Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Gitt, Sebastian
Master of Science - MSc
University of British Columbia
Cavity quantum electrodynamic systems hosted in photonic integrated circuits are being explored as a potential platform for realizing quantum computers, owing to their compact size and scalability. The qubits can be encoded in the electronic states of a localized defect centre in some otherwise uniform host material that comprises a microscopic optical cavity. A particular defect centre of much recent interest is the ⁷⁷Se⁺ donor in silicon, which has a dipole-allowed transition at 2.9 μm. The ground state of the transition is split due to a hyperfi ne interaction with the donor's nuclear spin, resulting in long-lived singlet and triplet states, separated by an energy of 6.87 μeV. These two states can be used to encode the states of a qubit. The fundamental building block of the photonic integrated circuit quantum computing platform involves placing the donor in an optical microcavity with a resonance frequency tuned to the transition frequency of only the singlet or triplet state to the excited state. The quantum state of the spin qubit then controls the transmitted and reflected photon statistics of the cavity in such a way that single photon detectors located in waveguides coupled to the qubit-containing cavities can be used to measure and control the spin state of the qubits via projective measurements. To entangle more than one qubit with high fidelity, photon loss must be minimized, a quantum engineering task that requires a rigorous treatment of open quantum systems. In this thesis, we develop a scattering-matrix based formalism to analyze the photon statistics of multiple defect centres embedded in microcavities evanescently coupled to waveguides. This formalism is used to quantify the properties of a single ⁷⁷Se⁺ donor in a photonic crystal cavity with an experimentally-accessible quality factor. More generally, we look at ensembles of atoms in cavities subject to inhomogenous broadening and dipole-dipole interactions. Modelling ensembles of ⁷⁷Se⁺ donors enables us to study and characterize these systems before site-selective single impurity implantation techniques are perfected.
Modelling Photon Statistics of CavityQED Systems for Integrated PhotonicQuantum ComputingbySebastian GittB.A.Sc., Queen’s University, 2018A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2021© Sebastian Gitt 2021The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the thesis entitled:Modelling Photon Statistics of Cavity QED Systems for Integrated PhotonicQuantum Computingsubmitted by Sebastian Gitt in partial fulfillment of the requirements for the degreeof Master of Science in Physics.Examining Committee:Jeff Young, Physics and AstronomySupervisorLukas Chrostowski, Electrical and Computer EngineeringAdditional ExamineriiAbstractCavity quantum electrodynamic systems hosted in photonic integrated circuits are beingexplored as a potential platform for realizing quantum computers, owing to their compactsize and scalability. The qubits can be encoded in the electronic states of a localized defectcentre in some otherwise uniform host material that comprises a microscopic optical cavity.A particular defect centre of much recent interest is the 77Se+ donor in silicon, whichhas a dipole-allowed transition at 2.9 µm. The ground state of the transition is split dueto a hyperfine interaction with the donor’s nuclear spin, resulting in long-lived singlet andtriplet states, separated by an energy of 6.87 µeV. These two states can be used to encodethe states of a qubit. The fundamental building block of the photonic integrated circuitquantum computing platform involves placing the donor in an optical microcavity with aresonance frequency tuned to the transition frequency of only the singlet or triplet state tothe excited state. The quantum state of the spin qubit then controls the transmitted andreflected photon statistics of the cavity in such a way that single photon detectors locatedin waveguides coupled to the qubit-containing cavities can be used to measure and controlthe spin state of the qubits via projective measurements.To entangle more than one qubit with high fidelity, photon loss must be minimized, aquantum engineering task that requires a rigorous treatment of open quantum systems. Inthis thesis, we develop a scattering-matrix based formalism to analyze the photon statisticsof multiple defect centres embedded in microcavities evanescently coupled to waveguides.This formalism is used to quantify the properties of a single 77Se+ donor in a photoniccrystal cavity with an experimentally-accessible quality factor. More generally, we lookat ensembles of atoms in cavities subject to inhomogenous broadening and dipole-dipoleinteractions. Modelling ensembles of 77Se+ donors enables us to study and characterizethese systems before site-selective single impurity implantation techniques are perfected.iiiLay SummaryRealizing a practical quantum computer requires reliably scaling up the number of so-calledqubits in a system, the quantum computer’s analogue to how a computer stores informationin 1s and 0s. Qubits can be implemented by placing atoms in optical cavities, where thestate of the qubit influences the light that gets transmitted and reflected from the cavity. Inthis way, the states of qubits can be determined by the transmitted spectrum of the cavityand qubit-qubit interactions can be facilitated with light. To do this, we must have a modeldescribing the interaction of light with an atom in a cavity that accounts for the varioussources of error introduced in an actual experimental implementation.ivPrefaceThis dissertation is original, unpublished, independent work by the author under the su-pervision of Dr. Jeff Young. The work presented is unpublished at the time of writing thisthesis.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction to Quantum Computing . . . . . . . . . . . . . . . . . . . . . 11.2 Different Implementation Schemes . . . . . . . . . . . . . . . . . . . . . . . 31.3 Integrated Photonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 A Protocol for Universal Quantum Computing . . . . . . . . . . . . . . . . 92 Cavity Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . 112.1 Cavity Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Light-Matter Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Extensions of the Jaynes-Cummings Model . . . . . . . . . . . . . . . . . . 162.4 Emitter Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5 Photon Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Cavity Transmission Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Input-Out Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 Scattering Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3.2 Scattering Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3.3 Calculating the Scattering Matrices . . . . . . . . . . . . . . . . . . 353.3.4 Calculating Transmission and Reflection . . . . . . . . . . . . . . . 373.3.5 Second-Order Correlation Function g(2)(t) . . . . . . . . . . . . . . 383.3.6 Third-order Correlations . . . . . . . . . . . . . . . . . . . . . . . . 393.4 Transmitted Photon Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 393.4.1 Inhomogeneous Broadening . . . . . . . . . . . . . . . . . . . . . . . 42viTable of Contents4 The Selenium Spin Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.1 Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2 Dipole-Allowed Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3 Parameters Related to the Dipole Coupling of Singly-Ionized 77Se+ to Radi-ation Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.1 Single 77Se+ Donor Coupled to a Cavity . . . . . . . . . . . . . . . . . . . 525.2 Many 77Se+ Atoms In a Cavity . . . . . . . . . . . . . . . . . . . . . . . . . 575.2.1 Inhomogeneous Broadening . . . . . . . . . . . . . . . . . . . . . . . 575.2.2 Dipole-dipole Interactions . . . . . . . . . . . . . . . . . . . . . . . . 596 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 61Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63AppendicesA Hamiltonian for Quantized Radiation . . . . . . . . . . . . . . . . . . . . . 68B Measuring Dipole Transition Strength from Absorption . . . . . . . . . 73viiList of Tables4.1 Character table for Td group [1]. . . . . . . . . . . . . . . . . . . . . . . . . 464.2 Direct product table for the Td double group, showing the decompositionsof products of irreducible representations within the group [1]. For example,an operator with symmetry E acting on a state with symmetry T1 will havesymmetry that can be represented as T1 ⊕ T2. . . . . . . . . . . . . . . . . 494.3 Summary of relevant optical transition parameters for the 77Se+ donor. . . 51viiiList of Figures1.1 The two basis states of a qubit can be represented as two antipodal pointson the surface of a unit sphere, where the rest of the surface represents thesuperposition of states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 (a) A hexagonal-lattice photonic crystal formed by etching cylinders in thetop layer of a silicon-on-insulator substrate. (b) Band structure of the pho-tonic crystal depicted in (a). A band gap is only present for the TE modes[2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 (a) A cartoon of an L3 cavity, which is a 3-hole defect in a hexagonal lattice ofcylinders with a different refractive index than their host substrate. (b) Themagnitude of the E-field for the fundamental cavity mode. (c) Component ofthe E-field in the xˆ direction. (d) Component of the E-field in the yˆ direction.The maximum mode amplitude coincides with the centre of the L3 cavity. . 81.4 (top) A single photon injected through the input port, if detected in theOutput 1(2) port will project the qubit into the spin-down(up) state. The actof measurement thus performs a Pauli Z measurement on the qubit. (bottom)A Pauli Z1Z2 parity measurement is performed on the two embedded qubits.The parity is positive (negative) if the single photon is detected in Output1(2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 The energy levels of a cavity are modified by the presence of a two-levelsystem on resonance with the cavity (ωe = ωc), resulting in unequally-spaceddressed states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 The eigenenergies for the first three rungs of the Tavis-Cummings model arepresented for up to three atoms. The eigenenergies become more equallyspaced as the number of atoms increases. Two-fold degeneracies are repre-sented by two thinner, closely spaced levels. . . . . . . . . . . . . . . . . . 172.3 a) Physical setup for a Hanbury-Brown Twiss interferometer. A signal issplit into two paths by a beamsplitter, which are then measured by separatedetectors. For a source emitting single photons, there will be no coincedentdetections. b) Second-order correlation functions for different optical sources.Only light from a quantum source can have a g(2)(τ) less than unity, known asanti-bunching. c) Generalization of the Hanbury-Brown Twiss interferometerfor measuring g(3)(t1, t2, t3). . . . . . . . . . . . . . . . . . . . . . . . . . . . 21ixList of Figures3.1 Schematic for a 2-sided cavity in the input-output formalism. The systemHamiltonian is comprised of at minimum the empty cavity, which interactswith outside bath Hamiltonians, consisting of continua, via linear couplingterms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 The transmission and reflection spectra for a bare optical cavity are given bya Lorentzian centered at the resonant frequency of the cavity and a widthdepending on the cavity coupling parameter κ. . . . . . . . . . . . . . . . . 273.3 Left: The transmission spectrum for an optical cavity versus increasing atom-cavity coupling g with Γc = 0, γ = κ/100 and ωe = 0. Right: Transmissionspectra for cooperativities of 0, 1 and 5 in blue, orange and green respectively.In the intermediate coupling regime C ≈ 1 a narrow dip appears in thetransmission spectrum. Beyond this, in the strong coupling regime, twoseparate peaks emerge with a polaritonic linewidth (κ+ γ)/2. . . . . . . . . 293.4 Left: Transmission spectrum for a cavity containing a single atom versusdetuning. Right: Transmission spectra for detuning of the atomic transitionfrequency ωe = {0, 0.2, 0.9}κ for the blue, red, and green spectra respectively. 303.5 (a) Second-order correlation function g(2)(0) as a function of detuning andatom-cavity coupling for a cavity containing a single atom on resonance withthe cavity. (b) Transmission and g(2)(0) spectrum for g = 2κ. (c) Trans-mission and g(2)(0) spectrum for g = 0.4κ. (d) Transmission and g(2)(0)spectrum for g = 0. For an empty cavity g(2)(0) = 1 for all frequencies,whereas large bunching appears at the cavity frequency for non-zero atom-cavity coupling as a consequence of reduced transmission through the cavity.In the strong coupling regime, g(2)(0) is less than 1, indicating that 2-photonstates are suppressed due to the anharmonicity of Jaynes-Cummings ladder. 403.6 (a) Second-order correlation function g(2)(0) and transmission spectra for acavity containing 1, 10, and 20 atoms on resonance with the cavity. Whenthe atoms have identical transition frequencies, the splitting increases as√Nwhere N is the number of atoms in the cavity. For increasing number ofatoms, minimum value of g(2)(0) < 1 increases. (b) Same as (a) but with theemitters detuned by ωe − ωc = 2κ. The detuning can enhance the photonblockade for one of the eigenstates of the cavity, decreasing the minimumvalue of g(2)(0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.7 (a) Left: Second-order correlation function g(2)(0) as a function of time for anemitter on resonance ωe − ωc = 0. Right: Transmission spectra and g(2)(0)are shown for times {0, 4, 7}κt. At later times, interference effects resultin fluctuations in g(2)(0). (b) Second-order correlation function g(2)(0) as afunction of time for a detuned emitter ωe−ωc = 2κ. Anti-bunching depth atthe higher-energy transmission peak is increased due to the detuning, whereasthe other dip fluctuates in time. . . . . . . . . . . . . . . . . . . . . . . . . . 423.8 Inhomogeneous broadening of atoms in a cavity with with parameters {g, γ, FWHM}= {1/100, 1/500, 10}κ. For a Gaussian distribution, strong coupling can beobserved if the collective Rabi frequency is larger than the FWHM of theemitter distribution. This is not the case for a Lorentzian distribution, whichhas far-reaching wings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43xList of Figures4.1 a) The 6 ellipsoidal conduction bands in silicon from effective mass theory.b) The indirect band gap of silicon with additional hydrogen-like states dueto the 77Se+ impurity. The band gap energy is labelled Eg. . . . . . . . . . 454.2 Including spin degeneracy, effective mass theory yields 12 degenerate 1sground states for the donor’s electronic wavefunction. However, the pres-ence of the substitutional donor alters the symmetry of the system, resultingin valley-orbit splitting. The states are further split due to a spin-valley in-teraction as well as a hyperfine interaction with the donor’s nuclear spins.This additional nuclear degree of freedom results in the two 1s:A states to besplit into a triplet and a singlet state which can be used to encode the stateof a qubit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.1 Schematic of the optical circuit for a 1-parity measurement, including thestates corresponding to the waveguides, cavity, atom, and loss channel. Theupper and lower waveguide couple to the cavity via κ1 and κ2, respectively,and Γc represents the coupling of the localized optical mode to radiationmodes not including those associated with the input and output from thewaveguides. Throughout this chapter, we restrict ourselves to systems whereκ1 = κ2 = κ/2. Here, reflection corresponds to the signal measured at theport for aout and transmission corresponds to the signal measured in the portby bout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2 (a) pmax as a function of κ and Γc for a single77Se+ donor, showing theoptimal waveguide-cavity for a given cavity. Pink and red crosses are at theoptimal coupling parameters for a cavity with a quality factor of 105 or 106,respectively. (b) On- and off-resonant transmission and p(ω − ωc) for theparameters at the pink cross (Γc = 0.43 µeV, κ = 3.9 µeV). (c) On- andoff-resonant transmission and p(ω − ωc) for the parameters at the red cross(Γc = 4.3 µeV, κ = 13 µeV). (d) Same as (a) but with effective mode volumeVeff = 0.1. (e) and (f) show close-ups of (a) and (d) near the origin. . . . . . 545.3 Transmission and reflection spectra for single 77Se+ donor with cooperativityC = 1.6 and Γc = κ/10 but different Veff and κ. (a) Veff = 1 and κ = 4.3 µeV.(b) Veff = 0.1 and κ = 43 µeV. (c) Veff = 0.01 and κ = 430 µeV. . . . . . . . 555.4 Transmission and reflection spectra and p(ω) for a single donor at the anti-node of a cavity with parameters κ = 3.9 µeV and Γc = 0.43 µeV . The caseof a donor with an off-resonant detuning ωe >> g is shown by the red dashedline for comparison. a) Veff = 1 b) Veff = 0.1 c) Veff = 0.01. . . . . . . . . . 565.5 (a) Transmission spectra for ensembles of 77Se+ donors with inhomogeneousbroadening given by a Gaussian with FWHM of 14 µeV, representing isotopicimpurities in NatSi. Donors are randomly distributed in an L3 cavity withκ = 43 µeV. The numbers in each figure correspond to the density of 77Se+donors per cm3. (b) Same parameters as (a) but in isotopically pure 28Siwithout inhomogeneous broadening. . . . . . . . . . . . . . . . . . . . . . . 57xiList of Figures5.6 Transmission spectra for 10 different ensembles of randomly distributed 77Se+donors in an L3 cavity with parameters ρ = 1012 atoms/cm3, κ = 3.9 µeVand Γc = 0.43 µeV, corresponding to a loaded quality factor QL = 100 000.Transition frequencies are drawn from a Gaussian distribution with a FWHMof 14 µeV, corresponding to broadening from isotopic inhomogeneities innatural silicon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.7 Transmission spectra with and without dipole-dipole interactions for increas-ing densities of 77Se+ donors without any inhomogeneous broadening present.The emitters are on resonance with a cavity with κ = 43 µeV and confinedto a cubic volume with a side length of 100 nm at the center of an L3 cavity. 59xiiChapter 1IntroductionAs one of the of the earliest proponents of quantum computing, Richard Feynman exploredthe limitations of simulating quantum systems using classical computers nearly forty yearsago [3]. While a classical computer requires exponentially increasing resources to simulatephysical systems increasing linearly in size, he explored the idea of instead performing thesimulation with a quantum system, in which case the computer would instead scale linearlywith the size of the simulated system. Thus, there exist sufficiently large systems that aclassical computer cannot simulate in a reasonable amount of time, yet may be tractablewith a quantum computer; this would demonstrate so-called quantum supremacy. Nearlyforty years later, Martinis et al in 2019 claimed to have achieved quantum supremacy bybuilding a processor that samples the random output from a system containing 53-qubits,a task that would take a classical computer thousands of years to perform [4]. However,it was soon shown that the classical computation could be reduced to a matter of days bystoring parts of the calculation with external memory [5]. One year later, Pan et al demon-strated a computational advantage by injecting up to 73 indistinguishable photons into aninterferometer and sampling the output statistics ∼1014 faster than achievable by classicalsimulations [6]. Nevertheless, it remains to be demonstrated that quantum computers canperform useful tasks that would replace their classical counterparts.Simulating physical systems for purposes such as drug development, developing usefulnew materials, and general research is itself a rich and promising field of applications, butquantum computers are also suited for tackling a variety of clasically-intractable problemssuch as the travelling salesman problem, which involves finding the shortest possible routethrough a set of different locations. An important application is factoring integers usingShor’s algorithm, which would render obsolete the RSA encryption protocol that is used toperform secure data transmissions online [7]. One of the reasons behind Shor’s algorithmefficiency is that it takes advantage of the quantum Fourier transform, which is much moreefficient than its classical counterpart [8]. These quantum algorithms manipulate the funda-mentally quantum properties of superposition and entanglement inherent to the quantumbits, or qubits, of a quantum computer.1.1 Introduction to Quantum ComputingThe distinguishing feature of quantum computing is that operations are performed usingqubits. Classical bits can only be in either a 0 or a 1 state, whereas quantum bits can bein a superposition of the two states until a measurement collapses the state into either a 0or a 1. The state of a qubit can be represented using Dirac notation as|ψ〉 = α |0〉+ β |1〉 , (1.1)11.1. Introduction to Quantum ComputingFigure 1.1: The two basis states of a qubit can be represented as two antipodal points onthe surface of a unit sphere, where the rest of the surface represents the superposition ofstates.where α, β are complex-number probability amplitude coefficients that satisfy the normal-ization condition|α|2 + |β|2 = 1. (1.2)Since the probability amplitudes can be complex, the two-level system represented by (1)spans a state space C2, which can be conveniently represented by the so-called Bloch sphere,shown in Figure 1.1, where α = cos(θ/2) and β = eiφsin(θ/2). A unique property of qubitsis that they can become entangled, a fundamentally quantum property of a multi-particlesystem. An example of 2-qubit entangled quantum state is|0102〉+ |1112〉√2, (1.3)where the subscripts label distinct qubits. A characteristic of an entangled state is that itcannot be written as a direct product of states associated with each individual qubit. Aconsequence of this is that measuring one of the qubits permits one to determine the stateof the other qubit: the two qubit states are correlated.Much like a classical computer, a quantum computer can be designed to consist of aset of gates that perform logical operations on input qubit states. A classical computeris said to be universal if it can map any input state to any output state, which can beaccomplished using just AND, OR and NOT gates. Similarly, a quantum computer is saidto be universal if it contains a set of gates that span the entire computational basis, allowingany possible operation to be performed. An example of such gates are the Hadamard, T,and Controlled-NOT (CNOT) gates, each of which demonstrate an aspect that is uniqueto quantum computing.The Hadamard gate acts on single qubits and its operator can be written in matrixnotation asH =1√2(1 11 −1), (1.4)21.2. Different Implementation Schemeswhere the states are represented as|0〉 =(01), |1〉 =(10). (1.5)Thus, the Hadamard gate puts states into superposition asH |0〉 → |0〉+ |1〉√2,H |1〉 → |0〉 − |1〉√2.The CNOT gate operates on 2-qubit input states, and its operator is given byCNOT =1 0 0 00 1 0 00 0 0 10 0 1 0 , (1.6)where the 2-qubit computational basis states are defined as|00〉 =1000 , |01〉 =0100 , |10〉 =0010 , |11〉 =0001 . (1.7)The CNOT gate has the effect of flipping the state of the second qubit if the first qubit isin state |1〉, and is necessary for creating entangled states. Finally, the T-gate is capable ofchanging the phase of a state, and is given byT =(1 00 eipi4). (1.8)Together, these gates fulfill the condition of universality, and given enough gates, are able tocreate arbitrary unitary operators to arbitrary precision. Crucially, however, the Solovay-Kitaev theorem states that a finite number of ideal gates rapidly becomes a very goodapproximation to an arbitrary unitary operator [9]. Currently, hardware is the biggestobstacle to achieving a quantum advantage, as there are difficulties associated with scalingreal systems that incorporate imperfect gates. There are numerous competing approaches,each with their own distinct advantages and disadvantages.1.2 Different Implementation SchemesIn 2000 DiVincenzo proposed five necessary criteria for implementing physical quantumcomputers that continue to guide current experimental efforts. In addition to having a uni-versal set of gates, he stipulated that a system must be scalable, initializable to a well-definedstarting state, have long decoherence times, and must have qubits that can be efficientlymeasured [10]. The following 20 years have brought us to the Noisy Intermediate-Scale31.2. Different Implementation SchemesQuantum (NISQ) technology era, where quantum computers are approaching 100s of qubits,and much attention is being given to potential applications for this intermediary stage [11].To go beyond this stage, the two most pressing criteria are maximizing decoherence timesand being scalable.In classical computers, random noise due to thermal fluctuations of the electric po-tential or current can change the states of classical bits, which would cause errors in thecomputation if not detected and corrected. Consequently, all classical computers employerror-correction schemes to reduce the actual error rate to some arbitrarily low value. Forinstance, a single classical bit can be encoded in three physical bits, where the majority ofthe three physical bits decides the state of the encoded bit; thus, even if one of the threephysical bits is flipped due to noise, the state of the encoded bit remains unchanged. Inquantum computers, an analogous source of error would be if some noise source flippedthe basis states, taking the qubit from some state α |0〉 +β |1〉 to α |1〉 +β |0〉. However,quantum computers are much more fragile, because any noise source that merely alters therelative phase of α and β is enough to spoil the computation if left uncorrected. Further-more, error-correction schemes involving redundant copies of physical bits do not work forqubits, since the no-cloning theorem forbids the copying of quantum information [12].The process whereby the defining probability amplitudes of a qubit state decay in anunwanted fashion is generally referred to as decoherence, and occurs due to physical in-teractions with a noisy environment. Qubits can lose information in one of two ways: (i)dephasing interactions, whereby only the relative phase of α and β change, which is charac-terized by its T2 time, and (ii) population decay, whereby the squared magnitudes of α andβ change with time, which is characterized by its T1 time. Since a population decay inter-action necessarily dephases the qubit state, 1/2T1 must be ≤ 1/T2. The precise manner inwhich particular dephasing processes introduce errors into a quantum computation dependson many factors that depend on the physical implementation. Bearing this in mind, we willreview some of the different approaches taken to implementing quantum computers.One of the oldest approaches consists of using an ion trap to localize ions for an indefinitelength of time at a specific location [13]. In this system, the qubit basis states are two distinctelectronic states of the ions that couple via the dipole moment operator, and hence are easilymanipulated by resonant laser pulses. Qubit-qubit interaction is facilitated via phononmodes that exist across the entire ensemble of atoms. Since only a certain number of atomscan be stored in each trap, scalability can only be achieved by connecting multiple trapsand moving atoms between them [14]. Preventing decoherence during this process, alongwith the short-lived nature of the collective phonon modes, present formidable challengesto this approach.Another popular approach is the superconducting qubit approach with Josephson junc-tions. Qubits can be created by small superconducting circuits that contain some combina-tion of inductive and capacitive elements and one or more Josephson junctions. The statesof the qubit can be encoded in the Josephson phase, the flux within the inductor, the chargeon the capacitor, or some combination of these. Qubits can be directly coupled togetherby circuits, limiting their interactions to nearest neighbours, but arbitrary couplings canbe facilitated by tuning qubits into resonance with an extended, intermediary cavity atresonance, allowing excitations to be transferred between them [8]. Whereas ion traps canoperate at room temperature, superconductor-based quantum computers currently must41.2. Different Implementation Schemesoperate at milli Kelvin temperatures. Though many recent materials science advancementshave made marked improvements to their decoherence time, they still decohere very quickly,and will therefore require significant error correction to be viable.In the aforementioned schemes, qubit decoherence was one of the largest obstacles torealizing a physical quantum computer. This can be avoided by using photonic qubits, wherethe state of the qubit is encoded in the photon’s spatial or temporal location, its frequency,or polarization. Photons can easily be initialized, operated on, and measured, but thedifficulty lies in creating 2-qubit gates, which requires a strong photon-photon interaction.Initial attempts to mediate a strong photon-photon interaction were based on the Kerreffect, utilizing third-order non-linearities in the host [15]. This effect is extremely weak,resulting in gates that only work 0.1% of the time, rendering it largely ineffectual. However,in 2000, the so-called KLM protocol was conceived, which generates an effective photon-photon interaction using only beam-splitters, photon detectors, and ancilla photonic qubits[16]. The photon-photon interactions in the KLM protocol are, however, fundamentallyprobabilistic, working only 25 percent of the time, but they can be parallelized for a successrate of n2(n+1)2where n is the number of gates. The major drawback to this implementationis that there is a large resource overhead required to reach an acceptable success rate.In a realistic implementation, as many as 100 physical qubits would be needed for eachactual qubit [17]. Connecting such a large number of physical qubits requires many opticalelements, such as waveguides and beamsplitters, to be integrated on a chip; since suchcomponents cannot be fabricated to perfectly confine photons to the chip, photon loss isinevitable, and error correction protocols capable of dealing with this loss of information arerequired. Like ion traps, pure photonic based approaches can operate at room temperature.An approach which aims to balance scalability with robustness to decoherence is inte-grated photonic computing with solid state qubits. In this approach, qubit states are chosenfrom a set of hyperfine-split atomic states or energy levels of an artificial ”quantum dot”.The qubit states are chosen based on their long T2 times, while also considering their opticalcoupling to higher-lying electronic states that can be used to probe the qubit states usingvisible or near-infrared wavelengths of light. By placing the qubits in an optical cavity atresonance with the transition to one of the two qubit states, light will only be transmittedthrough the cavity for one of the qubit states, creating an effective optical switch controlledby the quantum state of the qubit, thus allowing the state of the qubit to be determinedusing optical readout [18]. Additionally, Waks et al showed that entanglement can be gen-erated by cascading two cavity-qubits alongside two waveguides and probing them withnon-classical states of light, as the photons can take multiple paths that depend on thestates of both qubits [18]. A possible quantum computing architecture using these buildingblocks with the qubits encoded in the nuclear spin states of nitrogen vacancy (NV) centresin diamond microcavities, interconnected using optical fibre, was first proposed by Nemotoet al [19]. While this approach, like the all-photonic KLM scheme, suffers from inevitablephoton loss, it may be viable for creating a quantum computer in the NISQ-era, where werequire on the order of 100 qubits; the use of diamond and optical fibre, however, makesfurther scalability very difficult. Recently, it has been suggested that the NV centre couldbe replaced by a colour centre defect in isotopically pure silicon, and a similar architecturecould be implemented using integrated silicon photonic circuit technology to address thescalability of optical fibre interconnects [20] [21]. This system will be described in more51.3. Integrated Photonicsdetail below, in Section 1.4, but the next section will first provide some basic backgroundinformation about integrated photonic circuit technology.1.3 Integrated PhotonicsIntegrated photonic circuits consist of microstructures etched into semiconductor wafers tofacilitate routing and manipulating light in the plane of the wafer. Light is guided along thechip by waveguides, which typically consist of dielectric slabs that support electromagneticmodes that confine light to propagate only in a two-dimensional plane via total internalreflection. Optical cavities can be achieved by confining light in all three dimensions usingring resonators, micropillars, microdiscs, microtoroids, microspheres, or photonic crystals[22]. These various cavity geometries support a variety of 3D confined optical modes charac-terized primarily by their quality factor, Q, and their mode volume, Vm. The quality factoris defined in terms of the energy of a mode E and the energy dissipated per oscillation ∆EasQ =E∆E. (1.9)A cavity with a large quality factor will therefore sustain excitations for a longer timebefore losing them to the environment. To maximize the efficiency of the interaction oflight in the cavity mode with an embedded defect centre having a resonant dipole-allowedtransition, one also wants to minimize the mode volume. This is because the peak electricfield associated with one trapped photon of energy ~ω in a mode with volume Vm is [23]Epeak =√~ω2Vm(1.10)where  is the dielectric constant of the space occupied by the mode.Each cavity type has its own fabrication challenges, and, for various reasons, some typesare best suited to certain host substrates. For instance, micropillar cavities require two dif-ferent substrates with significantly different refractive indices but similar lattice constants,and are commonly fabricated using AlAs/GaAs layers, whereas there is no similar approachwith silicon. In this thesis we focus on photonic crystal cavities, which are a popular choicesince their parameters can be easily tuned to achieve different quality factors, mode volumes,and resonant frequencies in the popular silicon-on-insulator integrated photonic platform.Furthermore, their resonant frequencies can be tuned in controlled environments by theadsorption of inert gases such as nitrogen or xenon [24].Photonic crystals are structures with a periodically varying dielectric constant wherethe lattice constant is chosen such that certain frequencies of light destructively interfere,resulting in an electromagnetic band gap. They are commonly implemented in silicon usinga silicon-on-insulator substrate as depicted in Figure 1.3 (a), where total internal reflectionconfines the propagation of electromagnetic modes to a two-dimensional plane in a slabof silicon above a layer of silicon dioxide. The electromagnetic modes supported by thephotonic crystal must conform to the periodicity as the lattice and satisfy Bloch’s theoremEk(r, ω) = eik·ruk(r, ω), (1.11)61.3. Integrated Photonicswhere k is the wave vector labelling each mode, with a direction restricted to the two-dimensional plane of the slab containing the photonic crystal, and uk(r, ω) is a functionwith the periodicity of the lattice. Such a photonic crystal will support both transverse-electric-like (TE) and transverse-magnetic-like (TM) modes, which respectively have almostno electric and magnetic field components in the direction normal to the slab containingthe photonic crystal. The photonic band structure of a typical hexagonal lattice is depictedin Figure 1.3 (b), which only has a band gap for the TE modes. Superimposed on the bandstructure are light cones representing free radiation modes satisfying the dispersion relationω ≥ c|k|/n, where n is the refractive index of the material surrounding the slab.Figure 1.2: (a) A hexagonal-lattice photonic crystal formed by etching cylinders in the toplayer of a silicon-on-insulator substrate. (b) Band structure of the photonic crystal depictedin (a). A band gap is only present for the TE modes [2].Microcavities that confine modes in all 3 dimensions can be formed by introducing adefect within the 2D photonic crystal. The most common cavity types in hexagonal photoniccrystals are H- and L-type cavities, which are respectively circular and linear defects. An L3cavity, as well as the E-field associated with its fundamental mode, are depicted in Figure1.3. Traces of the periodically-etched cylinders are discernible in the E-field distributionof the cavity mode; the cylinders closest to the cavity are commonly modified in size andlocation to reduces the coupling to free space radiation modes, maximizing the qualityfactor as well as resulting in a smoother mode profile [25]. Machine learning techniquesand genetic algorithm optimization have been applied to modify more holes than just thenearest neighbours to yield theoretical quality factors well over a million, but these minoradjustments are typically not robust to the imperfections introduced by fabrication, and itis thus usually sufficient to only deal with the nearest holes [26]. These cavities are ableto support an abundance of different modes other than the fundamental, each with theirown resonance frequency, quality factor, mode volume and polarization [27]. In practice thecavities are designed so that the fundamental mode frequency of interest is well-separated71.3. Integrated Photonicsfrom the other modes, so it is a good approximation to treat them as single mode cavities.Figure 1.3: (a) A cartoon of an L3 cavity, which is a 3-hole defect in a hexagonal latticeof cylinders with a different refractive index than their host substrate. (b) The magnitudeof the E-field for the fundamental cavity mode. (c) Component of the E-field in the xˆdirection. (d) Component of the E-field in the yˆ direction. The maximum mode amplitudecoincides with the centre of the L3 cavity.One of the promising aspects of an integrated photonics implementation of quantumcomputing is that it takes advantage of the already mature field of integrated photonics, asopposed to having to develop the hardware from scratch. Another major advantage is thatthe system can be incredibly compact, fitting thousands of qubits on a single millimeter-sized chip.81.4. A Protocol for Universal Quantum Computing1.4 A Protocol for Universal Quantum ComputingRecently, an architecture for performing universal quantum computing was proposed thatuses a series of cavity-coupled emitters connected by waveguides that are interfaced tosources and detectors of single photons.[21]. 1- and 2-qubit parity measurements and localgates can be used to construct 3D cluster states. Once such a cluster state is prepared, acomputation is done by performing a sequential series of measurements on it.The protocol relies on the fact that the optical transition of the atom will only beresonant with the cavity for one of the qubit states. Therefore, parity measurements can beperformed by side-coupling the cavity qubit to two waveguides, as illustrated in Figure 3.1.If the qubit is in the up state, the optical transition from it to a higher-lying electronic statein the qubit is chosen to be resonant with the cavity mode, and a pulse of light enteringthrough the first input channel would exit via the second channel. If the qubit is in thedown state, the frequency at which it is optically coupled to the higher lying qubit stateis off resonance with the cavity mode, and the pulse of light will pass by the cavity andexit through the first channel. Hence, the state of the electronic-state encoded qubit canbe reliably measured by performing an optical transmission measurement. Qubits can beentangled using 2-qubit parity measurements, where the signal interacts with two cavityqubits before the output is measured.Figure 1.4: (top) A single photon injected through the input port, if detected in the Output1(2) port will project the qubit into the spin-down(up) state. The act of measurement thusperforms a Pauli Z measurement on the qubit. (bottom) A Pauli Z1Z2 parity measurementis performed on the two embedded qubits. The parity is positive (negative) if the singlephoton is detected in Output 1(2).By using active switches, qubits can be arranged in so-called unit clusters, where both 1-and 2-qubit parity measurements can be performed on adjoining qubits. These unit clusterscan then be joined together in a process known as fusion, allowing an arbitrarily largecluster state to be generated. Since this protocol relies on accurate parity measurements,it is important to have a detailed model of photonic transmission through cavity qubits,which is the topic of this thesis. In the next chapter, we will begin by modelling closed91.4. A Protocol for Universal Quantum Computingquantum systems, which will give insight into the light-matter interaction that takes placein the cavity. In Chapter 3, we will look at open quantum systems, as ultimately the qubitsmust be probed and their state measured, necessitating their interaction with an engineeredexternal environment consisting of a continuum of evanescently-coupled waveguide modesand free (out-of-plane) radiation modes associated with photon loss.10Chapter 2Cavity Quantum ElectrodynamicsCavity QED systems are a promising way to realize quantum computers due to their com-pact size and scalability. Qubits can be encoded in the occupancy of electronic or nuclearstates of a localized defect centre in some otherwise uniform host material that comprises amicrosopic optical cavity. If the encoded states of the defect centres have a dipole-allowedtransition to an excited electronic state at an optical frequency close to a resonant mode ofthe optical microcavity in which they reside, the states of nominally identical qubits, eachembedded in nominally identical microcavities, can be efficiently measured and entangledusing photons that propagate in waveguides connected to the microcavities, as proposedby Waks [18]. The first step towards modelling such a protocol is to first study the light-matter interaction between the cavity mode and the electronic state of the defect, which isthe subject of this chapter. We defer studying interactions with an outside environment toChapter 3.2.1 Cavity QuantizationBoth the electronic states of the defect centre and the electromagnetic field associated withthe cavity mode need to be treated quantum mechanically. The optically active electronicstates can often be approximated as those corresponding to the solution of the Schro¨dingerequation for a single electron in some crystal potential centred on the defect. The quantumtreatment of the electromagnetic field in the case of textured dielectric cavities mimics thetextbook quantization of the vacuum field, allowing for more general modes than planewaves. In particular, Appendix A shows how this formalism results in the following Hamil-tonian that represents the energy operator for photons in a multi-mode cavityHˆR =∑`~ω`(aˆ†`aˆ` +12), (2.1)where ~ is Planck’s constant, aˆ†` and aˆ` are the creation and annihilation operators corre-sponding to each mode `, and ω` is the frequency of mode `. For a single-mode cavity, theHamiltonian simplifies toHˆR = ~ω(aˆ†aˆ+12), (2.2)which describes a ladder of states separated by ~ω with a zero-point energy of ~ω/2. Thequantized field operators are given byAˆ(r, t) =∑`√~n32ω`λ30Veff[aˆ`e−iω`tu∗` (r) + aˆ†`eiω`tu`(r)], (2.3)112.2. Light-Matter InteractionEˆ(r, t) =∑`i√~ω`n32λ30Veff[aˆ`e−iω`tu∗` (r)− aˆ†`eiω`tu`(r)], (2.4)Bˆ(r, t) =∑`i√~n32ω`λ30Veff[aˆ`e−iω`tu∗` (r)− aˆ†`eiω`tu`(r)], (2.5)where n is the index of refraction, λ0 is the wavelength in vacuum, 0 is the permittivity offree space, u`(r) is the mode-shape function for each mode ` and the effective mode volumeVeff is defined asVeff =∫dr(r)|Eˆ(r)|2(r0)|Eˆ(r0)|2, (2.6)where (r) is the permittivity of the cavity, r0 is the location where (r)|Eˆ(r)|2 is maxi-mized, and the integration is performed over the volume where the E-field is non-zero. Thepermittivity is related to the refractive index n by (r) = n2(r).2.2 Light-Matter InteractionThe electromagnetic field interacts most strongly with an atom’s loosely bound valenceelectron via the Lorentz force, enabling changes in the state of the electron. To capture thedominant contributions to this interaction we follow the work of [23] and begin with thequantized Hamiltonian for an electron localized to some defect site with a spatially-varyingpotential V (r)Hˆ0 =12mPˆ 2 + V (r), (2.7)where m is the electron mass and Pˆ is the momentum operator. In the presence of anelectromagnetic field, the canonical momentum is modified by the vector potential and anadditional scalar potential acts on the electron as followsHˆ(r, t) =12m[Pˆ + eAˆ]2 + eΦ(r, t) + V (r). (2.8)Expanding the terms involving the vector potential, the Hamiltonian will involve morethan one interaction term. However, with a judicious gauge choice, the Hamiltonian can befurther simplified. The vector and scalar potentials are invariant under the transformationsΦ′(r, t) = Φ(r, t)− ∂χ(r, t)∂t, (2.9)A′(r, t) = A(r, t) +∇χ(r, t), (2.10)where χ is a gauge function corresponding to the gauge freedom of the potentials. As inthe previous section, we are interested in non-relativistic systems, so we are permitted towork in the Coulomb gauge. Then, the Hamiltonian becomesHˆ(r, t) =12m[Pˆ + e(Aˆ+∇χ)]2 + e∂χ∂t+ V (r). (2.11)122.2. Light-Matter InteractionNow the goal is to get rid of the quadratic vector potential terms by choosing an appro-priate gauge function. We can proceed by noting that the spatial dependence of the vectorpotential for free radiation is proportional to eik·r, and since the spatial extent of the atomis multiple orders smaller than an optical wavelength, we can approximate the vector po-tential as being uniform at the location of the atom Aˆ(r, t) ≈ Aˆ(r0, t), which is known asthe dipole approximation. Now by choosingχ(r, t) = −Aˆ(r0, t) · rˆ, (2.12)we get that∇χ(r, t) = −Aˆ(r0, t) (2.13)∂χ(r, t)∂t= −∂Aˆ(r0, t)∂t· rˆ = −Eˆ(r0, t) · rˆ, (2.14)and the Hamiltonian simplifies toHˆ(r, t) =12mPˆ 2 + V (r) + eEˆ(r0, t) · rˆ= Hˆ0(r, t) + eEˆ(r0, t) · rˆ= Hˆ0(r, t)− Eˆ(r0, t) · µˆ= Hˆ0(r, t) + HˆI(r0, t),(2.15)where we have used the definition of the dipole moment µˆ = −erˆ and represented theinteraction term as an interaction Hamiltonian HˆI(r0, t).The Jaynes-Cummings model treats the case wherein the solution of Hˆ0 for the looselybound electron possesses only a ground state and one excited state with an energy sepa-ration from the ground state close to the energy of an isolated cavity resonance ~ωc. Thecorresponding Hamiltonian isHˆ = HˆTLS + Hˆc + HˆI , (2.16)where for a single-mode cavity we haveHˆc = ~ωc(aˆ†aˆ+12). (2.17)The last remaining task is to describe the excitation energy of the two-level system, whichhas different energies corresponding to the ground state |g〉 and the excited state |e〉. Ifthe ground state is defined to be at zero energy, then the excited state will be at an energyproportional to the transition frequencyHˆTLS = ~ωe |e〉 〈e| . (2.18)Putting this all together, we getHˆ = ~ωe |e〉 〈e|+ ~ωc(aˆ†aˆ+ 12)− µˆ · Eˆ(r0). (2.19)132.2. Light-Matter InteractionWe can proceed further by representing the transition dipole operator in terms of the statesof the atom asµˆ = µ |g〉 〈e|+ µ∗ |e〉 〈g| . (2.20)Since we are dealing with a two-level system, it is convenient to use atomic raising andlowering operators, given byσˆ† = |e〉 〈g| , (2.21)σˆ = |g〉 〈e| . (2.22)The dipole transition operator can then be expressed asµˆ = µσˆ + µ∗σˆ†. (2.23)Using the expression for the quantized electric field in Equation 2.4 and working in theSchro¨dinger picture, we arrive at the so-called quantum Rabi Hamiltonian, where from nowon we drop the hats from our operators to simplify our notationH = ~ωeσ†σ + ~ωca†a− ~g(r)(a† + a)(σ† + σ), (2.24)where for simplicity we have assumed a real transition dipole moment and we have neglectedthe zero-point energy term of the radiation Hamiltonian since it does not affect the dynamicsof system. The generally position-dependent cavity-emitter coupling strength g(r) is givenbyg(r) = µ√ωcn32~λ30Veffu(r), (2.25)where ωe is the transition frequency of the two-level system, u(r) is the real-valued mode-shape function, and Veff is the unitless effective mode volume of the cavity as defined earlier.As it stands, the Hamiltonian does not conserve excitation number, as the terms a†σ† corre-sponds to the cavity and atom being simultaneously excited, while the term aσ correspondsto the cavity and atom being simultaneously de-excited; it has been shown that these termsonly become significant in the so-called ultra-strong coupling regime, when the Rabi fre-quency is on the same order as the pump frequency, so we discard them below in whatis known as the rotating wave approximation (RWA) to arrive at the Jaynes-CummingsHamiltonian1 [29]H = ~ωeσ†σ + ~ωca†a− ~g(a†σ + aσ†). (2.26)1It has also been shown that for a certain gauge choice these energy non-conserving terms can be elimi-nated without necessitating the RWA. Hence, in the ultrastrong coupling regime the choice of gauge has thecontradictory result of having a physical impact on the system, which has resulted in much confusion. Thishas been recently understood as being a direct consequence of the two-level approximation of the emitter,which results in the atomic potential becoming non-local [28]. The two-level approximation is thereforegauge-dependent, and its accuracy depends on the gauge choice. To properly treat the system in a gauge-invariant manner, the non-local potential must be written in terms of the canonical momentum, allowing itto also transformed according to the gauge transformation. We stress, however, that this treatment is onlynecessary in the ultrastrong regime, but it illustrates that the two-level approximation is on the same orderof approximation as the RWA.142.2. Light-Matter InteractionFigure 2.1: The energy levels of a cavity are modified by the presence of a two-level systemon resonance with the cavity (ωe = ωc), resulting in unequally-spaced dressed states.By consequence of the coupling between the two-level system and the cavity, the eigenstatesare no longer simply product states of the two-level system and cavity. Instead, they aresuperpositions of product states, known as ”dressed states”, given by|n,+〉 = cos ( θn2 ) |e, n− 1〉+ sin ( θn2 ) |g, n〉,|n,−〉 = cos ( θn2 ) |g, n〉 − sin ( θn2 ) |e, n− 1〉, (2.27)wheresin(θn2) =1√2[(∆2 + 4g2(n+ 1))1/2 −∆(∆2 + 4g2(n+ 1))1/2]1/2, (2.28)cos(θn2) =1√2[(∆2 + 4g2(n+ 1))1/2 + ∆(∆2 + 4g2(n+ 1))1/2]1/2, (2.29)and ∆ = ωc − ωe. Due to the exchange of energy between the emitter and cavity, theelementary excitations of the system can no longer be thought of as purely electronic orphotonic. Rather, the excitations of the system are quasiparticles known as polaritons, witheigenenergies corresponding toEn,± = ~ωcn± 12√∆2 + 4g2(n+ 1)), (2.30)from which it is clear that the presence of the atom in the cavity has the effect of shifting,or renormalizing each level of the cavity, as seen in Figure 2.1.152.3. Extensions of the Jaynes-Cummings Model2.3 Extensions of the Jaynes-Cummings ModelTavis-Cummings ModelThe Jaynes-Cummings model features only one atom, but can be straightforwardly extendedto the so-called Tavis-Cummings model by including an arbitrary number of atomsH = ωca†a+N∑n=1ωnσ†nσn −N∑n=1gn(rn)(a†σn + aσ†n). (2.31)where we have used Equation 2.25 to allow for atoms at different locations in the cavity tohave different atom-cavity coupling strengths. The eigenenergies of the Tavis-Cummingsmodel will then depend on the number of atoms in the system. For up to three atoms, weshow the rungs of the Tavis-Cummings model in Figure 2.2. For two atoms, insight can begained by making the following transformation, mapping the atomic operators onto a newbasisσ1 → σ′1 + σ′2√2, (2.32)σ2 → σ′1 − σ′2√2. (2.33)The transformed Hamiltonian can be represented asH ′ = ωca†a+g1 + g2√2(aσ′1† + σ′1a†) +g1 − g2√2(aσ′2† + σ′2a†)+2∑n=1ω1 + ω2√2(σ′n†σ′n) +ω1 − ω2√2(σ′1σ′2† + σ′2σ′1†),(2.34)which at first seems to have made things more complicated, but in the case where bothemitters are denegerate and close enough together to experience the same cavity field, theHamiltonian simplifies toH ′ = ωca†a+√2g(aσ′1† + σ′1a†) +2∑n=1√2ωe(σ′n†σ′n). (2.35)Now the antisymmetric σ′2 state is totally decoupled from the cavity, and will thus notinfluence its dynamics, and as a consequence it is known as a dark, or subradiant state.On the other hand, the symmetric σ′1 state now couples to the cavity with an increaseddipole transition moment, hence it is deemed a superradiant state. For more than twoatoms, it becomes convenient to work with a matrix representation of the Tavis-CummingsHamiltonianH =(a† σ†1 σ†2 . . . σ†n)ωc g1 g2 . . . gng1 ω1 0 . . . 0g2 0 ω2 . . . 0.......... . .gn 0 0 ωnaσ1σ2...σn . (2.36)162.3. Extensions of the Jaynes-Cummings ModelFigure 2.2: The eigenenergies for the first three rungs of the Tavis-Cummings model arepresented for up to three atoms. The eigenenergies become more equally spaced as thenumber of atoms increases. Two-fold degeneracies are represented by two thinner, closelyspaced levels.Then, a general unitary transformation to the superradiant and subradiant state basisbecomesU =1√N1 0 0 0 . . . 00 1 ξ1 ξ2 . . . ξN0 1 ξ2 ξ4 . . . ξ2N............. . .0 1 ξN ξ2N . . . ξN2 , (2.37)172.3. Extensions of the Jaynes-Cummings Modelwhere ξ = ei2pi/N . Performing this transformation on a Hamiltonian for an ensemble ofatoms with degenerate transition frequencies and Rabi frequencies we getH ′ = UHU † =ωc√Ngo 0 . . . 0√Ngo ωe 0 . . . 00 0 ωe . . . 0.......... . .0 0 0 ωe . (2.38)Similarly to the two atom case, when the atoms are degenerate the transformed Hamiltonianfeatures one superradiant state with a cavity coupling amplified by√N , while the otherstates are decoupled dark states. Hence, it is possible to achieve strong coupling in a cavityby increasing the number of emitters, provided they are degenerate. When the transitionfrequencies of the emitters are not the same, or if the emitters have different cavity-couplings,the dark states will become bright, affecting the dynamics.Dipole-dipole InteractionsIn the Tavis-Cummings model the emitters only directly interact with the cavity mode. Ifwe include interactions up to second order, however, it is possible for emitters to directlyinteract with each other by exchange of virtual photons. If a virtual photon is created atone emitter and annihilated at another emitter, an excitation can be transferred in a ”ra-diationless” manner, since the virtual photons cannot be detected directly. The amplitudeof the interaction between two dipoles in a uniform dielectric is given by [30]J =∑p〈f |Hint |ra; 1(p)〉 〈ra; 1(p)|Hint |i〉(Ei − Era)+∑p〈f |Hint |rb; 1(p)〉 〈rb; 1(p)|Hint |i〉(Ei − Erb), (2.39)which describes an initial state |i〉 with energy Ei being excited to a final state |f〉, whereHint is as given previously and |ra,b; 1(p)〉 are intermediate states involving a photon withwavevector p and have an energy Era,b . If we consider an empty cavity containing oneemitter in its ground state and one emitter in an excited state the initial state can be ex-pressed as |eg〉 |0〉 and the final state as |ge〉 |0〉, and the dipole-dipole interaction amplitudebecomesJ =∑p〈0| 〈ge|Hint |ee〉 a†(p) |0〉 〈0| a(p) 〈ee|Hint |eg〉 |0〉(−~pc− ~ωe)+∑p〈0| 〈ge|Hint |gg〉 a†(p) |0〉 〈0| a(p) 〈gg|Hint |eg〉 |0〉(~pc− ~ωe) ,(2.40)where ~pc is the energy of the virtual photon and ~ωe is the energy transferred betweenemitters. The first term corresponds to a virtual photon being absorbed by the groundstate emitter followed by a virtual photon being emitted by the initially-excited emitter,and involves an intermediate virtual state where both emitters are simultaneously excited.The second term represents the initially excited emitter emitting a virtual photon prior tothe other emitter absorbing a virtual photon, and involves an intermediate virtual statewhere both emitters are simultaneously in their ground states.182.3. Extensions of the Jaynes-Cummings ModelEvaluating Hint explicitly and performing some algebra, it can be shown that Equation2.39 can be written asJ = µa ·V(k,R) · µ∗b , (2.41)where µa,b is the transition dipole moment matrix element for each emitter, R = Ra −Rbis the separation between emitters andV(k,R) =eikR4pi0R3[(δij − 3RˆiRˆj)− (ikR)(δij − 3RˆiRˆj)− (kR)2(δij − RˆiRˆj)]. (2.42)The first term in V(k,R) is largest for kR << 1 and corresponds to near-field interactions,while the second and third terms correspond to intermediate- and far-field interactions,respectively. In a typical experiment, only the near-field term will be significant, which foridentical emitters can be written asJ =η4pi0R3(µa · µ∗b − 3(µa · Rˆ)(µ∗b · Rˆ)), (2.43)where η accounts for the spectral overlap between emitters. When the transition dipolemoment matrix elements are aligned and identical for both emitters, the interaction can besimplified toJ =η|µ2|4pi0R3(1− 3cos2θ), (2.44)where θ is the angle between the transition dipole moment and the interatomic distance.We can therefore account for this second-order effect by using the following extended Tavis-Cummings HamiltonianH = ωca†a+N∑n=1ωnσ†nσn −N∑n=1gn(a†σn + aσ†n) +N∑n6=mJ(σ†nσm + σ†mσn). (2.45)For the two atom case, the Hamiltonian isH = ωca†a+2∑n=1ωnσ†nσn −2∑n=1gn(a†σn + aσ†n) + J(σ†1σ2 + σ†2σ1), (2.46)and the eigenenergies can be solved analytically asE0 = ωe − J,E± =12(ωc + ωe + J ±√((ωe − ωc) + J)2 + 8g2)).(2.47)Hence, for the two atom case, the effect of the dipole-dipole interaction is the same as anatomic detuning, as has been previously noted [31]. For many atoms with the same dipole-dipole coupling J , the unitary transformation of the previous section can be employed toshow that the transformed Hamiltonian becomesH ′ = UHU † =ωc√Ngo 0 . . . 0√Ngo ωe + (N − 1)J 0 . . . 00 0 ωe − J . . . 0.......... . .0 0 0 ωe − J , (2.48)192.4. Emitter Typeswhich has the effect of blueshifting the superradiant state while redshifting the dark states.In general, the dipole-dipole interaction strengths for ensembles with more than two atomswill be different, and it will not be possible to find analytical expressions for the eigenenergiesof the system.2.4 Emitter TypesThe two main types of quantum emitters that have been seriously considered for use insolid-state based cQED applications are quantum dots and color centers. Quantum dotsare semiconductor structures confined in three dimensions, constraining the available statesof the electron and resulting in quantized energy levels. Quantum dots can be synthesizedusing a number of different methods, but the most common method is self-assemby using theStranski-Krastanov method, which uses molecular beam epitaxy on a lattice-mismatchedsubstrate to form islands of quantum dots [32]. This fabrication process constrains their useto certain materials, such as GaAs substrates. Indistinguishabilities of over 90% have beenreported between photons emitted from the same quantum dot, but a scalable quantumcomputer will require many quantum dots, and a major drawback of quantum dots is thatit is difficult to fabricate them with high uniformity [33]. Typically, the transition energiesfor quantum dots can range over 10 meV, whereas they have a homogeneous linewidth onthe order of 1-5 µeV [34]. Therefore, their transition frequencies will have to be tunedin-situ using methods such as changing the sample temperature, applying a magnetic field,or applying an AC Stark shift, all of which become extremely challenging for chips withmany emitters [35].On the other hand, color centers are point defects in a lattice that fluoresce at a certainwavelength, and can thus be introduced into substrates not compatible with quantum dots,such as silicon. These defects can be an atomic or molecular impurity, or a damage defectconsisting of vacancies or interstitial atoms in the lattice, and are commonly introduced byion implantation. An advantage of color centers is that each defect is intrinsically identical,and should therefore be indistinguishable and have the same transition frequency. One ofmore well-known colour centres in silicon is the G centre, which is a defect consisting ofan interstitial carbon atom, a substitutional carbon atom, and an interstitial silicon atom,and has a zero-phonon line at 1280 nm [36]. Due to the complexity of the center, minorvariations in the positions of the emitters results in a shifted transition energy, resultingin a 10 µeV range for different G centre configurations. By working with a substrate freeof impurities, such as isotopically purified 28Si, the inhomogeneous broadening of thesecentres can be reduced to 0.23-0.25 µeV, which is a little over twice the natural linewidth ofthe emitter [36]. However, a challenge with colour centres involving interstitial atoms andlattice vacancies is that these defects are mobile at room temperature, making them difficultto reliably store. This can be avoided by dealing with substitutional impurities, such asa singly-ionized substitutional 77Se+ impurity. 77Se+ has favourable decoherence lifetimesand a reasonably strong transition dipole moment, enabling it to be strongly coupled toan optical cavity [37]. Having an optical transition at 2.9 µm, it can be integrated with amid-infrared photonic circuit.202.5. Photon Correlations2.5 Photon CorrelationsTo understand the distinction between classical optics and quantum optics we must analyzethe photon statistics of the electromagnetic field. For a stationary quantum field, the generalnth order correlation function can be expressed in the Heisenberg picture as [38]g(n)(t1, t2 . . . tn) =〈a†(t1)a†(t2) . . . a†(tn)a(tn) . . . a(t2)a(t1)〉〈a†(t1)a(t1)〉 〈a†(t2)a(t2)〉 . . . 〈a†(tn)a(tn)〉 . (2.49)A particularly useful descriptor of light is its second-order coherence function, which actsas a tool for identifying if the electromagnetic field generated by a given source can beadequately considered classically or if it is fundamentally quantum mechanical.Figure 2.3: a) Physical setup for a Hanbury-Brown Twiss interferometer. A signal is splitinto two paths by a beamsplitter, which are then measured by separate detectors. For asource emitting single photons, there will be no coincedent detections. b) Second-ordercorrelation functions for different optical sources. Only light from a quantum source canhave a g(2)(τ) less than unity, known as anti-bunching. c) Generalization of the Hanbury-Brown Twiss interferometer for measuring g(3)(t1, t2, t3).g(2)(t1, t2) =〈a†(t1)a†(t2)a(t2)a(t1)〉〈a†(t1)a(t1)〉 〈a†(t2)a(t2)〉 . (2.50)212.5. Photon CorrelationsIf the field it generates can be considered stationary in a rigorous statistical sense, thesecond-order coherence function can be written asg(2)(τ) ≡ g(2)(0, τ) = 〈a†(0)a†(τ)a(τ)a(0)〉〈a†(0)a(0)〉2. (2.51)A classical field will always have a g(2)(τ) greater than or equal to unity, so if a field ismeasured to have a g(2)(τ) less than one, it can only be described quantum mechanically[39]. Some examples of different sources are plotted in Figure 2.3. Experimentally, g(2)(τ)can be measured using two detectors on each port of a beamsplitter in a setup commonlyknown as an intensity interferometer, first used by Hanbury-Brown and Twiss, which isillustrated in Figure 2.3. The intensity interferometer can be easily generalized to nth ordercorrelation functions by using a multiple beamsplitters, as shown in 2.3 (c).22Chapter 3Cavity Transmission Statistics3.1 IntroductionThe Jaynes-Cummings model, and its generalization to multiple atoms, assumes the cavityand atoms form a closed system. No cavity is perfectly isolated, as there will always besome leakage of the cavity mode into the radiation continuum due to imperfect ”mirrors”,and the excited state of the atom will always be able to decay into the same radiationcontinuum. Indeed, to use or to interrogate real atom-cavity systems, it is necessary tocouple some probing field from the external continuum the cavity, and to understand howthe resulting excitation of the atom-cavity system generates outward propagating radiationinto the surrounding environment. This formally defines a scattering problem, or how anincident electromagnetic field is scattered by an atom-cavity system that it can excite. Thischapter describes a scattering formalism that is used in subsequent chapters to analyze theproperties of 77Se+ donors in resonant photonic crystal microcavities.The problem of modelling a quantum system under influence of a an external field wasfirst developed using a master equation approach, where the input is treated as a heat bathof harmonic oscillators. In this approach, the total Hamiltonian of the model is comprisedof the system Hamiltonian defined by the dynamical variables treated within the JC model,the bath Hamiltonian that includes all of the external radiation modes, and the interactionHamiltonian that facilitates the interaction between the system and its environment. Then,two important approximations are made to allow the system dynamics to be solved for.First, the Born approximation assumes that the system and bath are initially uncorrelated,so that the total density matrix can be written as a product of the density matrices of thesystem and bath at some time in the past. Then, the degrees of freedom of the bath can betraced out, and the dynamics can be solved for in terms of the density matrix of the system.Second, the Markov approximation is typically made, which presumes that the bandwidthof the cavity-bath coupling is much larger than the bandwidth of the cavity. With these twoapproximations one obtains the so-called Lindblad master equation, which gives the timeevolution of the density matrix of the system. If the cavity is pumped coherently and is insteady-state, the resulting emission from the atom-cavity system can then be calculated bythe Wiener-Khintchin theorem, which gives the power spectrum of the system in terms of itssteady-state second order correlation functions. Higher-order correlation functions can becomputed with the quantum regression theorem, which approximates correlation functionsin terms of the expectation values of system operators.Another popular approach is the input-output method, which unlike the master equationapproach, treats the input and output fields explicitly. Like the master equation approach,the total Hamiltonian consists of the system, the bath, and the interaction. The dynamicsof the bath and system operators are solved for using the Heisenberg equations of motion to233.2. Input-Out Modelyield a relation between the input (the state of the bath field at some time in the distancepast), output (the state of the bath field at some time in the distant future), and systemoperators. Thus, the output fields can be solved for in terms of the input and systemoperators. An advantage of this approach is that scattering into different channels can becalculated explicitly, allowing the transmission and reflection of systems to be solved. Also,higher-order correlation functions can be calculated explicitly without use of the quantumregression theorem.Yet another approach is the modes-of-the-universe method. Unlike the master equationapproach, which treats optical cavities as having true modes that are phenomenologicallycoupled to their environment, this treatment uses inherently leaky ”quasimodes”, which aresuperpositions of modes of a larger cavity, the so-called universe [40]. It has been shownthat this approach is formally equivalent to the input-output method, making the decisionto use either one a matter of convenience [41].We have adopted the input-output formalism in this work as it most easily allows thetransmitted and reflected photon statistics from an arbitrary quantum system to be solved,and because it facilitates solving the photon statistics of a system in polynomial time, asopposed to the master equation approach which is solved in exponential time [42]. Hence,the input-output formalism can be used to numerically simulate more complicated systemswith only modest computational resources.3.2 Input-Out ModelThe input-output (IO) model is a method that allows fields propagated through quantumsystems to be solved for in terms of well-defined input fields [43]. The model consists ofa general system Hamiltonian linearly coupled to a continuum of photon modes, known asthe bath, given byH = Hsys +Hbath +Hint, (3.1)Hbath = ~∫ +∞−∞dωωb†(ω)b(ω), (3.2)Hint = i~∫ +∞−∞dωg(ω)[b†(ω)a− a†b(ω)], (3.3)where b(ω) is the continuous-variable bosonic annihilation operator, satisfying[b(ω), b†(ω′)] = δ(ω − ω′), (3.4)and a is the annihilation operator for a photon in the cavity. Since the interaction betweenthe bath and the cavity is localized to the input to the cavity, the spatial dependence ofthe bath modes can be ignored. The lower bounds of the integrals in 3.2 and 3.3 have beenset to minus infinity by the change of variable ω → Ω + ω′ where Ω >> 1 [44]. From thisHamiltonian, the Heisenberg equations of motion can be evaluated for the bath and systemoperators to beb˙(ω) = −iωb(ω) + g(ω)a, (3.5)a˙ = − i~[a,Hsys]−∫dωg(ω)b(ω). (3.6)243.2. Input-Out ModelEquation 3.5 can be solved to obtainb(ω, t) = e−iω(t−t0)b0(ω) + g(ω)∫ tt0e−iω(t−t′)a(t′)dt′, (3.7)where we have defined b0(ω) = b(ω, t0), which represents the bath state at some initial timet0 < t. Now we can solve for the dynamics of the system to geta˙ = − i~[a,Hsys] +∫g(ω)e−iω(t−t0)b0(ω)dω +∫g(ω)2∫ tt0e−iω(t−t′)a(t′)dt′dω. (3.8)To simplify things further, an approximation must be made. If the cavity linewidth is muchnarrower than the bandwidth of the coupling parameter g(ω), the coupling can be assumedto be frequency independent, and the the first Markovian approximation can be madeg(ω)2 =κ2pi. (3.9)Then Equation 3.8 simplifies toa˙ = − i~[a,Hsys] +√κbin(t)− κa(t)/2, (3.10)where we have defined the input fieldbin(t) = − 1√2pi∫e−iω(t−t0)b0(ω)dω. (3.11)We can repeat this process with the output field by again solving b(ω) but this time usingb1(ω) = b(ω, t = t1), representing the bath state at some future time t1 > t. Doing so, wegetb(ω, t) = e−iω(t−t1)b1(ω) + g(ω)∫ t1te−iω(t−t′)a(t′)dt′. (3.12)We can then define the output field asbout(t) =1√2pi∫e−iω(t−t1)b1(ω)dω, (3.13)and the time dynamics of the system can be written similarly asa˙(t) = − i~[a(t), Hsys]−√κbout(t) + κa(t)/2. (3.14)Combining Equations 3.10 and 3.14 we findbout(t) =√κa(t)− bin(t), (3.15)which is known as the input-output relationship. In practice we will often be working witha two-sided cavity like in Figure 3.2, so there will be two baths interacting with the system253.2. Input-Out ModelFigure 3.1: Schematic for a 2-sided cavity in the input-output formalism. The systemHamiltonian is comprised of at minimum the empty cavity, which interacts with outsidebath Hamiltonians, consisting of continua, via linear coupling terms.aout(t) =√κ1a(t)− ain(t),bout(t) =√κ2a(t)− bin(t).(3.16)We are now in a position to calculate transmission and reflection off a quantum system. Tobegin, we will deal with the case of an empty cavity, given by Equation 2.1 asHsys = ~ωc(12+ a†a). (3.17)The time dynamics of the cavity are then given by Equation 3.6 asa˙(t) = −iωca(t)− (κ1 + κ2)2a(t) +√κ1ain(t) +√κ2bin(t). (3.18)Since we wish to find transmission and reflection in terms of the input frequency, we candefine the Fourier components of the intracavity field by [44]a(t) =1√2pi∫ ∞−∞dωe−iω(t−t0)a(ω), (3.19)which gives us the following equationsi(ω − ωc)a(ω) = −(κ1 + κ2)2a(ω) +√κain(ω),aout(ω) =√κ1a(ω)− ain(ω),bout(ω) =√κ2a(ω)− bin(ω).(3.20)Now, assuming there is only one input into the two-sided cavity, we can set bin = 0 anddefine reflection and transmission coefficients asR(ω) = aout(ω)ain(ω)263.2. Input-Out ModelFigure 3.2: The transmission and reflection spectra for a bare optical cavity are given by aLorentzian centered at the resonant frequency of the cavity and a width depending on thecavity coupling parameter κ.andT (ω) = bout(ω)ain(ω) to getT (ω) =√κ1κ2κ1+κ2+Γc2 − iω, (3.21)R(ω) =12(κ1 − κ2 − Γc) + iω12(κ1 + κ2 + Γc)− iω. (3.22)which give the familiar Lorentzian transmission and reflection profiles of the resonances ofa classical Fabry-Perot cavity.We are interested in what happens when the cavity is populated by a two-level system,so we start with the Jaynes-Cummings modelH = ~ωeσ†σ + ~ωc(a†a+12)− g(a†σ + aσ†).Again, following the same procedure as before and using Equation 3.6 to solve for the cavitydynamics, we arrive at the set of equationsa˙(t) = −iωca(t) + igσ(t)− (κ1 + κ2 + Γc)2a(t) +√κ1ain(t),σ˙(t) = −iωeσ(t) + iga(t)− γrad2σ(t)− γnr2σ(t),aout(t) =√κ1a(t)− ain(t),bout(t) =√κ2a(t)− bin(t),(3.23)273.2. Input-Out Modelwhere we have implicitly defined an additional continuum interacting with the atom via γrad,representing a bath into which spontaneous emission can occur, as well as an additional losschannel due to non-radiative transitions via γnr. Additional decay of the cavity mode itself,due to scattering from the cavity into unwanted modes are represented by Γc, which can bethought of as the decay rate of an excited cavity mode when there is no atom present andthe waveguide-cavity coupling constants are zero. Taking the Fourier transform of theseequations and solving for transmission and reflection, we arrive atR(ω) =(γ2 − i(ω − δ))(κ1−κ2−Γc2 + iω)− g2(γ2 − i(ω − δ))(κ1+κ2+Γc2 − iω) + g2, (3.24)T (ω) =√κ1κ2(γ2 − i(ω − δ))(γ2 − i(ω − δ))(κ1+κ2+Γc2 − iω) + g2, (3.25)where δ = ωc−ωe is the cavity-atom detuning. If κ2 is set to zero, the reflection off a single-sided cavity can be obtained. When the atom and cavity are totally decoupled, we recoverthe transmission for the empty cavity. For a sufficiently large g, two distinct transmissionpeaks of equal linewidth emerge, corresponding to the dressed state eigenvalues given byEquation 2.30. This is the so-called strongly-coupled regime, where the excitation transfersbetween the cavity and the atom many times before being dissipated. In between thesetwo is the weak-coupling regime, where a narrow dip appears in the bare cavity linewidth.Since the degree of splitting depends on the Rabi frequency, cavity linewidth, and emitterlinewidth, it is helpful to introduce the dimensionless cooperativity factorC =g2κγ. (3.26)The transmission spectrum for a range of cooperativity values can be found in Figure 3.3.In the weak-coupling limit, the transmission spectrum consists of a peak consistent withthe linewidth of the bare cavity, with a dip characterized by γrad + γnr, the linewidth ofthe uncoupled emitter. This can be interpreted as an example of how placing the atomin the cavity effectively enhances the interaction strength of the exciting field with theatom, allowing high-extinction ratio spectroscopic studies of single atoms. As cooperativityincreases and the strong-coupling limit is reached, there are two distinct peaks associatedwith the polaritonic states of the system, having linewidths between those of the cavity andthe emitter, and the transmission at the cavity frequency approaches zero. Thus, if thecoupling is sufficiently strong, a single atom will completely reflect (transmit) light at thecavity frequency if it is on (off) resonant with the cavity. For a narrow-bandwidth source,this can be reliably done in the intermediate-coupling regime, so long as the transmissionamplitude reaches zero at the cavity frequency. For wider-bandwidth sources, the atom mustbe coupled strongly enough such that the gap between peaks is wider than the bandwidthof the source for the signal to be reliably reflected.Next, we address the effect of detuning the atom from the cavity. From the eigenvaluesof the Jaynes-Cummings Hamiltonian we see that introducing a detuning will shift theeigenvalues. We can see the effects of this by plotting the transmission vs. detuning,given in Figure 3.4. Once the detuning is sufficiently large, the cavity and the emitterare no longer coupled, the eigenstates of the Jaynes-Cummings Hamiltonian are no longer283.2. Input-Out ModelFigure 3.3: Left: The transmission spectrum for an optical cavity versus increasing atom-cavity coupling g with Γc = 0, γ = κ/100 and ωe = 0. Right: Transmission spectra forcooperativities of 0, 1 and 5 in blue, orange and green respectively. In the intermediatecoupling regime C ≈ 1 a narrow dip appears in the transmission spectrum. Beyond this,in the strong coupling regime, two separate peaks emerge with a polaritonic linewidth(κ+ γ)/2.in a superposition. Instead of the two equally wide peaks present for zero detuning, thelinewidth of one peak approaches that of the cavity while the linewidth of the other peakapproaches that of the emitter.To find the transmission and reflection for a cavity containing multiple emitters, we canrepeat the procedure for the Tavis-Cummings HamiltonianH = ωca†a+N∑n=1ωnσ†nσn −N∑n=1g(a†σn + aσ†n). (3.27)Solving the equations as before, we arrive atR(ω) =(κ1−κ2−Γc2 + iω)−∑Nn=1g2n( γ2−i(ω−δn))(κ1+κ2+Γc2 − iω) +∑Nn=1g2n( γ2−i(ω−δn)), (3.28)T (ω) =√κ1κ2(κ1+κ2+Γc2 − iω) +∑Nn=1g2n( γ2−i(ω−δn)). (3.29)We notice that when the atoms have the same transition frequency, we get the result asin Equation 2.48 where the situation is identical to a cavity containing a single atom witha modified Rabi frequency increased by the square root of the number of atoms in thecavity. When the atoms are detuned, however, we see that additional poles are introduced,corresponding to additional peaks and troughs in the transmission and reflection.293.3. Scattering FormalismFigure 3.4: Left: Transmission spectrum for a cavity containing a single atom versusdetuning. Right: Transmission spectra for detuning of the atomic transition frequencyωe = {0, 0.2, 0.9}κ for the blue, red, and green spectra respectively.While the basic input-output formalism has served us well up to this point, it is incon-venient to have to solve the system of equations for each new Hamiltonian. Furthermore,an analytical solution is not always possible. Also, we have been dealing with Hamiltoni-ans that conserve excitation number such has the Jaynes-Cummings model, but to studyhigher-order correlation functions we will need a framework to look at systems with varyingexcitation number. To do so, we recast input-output theory within a more general scatter-ing formalism, with the goal of being able to treat arbitrary Hamiltonians as well as modelarbitrary correlation functions.3.3 Scattering FormalismTo calculate higher-order correlation functions a more general treatment is required. Inthis section we frame the problem using a scattering formalism, allowing us to treat multi-photon scattering, which enables the calculation of higher-order correlation functions. Inthe limit of single-photon scattering, the formalism becomes equivalent to the input-outputmodel described previously. In this section, we will follow the work of [42].We will frame the problem as a homogeneous field in an ”input” waveguide continuumthat is coupled to and scatters from an arbitrary localized quantum system. We will expandthe excitation field state as a sum over continuum mode Fock states with different occupationnumbers and treat the scattering of each distinct photon number component individuallyby evaluating photon number specific scattering matrices [42]. These scattering matrices303.3. Scattering Formalismcan be written in terms of the effective Hamiltonian of the system, which incorporates lossvia non-waveguide modes. In the limit of one-photon scattering, the formalism becomesequivalent to the input-output formalism of the previous section.3.3.1 IntroductionIn the limit of the waveguide being much longer than the wavelength of the input, itslongitudinal modes become continuously spaced. Using continuous-mode operators, theinput state describing the field generated externally, independent of the state of the cavitysystem and evaluated at boundaries that define the interfaces between what is consideredthe cavity and the input or output continuum channels, can be expressed in the Schro¨dingerpicture as [45]|ψ〉in = |{α(ω)}〉 = exp(∫dω[α(ω)aˆ†(ω)− α∗(ω)aˆ(ω)])|0〉 , (3.30)which is known as a continuum coherent state, and can be used to represent pulsed andcontinuous-wave inputs. It can also be represented in the time domain as|{α(ω)}〉 ≡ |{α(t)}〉 = exp(∫dt[α(t)aˆ†(t)− α∗(t)aˆ(t)])|0〉 , (3.31)where t and ω are Fourier transform duals. If we express the state as|{α(ω)}〉 = exp[− 12∫dt|α(t)|2 +∫dtα(t)aˆ†(t)]|0〉 , (3.32)the second term in the exponential can be expanded as a series|{α(ω)}〉 = exp[− 12∫dt|α(t)|2][n=∞∑n=01n!∫ ∞t1,t2...tn=−∞n∏i=1dtiα(ti)a†(ti) |0〉]. (3.33)For a continuous-wave laser, α(t) is given by [45]αCW (t) = α0exp(−iωt+ iθ), (3.34)where ω is its frequency and θ is a phase factor. If we attempt to put this directly into3.33, the integral will be infinite, since a coherent continuous-wave state is not normalizable.Instead, we can work with a pulsed-coherent state defined byα(t) = α0exp(−iωt)rect( tτ), (3.35)where we have assumed the phase to be constant. In the limit of τ going to infinity, thesquare pulse shape becomes infinitely wide, approaching a continuous-wave coherent state.The input state then simplifies to|ψ〉in = exp(−|α0|2τ)[n=∞∑n=01n!∫ ∞t1,t2...tn=−∞n∏i=1dtiα(ti)a†(ti) |0〉]. (3.36)313.3. Scattering FormalismTo calculate the output state, we use the scattering matrix for continuous variable operators,where for example a one-photon state|ψ1(t′)〉in =∫ ∞t′=−∞dtα(t′)a†(t′) |0〉 (3.37)transforms as|ψ1(t)〉out =∫ ∞t=−∞S(t; t′) |ψ1(t′)〉in dt (3.38)=∫ ∞t=−∞∫ ∞t′=−∞S(t; t′)α(t′)a†(t′) |0〉 dt′dt, (3.39)which expresses a one-photon state entering the system at time t′ and being scattered outof the system at time t. The output state for a general coherent-state input can be writtenas|ψ〉out = exp(−|α0|2τ)[|0〉+n=∞∑n=11n!|ψn〉], (3.40)where|ψn〉 =∫ ∞t1,t2...tn=−∞∫ ∞t′1,t′2...t′n=−∞S(t1, t2...tn; t′1, t′2...t′n)n∏i=1α(ti)a†(ti)dtidt′i |0〉 . (3.41)The transmission through a system can then be calculated usingT (t, ω) =〈ψout | a†(t)a(t) |ψout 〉|α0|2 =e−|α0|2|α0|2∞∑n=0α2n0n!〈ψn| a†(t)a(t) |ψn〉 . (3.42)Taking the limit of weak driving, it is apparent that the contribution from the first termdominates, yieldinglimα0→0T (t, ω) = 〈ψ1| a†(t)a(t) |ψ1〉 . (3.43)This can now be evaluated using Equation 3.41 to yieldT (t, ω) =∣∣∣∣ ∫ ∞t′=−∞ST (t, t′)e−iωt′dt′∣∣∣∣2, (3.44)where ST is the one-photon transmission scattering matrix. Only the integral over t′ re-mains, since the integral over t is only non-zero at the time where the transmission ismeasured. Similarly, the reflection is given byR(t, ω) =∣∣∣∣ ∫ ∞t′=−∞SR(t, t′)e−iωt′dt′∣∣∣∣2, (3.45)where SR is the one-photon reflection scattering matrix. Higher-order correlation functionscan be computed by following the same procedure. For the second-order correlation function323.3. Scattering Formalismg(2)(t) we getg(2)(t1, t2;ω1, ω2) =〈ψout (ω)| a†(t1)a†(t2)a(t1)a(t2) |ψout (ω)〉〈ψout (ω)| a†(t1)a(t1) |ψout (ω)〉 〈ψout (ω)| a†(t2)a(t2) |ψout (ω)〉=e−|α0|2∑∞n=2α2n0n! 〈ψn| a†(t)a(t) |ψn〉[e−|α0|2∑∞n=1α2n0n! 〈ψn| a†(t1)a(t1) |ψn〉][e−|α0|2∑∞n=1α2n0n! 〈ψn| a†(t2)a(t2) |ψn〉].(3.46)Taking the limit of α0 → 0 for weak driving, we getlimα0→0g(2)(t1, t2;ω1, ω2) =〈ψ2(ω)| a†(t1)a†(t2)a(t1)a(t2) |ψ2(ω)〉2 〈ψ1(ω)| a†(t1)a(t1) |ψ1 (ω)〉 〈ψ1(ω)| a†(t2)a(t2) |ψ1(ω)〉 , (3.47)which as before can be represented using scattering matrices:limα0→0g(2)(t1, t2;ω) =∣∣∣∣ ∫∞t1,t2=−∞ S(t1, t2, ; t′1, t′2)e−iωt′1e−iωt′2dt′1dt′2∣∣∣∣2∣∣∣∣ ∫∞t1=−∞ S(t1, t′1)e−iωt′1dt′1∣∣∣∣2∣∣∣∣ ∫∞t2=−∞ S(t2, t′2)e−iωt′2dt′2∣∣∣∣2. (3.48)The transmission and second-order correlation function g(2)(t) can be calculated in termsof the scattering matrices, which will be explicitly evaluated in the following section. In thesame way, photon correlation functions of higher orders can be obtained. The expressionfor the third-order correlation function g(3)(t1, t2, t3;ω) requires the three-photon scatteringmatrix, and is given byg(3)(t1, t2, t3;ω)=∣∣∣∣ ∫∞t1,t2,t3=−∞ S(t1, t2, t3; t′1, t′2, t′3)e−iωt′1e−iωt′2e−iωt′3dt′1dt′2dt′3∣∣∣∣2∣∣∣∣ ∫∞t1=−∞ S(t1, t′1)e−iωt′1dt′1∣∣∣∣2∣∣∣∣ ∫∞t2=−∞ S(t2, t′2)e−iωt′2dt′2∣∣∣∣2∣∣∣∣ ∫∞t3=−∞ S(t3, t′3)e−iωt′3dt′3∣∣∣∣2,(3.49)which can be generalized to arbitrary order, givingg(n)(t1, t2 . . . tn;ω)=∣∣∣∣ ∫∞t1,t2...tn=−∞ S(t1, t2 . . . tn; t′1, t′2 . . . t′n)e−iω(t′1+t′2...t′n)dt′1dt′2 . . . dt′n∣∣∣∣2∏ni∣∣∣∣ ∫∞ti=−∞ S(ti, t′i)e−iωt′idt′i∣∣∣∣2.(3.50)3.3.2 Scattering MatricesIn this section we follow the work of [46]. Scattering matrices are unitary operators describ-ing the transformation of an incoming free particle state to an outgoing free particle state333.3. Scattering Formalism[47]. For instance, two particles at wave numbers k1 and k2 scattered to wave numbers p1and p2 can be represented using a scattering matrix asSp1p2,k1k2 = 〈p1p2|S|k1k2〉 , (3.51)where S is an evolution operator in the interaction picture that is taken to be defined from−∞ to ∞, given byS = limt0→−∞t1→∞UI (t1, t0)= limt0→−∞t1→∞eiH0t1e−iH(t1−t0)e−iH0t0 .(3.52)Then, defining [48]ain(k) ≡ eiHt0e−iH0t0akeiH0t0e−iHt0aout(k) ≡ eiHt1e−iH0t1akeiH0t1e−iHt1 ,(3.53)the scattering matrix can be rewritten asSp1p2,k1k2 = 〈0| aout (p1) aout (p2) a†in (k1) a†in (k2) |0〉 . (3.54)The ain(k) operator can then be transformed into the time domain byain(t) =1√2pi∫dkain(k)e−ikt. (3.55)which is equivalent to Equation 3.11 from the input-output model of the previous subsectionexcept written in wave number instead of frequency. However, the operators can be writtenin terms of frequency by linearizing the waveguide dispersion, which relates the operatorsby a(ω) = a(k)/√vg where vg is the group velocity of the light in the waveguide [46].Using Equation 3.55, the scattering matrix can then be written in terms of time-dependent operators asS(n)p;k = 〈p|S|k〉= 〈0| bout (p1) . . . bout (pn) b†in (k1) ..b†in (kn) |0〉= FT (2n) 〈0| bout (t1) ..bout (tn) b†in(t′1)..b†in(t′n) |0〉 .(3.56)where FT is the Fourier transform operator. The scattering matrix can also be defineddirectly in the time domain asS(t′1 . . . t′N ; t1 . . . tN ) ≡ 〈0|N∏i=1aout(t′i) N∏j=1a†in (tj) |0〉 , (3.57)Using the input-output relationsaout(t) + ain(t) =√κ1ac(t), (3.58)bout(t) + bin(t) =√κ2ac(t). (3.59)343.3. Scattering Formalismthe scattering matrix can then be expressed in terms of ac(t), the field inside the cavity.For the one-photon S-matrix, the input-output relations can be used to rewrite it in termsof system operators as follows [49]SR(t; t′) = 〈0| aout(t)a†in(t′) |0〉 (3.60)= 〈0|T[(√κ1a(t)− ain(t))a†in(t′)]|0〉 (3.61)= δ(t− t′) + 〈0|T[√κ1a(t)(√κ1a†(t′)− a†out(t′))]|0〉 (3.62)= δ(t− t′) + κ1 〈0|T [a(t)a†(t′)] |0〉 , (3.63)where T is a time-ordering operator that orders the operators by increasing time, and wehave used[ain(t), a†in(t′)] = δ(t− t′). (3.64)Since this S-matrix involves creating and annihilating a photon in the same continuum, itcorresponds to reflection. To calculate transmission, a photon must be created in a newchannel, which can be described asST (t; t′) = 〈0| bout(t)a†in(t′) |0〉 =√κ1κ2 〈0|T [a(t)a(t′)] |0〉 , (3.65)which differs from the reflection by the lack of delta function, and has a different couplingterm for each port. Similarly, the two-photon reflection S-matrix can be computed asSR(t1, t2; t′1, t′2) = 〈0| aout(t1)aout(t2)a†in(t′1)a†in(t′2) |0〉= κ21 〈0|T[a(t1)a(t2)a†(t′1)a†(t′2)]|0〉+ κ1 〈0|T[a(t1)a†(t′1)]|0〉 δ(t2 − t′2)+ κ1 〈0|T[a(t1)a†(t′2)]|0〉 δ(t2 − t′1) + κ1 〈0|T[a(t2)a†(t′1)]|0〉 δ(t1 − t′2)+ κ1 〈0|T[a(t2)a†(t′2)]|0〉 δ(t1 − t′1) + δ(t1 − t′1)δ(t2 − t′2) + δ(t1 − t′2)δ(t2 − t′1),(3.66)while the two-photon transmission S-matrix is given byST (t1, t2; t′1, t′2) = 〈0| bout(t1)bout(t2)a†in(t′1)a†in(t′2) |0〉= κ1κ2 〈0|T[a(t1)a(t2)a†(t′1)a†(t′2)]|0〉 . (3.67)3.3.3 Calculating the Scattering MatricesNow that the scattering operators are in terms of the creation and annihilation operatorsof the cavity, the time evolution can be calculated using the effective Hamiltonian of thesystem, which includes dissipative terms, as [48]a(t) = exp (iHefft) a exp (−iHefft) . (3.68)353.3. Scattering FormalismWe will follow the procedure in Trivedi et al. to calculate these matrix components in termsof the eigenvalues and eigenvectors of the so-called effective Hamiltonian of the system [42].We begin with the effective Hamiltonian for the Tavis-Cummings model which be writtenasHeff =(ωc − i (κ+ Γc)2)a†a+N∑n=1(ωn − iγn2)σ†nσn +N∑n=1gn(aσ†n + σna†), (3.69)where κ represents the coupling of the cavity mode to desired modes, such as waveguidemodes, Γc represents the coupling of the cavity mode to undesired loss modes, γn is thelinewidth of the nth emitter, and N is the number of emitters. These extra phenomenologicaldamping terms are due to coupling to other baths and arise naturally if those other bathsare included in the relevant equations of motion, as shown in Equation 3.23, but the netresult is an additional imaginary damping term in the effective Hamiltonian. Next, wedefine the number operator asNe =N∑nσ†nσ + a†a, (3.70)which has eigenvalues corresponding to the number of excitations for a given state. Sincethe number operator commutes with the Tavis-Cummings Hamiltonian[Heff, Ne] = 0, (3.71)it is apparent that the Hamiltonian conserves excitation number. Consequently, it is con-venient to work in the basis of the eigenstates of the number operator, which we can groupby excitation number. The eigenstate corresponding to zero excitations is the ground stateof the system {|0〉 |g1, g2...gN 〉}, representing a cavity with zero photons and N emitters intheir ground states. The one excitation subspace has N + 1 eigenstates, where the excita-tion takes the form of either one photon in the cavity or one of the N emitters being in theexcited state, and are given by {|1〉 |g1, g2...gN 〉, |0〉 |e1, g2...gN 〉... |0〉 |g1, g2...eN 〉}. The sec-ond excitation subspace has 2N + 1 eigenstates given by {|2〉 |g1, g2...gN 〉, |1〉 |e1, g2...gN 〉...|0〉 |g1, g2...eN−1, eN 〉}, and in general the number of eigenstates for the M excitation sub-space is N !M !(N−M)! . Then, in the basis of these eigenstates, the Hamiltonian can be writtenin block-diagonal form asHeff ≡H(0)eff 0 0 0 . . .0 H(1)eff 0 0 . . .0 0 H(2)eff 0 . . .0 0 0 H(3)eff · · ·............. . . , (3.72)where H(i)eff is the Hamiltonian for the ith excitation number. Since Heff is complex sym-metric, it can be written in terms of its eigenvectors and eigenvalues asH(i)eff = U(i)D{λ(i)}U(i)T, (3.73)363.3. Scattering Formalismwhere U is a matrix with columns given by the eigenvectors of Heff, D{v} is an otherwiseempty matrix with diagonal entries given by the vector v, and U(i)T is the transpose ofU(i). Hence, the time evolution can be written asexp(−iH(i)eff t)= U(i)D{exp(−iλ(i)t)}U(i)T. (3.74)The excitation-subspace Hamiltonians H(i)eff are connected by creation and annihilation op-erators with matrix representations of the forma ≡0 A0,1 0 0 . . .0 0 A1,2 0 . . .0 0 0 A2,3 . . .............. . . and a† ≡0 0 0 . . .A†0,1 0 0 . . .0 A†1,2 0 . . .0 0 A†2,3 . . ........... . . , (3.75)where Ai,i+1 annihilates a photon from the one-excitation subspace, bringing it to thezero-excitation subspace.3.3.4 Calculating Transmission and ReflectionAs shown previously, the single-photon scattering matrix for transmission is given byST (t; t′) =√κ1κ2 〈0|T [a(t)a(t′)] |0〉 , (3.76)where we take |0〉 to refer to the n-emitter ground state |0〉 |g〉⊗n. After using 3.68 torepresent the time dependence in terms of the Hamiltonian we getST (t; t′) =√κ1κ2 〈0| a exp(−iHeff)(t1 − t′1)a† |0〉 θ(t− t′). (3.77)Using Equations 3.74 and 3.75, the scattering matrix can be expressed as a matrix productgiven byST (t; t′) =√κ1κ2[A0,1U(1)D {exp (−iλ1 (t− t′))}U(1)TA†0,1] θ (t− t′) . (3.78)Finally, using Equation 3.45 the transmission can be written asT (ω) = κ1κ2∣∣∣∣A0,1U(1)D{ 1λ(1) − ω}U(1)TA†0,1∣∣∣∣2 , (3.79)which can be easily computed using matrix multiplication once the eigenvectors and eigen-values of Heff have been solved for. Similarly, using the one-photon reflection scatteringmatrix, the reflection is found to beR (ω) = κ1κ2∣∣∣∣1−A0,1U(1)D{ 1λ(1) − ω}U(1)TA†0,1∣∣∣∣2 . (3.80)373.3. Scattering Formalism3.3.5 Second-Order Correlation Function g(2)(t)The two-photon scattering matrix for transmission is given in the Schro¨dinger picture byST(t1, t2; t′1, t′2)= κ1κ2 〈0|T [a(t1)a(t2)a†(t′1)a†(t′2)] |0〉 . (3.81)Putting in the time-dependence as unitary transformations of the effective Hamiltonian asbefore, we getST (t1, t2; t′1, t′2) = −κ1κ2× 〈0| a exp(−iHeff(t1 − t2))a exp(−iHeff(t2 − t′1))a† exp(−iHeff(t′1 − t′2))a† |0〉(3.82)for t1 ≥ t2 ≥ t′1 ≥ t′2 andST (t1, t2; t′1, t′2) = −κ1κ2× 〈0| a exp(−iHeff(t1 − t′1))a exp(−iHeff(t′1 − t2))a† exp(−iHeff(t2 − t′2))a† |0〉(3.83)for t1 ≥ t′1 ≥ t2 ≥ t′2, otherwise ST (t1, t2; t′1, t′2) = 0. The first case corresponds to thesituation where the second photon arrives while the first photon is still in the cavity. In thesecond case, the second photon arrives after the first photon has left the cavity. Only thefirst case will contribute to photon bunching and a non-trivial g(2)(t). For the first case, wegetST(t1, t2; t′1, t′2)= −κ1κ2[A0,1U(1)D{exp(−iλ(1) (t1 − t2))}U(1)TA1,2U(2)×D{exp(−iλ(2) (t2 − t′1))}U(2)T A†1,2U(1)D{exp(−iλ(1) (t′1 − t′2))}U(1)TA†0,1] ,(3.84)and for the second case we getST(t1, t2; t′1, t′2)= −κ1κ2[A0,1U(1)D{exp(−iλ(1) (t1 − t′1))}U(1)TA†0,1A0,1×U(1)D{exp(−iλ(1) (t2 − t′2))}U(1)TA†0,1]. (3.85)From Equation 3.48, it can be shown that the second-order correlation function for thetransmitted state can be written asg(2)T (t1, t2;ω) =κ1κ2T 2 (ω)∣∣gT (ω) I+ (fT (ω)− gT (ω)) exp (−i (λ1 − ω) (t1 − t2))∣∣2 , (3.86)whereg (ω) = d{U(1)TA†0,1A0,1U(1)D{1λ(1)−ω}U(1)TA†0,1A0,1U(1)D{1λ(1)−ω}},f (ω) = d{U(1)TA1,2U(2)D{1λ(2)−2ω}U(2)TA†1,2U(1)D{1λ(1)−ω}U(1)TA†0,1A0,1U(1)}.(3.87)where d is the vector given by the diagonal of a matrix. g(2)(t) can thus be calculated usingmatrix algebra given the eigenvalues and eigenvectors of the first two excitation subspaceHamiltonians. g(2)(0) is given exclusively by f(ω), the scattering component from twophotons simultaneously occupying the cavity. For larger time differences, g(ω) becomes383.4. Transmitted Photon Statisticsdominant, which is the scattering component for each photon transmitting through thecavity in succession, meaning that the first photon exits the cavity prior to the arrival ofthe second photon; since both photons are never in the cavity at the same time, g(ω) onlydepends on the eigenvalues of the one-excitation subspace. For large time differences, g(2)(t)is given exclusively by g(ω), since the eigenvalues are complex, and the second-correlationapproaches unity, indicating that there is no bunching. From Equation 3.48 the reflectedstatistics are calculated asg(2)R (t1, t2;ω) =κ1κ2R (ω)2∣∣2− 4 t(ω) + gT (ω) I+ (fT (ω)− gT (ω)) exp (−i (λ1 − ω) (t1 − t2))∣∣2 ,(3.88)wheret(ω) =√κ1κ2A0,1U(1)D{1λ(1) − ωL}U(1)TA†0,1 (3.89)is the transmission amplitude, satisfying |t(ω)|2 = T (ω).3.3.6 Third-order CorrelationsWe can repeat the procedure in the previous section to find the third-order photon statistics.Using the three-photon scattering matrix for the transmitted component is given byST (t1, t2, t3; t′1, t′2, t′3) = 〈0| bout(t1)bout(t2)bout(t3)a†in(t′1)a†in(t′2)a†in(t′3) |0〉= (κ1κ2)3/2 〈0|T[a(t1)a(t2)a(t3)a†(t′1)a†(t′2)a†(t′3)]|0〉 . (3.90)We arrive at an expression for g(3)T (0)g(3)T (0) =− (κ1κ2)3/2A0,1A1,2A2,3U(3)D{1λ(3) − 3ω}U(3)T×A†2,3U(2)D{1λ(2) − 2ω}U(2)TA†1,2U(1)D{1λ(1) − ω}U(1)TA†0,1.(3.91)3.4 Transmitted Photon StatisticsWe now apply the formalism developed in the previous section to analyze the higher-orderphoton statistics of different systems. An immediate application is to use the second-order correlation function to investigate the bunching and anti-bunching properties whena weak coherent field with average photon number less than one interacts with variousconfigurations of atoms coupled to an optical cavity. The terms bunching and anti-bunchingrefer to the statistics that describe temporal spacing of photons in a given state of the field;a bunched state is one for which it is more likely to find photons overlapping in time thannot overlapping in time, whereas a perfectly anti-bunched state is one for which there is noprobability of measuring more than one photon at any given time [50]. A coherent state likethat considered as the external driving field in the above analysis has g(2)(t) = 1 for any α0.However, as is shown below, the state of the field in the transmitted waveguide generatedwhen such a field interacts with and scatters from an atom-cavity system, can in general393.4. Transmitted Photon Statisticshave g(2)(t) < 1. Hence, the atom-cavity system acts in some ways as a classical-to-quantumfield filter.We begin with the case of a single atom coupled to a cavity and see how the spectralbehaviour of the second-order correlation function changes with increased coupling to thecavity, seen in Figure 3.5. When the atom is totally decoupled we find that g(2)(0) = 1 for allFigure 3.5: (a) Second-order correlation function g(2)(0) as a function of detuning andatom-cavity coupling for a cavity containing a single atom on resonance with the cavity.(b) Transmission and g(2)(0) spectrum for g = 2κ. (c) Transmission and g(2)(0) spectrumfor g = 0.4κ. (d) Transmission and g(2)(0) spectrum for g = 0. For an empty cavity g(2)(0)= 1 for all frequencies, whereas large bunching appears at the cavity frequency for non-zeroatom-cavity coupling as a consequence of reduced transmission through the cavity. In thestrong coupling regime, g(2)(0) is less than 1, indicating that 2-photon states are suppresseddue to the anharmonicity of Jaynes-Cummings ladder.frequencies, which is consistent with a coherent source that is passively filtered when passingthrough a resonant cavity. For a non-zero Rabi frequency, there are two symmetrical localminima where g(2)(0) < 1 corresponding to photon anti-bunching. The value of g(2)(0) atthese minima decreases and tends to zero as the Rabi frequency gets larger. These minimacorrespond to the frequencies at which the first rungs of the Jaynes-Cummings ladder,shown in Figure 2.1, are resonant with the excitation frequency.Due to the anharmonicity of the eigenstates of the Jaynes-Cummings model, the eigen-values at the second rung of the ladder are not equal to twice the eigenvalues of the first403.4. Transmitted Photon Statisticsrung. Consequently, when the incident field state is in a mode that is resonant with theone-excitation eigenvalues of the atom-cavity system associated with the first rung of theJaynes-Cummings ladder, the one-photon component of the coherent excitation state canbe absorbed by the system, and later emitted into the transmitted channel in one of itsone-photon Fock states. However, under these circumstances, the energy of the two-photoncomponent of the incident field state will not be exactly equal to any of the eigenstatesof the two-excitation rungs of the system Hamiltonian’s eigenspectrum, so this componentcan’t excite the system, and no two-photon Fock state can be emitted into the transmissionchannel. Of course, when the broadening of the eigenstates is accounted for, there willalways be some ”leakage” of the two-photon component.Figure 3.6: (a) Second-order correlation function g(2)(0) and transmission spectra for acavity containing 1, 10, and 20 atoms on resonance with the cavity. When the atomshave identical transition frequencies, the splitting increases as√N where N is the numberof atoms in the cavity. For increasing number of atoms, minimum value of g(2)(0) < 1increases. (b) Same as (a) but with the emitters detuned by ωe − ωc = 2κ. The detuningcan enhance the photon blockade for one of the eigenstates of the cavity, decreasing theminimum value of g(2)(0).We now investigate the effects of increasing the number of on-resonance atoms on thesecond-order correlation function g(2)(0), shown in Figure 3.6 (a). The depth of the bunchinglessens as the number of atoms increases due to the eigenvalues of the Tavis-Cummingsmodel becoming more equally spaced. For emitter numbers greater than one, there arepeaks in g(2)(0) corresponding to the resonance of the second rung of the Tavis-Cummingsladder, which for up to three atoms is shown in Figure 2.2. In Figure 3.6 (b) we consideremitters detuned by ωe − ωc = 2κ. As a consequence of the detuning, the eigenstates areno longer equal superpositions of the cavity and atom eigenstates; instead, one eigenstateis more cavity-like and one is more atom-like. The atom-like eigenstate has a smallerlinewidth, which corresponds to a narrower transmission peak as well as a smaller value ofg(2)(0), since there is less overlap with the first rung of the Tavis-Cummings ladder.413.4. Transmitted Photon StatisticsNext we look at the time-dependence of the second-order correlation function, seenin Figure 3.7. When a single atom is on resonance, as in 3.7 (a), the regions that haveg(2)(0) < 1 monotonically reach g(2)(t) = 1 at a later time. At κt ≈ 5, a region whereg(2)(t) ≈ 0 appears due to interference between the 1- and 2-photon transmission ampli-tudes, which can be seen from Equation 3.86; consequently, an imbalanced Hanbury-BrownTwiss interferometer with this delay in one arm would see a reduction in coincedent photondetections for certain frequencies. A single atom off resonance by ωe − ωc = 2κ is shown in3.7 (b), where now only the dip in g(2)(0) corresponding to the atom-like eigenstate has amonotonically-increasing g(2)(t).Figure 3.7: (a) Left: Second-order correlation function g(2)(0) as a function of time foran emitter on resonance ωe − ωc = 0. Right: Transmission spectra and g(2)(0) are shownfor times {0, 4, 7}κt. At later times, interference effects result in fluctuations in g(2)(0).(b) Second-order correlation function g(2)(0) as a function of time for a detuned emitterωe−ωc = 2κ. Anti-bunching depth at the higher-energy transmission peak is increased dueto the detuning, whereas the other dip fluctuates in time.3.4.1 Inhomogeneous BroadeningIn a realistic experiment, each atom in a solid-state cavity will not have the exact sameenvironment due to chemical and isotopical impurities in the host material. Furthermore,etching the host substrate to form the photonic crystal microcavity will introduce a spatiallyvarying stress profile. The consequence of this is that the atom’s will experience slightlydifferent potentials, and as a result their transition frequencies will be shifted. The atomicensemble as a whole will therefore have an inhomogeneously broadened spectrum with a line-shape ranging from Lorentzian to Gaussian, determined by the type of broadening. Atomsin a solid-state environment are usually well-described by a Gaussian distribution, whilegaseous ensembles are usually modelled by the convolution of a Gaussian and a Lorentzianfunction, known as a Voigt lineshape. We can incorporate inhomogeneous broadening into423.4. Transmitted Photon Statisticsthe model by sampling atom transition frequencies from a given distribution. In Figure 3.8,we plot the transmission for a cavity with an increasing number of Gaussian-distributed in-homogeneously broadened atoms with parameters {g, γ, FWHM} = {1/100, 1/500, 10}κ,where FWHM is the full width at half maximum. Fano resonances are introduced to theFigure 3.8: Inhomogeneous broadening of atoms in a cavity with with parameters{g, γ, FWHM} = {1/100, 1/500, 10}κ. For a Gaussian distribution, strong coupling can beobserved if the collective Rabi frequency is larger than the FWHM of the emitter distribu-tion. This is not the case for a Lorentzian distribution, which has far-reaching wings.transmission peak due to the presence of the weakly-coupled, normally distributed atoms,which has the effect of broadening the Lorentzian transmission spectrum. As the collectiveRabi frequency reaches the same order of magnitude as the FWHM of the broadening,the broadened Lorentzian peaks emerge. Finally, as the collective Rabi frequency becomeslarger than the broadening FWHM, the peaks assume their original linewidth. This can beunderstood as the polaritonic states becoming completely decoupled from the dark states.In Figure 3.8 we also show emitters with transition frequencies given by a Lorentzian dis-tribution with the same FWHM. In this case, the distribution does not fall off as quicklyand we see that the peak broadening is more-or-less independent of emitter number, due tothe far-reaching wings of the Lorentzian distribution.43Chapter 4The Selenium Spin QubitSubstitutional impurities provide a promising route for implementing optically-accessiblespin interfaces in silicon. A recently proposed candidate is the 77Se+ donor, which has aground state split by a hyperfine interaction wherein both states are coupled to an excitedstate by a dipole-allowed transition at 2.9 µm [20]. Critically, this allows a platform based onthe donor to be integrated into a near-infrared silicon photonic network, taking advantage ofthe mature fabrication infrastructure for silicon [51]. We begin by looking at the electronicstructure of the 77Se+ donor.4.1 Electronic StructureIn covalently bonded crystals with large band gaps such as diamond, the electronic wave-functions are highly-localized around each lattice site and tight-binding models can beemployed to model defects with great success [52]. In contrast, the electronic wavefunc-tions for donors in semiconductors such as silicon are much more delocalized in real-space,and it is instead common to use so-called effective mass theory, where wavefunctions areconstructed using basis states in momentum-space [53]. Silicon has an indirect band gapwith 6 identical conduction band minima located in the 〈100〉 directions. If we look at theminimum in the zˆ direction, the electronic energy at the conduction band can be written asE(k) =~22m1(kz − k0)2 + ~22m2(k2x + k2y), (4.1)where k0 is difference in momentum between the valence band maximum and the conductionband minimum, and m1 = 0.98m and m2 = 0.19m are the effective masses, given in termsof the free electron mass m [54]. Hence, the electrons can be modelled as free electronswith some experimentally-determined effective mass, which represents how their motion isaffected by their environment. The constant energy surfaces for the six conduction bandminima are shown in Figure 4.1. The electronic wavefunctions for a donor can then befound by solving the following Schro¨dinger equation(− ~22m∇2 + V (r) + U(r))ψ(r) = E(k)ψ(r), (4.2)where V (r) is the periodic potential due to the lattice and U(r) is the difference in potentialdue to the replacement of a lattice site by a singly-ionized donor. Now, if silicon had a singleconduction band at the Γ point (k = 0), the solution to Equation 4.2 would be given byψ(r) = F (r)u(r), (4.3)444.1. Electronic StructureFigure 4.1: a) The 6 ellipsoidal conduction bands in silicon from effective mass theory.b) The indirect band gap of silicon with additional hydrogen-like states due to the 77Se+impurity. The band gap energy is labelled Eg.where u(r) is a function with the periodicity of the lattice, typically determined usingdensity functional theory, and F (r) satisfies the effective mass equation[− ~22m1∂2∂z2− ~22m2( ∂2∂x2+∂2∂y2)− U(r)− E)]F (r) = 0. (4.4)Taking the position of the donor to be the coordinate origin, the difference in potential dueto the donor, U(r), can be written for large r asU(r) = −e2r, (4.5)where  is the dielectric constant of silicon, accounting for the screening due to electronsoccupying the surrounding lattice. In the absence of an anisotropic effective mass, thesolution for F (r) is identical to the solution for the hydrogen atom. Consequently, lowestenergy state has the same functional dependence as the 1s state of the hydrogen atom andgoes like e−r. However, if we account for the anisotropy of the effective mass, the solutions toF (r) are no longer identical to the hydrogen atom, but can be solved for with the variationalmethod using Ansatz wave functions. Doing so, it is found that the solutions to F (r) stillhave the same symmetry as the hydrogenic wave functions, and are thus categorized as1s, 2s, 2p, 3s... states. Consequently, the interpretation of 4.3 is that, in the case of a singlevalley, the wave function is given by a hydrogenic envelope function F (r) that is periodicallymodulated by some function with the periodicity of the lattice u(r), where the ground statehas a 1s-like envelope function.Now, if we treat the actual six equivalent conduction band band minima of silicon at k ={(k0, 0, 0), (−k0, 0, 0), (0, k0, 0), (0,−k0, 0), (0, 0, k0), (0, 0,−k0)} where k0 ≈ 0.85(2pi/aSi) and454.1. Electronic StructureTd E E 8C3 8C3 3C2, 3C2 6S4 6S4 6σd, 6σd BasesA1 Γ1 1 1 1 1 1 1 1 1 x2 + y2 + z2A2 Γ2 1 1 1 1 1 -1 -1 -1E Γ3 2 2 -1 -1 2 0 0 0 (2z2 − x2 − y2, x2 − y2)T1 Γ4 3 3 0 0 -1 1 1 -1 (Rx, Ry, Rz)T2 Γ5 3 3 0 0 -1 -1 -1 1 (x,y,z)E1 Γ6 2 -2 1 -1 0√2 -√2 0E2 Γ7 2 -2 1 -1 0 -√2√2 0F Γ8 4 -4 -1 1 0 0 0 01s 6 6 0 0 2 0 0 2A1 + E + T2 6 6 0 0 2 0 0 2Table 4.1: Character table for Td group [1].aSi is the lattice constant for silicon, we get a six-fold degenerate ground state manifold givenbyψ(i)(r) = F (i)(r)φ(k(i), r), (4.6)where we have six solutions indexed by i = 1, 2...6 corresponding to each conduction bandminima, and φ(k(i), r) is the solution to Bloch’s equation at each conduction band minimagiven byφ(k(i), r) = uki(r)eiki·r, (4.7)where as before, u(r) is some function with the periodicity of the lattice. F (i)(r) is nowvalley dependent, as zˆ is defined to be in the direction of k(i), and is given by[− ~22m1∂2∂z2i− ~22m2( ∂2∂x2i+∂2∂y2i)− e2r− E)]F (i)(r) = 0. (4.8)However, this six-fold degeneracy is lifted if we consider the fact that, due to presence ofthe lattice, the symmetry of the impurity potential U(r) will not actually be spherical.The symmetry of the undoped silicon lattice has octahedral symmetry and belongs to theOh point group, which in the presence of a substitutional donor, is reduced to tetrahedralsymmetry about the donor, described by the Td point group.To see how imposing this symmetry constraint on the impurity potential lifts the six-fold degeneracy of the ground state we can use group theory. The character table for theTd group is shown in Table 4.1, which summarizes the symmetries of a system with Tdsymmetry. The first column of the table contains the irreducible representations of the Tdgroup in both Mulliken notation and denoted by the generic symbol Γ, which form a basisfor representing any system with Td symmetry. The foremost row of the table contains thedifferent symmetry operators of the group and the values in the table correspond to theresult of the symmetry operation on each of the irreducible representations. For spinless464.1. Electronic Structurerepresentations, we ignore the barred operators. C3 and C2 are rotations of 120◦ and 180◦,σd is a diagonal reflection, and S4 is a rotation of 90◦ with a reflection. E is a rotation of360◦, so for spinless representations it acts as the identity operator. Consequently, its entryin the character table denotes the degeneracy of a particular irreducible representation.Then, we can also see the result of these symmetry operations on the 6-fold degenerate 1sstates, which we have included as the second-last row in Table 4.1. The 1s states, which mustconform to tetrahedral symmetry, can then be decomposed into irreducible representationsof the Td group by finding a a combination of irreducible representations with the samesymmetry properties as the 1s states. In this case, it is easy to see that the combinationA1⊕E⊕T2 will add up to the same values as the 1s states. Hence, the ground state will besplit into a singlet state with A1 symmetry, a doublet E2 state, and a triplet T2 state. Thissymmetry can be imposed on the six degenerate solutions given by effective mass theory byadjusting the valley composition of each state. The following valley combinations achievethe desired symmetriesα(j)1 =1√6(1, 1, 1, 1, 1, 1)}(A1)α(j)2 =12(1, 1,−1,−1, 0, 0)α(j)3 =12(1, 1, 0, 0,−1,−1)}(E)α(j)4 =1√2(1,−1, 0, 0, 0, 0)α(j)5 =1√2(0, 0, 1,−1, 0, 0)α(j)6 =1√2(0, 0, 0, 0, 1,−1) (T2)(4.9)which we can use to construct six symmetry-allowed solutionsΨi(r) =6∑j=1α(j)i ψ(j) =6∑j=1α(j)i F(j)(r)φ(k(j), r). (4.10)indexed by i = 1, 2...6, each involving a distinct sum over different valley components. Thepotential experienced by the A1, E, and T2 states will in general be different. Since theA1 state is the only state without a node at the location of the donor, its energy will beshifted the most. The splitting of this six-fold degeneracy of states predicted by effectivemass theory is commonly referred to as a central cell correction or valley-orbit splitting, asthe states differ only in valley composition. The energy levels of these states, which havethe same 1s-like envelope function but have different valley compositions, are depicted inFigure 4.2.Until now, we have ignored the effects of spin, as they do not play a role in the valley-orbit interaction, which is the dominant correction to the states predicted by effective masstheory. However, there are additional shifts in energy due to a spin-orbit and hyperfineinteraction, which require us to account for the electronic spin degree of freedom. To doso, we use the character table for the Td double group, given by all eight rows in Table4.1, which includes the additional symmetries associated with a spin-12 particle. Namely, a720◦ rotation is required to bring a spin-12 particle to its original quantum state. Thus, ina double group representation, the identity operator E is a 720◦ rotation and not a 360◦rotation. Instead, a rotation of 360◦ is denoted by the E operator, and results in a change in474.1. Electronic Structuresign for the irreducible representations Γ6, Γ7 and Γ8, which represent spinors. Additionaloperators are given by adding a rotation of 360◦ to the existing operators, which is denotedby a bar.Now, to find the new representations for each state after including spin, we can take theproduct of the orbital and spin components for each stateΓ = Γv ⊗ Γs, (4.11)where Γv and Γs are the irreducible representations for the valley and spin components of thewavefunction, respectively, and we have neglected the symmetry of the orbital componentof the wavefunctions since we are only interested in 1s states, which have the same envelopefunction. We can evaluate such a product by using the direct product table for the Tddouble group given by Table 4.1, which gives the representation of a product of irreduciblerepresentations as a sum of irreducible representations. The From Table 4.1 we see that theappropriate representation for an electron is Γ6, so, including the valley-orbit interaction,our states will be transformed as E⊗Γ6 = Γ8, T2⊗Γ6 = Γ7⊕Γ8, and A1⊗Γ6 = Γ6. Thus,after including spin, the two degenerate 1s:E states become four degenerate 1s:E(Γ8) states,the 1s:A1 state becomes two degenerate 1s:A1(Γ6) states, and the three degenerate 1s:T2states become four 1s:T2(Γ8) and two 1s:T2(Γ7) states. The degeneracy of the 1s:T2(Γ7) and1s:T2(Γ8) states is broken by a spin-orbit interaction which is dominated by the potentialof the donor impurity and is given by [55]HSO =~22m∗2c21rdU(r)drL · S, (4.12)where m∗ is the effective mass, U(r) is difference in potential due to replacing a lattice siteby a donor, and L and S are the angular momentum and spin operators, respectively. Todetermine whether or not each state exhibits spin-orbit coupling, we can make the Ansatzthat each state can be expressed as a wavefunction that is separable in the spatial and spindegrees of freedom and then determine whether or not the expectation value of the spin-orbit Hamiltonian is zero. Consequently, the matrix elements for the spin-orbit interactionbetween two states can be written as∑i=x,y,z〈φ|Li |φ′〉 〈χ|Si |χ′〉 , (4.13)where 〈φ| and 〈φ′| are the spatial wavefunctions for two states with spinor wavefunctionsgiven by 〈χ| and 〈χ′|, respectively. Since the matrix element 〈φ|Li |φ′〉 is integrated overall space, it will only yield a non-zero value if it contains the totally symmetric irreduciblerepresentation, which for the Td group is Γ1. We can write the representation of the matrixelement asΓφ ⊗ ΓL ⊗ Γφ′ , (4.14)where Γφ,φ′ are the irreducible representations of the involved states and ΓL is the irreduciblerepresentation for the angular momentum operator L. From Table 4.1 we see that ΓL = Γ4.We then find the only non-zero matrix elements to be among the 1s:T2 states, as well asbetween the 1s:T2 and 1s:E states. The former is responsible for separating the energy levelsof the 1s:T2(Γ7) and 1s:T2(Γ8) states, whereas the latter introduces mixing between the484.2. Dipole-Allowed TransitionsTd Γ1 Γ2 Γ3 Γ4 Γ5 Γ6 Γ7 Γ8Γ1 Γ1 Γ2 Γ3 Γ4 Γ5 Γ6 Γ7 Γ8Γ2 Γ1 Γ3 Γ5 Γ4 Γ7 Γ6 Γ8Γ3 Γ1 ⊕ Γ2 ⊕ Γ3 Γ4 ⊕ Γ5 Γ4 ⊕ Γ5 Γ8 Γ8 Γ6 ⊕ Γ7 ⊕ Γ8Γ4 Γ1 ⊕ Γ3 ⊕ Γ4 ⊕ Γ5 Γ2 ⊕ Γ3 ⊕ Γ4 ⊕ Γ5 Γ6 ⊕ Γ8 Γ7 ⊕ Γ8 Γ6 ⊕ Γ7 ⊕ 2Γ8Γ5 Γ1 ⊕ Γ3 ⊕ Γ4 ⊕ Γ5 Γ7 ⊕ Γ8 Γ6 ⊕ Γ8 Γ6 ⊕ Γ7 ⊕ 2Γ8Γ6 Γ1 ⊕ Γ4 Γ2 ⊕ Γ5 Γ3 ⊕ Γ4 ⊕ Γ5Γ7 Γ1 ⊕ Γ4 Γ3+Γ4 ⊕ Γ5Γ8Γ1 ⊕ Γ2 ⊕ Γ3⊕2Γ4 ⊕ 2Γ5Table 4.2: Direct product table for the Td double group, showing the decompositions ofproducts of irreducible representations within the group [1]. For example, an operator withsymmetry E acting on a state with symmetry T1 will have symmetry that can be representedas T1 ⊕ T2.1s:T2 and 1s:E states, which can be neglected if the difference in energy levels is sufficientlylarge [55]. Since the 1s:E states are predicted to be 25 meV above the 1s:T2 states, andthe spin-orbit coupling is on the order of 2 meV, we can to a first approximation neglectthis mixing [56]. We note that the spin-orbit term is often interchangeably referred to as aspin-valley interaction, since it depends on the valley composition of each state [56].The next-most significant interaction is the hyperfine interaction with the donor’s nu-clear spin, which is non-zero due to the 77Se+ donor’s odd number of protons. The in-teraction is only significant for the A1 state since it is the only state without a node atthe location of the donor. Consequently, the two 1s:A1(Γ6) states are split into a singlet1s:A1(Γ6):S and a triplet 1s:A1(Γ6):T, which form the basis for a long-lived spin qubit [20].The energy levels of 77Se+ are shown in Figure 4.2.4.2 Dipole-Allowed TransitionsWhether or not a transition is forbidden or allowed can be determined by the symmetry ofthe initial and final states of the transition, as well as the operator connecting the states.The transition probability is proportional to a transition matrix element of the followingform〈Ψi|O |Ψf 〉 , (4.15)where Ψi and Ψf are the initial and final states and O is some transition-inducing operator,such as the dipole operator. For the transition to be allowed, the transition matrix elementmust be non-zero. Since the transition matrix element is integrated over all space, thisintegral can only be non-zero if the transition matrix element contains a totally symmetricrepresentation, which for the Td group is the irreducible group A1. Hence, for a transition494.2. Dipole-Allowed TransitionsFigure 4.2: Including spin degeneracy, effective mass theory yields 12 degenerate 1s groundstates for the donor’s electronic wavefunction. However, the presence of the substitutionaldonor alters the symmetry of the system, resulting in valley-orbit splitting. The statesare further split due to a spin-valley interaction as well as a hyperfine interaction with thedonor’s nuclear spins. This additional nuclear degree of freedom results in the two 1s:Astates to be split into a triplet and a singlet state which can be used to encode the state ofa qubit.operator with symmetry given by an unknown representation ΓO, the following productΓi ⊗ Γo ⊗ Γf (4.16)must contain A1, where Γi and Γf are the representations for the initial and final states,respectively. We can again evaluate such a product by using Table 4.1. Since the 1s:A1 and1s:E do not exhibit any spin-orbit coupling, we can evaluate the selection rules involvingthese states by considering their ordinary group representations. Doing so, we find thatthe product Γ1 ⊗ Γ4 ⊗ Γ4 does not contain the totally symmetric representation Γ1, and sothe 1s:A1 ↔ 1s:E transition is symmetry-forbidden. On the other hand, the 1s:T2 statesexhibit spin-orbit coupling, so we must evaluate their selection rules using double groups.Doing so, we find that the both the 1s:A1(Γ6) ↔ 1s:T2(Γ7) and 1s:A1(Γ6) ↔ 1s:T2(Γ8)transitions are symmetry-allowed, as well as the 1s:E(Γ8) ↔ 1s:T2(Γ7) and 1s:E(Γ8) ↔1s:T2(Γ8) transitions. The former transitions have been directly observed, whereas the lat-ter transitions have only been observed with shallow donors [56] [57]. The 1s:A1(Γ6) ↔1s:E(Γ8) transition has not been experimentally observed despite efforts to do so, indicat-ing that the mixing of the 1s:T2 and 1s:E states is insignificant and confirming that thetransition is symmetry forbidden [37].504.3. Parameters Related to the Dipole Coupling of Singly-Ionized 77Se+ to Radiation Modesµ γrad γ1.96 D 0.11 neV 0.086-0.12 µeVTable 4.3: Summary of relevant optical transition parameters for the 77Se+ donor.4.3 Parameters Related to the Dipole Coupling ofSingly-Ionized 77Se+ to Radiation ModesThe dipole transition moment of 77Se+ can be determined by measuring the absorptionof light at the transition frequency in a substrate containing many 77Se+ atoms. Usingthe Lorentz oscillator model, the attenuation of light at the transition frequency ωe and adistance L into such a medium can be expressed asexp[− 8pi2λ0nLRe[] + 23ρ2ωeµ2~γωe +4ω4eµ2√Re[]3~c3], (4.17)where ρ is the density of emitters, n and  are the substrate’s refractive index and complexpermittivity, and γ is a dissipative damping term. Such an experiment has determined thetransition dipole moment to be µ = 1.96 D [37]. We can then use the expression for theEinstein coefficient for spontaneous emission to relate the radiative linewidth to the dipoletransition moment [58]A21 =γrad2pi=ω3nµ23pi~0c3g1, (4.18)where g1 = 2 is the degeneracy of the ground state A1 and A21 is the Einstein coefficient forspontaneous emission in units [1/s] and γrad is the radiative linewidth in units of [rad/s].In energy, the linewidth of the of the A1 ↔ Γ7 transition is ~γrad = 1.4 neV. The radiativelifetime of the emitter is then found byTrad =1A21=2piγrad, (4.19)which for the A1 ↔ Γ7 transition yields a radiative lifetime of 0.5 µs. In addition to radiativedecay, however, the excited state can decay non-radiatively, which also contributes to thepopulation lifetime T1. The T1 time has been measured using a modulation-excitationexperiment to the be 7.7 ns, indicating that the non-radiative processes dominate. Thisvalue gives a homogeneous linewidth of 0.086 µeV, which is determined by both radiativeand non-radiative processesγ = γrad + γnon-rad. (4.20)This value is a lower bound, as the linewidth can be further broadened through interac-tion with thermal photons. This additional broadening becomes significant above 4 K,but a hole-burning experiment done at 1.5 K has set an upper bound to the homoge-neous linewidth of 0.12 µeV [20]. The relevant optical transition parameters for 77Se+ aresummarized in Table 4.3.51Chapter 5ResultsNow that we have developed a model for analyzing the photon statistics of two-level systemsin cavities and introduced the basic properties of the 77Se+ donor, we can investigate thefeasibility of using cQED to measure and entangle 77Se+ qubits with high fidelity. Wealso look at systems containing many 77Se+ donors, which will be useful for characterizingcavities before site-selective single impurity implantation techniques are perfected.A necessary part of the proposed cQED entangling protocol involving the 77Se+ donoris that the transmission and reflection statistics are significantly affected by the state ofthe qubit encoded in the hyperfine 1s:A1:T ↔ 1s:A1:S transition [21]. Since the 1s:A1:S↔ 1s:T2:Γ7 and 1s:A1:T ↔ 1s:T2:Γ7 transitions are separated by 6.87 µeV, it is possible totune a cavity into resonance with only one of these transitions. Consequently, whether lightis transmitted or reflected from the cavity at the resonant frequency of the cavity can becorrelated to the state of the qubit. However, it has been shown that the protocol is onlyfeasible if the 2-qubit parity measurement can be implemented with photon losses below≈ 1%. With this in mind, we devote the first section to investigating what kind of photonlosses are achievable with realistic cavity parameters.We then separately investigate the spectral response of cavities containing unknownnumbers of 77Se+ donors with different transition frequencies. As the number of ions in-troduced by a method like ion implantation is non-deterministic, this will be of use incharacterizing cavities with unknown numbers of donors. We also investigate the possibil-ity of achieving strong coupling in cavities containing ensembles of donors with and withoutthe presence of inhomogeneous broadening. As shown in Chapter 3, it is possible to achievestrong coupling in spite of inhomogeneous broadening if the collective Rabi frequency isgreater than the FWHM of the broadening. The collective Rabi frequency, however, isultimately limited by the number of donors that can be introduced without exceeding thedonor-specific solubility limit in silicon. We begin by analyzing a single 77Se+ donor insilicon.5.1 Single 77Se+ Donor Coupled to a CavityIdeally, the cavity should have near-unity reflection (transmission) when the 77Se+ donor ison (off) resonance. Then, the state of the qubit can be accurately determined by an opticalmeasurement. This requires minimizing the losses in the system, which can be present dueto the cavity being coupled to undesired radiation modes. Even in the limit of a cavitycoupled only to the desired input and output channels, losses can occur due to spontaneousemission and non-radiative transitions within the emitter. A schematic of the cavity withits input and output modes is shown in Figure 5.1. Throughout this chapter, we considersystems with κ1 = κ2 = κ/2, where the cavity-waveguide coupling to each waveguide is525.1. Single 77Se+ Donor Coupled to a Cavityidentical, and we use the same dipole transition moment of 1.96 D for the on- and off-resonant cases. We define the unloaded quality factor to be the quality factor of the cavityin the absence of coupling to the waveguides, determined solely by the coupling parameterΓc, which represents the coupling of the localized optical mode of the cavity to radiationmodes other than the designated input and output modes of the waveguide. In contrast,the loaded quality factor also accounts for the wave-guide cavity coupling parameters κ1and κ2. In general, the parameters yielding the maximum transmission when the emitterFigure 5.1: Schematic of the optical circuit for a 1-parity measurement, including the statescorresponding to the waveguides, cavity, atom, and loss channel. The upper and lowerwaveguide couple to the cavity via κ1 and κ2, respectively, and Γc represents the couplingof the localized optical mode to radiation modes not including those associated with theinput and output from the waveguides. Throughout this chapter, we restrict ourselves tosystems where κ1 = κ2 = κ/2. Here, reflection corresponds to the signal measured at theport for aout and transmission corresponds to the signal measured in the port by bout.is off resonance will not yield the maximum reflection when the atom is on resonance.Furthermore, the maxima in transmission and reflection will not necessarily occur at thesame frequency. Hence, it is useful to maximizep(ω) = T↑(ω)R↓(ω) (5.1)where T↑(ω) is the transmission when the atom is off resonance and R↓(ω) is the reflectionwhen the atom is on resonance. We refer to the maximum value of p(ω) as pmax, whichwhen close to unity implies near-lossless reflection (transmission) when the emitter is on(off) resonance. The frequency at which pmax occurs will not necessarily be precisely atthe resonant frequency of the cavity, since the the presence of an off-resonant atom in thedetuned case will shift the transmission peak. Now that we have a single figure of merit535.1. Single 77Se+ Donor Coupled to a Cavityfor the off- and on-resonant case, we can investigate its dependence on the cavity couplingparameters κ and Γc, taking the atomic parameters to be those experimentally determinedfor the 77Se+ impurity in Section 4.3.Figure 5.2: (a) pmax as a function of κ and Γc for a single77Se+ donor, showing the optimalwaveguide-cavity for a given cavity. Pink and red crosses are at the optimal couplingparameters for a cavity with a quality factor of 105 or 106, respectively. (b) On- and off-resonant transmission and p(ω − ωc) for the parameters at the pink cross (Γc = 0.43 µeV,κ = 3.9 µeV). (c) On- and off-resonant transmission and p(ω − ωc) for the parameters atthe red cross (Γc = 4.3 µeV, κ = 13 µeV). (d) Same as (a) but with effective mode volumeVeff = 0.1. (e) and (f) show close-ups of (a) and (d) near the origin.For an L3 cavity with effective mode volume Veff = 1 we show how pmax varies with κand Γc in Figure 5.2 (a). For an unloaded quality factor of 105, pmax = 0.3 can be achieved,corresponding to an on-resonant reflection of 0.53 and an off-resonant transmission of 0.56.For an unloaded quality factor of 106 it is possible to achieve pmax = 0.63, where the on-resonant reflection is 0.76 and the off-resonant transmission is 0.83. The transmission andreflection spectra and p(ω − ωc) are shown in (b) and (c) of Figure 5.2, where it can beseen that pmax occurs at the resonance frequency of the cavity. In the limit wherein the545.1. Single 77Se+ Donor Coupled to a Cavitycavity is only coupled to the waveguide modes, and not to any undesired free radiationmodes, pmax = 0.97 is achieved, which is ultimately limited by unavoidable losses due tospontaneous emission and non-radiative transitions. This corresponds to 99% transmissionand 98% reflection in the off- and on-resonant case, respectively. Since the fault tolerantthreshold for the protocol outlined in Section 1.4 is on the order of 1%, this suggests that newapproach to error correction must be taken to accommodate cavities with experimentallyachievable quality factors [21].For the L3 cavity we have an effective mode volume of approximately unity, but it ispossible to achieve mode volumes of Veff = 0.5 and Veff = 0.1, by using nanocavities [59][60].We illustrate the impact of lowering the mode volume in Figure 5.2 (d), where we comparepmax values for cavities with mode volumes of Veff = 1 and Veff = 0.1. Since decreasing themode volume by a factor of 10 results in the atom-cavity coupling g being increased by afactor of√10, we expect that the same pmax values are achieved when the cavity couplingvalues are 10 times larger, as their cooperativities are then equal. The only departure fromthis, as shown in the inset of Figure 5.2, is for value of Γc . 0.005 µeV, correspondingto unloaded quality factors on the order of 108. In Figure 5.3 we give an example ofFigure 5.3: Transmission and reflection spectra for single 77Se+ donor with cooperativity C= 1.6 and Γc = κ/10 but different Veff and κ. (a) Veff = 1 and κ = 4.3 µeV. (b) Veff = 0.1and κ = 43 µeV. (c) Veff = 0.01 and κ = 430 µeV.transmission and reflection spectra with the same cooperativity but different mode volumesand quality factors. It is clear that a potential benefit of using lower mode volume cavities555.1. Single 77Se+ Donor Coupled to a Cavityis that the transmission peak in the off-resonant case becomes much broader, which couldbe beneficial for inputs with a finite bandwidth. The reflection peak in the off-resonantcase is, however, unaffected. The impact of photon loss via coupling to the environmentby Γc is most obvious in Figure 5.3 (c), where it is clear that even in the absence of anemitter in the cavity, only ∼82% of the input to the cavity is transmitted while none isreflected. Hence, ∼18% of the input is transferred to leaky radiation modes. Finally, weFigure 5.4: Transmission and reflection spectra and p(ω) for a single donor at the anti-nodeof a cavity with parameters κ = 3.9 µeV and Γc = 0.43 µeV . The case of a donor with anoff-resonant detuning ωe >> g is shown by the red dashed line for comparison. a) Veff = 1b) Veff = 0.1 c) Veff = 0.01.demonstrate how varying the effective mode volume Veff can change the frequency for whichpmax is greatest. In Figure 5.4 we show transmission and reflection spectra and p(ω) fora a single donor at the anti-node of a cavity with coupling parameters κ = 3.9 µeV andΓc = 0.43 µeV . Mode volumes of Veff = 1, Veff = 0.1, and Veff = 0.01 are used for Figures5.4 a), b) and c), respectively. For mode volume Veff = 0.01 the frequency at which pmaxoccurs is significantly shifted from the resonance frequency of the cavity. As a reference, wealso include p(ω) for an emitter with an off-resonant detuning much greater than the cavitycoupling ωe >> g, shown as a red dashed line.565.2. Many 77Se+ Atoms In a Cavity5.2 Many 77Se+ Atoms In a CavityAs only a finite number of 77Se+ substitutional impurities can be introduced into a siliconsubstrate, for large quantities of emitters it makes more sense to work with densities asopposed to numbers of atoms. Pure silicon contains 5 · 1022 atoms/cm3 and has a solubilitylimit of approximately 1017 substitutional chalcogen atoms per cm3 [61]. In the followingsection, we use the mode profile of an L3 cavity given by Figure 1.3 to model ensemblesof emitters distributed throughout the cavity and investigate the requirements for strongcoupling, showing transmission spectra for realistic densities of emitters.5.2.1 Inhomogeneous BroadeningA significant source of inhomogeneous broadening for atoms in a silicon substrate is isotopicinhomogeneities. Natural silicon is composed of three main isotopes: 92.22% 28Si, 4.69% 29Si, and 3.09% 30Si, resulting in the broadening of an ensemble of 77Se+ atoms insuch a substrate to have a spectral FWHM of approximately 14 µeV [62]. We plot thetransmission for increasing emitter count in Figure 5.5 for κ = 43 µeV, corresponding to aloaded quality factor of 10 000 and compare it to the case of no inhomogeneous broadening.In the inhomogeneously broadened ensemble, strong coupling is achieved at a density ofFigure 5.5: (a) Transmission spectra for ensembles of 77Se+ donors with inhomogeneousbroadening given by a Gaussian with FWHM of 14 µeV, representing isotopic impurities inNatSi. Donors are randomly distributed in an L3 cavity with κ = 43 µeV. The numbers ineach figure correspond to the density of 77Se+ donors per cm3. (b) Same parameters as (a)but in isotopically pure 28Si without inhomogeneous broadening.ρ = 2.3 · 1015 atoms/cm3, where the collective atom-cavity coupling is larger than the575.2. Many 77Se+ Atoms In a Cavity14 µeV FWHM of the inhomogeneous broadening. For densities of ρ ≈ 2.3 · 1013, largedips can be seen in the transmission spectrum. In isotopically pure 28Si, however, a similardip in the transmission spectrum can be achieved with a donor density of ρ = 2.3 · 1011atoms/cm3, and a cooperativity of ≈ 1 is achieved for ρ = 2.3 · 1013 atoms/cm3. Next weFigure 5.6: Transmission spectra for 10 different ensembles of randomly distributed 77Se+donors in an L3 cavity with parameters ρ = 1012 atoms/cm3, κ = 3.9 µeV and Γc =0.43 µeV, corresponding to a loaded quality factor QL = 100 000. Transition frequencies aredrawn from a Gaussian distribution with a FWHM of 14 µeV, corresponding to broadeningfrom isotopic inhomogeneities in natural silicon.address the issue of cavity characterization. Since the transmission spectrum is sensitive tothe number of atoms in a cavity as well as their transition frequencies, it can potentiallybe used to determine the contents of a cavity containing an unknown number of atoms.Atoms with transition frequencies broadened due to inhomogeneous broadening will resultin dips in the transmission spectrum at their resonant frequency, thus making it possibleto determine how many atoms are present in the cavity. Furthermore, the magnitude ofthe dip in the transmission spectrum will be proportional to the coupling to the cavity,which provides an estimate of the proximity of the atom to the cavity. We consider donorsrandomly distributed in a substrate containing an L3 cavity with parameters κ = 3.9 µeVand Γc = 0.43 µeV, corresponding to a loaded quality factor of Q = 100 000. As the donorsare randomly distributed, there will be many donors located outside of the main anti-nodesof the cavity that will still be weakly coupled to the cavity, and the atom-cavity couplingcoefficients of the ensemble will be highly non-uniform. The mode profile for an L3 cavitywas shown in Figure 1.3, which we approximate to be constant in the direction normal tothe substrate surface, or that the atoms are located at a uniform depth in the cavity. For adensity of ρ = 1012 atoms/cm3, transmission spectra are shown in Figure 5.6 for 10 different585.2. Many 77Se+ Atoms In a Cavityensembles of randomly distributed atoms with transition frequencies given by a Gaussiandistribution with FWHM of 14 µeV.In this case, the inhomogeneous broadening is significantly broader than the cavity’slinewidth of 4.3 µeV, making it possible to determine the presence of atoms with differenttransition frequencies.5.2.2 Dipole-dipole InteractionsDonors having overlapping linewidths will be able to interact via virtual processes, resultingin dipole-dipole interactions between emitters as given by Equation 2.43. If we consider adensity of ρ = 1015 atoms/cm3, which is enough to observe strong coupling in NatSi and28Si, the dipole-dipole interactions are insignificant and their impact cannot be observedfrom the transmission statistics. It is possible for them to become observable if the donorsare concentrated within a small region of the cavity, such as may be the case with ion iml-pantation, since the interaction is proportional to the third power of their inverse distance.Furthermore, the dipole-interactions will be greater if there is no inhomogeneous broaden-ing present, as the emitters’ linewidths will have greater overlap. Thus, for the case of noinhomogeneous broadening and κ = 43 µeV, we show in Figure 5.7 the transmission spectrawith and without dipole-dipole coupling for various numbers of 77Se+ donors confined toa cubic region with a width of 100 nm at the center of an L3 cavity. Like inhomogeneousFigure 5.7: Transmission spectra with and without dipole-dipole interactions for increasingdensities of 77Se+ donors without any inhomogeneous broadening present. The emitters areon resonance with a cavity with κ = 43 µeV and confined to a cubic volume with a sidelength of 100 nm at the center of an L3 cavity.broadening, dipole-dipole interactions shift the frequencies of what would otherwise be dark595.2. Many 77Se+ Atoms In a Cavitystates with eigenvalues at the resonant frequency of the cavity, resulting in observable dipsin the transmission spectrum. Since the dipole-dipole interactions scale explicitly with thedensity of atoms, they only become significant for larger donor densities. Unlike inhomoge-neous broadening, the effects of dipole-dipole interactions persist into the strong-couplingregime.60Chapter 6Conclusion and Future WorkFirstly, we have used a scattering matrix formalism to model the transmitted and reflectedphoton statistics of systems composed of optical microcavities containing multiple quantumemitters. The model is valid in the limit of a weak coherent state input, which can bewritten in terms of 1- and 2-photon Fock states; the input-out formalism can then be usedto compute the scattering matrices for these states, which can be written in terms of theHamiltonian of the system. Hence, the transmission, reflection, and higher-order photonstatistics can be computed for arbitrarily Hamiltonians. We have modelled inhomogeneousbroadening of atomic transition frequencies with both Gaussian and Lorentzian profiles. ForGaussian-broadened ensembles, which is the most typical case for ensembles in solid-statehosts, strong coupling can be achieved when the collective atom-cavity coupling surpassesthe full-width half max of the inhomogeneous broadening, where there is no longer anyevidence of broadening.We have also investigated the feasibility of using cQED to measure and entangle 77Se+subsitutional donors in silicon with high fidelity. The ground state of the 77Se+ donor issplit due to a hyperfine nuclear interaction, resulting in a separation energy of 6.87 µeVbetween the triplet and singlet states, which can be used to encode the state of a qubit.Since the transmitted and reflected photon statistics are influenced by the donor’s detuningfrom the cavity, the state of the qubit can be determined by measuring the reflected ortransmitted photon statistics from the cavity. Furthermore, two such qubits can be en-tangled by cascading multiple cavities together via bus waveguides and universal quantumcomputation can be achieved via the measurement of generated cluster states. Such a pro-tocol requires minimizing photon losses that can occur due to non-radiative transitions inthe emitter, and due to the cavity being coupled to modes other than the input and outputmodes of the waveguide. Consequently, we have investigated the maximum reflection andtransmission achievable for a single 77Se+ donor being on and off resonance in cavities withexperimentally-achievable quality factors. We find that for an L3 cavity with a modestquality factor of 105, a transmission of 56% and reflection of 53% are achievable in the off-and on-resonant case, respectively. With a quality factor of 106, it is possible to achieve atransmission of 83% and reflection of 76%. For a cavity coupled only to the bus waveguides,without any direct coupling to undesired modes, 99% transmission and 98% reflection areachievable for the off- and on-resonant case, which are ultimately limited due to unavoidablelosses from spontaneous emission and non-radiative transitions.We have also modelled cavities containing inhomogeneously broadened ensembles of77Se+ donors, predicting that strong coupling should be achievable for a density of approx-imately 1015 77Se+ donors/cm3, which is well below the solubility limit of selenium donorsin silicon. We have used an extended Tavis-Cummings model to incorporate the effect ofdipole-dipole interactions, which we have shown to often be insignificant in the presence of61Chapter 6. Conclusion and Future Workinhomogeneous broadening. In the limit of no inhomogeneous broadening and a high con-centration of donors restricted to a small volume in the cavity, as might be done with preciseion implantation, the effect of dipole-dipole interactions can be observed in strong couplingregime, where they result in additional Fano resonances in the transmission spectrum.Furthermore, we have investigated the possibility of characterizing the contents of acavity based on its transmitted photon statistics. For 77Se+ donors broadened due to iso-topic impurities in NatSi, their presence can determined due to the appearance of resonancesin the transmission spectrum, with the magnitude of the resonance providing informationregarding the proximity of the donor to the mode maximum of the cavity.Future work to expand the analysis in this thesis includes dealing with pulsed lightsources and modelling cascaded systems of cavities, as well as comparing the theoreticalmodel to experimentally measured results. Furthermore, additional states of the 77Se+donor’s energy could be included in the model, such as might be the case if the 1s:A1:Ttriplet state degeneracy is broken due to the presence of a magnetic field, instead of mod-elling the transition as a two-level system.62Bibliography[1] S. Altmann and P. Herzig, Point-group Theory Tables. Oxford science publications,Clarendon Press, 1994.[2] S. Johnson, “Introduction to photonic crystals : Bloch ’ s theorem , band diagrams ,and gaps ( but no defects ),” 2003.[3] R. P. Feynman, “Simulating physics with computers,” International Journal of Theo-retical Physics, vol. 21, no. 6, pp. 467–488, 1982.[4] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas,S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro,R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney,M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho,M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri,K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Land-huis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandra`, J. R. McClean, M. McEwen,A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley,C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rief-fel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung,M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zal-cman, H. Neven, and J. M. Martinis, “Quantum supremacy using a programmablesuperconducting processor,” Nature, vol. 574, no. 7779, pp. 505–510, 2019.[5] E. Pednault, J. A. Gunnels, G. Nannicini, L. Horesh, and R. Wisnieff, “Leveragingsecondary storage to simulate deep 54-qubit sycamore circuits,” 2019.[6] H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu,X. Ding, Y. Hu, P. Hu, X.-Y. Yang, W.-J. Zhang, H. Li, Y. Li, X. Jiang, L. Gan,G. Yang, L. You, Z. Wang, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Quantumcomputational advantage using photons,” Science, vol. 370, no. 6523, pp. 1460–1463,2020.[7] P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete loga-rithms on a quantum computer,” SIAM Journal on Computing, vol. 26, no. 5, pp. 1484–1509, 1997.[8] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information:10th Anniversary Edition. USA: Cambridge University Press, 10th ed., 2011.[9] A. Y. Kitaev, “Quantum computations: algorithms and error correction,” RussianMathematical Surveys, vol. 52, pp. 1191–1249, dec 1997.63Bibliography[10] D. P. DiVincenzo, “The physical implementation of quantum computation,”Fortschritte der Physik, vol. 48, no. 9aˆ11, pp. 771–783, 2000.[11] J. Preskill, “Quantum Computing in the NISQ era and beyond,” Quantum, vol. 2,p. 79, Aug. 2018.[12] W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,” Nature,vol. 299, no. 5886, pp. 802–803, 1982.[13] C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, “Trapped-ion quantumcomputing: Progress and challenges,” Applied Physics Reviews, vol. 6, no. 2, p. 021314,2019.[14] E. National Academies of Sciences and Medicine, Quantum Computing: Progress andProspects. Washington, DC: The National Academies Press, 2019.[15] I. L. Chuang and Y. Yamamoto, “Simple quantum computer,” Phys. Rev. A, vol. 52,pp. 3489–3496, Nov 1995.[16] E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computationwith linear optics,” Nature, vol. 409, no. 6816, pp. 46–52, 2001.[17] T. Rudolph, “Why i am optimistic about the silicon-photonic route to quantum com-puting,” APL Photonics, vol. 2, no. 3, p. 030901, 2017.[18] E. Waks and J. Vuckovic, “Dipole induced transparency in drop-filter cavity-waveguidesystems,” Phys. Rev. Lett., vol. 96, p. 153601, Apr 2006.[19] K. Nemoto, M. Trupke, S. J. Devitt, A. M. Stephens, B. Scharfenberger, K. Buczak,T. No¨bauer, M. S. Everitt, J. Schmiedmayer, and W. J. Munro, “Photonic architec-ture for scalable quantum information processing in diamond,” Phys. Rev. X, vol. 4,p. 031022, Aug 2014.[20] K. J. Morse, R. J. S. Abraham, A. DeAbreu, C. Bowness, T. S. Richards, H. Riemann,N. V. Abrosimov, P. Becker, H.-J. Pohl, M. L. W. Thewalt, and S. Simmons, “Aphotonic platform for donor spin qubits in silicon,” Science Advances, vol. 3, no. 7,2017.[21] X. Yan, W. Asavanant, H. Kamakari, J. Wu, J. F. Young, and R. Raussendorf, “Aquantum computer architecture based on silicon donor qubits coupled by photons,”Advanced Quantum Technologies, vol. n/a, no. n/a, p. 2000011.[22] Resonator Optics, ch. 9, pp. 310–341. John Wiley Sons, Ltd, 2001.[23] C. Gerry and P. Knight, Introductory Quantum Optics. Cambridge University Press,2004.[24] S. Mosor, J. Hendrickson, B. C. Richards, J. Sweet, G. Khitrova, H. M. Gibbs,T. Yoshie, A. Scherer, O. B. Shchekin, and D. G. Deppe, “Scanning a photonic crystalslab nanocavity by condensation of xenon,” Applied Physics Letters, vol. 87, no. 14,p. 141105, 2005.64Bibliography[25] K. Maeno, Y. Takahashi, T. Nakamura, T. Asano, and S. Noda, “Analysis of high-qphotonic crystal l3 nanocavities designed by visualization of the leaky components,”Opt. Express, vol. 25, pp. 367–376, Jan 2017.[26] T. Asano and S. Noda, “Iterative optimization of photonic crystal nanocavity designsby using deep neural networks,” Nanophotonics, vol. 8, no. 12, pp. 2243 – 2256, 01Dec. 2019.[27] A. R. A. Chalcraft, S. Lam, D. OaˆBrien, T. F. Krauss, M. Sahin, D. Szymanski,D. Sanvitto, R. Oulton, M. S. Skolnick, A. M. Fox, D. M. Whittaker, H.-Y. Liu, andM. Hopkinson, “Mode structure of the l3 photonic crystal cavity,” Applied PhysicsLetters, vol. 90, no. 24, p. 241117, 2007.[28] O. Di Stefano, A. Settineri, V. Macr`ı, L. Garziano, R. Stassi, S. Savasta, and F. Nori,“Resolution of gauge ambiguities in ultrastrong-coupling cavity quantum electrody-namics,” Nature Physics, vol. 15, no. 8, pp. 803–808, 2019.[29] A. Frisk Kockum, A. Miranowicz, S. De Liberato, S. Savasta, and F. Nori, “Ultrastrongcoupling between light and matter,” Nature Reviews Physics, vol. 1, no. 1, pp. 19–40,2019.[30] D. L. Andrews and D. S. Bradshaw, “Virtual photons, dipole fields and energy transfer:a quantum electrodynamical approach,” European Journal of Physics, vol. 25, pp. 845–858, oct 2004.[31] Y.-Q. Zhang, L. Tan, and P. Barker, “Effects of dipole-dipole interaction on the trans-mitted spectrum of two-level atoms trapped in an optical cavity,” Phys. Rev. A, vol. 89,p. 043838, Apr 2014.[32] S. Buckley, K. Rivoire, and J. Vucˇkovic´, “Engineered quantum dot single-photonsources,” Reports on Progress in Physics, vol. 75, p. 126503, nov 2012.[33] M. Reindl, J. H. Weber, D. Huber, C. Schimpf, S. F. Covre da Silva, S. L. Portalupi,R. Trotta, P. Michler, and A. Rastelli, “Highly indistinguishable single photons fromincoherently excited quantum dots,” Phys. Rev. B, vol. 100, p. 155420, Oct 2019.[34] R. B. Patel, A. J. Bennett, I. Farrer, C. A. Nicoll, D. A. Ritchie, and A. J. Shields,“Two-photon interference of the emission from electrically tunable remote quantumdots,” Nature Photonics, vol. 4, no. 9, pp. 632–635, 2010.[35] S. Sun, H. Kim, G. S. Solomon, and E. Waks, “Strain tuning of a quantum dot stronglycoupled to a photonic crystal cavity,” in CLEO: 2014, p. FF1K.7, Optical Society ofAmerica, 2014.[36] C. Chartrand, L. Bergeron, K. J. Morse, H. Riemann, N. V. Abrosimov, P. Becker, H.-J. Pohl, S. Simmons, and M. L. W. Thewalt, “Highly enriched 28Si reveals remarkableoptical linewidths and fine structure for well-known damage centers,” Phys. Rev. B,vol. 98, p. 195201, Nov 2018.65Bibliography[37] A. DeAbreu, C. Bowness, R. J. Abraham, A. Medvedova, K. J. Morse, H. Riemann,N. V. Abrosimov, P. Becker, H.-J. Pohl, M. L. Thewalt, and S. Simmons, “Characteri-zation of the Si:se+ spin-photon interface,” Phys. Rev. Applied, vol. 11, p. 044036, Apr2019.[38] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics. Cambridge UniversityPress, 1995.[39] R. Loudon, The Quantum Theory of Light. Oxford: Clarendon Press, 1973.[40] R. Lang, M. O. Scully, and W. E. Lamb Jr, “Why is the laser line so narrow? a theoryof single-quasimode laser operation,” Physical Review A, vol. 7, no. 5, p. 1788, 1973.[41] J. Gea-Banacloche, “Space-time descriptions of quantum fields interacting with opticalcavities,” Physical Review A, vol. 87, no. 2, p. 023832, 2013.[42] R. Trivedi, M. Radulaski, K. A. Fischer, S. Fan, and J. Vucˇkovic´, “Photon blockadein weakly driven cavity quantum electrodynamics systems with many emitters,” Phys.Rev. Lett., vol. 122, p. 243602, Jun 2019.[43] C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems:Quantum stochastic differential equations and the master equation,” Phys. Rev. A,vol. 31, pp. 3761–3774, Jun 1985.[44] D. F. Walls and G. J. Milburn, Quantum optics / D.F. Walls, Gerard J. Milburn.Springer Berlin, 2nd ed. ed., 2008.[45] K. J. Blow, R. Loudon, S. J. D. Phoenix, and T. J. Shepherd, “Continuum fields inquantum optics,” Phys. Rev. A, vol. 42, pp. 4102–4114, Oct 1990.[46] S. Fan, S. E. Kocabas¸, and J.-T. Shen, “Input-output formalism for few-photon trans-port in one-dimensional nanophotonic waveguides coupled to a qubit,” Phys. Rev. A,vol. 82, p. 063821, Dec 2010.[47] J. Taylor, Scattering Theory: Quantum Theory on Nonrelativistic Collisions. Wiley,1972.[48] R. Trivedi, K. Fischer, S. Xu, S. Fan, and J. Vuckovic, “Few-photon scattering andemission from low-dimensional quantum systems,” Phys. Rev. B, vol. 98, p. 144112,Oct 2018.[49] T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, J. I. Cirac, and D. E. Chang,“Quantum dynamics of propagating photons with strong interactions: a generalizedinput–output formalism,” New Journal of Physics, vol. 17, p. 113001, oct 2015.[50] X. T. Zou and L. Mandel, “Photon-antibunching and sub-poissonian photon statistics,”Phys. Rev. A, vol. 41, pp. 475–476, Jan 1990.[51] Y. Zou, S. Chakravarty, C.-J. Chung, X. Xu, and R. T. Chen, “Mid-infrared siliconphotonic waveguides and devices,” Photon. Res., vol. 6, pp. 254–276, Apr 2018.66[52] M. W. Doherty, F. Dolde, H. Fedder, F. Jelezko, J. Wrachtrup, N. B. Manson, andL. C. L. Hollenberg, “Theory of the ground-state spin of the nv− center in diamond,”Phys. Rev. B, vol. 85, p. 205203, May 2012.[53] S. T. Pantelides, “The electronic structure of impurities and other point defects insemiconductors,” Rev. Mod. Phys., vol. 50, pp. 797–858, Oct 1978.[54] W. Kohn and J. M. Luttinger, “Theory of donor states in silicon,” Phys. Rev., vol. 98,pp. 915–922, May 1955.[55] T. G. Castner, “Orbach spin-lattice relaxation of shallow donors in silicon,” Phys. Rev.,vol. 155, pp. 816–825, Mar 1967.[56] H. G. Grimmeiss, E. Janze´n, and K. Larsson, “Multivalley spin splitting of 1s states forsulfur, selenium, and tellurium donors in silicon,” Phys. Rev. B, vol. 25, pp. 2627–2632,Feb 1982.[57] R. L. Aggarwal and A. K. Ramdas, “Optical determination of the symmetry of theground states of group-v donors in silicon,” Phys. Rev., vol. 140, pp. A1246–A1253,Nov 1965.[58] R. C. Hilborn, “Einstein coefficients, cross sections, f values, dipole moments, and allthat,” American Journal of Physics, vol. 50, no. 11, pp. 982–986, 1982.[59] J. Zhou, J. Zheng, Z. Fang, P. Xu, and A. Majumdar, “Ultra-low mode volume on-substrate silicon nanobeam cavity,” Opt. Express, vol. 27, pp. 30692–30699, Oct 2019.[60] P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. LonA¨ar, “Highquality factor photonic crystal nanobeam cavities,” Applied Physics Letters, vol. 94,no. 12, p. 121106, 2009.[61] M. A. Sheehy, B. R. Tull, C. M. Friend, and E. Mazur, “Chalcogen doping of silicon viaintense femtosecond-laser irradiation,” Materials Science and Engineering: B, vol. 137,no. 1, pp. 289 – 294, 2007.[62] M. Steger, A. Yang, M. L. W. Thewalt, M. Cardona, H. Riemann, N. V. Abrosimov,M. F. Churbanov, A. V. Gusev, A. D. Bulanov, I. D. Kovalev, A. K. Kaliteevskii,O. N. Godisov, P. Becker, H.-J. Pohl, E. E. Haller, and J. W. Ager, “High-resolutionabsorption spectroscopy of the deep impurities s and se in 28Si revealing the 77Sehyperfine splitting,” Phys. Rev. B, vol. 80, p. 115204, Sep 2009.[63] G. Grynberg, A. Aspect, C. Fabre, and C. Cohen-Tannoudji, Introduction to QuantumOptics: From the Semi-classical Approach to Quantized Light. Cambridge UniversityPress, 2010.[64] L. C. Andreani, G. Panzarini, and J.-M. Ge´rard, “Strong-coupling regime for quantumboxes in pillar microcavities: Theory,” Phys. Rev. B, vol. 60, pp. 13276–13279, Nov1999.67Appendix AHamiltonian for QuantizedRadiationTo quantize a system of massive particles, the problem would first be formulated classicallyusing Hamilton’s equations. The Hamiltonian, representing the total energy of the system,would be written in terms of the canonical conjugate variables position and momentumof the particles. The corresponding quantum Hamiltonian simply involves changing theclassical position and momentum operators to quantum operators. The dynamics of thesystem can then be solved using the Schro¨dinger equation. In order to quantize and solvethe dynamics of the electromagnetic field, we must first identify proper conjugate variables.Following [63], we begin with Maxwell’s Equations~∇×E(~r, t) = − ∂∂tB(~r, t), (A.1)~∇×B(~r, t) = 1c2∂∂tE(~r, t), (A.2)~∇ ·B(~r, t) = 0, (A.3)~∇ ·E(~r, t) = 0. (A.4)The set of vector field components of the E and B field constitute a complete set of dynamicalvariables. However, quantization will be difficult due to the continuous nature of the fields.Instead, it will be easier to work with the spatial Fourier expansion of the fields, wherewe have a countable infinity of normal modes. For simplicity, we will deal with a cube ofvolume V with sides of length L, which will be valid for radiation in free space. The spatialFourier components of a field occupying this volume areEn =1L3∫Vd3rE(r, t)e−ikn·r. (A.5)The field can then be written in terms of the Fourier components asE(r, t) =∑nEn(t)e−ikn·r, (A.6)wherekn =(nx2piL, ny2piL, nz2piL), (A.7)and Σn is an abbreviation for summing over nx, ny and nz. To explicitly include polarization,we note that by Equation 2.4kn ·En(t) = 0. (A.8)68Appendix A. Hamiltonian for Quantized RadiationHence, En(t) is a transverse field with components in two mutually orthogonal directionsn,1 and n,2, so we can writeEn(t) =2∑sEn,s(t)n,s = En,1(t)n,1 + En,2(t)n,2. (A.9)Furthermore, we can use a new index ` to include polarization` = (nx, ny, nz, s), (A.10)so we can writeE(r, t) =∑`E`(t)e−ik`·r`. (A.11)The B-field, which is orthogonal to the E-field, can be written asB(r, t) =∑`B`(t)e−ik`·r′`, (A.12)where′` =k`k`× `. (A.13)It will be simplest to quantize the fields in terms of the E-field and the vector potential, thelatter of which is described byB(r, t) = ∇×A(r, t), (A.14)E(r, t) =∂A(r, t)∂t−∇U(r, t), (A.15)where U(r, t) is the scalar potential. Working in the Coulomb gauge we have∇ ·A(r, t) = 0. (A.16)And the vector potential can be expanded in a similar fashion as E(r, t)A(r, t) =∑`A`(t)e−ik`·r`. (A.17)From (12) we get the relationB`(t) = ikA`(t), (A.18)and the time evolution of A(t) and E(t) can be found using Equation 2.2ddtA`(t) = −E`(t), (A.19)ddtE`(t) = ω2`A`(t), (A.20)whereω` = ck`. (A.21)69Appendix A. Hamiltonian for Quantized RadiationBy defining the quantitiesα` =12ε`(ω`A` − iE`), (A.22)β` =12ε`(ω`A` + iE`), (A.23)Equations A.22 and A.23 can be decoupled into two first-order differential equations.α˙` + iω`α` = 0, (A.24)β˙` − iω`β` = 0, (A.25)with solutions given byα`(t) = α`(0)e−iω`t, (A.26)β`(t) = β`(0)eiω`t. (A.27)Then, E(t) and A(t) can be written in terms of them asA`(t) =`ω`[α`(t) + β`(t)], (A.28)E`(t) = i`[α`(t)− β`(t)]. (A.29)Finally, the full solutions can be written asA(r, t) =∑``ω`[α`(t) + β`(t)]e−ik`·r`, (A.30)E(r, t) =∑`i`[α`(t)− β`(t)]e−ik`·r`. (A.31)Noting from Equations A.22 and A.23 thatβ∗` (t) = α−`(t), (A.32)we can simplify them toA(r, t) =∑``ω`[α`(t)eik`·r + α∗` (t)e−ik`·r]`, (A.33)E(r, t) =∑`i`[α`(t)eik`·r − α∗` (t)e−ik`·r]`, (A.34)where the fact that ` is a dummy index has been used to replace −` with `. Using EquationsA.12 and A.18, we also obtainB(r, t) =∑`i`c[α`(t)eik`·r − α∗` (t)e−ik`·r]′`. (A.35)We can now substitute these into the expression for the classical energy of a free electro-magnetic fieldHR =02∫Vd3r[E2(r, t) + c2B(r, t)], (A.36)70Appendix A. Hamiltonian for Quantized Radiationwhich then simplifies toHR = 20L3∑`2` |α`|2, (A.37)meaning that the total energy is a sum over the energy of each normal mode. To quantizethis, we start by defining canonical conjugate variables in terms of α`. First, we note wecan rewrite the first-order differential equations for α` asddtRe[α`] = ω`Im[α`], (A.38)ddtIm[α`] = −ω`Re[α`], (A.39)which are equivalent to the Hamiltonian equationsdQ`dt=∂H`∂P`, (A.40)dP`dt= −∂H`∂Q`, (A.41)whereQ` =√40L3ω``Re[α`], (A.42)P` =√40L3ω``Im[α`]. (A.43)Then, since these are canonical conjugate variables, they satisfy the following commutationrelationships: [Qˆ`, Pˆ`′]= i~δ``′ , (A.44)[Qˆ`, Qˆ`′]=[Pˆ`, Pˆ`′]= 0, (A.45)and α` then becomes a quantum operator aˆ` given byQˆ` + iPˆ` =√40L3ω``aˆ`. (A.46)Now by setting` =√~ω`20L3, (A.47)the commutation relations for aˆ` become[aˆ`, aˆ†`] = δ`,`′ , (A.48)[aˆ`, aˆ`] = 0. (A.49)71Appendix A. Hamiltonian for Quantized RadiationNow, the classical Hamiltonian for free radiation can be written asHR =∑`~ω`|α`|2 =∑`ω`2(Q2` + P2` ), (A.50)with the corresponding quantized HamiltonianHˆR =∑`ω`2(Qˆ2` + Pˆ2` ) =∑`~ω`(aˆ†`aˆ` +12). (A.51)For a single-mode cavity, the Hamiltonian simplifies toHˆR = ~ω(aˆ†aˆ+12), (A.52)which describes a ladder of states separated by ~ω with a zero-point energy of ~ω/2. Finally,the quantum field operators for radiation in free space are given byAˆ(r, t) =∑`√~20L3ω`[aˆ`eik`·r−ω`t + aˆ†`e−ik`·r+ω`t]`, (A.53)Eˆ(r, t) =∑`i√~ω`20L3[aˆ`eik`·r−ω`t − aˆ†`e−ik`·r+ω`t]`, (A.54)Bˆ(r, t) =∑`i√~20L3ω`[aˆ`eik`·r−ω`t − aˆ†`e−ik`·r+ω`t]′`. (A.55)To model the fields of microcavities with arbitrary mode profiles, we can repeat the analysiswith a general mode profile u`(r) for each mode `. Then, the E-field of the cavity can bewritten asE(r, t) =∑`E`(t)u`(r). (A.56)Following the same procedure as before, we arrive atAˆ(r, t) =∑`√~n32ω`λ30Veff[aˆ`e−iω`tu∗` (r) + aˆ†`eiω`tu`(r)], (A.57)Eˆ(r, t) =∑`i√~ω`n32λ30Veff[aˆ`e−iω`tu∗` (r)− aˆ†`eiω`tu`(r)], (A.58)Bˆ(r, t) =∑`i√~n32ω`λ30Veff[aˆ`e−iω`tu∗` (r)− aˆ†`eiω`tu`(r)], (A.59)where  is the permittivity of the material and we have used the dimensionless effectivemode volume [64]Veff =∫V dV |E|2max(|E|2) = Vn3λ30, (A.60)which can be evaluated for arbitrary mode profiles.72Appendix BMeasuring Dipole TransitionStrength from AbsorptionDipole transition strength can be measured indirectly by absorption, since the macroscopicabsorption of light is depends on the dipole transition moment. Absorption in a materialcan be accounted for by a complex refractive index n′ = n + iκ, where κ is known as theextinction coefficient. Then, the electric field of an electromagnetic plane wave travelling inthe z-direction through the material is given byE(z, t) = Re[E0ei(kz−ωt)] = Re[E0ei(2pi(n+iκ)zλ0−ωt)] = e−2piκzλ0 Re[E0ei(kz−ωt)]. (B.1)Hence, the signal is attenuated by a factor proportional to κ. Written in terms of intensity,it isI(z) = I(0)e−4pizIm[√]λ0 Re[E0ei(kz−ωt)], (B.2)where κ has been written in terms of the imaginary component of the dielectric constant ofthe material. If the extinction coefficient is not on the order of the real part of the refractiveindex, the following approximation can be made√ = (Re[n])(1 +4piIm[χ]Re[])1/2 ≈ Re[n] + i2piIm[χ]Re[n]. (B.3)And the imaginary component can be written asIm[√] ≈ 2piIm[χ]Re[n]. (B.4)To get an appropriate value for χ, local field effects due to the presence of many dipoles inthe material must be accounted for. This can be done using the Claussius-Mossotti relationχ3 + χ=ρα3, (B.5)where ρ is the density of two-level systems in the material and α is the atomic polarizability.In a more convenient form,Im[χ] =Re[] + 23Im[α]ρ+Im[] + 23Re[α]ρ. (B.6)Next, the atomic polarizability α can be written asα =2ωeµ2~ω2e − ω2 − iΓω − i4√Re[]ω3ωeµ23~, (B.7)73Appendix B. Measuring Dipole Transition Strength from AbsorptionwhereIm[α] =2ωeµ2~Γωe +4ω4eµ2√Re[]3~c3, (B.8)andRe[α] = 0. (B.9)Hence, the attenuation at a distance L into the material can be written asexp[− 8pi2λ0nLRe[] + 23ρ2ωeµ2~Γωe +4ω4eµ2√Re[]3~c3](B.10)or in terms of µ2,µ2 = ln(I(L)I(0))3~λ0∆ωFWHM√Re[]16pi2Lρ(Re[] + 2)). (B.11)where ∆ωFWHM is the linewidth of the oscillator from the atomic polarizability, given by∆ωFWHM = Γωe +4ω4eµ2√Re[]3~c3. (B.12)74
Thesis/Dissertation
eng
Physics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Young, Jeff F.
Modelling photon statistics of cavity QED systems for integrated photonic quantum computing
Text