depending on their production process. QD nanotechnology has successfully en- tered numerous electronic and biomedical industries due to their wide variety of applications such as single-electron transistors [61], photovoltaic solar cells [62], and medical imaging [63], just to name a few. Their small size causes the motion of conduction band electrons and valence band holes, or excitons (bound state of an electron and electron hole) to be confined in all three spatial dimensions [64]. This confinement results in discrete energy states which are characteristic of individual atoms (Figure 1.2), which is why QDs are commonly referred to as “artificial atoms”. These atom-like energy states contribute to the QD’s special optical properties, such as the emission of photoluminescence light.

3D Material 2D Material 1D Material 0D Material Bulk Quantum Well Quantum Wire Quantum Dot (No confinement) (1D confinement) (2D confinement) (3D confinement)

1 ∝ 퐸 ∝ 퐸0 = 푐표푛푠푡. ∝ ∝ 훿 퐸

퐸

Density Density States of

Density Density ofStates Density Density States of Energy Energy Energy Density States of Energy Figure 1.2: Simpliﬁed schematic illustration of the electronic density of states in semiconductors with 3, 2, 1, and 0 degrees of freedom.

Photoluminescence occurs when the light is absorbed by the quantum system exciting the electron to a higher unstable energy state and, after some time, the electron decays via spontaneous emission as shown in Figure 1.3. When this process involves a change in the electron’s spin multiplicity, it results in a longer lifetime of the excited state (seconds to minutes) producing phosphorescent light. However, when no such spin change is involved, the absorption/emission process occurs almost instantaneously (<10−5 s) emitting ﬂuorescent light [65]. Typically, the energy of the emitted photon is lower than that of the absorbed

7 photon, however, this study focuses on one particular type of ﬂuorescence emission where the frequency of the absorbed radiation corresponds almost exactly to the natural frequency of the QD, known as resonance ﬂuorescence. Essentially, this makes the QD a system consisting of only two energy levels: ground and excited states.

Nonradiative transition

푆1 푆1 Intersystem crossing (change in spin multiplicity) 푇1

Absorption Absorption 퐸 Resonance Fluorescence Vibrational Fluorescence Phosphorescence Levels

푆0 푆0

Figure 1.3: Simplified Jablonski diagram illustrating different types of photoluminescence processes. Resonance fluorescence occurs when the frequency of the absorbed photon matches exactly the natural frequency of the system. Fluorescence is the process where the excited system undergoes a non-radiative relaxation where the energy is dissipated through non-radiative transitions before emitting a photon of lower energy than the absorbed radiation. Phosphorescence occurs when the molecule undergoes an intersystem crossing from a singlet spin state S1 to the triplet spin state T1 before emission.

As a consequence of these fascinating optical properties, QDs have been exploited for a wide range of applications including single-electron transistors [66], quantum computing [67], lasers [68], second-harmonic generation [69], LEDs [70], photovoltaic solar cells [62], medical imaging [71], and single-photon sources [72]. In an attempt to improve feasibility for large scale production and reduce fabrication defects, scientist continue to search for diﬀerent processes to manufacture QDs such as colloidal synthesis [73], and chemical vapour deposi-

8 tion [74]. This work focuses on light generated from InAs QDs grown by molecular-beam epitaxy. QDs grown epitaxially in the Stranski-Krastanov growth mode (or “self-assembly” effect) [75] consist on the deposition of a so-called “wetting layer” (successive atomic layers) of a material with a small band gap, such as indium arsenide (0.36 eV at 300 K), on substrate with a larger band gap such as gallium arsenide (1.43 eV at 300 K)[76]. The diﬀerence in lattice structures between the InAs layers and the GaAs substrate begins a phase transition once the critical thickness of the wetting layer is achieved. Due to the compressive strain, it becomes energetically favorable for the InAs layer to form nano-scale islands, aka QDs. Finally, to preserve the QDs from the external environment, a capping layer of the substrate is deposited over the wetting layer.

(d) Conduction band InAs (a) wetting 푃푒 layer 푒− 푆 GaAs 푒 InAs QD (b) islands

GaAs band gap InAs (c) gap band GaAs + GaAs ℎ 푆ℎ capping InAs 푃 layer ℎ GaAs GaAs Valence band Figure 1.4: Schematic illustration of the epitaxial growth of self-assembled QDs: (a) Deposition of a thin layer of InAs on top of a GaAs substrate leading to an accumulation of biaxial strain due to the lattice mismatch. (b) Once the critical thickness of the wetting layer is reached, QDs islands are formed to minimize the total energy of the system. (c) Deposition of additional GaAs serving as a capping layer to avoid interaction with the external environment. (d) Band structure scheme illustration for self-organized quantum dots.

1.6 Overview

In this work, we aim to understand the spectral and time-correlation properties of a light scattered by a strongly driven semiconductor quantum dot. Chapter2 is devoted to a review on the theoretical study of the interaction between a two-level system and a monochromatic

9 resonant (or near-resonant) laser field. In Section 2.1, treat the laser field as a single-mode quantum field to clearly identify the elementary processes of the spontaneous emission for fluorescence photons. We start by neglecting the interaction between the two-level system (TLS) and the laser photon system, such that the two systems are treated separately as a superposition of one another. This way the processes of absorption and stimulated emission of laser photons appear simply in the basis of uncoupled states which describes the two- level system driven under low power laser illumination. Then, we treat them as a coupled- system such that the two-level system is “dressed” by the laser photons and the process of spontaneous emission appear now in the basis of dressed states which describes the two-level system driven under high power laser illumination. Finally, we derive the optical Bloch equations which compose the master equation of the driven two-level model. In Chapter3, we explore the theory behind first-order correlations and the power spectrum of a strongly-driven two-level system, and provide the dressed-state picture to illustrate the origin of the features present in the correlation spectra. In Section 3.2 we discuss the experimental aspect of such measurements and include the introduction of non-idealities due to filter bandwidth and spectral diffusion into theoretical simulations. Chapter4 is devoted to the second-order correlations for the arrival times of the photons emitted by the coupled-system. First, we discuss the temporal aspect of the two-photon correlations and how it is used to characterize the type of emission process from a light source. Then, in Sections 4.1 and 4.2, we make use of the sensing method to explore the frequency- resolved two-photon spectrum and its underlying emission process down the dressed-states ladder. In Chapter5, we have performed an experiment to further our knowledge in the nonclassical aspect of the scattered light emitted by a resonantly-driven quantum dot. In Sec- tion 5.1, we show the experimental results of the third-order auto-correlations measurements, i.e., correlations between the arrival times of three photons detected at the same frequency. The results of this investigation were published in a peer-reviewed journal: “Third-order

10 frequency-resolved photon correlations in resonance fluorescence” [Nieves et al., Phys. Rev. B 98, 165432 (2018), 77]. In Section 5.2, we show the experimental results of the third-order cross-correlations measurements, i.e., correlations between the arrival times of three photons filtered at different frequencies. The results of this investigation were published in a peer- reviewed journal: “Third-order photon cross-correlations in resonance fluorescence” [Nieves et al., Phys. Rev. B 102, 155418 (2020), 78].

11 Chapter 2 Light-Matter Interaction

Quantum dots are often referred to as “artificial atoms” due to their well-defined spectrum of energy levels [79]. The pumping of photons from a monochromatic laser excites the electron in the quantum dot which later goes through a relaxation process from a higher energy level back to the lower energy state. This results in the excited quantum dot to emit light via spontaneous emission, known as fluorescence. When the driving laser closely matches the natural frequency of the quantum dot, this fluorescence light is said to be in resonance with the quantum system. This so called resonance fluorescence is a fundamental source of light that has been particularly fit for the study of correlations with combined time and frequency information. This process of fluorescent emission from the quantum dot results in a simple model know as a two-level system which resembles the electronic shells in atoms featuring exceptional optical properties at cryogenic temperatures [80].

2.1 The Driven Two-Level Model

A two-level system can be described by a 2-dimensional complex Hilbert space consisting

1 of two independent quantum states: a ground state, |gi = ( 0 ), with energy ~ωg, and an

0 excited state, |ei = ( 1 ), with energy ~ωe. The atomic operator of the system can be deﬁned

12 by a linear combination of the four basis operators:

2.1.1 Non-Interacting Hamiltonian

We begin by neglecting the interaction between the two-level system and the laser

photons, the Hamiltonian for this system is then in the form of H = H0 + HL where

H0 = ~ωgSgg + ~ωeSee describes the Hamiltonian of the unperturbed two-level system, and

HL = ~ωL(See + 1/2) represents the Hamiltonian of the laser mode with frequency ωL. Therefore the global Hamiltonian of the system is given by:

1 H = ω S + ω S + ω S + . (2.2) ~ g gg ~ e ee ~ L ee 2

1 0 Since Sgg + See = 1, where 1 is the identity matrix ( 0 1 ), the Hamiltonian above can be

expressed in terms of the system’s resonance frequency, ω0 = ωe − ωg, by subtracting the

term ~ωg1 eﬀectively moving the reference point. The Hamiltonian of the system then becomes: 1 H = ω S + ω S + , (2.3) ~ 0 ee ~ L ee 2

13 The eigenstates of H are labeled by two quantum numbers: (1) the two-level system quantum numbers, g or e, and (2) the number of laser photons, N. The states |g, Ni and |e, N − 1i are close to each other when the detunning of the laser central frequency, ∆, is

very small compared to the resonance frequency of the two-level system, |∆| ω0, where the detunning is given by:

∆ = ωL − ω0 . (2.4)

This superposition between the two-level system and monochromatic electric ﬁeld is illustrated in Figure 2.1.

Laser Photon System Uncoupled States |g, N + 2i |N + 2i ~∆ E(N + 2) |e, N + 1i

~ωL

|g, N + 1i Two-Level System |N + 1i ~∆ E(N + 1) |ei |e, Ni N ~ωL ~ω0 ~ωL = ~ω0

|gi |g, Ni |Ni ~∆ E(N) |e, N − 1i

~ωL

|g, N − 1i |N − 1i ~∆ E(N − 1) |e, N − 2i

Figure 2.1: Schematic representation of the superposition of a two-level system, with a resonance frequency ω , driven at near-resonant condition by a monochromatic light ﬁeld with frequency ω , 0 n o L resulting in a manifold of two uncoupled states (e.g., E(N) = |g, Ni , |e, N − 1i ). The energy diﬀerence between the two states of each doublet depends on the detunning of resonance frequencies between the laser and the two-level system, ∆ = ωL − ω0. The order between the doublet states change depending on the sign of ∆ (e.g., if ω0 > ωL, |e, Ni is placed above |g, N + 1i).

14 2.1.2 Coupled Hamiltonian and the Dressed State Picture

The Hamiltonian for the coupling between the two-level system and monochromatic elec-

tric ﬁeld is in the form H = H0 + HI , where the interaction Hamiltonian HI in the electric dipole approximation is given by:

HI = −µE(Seg + Sge) , (2.5)

where µ is the transition dipole moment of the system, and E is the applied electric ﬁeld

assumed to oscillate harmonically around the central frequency, ωL, that is:

E0 E(t) = eiωLt + e−iωLt . (2.6) 2

The next stage is to find a modified effective Hamiltonian in the rotating frame, so that it satisfies the Schrödinger equation in all frames. The unitary matrix required for the

1 0 transformation to the interaction frame is U = 0 e−iωLt . If we define a modified effective † † dU Hamiltonian in the rotating frame as Hrot = UHU −iU dt , it can be easily shown that this results in the same form for the Schrödinger equation in both the stationary and rotating frames. Thus, the corresponding matrix for the effective Hamiltonian in the rotating frame has the form:   0 − ΩR − ΩR e−2iωLt  2 2  Hrot =   , (2.7) ΩR ΩR 2iωLt − 2 − 2 e −∆

µE0 where ΩR = is the induced Rabi frequency of the laser mode which plays a big part in ~ the separation between the uncoupled states shown in Figure 2.1. Finally, we can get rid of the time dependency since, in a typical atomic system, the fast oscillating terms (i.e., those that oscillate at twice the optical frequency in the eﬀective Hamiltonian) lead to weak excitations that can be neglected. This is known as the rotating-

15 wave approximation which yields a new time-independent Hamiltonian in the form:

  ΩR 0 2 H = −~  . (2.8)  ΩR  2 ∆

This Hamiltonian couples the states of each doublet in Figure 2.1 to each other such that, when this coupling is taken into account, the doublets become perturbed. These states are commonly referred to as “dressed” by the laser [81]. When the driven laser is at near- resonant condition, the separation between dressed states is dictated by the generalized Rabi frequency: q 2 2 Ω = ∆ + ΩR . (2.9)

For simplicity, we use integers ...0, 1, 2... to refer to the coupled states, and the quantum numbers + and − to label the upper and lower state of each doublet, as shown in Figure 2.2.

Dressed States Uncoupled States |2, +i

E(N + 2) ~∆ ~Ω |2, −i

~ωL |1, +i

E(N + 1) ~∆ ~Ω |1, −i

~ωL |0, +i

E(N) ~∆ ~Ω |0, −i

Figure 2.2: Dressed state picture as a consequence the interaction between the two-level system and monochromatic laser ﬁeld where the two states of each doublet in Figure 2.1 become “coupled” by the Rabi frequency of the electric ﬁeld.

16 2.1.3 Optical Bloch Equations

When dealing with mixed states it is specially helpful to describe the statistical state of a quantum system using a density matrix. The mixed state, |ψi, of a quantum system is represented to be in a statistical ensemble of diﬀerent pure state vectors, |ψni. For a ﬁnite-dimensional system, the most general density operator is in the form:

X ρ = Pn |ψnihψn| , (2.10) n

where Pn is the probability that the system is in the pure state |ψni. Let S be an observable of the system, the expectation value of the measurement can be found as:

hSi = tr[Sρ] . (2.11)

The density operator in Equation (2.10) can be written as a linear combination of the atomic operators in Equation (2.1) as:

  x (t) x (t)  3 2  ρ(t) = x1(t)Seg + x2(t)Sge + x3(t)Sgg + x4(t)See =   . (2.12) x1(t) x4(t)

Using Equation (2.11), one can ﬁnd the expectation values of the atomic operators:

hSegi = tr[Segρ(t)] = x2(t) ,

hSgei = tr[Sgeρ(t)] = x1(t) , (2.13)

hSggi = tr[Sggρ(t)] = x3(t) ,

hSeei = tr[Seeρ(t)] = x4(t) .

17 From these we obtain the density operator:

  hS i hS i  gg eg  ρ(t) = hSgei Seg + hSegi Sge + hSggi Sgg + hSeei See =   . (2.14) hSgei hSeei

The master equation describing the evolution of the density matrix can be described by a generalization of the quantum Louisville equation, known as the Lindbladian equation [82]:

dρ(t) 1 X κ X = − [H, ρ(t)] + κ S ρ(t)S† − S† S ρ(t) + ρ(t)S† S , (2.15) dt i m n 2 m n m n ~ m,n m,n

where the ﬁrst term is the usual Schrödinger term that generates the unitary evolution for a closed system, the second term describes the possible quantum jumps due to a radiative decay at a rate κ that the system may undergo due to interactions with the environment, and the third term is needed to normalize properly the case in which no jumps occur. By substituting Equations (2.8) and (2.14) into Equation (2.15), and making use of the deﬁnitions in Equation (2.1), we obtain a set of four equations known as the optical Bloch equations: d iΩ iΩ hS i = −κ hS i − R hS i + R hS i , dt ee ee 2 eg 2 ge d iΩR κ iΩR hSegi = − hSeei + i∆ − hSegi + hSggi , dt 2 2 2 (2.16) d iΩ κ iΩ hS i = R hS i + −i∆ − hS i − R hS i , dt ge 2 ee 2 ge 2 gg d iΩ iΩ hS i = κ hS i + R hS i − R hS i . dt gg ee 2 eg 2 ge The optical Bloch equation in the rotating-wave approximation given in Equation (2.16) may be rewritten in matrix form as:

d R(t) = M · R(t) , (2.17) dt

18 where R contains the expectation values of the observables and M is a single matrix capturing the parameters of the coupled system which may be used to ﬁnd its dynamical evolution:

    ΩR ΩR hSeei −κ −i i 0    2 2     ΩR κ ΩR  hSegi −i i∆ − 0 i     2 2 2  R(t) =   , and M =   , (2.18) hS i  i ΩR 0 −i∆ − κ −i ΩR   ge   2 2 2      ΩR ΩR hSggi κ i 2 −i 2 0

The solution of Equation (2.17) is important as it leads to the correlation functions required in the study of light-matter interactions in a two-level system.

19 Chapter 3 First-Order Photon Correlations

In the pursuit of the applications described in Section 1.3, the science of correlations between subsequently emitted photons from a light source has become a powerful investigative tool in the ﬁeld of quantum optics. The concept of the N-photon spectrum,

(N) g (t1, ω1; ...; tN , ωN ), denotes the joint probability density of detecting photons at times

ti with frequencies ωi, for i = 1, 2, ..., N [83–85]. Starting at the simplest case where N = 1, the seminal work from Benjamin Mollow provided the ﬁrst complete theoretical description of the spectrum of light from resonance ﬂuorescence.

3.1 The Mollow Triplet

The ﬁrst-order correlation function is proportional to the correlation of atomic operators

(1) † when the two-level system is in equilibrium with the driving field, g (t1) ≡ a (t)a(t + t1) . In 1969, Mollow showed that the power spectrum (the distribution of the signal’s power in frequency bins) of resonance fluorescence can be obtained by performing a Fourier transform of the first-order correlation function, that is:

Z ∞ (1) (1) iω1t1 g˜ (ω1) ≡ g (t1)e dt1 . (3.1) −∞

The expression for the “one-photon spectrum” is then given explicitly by Mollow [86]:

2 2 2 (1) 2 2 (ω1 − ωL) + (ω1/2 + κ ) g˜ (ω1) = 2π|α∞| δ(ω1 − ωL) + n∞κω1 2 (3.2) |f(i(ω1 − ωL))|

20 where the steady-state probability of ﬁnding the two-level system in its excited state n∞, the quantum mechanical expectation value of the two-level coherence α∞, and the third-order polynomial f(s) are given by: 2 1 κ + 2i∆ n∞ = ω1 2 2 2 , α∞ = iω1 2 2 2 , 4∆ + κ + 2ω1 4∆ + κ + 2ω1

3 2 2 2 2 2 2 f(s) = s + (ω1 + ∆ + 5κ /4)s + κ(ω1/2 + ∆ + 2κ + κ /4) .

For a two-level system under low incident field power compared to the natural decay rate (that is, Ω ≤ κ), the spectral lineshape of the power spectrum consists of a single peak situated at the frequency of the driving laser ωL. Meanwhile, for a two-level system under an intense driving field (i.e., Ω κ), the fluorescence profile consists of a peak centered at the laser frequency, ωL, and peaks at the displaced frequencies ωL ± Ω, with widths proportional to the decay rate κ, as shown in Figure 3.1 for a system with decay rate κ = 0.2 GHz.

Ω=κ Ω=5κ Ω= 10κ ( ω )( arb. units ) ( 1 ) g

-3-2-1 0 1 2 3 (ω-ω L)/2π (GHz)

Figure 3.1: Power spectrum for a two-level system driven exactly on resonance (∆ = 0), with decay rate κ = 0.2 GHz, for diﬀerent Rabi frequencies as given by Benjamin Mollow [86] through Equation (3.2).

These peaks, commonly known as the “Mollow triplets”, are a consequence of the Autler- Townes eﬀect (or AC Stark eﬀect). For a system under low power illumination (Ω ≤ κ) driven

21 exactly at resonance (∆ = 0), each doublet in the resulting manifold shown in Figure 2.1 becomes a superposition of the uncoupled states, allowing the system only one possible transition at the laser frequency. Meanwhile in the high power regime, that is, when the Rabi frequency far exceeds the natural relaxation rate of the system (Ω κ), the light emission process may then be viewed as a cascade down the ladder of the dressed states where the Mollow triplet emerges out of the only four possible spontaneous transitions between successive doublets.

Uncoupled States Dressed States (Ω ≤ κ) (Ω κ) |1, +i E(N + 1) Ω |1, −i ω L Ω Ω + − L ω |0, +i E(N) Ω ωL |0, −i ωL

Figure 3.2: Dressed state picture of a resonantly driven two-level system generating resonance ﬂuorescence with a decay rate κ. In the low power regime (Ω ≤ κ), when ∆ = 0, only one possible transition is allowed at the laser frequency. At high ﬁeld intensities, Ω κ, the four possible transitions between successive doublets result in the Mollow triplet lineshape of the power spectrum.

As shown in Figure 3.2, the resonance ﬂuorescence process originating from the lower

energy (lower frequency) peak of the Mollow triplet, ωL − Ω, occur due to the transition from the lower state of a top doublet to the upper state of a bottom doublet of the manifold, |1, −i −→|0, +i. Meanwhile, the photons originating from the higher energy (higher frequency) peak can be viewed as a transition from the upper state of a top doublet to the lower state of a bottom doublet, |1, +i −→|0, −i. Finally, the central peak is a consequence of the doubly degenerate transitions from the upper state of the top doublet to the upper state of the bottom doublet, |1, +i −→|0, +i, and transitions from the lower state of the top doublet to the lower state of the bottom doublet, |1, −i −→|0, −i.

22 3.2 Experimental One-photon Spectrum

Experimentally, the frequency-resolved ﬁrst-order photon correlation function, g(1)(t , ω ), Γf 1 f of a light source is readily obtained interferometrically by using a spectral ﬁlter (such as a

Fabry-Perot interferometer) with bandwidth Γf . The ﬁrst-order photon correlation function is then proportional to the count rate of one photo detector as a function of the ﬁltered frequency, ωf .

Two-level Photo System Fabry-Perot Etalon (Γ푓) Detector 푒ۧ| ℏ휔푓

ℏ휔0 ℏ휔퐿 Filtered 푔ۧ| Resonance Signal Fluorescence

Figure 3.3: Schematic representation for the experimental setup for the measurement of the power spectrum from a resonantly driven two-level system system.

In practice, however, complications such as the stabilization and uniformity of the properties of the spectral filter may introduce some non-idealities into the measurements. For instance, the filter’s resonance frequency will inevitably fluctuate during the measurement as will the filter bandwidth. In addition, the precise width of each spectral filter may be difficult to set to an arbitrary independent value, and to maintain over time. Small variations in filter bandwidths are particularly consequential when the filter bandwidth is near the natural linewidth of the emitter. In fact it was recently shown that, even in the weak excitation regime, the photon statistics of a filtered single QD’s resonance fluorescence depend dramatically on the filter bandwidth [87]. Therefore, the spectral bandwidth of the filter must be taken into account in the theoretical simulations. Given the solution of Equation (2.17) can be written as:

Mt1 R(t + t1) = e · R(t) , (3.3)

23 and making use of the quantum regression theorem [88], which states that the time evolution of an operator O whose expectation value is known is given by:

X hO(t + t1)i = aj(t1) hOj(t)i , (3.4) i

we can calculate the ﬁrst-order correlation function as:

(1) Mt1 Mt1 ∗ −iωLt1 g (t1) = e |2,2n∞ + e |2,4α∞ e . (3.5)

The resolution of the Fabry-Perot interferometer, Γf , is introduced into Equation (3.5) by

multiplying the correlation function by e−t1Γf /2 and ﬁnding its Laplace transform. The expression for the spectrally ﬁltered “one-photon spectrum” is given by Konthasinghe et al. [89]:

( Γ −1 g(1)(ω ) = 2 Re i(ω − ω ) + f 1 − M n Γf 1 L 1 ∞ 2 (2,2) −1 ) Γf ∗ + i(ωL − ω1) + 1 − M α∞ , (3.6) 2 (2,4)

where 1 is the 4 × 4 identity matrix, M is the matrix capturing the inputs of the equation of motion for the system’s observables given in Equation (2.18), and [··· ](i,j) denotes the element (i, j) of the resulting matrix. Additionally, the transition frequency of the two-level system being probed will fluctuate to some extent. This “spectral diffusion” results in inhomogeneous broadening, i.e., an averaging over different resonance frequencies. To account for the fluctuations of the resonance frequency, ω0, due to spectral diffusion [90, 91], as in Ref. [89], we assume a Gaussian distribution of the detuning between the QD resonance frequency and the laser frequency, ∆

(centered at ∆µ = 0 with a standard deviation ∆σ = 2 GHz), and integrating Equation (3.6)

24 over all possible random detunings:

2 Z −(∆−∆µ) (1) (1) 2 g˜ (ω ) ∝ g (ω , ∆)e 2∆σ d∆ . (3.7) Γf f Γf f

This results in a theoretical simulation of the Mollow triplet in the one-photon spectrum that demonstrates a closer interpretation to experimental results as shown in ﬁg. 3.4.

105

1,+ 1,- )

- 1 0,+ 104 0,- Count Rates ( s

103 Experiment Theory

ωL-Ω ωL ωL+Ω Filter frequency,ω f

Figure 3.4: Theoretical and experimental one-photon spectrum of the resonance ﬂuorescence from a single QD. The allowed spontaneous transitions in the dressed-state picture are shown at their corresponding emitted frequency.

Although the ﬁrst-order correlations provide invaluable information about the light and the mechanism underlying its generation, this measurement alone reveals little about the source itself. For example, a light source consisting of a single radiating atom, and a light source consisting of an ensemble of non-interacting radiating atoms could have identical g(1)(ω ). To distinguish the two, a second-order photon correlation measurement is re- Γf f quired.

25 Chapter 4 Second-Order Photon Correlations

(2) The second-order correlation function, g (t1, t2), measures the time correlation between

two photons emitted by a continuous source arriving at the detectors at times t1 and t2

(2) respectively. The normalized g (t1, t2) can be expressed on the form of the photon creation and annihilation operators as:

† † (2) a (t1)a (t2)a(t1)a(t2) g (t1, t2) = † † (4.1) ha (t1)a(t1)i ha (t2)a(t2)i

From an experimental point of view, photon arrival times can be measured using the Hanbury-Brown and Twiss (HBT) setup [5], wherein a beamsplitter is used to divide the direction of the emitted photons into two pathways, and each path is equipped with a detector connected to electronics that records the arrival time of each photon as shown in Figure 4.1. The second-order correlations between two photons has attracted signiﬁcant attention in the

Detector 1

Photons emitted (1) from a light source (2)

Time-Tagging Electronics Detector 2

Figure 4.1: Schematic representation of a Hanbury Brown & Twiss setup for the measurement of the time-delay between photons, τ = t2 − t1. An incoming light is split in two pathways using a 50:50 non-polarizing beamsplitter, then photon arrival times are recorded by a detector at each path. A histogram is build from the time diﬀerence between detected photons to obtain the measurement of the second-order photon correlations, g(2)(τ).

26 ﬁeld of quantum optics as it serves as the primary tool to distinguish between coherent, incoherent, and quantum light sources. To distinguish the type of light source one most deal with the time-delay between subsequently emitted photons. The second-order photon correlation, g(2)(τ), is measured by creating a histogram corresponding to the frequency of the time diﬀerence between photon

(2) arrival times, τ = t2 − t1. The lineshape of g (τ) contains unique features that are directly dependent on the type of photon emission process. For example, photons originating from incoherent light sources (blackbody radiation sources such as stars or incandescent light bulbs) have the tendency of arriving together at the detectors, exhibiting a peak within a short time-delay interval due to the “bunching” of photons, g(2)(0) > 1. Meanwhile, photons emitted from a coherent light source (such as a laser) tend to arrive at the detectors randomly, exhibiting a ﬂat line in the second-order correlations, g(2)(τ) ≈ 1. On the other hand, photons emitted from a typical quantum source (a two-level system such as an atom, molecule, or quantum dot) tend to be well spaced, arriving at the detectors one at a time producing a characteristic "anti-bunching" dip in the second-order correlation function, g(2)(0) < 1. This pattern makes clear the importance of the behaviour of the second-order photon corre-

Incoherent 2 Light Sources: ( τ )

( 2 ) Incoherent Coherent Single-photon Coherent

Single-photon Second - order correlations, g 0 0 τ Arrival time Time-delay,τ

Figure 4.2: Second-order photon correlations for incoherent (square), coherent (triangle), and single- photon (circle) sources.

27 lations at τ = 0, however, the temporal aspect alone cannot determine whether photons are indistinguishable. It is important for quantum applications that photons remain indistinguishable from one another in order to avoid nullifying the quantum eﬀects and reducing the emitted photons to a mere classical stream of bits. This entails that the frequency is to be resolved in the second-order correlation measurements. Recent theoretical work by Del Valle et al. has laid out a framework for studying these seemingly hidden correlations in a way that can be compared in a straightforward manner with experiments [92]. Following the supplemental material in Ref. 92, to compute the correlation between

† atomic operators a (t)a(t + t1) , a vector vX,Y (t1) is deﬁned for any two operators X and Y acting on the Hilbert space:

  hX(0)Y (0)i      hX(0)a(t1)Y (0)i    vX,Y (t) =   . (4.2)  X(0)a†(t )Y (0)   1    † X(0)(a a)(t1)Y (0)

The dynamical evolution of vX,Y (t1) is ruled by the matrix M:

  0 0 0 0     − Ω − κ − i∆ 0 Ω   2 2  ∂tvX,Y (t1) = MvX,Y (t1); where M =   , (4.3) − Ω 0 − κ + i∆ Ω   2 2    Ω Ω 0 − 2 − 2 −κ so that the steady-state observables for the two-level system are given by:

1 ss Mt1 0 v = lim v1,1(t1) = lim e 0 . (4.4) t→∞ t→∞ 0

28 † Two matrices T± are then deﬁned which introduce an extra a for T+ and an extra a for T− between X and Y when acting on vX,Y (t1):

  † X(0)a (t1)Y (0)    †   X(0)(a a)(t1)Y (0)    T+vX,Y (t1) =   , (4.5)  X(0)a†2(t )Y (0)   1    †2 X(0)(a a)(t1)Y (0) and   hX(0)a(t1)Y (0)i    2   hX(0)a (t1)Y (0)i    T−vX,Y (t1) =   . (4.6)  X(0)(a†a)(t )Y (0)   1    † 2 X(0)(a a )(t1)Y (0)

Since a†µaν = 0 for any µ or ν > 1, these matrices read:

    0 0 1 0 0 1 0 0         0 0 0 1 0 0 0 0     T+ =   and T− =   . (4.7) 0 0 0 0 0 0 0 1         0 0 0 0 0 0 0 0

With Equations (2.18), (4.4) and (4.7), the one-photon spectrum may be obtained

† through the Fourier transform of a (t)a(t + t1) . As given by Del Valle et al., the power spectrum in its integral form is given by:

" # (1) 1 −1 ss g (ω1) = < T+ T−v , (4.8) Γ1 Γ1 π M + −iω1 − 1 2 1

where [··· ]1 denotes the ﬁrst element of a vector. However, while the one-photon spectrum can be readily obtained by Equation (4.8), the computation of the correlation function becomes challenging even at the second order [93]. The pioneering theoretical work of Del

29 Valle et al. has introduced a method that allows to compute this quantity without the approximations performed before, making calculations manageable even at high orders.

4.1 Sensing Method

The correlations between N photons can be computed exactly using the “sensors method” introduced by Del Valle et al. [92]. This method consist in the introduction of a set of sensors weakly coupled to the dynamics of the open quantum system through the resonance ﬂuorescence as shown in Figure 4.3.

Sensor 1

ω1 ε1 Γ |ei TLS 1 ωL εi ω0 |gi Sensor N εN ωN

ΓN

Figure 4.3: Schematic representation of the sensors theory of frequency-resolved photon correlations. This theory consists on the introduction of N sensors, each with bandwidths Γi, and transition frequencies ωi, weakly coupled to the two-level system (TLS) its resonance ﬂuorescence with strength εi assumed small enough such that the sensors leave the TLS undisturbed.

In the simplest case, each sensor ni (for i = 1, 2, ...N) is modeled as a two-level system

† with resonance frequency ωi, bandwidth Γi, and creation and annihilation operators ςi and D † E ςi such that hni(ti)i = ςi ςi(ti) . Sensors correlations are then computed in the limit of their

30 vanishing coupling εi, by introducing a sensing vector w of steady-state correlators:

 D E  †µ1 ν1 †µN νN ς1 ς1 (t1) . . . ςN ςN (tN )  D E   †µ1 ν1 †µN νN   ς1 ς1 (t1) . . . ςN ςN (tN )a  w =   , (4.9) [µ1ν1,...,µN νN ]  D E   ς†µ1 ςν1 (t ) . . . ς†µN ςνN (t )a†   1 1 1 N N N  D E †µ1 ν1 †µN νN † ς1 ς1 (t1) . . . ςN ςN (tN )a a

where the indices µi and νi take values of 0 or 1. In the supplemental material to Ref. 92,

Del Valle et al. provided the system’s equation of motion in the couplings εi derived from a master equation, which reads:

" N # X Γi ∂ w = M + 1 (µ − ν )iω − (µ + ν ) w t [µ1ν1,...,µN νN ] i i i i i 2 [µ1ν1,...,µN νN ] i=1 (4.10) N X + iεiµiT+w[0νi,...,µN νN ] − iεiνiT−w[µi0,...,µN νN ] , i=1

(N) Using this equation of motion, any relevant correlation function, g (t1, ω1; ...; tN , ωN ), can be obtained by applying the quantum regression theorem. For example, recursively comput-

ing the steady-state solutions (solutions at zero-time delay, that is t1 = t2 = ... = tN = 0), Equation (4.10) becomes:

! ss 1 w[µ ν ,...,µ ν ] = 1 1 N N PN Γi −M − I i=1 (µi − νi)iωi − (µi + νi) 2 N X × iε µ T wss − iε ν T wss , (4.11) i i + [0νi,...,µNνN] i i − [µi0,...,µNνN] i=1

ss ss down to the vector w[00,...,00] = v . The one-photon spectrum through the sensors method can then be calculated by: g(1)(ω ) = < wss . (4.12) Γ1 1 [11] 1

31 Meanwhile, the steady-state two-photon spectrum may be easily computed by:

" ss # (2) w[11,11] gΓ Γ (ω1, ω2) = < . (4.13) 1 2 g(1)(ω )g(1)(ω ) Γ1 1 Γ2 2 1

Furthermore, the steady-state N th-photon spectrum for N ≥ 2 can then be found straight- forwardly as: " ss # (N) w[11,...,11] gΓ ...Γ (ω1, ..., ωN ) = < . (4.14) 1 N QN g(1)(ω ) n=1 Γn n 1 The importance of being able to observe photon correlations in both time and frequency, led us to the discovery of a new type of photon emission which had remained unnoticed.

4.2 The Two-photon Spectrum

Experimentally, the frequency-resolved second-order photon correlations, g(2) (τ, ω , ω ), Γ1Γ2 1 2 associated with the resonance ﬂuorescence scattered light can be measured by altering the

HBT setup in Figure 4.1. Two ﬁlters, with bandwidths Γ1 and Γ2 respectively, are placed at each pathway such that the detected photons will have known frequencies, ω1 and ω2, as shown in Figure 4.4. The “two-photon spectrum” can then be obtained through a histogram of the photon arrival times for a matrix of the ﬁltered photons at frequencies ω1 and ω2 [94].

Filter (Γ1) Detector 1

Photons emitted (1) from a light source (2) Filtered Signal Filter (Γ2) (휔1) Filtered Signal Time-Tagging (휔 ) 2 Electronics Detector 2

Figure 4.4: Schematic representation of the modiﬁed Hanbury Brown & Twiss setup for the measurement of correlations between two ﬁltered photons.

32 The two-photon spectrum measures the correlations between photons detected at τ = 0

for all possible combinations of frequencies ω1 and ω2. This results in a rich two-dimensional landscape as shown in Figure 4.5, wherein the color code indicates the correlation statistics for bunched photons g(2) > 1 (red), anti-bunched photons g(2) < 1 (blue), and no correlations between photons g(2) = 1 (white). This spectrum revealed that most of the interesting quantum emission processes arise away from the Mollow triplet peaks in the one-photon spectrum.

Figure 4.5: Two-photon resonance fluorescence spectrum, for photons filtered at frequencies ω1 and ω2 respectively, arriving simultaneously (τ = t2 − t1 = 0) at the detectors. Theoretical simulation was performed using Equation (4.13) with filter bandwidths Γ1 = Γ2 = 0.33 GHz for a system with decay rate κ = 0.2 GHz driven at a Rabi frequency of Ω = 1.88 GHz and assuming no detunning ∆ = 0.

Before further scrutinizing the origin of the features in the two-photon spectrum, it is helpful to compare it with its predecessor. A direct comparison between g(1)(ω ) and Γ1 1 g(2) (ω , ω ) can be observed by taking the auto-correlations of the two-photon spectrum, Γ1Γ2 1 2 that is, correlations between photons emitted at the same frequency corresponding to the

diagonal of Figure 4.5 at ω1 = ω2.

33 105 (a) 1,+ 1,- )

f 0,+ 104 0,- = ω 1 ( ω 1 ( 1 ) Γ g

103

101 (b) 2,+ 2,-

1,+ )

f 1,- = ω

2 0,+ 0,-

= ω 0

1 10 ( ω 2 Γ 1 ( 2 ) Γ g

10-1 -Ω -Ω/2 0 Ω/2 Ω Relative filter frequency,ω f-ωL

Figure 4.6: Theoretical and experimental measurements between (a) the one-photon spectrum and (b) the two-photon auto-correlation spectrum. The two-photon auto-correlations (i.e., correlations between photons emitted at the same frequency ω1 = ω2) corresponds to the diagonal of the two- photon spectrum in Figure 4.5. The allowed spontaneous dressed-state transitions are shown, each at their corresponding emitted frequency.

Figure 4.6 shows a direct comparison between (a) the ﬁrst-order and (b) second-order auto-correlations measurements. In this juxtaposition, we can clearly observe the features in the two-photon auto-correlations, g(2) (τ = 0, ω = ω ), occurring at the frequency of Γ1Γ2 1 2 the Mollow triplet peaks. The origin of these features can be illustrated in the dressed state picture for the two-photon resonance ﬂuorescence (Figure 4.7). In this picture, disconnected pathways in the transitions between doublets result in photon anti-bunching since the system requires a change of its internal state to shift levels before a second transition can occur. This anti-bunching behaviour is seen with emissions

34 from the Mollow triplet sidebands at frequencies ±Ω. On the other hand, photon bunching from the central peak occurs due to successive transitions illustrated as connected pathways between states. In the full picture, however, Figure 4.6 reveals this bunching sitting at a local maximum. The two-photon spectrum revealed some interesting features in the quantum emission process occurring away from the frequencies of the Mollow triplet peaks. Although the

(2) central peak is indeed bunched, this comes in a region of suppressed bunching, g ' 1, meaning simultaneous transitions are likely, but at a low repetition rate. This occurs because the ﬁrst transition simply reached a state in which is favorable for the second transition to take place. While photon bunching and anti-bunching features are observed at the Mollow triplet central peak and sidebands respectively, the most notable feature in the two-photon auto-correlation spectrum are the high resonances observed at the region in-between these peaks. The high resonance between photons emitted in the region in-between peaks occurs due to another class of transition in the same ladder commonly known as “leapfrog transitions” [95].

Figure 4.7: Dressed-state picture formalism for the quantization of energy states by two-photons.

35 Rather than simply transitioning to a neighbouring doublet, leapfrog transitions in the ladder of states jump over one of these doublets. This happens because the second photon lifts the

E quantization of the spectrum to another state E(N + 2) with frequency = ω1 + ω2. This ~ allows for new transitions in the dressed-states picture to occur through an intermediate “virtual” (highly unstable) state as shown in Figure 4.7. The high instability of the virtual state then allows for subsequent photons to be emitted instantaneously exhibiting photon

(2) “superbunching” in the two-photon spectrum, g (ωf ) 1. Similar results are generalized when turning to higher orders. While the correlations of the peaks retain the same qualitative behaviours, new features associated to the leapfrog transitions thus appear away from the peaks [95]. This have led scientists to explore correlations at higher orders.

36 Chapter 5 Third-Order Photon Correlations of Resonance Fluorescence from a Quantum Dot

Single-photon sources are at the core of the highly sought after quantum technology, utilizing photons as the information carriers. One of the cornerstones of quantum theory is the principle that any property of an object cannot be measured without aﬀecting the object itself. In quantum optics, measuring the light ﬁeld means destroying the photon leaving only vacuum behind. This principle is an important feature in quantum cryptography as it removes the possibility of an eavesdropper. However, for other applications such as ghost imaging and quantum teleportation, it is useful to employ a bundle of N-photons. With a bundle of N-photons, one photon from the bundle may be measured to ‘herald’ the existence of the others. For this purpose, there has been increasing interest in correlation measurements which go beyond the second order. In Chapter4, we have seen how a single-photon source is characterized by determining

(2) the value of the second-order photon correlations, g (τ), at τ = t2 − t1 = 0 to evaluate the suppression of multi-photon states. We also discussed how the frequency-resolved two-photon spectrum, g(2) (τ, ω , ω ), reveals a new type of photon emission where the generation of a Γ1Γ2 1 2 bundle of photons is also possible through the ﬁltered resonance ﬂuorescence from a two- level system. However, just like the two-photon correlations are necessary to measure the quality of detection of single-photons, three-photon correlations measurements are crucial to produce pure quantum states of light associated with two-photon generators.

37 (3) The third-order photon correlation function, g (t1, t2, t3), measures the correlation between the arrival times involving three photons emitted by a continuous source. Early work using a streak camera showed strong photon bunching statistics in third-order correlation measurements of microcavity laser light [96, 97]. The third-order correlation function was shown to provide more reﬁned information about non-classical light sources such as multiphoton diﬀerentiation [98] and the ability to analyze components [99]. The aim of this chapter is to provide the initial steps towards constructing the full three-photon spectrum, g(3) (τ , τ ; ω , ω , ω ), which constitute a coincidence “cube” of photons detected simulta- Γ1Γ2Γ3 1 2 1 2 3

neously (i.e., τ1 = τ2 = 0 where τ1 = t2 − t1 and τ2 = t3 − t1) as shown in Figure 5.1. Here we explore three-photon correlations which make up the three-photon spectrum from single quantum dot resonance fluorescence under strong monochromatic laser illumination. It is worth mentioning these measurements are not specific to quantum dots, but rather are a characteristic of the resonance fluorescence from a two-level system.

Figure 5.1: Three-photon resonance fluorescence spectrum for photons filtered at frequencies ω1, ω2 and ω3 respectively, arriving simultaneously (τ1 = t2 − t1 = 0 and τ2 = t3 − t1 = 0) at the detectors. Theoretical simulation was performed using Equation (4.14) with filter bandwidths Γ1 = Γ2 = Γ3 = 0.33 GHz for a system with decay rate κ = 0.2 GHz driven at a Rabi frequency of Ω = 1.88 GHz and assuming no detunning ∆ = 0.

38 5.1 Three-photon Auto-correlations Measurements

From an experimental point of view, photon arrival times between three photons can be measured from a setup based on a straightforward extension of the Hanbury-Brown and Twiss type measurement wherein light from a source is split into 3-channels, each channel equipped with a spectral filter and detector. This work focuses on InAs Quantun Dots grown by molecular-beam epitaxy held in a cryostat at a base temperature of 4 K and interacting with a resonant wave-guided monochromatic laser beam. The simplest measurements to obtain from the three-photon spectrum are the three- photon auto-correlations, that is, measurements involving a single filter. In this section, we focus exclusively on third-order auto-correlation measurements which constitute a small subset of the data required to reconstruct the complete three-photon spectrum. After briefly describing our experimental setup in Section 5.1.1, we report our measurements in a juxtaposition of the extracted N-Photon spectra of auto-correlations for N = 1, 2 and 3, and present the raw time third-order correlation density maps for the filtering frequencies ωf − ωL ≈ Ω/2 and ωf − ωL ≈ Ω in Section 5.1.2. Furthermore, in Section 5.1.3, theoretical simulations are presented for the ideal case of a two-level system under a more intense driving field, and provided a simple dressed-states interpretation to further understand the origin of its spectral features.

5.1.1 Experimental Setup

Here we probed epitaxially grown InAs QDs held in a cryostat (at a base temperature of 4 K) and strongly interacting with a resonant wave-guided monochromatic laser beam. Our setup, depicted in Figure 5.4, is optimized to efficiently collect the light scattered by a single QD while minimizing unwanted background laser scattering. This optimization was performed by using an orthogonal excitation/detection geometry where QDs were grown between two distributed Bragg reflectors of moderate reflectivity as shown in Figure 5.2[100].

39 The resonance ﬂuorescence from the QD is then collected by lens and a single spectral ﬁlter was placed in the path of the QD scattered light.

Figure 5.2: Photograph of the inside of the cryostation illustrating the orthogonal excitation/detection geometry of the InAs quantum dots samples in a planar microcavity containing alternating layers of GaAs/AlAs Bragg mirrors.

The home-made optical filter consists of a thick solid ethalon, a monolithic transparent cuboid made of fused silica with parallel surfaces reflectively coated [101]. When inserted into the path of the electromagnetic radiation, the etalon acts as an optical cavity with the transmission periodically varying with the optical frequency. In resonance, the reflections from the two surfaces cancel each other via destructive interference. The resonance frequency

of the ﬁlter, ωf , can be adjusted by controlling its temperature to change the index of refraction of the ethalon. For optimal heat distribution, the ethalon is placed between two

Single-stage TEC element

Filtered light In-coming light Reflective surfaces

Figure 5.3: Fabry-Perot etalon with adjustable resonance frequency between two thermoelectric coolers (TEC) for optimal heat distribution.

40 thermoelectric coolers (as shown in Figure 5.3). The bandwidth of the filter, Γf , was fixed to approximately match the radiative decay rate of InAs QDs by adjusting the incidence angle of the laser in the cuboid. The filtered QD scattered light is then split in three pathways using two non-polarizing beam-splitters. The filtered resonance fluorescence is then detected by single photon detector modules (Excelitas SPCM-AQRH-14). Tagging of photon arrival times was performed for each detection channel independently with the help of a router (PicoQuant PHR 800 router with 100 ns cable delay to avoid the router dead-time window) and time-tagging electronics (PicoQuant PicoHarp 300) at an overall detector-limited resolution of about 400 ps. Note that the router is used here for no purpose other than as a replacement for a more expensive multichannel

time-tagging system. Ch.1

Excitation Photon Ch.1

Laser Counter

Time

Ch.2

Router

- Tagging

InAs Ch.3

QD Filter Electronics (ω푓, Г푓) Non-polarizing Beamsplitters ~100푛푠 Ch.4

Delay Ch.0 Cryostat GaAs (4K) Substrate Figure 5.4: Schematic representation of the experimental setup for the measurements of third-order photons auto-correlations. The light near-resonantly scattered by a strongly-driven QD excitonic two-level system was collected by a lens, spectrally ﬁltered using a Fabry-Perot etalon, split in three ways by two non-polarizing beam splitters, and detected at an overall detector-limited resolution of about 400 ps. Time-tagging of photon arrival times was performed on each channel independently.

41 5.1.2 Auto-correlation Measurements

Similarly to its one-photon counterpart, a two-photon source can be characterized by the third-order photon auto-correlations, g(3)(ω , τ , τ ). The third-order photon auto-correlations Γf f 1 2

measures the time difference between the first and second detected photons, τ1 = t2 − t1, and the first and third detected photons, τ2 = t3 − t1, for all three photons filtered at the same frequency, ωf , with a filter resolution Γf . Due to the long exposure times needed to collect sufficiently third-order coincidence events (on the order of days), we focus here on two specific measurements, each with a different filter frequency setting. From the two-photon auto-correlation spectrum, it is evident the areas of interest are located at the bunching, and antibunching, regions of the Mollow triplet (i.e., the regions in- between peaks, and at the sidebands, respectively). For this reason, we focused specifically on measurements of the third-order auto-correlations at the filter frequencies ωf − ωL = Ω/2 and ωf − ωL = Ω. Figure 5.5, shows a juxtaposition of the theoretical and experimental correlations between one, two, and three photons detected at the same frequency window using a fixed Rabi frequency Ω/2π = 1.88 GHz and a filter bandwidth Γf /2π = 0.33 GHz. The radiative decay rate of the QD and the laser-QD detunning are assumed to be κ/2π = 0.2 GHz and ∆ = 0 respectively. Figure 5.5 then reveals the three-photon spectrum exhibits significantly more pronounced photon anti-bunching at ωf − ωL = Ω compared to its two-photon counterpart. Meaning the likelihood of the system emitting three photons simultaneously is far lower than the probability of emitting two photons concurrently at this frequency configuration. In addition, three-photon pathways through virtual states are found to lead to more strongly correlated emission than at second order at ωf − ωL = Ω/2. To better appreciate the two experimental data points in Figure 5.5c, we turn to the raw three-photon time correlations illustrated by creating a histogram of the time-delays τ1 and τ2. These histograms are best visualized as a three-dimensional density plot as shown in Figure 5.6.

42 105 (a) 1,+ 1,-

0,+ ) f 104 0,- ( ω f ( 1 ) Γ g

103

2,+ 101 (b) 2,-

1,+ 1,-

) 0,+ f 0,- ( ω 0

f 10 ( 2 ) Γ g

10-1

102 3,+ (c) 3,- 2,+ 101 2,- 1,+ 1,- ) f

( ω 0 0,+

f 10 Γ ( 3 ) 0,- g

10-1

10-2 -Ω -Ω/2 0 Ω/2 Ω Relative filter frequency,ω f-ωL

Figure 5.5: First (a), second (b), and third (c) order filtered resonance fluorescence spectrum at time- delays τ1 = τ2 = 0. Dressed-state diagrams illustrate the allowed transitions at each corresponding filter frequency.

43 In Figure 5.6(a-b), the experimental and theoretical three-photon time correlation maps

(with a 1 ns bin width) are shown for a filter frequency ωf − ωL ≈ Ω/2. These maps reveal a strong coincidence peak at τ1 = τ2 = 0 demonstrating that, for this filter configuration, photons are more likely to be detected in bunches than spread over time. In addition, the map also reveals diagonal ridges which indicate events in which two photons are detected simultaneously and one photon is accidental are more frequent than events in which all three detectors “click” accidentally at different times. In a second configuration, namely for a filter frequency ωf −ωL ≈ Ω, dispersing correlations are detected producing anti-bunching troughs shown in Figure 5.6(c-d).

Figure 5.6: Third-order correlation maps are shown for experimental (a) and theoretical (b) data for a filtering frequency ωf − ωL ≈ Ω/2, and for experimental (c) and theoretical (d) data for a filtering frequency ωf − ωL ≈ Ω. Measurements were taken using a fixed Rabi frequency Ω/2π = 1.88 GHz, a quantum dot radiating fluorescence at a decay rate of κ/2π = 0.2 GHz, and a filter bandwidth Γf /2π = 0.33 GHz. The detunning between laser and two-level system of is assumed to be ∆ = 0.

44 The addition of a third photon into the system increases the quantization of the spectrum

E to yet another state E(N + 3) with frequency = ω1 + ω2 + ω3. Similar to its two-photon ~ counterpart, a new peak was expected to arise in the three-photon spectrum at the filter frequency ωf − ωL ≈ ±Ω/3 [95]. The absence of this feature in our experiment can be attributed to the low resolution of the used filter relative to the Rabi frequency. More precisely, a ratio between filter bandwidth and Rabi frequency in our experiment of Γf /Ω = 0.18. This can be remedied by either decreasing the filter bandwidth or increasing the Rabi frequency in our experiment, such that the ratio between the filter resolution and the frequency of the laser Rabi oscillations is at least Γf /Ω ≤ 0.1 when the feature at ±Ω/3 initiates to become evident according to theory.

5.1.3 Discussion

To further scrutinize the three-photon auto-correlation measurements in Figures 5.5 and 5.6, we turn to simulated correlations where the only difference is an increased Rabi frequency, specifically Ω/2π = 15 GHz. With the increased laser power, the ratio between the filter resolution and the Rabi frequency of the driving field is now Γf /Ω = 0.02. Un- der this condition, g(3)(ω ) then reveals that, what seemed to be single correlation peaks Γf f in Figure 5.5c, are actually two pairs of correlation peaks as shown in Figure 5.7b.

One set of peaks, at ωf − ωL = ±Ω/2, corresponds to emissions in which only two of the three photons are entangled by a transition through a virtual intermediate state while a third photon originates from a simultaneous but independent transition. This occurs due to the latter being left in a highly unstable state. Meanwhile, the relevant emission process are those occurring between states separated by two rungs were all three transitions are entangled. The new set of peaks on the three-photon auto-correlation spectra originating at the frequencies ωf − ωL = ±Ω/3 are associated with transitions in which all three photons are entangled. At these frequency windows, leap-frog transitions occur over three doublets in the dressed-states ladder via two virtual intermediate states, as shown in Figure 5.8.

45 100 (a)

10-1 )[ Norm. ] f ( ω f

( 1 ) Γ -2 g 10

103 (b)

102 ) f 101 ( ω f ( 3 ) Γ g 100

10-1

-Ω -Ω/2 0 Ω/2 Ω Relative filter frequency,ω f-ωL

Figure 5.7: Simulation for the normalized one-photon spectrum (a) the auto-correlated three-photon spectrum at zero time-delay (b) for which all parameters are identical to those in Figure 5.5 except for and increased Rabi frequency, Ω/2π = 15 GHz.

The larger Rabi frequency relative to the filter bandwidth is the reason that the peaks can be separably viewed in the three-photon spectrum of Figure 5.7b but not in that of Fig- ure 5.5c. While it is straightforward to increase the Rabi frequency in our experiments by increasing the power of the applied laser, the emission rate then becomes low enough that the overall recording time becomes impractical. This problem can likely be remedied by improving the collection and propagation efficiency, as well as the detector quantum efficiency in our experiments (combined these amount to ≈1% here). In addition, other more complex approaches have been proposed to overcome the reduced emission rate with increased

46 Rabi frequency, such as the use of photonic nanowires [102], microlenses [103], broadband enhancement solution using bullseye structures [104], and using the Purcell eﬀect of cavity quantum electrodynamics [105] to enhance the rate of emission. Then, according to the simulations of Figure 5.7b, correlations larger by many orders of magnitude than in the current experiments may be obtained. For the presently considered case of three photons with the same frequency, the distinguishing pathways are the ones from the upper (lower) state of the top rung to the lower (upper) state of the bottom doublet which necessarily must proceed via two virtual interme-

diate states. These cascades require the photons to have frequencies of ωL±Ω/3 and ωL±Ω/2 as shown in Figure 5.8. To better understand the emission process in these cascades, we turn to simulated third-order time correlation maps. Figure 5.9 shows the time correlation maps for the frequencies associated with the three- photon cascades involving virtual states corresponding to the set of peaks at Ω/2 and Ω/3. Accordingly, the transitions involving virtual states at Ω/2 are associated with correlation maps exhibiting bunching ridges at times in which only two out of three photons are detected simultaneously while the third photon is accidental (i.e., at τ1 = 0, τ2 = 0, and at τ1 = τ2) as shown in Figure 5.9a. Meanwhile, the transitions involving virtual states at Ω/3 are associated with correlations maps in which the ridges are nearly vanishing relative to the coincidence peak at τ1 = τ2 = 0 as shown in Figure 5.9b. The vanishing of the horizontal, vertical and diagonal ridges are consequence of the three-photon time-energy entanglement in which two-photon correlations are restricted. An interesting prospect emerging from the N-photon spectrum measurement and analysis is the possibility of generating heralded emission of groups of photons of any desirable number. For example, when considering the pathways analyzed in Figure 5.9b which consist of three photons emitted subsequently, it is clear that no second such sequence is allowed immediately following the ﬁrst since the system is left in a lower dressed state. Therefore, there is a tendency to emit in bundles in between other emission events. This feature has

Figure 5.8: Dressed state picture formalism for the quantization of energy states by three-photons.

been analyzed in detail theoretically and was explicitly veriﬁed using Monte Carlo simulations [92]. With the use of independently tunable ﬁlters, a large parameter space exists to select pathways for the generation of heralded N-photon sequences [95].

5.1.4 Conclusion

In conclusion, spectrally-filtered third-order photon auto-correlations were recorded for resonance fluorescence from an individual semiconductor QD. Only a single filter was used, thus yielding the auto-correlation cross-section of the three-photon spectrum as introduced by Del Valle et al. Our measurements establish a firmer foothold of this concept and show its utility for characterizing non-classical light sources.

48 (a) (b)

Figure 5.9: Simulation for the third-order auto-correlation maps for the ﬁlter frequencies (a) ωf − ωL = Ω/2, and (b) ωf − ωL = Ω/3 corresponding to the pair of superbunching peaks in Figure 5.7b.

Comparing correlations measured for second and third order it can nevertheless already be seen the advances of higher-order correlations. A three-photon spectrum provides more pronounced photon anti-bunching compared to the two-photon spectrum, when the filter frequency matches that of the Mollow triplet sidebands. In addition, three-photon transitions through virtual states corresponding to photons filtered at frequencies between the sidebands and the central peak are found to lead to more strongly correlated emission than at second order adding a new dimension to potential communication schemes in quantum information science applications [95]. Additionally, we discussed how three-photon frequency entanglement could be obtained under high resolution spectroscopy of the resonance fluorescence from a strongly driven two-level system. This experimental investigation have resulted in a peer-reviewed journal: “Third-order frequency-resolved photon correlations in resonance fluorescence” [Nieves et al., Phys. Rev. B 98, 165432 (2018), 77]. In the next section we build up on this investigation through the measurements of photons cross-correlations at third-order, i.e. the use of independently tunable filters for each channel which brings about new experimental challenges.

49 5.2 Three-photon Cross-correlations Measurements

In Section 5.1, we focused on auto-correlation measurements of the three-photon spectrum of resonance fluorescence from a quantum dot, g(3)(τ , τ ; ω ). In other words, three-photon Γf 1 2 f correlations made using only one filter, such that all three emitted photons are detected at the same frequency, ω1 = ω2 = ω3 = ωf . However, the dressed-state picture in the simulation of Figure 5.9(a) shows that a three-photon coincidence can be measured while only two of the three photons are entangled. Therefore, although the photons reaching the detectors in this experiment are filtered at identical frequencies, it is possible to observe interference even if the photons are distinguishable. To measure correlations between distinct photons, one most be able to distinguish the resonance frequencies of each photon and build a histogram of photon arrival times in order to obtain the third-order cross-correlations maps, g(3) (τ , τ ; ω , ω , ω ). That is, correlation Γ1Γ2Γ3 1 2 1 2 3 measurements using three independently tunable filters, each with bandwidth Γi, such that the detected photons have independently known frequencies ωi, where i = 1, 2, 3 indicates each of the independent filters. In this section, we focus exclusively on third-order cross-correlation measurements which constitute a small subset of the data required to reconstruct the complete three-photon spectrum shown in Figure 5.1.

5.2.1 Experimental Setup

The three-photon cross-correlation maps can be measured by altering the setup depicted in Figure 5.4 to include three independent tunable ﬁlters (instead of just one) in order to distinguish between detected photons. This extension is shown in Figure 5.10, both schematically (top) and the actual setup (bottom). A polarized beam-splitter was used to adjust the power of the excitation laser, and therefore manipulate the Rabi frequency to increased the separation between the Mollow triplet peaks.

50 Ch.4

Excitation (ω3, Г3) ~100푛푠 Ch.1

Delay Time

Laser Ch.3

Router

- Tagging (ω2, Г2)

InAs Ch.2

Electronics Ch.1 (ω1, Г1)

Cryostat Ch.0 Non-polarizing Filters (4K) Photon GaAs Beamsplitters Counters Substrate

Polarized Beam-Splitter Lens

Closed-cycle Lens cryostat Filter 3 휔 , Γ Excitation 2 3 3 Laser Non-polarizing 3 Beamsplitters Filter 2 휔2, Γ2 Resonance Fluorescence 1 from QD Filter 1 휔1, Γ1

Figure 5.10: Schematic (top) and actual (bottom) experimental setup for the measurement of three- photon cross-correlations.

Here we use three home-made ﬁlters, similar to the one used in Figure 5.4, consisting of Fabry-Perot etalons inserted between two thermo-electric coolers (as shown in the inset

in Figure 5.10) in order to control the ﬁlter’s resonance frequencies ω1, ω2, and ω3. The

bandwidth of each filter, Γ1/2π = 0.5 GHz, Γ2/2π = 0.75 GHz and Γ3/2π = 0.67 GHz, were fixed to approximately match the linewidths of the peaks in the Mollow triplet by adjusting the incidence angle of the laser in the etalon. Similar to the setup described in Section 5.1.1, the filtered resonance fluorescence is then detected by single photon detector modules while tagging of photon arrival times is performed for each detection channel independently at an overall detector-limited resolution of about 400 ps.

51 5.2.2 Cross-correlation Measurements

As with the auto-correlation measurements in Figure 5.6, cross-correlation maps were recorded by creating a histogram of photon arrival times from each channel, only this time for a set of conﬁgurations from ﬁlter frequencies (ω1, ω2, ω3). The three-photon cross-correlation maps are shown at the bottom of Figure 5.11 where the color code is chosen so that the anti-bunching statistics (g(3) < 1) are shown in blue, the classical correlations (g(3) ≈ 1) in white, and the bunching statistics (g(3) > 1) are indicated in red.

Figure 5.11: Experimental three-photon correlation maps as a function of time delays τ1 = t2 − t1 and τ2 = t3 − t1 with fixed Rabi frequency of Ω/2π ≈ 1.88 GHz for different filtering frequency configurations with filter bandwidths of Γ1/2π = 0.5 GHz, Γ2/2π = 0.75 GHz, and Γ3/2π = 0.67 GHz. Numerals in (a) show areas with different permutations of a fixed order of photon detection events (e.g., quadrant I includes detection events for which t2 < t1 < t3).

Experimentally, we obtained g(3) (τ , τ ; ω , ω , ω ) by measuring the total coincidences Γ1Γ2Γ3 1 2 1 2 3 at τ1 and τ2 (i.e., count of simultaneous detections) and normalizing by the total of accidentals in the same time bin (i.e., the coincidences between a time-delay interval in which detections

52 are uncorrelated, for example, the average of the total coincidences in the anti-diagonal

region where τ1 = −τ2, away from τ1 = τ2 = 0):

Total Coincidences at τ , τ g(3) (τ , τ ; ω , ω , ω ) = 1 2 . (5.1) Γ1,Γ2,Γ3 1 2 1 2 3 Average of Total Accidentals at τ1 = −τ2 6= 0

The spectral lineshapes above the correlation maps in Figure 5.11 show the filter frequency configuration used for each correlation map relative to the power spectrum, wherein the resonance frequency of one filter is set to match one of the sidebands of the Mollow triplet while the other two filters are set to match the opposite sideband. In contrast to the auto-correlation maps (Figure 5.6), the cross-correlation maps in Fig- ure 5.11 contain a mixture of both bunching and anti-bunching statistics. It can be seen that correlations from photons originating from the same sideband remain as anti-bunching troughs, that is, the diagonal τ1 = τ2 in Figure 5.11a, the horizontal at τ2 = 0 in Figure 5.11b,

and the vertical at τ1 = 0 in Figure 5.11c. Meanwhile, the remaining features correspond to coincidence ridges due to two-photon bunching from cross-correlated photons.

Furthermore, it can be seen that the nonclassical character at τ1 = τ2 = 0 in the auto- correlation maps in Figure 5.6, have turned into Poissonian statistics due to the superposition of the second-order auto-correlated and cross-correlated photon statistics. Table 5.1 provides a quantitative comparison of the two and three-photon correlations corresponding to the central feature from each correlation map.

Table 5.1: Raw data for the indicated correlation maps showing the normalized 3rd and 2nd-order correlations at τ1 = τ2 = 0.

Figure g(3) (0, 0; ω , ω , ω ) g(2) (0; ω , ω ) g(2) (0; ω , ω ) g(2) (0; ω , ω ) Γ1Γ2Γ3 1 2 3 Γ1Γ2 1 2 Γ1Γ2 1 3 Γ1Γ2 2 3 5.6a 10.53±0.25 2.97 3.11 3.14 5.6c 0.13±0.04 0.50 0.50 0.49 5.11a 1.14±0.08 1.46 1.51 0.51 5.11b 1.14±0.14 1.51 0.48 1.48 5.11c 1.22±0.13 0.48 1.47 1.60

53 5.2.3 Discussion

To better understand the experimental maps presented here, we may divide them into six regions corresponding to a ﬁxed order of photon detection events as illustrated in Fig- ure 5.11a. For example, the quadrant I (at the upper-left corner) corresponds to a photon

detection order t2 < t1 < t3, meaning that a photon from channel 2 is detected ﬁrst, followed by a photon detected in channel 1, and the last photon detected will result from channel 3. Meanwhile, the quadrant II (at the lower-right corner) corresponds to a channel detection order for which t3 < t1 < t2, meaning a photon in channel 3 is detected ﬁrst, followed by a photon in channel 1, and then by a photon from channel 2. The remaining octants at the top-right and bottom-left corners correspond to the remaining orders of channel detection events t1 < t2 < t3 for octant III, t1 < t3 < t2 for octant IV, t3 < t2 < t1 for octant V, and t2 < t3 < t1 for octant VI. This order is the same for all correlation maps shown here. The locations of interest, however, are at the boundaries between each region, at which two out of three photons are detected simultaneously, or right at the center of the map at

τ1 = τ2 = 0, at which all three photons are detected at the same time. At these boundaries, the presence of bunching or anti-bunching depends on whether it corresponds to emission from alternating sidebands, or if the emission process involves consecutive emission from the same sideband. As done in previous chapters, we turn to the dressed-states formalism to further understand the origin of the features in the cross-correlation maps in Figure 5.11. In the dressed-states picture shown in Figure 5.12, the resonance ﬂuorescence process originating from the “long” sideband of the Mollow triplet, ωL +Ω, can be viewed as a transition from the upper state of a top doublet to the lower state of a bottom doublet (e.g., |1, +i −→|0, −i).

Meanwhile, the photons originating from the “short” sideband, ωL − Ω, occur due to the transitions from the lower state of a top manifold to the upper state of a bottom manifold (e.g., |1, −i −→|0, +i). The two-photon and three-photon emission can then be visualized as a cascade down this ladder as illustrated in Figure 5.12.

54 The vertical, horizontal, and diagonal features of each correlation map in Figure 5.11 are an aftereﬀect of the two-photon cascades (Figure 5.12b). The two-photon cascade from cross-correlated photons accounts for the bunching that occurs due to connected transitions between states. On the other hand, disconnected pathways, and thus photon anti-bunching, are associated with the two-photon cascade from auto-correlated transitions, i.e., second- order events involving photons from the same sideband. Meanwhile, the three-photon correlations consist of three possible cascades which contain a combination of disconnected and connected pathways (Figure 5.12a). The cascades com- posed of disconnected transitions originate when either the ﬁrst or third photon originate

(a) Three-photon Cascades (antibunching) (bunching) |3, +i Ω |3, −i Ω Ω + Ω − − (b) L L L Two-photon Cascades ω ω ω (bunching) (antibunching) |2, +i Ω |2, −i ω ω ω L L L Ω Ω − − Ω + + Ω − − Ω Ω L L L ω ω ω |1, +i Ω |1, −i ω ω ω L L L Ω Ω − − Ω + + Ω − − Ω Ω L L L ω ω ω |0, +i Ω |0, −i

Figure 5.12: Dressed-state picture formalism for (a) three-photon correlations corresponding to the origin of central feature of all correlation maps in Figure 5.11, and (b) two-photon correlations corresponding to the origin of the vertical, horizontal, and diagonal features of each correlation map.

55 from the long sideband of the Mollow triplet. This results in the need of simultaneous emission of photons originating from the short sideband, however, since the internal state of the system must change before a second emission from the same sideband can take place, these cascades result in photon anti-bunching. In turn, it is more favorable for the system that the secondly emitted photon originates from the long sideband allowing the bunching of three photons. As such, the three-photon correlations corresponding to the the central feature of all cross-correlation maps in Figure 5.11, that is g(3) (0, 0; ω , ω , ω ), is a consequence of Γ1Γ2Γ3 1 2 3 the superposition of all three possible cascades. This results in the suppressed bunching,

(3) g ' 1, as seen in Table 5.1.

These central features at τ1 = τ2 = 0 of the experimental maps constitute a small subset of the data required to reconstruct the complete three-photon spectrum of resonance fluorescence. Specifically, by repeating such measurements for a matrix of frequency triplets (ω , ω , ω ) and logging the normalized coincidence rate, an entire g(3) (0, 0; ω , ω , ω ) 1 2 3 Γ1,Γ2,Γ3 1 2 3 coincidence “cube” would be obtained as shown in Figure 5.1. It is worth mentioning these measurements are not specific to quantum dots, but rather are a characteristic of the resonance fluorescence from a two-level system. In practice, however, many hurdles remain before such a quantity can be measured. An obvious obstacle in the measurement of three-photon correlations are the impractically long recording times that would be required. This limitation could at least in part be remedied by improving the collection and propagation efficiency, as well as the detector quantum efficiency in our experiments, or by using more complex approaches, for instance those which employ photonic nanowires [102], microlenses [103], broadband enhancement solutions using bullseye structures [104] or in general make use of the Purcell effect of cavity quantum electrodynamics [105]. Another complication that may introduce non-idealities is the stabilization and uniformity of the properties of the spectral filters. For instance, the filters resonance frequencies will inevitably fluctuate during the measurement as will the filter bandwidth. In addition,

56 the precise width of each spectral filter may be difficult to set to an arbitrary independent value, and to maintain over time. Small variations in filter bandwidths are particularly consequential when the filter bandwidth is near the natural linewidth of the emitter. In fact it was recently shown that, even in the weak excitation regime, the photon statistics of a filtered single QD’s resonance fluorescence depend dramatically on the filter bandwidth [87]. In the measurements presented here, the filters had slightly different bandwidths, which explains in part the variations between the three panels in Figure 5.11. Were it not for slightly different properties of individual filters the three configurations would be equivalent to those obtained by simply switching electric cables between detectors and time-tagging electronics. Lastly, the transition frequency of the QD being probed will fluctuate to some extent. This “spectral diffusion” results in inhomogeneous broadening, i.e., an averaging over different QD resonance frequencies. To examine the consequences of spectral diffusion under various filter bandwidths, we turn to theoretical simulations of the one-photon spectrum using Equation (3.7). Figure 5.13 shows the normalized power spectrum obtained experimentally by scanning a filter with bandwidth Γ /2π = 0.33 GHz, and the theoretical simulations for g(1)(ω ) using f Γf f the same bandwidth as the experimental trace, and for Γf /2π = 0.75 GHz which matches the largest bandwidth involved in the cross-correlation maps in Figure 5.11. The theoretical power spectrum in the absence of spectral diffusion, obtained directly using Equation (3.6), is also included in order to observed its impact in the experimental measurements. From Figure 5.13 it is clear that spectral diffusion substantially impacts the measured correlations. In particular, it leads to a greater central peak magnitude relative to the sideband magnitude compared to the case of a purely radiatively broadened two-level system. In that sense spectral diffusion causes an effectively reduced interaction strength since, during part of the measurement, the laser is off-resonance with the QD. In addition, Figure 5.13 reveals how a larger filter bandwidth increases the spectral over- lap between the tails of the central peak and the sidebands, even more so in the presence of

57 0.3 1.2 Experiment:Γ f/2π= 0.33GHz Γf/2π= 0.33GHz(with SD)

Γf/2π= 0.33GHz 1.0 0.2 Γf/2π= 0.75GHz(with SD)

Γf/2π= 0.75GHz 0.8 0.1 )( arb. units ) f 0.6 ( ω f ( 1 ) Γ 1 2 0.4 Norm. g 0.2

0.0 -3-2-1 0 1 2 3 (ωf -ω L)/2π(GHz)

Figure 5.13: Experimental and theoretical Mollow triplet lineshape with Rabi frequency Ω/2π = 1.88 GHz for filter bandwidths Γf /2π = 0.33 GHz (black) and Γf /2π = 0.75 GHz (red). Theoret- ical traces with (solid) and without (dashed) spectral diffusion (SD) are based on Equations (3.6) and (3.7) respectively, with a radiative decay rate κ/2π = 0.2 GHz. spectral diffusion. This source of “contamination” may be in principle avoided by increasing the Rabi frequency, which was not possible in our current experimental configuration. Altogether, variations over time in the QD resonance frequency and in the filter properties reduce the degree of bunching and anti-bunching in the maps of Figure 5.11.

5.2.4 Conclusion

In summary, spectrally-filtered third-order photon cross-correlations were recorded for resonance fluorescence from an individual semiconductor quantum dot under strong laser illumination. In this section, we presented considerably more challenging measurements involving three separate filter configurations. Namely, those for which the resonance frequency of two of the filters matches the lower frequency sideband of the Mollow triplet, while the remaining filter is set at the opposite sideband. The resulting correlation maps provide a

58 stepping-stone towards constructing a complete three-photon spectrum. Furthermore, the revelation of a rich landscape of bunching and antibunching features opens the way for potential new quantum devices capable of producing two-photon bunching or antibunching on demand. Additionally, our measurements highlight how, in practice, these measurements are sensitive to filter bandwidths and spectral fluctuations of the QD resonance frequency over time. Future progress will likely hinge upon technical filter improvements and operation at higher Rabi frequency. A complete three photon spectrum may be mapped for quantum dot resonance fluorescence, but also any other light source, so long as photon emission rates are large enough. This experimental investigation have resulted in a peer-reviewed journal: “Third-order photon cross-correlations in resonance fluorescence” [Nieves et al., Phys. Rev. B 102, 155418 (2020), 78]. Future directions include measurements of the three-photon spectrum at different frequency configurations. Additionally, one order up, measurements for the fourth-order photon correlations, g(4) (ω , ω , ω , ω ), are possible through a simple τ1,τ2,τ3;Γ1Γ2Γ3Γ4 1 2 3 4 extension of the setup in Figure 5.10 by using another beamsplitter in order to divide the light scattered by the QD in four pathways. This will allow us to move towards the first step of detecting pure three-photon states and provide more clear indication of the nonclassical character of the light emitted due to resonance fluorescence of a two-level system.

59 Chapter 6 Concluding Remarks

The resonance fluorescence originating from a two-level system has been shown to provide a unique opportunity to study a host of fascinating phenomena. The ability to control the photon emission from resonance fluorescence has sparked an increasing interest in the field of quantum optics due to their potential applications in the development of highly sought after quantum technologies. As such, the first-order and second-order photon correlations continue to be researched extensively providing invaluable information about the light and the mechanism underlying its generation. The main goal of this study was to investigate the majorly unexplored frequency-resolved third-order photon correlations of the resonance fluorescence from two-level system. This work presented experimental progress towards constructing the complete spectrally filtered three-photon spectrum from a single semiconductor quantum dot under strong monochromatic light illumination. First, we investigated third-order autocorrelation measurements in which photons were identically filtered at the two regions of the Mollow triplet where the two-photon auto- correlation spectrum showed fascinating photon statistics. A comparison with the correlation maps computed using the “sensors method" introduced by Del Valle et al. [92] reveals faithful agreement with theory. The three-photon autocorrelation measurements demonstrated significantly more pronounced photon antibunching compared to its two-photon counterpart. In addition, three-photon pathways through virtual states are found to lead to more strongly correlated emission than at second order. We characterized the correlations asso-

60 ciated with these virtual transitions, and compared them with correlations at other Mollow triplet frequency windows. Additionally, we investigated third-order cross-correlations, where two out of three photons originates from one of the Mollow triplet sidebands while the third photon originate from the opposite sideband of the power spectrum. The three-photon cross-correlation measurements resulted in correlation maps with antibunching features as a consequence of correlations arising from the same sideband, and bunching ridges due to opposite sideband correlated photons. The revelation of a rich landscape of bunching and antibunching features opens the way for potential new quantum devices capable of producing two-photon bunching or antibunching on demand. Furthermore, our measurements highlight how, in practice, these measurements are sensitive to filter bandwidths and spectral fluctuations of the QD resonance frequency over time. Future progress will likely hinge upon technical filter improvements and operation at higher Rabi frequency. A complete three-photon spectrum may be mapped for quantum dot resonance fluorescence, but also any other light source, so long as photon emission rates are large enough.

61 Bibliography

[1] A. A. Michelson and E. W. Morley. “On the relative motion of the earth and the luminiferous ether”. In: Am. J. Sci. 34.203 (1887). url: https://history.aip.org/ exhibits/gap/PDF/michelson.pdf. [2] A. Einstein et al. “On the electrodynamics of moving bodies”. In: Annalen der physik 17.10 (1905), pp. 891–921. url: http://chiuphysics.cgu.edu.tw/yun-ju/cguweb/ PhyChiu/H305Relativity/EinsteinSpecialRelativity.pdf. [3] M. Planck. “On the Law of Distribution of Energy in the Normal Spectrum”. In: Ann. Phys. 4 (1901), 553ﬀ. url: http://strangepaths.com/files/planck1901.pdf. [4] A. Einstein. “On a heuristic point of view about the creation and conversion of light”. In: Ann. Phys. 17.6 (1905), pp. 132–148. url: https://einsteinpapers.press. princeton.edu/vol2-trans/100. [5] R. H. Brown and R. Q. Twiss. “Correlation between photons in two coherent beams of light”. In: Nature (London) 177.4497 (1956), pp. 27–29. url: http://einstein. drexel.edu/~bob/PHYS518/brown-twiss.pdf. [6] I. Silva and O. Freire. “The Concept of the Photon in Question: The Controversy Surrounding the HBT Eﬀect circa 1956–1958”. In: Hist. Stud. Nat. Sci. 43.4 (2013), pp. 453–491. url: https://www.jstor.org/stable/pdf/10.1525/hsns.2013.43. 4.453.pdf?refreqid=excelsior%3Af4efc769cea735f66d4527a3ea6b3fbd. [7] E. M. Purcell. “The question of correlation between photons in coherent light rays”. In: Nature 178.4548 (1956), p. 1449. url: https://doi.org/10.1038/1781449a0. [8] C. K. Hong, Z. Y. Ou, and L. Mandel. “Measurement of subpicosecond time intervals between two photons by interference”. In: Phys. Rev. Lett. 59 (18 1987), pp. 2044– 2046. url: https://link.aps.org/doi/10.1103/PhysRevLett.59.2044. [9] R. J. Glauber. “Nobel Lecture: One hundred years of light quanta”. In: Rev. Mod. Phys. 78.4 (2006), p. 1267. url: https://journals.aps.org/rmp/pdf/10.1103/ RevModPhys.78.1267. [10] S. Bounouar, M. Strauß, A. Carmele, P. Schnauber, A. Thoma, M. Gschrey, J.-H. Schulze, A. Strittmatter, S. Rodt, A. Knorr, and S. Reitzenstein. “Path-Controlled Time Reordering of Paired Photons in a Dressed Three-Level Cascade”. In: Phys. Rev. Lett. 118 (23 2017), p. 233601. url: https://link.aps.org/doi/10.1103/ PhysRevLett.118.233601. [11] B. Lounis and M. Orrit. “Single-photon sources”. In: Rep. Prog. Phys. 68.5 (2005), p. 1129. url: https://iopscience.iop.org/article/10.1088/0034-4885/68/5/ R04.

62 [12] M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov. “Invited review article: Single-photon sources and detectors”. In: Review of scientific instruments 82.7 (2011), p. 071101. url: https://doi.org/10.1063/1.3610677. [13] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn. “Linear optical quantum computing with photonic qubits”. In: Rev. Mod. Phys. 79 (1 2007), pp. 135–174. url: https://link.aps.org/doi/10.1103/RevModPhys.79. 135. [14] P. W. Shor. “Algorithms for quantum computation: Discrete logarithms and factor- ing”. In: Proceedings 35th annual symposium on foundations of computer science. IEEE. 1994, pp. 124–134. url: https://ieeexplore.ieee.org/document/365700. [15] E. Knill, R. Laflamme, and G. J. Milburn. “A scheme for efficient quantum computation with linear optics”. In: Nature 409.6816 (2001), p. 46. url: https://www. nature.com/articles/35051009.pdf. [16] W.-B. Gao, P. Xu, X.-C. Yao, O. Gühne, A. Cabello, C.-Y. Lu, C.-Z. Peng, Z.-B. Chen, and J.-W. Pan. “Experimental Realization of a Controlled-NOT Gate with Four-Photon Six-Qubit Cluster States”. In: Phys. Rev. Lett. 104 (2 2010), p. 020501. url: https://link.aps.org/doi/10.1103/PhysRevLett.104.020501. [17] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien. “Quantum computers”. In: Nature (London) 464.7285 (2010), p. 45. url: https : //www.nature.com/articles/nature08812.pdf. [18] A. K. Ekert. “Quantum cryptography based on Bell’s theorem”. In: Phys. Rev. Lett. 67 (6 1991), pp. 661–663. url: https://link.aps.org/doi/10.1103/PhysRevLett. 67.661. [19] W. T. Buttler, R. J. Hughes, P. G. Kwiat, S. K. Lamoreaux, G. G. Luther, G. L. Morgan, J. E. Nordholt, C. G. Peterson, and C. M. Simmons. “Practical Free-Space Quantum Key Distribution over 1 km”. In: Phys. Rev. Lett. 81 (15 1998), pp. 3283– 3286. url: https://link.aps.org/doi/10.1103/PhysRevLett.81.3283. [20] W. Tittel, J. Brendel, H. Zbinden, and N. Gisin. “Violation of Bell Inequalities by Photons More Than 10 km Apart”. In: Phys. Rev. Lett. 81 (17 1998), pp. 3563–3566. url: https://link.aps.org/doi/10.1103/PhysRevLett.81.3563. [21] A. Soujaeff, T. Nishioka, T. Hasegawa, S. Takeuchi, T. Tsurumaru, K. Sasaki, and M. Matsui. “Quantum key distribution at 1550 nm using a pulse heralded single photon source”. In: Opt. Express 15.2 (2007), pp. 726–734. url: https : / / www . osapublishing.org/oe/abstract.cfm?uri=oe-15-2-726. [22] C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin. “Experimental quantum cryptography”. In: Journal of Cryptology 5.1 (1992), pp. 3–28. issn: 1432- 1378. url: https://doi.org/10.1007/BF00191318. [23] H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller. “Quantum Repeaters: The Role of Imperfect Local Operations in Quantum Communication”. In: Phys. Rev. Lett. 81 (26 1998), pp. 5932–5935. url: https://link.aps.org/doi/10.1103/PhysRevLett. 81.5932.

63 [24] R. Van Meter and J. Touch. “Designing quantum repeater networks”. In: IEEE Com- munications Magazine 51.8 (2013), pp. 64–71. url: https://ieeexplore.ieee. org/stamp/stamp.jsp?tp=&arnumber=6576340. [25] T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko. “Optical imaging by means of two-photon quantum entanglement”. In: Phys. Rev. A 52 (5 1995), R3429– R3432. url: https://link.aps.org/doi/10.1103/PhysRevA.52.R3429. [26] D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih. “Observation of Two-Photon “Ghost” Interference and Diffraction”. In: Phys. Rev. Lett. 74 (18 1995), pp. 3600–3603. url: https://link.aps.org/doi/10.1103/PhysRevLett.74.3600. [27] C. H. Monken, P. H. Souto Ribeiro, and S. Pádua. “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion”. In: Phys. Rev. A 57 (4 1998), pp. 3123–3126. url: https://link.aps.org/doi/10.1103/PhysRevA.57. 3123. [28] D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger. “Experimental quantum teleportation”. In: Nature (London) 390.6660 (1997), p. 575. url: https://www.nature.com/articles/37539.pdf. [29] D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu. “Experimental Re- alization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels”. In: Phys. Rev. Lett. 80 (6 1998), pp. 1121–1125. url: https://link.aps.org/doi/10.1103/PhysRevLett.80.1121. [30] M. Riebe, H. Häffner, C. Roos, W. Hänsel, J. Benhelm, G. Lancaster, T. Körber, C. Becher, F. Schmidt-Kaler, D. James, et al. “Deterministic quantum teleportation with atoms”. In: Nature (London) 429.6993 (2004), p. 734. url: https://www.nature. com/articles/nature02570.pdf. [31] T. Müller, M. Anderson, J. Huwer, J. Skiba-Szymanska, A. B. Krysa, R. M. Steven- son, J. Heffernan, D. A. Ritchie, and A. J. Shields. “Quantum teleportation using coherent emission from telecom C-band quantum dots”. In: Quantum Information and Measurement. Optical Society of America. Washington, DC, 2019, S2A–S24. url: https://doi.org/10.1364/QIM.2019.S2A.4. [32] V. Giovannetti, S. Lloyd, and L. Maccone. “Quantum Metrology”. In: Phys. Rev. Lett. 96 (1 2006), p. 010401. url: https://link.aps.org/doi/10.1103/PhysRevLett. 96.010401. [33] J. Joo, W. J. Munro, and T. P. Spiller. “Quantum Metrology with Entangled Coherent States”. In: Phys. Rev. Lett. 107 (8 2011), p. 083601. url: https://link.aps.org/ doi/10.1103/PhysRevLett.107.083601. [34] G. Tóth and I. Apellaniz. “Quantum metrology from a quantum information science perspective”. In: J. Phys. A-Math. Theor. 47.42 (2014), p. 424006. url: https:// doi.org/10.1088/1751-8113/47/42/424006. [35] W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus. “Improved quantum metrology using quantum error correction”. In: Phys. Rev. Lett. 112.8 (2014), p. 080801. doi: 10.1103/PhysRevLett.112.080801.

64 [36] R. Ghosh and L. Mandel. “Observation of nonclassical effects in the interference of two photons”. In: Phys. Rev. Lett. 59 (17 1987), pp. 1903–1905. url: https://link. aps.org/doi/10.1103/PhysRevLett.59.1903. [37] Y. H. Shih and C. O. Alley. “New Type of Einstein-Podolsky-Rosen-Bohm Experiment Using Pairs of Light Quanta Produced by Optical Parametric Down Conversion”. In: Phys. Rev. Lett. 61 (26 1988), pp. 2921–2924. url: https://link.aps.org/doi/ 10.1103/PhysRevLett.61.2921. [38] P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih. “New high-intensity source of polarization-entangled photon pairs”. In: Physical Re- view Letters 75.24 (1995), p. 4337. url: https://journals.aps.org/prl/abstract/ 10.1103/PhysRevLett.75.4337. [39] E. Waks, E. Diamanti, and Y. Yamamoto. “Generation of photon number states”. In: New J. Phys. 8.1 (2006), p. 4. url: https://doi.org/10.1088/1367-2630/8/1/004. [40] A. B. U’Ren, C. Silberhorn, K. Banaszek, and I. A. Walmsley. “Efficient Conditional Preparation of High-Fidelity Single Photon States for Fiber-Optic Quantum Net- works”. In: Phys. Rev. Lett. 93 (9 2004), p. 093601. url: https://link.aps.org/ doi/10.1103/PhysRevLett.93.093601. [41] J. Brendel, N. Gisin, W. Tittel, and H. Zbinden. “Pulsed Energy-Time Entangled Twin-Photon Source for Quantum Communication”. In: Phys. Rev. Lett. 82 (12 1999), pp. 2594–2597. url: https://link.aps.org/doi/10.1103/PhysRevLett.82.2594. [42] L. J. Wang, C. K. Hong, and S. R. Friberg. “Generation of correlated photons via four- wave mixing in optical fibres”. In: Journal of Optics B: Quantum and Semiclassical Optics 3.5 (2001), pp. 346–352. url: https://doi.org/10.1088%2F1464-4266% 2F3%2F5%2F311. [43] E. A. Goldschmidt, M. D. Eisaman, J. Fan, S. V. Polyakov, and A. Migdall. “Spec- trally bright and broad fiber-based heralded single-photon source”. In: Phys. Rev. A 78 (1 2008), p. 013844. url: https://link.aps.org/doi/10.1103/PhysRevA.78. 013844. [44] B. Fan, Z. Duan, L. Zhou, C. Yuan, Z. Y. Ou, and W. Zhang. “Generation of a single- photon source via a four-wave mixing process in a cavity”. In: Phys. Rev. A 80 (6 2009), p. 063809. url: https://link.aps.org/doi/10.1103/PhysRevA.80.063809. [45] A. Ulhaq, S. Weiler, S. M. Ulrich, R. Roßbach, M. Jetter, and P. Michler. “Cascaded single-photon emission from the Mollow triplet sidebands of a quantum dot”. In: Nat. Photonics 6.4 (2012), p. 238. doi: 10.1038/nphoton.2012.23. [46] H. J. Kimble, M. Dagenais, and L. Mandel. “Photon Antibunching in Resonance Fluorescence”. In: Phys. Rev. Lett. 39 (11 1977), pp. 691–695. url: https://link. aps.org/doi/10.1103/PhysRevLett.39.691. [47] P. Grangier, G. Roger, and A. Aspect. “Experimental evidence for a photon anticor- relation effect on a beam splitter: a new light on single-photon interferences”. In: EPL (Europhysics Letters) 1.4 (1986), p. 173. url: https://doi.org/10.1209/0295- 5075/1/4/004.

65 [48] M. Hennrich, T. Legero, A. Kuhn, and G. Rempe. “Photon statistics of a non- stationary periodically driven single-photon source”. In: New Journal of Physics 6.1 (2004), p. 86. url: https://doi.org/10.1088/1367-2630/6/1/086. [49] J. McKeever, A. Boca, A. D. Boozer, R. Miller, J. R. Buck, A. Kuzmich, and H. J. Kimble. “Deterministic Generation of Single Photons from One Atom Trapped in a Cavity”. In: Science 303.5666 (2004), pp. 1992–1994. issn: 0036-8075. url: https: //science.sciencemag.org/content/303/5666/1992. [50] T. Basché, W. E. Moerner, M. Orrit, and H. Talon. “Photon antibunching in the ﬂuorescence of a single dye molecule trapped in a solid”. In: Phys. Rev. Lett. 69 (10 1992), pp. 1516–1519. url: https://link.aps.org/doi/10.1103/PhysRevLett. 69.1516. [51] B. Lounis and W. E. Moerner. “Single photons on demand from a single molecule at room temperature”. In: Nature 407.6803 (2000), p. 491. url: https://doi.org/10. 1038/35035032. [52] M. Steiner, A. Hartschuh, R. Korlacki, and A. J. Meixner. “Highly eﬃcient, tunable single photon source based on single molecules”. In: Appl. Phys. Lett. 90.18 (2007), p. 183122. url: https://doi.org/10.1063/1.2736294. [53] C. Maurer, C. Becher, C. Russo, J. Eschner, and R. Blatt. “A single-photon source based on a single Ca+ ion”. In: New Journal of Physics 6 (2004), pp. 94–94. url: https://doi.org/10.1088%2F1367-2630%2F6%2F1%2F094. [54] M. Keller, B. Lange, K. Hayasaka, W. Lange, and H. Walther. “Continuous generation of single photons with controlled waveform in an ion-trap cavity system”. In: Nature 431.7012 (2004), p. 1075. url: https://doi.org/10.1038/nature02961. [55] J. Kim, O. Benson, H. Kan, and Y. Yamamoto. “A single-photon turnstile device”. In: Nature 397.6719 (1999), p. 500. url: https://doi.org/10.1038/17295. [56] C. Kurtsiefer, S. Mayer, P. Zarda, and H. Weinfurter. “Stable Solid-State Source of Single Photons”. In: Phys. Rev. Lett. 85 (2 2000), pp. 290–293. url: https://link. aps.org/doi/10.1103/PhysRevLett.85.290. [57] A. Beveratos, R. Brouri, T. Gacoin, J.-P. Poizat, and P. Grangier. “Nonclassical radiation from diamond nanocrystals”. In: Phys. Rev. A 64 (6 2001), p. 061802. url: https://link.aps.org/doi/10.1103/PhysRevA.64.061802. [58] R. Alléaume, F. Treussart, G. Messin, Y. Dumeige, J.-F. Roch, A. Beveratos, R. Brouri-Tualle, J.-P. Poizat, and P. Grangier. “Experimental open-air quantum key distribution with a single-photon source”. In: New Journal of physics 6.1 (2004), p. 92. url: https://doi.org/10.1088/1367-2630/6/1/092. [59] E. Wu, J. Rabeau, G. Roger, F. Treussart, H. Zeng, P. Grangier, S. Prawer, and J.-F. Roch. “Room temperature triggered single-photon source in the near infrared”. In: New J. Phys. 9.12 (2007), p. 434. url: https://doi.org/10.1088/1367- 2630/9/12/434.

66 [60] S. Buckley, K. Rivoire, and J. Vučković. “Engineered quantum dot single-photon sources”. In: Reports on Progress in Physics 75.12 (2012), p. 126503. url: https: //doi.org/10.1088/0034-4885/75/12/126503. [61] S. Ihara, A. Andreev, D. Williams, T. Kodera, and S. Oda. “Quantum dots in single electron transistors with ultrathin silicon-on-insulator structures”. In: Applied Physics Letters 107.1 (2015), p. 013102. url: https://doi.org/10.1063/1.4926335. [62] A. J. Nozik, M. C. Beard, J. M. Luther, M. Law, R. J. Ellingson, and J. C. Johnson. “Semiconductor quantum dots and quantum dot arrays and applications of multiple exciton generation to third-generation photovoltaic solar cells”. In: Chemical reviews 110.11 (2010), pp. 6873–6890. url: https://doi.org/10.1021/cr900289f. [63] M. N. Rhyner, A. M. Smith, X. Gao, H. Mao, L. Yang, and S. Nie. “Quantum dots and multifunctional nanoparticles: new contrast agents for tumor imaging”. In: (2006). url: https://doi.org/10.2217/17435889.1.2.209. [64] T. Edvinsson. “Optical quantum conﬁnement and photocatalytic properties in two-, one-, and zero-dimensional nanostructures”. In: R. Soc. Open Sci. 5.9 (2018), p. 180387. url: https://doi.org/10.1098/rsos.180387. [65] B. Valeur and M. N. Berberan-Santos. “A brief history of ﬂuorescence and phosphorescence before the emergence of quantum theory”. In: J. Chem. Educ. 88.6 (2011), pp. 731–738. url: https://doi.org/10.1021/ed100182h. [66] D. L. Klein, R. Roth, A. K. Lim, A. P. Alivisatos, and P. L. McEuen. “A single- electron transistor made from a cadmium selenide nanocrystal”. In: Nature 389.6652 (1997), pp. 699–701. url: https://doi.org/10.1038/39535. [67] D. Loss and D. P. DiVincenzo. “Quantum computation with quantum dots”. In: Phys. Rev. A 57 (1 1998), pp. 120–126. url: https://link.aps.org/doi/10.1103/ PhysRevA.57.120. [68] V. Klimov, A. Mikhailovsky, S. Xu, A. Malko, J. Hollingsworth, a. C. Leatherdale, H.-J. Eisler, and M. Bawendi. “Optical gain and stimulated emission in nanocrystal quantum dots”. In: Science 290.5490 (2000), pp. 314–317. url: https://doi.org/ 10.1126/science.290.5490.314. [69] M. Jacobsohn and U. Banin. “Size dependence of second harmonic generation in CdSe nanocrystal quantum dots”. In: J. Phys. Chem. B 104.1 (2000), pp. 1–5. url: https://doi.org/10.1021/jp9925076. [70] N.-M. Park, T.-S. Kim, and S.-J. Park. “Band gap engineering of amorphous silicon quantum dots for light-emitting diodes”. In: App. Phys. Lett. 78.17 (2001), pp. 2575– 2577. url: https://doi.org/10.1063/1.1367277. [71] A. M. Smith, S. Dave, S. Nie, L. True, and X. Gao. “Multicolor quantum dots for molecular diagnostics of cancer”. In: Expert Rev. Mol. Diagn. 6.2 (2006), pp. 231–244. url: https://doi.org/10.1586/14737159.6.2.231. [72] P. Senellart, G. Solomon, and A. White. “High-performance semiconductor quantum- dot single-photon sources”. In: Nat. Nanotechnol. 12.11 (2017), p. 1026. url: https: //doi.org/10.1038/NNANO.2017.218.

67 [73] Y. Pu, F. Cai, D. Wang, J.-X. Wang, and J.-F. Chen. “Colloidal synthesis of semiconductor quantum dots toward large-scale production: a review”. In: Ind. Eng. Chem. Res. 57.6 (2018), pp. 1790–1802. url: https://doi.org/10.1021/acs.iecr. 7b04836. [74] N. Ledentsov, M. Maximov, D. Bimberg, T. Maka, C. S. Torres, I. Kochnev, I. Krest- nikov, V. Lantratov, N. Cherkashin, Y. M. Musikhin, et al. “1.3 µm luminescence and gain from defect-free InGaAs-GaAs quantum dots grown by metal-organic chemical vapour deposition”. In: Semicond. Sci. Technol. 15.6 (2000), p. 604. url: https : //doi.org/10.1088/0268-1242/15/6/320. [75] W. Seifert, N. Carlsson, M. Miller, M.-E. Pistol, L. Samuelson, and L. R. Wallenberg. “In-situ growth of quantum dot structures by the Stranski-Krastanow growth mode”. In: Progress in Crystal Growth and Characterization of Materials 33.4 (1996), pp. 423– 471. url: https://doi.org/10.1016/S0960-8974(96)00090-3. [76] G. Costantini, A. Rastelli, C. Manzano, P. Acosta-Diaz, R. Songmuang, G. Katsaros, O. G. Schmidt, and K. Kern. “Interplay between Thermodynamics and Kinetics in the Capping of InAs/GaAs(001) Quantum Dots”. In: Phys. Rev. Lett. 96 (22 2006), p. 226106. url: https://link.aps.org/doi/10.1103/PhysRevLett.96.226106. [77] Y. Nieves and A. Muller. “Third-order frequency-resolved photon correlations in resonance fluorescence”. In: Phys. Rev. B 98 (16 2018), p. 165432. url: https://link. aps.org/doi/10.1103/PhysRevB.98.165432. [78] Y. Nieves and A. Muller. “Third-order photon cross-correlations in resonance fluorescence”. In: Phys. Rev. B 102 (15 2020), p. 155418. url: https://link.aps.org/ doi/10.1103/PhysRevB.102.155418. [79] M. A. Kastner. “Artificial atoms”. In: Physics today 46 (1993), pp. 24–24. url: https: //doi.org/10.1063/1.881393. [80] L. Allen and J. H. Eberly. Optical resonance and two-level atoms. Vol. 28. Courier Corporation, 1987. isbn: 978-0486655338. url: https://books.google.com/books? vid=ISBN9780486655338. [81] C. Cohen-Tannoudji and S. Reynaud. “Atoms in strong light-fields: Photon antibunching in single atom fluorescence”. In: Phil. Trans. R. Soc. Lond. A 293.1402 (1979), pp. 223–237. doi: 10.1098/rsta.1979.0092. [82] G. Lindblad. “On the generators of quantum dynamical semigroups”. In: Communi- cations in Mathematical Physics 48.2 (1976), pp. 119–130. url: https://doi.org/ 10.1007/BF01608499. [83] L. Knöll and G. Weber. “Theory of n-fold time-resolved correlation spectroscopy and its application to resonance fluorescence radiation”. In: J. Phys. B: At. Mol. Phys. 19.18 (1986), p. 2817. url: http://iopscience.iop.org/article/10.1088/0022- 3700/19/18/012/meta. [84] J. Cresser. “Intensity correlations of frequency-filtered light fields”. In: Journal of Physics B: Atomic and Molecular Physics 20.18 (1987), p. 4915. url: https://doi. org/10.1088/0022-3700/20/18/027.

68 [85] L. Knöll, W. Vogel, and D.-G. Welsch. “Spectral properties of light in quantum optics”. In: Phys. Rev. A 42 (1 1990), pp. 503–522. url: https://link.aps.org/doi/10. 1103/PhysRevA.42.503. [86] B. R. Mollow. “Power Spectrum of Light Scattered by Two-Level Systems”. In: Phys. Rev. 188 (5 1969), pp. 1969–1975. url: https://link.aps.org/doi/10.1103/ PhysRev.188.1969. [87] C. L. Phillips, A. J. Brash, D. P. S. McCutcheon, J. Iles-Smith, E. Clarke, B. Royall, M. S. Skolnick, A. M. Fox, and A. Nazir. “Photon Statistics of Filtered Resonance Fluorescence”. In: Phys. Rev. Lett. 125 (4 2020), p. 043603. url: https://link.aps. org/doi/10.1103/PhysRevLett.125.043603. [88] M. O. Scully and M. S. Zubairy. Quantum Optics. Cambridge University Press, 1997. doi: 10.1017/CBO9780511813993. [89] K. Konthasinghe, J. Walker, M. Peiris, C. K. Shih, Y. Yu, M. F. Li, J. F. He, L. J. Wang, H. Q. Ni, Z. C. Niu, and A. Muller. “Coherent versus incoherent light scattering from a quantum dot”. In: Phys. Rev. B 85 (23 2012), p. 235315. url: https://link. aps.org/doi/10.1103/PhysRevB.85.235315. [90] M. Pelton, D. G. Grier, and P. Guyot-Sionnest. “Characterizing quantum-dot blinking using noise power spectra”. In: Appl. Phys. Lett. 85.5 (2004), pp. 819–821. doi: 10. 1063/1.1779356. [91] S. F. Lee and M. A. Osborne. “Brightening, blinking, bluing and bleaching in the life of a quantum dot: friend or foe?” In: ChemPhysChem 10.13 (2009), pp. 2174–2191. doi: 10.1002/cphc.200900200. [92] E. del Valle, A. González-Tudela, F. P. Laussy, C. Tejedor, and M. J. Hartmann. “Theory of Frequency-Filtered and Time-Resolved N-Photon Correlations”. In: Phys. Rev. Lett. 109 (18 2012), p. 183601. url: https://link.aps.org/doi/10.1103/ PhysRevLett.109.183601. [93] R. Centeno Neelen, D. M. Boersma, M. P. van Exter, G. Nienhuis, and J. P. Wo- erdman. “Spectral ﬁltering within the Schawlow-Townes linewidth as a diagnostic tool for studying laser phase noise”. In: Opt. Commun. 100.1-4 (1993), pp. 289– 302. url: https : / / www . sciencedirect . com / science / article / abs / pii / 003040189390591R. [94] M. Peiris, B. Petrak, K. Konthasinghe, Y. Yu, Z. C. Niu, and A. Muller. “Two-color photon correlations of the light scattered by a quantum dot”. In: Phys. Rev. B 91 (19 2015), p. 195125. url: https://link.aps.org/doi/10.1103/PhysRevB.91.195125. [95] J. C. López Carreño, E. del Valle, and F. P. Laussy. “Photon Correlations from the Mollow Triplet”. In: Laser Photonics Rev. 11.5 (2017), p. 1700090. doi: 10.1002/ lpor.201700090.

69 [96] J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kist- ner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel. “Direct observation of correlations between individual photon emission events of a microcavity laser”. In: Nature (London) 460.7252 (2009), p. 245. doi: 10.1038/nature08126. [97] M. Aßmann, F. Veit, M. Bayer, M. van der Poel, and J. M. Hvam. “Higher-order photon bunching in a semiconductor microcavity”. In: Science 325.5938 (2009), pp. 297– 300. doi: 10.1126/science.1174488. [98] D. Elvira, X. Hachair, V. B. Verma, R. Braive, G. Beaudoin, I. Robert-Philip, I. Sagnes, B. Baek, S. W. Nam, E. A. Dauler, I. Abram, M. J. Stevens, and A. Beveratos. “Higher-order photon correlations in pulsed photonic crystal nanolasers”. In: Phys. Rev. A 84 (6 2011), 061802(R). url: https : / / link . aps . org / doi / 10 . 1103 / PhysRevA.84.061802. [99] M. J. Stevens, S. Glancy, S. W. Nam, and R. P. Mirin. “Third-order antibunching from an imperfect single-photon source”. In: Opt. Express 22.3 (2014), pp. 3244–3260. doi: 10.1364/OE.22.003244. [100] A. Muller, E. B. Flagg, P. Bianucci, X. Y. Wang, D. G. Deppe, W. Ma, J. Zhang, G. J. Salamo, M. Xiao, and C. K. Shih. “Resonance Fluorescence from a Coherently Driven Semiconductor Quantum Dot in a Cavity”. In: Phys. Rev. Lett. 99 (18 2007), p. 187402. url: https://link.aps.org/doi/10.1103/PhysRevLett.99.187402. [101] B. Petrak, M. Peiris, and A. Muller. “Solid optical ring interferometer for high- throughput feedback-free spectral analysis and filtering”. In: Rev. Sci. Instrum. 86.2 (2015), p. 023104. url: https://doi.org/10.1063/1.4907403. [102] J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J.-M. Gérard. “A highly efficient single-photon source based on a quantum dot in a photonic nanowire”. In: Nat. Photonics 4.3 (2010), p. 174. doi: 10.1038/nphoton.2009.287x. [103] M. Gschrey, A. Thoma, P. Schnauber, M. Seifried, R. Schmidt, B. Wohlfeil, L. Krüger, J.-H. Schulze, T. Heindel, S. Burger, et al. “Highly indistinguishable photons from deterministic quantum-dot microlenses utilizing three-dimensional in situ electron- beam lithography”. In: Nat. Commun. 6 (2015), p. 7662. doi: 10.1038/ncomms8662. [104] L. Sapienza, M. Davanço, A. Badolato, and K. Srinivasan. “Nanoscale optical posi- tioning of single quantum dots for bright and pure single-photon emission”. In: Nat. Commun. 6 (2015), p. 7833. doi: 10.1038/ncomms8833. [105] C. Sánchez Muñoz, F. P. Laussy, E. del Valle, C. Tejedor, and A. González-Tudela. “Filtering multiphoton emission from state-of-the-art cavity quantum electrodynamics”. In: Optica 5.1 (2018), pp. 14–26. doi: 10.1364/OPTICA.5.000014. [106] K. Konthasinghe, J. Gomez Velez, M. Peiris, Y. Nieves, L. T. M. Profeta, and A. Muller. “Dynamics of light-induced thermomechanical mirror deformations in high- finesse Fabry-Perot microresonators”. In: J. Opt. Soc. Am. B: Opt. Phys. 35 (2 2018), pp. 372–379. url: http://josab.osa.org/abstract.cfm?URI=josab-35-2-372.

70 [107] K. Konthasinghe, J. Gomez Velez, A. J. Hopkins, M. Peiris, L. T. M. Profeta, Y. Nieves, and A. Muller. “Self-sustained photothermal oscillations in high-ﬁnesse Fabry- Perot microcavities”. In: Phys. Rev. A: At. Mol. Opt. Phys. 95 (1 2017), p. 013826. url: https://link.aps.org/doi/10.1103/PhysRevA.95.013826. [108] R. J. Muller, J. C. Cersosimo, R. Rodriguez, E. Franco, M. Rosario, M. Diaz, Y. Nieves, and B. S. Torres. “Observation Report for the Year 2010, Humacao University Observatory”. In: Journal of Double Star Observations 10 (2 2014), pp. 111–117. url: http://www.jdso.org/volume10/number2/April2014.pdf#page=8. [109] Y. Nieves and A. Muller. “Measurements of Frequency-Resolved Third-Order Corre- lations in Quantum Dot Resonance Fluorescence”. In: CLEO: Applications and Tech- nology. Optical Society of America. 2019, JTu2A–23. url: https://doi.org/10. 1364/CLEO_AT.2019.JTu2A.23. [110] Y. Nieves, J. S. Gómez, J. Dupere, D. Clark, and M. Kim. “Non-Scanning Hologra- phy Fluorescence Microscopy”. In: Biomedical Optics. paper JM3A.8 (online). Optical Society of America, 2016. url: http://www.osapublishing.org/abstract.cfm? URI=BRAIN-2016-JM3A.8.

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License Number: RNP/20/DEC/034225 License date: 18-Dec-2020 DO I: 10.1103/PhysRevB.102.155418 Title: Third-order photon cross-correlations in resonance fluorescence Author: Yamil Nieves and Andreas Muller Publication: Physical Review B Publisher: American Physical Society Cost: USD $ 0.00

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Name: Yamil Nieves Affiliation: Individual Email Id: [email protected] Country: United States

Page 1 of 2 Appendix B Author’s Work

B.1 List of Publications

• Y. Nieves and A. Muller. “Third-order photon cross-correlations in resonance fluorescence”. In: Phys. Rev. B 102 (15 Oct. 2020), p. 155418. url: https://link.aps. org/doi/10.1103/PhysRevB.102.155418 • Y. Nieves and A. Muller. “Third-order frequency-resolved photon correlations in resonance fluorescence”. In: Phys. Rev. B 98 (16 Oct. 2018), p. 165432. url: https: //link.aps.org/doi/10.1103/PhysRevB.98.165432 • K. Konthasinghe, J. Gomez Velez, M. Peiris, Y. Nieves, L. T. M. Profeta, and A. Muller. “Dynamics of light-induced thermomechanical mirror deformations in high- finesse Fabry-Perot microresonators”. In: J. Opt. Soc. Am. B: Opt. Phys. 35 (2 2018), pp. 372–379. url: http://josab.osa.org/abstract.cfm?URI=josab-35-2- 372 • K. Konthasinghe, J. Gomez Velez, A. J. Hopkins, M. Peiris, L. T. M. Profeta, Y. Nieves, and A. Muller. “Self-sustained photothermal oscillations in high-finesse Fabry- Perot microcavities”. In: Phys. Rev. A: At. Mol. Opt. Phys. 95 (1 2017), p. 013826. url: https://link.aps.org/doi/10.1103/PhysRevA.95.013826

• R. J. Muller, J. C. Cersosimo, R. Rodriguez, E. Franco, M. Rosario, M. Diaz, Y. Nieves, and B. S. Torres. “Observation Report for the Year 2010, Humacao University Observatory”. In: Journal of Double Star Observations 10 (2 2014), pp. 111–117. url: http://www.jdso.org/volume10/number2/April2014.pdf#page=8

B.1.1 Conference Proceedings • Y. Nieves and A. Muller. “Measurements of Frequency-Resolved Third-Order Cor- relations in Quantum Dot Resonance Fluorescence”. In: CLEO: Applications and Technology. Optical Society of America. May 2019, JTu2A–23. url: https://doi. org/10.1364/CLEO_AT.2019.JTu2A.23 • Y. Nieves, J. S. Gómez, J. Dupere, D. Clark, and M. Kim. “Non-Scanning Holography Fluorescence Microscopy”. In: Biomedical Optics. paper JM3A.8 (online). Optical Society of America, 2016. url: http://www.osapublishing.org/abstract.cfm? URI=BRAIN-2016-JM3A.8

74 About the Author

Yamil A. Nieves González was born in Humacao, Puerto Rico. He attended the Uni- versity of Puerto Rico-Humacao as an undergraduate where he majored in Physics Applied to Electronics. In 2014, he aﬃliated with the American Physical Society (APS) Bridge Program at the Physics department of the University of South Florida, where he joined the Digital Holography & Microscopy laboratory and received the Masters (M.S.) degree in Applied Physics in August 2016. The following Fall, he joined the Solid State Quantum Optics laboratory mainly studying resonant light-matter interactions in epitaxially-grown semiconductor quantum dots.