Generation of Pulsed Quantum Light

GENERATION OF PULSED QUANTUM LIGHT

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

Kevin Fischer August 2018

This dissertation is online at: http://purl.stanford.edu/xd125bf7704

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.

Shanhui Fan

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

At the nanoscale, light can be made to strongly interact with matter, where quantum mechanical effects govern its behavior and the concept of a photon emerges—the branch of physics describing these phenomena is called quantum optics. Throughout the history of quantum optics, experiments and theory focused primarily on understanding the internal dynamics of the matter when interacting with the light field. For instance, in an ion trap quantum computer the motional states of the ions are of interest while the state of the light field is secondary. However, a new paradigm for information processing is becoming experimentally accessible where many quantum-optical devices are integrated in nanophotonic circuits. Then, the relevant information is encoded in the quantum states of the photon wavepackets as they fly around the nanophotonic chip. Hence, the converse problem is now important: the way the state of the photon field changes after interacting with the piece of matter. The effect of an individual device on a photon can now be formally described in a so-called scattering experiment. First, a photon wavepacket is prepared far away from the device. It travels freely towards the device and as it hits the interaction region forms a strongly entangled state of light and matter. Afterwards, the character of the interaction is imprinted on the outgoing wavepacket. This concept of a scattering experiment was originally envisioned in high-energy field theories with static Hamiltonians, and this thesis shows the concept adapted to nanophotonic devices driven by pulses of laser light (corresponding to time-dependent or energy nonconserving Hamiltonians). The formalism developed here will thus be of relevance to design and analysis of quantum information processing systems in which the information is encoded in the state of the photonic field, with the piece of matter implementing either a source of photons or implementing a unitary operation on the photonic state. In summary, for this dissertation I show experiments and theory that fully describe how the quantum state of a laser pulse changes when interacting with nano-sized pieces of matter. In particular, this work covers both a general formalism for photon scattering as well as two standard physical

iv systems as experimental examples that generate quantum light: the quantum two-level system and the Jaynes–Cummings system.

v Acknowledgments

I would like to dedicate this dissertation to my wonderful wife, Irena Fischer-Hwang. The PhD can sometimes seem surmountable, but with Irena as a partner in my life, I was able to face all its challenges. She constantly challenged me to be better and helped ground my life—without her none of this work would have been possible. Of course, no PhD is possible without a supportive adviser and colleagues. Prof. Jelena Vuˇcković easily provided one of the best scientific environments to learn and grow, that any student could have possibly hoped for. I believe one of Jelena’s most important qualities is the trust that she places in her students, providing critical guidance and direction but allowing us the proper space to learn and occasionally fail. In particular, I am grateful that Jelena sent me to conferences early and often, and even to give invited talks in her stead. She placed the trust in me that I could manage and handle some of the most complicated and expensive modifications to our lab. Most importantly, she trusted me with academic freedom to pursue the topics that were most interesting to me, helping to provide the context in framing their discussions. When finding my initial project, Dr. Kai Müllerserved as a secondary research advisor, with whom I have built one of the most successful working relationships in my career. Our collaboration began during our time at Stanford, but has lasted many impactful projects and publications into his time as a group leader at the Technical University of Munich. Kai is particularly calming in his ability to provide unbiased guidance for any difficult circumstance, technical or otherwise. In my overview in Ch. 1, I will acknowledge the very productive working relationships I have had with colleagues in the Vuckovic group, but in particular I’d like to mention Rahul Trivedi. Rahul constantly challenged me technically and would never take hand waving for granted. Working with Rahul closely on the photon scattering theory, I felt that my knowledge grew exponentially every day. Towards the end of my PhD career, I am grateful to the guidance of Tina Seelig and Fern

vi Mandelbaum, as well as my Accel Innovation Scholars cohort. They have all helped provide a critical piece of my education, and helped me to eﬀectively translate my skills onto life after the PhD.

vii Contents

Abstract iv

Acknowledgments vi

1 Introduction 1 1.1 Cavity QED with a single quantum emitter ...... 3 1.2 Strong coupling with quantum dots ...... 3 1.3 The Jaynes–Cummings model ...... 4 1.4 Observing strong coupling ...... 5 1.5 Basics of nonclassical light generation ...... 10 1.6 Overview ...... 11

2 Second-order quantum coherence 15 2.1 Hanbury–Brown and Twiss interferometer ...... 17 2.2 Connection to correlations of instantaneous ﬁelds ...... 21 2.3 Time-dependent quantum regression theorem ...... 23

3 Coherent drive in the Jaynes–Cummings system 25 3.1 Heisenberg picture ...... 27 3.2 Addition of coherent state drive ...... 29 3.3 Analytic analysis of homodyne interference ...... 31 3.4 Visualizing the homodyning in transmission spectra ...... 33

4 Cavity QED beyond the Jaynes–Cummings model 35 4.1 Dissipative structure ...... 36 4.2 Emitter-cavity detuning ...... 39

viii 4.3 Excitation pulse length ...... 42 4.4 Electron-phonon interaction ...... 44 4.5 Blinking of the quantum emitter ...... 47 4.5.1 Effects of blinking on transmission spectra ...... 48 4.5.2 Effects of blinking on nonclassical light generation ...... 50 4.6 Self-homodyne interference ...... 52 4.6.1 Effects of self-homodyne interference on emission spectra ...... 54 4.6.2 Effects of self-homodyne interference on nonclassical light generation . . . . . 56 4.7 Rabi oscillations on a cavity QED polariton ...... 60

5 Photon scattering formalism 61 5.1 Problem deﬁnition ...... 63 5.1.1 System Hamiltonian ...... 63 5.1.2 Bath Hamiltonian ...... 64 5.1.3 Waveguide-system coupling ...... 67 5.1.4 Interaction-picture Hamiltonian ...... 69 5.1.5 Scattering matrices ...... 70 5.2 Formalism ...... 71 5.2.1 Coarse-graining of time ...... 72 5.2.2 A general solution ...... 74 5.2.3 Extension to multiple output waveguides ...... 75 5.2.4 Connection to quantum trajectories and measurement theory ...... 77

6 Photon emission from a two-level system 79 6.1 Traditional theory of a single-photon source ...... 81 6.2 General theory of photon emission ...... 82 6.3 Short pulse regime ...... 86 6.3.1 Theory ...... 86 6.3.2 Experiment ...... 93

7 Conclusions and outlook 98 7.1 Homodyne interference ...... 98 7.2 Single-emitter cavity QED ...... 99 7.3 Photon scattering ...... 100

ix 7.4 Quantum two-level system ...... 100 7.5 Re-excitation in single-photon sources ...... 102 7.6 Indistinguishability ...... 102

x List of Tables

6.1 Counting statistics for a driven quantum two-level system ...... 91

xi List of Figures

1.1 AFM image of self-assembled InGaAs quantum dots ...... 2 1.2 Emission spectra from quantum dots ...... 2 1.3 Schematic of a strongly-coupled system based on a quantum dot embedded within a photonic crystal cavity ...... 4 1.4 Observation of strong coupling in either transmission or spontaneous emission spectra6 1.5 Cross-polarized reﬂectivity for observing strong coupling experimentally ...... 9 1.6 Principle of photon blockade and photon tunneling ...... 11

2.1 Schematic of the Hanbury–Brown and Twiss interferometer ...... 18 2.2 Hanbury–Brown and Twiss histogram of pulsed single-photon emission ...... 20

3.1 Input-output schematic of a Jaynes–Cummings system ...... 25 3.2 Difference between atom and cavity drive in the Jaynes–Cummings system . . . . . 26 3.3 Schematic of the Jaynes–Cummings system with homodyned output field ...... 31 3.4 Schematic of a Jaynes–Cummings drop filter ...... 33 3.5 Theoretical transmission spectra through a Jaynes–Cummings drop filter ...... 34

4.1 Climbing the Jaynes–Cummings ladder with dissipation ...... 38 4.2 Detuned photon blockade, experiment and theory ...... 41 4.3 Pulse-length dependence of photon blockade, experiment and theory ...... 43 4.4 Experimental eﬀects of electron-phonon interaction on polariton lifetimes ...... 44 4.5 Experimental polariton-phonon transfer process in photon blockade ...... 46 4.6 Fit to experimental transmission spectra ...... 49 4.7 Eﬀects of blinking on experimental transmission from a strongly-coupled system . . 51 4.8 Experimental origin of self-homodyne interference ...... 53

xii 4.9 Experimental effect of self-homodyne interference on resonance fluorescence from a dissipative strongly-coupled system ...... 55 4.10 Effects of strong emitter blinking on a resonant strongly-coupled system ...... 57 4.11 Perfectly modeled photon blockade and tunneling experiments under pulsed excitation 58 4.12 Detuned photon blockade as a function of excitation power ...... 59 4.13 Experimental Rabi oscillations and second-order coherence from a strongly-coupled system ...... 60

5.1 Schematic of photon scattering from a low-dimensional quantum system into a unidirectional waveguide ...... 62 5.2 An intuitive picture of the coarse-grained dynamical map for photon scattering . . . 74 5.3 Sketch of the path integral representing photon emission ...... 76

6.1 Driven quantum two-level system scattering photons into a waveguide ...... 79 6.2 Sketch of the path integral representing photon emission from a two-level system . . 80 6.3 Complete mathematical characterization of photon emission under Rabi oscillations 87 6.4 Simulations showing time-resolved single-photon and two-photon emission from a driven quantum two-level system ...... 92 6.5 Experimental details of a quantum two-level system based on an InGaAs quantum dot 94 6.6 Experimental quantum two-level system excited by a short laser pulse ...... 95 6.7 Experiments showing two-photon emission from a quantum two-level system . . . . . 96

7.1 Principle of using a quantum two-level system as an entangled photon pair source . . 101

xiii Chapter 1

Introduction

Many 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 [1]. Such sources are expected to have high efficiencies, rarely emit a wrong number of photons, and produce indistinguishable photons that interfere with each other. The majority of scientific efforts toward creating sources of nonclassical light have so far been focused on engineering single-photon sources [2], and the basic idea used to generate single photons on demand in state-of- the-art approaches is very simple: a single quantum emitter is excited with a pulsed source and its emission is filtered in order to isolate a single-photon with the desired properties [3]. For example, an optical or electrical pulse can generate carriers–electrons and holes–inside a quantum dot that recombine to produce several photons at different frequencies. Subsequent spectral filtering can be used to isolate a single-photon [4, 5]. Although these systems are already characterized by high multi-photon probability suppression, both the efficiency and indistinguishability of such a source can be further improved by embedding the quantum emitter into a cavity that has a high quality factor and a small mode volume [6], or by resonant excitation approaches such as photon blockade [7, 8]. Epitaxial, III-V semiconductor quantum dots have been a leading platform for nonclassical light generation experiments, and hold the record parameters for the implementation of many quantum technologies, including networks [9, 10] or quantum simulators [11–14]. Namely, a stream of indistinguishable single photons from one or more quantum emitters could be employed to generate larger photon number states or entangled photon states by quantum interference. In particular, InAs quantum dots in GaAs have generated the best results for single-photon sources (in terms of

1 CHAPTER 1. INTRODUCTION 2

Figure 1.1: AFM image of a 1x1 micron square array of uncapped, self-assembled InGaAs quantum dots.

( a ) ( b ) 8 0 0 7 0 0 7 0 0 6 0 0 6 0 0 ) ) . 5 0 0 . b b

r r 5 0 0 a a ( (

y 4 0 0 y t t

i i 4 0 0 s s n n

e 3 0 0 e

t t 3 0 0 n n I I 2 0 0 2 0 0

1 0 0 1 0 0

0 0 9 1 2 9 1 4 9 1 6 8 8 8 8 9 0 8 9 2 W a v e l e n g t h ( n m ) W a v e l e n g t h ( n m )

Figure 1.2: Spectra of quantum dots, (a) of a neutral excitonic transition and (b) of an ensemble of excitonic transitions from many quantum dots that feature a large inhomogeneous broadening. both rate and indistinguishability) for quantum optics and cavity QED experiments for the past 20 years [2, 15, 16]. Such quantum dots are formed by self-assembly during the growth process called molecular beam epitaxy (MBE), as a result of lattice mismatch between InAs and GaAs. Since quantum dot islands are formed by self-assembly, their locations are random and they have a distribution of sizes and shapes (Fig. 1.1), leading to variation in the transition energies of different quantum dots on the same wafer (inhomogeneous broadening) (Fig. 1.2). Because of a shallow confining potential, these quantum dots require operation at cryogenic temperatures, lower than 70 K and typically around 4 K. In this chapter, I present a brief introduction to nonclassical light generation with cavity quantum electrodynamical sources using a single InAs quantum dot. This perspective helps frame my thesis work and provides an entry-point to understanding my contributions in the field. CHAPTER 1. INTRODUCTION 3

1.1 Cavity QED with a single quantum emitter

Cavity quantum electrodynamical systems hold great promise for exploring fundamental light- matter interactions, generating novel quantum states of light, and building quantum information networks [1]. Originally, such investigations were centered on atomic systems where both the quantum matter and light conﬁning components of the cavity QED system possessed intrinsically narrow linewidths [7]. There, a phenomenon called strong coupling was ﬁrst observed. Strong coupling is the hallmark signature of strong light-matter interaction, where new hybridized states of light and matter emerge. The hybridized states are anharmonic, resulting in giant nonlinearities at the single- photon level. However, these atomic systems will unlikely serve as practical elements in quantum networks due to their slow interaction rates and lack of scalability. In the 2000’s, major advances towards the study of light-matter interaction in systems based on InGaAs quantum dots and photonic crystal nanocavities enabled the observation of strong coupling in an optical solid-state system [17, 18]. Since then, the boundaries of quantum-dot-based cavity QED work have been pushed from ultra-low threshold lasers [19] and ultra-fast single-photon sources [20], to photon blockade [8, 21] and single-photon phase gates [22]. More recently, group-IV solid-state quantum emitters, in the form of the popular nitrogren [23, 24] and silicon [25] vacancy-related complexes, have generated strong interest for future cavity QED experiments with their demonstrations of Purcell enhancement, single-photon emission, and exceptionally small inhomogeneous broadening. This chapter gives an introduction to the basic concepts relevant to nonclassical light generation with a single-emitter cavity QED system.

1.2 Strong coupling with quantum dots

One of the most successful and established platforms for single-emitter cavity QED is the InGaAs quantum dot embedded within an L3 photonic crystal cavity [26, 27]. Self-assembed InGaAs quantum dots are widely used as quantum emitters due to their excellent optical quality, and are particularly useful for strong coupling experiments due to their narrowband optical transitions and ability to be integrated into planar photonic crystal cavities [28]. Planar photonic crystal resonators provide extremely small mode volumes (V ) that enable a large enhancement of the light-matter interaction √ strength with any embedded quantum emitters, g ∝ 1/ V [29]. A schematic depiction of such a system is shown in Fig. 1.3, with the L3 photonic crystal cavity shown in the upper left (characterized by energy decay rate κ) and the quantum dot shown in the bottom right (characterized by CHAPTER 1. INTRODUCTION 4

κκκ (cavity decay rate)

g ∝ 1/ V (dot-cavity coupling strength)

Γ (dot decay rate)

Figure 1.3: Schematic depiction of a strongly-coupled system based on a single InGaAs quantum dot embedded within an L3 photonic crystal cavity. The red partial-dome in the cross-section (middle) represents the quantum dot and a TEM image is shown of a typical dot in the bottom right. TEM adapted with permission from Ref. [33]. energy decay rate γ). The quantum dot, illustrated as the red half-dome, is either probabilistically or deliberately positioned at the center of the crystal where it can maximally couple to the fundamental mode of the cavity for optimal interaction rate g [30]. Ideally, the quantum dot behaves like a quantum two-level system, and it can be placed inside of a diode structure to control which speciﬁc excitonic transition forms the two-level system [31, 32]. However, most of the work on strongly- coupled systems in the solid-state has been performed without such a structure, and usually either the neutral or singly charged exciton couples to the cavity mode.

1.3 The Jaynes–Cummings model

At its simplest, a strongly-coupled system is modeled by the Jaynes–Cummings (JC) Hamiltonian [34] † † † † HJC = ωaa a + (ωa + ∆e) σ σ + g a σ + aσ , (1.1) which represents a single cavity mode coupled to a single quantum two-level system. Here, a is the † operator which represents the cavity’s mode [a, a ] = 1, ωa is the mode’s frequency, σ is the dipole † operator representing a single two-level subsystem of a quantum emitter {σ, σ } = 1, and ∆e is the detuning of the quantum emitter’s transition frequency from that of the cavity. As we will discuss CHAPTER 1. INTRODUCTION 5

in subsequent sections, speciﬁc solid-state considerations will have important implications in exactly modeling such a system, but for now we will consider the idealized JC model. Because of the coherent interaction between the bosonic cavity mode and fermionic emitter mode, a new set of eigenstates arises [34]. The optimal basis is no longer the bare states of the system, the direct product of individual emitter (ground |gi or excited |ei) and cavity (photon number |ni) states, but rather a so-called dressed state basis that comprises states known as polaritons. These polaritons are entangled states of light and matter, with eigenstates

q n √ 2 2 E± = nωa + ∆e/2 ± ng + (∆e/2) . (1.4)

Consider the ﬁrst two polaritons of non-zero energy (n = 1), called the ﬁrst upper and lower polaritons: when the cavity and emitter are tuned on resonance (∆e = 0), then the polaritons each contain equal contributions of a single electronic or a single photonic excitation. However, as the emitter is detuned from the cavity (by increasing ∆e) then the polaritons trend towards either the electronic or photonic characters.

1.4 Observing strong coupling

The evolution of the system dynamics is governed by a Liouville equation ∂tρ(t) = Lρ(t), which accounts for the non-unitary evolution induced by the cavity and emitter dissipation [35]. Speciﬁcally,

κ Γ L ρ(t) = i [ρ(t),H ] + D[a]ρ(t) + D[σ]ρ(t), (1.5) JC JC 2 2 where D[c]ρ(t) = 2cρ(t)c† − c†cρ(t) − ρ(t)c†c is a super-operator called the Lindblad dissipator and c is an arbitrary system operator. Now, we can investigate how strong coupling behavior can be experimentally explored. The most common way to initially observe the eﬀects of strong coupling is to directly measure the splitting of CHAPTER 1. INTRODUCTION 6

1 . 0 T r a n s m i s s i o n S p o n t a n e o u s L o w e r U p p e r e m i s s i o n p o l a r i t o n p o l a r i t o n ) . b r a (

y t i 0 . 5 s n e t n I

0 . 0 - 2 - 1 0 1 2 F r e q u e n c y ( g )

Figure 1.4: Observation of strong coupling by measuring either transmission or spontaneous emission spectra for zero emitter detuning (∆e = 0). Typical solid-state cavity QED parameters were used, where κ ≈ 3/4g and γ is small enough to be insigniﬁcant (γ g, κ). the two lowest energy polaritons either in a transmission experiment [17] or spontaneous emission experiment [36], as shown in Fig. 1.4. First, we discuss a spontaneous emission experiment. Here, the emitter is incoherently pumped by exciting charge carriers in the semiconductor to higher energy than the emitter’s transition (e.g., in a quantum dot’s multi-excitonic states, quasi-resonant levels, or above the band-gap [37]), where they relax into the transition coupled to the cavity. Subsequently, the system relaxes through cavity emission. This process can easily be modeled by adding the term

Pσ † 2 D[σ ]ρ(t) to LJCρ(t), where Pσ is the incoherent pumping rate of the quantum emitter [38]. To get an idea of whether the emission occurs through the cavity or the emitter decay channel, consider their typical emission rates. Typical rates for a well-performing cavity QED system are approximately γ/2π ≈ 0.2 GHz (which is often further suppressed by a photonic bandgap) and κ/2π ≈ 10 GHz [20]. (There is additional non-radiative broadening for the emitter, which we will explore further in Sec. 4.4 but will ignore in the introduction.) Because solid-state systems work in the bad-cavity limit where κ γ, the photons are almost exclusively emitted by the cavity. Therefore, to obtain the measured spectrum of spontaneous emission (e.g., through an ideal spectrometer or scanning Fabry-Perot cavity), we compute the spectrum of the cavity mode operator a [38] Z ∞ † −iωτ Sa(ω) = lim dτ ha (t + τ)a(t)ie . (1.6) t→∞ −∞ The astute reader may also notice that it would be possible to incoherently excite the cavity

Pa † mode using the term 2 D[a ]ρ(t), where Pa is the cavity’s incoherent pumping rate. While this conﬁguration is diﬃcult to realize experimentally, the resulting incoherent spectrum of emission CHAPTER 1. INTRODUCTION 7

is formally equivalent to a transmission spectrum for arbitrarily small excitation powers. When the incoherent excitation rate is much slower than the decay rates of the ﬁrst-rung polaritons, D[a†] randomly initializes the system with a maximum of one photon in the cavity mode. Thus, calculating a spectrum under incoherent cavity excitation is equivalent to calculating a one-photon spectrum [39], which represents the linear impulse response of the system and hence its transmission spectrum. Additionally, the computational complexity of this approach for calculating a transmission spectrum is much lower, especially for cavity QED systems with multiple emitters. On the other hand in a transmission experiment, a weak continuous-wave (CW) laser incident on the cavity is scanned in frequency and the transmitted light is measured. Now, as opposed to adding the excitation source as a dissipator, the input light is modeled as a coherent state coupled to the

iωLt † −iωLt cavity [35]. This is represented by a Hamiltonian driving term Hdrive = E a e + a e , where

E is the real-valued driving strength (proportional to the incident ﬁeld) and ωL is the coherent † † state frequency. For simplicity, a rotating-frame transformation H˜ = UHU + i (∂tU) U with † † iωL(a a+σ σ)t U = e is used to remove the time-dependence of the excitation term [40] so that H˜drive = E a + a†, and written as a super-operator

h i L˜driveρ˜(t) = i ρ˜(t), H˜drive . (1.7)

Similarly, the Jaynes–Cummings Liouvillian transforms according to

h i κ Γ L˜ ρ˜(t) = i ρ˜(t), H˜ + D[a]˜ρ(t) + D[σ]˜ρ(t) (1.8) JC JC 2 2 with ˜ † † † † HJC = (ωa − ωL) a a + (ωa + ∆e − ωL) σ σ + g a σ + aσ . (1.9)

Calculating the transmission spectra is then performed ﬁrst by calculating the steady-state density matrix through solving ˜ ˜ LJC(ωL) + Ldrive ρ˜ss(ωL) = 0 (1.10)

† † and then usingρ ˜ss(ωL) to obtain the transmission spectrum ST(ωL) = ha ai(ωL) = Tr{a a ρ˜ss(ωL)}. In this way, only light transmitted through the cavity is simulated rather than including interference eﬀects of the reﬂected light. Although the focus is on numerical techniques in this section, analytic solutions to the spectra under incoherent and coherent pumping may be found in [41] and [42], respectively. From these analytic forms with dissipation, one can see that the transmission CHAPTER 1. INTRODUCTION 8

eigenstates become non-degenerate when g > |κ − γ|/4 [43]. However, reaching the strong coupling regime requires the more stringent criteria that g > κ/4 and g > γ/4, since the coherent coupling needs to overcome all dissipation in the system [17]. Although diﬃcult to realize in planar two-dimensional photonic crystal structures, other con- ﬁgurations such as nanobeam cavities [44] allow easy excitation of the quantum emitter through radiation modes orthogonal to the cavity. This translates into a similar transmission experiment,

iωLt † −iωLt but where the emitter is driven through Hdrive = Ω σ e + σ e and Ω incorporates the driving field amplitude, dipole moment, and mode matching. For similar reasons as incoherent excitation of the cavity mode, weak incoherent excitation of the quantum emitter can equivalently be viewed as monitoring cavity transmission when coherently driving the emitter. However, due to the difficulty in observing this phenomenon in our photonic crystals with planar geometry, in the first two chapters we will mean a ‘transmission experiment’ to refer exclusively to cavity drive and monitoring. Returning to the traces in Fig. 1.4, one can immediately identify that both the transmission and emission spectra clearly identify the correct splitting of the first two polaritons as 2g. However, both the correct linewidths of the polaritons and their precise locations (which are slightly shifted by the effects of loss) are only captured in the spontaneous emission spectrum; we will derive this effect in Sec. 4.1. In transmission, because the two polaritons have cavity components that are shifted by π phase, they interfere destructively at frequencies between the polaritons and artificially decrease the apparent polariton linewidths. In fact, this interference occurs for systems not even in strong coupling and is referred to as dipole-induced transparency [42]. Nevertheless, transmission experiments are better for identifying strong coupling due to the fact that incoherently exciting the quantum emitters causes experimental non-idealities. For instance, it generates excess carriers that can induce effects such as field noise [45], and specifically in quantum dot samples with randomly positioned emitters, the carriers can cause nearby dots to indirectly pump the cavity [46]. Now that we have discussed transmission experiments, we compare with experiments performed in a configuration called cross-polarized reflectivity [17] (Fig. 1.5a). Here, both the excitation and detection light traverse along (approximately) the same physical path into and out-of the cavity. However, this setup is still mathematically equivalent to a transmission experiment due to the polarization degree of freedom and the configuration of polarizers [17]. The linearly polarized cavity is rotated 45◦ relative to the linear polarization of the incident light. This way, the incident light projects onto the cavity polarization with 1/2 efficiency. Meanwhile, the orthogonal polarization CHAPTER 1. INTRODUCTION 9

( a ) ( b ) 9 2 1 . 4 C r o s s - p o l a r i z e d r e f l e c t i v i t y H 9 2 1 . 6 L i n e a r C a v i t y

P B S ) p o l a r i z e r m n (

h P B S t g

n 9 2 1 . 8 e l

V e v a

W a v e p l a t e s W 9 2 2 . 0 M i c r o s c o p e Q u a n t u m d o t x 1 0 0 o b j e c t i v e

9 2 2 . 2 C r y o 3 5 3 4 3 3 3 2 3 1 3 0 2 9 2 8 2 7 C a v i t y a t 4 5 ° T e m p e r a t u r e ( K )

Figure 1.5: A typical cross-polarized reflectivity measurement used to observe strong coupling. (a) Experimental setup of a reflectivity measurement that is equivalent to transmission [17]. The linear polarizer and two polarizing beamsplitter cubes perform the polarization selection, while the wave- plates allow for polarization rotation and correction for bi-refringence in the optical elements [47]. (b) Typical reflectivity experiment on a strongly-coupled system, where the emitter detuning relative to the cavity frequency is controlled by tuning the temperature of the sample. An anticrossing between the bare cavity (red) and dot (green) states is observed. of the output light is filtered via the polarizing beamsplitters. Therefore, the input and output channels are strictly orthogonal and the reflectivity experiment is equivalent to transmission, to within a scaling factor of 1/2. This scaling factor does not often matter in realistic experiments because the collection losses are typically at least an order of magnitude larger [17]. As a result, normalized quantities are typically used to study the systems. A typical cross-polarized reflectivity measurement revealing the strong coupling of an InGaAs- based system is shown in Fig. 1.5b. The emitter detuning ∆e is often controlled relative to the cavity frequency ωa by changing the sample temperature. The quantum dot has a stronger dependence on temperature than the cavity due to the bandgap’s quadratic temperature dependence compared to the cavity’s linear change in permittivity [48]. A clear avoided crossing can be seen between the two polaritons near 32.5 K, as compared to the bare dot (green) and cavity (red) states. Here, the character of each of the polaritons switches between electronic and photonic, providing clear evidence for strong coupling between the cavity and dot. This crossing matches the change in character present in the eigenstate equations for |n, ±i(∆e). CHAPTER 1. INTRODUCTION 10

1.5 Basics of nonclassical light generation

All of the above dynamics in weak excitation regimes can be fully captured by linear theories, however, the generation of nonclassical light is inherently non-linear. In a cavity QED system, nonclassical light is generated by ﬁltering a stream of incident coherent light through a single strongly-coupled system [8, 20, 49]. Owing to the highly nonlinear character of the interaction between the input light and a strongly-coupled system, the admission of a single-photon into the cavity may enhance (photon tunneling) or diminish (photon blockade) the probability for a second photon to enter the cavity. Ideal photon blockade is depicted schematically in Fig. 1.6a, where the upward blue arrows represent the laser tuned for photon blockade (left side). Absorption of a photon into state |1, +i blocks the admission of a second photon because no state is present to absorb the second photon. In this sense, ideal photon blockade is indistinguishable from transmission of light through an equivalent two-level system comprised of the ground state and the tuned polariton. Meanwhile, in photon tunneling (depicted by the upward red arrows on the right side), two photons are absorbed together in a multiphoton transition to directly excite the state |2, +i. Following these interactions, a light beam exits the cavity with the nonlinear action imprinted on its quantum character. The quality of the nonclassical light is typically characterized by the measured degree of second- (2) order coherence g [0], as discussed in [50]. If a given pulse results in a photocount distribution Pm, then E[m(m − 1)] P P m(m − 1) g(2)[0] = = m m , (1.11) E[m]2 P 2 ( m Pmm) which is a normalized second-order factorial moment. Notably, g(2)[0] = 1 for a coherent pulse with Poissonian counting statistics, while g(2)[0] < 1 for sub-Poissonian or g(2)[0] > 1 for super-Poissonian counting statistics; for ideal photon-blockade g(2)[0] = 0. Importantly, this statistic is completely independent of collection losses so it gives an accurate representation of the internal system dynamics in almost any experiment. Although complete characterization of the photocount distribution would require measurement of all factorial moments of the photocount distribution, g(2)[0] is an important measure in determining whether the source is acting as a single- or multi-photon source, and is the most readily accessible in experiments [51]. Therefore, we will focus on the measured degree of second-order coherence in this section. In Fig. 1.6b, we present experimental evidence of photon blockade and photon tunneling in a series of typical experiments measuring g(2)[0] on emission from an InGaAs-based strongly-coupled system. When the laser detuning is near the upper and lower polaritons (states |1, ±i), g(2)[0] < 1 CHAPTER 1. INTRODUCTION 11

B a r e c a v i t y L o w e r p o l a r i t o n U p p e r p o l a r i t o n ( a ) ( b ) f r e q u e n c y 2 , 1 . 6 T u n n e l i n g ( h i g h e r m a n i f o l d s ) ω √ 2 a 2 g 2 1 . 4 2 , ] B l o c k a d e 0

[ s t )

1 , 2 1 . 2 ( 1 m a n i f o l d ) ( ω g a 2 g 1 . 0 1 ,

0 . 8 G r o u n d 0 - 2 - 1 0 1 2 L a s e r d e t u n i n g ( g )

Figure 1.6: Photon blockade and photon tunneling. (a) Schematic depiction of the Jaynes–Cummings ladder in the dressed basis |n, ±i when the emitter and cavity are resonant [51]. The upward blue arrows (on the left side) depict photon blockade, while the upward red arrows (on the right side) depict photon tunneling. (b) Measured degree of second-order coherence g(2)[0] for an InGaAs-based resonant system, illustrating tunable nonclassical light generation from a strongly-coupled system (with ﬁtted system parameters g/2π = 10.9 GHz and κ/2π = 10 GHz). anti-bunches for photon blockade. The precise location of photon blockade occurs slightly detuned from the polaritons due to the strong dissipation in III-V systems [20], which will be thoroughly discussed in the next chapter. Meanwhile, when the laser is tuned in-between the polaritonic rungs, multi-photon transitions are emphasized such that photon tunneling occurs and the light bunches causing g(2)[0] > 1—the bunching occurs because a large component of the vacuum state is present in conjunction with super-Poissonian statistics, which results in dividing by a small photon intensity P 2 squared ( m Pmm) [51]. However, because of the strong dissipation in the system (since g ≈ κ), the second-order coherence statistics do not deviate much from the laser statistics of g(2)[0] = 1, as discussed in [20]. In the next chapter, we will explore several nonidealities that result in this relatively poor performance and ways to leverage these nonidealities for interesting physics and better nonclassical light generation.

1.6 Overview

In the ﬁrst chapter, I have now presented an introduction to nonclassical light generation with cavity quantum electrodynamical systems. This corresponded to state-of-the-art nonclassical light generation from such strongly coupled systems at the beginning of my PhD work. Further, until my PhD work experimentally relevant solid-state systems were modeled with steady-state dynamics despite CHAPTER 1. INTRODUCTION 12

being driven by short pulses. Hence, my main contribution is to understand how to correctly model the generation of nonclassical light when coherent laser pulses drive a low-dimensional quantum system such as a quantum dot. Understanding this physics required a reinvention of the tools and techniques available in quantum optics to understand low-dimensional systems coupled to photonic ﬁelds, and my thesis outlines this story of discovery. Below, I summarize the contributions of each chapter.

• Chapter 2: The investigation into strongly coupled systems motivated many fundamental inquires for my research into nonclassical light generation. The ﬁrst of these was to understand the connection between experimental measurements of the second-order coherence and the quantum evolution equations for the system. These results are summarized in Ch. 2 with excepts from Ref. [50].

• Chapter 3: In Sec. 1.4, we briefly discussed the difference between driving the cavity or the emitter with a laser. This chapter is about formalizing the similarity between the two cases and is based mostly on Ref. [52]. Together with Shuo Sun, Daniil Lukin, and Yousif Kelaita, I discovered the intricate connections between atom and cavity driving. Specifically, the cavity and emitter drives are related via a trivial transformation in the system dynamics, and the output statistics can be changed from one to the other by the addition of a coherent state. As a result, we could show how a homodyning technique can be used to access an effective atomic driving while actually driving the cavity, which is especially important for nanophotonic implementations of Jaynes–Cummings systems.

• Chapter 4: This chapter details a combined experimental and theoretical investigation of a strongly-coupled system. As discussed in Sec. 1.5, despite the promise of tunable photon statistics from strongly-coupled systems, the experimental work to date had shown only small deviations from Poissonian counting statistics. Certainly, an ideal JC system would provide the desired tunability, but there appeared to be some experimental limitations with our platform. When I joined Prof. Vuˇcković’sgroup, one of my first projects was to understand what were all of these limitations. Working closely with Armand Rundquist, Kai Müller,and later Constantin Dory, we thoroughly investigated all the mechanisms and interactions present in InAs quantum dots coupled to planar photonic crystal cavities. Through our exploration, we found a host of new interactions and were able to utilize them to demonstrate high quality nonclassical light generation from strongly coupled systems for the first time. Particularly CHAPTER 1. INTRODUCTION 13

important was our detailed understanding of second-order coherence measurements (Ch. 2) and driving the Jaynes–Cummings system (Ch. 3), which helped us experimentally realize atom-like driving through a self-homodyne interference. A summary of some of our exploits with an experimental strongly-coupled system is provided in Ch. 4, and is adapted largely from the book chapter Ref. [53] and reviewing key concepts from our work in Refs. [20, 51, 54–58].

• Chapter 5: Ideologically, quantum mechanics is typically treated as a problem of solving state vectors in a Hilbert space. However, for open quantum systems or quantum optics where there is a low-dimensional quantum system like a quantum dot coupled to a photonic field, the state vectors representing the photons are almost always averaged out (traced over). Then, the output photonic state is only accessible through correlators of the internal dynamics due to a boundary theory between the system and the field. To me this was a bit of a conceptual Rube Goldberg, and I was interested in fitting the open quantum system problem into a framework where only state vectors were computed. This story is the subject of Ch. 5 and based on Ref. [59], wherein I developed a new framework with Rahul Trivedi and Vinay Ramasesh to understand the behavior of open quantum systems.

• Chapter 6: With my newfound understanding of the quantum two-level system coupled to a photonic field, I revisited a many decades-old problem of exploring the effect of a coherent pulse on driving such a system. Initially my goal was to understand the sources of error limiting quantum two-level systems as single-photon sources. Again with Dr. Kai Müller,and this time in collaboration with his and Prof. Jonathan Finley’s experimental group at the Technical University of Munich, we explored the effect of short laser pulses on the quantum two-level system. Lukas Hanschke performed the experiments in Munich and did a fantastically thorough job. We additionally found the surprising result that a short laser pulse can drive not just single-photon emission from a two-level system, but also two-photon emission. I present an analytic analysis followed by these experimental results, which represent the first complete exploration of a few-photon source that is subject to re-excitation by the laser pulse. This chapter is based on Refs. [59–61].

This thesis covers only a small fraction of the work performed during my time at Stanford with Prof. Jelena Vuˇckovi´c. For the sake of brevity I have chosen to discuss only a few topics in my thesis that represent broad concepts which motivated my studies at Stanford. In particular, I hope this thesis provides an introduction into my fundamental explorations of the generation of pulsed quantum light, which show the co-development of both theory and experiment together. CHAPTER 1. INTRODUCTION 14

Regarding the remaining work, I was fortunate enough to have amazingly productive working relationships with colleagues in Jelena’s group that led to signiﬁcant work on photon scattering with Konstantinos Lagoudakis [62], site-controlled quantum dots with Konstantinos [63, 64], coherent state control with Konstantinos, Constantin Dory, and Linda Zhang [65–67], nanometallic cavity quantum electrodynamics with Yousif Kelaita and Tom Babinec [68], color-center work with Marina Radulaski [69], Shuo Sun, and Linda [70, 71], as well as dissipative state preparation with my international collaborator Prof. Jonathan Finley and his student Per-Lennart Ardelt [72, 73]. Chapter 2

Second-order quantum coherence

As discussed in Ch. 1, an ideal pulsed single-photon source would contain only a single-photon per pulse and hence only a single quantum of energy in its wavepacket—in this chapter we will fully explore this concept. However, the second-order coherence is usually used a metric for judging the quality of single-photon generation due to its property that g(2)[0] = 0 for a single-photon source, g(2)[0] = 1 for a laser pulse, and is independent of losses from the system until the measuring device. Stemming from my work on quantum correlation measurements, one of my first theoretical interests was in establishing the connection between the dynamical evolution equations of the open quantum system and the measured degree of second-order coherence. First, I review the theory of photon counting which describes how an ideal photon detector sees a quantum light field. Consider photon flux incident on a photon counter with finite timing resolution. Mathematically, the probability that the flux results in a detection event between t and t + dt is

p(t) dt = ηhfˆ(t)i dt , (2.1) where η is the total detection and collection eﬃciency and fˆ(t) is the instantaneous photon ﬂux operator (analogous to classical intensity) [74]. Notably, the instantaneous detection probability ηhfˆ(t)i must be negligible such that multiple photoionizations in the detector cannot occur. Inte- grating over the photon pulse duration, T , there exists a classical probability distribution Pm(T )

15 CHAPTER 2. SECOND-ORDER QUANTUM COHERENCE 16

that governs the number of expected photocounts. A single-photon source requires

 1 − η if m = 0   Pm(T ) = η if m = 1 , (2.2)   0 otherwise where m ∈ {Z ≥ 0} is a classical random variable that represents the number of photocounts. Thus, it is suﬃcient to characterize a single-photon source by a measure which roughly corresponds to the probability of two events occurring within T : the second-order factorial moment of its photocount distribution [75] G(2)[0] ≡ E[m(m − 1)] . (2.3)

(Because the photocount distribution is over the classical random variable m, we use the E [...] notation to denote the classical expectation value.) While the first detection still depends on m, the second detection depends on m − 1 because the first detection subtracts a quantum of energy from the distribution [74]. Therefore, for an ideal pulsed single-photon source G(2)[0] = 0. Unfortunately, the detection and collection efficiency η can be quite difficult to experimentally measure, which makes this correlation challenging to directly estimate. Instead we will focus on determining its normalized version

G(2)[0] E[m(m − 1)] g(2)[0] ≡ = . (2.4) E[m]2 E[m]2

This normalized second-order factorial moment of Pm(T ) is referred to as the measured degree of second-order coherence at zero time delay and doesn’t depend on the eﬃciency η [74]. This quantity is useful because any value g(2)[0] < 1 is disallowed by classical physics [76] and because g(2)[0] approximates the error probability of the source to produce two detection events relative to the number of single detection events. Because the photocount distribution can only be estimated from the outcome of many pulse-wise experiments, a typical experimental cycle will involve periodic generation of the photon wavepacket and its subsequent detection every tr seconds. The number of photodetections from a given pulse at the time bin ntr (with n ∈ Z) can then be represented by the classical random variable m[ntr] = CHAPTER 2. SECOND-ORDER QUANTUM COHERENCE 17

{Z ≥ 0}. We can then extend the pulse-wise deﬁnition of G(2)[0] to non-zero time delays, i.e.

(2) G [ktr] = E [m[0]m[ktr]] for k ∈ {Z > 0}, (2.5)

which we refer to as the un-normalized second-order intensity correlation at time delay ktr. We note that its time origin is irrelevant due to the quasi-stationary nature of the pulses. If the probability for three photodetections over a given pulse is negligible, then this correlation represents the probability that two photons are detected between pulses separated by the time difference ktr. From this definition, we can arrive at another definition of g(2)[0] which will turn out to be experimentally most useful: G(2)[0] g(2)[0] = for k ∈ { > 0}, (2.6) (2) Z G [ktr] provided that we have correctly chosen tr longer than the correlation time of any system operators (2) 2 (where G [ktr] = E[m] ). Experimental realities may sometimes inhibit the realization of this criterion and one can then simply take

G(2)[0] g(2)[0] = lim . (2.7) (2) k→∞ G [ktr]

While this deﬁnition may seem a trivial extension, this form is most useful (due to the limitations of legacy measurement instrumentation) in discussing experimental setups that estimate g(2)[0]. Thus, we have discussed how experimental single-photon sources can be readily characterized even in the face of unknown detection and collection eﬃciencies.

2.1 Hanbury–Brown and Twiss interferometer

While one can easily estimate g(2)[0] from the detection record of ideal photon counters, this is not possible for most experimental detectors and single-photon sources. Experimental photodetectors have a so-called “dead time”, for which the detector cannot register a second count following the ﬁrst, that usually is much longer than the temporal length of the photon pulse. With a maximum of one registered count per pulse, such a detector could never distinguish between single- and multi- photon sources. Fortunately, it turns out that under the right approximations, an experimental setup known as the Hanbury–Brown and Twiss interferometer (Fig. 2.1) is still capable of precisely estimating g(2)[0] by using two photon counters, and hence is still capable of characterizing realistic single-photon sources [77]. CHAPTER 2. SECOND-ORDER QUANTUM COHERENCE 18

Correlator

vac

Figure 2.1: Schematic of the Hanbury–Brown and Twiss interferometer: The source’s emission (periodic every tr) is path-entangled by a beamsplitter and measured by two detectors. A digital recorder then correlates the detection times and computes hHBT[ktr]. Here, E(t) indicates the coherent drive, LJC represents the dynamics of the source, b(t) represents the output of the source, and c(t) and d(t) represent the ﬁelds at the inputs to the detectors.

Because our focus is on pulsed single-photon sources, we will only discuss the Hanbury–Brown and Twiss (HBT) interferometer operated in a pulsed mode of operation, where we repeatedly excite our source every tr seconds [77]. In the HBT setup, every pulse is path-entangled between two channels by a 50 : 50 beamsplitter and each channel is fed into a single-photon detector. The periodic photon absorption events registered by the two detectors can be represented as the classical random variables mc[ntr] = {0, 1} and md[ntr] = {0, 1} where n ∈ {Z > 0}; They may take on unity values at times ntr to represent photon detections. Here, the square brackets will indicate the discrete nature of the random variables and their intensity correlations, due to both their periodic pulsed nature and any timing uncertainty in their detection (either explicit by detector jitter or implicit through purposeful erasure of timing information). Although an estimate for Pm(T ) could be built up through many periodic trials and used to directly compute g(2)[0], traditionally g(2)[0] has been extracted from a histogram of the time-correlated detection records, i.e.

Tint b t c Xr hHBT[ktr] = mc[ntr]md[(n + k)tr], (2.8) n=0 where k ∈ Z, Tint is the integration time, and ktr is the time diﬀerence between detections [37].

Each histogrammed time-bin hHBT[ktr] is an independent and binomially-distributed random vari- p p able, whose standard deviation estimator is given by hHBT[ktr ](1 − η) ≈ hHBT[ktr] for most experiments (η is again the combined detection and collection eﬃciency). After careful consideration of the detector nonidealities such as dead time and dark counts, as CHAPTER 2. SECOND-ORDER QUANTUM COHERENCE 19

well as of the expected statistics of a single-photon source [37], one can arrive at the approximation

2 η jTint k (2) E[hHBT[ktr]] ≈ G [ktr]. (2.9) 4 tr

1 Here, the factor of 4 accounts for the action of the beamsplitter to halve the signal at each detector. In making the connection back to the second-order factorial moment at zero delay that was discussed in the previous section, consider

b Tint c tr (2) 4 1 X G [0] ≈ E[mc[ntr]md[ntr]] . (2.10) η2 b Tint c tr n=0

While it may be surprising that this expression is equivalent to G(2)[0], i.e. E [m(m − 1)], the key insight is that the random variables mc[ntr] and md[ntr] are not independent because a detection event by either detector pulls one quantum of energy from the total path-entangled field. As previously mentioned, however, it is difficult to experimentally estimate G(2)[0] due to un- (2) known setup efficiencies. Fortunately, since all correlations in G [ktr] are lost at long times such that (2) 2 lim G [ktr] = E [m] , (2.11) k→∞ then limk→∞ E[hHBT[ktr]] can serve as an intensity reference. Thus, we can obtain an estimate for the measured degree of second-order coherence from the following ratio

s ! h [0] 1 1 gˆ(2)[0] = lim HBT 1 ± + . (2.12) k→∞ hHBT[ktr] hHBT[0] hHBT[ktr]

Importantly, several approximations were required to arrive at this result [37]:

1 • The net detection probability per pulse 2 ηE[m] must be very small relative to the dead time.

1 • The net detection probability per pulse 2 ηE[m] must be very small relative to the repetition

time tr.

• The probability of many-photon detection must be moderately low; Analogously, the higher- order factorial moments of the photocount distribution must be of order unity, i.e. g(n)[0] ∼ 1 for n > 2. CHAPTER 2. SECOND-ORDER QUANTUM COHERENCE 20

1 5 0 0 ] s t n u

o 1 0 0 0 c [

T B

H 5 0 0 h

0 - 3 - 2 - 1 0 1 2 3

T i m e d e l a y [ t r ]

Figure 2.2: Example Hanbury–Brown and Twiss histogram generated using pulsed single-photon emission from a quantum-dot-based cavity quantum electrodynamical system. An estimate for the measured degree of second-order coherence can be obtained by taking the ratio of the counts at zero (2) delay, hHBT[0], to the counts at delay tr, hHBT[tr]. This appropriately estimates g [0] because the pulse-wise correlations have already disappeared after one repetition time.

• The detectors’ dark count rates dn are minimal, such that

1 1 G(2)[kt ] d2 + E[m] d. (2.13) r η2 η

As an example of such a histogram, consider a correlation measurement from Ref. [20] in Fig. 2.2. In this experiment, a single-photon source was run in pulsed mode and time-correlated using an HBT setup. Because all pulse-wise correlations have decayed already after one repetition time in this experiment,

E[hHBT[tr]] = lim E[hHBT[ktr]] (2.14) k→∞ and therefore s ! h [0] 1 1 gˆ(2)[0] = HBT 1 ± + , (2.15) hHBT[tr] hHBT[0] hHBT[tr] which yieldsg ˆ(2)[0] = 0.29 ± 0.04.

Finally, we note that many experimental HBT interferometers do not exactly histogram hHBT[ktr].

Rather, they only approximate hHBT[ktr] by electronically time-correlating detections on-the-ﬂy. This is done by taking the ﬁrst detector as the signal to start timing and the second detector as the signal to stop timing: Each start-stop sequence generates a count in the time-bin of the timer value. This way, the histogram is built up in real-time as new measurement correlations are recorded. The downside to this method is that each successively longer time-bin requires more failed detections to CHAPTER 2. SECOND-ORDER QUANTUM COHERENCE 21

b Tint c tr k−1 ˆ X Y hHBT[ktr] = mc[ntr] (1 − md[(n + l)tr])md[(n + k)tr]. (2.16) n=0 l=0

1 k Here the product term means that at long time-correlations, E [h[ktr]] ∝ (1 − 2 ηE[m]) approximately and hence it decays to zero. Therefore, this electronic method of time-correlating to estimate (2) (2) g [0] may not always work well if the G [ntr] shows correlated behavior for large n. Now that we have a good understanding of experimentally how to characterize the photocount distribution of a single-photon source, we will discuss the theoretical connection to the instantaneous correlations of the photon wavepacket’s ﬁelds.

2.2 Connection to correlations of instantaneous ﬁelds

From a numerical modeling perspective, estimating g(2)[0] for a given system Liouvillian is fairly straightforward using Monte–Carlo wavefunction techniques [78]. Here, evolution of the system wavefunction is modeled, conditioned on the detection events of an ideal single-photon detector with infinite bandwidth. Such a detector may absorb one quantum of energy from the emitted field at a time, in proportion to the photon flux at a given instant, and can distinguish detection events with infinite timing resolution. Through this process, detection records are generated that build (2) up an estimate for Pm(T ) and hence g [0]. However, the Monte–Carlo method often requires an extremely large number of simulated evolutions (trajectories) to obtain an acceptable approximation of the measured degree of second-order coherence. Fortunately, for systems with reasonably-dimensioned Hamiltonians, there exists a faster and more intellectually satisfying way of computing g(2)[0]—as used for pulsed nonclassical light sources, this algorithm was first implemented numerically in our previous work [20]. This method directly relies on using the quantum regression theorem (Sec. 2.3), to compute correlations between the instantaneous system operators such as the Heisenberg cavity operator a(t) previously discussed. These correlations in turn are related to the instantaneous correlations of the continuous-mode free- field operator b(t) that describes the emitted photon-field flux, as is described by [b(t), b†(t0)] = δ(t − t0) and b(t) |vaci = 0. (Notably, b(t) is a Schrödinger-picture operator and t is an index.) In this section, we now fully elaborate on this explicit connection of the measured degree of coherence with its associated instantaneous field correlations. CHAPTER 2. SECOND-ORDER QUANTUM COHERENCE 22

We begin with a more directly quantum mechanical model of our ideal single-photon detector that formalizes and combines the stories from Refs. [74, 79] and [77]. In analogy to the classical integrated mean intensity, the quantum mechanical operator

Z T Z T Mˆ (T ) = dt fˆ(t) = dt b†(t)b(t) (2.17) 0 0 represents the total photon number arriving at an ideal detector over the time interval t ∈ [0,T ] (where T is again over the duration of the photon pulse). As such, its quantum mechanical expectation value yields hMˆ (T )i = E [m]. Comparing our semiclassical deﬁnition of G(2)[0] to the quantum mechanical operator Mˆ (T ) we have

G(2)[0] = hMˆ (T )(Mˆ (T ) − 1)i or G(2)[0] = h: Mˆ (T )2 :i. (2.18)

Here, h: Mˆ (T )2 :i denotes the quantum mechanical expectation value of the normally-ordered second moment of the photon number operator Mˆ (T ). Writing out this moment explicitly

Z T Z T (2) 0 † † 0 0 G [0] = dt dt hT−[b (t)b (t )]T+[b(t )b(t)]i, (2.19) 0 0 where the operators T± again indicate the time-ordering required of a physical measurement [80] (operators with higher time indices towards the center of the expression). We can also compute the quantum mechanical version of the normalized second-order moment g(2)[0], with G(2)[0] h: Mˆ (T )2 :i g(2)[0] = = (2.20) hMˆ (T )i2 hMˆ (T )i2 and its explicit form

R T R T 0 † † 0 0 dt dt hT−[b (t)b (t )]T+[b(t )b(t)]i g(2)[0] = 0 0 . (2.21) R T † 2 ( 0 dt hb (t)b(t)i)

Written another way by deﬁning

(2) 0 † † 0 0 G (t, t ) ≡ hT−[b (t)b (t )]T+[b(t )b(t)]i, (2.22) then R T R T dt dt0 G(2)(t, t0) g(2)[0] = 0 0 . (2.23) hMˆ (T )i2 CHAPTER 2. SECOND-ORDER QUANTUM COHERENCE 23

Therefore, the measured g(2)[0] actually represents the sum of all field-flux correlations G(2)(t, t0) over the detection time. This expression agrees with intuition when higher-order correlations are negligible, as it then represents the probability to detect two photons at every possible pair of times, normalized to the total photon number squared. Then, input–output theory provides a direct connection between the internal system operators and external mode operators such that measuring zero delay flux correlations is equivalent to calculating the mode correlators. Moving to a Heisenberg picture, note that the expectations are taken over the state vector at some time t1. From this perspective, the time argument in b(t) is again an index not a time evolution, and b(t) for all t can be used to form a complete basis of waveguide states (as will become very important in Ch. 5). The Heisenberg picture version of b(t) at some time t1 long after the photon emission has taken place is typically labeled bout(t), which is related to √ the cavity mode’s Heisenberg operator via the input-output relation bout(t) = bin(t) + κa(t). The input operator bin(t) is the Heisenberg field operator at time t0, and has zero mean. Thus, we may simply calculate R T R T 0 † † 0 0 dt dt hT−[a (t)a (t )]T+[a(t )a(t)]i g(2)[0] = 0 0 . (2.24) R T † 2 ( 0 dt ha (t)a(t)i) Hence, we have finished outlining the relationship between second-order coherence and the Hanbury– Brown and Twiss interferometer.

2.3 Time-dependent quantum regression theorem

In this section, we consider precisely how to calculate the correlations involved in the expression of the measured degree of second-order coherence, i.e. for Eq. 2.24. Single-photon sources are characterized by potentially non-trivial ﬁrst- and second-order optical coherences [50]. Therefore, we will be interested to use the system dynamics to calculate correlations of the form

G(t, τ) = hA(t)B(t + τ)C(t)i, (2.25) where A, B, and C are each some combination of a, a†, and the identity matrix. Although this calculation is often performed with the quantum regression theorem [76], its formal statement usually excludes time-dependent Liouvillians. Yet, when studying a system under pulsed excitation, a time- dependent Liouvillian invariably arises due to the dynamical driving. Therefore, we have extended the quantum regression theorem to time-dependent Liouvillians below. CHAPTER 2. SECOND-ORDER QUANTUM COHERENCE 24

When the quantum regression theorem is applied for time-dependent Liouvillians, it inherits the approximations from the quantum-optical master equation [81] and yields the following result

hA(t)B(t + τ)C(t)i = Trsys{BΛ(t, t + τ)}, (2.26) where Λ(t, t + τ) is governed by the evolution equation

∂τ Λ(t, t + τ) = L(t + τ)Λ(t, t + τ) (2.27) and is subject to the initial condition

Λ(t, t) = Cρ(t)A. (2.28)

Although this equation only can evolve forward in time, the correlators discussed in this paper either give rise to physical measurements and inherently require τ > 0 [82] or possess a conjugate time-reversal symmetry about τ [83]. In the course of my PhD, I added this algorithm as a routine to QuTiP along with various tutorial notebooks, so a simple call to the correlators automatically calculates two-time correlations with time-dependent Liouvillians. This evolution equation has a particularly nice interpretation when the probability for three successive photodetection events is negligible and

hA(t)B(t + τ)C(t)i = ha†(t)a†(t + τ)a(t + τ)a(t)i, (2.29) which then represents the probability density of detecting an excitation at time t followed by another excitation at time t + τ. The quantum regression theorem clearly captures this through: ﬁrst a reduction of the density of matrix by one excitation

Λ(t, t) = aρ(t)a† (2.30) and by then computing the probability density of a second reduction

† Trsys{a aΛ(t, t + τ)}. (2.31) Chapter 3

Coherent drive in the Jaynes–Cummings system

In this chapter, I present a study of the Jaynes–Cummings (JC) Hamiltonian driven by a coherent state. The JC system consists of a single bosonic mode (e.g., an electromagnetic cavity) coherently interacting with a single fermionic mode (e.g., an electron or two-level atom). These modes are radiatively coupled to external reservoirs, which can cause the system to both leak or absorb energy depending on their state. As discussed in Ch. 1, preparing the reservoirs with coherent light is one of the most common ways of pumping energy into the system, and provides an excellent way to understand its quantum-mechanical responses. Despite the simplicity of the JC model itself (Fig. 3.1), it gives rise to a variety of complex phenomena when driven. Beyond photon blockade, photon tunneling, and Mollow triplets (discussed in Chs. 1 and 4), for example, vacuum Rabi splitting [44, 84], bistability [85, 86] or symmetry breaking [87], squeezed states [88], state dressing

Figure 3.1: Schematic of a Jaynes–Cummings system, containing a two-level quantum system coherently interacting with a cavity mode. The cavity couples to the reservoir b, while the two-level atom couples to reservoir c.

25 CHAPTER 3. COHERENT DRIVE IN THE JAYNES–CUMMINGS SYSTEM 26

Figure 3.2: Energy-level structure of a Jaynes–Cummings system when the atom and cavity have the same natural oscillation frequency. Blue arrows denote allowed transitions from the reservoir-cavity coupling, while red arrows denote allowed transitions from the reservoir-atom coupling.

[89], and a rich structure of multi-photon resonances [35, 90] have all been observed, as well as finding use in the readout of qubits [91, 92]. There are multiple ways to drive the JC system, as shown schematically in Fig. 3.1. In particular, the coherent state could be prepared in the reservoirs coupled to the cavity or coupled to the atom [53] (see Sec. 1.4). Almost all of the above studies have shown substantive differences in observed radiation from the JC system depending on the driving configuration, such as the transmission differences in Fig. 1.4. With many of these works, observing the differences between reservoir initialization is a primary portion of the result. For example, in a very comprehensive experimental study, [93] observed photon correlations on whether the cavity or atom was driven by a laser. To understand the perceived importance of which reservoir contains the coherent state, reconsider the JC level structure in Fig. 3.2—here we refine the description of the distinction between cavity and atom driving. To summarize the previous statements, the two-level atom coherently interacts with the cavity at rate g, breaking the harmonic ladder of uncoupled levels. The new eigenstates of the system are called polaritons because they contain equal superpositions of light and matter, and are indexed by the labels |±, ni. The anharmonicity of these eigenstates causes the JC system to have a highly nonlinear response to input light, leading to its wide variety of quantum effects. However, the magnitude of the nonlinearity experienced by light driving the system through the cavity or atom appears to be different in this picture. Notably, the allowed transitions through the ladder are different depending on the reservoir coupling. Allowed transitions through the cavity keep the same polariton symmetry |±, ni ↔ |±, n + 1i while the allowed transitions through the atom alternate |±, ni ↔ |∓, n + 1i. The anharmonicity between successive eigenstates of the same symmetry is lower than for successive eigenstates of alternating symmetry. Consequently, the effective nonlinearity in this picture is higher for light driving the atom than the cavity. From this CHAPTER 3. COHERENT DRIVE IN THE JAYNES–CUMMINGS SYSTEM 27

perspective, one might easily imagine that driving the atom is a far more efficient way to obtain a strong nonlinearity than driving the cavity. In the steady state, however, it is known that the dynamics under drive through the cavity or atom can be related via a trivial transformation [87, 88, 94]. For this work, we extended the analysis to an arbitrary input coherent pulse and demonstrated there is a simple transformation between driving with the cavity or atom in the general case. For this chapter, I summarize our work in formalizing this transformation using an approach based on Heisenberg–Langevin equations. Our approach allows us to show the full equivalence between driving with the cavity or atom for an arbitrary input coherent pulse. Specifically, we showed that driving the cavity can be seen simply as building up a coherent state in the cavity as if it were empty, and that coherent state drives the atom through the atom-cavity coupling. This result is important for quantum information processing, where cavity-atom systems typically manipulate pulses of laser light rather than steady-state quantities [1], and is what enabled us to achieve an atom-like response from our strongly-coupled system (Sec. 4.6). Experimentally, it is often far easier to drive the cavity than the atom, which posed a major impediment to observing a strong nonlinear response in early JC systems based on solid-state platforms (e.g., see Refs. [8] or [49]) as discussed in Ch. 1. However, we used a homodyning effect to remove the unnecessary coherent state output under cavity driving and achieve an atom-like driving response (see Sec. 4.6) [55–57, 92].

3.1 Heisenberg picture

To begin, we discuss a mathematical model for the Jaynes–Cummings system. As a reminder, the model describes a single cavity mode and a single two-level atom, which coherently exchange energy at the rate g. For convenience, we rewrite the resulting Hamiltonian (Eq. 1.1) as

† † † † H = ωaa a + ωσσ σ + ig a σ − σ a , (3.1)

where ωa and ωσ are the natural frequencies of the cavity mode a and two-level atom σ, respectively. (The bosonic cavity obeys [a, a†] = 1, while the fermionic atom obeys {σ, σ†} = 1.) For simplicity, we consider that the cavity and atom couple to independent reservoirs, b and c, respectively (Fig. 3.1). The cavity couples to the photonic reservoir b through the operator a at a rate κ, while the atom couples to the reservoir c through σ at rate γ. This coupling leads to spontaneous emission, but also CHAPTER 3. COHERENT DRIVE IN THE JAYNES–CUMMINGS SYSTEM 28

allows for energy to be injected into the system from a laser pulse. These processes can be described by the Heisenberg–Langevin equations [95] for an arbitrary system operator x

x˙ = −i [x, H] κ √ κ √ − x, a† a + κb (t) + a† + κb† (t) [x, a] 2 in 2 in γ √ γ √ − x, σ† σ + γc (t) + σ† + γc† (t) [x, σ] , 2 in 2 in (3.2) where the input and output operators are deﬁned for reservoir mode o (b or c here) as

1 Z −iω(t−t0) oin(t) = √ dω e o0(ω) (3.3a) 2π 1 Z −iω(t−t1) oout(t) = √ dω e o1(ω). (3.3b) 2π

The operators o0(ω) and o1(ω) are the Heisenberg operators o(ω) at times t0 and t1, respec- † tively. Consequently, the input and output modes obey the relations [oin(t), oin(s)] = δ(t − s) † and [oout(t), oout(s)] = δ(t − s). Further, operators between different reservoir modes commute. The input modes represent the Fourier transformed fields at time t0 forward propagated to time t, while the output modes represent the transformed fields at time t1 backward propagated to time t. Applying Eq. 3.2 for the Jaynes–Cummings system yields a closed set of Langevin equations

κ √ a˙ = − iω + a + gσ − κb (3.4a) a 2 in γ √ σ˙ = − iω + σ + σ ga + γc (t) (3.4b) σ 2 z in ˙ † √ † √ † N = −γN − σ ga + γcin(t) − ga + γcin(t) σ, (3.4c)

† † where σz = σ σ − σσ and N = (σz + 1)/2. The equation of motion 3.4a for the cavity mode is linear, but the two-level system can only store a ﬁnite amount of energy, resulting in the nonlinear Eqs. 3.4b-3.4c. Lastly, there is an important set of boundary conditions between the input and output ﬁelds

√ bout(t) = bin(t) + κa(t) (3.5a) √ cout(t) = cin(t) + γσ(t), (3.5b) CHAPTER 3. COHERENT DRIVE IN THE JAYNES–CUMMINGS SYSTEM 29

which explains the output field in a reservoir is the linear combination of the input field and the radiated field into that channel. There also exists a causality relationship between the system operators and the input/output fields at different times, however we will not make use of those commutators in this work.

3.2 Addition of coherent state drive

We will next explore the eﬀects on adding input coherent states to the modes bin or cin. Our goal is to show that the expectation values of any Heisenberg operators of the system are governed by the same dynamics, regardless of whether driven by the cavity or atom. For any Heisenberg-picture operator

† hΨ0|U s1(t1)s2(t2) . . . sn(tn)U|Ψ0i = hΨ0|s˜1(t1)˜s2(t2) ... s˜n(tn)|Ψ0i . (3.6)

This relation follows froms ˜(t) = U †s(t)U and U †U = 1. In the case where we let U be a displacement operator D that creates coherent fields in the input photonic modes, then this transformation is called the Mollow transformation. Hence, it is only necessary to find each unique transformed operator in order to fully specify the output field of the system. To be more concrete, explicitly let U be some combination of displacement operators

Z † ∗ Do[δ] = exp dt δ(t)oin(t) − δ (t)oin(t) , (3.7)

where Do[δ] creates the coherent state δ in the photonic reservoir o, which could potentially represent b or c in our model. Hence, the displacement operators commute with every other input ﬁeld and Schr¨odinger-picturesystem operator. The action of the Mollow transformation on input operators is given by

† Do[δ]oin(t)Do[δ] = oin(t) + δ(t). (3.8)

Consequently, the eﬀect of a Mollow transformation on the total system dynamics can be understood CHAPTER 3. COHERENT DRIVE IN THE JAYNES–CUMMINGS SYSTEM 30

as follows. For all operators, the transformation would yield identical dynamics as Eqs. 3.4-3.5, except the input operators need to be modified oin(t) → oin(t) + δ(t) to account for all potential input fields {δ}. Now, consider two specific cases for driving the Jaynes–Cummings system that correspond to relevant experimental situations:

1. Only the emitter is driven by a coherent state ζ

U = Dc[ζ]. (3.9)

Then, in the Mollow transformation all of the system operators and output ﬁelds obey equivalent dynamics as Eqs. 3.4-3.5 except

√ σ˙ → σ˙ − γσzζ(t) (3.10a) √ √ N˙ → N˙ + γσ†ζ(t) + γζ∗(t)σ (3.10b)

cout(t) → cout(t) − ζ(t). (3.10c)

This corresponds to the situation where the laser pulse in mode c comes in and imparts energy to the emitter, and causes the system’s dynamics to be imprinted onto the output ﬁelds. After interaction, the input pulse continues traveling along mode c and is found in the output channel

cout.

2. Only the cavity is driven by a coherent state β

U = Db[β]. (3.11)

Then, the Mollow transformation yields equivalent dynamics as Eqs. 3.4-3.5 except

√ a˙ → a˙ + κβ(t) (3.12a)

bout(t) → bout(t) − β(t). (3.12b)

Now the laser pulse in mode b imparts energy to the cavity. After interaction, it is found in

the output channel bout.

(When these coherent states drive the Jaynes–Cummings system, the Mollow transformations are equivalent to inclusion of the diﬀerent driving Hamiltonians discussed in Ch. 1.) CHAPTER 3. COHERENT DRIVE IN THE JAYNES–CUMMINGS SYSTEM 31

Figure 3.3: Schematic of the Jaynes–Cummings system with the reservoir b homodyned with ﬁeld e using a beamsplitter.

Importantly, it is possible to show that driving the cavity is equivalent to building up a coherent state α(t) in the cavity, which drives the atom through the coupling g. Consider the transformation a(t) → a(t) + α(t) applied to Eqs. 3.4-3.5, where α(t) = −(β ∗ f)(t) and

√ f(t) = κ e−(iωa+κ/2)tΘ(t) (3.13) is the linear filter response of the cavity and Θ is the Heaviside function. The coherent field α(t) corresponds to a filtered version of the input pulse, i.e. passing through a bare version of the cavity without the atom. Then, the Mollow transformation gives equivalent dynamics as Eqs. 3.4-3.5 except

σ˙ → σ˙ − gσzα(t) (3.14a)

N˙ → N˙ + gσ†α(t) + gα∗(t)σ (3.14b) √ bout(t) → bout(t) − β(t) − κα(t). (3.14c)

This has the remarkable consequence that we drove the cavity ﬁeld, but found a transformation that turned the Langevin equations into those of the atom being driven. This shows how the two driving cases are, in fact, intimately related to each other.

3.3 Analytic analysis of homodyne interference

Equation 3.14c shows that the output field resulting from driving the cavity is equivalent to directly √ driving the atom plus an offset of a coherent field β(t) + κα(t). This suggests that by homodyning the output on a beamsplitter with a matching auxiliary field, it is possible to relate the two driving cases perfectly. Hence, imagine we add a beamsplitter (Fig. 3.3) that mixes the cavity output with CHAPTER 3. COHERENT DRIVE IN THE JAYNES–CUMMINGS SYSTEM 32

a new ﬁeld ein. The unitary beamsplitter transformation on the ﬁelds is given by

1 eout(t) = √ (−ein(t) + bout(t)) (3.15a) 2 1 dout(t) = √ (ein(t) + bout(t)) . (3.15b) 2

Additionally, the deﬁnition of vacuum is expanded to include the new ﬁeld.

Now, the cavity is driven by the original coherent state β and the beamsplitter channel ein is fed a compensatory coherent ﬁeld ξ

U = Db[β]De[ξ]. (3.16)

Again, let α(t) = −(β ∗ f)(t)—then under the replacement a(t) → a(t) + α(t), while identifying √ √ ζ(t) = gα(t)/ γ and ξ(t) = β(t)+ κ α(t), the Mollow transformation results in equivalent dynamics as Eqs. 3.4-3.5 and 3.15 except

√ σ˙ → σ˙ − γσzζ(t) (3.17a) √ √ N˙ → N˙ + γσ†ζ(t) + γζ∗(t)σ (3.17b)

bout(t) → bout(t) − ξ(t) (3.17c)

dout(t) → dout(t) − 2ξ(t). (3.17d)

Hence, by our appropriate compensation ξ(t), the output field after the beamsplitter eout then looks identical to the case where the atom is actually driven by a real input field ζ(t). Here, a compensatory field on a beamsplitter can perfectly cancel the coherent state leaking from the JC system and the output looks as though only the atom were driven! Because this compensatory field involves the input pulse convolved with the filter response of the bare cavity, ξ(t) could be generated by using a reference cavity (Fig. 3.4). The pulse β(t) would be split between the reference cavity and the Jaynes–Cummings system, and then recombine on a second beamsplitter. A similar configuration, where the cavities are double-sided, has previously been explored for quantum light generation and photonic gates [96–102]. However, these studies did not identify the root cause of the benefit from the interference, which we explain here as removing an unwanted coherent state that builds up in a Jaynes–Cummings system under cavity drive. CHAPTER 3. COHERENT DRIVE IN THE JAYNES–CUMMINGS SYSTEM 33

Figure 3.4: Schematic of a drop filter that automatically generates the appropriate coherent state to compensate the coherent field that builds up in the Jaynes–Cummings system. The first beamsplitter divides the incident pulse and sends it to the JC system and the bare cavity, while the second beamsplitter homodynes the signals together. The response out of the homodyne channel of the drop filter eout is the same as if the JC system were instead driven through the atom by the input coherent field convolved with the impulse response of the cavity.

3.4 Visualizing the homodyning in transmission spectra

The results presented in this chapter thus far have been fairly abstract, in the sense that the dynamics of various Heisenberg operators have been compared. Because the Heisenberg operators are quite complicated objects, i.e. matrices acting on the total Hilbert space of the internal system and external fields, visualizing what this means intuitively can be somewhat challenging. Here, I show clearly the effect on transmission spectra through the system, revealing the relationship derived here exactly. Of course, the transmission spectra only look at first-order zero-delay correlators of the fields, and hence only provide a partial picture of what is happening. Nevertheless, I find that visualizing these spectra may help provide some intuition. (In Sec. 4.6.2, I present experimental and theoretical examples of how homodyne interference affects the photon statistics.) The transmission spectra are shown in Fig. 3.5, which contains several traces for different g/κ ratios spanning weak coupling to strong coupling (left to right, respectively). In (a), the green trace shows a JC cavity transmission spectrum given cavity drive by coherent state β(ωL), and ωL is again the coherent state driving frequency. This spectrum reveals dipole induced transparency, where the dip has a width given by the emitter decay rate (set to γ = 0.025κ). Once strong coupling is reached [(b) and (c)], the green traces dip to zero when the laser is resonant with the cavity frequency, an interference characteristic of strong coupling. For each plot, the dotted line shows the transmission through a bare cavity uncoupled to any emitter for reference. Next in Fig. 3.5 are the blue traces, which represent cavity transmission under emitter drive by co- √ √ herent state ζ(ωL) = gα(ωL)/ γ. Note in the steady state that α(ωL) = κβ(ωL)/ (i(ωL − ωa) + κ/2), where the laser detuning is hence defined as ωL − ωa. Finally, the dashed red traces represent the CHAPTER 3. COHERENT DRIVE IN THE JAYNES–CUMMINGS SYSTEM 34

Figure 3.5: Transmission spectra through a Jaynes–Cummings system for visualization of matched homodyne interference and diﬀerent g/κ ratios. Green traces represent transmission through the cavity of the JC system under cavity drive by coherent state β(ω ), while blue traces represent L √ transmission under emitter drive by coherent state ζ(ωL) = gα(ωL)/ γ. The dashed red traces represent the output eout(ωL) of the drop ﬁlter shown in Fig. 3.3. The dotted blue curves represent transmission through a bare cavity uncoupled to the emitter.

output eout(ωL) of the drop filter shown in Fig. 3.3. Notably, the drop filter’s output, which is cavity driven, matches the cavity transmission under emitter drive by ζ(ωL). This shows visually how the drive under cavity can be thought of as emitter drive plus a coherent offset in the output, and the internal dynamics are really governed by the equivalent emitter drive. The careful reader may notice the three-peaked structure in Fig. 3.5c and initially find this result surprising. In fact, there are only two lower polaritons to drive from the JC ladder, which was shown in the transmission spectrum of Fig. 1.4. What is different here is that previously the comparison was made between scanning a laser of constant amplitude while varying frequency for either the emitter or cavity drive. However, because the emitter drive is related to cavity drive via the coherent state building inside the cavity, it is related partially via the frequency response of the cavity. Hence, the equivalent emitter drive for fixed amplitude cavity drive versus frequency is strongest at zero detuning, but decays as the laser moves off resonance. This competition between the decreasing equivalent drive of the emitter and the increasing amplitude from moving towards the polariton resonances results in the three-peaked structure. Chapter 4

Cavity QED beyond the Jaynes–Cummings model

In Chs. 2 and 3, I presented theoretical models important for understanding Jaynes–Cummings systems and measuring their outputs. For this section, I provide a comprehensive study on the non- idealities facing strongly-coupled systems in the solid-state. There are several key considerations that alter the performance of a solid-state cavity QED system from that of an ideal Jaynes–Cummings system:

1. Strong dissipation caused by cavity loss rates comparable to the coherent interaction strength; dissipation decreases the ﬁdelity of nearly all cavity QED eﬀects [20, 54].

2. Detuning of the emitter from the cavity; this is an important tool in improving the nonclassical light generated from a dissipative JC system [20].

3. Pulse-wise experiments, which are potentially more interesting for applications in quantum networks and are anyway required by the timing resolution of most single-photon detectors [50]; the pulse length and pulse shape have a strong inﬂuence on the ﬁdelity of nonclassical light generation [54].

4. The solid-state environment results in an important interaction with phonons [103]; this dissipation can both be a detrimental and a positive inﬂuence towards nonclassical light generation [54].

35 CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 36

5. Temporal variation of the ground state of a quantum emitter, a phenomenon known as blinking [104]; when the emitter blinks, it either decouples from the cavity or is so far spectrally detuned that the system is eﬀectively no longer strongly coupled [49].

6. Interferometric eﬀects owing to the photonic crystal’s background density of states; the light transmitted through the cavity cannot simply be modeled as a single cavity mode, but also requires the modeling of a continuum scattering channel [55–57]. These eﬀects enable an atom-like driving response despite driving the cavity (as theoretically examined in Ch. 3).

In carefully studying these eﬀects, we have learned to signiﬁcantly improve both the modeling of a solid-state strongly-coupled system and its ability to generate high-purity states of nonclassical light. Although this chapter details these nonidealities using investigations with III-V quantum emitters due to their technological maturity, the physics and the theory will be equally applicable to future experiments with other quantum emitters such as color centers found in group-IV systems.

4.1 Dissipative structure

The first nonideality that we explore in detail is the large dissipation present in solid-state cavity QED systems. Due to limitations in current material fabrication technologies for III-V systems, the cavity dissipation rates are comparable to the coherent coupling rates. We explore this point through the language of non-Hermitian effective Hamiltonians [105, 106]. By using a non-Hermitian Hamiltonian, the effects of decay may be incorporated in a linear manner, but at the cost of generating an evolution equation that leaks probability. This non-Hermitian evolution plays an important role in the Monte- Carlo wavefunction or trajectory approach to quantum simulation [75, 83]: it governs the evolution of the wavefunction in-between photon emission events. Further, it has a similarly critical role in calculating the scattered photonic wavefunction, as will be summarized in Ch. 5. Thus, the non- Hermitian Hamiltonian is useful because after a state has been prepared, its initial evolution and hence decay rate is set by the effective Hamiltonian. For a given system, such a Hamiltonian is

X γk H = H − i c† c (4.1) eﬀ 0 2 k k k CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 37

where ck is an arbitrary system operator and γk is its decay rate. For the Jaynes–Cummings system speciﬁcally, this takes the form

κ γ H = H − i a†a − i σ†σ. (4.2) eﬀ JC 2 2

In a similar manner to diagonalizing HJC to obtain the dressed states in Sec. 1.3, Heﬀ can be diagonalized to additionally obtain the loss rates of the dressed states [35]. Diagonalization yields the complex eigenenergies for the n-th rung of the system to be

s 2 ∆ (2n − 1)κ + γ √ 2 κ − γ ∆ En = nω + e − i ± ng − + i e , (4.3) ± a 2 4 4 2

n where the full-width half-maxes of the linewidths are given by the imaginary parts ±2 Im{E±}. These n n complex eigenenergies are depicted in Fig. 4.1a, with the bounding lines showing Re{E+}± Im{E+} n n and Re{E−} ± Im{E−}. One can derive the bounds for non-degeneracy of the levels in a given rung n n by solving for when ∆e = 0 and Re{E+}= 6 Re{E−}, which occurs when g > |κ − γ|/4.

For large emitter detunings ∆e, the statement made in Sec. 1.3 that the polaritons take on primarily an electronic or photonic character can be understood even more intuitively: here, one polariton has approximately the emitter lifetime γ and the other the cavity lifetime κ. At zero detuning, the upper (UP) polaritons have equivalent character to the lower (LP) polaritons. From this plot alone, we can already suspect a way to improve photon blockade in highly dissipative systems. Considering the black arrows for photon blockade at zero detuning where the laser is tuned in resonance with UP1, then the second photon is nearly resonant with UP2 due to the finite linewidths. Notably, this problem arises because the anharmonicity of the Jaynes–Cummings √ ladder scales as n, while the decay rates scale as n. However, by introducing a small detuning, the effective non-linearity of the system is much higher. Now consider the red arrows for detuned photon blockade, where UP2 is much further off resonance in comparison. Thus, we would expect better quality photon blockade to occur for the detuned system. This information can also be visualized in an alternative manner, as shown in Fig. 4.1b. Here, the energies to climb the dressed-states ladder one-by-one [35] are shown, with their linewidths, by plotting n n n−1 n n−1 ∆E++ = Re{E+} − Re{E+ } ± Im{E+} + Im{E+ } (4.4) CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 38

( a ) ( b ) 4 C a v i t y f o r b i d d e n G N D → U P 1

E m i t t e r ) U P 1 → U P 2 g

( → 2 U P 2 U P 3

U P 2 y g ω r 2 a e n a p p r o x i m a t e l y E 0 ∆

} h a r m o n i c X t n e X i s - 2 G N D → L P 1 n

a L P 1 → L P 2 L P 2 r

T L P 2 → L P 3 - 4 f o r b i d d e n y

g - 4 - 2 0 2 4 r

e E m i t t e r d e t u n i n g , ∆ ( g ) n U P 1 e

E ω a ( c ) 4 G N D → U P 1 )

g G N D → U P 2 ( → y 2 G N D U P 3 g r e

L P 1 n E

∆ 0

n o t o

h - 2 G N D → L P 1 p i

t → G r o u n d s t a t e l G N D L P 2

0 u G N D → L P 3 M - 4 - 4 - 2 0 2 4 - 4 - 2 0 2 4 ∆ ∆ E m i t t e r d e t u n i n g , e ( g ) E m i t t e r d e t u n i n g , e ( g )

Figure 4.1: Energy-level structure of a strongly-coupled system. (a) Dressed states of the Jaynes– Cummings ladder and their linewidths. Note: the separation between the rungs of the dressed-states ladder is not to scale. (b) Transient energies to climb the Jaynes–Cummings ladder rung-by-rung. (c) Resonant laser frequencies for multi-photon transitions. (a-c) The heights of the bounding regions represent the full-width half-maxes of the linewidths for each transition. The g/κ = 5 ratio was used in order to cleanly illustrate the trends in each plot (which is now becoming achievable with state- of-the-art parameters for solid-state systems). UPn and LPn label the upper and lower polaritons, respectively. and n n n−1 n n−1 ∆E−− = Re{E−} − Re{E− } ± Im{E−} + Im{E− } . (4.5)

0 n n (Note, we define E± = 0). The linewidths add for the eigenstates involved in E++ and E−− because the transient energies are used to consider a resonant process that excites the system rung-by-rung. For small detunings, the jumps to the second (dashed) and third (dotted) lines are almost on top of the first jump from the ground state. Thus, we would not expect good quality nonclassical light generation to come from the system. On the other hand, for non-trivial detunings the effective anharmonicity increases linearly because the jump to the first rung differentially increases relative to the second-rung with detuning. After the first jump, however, the ladder is relatively harmonic since the higher jumps are centered around ∆E = 0. Importantly, these jumps occur along either the upper or lower branches, but not between branches. Jumps between branches are disallowed for the same reason the transmission dips to zero in-between the upper and lower polaritons (as seen CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 39

n n n−1 in Fig. 1.4). Specifically, the transitions between branches, ∆E+− = Re{E+} − Re{E− } and n n n−1 ∆E−+ = Re{E−} − Re{E+ }, are forbidden because the states have cavity components that are shifted by π phase and hence zero dipolar overlap between hn − 1, ∓|a|n, ±i = 0. We have plotted the forbidden transitions as dashed/dotted lines with no bounding regions. These transitions occur through emission by the quantum emitter, i.e. hn − 1, ∓|σ|n, ±i= 6 0, but at a negligible rate due to typical solid-state cavity QED parameters. Finally, we discuss the multi-photon structure of the dissipative Jaynes–Cummings ladder. Just like for the photon blockade argument where the detuning increased the effective nonlinearity between the first and second rungs of the dressed-states ladder, increasing the emitter detuning has the effect to separate the multi-photon transitions. This can be seen in Fig. 4.1c, where the multiphoton transitions fan out with detuning. Interestingly, the absorption linewidths for multi-photon emission are all κ after the first blockaded rung, because the multi-photon transitions occur between the ground state and an upper dressed state in an idealized model that assumes the intermediate levels remain unpopulated. (Certainly, this approximation breaks down when the multi-photon transitions strongly overlap, but then it’s difficult to identify linewidths for distinctive multi-photon processes anyway.) In the idealized model, the target dressed state determines the linewidth since the ground state has approximately zero dephasing. Although the dephasing rate of the nth dressed state scales with n, the n-photon absorption linewidth scales with 1/n because the laser detuning is compounded by the number of photons involved in the process. Thus, the multi-photon absorption linewidths are left at κ, so we plot n Re{E+}/n − ωa ± κ/2 (4.6) and n Re{E−}/n − ωa ± κ/2. (4.7)

Note: it’s possible to derive an eﬀective multi-photon absorption Hamiltonian based on adiabatic elimination of the unpopulated states which rigorously supports this analysis [107].

4.2 Emitter-cavity detuning

In this section, we experimentally and numerically consider the eﬀects of photon blockade by measuring g(2)[0] as a function of the excitation laser frequency, for multiple emitter-cavity detunings. We excited an InGaAs-based strongly-coupled system with short Gaussian pulses and experimentally measured the degree of pulse-wise second-order coherence, as shown in Fig. 4.2a. [For the tuned CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 40

system (red), the correlation statistics are similar to the ones presented in Fig. 1.6b.] However, when the emitter detuning ∆e is increased to 4g, the quality of the photon blockade is increased dramatically. This can be seen in the significantly decreased value of g(2)[0] at the vertical red line compared to the vertical blue line. This provides experimental evidence for our discussion in Sec. 4.1 of how the emitter detuning modulates the effective anharmonicity of the system. (We note that in this set of experiments the cross-polarized suppression was not optimized for large detunings so in the tunneling region, where less light is emitted, the statistics were dominated by unwanted coherently scattered light [20]). To ensure that we fully understand the experimental behaviors, we discuss a numerical model for capturing the observed trend in photon blockade. As mentioned in Sec. 1.5, the measurements of the degree of second-order coherence are performed in a pulsed manner, and hence we must adjust our master equation accordingly. Specifically, the driving Hamiltonian changes to Hdrive = E(t) a eiωLt + a†e−iωLt, where E(t) is the time-dependent driving strength. As before, we use the same rotating frame transformation to remove the time-dependence of the excitation term so that † H˜drive(t) = E(t) a + a and written as a super-operator

h i L˜drive(t)˜ρ(t) = i ρ˜(t), H˜drive(t) . (4.8)

2 2 q4 −t /2τp 2 Typically, a Gaussian pulse represents experiment, where E(t) = E0e / πτp and τp = √ 2 τFWHM/2 ln 2 is the Gaussian pulse parameter with τFWHM as the pulse width of |E(t)| . The overall system evolution is governed by the Liouvillian L˜(t) = L˜JC + L˜drive(t). This Liouvillian can be used to calculate the measured degree of second-order coherence in two ways. The ﬁrst is to unravel L˜(t)˜ρ(t) into a quantum trajectory equation, approximate the expected photocount distribution Pm over the entire pulsed emission from an ensemble of trajectories, and estimate g(2)[0] [75]. The second way is to use a time-dependent form of the quantum regression theorem (Sec. 2.3) to calculate

R T R T 0 † † 0 0 (2) dt dt hT−[a (t)a (t )]T+[a(t )a(t)]i G [0] g(2)[0] = 0 0 ≡ , (4.9) 2 2 R T † E[m] 0 dt ha (t)a(t)i where the time range 0 → T encompasses the entire emission pulse and the operators T± indicate the time-ordering required of a physical measurement (operators with higher time indices towards the center of the expression) [50]. The second method is typically more computationally eﬃcient CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 41

( a ) 2 . 0 ∆ : D o t d e t u n i n g e 1 . 5 0 g

] 4 g 0 [

) 1 0 g

2 (

g 1 . 0

τ = 3 0 p s 0 . 5 F W H M ( b ) 2 . 0

1 . 5 ] 0 [ )

2 (

g 1 . 0

0 . 5 ( c ) 1 . 0 1

P 0 . 5

0 . 0 - 2 0 2 4 6 8 1 0 1 2 L a s e r d e t u n i n g , ∆ω ( g )

Figure 4.2: Detuned photon blockade, experiment and theory. (a) Measured second-order coherence g(2)[0] as a function of laser detuning for both a tuned and detuned strongly-coupled system based on an InGaAs quantum dot. Horizontal black dashed line represents statistics of the incident laser pulses. (b) Theoretical g(2)[0] as a function of laser detuning for one resonant and two detuned strongly-coupled systems. (c) Single-photon detection probability P1 of the photocount distribution Pm, calculated using a quantum trajectory approach. The theoretical model for the simulations is a pure Jaynes–Cummings system with a time-dependent driving term. and has an intellectually satisfying connection to instantaneous two-time correlations. Using this method and a best ﬁt to the Jaynes–Cummings model with no dephasing (yielding

{g/2π, κ/2π} = {10.9 GHz, 10 GHz}), we use the driving strength E0 as a fitting parameter for the observed photon blockade regions (Fig. 4.2b). Just as experimentally measured, the blockade dip grows with increasing detuning from ∆e = 0g to ∆e = 4g, which further supports the pictorial description of detuned blockade from the Sec. 4.1. The simulations do show a difference compared to this intuition for the very large detuning of ∆e = 10g: the blockade dip saturates. With increasing detuning, the oscillator strength of the emitter-like polariton decreases until its emission strength is comparable to off-resonant transmission through the cavity-like polariton. The light from the cavity-like polariton then begins to destroy the photon blockade. Because the emitter-like polariton has a smaller oscillator strength with increasing detuning, one might be concerned with how the efficiency of single-photon generation is affected by changing the emitter detuning [20]. To answer this question, we simulated the probabilities of single-photon CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 42

generation P1 using quantum Monte-Carlo techniques and present these data in Fig. 4.2. Surpris- ingly, the probability for single-photon generation actually increases for small emitter detunings by approximately a factor of two over the case in resonant photon blockade (compare P1 at the red and blue lines). This occurs because the highly dissipative nature of the strongly-coupled system spoils the blockade so completely on resonance, that with increasing detuning there are plenty of multi-photon photocounts that can be suppressed and converted to single-photon counts. Of course, this effect wears out with large enough detuning and the P1 for 10g is noticeably less efficient. Thus, emitter detuning has been shown both intuitively, experimentally, and theoretically to be a valuable mechanism for enhancing nonclassical light generation for strongly coupled systems. We make two brief comments on the figure:

1. In the photon tunneling regions, the bunching values of g(2)[0] are largest when the minimum amount of light is transmitted. This can easily be seen by comparing the maximum simulated

bunching in the blue curve to the local minimum value of P1 near the blockade region. Such bunching behavior is consistent with having a photocount distribution that has multi-photon components that are emphasized over a coherent state, but additionally has a large vacuum

component [51]. For instance, although the photocount distribution P2 = 1 anti-bunches with (2) (2) g [0] = 0.5, the distribution {P0,P2} = {0.75, 0.25} bunches with g [0] = 2. This discussion further suggests that a highly dissipative Jaynes–Cummings system by itself is not necessarily ideal for the generation of multi-photon states.

2. The modeled photon blockade is actually weaker for the ideal Jaynes–Cummings system than experimentally measured, even for arbitrarily low powers. Thus, even though the theory and experiment do not perfectly match, we trust the strength of our general arguments in this section. This additionally suggests missing physics from the model of the solid-state strongly- coupled system, which will be thoroughly discussed in the latter sections of this chapter.

4.3 Excitation pulse length

As has recently been shown in depth [54], the length of an excitation pulse is critically important to optimizing emission from photon blockade. Specifically, optimizing photon blockade in a detuned strongly-coupled system by changing the pulse length is a careful trade-off between avoiding excitation of the higher rungs for short pulses or re-excitation of the first polariton for long pulses. Here, we show experimental results from an InGaAs-based system in Ref. [54] (Figs. 4.3a and 4.3c) CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 43

( a ) 2 . 2 ( b ) 2 . 2 L P 1 U P 1 L P 1 U P 1 1 . 8 1 . 8

] 1 . 4 ] 1 . 4 0 0 [ [ ) ) 2 2 ( (

g 1 . 0 g 1 . 0 ∆ = 4 g ∆ = 4 g 0 . 6 e 0 . 6 e τ τ F W H M = 3 4 p s F W H M = 3 4 p s 0 . 2 0 . 2 - 2 - 1 0 1 2 3 4 5 6 - 2 - 1 0 1 2 3 4 5 6 L a s e r d e t u n i n g , ∆ω ( g ) L a s e r d e t u n i n g , ∆ω ( g )

( c ) 2 . 2 ( d ) 2 . 2 L P 1 U P 1 L P 1 U P 1 1 . 8 1 . 8

] 1 . 4 ] 1 . 4 0 0 [ [ ) ) 2 2 ( (

g 1 . 0 g 1 . 0 ∆ = 4 g ∆ = 4 g 0 . 6 e 0 . 6 e τ τ F W H M = 1 1 0 p s F W H M = 1 1 0 p s 0 . 2 0 . 2 - 2 - 1 0 1 2 3 4 5 6 - 2 - 1 0 1 2 3 4 5 6 L a s e r d e t u n i n g , ∆ω ( g ) L a s e r d e t u n i n g , ∆ω ( g )

Figure 4.3: Pulse-length dependence of photon blockade. Experimentally measured degrees of second-order coherence g(2)[0] as a function of laser detuning for a detuned strongly-coupled system based on an InGaAs quantum dot under short (a) and long (c) pulses. Theoretical g(2)[0] as a function of laser detuning for a detuned strongly-coupled system under (b) short and (d) longer pulses. The theoretical model for the simulations is a pure Jaynes–Cummings system with a time-dependent driving term. Horizontal black dashed lines represent statistics of the incident laser pulses. and discuss their comparison with simulated photon blockade in a pulsed regime from a Jaynes– Cummings system (Figs. 4.3b and 4.3d). The blockade dip has two important characteristics: its depth and its width. Here, the better pulse length of the two is seen to be 110 ps since this pulse length better optimizes between higher rung excitation and lower rung re-excitation. Regarding the blockade dip, it is widest for short pulses but narrowest for long pulses; because the detuned polariton has a relatively long lifetime, the width of the blockade dip is roughly determined by the spectral width of the laser pulse. These trends are clearly observable in both experiment and theory. Although the trends of the blockade dips are again reproduced well by the pulsed Jaynes– Cummings model, the agreement is only qualitative. For instance, the blockade dips are again smaller in simulation than in experiment, but now shown for multiple pulse lengths. This suggests that the disparity is not simply an error in the pulse length but rather additional physics. Especially in this set of experimental data where the cross-polarized setup was better optimized for suppression than in Fig. 4.2a, the simulated tunneling regions quite poorly agree with the experimental data. While we supposed initially that the experimental diﬀerence was potentially an imprecision in our ability to accurately determine the laser detuning or an experimental drift, further investigation has CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 44

( a ) 9 2 1 . 4 ( b ) ( c ) D o t l i f e t i m e 1 0 n s ( s u p p r e s s e d ) 8 0 L P 1 U P 1 ) 9 2 1 . 6 C a v i t y l i f e t i m e 9 0 0 2 2 K m ) ) n

( e n e r g y ) 8 p s s s

( 6 0

p p ( ( h

t e e g 6 0 0 n 9 2 1 . 8 P h o n o n s m m i i e t t

l 4 0 e e e f f i i v L L a

W 9 2 2 . 0 P o l a r i t o n 3 0 0 2 0 l i f e t i m e 1 6 p s 3 1 K 9 2 2 . 2 0 0 3 5 3 4 3 3 3 2 3 1 3 0 2 9 2 8 2 7 - 1 2 - 6 0 6 1 2 0 3 6 9 1 2 T e m p e r a t u r e ( K ) ∆ ∆ E m i t t e r d e t u n i n g , e ( g ) E m i t t e r d e t u n i n g , e ( g )

Figure 4.4: Effects of electron-phonon interaction on polariton lifetimes. (a) Annotated avoided- crossing plot that matches transmission spectra to measured lifetimes. (b) Lifetimes as calculated from the cavity and dot dissipation rates alone in the Jaynes–Cummings model. (c) Measured lifetimes from a typical solid-state cavity QED system, which are an order of magnitude shorter for large detunings than those predicted by a pure Jaynes–Cummings model. shown the difference to be the result of an unexpected interference effect. We will discuss this effect later.

4.4 Electron-phonon interaction

In this section we discuss the first true non-ideality to our model for the strongly-coupled systems: the effect of phonons on solid-state cavity QED platforms. Any subsystem embedded within a solid- state environment, at any temperature, has the potential to feel the effects of interactions with the phonon bath. In particular, the electron-phonon interaction between solid-state emitters and their environment can lead to a variety of incoherent phenomena—the physically large size of quantum dots lends well towards coupling of only acoustic phonons [103]. In this section, we discuss how phonon coupling manifests in single-emitter cavity QED systems using an InGaAs-based device. First, consider the avoided crossing spectrum of a typical InGaAs-based system [54], reproduced in Fig. 4.4a. Now, the spectra are annotated according to the state lifetimes. The measured cavity lifetime is 1/κ = 8 ps, while the suppressed dot lifetime is expected to be approximately 10 ns, and hence the maximally entangled polariton has a lifetime of 16 ps since it is half electronic and half photonic character. As a function of the emitter detuning, the expected lifetimes of the LP1 and UP1 polaritons from the Jaynes–Cummings model are plotted in Fig. 4.4b. However, from experimental measurements performed on a streak camera the lifetimes are an order of magnitude shorter for large detunings (Fig. 4.4c). This difference results from the electron-phonon interaction in solid-state cavity QED systems. CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 45

While the primary effect of electron-phonon interaction for quantum dots embedded in bulk GaAs is to generate a power-dependent damping, the effect in a cavity QED system is dramatically different. Because only acoustic phonons couple to the dots, there is an arbitrarily small density of phononic states to couple to under weak driving in bulk. However, in a cavity QED system the dressed ladder provides a constant energy difference between polaritons whereby the electron-phonon interaction samples the density of states, resulting in constant phonon emission and absorption. Three models have primarily been used to explore this effect in the solid-state:

1. Non-markovian models such as a path-integral form of the system dynamics [108]. This formalism includes phonon eﬀects to all orders.

2. The polaron master equation, which is a powerful formalism for including phonon eﬀects to all orders in a Markovian model [103].

3. The effective phonon master equation, which only includes first-order phonon effects [54, 109]. This model is by far the simplest, but readily captures most experimental effects observed thus-far in solid-state cavity QED systems.

For all of the nonclassical light generation phenomenon we have observed thus far, we have found the effective phonon master equation sufficient, and we will briefly review its findings here. For a cavity-driven system only (driven by Hdrive), then the effects of phonons are captured by the addition Γf † Γr † of two incoherent dissipators 2 D[a σ] and 2 D[σ a], i.e. with the addition of the new Liouvillian term Γ Γ L˜ = f D[a†σ] + r D[σ†a] (4.10) phonon 2 2 where Γf and Γr stand for forward and reverse phonon transfer rates, respectively. At the operating temperatures of 25 K and small emitter detunings used in our experiments, these rates are each −1 approximately Γf,r ≈ (80 ps) . Importantly, these rates vary with detuning and fall off rapidly at large detunings; full details for the extracted rates can be found in Ref. [54]. Because the dissipators in equation (4.10) transfer excitations between the cavity and the dot, at large detunings they result in an effective transfer of population between polaritons. The transfer rates then dominate the lifetime of the emitter-like polariton for large detunings because the system emits through a phonon-induced transfer (Fig. 4.4a red arrows) and cavity emission over dot emission. Specifically, the phonon-involved pathway has a lifetime of ≈ 80 ps + 8 ps, which is much shorter than the dot’s spontaneous lifetime of ≈10 ns. CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 46

(a) (b)

Energy (arb. units) (arb. Energy Intensity (arb. units) (arb. Intensity

921.2 921.6 922.0 -4 -2 0 2 4

Wavelength (nm) Detuning (g)

2500 300

(2) (2)

g (0) g (0)

D D

2000

200

1500

1000

100 Intensity (counts) Intensity (counts) Intensity 500

0 0

-60 -30 0 30 60 -60 -30 0 30 60

Time delay (ns) Time delay (ns)

(e) (f) 300

(2) (2)

g (0) g (0)

D D

1200

200

800

100

400 Intensity (counts) Intensity (counts) Intensity

0 0

-60 -30 0 30 60 -60 -30 0 30 60

Time delay (ns) Time delay (ns)

Figure 4.5: Photon-blockade statistics illustrating phonon-induced polariton transfer in Ref. [58]. (a) Spectrum of the strongly coupled system at ∆ = 3.5g and resonant excitation of UP1 with a 25 ps long pulse. (b) Illustration of the JC-ladder. The excitation laser is resonant with UP1 (depicted with a solid blue upward arrow) but not resonant with higher climbs up the ladder. Following excitation of UP1, possible recombination channels are from UP1 to the ground state (solid blue downward arrow) or from LP1 to the ground state (dashed red arrow) via a phonon-assisted population transfer from UP1 to LP1 (curved orange arrow). (c-f) Correlation measurement of the (c) unfiltered signal, revealing g(2)[0] = 0.263 ± 0.008, (d) filtered emission from UP1 (indicated in blue on the left in (a), revealing g(2)[0] = 0.162 ± 0.016, (e) filtered emission from LP1 [indicated in red on the right in (a)], revealing g(2)[0] = 0.063 ± 0.010 and (f) cross-correlation between UP1 and LP1, revealing g(2)[0] = 0.079 ± 0.018. CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 47

The polariton transfer effect can be directly observed in both streak camera images [54] and in terms of frequency filtered photon correlations [58]. Here, I present one example from our work showing how the transfer manifests in photon blockade (Fig. 4.5). A typical transmission spectrum for detuned blockade is presented in Fig. 4.5a. The signal is composed of two main contributions: emission from the resonantly excited UP1 (schematically illustrated by a solid blue arrow) and phonon-assisted emission from LP1 (schematically illustrated as a dashed red arrow). The relative intensities of UP1 and LP1 are mainly given by the ratio of the (detuning dependent) radiative transition rate ΓUP1 and the phonon-assisted transfer rate Γf (illustrated by a curved orange arrow). We investigated the quantum character of the emission through measurements of the degree of second-order coherence, and in particular by frequency filtering different components of the emission. The experiments all show highly anti-bunched emission, where filtered emission from either UP1 or LP1 antibunches independently. Further, a photon cross correlation measurement (Fig. 4.5f) shows that emission from UP1 and LP1 are anti-correlated. These measurements together definitively show that once the UP1 is excited, it can decay either by emitting one photon or by emitting one photon and one phonon, hence illustrating the phonon transfer process. At this point, I briefly comment that although we thoroughly investigated frequency filtered correlations both experimentally and theoretically, including demonstrations of highly indistinguishable photon generation [55, 58], I will generally avoid discussions of frequency filtering and photon distinguishability in this thesis. These concepts require their own theoretical framework [50, 110] which is a bit different than the main themes in my thesis.

4.5 Blinking of the quantum emitter

While we hope the quantum emitter behaves as an ideal two-level system and is described by a single dipole operator σ, in reality it possesses a very complicated level structure. Consider InGaAs quantum dots: when placed in a high-quality electrical diode to control their precise ground states through controlling the local charge environment, they have been shown to behave as nearly ideal two-level systems [45]. However, this technology was later introduced into devices with planar photonic crystal cavities [31, 32, 111], and therefore, much of the work on solid-state strongly- coupled systems still shows residual effects of a slightly unstable charge environment. Because the different charge configurations of the quantum dot have different binding energies, and hence emission frequencies, when the quantum dot sequentially absorbs a charge from the environment [104] its resonant frequency is highly detuned from the cavity and the strong coupling is no longer CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 48

visible [49, 51]. During this time, the system behaves like the bare cavity by itself and coherently scatters the incident light with a Lorentzian lineshape. When this process periodically occurs, slowly modulating the transmission spectrum between that of a resonant strongly-coupled system and a bare cavity, the phenomenon is referred to as blinking.

4.5.1 Eﬀects of blinking on transmission spectra

While it’s certainly possible to provide a complete description of the charge states of a quantum emitter, in many situations it is suﬃcient to model the system as behaving as the bare cavity for some fraction of the time fblink and like a strongly-coupled system for the rest. The statistical independence of these two situations holds because the blinking timescales are at least 10’s of nanoseconds [112], while the emission timescales are 10’s of picoseconds. When the system blinks, we calculate the steady-state density matrix by solving

˜ ˜ 0 Lblink(ωL) + Ldrive ρ˜ss(ωL) = 0, (4.11) where the bare cavity dynamics are represented by

˜ 0 0 † Lblinkρ˜ss = i ρ˜ss, (ωa − ωL) a a . (4.12)

† 0 Then, the transmission spectrum while blinking is Sblink(ωL) = ha ai(ωL), usingρ ˜ss(ωL). To incorporate the experimental strongly-coupled system that is often modeled as a Jaynes– Cummings system with phonons, we calculate the steady-state density matrix by solving

˜ ˜ ˜ LJC(ωL) + Lphonon + Ldrive ρ˜ss(ωL) = 0. (4.13)

Usingρ ˜ss(ωL), we calculate the transmission spectrum of the strongly-coupled system Ssc(ωL) = † ha ai(ωL). Finally, we weight and combine the two transmission proﬁles to obtain

ST(ωL) = (1 − fblink) Ssc(ωL) + fblinkSblink(ωL). (4.14)

With this quantum-optical model, we are ready to ﬁt realistic transmission spectra with ST(ωL). Experimental transmission spectra from an InGaAs quantum dot strongly coupled to an L3 photonic crystal cavity mode are shown in Fig. 4.6; in Fig. 4.6a the system is tuned nearly in resonance, and in CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 49

( a ) ( b ) 1 . 0 1 . 0 J C + p h o n o n s + b l i n k i n g L P 1 U P 1 L P 1 J C + b l i n k i n g J C

) ) b l i n k i n g . . b b r r a a ( (

y y t t i 0 . 5 i 0 . 5 s s n n e e t t n n I I ∆ ∆ e = 0 g e = 5 . 8 g τ U P 1 τ F W H M = C W F W H M = C W

0 . 0 0 . 0 - 5 0 5 - 5 0 5 1 0 L a s e r d e t u n i n g , ∆ω ( g ) L a s e r d e t u n i n g , ∆ω ( g )

Figure 4.6: Transmission spectra from an InGaAs quantum dot strongly coupled to an L3 photonic crystal cavity mode, taken in cross-polarized reﬂectivity. (a) Spectrum at nearly zero dot detuning. (b) Spectrum at a large dot detuning. The spectra were taken using weak broad-band diode, which is equivalent to transmission in the linear regime of cavity QED. Simulated spectra were convolved with the response function of the spectrometer (linewidth ΓFWHM/2π = 4.5 GHz), which is why the spectra for pure Jaynes–Cummings transmission do not dip to zero between LP1 and UP1.

Fig. 4.6b the system is tuned signiﬁcantly oﬀ resonance. In this model, we used the phonon-induced dot-cavity transfer rates extracted from the data in Fig. 4.4, while g, κ, and fblink were taken as

fitting parameters. From the fits, values of {g/2π, κ/2π} = {9.2 GHz, 12.3 GHz} and fblink = 0.09 were extracted; by fitting one spectra on- and one spectra off-resonance κ and g can be extracted almost independently of one another. We discuss several interesting features in these transmission spectra. Building the spectra component by component, we first show the blinking spectra and pure Jaynes–Cummings spectra as the dashed green and blue lines, respectively. The simulated spectra were convolved with the response function of the spectrometer (linewidth ΓFWHM/2π = 4.5 GHz), which is why the spectra for pure Jaynes–Cummings transmission do not dip to zero between LP1 and UP1. Then, the two spectra were added together (cyan lines) to show the effects of the blinking term: blinking decreases the visibility of the strong coupling dip in the resonant case and increases the height of the cavity-like polariton in the off-resonant case. Finally, we added the effects of electron-phonon interaction to fully model our strongly-coupled system. The effect of phonons is to decrease the depth of the transmission dip in the resonant case due to the additional incoherent dephasing. With this dephasing also comes a small increase in the linewidths of the polaritons. Meanwhile in the off-resonant case, the effect of the electron-phonon interaction is to reduce the lifetime of the emitter-like polariton (small peak) so that the polariton can emit at a faster rate, resulting in a higher count rate than without the interaction. CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 50

4.5.2 Eﬀects of blinking on nonclassical light generation

Although the effects of blinking resulted in simply adding the strongly-coupled and blinking spectra together, this approach is not sufficient for photon statistics. To understand the effect of blinking on nonclassical photon statistics, consider the standard setup for measuring pulsed g(2)[0], as thoroughly discussed in Ref. [50]. Using a Hanbury–Brown and Twiss interferometer operated in a pulsed manner a binned temporal coincidence histogram is built up, with peaks separated by the pulse reptition rate τr, i.e. hHBT[nτr] where n ∈ {Z ≥ 0}. The measured degree of second-order coherence is estimated by taking the ratio h [0] gˆ(2)[0] = HBT , (4.15) hHBT[τr] where each histogrammed time-bin hHBT[nτr] is an independent and binomially-distributed random variable. Here, we additionally consider the effects of blinking. Often the blinking time-scale is very long compared to τr, where the counts due to transmission through the strongly-coupled system or the blinking system are modeled by adding their individual histograms together, i.e. hHBT[nτr] = hSC[nτr] + hblink[nτr]. Hence, h [0] + h [0] gˆ(2)[0] = SC blink . (4.16) hSC[τr] + hblink[τr] Because the statistics of the transmitted light are directly inherited from the laser when the system 2 2 R T † blinks, hblink[nτr] ∝ hNblinki = 0 dt ha (t)a(t)i and is calculated using the Liouvillian L(t) =

L˜blink + L˜drive(t). Then, in terms of the instantaneous correlations of the system and by extension of Eq. 4.9 (2) 2 2 (2) (1 − fblink) gsc [0]hNsci + fblinkhNblinki g [0] = 2 2 , (4.17) (1 − fblink) hNsci + fblinkhNblinki

(2) where hNsci and gsc [0] are calculated using the Liouvillian

Lsc(t) = L˜JC + L˜phonon + L˜drive(t). (4.18)

The strongly-coupled system examined in this section had a relatively low fraction of blinking time, at only fblink = 9 %. However, many quantum dots may have higher blinking fractions due to variation in their local charge environments. Here, we explore the effects of stronger blinking, both on the transmission spectrum and on the second-order coherence statistics, as shown in Fig. 4.7. Consider the transmission spectra (Fig. 4.7a): the doublet that signifies strong coupling dis- appears with increasing blinking. Such a result is easily controlled in theory via changing fblink, however, here we modified the local charge environment with a broad-band diode. While collecting CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 51

( a ) ( b ) 2 . 0 ∆ ∆ e = 0 g e = 0 g

τ f b l i n k = 0 τ F W H M = C W F W H M = 3 0 p s 1 . 0 1 . 5 0 . 0 9 ] n [ 0 . 3 E

]

, 0 . 5 0 y [ t

) 0 . 7 i

2 1 . 0 (

s 0 . 9 g n e t 0 . 5 n I 0 . 5

L P 1 U P 1 L P 1 U P 1 0 . 0 0 . 0 - 2 - 1 0 1 2 - 2 0 2 L a s e r d e t u n i n g , ∆ω ( g ) L a s e r d e t u n i n g , ∆ω ( g )

Figure 4.7: Effects of blinking on a resonant strongly-coupled system. (a) Normalized CW transmission spectra with increasing blinking rate (black to orange). (b) Pulse-wise second-order coherence g(2)[0] as a function of laser detuning with increasing blinking rate. Horizontal black dashed line represents statistics of the incident laser pulses. information about the transmission spectrum, the diode power was increased in order to encourage blinking in the system. Note: this effect is different than saturation of the strongly-coupled system, which also destroys the signature of strong coupling. As a Jaynes–Cummings system is driven into saturation, the separation between the doublet decreases until the spectra is comparable to that of the bare cavity, while no such linewidth narrowing is present here [22]. Of course, if the g/κ ratio were larger then the addition of blinking would be easy to identify as a triplet in the transmission spectrum [113]. Meanwhile in the second-order coherence statistics (Fig. 4.7b), blinking most strongly affects the tunneling region because this is where the blinking spectrum is maximized, i.e. at the bare cavity frequency. Here, the effect to push down the g(2)[0] near zero laser detuning manifests itself as a broader tunneling peak and even a slight dip at zero detuning. For the case of the resonant system, the agreement between experiment and theory is excellent. Blinking has a large influence due to the spectral proximity of the strongly-coupled system’s emission to the bare cavity peak. By extension, it has negligible effect on the detuned spectra and cannot be used to explain the disparity between the detuned tunneling experiments and theory in Sec. 4.2. Finally, the disparity in the blockade region of Fig. 4.7b will be addressed in the next section. CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 52

4.6 Self-homodyne interference

Previously, we discussed how both the experimental blockade and tunneling data seemed to reveal stronger nonclassical correlations than an ideal Jaynes–Cummings model would predict. In this section, we explain the missing physics that allows for better performance through enhancement of the nonclassical light emission [52, 56, 57]. It is very tempting to model the physics of an L3 photonic crystal cavity (Fig. 4.8a) as a single mode of a harmonic oscillator, i.e. with a single Heisenberg mode operator a(t) which we will refer to as the discrete scattering channel. Under certain cross- polarized reflectivity conditions, such as those used in Fig. 4.8b, this model accurately captures the system dynamics. In this scenario, the detuned transmission profile nearly resembles the bare cavity’s Lorentzian profile at high excitation powers. Here, an experimental transmission spectrum through a strongly-coupled system is shown and fitted with the same quantum optical model as in Fig. 4.6 (red line). The green and blue decompositions will be discussed later. However, the L3 photonic crystal has a rich mode structure, as shown in Fig. 4.8c, that is not necessarily well-approximated by a single mode operator a(t). Instead, a better approximation is to consider an additional scattering channel that is due to a roughly constant background density of photonic states; this scattering pathway is referred to as the continuum channel. The discrete and continuum channels can interfere with one another to generate a lineshape that is closely related to a Fano resonance, and this is modeled with the operator A(t) → a(t) + α(t) instead of just a(t) where α(t) = αE(t)/E0, E(t) is again the Gaussian pulse, and α is a c-number [56, 57]. The operator α(t) represents the laser light reflected into the cross-polarized output channel via the continuum modes of the photonic crystal. The lineshape can be changed between Lorentzian-like or Fano-like in cross-polarized reflectivity by altering the focal spot size and the excitation/detection polarizations, an effect which has been theoretically verified in L3 photonic crystal cavities through a rigorous scattering-matrix formalism [114]. While the difference between these two transmission profiles may initially appear small from a comparison between these two lineshapes, the effects on nonclassical light generation are dramatic and manifest in the grey boxed regions that represent the frequency of the emitter-like polariton. In fact, the mixing action of combining the reflected laser light with light scattered by the strongly- coupled system is a type of homodyne measurement, which has the power to emphasize the incoherent or nonclassical portion of the scattered light over the coherent or classical portion. Because the mixing occurs at the level of the photonic crystal, this effect is named a self-homodyne interference (SHI) [56]. To further explore this point, the transmission profiles in Figs. 4.8b and 4.8d were CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 53

( a ) ( b ) 1 . 0 0 I t o t I

) c o h

. 0 . 7 5 b

r I i n c a (

y t

i 0 . 5 0 s n e t

n ∆ I 0 . 2 5 e = 2 . 6 g

τ = C W F W H M 0 . 0 0 - 1 5 - 1 0 - 5 0 5 1 0 1 5 L a s e r d e t u n i n g , ∆ω ( g ) ( c ) ( d ) 1 . 0 0 )

. 0 . 7 5 b y r c a ( n

e y t u i 0 . 5 0 s q ∆ = 2 . 6 g n e e r e t F τ n F W H M = C W I 0 . 2 5

0 . 0 0 Γ Μ Κ Γ - 1 5 - 1 0 - 5 0 5 1 0 1 5 L a s e r d e t u n i n g , ∆ω ( g )

Figure 4.8: Origin of self-homodyne interference. (a) Schematic of a planar L3 photonic crystal cavity. (b) Transmission spectra under excitation conditions that produce a Lorentzian-like profile, fit to a quantum optical model. (c) Complicated mode structure of a planar L3 Photonic crystal cavity, calculated using the MIT Photonic-Bands (MPB) package. Red and black horizontal lines depict the cavity’s fundamental and higher order modes, respectively. Curved lines represent photonic crystal guided modes. Grey region indicates leaky modes that are above the light line. (d) Transmission spectra under excitation conditions that produce a Fano-like profile, fit to a quantum optical model. Simulated spectra were convolved with the response function of the spectrometer

(linewidth ΓFWHM/2π = 4.5 GHz). decomposed into their incoherent and coherent portions of emission. The coherent portions (blue) are primarily due to the classically scattered light from a subset of almost harmonically spaced dressed states or the continuum modes and hence look predominantly like the Lorentzian or Fano lineshapes. † This light is due to the mean of the electric field, i.e. Icoh ∝ ha ihai. Meanwhile, the incoherent portions of the emissions (green) are the result of the nonlinearity in the Jaynes–Cummings system and hence from the nonclassically scattered light. This light is due to the fluctuations of the electric † † field, i.e. Iinc ∝ ha ai − ha ihai. Now, we revisit Figs. 4.8b and 4.8d and compare the coherent and incoherent portions in the insets. Under the Lorentzian-like conditions (Fig. 4.8b), the coherent portion of the transmitted light (blue) dominates the incoherent portion (green). However, under the Fano-like conditions (Fig. 4.8d), the incoherent portion of the transmitted light dominates and over 90 % of the coherently CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 54

scattered portion is suppressed at the frequency of the emitter-like polariton. In this way, the eﬀect of the quantum nonlinearity in the Jaynes–Cummings ladder is emphasized, with great potential to allow highly dissipative systems to still exhibit robust signatures of nonclassical light generation, as discussed in Ch. 3 theoretically.

4.6.1 Eﬀects of self-homodyne interference on emission spectra

As our first experimental example of the dramatic influence of self-homodyne interference on em- phasizing nonclassical light generation, we review data from Ref. [56]. In Fig. 4.9, we consider the effect of self-homodyne interference on the spectra of pulsed resonance fluorescence from a solid-state strongly-coupled system. The ideal spectrum of pulsed resonance fluorescence for an operator A(t) is calculated with ZZ S(ω) = dt dτ e−iωτ hA†(t + τ)A(t)i. (4.19) 2 R The spectrum for resonance fluorescence from a Jaynes–Cummings system (even with phonons) is given when A(t) → a(t) and the spectrum with self-homodyne interference is given when A(t) → a(t) + α(t). The two-time correlations are again computed with a time-dependent version of the quantum regression theorem (Sec. 2.3), using the system Liouvillian

Lsc(t) = L˜JC + L˜phonon + L˜drive(t). (4.20)

Note: although we do not include blinking in this subsection, it has very minimal effects on the results since the dot is detuned from the bare cavity. In the experiment, the focal conditions were first tuned to the Lorentzian-like conditions (Fig. 4.9a) and we then excited the emitter-like polariton (denoted by the grey box) with a high-power τFWHM = 100 ps pulse. The spectrum of resonance fluorescence was measured on a spectrometer and is shown in Fig. 4.9b. Notably, the spectrum simply shows a single peak with a linewidth determined by the laser pulse (black). Simulating the spectrum with a Jaynes–Cummings model including phonon effects, i.e. with the Liouvillian Lsc(t), the singly peaked spectrum can be reproduced (red). Like the transmission spectra, the fluorescence spectrum can also be decomposed into the coherent and incoherent potions by making the replacement of hA†(t + τ)A(t)i → ha†(t + τ)iha(t)i and hA†(t + τ)A(t)i → ha†(t + τ)a(t)i − ha†(t + τ)iha(t)i, respectively, in equation (4.19). These decompositions are shown in the blue and green lineshapes, and they show that the coherently scattered light completely dominates the emission spectrum at high powers. Again, we emphasize that this CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 55

τ = C W E x p e r i m e n t S i m u l a t i o n τ = 1 0 0 p s ( a ) F W H M ( b ) F W H M 1 . 0 ∆ 1 . 0 ∆ e = 2 . 6 g e = 2 . 6 g I c o h ) ) s . . n b b r r o a a n ( (

o y y t t h

i 0 . 5 i 0 . 5 p s s

n n +

e e t t C n n I I J 0 . 0 0 . 0 - 1 0 - 5 0 5 1 0 - 3 0 3 - 3 0 3 - 3 0 3 6 L a s e r d e t u n i n g , ∆ω ( g ) E m i s s i o n f r e q u e n c y ( g )

e ( c ) ( d ) c

n 1 . 0 ∆ 1 . 0 ∆ e = 2 . 6 g e = 2 . 6 g e I i n c r ) ) e . . f b b r r r e a a t ( (

n i y y

t t

i 0 . 5 i 0 . 5 e s s n n n y e e t t d n n o I I m

o 0 . 0 0 . 0 h - f l

e - 1 0 - 5 0 5 1 0 - 3 0 3 - 3 0 3 6 - 3 0 3 6 s ∆ω

+ L a s e r d e t u n i n g , ( g ) E m i s s i o n f r e q u e n c y ( g )

Figure 4.9: Effect of self-homodyne interference on resonance fluorescence from a dissipative strongly- coupled system. (a) Transmission profile under excitation conditions that produce a Lorentzian- like profile. (b) Experimental and simulated resonance fluorescence under drive on the emitter- like polariton [grey box in (a)] by a high-power τFWHM = 100 ps pulse. (c) Transmission profile under excitation conditions that produces a Fano-like profile. (d) Experimental and simulated resonance fluorescence under drive on the emitter-like polariton [grey box in (c)] by a high-power

τFWHM = 100 ps pulse, including self-homodyne interference. (a-d) Quantum optical ﬁts in red, and decomposition into coherent (Icoh) and incoherent (Iinc) components in blue and green, respectively. All simulated spectra were convolved with the response function of the spectrometer (linewidth

ΓFWHM/2π = 4.5 GHz). is a standard feature of highly dissipative but strongly-coupled Jaynes–Cummings systems, that well- known cavity QED effects can be unobservable! On the other hand, the incoherent portion shows a very interesting quadruplet structure that is closely related to the Mollow triplet, a hallmark of quantum-mechanically scattered light. This quadruplet structure was discussed thoroughly in Ref. [56] and its origins will not be discussed here; suffice to say that it is a structure arising from the nonclassical light emission of the cavity QED system. In order to experimentally observe the quadruplet, self-homodyne interference was critical. We next tuned to the Fano-like conditions (Fig. 4.9b) and again excited the emitter-like polariton (denoted by the grey box) with the same high-power τFWHM = 100 ps pulse. Now, the optimally tuned self-homodyne interference experimentally reveals the interesting quadruplet structure, which is completely unobservable otherwise. The interference was also included theoretically in simulation to reveal the quadruplet (Fig. 4.9). We make two practical notes here regarding the modeling of these experiments. First, an entire power-series of spectra was necessary to accurately fit excitation CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 56

powers (see Fig. 3 in Ref. [56]). Second, the experimental data shows a better defined quadruplet, with more energy in the sidebands than the model. We expect this difference is due to an inaccuracy of the effective phonon master equation at large driving powers.

4.6.2 Eﬀects of self-homodyne interference on nonclassical light generation

As our second application of self-homodyne interference, we consider its effect on nonclassical light generation. As we saw in the Sec. 4.6.1, an optimally tuned self-homodyne interference (SHI) has the ability to remove unwanted coherently scattered light from the system (discussed theoretically in Ch. 3). On a similar principle, we would expect the interferometric technique to be capable of enhancing nonclassical light generation by isolation of the quantum-mechanical signal. In prior work, we explored the precise mechanism for how SHI is capable of enhancing both single- and multi- photon emission in dissipative Jaynes–Cummings systems [52, 57]. In this section, we will examine the enhancement under pulsed excitation and in doing so will almost perfectly fit the experimental data from Refs. [20] and [58]. First, we discuss our complete model that we believe captures nearly all experimental effects relevant for photon blockade and photon tunneling. It is again based off of the system Liouvillian

Lsc(t) = L˜JC + L˜phonon + L˜drive(t), (4.21) but now we calculate g(2)[0] with a self-homodyne interference. Speciﬁcally, that means to calculate

R T R T 0 † † 0 0 (2) dt dt hT−[A (t)A (t )]T+[A(t )A(t)]i G [0] g(2)[0] = 0 0 ≡ AA , (4.22) 2 2 R T † hNAi 0 dt hA (t)A(t)i where A(t) → a(t) + α(t) and again α(t) = αE(t)/E0. To incorporate blinking, we simply change Eq. 4.17 into (2) 2 (2) (1 − fblink) GAA[0] + fblinkhNblinki g [0] = 2 2 . (4.23) (1 − fblink) hNAi + fblinkhNblinki

R T † ˜ ˜ Now, hNblinki = 0 dt hA (t)A(t)i and is calculated using the Liouvillian L(t) = Lblink + Ldrive(t); again, A(t) → a(t) + α(t). With the necessary theoretical machinery established, we are now ﬁnally ready to fully model CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 57

( a ) 2 . 0 ( b ) 3 . 0 ∆ 0 . 9 ∆ e = 0 g e = 0 g τ = 3 0 p s f = 0 τ = 3 0 p s F W H M b l i n k 2 . 5 0 . 7 F W H M 1 . 5 0 . 0 9 0 . 5 ]

n 2 . 0 0 . 3 [ 0 . 3 E

] 0 . 5 , 0 y [ 0 . 7 0 . 0 9 t ) i

2 1 . 0 1 . 5 (

0 . 9 s g n

e f = 0 t b l i n k n

I 1 . 0 0 . 5 0 . 5

L P 1 U P 1 L P 1 U P 1 0 . 0 0 . 0 - 2 0 2 - 2 0 2 L a s e r d e t u n i n g , ∆ω ( g ) L a s e r d e t u n i n g , ∆ω ( g )

Figure 4.10: Effects of strong emitter blinking on a resonant strongly-coupled system. (a) Second- order coherence g(2)[0] as a function of laser detuning. The ideal fit to g(2)[0] is given by the brown dashed line. Horizontal black dashed line represents statistics of the incident laser pulses. (b) Pulsed transmission spectrum, with quantum simulation only. In both sub-figures, the statistics or intensities are given for different blinking fractions. the g(2)[0] versus laser detuning scans. First, we revisit the data from Fig. 4.7, but with self- homodyne interference, in Fig. 4.10a. The optimal fit is given by the dashed brown line, which almost perfectly matches the data with the addition of an optimized SHI. The exact same trends are visible for blinking with or without the interference, where the tunneling regions are affected first with increasing blinking while the blockade regions are relatively unaffected. Now, we additionally show the simulated transmission plots for the incident pulses (Fig. 4.10b). We emphasize here that when the system blinks the SHI still occurs, and hence the system blinks with a Fano lineshape.

Next, we consider detuned blockade and tunneling with an emitter detuning of ∆e = 3.2g. Comparing the resonant and detuned cases in Fig. 4.11, one can again observe the general trend of enhanced photon blockade with increasing detuning, as was discussed in Sec. 4.2. (We provide a brief technical note that, just as with the resonance fluorescence experiments in the Sec. 4.6.1, before performing any g(2)[0] scan the photon blockade region was optimized. At that point, a transmission experiment would reveal a Fano-like lineshape, incorporating SHI; these interference conditions were held constant over the course of the g(2)[0] scan.) Back to both subfigures, the quantum-optical model is broken down into several different lines, to separate out the effects of blinking, electron-phonon interaction, and self-homodyne interference. The primary effects are as follows:

1. Blinking has the strongest effect in the zero dot detuning case and minimal effect in the detuned case. For zero detuning, the effect of blinking is not just to decrease the maximum achievable CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 58

( a ) 2 . 0 ( b ) 2 . 0 ∆ ∆ e = 0 g e = 3 . 2 g τ τ F W H M = 3 0 p s F W H M = 6 0 p s 1 . 5 1 . 5 ] ] 0 0 [ [ ) )

2 1 . 0 2 1 . 0 ( ( g g

J C + b l i n k i n g ( ) + 0 . 5 J C + b l i n k i n g ( ) + p h o n o n s ( ) + 0 . 5 p h o n o n s ( ) + s e l f - h o m o d y n e i n t e r f e r e n c e ( ) + s e l f - h o m o d y n e i n t e r f e r e n c e ( ) + e f f e c t s o f b l i n k i n g ( ) e f f e c t s o f b l i n k i n g ( , , ) 0 . 0 0 . 0 - 2 - 1 0 1 2 - 1 0 1 2 3 4 L a s e r d e t u n i n g , ∆ω ( g ) L a s e r d e t u n i n g , ∆ω ( g )

Figure 4.11: Perfectly modeled photon blockade and tunneling under pulsed excitation. Second- order coherence g(2)[0] as a function of laser detuning on an (a) resonant and (b) detuned system. The legends discuss the complete decomposition of the quantum-optical model into its constituent eﬀects. Green and blue dashed lines represent the LP1 and UP1, respectively. Horizontal black dashed lines represent statistics of the incident laser pulses. A laser power of 5 nW was used.

value of g(2)[0] in the tunneling region, but also to broaden the width of the tunneling region. We note that interestingly, none of the other tunable parameters are able to control this effect— without blinking, the resonant tunneling region could never be fitted properly. Additionally, the blinking fraction was taken from the fit in Sec. 4.5.

2. Electron-phonon interaction has almost negligible effects on photon blockade for either the resonant or detuned system, because the effect of the dissipation is only to change the frequency and coherence of any emitted single photons [55]. In the tunneling region, phonon-induced transfers have the largest effect on the detuned system since the g(2)[0] values in the tunneling region are much larger due to the decrease in photon transmission, as discussed in Sec. 4.2.

3. Self-homodyne interference has the strongest effect on the detuned system, though it’s certainly important in the resonant case as well. For both system configurations, SHI enables much lower values of g(2)[0] in the blockade regions, thus significantly improving the quality of single-photon emission. Because SHI occurs on one side of the cavity profile or the other, in the resonant case the LP1 blockade is worsened while the UP1 blockade is improved. This evidence of enhanced photon blockade fully suggests that in some previous experiments with strongly-coupled systems where the simulations were surprisingly unable to perform as well as the experiments (e.g., those in Refs. [20], [49], [54], and [115]), researchers may have unknowingly utilized SHI.

To further discuss the eﬀect of SHI, we note that the detuned tunneling region is dramatically CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 59

1 . 2 5 ∆ e = 3 . 2 g τ F W H M = 6 0 p s 1 . 0 0

] 0 . 7 5 0 [ )

2 ( g 0 . 5 0

0 . 2 5 B o u n d f o r J C + p h o n o n s + b l i n k i n g ( ) + i n t e r f e r e n c e ( ) 0 . 0 0 1 1 0 1 0 0 E x c i t a t i o n P o w e r ( n W )

Figure 4.12: Detuned photon blockade as a function of excitation power. Data and fit parameters identical to those in Fig. 4.11. Horizontal black dashed line represents statistics of the incident laser pulses. different with SHI, where the peak of the g(2)[0] scan no longer occurs at the point of minimal photon transmission. Because the unwanted coherently scattered light is removed with SHI, the tunneling region is now a much stronger indicator of the multi-photon processes occurring in the detuned strongly-coupled system [58]. Since no fitting parameter other than SHI was able to enhance the tunneling region, it is clear that SHI plays an important role in nonclassical light generation. Next, we explore detuned photon blockade as a function of excitation power, both by presenting experimental data and with quantum-optical fits (Fig. 4.12). This power-dependent data and fitting is important in verifying that we have not over-fit our experimental data and in reaffirming the strength of the self-homodyne interference to improve photon blockade. In performing the power scan of g(2)[0], the experimental uncertainty in the precise laser detuning is given by the spectrometer (2) linewidth of ΓFWHM ≈ g/3 (though there is little uncertainly in the laser detuning for the g [0] versus laser detuning scans due to the high relative precision of the experimental pulse shaper). Therefore, we have simulated both the minimum g(2)[0] values in photon blockade and the values under a system that was imprecisely tuned by ΓFWHM/2. This procedure pictorially shows the potential uncertainty in the correct laser detuning for the simulated values. Using the optimal fit from Fig. 4.11b, the green simulated blockade region matches almost perfectly with the experimental values. Meanwhile if SHI is excluded, then the blockade values are not just worse at the minimum, but much more sensitive to any possible imprecision in laser detuning. Because both the blockade and tunneling regions are sensitive to the excitation power, the strong fit with experiment helps confirm our complete model of an experimental strongly-coupled system. With such a complex model, one must be wary of over-fitting the data. However, each of the CHAPTER 4. CAVITY QED BEYOND THE JAYNES–CUMMINGS MODEL 60

( a ) ) ( b ) 1 5 0 s t

i M e a s u r e d

n F i t u

s 1 0 0 b t r n a u (

o y t C i P u l s e l e n g t h : 1 6 p s 5 0 s

n ∆: 4 . 8 g e t n I 0 0 1 0 2 0 3 0 4 0 5 0 - 5 0 0 5 0 P o w e r 0 . 5 ( n W ) 0 . 5 T i m e d e l a y ( n s )

Figure 4.13: Strongly coupled system excited by a short laser pulse. (a) Rabi oscillation observed upon exciting UP1. (b) Measurement of the second-order coherence for exciting UP1 with a π pulse yielding g(2)[0] = 0.03 ± 0.01. elements has been independently verified and fitted through a large series of a experimental data to extract the Jaynes–Cummings dissipation, electron-phonon effects, blinking, and self-homodyne interference. Taken as a whole, each of the non-ideal effects almost independently tunes different aspects of the emission statistics and hence we believe we have identified the appropriate number of model parameters.

4.7 Rabi oscillations on a cavity QED polariton

As previously discussed, the fundamental idea behind photon blockade is to use a two-level system comprising the ground state and one polariton of the strongly-coupled Jaynes–Cummings system to filter just one photon from the laser pulse. If the system were performing well one would therefore expect that Rabi oscillations, a characteristic signature of driving a two-level system with a laser pulse, should be visible. Upon applying self-homodyne interference, a small cavity-dot detuning, and frequency filtering the emission on the LP1, Rabi oscillations between the ground state and UP1 can indeed be seen (Fig. 4.13a). These Rabi oscillations can further be used to set a normalized power for experiments, where the first peak in the oscillation should ideally correspond to scenario with the best single-photon emission. This is referred to as a π pulse, since the Rabi oscillations have undergone half a period. In this case, the second-order coherence was measured to be extremely low, resulting in excellent single-photon emission (Fig. 4.13b). Chapter 5

Photon scattering formalism

A central object of study in quantum optics is a finite-dimensional quantum system (e.g., the quantum dots and Jaynes–Cummings systems in the previous four chapters) coupled to a bath with an infinite number of degrees of freedom. In this chapter, we consider the bath more directly, and take it to be a unidirectional waveguide with photonic modes which exchange energy with the quantum system. Historically, the infinite dimensionality of this problem led many to believe that general solutions for the evolution operator of the composite system and hence state vector of the combined low-dimensional plus photonic system were intractable. Thus, quantum optical methods initially focused on the dynamics of the system, i.e. by tracing out the state of the waveguide. Nevertheless, from an important contribution by Gardiner and Collett [116], it is still possible to describe the entire photonic state based on computing all the correlation functions [117, 118] associated with the system’s coupling operators. For example, if the system operator a linearly couples to a waveguide, then the total state of the field could be extracted from the entire family of correlations 0 0 † † 0 0 G(t1, . . . , tn, t1, . . . , tm) = ha (t1) ··· a (tn)a(t1) ··· a(tm)i . (5.1)

This technique has been used, to much success, in understanding fields in quantum-optical problems. In our case, we used the second-order correlator as a metric for the quality of a single-photon source in Chs. 1-3. However, it has two major drawbacks: first, if the field is in a pure state, many correlations contain redundant information; second, an arbitrarily large number of correlations may be needed to determine the N-photon state of the field. More recently, theorists have made rapid progress in exploring alternative ways to arrive at the

61 CHAPTER 5. PHOTON SCATTERING FORMALISM 62

Figure 5.1: The general problem we solve in this chapter is to compute the field scattered into unidrectional (chiral) waveguide(s) from an energy-nonconserving system Hamiltonian with a unique ground state. This class of Hamiltonian is often used to represent coherent laser pulses scattering off quantum-optical systems such as a two-level system, Jaynes-Cummings system, or entangled photon pair source. First, we discuss just a single waveguide (a) and later extend to multiple waveguides (b). state of the field in the waveguide. I briefly cite a few of the many excellent techniques developed, based on Heisenberg picture approaches [119, 120], diagrammatic summations [121–123], Green’s functions [124–135], quantum integrability [136–141], Bethe ansatz [126, 127, 142–144], pure-state wavefunctions [145–147], stochastic master equation [132, 148, 149], generalized master equations [132, 150, 151], Matrix Product State (MPS) or continuous Matrix Product State (cMPS) [152–156]. Quite remarkably, most of these results were discovered within the last ten years. As an example of the types of calculations enabled by these new methods, we point the interested reader to recent works calculating field states in quantum feedback problems [154, 155, 157]. Here, we took a different approach to solve for the scattered fields, presenting a general technique for directly integrating the total state vector for the combined waveguide(s) and quantum-optical system. I was primarily interested in the scattered field here, and hence mostly ignored the state vector during the emission process. We showed how to overcome the singularities posed by the infinite dimensionality of the baths to find a general solution for the composite evolution operator of the bath(s) and system [59]. We consider the technique first for a single waveguide coupled to a low-dimensional system (Fig. 5.1a) and then extend the formalism to handle multiple waveguides (Fig. 5.1b). This technique, to our knowledge, is the first to provide a general method of obtaining the scattered state vector in the case where coherent laser pulses are incident on a system and source the energy of the scattered photons. My interest in this problem was to understand the dynamics of few-photon sources as potential state generators for quantum communication or computation applications [1] and to enable analytic calculations of source behavior using photonic state vectors. CHAPTER 5. PHOTON SCATTERING FORMALISM 63

Here, our main contributions were to:

1. Present a complete introduction to the scattering problem between a local system and a set of waveguides.

2. Develop a general treatment of using temporal modes to describe the waveguide ﬁeld and the system’s coupling to the waveguides.

3. Allow the local system to be driven over some time interval t ∈ (0,TP ), and hence the system does not conserve energy there (so the problem is not quantum integrable).

Our pedagogical choice was to present the introductory material without the use of quantum noise approaches, stochastic calculus, or input-output theory, and we directed the reader to, e.g., Refs. [95, 130, 149, 151, 158] and references therein to learn about these connections. The chapter is structured as follows. In Sec. I, a general framework for scattering theory between temporal waveguide modes is described, including when the system is driven over the interval t ∈ (0,TP ) and hence brieﬂy does not conserve energy. Then in Sec. II, our general solution for a low-dimensional quantum system scattering photons into waveguide(s) is summarized—there we arrive at the central result of this chapter.

5.1 Problem deﬁnition

We consider a system described (in the Schr¨odingerpicture) by the time-dependent Hamiltonian

HS(t), coupled to a bath of modes described by the static Hamiltonian H0B via the coupling operator V . We ﬁrst discuss the three components of the total Hamiltonian and their properties separately.

5.1.1 System Hamiltonian

The system Hamiltonians we will consider take the form

 H0S + H1S(t) if 0 < t < TP HS(t) = (5.2) H0S otherwise.

While the above form of Hamiltonian is applicable to general systems which are driven from time t = 0 to t = TP , we place four additional restrictions on the system: CHAPTER 5. PHOTON SCATTERING FORMALISM 64

1. There exists an operator NS counting the total number of excitations in the system, which is

conserved in the absence of the drive, i.e. [NS,H0S] = 0, which is satisﬁed as long as H0S is Hermitian and time-independent.

2. No internal phase evolution occurs when the system is in its state with zero excitations. Practically, this can almost always be arranged by applying the appropriate transformation to

H0S.

3. The spacing between consecutive eigenfrequencies of H0S either be close to 0 or be clustered

near some optical frequency ω0.

4. The magnitude of HS(t) is small compared with ω0, a standard assumption in quantum optics.

Notably, the time-dependent part of the Hamiltonian, while Hermitian, is not required to conserve excitation number, i.e. in general [NS,H1S(t)] 6= 0. As a result, we can think of HS(t) as conserving excitation number except for a brief period when some external interaction causes the system’s

Hamiltonian to acquire a time-dependence—we emphasize that HS(t) acts only on the system of interest, not the bath.

A natural basis for the Hilbert space of the system HS is a number basis, with vectors

X n~ ≡ |n1, . . . , npi and NS |n~ i = nq |n~ i (5.3) q

th where nq is the number of excitations in the q degree of freedom, and we assume the number of degrees p is ﬁnite. For example, these degrees may represent: a cavity and atom in the case of a Jaynes-Cummings system, multiple cavity modes in spontaneous parametric downconversion (SPDC) or spontaneous four-wave mixing (SFWM), many atomic degrees in multi-emitter cavity systems, or just a single cavity or two-level system. Including the state with zero excitations |0Si ≡ 0 |01,..., 0pi where NS |0Si = 0, the number states form a complete orthonormal basis hn~ |n~ i = δn~ n~ 0 , with δn~ n~ 0 as the Kronecker-delta function. Then, the system’s wavefunction is expressed as |ψSi = P n~ hn~ |ψSi |n~ i.

5.1.2 Bath Hamiltonian

Consider the bath to represent a single chiral channel of a waveguide, i.e. a waveguide with a single transverse spatial proﬁle that carries energy only along one direction [131]. This type of bath forms the basis for more complicated waveguide geometries and is easily extensible to multiple channels. CHAPTER 5. PHOTON SCATTERING FORMALISM 65

R ∞ † Such a channel is described by the Hamiltonian (with ~ = 1) H0B = 0 dβ ω(β) bβbβ, where ω(β) is the waveguide’s dispersion relation and bβ is the annihilation operator for a delta-normalized plane- † 0 wave excitation with wavevector β. The bβ obey the commutation relations [bβ, bβ0 ] = δ(β − β ) and

[bβ, bβ0 ] = 0.

To transition to the standard annihilation operators bω, i.e. for modes labeled by their frequency, p we need the relationship bβ = bω(β)/ dω / dβ. The normalization is by the group velocity vg = dω / dβ (which can either be written with a wavevector or frequency dependence). This results in

Z ∞ † H0B = dω ω bωbω, (5.4) 0 where the mode operators similarly obey

† 0 [bω, bω0 ] = δ(ω − ω ) and [bω, bω0 ] = 0. (5.5)

We comment Eq. 5.4 is exact for a complete basis of waveguide modes when ω(β) is one-to-one.

Frequency mode basis

R ∞ † Because the bath’s excitation number operator NB = 0 dω bωbω commutes as [NB,H0B] = 0, the waveguide’s state with zero excitations is deﬁned by NB |0Bi = 0, and a natural basis for HB is then

√ |ω~ (m)i ≡ b† ··· b† |0 i / m! with N |ω~ (m)i = m |ω~ (m)i . (5.6) ω1 ωm B B

We deﬁne the m-dimensional vector that parameterizes the inﬁnite-dimensional states in HB as (m) (1) ω~ = {ω1, . . . , ωm} with 0 ≤ ω1, . . . , ωm. Our notation is meant to include the vectors ω~ ≡ {ω1} and ω~ (0) ≡ {∅}, and hence the states |{ω }i ≡ b† |0 i and |{∅}i ≡ |0 i for completeness. 1 ω1 B B Also from applying Eq. 5.5, these states form a complete orthogonal basis that is delta normalized (m) 0(m0) (m) 0(m) hω~ |ω~ i = δmm0 δ(ω~ − ω~ ). We can then write the bath’s wavefunction as a projection onto all |ω~ (m)i as ∞ Z X (m) (m) (m) |ψBi = dω~ hω~ |ψBi |ω~ i (5.7) m=0

R (m) R ∞ R ∞ R ∞ R R ∞ R where dω~ ≡ 0 dω1 0 dω2 ··· 0 dωm . To clarify two cases d{ω1} ≡ 0 dω1 and d{∅} ≡ 1 . P∞ R (m) (m) 2 Finally, we note the normalization of the wavefunction requires m=0 dω~ hω~ |ψBi = 1. CHAPTER 5. PHOTON SCATTERING FORMALISM 66

Temporal mode basis

At this point, we make a standard approximation in quantum optics that only modes near the characteristic frequencies of the system ω0 will be occupied [131]. Thus, we are free to define fictitious waveguide modes of negative frequency and extend the limit of ω from 0 → −∞ in Eqs. 5.4-5.7. This extension conveniently allows for convergence of integrals over all frequencies, and hence it is possible to define the Fourier transformed operator of bω

Z ∞ dω −iωτ bτ ≡ √ e bω, (5.8) −∞ 2π whose the inverse transform is also well-deﬁned. Then,

† 0 [bτ , bτ 0 ] = δ(τ − τ ) and [bτ , bτ 0 ] = 0 (5.9)

so we can interpret bτ as the annihilation operator for a ‘temporal’ mode [74, 159]. Notably, τ is (m) an index of the operators bτ not an actual time evolution (see Sec. 5.1.2). Because |ω~ i form a complete basis for HB and the Fourier transform is unitary,

|τ~ (m)i ≡ b† ··· b† |0 i with N |τ~ (m)i = m |τ~ (m)i , (5.10) τ1 τm B B

also form a complete, orthogonal, and delta-normalized basis in HB. Although the τ’s can take on any real values, for convenience we will only consider waveguide states for which they are positive and ordered (m) τ~ = {τ1, . . . , τm} with 0 < τ1 < ··· < τm. (5.11)

We are free to impose the order without loss of generality due to Eq. 5.9, though the ordered √ and non-ordered basis states diﬀer by a normalization of m!. Although this basis excludes states having multiple photons with the same time index, the subset of the full Hilbert space described by Eq. 5.11 forms a complete basis for the scattered states given a linear system-waveguide interaction Hamiltonian (described in Sec. 5.2). For linear interactions with the waveguide, the amplitude of scattering two photons precisely at the same time instant vanishes because the choice of system-bath interaction involves only single excitation exchange [59]. CHAPTER 5. PHOTON SCATTERING FORMALISM 67

The bath wavefunction may thus alternatively be written as a projection onto all |τ~ (m)i as

∞ Z X (m) (m) (m) |ψBi = dτ~ hτ~ |ψBi |τ~ i (5.12) m=0

R (m) R ∞ R ∞ R ∞ R R ∞ where dτ~ ≡ dτ1 dτ2 ··· dτm . To again clarify two cases d{τ1} ≡ dτ1 and 0 τ1 τm−1 0 R d{∅} ≡ 1 . Similar as before, the normalization of the wavefunction requires

∞ Z 2 X (m) (m) dτ~ hτ~ |ψBi = 1. m=0

(m) (m) The temporal mode weighting functions hω~ |ψBi and hτ~ |ψBi are related via m-dimensional (0) (0) symmetric Fourier transforms, with the trivial relation hω~ |ψBi = hτ~ |ψBi = h0B|ψBi. When the bath is in a number eigenstate other than the ground state, it is said to be in an m-photon Fock state (m) if h(·)|ψBi is separable into like functions, i.e. when hω~ |ψBi = φ(ω1) ··· φ(ωm) or equivalently (m) hτ~ |ψBi = φ(τ1) ··· φ(τm). Notably, in the continuous case an m-photon Fock state is not unique. Please see Ref. [59] for a precise discussion on the origin of the term ‘temporal mode’.

Free waveguide evolution

Under a unitary transformation that removes the free evolution of the bath, the frequency-indexed

iH0Bt −iH0Bt −iωt annihilation operators acquire phase based on bω(t) = e bωe = e bω. As a result, the free

iH0Bt −iH0Bt evolution causes the temporal mode operators to translate in time bτ (t) = e bτ e = bτ+t. This transformation has the consequence that

bτ=0(t) = bτ=t, (5.13) but it is important to keep in mind they are operators in diﬀerent pictures. For instance, it’s common to use a basis for |ψBi formed with the bτ operators and a Hamiltonian from the b0(t) † operators. Hence, the commutation relations between the two are important b0(t), bτ = δ(t − τ) and [b0(t), bτ ] = 0.

5.1.3 Waveguide-system coupling

Suppose the system couples to the waveguide via some operator A~ = ~λa† + ~λ∗a, acting on a sector ~ of the system’s Hilbert space HS. The vector λ has a value and interpretation determined by the CHAPTER 5. PHOTON SCATTERING FORMALISM 68

speciﬁc realization of the system, e.g., it represents a two-level system’s dipole moment or a cavity’s electric ﬁeld. We consider the case where a is an annihilation operator that acts on the system’s 1-st degree of freedom, potentially truncated to some excitation number nmax. Thus, we write its action √ † on the system’s basis states as a |n~ i = n1 |n1 − 1i hn1|n~ i with a |0, n2, ··· , npi = 0, and a |n~ i = √ † nmax+1 n1 + 1 |n1 + 1i hn1|n~ i with (a ) |0, n2, ··· , npi = 0. Thus, it obeys the commutation

nmax−1 † X a, a = |n1i hn1| − nmax |nmaxi hnmax| , (5.14)

n1=0 with an implicit identity operator in the remaining system degrees of freedom. For a cavity nmax →

∞ and for a two-level system nmax = 1. The system, placed at position ~r = 0, is linearly coupled to the bath via the overlap between A and the waveguide’s electric ﬁeld operator E(~ ~r). The positive frequency part of E(~ ~r = 0) is given by

∞ Z p E~(0) = i dβ ω(β)~uβ(0)bβ, (5.15) 0 where ~uβ(~r) represent the orthonormal spatial modes of the waveguide. Transforming again to a sum over modes indexed by their frequency,

Z ∞ r ω E~(0) = i dω ~uω(0)bω. (5.16) 0 vg(ω)

Therefore, we write the coupling Hamiltonian as

V = −A~ · E(0)~ (5.17a) Z ∞ † † = dω κ(ω) a + a ibω − ibω , (5.17b) 0

q with the coupling rate κ(ω) = ~λ · ~u(0) ω , and we have chosen ~λ · ~u(0) as real-valued without vg(ω) loss of generality. At this point, we note that the system will only excite waveguide modes within a narrow band near its natural frequency ω0, which requires κ(ω0) ω0, and then we make the standard three approximations in one [74, 160]:

1. Ignore terms in V that do not conserve the composite system’s total excitation number N =

NS + NB (in a rotating-wave approximation). CHAPTER 5. PHOTON SCATTERING FORMALISM 69

2. Make a standard Markovian approximation, whereby we assume a ﬂat coupling constant be- p γ tween the system and the waveguide κ(ω0) → 2π .

3. Again, extend the lower limit of integration over frequency.

Z ∞ √ dω † † V ≈ i γ √ bωa − bωa (5.18a) −∞ 2π √ † † = i γ b0(0)a − b0(0)a . (5.18b)

Because [N,V ] = 0, the zero excitation state of the composite system in HS ⊗ HB is deﬁned by

(NS + NB) |0Si ⊗ |0Bi = 0, which we define as |0i ≡ |0Si ⊗ |0Bi. Both a |0i = 0 and bτ |0i = 0, and a natural set of basis vectors for the composite system is formed from |n~ , τ~ (m)i ≡ |n~ i ⊗ |τ~ (m)i with (m) 0 0(m0) (m) 0(m) hn~ , τ~ |n~ , τ~ i = δmm0 δn~ n~ 0 δ(τ~ − τ~ ). We now present our final requirement in this paper, that the system have only one ‘ground’ state in the parlance of typical quantum theory. In our context this means that the system-waveguide state factorizes as t → ∞ and that the system’s final state is unique, so we ask

|Ψ(∞)i = |ψS(t → ∞)i ⊗ |ψB(t → ∞)i with |ψS(∞)i = |0Si . (5.19)

This will be true as long as any energy put in the system will leak out into the waveguide (see our work in Refs. [110] and [135] for mathematical criteria).

5.1.4 Interaction-picture Hamiltonian

The Schr¨odinger-pictureHamiltonian is given by the sum of the bare Hamiltonians and their coupling ∂ Hamiltonian H(t) = HS(t) + V + H0B, with the time-evolution of the state vector i ∂t |Ψ(t)i = H |Ψ(t)i. Our ﬁrst step towards solving this equation is to transform to an interaction-picture with respect to the free waveguide evolution [155].

iH0Bt −iH0Bt The Hamiltonian transforms as HI(t) = HS(t) + e V e , obeying the evolution equation

∂ i |Ψ (t)i = H (t) |Ψ (t)i . (5.20) ∂t I I I

−iH0Bt The Schr¨odinger-and interaction-picture state vectors are related via |Ψ(t)i = e |ΨI(t)i . We CHAPTER 5. PHOTON SCATTERING FORMALISM 70

then ﬁnd the transformed Hamiltonian

√ † † HI(t) = HS(t) + V (t) with V (t) = i γ bτ=0(t)a − bτ=0(t)a , (5.21) and making use of Eq. 5.1.2. The Hamiltonian possesses a few notable properties. First, no phase evolution occurs in the ground state during the time-periods with energy conservation HI(t) |0i = |0i for {t ≤ 0, t ≥ TP }. √ Second, due to the commutation relations of the temporal mode operators [bτ ,HI(t)] = i γa δ(t−τ), giving the interpretation that the Hamiltonian only interacts locally with one temporal mode at a time before visiting the next. The formal solution to Eq. 5.20 is given by

|ΨI(t1)i = UI(t1, t0) |ΨI(t0)i (5.22)

R t1 −i dt HI(t) with the unitary time-evolution operator UI(t1, t0) = T e t0 and T again indicates chrono- logical ordering. The time-evolution operator has the property that it can be partitioned into arbitrary time-intervals UI(t2, t0) = UI(t2, t1)UI(t1, t0).

5.1.5 Scattering matrices

The scattering matrix is formally deﬁned in the interaction picture [161] as

Sˆ = lim UI(t1, t0) (5.23a) t0→−∞ t1→+∞ −i R +∞ dt H (t) = T e −∞ I . (5.23b)

ˆ ˆ † ˆ This matrix can be decomposed into the Moller wave operators as S = Ω−Ω+ with

Ωˆ + = lim UI(0, t0) and Ωˆ − = lim UI(0, t1). (5.24) t0→−∞ t1→+∞

(We will drop further limit notation for compaction.) Traditionally, the scattering matrix has been used to calculate the overlap between initial and (m) ﬁnal states like |ω~ i |0Si and also under the constraint that HS(t) conserves excitation number

[126–132] (for our hypothetical system this would correspond to the case where HS(t) = H0S at all times). This method precludes preparing the system in an excited state before scattering. CHAPTER 5. PHOTON SCATTERING FORMALISM 71

When excitation number is conserved, the initial and ﬁnal states must have the same number of (m) ˆ † energy excitations, and hence matrix elements like hω~ | Ω− conserving |0i = 0 for m > 0. However, in our work we explicitly are considering the case where HI(t) does not conserve energy while 0 < t < TP , as would be the case for a coherent pulse driving, for example, a two-level system or Jaynes–Cummings (m) ˆ † system. Hence, for this type of Hamiltonian h0S| hω~ | Ω− |0i= 6 0 and we wish to calculate elements of this type. Our next change relative to standard scattering theory is to use the temporal mode basis all the way through, from the beginning to end of our calculations [149], and to compute the elements

ˆ † (m) ˆ † hΩ−iτ~ (m) ≡ h0S| hτ~ | Ω− |Ψ(t = 0)i , (5.25)

with |Ψ(0)i = |ψS(0)i |0Bi. Specifically, in our calculations we assumed a system Hamiltonian with a unique ground state so that hψS(t → ∞)| = h0S|. Furthermore, we will only consider cases with the waveguide prepared in its zero-excitation state, but the system may be initially excited. † + By defining τ+ as the first time after τ where [bτ , bτ + ] = 0, it can be shown that [bτ ,UI(+∞, τ )] = 0 and hence

ˆ † (m) hΩ−iτ~ (m) = h0S| hτ~ | UI(τmax, 0) |0Bi |ψS(0)i , (5.26)

where only times before τmax = max(TP , τm) matter [59]. We note that if the Hamiltonian is energy- conserving for all time, i.e. TP = 0, then a solution to this scattering problem reduces to solving for the emitted photon wavepacket under spontaneous emission.

5.2 Formalism

Practically integrating Eq. 5.20 is not straight forward because of the singularity at time t in HI(t). Although realistically the singularity is regularized through the coupling rate κ(ω), we anyways want to obtain a result that is independent of the precise form of κ(ω). Thus, we coarse-grained the temporal dynamics at a scale of ∆t [59], in eﬀect averaging over the singularity. We solved the dynamics in a coarse-grained basis, using a technique that is conceptually similar to manually performing a path integral between our initial and ﬁnal states—quantum-optical systems are simple enough that it is not necessary to use the formal machinery of path integrals. Then, we returned to the continuous-mode basis by taking the limit as ∆t → 0. CHAPTER 5. PHOTON SCATTERING FORMALISM 72

5.2.1 Coarse-graining of time

We discuss a coarse-grained propagator U[k+1, k] that maps the wavefunction from tk → tk+1 (with tk = k∆t), deﬁned as

|ΨI[k + 1]i = U[k + 1, k] |ΨI[k]i (5.27) and by extension

U[n, m] ≡ U[n, n − 1] U[n − 1, n − 2] ··· U[m + 1, m] (5.28a) ←−− n−1 Y = U[k + 1, k], (5.28b) k=m with the

lim U[bt1/∆tc, bt0/∆tc] = UI(t1, t0). (5.29) ∆t→0

As long as U[k + 1, k] is accurate to O(∆t) and the norm of the wavefunction is conserved, i.e. hΨI(t)|ΨI(t)i = 1 for all time, this limit will hold. Note: we only deﬁne this map in the interaction picture and hence drop the labels (·)I for all coarse-grained operators. Such a map is [59]

U[k + 1, k] ≡ US[k + 1, k]Uswap[k + 1, k] (5.30)

h R (k+1)∆t i h R (k+1)∆t i with US[k +1, k] = exp −i k∆t dt HS(t) and Uswap[k +1, k] = exp −i k∆t dt V (t) , which is still explicitly unitary. The leading error in this approximation is still O(∆t3/2) because the unitaries US[k + 1, k] and Uswap[k + 1, k] commute to order ∆t in the series expansion of Eq. 5.30. Plugging in the expression for V (t) from Eq. 5.21, we can rewrite

hp † †i Uswap[k + 1, k] ≡ exp γ∆t ∆B0[k]a − ∆B0[k]a , (5.31)

(k+1)∆t where we defined a coarse-grained interaction-picture operator ∆B [k] = √1 R dt b (t). We 0 ∆t k∆t 0 (j+1)∆t can similarly define a Schrödinger-pictureoperator ∆B = √1 R dτ b , and combined with j ∆t j∆t τ h † i Eq. 5.13 we see that ∆Bj = ∆B0[j]. Hence, these operators obey the commutation ∆Bj, ∆B0[k] = δ[j − k], where δ[l] is also the discrete Kronecker-delta function. Therefore, unlike the singular † † operator bτ , our new operator ∆Bj can be interpreted as creating a properly normalized excitation of a harmonic oscillator, which is indexed by the temporal-mode bin j. CHAPTER 5. PHOTON SCATTERING FORMALISM 73

Additionally, the waveguide mode operators commute with the dynamical map for nonequal bins. The continuum temporal mode operators are recovered in the limit

∆Bbτ/∆tc lim √ = bτ , (5.32) ∆t→0 ∆t and the subtleties of taking this limit have been discussed elsewhere [152].

Hilbert space

Consider the Hilbert space of the combined internal system HS and the coarse-grained waveguide coarse coarse coarse ﬁeld HB is H = HS ⊗ HB . Our new coarse-grained wavefunction |Ψ[k]i lives in this space. Here, we are representing the Hilbert space of the waveguide as a combined space of each th coarse N+∞ n harmonic oscillator representing a coarse-grained temporal mode HB = n=−∞ Hn. The coarse P+∞ † new number operator in H given by NC = j=−∞ ∆Bj ∆Bj commutes with US. The total excitation number operator NC + NS is conserved when t < 0 and t > TP (or equivalently k < 0 coarse and k > TP /∆t). Hence, the state |0Bi now represents zero excitations in HB , and the global ground state is (again) the elementary direct product

+∞ O |0i ≡ |0Si ⊗ |0Bi with |0Bi = |0ni . (5.33) n=−∞

The new orthonormal basis states for the ﬁeld modes are

|~j(m)i ≡ ∆B† ··· ∆B† |0 i (5.34) j1 jm B O = |1j1 i ⊗ · · · ⊗ |1jm i ⊗ |0ni , (5.35)

n6={j1,...,jm}

~ (m) and j = {j1, . . . , jm}. Like for the τ’s, we only consider the cases where 0 < j1 < ··· < jm. When these states are taken in a similar limit as Eq. 5.32, they form a complete basis for the waveguide state in the continuum.

At this point, it is clear that US[k + 1, k] acts on HS and Uswap[k + 1, k] acts on HS ⊗ Hn=k.

Furthermore, Uswap[k + 1, k] is the standard swap Hamiltonian between two modes. This leads to an intuitive picture (Fig. 5.2):

1. At every time step the internal system performs a partial unitary swap operation with the kth waveguide bin, CHAPTER 5. PHOTON SCATTERING FORMALISM 74

Time-local swap with waveguide bin System evolution only

Figure 5.2: An intuitive picture of the coarse-grained dynamical map. Here, we use half indices to indicate the intermediate states after application of the swap but before the system evolution. This procedure is similar to the Trotterization in quantum-mechanical path integrals, although here the Hamiltonian is also varying in time.

2. Following this swap, the system is brieﬂy evolved in HS for ∆t,

3. The time index is shifted k → k + 1 and the process repeats.

5.2.2 A general solution

Now, we deﬁne a coarse-grained scattering operator

ˆ † ~ (m) hΩ−i~j(m) = h0S| hj | U[max(jm + 1,P ), 0] |0Bi |ψS(0)i . (5.36)

From Eqs. 5.29, 5.32, and 5.34, the scattering operator for continuum modes can be recovered in the limit m ˆ † 1 ˆ † hΩ−i (m) = lim √ hΩ−i~(m) (5.37) τ~ ∆t→0 ∆t j

~ (m) with j → {bτ1/∆tc,..., bτm/∆tc} and P → bTP /∆tc. Taking this limit yields a solution to the dynamics of the emitted photons [59], in terms of the CHAPTER 5. PHOTON SCATTERING FORMALISM 75

Moller scattering operator

←−m ˆ † Y√ hΩ−iτ~ (m) = h0S| Ueﬀ(τmax, τm) γa Ueﬀ(τq, τq−1) |ψS(0)i , (5.38) q=1 where τ0 = 0, τmax = max(TP , τm), and

" # Z τq Ueﬀ(τq, τq−1) = T exp −i dt Heﬀ(t) (5.39) τq−1

γ † is the propagator corresponding to the effective Hamiltonian Heff(t) = HS(t) − i 2 a a where HS(t) is again the system Hamiltonian from Eq. 5.2. It should be noted that Ueff(τq, τq−1) is not unitary since the effective Hamiltonian Heff is not Hermitian. The m-photon temporal mode functions are given, for the interaction-picture wavefunction, by

(m) ˆ † h0S, τ~ |ΨI(t → ∞)i = hΩ−iτ~ (m) . (5.40)

Hence, our ﬁnal result for the scattered ﬁeld is

∞ Z X (m) ˆ † (m) |ψB,I(t → ∞)i = dτ~ hΩ−iτ~ (m) |τ~ i . (5.41) m=0

We again emphasize that if the system Hamiltonian conserves energy for all times, i.e. TP = 0 so

HS(t) = H0S, this result reduces to the solution for the system spontaneously emitting its energy into the waveguide after being prepared in |ψS(0)i. Remarkably, the scattered ﬁeld state after the system has ﬁnished decaying depends only on operators that act on the quantum-optical system’s

Hilbert space. The result has a very intuitive form, where m evolution periods are governed by Ueﬀ and hence by the eﬀective non-Hermitian Hamiltonian, which represent the amplitudes the system does not emit a photon in those periods (Fig. 5.3).

5.2.3 Extension to multiple output waveguides

The general framework outlined above can easily be extended to problems where a quantum-optical system couples to multiple (M) waveguides. The bath Hamiltonian and the bath-system interaction CHAPTER 5. PHOTON SCATTERING FORMALISM 76

Energy non-conserving Energy conserving

Time t

Figure 5.3: Sketch of a path formalized by Eqs. 5.38–5.40, specifically for the path amplitude (3) with a realization of the photon emission times τ~ = {τ1, τ2, τ3} and TP < τ3. Through a series of commutations, we reduced the problem of computing the scattered waveguide field to one of (m) computing many system evolutions for different τ~ , subject to Heff(t), and punctuated by state (m) collapses at times τ~ . This non-Hermitian Hamiltonian causes |ψS(t)i to leak probability and at the end of the evolution we calculate the overlap with the remaining amplitude in the ground state, i.e. with h0S|ψS(t)i. But, since the system conserves energy after TP , we can compute the remaining + ground state amplitude just by following the evolution to the greater of τm or TP , i.e. h0S|ψS(τm)i or h0S|ψS(TP )i, respectively.

Hamiltonian for such a system is given by

M Z M Z X † X √ dω † † H0B = dω ω bi,ωbi,ω and V = i γi √ (bi,ωai − bi,ωai ), (5.42) i=1 i=1 2π

th where bi,ω are the annihilation operators for the delta-normalized plane-wave modes in the i waveguide and ai are the system operators through which the quantum-optical system interacts th with the i waveguide. Note that ai need not be distinct operators, in which case they correspond to multiple waveguides coupling to the quantum-optical system through the same operator. This scenario is comparable to passing the emission of one waveguide through a multi-port beamsplitter. A complete basis for the bath states can be constructed taking the tensor products of the bases for individual waveguides |τ~ (m1), τ~ (m2),..., τ~ (mM )i = QM b† b† ··· b† |0i, where τ (mi) = 1 2 M i=1 i,τi,1 i,τi,2 i,τi,mi i th {τi,1, τi,2, . . . , τi,mi } parametrizes the state of the i waveguide. One important point to note is regarding the time ordering of the indices. Since photons at the time index in the same waveguide are identical, it is possible to impose the ordering τi,1 < τi,2 < ··· < τi,mi . However, photons in diﬀerent waveguides are not identical and hence, it is not possible to impose an ordering on the temporal indices across diﬀerent waveguides. Following a procedure similar to the problem of a single waveguide coupled to the local system, CHAPTER 5. PHOTON SCATTERING FORMALISM 77

we obtained a generalization of Eq. 5.38 for multiple waveguides [59]

ˆ † ˆ † hΩ−iτ˜(N) ≡ hΩ−i (m1) (m2) (mM ) (5.43a) τ~1 ,τ~2 ,...,τ~M ←− N Y√ = h0S| Ueff(τmax, τÑ ) γQ[q]aQ[q]Ueff(˜τq, τ˜q−1) |ψS(0)i (5.43b) q=1

(m1) (m2) (mM ) as a projection onto |τ~1 , τ~2 ,..., τ~M i. Here, we unpack several new deﬁnitions:

• The total number of photons scattered is N = m1 + m2 + ··· + mM .

(N) • The time indicesτ ˜q ∈ τ˜ with 0 < τ˜1 ≤ τ˜2 ≤ · · · ≤ τ˜N and τmax = max(TP , τ˜N ).

(N) (mi) (N) (m1) • τ˜ is a chronologically sorted set of all time indices from the τ~i ’s as τ˜ = sort{τ~1 +

(m2) (mM ) τ~2 + ··· + τ~M }.

• Q[q] is the index of the waveguide corresponding to the photon scattered atτ ˜q.

• Ueff(τq, τq−1) is the propagator generated by the effective Hamiltonian Heff(t) evolution, where

PM γi † Heﬀ(t) = HS(t) − i i=1 2 ai ai.

Finally, a brief note about the numerical applicability of this approach. Suppose that the waveguides are each discretized into B bins. Then, the total computational complexity scales polynomially with the number of bins, given that we need to compute no more than (MB)N /N! scattering amplitudes, where N is the number of photons in the state and M is the number of waveguides. In most applications involving the generation of pulsed quantum light, B M,N and hence this problem is tractable. Further, the computation can be executed in a highly eﬃcient manner: only B2 propagators must be computed and then the state amplitudes can all be assembled in parallel on a GPU.

5.2.4 Connection to quantum trajectories and measurement theory

In this section, we show a connection between our derived scattering amplitudes and quantum measurement theory. For a traditional approach to understanding how a system HS interacts with its environmental baths, a stochastic Schrödingerequation is arrived at for the system’s wavefunction [162]. This equation gives a pure-state evolution of the system under the influence of the M baths, at the expense of turning the system’s wavefunction into a stochastic process. A complete realization of the stochastic process (from t = 0 to t → ∞) is uniquely identified by a sequence of collapse CHAPTER 5. PHOTON SCATTERING FORMALISM 78

events, called the ‘measurement record’. The wavefunction conditioned on the measurement record (N) undergoes a series of discontinuous jumps at times τ˜ = {τ˜1,..., τ˜N }. Each jump collapses the (N) wavefunction at timeτ ˜q ∈ τ˜ according to the operator aQ[q], where Q[q] determines which bath ‘caused’ the jump. This type of evolution is also equivalent to modeling measurement of the system by M detectors, each coupled to the system via aQ[q]. Then, for the wavefunction conditioned on a measurement record [95]

dψc(t) ≡ ψc(t + dt) − ψc(t) (5.44)  N ! N    X γQ[q] † X aQ[q] =  −iHeﬀ(t) + ha aQ[q]ic dt +  − 1 dNq(t) ψc(t), 2 Q[q] q † q=1 q=1 haQ[q]aQ[q]ic

where h...ic ≡ hψc(t)| ... |ψc(t)i. The Poisson increment dNq(t) represents a counting process that increments for each collapse registered by any detector, up to N for a trajectory parameterized by τ˜(N). Using techniques from stochastic calculus, one can arrive at the probability density for a given trajectory [162, 163]. Critically, this probability density is equivalent to the modulus square of our results in Sec. 5.2.3

2 (τ˜(N)) = hΩˆ † i . (5.45) P − τ˜(N)

Hence, we showed the intricate connection between photon emission probabilities in a stochastic Schr¨odingerequation and our microscopic scattering theory based on temporal waveguide modes. We recently expanded upon these connections in a new work [110]. On a historical note, Eq. 5.44 was often postulated or ‘unravelled’ from the unconditioned dynamics of the system’s density matrix [75]. As a consequence, these stochastic Schr¨odingermethods were originally derived by tracing over all bath degrees of freedom. Hence, there still exists a mis- conception that quantum trajectories are only the conditional dynamics of the system based on measurement records. In other words, the sequence of collapse events is often interpreted to exist in an abstract ‘detection Fock space’. We have shown here, by deriving the photon scattering amplitudes based on a microscopic theory of the system-bath interaction, without any trace operation over the waveguide modes, that the emission amplitudes from measurement theory have a physical connection to the waveguides’ photonic state. We of course note that modern understanding of Eq. 5.44 also yields a similar interpretation of photon emission probabilities [148, 158]. Chapter 6

Photon emission from a two-level system

In Ch. 4, we discussed how photon blockade acts as a single-photon source by filtering a single photon from a laser pulse. We saw how the fundamental physics behind the effect is the same as a quantum two-level system, and hence after understanding the formalism for how photons scatter into a waveguide (Ch. 5), the first system I targeted with the formalism was the quantum two-level system coupled to a chiral photonic channel (shown schematically in Fig. 6.1). The quantum two-level system, which models a single discrete atomic transition, is one of the most fundamental building blocks of quantum optics [164]. This type of system has been behind fundamental discoveries such as photon anti-bunching [165, 166], Mollow triplets [167, 168], and quantum interference of indistinguishable photons [4, 169]. After almost two decades of development in a solid-state environment, the quantum two-level system is now poised to serve the pivotal role of

Figure 6.1: Quantum two-level system driven by a laser pulse with driving strength Ω(t), and coupled to a unidirectional (chiral) mode with rate γ. The system emits a wavepacket that travels towards the right.

79 CHAPTER 6. PHOTON EMISSION FROM A TWO-LEVEL SYSTEM 80

Energy non-conserving Energy conserving

Time t

Figure 6.2: Sketch of a path for two-level system evolution, formalized by Eq. 6.9, and specifically (3) for the path amplitude with a realization of the photon emission times τ~ = {τ1, τ2, τ3} and τ2 < TP < τ3. Compared to an arbitrary system, the two-level system has the interesting property that it can only store one excitation at a time. As a result, every photon emission collapses the system in its ground state |0Si. The energy non-conserving phase can correspond to, e.g., optical pumping of the two-level system by a laser pulse. Also note: a maximum of only one emission may occur in the energy conserving phase when the system is not pumped. an on-demand single-photon source [4–6, 165, 170–177]—by converting laser pulses with Poissonian counting statistics to single photons—for quantum networks [1, 28, 178]. More recently, multi-photon quantum state generators have found strong interest as replacements for the single-photon source in many quantum applications [51, 179, 180]. Within this context, my goal was to understand exactly how single-photon generation from a quantum two-level system operates and how it filters just a single photon from a Gaussian laser pulse that has Poissonian counting statistics. In the process, we found the unexpected result that even for arbitrarily short driving laser pulses, two-photon emission can dominate over single-photon emission for certain driving powers. In this chapter, I summarize all of our insight into how a quantum two- level system interacts with both laser pulses and external photonic fields. This mechanism not only suggests a new type of interesting quantum state to explore, but also places a fundamental limit on the two-photon error rate from a two-level system acting as a single-photon source based on the pulse length and spontaneous decay time of the system. The single-photon generation capability of the two-level system is based on the property that it can only store one excitation at a time (Fig. 6.2). As a result, the system system operator a is truncated to one excitation, a → σ, where σ is the atomic dipole operator. It is defined by its † action on the system’s Hilbert space, which is spanned by |0Si, σ |0Si = 0, and |1Si = σ |0Si. The CHAPTER 6. PHOTON EMISSION FROM A TWO-LEVEL SYSTEM 81

Hamiltonian for the bare two-level system has the simple form

† H0S = ωσσ σ. (6.1)

6.1 Traditional theory of a single-photon source

To understand how an ideal quantum two-level system acts as a single-photon source, suppose the system is instantaneously prepared in the superposition of its ground |0Si and excited |1Si states

p p |ψS(0)i = 1 − Pe |0Si + Pe |1Si , (6.2)

where Pe is the probability of having initialized the system in |1Si. From this point, spontaneous emission at a rate of γ governs the remaining system dynamics and a single photon is coherently emitted with probability Pe, while no photon is emitted with probability 1 − Pe. This emission process can be computed from our theory in Ch. 5. To model the case where the system is instantaneously excited, we have the two-level system conserve energy for all time, i.e. there is no laser drive and TP → 0. From Eq. 5.2, we then have the time-dependent part of HS as

H1S(t) = 0. (6.3)

The eﬀective non-Hermitian Hamiltonian that governs the evolution is given by

γ H = H − i σ†σ. (6.4) eﬀ 0S 2

Because energy is conserved and quantum mechanics is a linear theory, we know that the output field will be a superposition of field states with either one photon or the vacuum state. Furthermore, † due to the time-independence of Heff, we can simplify the integral operator in hΩˆ i from Eq. 5.38 − τ1 to an exponential one

√ R τ1 † −i dt Heff hΩˆ i = γ h1S| T e 0 |1Si (6.5a) − τ1 √ −iHeffτ1 = γ h1S| e |1Si (6.5b) √ = γ e−iωσ τ1 e−γτ1/2. (6.5c) CHAPTER 6. PHOTON EMISSION FROM A TWO-LEVEL SYSTEM 82

Hence, the ﬁnal state of the waveguide after spontaneous emission can be written as

Z ∞ p p −iωσ τ1 −γτ1/2 |ψB,I(∞)i = 1 − Pe |0i + γPe dτ1 e e |τ1i . (6.6) 0

Detected by an ideal photon counter, this state results in the photocount distribution

Pm = {P0,P1,P2, ...} = {1 − Pe, Pe, 0, ...}, (6.7)

where Pm is the probability to detect m photons in the emitted pulse. It is on this principle that most indistinguishable single-photon sources based on solid-state quantum emitters operate [50, 171], ignoring the precise mechanism that excites the system at time t = 0, as in Eq. 6.2.

6.2 General theory of photon emission

However, to fully understand the two-level system as a photon source, it’s important to consider the excitation dynamics of the system. The way to generate the highest purity single-photon source from a two-level system is to excite the system with a coherent laser pulse [50]. Here, the coherent pulse drives Rabi oscillations in the two-level system, which are terminated when the system is maximally excited. Recently, we investigated these dynamics for two-photon emission [60, 61]— both to characterize the two-photon errors that spoil single-photon emission and the potential to use the two-photon state as a quantum resource. In this scenario, the laser will inject energy into the system and hence we set

|ψS(0)i = |0Si . (6.8)

Because the laser is injecting energy, we may need a very large number of scattering matrix elements with different photon number from Eq. 5.38 in general. Fortunately, the scattering matrix elements for the two-level system simplify significantly. In particular, each evolution between emissions, i.e. from each τq−1 to τq, is uncorrelated. Hence, the expectations may be evaluated independently. In this section, we will be evaluating Ueff by inspection, so it is easier to use the definition from Eq. CHAPTER 6. PHOTON EMISSION FROM A TWO-LEVEL SYSTEM 83

5.39 to write the solution in the form (deﬁning τ0 = 0)

ˆ † hΩ−iτ~ (m) =

 T R τq √ R P m −i dt Heff(t) m −i τ dt Heff(t) Q τq−1 ( γ) h0S| T e m |0Si q=1 h1S| T e |0Si if τm < TP   −i R τq dt H (t) √ m Qm τ eff ( γ) h1S| T e q−1 |0Si if τm−1 < T < τm , (6.9)  q=1  0 otherwise

where energy conservation dictates that only one emission can occur after TP due to the fact that the system holds a maximum of one excitation. Also, note: unlike in Eq. 5.38, the products are no longer ordered because of their statistical independence.

Now, consider the speciﬁc form of H1S(t). If we consider the frequency of the driving laser pulse to be resonant with the two-level system, then we have the time-dependent part of HS as

−iωσ t † ∗ iωσ t H1S(t) = f(t)e σ + f (t)e σ, (6.10) where f(t) is an arbitrary temporal function that contains the pulse shape and the overlap of the system’s dipole moment with the pulse’s electric ﬁeld. Then, the non-Hermitian Hamiltonian that governs the evolution is also time-dependent

γ H (t) = H + H (t) − i σ†σ. (6.11) eﬀ 0S 1S 2

We choose a simple square shape for the pulse f(t), so that we may arrive at nice analytic expressions for the scattered ﬁelds (though one could easily numerically integrate Eq. 6.9 for more complicated pulse shapes)

 -iωσ t † iωσ t Ω ie σ − ie σ if 0 < t < TP H1S(t) = (6.12) 0 otherwise, where Ω is the Rabi frequency (and has no relation to the Moller scattering operators). Now, the CHAPTER 6. PHOTON EMISSION FROM A TWO-LEVEL SYSTEM 84

expectations in Eq. 6.9 that end with a photon emission simplify into two categories

where conservation of energy for t > TP allows us to insert the projector |1Si h1S|. Computing these expectations, we arrive at

−iHeff(∞)τ −iωσ τ −γτ/2 h1S| e |1Si = e e , (6.14) like in Eq. 6.5c, and

+ Ω h1 | e−iHeff(0 )τ |0 i = e−iωσ τ e−γτ/4 sin (Ω0τ) (6.15) S S Ω0 with the (potentially complex) Rabi frequency

r γ 2 Ω0 ≡ Ω2 − . (6.16) 4

Then, for emission paths that end during the energy-nonconserving phase, we also need the waiting integral

R TP −i dt Heff(t) h0S| e τm |0Si

+ −iHeff(0 )(TP −τm) = h0S| e |0Si (6.17a) 0 γ sin (Ω (TP − τm)) = e−γ(TP −τm)/4 cos (Ω0(T − τ )) + , (6.17b) P m 4 Ω0 which calculates the amplitude no photon emission occurs between τm and TP . Combining these together, we arrive at the general solution for the scattered ﬁeld from a two-level CHAPTER 6. PHOTON EMISSION FROM A TWO-LEVEL SYSTEM 85

system driven by a square pulse

ˆ † hΩ−iτ~ (m) = (6.18)  0 √ sin (Ω (TP −τm)) m ( γ)me−iωσ τm e−γTP /4 cos (Ω0(T − τ )) + γ Q Ω sin (Ω0(τ − τ )) if τ < T  P m 4 Ω0 q=1 Ω0 q q−1 m P   √ m −iωσ τm −γ(τm−TP )/2 −γTP /4 Ω 0 Qm−1 Ω 0 . ( γ) e e e Ω0 sin (Ω (TP − τm−1)) q=1 Ω0 sin (Ω (τq − τq−1)) if τm−1 < TP < τm   0 otherwise

In either the strong driving limit Ω γ (previously identiﬁed by Mollow [145]) or the short pulse 1 limit TP γ , the scattering elements have a particularly simple form

ˆ † hΩ−iτ~ (m) ≈ (6.19)  √ m −iωσ τm −γTP /4 Qm ( γ) e e cos (Ω(TP − τm)) q=1 sin (Ω(τq − τq−1)) if τm < TP   √ m −iωσ τm −γ(τm−TP )/2 −γTP /4 Qm−1 ( γ) e e e sin (Ω(TP − τm−1)) sin (Ω(τq − τq−1)) if τm−1 < TP < τm .  q=1  0 otherwise

0 Ω 0 For strong driving Ω ≈ Ω, and for weak short pulses Ω0 sin (Ω τ) ≈ sin (Ω τ). From these scattering elements, it is quite simple to identify the origin of photon antibunching (similarly did Mollow [145] and Pletyukhov [122]). Photon antibunching is a statement that the √ intensity correlation between two different points in space or time is zero. If we approximate ω ≈ √ ωσ in the intensity operator since γ ωσ, we may equally look at photon-number correlations. Then, consider the second-order coherence function of the field at position ~r = 0+ [74], which is a special case of Eq. 5.1. This is the correlation that would be measured were a photon-number- resolving detector placed right in front of the two-level system and measuring the scattered field

(2) † † G (t1, t2) = hψB,I(∞)|b0(t1)b0(t2)b0(t2)b0(t1)|ψB,I(∞)i . (6.20)

(This is a slight abuse of notation, because the actual state has propagated out to inﬁnity. This statement is anyways true since the interaction of the system with the waveguide is spatially and temporally localized.) ˆ † To understand how the form of hΩ−iτ~ (m) requires antibunching, consider the operator

† † b0(t1)b0(t2)b0(t2)b0(t1). CHAPTER 6. PHOTON EMISSION FROM A TWO-LEVEL SYSTEM 86

It removes two photons from both the initial and ﬁnal states, so neither zero-photon nor single- ˆ † photon basis states contribute. Further, the expectation is a sum of terms, each based on hΩ−iτ~ (m) , where terms with diﬀerent numbers of photons m do not interfere [59].

Then, look at only b0(t2)b0(t1) |ψB,I(∞)i, and the interaction-picture state is most easily written in the temporal mode basis. We want to commute the two annihilation operators towards the right into the state. Each commutation of an annihilation operator with a temporal mode operator will yield a delta function. Thus, each nonzero term in the photon-subtracted initial state will contain a product of two delta functions like δ(t1 − τq)δ(t2 − τp), where q 6= p and p, q ∈ {1, . . . , m}. When ˆ † t1 → t2, then at least two time indices in each term of hΩ−iτ~ (m>1) will approach each other and be ˆ † the same. When two time-indices approach each other, hΩ−iτ~ (m>1) → 0 for the two-level system from Eq. 6.18, and there is perfect anti-bunching to all orders of photon number in the scattered ﬁeld state!

6.3 Short pulse regime

We now consider the above dynamics speciﬁcally in the regime of short laser pulses driving the quantum two-level system.

6.3.1 Theory

1 The short pulse regime TP γ , corresponds to the previously identified single-photon and two- photon emission regimes [60, 61]. There, the system undergoes somewhat high-fidelity Rabi oscillations between its ground |0si and excited states |1si as a function of the interacted pulse area AR, with the Rabi frequency defined as A Ω = R . (6.21) 2TP

In this section, we will keep all terms to ﬁrst order in γTP to arrive at nice analytic forms for various quantities of interest. First, consider the case where no photons are scattered. Then,

R TP −i dt Heff(t) h0|Ψ(t → ∞)i = h0S| T e 0 |0Si (6.22a) 0 γ sin (Ω TP ) = e−γTP /4 cos (Ω0 T ) + (6.22b) P 4 Ω0 γTP ≈ e−γTP /4 cos (A /2) + sin (A /2) . (6.22c) R 2A R CHAPTER 6. PHOTON EMISSION FROM A TWO-LEVEL SYSTEM 87

( a ) 1 . 0 ( b ) 1 . 0

0 . 8 A = π ] R

2 0 . 5 / 2 1 γ P [

1 ,

0 . 6 τ 0 π 1 A = 2 ω

i R - P 0 . 0

e , ) 0 0 . 4 1 τ P π ( A R = 3 1

φ - 0 . 5 0 . 2

0 . 0 - 1 . 0 0 1 2 3 4 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 π τ 1/γ P u l s e a r e a , A R [ ] T i m e i n d e x , 1 [ ]

( c ) 1 . 5 ( d ) 6 5 ] γ /

1 1 . 0 4 [ ]

0 2 [ t )

2 3 ( , e g m

i 0 . 5 2 T 1 0 . 0 0 0 . 0 0 . 5 1 . 0 1 . 5 0 1 2 3 4 1/γ π T i m e , t 1 [ ] P u l s e a r e a , A R [ ]

Figure 6.3: Statistical characterization of photon emission under Rabi oscillations—a two-level system is driven by a short optical pulse TP = 0.2/γ. (a) Signatures of Rabi oscillations in the emitted photon numbers versus pulse area, showing vacuum P0, single-photon P1, and two-photon P2 contributions. Dotted lines show ideal values for arbitrarily short pulses. (b) Envelopes of single- iωσ τ1 (2) photon amplitudes for diﬀerent pulse areas hτ1|φ1i e . (c) Second-order coherence G (t1, t2) for (2) AR = 6π. (d) Pulse-wise second-order coherence g [0], black shows case of TP = 0.2/γ, while red arrows depict the singularities for even-π pulses of arbitrarily short length.

Hence, the probability of scattering zero photons is given by

2 P0 ≡ |h0|Ψ(t → ∞)i| (6.23a) γT 2 −γTP /2 P ≈ e cos (AR/2) + sin (AR/2) , (6.23b) 2AR which is shown as the black curve in Fig. 6.3a for a short pulse. CHAPTER 6. PHOTON EMISSION FROM A TWO-LEVEL SYSTEM 88

Second, consider the cases where one photon is scattered. Then,

h0S, τ1|Ψ(t → ∞)i

√ R TP R τ1 −i dt Heff(t) −i dt Heff(t) = γ h0S| T e τ1 |0Si h1S| T e 0 |0Si (6.24a)  0 √ sin Ω (TP −τm) −iωσ τ1 −γTP /4 0 γ ( ) Ω 0  γ e e cos (Ω (TP − τm)) + 4 Ω0 Ω0 sin (Ω τ1) if τ1 < TP = √  −iωσ τ1 −γ(τ1−TP )/2 −γTP /4 Ω 0  γ e e e Ω0 sin (Ω TP ) if TP < τ1 √ −iωσ τ1 −γTP /4 AR AR  γ e e cos (TP − τ1) sin τ1 if τ1 < TP ≈ 2TP 2TP . (6.24b) √ −iωσ τ1 −γ(τ1−TP )/2 −γTP /4  γ e e e sin (AR/2) if TP < τ1

Hence, the single-photon part of the bath wavefunction is written as

This amplitude shows Rabi oscillations during the pulse period 0 < τ1 < TP , followed by exponential decay (Fig. 6.3b). Quite interesting, is that the single-photon amplitude is zero outside of the pulse period when AR = 2π to second-order in γTP . The total probability of emitting one photon is then given by

P1 ≡ hφ1|φ1i (6.26a) 1 γT sin (A ) −γTP /2 P R ≈ e 1 − cos (AR) + 1 − cos (AR)/2 − , (6.26b) 2 2 2AR which is shown as the blue curve in Fig. 6.3a for a short pulse. CHAPTER 6. PHOTON EMISSION FROM A TWO-LEVEL SYSTEM 89

Third, consider the cases where two photons are scattered. Then,

h0S, {τ1, τ2}|Ψ(t → ∞)i  R TP R τ2 τ −i dt Heff(t) −i dt Heff(t) −i R 1 dt H (t) h0S| T e τ2 |0Si h1S| T e τ1 |0Si h1S| T e 0 eff |0Si if τ2 < TP = R τ2 τ −i dt Heff(t) −i R 1 dt H (t) h1S| T e τ1 |0Si h1S| T e 0 eff |0Si if τ1 < TP < τ2  √ 2 −iωσ τ1 −γTP /4 AR AR AR ( γ) e e cos 2T (TP − τ2) sin 2T (τ2 − τ1) sin 2T τ1 if τ2 < TP  P P P  √ 2 −iωσ τ1 −γ(τ2−TP )/2 −γTP /4 AR AR ≈ ( γ) e e e sin (TP − τ1) sin τ1 if τ1 < TP < τ2 .  2TP 2TP  0 otherwise

Hence, the two-photon part of the bath wavefunction is written as

Z ∞ Z ∞ |φ2i ≈ dτ1 dτ2 h0S, {τ1, τ2}|Ψ(t → ∞)i |τ1, τ2i (6.27a) 0 τ1 √ = ( γ)2 e−γTP /4 × (6.27b) Z TP Z TP −iω0τ2 AR AR AR dτ1 dτ2 e cos (TP − τ2) sin (τ2 − τ1) sin τ1 |τ1, τ2i 2TP 2TP 2TP 0 τ1 Z TP Z ∞ −iω0τ2 −γ(τ2−TP )/2 AR AR + dτ1 dτ2 e e sin (TP − τ1) sin τ1 |τ1, τ2i . 2TP 2TP 0 τ1

The total probability of emitting two photons is then given by

P2 ≡ hφ2|φ2i (6.28a) Z TP A A −γTP /2 R R ≈ γ e dτ1 sin (TP − τ1) sin τ1 (6.28b) 0 2TP 2TP γT sin (A ) P −γTP /2 R = e 2 + cos (AR) − 3 , (6.28c) 8 AR

2 where we used only the second integral of Eq. 6.27b because the ﬁrst integral is O((γTP ) ). This probability is shown as the red curve in Fig. 6.3a for a short pulse. Quite interesting are the points where P2 > P1, which may not have been naively expected for the two-level system. We also note that one can easily use our solution in Eq. 6.18 and compute its integrals numerically for higher values of m, to directly calculate the oscillating Pm extracted from photon correlations in Ref. [181]. CHAPTER 6. PHOTON EMISSION FROM A TWO-LEVEL SYSTEM 90

In experiment, the two-photon wavefunction is often characterized from the second-order coher- 2 ence (Eq. 6.20). For short pulses, |φ3i is O((γTP ) ) and hence

(2) + G (t1, t2) 2 2 −γTP /2 AR AR AR = γ e cos (TP − t2) sin (t2 − t1) sin t1 Θ(0 < t1 < t2 < T ) + 2TP 2TP 2TP 2 2 −γTP /2 −γ(t2−TP ) AR AR γ e e sin (TP − t1) sin t1 Θ(0 < t1 < T < t2), 2TP 2TP (6.29)

in the positive half plane where t1 < t2. The second-order coherence is symmetric with respect to exchange of t1 and t2 so

(2) (2) + (2) + G (t1, t2) = G (t1, t2) + G (t2, t1) , (6.30)

which is shown in Fig. 6.3c. Signatures of the Rabi oscillations are seen when 0 < t1 < TP or

0 < t2 < TP , followed again by exponential decay to long times at a rate of γ: we will discuss the intuitive reasoning behind these multi-lobed correlations in Sec. 6.3.2 [60]. Anyways, we comment that the separation of timescales—one photon order TP and one order 1/γ—suggests the two-photon state is mostly separable and may be useful as a source of entangled photon pairs, though more investigation is required here. Such investigation would require computing the entanglement entropy between the two photons (after frequency ﬁltering to route them into separate waveguides [182–184]), as done for standard photon-pair sources. Our formalism should allow for such a calculation.

Experimentally accessing these temporal correlations or photocount distributions Pm is quite challenging, and more typically a quantity called the pulse-wise second-order coherence is used [50]

P (2) m m(m − 1)Pm g [0] = P 2 (6.31a) ( m mPm) 2P2 ≈ 2 (6.31b) (P1 + 2P2) for emission from a two-level system excited by a short pulse. This quantity has the property that g(2)[0] = 1 for a coherent laser pulse, g(2)[0] = 0 for a single-photon wavepacket, and g(2)[0] 1 for a two-photon wavepacket superposed with a strong |0Bi component. Hence, it periodically oscillates, antibunches roughly when P1 > P2, and bunches roughly when P1 < P2. We plot this quantity for a short pulse in Fig. 6.3d, which we used experimentally to verify the existence of these photon CHAPTER 6. PHOTON EMISSION FROM A TWO-LEVEL SYSTEM 91

γTP /2 γTP /2 Area, A P1(A)e P2(A)e 3 1 π 1 + 8 γTP 8 γTP 1 3 2π 8 γTP 8 γTP

Table 6.1: Counting statistics for single- and two-photon emission probabilities when an ideal quantum two-level system is driven by a square pulse of length TP . number oscillations (Sec. 6.3.2).

Notably, because P1 and P2 are roughly periodic, they have very simple expressions for areas that are multiples of π (Table 6.1). As a result

P (A = 2π) 2 = 3, (6.32) P1(A = 2π) which is independent of the pulse length or system-waveguide coupling for short pulses! Hence the pulse-wise second-order coherence has the simple expression

e+γTP /2 g(2)[0](A = 2π) ≈ (6.33) γTP that diverges at even areas (shown as the red arrows in Fig. 6.3d) for arbitrarily short pulses. These results provide nice formalism and rigor to our studies on photon emission from two-level systems driven by short optical pulses [50, 60, 61]. As suggested earlier, two-photon emission from a quantum two-level system under excitation by a short laser pulse comes as an ordered pair, where the first emission event within the pulse excitation window triggers the absorption and subsequent emission of a second photon. Unlike the first photon, the second photon has the entire excited-state lifetime to leave. This statement can be visualized by investigating the time-resolved probability mass functions for photodetection [74] defined by 2 2 p1(t1) = | ht1|φ1i | and p2(t1, τ) = | ht1, t1 + τ|φ2i | . (6.34)

The mass function p1(t1) represents the probability density for emission of a single photon at time t1 with no subsequent emissions, while p2(t1, τ) represents the joint probability density for emission of a single photon at time t1 with a subsequent emission at time t1 + τ. Additionally, p2(t1, τ) can be integrated along t1 or τ to yield p2(τ) or p2(t1), which give the probability density for waiting

τ between the two emission events or detecting a photon pair with the ﬁrst emission at time t1, CHAPTER 6. PHOTON EMISSION FROM A TWO-LEVEL SYSTEM 92 ] Γ [

) τ ( 2 p 0 . 3 a ]

Γ 0 . 2 / 1 [

t 0 . 1

0 . 0 0 . 3 b ]

Γ 0 . 2 / 1 [

t 0 . 1

0 . 0 0 . 3 c ]

Γ 0 . 2 / 1 [

t 0 . 1

0 . 0 0 1 0 1 2 Γ τ Γ p 1 ( t 1 ) , p 2 ( t 1 ) [ ] [ 1 / ]

Figure 6.4: Simulations showing time-resolved single-photon and two-photon emission from an ideal quantum two-level system under excitation by a Gaussian shaped pulse. (a)-(c) Probability mass functions for single-photon (green) and two-photon (red) detection showing the internal temporal structure of the photon pairs, for excitation by a 2π-pulse (a), 4π-pulse (b), and 6π-pulse (c). Colour plots show p2(t1, τ), while the traces to the left show p1(t1) and p2(t1), and the traces to the top show p2(τ). respectively. We explore these temporal dynamics for excitation by a 2π-, 4π-, and 6π-pulse in Figs. 6.4a, 6.4b, and 6.4c, respectively. First, consider excitation by the 2π-pulse: p2(t1, τ) captures the dynamics already discussed through having a high probability of the ﬁrst emission at time t1 only within the pulse window of 0.1/γ, but the second emission occurs at a delay τ later within the spontaneous emission lifetime τe. This eﬀect is most clearly seen in p1(t1), p2(t1), and p2(τ) [traces to the left and top of the colour plots in Fig. 6.4], where the density of photon pair emission being triggered at time t1, i.e. p2(t1), is maximized after π of the pulse has been absorbed (t1 = 0.15/γ) and reaches nearly unity. The second photon of the pair then has the entire lifetime to leave, as seen in the long correlation time for p2(τ). Meanwhile, the enhancement in photon pair production leads to a corresponding decrease in density of single-photon emissions p1(t1) around t1 = 0.15/γ.

Next, consider excitation by the 4π-pulse: p2(t1, τ) shows the eﬀects of an additional Rabi CHAPTER 6. PHOTON EMISSION FROM A TWO-LEVEL SYSTEM 93

oscillation that the system undergoes during interaction with the pulse. If the first emission occurs after π of the pulse has been absorbed, a remaining 3π can result in a second emission in two different ways: either after absorption of π additional energy or after absorption of 3π additional energy. On the other hand, if the first emission occurs after 3π of the pulse has been absorbed, only a remaining

π can be absorbed, resulting in a monotonic region of p2(t1, τ) just like for the 2π case. In either scenario, the probability of two emissions is most likely (but three emissions almost never happen) because a single emission converts an even-π-pulse into an odd-π-pulse, which anti-bunches the next emission. The high ﬁdelity of this conversion process can clearly be observed in p1(t1) and p2(t1).

The pair production is most likely after either π or 3π of the pulse has been absorbed (times t1 = 0.1 and t1 = 0.3, respectively), and it occurs with almost unity probability density. This means that if the ﬁrst photon is emitted at time t1 = 0.1 or t1 = 0.3, then the conditional probability to emit a second photon is near unity. As a result, p1(t1) and p2(t1) almost look like they were just copied a second time from the 2π-pulse scenario, conﬁrming our intuitive interpretation of the 4π-pulse scenario. These ideas trivially extrapolate to the 6π-pulse, where three complete Rabi oscillations occur, and the projections p1(t1) and p2(t1) are copied once more along t1.

Looking at the oscillations in p2(τ) for increasing pulse areas, one may notice a qualitative resemblance to the photon bunching [185] behind a continuous-wave Mollow triplet [83]. In fact, the underlying process where a photon emission collapses the system into its ground state, restarting a Rabi rotation, is responsible for the dynamics in both cases. However, our observed phenomenon has a very important diﬀerence: after a photodetection the expected waiting time for the second, third, and m-th photon emissions is identical in the continuous case, while our observed process dramatically suppresses P3.

6.3.2 Experiment

After having theoretically understood the process through which a quantum two-level system emits two-photon bound states through a complex many-body scattering phenomenon, we found experimental signatures of this two-photon process using a single transition from an artificial atom [60]. Our artificial atom of choice is an InGaAs quantum dot, due to its technological maturity and good optical quality [2]. The dot is embedded within a diode structure in order to minimize charge and spin noise (Fig. 6.5a and 6.5b), resulting in a nearly transform-limited optical transition [45]; lumi- nescence experiments as a function of gate voltage (Fig. 6.5c) reveal the charge-stability region in - which we operate [31] (Vg = 0.365 V). We used the X transition due to its lack of fine-structure, CHAPTER 6. PHOTON EMISSION FROM A TWO-LEVEL SYSTEM 94

(a) (b) (c) Au Ti CB 910 QDs GaAs X0 EF

λ cavity n+ GaAs 912 AlGaAs 35nm GaAs VB X- 914 Wavelength [nm] RF Intensity 0.34 0.39

18 pair DBR 916 0.2 0.4 Gate voltage [V]

Figure 6.5: Experimental details of the sample used as a quantum two-level system coupled to radiation modes. (a) Schematic of the weak cavity formed between the sample surface and an 18 pair distributed Bragg reflector (DBR)—the cavity’s sole purpose is to enhance collection efficiency and is only weakly coupled to the quantum dots. A Schottky diode is formed between the transparent titanium oxide gate and the n+ region of the diode. (b) Schematic of the energy-level diagram of the Schottky diode. The electrical bias on the diode controls the Fermi energy EF relative to the confined quantum dot levels. When EF is on resonance with the lowest-energy conduction-band (CB) quantum dot level, an extra charge tunnels from the Fermi reservoir into the dot, negatively charging it (X-). The excited state is then the trion complex which involves two electrons in the CB of the quantum dot and one hole in the valence band of the dot. (c) Photoluminescence versus applied gate bias and wavelength showing the charge-stability region of the X- transition. Inset: Voltage dependence of resonance fluorescence intensity under pulsed excitation, with optimal resonance fluorescence signal occurring at Vg = 0.365 V . which results in a true two-level system (at zero magnetic field) with an excited-state lifetime of

τe = 602 ps.

Single-photon source error rate

Exciting the system with laser pulses, we drove Rabi oscillations between its ground and excited states (Fig. 6.6a). However, because the artiﬁcial atom resides in a solid-state environment, it possesses several non-idealities that slightly decrease the ﬁdelity of the oscillations: a power-dependent dephasing rate arising from electron-phonon interaction [186] and an excited-state dephasing due to spin or charge noise [45]. Additionally, the quantum systems are very sensitive to minimal pulse chirps arising due to optical setup non-idealities [187]. As we saw in Sec. 6.3.1 and Table 6.1, when a quantum two-level system is excited by a short π pulse, it is most likely to emit only one photon. Occasionally, however, the system will emit two- photons together—this behavior is unwanted in a single-photon source and would cause errors if used in a quantum network. In particular, many types of quantum networks have extremely stringent CHAPTER 6. PHOTON EMISSION FROM A TWO-LEVEL SYSTEM 95

B (a) (b) (c) π 1 1 3π 5π 7π 0.1

[0] [0] 0.1 (2) (2) g g 8π 0.01 6π 0.01 4π

Intensity (arb. u.) 2π 1E-3 0.01 0.1 1 10 100 0.01 0.1 1 10 Pulse area (arb. u.) Pulse width [γT] Pulse width [γT]

Figure 6.6: Quantum two-level system excited by a short laser pulse. (a) Experiments showing Rabi oscillations between the ground and excited states of the system, excited by a short laser pulse around 10 ps. Data shows resonance fluorescence signal. (b),(c) Quantum two-level system driven by a laser pulse with area A = π as a single-photon source. Theoretical (b) and experimental (c) measured degrees of second-order coherence. multi-photon requirements [188, 189], and hence understanding the limitations of the quantum two- level system as a single-photon source is incredibly important. As discussed previously, the measured degree of second-order coherence g(2)[0] is used an experimental metric for a single-photon source since for short pulses it is closely related to P2, which is the source error rate of interest. From the analytic calculations in Sec. 6.3.1, I now show g(2)[0] as a function of laser pulse length in Fig. 6.6b. As we saw from Table 6.1, the error rate for short pulses is proportional to γTP , where TP is the pulse width. This trend is repeated in the plot of the coherence, except for long pulses the coherence levels to 1, representing a return to Poissonian counting statistics. By exciting our quantum dot two-level system with laser pulses in a cross-polarized reflectivity setup, we were able to completely reproduce this trend (Fig. 6.6c). Therefore, we have established an important limit that a quantum two-level system used as a single-photon source always has a finite error rate, even for arbitrarily short pulses.

Photon number oscillations

Next, we explored the power-dependence of the second-order coherence statistic for a fixed pulse length. For these measurements, we chose a longer fixed pulse length that experimentally optimized for the strongest signal. Specifically, we repeated the Rabi oscillation measurement but with longer pulses, τFWHM = 80 ps (Fig. 6.7a). Using power-dependent dephasing, excited state dephasing, and pulse chirp as fitting parameters [60], we obtained near perfect agreement between our quantum- optical model (blue) and the experimental Rabi oscillations. Then, we measured the g(2)[0] values of the emitted wavepackets, in order to study the photon bunching effects outlined in Sec. 6.3.1. CHAPTER 6. PHOTON EMISSION FROM A TWO-LEVEL SYSTEM 96

a b 1500 c 1.00 900 1200 0.75 900 600

[ n ] 0.50 E Counts 600 Counts 300 0.25 300

0.00 0 0 0 2 4 6 -100 -50 0 50 100 -100 -50 0 50 100 Pulse area [π] Time [ns] Time [ns]

d e 4 2 3 [0] [0]

(2) (2) 2 g 1 g

0 0 0 2 4 6 0 100 200 300 Pulse area [π] Pulse length [ps]

Figure 6.7: Experiments showing two-photon emission from a single artificial atom’s transition. (a) Experimental resonance fluorescence signal showing Rabi oscillations, scaled to quantum simulations of emitted photon number (blue). (b),(c) Under excitation by π- and 2π-pulses (green and red triangles in [a]), Hanbury-Brown and Twiss data respectively show anti-bunching (g(2)[0] < 1) [b] and bunching (g(2)[0] > 1) [c]. The measured values are g(2)[0] = 0.096 ± 0.009 and g(2)[0] = 2.08 ± 0.13, respectively. (d) Experimental second-order coherence measurements g(2)[0] versus pulse area showing oscillations between anti-bunching (at odd π-pulses) and bunching (at even-π- pulses). Blue curve represents quantum simulations of the time-integrated correlations g(2)[0] from the experimental system. Dashed black line represents statistics of the incident laser pulse. (e) Experimental second-order coherence measurements g(2)[0] versus pulse length (using 2π-pulses for excitation). Optimal bunching, and hence two-photon pair generation, occurs for the 80 ps pulse. Solid blue line represents emission from an ideal two-level quantum system, long dashed blue line represents inclusion of dephasing, short dashed blue line represents addition of a 2.7 % chirp in bandwidth, and short dotted blue line represents addition of a further 2.7 % chirp in bandwidth. (2) Again, dashed black line represents statistics√ of the incident laser pulse. Note, the errors in g [0] values for (d) and (e) are the standard m fluctuations in the photocount distribution [50]. The error in pulse area accounts for power drifts during the experiment and the errors in pulse lengths are least squares fitting errors to the pulse spectra. CHAPTER 6. PHOTON EMISSION FROM A TWO-LEVEL SYSTEM 97

Two typical experiments are presented in Figs. 6.7b and 6.7c, showing g(2)[0] ≈ 0 (anti-bunching) and g(2)[0] > 1 (bunching) for π- and 2π-pulses, respectively. A complete data-set is shown in Fig. 6.7d, with oscillations between anti-bunching at odd-π pulses and bunching at even-π-pulses. Using the same fitting parameters as in Fig. 6.7a, the correlation data are almost perfectly matched with our full quantum-optical model. Hence, we have found experimental evidence that suggests the artificial atom is affected by the predicted many-body two-photon scattering process that causes P2 oscillations out of phase from the Rabi oscillations. Finally, we investigated how non-idealities affect the bunching values (Fig. 6.7e) by experimentally characterising the emission at four pulse lengths. From our quantum-optical model, we see that bunching is strongest for the ideal case (blue line), and decreases for every added non-ideality (long dashed blue for dephasing, short dashed blue for additional 2.7% chirp in bandwidth, and short dotted blue for further 2.7% chirp in bandwidth), yielding excellent agreement with the data. Discussing these effects further, an enhanced pulse chirp decreases the fidelity of photon bunching due to the function of a large chirp to adiabatically prepare the system in its excited state [73], which decays with a single-photon emission. The minimal chirp that we observed can be removed with pulse compressors in future experiments to achieve an even better match with the ideal photon bunching curve. Additionally, at short pulse lengths the power-dependent dephasing results in anti-bunching. Due to the higher amplitude of shorter pulses (with fixed area), the dephasing rate diverges and the system acts as an incoherently pumped single-photon source. Thus, when including non-idealities we found the optimum bunching to occur at a pulse length of approximately 80 ps, which indicates where the two-photon process is strongest experimentally. Although we expect the two-photon emission is dominant, the non-idealities of the solid-state system could result in non- negligible P3 or higher Pm. This scenario is unfortunately not distinguishable through measuring g(2)[0] alone, but it is recently becoming possible to measure higher-order photon correlations that could help definitively identify regimes of operation where P2 P3 [51, 190]. Chapter 7

Conclusions and outlook

This dissertation has covered a broad range of topics surrounding the generation of pulsed nonclassical light. In particular, the contributions of this thesis are focused on the pulsed aspect, and represent a new way to view and understand how pulsed quantum light is created. I hope these frameworks can be applied in the future to problems beyond the standard Jaynes–Cummings system and the quantum two-level system. Below, I summarize some of the main take-aways and outlooks.

7.1 Homodyne interference

In Ch. 3, I presented a theoretical investigation of the differences between driving the cavity versus driving the atom of a Jaynes–Cumming system. Traditionally driving the atom has been perceived as providing a much greater nonlinear response from the system. However, we proved that driving the cavity can be thought of as driving the atom through the atom-cavity coupling g, with the convolution of the input coherent state with the cavity’s linear response. This is important especially for integrated nanophotonic implementations, where driving the atom is much more difficult than driving the cavity. Hence, potentially a drop filter could be used in nanophotonic devices to effectively observe the full nonlinear response from driving the atom [98]. Alternatively, the homodyning can be performed using a Fano-like structure [57], using the full mode structure of a photonic crystal cavity [56] or off-chip as well [92]. Finally, the type of analysis presented holds in any situation where an arbitrary nonlinear system is linearly coupled to a set of cavity modes, e.g., when analyzing a cavity linearly coupled to multiple quantum dots [191] or a multi-level system [192].

98 CHAPTER 7. CONCLUSIONS AND OUTLOOK 99

7.2 Single-emitter cavity QED

In Ch. 4, I presented a complete model for photon blockade and tunneling in III-V quantum dot cavity QED systems. We found that the pure Jaynes–Cummings model was incapable of accurately modeling either the spectra or photon statistics in transmission or emission from a strongly-coupled system based on InGaAa quantum dots. However, by including dissipation, dot-cavity detuning, pulsed dynamics, effects of phonons, blinking, and a new effect called self-homodyne interference we were able to almost perfectly model the nonclassical light generation from our cavity QED system. We showed experimentally how self-homodyne interference is a promising way to realize atom-like driving of a Jaynes–Cummings system in nanophotonics, as discussed theoretically in Ch. 3. By incorporating frequency filtering of the emission, we further showed highly indistinguishable photon generation and evidence for two-photon generation from such a strongly-coupled system [55, 58]. Looking towards future experiments, by adapting the self-homodyne interference technique to on-chip photonic crystal waveguide devices, we expect that this work could easily become a standard feature of optical solid-state platforms in enhancing nonclassical light generation [57]. This technique should also enable the first direct observation of a solid-state system’s higher-order Jaynes–Cummings structure and the efficient generation of N-photon states. Moving towards more interesting and complex level structures will allow for a much richer set of dynamics and possibilities for nonclassical light generation. For instance, charged III-V quantum dots in a magnetic field [32], III-V quantum dot molecules [193], and group-IV color-centers [25] may allow for the exploration of cavity QED with a single, multi-level quantum emitter coupled to a cavity [194]. These possess untapped level structures for improving photon blockade and studying multiphoton transitions, and they may allow experimentalists to more readily probe quantum nonlinearities in a solid-state environment. One of their most promising applications is to realize arbitrary single- photon generation in a solid-state nanocavity [195, 196]. In this scheme, the cavity mediates the generation or annihilation of an arbitrarily shaped single-photon through a Raman transition [197]. Such devices form the backbone of spin-photon interfaces in some theoretically proposed quantum networks [198]. Unlike the already-demonstrated spin-photon interfaces that rely only on weak cavity coupling, it is possible for a flying photonic qubit to be perfectly absorbed by a cavity QED device operating in the Raman single-photon regime. CHAPTER 7. CONCLUSIONS AND OUTLOOK 100

7.3 Photon scattering

In Ch. 5, I presented a powerful technique to integrate the Schrödingerequation for waveguide(s) coupled to a low-dimensional quantum system based on a coarse-graining of the temporal waveguide modes. Our technique works even in the presence of singularities and time-dependent system Hamil- tonians. The ease with which this method allows direct solutions to the Schrödingerequation for problems with infinite dimensions suggests that temporal modes may be an ideal basis to consider quantum-optical problems generally. This viewpoint seems to be gaining popularity in the field. The theory described in this work applied only to low-dimensional systems with a single ‘ground’ state, and the waveguides initially in the vacuum state. An obvious extension of this work is to treat systems with multiple ground states which, for example, could lead to a more thorough understanding of spin-photon entanglement [199] or optically-controlled single-photon phase gates [200]. Some of these scenarios might include cases in which a single photon impinges on an energy-nonconserving system [150], a situation we believe to be ideally suited for our formalism. We recently undertook this theoretical effort successfully with interesting results by considering the problem in the Heisenberg picture [135]. Lastly, I note that while this theory only considered situations in which the system of interest underwent Hamiltonian evolution, an exciting avenue for further research would be the possible extension of this work to directly computing N-photon scattering super-operators under the addition of dissipation. We recently developed this theory in Ref. [110]. This theory may be useful because, e.g., phonon-induced dephasing is an important consideration in the solid state [54, 103, 201], where a computation of the single-photon density matrix of the output field directly in the presence of this dephasing would be quite valuable. I believe that our framework constitutes a compelling argument to reconsider the dynamics of photon emission in a temporal mode basis and provides a way to answer many interesting questions going forwards.

7.4 Quantum two-level system

In Ch. 6, we discussed how a short laser pulse can induce two-photon emission over single-photon emission from a quantum two-level system. I expect future investigations on exploring optimal pulse shapes to enable much more eﬃcient and higher purity two-photon emission both from ideal and experimental two-level systems [61]. For example, the ideal 2π pulse shape is two extremely short π pulses separated by a brief delay. This results in P2/P0 → ∞. CHAPTER 7. CONCLUSIONS AND OUTLOOK 101

(a) temporal filter

HSPS

(b) frequency filter

HSPS

Figure 7.1: Principle of using a quantum two-level system as an entangled photon pair source. A two-photon state is created by driving the system with a short 2π pulse. The two photons are separable in both time and frequency, which can be used to demultiplex the photons into diﬀerent spatial channels either temporally (a) or in frequency (b). At this point, the output state can be used as an entangled pair source. If the application is sensitive to the shape of the photons, a dispersive element can be used to mode match each channel (not shown). Optional addition of a detector on one output channel (gray box) allows for the system to act as a heralded single-photon source (HSPS) conditioned on the current pulse found in i(t).

Interestingly, I note that the state appears mostly (if not completely) separable in either time or frequency. This means that the emitted state looks approximately like

p p ZZ |ψB,I(∞)i ≈ P0 |0i + P2 dτ1 dτ2 f(τ1)g(τ2) |τ1, τ2i (7.1) where f is a function with length on the order of the laser pulse and g is a function with length on the order of the spontaneous lifetime of the two-level system. Notably, this is very similar to the entangled photon pair produced by spontaneous parametric downconverison, for example [202]. However, the pair might be most useful if the two photons were separated into spatial channels, which could be done either temporally or in frequency (Fig. 7.1). Once the photons are spatially separated, a unitary mode converter or dispersive element could operate on the temporal mode profiles so the photons match their shape. This configuration could potentially allow for much higher-efficiency entangled photon pair production than spontaneous mixing processes, with a much higher rate as well. Exact numerical investigations into this idea such as computing the separability with Schmidt number and the effects of different filtering schemes will be left as a direction for future studies. CHAPTER 7. CONCLUSIONS AND OUTLOOK 102

7.5 Re-excitation in single-photon sources

The ability of a laser pulse to drive re-excitation in a two-level system fundamentally limits the error rate of its behavior as a single-photon source, as discussed in Ch. 6. As a result, we began looking for a system that was both experimentally accessible with solid-state systems and would show suppressed re-excitation. We recently found this behavior in the biexcitonic cascade of an InAs quantum dot. Like with the two-level system, we performed experiments and further developed photon emission theory for analyzing the biexcitonic system, which explains how it suppresses re-excitation and hence the error rate [203]. However, ﬁnding experimentally realizable systems that perform well on all fronts including indistinguishability and collection eﬃciency is still an open challenge.

7.6 Indistinguishability

Conspicuously absent from this thesis is a discussion of indistinguishability for single-photon sources. The reason for this choice is that fundamentally the ideal quantum two-level system always emits a pure state. Therefore, it is very simple to use some distance metric between two different single- photon states generated by a two-level system to compare the distinguishability. On the other hand, when any sort of nonideality enters, the discussion on indistinguishability becomes quite complicated. I believe going into these details would not fit succinctly in this thesis. However, I would like to briefly note that during my PhD, we did perform significant experimental and theoretical investigations regarding indistinguishable photon generation. We looked at theoretical performance of the Hong–Ou–Mandel and Mach–Zhender interferometers for measuring indistinguishably (and their relation to each other) [50]. We additionally performed measurements of the photon indistinguishably from a strongly-coupled system based on a quantum dot in a photonic crystal cavity [55]. Through these investigations, we understood that the Hong–Ou–Mandel or Mach–Zhender interferometers can only provide a heuristic for indistinguishability [110]. To me, there are many remaining and important questions. For example: how is the photonic density matrix computed in the presence of a driven few-level system interacting with a non-Markovian phonon bath, and how does this interaction affect the indistinguishability of a pulsed photon source as measured using some metric like trace purity? I’m sure answering this question, of great practical importance for single-photon sources, would be an interesting theoretical endeavor. Bibliography

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