On-Chip Generation of Non-Classical States of Light Via Quantum Dots Coupled to Photonic Crystal Nanocavities

ON-CHIP GENERATION OF NON-CLASSICAL STATES OF LIGHT VIA QUANTUM DOTS COUPLED TO PHOTONIC CRYSTAL NANOCAVITIES

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF ELECTRICAL ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Armand Rundquist June 2015

This dissertation is online at: http://purl.stanford.edu/sd515wp4826

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Jelena Vuckovic, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Hideo Mabuchi

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

David Miller

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

iii Abstract

Cavity quantum electrodynamics has enabled unprecedented control over the fundamental interaction of matter and light—the coupling between individual atoms and single photons. At the same time, semiconductor electronics is quickly approaching a regime in which the strange effects of quantum mechanics can no longer be ignored. New types of on-chip optical technologies that exploit the quantum mechanical nature of light have the potential to open up an entirely new direction for semiconductor devices, combining the fine control of cavity quantum electrodynamics with the con- venience of the semiconductor platform. However, the practical implementation of quantum technologies on a chip will require an on-demand source of non-classical states of light, such as pulses with a well-defined number of photons. In this dissertation, I will present the development of a semiconductor non-classical light source based on coupling artificial atoms (quantum dots) to small mode-volume optical resonators (photonic crystal nanocavities). The strong coupling we achieve between a quantum dot and a photonic crystal nanocavity produces a hybridization of the quantum dot excitation with the optical field confined inside the cavity. I will demonstrate how the rich energy structure exhibited by this system enables us to control the statistics of photons in a transmitted laser beam, moving between sub-Poissonian (in which photons are more evenly spaced in each pulse) and super- Poissonian (in which photons are more likely to arrive bunched together) on demand. I will further discuss how these non-classical states of light can be characterized by examining the higher-order photon correlations measured via a generalized Hanbury Brown and Twiss type interferometer. In addition, I will show that by detuning the quantum dot resonance away from the cavity resonance, we can improve both the

iv purity and the efficiency of single-photon generation in this system. This approach allows us to combine the high fidelity of single quantum emitters with the high repetition rate and accessibility of optical cavities. Finally, I will explore methods for scaling up this system by fabricating multiple photonic crystal nanocavities in such a way that they couple to each other. I will present the experimental realization of a photonic molecule—two coupled photonic crystal nanocavities—that is strongly coupled to a quantum dot contained inside one of the component cavities. I will also examine the fabrication of coupled optical cavity arrays in this photonic crystal platform. Our experimental findings demonstrate that the coupling between the cavities is significantly larger than the fabrication-induced disorder in the cavity frequencies. Satisfying this condition is necessary for using such cavity arrays to generate strongly correlated photons, which could potentially be used for the quantum simulation of many-body systems. These on-chip sources of non-classical light represent a significant step in the ad- vancement of semiconductor quantum optical systems. Their capability to produce single photons of high purity at high repetition rate is essential for quantum key distribution and for generating highly-entangled states in quantum metrology. Fur- thermore, the photonic crystal platform represents an ideal way to move beyond the single resonator and begin the development of integrated quantum optical networks.

v Acknowledgements

First, I would like to thank my advisor Prof. Jelena Vuˇckovićfor all of her incredible support. When she brought me into her lab during my first year, I had no idea how lucky I was to find such an excellent supervisor and mentor. I particularly appreciate the way in which she supports her students’ independence and capacity to grow while still providing feedback to guide their development. I’ve learned so much from arguing about technical points and scientific ideas with her, even when I eventually come around to her point of view in the end. In addition to her invaluable research experience, Jelena is also an incredible teacher; I want to thank her for giving me the opportunity to TA so many of her classes and learn from her expertise. I also want to thank my reading and defense committee, Prof. David A. B. Miller, Prof. Hideo Mabuchi, and Prof. Shanhui Fan for their guidance and advice. I’ve taken classes from each of them that have left their mark not only on my knowledge of the field but also on the way I look at science. In addition, I’m grateful to Prof. Monika Schleier-Smith for chairing my defense. Science is a highly collaborative enterprise, and none of this work would have been possible without the constant support and friendship of the other members of the Nanoscale and Quantum Photonics Laboratory. Before I even came to Stanford, Prof. Edo Waks (a former Vuˇckovićgroup member) sparked my interest in nanophotonics through a summer research program I participated in at the University of Maryland. Then, during my first year at Stanford, I took a lab class in which Kelley Rivoire recommended to Jelena that I join their lab, so I am especially grateful for her willingness to take a chance on me. At first, I worked closely with Yiyang Gong, and I greatly appreciate his patience for putting up with my near-constant confusion.

vi I later worked primarily with Arka Majumdar and Michal Bajcsy, both of whom shaped my scientific development immensely. I’m also thankful for the mentorship and support of the other senior members of the lab: Bryan Ellis, Erik Kim, Jesse Lu, and Gary Shambat. Along the way, Sonia Buckley was an unforgettable friend and colleague, and I benefited so much from her valuable help, hard work, and infectious laughter. I’ve learned an incredible amount from working with our current post-docs Tom Babinec, Konstantinos Lagoudakis, Kai Müller,and Tomas Sarmiento, each of whom I’ve collaborated with on one project or another. And I’ve been extremely fortunate these past few years to enjoy the camaraderie and scientific insight of Jan Petykiewicz, Marina Radulaski, Kevin Fischer, Yousif Kelaita, and Alex Piggott. Even as we move on to other endeavors, I know the future of these projects is in the capable hands of Tori Borish, Constantin Dory, Vincent Sebag, and Linda Zhang, and I’m excited to see what brilliant ideas they’ll come up with in the years to come. I’m also very grateful for the tireless work of our administrative assistant Ingrid Tarien, without whom life in graduate school would have been a lot more painful and bureaucratic. Of course, this work depends a great deal on the quantum dot material provided by Hyochul Kim and Prof. Pierre Petroff at UC Santa Barbara, by Vaishno Dasika and Prof. Seth Bank at UT Austin, and by our own Tomas Sarmiento originally from Prof. Jim Harris’s group here at Stanford. We’ve also had a variety of collaborators from other universities whose skills and expertise have expanded what our group was capable of: Per Kaer at TU Denmark; Kassem Alassaad and Prof. Gabriel Ferro at the Universite de Lyon; Carlos Sánchez-Muñoz,Elena del Valle, and Fabrice P. Laussy at the Universidad Autónomade Madrid; and Prof. Jonathan Finley at TU München, who was kind enough to invite me to visit his lab and has been incredibly supportive of my research efforts. Much of this research involved a great deal of time in the cleanrooms of the Stanford Nanofabrication Facility (SNF) and the Stanford Nano Center, and I want to express my gratitude to the many staff members who helped me out: James Conway for first introducing me to the pleasures of e-beam lithography; the whole etcher team—Jim Kruger, Elmer Enriquez, Nancy Latta, and Jim McVittie—for keeping

vii those machines running throughout the years; Uli Thumser who is the true heart of the SNF even when she has a funny way of showing it; J Provine and Tom Carver who have been indispensable on a variety of projects; and Rich Tiberio for working with us to bring the new JEOL e-beam system to bear on our designs. I also want to thank the entire Stanford photonics community, from the Stanford Optical Society and Stanford Photonics Research Center for running so many fun and useful events (in particular, the Stanford University Photonics Retreat that I’ve enjoyed for the past ﬁve years) to all the groups in the Ginzton Laboratory for their support, advice, and willingness to help when needed. They have all made this a great community to be a part of. Outside of my academic community, I’ve been fortunate to have the support of a great group of friends who have helped make my time at Stanford both enjoyable and meaningful: Lewis Marshall, Sarah Edwards and Colin Thom, Josh and Pooja Loftus, Will Pfalzgraﬀ, Aaron Silverman, Esra Burak Ho, Taylor Newton, and Chris and Beth Crapo. Every one of them has been a wonderful source of strength, sympathy, inspiration, and (at times) much-needed distraction. Finally, I would like to thank my family: my mother Patricia and grandmother Doris, who have made me who I am; my siblings—Ariana, Trystan, Kyra (and Gus), and Nils—who have shaped my life more than I will ever be able to express; and my traveling companion and life partner, Molly King. Meeting her is the most important thing that happened to me during graduate school, and this thesis would not have been possible without her love and support. The future is an adventure, Molly—thank you for chasing this particular metaphorical squirrel with me.

viii Contents

Abstract iv

Acknowledgements vi

1 Introduction 1 1.1 Quantum light sources ...... 2 1.2 Dissertation outline ...... 3

2 Strong interaction between a quantum dot and a photonic crystal nanocavity 4 2.1 Quantum dots ...... 4 2.2 Photonic crystal nanocavities ...... 6 2.3 Strong coupling between a QD and an optical cavity ...... 7 2.4 Transmission through a strongly coupled QD–cavity system ...... 9 2.4.1 Optical spectroscopy ...... 10 2.4.2 Extraction of parameters ...... 10

3 Higher-order photon correlations of non-classical light 12 3.1 Background ...... 12 3.2 Three-photon correlations in a weakly coupled QD–cavity system . . 16 (3) 3.2.1 Extraction of g (τ1, τ2) ...... 18 3.3 Two-photon correlations in a strongly coupled QD–cavity system . . 20 3.4 Three-photon correlations in a strongly coupled QD–cavity system . . 23 3.5 Frequency dependence of the multiple-photon correlations ...... 24

ix 3.6 Four-photon correlations in a strongly coupled QD–cavity system . . 26 3.7 Theoretical modeling ...... 27 3.7.1 Relation between g(2)(0) and correlation measurements . . . . 31 3.8 Outlook ...... 33

4 Photon blockade in a detuned quantum dot–cavity system 34 4.1 Background ...... 35 4.2 Effect of QD–cavity detuning ...... 37 4.3 Photon blockade in a detuned system ...... 38 4.4 Photon-number components of the generated non-classical light . . . 41 (2) 4.4.1 Details on the simulations of g (0) and Pn ...... 44 4.5 Optimum detuning for efficient generation of single photons . . . . . 46 4.6 Influence of the pulse length ...... 48 4.7 Outlook ...... 49

5 A quantum dot coupled to a photonic molecule 51 5.1 Background ...... 51 5.2 Theoretical analysis of a photonic molecule ...... 53 5.3 Photonic molecule characterization ...... 56 5.4 Strong coupling between a photonic molecule and a single QD . . . . 58 5.4.1 Feasibility of non-classical light generation ...... 61 5.5 Oﬀ-resonant coupling in a photonic molecule ...... 62 5.6 Outlook ...... 63

6 Beyond isolated systems: coupled cavity arrays 64 6.1 Background ...... 64 6.2 Spectra of coupled cavity arrays ...... 66 6.3 Numerical estimation of coupling and disorder ...... 70 6.4 Statistical study of CCA mode separations ...... 74 6.4.1 Dependence of mode separations on hole radius ...... 75 6.5 Outlook ...... 77

x 7 Conclusions 78 7.1 Summary of contributions ...... 78 7.2 Future work ...... 79 7.2.1 Accessing higher rungs of the Jaynes–Cummings ladder . . . . 79 7.2.2 Phonon-assisted transitions ...... 81 7.2.3 Four-level emitter coupled to a cavity ...... 82 7.2.4 Non-classical light from coupled cavities ...... 83

Bibliography 85

xi List of Tables

6.1 Statistics of cavity array mode separations ...... 75

xii List of Figures

2.1 InAs QDs embedded in a GaAs membrane ...... 5 2.2 Optical nanocavity in a 2D photonic crystal membrane ...... 6 2.3 Cross-polarized reﬂectivity spectrum of a strongly-coupled system . . 9

3.1 Experimental setup for detecting photon correlations ...... 15 3.2 Three-photon correlations from QD ﬂuorescence ...... 17 3.3 Two- and three-photon correlations in a resonantly probed strongly coupled QD–cavity system ...... 22 3.4 Comparison of autocorrelation functions ...... 25 3.5 Creating and detecting two-photon states inside a cavity containing a quantum emitter ...... 30

4.1 Transition energies of a strongly coupled system ...... 36 4.2 Measured and simulated g(2)(0) as a function of both laser and QD– cavity detunings ...... 40 4.3 Comparison of the multi-photon probability for resonant and detuned systems ...... 42 4.4 Comparison of the quantum regression theorem and quantum jump method for g(2)(0) simulations ...... 45 4.5 Eﬀect of detuning on single-photon purity and eﬃciency ...... 47 4.6 Choosing a detuning-dependent pulse length ...... 49

5.1 Numerically calculated cavity transmission spectra when the QD resonance is tuned across the two cavity resonances ...... 55

xiii 5.2 Photoluminescence spectra of the coupled cavities for diﬀerent hole spacings between two cavities ...... 57 5.3 Normalized PL intensity plotted for temperature tuning the QD across the photonic molecule resonances ...... 59 5.4 QD–photonic molecule spectra and numerical simulations ...... 61 5.5 Oﬀ-resonant interaction between a photonic molecule and a QD . . . 63

6.1 Coupled cavity array SEMs and FDTD simulations ...... 67 6.2 Photoluminescence spectra of coupled cavity arrays ...... 69 6.3 Eﬀect of disorder on the observed mode separations in a photonic molecule ...... 71 6.4 Numerically calculated eigenspectra of the coupled cavities ...... 73 6.5 Mode separations as functions of the photonic crystal hole radius . . 76

xiv Chapter 1

Introduction

In much the same way that electrons can be used to carry information, to process calculations, and to store data, the individual elements of light (photons) can be manipulated to perform similar tasks. Coined analogously to the term “electronics,” the field of “photonics” has received considerable recent attention due to its potential to revolutionize traditional electronic devices. For example, fiber-optic cables have now almost universally replaced copper cables for our long-haul Internet traffic. Many scientists foresee a similar transition to photonics for information transfer at smaller and smaller size scales, down to the minuscule interconnects on a computer chip [1, 2]. But this is not the only way that photonics can improve on electronics. By taking advantage of the unique quantum-mechanical properties of photons and using them to complement traditional electronic computation and communication, it is possible to implement novel technologies that combine the best of both worlds. However, many proposed quantum technologies, including quantum key distribution and photonic- qubit based quantum computation, require on-demand sources of light that produce pulses containing a well-defined number of photons [3]. In order to realize these opportunities, we must be able to manipulate the properties of individual photons in a semiconductor device, which is the primary focus of my dissertation research. Using the same techniques developed for fabricating computer chips, we can drill an array of holes into a semiconductor membrane to create

1 CHAPTER 1. INTRODUCTION 2

a structure called a “photonic crystal cavity.” This structure forms a type of optical resonator, confining light of a particular frequency into a very small volume at high intensities. By introducing a patch of a different semiconductor material inside the resonator we can make an artificial atom (called a “quantum dot”) that couples very strongly to the light field, producing a hybrid light–matter state that exhibits a strong nonlinearity even at the single-photon level [4, 5]. This phenomenon enables the development of optical devices whose operation depends fundamentally on the strange laws of cavity quantum electrodynamics (QED).

1.1 Quantum light sources

Self-assembled quantum dots (QDs) are promising candidates for quantum light sources due to their strong interaction with light and ease of integration into optoelectronic devices [6]. High-fidelity single-photon generation from QDs for off-chip applications has been demonstrated under both non-resonant [7] and resonant [8, 9, 10] excitation. Some of these experiments have employed micropillar cavities [11] or photonic nanowires [12, 13] for enhanced off-chip extraction efficiency. On the other hand, photonic crystal cavities provide a promising on-chip route toward optoelectronic integration of QDs due to the established set of integrated waveguide and detector structures [14, 15]. Such structures will be able to exploit strong light–matter coupling with QDs for the generation of a variety of on-chip non-classical light states by quantum-electrodynamical methods [16, 17, 18]. Recent exotic proposals have even explored the possibility of releasing energy exclusively in bundles of n photons [19]. However, these devices are currently limited to single resonators operating in isola- tion. With this in mind, part of my research has explored the ways we can scale up our ability to manipulate individual photons. By building networks of coupled photonic crystal cavities, it may be possible to take full advantage of the quantum interactions between different resonators, allowing even better control over the transport, storage, and manipulation of individual photons [20]. The integration of multiple interacting photonic components on the same chip has the potential to drive the full-scale development of photonic technologies. CHAPTER 1. INTRODUCTION 3

1.2 Dissertation outline

Chapter 2 provides a brief overview of the solid-state platform we use to generate non- classical states of light, consisting of a QD coupled to a photonic crystal nanocavity. In this system, the optical confinement provided by the photonic crystal resonator can enhance the light–matter interaction into the “strong coupling” regime, which results in a hybridization of QD and cavity excitations. A strongly coupled QD–cavity system is capable of emitting non-classical states, but the nature of such light is more complex than the ideal single-photon stream emitted by a QD alone. Therefore, we must measure higher-order autocorrelation functions in order to characterize this non-classical light. To that end, in chapter 3, we report the observation of non-classical third- and fourth-order photon correlations in an originally coherent probe after it was transmitted through a photonic crystal nanocavity strongly coupled to a single QD [21]. In chapter 4, we demonstrate how single-photon generation from a strongly coupled system can be improved by detuning the QD resonance away from the cavity resonance [22]. With this approach, we find that high-quality single-photon generation is possible with current state-of-the-art samples. Chapter 5 presents an alternative approach to non-classical light generation: coupling the QD not to a single optical cavity but to a pair of coupled photonic crystal cavities, also called a “photonic molecule” [23]. We extract the relative contributions of the coupling strength and nanofabrication-induced disorder in such a system, and simulate its use as a non-classical light source. We conclude that the photonic molecule represents a potential building block for an integrated quantum optical network. To explore this idea further, in chapter 6 we examine larger arrays of coupled cavities, and find that the coupling strengths between the cavities are great enough to overcome the frequency variation due to fabrication imperfections [24]. Hence, our coupled cavity arrays hold promise for generating strongly-correlated photons that can be used for quantum simulation. Finally, chapter 7 presents some conclusions and possible future research directions for the development of this solid-state nanophotonic platform for cavity QED. Chapter 2

Strong interaction between a quantum dot and a photonic crystal nanocavity

The building blocks of our non-classical light source are quantum dots and photonic crystal nanocavities. In this chapter, I will discuss each of these elements and describe how they interact.

2.1 Quantum dots

In order to obtain an optically-addressable, semiconductor-based quantum emitter, we use molecular beam epitaxy (MBE) to grow a layer of self-assembled InAs QDs embedded in a GaAs membrane. The MBE-grown structure consists of a 900 nm ∼ thick Al . Ga . As sacrificial layer followed by a 150 nm thick GaAs layer that 0 8 0 2 ∼ contains a single layer of InAs QDs. Our growth conditions result in a typical QD density of 60 80 µm−2, as shown in the atomic force microscope (AFM) image in − figure 2.1(a). Due to the difference in band gaps between InAs and GaAs, the QD forms a confining potential in all three directions that results in atom-like discrete energy levels, as depicted in figure 2.1(b). An electron-hole pair or “exciton” generated via

4 CHAPTER 2. STRONG INTERACTION 5

above-band or resonant optical pumping populates these energy levels before recom- bining and emitting a photon with a well-defined frequency. The three-dimensional confinement of the QD, combined with its relatively long lifetime ( 1 ns) results in ∼ very narrow optical lines corresponding to these transitions, as shown in figure 2.1(c). Our QDs emit in the 900 940 nm range and operate best at low temperatures; − above 50 K, the QDs lose confinement and cease photoluminescence. The wide ∼ range of wavelengths (i.e., inhomogeneous broadening) is due to variation in the size of the QDs resulting from the random self-assembly process that occurs during MBE growth.

(a) (b) Conduction (c) 2e band 1e

emission pump 250 nm 1h 2h Valence band  [nm]

Figure 2.1: InAs QDs embedded in a GaAs membrane. (a) AFM image of randomly-distributed uncapped QDs (1 µm 1 µm region). (b) Schematic of the energy structure of a QD, including photoluminescence× emission following above-band excitation. (c) Optical spectrum of bulk QDs.

On its own, a single QD can be thought of as a two-level system, consisting of a ground state (in which no electrons or holes are trapped inside the potential well) and an excited state (in which an exciton is conﬁned inside the QD). While additional states exist (e.g., the bi-exciton transition and the higher p-shell energy levels), they can be ignored as long as we choose the frequency and power of our pump to access only the single-exciton state, driving the QD well below saturation. Under these conditions, the QD operates as a nearly ideal quantum emitter, producing individual photons one at a time. However, it is diﬃcult to optically address a single dot in a bulk sample, and the production of single photons is relatively slow (due to the CHAPTER 2. STRONG INTERACTION 6

QD’s long lifetime). These challenges suggest that we need a method of enhancing the optical interaction. We can do this using an optical resonator.

2.2 Photonic crystal nanocavities

By analogy with the free-space Fabry–Pérotcavities used to enhance the interaction of photons with an isolated atom [25, 26, 27], we fabricate an optical resonator in the semiconductor membrane containing our QDs in order to confine the electromagnetic field and increase the strength of the field’s interaction with the QD transitions. Although a variety of solid-state cavities can be fabricated for this purpose [28], we focus on the photonic crystal nanocavity due to its small mode volume (V ) and high quality factor (Q), properties that are essential to reaching the strong coupling regime of cavity QED. The photonic crystal platform is also attractive for the ease of integration with additional optical components, such as waveguides or other cavities (as will be discussed in chapters 5 and 6).

(a) (b)

(c)

Figure 2.2: Optical nanocavity in a 2D photonic crystal membrane. (a) SEM image of a 2D photonic crystal in a suspended membrane. (b) SEM image of the defect region in the center of the photonic crystal that forms the optical cavity. Note that several of the holes near the cavity have been shifted to improve the quality factor. (c) Electric ﬁeld intensity proﬁle of the cavity resonance, as calculated from FDTD simulations. CHAPTER 2. STRONG INTERACTION 7

We fabricate two-dimensional (2D) photonic crystals by etching an array of holes into the semiconductor membrane containing our QDs. This patterning allows light to be confined in-plane through distributed Bragg reflection and out-of-plane through the index contrast provided by the membrane [29]. The photonic crystals are fabricated using 100 keV e-beam lithography with ZEP resist, followed by reactive ion etching and removal of the sacrificial Al0.8Ga0.2As layer in hydrofluoric acid [4]. The photonic crystal lattice constant is a = 246 nm and the hole radius is r 60 nm. Figure ≈ 2.2(a) shows a scanning electron microscope (SEM) image of the completed photonic crystal. The undercut of the sacrificial layer leaves the photonic crystal in a suspended membrane of GaAs, with the thin layer of QDs just visible at the center of the slab. For the majority of our experiments, we employ a linear three-hole defect (L3) cavity [30]. To improve the cavity quality factor, several holes near the cavity were scaled and shifted slightly [31, 32]. This defect region can be seen in the top-down SEM in figure 2.2(b). Finite-difference time-domain (FDTD) simulations are used to calculate the cavity field profile [figure 2.2(c)] and optimize the photonic crystal parameters. From these simulations, we find that the mode volume of our photonic crystal cavities can be as low as V 0.5(λ /n)3, where λ is the cavity wavelength in ∼ 0 0 free space and n 3.4 is the refractive index of GaAs. The simulated quality factor ≈ can be greater than Q 105, though in practice, fabrication imperfection limits us ∼ to experimental values of Q 10, 000 20, 000. ∼ −

2.3 Strong coupling between a QD and an optical cavity

The performance of our QD–cavity system is limited by the loss rates for both the QD (denoted by γ) and cavity ﬁeld (denoted by κ). The (radiative) decay rate of a bulk QD is γ/2π 0.1 GHz, while in the photonic crystal the decay into continuum ∼ (non-cavity) modes is suppressed further to γ/2π 0.01 GHz due to the photonic ∼ band gap [33, 34]. Experimentally, the linewidth of the QD transitions is assumed to CHAPTER 2. STRONG INTERACTION 8

be γ/2π 1 GHz due to non-radiative effects such as spectral diffusion and phonon- ∼ induced dephasing [35, 18]. The cavity field decay rate κ = ω0/2Q (where ω0 is the cavity frequency) for our GaAs photonic crystal cavities is generally in the range of κ/2π 8 16 GHz. This puts our system in the “bad cavity” limit (κ γ) of cavity ≈ − QED, in which loss occurs primarily through radiation from the cavity mode. If the coupling strength g between the QD and the optical cavity exceeds these loss rates, the system is said to be “strongly coupled.” In this regime, energy may be reversibly exchanged between the emitter (in this case, a single QD) and the cavity, a process referred to as “vacuum Rabi oscillations” [36, 37, 38]. The energy structure of a strongly coupled system is typically described by the Jaynes–Cummings (JC) Hamiltonian [39, 40],

† † † † H = ω0a a + (ω0 + ∆)σ σ + g(a σ + aσ ), (2.1)

where ω0 denotes the frequency of the cavity, a the annihilation operator associated with the cavity mode, σ the lowering operator of the quantum emitter, ∆ the detuning between quantum emitter and cavity, and g the emitter–cavity ﬁeld coupling strength. Note that by convention we express the Hamiltonian in angular frequency units rather than energy, thus neglecting factors ofh ¯. Diagonalizing this Hamiltonian results in new eigenstates (“dressed” states) which are a superposition of both QD and cavity excitation (also referred to as “polaritons”). For the case of n excitations (n 1), ≥ the two dressed states have frequencies given by

∆ ∆ 2 2 ω± = nω + n g + . (2.2) 0 2 ± | | 2 s Thus, it can be seen that the presence of a QD splits the empty cavity resonance into two “polariton” peaks, corresponding to each of these eigenstates. This splitting is depicted schematically in ﬁgure 2.3(a). Furthermore, since the splitting between the dressed states increases as √n, rather being constant for each level of excitation (“rung”) of the JC ladder, the sequence of energy levels is no longer evenly spaced (as it would be for the harmonic energy levels of an empty cavity). This anharmonicity is CHAPTER 2. STRONG INTERACTION 9

the key for enabling highly nonlinear phenomena in this system, such as the generation of non-classical light (as will be discussed in more detail in chapter 3).

(a) (b) (c)

Cavity

Cavity states Dressed states

Figure 2.3: Cross-polarized reflectivity spectrum of a strongly coupled system. (a) Harmonic ladder of bare cavity states (left) compared with the anharmonic dressed states ladder (right) corresponding to the strongly coupled QD–cavity system. (b) Anticrossing between the QD and cavity resonances as the temperature is increased. (c) Spectrum of a strongly coupled QD–cavity system at a fixed temperature (T 32 K), with a fit (see section 2.4.2) in red. Note the two polariton peaks. ≈

2.4 Transmission through a strongly coupled QD– cavity system

Experimentally, the strong coupling manifests as an avoided crossing or “anticrossing” between the QD and cavity modes as they move into and out of resonance (by temperature tuning, for example). Since the relative band-gap diﬀerence between InAs and GaAs changes as a function of temperature, so will the QD transition frequency (the cavity frequency shifts as well due to the change in refractive index, but this occurs much more slowly). Hence, the detuning between the QD and cavity resonance can be controlled by the lattice temperature [37, 41]. This anticrossing can be observed using a cross-polarized reﬂectivity setup that suppresses the scattered light from the pump laser while detecting light that has been transmitted through CHAPTER 2. STRONG INTERACTION 10

the QD–cavity system [4]. As seen in figure 2.3(b), the cross-polarized reflectivity signal (which is equivalent to a transmission signal) from a strongly-coupled system exhibits an anticrossing as the QD is temperature-tuned across the cavity resonance. When the QD is on resonance with the cavity frequency, the transmission of a weak probe through the system is primarily governed by the dressed states of the first rung of the JC ladder. Hence, in figure 2.3(c) we observe the two polariton peaks. By fitting this spectrum (see section 2.4.2 below for details), we are able to extract the relevant system parameters. For the reflectivity spectrum shown in figure 2.3(c), we calculate the cavity field decay rate as κ/2π = 9.6 GHz and the QD–cavity coupling strength as g/2π = 11.5 GHz, confirming that this system operates in the strong-coupling regime.

2.4.1 Optical spectroscopy

All optical measurements were performed with a liquid helium flow cryostat at temperatures in the range 10 40 K. For excitation and detection a microscope objective − with a numeric aperture of NA = 0.75 was used. Cross-polarized reflectivity measurements were performed using a polarizing beam splitter to excite the system with a laser polarization orthogonal to the collected polarization [4]. To further enhance the extinction ratio, additional thin film linear polarizers were placed in the excitation/detection pathways and a single-mode fiber was used to spatially filter the detection signal. Furthermore, two wave plates were placed between the beamsplitter and microscope objective: a half-wave plate to rotate the probe polarization relative to the cavity orientation and a quarter-wave plate to correct for the birefringence (i.e., polarization-dependent refractive index) of the optics and the sample itself.

2.4.2 Extraction of parameters

To describe the QD’s blinking and spectral diffusion, we use a simplified model in which we assume the QD is in a bright state and resonant with the cavity for pbright fraction of the time and does not interact with the cavity for pdark = 1 pbright − fraction of the time, when it either goes dark [42] or has jumped to a far off-resonant CHAPTER 2. STRONG INTERACTION 11

state. A more detailed analysis of the dark state dynamics can be obtained using a rate equation approach [43], but here we simply assume that blinking results in a transmission spectrum which is an average between the double-peaked dressed-states spectrum and the single-peaked bare-cavity Lorentzian. Hence, to extract the decay rate of the cavity ﬁeld κ and the QD–cavity coupling rate g, we perform a least-squares ﬁt of

Ftotal = Iin [pbrightFdressed + (1 pbright) Fcav] + Ibg (2.3) − to the observed transmission spectrum [ﬁgure 2.3(c)] of the strongly coupled system.

Here, Iin is the intensity of the laser coupled into the cavity, Ibg is the background laser signal unsuppressed by the cross-polarization setup,

2 κ[γtot + i(ωQD ωl)] Fdressed = − (2.4) [κ + i(ω ω )][γ + i(ω ω )] + g2 0 l tot 0 l − − is the transmission spectrum of a strongly coupled QD–cavity system in the weak probe approximation [4], κ 2 F = (2.5) cav κ + i(ω ω ) 0 l − is the Lorentzian transmission spectrum of an empty cavity, γ = γ + γ , ω and tot d 0 ωQD are (respectively) the resonant frequencies of the cavity and the dot, and ωl is the frequency of the probe laser. Based on previous estimates [18], we assume the

QD dipole decay and pure dephasing rates to result in (γ + γd) /2π 1 GHz (for ≈ simplicity we neglect any temperature dependence of the dephasing), and we used pbright, Iin, Ibg, κ, g, ω0, and ωQD as the ﬁtting parameters. Chapter 3

Higher-order photon correlations of non-classical light

In this chapter, I present experiments that use the second- and third-order autocor- (2) (3) relation functions g (τ) and g (τ1, τ2) to detect the non-classical character of the light transmitted through a photonic crystal nanocavity containing a strongly coupled quantum dot probed with a train of coherent light pulses [21]. We contrast the value of g(3)(0, 0) with the conventionally used g(2)(0) and demonstrate that, in addition to being necessary for detecting two-photon states emitted by a low-intensity source, g(3) provides a more clear indication of the non-classical character of a light source. We also present preliminary data that demonstrates bunching in the fourth-order au- (4) tocorrelation function g (τ1, τ2, τ3) as the ﬁrst step toward detecting three-photon states.

3.1 Background

A strongly coupled QD–cavity system can produce non-classical light by ﬁltering the input stream of photons coming from a classical coherent light source through mechanisms described as “photon blockade” [16, 44] and “photon-induced tunneling” [16, 45]. The anharmonic nature of the dressed states ladder (discussed in section 2.3 above) results in large optical nonlinearities at the single-photon level, where the

12 CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 13

admission of a single photon into the cavity may enhance (photon-induced tunneling) or diminish (photon blockade) the probability for a second photon to couple into the system. Recent proposals [18, 19] have extended the concept of photon blockade from single photons to two-photon Fock state generation by coupling the probe laser to the second manifold of the Jaynes–Cummings ladder via a two-photon transition [46]. This approach can potentially be further generalized to create three- and higher-photon number states inside the cavity through multi-photon transitions to the corresponding manifold. Here, we explore the probing of these multi-photon transitions into the higher manifolds of the JC ladder of a strongly coupled QD–photonic crystal nanocavity system [16] by measuring the third-order autocorrelation function (3) [g (τ1, τ2)] of a probe laser transmitted through such a system. One of the benchmarks used to characterize a source of single photons is the

(2) ha†a†(τ)a(τ)ai measurement of the the second-order autocorrelation function g (τ) = 2 ha†ai [47] at τ = 0, which quantifies the suppression of multi-photon states (for details see section 3.7.1). In actual experiments, the value of g(2)(0) for a light source is usually estimated from a Hanbury Brown and Twiss (HBT) setup that measures coincidence counts between two single photon counting modules (SPCMs). A classical coherent light source will produce a photocount distribution with Poisson statistics [g(2)(0) = 1], while a source whose output contains at most one photon at a time will produce g(2)(0) = 0. More generally, photons from a “sub-Poissonian” light source— in which the single-photon component dominates over the multi-photon states—will be antibunched and yield g(2)(0) < 1. In theory, the second-order autocorrelation function can also be used to identify a two-photon source, since a pure two-photon Fock state will have g(2)(0) = 1/2. However, most experimental demonstrations of non-classical light sources result in low-intensity (sparse) output; i.e., the source is outputting zero photons most of the time. While this does not affect the value of g(2)(0) for a single-photon source, a perfect but low-intensity two-photon source outputting the state ψ √1 ε2 0 + ε 2 , ≈ − | i | i with ε 1, will give g(2)(0) 1/2ε2 (see section 3.7). A similar argument can ≈ be made for any perfect but sparse n-photon source, which illustrates the difficulty CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 14

of quantitatively distinguishing between various multi-photon Fock states in an experiment relying on a two-detector measurement. In particular, photon bunching [g(2)(0) > 1] will be observed for low-intensity non-classical light sources in which the presence of the vacuum state is stronger than that of the photon-number Fock state [16]. To resolve the presence of a particular Fock state, it is necessary to evaluate higher-order photon autocorrelation functions and compare them with lower-order ones. For example, a low-intensity non-classical light source with a dominant two- photon component will show g(2)(0) > 1 and g(3)(0, 0) < 1 (section 3.7). Here, the value of the third-order autocorrelation function [47]

a†a†(τ )a†(τ + τ )a(τ + τ )a(τ )a g(3)(τ , τ ) = h 1 1 2 1 2 1 i (3.1) 1 2 a†a 3 h i can be estimated with a generalized form of HBT setup that monitors coincidences between three SPCMs [see figure 3.1(b)] and thus allows one to measure the suppression of simultaneous three- and higher-photon events [46]. It is worth noting that the use of photon-number-resolving detectors [48, 49], especially transition-edge sensors [50] or superconducting nanowires [51, 52, 53], could provide an alternative technique for the characterization of multi-photon Fock states; however, these devices are still under experimental investigation and are not yet widely available. Prior to this work, higher-order photon correlations had been measured for thermal [54, 55, 56, 57] and laser [58] sources, relying on the strong excitation and high count rates available in these systems. Contemporaneous with our work, g(3) measurements of the fluorescence from a single QD weakly coupled to a microcavity had been reported as well [59]. However, in the low-intensity, strongly coupled regime of cavity quantum electrodynamics, such correlations had only been measured in an atomic system [60]. Therefore, this work constitutes a significant step towards implementing a solid-state non-classical light source of photon number states. Our system consists of an InAs QD embedded in an L3-defect nanocavity [30] in a 2D GaAs photonic crystal, fabricated [4] as described in section 2.2. The QD–cavity system is maintained at cryogenic temperatures (between 4 K and 50 K) using a continuous-flow liquid helium cryostat. We excite this system with focused pulses fteotu ed h ytmhsparameters [ has system system two-level The effective field. an output form the eigenstates of first-order the of eda h nest ftelsrpulse laser the of intensity the as field κ ope tteiptpr r otoldsc htteotu edhspiaiyasnl-htncmoet b omlzdFock-state Normalized (b) component. single-photon a primarily has field output the that such controlled are port ( input coefficients the at coupled eiei prtd oee,teotu edi o na be in can it not basis, is state the Fock field time a as in output every expressed and the detector state, However, single-photon the single pure operated. monitored. by a is is registered source, device output be single-photon the should deterministic at click a detected and for cavity clicks Ideally, the of to coupled each number is For the pulse detector. laser single-photon a trajectory, ideal quantum an using by analyzed EEAINO OCASCLSAE FLIGHT OF STATES NONCLASSICAL OF GENERATION eetdpoosa h uptwe unn ag number large a example, For running trajectories. when of output the at photons detected ftecoefficients the of where h uptfil stedrto ftelsrple( pulse laser the of duration the as field output the aeo h ytmi elgbe h omlzdvle( value normalized The negligible. dephasing is the system that the considering state, of pure rate a as state output the ls spsil oasnl-htnsae hntesimulation that the such then optimized as be state, be should single-photon should parameters a state to output possible desired as the close If output. the which at for trajectories detected of number relative the by nFig. in g/ / 2 2 h ocasclsaeo ih mte tteotu is output the at emitted light of state nonclassical The pulse laser the of properties The blockade. photon the in operation pulsed via generation state Nonclassical (a) online) (Color 2. FIG. h xeietlcngrto osdrdhr sa shown as is here considered configuration experimental The π π = = ϕ GHz, 1 0GHz. 30 n 2(a) stecefiin fteFc state Fock the of coefficient the is | .Thecavityhastwomirrors,withdecayrates c a (b) (a) c (d) (c) 0 2 | γ 2 |c |

/ n o aumstate, vacuum for 2 0.5 | π ϕ 0 1 | 0.2 n b = | out 2 0 | a eetmtdfo h ubrof number the from estimated be can " a . H,and GHz, 1 = in (t) 0.3 " n | ∞ = c 〉 n 0 | ϕ 2 n = | c R | 0.4 1 n g/ | ! in " ! 2 2 | CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 15 , ϕ i 0 o igepoo tt,and state, single-photon for π τ | n ϕ |

smdfidadteplewdhi etcntn at constant kept is width pulse the and modified is 2 κ i = | | | | 2 c g e | 0.5 1 〉 〉 n 0Gz d h aesmlto si ae b u o ytmwith system a for but (b) panel in as simulation same The (d) GHz. 40 swl estimated well is | " 2 κ GENERATION OF NONCLASSICAL STATES OF LIGHT ... PHYSICAL REVIEW A 81,033838(2010) R .Herewewrite / smaximized. is n 2 out π τ onswere counts smdfidwhile modified is ) ~ = 0.6 (a) (a)|1 (b) GHz, 1 |! 〉 2

... 1 |c | |c |c |c ! 0 |

Two-levelmulti 1 0 quantum emitter inside a cavity Multi-photon State Generation from Strongly Coupled Quantum Dot-Cavity System c " | | HBT setup 2 2 2 a (t) n | |in 〉 |c |

(9) !

| 1 |

033838-3 Arka Majumdar, 2 2 ∗ Michal Bajcsy, and Jelena Vuˇckovi´c |e〉 ~|1〉 2 ) HWP E.L.Ginzton Laboratory, Stanford University, Stanford, CA- γ |c94305 | multi 2 18 / We investigate the photon induced tunneling phenomena in a photonic crystal cavity containing a | 2

n strongly coupled quantum dot and describe how this tunneling can be used to generate photon states oRb siltosaeosre ntesnl-htncharacter single-photon the in observed are oscillations Rabi so ] π consisting mainly of a particular Fock state. Additionally, we present initial experimental data of

2 2 |c κ htnbokd ne usddiig efis nlz a analyze first we parameters driving, with pulsed system under blockade photon the at field nonclassical the output. configuration of is cavity This collection field mirror. efficient output lossier for the allows the and from reflectivity collected higher mainly with mirror the on aeo h aiyis cavity the of rate eopoo,snl-htn swl smlihtnpopulation multiphoton as well to as set single-photon, zero-photon, is width pulse the ihtetasto otefis-re aiod( manifold first-order resonance on the set to is transition field driving the the with of frequency center The and oacvt ihaqaiyfco of factor quality a with cavity a to osoeaigaon 3 m u tl ihntetheo- the within g/ [ still material but this rate nm, for of 930 limit values retical around state-of-the-art operating the dots than larger Q times five about unu o r sue ob nrsnne( resonance on the be and to cavity assumed the simulation, are is this dot it For quantum achievable. techniques be for fabrication values will the higher and that system expected material the in g 0.5the photon-induced tunneling as a function of excitation laser power and frequency and show the

| |2,+ =

1 |c | |c | 〉 c n n2g spectrometersignature of second rung of the Jaynes-Cummings Hamiltonian in the observed photon-statistics. multi g

2 | 〉 oilsrt h eairo h ytmoeaigin operating system the of behavior the illustrate To bevdi ascvte ihculdIA quantum InAs coupled with cavities GaAs in observed 0.5 0.5 0 and ! π |"! Multi-photon State Generation from Strongly Coupled Quantum Dot-Cavity System g/ . 1 H,and GHz, 1 obj.2 2 0 ! 1 2 0 1 0 g PBS 2 2g (κ | 4 γ) which assumes the rotating wave approximation = 2 − − 0 0 ,t h et y p i c a lv a l u e sm e a s u r e ds of a ra r ea r o u n d 2 = |g,2〉 !," |e,1〉 A single optical mode confinedArka inside Majumdar, an optical∗ Michal Bajcsy,(RWA) and Jelena and aVuˇcković frame of reference rotating with the = π κ R R cavity behaves like a simpleE.L.Ginzton harmonic Laboratory, oscillator, Stanford University,frequency Stanford, of the CA- laser field 5Gz[ GHz 25 94305 ωl. Here ∆a = ωa ωl 0Gzwskp osat h ytmhsparameters has system The constant. kept was GHz 10 in out 2 m1 m1 m1|2,- where all the energym1 levels are equally spaced. and∆c = ωc ωl are respectively the detuning of− ! 〉

= We investigate the photon induced tunneling phenomena in a photonic crystal cavity containing a When this cavity0 mode is strongly coupled to a the QD resonant− frequency strongly coupled quantum dot and describe how this tunneling can be used to generate photon statesωa and the cavity reso- uhthat such

n ∞ two-level quantumconsisting0 emitter mainly of such a particular as20 a quantum Fock state. dot Additionally,40 nance we present frequency initial60 experimentalωc with datathe laser,of80 g is the coher- 0Gz h value The GHz. 40 = (QD), the energythe photon-induced structure of tunneling the coupled as a function sys- of excitation laserent couplingpower and frequency strength and between show the the QD and the

2 cavity 100 g tem becomessignature anharmonic. of second This rung anharmonicity, of the Jaynes-Cummings Hamiltonian in the observed photon-statistics.κP (t)

20 |1,+ (GHz)cavity mode, | (b) (c) 〉 Ω (t)= is the slowly varying 0 ωc c known as Jaynes-Cummings ladder, can generate E g/ QD envelope of the coherent driving field with power n timing nonclassical correlationsT increase between photons trans- | 21 ! 2 1 2 P (t) incident onto the cavity, and 2 sync 2 a is the annihila- 2 mittedg through(κ theγ) cavity and result in funda- which assumes the rotating wave approximation module(c) (d) − 4 − tion operator for the cavity mode. If the excited π |g,1〉 |e,0 mental phenomena of photon blockade and pho- o utpoo rbblt)frteoutput the for probability) multiphoton for " A single optical mode confined inside an optical (RWA) and a frame of reference rotating with the .Hwvr ihfrhrimprovements further with However, ]. 〉 and ground state of the QD are denoted by HSCLRVE A REVIEW PHYSICAL Ω Ω TTL toncavity induced behaves tunneling. like a simple These harmonic cavity quantum oscillator, frequency of the laser field e κ |1,- = 2 and ω . Here

κ g l ∆ | Stop 〉 then σ = g e and σ+ = e g . a = ωa ωl SPCM 3 1 |c | electrodynamicwhere all the energy1 (cQED) levels e areﬀects, equally which spaced. have been and∆| = − | − 0 ω | 0 c c ωl are respectively the| detuning| of 1 0 For the rest− of our analysis, we will assume the ≈ experimentally observed both in solid state [ (GHz) (GHz) 200

τ When this cavity mode is strongly coupled to a the QD resonant frequency 0Gz c oksaecefiinsfor coefficients Fock-state (c) GHz. 40 1] QD and the cavity are resonantω (a and the cavity reso- 40 τ # 2 ∆ κ Start and atomic systems [ 2], were recently used to c = ∆a = ∆). |c | two-level quantum emitter such as a quantum dot Twonance main frequency loss mechanismsωc with the in this laser, systemg is are the the coher- = / κ 1 probe(QD), higher the energy order structure dressed states of the coupledin the Jaynes- sys- ent coupling strength between the QD and the = cavity field decay rate κ 2 = ωc/2Q (Q 1 TTL Cummings Hamiltonian [ is the qual- κ 2 tem becomes anharmonic. This3, 4 anharmonicity,]. However, pho- κP (t) 0 (3) itycavity factor mode, of the resonator) and QD spontaneous .Thedrivinglaserisincident π SPCM 2 Stop |c | ton blockade and photon-induced tunneling can (t)= is the slowly varying 2 1 2 known as Jaynes-Cummings ladder, can generate ωc . 0 g (! , ! ) multi emission rate E 2 γ. These losses can be incorporated 2 45 .Effectively,thetotaldecay be used for applications beyond cQED, includ- envelope of the coherent driving field with power | |

. nonclassical correlations between photons trans- =

20 by the Master equation Q 45

n P (t) incident onto the cavity, and Start n ingmitted generation through ofthe non-classical cavity and statesresult inof funda-light [ a is the annihila- κ / 5], 2 |g,0 quantum simulation of complex many-body sys- tion operator for the cavity mode. If the excited |c κ 〉 mental phenomena13 of photon blockade and pho- |c | |c |g,0 0.5 dρ / .Rgrigtecoupling the Regarding ]. 0.5 〉 GHz, 1 300 / Reflectivity (a.u.) (a.u.) Reflectivity and tems [ and ground state= of thei[H, QD0 are denoted by 60 QD 6], quantum information processing, and ρ]+κ [a]+γ [ h rudsaeadone and state ground The . light from ton induced tunneling. These cavity quantum σ], (2)e 2 TTL κ and dt − L L SPCM 1 Stop high precision sensing and metrology [ g then σ = g e and2 σ+ = e g . | π electrodynamic (cQED) e ffects, which have7]. been | − .Figure experiment For the rest of our analysis,| |c| | we will assume| | the experimentallyIn this paper, we observed explore theboth utility in solid of statethe [ where ρ is the density matrix1 of the coupled QD- time time differences 1] Q QD and the cavity are resonant (

binning binning of arrival- cavity system and g = Start photonand atomic induced systems tunneling [ 2], and were blockade recently for used non- to [D] is the2 Lindblad∆c operator= ∆a = ∆). |c |c |c correspondingTwo main loss mechanisms to operatorL|c in| this system are the ,asconsideredhere, |c |c |c classicalprobe higher photon order state dressed generation states and in the dressed Jaynes- multi D, deﬁned as multi 1 0 cavity ﬁeld decay rate = multi 1 0 1GHzcorresponds state probing. First, we provide numerical simu- κ = ω /2 | | Cummings Hamiltonian [ c Q (Q is the qual- 2 2 | | 3, 4]. However, pho-

2 2 ity factor of the resonator) and QD spontaneous γ lation data showing that photon induced tunnel- κ ton blockade and photon-induced tunneling can

| emission rate 6 0.Ti is This 000. 160 400 2 [D]=2γ. TheseD losses can be incorporated | ! ρD† / (nm)ing can be used to preferentially generate speciﬁc D†D 81 / be used for applications beyond cQED, includ- ρ ρD†D. (3) 2 0 80 0 by the MasterL equation − − 2 2 multi-photon states. Following this, we present

2(b) ing generation of non-classical states of light [ 5], When the coupling strength π ,033838(2010) π 0.2 0.3 0.4 0.5 0.6 experimental data0 demonstrating100 the transition 200 300 400g is greater than quantum simulation of complex many-body sys- κ dρ from blockade to tunneling regime in a strongly 2 and γ, the system is in the strong coupling ω tems [ 6], quantum information processing, and = i[H, ρ]+κ [a = regime [ ]+γ [σ], (2) = 8, 9dt]. In this regime, energy eigenstates τ κ coupledhigh precision QD-cavity sensing system and metrology and show the [ signa-Ω (GHz) − L L c 7]. 0 are grouped in two-level manifolds with eigen- hw the shows ω ture of higher order dressed states observed in 5GHzand In this paper, we explore the utility of the where ρ is the density matrix of the coupled QD- + 0 energies given by nω the measured photon statistics. cavity system and c g√n (for ωa = ωc), where c Figure 3.1: Experimental setup for detecting photon correlations.photon induced tunneling and blockade(a) for non- An [D±] is the Lindblad operator . (d)n is the number of energyL quanta in the coupled GHz, 1 classicalThe dynamics photon ofstate a coupled generation QD-cavity and dressed system, corresponding to operator g = QD-cavity system. The eigenstatesD, defined can be as writ- FIG. 2. (Color online) (a) Nonclassical state generation via pulsedcoherentlystate operation probing. driven First, by we ain laser provide the field, numerical is photon well described simu- blockade. The properties of the laser pulse )and ten as: SEM image of the photonic crystal nanocavity (left) and the schematicsbylation the Jaynes-Cummings data showing that photon Hamiltonian of induced the of tunnel- the cross- form ω coupled at the input port are controlled such that the output field hasing can beprimarily used to preferentially a single-photon generate specific component.[D]=2 (b)DρD Normalized† D†Dρ ρD†D. Fock-state(3) a L − − multi-photon states. Following(e) this,(c) we present g, n ). 2 2 2 2 + e, n 1 polarized microscopy setup. The polarizing beam splitter (PBS)Hexperimental= ∆ inσ σ + combination data demonstrating the transition withWhen the couplingn, + = strength| | −g is greater than(4) coefficients ( c for vacuum state, c for single-photon state, and a c+ ∆ca†a+ig(a†σ ∞ aσ c for multiphoton probability) for the output 0 1 multi− n 2 +)+n (t)a+ ∗(t)a†, κ | √2 from blockade to tunneling regime− − in a stronglyE E 2 and γ, the system is in the strong coupling | | | | | | = = | | (1) regime [ 8, 9]. In this regime,g, n energye, n 1 eigenstates field as the intensity of the laser pulse !0 is modified and the pulsecoupled width QD-cavity is system kept and show constant the signa- at τ 0.45/κn,. The= | ground−| − state and(5) one the half-wave plate (HWP) allows us to filter and select only theture of lighthigher order dressed that statescirculated observed in are grouped in| two-level− manifolds√2 with eigen- ! =energies given by the measured photon statistics. nωc g√n (for ωa = ωc), where of the first-order eigenstates form an effective two-level system [18] so Rabi oscillations are observedn Theis the splitting number in thebetween of energy single-photon± the quanta energy in the eigenstates coupled characterin ∗ElectronicThe dynamics address: [email protected] of a coupled QD-cavity system, eachQD-cavity manifold system. has a non-linear The eigenstates dependence can be writ- on inside the cavity. Schematics of the generalized HBT setupcoherently used driven to by a laser detect field, is well described arrival n (b)of the output field. The system has parameters κ/2π 1 GHz, γ /2π 0.1 GHz, and g/2π 40ten GHz. as: (c) Fock-state coefficients for = by the Jaynes-Cummings= Hamiltonian of the form= the output field as the duration of the laser pulse (τ) is modified while !0 10 GHz was kept constant. The system has parameters times of three-photon events, from which the third-order autocorrelation function g, n + e, n 1 = n, + = | H = ∆aσ+σ +∆ a†a+ig | − (4) κ/2π 1 GHz, γ /2π 0.1 GHz, and g/2π 40 GHz. (d) The same simulationc as(a†σ ina panelσ+)+ (t)a (b)+ ∗(t but)a†, for a system| with√κ2 /2π 5GHzand (3) − − − E E Fig. 2 = = = (1) g, n = g (τ , τ ) is then extracted. As the QD is temperature tuned across the resonance e, n 1 1 2 g/2π 30 GHz.(c) n, = | −| − (5) = | − √2 The splitting between the energy eigenstates in of the cavity, an anticrossing is observed in the system’s spectrum∗Electronic (the address: [email protected] cross-polarizedeach manifold has a non-linear dependence on n reflectivity curves are obtained by using a superluminescent broadband diode as the The nonclassical state of light emitted at the output is κ1 and κ2 such that κ1 κ2.Effectively,thetotaldecay # source). analyzed by using an ideal single-photon detector. For each rate of the cavity is κ κ1.Thedrivinglaserisincident quantum trajectory, a laser pulse is coupled to the cavity and on the mirror with higher≈ reflectivity and the output field is the number of clicks detected at the output is monitored. mainly collected from the lossier mirror. This configuration from a mode-lockedIdeally, Ti:sapphire for a deterministic laser tuned single-photon near source, the bare a single cavityallows resonance, for efficient and collection the of the nonclassical field at the click should be registered by the detector every time the cavity output. emitted light wasdevice collected is operated. with However, a high the numerical output field aperture is not in a objectiveTo illustrate lens the (NA behavior = of the system operating in pure single-photon state, and in a Fock state basis, it can be photon blockade under pulsed driving, we first analyze a 0.75). We employexpressed the cross-polarized as reflectivity technique (inputsystem source with orthogonalparameters κ/2π 1 GHz, γ /2π 0.1 GHz, and g/2π 40 GHz. The value=κ/2π 1GHzcorresponds= ◦ to the collected reflected signal, and at∞ 45 relative to the cavityto mode) a cavity= that with a mimics quality factor of Q= 160 000. This is bout ϕn n , (9) | " = | " about five times larger than the state-of-the-art= values of n 0 the results of a transmission measurement"= [4], as discussed inQ sectionobserved 2.4.1. in GaAs This cavities with coupled InAs quantum where ϕn is the coefficient of the Fock state n .Herewewrite | " dots operating around 930 nm, but still within the theo- experimental setupthe is output depicted state as ain pure figure state, considering 3.1(a). that the dephasing retical limit for this material [20]. Regarding the coupling 2 rate of the system is negligible. The normalized value ( cn ) rate g,thetypicalvaluesmeasuredsofararearound 2 | | of the coefficients ϕn can be estimated from the number of | | g/2π 25 GHz [21]. However, with further improvements detected photons at the output when running a large number in the= material system and the fabrication techniques it is 2 2 ϕn of trajectories. For example, cn | ϕ| 2 is well estimated expected that higher values for Q and g,asconsideredhere, | | = i i | | n will be achievable. For this simulation, the cavity and the by the relative number of trajectories for! which counts were detected at the output. If the desired output state should be as quantum dot are assumed to be on resonance (ωc ωa). = close as possible to a single-photon state, then the simulation The center frequency of the driving field is set on resonance 2 parameters should be optimized such that c is maximized. with the transition to the first-order manifold (ωc g)and 1 + The experimental configuration considered| | here is as shown the pulse width is set to τ 0.45/κ.Figure2(b) shows the = in Fig. 2(a).Thecavityhastwomirrors,withdecayrates zero-photon, single-photon, as well as multiphoton population

033838-3 CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 16

3.2 Three-photon correlations in a weakly coupled QD–cavity system

To demonstrate the measurement of the third-order autocorrelation function from a (3) solid-state system, we first measured g (τ1, τ2) from a single-photon source based on the spontaneous emission from an individual QD coupled to a low quality-factor cavity (Q 2, 000). This particular QD–cavity system was in the weakly coupled ≈ regime, with the cavity only serving to improve the photon collection efficiency. The QD was temperature-tuned to be on resonance with the cavity at 921 nm. The system was then illuminated with focused pulses from a mode-locked Ti:sapphire laser tuned to a higher-order mode of the L3 cavity ( 889 nm), which allowed us to excite the ∼ dot through a higher-order state [61]. The pulses were 3 ps long with a repetition ∼ rate of 80 MHz. The light emitted by the QD was collected with a high numerical ∼ aperture objective (NA = 0.75) and passed through a 1 nm FWHM band-pass filter ∼ to reject the scattered light from the excitation pulses and to suppress any undesired fluorescence. Figure 3.2(a) shows a subset of the raw data for the three-photon coincidence histogram collected from the QD fluorescence with our three-channel setup (as discussed in more detail in section 3.2.1 below). Even before any further time-binning, we can see qualitative features of the system that are characteristic of the emission from a single quantum emitter. In particular, the noticeable lines of suppressed peaks correspond to the number of three photon events in which (i) the first and second photons arrive simultaneously (the vertical line with τ1 = 0), (ii) the second and third photons arrive simultaneously (the horizontal line with τ2 = 0, shown in figure 3.2(d) in more detail), and (iii) the first and third photons arrive simultaneously (the diagonal line with τ1 + τ2 = 0). At the same time, the missing peak at (τ1, τ2) = (0, 0) corresponds to the number of events in which all three photons arrive simultaneously. CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 17

(a) (c) —(3) G∞ Three-photon coincidences ! /Trep (d)

~ ~ (3) (b) G0 Three-photon coincidences

! /Trep (e)

Two-photon coincidences ~ (2) G0

! /Trep

Figure 3.2: Three-photon correlations from QD fluorescence. (a) Three- photon coincidences (a subset of the raw data) detected in the fluorescence from a quasi-resonantly excited QD weakly coupled to a photonic crystal cavity. Here, τ1 is the time interval between the arrival of the first and second photons, while τ2 is the time interval between the arrival of the second and third photons. (b) ˜(3) G (τ1, τ2), the unnormalized values of the three-photon correlations, obtained by integrating the counts under the coincidence peaks in (a). (c) Diagonal elements of ˜(3) G (τ1, τ2), with τ1 = τ2 = τ. Trep 12.5 ns is the repetition period of the pulse train from the Ti:sapphire laser used to≈ excite the QD and the dashed red line marks the (3) normalization level, G¯∞ = 819 34, for the autocorrelation function. (d) Coincidence ˜(3) ± counts G (τ1, τ2), with τ2 = 0, corresponding to the three-photon events in which the system emits the second and third photons simultaneously. Note the different scale of the vertical axes compared to (c). (e) Two-photon coincidence counts G˜(2)(τ) (2) 5 observed from the QD. The normalization level here is G¯∞ = (4.26 0.01) 10 . ± × CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 18

(3) 3.2.1 Extraction of g (τ1, τ2)

(3) For the measurement of g (τ1, τ2), we implemented a three-channel photon arrival detecting setup [shown in figure 3.1(b)] that records the arrival times of individual photons at each of the three SPCMs using FPGA-based timing electronics. We time- bin the resulting data files containing the arrival sequence in order to produce a time- of-arrival histogram for three-photon events. Similar to the process used to extract the number of two-photon coincidences G(2)(τ) from a two-detector measurement, our time-binning algorithm uses a photon detected by SPCM1 as a start signal, the detection of a photon by SPCM2 as the first stop (τ1), and the detection of a photon (3) by SPCM3 as the second stop (τ1 +τ2). This results in a 2D histogram for G (τ1, τ2) with a grid of peaks with spacing given by the repetition rate of the Ti:sapphire pulses, as shown for instance in figure 3.2(a). The width of the peaks in the grid is in this case mostly given by the time jitter of the TTL output from the SPCMs, so integrating each peak will result in a time-binned average number of three-photon ˜(3) events, which we denote by G (τ1, τ2) [examples plotted in figure 3.2(b–d)]. (3) To obtain the normalized pulse-averaged third-order autocorrelationg ¯ (τ1, τ2), we generalize the procedure for extracting the value of the second-order autocorrelation function [16]; specifically, we rescale the data such thatg ¯(3)(τ , τ 1 → ∞ 2 → (3) (3) ) = 1. This is done by dividing G˜ (τ , τ ) by G¯∞ , which we obtain by fitting the ∞ 1 2 histogram of G˜(3)(τ, τ) with the function

˜(3) ¯(3) ¯(3) −mTrep/Tdecay ¯(3) G (mTrep, mTrep) = G (Trep,Trep) G e +G (3.2) − ∞ ∞ (3) to remove the eﬀects of probe-induced blinking [62] on G¯∞ . The error ranges given on our ﬁnal values forg ¯(2)(0) andg ¯(3)(0, 0) take into account ˜(n) both the standard deviation σ0 = G0 of the number of detection events in the given pulse (at zero time-delay) as derivedq from Poissonian statistics and the uncertainty CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 19

¯(n) σ∞ in the normalization value G∞ . Combining these eﬀects,

(n) g¯ (τ1 = 0, τ2 = 0, ..., τn−1 = 0)

(n) (n) (n) (n) G˜ G˜ G˜ G˜ σ σ 2 0 ± 0 = 0 ± 0 1 ∞ + ∞ ¯(n) q ¯(qn) ¯(n) ¯(n) ≡ G∞ σ∞ G∞ ∓ G∞ O G∞ ! ± (n) (n) (n) (n) ¯ ˜ ˜ G˜ G∞ + σ∞ G0 + σ∞G0 0 , (3.3) ≈ ¯(n) ± q(n) 2 G∞ G¯∞ which provides a method of easily computing both the nominal values forg ¯(2)(0) and g¯(3)(0, 0), as well as their expected variation. Each peak in the histogram in ﬁgure 3.2(a) represents the unnormalized value of the third-order autocorrelation, spread over the duration of the excitation pulse. Since the width of the peaks is an artifact of the SPCM timing jitter, we sum the events under each peak into a single time-bin to obtain the average number of three- ˜(3) ˜(3) photon events G (τ1, τ2) (here G means we have time-binned the raw detection events but not yet normalized them), in which the peaks are spaced by τ1 = mTrep and

τ2 = nTrep [figure 3.2(b–d)]. We find that the normalized third-order autocorrelation (3) (3) (3) at (τ , τ ) = (0, 0) is given byg ¯ (0, 0) = G˜ (0, 0)/G¯∞ = 0.016 0.005; i.e., the 1 2 ± simultaneous arrival of three photons is almost completely suppressed (we use the notationg ¯(3) to indicate we have both time-binned and normalized the raw coincidence counts, so this represents our experimental measurement of the theoretical value g(3)). For comparison, figure 3.2(e) then plots the two-photon coincidence counts G˜(2)(τ) detected between SPCM1 (start) and SPCM2 (stop), from which we extract (2) (2) (2) g¯ (0) = G˜ (0)/G¯∞ = 0.126 0.001. Note that for this case of quasi-resonant ± excitation we observe the blinking-related decay of two-photon coincidences with (3) (2) Tdecay 1.37 µs. The non-zero values of bothg ¯ (0, 0) andg ¯ (0) are the result ≈ of imperfect spectral filtering of the background photoluminescence (PL) from the sample. We also excited the system through the wetting layer of the QDs (860 nm), which resulted in additional PL noise, worsening the single-photon behavior (2) of the system. In this case,g ¯ (0) = 0.795 0.002 (with Tdecay 0.3 µs), while ± ≈ CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 20

g¯(3)(0, 0) = 0.59 0.02; i.e., the number of events in which three photons arrive ± simultaneously is still significantly lower than the number of events in which two photons arrive simultaneously. It is also worth mentioning that since we extract the correlations from a complete list of photon arrival times instead of the more conventional approach of detecting two photons and binning the difference of their arrival times, our recorded correlations at τ Trep are not affected by the exponential decay that otherwise arises as a consequence of the single-stop binning technique [63].

3.3 Two-photon correlations in a strongly coupled QD–cavity system

We next examine a different system in which the QD is strongly coupled to the photonic crystal nanocavity. We verify the strong coupling between the cavity and the QD by observing an anticrossing in the reflectivity (taken using a superluminescent broadband diode as a source) when the QD is temperature tuned through resonance with the cavity [figure 3.1(c)]. By fitting the observed spectrum [figure 3.3(a)] as described in section 2.4.2, we extract the experimental parameters of the system— the decay rate of the cavity field κ/2π 26 GHz (corresponding to a quality factor ≈ Q 6, 200), the QD–cavity coupling rate g/2π 21 GHz, and the fraction of the ≈ ≈ time the dot spends in a dark state (not interacting with the cavity) due to blinking, pdark 0.38. As mentioned earlier, when probed near resonance with coherent light ≈ from a laser, such a system can act as an adjustable photon number filter thanks to the anharmonic JC ladder [18]. Prior to this work, only measurements of the second-order autocorrelation function had been performed on such a system [16, 45]. The relevant features of the photon correlations in this system occur at a time- scale given by the lifetime of a photon inside the cavity [16], which is significantly shorter than the time resolution of the SPCMs. To resolve these features, we sample the correlations by a train of pulses from the mode-locked Ti:sapphire laser ( 80 ∼ MHz repetition rate). The linewidth of the original 3 ps pulses was reduced to ∼ CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 21

∆λFWHM 0.04 nm (corresponding to a bandwidth of roughly 14 GHz) by passing ≈ the pulses through a monochromator. This allows us to resolve the relevant spectral features of the system while retaining the fast sampling. The average optical power ¯ in the pulse train was measured to be Pprobe 0.2 nW in front of the objective lens, ≈ which at an frep 80 MHz repetition rate and with a coupling efficiency of η 0.01 ≈ ∼ corresponds to an approximate intracavity photon number (during the on-time of the ¯ pulse) of ηPprobe 0.12. frep¯hω ≈ We first tune the pulses to be on resonance with the QD–cavity system [red arrow in figure 3.3(a)] and record the arrival times of the transmitted photons using the three-SPCM setup from figure 3.1(b), with the system held at a temperature of T 30 K. From this data we can extract the second-order autocorrelation func- ≈ tion g(2)(τ) via the two-photon coincidence counts G˜(2)(τ) detected between SPCM1 (start) and SPCM2 (stop) [figure 3.3(b)]. In addition to two-photon bunching caused by photon-induced tunneling, this data also reveals the presence of classical bunching resulting from QD blinking [16, 62]. The latter manifests itself as an exponential decay of coincidence counts G˜(2)(τ) for increasing τ. The time constant of this decay,

Tdecay 0.5 µs, is much longer than the decay of the bunching from photon-induced ≈ (2) tunneling, and we extract it together with the normalization constant G¯∞ by ﬁtting

˜(2) ¯(2) ¯(2) −mTrep/Tdecay ¯(2) the histogram with the function G (mTrep) = (G (Trep) G∞ ) e +G∞ . − Note that this decay time is determined by the mean switching rate between the bright and dark states, and is independent of the fraction of time pdark the QD actually spends in the dark state. Unfortunately, because of the much faster time-scales of our system compared to conventional atom–cavity experiments [60], we cannot resolve the decay rate of photon bunching caused by photon-induced tunneling, as this happens within the time scale of the individual pulses. After normalization, the (2) (2) (2) second-order autocorrelation isg ¯ (0) = G˜ /G¯∞ = 1.141 0.003. 0 ± CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 22

(a) (b) ~ (2) G0

—(2) G∞

! /Trep Two-photon coincidences

(d) ~ (3) G0 ~ (c) ~

—(3) G∞

! /Trep

(e) ~ (3) G0 ~

—(3) G∞

!1 /Trep

FigureFig. 3.3: 3Two- (v3b), and g3=1.45 three-photon \pm 0.02 correlations in a resonantly probed strongly coupled QD–cavity system. (a) The transmission spectrum (blue) of the strongly coupled system in which the QD was tuned into resonance with the cavity. For comparison purposes, the dashed gray curve plots the calculated transmission of an empty cavity with Q = 6, 200, while the red dotted curve represents the spectrum of the probe pulse (whose center frequency is tuned into resonance with the dot and the cavity, as marked by a red arrow). (b) Two-photon coincidence counts G˜(2)(τ) observed in the transmission of the strongly coupled system. Notice the classical bunching caused by QD blinking. The second-order autocorrelation is (2) ˜(2) ¯(2) ˜(3) g¯ (0) = G0 /G∞ = 1.141 0.003. (c) Three-photon coincidence counts G (τ1, τ2) observed in the photons transmitted± through the resonantly probed system. (d) Di- ˜(3) agonal elements of G (τ1, τ2), with τ1 = τ2 = τ. The QD blinking again results in classical bunching that decays with the same time scale as for that observed in ˜(2) ˜(3) G (τ). (e) G (τ1, τ2), with τ2 = 0, corresponding to the three-photon events in which the system emits the second and third photons simultaneously. CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 23

3.4 Three-photon correlations in a strongly coupled QD–cavity system

In a process analogous to obtaining the second-order autocorrelation function, we (3) now extract the third-order temporal auto-correlation functiong ¯ (τ1, τ2) via the ˜(3) three-photon coincidence counts G (τ1, τ2) shown in figure 3.3(c). The observed correlations in this plot are in agreement with the previously reported measurements [16] of g(2)(τ) in the photon-induced tunneling regime of a strongly coupled QD– cavity system. In particular, the noticeable lines of enhanced peaks correspond to the number of three photon events in which (i) the first and second photons arrive simultaneously (the vertical line with τ1 = 0), (ii) the second and third photons arrive simultaneously (the horizontal line with τ2 = 0, shown in figure 3.3(e) in more detail), and (iii) the first and third photons arrive simultaneously (the diagonal line with τ1 + τ2 = 0). At the same time, the highest peak at (τ1, τ2) = (0, 0) corresponds to third-order temporal bunching in transmitted photons withg ¯(3)(0, 0) = 1.45 0.04, ± which is noticeably larger than the value obtained forg ¯(2)(0). Note that as discussed in section 3.2.1, the range of variation in this value has contributions both from the standard deviation of the number of detection events for a given pulse (as determined by a Poisson distribution) and from the uncertainty in the normalization constant. These results are in marked contrast with the g(3) and g(2) measurements reported in section 3.2 for the photoluminescence from a QD weakly coupled to a photonic crystal nanocavity. In that system we discovered thatg ¯(3)(0, 0) < g¯(2)(0) < 1; i.e., there were significantly fewer simultaneous three-photon detection events than simultaneous two-photon detection events, as expected for an antibunched imperfect single-photon source [59]. The strongly coupled system, on the other hand, exhibits photon-induced tunneling when probed at the bare cavity resonance frequency, and hence shows the signature of bunching:g ¯(3)(0, 0) > g¯(2)(0) > 1. CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 24

3.5 Frequency dependence of the multiple-photon correlations

We repeated the autocorrelation measurements for a set of probe laser frequencies to (3) map the spectral dependence ofg ¯ (τ1, τ2). Because the cavity had slightly shifted in frequency, the measurement was now performed with the sample kept at a higher temperature ( 40 K instead of 30 K), which negatively affected the amount of ∼ ∼ detectable photon bunching. The QD was also slightly red-detuned from the cavity resonance. Nevertheless, the frequency scan in figure 3.4(a) shows that as the probe is tuned, the third-order autocorrelationg ¯(3)(0, 0) of the transmitted photons exhibits either antibunched or bunched behavior as the system transitions from the photon blockade to the photon-induced tunneling regime (for a probe red-detuned from the cavity resonance). For comparison, figure 3.4(a) also shows the values ofg ¯(2)(0) obtained for the same frequency scan. In the tunneling regimeg ¯(3)(0, 0) > g¯(2)(0), i.e., the simultaneous arrival of three photons is enhanced compared to simultaneous two-photon arrivals, which is in qualitative agreement with numerical simulations (explained further in section 3.7). During the experiment, we kept the probe power ¯ constant at Pprobe 0.3 nW (corresponding to an approximate intracavity photon ≈ number of 0.17) and the coupling of the probe into the cavity was re-optimized for every data point. The data in figure 3.4 show a good agreement with numerical simulations of the values of g(2)(0) and g(3)(0, 0) as a function of probe detuning, given the system parameters measured earlier [g/2π = 21 GHz, κ/2π = 26 GHz, and (γ + γd) /2π ≈ 1 GHz], a probe driving strength of /2π = 10 GHz, a QD–cavity detuning of E0 ∆ = 20 GHz, and the fraction of QD “dark state” time pdark 0.9 (note that this ≈ is significantly higher than our earlier estimate based on the reflectivity spectrum of the QD–cavity system, probably due to the higher temperature needed to bring the dot into resonance with the cavity during this measurement). Importantly, the experimental data and numerical simulations show that the bunching in g(2)(0) and g(3)(0, 0) when the probe is on resonance with the QD and the cavity drops off as a function of probe power [the inset of figure 3.4(a)], leveling out when the intracavity CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 25

sync timing Fig. 4 (c) module (b) (4) TTL g (!1, !2, !3) SPCM 4 Stop 1.4 Start 1.35 TTL SPCM 3 Stop (a) 1.3 _ (2) _ g (!) gg(3)(3)(0,0)(0,0) Start 1.25 (3) g (!1, !2) _ 1.2 light from TTL _ SPCM 2 Stop (4) (3)(3) experiment g (!1, !2, !3) g g (0,0)(0,0) 1.15 __ (2)(2) Correlation 1.1 gg (0)(0) Start binning binning of arrival 1.05 differences-time SPCM 1 TTL Stop 1 (3) 0 0.5 1 1.5 2 2.5 3 g (0,0) Start Intra-cavity photon number

1.45 _ _ g(2)g(2)(0)(0) (d) g(2)(0) 1.35 0) ( (n) g 1.15

1.05

2 3 n 4

Figure 3.4: Comparison of autocorrelation functions. (a) A frequency scan of ¯ the autocorrelation measurements with Pprobe 0.3 nW, corresponding to intracavity photon number 0.17. The inset shows a comparison≈ of the second- and third-order ∼ ¯ autocorrelations for various levels of Pprobe, when the probe is on resonance with the QD and the cavity. The dashed (solid) lines plot the result of a numerical simulation for the second- (third-) order correlations. (b) A visualization of the time-binned and (4) normalized fourth-order autocorrelation functiong ¯ (τ1, τ2, τ3). To guide the eye, the value of each peak is represented both by color and size of the plotted data point. (c) Schematics of the expanded HBT setup used to detect arrival times of up to four- photon events used to obtain autocorrelation functions up to the fourth-order, i.e., (4) g¯ (τ1, τ2, τ3), shown in (b). (d) Increasing values of the autocorrelation functions g¯(n) at zero time-delay, plotted as a function of their order n (the red squares with error bars represent experimental data, while the green diamonds plot the results of a numerical simulation). photon number nears 1. This power dependence indicates that the bunching we observe is indeed due to photon-induced tunneling and not classical bunching due to blinking; at higher powers, the QD saturates [64], which in turn diminishes the polariton dip [5] and reduces the amount of bunching observed in the light transmitted through the system, an effect which is well reproduced by our numerical simulations. There are several factors that account for the difference between the theoretically (2) (3) predicted values of the second- and third-order autocorrelationsg ¯ (τ) andg ¯ (τ1, τ2) and our experimentally observed values: background light due to imperfect extinction of the uncoupled probe in the cross-polarization setup, QD blinking and spectral diffusion, temperature-dependence of the QD dephasing rate, and the non-negligible bandwidth of the probe pulses. In particular, the QD blinking and background light CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 26

cause the observed signal to have a larger coherent-state component than desired. The slight asymmetry of the data is due to the nonzero detuning between the QD and the cavity resonance, an eﬀect that will be explored in detail in chapter 4.

3.6 Four-photon correlations in a strongly coupled QD–cavity system

We observe that the theoretically predicted values of g(3)(0, 0), the third-order photon correlations in a laser beam transmitted through a strongly coupled QD–cavity system, differ more significantly than the values of g(2)(0) from the unity expected (3) from coherent (laser) light. This confirms that g (τ1, τ2) is a more sensitive diag- nostic tool for observing non-classicality in the measured photon statistics, as it can more clearly be distinguished from the signature of coherent light. This approach— increasing the order of the correlations in order to get a more clear signature of non-coherent light—is further illustrated in figure 3.4(b–d), showing the preliminary results of our measurements of the fourth-order autocorrelation function

a†a†(τ )a†(τ + τ )a†(τ + τ + τ )a(τ + τ + τ )a(τ + τ )a(τ )a g(4)(τ , τ , τ ) = h 1 1 2 1 2 3 1 2 3 1 2 1 i. 1 2 3 a†a 4 h i (3.4) The four-photon correlations [figure 3.4(b)] were obtained by adding another photon counter to the generalized HBT setup [figure 3.4(c)], and then binning and nor- malizing the four-photon coincidences in the transmitted light through this QD–cavity system probed with resonant [figure 3.3(a)] laser pulses, a process equivalent to that (2) (3) described earlier for g (τ) and g (τ1, τ2). Figure 3.4(d) then shows how the value of the autocorrelation functions at zero time-delay keeps increasing with the order of the autocorrelation function for light transmitted through the QD–cavity system in the photon-induced tunneling regime. To obtain a sufficient number of four-photon coincidences over a reasonable data collection time, the system was probed with ¯ Pprobe 1.0 nW, which partially saturated the dot and resulted in lower observed ≈ CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 27

values ofg ¯(3)(0, 0) andg ¯(2)(0) for this particular measurement. This highlights a significant drawback of this approach: the increasing integration time required to collect enough events for a meaningful statistical analysis makes the measurement of higher- order autocorrelations difficult or even impossible in some experimental systems. In our setup, the single-photon count rate on each SPCM was roughly 7 105 × counts per second for the g(4) measurements shown in figure 3.4. After identifying the multiple-photon correlations and time-binning the delays, the coincidence counts are distributed into a histogram consisting of a series of discrete peaks (corresponding to different time-delays between photon arrivals in increments of 12.5 ns, the repetition rate of the driving laser), as discussed in section 3.2.1. On average (away from the zero time-delay peak), each two-photon coincidence peak accumulated events at a rate of 22,000 to 42,000 counts per second (depending on the particular SPCM configuration), while three-photon coincidence peaks accumulated events at a rate of 27 to 42 counts per second, and four-photon coincidence peaks accumulated events at a rate of only 0.25 counts per second. We observe that moving to the next higher-order autocorrelation function decreases the count rate (and hence increases the integration time) by roughly a factor of 100. Since at minimum several hundred counts are needed to make a reliable measurement, this puts the total integration time for g(4) measurements (in our setup) on the order of hours.

3.7 Theoretical modeling

We model the dynamics of a coupled QD–cavity system (coherently driven by a laser ﬁeld) with the JC Hamiltonian of the form

† † ∗ † H = ∆QDσ σ− + ∆cava a + g(a σ− + aσ ) + (t)a + (t)a , (3.5) + + E E which assumes the rotating-wave approximation (RWA) and a frame of reference rotating with the frequency of the laser ﬁeld ωl. Here ∆QD = ωQD ωl and ∆cav = − ω ωl are respectively the detuning of the QD resonant frequency ωQD and the 0 − cavity resonance frequency ω0 from the laser, g is the coherent coupling strength CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 28

between the QD and the cavity mode, (t) = κP (t) is the slowly-varying envelope E ¯hω0 of the coherent driving ﬁeld with power P (t)q incident on the cavity, and a is the annihilation operator for the cavity mode. If the excited and ground states of the QD are denoted by e and g , then σ− = g e and σ = e g . | i | i | ih | + | ih | Three main loss mechanisms of this system (the cavity ﬁeld decay rate κ = ω0/2Q where Q is the quality factor of the resonator, QD spontaneous emission rate γ, and pure dephasing of the QD γd) are incorporated in the master equation,

dρ = i[H, ρ] + κ [a] + γ [σ] + γd [σ σ−], (3.6) dt − L L L + where ρ is the density matrix of the coupled QD–cavity system and [D] is the L Lindblad operator corresponding to a collapse operator D, deﬁned as

[D] = 2DρD† D†Dρ ρD†D. (3.7) L − −

We deﬁne the values of the second- and third-order autocorrelation functions (2) (3) g (τ1) and g (τ1, τ2) [47] as

a†a†(τ )a(τ )a g(2)(τ ) = h 1 1 i, 1 a†a 2 h i a†a†(τ )a†(τ + τ )a(τ + τ )a(τ )a g(3)(τ , τ ) = h 1 1 2 1 2 1 i, (3.8) 1 2 a†a 3 h i where τ1 is the time between the arrival of the ﬁrst and second photons and τ2 is the time between the arrival of the second and third photons. Thus, if we assume that the output photon state transmitted through the system can be expressed as a superposition of the photon number (Fock) states ψ = n cn n where the probability 2 | i | i of the nth Fock state is P (n) = cn , it follows that | | P n(n 1)P (n) g(2)(0) = n − , [ nP (n)]2 P n n(n 1)(n 2)P (n) g(3)(0, 0) = Pn − − . (3.9) [ nP (n)]3 P n P CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 29

In particular, this means that g(2)(0) = 0 for a single-photon pulse train (whether perfect or sparse), while g(2)(0) = 1/2 for a perfect two-photon pulse train but g(2)(0) ≈ 1/2ε2 for a sparse two-photon pulse train (defined as ψ √1 ε2 0 + ε 2 ). | i ≈ − | i | ni Note that a classical light source producing a coherent state α = √α n can | i n n! | i also have a particular Fock state m to be the state with the highest probability of | i P occurrence P (m), if α is chosen such that m < α2 < m + 1. Thus to evaluate the non-classicality of a Fock-state source (and to distinguish it from a coherent source), P 2(m) we define the ratio rm = P (m−1)P (m+1) [18], which contrasts the probability of state m with the probabilities of the neighboring Fock states m 1 and m + 1 in a | i | − i | i given light source. For a coherent state, the ratio rm = 1 + 1/m remains a constant for a given m, which cannot be optimized by adjusting the value of α. This, in turn, leads to g(n)(0) = 1 at all orders n for an m-photon source based on attenuation of coherent light (from the simple application of the expressions above for g(2) and g(3)). Thus, the level of non-classicality of a weak m-photon Fock-state source, such as one based on the generalized photon blockade studied here, can be quantified either by how much its value of rm differs from the classical limit [figure 3.5(d–f)] or by looking at the values of its g(m)(0) and g(m+1)(0) [figure 3.5(c)]. Hence, the measurement of higher-order photon correlations is of critical importance for distinguishing non- classical states from coherent light. Figure 3.5(a) shows the theoretical technique for addressing higher-order manifolds of the JC ladder of a strongly coupled QD–photonic crystal nanocavity system [16], while figure 3.5(b–f) plot the numerical simulation results that show the response of the system to different probe laser frequencies. These simulations were performed by numerically integrating the quantum master equation using the QOToolbox originally developed by Tan [65]. The parameters used for the simulation are those of an ideal QD–cavity system that is currently within experimental reach and that is driven with laser pulses comparable to those used in our experiment. As expected, the system behaves as a highly non-classical source in the single-photon regime [blue arrows in figure 3.5(a)]. The single-photon component dominates the vacuum, as well as all multi-photon Fock states [figure 3.5(b)], resulting in g(2)(0) 0.4 [figure 3.5(c)] and ≈ r 25 [figure 3.5(d)] for ∆/g 1.1. Accessing the two-photon regime requires 1 ≈ ≈ CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 30

(a) (b) two-photon ...... |2, +! dominance (d)

_ P(0) 2!c 2g!2 P(1) P(n) cavity transmission P(2) cavity transmission (ﬁrst and second (ﬁrst and second |2, -! manifolds) P(3) manifolds) _ (e) !c+g(!2-1) P(n) or cn P(n) or cn vs vs |1, + frequency frequency ! (c) (dream system) (realistic system)

!c 2g

|1, -! g2 and g3 g2 and g3 vs vs (f) !c+g _ frequency frequency (dream system) (realistic system) !c+g/!2 _ !c+g/!3 Correlations 0 |0!

Figure 3.5: Creating and detecting two-photon states inside a cavity containing a quantum emitter. (a) Energy diagram of a strongly coupled QD–cavity system showing the spacings between the levels of the first- and second-order manifolds. Because the energy differences between consecutive(i) levels(ii) are not constant, photon photon (c) the system can act as a photon-number filter when the frequencycounter of the probecounter laser Stop Stop is tuned properly. Here the blue arrow represents thephoton probe Start arrival frequency photon atStart whicharrival only counter time counter time individual photons couple into the system. The red(iii) arrow represents(iv) the probe fre- photon photon quency at which photons couple in pairs via a two-photon transition,counter whilecounter the green arrow represents the frequency at which a three-photon transitionStop is addressed.Stop (b) photon Start arrival photon Start arrival Probability of n-photon state, P (n), inside the QD–cavitycounter time systemcounter as a functiontime of

timing laser-cavity detuning ∆ and (c) the corresponding second- and third-ordersync correla-module

TTL tion functions (plotted in log scale). The photon statistics were numericallySPCM calculated3 Stop Start

TTL for a system driven by Gaussian pulses with duration τ 25 ps. TheSPCM simulation 2 Stop (3) p g (!1, !2) parameters for both (b) and (c) are g/2π = 40 GHz, κ/2π≈= 4 GHz, and /2πStart = 9 light oTTL of arrival times

SPCM 1 Stop post-processing from experiment E GHz, close to the highest achievable g with this type of system and to the highestStart quality factor (Q 25, 000) measured in our laboratory; here, γ/2π = 1 GHz and ≈ pure QD dephasing γd is neglected. The magniﬁed sections of the plots (both in linear scale) show the frequency range (marked by the vertical dashed lines) in which the two-photon state is dominant over the other states [P (2) > P (1) + P (3)], where 2 2 (2) (3) P (1) P (2) g (0) > 1 and g (0, 0) < 1. The ratios (d) r1 = P (0)P (2) , (e) r2 = P (1)P (3) , and (f) P 2(3) r3 = P (2)P (4) , as functions of the frequency of the probe laser. Each plot is contrasted with r , r , and r for a coherent light source with a†a = 2 (dashed gray lines). 1 2 3 h i CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 31

addressing the levels of the second manifold of the JC ladder [red arrows in figure 3.5(a)]. This addressing is more difficult to do precisely, given that the linewidth of the levels in the second manifold is roughly twice as big as the linewidth of the levels in the first manifold [66]. As a result, the frequency for the maximum probability of a two-photon state [∆/g = 0.94, figure 3.5(b)] does not fully coincide with the maximum non-classicality of the two-photon regime [r 4.4 at ∆/g 0.9, figure 2 ≈ ≈ 3.5(e)], and both are actually outside of the frequency region in which the two-photon state dominates over the other non-zero Fock states [g(3)(0, 0) < 1 and g(2)(0) > 1, magnified section of figure 3.5(c)]. The detrimental effect of the increasing level broadening of the higher-order manifolds fully takes over when one tries to access the three-photon regime [green arrows in figure 3.5(a)]. While P (3) is maximized at ∆/g 0.7 [figure 3.5(b)], the three-photon state is far from dominant and r only ≈ 3 reaches 2 [figure 3.5(f)]. ∼ These results illustrate the main limit of this scheme for non-classical light generation, which comes from the unresolvability of the higher-order manifolds in the currently achievable GaAs L3 cavities with self-assembled QDs. The full potential of this scheme, in particular its photon-number filtering capabilities, could nevertheless be utilized in optical systems with higher quality factors (such as silicon-based L3 cavities) or smaller mode volumes (which can potentially result in higher coupling strengths), or in circuit cavity-QED systems.

3.7.1 Relation between g(2)(0) and correlation measurements

(2) ha†a†aai The expression g (0) = 2 is commonly used to describe the second-order ﬁeld ha†ai autocorrelation of a light beam. However, in the context of our pulsed excitation scheme this expression is somewhat incomplete [67]. A semiclassical treatment of photodetection deﬁnes the measured degree of second-order coherence at zero time- delay asg ¯(2)(0) hm(m−1)i , which is the normalized second-order factorial moment of ≡ hmi2 the time-integrated photocount distribution Pm(T ) for m photons in a detection window of width T . By analogy to the classical integrated mean intensity, the quantum CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 32

mechanical operator T Mˆ (T ) = dt b†(t)b(t) (3.10) Z0 represents the total photon number arriving at an ideal detector over the time interval t [0,T ], such that m = Mˆ (T ) . Here, b(t) is the instantaneous field mode ∈ h i h i operator describing the field flux, as distinct from the cavity field mode operator a(t): b(t) describes a field flux while a(t) describes a field. Comparing our semiclassical definition of g(2)(0) to the quantum mechanical operator Mˆ (T ), we have

Mˆ (T )(Mˆ (T ) 1) : Mˆ (T )2 : g¯(2)(0) = h − i = h i, (3.11) Mˆ (T ) 2 Mˆ (T ) 2 h i h i where : Mˆ 2(T ): denotes the quantum mechanical expectation value of the normally h i ordered second moment of the photon number operator. Writing out this expression explicitly, we obtain

T T dtdt0 b†(t)b†(t0)b(t0)b(t) g¯(2)(0) = 0 0 h i, (3.12) ( T dt b†(t)b(t) )2 R R 0 h i and substituting the unnormalized correlationR moments for the expectation values, we arrive at T T dtdt0 G2 (t, t0) g¯(2)(0) = 0 0 bb . (3.13) m 2 R R h i Therefore, the measuredg ¯(2)(0) actually represents a quantity other than that which

(2) ha†a†aai g (0) = 2 would suggest. Instead of representing the correlation between ha†ai two Fock states—states that incidentally do not exist in free space—¯g(2)(0) repre- 2 0 sents the sum of all field flux correlations Gbb(t, t ) over the detection time. In other ha†a†aai words, 2 operates only on states which are well defined in a cavity whereas ha†ai the correlation measurements are performed on fluxes. However, input-output theory provides a direct connection between the internal and external mode operators such that measuring zero time-delay flux correlations is equivalent to calculating R T R T 0 0 dtdt0ha†(t)a†(t0)a(t0)a(t)i R T 2 inside the cavity. ( 0 dtha†(t)a(t)i) CHAPTER 3. HIGHER-ORDER PHOTON CORRELATIONS 33

3.8 Outlook

To achieve efficient generation of photon pairs and other higher-order Fock states using this method, a system with a better dot–cavity coupling strength g and higher cavity quality factor would be needed (as indicated by the numerical simulations presented in figure 3.5 above). Alternatively, control over the dot–cavity detuning with current experimental parameters could also be employed to improve photon blockade and as an alternative scheme for generating higher-order Fock states, as proposed by Sánchez-Muñoz et al. [19]. This approach will be explored further in chapter 4. Finally, these higher-order autocorrelations have the potential to be used for monitoring phase transitions in condensed-matter simulations based on photon gases [68]. The possibility of using an array of coupled photonic crystal cavities for such a simulation will be discussed in more detail in chapter 6. Chapter 4

Photon blockade in a detuned quantum dot–cavity system

The on-chip generation of non-classical states of light is a key requirement for future quantum optical hardware. As I’ve shown in the previous chapters, such non-classical light can be generated from self-assembled quantum dots strongly coupled to photonic crystal cavities. These systems have the advantage that the degree of non-classicality can be controlled simply by varying the probe frequency, as discussed in chapter 3. However, the photon blockade observed in such resonant systems is quite limited compared to what is achievable with single QDs alone. In this chapter, I present a method of improving the suppression of multi-photon states by controlling the detuning between the QD transition and the cavity resonance. In particular, we demonstrate the feasibility of performing photon blockade at signiﬁcant detuning, and indeed the importance of doing so for high-purity and high-eﬃciency operation [22]. We show that by detuning the QD and cavity resonances while operating in the photon-blockade regime, the second-order autocorrelation function [g(2)(0)] of the light transmitted through the cavity decreases from g(2)(0) = 0.9 0.05 to g(2)(0) = 0.29 0.04. Simulations of the second- and third- ± ± order autocorrelation functions for our system are in excellent agreement with the measurements, and they reveal that not only does the quality of the single-photon

34 CHAPTER 4. DETUNED PHOTON BLOCKADE 35

stream increase, but that the absolute probability of obtaining a single photon increases by a factor of 2. Furthermore, we show that the values we obtain for g(2)(0) ∼ are only limited by the system parameters (QD–cavity field coupling strength g and cavity field decay rate κ), and that high-quality single-photon emission is within reach for current state-of-the-art samples with specific QD–cavity detunings.

4.1 Background

The phenomena of photon-induced tunneling and photon blockade in strongly coupled systems have been experimentally demonstrated both for the case of the QD on resonance [16, 17, 18] and near resonance [45] with the cavity (as well as for resonant atom–cavity systems [44]). However, in the case of large detuning between the QD and cavity resonance these eﬀects have only been investigated theoretically [69]. The energy structure of a quantum emitter strongly coupled to a cavity is well described by the Jaynes–Cummings Hamiltonian introduced in section 2.3:

† † † † H = ω0a a + (ω0 + ∆)σ σ + g(a σ + aσ ), (4.1)

where ω0 denotes the frequency of the cavity, a the annihilation operator associated with the cavity mode, σ the lowering operator of the quantum emitter, ∆ the detuning between quantum emitter and cavity, and g the emitter–cavity ﬁeld coupling strength. After introducing dissipation into the JC system we obtain a quantum Liouville equation with complex eigenenergies [69]:

∆ (2n 1)κ + γ ∆ κ γ 2 En = nω + i − (√ng)2 + + i − , (4.2) ± 0 2 − 2 ± 2 2 s

n n where E± corresponds to the nth rung of the system. The real part of E± yields the energies of the states whereas the imaginary part yields their linewidths. Interestingly, all four parameters κ, γ, ∆, and g contribute to the polariton splittings and linewidths. The resulting eigenenergies, the JC ladder dressed states, are illustrated in ﬁgure

4.1(a). For n photons in the cavity the energy is nω0 (red lines), and the energy of the CHAPTER 4. DETUNED PHOTON BLOCKADE 36

( a ) ( b ) ( c ) 4 f i r s t r u n g s e c o n d r u n g

U P 2 ) 2 t h i r d r u n g

3 8 K g (

g 0 r ) e s t n i ) L P 2 - 2 n u

X E

u . X

. b r

b - 4 r a ( a

( y U P 1 ( d ) y g 4 t r i e s ) n n g (

e 2 E

t y n I L P 1 g r 0 e n

E - 2

g r o u n d ∆ s t a t e 3 1 K - 4

9 2 1 . 6 9 2 1 . 9 9 2 2 . 2 - 4 - 2 0 2 4 ∆ ∆ ( a r b . u ) W a v e l e n g t h ( n m ) ( g )

Figure 4.1: Transition energies of a strongly coupled system. (a) JC ladder obtained from equation 4.1, showing the lower polariton (LP) and upper polariton (UP) branches. (b) Cross-polarized reflectivity spectrum of the coupled QD–cavity system obtained for tuning the QD through the cavity resonance. (c) Relative energies for exciting the nth rung of the JC ladder in an n-photon process. (d) Relative transient energies for climbing the JC ladder rung by rung. Transitions from upper and lower polaritons are color coded in blue and green, respectively. quantum emitter (orange) varies with a detuning parameter. Due to the coupling, the resulting energy eigenstates are the anticrossing polariton branches. At resonance, the splitting is given by 2g√n (with n being the index of the rung). While this dissertation explicitly discusses the case of a QD in a photonic crystal cavity, the same physics holds for a large number of systems such as those formed by atoms [70, 71] or superconducting circuits [72]. As discussed in chapter 2, our sample consists of a layer of low-density InAs QDs grown by molecular beam epitaxy and embedded in a photonic crystal L3 cavity [30]. For QDs, the anticrossing that results from the coupling to a cavity can be efficiently studied in optical spectroscopy experiments, where the detuning between the QD and cavity resonance is controlled by the lattice temperature [37, 41]. Figure 4.1(b) shows the transmission through the cavity measured in a cross-polarized reflectivity configuration [4] in the temperature range T = 31 38 K. A clear anticrossing provides − evidence of strong coherent coupling between QD and cavity. A fit (as described in section 2.4.2) reveals a coupling strength of g/2π = 10.9 GHz and a cavity field decay CHAPTER 4. DETUNED PHOTON BLOCKADE 37

rate κ/2π = 10.0 GHz for this system. Due to the unequal energy spacing (anharmonicity) of the JC ladder, transmission of a laser through the cavity affects the beam’s photon statistics and introduces strong photon correlations [16, 45]. This is schematically illustrated by the solid blue arrows in figure 4.1(a); if the laser is tuned into resonance with one of the polariton branches of the first rung, it cannot excite the system to the second rung due to the ladder anharmonicity. Therefore, in this regime the transmitted beam consists of a series of single photons as a result of this “photon blockade.” However, the fidelity of this process is inherently limited by the transition linewidths, given by the cavity field κ and quantum emitter γ decay rates. In particular, due to final state broadening and the shorter lifetime of higher excited states, transitions to higher rungs have larger linewidths, further reducing the probability of generating single photons.

4.2 Eﬀect of QD–cavity detuning

Surprisingly, operating the system at a significant QD–cavity detuning can lead to higher-purity single-photon emission. We consider two cases to support this conclusion: the excitation of a higher rung in a “simultaneous” multi-photon process and “subsequent” photon-by-photon excitation. In order to investigate both cases in detail, we plot the energies for an n-photon excitation of the nth rung in figure 4.1(c) and the transient energies from one rung to the next in figure 4.1(d). Clearly, at zero detuning the energies for exciting the first and higher rungs are close together [figure 4.1(c)], and their separation strongly increases for the upper (lower) polariton branch for positive (negative) detunings of the quantum emitter. Hence, for a laser in resonance with the first rung, the probability of n-photon excitation of higher rungs decreases with increased detuning. Similar scenarios can be found for subsequent climbs up the ladder; figure 4.1(d) shows the transition energies from the ground state to the first rung, the first to the second rung, and the second to the third rung as solid, dashed, and dotted lines, respectively. Transitions from an upper (lower) polariton branch to higher rungs are color coded in blue (green). Near resonance CHAPTER 4. DETUNED PHOTON BLOCKADE 38

the first and second transitions are close in energy but their separation strongly increases with the detuning of the quantum emitter [c.f. dashed blue arrows in figure 4.1(a)]. The close proximity of the first rung to the outer higher order transitions for large detunings does not reduce the single-photon emission character, since these transitions occur from the other polariton branch as can be seen from the different colors. Therefore, a detuning between quantum emitter and cavity is also expected to improve the purity of single-photon generation under photon blockade for subsequent rung excitation. Detuning also affects the linewidths of the states in such a way that the linewidth of a polariton branch that evolves towards the bare QD (bare cavity) transition decreases (increases). Equation 4.2 shows that the detuning between different rungs scales with √n while the linewidth scales with n. This can be easily understood— the decay rate of a certain rung is proportional to the number of photons involved. Most strikingly, for the case of QDs and photonic crystal cavities with κ γ, the linewidth of a given polariton branch closer to the bare QD (bare cavity) at nonzero detuning is smaller (larger) than at resonance. This is particularly important for photon-blockade phenomena that rely on the absence of an energy overlap between transitions from the crystal ground state to the first rung and transitions from the first rung to higher rungs. Therefore, detuning the QD and cavity resonances not only increases the energy difference between transitions involving different rungs but also decreases the linewidth of the polariton branches closer to the bare QD transition. This further reduces the overlap of transitions involving different rungs of the JC ladder and increases the fidelity of photon blockade.

4.3 Photon blockade in a detuned system

To quantify the quantum character of light, we return to the second-order autocorre-

(2) ha†a†(τ)a(τ)ai lation function g (τ) = 2 [47] at τ = 0. For a pulsed source, this can be ha†ai expressed as m(m 1) g(2)(0) = h − i, (4.3) m 2 h i CHAPTER 4. DETUNED PHOTON BLOCKADE 39

where m signifies a number of detections in the photocount distribution [67]. As discussed in section 3.7, this results in g(2)(0) = 1 for a coherent source and g(2)(0) = 0 for a perfect stream of single photons. To test our expectation that the purity of single-photon generation under photon blockade can be improved by detuning the QD and cavity resonances, we measured g(2)(0) from the output correlations of a laser beam transmitted through the cavity using a Hanbury Brown and Twiss setup (see section 3.1 for more details). The left part of figure 4.2 shows g(2)(0) as a function of the laser detuning for six different

QD–cavity detunings. The data were recorded under pulsed excitation with τp = 30 ps pulses. This pulse duration was chosen as a compromise between frequency resolution and avoiding re-excitation of the system. In the case of ∆ 0, the form of ≈ g(2)(0) is nearly symmetric with photon-induced tunneling generating a maximum of g(2)(0) = 1.45 0.05 in the center, and photon blockade generating a minimum dip ± of g(2)(0) = 0.85 0.05 [g(2)(0) = 0.92 0.05] at the laser detuning of 1.5g [ 1.5g]. ± ± − When detuning the QD, the maximum of g(2)(0) shifts such that it stays between the polariton branches before it disappears for detunings greater than 4 g. The dip ∼ of g(2)(0) both moves with and shifts toward the polariton branch that is closer to the bare QD transition. Most strikingly, the depth of the dip increases and reaches a value as low as g(2)(0) = 0.45 0.05 for the detunings of ∆ = 2.7g and ∆ = 4.4g. ± This value is lower than 0.5, indicative of strong single-photon character, and lower than g(2)(0) measured in any prior photon-blockade experiments in the solid state. We note here that since the lifetime of the polariton branch closer to the bare QD transition increases with detuning (for details see section 4.6), excitation with 70 ps pulses was possible without re-exciting the system at detunings of ∆ = 3g 5g, − further reducing the minimum antibunching observed to g(2)(0) = 0.29 0.04 (see ± figure 4.6). Small asymmetries in the experimental measurements result from the wavelength dependence of the cross-polarized laser suppression, asymmetries in the spectral shape of the laser pulse, drift of the QD–cavity detuning, and temperature differences between curves. To support our findings, we performed quantum optical simulations using a quantum trajectory method (see section 4.4.1). Overall, the results of these simulations CHAPTER 4. DETUNED PHOTON BLOCKADE 40

1 . 6 ∆= - 2 . 8 g 1 . 6 U P 1 ) U P 2 - U P 1 0 (

2 U P 2 / 2

g 1 . 0 1 . 0

0 . 4 0 . 4

1 . 6 ∆= - 1 . 4 g 1 . 6 ) 0 ( 2

g 1 . 0 1 . 0 0 . 4 0 . 4

1 . 6 ∆= 0 g 1 . 6 ) 0 ( 2

g 1 . 0 1 . 0 0 . 4 0 . 4

1 . 6 ∆= 1 . 8 g 1 . 6 ) 0 ( 2

g 1 . 0 1 . 0 0 . 4 0 . 4

1 . 6 ∆= 2 . 7 g 1 . 6 ) 0 ( 2

g 1 . 0 1 . 0 0 . 4 0 . 4 ∆ L P 1 1 . 6 = 4 . 4 g 1 . 6 ) L P 2 - L P 1 0 (

L P 2 / 2 2

g 1 . 0 1 . 0 0 . 4 0 . 4 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 L a s e r d e t u n i n g ( g ) L a s e r d e t u n i n g ( g )

Figure 4.2: Measured (left) and simulated (right) g(2)(0) as a function of the laser detuning, for a set of diﬀerent QD–cavity detunings. With increased detuning the depth of the antibunching is more pronounced. Vertical lines represent the relevant transition energies of the JC ladder as described in ﬁgure 4.1. CHAPTER 4. DETUNED PHOTON BLOCKADE 41

(presented on the right side of ﬁgure 4.2) are in excellent qualitative agreement with the measurements and also quantitatively resemble the values measured in the photon- blockade regime. Only small diﬀerences exist; for example, the measured maximum values of g(2)(0) are slightly lower than the simulated ones. This can be explained by blinking of the quantum emitter [45], which was not included in the simulations.

4.4 Photon-number components of the generated non-classical light

As discussed in chapter 3, higher-order autocorrelations are necessary to characterize the multi-photon nature of non-classical light [21]. Thus, to further investigate the single-photon character of the light transmitted through the cavity we performed measurements of the third-order autocorrelation function g(3)(0, 0) = hm(m−1)(m−2)i . hmi3 Figure 4.3(a) shows both g(2)(0) and g3(0) as functions of the laser detuning for the case of QD–cavity detuning of ∆ = 0 (left) and ∆ = 2.8g (right). Clearly, g3(0) shows the same qualitative shape as g(2)(0) but with stronger non-classical values. Simulations of these autocorrelation functions [figure 4.3(b)] show good agreement with the measurements. In particular, in the photon-blockade regime the values of g3(0) are lower than those of g(2)(0), indicating that g(2)(0) is mainly limited by two-photon events and not events corresponding to higher photon numbers. Since the agreement with the measured autocorrelation functions is very good, we can rely on the simulations to explicitly evaluate quantities only within reach of theory, such as the probabilities Pn of transmitting n photons per excitation pulse through the cavity. These probabilities are presented in figure 4.3(c) for n = 0 3 − under the same conditions as the data shown in figure 4.3(a). Interestingly, we find that in the case of zero QD–cavity detuning [figure 4.3(c) left], we see significant contributions of one-, two- and three-photon events for all laser detunings. In fact, the probability of two-photon events (blue) actually exceeds the probability of single photons (red) in the case of the best photon blockade. In contrast, for a QD–cavity detuning of ∆ = 2.8g [figure 4.3(c) right] operating in the photon-blockade regime, CHAPTER 4. DETUNED PHOTON BLOCKADE 42

∆ = 0 g ∆ = 2 . 8 g

2 . 5 2 2 . 5 2 ( a ) g ( 0 ) g ( 0 ) 2 . 0 g 3 ( 0 ) 2 . 0 g 3 ( 0 ) ) 0

( 1 . 5 1 . 5 3 / 2 1 . 0 1 . 0 g 0 . 5 0 . 5 0 . 0 0 . 0 3 3 ( b ) g 2 ( 0 ) g 2 ( 0 ) g 3 ( 0 ) g 3 ( 0 ) )

0 2 2 ( 3

/ 2

g 1 1 0 0 1 . 0 1 . 0 ( c ) P P 0 . 8 0 0 . 8 0 n P 1 P 1 P 0 . 6 0 . 6 P 2 P 2

0 . 4 P 3 0 . 4 P 3 0 . 2 0 . 2 0 . 0 0 . 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 0 1 2 3 L a s e r d e t u n i n g ( g ) L a s e r d e t u n i n g ( g )

Figure 4.3: Comparison of the multi-photon probability for resonant and detuned systems. (a) Measured second- and third-order autocorrelation functions as a function of the laser frequency for a QD–cavity detuning of ∆ = 0 (left) and ∆ = 2.8g (right). (b) Simulated g(2)(0) and g(3)(0, 0) for the experimental conditions presented in (a). (c) Simulated probabilities for having n = 0 3 photons in the output of a pulse transmitted through the cavity. Clearly, for a detuning− of ∆ = 2.8g (right), P1 (red) exhibits a pronounced peak. Vertical dashed lines represent the bare cavity (red) and ﬁrst-rung upper polariton (black) frequencies. CHAPTER 4. DETUNED PHOTON BLOCKADE 43

single-photon events (red) strongly dominate over two-photon events (blue) and the probability of three-photon events (purple) becomes negligible. Most strikingly, in the detuned case, not only does the quality of the single-photon stream increase, but the absolute probability of finding a single photon in the transmitted laser pulse increases by a factor of 2. In addition to the agreement between measured and ∼ simulated values for both g(2)(0) and g(3)(0, 0), the experimental count rates support this finding. Since the overall count rate is proportional to n nPn, it does not directly correspond to the overall single-photon efficiency. However,P we can still calculate the ratio of the count rates at different detuning conditions in order to support our claim of increased single-photon efficiency. Comparing the resonant and detuned cases at the points of best photon blockade, our simulated count rates result in a ratio of 2.27 : 1.05 = 2.16, which is in very good agreement with the measured ratio of 4 104 : 1.8 104 = 2.22. × × This counterintuitive finding that the efficiency of single-photon generation increases when detuning QD and cavity resonances can be understood in the following way. Photon blockade is obtained if the first rung of the JC ladder is excited while the overlap of the laser with higher rungs is suppressed. When the QD and cavity are on resonance, this suppression is inherently limited by the linewidths of the transitions, which scale with n since the decay rate of a rung is proportional to the number of photons. Meanwhile, the energy difference of subsequent rungs scales with √n (see section 4.2 above). Therefore, for any system parameters κ and g that can be achieved with the QD and cavity in resonance, there will always be an overlap between the transition to the first rung and transitions corresponding to higher climbs up the ladder. As a result, the strongest photon blockade at zero QD–cavity detuning is not observed for the laser exactly on resonance with the first rung of the JC ladder ( g), but rather with the laser off resonance and detuned to 1.5g (c.f. figure ∼ ± ∼ ± 4.3, left). In contrast, if the separation between different JC rungs is enhanced by detuning the QD and cavity, the strongest photon blockade is obtained with the laser on resonance with the polariton branch, making photon blockade more efficient than in the zero detuning case. Therefore, not only the purity but also the efficiency of single-photon generation improves given the correct QD–cavity detuning. CHAPTER 4. DETUNED PHOTON BLOCKADE 44

(2) 4.4.1 Details on the simulations of g (0) and Pn

As in section 3.7 above, under RWA and in a rotating reference frame relative to the laser ﬁeld (frequency ωl) the JC Hamiltonian describing the interaction of our strongly coupled system with a laser pulse takes the form

† † † H = ∆QDσ σ− + ∆cava a + g(a σ− + aσ ) + (t)(a + a ), (4.4) + + E where σ± are the dipole operators, a is the cavity mode operator, and (t) is the E slowly-varying field–cavity coupling amplitude. The self-energies and their shifts in the rotating frame are defined by ∆QD = ωQD ωl and ∆cav = ω ωl, where ωQD − 0 − and ω0 are the QD and cavity energies, respectively. We take the laser pulse as a t2 Gaussian, well described by (t) = 0 exp( 2 ), where τp = 30 ps is the pulse length. E E − 2τp In simulations, we find that results in correlations that closely match the observed E0 experimental data. Additionally, we assume that the cavity field decay (κ) dominates the loss dynamics because the QD lifetime is almost two orders of magnitude larger than the cavity lifetime. The simulations presented in this chapter were performed using the Quantum Optics Toolbox in Python (QuTiP) [73]. To simulate the second order correlation function, two different approaches are used: the quantum regression theorem and the quantum jump method. Application of the quantum regression theorem to a problem of the form A(t)B(t + τ)C(t) (4.5) h i yields the result A(t)B(t + τ)C(t) = Tr BΛ(t, t + τ) , (4.6) h i sys{ } where Λ(t, t + τ) is governed by the evolution equation

∂rΛ(t, t + τ) = (t + τ)Λ(t, t + τ) (4.7) L CHAPTER 4. DETUNED PHOTON BLOCKADE 45

subject to the initial condition

Λ(t, t) = Cρ(t)A. (4.8)

After careful time-ordering [74], equation 3.12 has the same form as equation 4.5 and thus the implementation of the quantum regression theorem for the calculation of g(2)(0) is straightforward. In the quantum jump method, quantum trajectories that are solutions for our system’s associated stochastic Schrödingerequation are calculated. Monitoring collapse events of the cavity field over many trajectories generates the expected photocount distribution that would be measured by an ideal, infinite bandwidth detector placed directly at the cavity output. From this distribution, we directly compute the normalized second-order factorial moment in order to simulate our experimentally measured g(2)(0) = hm(m−1)i . hmi2

( a ) 2 . 2 ( b ) 2 . 2 ∆ = 0 g ∆ = 2 . 8 g

) 1 . 6 ) 1 . 6 0 0 ( ( ) )

2 2 ( ( g 1 . 0 g 1 . 0 Q u a n t u m r e g r e s s i o n t h e o r e m Q u a n t u m r e g r e s s i o n t h e o r e m Q u a n t u m j u m p m e t h o d Q u a n t u m j u m p m e t h o d 0 . 4 0 . 4 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 L a s e r d e t u n i n g ( g ) L a s e r d e t u n i n g ( g )

Figure 4.4: Comparison of the quantum regression theorem and quantum jump method. Simulated g(2)(0) obtained using the quantum regression theorem (solid lines) and quantum jump method (red triangles) for a QD–cavity detuning of (a) ∆ = 0 and (b) ∆ = 2.8g.

While using the quantum regression theorem results in a faster simulation of g(2)(0), the quantum jump method additionally provides the explicit probabilities for the transmission of n photons per pulse. To validate the performance of the quantum jump method for our case and to ensure that we have simulated a sufficiently large number of quantum trajectories, we present in figure 4.4 simulated values of g(2)(0) as a function of the laser detuning for two different QD–cavity detunings (c.f. figure CHAPTER 4. DETUNED PHOTON BLOCKADE 46

4.2), as obtained from the quantum regression theorem and quantum jump method (black solid lines and red triangles, respectively). Clearly, the two methods are in excellent agreement. Furthermore, because the correlation functions describing photon statistics are robust to loss and our detectors have finite detection probability paired with low photon-coincidence probability, the experimentally measured and normalized coincidences from the HBT setup should very closely match these simulated values. For the simulations presented throughout this chapter, we used the experimentally extracted cavity field decay rate κ/2π = 10.0 GHz and coupling strength g/2π = 10.9 GHz. The QD decay γ plays almost no role for the detunings investigated here as it is negligible due to a low intrinsic decay rate that is even further Purcell suppressed by the photonic band gap (as discussed below). Therefore, the only parameter that cannot be measured directly is the excitation power (coherent scattering of the laser at high powers prevents reference to a saturation value). Thus, the power has been manually fitted using the measured values of g(2)(0), g3(0), and the relative count rates for different excitation conditions.

4.5 Optimum detuning for eﬃcient generation of single photons

It is important to note that for detuned strongly coupled systems, the emission from the emitter-like polariton branch occurs almost exclusively through the cavity mode. Since the QD is embedded in a photonic band-gap material, its emission into free- space modes through non-cavity channels is strongly Purcell suppressed. For the photonic crystal geometries used in our work, measured values of this emission rate are suppressed to approximately one tenth of their bulk value [33, 34]. As a result of this extreme time-scale disparity, the emission of the QD through the detuned cavity mode obtained from equation 4.2 dominates over all other channels for detunings up to 60 g. This limit is much larger than the detunings of a few g that are investigated ∼ here. Hence, the emission occurs almost exclusively through the cavity. CHAPTER 4. DETUNED PHOTON BLOCKADE 47

( a ) 2 D e t u n i n g ∆: 0 g 3 g )

0 1 0 g (

g 1

- 2 0 2 4 6 8 1 0 1 2

( b ) 1 . 0 D e t u n i n g ∆: 0 g 3 g

1 1 0 g

P 0 . 5

0 . 0 - 2 0 2 4 6 8 1 0 1 2 D e t u n i n g ( g )

Figure 4.5: Eﬀect of detuning on single-photon purity and eﬃciency. Simu- (2) lated (a) g (0) and (b) P1 as functions of the laser detuning for QD–cavity detunings of 0g (black), 3g (red) and 10g (blue). The position of the minimum g(2)(0) in each case is indicated by the corresponding vertical line. CHAPTER 4. DETUNED PHOTON BLOCKADE 48

This means that with increasing detuning between the QD and cavity resonances, the oscillator strength of the QD-like polariton branch decreases. Therefore, when the detuning becomes too large, the efficiency decreases, resulting in a decreased likelihood of single-photon generation beyond detunings of a few g. In section 4.4 we compared the probability of finding n = 0 3 photons in a pulse transmitted through − the cavity. In particular, we found that the probability for obtaining a single photon increases when the detuning is increased from 0g to 3g. However, as the oscillator strength of the QD-like polariton branch decreases with increasing detuning, for too- large detunings the probability of finding a single photon in the output decreases. To (2) visualize this, we present in figure 4.5(a) g (0) and in figure 4.5(b) P1 as functions of the laser frequency for a QD–cavity detuning of 0g (black), 3g (red), and 10g (blue). (2) While for ∆ = 10g g (0) is close to the value obtained at ∆ = 3g, P1 (and thus the efficiency for single photon generation) decreases to one eighth of the value at ∆ = 3g. This clearly demonstrates the existence of an optimum detuning for efficient single photon generation.

4.6 Inﬂuence of the pulse length

Since the cavity lifetime is much shorter than the timing jitter of the single photon counters, measurements of g(2)(0) can only be performed with pulsed excitation. As discussed in the section 4.3, the choice of pulse length forces a compromise between frequency resolution (reducing the overlap of diﬀerent rungs) and re-excitation of the system. In other words, if the laser pulses are too long the system will be re-excited during the interaction with a single pulse, reducing the non-classical character of the transmitted light. On the other hand, pulses with a shorter duration are spectrally broader, resulting in a larger overlap with higher rungs. The dependence of the radiative polariton lifetimes on the QD–cavity detuning is calculated by using equation 4.2 with the measured values of κ and g [ﬁgure 4.6(a)]. With increasing detuning between the QD and the cavity, the lifetime of the QD-like (cavity-like) polariton branch increases (decreases). Therefore, in order to obtain the strongest photon CHAPTER 4. DETUNED PHOTON BLOCKADE 49

blockade, the pulse length has to be chosen according to the detuning. To demonstrate the existence of an optimum pulse length for given QD and cavity frequencies, we present in figure 4.6(b) simulations of g(2)(0) as a function of the pulse length and laser detuning for a QD–cavity detuning of ∆ = 3g. With longer excitation pulses the photon-blockade dip narrows and g(2)(0) decreases, reaching its lowest values for pulse lengths of 70 110 ps. As shown in figures 4.6(c) and 4.6(d), for detunings − of ∆ = 3g 5g we were able to obtain a significantly improved photon blockade of − g(2)(0) 0.3 using a pulse duration of 70 ps instead of 30 ps. ≈ ( a ) ( b ) 3 . 9 ( c ) 7 ( d ) 1 . 0 0 0 ∆ L P 1 ) = 3 g ∆= 5 g

3 . 8 s 6 t 2 0 0 ) ( 2 ) ( 2 ) g ( 0 ) = 0 . 3 2 ± 0 . 0 2 2 g ( 0 ) = 0 . 2 9 ± 0 . 0 4 n U P 1 g ) ( u 3 . 7 s 5 o g p c ( n

3 . 6 3 4 n e 0

1 0 0 u 1 t

m 0 . 6 5 0 0 i x e t 3 . 5 3 ( 1 d e

f i y r t i L 3 . 4 e 2 s s n 0 a 3 . 3 e L 1 t n 3 . 2 0 . 3 0 0 0 I 0 0 - 4 - 2 0 2 4 3 0 6 0 9 0 1 2 0 1 5 0 - 2 5 0 2 5 - 2 5 0 2 5 D e t u n i n g ( g ) P u l s e l e n g t h ( p s ) T i m e d e l a y ( n s ) T i m e d e l a y ( n s )

Figure 4.6: Choosing a detuning-dependent pulse length. (a) Radiative polariton lifetime as a function of the QD–cavity detuning, as calculated using the measured values of κ and g. (b) Simulation of g(2)(0) as a function of the pulse length and laser detuning for a QD–cavity detuning of ∆ = 3g. (c–d) HBT measurements of g(2)(0) using 70 ps long pulses and a detuning of (c) ∆ = 3g and (d) ∆ = 5g.

4.7 Outlook

In conclusion, QD–cavity detuning is a key ingredient for high-purity generation of non-classical light from strongly coupled systems. We have shown that detuning strongly reduces the spectral overlap with higher rungs of the JC ladder and hence greatly improves the generation of single photons by photon blockade. This approach to photon blockade has strong potential for single-photon generation under currently achievable system parameters. Improvements in the spatial alignment of the QD and cavity ﬁeld have enabled the coupling strength to reach values up to g/2π = 40 CHAPTER 4. DETUNED PHOTON BLOCKADE 50

GHz [75]. Recent nanofabrication improvements have allowed for experimental GaAs photonic crystal cavity loss rates as low as κ/2π = 4.0 GHz [76]. When using these parameters in our simulations we obtain g(2)(0) = 0.1 in the photon-blockade regime, and an absolute probability of over 90% for single-photon emission, demonstrating that high-quality single-photon streams generated by photon blockade are within reach. The generation of single photons by photon blockade might have additional advantages over other techniques. First, the use of high quality photonic crystal cavities promises a method of on-chip routing of the photons by coupling them to photonic crystal waveguides (with high efficiency) [16]. Second, the cavity emission rate is at least one order of magnitude faster than the bare QD emission rate, resulting in a comparable increase in the maximum single-photon generation rate while maintaining potential advantages from resonant excitation. Furthermore, the successful experimental demonstration of photon blockade in the detuned light–matter configuration demonstrates the feasibility of operating cavity QED in such an extreme regime and paves the way for a wealth of other quantum light sources, including those generating n-photon states [19]. A source of such higher-order photon states could potentially be used for efficient generation of the highly entangled N00N states, which are particularly interesting for quantum metrology and high-resolution quantum lithography and sensing [77]. Chapter 5

A quantum dot coupled to a photonic molecule

In this chapter, I present a demonstration of the eﬀects of cavity quantum electrodynamics for a quantum dot coupled to a photonic molecule consisting of a pair of coupled photonic crystal cavities [23]. We show an anticrossing between the QD and the two super-modes of the photonic molecule, signifying achievement of the strong coupling regime. From the anticrossing data, we estimate the contributions of both mode-coupling and intrinsic fabrication-dependent detuning to the total separation between the super-modes. We also show signatures of oﬀ-resonant phonon-mediated interaction between the two cavity modes, a phenomenon recently explored in several solid-state cavity systems [64].

5.1 Background

As discussed in chapter 2, a single quantum emitter coupled to a resonator constitutes a cavity-QED system for studying strong light–matter interaction. With an eye toward achieving a scalable quantum computer [78] and more recently a photonic quantum simulator [79, 80, 81], signiﬁcant progress has been achieved in optical cavity QED as well as its microwave counterpart circuit QED. The cavity-QED system consisting of a single QD coupled to a photonic crystal cavity has recently emerged

51 CHAPTER 5. A QD COUPLED TO A PHOTONIC MOLECULE 52

as an attractive candidate for integrated nanophotonic quantum information processing devices [20]. This solid-state cavity-QED system is of considerable interest to the quantum optics community for the generation of non-classical states of light [16, 18, 21], for its application to all-optical [5, 82] and electro-optical switching [83], and due to unusual effects like the off-resonant dot-cavity interaction due to electron- phonon coupling [84]. However, all of the cavity-QED effects demonstrated so far in this system involve a single cavity. Although numerous theoretical proposals employ- ing multiple cavities coupled to single quantum emitters exist in the cavity-QED and circuit-QED literature [79, 80, 81, 85], experimental development in this direction is rather limited. Most of these proposals—for example, observing the quantum phase transition of light—require a nonlinearity in each cavity, which is a formidable task with current technology. However, several proposals involving a single quantum emitter coupled to multiple cavities also predict novel quantum phenomena—for example, generation of bound photon-atom states [86] or sub-Poissonian light generation in a pair of coupled cavities (a “photonic molecule”) containing a single QD [87, 88]. This photonic molecule, coupled to a single QD, forms the first step toward building an integrated cavity network with coupled QDs. Photonic molecules made of photonic crystal cavities have been studied previously [89, 90] to observe mode-splitting due to coupling between the cavities. In those studies, a high density of QDs was used merely as an internal light source to generate photoluminescence (PL) under above-band excitation; no quantum properties of the system were studied. In another experiment, a photonic molecule consisting of two micropost cavities was used along with a single QD to generate entangled photons via exciton-biexciton decay, but the QD–cavity system was in the weak coupling regime and the only cavity-QED effect observed was Purcell enhancement [91]. We have achieved strong coupling of a photonic molecule with a single QD. We note that even though the photonic molecule described here is based on photonic crystal optical nanocavities, one could also employ Fabry–Pérotcavities, distributed- Bragg-reflector microcavities, microposts, or microwave cavities [92, 93] for this task. However, prior to this work, coupling multiple cavities to single quantum emitters had not been shown in any cavity-QED system. We show a clear anticrossing between CHAPTER 5. A QD COUPLED TO A PHOTONIC MOLECULE 53

the QD and the two super-modes formed in the photonic molecule. In general, the exact coupling strength between two cavities in a photonic molecule is diﬃcult to calculate, as the observed separation between the two modes has contributions both from the cavity coupling strength as well as from the mismatch between the two cavities due to fabrication imperfections. However, by monitoring the interaction between a single QD and the photonic molecule we can exactly calculate the coupling strength between the cavities and separate the contribution of the bare detuning due to cavity mismatch. In addition to the photonic molecule, the fabrication of large-scale arrays of coupled cavities (without any quantum emitters) has been reported both in a photonic crystal platform [94] and in circuit QED [92], revealing the widespread interest in exploring the scalability of these systems. We will turn our attention to such coupled cavity arrays in chapter 6.

5.2 Theoretical analysis of a photonic molecule

Let us consider a photonic molecule consisting of two cavities with annihilation operators for their bare (uncoupled) modes denoted by a and b, respectively. We assume that a QD is placed in and resonantly coupled to the cavity described by operator a. The Hamiltonian describing such a system is given by

† † † † † H = ∆0b b + J(a b + ab ) + g(a σ + aσ ), (5.1)

where ∆0 is the detuning between the two bare cavity modes; J and g are, respectively, the intercavity and dot-cavity coupling strength; σ is the QD lowering operator; and the resonance frequency ω0 of the cavity with annihilation operator a is assumed to be zero. We now transform this Hamiltonian by mapping the cavity modes a and b to the bosonic modes α and β introduced as a = cos(θ)α+sin(θ)β and b = sin(θ)α cos(θ)β. − This mapping maintains the appropriate commutation relations between operators a and b. Under these transformations we can decouple the two cavity modes (α and β) CHAPTER 5. A QD COUPLED TO A PHOTONIC MOLECULE 54

for the following choice of θ: 2J tan(2θ) = . (5.2) −∆0 Under this condition the transformed Hamiltonian becomes:

† 2 † † H = α α(∆0 sin (θ) + J sin(2θ)) + g cos(θ)(α σ + ασ ) + β†β(∆ cos2(θ) J sin(2θ)) + g sin(θ)(β†σ + βσ†). (5.3) 0 −

Therefore, a QD coupled to a photonic molecule has exactly the same eigenstructure as two detuned cavities with the QD coupled to both of them (from the equivalence of the two expressions above for the Hamiltonian H). The super-modes of the transformed 2 2 Hamiltonian α and β will be separated by ∆ = ∆0 + 4J (obtained by subtracting † † the terms multiplying α α and β β in equationp 5.3, under the conditions of equation 5.2), and the interaction strength between the QD and each super-mode will be 2 2 g1 = g cos(θ) and g2 = g sin(θ). Hence, g = g1 + g2, θ = arctan(g2/g1), and so tan(2θ) = 2g g /(g2 g2) = 2J/∆ . 1 2 1 − 2 − 0 p If the two cavities are not coupled (J = 0 and so θ = 0), we can still observe two different cavity modes in the experiment due to ∆0, the intrinsic detuning between the two bare cavities. However, if we tune the QD across the two cavities in this case, we will observe QD–cavity interaction only with one cavity mode (in this case α, as the term coupling β to the QD in the transformed Hamiltonian will vanish as a result of sin(θ) = 0). In other words, in this case the QD is spatially located in only one cavity and cannot interact with the other, spatially distant cavity. Figure 5.1 shows the numerically calculated cavity transmission spectra (proportional to a†a + b†b ) h i h i when the QD is tuned across the two cavity resonances. When the two cavities are coupled (J = 0), we observe anticrossings between each cavity mode and the QD 6 [figure 5.1(a)]. However, only one anticrossing is observed when the cavities are not coupled [figure 5.1(b)]. CHAPTER 5. A QD COUPLED TO A PHOTONIC MOLECULE 55

(a) 7 (b) 7 6 6

5 5

4 4

3 3

2 2

Normalized Cavity Output Cavity Normalized 1 1

0 0 -5 0 5 10 -5 0 5 10 (ωp - ω0 ) / κ (ωp - ω0 ) / κ

Figure 5.1: Numerically calculated cavity transmission spectra when the QD resonance is tuned across the two cavity resonances. (a) An anticrossing is observed between the QD and both cavity modes when the two cavities are coupled (the coupling rate between the two cavities is taken to be J/2π = 80 GHz). (b) When the two cavities are not coupled (J = 0), we observe an anticrossing in only one cavity. Parameters used for the simulation: cavity decay rate κ/2π = 20 GHz (for both cavities); QD dipole decay rate γ/2π = 1 GHz; QD–cavity coupling strength g/2π = 10 GHz; intrinsic detuning between the bare cavity modes ∆0/2π = 40 GHz for (a) and ∆0/2π = 120 GHz for (b). The plots are vertically oﬀset for clarity. The horizontal axis corresponds to the detuning of the probe laser frequency ωp from the cavity a resonance (ω0) in units of cavity ﬁeld decay rate. Note that here the QD is taken to be coupled to the higher-energy (cavity b) resonance. CHAPTER 5. A QD COUPLED TO A PHOTONIC MOLECULE 56

5.3 Photonic molecule characterization

The actual experiments are performed with self-assembled InAs QDs embedded in GaAs, and the whole system is kept at cryogenic temperatures ( 10 25 K) in a ∼ − helium-flow cryostat. The cavities used are L3-defect GaAs photonic crystal cavities coupled via spatial proximity. The photonic crystal is fabricated from a 160 nm thick GaAs membrane, grown by molecular beam epitaxy on top of a GaAs (100) wafer. A low-density layer of InAs QDs is grown in the center of the membrane (80 nm beneath the surface). The GaAs membrane sits on a 918 nm sacrificial layer of Al0.8Ga0.2As. Under the sacrificial layer, a 10-period distributed Bragg reflector, consisting of a quarter-wave AlAs/GaAs stack, is used to increase collection into the objective lens. The photonic crystal was fabricated using electron beam lithography, dry plasma etching, and wet etching of the sacrificial layer in diluted hydrofluoric acid [4], as described in section 2.2. We fabricated two different types of coupled cavities: in one case, the two cavities are offset at a 60◦ angle [inset of figure 5.2(a)] and in the other the two cavities are laterally coupled [inset of figure 5.2(b)]. In the first case the coupling between the cavities is stronger, since the overlap between the cavity electromagnetic field profiles is larger along the 60◦ angle. Figures 5.2(a) and 5.2(b) show the typical PL spectra of these two different types of coupled cavities for different spacings between the cavities. A clear decrease in the frequency separation between the cavities is observed with increasing spatial separation. Note that the consistency of this trend between different fabrication runs already indicates that this frequency separation cannot be purely due to the fabrication-related intrinsic detuning between the two cavities. Nevertheless, it is very difficult to quantify how much of the separation is due to coupling (J) and how much is due to intrinsic detuning (∆0) of the cavity resonances. However, we will show that by observing the anticrossing between the

QD and the two modes we can conclusively determine both J and ∆0. CHAPTER 5. A QD COUPLED TO A PHOTONIC MOLECULE 57

Figure 5.2: Photoluminescence spectra of the coupled cavities for diﬀerent hole spacings between two cavities. (a) The cavities are separated at an angle of 60◦ (see the inset for an SEM image). (b) The cavities are laterally separated (see the inset for SEM). A decrease in the wavelength separation between the two cavity modes is observed with increasing spatial separation between the cavities (i.e., with increasing number of holes inserted between the two cavities). A much larger separation is observed in (a) when the cavities are coupled at an angle compared to the lateral coupling in (b). CHAPTER 5. A QD COUPLED TO A PHOTONIC MOLECULE 58

5.4 Strong coupling between a photonic molecule and a single QD

First, we investigate the strong coupling between a single QD and the photonic molecule in PL. For this particular experiment, we used a photonic molecule consisting of cavities separated by four holes along the 60◦ angle. In practice it is not trivial to tune the QD over such a long wavelength range as required by the observed separation of the two cavity peaks. Hence, we use two different tuning techniques: we tune the cavity modes by depositing nitrogen on the cavity [95], and then we tune the QD resonance across the cavity modes by changing the temperature of the system. For the spectra shown in figure 5.3(a), there is no nitrogen deposition and the temperature is changed from 20 K to 30 K to tune the QD across the longer-wavelength cavity mode. To obtain the spectra in figure 5.3(b), we deposited nitrogen (to red- shift the cavity modes) and changed the temperature from 20 K to 40 K, showing the anticrossing between the QD and the shorter-wavelength cavity mode (all these temperatures are measured at the cold-finger of the cryostat). We observe less shift of the cavity resonance with temperature after nitrogen deposition, most likely because the nitrogen starts evaporating with increasing temperature, causing the cavity to blue-shift and thus compensating for the red-shift caused by the increased temperature. We fit the PL spectra (as described in section 2.4.2) to find the resonances of the coupled QD–cavity system, and plot the peak locations in the top panels of figures 5.3(a) and 5.3(b). We observe clear anticrossings for both coupled modes. The nitrogen and the temperature tuning do not cause a significant change in the coupling and the detuning between the cavities, as confirmed by the measurements described below. We note that there is another QD [shown by the red arrow in figure 5.3(a)] in the vicinity of the cavity resonances; however, it is only weakly coupled to the cavity as no anticrossing between that QD and the cavity mode is observed. Within the cavity defect region, we typically have about four QDs that are spectrally distributed over a wide range. Spatially, they are also randomly distributed within the cavity and, thus, have different coupling strengths with the cavity mode. Tuning the system over a large range of wavelengths (as we do in this experiment, with temperature and CHAPTER 5. A QD COUPLED TO A PHOTONIC MOLECULE 59

923.9 923.8 Temperature Temperature (nm) λ 923.6 923.7 (a) (b) Normalized PL IntensityNormalized

923 924 923 924 925 λ (nm) λ (nm)

Figure 5.3: Normalized PL intensity plotted for temperature tuning the QD across the photonic molecule resonances. (a) Spectra before nitrogen deposition (i.e., the QD is temperature tuned across the longer wavelength resonance). (b) Spectra after nitrogen deposition (which red-shifts the cavity resonances and allows us to temperature tune the QD across the shorter wavelength resonance). In both cases, the temperature is increased from bottom to top (the plots are vertically oﬀset for clarity). Clear anticrossings are observed between the QD and both cavity super- modes; the peaks of the two anticrossing resonances (as extracted from curve ﬁtting) are plotted in the top panels. The red arrow in (a) indicates the presence of an additional, weakly coupled QD, which does not contribute to the anticrossing. CHAPTER 5. A QD COUPLED TO A PHOTONIC MOLECULE 60

nitrogen tuning) can bring into resonance some of these other QDs, as shown by this additional peak. However, no signature of coupling between this QD and the QD under study for anticrossing is observed, and the presence of this additional weakly coupled dot does not change the conclusions of the experiment. Furthermore, the additional QD is not visible in figure 5.3(b), as it is far-detuned from both cavity modes. We perform curve fitting for the PL spectra when the QD is resonant with the cavity super-modes and estimate the system parameters [figures 5.4(a) and 5.4(b)]. The shorter-wavelength super-mode is denoted as sm1, while the longer-wavelength super-mode is denoted sm2. As the detuning between the super-modes is much larger than the vacuum Rabi splitting caused by the QD, we can assume that when the QD is resonant to sm1, its interaction with sm2 is negligible, and vice versa. Therefore, we can fit the PL spectra of each super-mode exhibiting vacuum Rabi splitting individually. For sm1, we extract from the fit the field decay rate κ1/2π = 22.4 GHz and the

QD–ﬁeld interaction strength g1/2π = 14.2 GHz [ﬁgure 5.4(b)]; for sm2, κ2/2π = 16.7

GHz and g2/2π = 23.7 GHz [figure 5.4a]. We observe from the extracted κ values that the coupled cavity modes have very high quality factors (Q 7, 000 10, 000). We ∼ − also estimate the total separation between the two observed modes as ∆/2π 246.5 ≈ GHz and ∆/2π 253.5 GHz before and after nitrogen tuning, respectively. This ≈ minimal difference in ∆ resulting from the nitrogen tuning does not impact our further analysis, and we take ∆ to be the average of these two values. The change in the cavity field decay rates arising from the nitrogen deposition is also minimal. From these data, we use the relations θ = arctan(g /g ), tan(2θ) = 2J/∆ , and 2 1 − 0 ∆ = ∆2 + 4J 2 (derived in section 5.2 above) to obtain J/2π 110 GHz and 0 ≈ ∆ /2π 118 GHz. The intercavity coupling strength J could be increased further 0 p≈ by placing the cavities closer together (though this would also make observation of both anticrossings more difficult). CHAPTER 5. A QD COUPLED TO A PHOTONIC MOLECULE 61

1200 3 (0)

1000 2 g 1 800 -5 0 5 (ωp - ω0 ) / κ 600 1.5 (c)

400 1

0.5 200

Cavity Transmission Cavity 0 923 924 923 924 925 -10 0 10 20 λ (nm) λ (nm) (ωp- ω0 ) / κ (a) (b) (d)

Figure 5.4: QD–photonic molecule spectra. (a) PL intensity when the QD is resonant with super-mode sm2. (b) PL intensity when the QD is resonant with super- mode sm1 (after nitrogen deposition). From the ﬁts (shown in red) we extract the system parameters. Numerically simulated (c) second order autocorrelation g(2)(0) and (d) transmission through cavity b, as a function of probe laser frequency ωp, with the experimental system parameters that were extracted from the ﬁts.

5.4.1 Feasibility of non-classical light generation

We now numerically simulate the performance of such a QD–photonic molecule system for the generation of sub-Poissonian light using the quantum optical master equation approach [96]. Two bare cavity modes are separated by ∆0/2π = 118 GHz; a QD is resonant and strongly coupled to one of the modes (cavity a) with interaction strength 2 2 g/2π = 27.6 GHz (as discussed earlier, g = g1 + g2, where g1 and g2 are the two values of QD–cavity interaction strengths obtainedp by ﬁtting the PL spectra for each super-mode). We drive mode b (the empty cavity) with a probe laser of frequency

(2) hb†b†bbi ωp, and calculate the second order autocorrelation g (0) = 2 of the transmitted hb†bi light [88]. We also assume the two cavities to have the same cavity decay rate, which is an average of the cavity decay rates measured from the two super-modes (as the decay rates of the bare cavity modes a and b are experimentally inaccessible). Note, however, that having slightly different decay rates does not significantly affect the CHAPTER 5. A QD COUPLED TO A PHOTONIC MOLECULE 62

performance of the system. The numerically simulated cavity b transmission and g(2)(0) of the transmitted light are shown in figure 5.4(c–d). From these simulations, we observe that with our system parameters we should be able to achieve strongly sub-Poissonian light with g(2)(0) 0.03. Unfortunately, in practice it is very difficult ≈ to drive only one cavity mode without affecting the other mode due to the spatial proximity of the two cavities. This individual addressability is critical for satisfactory performance of the system [88], and to retain such a capability in a photonic molecule the cavities should be coupled via a waveguide [97].

5.5 Oﬀ-resonant coupling in a photonic molecule

As a further demonstration of cavity-QED effects in the photonic molecule, we investigated the off-resonant interaction between the coupled cavities and the QD, similar to the observations of this phenomenon in a single L3-defect cavity [64] and a nanobeam photonic crystal cavity [98]. This experiment was performed on a different QD–photonic molecule system than the one in which we observed strong coupling. Figure 5.5 shows the spectra indicating off-resonant coupling between the photonic molecule and the QD. Under resonant excitation of the longer-wavelength super- mode (sm2, top panel), we see pronounced emission from both sm1 and a nearby QD. Similarly, under resonant excitation of sm1 (bottom panel), we see emission from sm2, although the emission is much weaker. We exclude the presence of any nonlinear optical processes by performing a power-dependent study of the cavity emission, which shows a linear dependence of the cavity emission on the laser power. Hence, we can conclude that this off-resonant emission is a result of a phonon-assisted transfer of population between the two super-modes of the photonic molecule [96]. CHAPTER 5. A QD COUPLED TO A PHOTONIC MOLECULE 63

sm1 Laser Tuning

sm2

928.6 929.8 928.6 929 929.4 929.8 λ (nm) λ (nm)

Figure 5.5: Oﬀ-resonant interaction between a photonic molecule and a QD. We scan the laser across both coupled modes; when one super-mode is excited, we observe emission from the oﬀ-resonant super-mode. A spectral slice for each resonance shows the relative position of the laser and the cavity modes.

5.6 Outlook

In summary, we demonstrated strong coupling of a single QD to a photonic molecule in a photonic crystal platform. Clear anticrossings between the QD and both super- modes of the photonic molecule were observed, showing conclusive evidence of intercavity coupling. From the anticrossing data we were able to separate the contributions of the intercavity coupling and the fabrication-caused intrinsic detuning to the cavity mode splitting, and discovered that the two eﬀects were of comparable magnitude for reasonably-separated cavities. We have also observed oﬀ-resonant cavity–cavity and QD–cavity interaction in this type of system. These photonic molecules could be employed for non-classical light generation (as theoretically studied in section 5.4.1), and also represent a building block for an integrated nanophotonic network in a solid- state cavity-QED platform. Such a network could be implemented by scaling up the number of coupled cavities in our system, a possibility which I explore further in chapter 6. Chapter 6

Beyond isolated systems: coupled cavity arrays

In this chapter, I will explore the effects of coupling multiple photonic crystal cavities together. Specifically, I present an experimental study of coupled optical cavity arrays in a photonic crystal platform [24]. We find that the coupling between the cavities is significantly larger than the fabrication-induced disorder in the cavity frequencies. Satisfying this condition is necessary for using such cavity arrays to generate strongly correlated photons, which have potential application to the quantum simulation of many-body systems.

6.1 Background

Solving strongly correlated quantum many-body systems is a formidable task. One promising approach is to mimic such complicated systems using another simpler and more easily controllable quantum system, as envisioned by Feynman [99]. To that end, the first demonstration of quantum phase transitions with ultra-cold atoms in an optical lattice [100] has sparked a significant amount of research on quantum simulation with atomic systems [101]. Another very promising possibility is to use photons themselves as the interacting particles [68, 80, 79]. The central aim of this approach is to obtain a correlated “quantum fluid of light” [68] by building a coupled network

64 CHAPTER 6. COUPLED CAVITY ARRAYS 65

of nonlinear electromagnetic cavities. The photons can hop between cavities due to the electromagnetic coupling and can repel each other in the same cavity due to the intracavity nonlinearity. Such a coupled cavity network exhibits rich physics—such as topologically protected optical delays [102]—even without any nonlinearity, although having nonlinear cavities opens up many more avenues of research. Obviously, the optical nonlinearity required for signiﬁcant repulsion at low photon number is very high, and in current technology, only two-level systems (atoms, single quantum emitters such as QDs, or superconducting transmon qubits) strongly coupled to a cavity provide such strong nonlinearity (in the photon-blockade regime). Such a platform consisting of coupled nonlinear cavities is useful not only for quantum simulation but also for quantum error correction [103], as well as for classical optical signal processing [104]. Although plenty of theoretical proposals for simulating interesting physics in such a coupled cavity array (CCA) are present in the literature, the experimental progress in this direction is rather limited. As it is necessary to have many cavities for this proposal, a solid-state system is obviously an ideal choice. However, due to imperfect nanofabrication, solid-state cavities have inherent disorder, resulting in diﬀerent resonance frequencies than originally designed. Such disorder might limit the utility of CCAs for quantum simulation. However, Underwood et al. recently argued that as long as the coupling strengths are much larger than the disorder, the CCAs can be used for quantum simulation, and it was shown that microwave transmission line cavities for circuit QED satisfy this condition [92]. In this work, we demonstrate high-Q 2D CCAs based on photonic crystals fabricated in GaAs with embedded high-density self-assembled InAs QDs. Although the photonic molecule has been well studied [23, 90, 105, 106, 107, 108], relatively few experiments have been conducted with CCAs. A 2D CCA of photonic crystals in GaAs (with multiple quantum wells as the active material) has been studied previously for increasing the output intensity from nanolasers or slowing down light [109, 110, 111, 112], but the quality factors of the cavities were too low to identify individual cavity modes. A long chain of high-Q coupled cavities has been studied in silicon [94], but the physical phenomena observable in such a one-dimensional (1D) CHAPTER 6. COUPLED CAVITY ARRAYS 66

system are rather limited. While a 1D array [113] has been studied as a platform to simulate the physics of Bose glass [114] and Tonks-Girardeau gas [115], a 2D array is a more suitable candidate for simulating many other systems, including topologically nontrivial states such as the fractional quantum Hall state [116, 117, 118]. Although an extensive treatment of disorder in the context of circuit quantum electrodynamics has already been reported [92], our optical cavity-QED system is capable of achieving much larger coupling strengths ( THz) between the cavities. Our experimental ﬁnd- ∼ ings on the nature of coupling strengths and disorder in this system provide a more realistic picture for exploring the utility of optical CCAs for quantum simulation.

6.2 Spectra of coupled cavity arrays

We employ an array of L3-defect photonic crystal cavities, as typically studied in single QD–cavity QED experiments [4, 41]. The fundamental mode of such a cavity is (to some extent) linearly polarized in the direction orthogonal to the cavity axis; in our proposed CCA geometry, all the cavities have parallel axes, and their modes therefore have the same polarization. As in our single-cavity work [4], the photonic crystal CCA (with photonic crystal hole radius 60 nm and lattice periodicity 246 ∼ nm) is fabricated in a 164 nm thick GaAs membrane (with self-assembled InAs QDs embedded at a depth of 82 nm from the surface) using electron-beam lithography and reactive ion etching (see section 2.2). Three different CCAs are designed consisting of 4, 9, and 16 cavities. Scanning electron micrographs of the fabricated structures are shown in figure 6.1(a–c). We assume each cavity couples primarily to their closest neighbours in each direction. These cavities are coupled to each other by three different coupling strengths depending on their relative orientation and separation. When two cavities are ori- ented at an angle of 60◦ [figure 6.1(d–e)], the coupling strength t is strongest; for vertically stacked cavities [figure 6.1(h–i)], the coupling strength J1 is smaller than t; and for horizontally adjacent cavities [figure 6.1(f–g)], the coupling strength J2 is much smaller than t and J1 (the difference in coupling strengths is a result of the different radiation patterns of the cavity modes and their different overlaps in various CHAPTER 6. COUPLED CAVITY ARRAYS 67

directions). From finite-difference time-domain (FDTD) simulations, we can calculate the field profiles of the pairwise coupled cavities [figure 6.1(d–i)] and estimate the coupling strengths from the separation of the super-modes in the simulated spectra, assuming cavity operation in the range of QD emission (wavelengths of 900 930 ∼ − nm). For a hole radius r varying from 70 nm down to 50 nm (to account for fabrication variation), with photonic crystal lattice constant a = 246 nm, we find that t/2π 0.8 1.3 THz, J /2π 0.4 0.8 THz, and J t, J . ∼ − 1 ∼ − 2 1

(d)

1µm (e) 1 (a) 0 (f)

-1

1µm (g)

(b)

1µm

Figure 6.1: SEM images of CCAs with (a) 4 cavities, (b) 9 cavities and (c) 16 cavities. The simulated electric field profiles for each of the two super-modes of the different types of coupled cavities are shown: (d), (e) for 60◦ coupled cavities; (f), (g) for laterally coupled cavities; (h), (i) for vertically coupled cavities. The frequency splitting of these super-modes specifies the coupling strength in each case. CHAPTER 6. COUPLED CAVITY ARRAYS 68

We characterize the resonances of the coupled cavity array by photoluminescence (PL) studies, where the large density of embedded QDs ( 200/µm2 in this case) acts ∼ as an internal light source. Figures 6.2(a–c) show the PL spectra obtained from the CCAs at 30 K, under excitation of an 820 nm continuous-wave laser. The excitation power is 1 nW measured in front of the objective lens. Figure 6.2(d) shows the PL ∼ spectrum collected from the bulk QDs, which provide the background illumination for the cavity spectra. The quality factors of the observed modes are Q 1, 000 3, 000 ∼ − [figure 6.2(e)], and all the modes have a similar polarization axis. We identify the set of higher-Q resonances in figure 6.2 with the coupled fundamental modes of the L3 cavities, shown in figure 6.1. Note that these modes are not necessarily in the same wavelength range for different sizes of the arrays, as the structures were defined during the fabrication process with different exposure times in e-beam lithography, and thus different photonic crystal devices may have different parameters (in this case, slightly different hole radii). We also point out that the number of modes observed in PL should be the same as the number of cavities in the CCA, irrespective of whether the cavities are coupled or not (assuming any degeneracy is lifted due to fabrication imperfection, as discussed in chapter 5). Without coupling between the cavities, the observed modes would be randomly placed spectrally and no specific order between the modes should be observed. On the other hand, if the cavities are coupled, the cavity modes are expected to be spaced in frequency at a specific order determined by the coupling strengths. However, because of the disorder introduced during the nanofabrication process, the exact distribution of the cavity resonance frequencies will be perturbed. Hence, the ratio between the cavity coupling strengths and the disorder in the cavity resonances represents a measure of the relative magnitude of these effects, which can be estimated by examining the differences in the cavity resonance frequencies. One could instead estimate the disorder in the cavity resonances from the actual cavity frequencies rather than the differences. However, cavity arrays written on different parts of the chip are more susceptible to fabrication variation, and might suffer an overall frequency shift of their entire set of modes. Thus, the mode separations provide a better measure of the disorder present within each CCA while allowing us to CHAPTER 6. COUPLED CAVITY ARRAYS 69

1 (d) ∆1 (a) 3000 QD 0.5

Wetting 0 PL signal 1000 910 920 930 940 950 Layer 1 (b) 850 900 950 ∆2 ∆3 ∆ λ (nm) 0.5 4 (e) 16 cavities 9 cavities 0 4 cavities 920 930 940 950 960 1 (c) ∆5 0.5 ∆6 Photoluminescence Signal Photoluminescence 0 910 920 930 940 950 912 916 920 924 λ (nm) λ (nm)

Figure 6.2: The PL spectra of the CCA for (a) 4; (b) 9 and (c) 16 cavities. We can clearly identify all the cavity array modes. We focus on several specific separations between the CCA modes labeled ∆1 through ∆6 in the plots and perform statistical analysis. We also observed several low-Q modes at long wavelengths for several cavity arrays, as can be seen in part (a). These modes are not the actual coupled cavity modes under study, which is confirmed by monitoring the resonance frequencies of single (uncoupled) L3 cavities fabricated in the same chip. (d) PL spectra collected from the bulk QDs. (e) Lorentzian fits to the highest frequency modes [shaded in (a), (b), and (c)] to estimate the quality factors. The estimated quality factors for the highest frequency mode in the 4-, 9-, and 16-cavity CCAs are 1,800, 2,800, and 2,500, respectively. ∼ ∼ ∼ CHAPTER 6. COUPLED CAVITY ARRAYS 70

gather statistics from several CCAs for comparison. We also tried performing the disorder analysis using mode separations that have been normalized by the bare cavity frequency in order to exclude any contribution from the overall frequency shift. We found that normalized mode separations and unnormalized mode separations provide very similar results, so in the interests of clarity, we provide numerical estimates and statistical data for the raw mode separations only.

6.3 Numerical estimation of coupling and disorder

Using the coupling strengths derived from FDTD simulations (t/2π = 1.2 THz, J /2π = 0.8 THz, J 0), we calculate the eigenstates of the CCA by diagonal- 1 2 ≈ izing the Hamiltonian

† † † H = ∆iai ai + gij(ai aj + ajai), (6.1) i X Xhi,ji

th where ∆i is the bare resonance frequency of the i cavity due to fabrication imper- th th fection, and gi,j is the coupling strength between the i and j cavity. The cavity frequencies ∆i are randomly chosen from a Gaussian distribution with zero mean and standard deviation σf [92]:

∆2 − 0 1 2σ2 f P r(∆0) = e . (6.2) √2πσf

The zero for the eigenfrequencies of the coupled-cavity system is set at the frequency of an uncoupled cavity with no disorder. We note that for photonic crystal cavities, the disorder affects both the bare cavity frequencies and the coupling strengths; however, it is very difficult to separate the two effects. Depending on the spatial positioning of the disorder, the effect on the bare mode separation and the coupling strengths can vary. Figure 6.3 shows the finite-difference time domain simulation of the effect of disorder on the coupling between cavities in a photonic molecule. We consider two different CHAPTER 6. COUPLED CAVITY ARRAYS 71

Perturbed side hole (a) 2. 2 Mode (THz) Separation 1. 8 20 40 60 80 100 (b) Hole Radius (nm)

Figure 6.3: FDTD simulation of the eﬀect of disorder on the observed mode separations in a photonic molecule. (a) Perturbation where a side hole in the cavity is changed. (b) Perturbation where a hole in between two cavities is changed. (c) The observed mode separations as a function of the perturbed hole radius for the two cases shown in (a) and (b). The unperturbed hole radius is 60 nm. CHAPTER 6. COUPLED CAVITY ARRAYS 72

perturbations in the cavities: in one case, a side hole of one of the cavities is perturbed [figure 6.3(a)], and in the other case, a hole between two cavities is perturbed [figure 6.3(b)]. The unperturbed hole radius is 60 nm, and the perturbed hole radius changes from 30 to 90 nm. We find that the mode separations in the numerically simulated spectrum vary differently [figure 6.3(c)]: for the side hole perturbation, the mode separation changes by 500 GHz, whereas for the perturbed hole between ∼ two cavities, the mode separation changes by 800 GHz. In the latter case, the ∼ perturbation affects the coupling strength strongly, but the effect in intrinsic (bare) detuning is not significant (as both cavities sense the perturbation). However, for the side hole perturbation, the coupling strength is not affected much, but the bare detuning changes more. Hence, the experimentally measured variations have contributions from randomness both in the bare mode separations as well as the coupling strengths. Nevertheless, in order to simplify our analysis, we assume that the total variance originates from just the bare mode separations, and the coupling strengths are constant. The mean values of the eigenfrequencies (averaged over 10,000 instances) are ∼ plotted in figure 6.4(a–c) as a function of increasing σf . We observe that the relative separations between the modes follow a specific pattern when the disorder is small. However, with increasing disorder any specific distribution of the cavity modes disappears. This can be observed more clearly in figure 6.4(d–f), where the differences in the mode frequencies are plotted as a function of σf . At high disorder, the differences become similar, and increase linearly with σf . We note that the mean µ of the mode separations is a combination of the coupling strength and the disorder, whereas the standard deviation σ of the mode separations mostly depends on the disorder alone. To elaborate further, we can consider the simple example of a photonic molecule (two coupled cavities), where the observed separation ∆ between two modes is given 2 2 by ∆0 + 4J , where ∆0 is the random bare detuning between the cavities due to fabricationp imperfection and J is the the coupling strength (see section 5.2 above). Under the approximation of a Gaussian distribution for the bare cavity detunings (as in equation 6.2), we can calculate the mean µ and standard deviation σ of the mode separation ∆ (considered as an absolute value). If the coupling is weak compared CHAPTER 6. COUPLED CAVITY ARRAYS 73

8 3 ∆1 10 (a) ∆1 1 0 4 0 0.4 in THz)

/2π (d) -10 0 ∆ω in THz) 6 2 ∆2 ∆

/2π (b) ∆ 3 10 2 ∆ ω 1 4 ∆3 4 0 0 0.4 ∆4 2 -10 (e) 0 ∆5 (c) ∆ 1 ∆6 5 4 10 ∆6 Eigen Frequency ( Eigen Frequency 0 2 0 0.4 -10

Di erence in Eigen Frequency ( in Eigen Frequency Di erence (f) 0 0 5 10 0 5 10 σf /2π (in THz) σf /2π (in THz)

Figure 6.4: Numerically calculated eigenspectra of the coupled cavities. The eigenfrequencies are shown as a function of the disorder standard deviation σf for (a) 4, (b) 9, and (c) 16 cavities in the arrays. The spacings between the cavities are the same as the structures shown in the SEM images in ﬁgure 6.1(a–c). The diﬀerences in subsequent eigenvalues are shown as a function of σf for (d) 4, (e) 9, and (f) 16 cavities in the arrays. These mode separations increase linearly with increasing σf when σf is much greater than the coupling strengths, as found from the theory for a simple photonic molecule. Insets magnify the region of low disorder, and we identify the mode separations ∆ ∆ . 1 → 6 CHAPTER 6. COUPLED CAVITY ARRAYS 74

2 to the disorder (J = 0 or σf /J 1), we find that µ = σf and σ σf . If π ≈ instead the disorder is weak compared to the coupling (i.e.,qσf /J 1), we have µ = 2J + σ2/4J + (σ4) and σ (σ2/J). Similar analysis can be performed for f O f ∼ O f CCAs with more than two cavities, although the formulae for the mean and standard deviation become complicated, and a simple closed-form expression is difficult to obtain. Nevertheless, as seen for the photonic molecule, the ratio of the standard deviation to the mean gives us an estimate of the relative contribution of the disorder and the coupling to the total mode separations. Curiously, several modes are spaced very closely at weak disorder, indicating a lesser contribution of the coupling to such mode separations [this can be seen by the clustering of mode separations near zero in figure 6.4(d–f)]. On the other hand, several mode separations are large compared to the others (denoted by ∆ ∆ ), 1 → 6 signifying a large contribution from the coupling strengths to the mode separations. We observe that the relative positions of the cavity modes agree qualitatively with our experimental findings. We can identify the same specific separations ∆ ∆ 1 → 6 between the modes in the experimental results of figure 6.2 as in the theoretical calculations of figure 6.4. Clearly, the fabricated structures are in the regime where the coupling strengths are greater than the disorder. This regime is magnified in the inset of figure 6.4(d–f) and the mode separations ∆ ∆ are identified. 1 → 6

6.4 Statistical study of CCA mode separations

We find a consistent order between the experimentally observed modes of different CCAs (figure 6.2), indicating the cavities are coupled. Next we analyze all the separations between subsequent cavity modes. To do this, we fabricated 30 copies of ∼ each of the three types of cavity arrays, and we calculated the mean µ and standard deviation σ for all of the observed mode separations. In our fabricated CCAs, we find that the ratio σ/µ 1 for almost all the mode separations, indicating the presence of strong intercavity coupling; otherwise, for no coupling, the ratio would be equal to π/2 1 1. Table 6.1 shows the data − ∼ for specific separations (∆ ∆ ) between the modes in the cavity array spectra 1 → 6 p CHAPTER 6. COUPLED CAVITY ARRAYS 75

in detail. As seen from the numerical simulations presented above (figure 6.4), the separations are not all equally influenced by the coupling strengths, and the chosen separations (indicated in figures 6.2 and 6.4) are the ones that are most heavily dependent on the coupling strengths.

Table 6.1: The mean mode separations (µ) and standard deviation (σ) measured over 30 cavity arrays, with similar hole radii (see ﬁgure 6.2 for the deﬁnition of the separations).∼ ∆ µ (THz) σ (THz) σ/µ

∆1 2.33 0.25 0.1 ∆2 3.22 0.13 0.04 ∆3 2.35 0.14 0.06 ∆4 1.19 0.19 0.16 ∆5 1.94 0.2 0.1 ∆6 2.35 0.14 0.06

6.4.1 Dependence of mode separations on hole radius

As a further proof of the fact that the separations between the observed cavity array modes are mostly due to the coupling between the cavities, and not the disorder, we repeated the fabrication of sets of 30 cavities for different values of air-hole radius ∼ for all three types of CCAs. A decreasing trend in the separation is observed with increasing hole radius (figure 6.5). A similar trend is observed in simulation for a photonic molecule with diagonally placed cavities, as a function of the hole radius (inset of figure 6.5). The decrease in the mode separation with an increase in the hole radius can be explained by the accompanying increase in the size of the photonic band gap (and thus the larger reflectivity of the mirror layers separating cavities, which reduces the cavity couplings). Consequently, such a trend indicates that the separations are mostly due to the coupling between cavities, as a detuning due solely to disorder would have a much weaker dependence on the photonic crystal hole size. CHAPTER 6. COUPLED CAVITY ARRAYS 76

∆1 ∆ 3.5 4 1.2 ∆2 ∆5 1 ∆3 ∆6 3 in THz) 0.8

Coupling Strength Coupling (THz) 50 60 70 /2π Hole Radius (nm)

∆ω 2.5

1.5

Di erence in Eigen Frequency ( in Eigen Frequency Di erence 0.5 55 60 65 70 75 80 Hole Radius (nm)

Figure 6.5: The observed mode separations ∆1 ∆6 as functions of the photonic crystal hole radius. A decreasing trend→ in the separation is observed with increasing hole radius (the photonic crystal lattice periodicity a is 246 nm). For comparison, the inset shows the numerically simulated (FDTD) coupling strength between two cavities placed diagonally as a function of the hole radius. CHAPTER 6. COUPLED CAVITY ARRAYS 77

6.5 Outlook

We have shown the signature of large coupling strengths between photonic crystal cavities in a coupled cavity array. The coupling strengths are signiﬁcantly larger than the disorder introduced during nanofabrication, which is a necessary condition for em- ploying such cavity arrays in quantum simulation with correlated photons. However, the challenge of achieving a signiﬁcant nonlinearity in each cavity still remains. Chapter 7

Conclusions

The quantum dot strongly coupled to a photonic crystal nanocavity provides a rich and rewarding platform for studying cavity-QED eﬀects in the solid state. In this dissertation, I have described several contributions that explore the generation of non-classical light and the ways in which this system can be scaled up for potential use in integrated quantum optical networks.

7.1 Summary of contributions

First, we demonstrated the measurement of higher-order autocorrelation functions [g(2)(0), g(3)(0, 0), and g(4)(0, 0, 0)] and showed how such measurements can be used to characterize the non-classical light emitted from a strongly coupled QD–cavity system. While this system exhibits both photon blockade and photon-induced tunneling, the quality of the single-photon stream produced by photon blockade is severely limited (g(2)(0) 0.9) when the QD is on resonance with the cavity. To address this issue, we ≈ proposed detuning the QD away from the cavity resonance and optimizing the pulse length to compromise between spectral overlap with unwanted higher-rung transitions and re-excitation of the cavity. With these improvements, we were able to produce single photons both with a much higher degree of purity (g(2)(0) 0.3) and at ≈ a much greater eﬃciency (roughly doubling the absolute probability of the single- photon component).

78 CHAPTER 7. CONCLUSIONS 79

Of course, one of the advantages of working in the solid state is that we are not limited to single cavities; we can explore the effects of integrating multiple systems together on one chip. With this in mind, we demonstrated the first cavity-QED effects with a QD coupled to a photonic molecule, measuring anticrossings between the QD resonance and both photonic molecule super-modes. These measurements allowed us to extract the degree to which the two cavities in the photonic molecule were actually coupled together as well as the influence of disorder on their frequencies. Following this line of thinking to its natural conclusion, we fabricated larger arrays of four, nine, and sixteen coupled photonic crystal cavities and investigated their coupling strengths and disorder. Here too, we found that the coupling between the cavities is large enough to overcome their frequency variation due to fabrication-induced disorder, suggesting the possibility of using these arrays to generate strongly-correlated photons for quantum simulation.

7.2 Future work

While we have achieved signiﬁcant improvements in using the QD–photonic crystal cavity platform as a non-classical light source, a great deal of research remains to be done in order to make this system a practical resource for various quantum optical applications. In particular, moving beyond single photons to on-demand pulses of n photons would lay the groundwork for novel concepts in quantum information processing and metrology.

7.2.1 Accessing higher rungs of the Jaynes–Cummings ladder

Coupling a QD to a high quality-factor cavity (such as our photonic crystal nanocavity) produces a rich energy structure that can be used as an adjustable photon-number ﬁlter [18]. However, in order to generate a speciﬁc state with n photons using this method, we must be able to precisely select the corresponding higher-order manifold. Unfortunately, as discussed in section 3.7, the increase in linewidths makes CHAPTER 7. CONCLUSIONS 80

the higher-order states that produce multi-photon transitions unresolvable with our current experimental parameters. There are several possibilities for addressing these challenges. First, we can improve the quality factor of our photonic crystal cavity in order to shrink the linewidths of all of the dressed states, and hence reduce their overlap with each other. In addition to the ongoing process of seeking improvements in fabrication, there are some opportunities for improving the cavity design as well, by using optimized hole shifts derived from genetic algorithms [32]. While the improved cavity design may enable the generation of few-photon states with comparable efficiency to the current state-of-the-art with single photons, it is unlikely to solve the problem of generating n-photon states completely. In particular, even though in this configuration an excitation of the nth rung of the JC ladder is possible, the dynamics of emission from subsequent transitions down the ladder and their coherence properties are rather complicated, making it likely that the final state will be composed of many different photon-number states of various probabilities rather than being dominated by a single n-photon component. Another approach is suggested by the recent proposal of generating n-photon “bundles” in a strongly-driven detuned QD–cavity system via “super-Rabi oscillations” [19]. In this scheme, instead of probing the energy levels that result from the strong coupling between the QD and the cavity with a weak laser beam, a strong laser beam interacting with the QD dresses its states and the cavity is used to probe the system. While this process produces pulses which are not exactly n-photon Fock states, their photon number is still well-defined within a given time window, and the composite photons exhibit a high degree of correlation. Hence, the bundle may still be useful for applications in quantum sensing and lithography. From a practical point of view, the generation of these n-photon bundles also requires the development of slightly higher quality-factor cavities, and so this technique must be complemented by the design improvements discussed above. Fortunately, the requirements on the cavity quality factor are somewhat less strict in this proposal, and simulations of the process have shown that close to 100% of two-photon emission is within reach of current state-of-the-art samples. CHAPTER 7. CONCLUSIONS 81

7.2.2 Phonon-assisted transitions

In contrast to atomic cavity-QED systems, our QDs are embedded in a crystalline host matrix, resulting in strong interactions with phonons. While in many circumstances such interactions are undesirable (resulting in dephasing of the QD), it is also possible to carefully design experiments which benefit from the presence of phonons [119, 120, 121]. Solid-state systems are becoming increasingly relevant for their potential in building quantum optical networks, and so it is essential to develop a thorough understanding of the influence that phonons have on the system dynamics. As discussed in section 5.5, the interaction with phonons can enable off-resonant transfer of population between QDs and cavity modes. Recently, we have begun to explore how this effect can be exploited to improve photon blockade. In particular, by pumping one polariton branch and collecting the emission of the other polariton in a detuned strongly coupled QD–cavity system, we observe a phonon-assisted transfer between the two states [122]. Hence, by varying the power of the driving laser, we can achieve direct coherent control of the population in each state. This means that at positive detunings (where the QD-like polariton is at a higher energy than the cavity-like polariton), we can excite the system through the upper polariton, and wait for the phonon-assisted decay into the lower polariton state. Then by collecting only the emission from the lower polariton, we can dramatically improve the quality of single-photon generation (since this emission occurs at a different frequency, it is easy to filter away the background from our driving laser). This process has allowed us to achieve a photon blockade as low as g(2)(0) = 0.03 0.01, rivaling the best ± values obtained from bulk QDs. Unfortunately, since the phonon-assisted transfer is an incoherent process, the resulting photons are not completely indistinguishable, which removes one of the key motivations for coupling a QD to an optical cavity in the first place. However, this method still retains the other advantages provided by a cavity: the very short lifetime that enables a high photon generation rate and the potential for integration into on-chip optical networks. Thus, for applications that do not require perfectly indistinguishable photons but simply depend on the generation rate of high-purity CHAPTER 7. CONCLUSIONS 82

single photons (such as the BB84 quantum key distribution protocol [123]), phonon- assisted photon blockade in a detuned QD–cavity system is a promising technology.

7.2.3 Four-level emitter coupled to a cavity

There may also be benefits from replacing our QD (considered here as a two-level system) with a four-level emitter. From a theoretical study of a four-level emitter coupled to a cavity with realistic parameters [66], we discovered that this system can achieve a significantly improved photon blockade even outside of the strong coupling regime. Essentially, in this system the frequency of maximum transmission for the first photon coincides with a local minimum in the transmission of the second photon. This arrangement stands in contrast with the more overlapping probabilities of multi- photon transmission that result from the two-level emitter (observed, for example, in figures 3.5 and 4.3). Of course, higher cavity quality factors would still be beneficial in either case to reduce the dressed-state linewidths and suppress off-resonant coupling of additional photons, but even for current experimental parameters the four-level emitter results in significantly lower values of g(2)(0). Such a four-level emitter could be realized in the photonic crystal platform by using a charged QD in a strong magnetic field, which we have also investigated for use as a spin-photon interface [124]. However, this approach requires strong magnetic fields, and also results in an energy splitting between the pairs of transitions which is not desirable for non-classical light generation. A more promising potential candidate for the solid-state emitter in this scheme would be a negatively charged QD molecule [125]. Such a molecule exhibits two energetically close-lying levels in which a single electron resides in one (the “upper”) or the other (“lower”) of the two QDs. These levels are linked by allowed optical transitions to a pair of excited states where one hole and one electron are located in the upper dot, and the second electron is located in the upper or lower dot, respectively. At a specific electric field ( 10 kV/cm), the ∼ transitions within each of these pairs of levels have the same energies, leading to the desired ideal four-level system necessary for the implementation of photon blockade with a photonic crystal nanocavity. CHAPTER 7. CONCLUSIONS 83

7.2.4 Non-classical light from coupled cavities

As discussed in detail in chapter 5, more exotic methods for the generation of non- classical states of light can be realized by using systems of coupled cavities. The photonic molecule we have investigated has the potential to produce strongly sub- Poissonian light with current experimental parameters, but only if we can individually address the component optical cavities [88]. Since our photonic crystal cavities are proximity coupled, we are limited by the spot size of our laser. One potential solution to this problem would be to couple the cavities via a waveguide instead [97]. Such a system should be achievable in the photonic crystal platform, and would allow each of the cavities to be addressed individually. However, some optimization in design and fabrication will be required in order to tune the coupling of each cavity to the waveguide so as to achieve a similar regime of coupled operation. An alternative design would be to couple a QD equally to two separate modes of a single photonic crystal nanocavity. This kind of setup can be achieved in a “bimodal” cavity (such as the single-hole H1-defect photonic crystal cavity) which has two nearly-degenerate resonances of orthogonal polarizations. In such a system we have learned that strongly sub-Poissonian light results from interference between the coherent light transmitted through the resonant cavity and the super-Poissonian light generated by photon-induced tunneling [126]. Peculiarly, this eﬀect is most pronounced in a regime where the coupling strength g and cavity decay rate κ have comparable values, and it vanishes completely in the absence of cavity loss. Thus, this approach does not require extremely high quality factors. However, this scheme is sensitive to the frequency diﬀerence between the cavity resonances, which is the inevitable result of disorder introduced in the fabrication process. To counteract this complication, the cavity quality factor can actually be reduced to increase the overlap of the two resonances. We have also explored various post-fabrication techniques for tuning the cavity frequencies back into degeneracy with each other [127, 128, 129], which could potentially be used to improve the performance of the photonic molecule as well. Finally, scaling up the number of coupled cavities even further (as discussed in chapter 6) opens up entirely new possibilities for quantum simulation and quantum CHAPTER 7. CONCLUSIONS 84

optical networks. In order to generate a strongly-correlated quantum fluid of light [68] using such a coupled cavity array, the coupling strengths between the various cavities need to be large enough to overcome any variation in their frequency due to disorder. We showed that this condition was met for our photonic crystal coupled cavity array. However, the other requirement of achieving a sufficiently strong nonlinearity in each cavity remains more challenging. In most of the applications relating to quantum simulation, single quantum emitters must be deterministically positioned in each of the cavities, which is very difficult to achieve with even state-of-the-art solid- state technology. Site-controlled QDs [119, 130, 131, 132, 133] represent one potential solution to this problem, while other possibilities include bulk or quantum-well nonlinearities [85, 134]. Recently, there has also been significant interest in coupling 2D materials (such as graphene or WSe2) to photonic crystal cavities [135, 136], which could also provide the requisite nonlinearity. Naturally, a great amount of work still needs to be done in reducing the losses inherent in these systems and demonstrating their reliability for cavity QED. Nevertheless, coupled photonic crystal cavities hold great promise for implementing a scalable, integrated, quantum optical network in a semiconductor platform. Bibliography

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