SILICON CARBIDE AND COLOR CENTER QUANTUM …SILICON CARBIDE AND COLOR CENTER QUANTUM PHOTONICS A...

SILICON CARBIDE AND COLOR CENTER

QUANTUM PHOTONICS

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF APPLIED PHYSICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Marina Radulaski

March 2017

I certify that I have read this dissertation and that, in my opinion, it is fully adequate

in scope and quality as a dissertation for the degree of Doctor of Philosophy.

(Dr. Jelena Vuckovic) Principal Adviser



(Dr. Shanhui Fan)



(Dr. Martin Fejer)

Approved for the Stanford University Committee on Graduate Studies

Abstract

Quantum photonics has demonstrated unprecedented ways of manipulating interaction

of light and matter at the nanoscale. Using effects such as confinement of light to subwave-

length volumes and the generation of hybridized light-emitter states, one can reduce power

and increase speed in optical networks, manipulate quantum information, and sense biolog-

ical parameters. A preference for narrow emission linewidths and long spin-coherence has

drawn special interest to solid-state systems that incorporate color centers. These platforms

hold a promise of novel cavity quantum electrodynamical effects, integrated photonics and

spintronics. While a majority of color center findings has been obtained using the nitrogen-

vacancy center in diamond, recent research efforts have focused on discovering new emitters

in wide band gap substrates. Therein, silicon carbide has emerged as a color center host

with outstanding optical properties.

This thesis presents the development of silicon carbide and hybrid silicon carbide-diamond

color center quantum photonic platforms, studied through modeling, nanofabrication, and

confocal spectroscopy. This includes our pioneering demonstrations of high quality factor

and small mode volume microresonators in cubic silicon carbide (3C-SiC), such as photonic

crystal cavities at telecommunication wavelengths and microdisks in the visible and near

infra-red. Next, we have developed a scalable and efficient photonic design for spintronics

at the single emitter/quantum bit level implemented in hexagonal silicon carbide (4H-SiC).

We have also utilized the refractive index similarity between diamond and silicon carbide

to enhance silicon-vacancy and chromium center emission in nanodiamond. Finally, the

thesis introduces the model of a nanocavity containing multiple color centers that exhibits

collective cavity quantum electrodynamical effects, and discuss how this regime could be

reached experimentally.

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Acknowledgments

I am thankful to many people and organizations for making my graduate education

possible.

First and foremost, I am grateful to my supervisor Jelena Vuckovic for taking a chance

on me and building a relationship of trust. I have learned incredible concepts under her

instruction, as well as though the resources she made possible, including international and

local collaborations, state-of-the-art laboratory equipment, and opportunities to attend

conferences. I am also grateful to my commencement cheerleader Zoe Jovanovic who has

been a great partner in science outreach.

Being a part of the Stanford community has been a life-changing experience. I am grateful

to the Applied Physics department for selecting me for their graduate program and to the

Stanford Graduate Fellowship for providing financial support, which I also appreciated as

an act of inclusion from a perspective of an international student.

I am grateful to the professors who advised my research: Steven Chu for the inspiring

discussions in our collaboration, and for chairing my thesis defense; Shanhui Fan for spark-

ing my interest in nanophotonics, collaborating on student organizing as the head of the

Ginzton Laboratory, and agreeing to be a reader of my thesis; Martin Fejer for teaching me

about nonlinear optics, discussing practical experimental issues, and also agreeing to be my

thesis reader; David Miller for mentoring me on becoming an effective teacher, providing

new angles of looking at physics, and joining my thesis committee; Harold Hwang for being

my academic advisor and discussing my future plans; Robert Byer for providing academic

career advice and helping me become a better scientist; Stephen Harris for encouraging my

academic growth and taking interest in the details of my research; Hideo Mabuchi for ad-

vising me on cavity electrodynamics and being a great department chair; Zhi-Xun Shen and

Nicholas Melosh for introducing me to material science challenges in our diamond collabo-

ration; W. E. Moerner for helping me understand what photonics can offer in bio-related

research; Audrey Bowden for support and academic career advice; Amir Safavi-Naeini for

nanofabrication and career advice; Mark Brongersma for introducing me to plasmonics and

spin-off companies; Karl Wieman for insights on modern science teaching methods; Sheri

Sheppard for organizing a movement of women STEM students and motivating us to reach

our goals; Dave Evans for an incredibly insightful course on life planning; Dan Klein for

introducing me to improvisation which helped me develop a new teaching style; Lauren

Tompkins and Monika Schleier-Smith for being young-faculty role models.

I would like to thank my international collaborators: Gabriel Ferro (University of Lyon,

France) for collaboration on cubic silicon carbide projects; Jorg Wrachtrup (Univeristy of

v

Stuttgart, Germany) and his group memebers Matthias Widmann, Matthias Niethammer,

Sang-Yun Lee and Torsten Rendler, Erik Janzen and Nguyen Son (Linkoping University,

Sweden), and Takeshi Ohshima (QST, Japan) for collaboration on hexagonal silicon carbide

project; Marko Loncar and Michael Burek (Harvard University) for collaboration on silicon

carbide angled etching.

I am thankful to the Vuckovic group members for sharing the hurdles and the excite-

ment of developing new photonics. Kelly Rivoire and Sonia Buckley got me started with

nonlinear optics and photonic crystal cavities and made my decision to join the group very

smooth. I had a lot of fun working alongside them and they kept providing a valuable per-

spective even post-graduation. Michal Bajcsy and Arka Majumdar have offered a healthy

dose of cynicism which was useful for the rethinking our research direction, as well as made

the lab dynamics more interesting. Jesse Lu has influenced my interests in other research

topics through the memorable journal club presentations. Gary Shambat encouraged me

to join Stanford Optical Society which has been a bigger and a more rewarding part of

my graduate school than I had anticipated. Armand Rundquist contributed a lot to the

group with his wide-ranging interests and I was lucky to overlap with him for four years.

Jan Petykiewicz is always open for a discussion about the captivating scientific phenomena

and I enjoyed sharing that passion with him. Tom Babinec taught me a lot about color

centers and shared the excitement of obtaining our initial findings in silicon carbide. His

interest in arts and taste in music contributed to the atmosphere during the lengthy spec-

troscopic measurements. Konstantinos Lagoudakis and Kai Muller did an incredible job on

teaching me to build and use optical setups and have advised me on all kinds of scientific

topics. I feel fortunate to have been able to interact with them, despite not being assigned

to the same project at first. Tomas Sarmiento has been a great person to discuss material

science challenges with. Kevin Fischer helped me get started with QuTiP modeling and

his attention to detail has been invaluable in our theoretical and experimental discussions.

Yousif Kelaita has been a great companion in clean room and showed us all how to make

great-looking publication figures. Alex Piggott shared a lot of late evenings from the neigh-

boring cubicle and helped out with simulation troubleshooting. Working with Linda Zhang

on color centers has been a great experience, she is very dedicated and skilled in the lab.

Constantin Dory has been another lab whiz who has transferred his quantum dot skills to

our project. Neil Sapra designed the best lab t-shirts. Logan Su, Rahul Trivedi and Yijun

Jiang have been proposing insightful ideas for the budding projects. Shuo Sun has been a

great addition to our color center team and is a quick and a valuable contributor. I have

enjoyed discussing science with Dries Vercruysse. It has also been great working with our

undergraduate and visiting students Olivia Grubert, Zain Zaidi, Vincent Sebag and Armin

Regler. Ingrid Tarien has provided incredible administrative support.

vi

I am grateful to the colleagues who helped discuss new ideas or try them out in the labo-

ratory and the clean room – Yan-Kai Tzeng, Hitoshi Ishiwata, Peter Maurer, Lucien Weiss,

Nikolas Tezak, Dmitri Pavlichin, Jeff Hill, J Provine, and all Stanford Nanofabrication

Facility and Stanford Nanopatterning Center staff.

I would like to thank Ray Beausoleil, Ranojoy Bose and Dave Kielpinski from Hewlett

Packard Labs for a great summer internship experience where I learnt how photonic devices

are scaled up into integrated systems.

The projects I have worked on have been generously supported by the National Science

Foundation and the Air Force Office of Scientific Research.

My experience at Stanford has included many additional activities thanks to the efforts

of the Stanford Optical Society, ME+ Women, Women in Science and Engineering (WISE),

Hume Center for Writing and Speaking, Bridging Education, Ambition and Meaningful

Work (BEAM) Center, Women in Electrical Engineering (WEE), Women in Physics, Stan-

ford Photonics Research Center (SPRC), d.School, and Stanford Graduate Summer Insti-

tute (SGSI). I am especially grateful for the opportunities to co-chair the outreach team

and serve as a co-president of the Stanford Optical Society with Patrick Landreman and

Nikolas Tezak.

Finally, I am forever indebted to my family and my life partner for supporting my passion

for science.

vii

Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Quantum Photonic Substrates 5

2.1 Photonic Substrate Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Overview of Photonic Substrates . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Color Centers 9

3.1 Definition and Optical Properties . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Experimental Characterization . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2.1 Optical Characterization . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.2 Spin Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 Diamond Color Centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.4 Silicon Carbide Color Centers . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Optical Microresonators 15

4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Photonic Crystal Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.3 Microdisk Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.4 Purcell Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Telecommunication Wavelength Photonic Crystal Cavities in 3C-SiC 19

5.1 Fabrication of Diverse Photonic Crystal Cavities in 3C-SiC Films . . . . . . 19

5.2 Planar SiC Photonic Crystal Cavities at Telecommunication Wavelengths . 21

5.3 Nanobeam Photonic Crystal Cavities at Telecommunication Wavelengths . 25

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6 Visible Wavelength Microdisk Resonators in 3C-SiC 27

6.1 Fabrication of 3C-SiC Microdisks . . . . . . . . . . . . . . . . . . . . . . . . 27

6.2 Simulated Whispering Gallery Modes . . . . . . . . . . . . . . . . . . . . . . 28

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Contents

6.3 Characterization of Visible Resonances in 3C-SiC Microdisks . . . . . . . . 30

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7 Hybrid Silicon Carbide-Nanodiamond Cavity Color Center Emission

Enhancement 35

7.1 Hybrid Microdisk Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7.2 Growth and Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7.3 Optical Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

8 Photonic Design for Color Center Spintronics 47

8.1 The Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

8.2 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

8.3 Collection Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

8.4 Single Photon Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

8.5 Single Spin Optically Detected Magnetic Resonance at Room Temperature 54

8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

9 Multi Emitter-Cavity Quantum Electrodynamics 57

9.1 Cavity Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . 57

9.1.1 Cavity QED Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 57

9.1.2 Determining Cavity QED Parameters . . . . . . . . . . . . . . . . . 59

9.2 Multiple Emitters in a Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . 60

9.3 The Tavis–Cummings Model . . . . . . . . . . . . . . . . . . . . . . . . . . 61

9.4 Strong-Coupling Cavity QED with an Ensemble of Color-Centers . . . . . . 63

9.5 Effective Hamiltonian Approach to Multi-Emitter Cavity QED . . . . . . . 66

10 Nonclassical Light Generation in Multi Emitter Cavity QED Platform 73

10.1 Resonant Photon Blockade . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

10.2 Detuned Photon Blockade in the Multi-Emitter System . . . . . . . . . . . 75

10.3 Effective Hamiltonian Approach to n-Photon Generation . . . . . . . . . . . 79

10.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

11 Outlook 87

11.1 Biophotonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

11.2 Nonlinear Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

11.3 Spin Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Bibliography 91

x

List of Figures

3.1 (a) Elements of color center emission; figure adapted from [33]. (b) Energy

levels of VSi in 4H-SiC in zero magnetic field. . . . . . . . . . . . . . . . . . 10

3.2 2D scanning confocal photoluminescence setup. Gray region corresponds to

a common path in charge of the scanning functionality achieved through a

rotating mirror and 4-f confocal setup. Green (blue) region provides contin-

uous wave and pulsed input, and collection path for emission from diamond

(silicon carbide) color centers. . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 Photoluminescence spectra. (a) Silicon-vacancy center in diamond exhibiting

four ZPLs caused by spin-orbit coupling in each system. (b) Silicon vacancy

center in 4H-SiC exhibiting two ZPLs from two non-equivalent positions in

the lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.1 Microcavity design and Ey component of the resonant electric field. (a) L3

photonic crystal cavity design with a linear three-hole defect that localizes

light. (b) Nanobeam photonic crystal cavity with tapered holes in the central

region that localizes light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2 Absolute value of the electric field in the whispering gallery modes. (a) WGM

(1,13). (b) WGM (2,10). The orange line shows the edge of the microdisk. . 17

5.1 The process flow to generate suspended photonic crystal structures in 3C-SiC

films. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.2 Photonic crystal cavities fabricated from 3C-SiC films on a silicon substrate

using the process flow in Fig. 5.1. (a) Photonic crystal waveguides. (b)

Planar L3 photonic crystal cavity. (c) 1D nanobeam photonic crystal cavity.

(d) Crossbeam photonic crystal cavity. . . . . . . . . . . . . . . . . . . . . . 20

5.3 SEM images of fabricated 3C-SiC shifted L3 cavity from (a) vertical inci-

dence, and (b) 40-degree tilt. (c) FDTD simulation of the dominant (Ey)

electric field component of the fundamental mode, with the outlines of the

holes indicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

xi

List of Figures

5.4 (a) Cross-polarized reflectivity setup scheme, (b) characterized fundamental

mode resonance at telecommunications wavelength, blue dots represent data

points, red line shows fit to a Fano-resonance with quality factor of Q ≈ 800,

(c) resonance intensity dependence on polarization angle of incoming light,

blue dots represent data points, red line shows fit to sin(4θ), where θ is

half-wave plate rotation angle. . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.5 Theoretical analysis of the wavelengths and the quality factors of SiC L3

photonic crystal cavities as a function of the hole radius (r) shown in green

triangles and slab thickness (t) shown in yellow squares. The optimal pa-

rameters (t = 210 nm, r = 0.30a) are represented by the red dot and the

corresponding experimental result is indicated by the black star. . . . . . . 24

5.6 (a) Fundamental mode resonance scaling with lattice constant a, shown with

Fano-resonance fit with quality factors in range 630 to 920; all cavities are

shifted L3 cavities made in 210 nm thick slab of 3C-SiC. (b) Experimentally

measured linear dependence of resonant wavelength on lattice constant. Red

circles are measured data points and black line is a linear fit to the data. . . 25

5.7 (a) FDTD simulated Ey field of the fundamental mode in a 3C-SiC nanobeam

cavity. (b) SEM image of a fabricated 3C-SiC nanobeam cavity. (c) Charac-

terized resonances across the telecommunications range for different values

of the lattice constant a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6.1 (a) 3C-SiC microdisk fabrication process flow. (b) Fabricated 10 x 10 arrays

of microdisks and 50 µm x 50 µm square structures. (c) The smallest fabri-

cated microdisk with diameter 1.1 µm on a 100 nm wide pedestal. (d) Array

of robust 1.9 µm diameter SiC microdisks on 900 nm wide Si pedestals. . . 28

6.2 FDTD simulations of microdisks. (a) Radiative quality factor of TM and TE

first radial order (n = 1) modes supported in a 1.7 µm diameter disk on a

700 nm pedestal; free spectral range between subsequent modes is marked

by arrows. (b) |Ez| component of electric field in plane of the microdisk and

in vertical plane of the structure, for whispering gallery TM modes (13,1),

(10,2), (7,3) at wavelengths 682 nm, 675 nm, 662 nm, respectively, with

quality factors noted beneath. (c) Whispering gallery (12,1) mode wavelength

scaling with disk size for TM and TE polarization. . . . . . . . . . . . . . . 29

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List of Figures

6.3 (a) 2D green/red confocal photoluminescence imaging setup used to optically

address 3C-SiC microdisks. (b) Optical micrograph of the microdisks sample.

(c) 2D photoluminescence scan showing increased signal in the undercut re-

gions of microdisks; maximum signal corresponds to several hundred counts

for 0.025 s integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.4 (a) Room temperature photoluminescence signal (linear scale) collected from

corresponding microdisks with diameters 1.7 µm, 1.9 µm and 2.5 µm, on 700

nm, 900 nm and 1.5 µm wide pedestals, respectively; red triangles and blue

crosses mark identified TM and TE mode series. (b) Maximal quality factor

resonance for each disk size. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.5 (a) Comparison between simulation results and experimentally obtained free

spectral range for TE and TM polarizations. (b) Comparison between simu-

lation results and experimentally characterized resonances in 1.7 µm diameter

microdisks, assuming the angular modes mTM = 14 and mTE = 15 of the

shortest wavelength peaks. (c) Comparison between radiative, absorptive

and experimentally measured quality factor for TM (13,1) mode in 1.7 µm

diameter microdisks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7.1 (a) Model of the hybrid silicon carbide-nanodiamond microresonator. (b)

The analysis of different sized nanodiamonds influence on a representative

whispering gallery mode and its parameters in the hybrid microresonator,

simulated by the FDTD method. c) Growth and fabrication processes for gen-

erating arrays of hybrid silicon carbide-nanodiamond microresonators with

preferential positioning of color centers, as described in the text. . . . . . . 36

7.2 Three generations of hybrid silicon carbide-nanodiamond microdisk arrays

with variable growth time tG. . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7.3 (a) 2D scanning photoluminescence setup; PL: photoluminescence, LP: long-

pass filter, BP: bandpass filter, SM: single mode, MM: multimode, CW:

continuous wave, NA: numerical aperture, G: green, B: blue, R: red. Images

show SEM image and 2D PL maps without and with a 740 nm band pass

filter. (b) Photoluminescence from hybrid microresonators with 80 nm, 150

nm and 330 nm nanodiamond size, showing SiV− and Cr-related color center

zero-phonon lines at T = 10 K in hybrid microresonators, as well as ∼ 1 nm

linewidth WGMs in both SiC microdisks and hybrid microresonators. . . . 39

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List of Figures

7.4 Photoluminescence spectra at 10 K on a hybrid microresonator (shown in the

SEM image) with a 60 nm nanodiamond, taken at the location of the nanodi-

amond (red) and away from the nanodiamond (black), exhibiting whispering

gallery mode resonances and two SiV− zero phonon lines at 736 nm and 740

nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

7.5 Temperature dependent photoluminescence spectra of two free diamonds

showing no distinction between individual emitters at room temperatures.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

7.6 (a) Time resolved measurements of SiV− (left) and Cr-related (right) color

centers in nanodiamonds collected at 1.5 nm band filtered peak wavelengths.

Lifetimes obtained from exponential decay fits are τSiV = 0.9 ns and τCr =

0.6 ns. (b) Spectral variation in color center lifetime at room temperature. 42

7.7 Hybrid disk and neighboring free nanodiamond photoluminescence at room

temperature. Hybrid disks shows resonant enhancement of a) SiV− and b)

Cr-related color center emission, detail shown inset. Red (yellow) circles

correspond to red (black) plot curves. Uncoupled WGMs are also present in

the spectrum at expected free spectral range. . . . . . . . . . . . . . . . . . 43

7.8 Temperature scans of (a) SiV− and (b) Cr-related center in resonance with

WGMs (spectra are normalized to a mean number of counts for comparison);

(c) Hybrid microdisk emission indicating an on-resonant (red circle) and two

off-resonant (dark and light blue circles) ZPLs; (d) Color-coordinated ZPL

peak intensities obtained through the Lorentzian fit to the emission spec-

tra in (c) fitted with the constant and non-constant radiative rate intensity

dependencies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

7.9 (a) Photoluminescence spectra taken from two hybrid microdisks and two

free nanodiamonds. (b) Spectrally resolved lifetime measurements, color-

coordinated with (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

8.1 (a) Lithographic process for fabrication of nanopillars in 4H-SiC. (b) SEM

images of fabricated array of nanopillars with variable diameters. (c) SEM

image of a 600 nm wide, 800 nm tall nanopillar. . . . . . . . . . . . . . . . . 49

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List of Figures

8.2 (a) Collection efficiency from a dipole emitter positioned at the center of

800 nm tall SiC nanopillar (indicated by inset drawing), with 4o offset from

z-axis (red), plotted with its vertical (yellow) and radial (blue) components.

(b) Finite Difference Time Domain model of electric field propagation from

a dipole emitter placed in the center of a 600 nm wide nanopillar. (c) Far

field emission profile of (b) with light line (white) and NA line (red). (d)

Color center depth dependence of collection efficiency from a dipole emitter

in bulk SiC, with components marked as in (a); inset drawing illustrates

the direction of increasing depth. (e) Diameter dependence of optimal (red)

and centrally (blue) positioned color center in nanopillar, showing up to two

orders of magnitude higher collection efficiency compared to bulk value of

0.0017. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

8.3 Collection efficiency from VSi at the center of the nanopillar (r = 0, h = 400

nm) into air objective with NA = 0.65 and NA = 0.95. . . . . . . . . . . . . 52

8.4 (a) Collection efficiency dependence on vertical position along the central

axis (r = 0), where negative values correspond to the top of the nanopillar,

and positive ones to the bottom, plotted for various diameters. (b) Collection

efficiency dependence on horizontal position along the radius at the central

heights (h = 400 nm), plotted for various diameters. . . . . . . . . . . . . . 52

8.5 (a) SEM image of a pattern with nanopillars and unetched bulk regions. (b)

Photoluminescence measurements at T = 7 K, showing same linewidth for

nanopillars as in the bulk unetched region, as well as the increased collection

efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

8.6 (a) Photoluminescence intensity map of a block of pillars. (b) PL intensity

of a single VSi as a function of the optical power, saturating at 22 kcps. The

red line shows a fit to the data. (c) Diameter dependent collected count

rate. (d) Typical room temperature PL spectrum of a single VSi emitter

in a nanopillar. (e) Typical second order correlation function showing anti-

bunching at zero time delay, a clear evidence for single photon emission. The

red line shows a double exponential fit to the data. (f) ODMR signal from a

single VSi in a nanopillar. The red line shows a Lorentzian fit to the data. 54

9.1 Empty cavity (blue) transmission spectrum vs. strongly coupled emitter-

cavity system (red) featuring two polariton peaks. Parameters are {g, κ, γ}/2π =

{15, 25, 0.1} GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

9.2 Illustration of N non-identical emitters coupled to a cavity mode. . . . . . . 62

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List of Figures

9.3 Dressed ladder of states for N = 1, 2, 3 identical emitters coupled to a single

cavity mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

9.4 (a) Transmission spectra of N identical and equally coupled emitters. (b)

Transmission spectra showing the emerging subradiant peak as the second of

N = 2 emitters becomes off-resonant. . . . . . . . . . . . . . . . . . . . . . . 66

9.5 Randomly generated transmission spectra forN = 4, {κ/2π,Γ/2π, gmax/2π} =

{25GHz, 0.1GHz, 10GHz} and δ/2π = (a) 1 GHz, (b) 10 GHz, and (c) 100 GHz. 67

9.6 Polariton spacing of randomly generated strongly coupled systems vs. ex-

pected collective coupling rate forN = 4, {κ/2π,Γ/2π, gmax/2π} = {25GHz, 0.1GHz, 10GHz}and δ/2π = (a) 1 GHz, (b) 10 GHz and (c) 20 GHz. The red line corresponds

to the system with identical emitters. . . . . . . . . . . . . . . . . . . . . . . 67

9.7 Polariton spacing calculated using the effective Hamiltonian diagonalization

vs. expected collective coupling rate for N = 4, {κ/2π,Γ/2π, gmax/2π} =

{25GHz, 0.1GHz, 10GHz} and δ/2π = (a) 1 GHz, (b) 10 GHz and (c) 20 GHz.

The red line corresponds to the system with identical emitters. . . . . . . . 69

9.8 A comparison between the transmission spectra calculated using quantum

master equation (QME) and the first effective Hamiltonian approximation

(SEFFI ) for aN = 4 multi-emitter system {κ/2π,Γ/2π, gmax/2π} = {25GHz, 0.1GHz, 10GHz}

and emitter frequencies sampled for δ/2π = (a) 1 GHz, (b) 10 GHz, and (c)

20 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

9.9 A comparison between the transmission spectra calculated using quantum

master equation (QME) and the second effective Hamiltonian approxima-

tion (SEFFII ) for a N = 4 multi-emitter system {κ/2π,Γ/2π, gmax/2π} =

{25GHz, 0.1GHz, 10GHz} and emitter frequencies sampled for δ/2π = (a)

1 GHz, (b) 10 GHz, and (c) 20 GHz. . . . . . . . . . . . . . . . . . . . . . . . 71

9.10 Transmission spectra calculated by the second effective Hamiltonian ap-

proximation for a N = 100 multi-emitter system {κ/2π,Γ/2π, gmax/2π} =

{25GHz, 0.1GHz, 10GHz} and emitter frequencies sampled for δ/2π = (a)

10 GHz, (b) 40 GHz, and (c) 80 GHz. . . . . . . . . . . . . . . . . . . . . . . 71

10.1 Photon blockade trends with an increasingN for parameters {∆e,n/2π,Γ/2π, gn/2π} =

{0GHz, 0.1GHz, 10GHz} and κ/2π = (a) 25 GHz (GN < κ), (b) 15 GHz

(GN ∼ κ), and (c) 8 GHz (GN > κ). Dashed lines illustrate the transmission

spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

10.2 Photon blockade trends forN = 3, {κ/2π,Γ/2π,∆e,n/2π} = {25GHz, 0.1GHz, 0GHz}and the variable coupling parameters {g1/2π, g2/2π, g3/2π} noted in the legend. 76

xvi

List of Figures

10.3 Zero-time second-order coherence minima for a detuned system with N = 2

emitters and {κ/2π,Γ/2π, g1/2π} = {25GHz, 0.1GHz, 10GHz}; g2 is quoted

in each panel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

10.4 (a) Zero-time second-order coherence for N = 2, {κ/2π,Γ/2π, g1,2/2π} =

{25GHz, 0.1GHz, 10GHz} and the different emitter detunings {∆e,1/2π,∆e,2/2π}noted in the legend; plotted with transmission spectra (dashed lines). (b)

Dressed ladder of states calculated with our effective Hamiltonian approach

for {∆e,1/2π,∆e,2/2π} = {26GHz, 20GHz}, plotted with their linewidths;

black vertical arrows illustrate first and second excitation at the point of

the best photon blockade. Energy spacing between the rungs is significantly

reduced for clarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

10.5 (a) An illustration of the multi-emitter cavity QED system with two noniden-

tical emitters coupled to a common cavity mode. (b) The ladder of dressed

states, frequencies and unnormalized eigenvectors, for the case of two identi-

cal and equally coupled emitters, ∆C = ∆E = 0 and g1 = g2 = g. . . . . . . 79

10.6 (a) Emission spectrum, (b) second- and (c) third-order coherence of transmit-

ted light calculated by the quantum master equation. (d) Frequency overlap

between the k-th order rungs of the dressed ladder of states of Heff, presented

as ωk/k − ωC with indicated linewidths; ∆C/2π takes values of (i) 0, (ii) 20

and (iii) 30 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

10.7 Prevalent character of the eigenstates of (a) the first and (b) the second rung

of Heff for ∆C = 20 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

10.8 (a) Zero-delay second- and third-order coherences for ∆C ,∆E/2π = 30, 5

GHz system as a function of laser detuning from the cavity; emission spec-

trum in dashed lines illustrates the relationship between features. (b) g(2)(τ)

at frequencies of the corresponding g(2)(0) minima from (a). . . . . . . . . . 83

10.9 Comparison between second-order coherence in N = 2 and N = 1 emitter-

cavity systems; emission spectrum in dashed lines illustrates the relationship

between features. Each emitter is characterized by frequency and coupling

rate, {ωE1, ωE2, g}/2π = {20, 26, 10} GHz. . . . . . . . . . . . . . . . . . . . 84

11.1 SEM images of HeLa cells grown on 4H-SiC substrate patterned with array

of nanopillars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

11.2 SEM images of HeLa cells seeded on an array of 3C-SiC on Si pedestals. . . 88

xvii

List of Figures

11.3 (a) Two views of a zincblende lattice that are simultaneously a mirror image

and 90o rotation relative to one another (adjusted from [116]). (b) Quasi-

phasematching domains in a 3C-SiC microdisk for the second harmonic gen-

eration process ω2 = 2ω1 illustrated inset. . . . . . . . . . . . . . . . . . . . 89

xviii

List of Tables

2.1 Properties of substrates commonly used in photonics. . . . . . . . . . . . . . 6

9.1 Relationship between quality factor Q and cavity linewidth (energy decay

rate) κ for cavity frequency ωa/2π = 400 THz. . . . . . . . . . . . . . . . . 59

9.2 Temperature dependence of the emitter linewidth γ for SiV center in dia-

mond [66]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

9.3 Relationship between the cavity mode volume factor v and the maximal

emitter-cavity coupling rate g cited for λ = 738 nm, τ = 1 ns and two values

of ρZPL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

xix

1 Introduction

The pioneering experiments on light and matter interaction date to Isaac Newton’s explo-

ration of refraction of light through a prism and Michael Faraday’s magneto-optic phe-

nomenon of rotated polarization. Bringing together physics, chemistry, material science,

and engineering, the studies of light and matter interaction have come far from those mod-

est beginnings and now define modern and future technologies. The applications are as

wide as alternative energy sources (photovoltaics), novel surgery techniques (lasik), en-

ergy efficient lighting (Laser Emitting Diodes – LEDs), and presentation technologies (laser

pointers, quantum dot based screens). Quantum photonics, in particular, is one of the

growing areas that target low power and high speed telecommunications and information

processing, novel computation paradigms, and nanoscale biosensing.

1.1 Motivation

As the growth of personal and super- computing is nearing the limits of the so-called Moore’s

law [1], the paradigm where electronic information carriers are replaced with photons opens

up new frontiers. Light-based computing holds the promise of achieving higher clock speeds

through faster modulation of signal [2], reduced power consumption via harnessing of the

strong light-matter interaction [3], exploration of novel architectures based on light inter-

ference effects such as the neuromorphic computing [4] and Ising solvers [5], and developing

quantum computation and quantum cryptography platforms based on electron and nuclear

spin-qubits [6] that would reduce algorithm complexity [7] and increase the safety of com-

munication channels [8]. In parallel, these technologies allow for in vivo sensing of local

index of refraction [9] and temperature [10], providing unprecedented nanoscale resolution

measurements.

In order to engineer optical quantum systems and devices, one first needs to develop

reliable photonic elements: waveguides to carry the signal [11], low-loss microresonators

to enhance the local electromagnetic field intensity [12] and serve as nonlinear elements

for switching [3] and frequency conversion [13], and narrow linewidth quantum emitters

to serve as single photon sources or qubits [14]. Many of these component have been

demonstrated in atomic optical systems harnessing cavity quantum electrodynamic effects,

1

1 Introduction

often by coupling rubidium atoms to a high finesse cavity [15]. However, the coupling rates

in atomic systems are in the MHz range, the operation requires cryogenic temperatures and

high vacuum. A leap forward has been achieved with implementations in III-V solid-state

platforms incorporating quantum wells and quantum dots coupled to Distributed Bragg

Reflector (DBR) and Photonic Crystal (PC) cavities [16]. These systems reach up to 25

GHz operating speeds and have on-chip footprint in tens of micrometers.

More recently, color centers in diamond and silicon carbide have taken a prominent po-

sition among solid-state quantum emitters, due to their narrow inhomogeneous (ensemble)

linewidth, up to room temperature operation, and lattice-point size [14]. These properties

make them ideal candidates for multi-emitter cavity quantum electrodynamics, a regime

that has not yet been thoroughly explored in solid-state systems and could potentially in-

crease the strength of interaction – and therefore the speed of optical devices – by an order

of magnitude [17]. Additionally, color centers provide optical readout of their electron spin,

as well as microwave spin control with coherence times up to several seconds [18]. This

property has been explored for the quantum bit (qubit) applications, as well as for the

nanoscale magnetic and temperature measurements.

Color centers incorporated into photonic devices such as microcavities or waveguiding

elements experience modified density of states that the photons are emitted to [12]. De-

pending on the strength of interaction, such coupling can result in faster and spectrally

better defined single photon emission, or radically change the emission spectrum and statis-

tics. Experimental realization of such devices requires the development of advanced growth

methods and nanofabrication techniques.

The goal of this thesis is the development of methods and devices in silicon carbide and

hybrid silicon carbide-nanodiamond platforms that modify the emission of color centers for

applications in optical networks and high sensitivity nanoscale sensing.

1.2 Contribution

The results in this thesis represent some of the pioneering demonstrations of nanoscale

systems in the field of silicon carbide photonics, and integration of developed devices with

the silicon carbide and diamond color centers.

From the technological standpoint, I have developed two nanofabrication procedures for

generating nanoscale features in either bulk or heteroepitaxial silicon carbide, as well as for

releasing the underlying silicon in the latter case. I have built a 2D scanning photolumines-

cence setup with pulsed and CW excitations at visible and near infra-red (IR) wavelengths

used for characterization of color centers and photonic devices at room and cryogenic tem-

peratures. Furthermore, I have designed, fabricated and characterized high quality factor

2

1.3 Thesis Outline

photonic crystal cavities tunable across a telecommunications range, as well as the smallest-

to-date microdisk resonators at visible and near IR in cubic silicon carbide. Subsequently, I

have demonstrated the first cavity coupling to the intrinsic luminescence in silicon carbide

at visible wavelengths. This work has been performed in collaboration with Gabriel Ferro

(University of Lyon, France), whose group provided the samples of heteroepitaxially grown

silicon carbide on silicon. Based on our results, I have modeled novel designs of hybrid

silicon carbide-nanodiamond cavity which takes advantage of the similarity of the indices of

refraction between the two materials to resonate a shared mode and enhance the diamond

color center emission. This project has been realized within the Stanford Diamond collab-

oration between the groups of Jelena Vuckovic, Steven Chu, Zhi-Xun Shen and Nicholas

Melosh, in which our collaborators developed a method to grow color center rich nanodia-

mond on pre-fabricated silicon carbide microresonators. In an international collaboration

on silicon carbide between the groups of Jelena Vuckovic, Jorg Wrachtrup (University of

Stuttgart, Germany), Erik Janzen (University of Linkoping, Sweden) and Takeshi Ohshima

(Japan Atomic Energy Agency), I have initiated, modeled, fabricated and performed a

part of optical characterization of a scalable color center based spin-qubit platform that

improves the device footprint by two orders of magnitude and significantly simplifies the

optical interfacing with the qubits.

From the intellectual standpoint, I have identified advantages and limitations of silicon

carbide as a photonics substrate, as well as an alternative host of color centers to dia-

mond. I have explored integration of color centers to on-chip photonic devices, which serves

as a solid-state parallel to atomic and molecular optics experiments. The advantages of

this platform are light confinement to subwavelength mode volumes which renders strong

light matter interaction at GHz operating rates, potential room temperature operation, and

miniaturization. I have studied the artifacts of the small discrepancies in emission wave-

length within an ensemble of color centers to the electrodynamical effects in the cavity they

are coupled to, laying out the theoretical framework for solid-state multi-emitter cavity

quantum electrodynamics. Within this pursuit, I have discovered novel interference ef-

fects that provide outstanding opportunities for nonclassical light generation in color center

systems.

1.3 Thesis Outline

The organization of the thesis is the following: basic terms of quantum photonics are

introduced, followed by the experimentally realized devices in silicon carbide platforms, after

which the model of the multi-emitter cavity quantum electrodynamics system is presented

together with the predicted effects. Chapter 1 provides the introduction to the thesis topic

3

1 Introduction

and contributions. Chapters 2 to 4 give an overview of the photonics substrates, emitters

and optical microresonators studied in the field of color center quantum photonics. Chapters

5 to 8 present design, fabrication and characterization of photonic devices in silicon carbide

and hybrid silicon carbide-diamond platforms. Theoretical considerations and predicted

applications of a multi-emitter cavity quantum electrodynamics system are presented in

Chapters 9 and 10, followed by the outlook in Chapter 11.

4

2 Quantum Photonic Substrates

The field of photonics is closely tied to engineering and material science. The choice of

substrate defines the physical processes in the nanoscale optical device, and imposes the

fabrication limitations under the current processing techniques. In this chapter, I will

present an overview of different materials traditionally used in quantum photonics, and

provide more details about the substrates studied in this thesis: silicon carbide and diamond.

2.1 Photonic Substrate Properties

Photonic substrates are expected to provide mechanisms for light confinement, light gen-

eration and frequency conversion. Light confinement into a 2D slab is often facilitated by

the total internal reflection (TIR), which relies on index of refraction contrast, ∆n, between

the substrate and its environment. For example, photonics in silicon-on-insulator (SOI) is

based on the TIR between silicon, nSi = 3.5, silicon dioxide, nSiO2 = 1.5, from its bottom

side, and air, nair = 1, from the top. To be able to confine light with low absorption, photon

energy should be below the substrate band gap. Consequently, wider band gap materials

offer a multitude of operational frequencies, including visible wavelengths, which are of great

interest for biological applications due to the overlap with the water optical transparency

window at the wavelength range 300–700 nm.

Photonics applications such as quantum cryptography — which relies on single photon

operation [8]; and in vivo bio labeling — which requires bright point sources of light [19], can

be implemented using solid-state quantum emitters. Quantum dots and color centers have

been studied for these purposes. For example, 20 nm x 20 nm x 5 nm InAs quantum dots

grown inside GaAs slabs confine electrons and holes inside the band gap that is narrower

than the the band gap of the surrounding substrate [20]; the electrons and holes recombine to

valence band, emitting single photons at near-infrared (NIR) wavelengths (though visible

wavelengths are present in different quantum dot systems). This system is operational

only at cryogenic temperatures. Color centers, which are point defects in the lattice of

a substrate, provide a different set of emission properties. They are often operable up

to, and in some cases even beyond, room temperature, at a variety of wavelengths across

the visible and NIR spectrum. For example, the negatively charged nitrogen-vacancy in

5

2 Quantum Photonic Substrates

diamond, usually called the NV center, occurs as a lattice defect in which two carbon

atoms are replaced by a nitrogen atom and a vacancy [18]. The defect localizes electronic

orbitals with energies within the diamond band gap, and electron transitions between these

levels are tied to photon absorption and emission. Finally, quantum emitters usually have

optically addressable spins, which can be exploited for quantum bit (qubit) and nanosensing

applications.

The capability of a substrate to convert one color of light to another depends on the

lattice symmetry. Frequency conversion is needed to connect parts of optical networks that

operate at different wavelengths, as well as for light generation and detection at a new set of

wavelengths [13]. Such frequency-mixing processes are nonlinear, and depend polynomially

on the input power of light. Photonic applications often rely on low power consumption,

which makes quadratic processes, such as the second harmonic generation utilized in green

laser pointers, preferable in terms of efficiency. To facilitate these effects, the substrate

has to exhibit second order nonlinearity χ(2), which is exhibited in non-centrosymmetric

materials.

2.2 Overview of Photonic Substrates

Table 2.1 summarizes the properties of commonly used photonic substrates.

Si GaAs diamond 6H/4H-SiC 3C-SiC

Index 3.5 3.4 2.4 2.6 2.6

Band gap [eV] 1.12 1.42 5.5 3.05/3.23 2.36

Quantum emitter - InAs QD color centers color centers color centers

Room T emission - no yes yes yes

Wafer-scale substrates yes yes no yes yes

Sacrificai layer SOI AlGaAs - - Si

Table 2.1: Properties of substrates commonly used in photonics.

Silicon is a ubiquitous substrate for many applications including electronics, microelec-

tromechanical systems (MEMS) and photonics. Its availability in silicon-on-insulator (SOI)

platform where a layer of silicon dioxide is sandwiched between a thin slab of silicon and a

silicon wafer, and a variety of the industrially developed processing techniques, make silicon

photonics possible [21]. This includes implementation of waveguides, microresonators, as

well as optical modulators in many different geometries. Silicon’s index of refraction pro-

vides high contrast to both air and glass. These devices are operable at NIR wavelengths

6

2.2 Overview of Photonic Substrates

due to the narrow band gap, and there is no χ(2) nonlinearity. Silicon is usually assumed

not to host quantum emitters, however, there have been recent studies of color centers in

silicon operating at longer wavelengths (2.9 µm) and cryogenic temperatures featuring long

spin coherence times [22]. The lack of light sources and detectors at these wavelengths, as

well as their increased absorption in glass or native oxides, make the use of this platform

technologically challenging.

III-V substrates such as GaAs and GaP have high index of refraction and χ(2) nonlin-

earity, they are hosts of quantum dots, and can incorporate sacrificial layers. InAs/GaAs

quantum dot platform with AlGaAs sacrificial layer has been prominent in photonics re-

search, for example for implementing suspended photonic crystal cavities with embedded

quantum dots for cavity quantum electrodynamics [3] or interfacing telecommunication

wavelengths with quantum dots operating at 900 nm [23]. GaP, on the other hand, has

a wider band gap and provides opportunity for operation at visible wavelengths with low

absorption [24]. III-V substrates are not considered CMOS-compatible due to the poten-

tially unwanted doping of group-IV substrates, as well as due to the lack of high quality

heteroepitaxial growth on silicon.

Diamond, which is considered an insulator due to its exceptionally wide band gap, has

entered photonics as a host of visible and NIR color centers [25], though more recently

it has been also studied in opto-mechanics as well [26]. Its color centers are active up to

room temperatures, and can have spin coherence times of up to a second [18]. Methods

for growing color center rich nanodiamond have been developed; centimeter scale single-

crystal diamond substrates are also commercially available. Diamond is chemically inert

and therefore nontoxic to cells, which has allowed for applications in biosensing [10]. The

platform does not feature a sacrificial layer nor χ(2) nonlinearity.

Silicon carbide has been present in power electronics and the MEMS industry for

decades [27], but has entered photonics only in 2011, when its color centers [28] and χ(2)

processes [29] have been studied at the nanoscale. This material occurs in about 250 differ-

ent polytypes [30], however, three of them have been prominent in industry and research:

bulk hexagonal polytypes 4H-SiC and 6H-SiC, and cubic 3C-SiC. Silicon carbide combines

the desired properties of all the previously mentioned photonics materials: wide band gap,

room temperature active color centers, long coherence times, χ(2) nonlinearity, and good

opto-mechanical properties. It is available in many forms, from nanoparticles to 6-inch

wafers. Due to its symmetry, 3C-SiC can be heteroepitaxially grown on silicon [31], which

provides a direct sacrificial layer. 4H- and 6H-SiC layers can be bonded to silicon dioxide

through the so-called Smart Cut step [29].

In this thesis, we will explore color centers and nanofabrication techniques in silicon

carbide and diamond.

7

3 Color Centers

Color centers have gained prominence as solid state quantum emitters due to their excep-

tional brightness, room temperature operation, optical control of electron and nuclear spins

and angstrom-scale dimensions. Their applications range from single photon sources to

nanoscale sensors.

3.1 Definition and Optical Properties

A color center is a point defect in a lattice of a semiconductor that has optically addressable

electronic transitions [32]. They can be intrinsic, e.g. a vacancy or a misplaced atom, or

extrinsic, when an atom of a different species is incorporated into the lattice.

Point defects in a lattice break its translation symmetry which means the local electronic

levels are not solutions to the Bloch equation and that electrons can be confined into

localized orbitals. If ground and excited electronic state are both within the band gap of the

host material, they represent an isolated two-level system and the electron can be optically

addressed using sub-band gap frequencies. The emission spectrum consists of two features

shown in Figure 3.1a: the narrow lorentzian peak called zero-phonon line (ZPL), arising

from the zero-phonon transition, and a phonon side-band, which consists of the transitions

to other vibronic levels of the ground state. Emission branching into ZPL (Debye-Waller

factor) increases at cryogenic temperatures. Color centers are emitters of single photons,

both at cryogenic and at room temperatures. This makes them excellent candidates for

integrated photonics applications.

Electron spin plays an important role in optical processes in color centers. Understanding

spin-dependent electronic levels and their symmetry helps explain the emission peaks and

their intensities in the experimental spectra. Here, we will use an example of the silicon

vacancy (VSi) color center in 4H-SiC [34] whose energy levels are illustrated in Figure 3.1b.

Due to its electron spin of S = 32 , the ground and excited level are split into ±1

2 and

±32 spin projections in zero magnetic field. Direct optical transitions between ground and

excited state are spin-preserving. The intermediate shelving state that is non-radiatively

coupled through spin-orbit interaction to one of the excited states, provides the mixing of

the polarizations with preferential direction toward the ±32 projection, therefore enabling

9

3 Color Centers

Figure 3.1: (a) Elements of color center emission; figure adapted from [33]. (b) Energy levels of VSi

in 4H-SiC in zero magnetic field.

a polarization mechanism. The shelving state, also called dark state, allows an electron

to change its spin in the de-excitation process while emitting a photon of lower energy

which can be filtered out in the experiment. The spin state, also called dark state, that is

more prone to the spin-flip exhibits lower intensity of photoluminescence, which provides a

readout mechanism for the spin orientation. Applying microwave pulses at the frequency

of the ground state splitting causes the spin to precess from one orientation to the other

in a coherent fashion. These are the basic principles of initialization, readout and coherent

control of electron spins in color centers used for spin qubits and magnetic sensing. In the

case of qubit applications, states |0〉 and |1〉 are represented by energy states | ± 32〉 and

| ± 12〉, respectively.

3.2 Experimental Characterization

Color centers can be naturally occurring, generated by incorporation of other atoms during

growth of the substrate [35], or by electron [36] or neutron [37] irradiation of samples. Strain,

electric field, and other non-uniformities in the substrates cause color centers in an ensemble

to experience different environments which is reflected in a variation of their properties such

as emission wavelength. Spin-coherence time is affected by the nuclear bath, which can be

controlled through isotopic purification of the sample where only nuclear spin-zero isotopes

are used during the epitaxial growth of the substrate.

To characterize the properties of the sample, we have used Photoluminescence and Opti-

cally Detected Magnetic Resonance measurements.

10

3.2 Experimental Characterization

3.2.1 Optical Characterization

The Photoluminescence (PL) is a method where the absorption of a photon excites the

system which then spontaneously luminesces. The frequency of the emitted peaks provides

the identification of the color center system. The excitation is performed with an above-band

laser delivered to the sample through a confocal microscope configuration and filtered from

the collected signal by dichroic mirrors and long-pass filters. Due to random positioning of

color centers in the substrate the 2D scanning capability is beneficial in characterization of

the sample. This is realized trough a rotating mirror followed by a 4f confocal setup and

high numerical aperture objective lens. Our setup is shown in Figure 3.2.

Figure 3.2: 2D scanning confocal photoluminescence setup. Gray region corresponds to a common

path in charge of the scanning functionality achieved through a rotating mirror and 4-f

confocal setup. Green (blue) region provides continuous wave and pulsed input, and

collection path for emission from diamond (silicon carbide) color centers.

To characterize the emitter lifetime τ , the sample is pumped with the pulsed excitation

and the signal is sent to the single photon counting module synchronized with the laser

repetition rate. The measurement builds up the statistics of the photon detection in terms of

the delay relative to the triggering signal, which corresponds to the excited state occupation

probability proportional to e−tτ .

3.2.2 Spin Characterization

The Optically Detected Magnetic Resonance (ODMR) measurement provides an optical

readout of the spin resonance. Here, microwave (MW) excitation is used to rotate the

electron-spin and PL is used to readout the state of the system [36]. For color centers

that possess a spin-polarizing dark state the ODMR signal represents a resonance in the

11

3 Color Centers

optical signal as a function of MW frequency. The resonance occurs when the microwave

frequency corresponds to the energy splitting between the ground states with different spin

projections. Once this frequency is determined, MW pulses of arbitrary length can be used

to perform rotations on Bloch sphere needed for qubit applications.

3.3 Diamond Color Centers

The negatively charged nitrogen-vacancy (NV) center in diamond has, until recently, been

the dominant color-center system in photonics [25]. Its ZPL at 637 nm contains 3–5% of the

emission. Single-photon generation rates are in the tens of kHz and have been enhanced by

the use of nanopillar structures to 168 kHz [38]. The NV center in diamond has been utilized

for quantum cryptography [39], and integrated systems with silicon nitride waveguides have

been demonstrated [40]. This color center exhibits S = 1 ODMR resonance which has been

utilized to develop nanoscale magnetic sensor with unprecedented spatial resolution [41].

Temperature sensitivity of the ODMR has been utilized for in vivo thermometry [10]. Long

spin coherence times of up to a second [18] are enabling long-lived spin-qubits. One of the

challenges in working with the NV center is its spectral instability [42], which has motivated

a search for novel color-center systems [43].

The negatively charged silicon vacancy (SiV−) in diamond has emerged as an emitter

with a large (70%) emission ratio into its four ZPLs at 738 nm (Fig. 3.3a). Due to electron

spin S = 12 the ground and excited state are split by 50 GHz and 250 GHz, respectively,

by spin-orbit interaction. In addition, the ensemble linewidths are as narrow as 400 MHz

at cryogenic temperatures [14], which qualifies these systems as nearly identical solid state

quasi-atoms. An order of magnitude increase in the ZPL extraction and two to three orders

of magnitude reduction in the ZPL linewidth compared to the NV center makes SiV a more

attractive center for scalable systems. This is a result of the central symmetry of SiV−

leading to the absence of its permanent dipole moment and to the insensitivity to external

electric fields. Additionally, reactive ion etching can be performed while maintaining the

SiV’s optical properties, which has been demonstrated with an array of nanopillars [35] and

in nanobeam cavities [44]. These recent results pave a promising path for integrating nearly

identical quantum emitters in the suspended photonic structures needed for cavity QED.

Chromium-related color centers have recently emerged as single-photon emitters at wave-

lengths around 760 nm [45]. Their structure has not been completely understood, however

there seem to be several types of ZPL transitions. These color centers have remarkable

brightness of 500 kcounts/s.

12

3.4 Silicon Carbide Color Centers

Figure 3.3: Photoluminescence spectra. (a) Silicon-vacancy center in diamond exhibiting four ZPLs

caused by spin-orbit coupling in each system. (b) Silicon vacancy center in 4H-SiC

exhibiting two ZPLs from two non-equivalent positions in the lattice.

3.4 Silicon Carbide Color Centers

Silicon carbide hosts numerous color center systems across its polytypes [46]. Emitters in

3C-SiC, such as the carbon-anti-site-vacancy (VCCSi) [47] and oxidation-induced centers

[48] have exceptional brightness of around 1 MHz count rates. The divacancy (VCVSi) in

4H-SiC gives rise to six ZPL transitions in the range of 1,100–1,200 nm, two of which are

active and can be coherently controlled at room temperatures [28]. The silicon vacancy

(VSi) in 4H-SiC and 6H-SiC emits into 25–30 GHz narrow ZPLs capturing several percent

of emission at cryogenic temperatures in their two and three lines [49], respectively, in the

wavelength region 860–920 nm. These centers have been used as qubits [50] and magnetic

field sensors [51]. In particular, we are interested in the 916 nm ZPL (Fig. 3.3b) from VSi

in 4H-SiC that has been demonstrated as a qubit at the level of a single color center [50].

13

4 Optical Microresonators

4.1 Overview

Optical microresonators confine light to small mode volumes that are proportional to or

smaller than a cubic wavelength of light in the medium [52].

V .

(λ

n

)3

Such confinement results in increased interaction of light and matter which can be utilized

for cavity quantum electrodynamics [16], nonlinear optics [13] and opto-mechanics [26]. Two

geometries are explored in this thesis: photonic crystal cavities and microdisk resonators.

The resonant fields are modeled using the Finite-Difference Time-Domain method [53]. In

this chapter, we will also introduce Purcell Enhancement effect whose occurrence relies on

resonators with high quality factors and small mode volumes.

4.2 Photonic Crystal Cavities

Photonic crystal cavities confine light by generating a photonic band gap around the region

of light localization. A photonic band gap is generated by the periodic alteration of the

index of refraction, a structure also known as distributed Bragg reflector (DBR) and usually

achieved by etching holes in a geometrically ordered pattern [54], or growing alternate

layers of material with differing index of refraction [55]. The propagating light is partially

transmitted and partially reflected off of each boundary generating backward propagating

waves that interfere with the forward propagating ones. For a range of wavelengths, close

to twice the DBR pitch, this interference results in an exponentially decaying field that

prohibits propagation light in the structure, thus forming the first-order photonic band

gap. Higher order photonic band gaps are also present at shorter wavelengths.

A defect in the periodicity of the alternation of the refractive index, such as omitting

several holes or a change in the pitch, localizes light to that region, while the rest of the

structure functions as a highly reflective mirror. This way, light can be confined with high

quality factors in a subwavelength mode volume. Typical nanophotonics implementations

use 1D or 2D photonic crystal cavity designs. Hereby, a thin slab (∼ 200 nm) of dielectric

15

4 Optical Microresonators

is pierced and suspended in air, while the light in the out-of-the-plane direction (and in the

y-direction in the case of a 1D photonic crystal cavity) is confined through the total internal

reflection.

The designs explored in this thesis are L3 photonic crystal cavity and nanobeam photonic

crystal cavity shown in Figure 4.1. Typical mode volumes are V ≈ 0.75(λn

)3. The designs

can be scaled to operate at different wavelengths.

Figure 4.1: Microcavity design and Ey component of the resonant electric field. (a) L3 photonic

crystal cavity design with a linear three-hole defect that localizes light. (b) Nanobeam

photonic crystal cavity with tapered holes in the central region that localizes light.

The quality factor of photonic crystal cavities depends on the index contrast between the

substrate and its surroundings, therefore, for the same design, silicon and III-V materials

(n ≈ 3.5) would have higher confinement than diamond (n = 2.4), silicon carbide (n = 2.6)

and silicon dioxide (n = 1.5). Highest experimentally demonstrated quality factor is Q ∼106 [56].

Fabrication issues such as the over-etching or the lack of etching directionality cause

imperfect feature profiles which both shift the cavity resonance and reduce quality factors.

4.3 Microdisk Resonators

Microdisk resonators confine light in the so-called whispering gallery modes (WGMs) which

are located at the periphery of the disk. The WGMs are defined by radial and azimuthal

numbers (r, l) which identify the number of lobes along the radius and around the perimeter.

Example are shown in Figure 4.2.

Microdisk resonators can achieve mode volumes as low as several cubic wavelengths. In

theory, their loss mechanism is mainly in-plane and depends on the curvature radius of

the circumference relative to the resonant wavelength. In the nanofabricated devices, the

imperfections in the fabrication additionally contribute to the radiation losses.

16

4.4 Purcell Effect

Figure 4.2: Absolute value of the electric field in the whispering gallery modes. (a) WGM (1,13).

(b) WGM (2,10). The orange line shows the edge of the microdisk.

4.4 Purcell Effect

When a quantum emitter overlaps with a resonant mode of a cavity the density of states

of the radiative transition changes compared to the one in vacuum. This causes the en-

hancement of the spontaneous emission rate called Purcell effect [12]. The enhancement

occurs for high quality factor resonators with small mode volume. For a dipole emitter

positioned at the maximum of the field with polarization aligned to the cavity polarization

the enhancement ratio (Purcell factor) is equal to

Fmax =3

4π2

λ3

n2

Q

V.

If the angle between the emitter and cavity polarizations is φ and the emitter experiences

electric field E(r) Purcell enhancement is given as

F (r, φ) = Fmax

∣∣∣∣ E(r)

Emax

∣∣∣∣2 cos2(φ).

Experimentally, Purcell enhancement is confirmed by emitter lifetime reduction in time-

resolved measurements.

17

5 Telecommunication Wavelength

Photonic Crystal Cavities in 3C-SiC

In this chapter, we present the design, fabrication, and characterization of high quality

factor (Q ∼ 103) and small mode volume (V ∼ 0.75× (λn)3) planar photonic crystal cavities

from cubic (3C) thin films (thickness ∼ 200 nm) of silicon carbide (SiC) grown epitaxially on

a silicon substrate [57]. We demonstrate cavity resonances across the telecommunications

band, with wavelengths from 1.25-1.6 µm. We also discuss possible applications in nonlinear

optics, optical interconnects, and quantum information science.

5.1 Fabrication of Diverse Photonic Crystal Cavities in

3C-SiC Films

To fabricate photonic crystal cavities in 3C-SiC, we developed a new nanofabrication process

capable of generating features at the 100 nm scale. Figure 5.1 shows the process flow

we developed to realize a diverse set of 3C-SiC photonic crystal structures. The starting

material consists of a 3C-SiC layer grown epitaxially on a 35 mm diameter Si substrate using

a standard two-step chemical vapor deposition (CVD) process [31]. The film thickness

d ≈ 210 nm and refractive index (n ≈ 2.55 at 1,300-1,500 nm) were confirmed using a

Woollam spectroscopic ellipsometer before proceeding to subsequent masking and etching

steps. A layer of SiO2 was then deposited as a hard mask using a low-pressure CVD furnace

followed by SiO2 thinning in a 2% HF solution to the target thickness of ∼ 200 nm.

Next, a second mask layer (Fig. 5.1a) of ∼ 400 nm of electron beam lithography resist

(ZEP 520a) was spun. Photonic crystal cavities were then patterned at the base-dose range

250-400 µC/cm2 using a 100 keV electron beam lithography tool (JEOL JBX 6300) and then

developed using standard recipes (Figs. 5.1b-c). The initial pattern transfer from resist into

SiO2 (Fig. 5.1d) was performed using CF4, CHF3, and Ar chemistry in a plasma etching

system (Advanced Materials, p5000). Etch tests on bare (un-patterned) chips indicated

that the selectivity of this etch was approximately 1:1 (SiO2 to ZEP). Residual electron

beam resist was then completely removed in Microposit remover 1165 (Fig. 5.1e), and a

second pattern transfer from SiO2 into SiC using HBr and Cl2 chemistry (Fig. 5.1f) was

19

5 Telecommunication Wavelength Photonic Crystal Cavities in 3C-SiC

Figure 5.1: The process flow to generate suspended photonic crystal structures in 3C-SiC films.

Figure 5.2: Photonic crystal cavities fabricated from 3C-SiC films on a silicon substrate using the

process flow in Fig. 5.1. (a) Photonic crystal waveguides. (b) Planar L3 photonic crystal

cavity. (c) 1D nanobeam photonic crystal cavity. (d) Crossbeam photonic crystal cavity.

20

5.2 Planar SiC Photonic Crystal Cavities at Telecommunication Wavelengths

performed in the same system. Etch tests on bare chips indicated that the selectivity was

approximately 2:1 (SiC to SiO2 ). Next, undercutting of the SiC device layer was achieved

by isotropic etching of the underlying Si (Fig. 1(g)) with a XeF2 vapor phase etcher (Xactix

e-1). Finally, the remaining SiO2 hard mask layer was removed in 2% HF solution (Fig.

5.1h).

This fabrication procedure is suitable for various photonic crystal structures. Several

different geometries are presented in Fig. 5.2. For example, Fig. 5.2a shows a realization

of a 2D photonic crystal waveguide and Fig. 5.2b a planar photonic crystal cavity, which

may be integrated with color centers in the future, similarly to the development of quantum

photonic networks based on InAs quantum dots in GaAs [58]. Fig. 5.2c shows an example

of a tapered 1D nanobeam cavity for applications in opto-mechanics [59]. Finally, Fig. 5.2d

shows an example of a crossbeam cavity which may, in principle, support orthogonal modes

with high spatial overlap for frequency conversion in nonlinear optics [60].

5.2 Planar SiC Photonic Crystal Cavities at

Telecommunication Wavelengths

In order to demonstrate properties of 3C-SiC platform for photonics applications, we pro-

ceed to design, fabricate and characterize low mode-volume, high quality factor, resonances

at telecommunications wavelengths. In particular, the devices examined here are planar

photonic crystal cavities implementing a shifted linear three-hole defect (L3). These cavi-

ties (Fig. 5.3) have their three central holes removed and holes adjacent the cavity center

displaced outwards by 15% of the lattice constant a. The structures are also characterized

by a device layer thickness d ≈ 210 nm, silicon carbide refractive index n ≈ 2.55, and hole

radius r = 0.3a. Finally, for our initial purposes of prototyping devices in the 3C-SiC films,

we choose a simple and robust fabrication design where all photonic crystal holes are the

same size. Finite-difference time-domain (FDTD) simulations indicate that these struc-

tures possess theoretical mode volume values V ∼ 0.75 × (λn)3 and cavity quality factors

Q ∼ 2, 000.

The fabricated 3C-SiC photonic crystal cavities were characterized using a cross-polarization

measurement technique with apparatus presented in Fig. 5.4a. In this experiment, a halo-

gen lamp (Ocean Optics) provided broadband excitation of the L3 cavity resonances in the

wavelength region 1,250-1,600 nm and a high numerical aperture (NA = 0.5) objective lens

was used to optically address the cavity with high efficiency. Incoming light was first polar-

ized vertically (y-direction in Fig. 5.3c) and reflected off the polarizing beam splitter (PBS)

towards the sample of L3 cavities oriented horizontally and with cavity resonance polarized

vertically. A half wave plate (HWP), with optical axis oriented at θ with respect to the

21


Figure 5.3: SEM images of fabricated 3C-SiC shifted L3 cavity from (a) vertical incidence, and (b)

40-degree tilt. (c) FDTD simulation of the dominant (Ey) electric field component of

the fundamental mode, with the outlines of the holes indicated.

vertical, was positioned between the PBS and objective. Resonance spectra were moni-

tored via transmission through the PBS on a liquid nitrogen-cooled InGaAs spectrometer

(Princeton Instruments).

Rotating the HWP modifies the cavity coupled light, as well as transmission through

the PBS, in order to verify the polarization properties of the L3 photonic crystal cavity.

For example, even though the cavity coupling is maximal for a half wave plate oriented at

θ = 0o, signal transmitted through the PBS is still zero. At θ = 45o, both cavity coupling

as well as signal transmitted through the PBS is zero. A high signal/noise ratio of cavity

coupled light to uncoupled background is achieved when the HWP at θ = 22.5o rotates

the polarization of the incoming beam at 45o from the vertical. Fig. 5.4b shows one such

spectrum of the fundamental mode of a representative 3C-SiC cavity at 1,530 nm with

quality factor of 800, obtained from a fit to Fano-resonance [61]. The peak intensity of the

cavity resonance has sin(4θ) dependence and the polarization-dependent signal presented

in Figure 5.4c shows good agreement with this theory.

The observed resonances are blue-shifted relative to the theoretically predicted value by

90 nm, while the Q factor is decreased by a factor of 1.75.To explain these differences, we

analyze the influences of material and fabrication imperfections, by simulating the resonant

wavelengths and quality factors of photonic crystal cavities with parameters in the same

range as those fabricated on the chip. We analyzed up to 30% hole radius increase, which

could be caused in fabrication by over-etching, and the same percentage of slab thickness

decrease, as the damaged interfacial layer leads to uncertainty in device layer thickness.

The results are shown in Fig. 5.5 in green triangles and yellow squares, respectively. Both

of these effects reduce quality factors and blue-shift the resonances, which is consistent with

our experimental observations. Additional deviations may be caused by factors such as

imperfect sidewall profile [62].

For most applications, it is necessary to be able to controllably target resonance wave-

lengths of high-Q cavities both over a large range and to specific wavelengths of interest

22

5.2 Planar SiC Photonic Crystal Cavities at Telecommunication Wavelengths

Figure 5.4: (a) Cross-polarized reflectivity setup scheme, (b) characterized fundamental mode reso-

nance at telecommunications wavelength, blue dots represent data points, red line shows

fit to a Fano-resonance with quality factor of Q ≈ 800, (c) resonance intensity depen-

dence on polarization angle of incoming light, blue dots represent data points, red line

shows fit to sin(4θ), where θ is half-wave plate rotation angle.

23


Figure 5.5: Theoretical analysis of the wavelengths and the quality factors of SiC L3 photonic crystal

cavities as a function of the hole radius (r) shown in green triangles and slab thickness

(t) shown in yellow squares. The optimal parameters (t = 210 nm, r = 0.30a) are

represented by the red dot and the corresponding experimental result is indicated by the

black star.

24

5.3 Nanobeam Photonic Crystal Cavities at Telecommunication Wavelengths

(e.g. emission lines of optical emitters). In order to generate cavities that are tunable across

the telecommunications band, we therefore varied the lattice constant a of the L3 cavities

in the range 500-700 nm. We again monitored the resonant wavelength and quality factor

of the fundamental modes of these structures via the cross-polarized reflectivity technique.

The results of this experiment, which are presented in Fig. 5.6, show that resonances of L3

cavities of Q ∼ 103 may be achieved across wavelengths 1,250-1,600 nm.

Figure 5.6: (a) Fundamental mode resonance scaling with lattice constant a, shown with Fano-

resonance fit with quality factors in range 630 to 920; all cavities are shifted L3 cavities

made in 210 nm thick slab of 3C-SiC. (b) Experimentally measured linear dependence

of resonant wavelength on lattice constant. Red circles are measured data points and

black line is a linear fit to the data.

5.3 Nanobeam Photonic Crystal Cavities at

Telecommunication Wavelengths

For certain applications, such as nonlinear frequency conversion and strong coupling to

emitters in 3C-SiC, it is advantageous to realize optical cavities with even higher quality

factor (Q ∼ 104). For these purposes we designed nanobeam (1D) photonic crystal cavities.

Figure 5.7a shows an FDTD simulated nanobeam fundamental mode with Q 40, 000. An

SEM image of a fabricated structure is shown in Figure 5.7b. The cross-polarized reflectivity

method for characterization of these devices is very sensitive to the setup alignment, but it

provides an accurate quality factor measurement. In practice, it is very helpful to know the

resonant wavelength before optimizing the optical path. In order to locate the wavelength

of the resonances, we first characterized the nanobeams with a tapered fiber connected to

an optical spectrum analyzer. Figure 5.7c shows resonances 1.3 – 1.55µm, with maximal

quality factor Q = 1, 500. While these quality factors represent an improvement over L3

cavities, they are an order of magnitude below the modeled value, which we attribute to

25


the fabrication imperfections as opposed to a fundamental limit.

Figure 5.7: (a) FDTD simulated Ey field of the fundamental mode in a 3C-SiC nanobeam cavity.

(b) SEM image of a fabricated 3C-SiC nanobeam cavity. (c) Characterized resonances

across the telecommunications range for different values of the lattice constant a.

5.4 Summary

In summary, we have demonstrated a robust fabrication routine that has allowed us to realize

a diverse set of devices such as 2D planar photonic crystal cavities, 1D nanobeam photonic

crystal cavities, and crossbeam photonic crystal cavities from cubic (3C) silicon carbide

films. We have also shown that the procedure may be used to realize high quality factor

(Q ∼ 103 ) resonances at telecommunications wavelengths. The demonstrated scalable

fabrication process will enable wider use of 3C-SiC nanophotonic classical and quantum

information processing.

26

6 Visible Wavelength Microdisk

Resonators in 3C-SiC

In this chapter, we present the design, fabrication and characterization of cubic (3C) silicon

carbide microdisk resonators with high quality factor modes at visible and near infrared

wavelengths (600-950 nm) [63]. Whispering gallery modes with quality factors as high

as 2,300 and corresponding mode volumes V ≈ 2(λn)3 are measured using laser scanning

confocal microscopy at room temperature. We obtain excellent correspondence between

transverse-magnetic (TM) and transverse-electric (TE) polarized resonances simulated us-

ing Finite Difference Time Domain (FDTD) method and those observed in experiment.

These structures based on ensembles of optically active impurities in 3C-SiC resonators

could play an important role in diverse applications of nonlinear and quantum photonics,

including low power optical switching and quantum memories.

6.1 Fabrication of 3C-SiC Microdisks

We developed a new fabrication process illustrated in Figure 6.1 to generate state-of-the-art

3C-SiC microdisks. We begin with a 3C-SiC film of approximate 210 nm thickness, grown

on Si by a standard two-step chemical vapor deposition technique [31]. In our previous

demonstration of telecom wavelength photonic crystal nanocavities in the same material

[57], we used positive tone resist to define holes in the pattern. Here, the microdisk fabri-

cation was based on the use of negative tone e-beam resist Microposit ma-N 2403, followed

by patterning with a 100 keV electron beam tool (JEOL JBX 6300) with a 375 µC/cm2

dose. Development was performed with standard Microposit MF-319 developer. To in-

crease directionality of our SiC etch, in this work we transferred the HBr-Cl2 process to

a TCP/RIE Lam Research (9400 TCP Poly Etcher) tool, which provides a uniform, high

density transformer-coupled plasma. Acetone sonication was used to remove the residual

resist from the structures, after which the underlying silicon was etched for three 20 second

cycles at 800 mTorr XeF2 and 50 Torr N2 gas pressures in Xactix e-1 etcher, to release the

outer 500 nm along the microdisk radius.

Microdisks were fabricated with diameters varying between 1.1 µm and 2.5 µm, in 10 x

27

6 Visible Wavelength Microdisk Resonators in 3C-SiC

Figure 6.1: (a) 3C-SiC microdisk fabrication process flow. (b) Fabricated 10 x 10 arrays of microdisks

and 50 µm x 50 µm square structures. (c) The smallest fabricated microdisk with

diameter 1.1 µm on a 100 nm wide pedestal. (d) Array of robust 1.9 µm diameter SiC

microdisks on 900 nm wide Si pedestals.

10 arrays, alongside 50 µm x 50 µm square structures, later used as a reference in photo-

luminescence measurements (Fig. 6.1b). Additionally, prior to the optical measurement,

a cleaning process in 3:1 piranha solution was performed to remove any remaining organic

residue. Microdisks of diameter 1.7 µm and higher (Fig. 6.3d) were robust, with a satisfac-

tory etch profile and undercut area that can support first-order radial whispering gallery

modes. Figure 6.1c shows the smallest fabricated microdisk of 1.1 µm diameter on a 100

nm thin pedestal (although this one could not be characterized, as it turned out too fragile

and fell off the pedestal during the piranha solution cleaning).

6.2 Simulated Whispering Gallery Modes

We used the Finite Difference Time Domain (FDTD) method to simulate whispering gallery

modes supported in the microdisks. Figure 6.2a shows quality factors and wavelengths of

simulated modes supported in a SiC microdisk (index n = 2.6) with diameter 1.7 µm and

thickness 210 nm, standing on a Si (n = 3.77) cylindrical pedestal with diameter 700 nm and

height 500 nm. Material absorption was not included in the model, therefore the quality

factors presented have purely radiative character (Qrad), which exponentially decays with

mode wavelength. For wavelengths shorter than 700 nm, multiple mode overlap introduced

error in the extraction of Qrad, therefore the plotted values represent its lower boundary.

28

6.2 Simulated Whispering Gallery Modes

We identified whispering gallery modes with radial mode and azimuthal (angular) modes

for transverse-magnetic polarization (TM, electric field orthogonal to the plane of the disk)

and for transverse-electric polarization (TE, electric field in the plane of the disk). The

quality factors were as high as , and the mode volumes were in the range V ∈ [1.8, 4.6]×(λn)3.

Neighboring (m, 1) modes were frequency (f = c/λ) separated by free spectral range ∆TM =

20.0 THz for TM polarization, and ∆TE = 22.8 THz for TE polarization. For 1.9 µm

diameter disks, these values were ∆TM = 18.1 THz and ∆TE = 19.8 THz, while for 2.5

µm diameter disks, they were ∆TE = 13.2 THz and ∆TE = 15.5 THz, with uncertainty

of 3%. The polarization dependence of the free spectral range is caused by the different

mode confinement in TM and TE modes, where the modal region outside of the disk volume

lowers the effective index of the mode, and therefore changes its group velocity.

Figure 6.2: FDTD simulations of microdisks. (a) Radiative quality factor of TM and TE first radial

order (n = 1) modes supported in a 1.7 µm diameter disk on a 700 nm pedestal; free

spectral range between subsequent modes is marked by arrows. (b) |Ez| component

of electric field in plane of the microdisk and in vertical plane of the structure, for

whispering gallery TM modes (13,1), (10,2), (7,3) at wavelengths 682 nm, 675 nm, 662

nm, respectively, with quality factors noted beneath. (c) Whispering gallery (12,1) mode

wavelength scaling with disk size for TM and TE polarization.

29


Higher order radial modes were not supported with high quality factor, due to their

leakage into the pedestal. Figure 6.2b shows examples of 1st, 2nd and 3rd order radial modes

at nearby wavelengths and illustrate the dramatic three orders of magnitude deterioration

of quality factor to Q ∼ 102 for higher order modes. To illustrate the tuning of the mode

wavelength with the size of the disk, Figure 6.2c shows whispering gallery (12,1) modes for

both TM and TE polarization and the range of fabricated microdisk diameters. We observe

a steeper scaling for TE polarized modes, indicating that for the same set of wavelengths

larger microdisks support higher angular order modes, and that TE modes have higher free

spectral range than TM modes.

6.3 Characterization of Visible Resonances in 3C-SiC

Microdisks

In this experiment, we use a 2D scanning photoluminescence (PL) setup shown on Fig. 6.3a

to characterize the 3C-SiC sample and observe microdisk resonances at room temeprature.

A strong (3 mW) green pump laser is reflected off of a dichroic mirror (Semrock LM01-552-

25) towards a 2-axis scanning voice-coil mirror (Newport model FSM-300-01). From here,

the deflection of the green pump beam is imaged onto a high numerical aperture objective

lens (Olympus SPLAN 100X, NA = 0.95) in a 4-f imaging configuration. Angular deviation

of the scanning mirror is converted to real space deflection on the chip. Emitted fluorescence

is collected back through the objective, then transmitted through the dichroic mirror and

collected in a single mode fiber acting as a confocal pinhole. An additional 532 nm notch

filter rejects reflected green pump laser light and a 600 nm long pass filter assures collection

of the signal only in the red part of the spectrum. An avalanche photodiode (APD) detects

the emitted photons in order to build up a scanned image, while a spectrometer with a a

liquid nitrogen cooled Si CCD is used to spectrally resolve the fluorescence.

In order to investigate any low-level fluorescence from the sample we measure the PL

spectrum of the chip at different locations on the sample. A scanning confocal microscope

image corresponding to our 3C-SiC microdisk structures (Fig. 6.3b) is shown in Fig. 6.3c.

We observe an increase in PL signal from the freestanding 3C-SiC film relative to the

background with a broad peak at 700 nm. This signal was used to characterize the designed

microdisk resonances in an active measurement.

We studied 100 keV e-beam irradiation as a possible source of the broad 700 nm PL peak,

that may have occurred during SiC irradiation through the negative tone resist during e-

beam lithography. As a control sample, we fabricated nonirradiated suspended membranes

using positive resist. The spectra in the two approaches were identical at both room tem-

peratures and at 10 K, therefore we rulled out the option of the signal being induced by

30

6.3 Characterization of Visible Resonances in 3C-SiC Microdisks

Figure 6.3: (a) 2D green/red confocal photoluminescence imaging setup used to optically address

3C-SiC microdisks. (b) Optical micrograph of the microdisks sample. (c) 2D photolumi-

nescence scan showing increased signal in the undercut regions of microdisks; maximum

signal corresponds to several hundred counts for 0.025 s integration.

e-beam irradiation. The likely cause of this PL is the G-band emission from 3C-SiC donor-

acceptor recombination in defects close to Si interface [64], or alternatively emission due to

fabrication and processing (etching effects or residual organics).

Fig. 6.4a shows PL spectra for microdisk resonators of diameter 1.7 µm, 1.9 µm and 2.5

µm. Resonances with quality factor from 500 to 2,300 are visible along the broad PL peak,

with maxima shown in Fig. 6.4b. The obtained spectra were closely reproducible across

microdisks in the same array, with wavelength shifts of less than 1 nm. Red triangles and

blue crosses mark sets of resonances with frequency spacings closely resembling free spectral

range values simulated for TM and TE polarization, respectively. We conclude these mode

spacings indeed represent free spectral range in sets of TM and TE polarized resonances in

experimental spectra. For 1.7 µm diameter disks, these values were found to be ∆TM = 20.6

THz and ∆TE = 22.9 THz across the spectrum. For 1.9 µm they were ∆TM = 17.4 THz

and ∆TE = 19.8 THz, while for 2.5 µm diameter disks, they ∆TM = 13.2 THz and were

∆TE = 15 THz, with 5% uncertainty. A comparison between the measured and simulated

free spectral range yields 95% agreement and is shown in Fig. 6.5a. We note that values in

simulation and experiment were obtained independently of one another.

Next, we tried to deduce the (m,n) orders of the characterized whispering gallery modes.

As discussed under simulation results, the radial mode is determined to be n = 1 for all

resonances in the series, due to the value of their quality factors. To determine the angular

order m of the experimentally obtained TM and TE series of resonances, we found the

closest wavelength match to the simulation results. Figure 6.5b shows the best fit for 1.7 µm

diameter microdisks, where the shortest wavelength experimental resonances correspond to

31


Figure 6.4: (a) Room temperature photoluminescence signal (linear scale) collected from correspond-

ing microdisks with diameters 1.7 µm, 1.9 µm and 2.5 µm, on 700 nm, 900 nm and 1.5

µm wide pedestals, respectively; red triangles and blue crosses mark identified TM and

TE mode series. (b) Maximal quality factor resonance for each disk size.

modes with mTM = 14 and mTE = 15. For 1.9 µm diameter disks the best fit was obtained

for mTM = 15 and mTE = 16, while for 2.5 µm diameter disks it was mTM = 20 and

mTE = 21. As expected, larger disks have higher angular order modes at the same set of

wavelengths. We believe we have successfully identified TM and TE polarized modes in the

microdisks, which is of special interest for coupling to the impurities of known orientation

in the lattice.

The possible sources of quality factor reduction in experiment, compared to the simu-

lation, may be the imperfect etch profile, absorption of silicon carbide at the lattice mis-

matched interface with silicon, or absorption in the silicon pedestal whose band gap is below

the energy of the mode. To study the contribution of absorption in the pedestal, we ran

additional FDTD simulations to analyze the absorptive loss, assuming the extinction co-

efficient value of k = 0.012. Figure 6.5c compares the values of calculated radiative and

absorptive quality factor, alongside with the experimentally obtained value, for a sample

mode in a 1.7 µm diameter disk. We conclude that the pedestal absorption is not the

leading source of loss in the system.

32

6.4 Summary

Figure 6.5: (a) Comparison between simulation results and experimentally obtained free spectral

range for TE and TM polarizations. (b) Comparison between simulation results and

experimentally characterized resonances in 1.7 µm diameter microdisks, assuming the

angular modes mTM = 14 and mTE = 15 of the shortest wavelength peaks. (c) Compar-

ison between radiative, absorptive and experimentally measured quality factor for TM

(13,1) mode in 1.7 µm diameter microdisks.

6.4 Summary

In summary, we demonstrate the high quality visible and near-IR resonances coupled to

the intrinsic 3C-SiC luminescence in 1.7-2.5 µm diameter 3C-SiC microdisks supported by

a silicon pedestal. The origin of the luminescence is still under investigation, but possible

origins include donor-acceptor G-band recombination. Pushing the fabrication limits of

3C-SiC microdisk resonators down to the record diameters (sub 2 µm), we obtained a

new generation of structures capable of confining visible light in high quality resonances

previously unobserved in this material. Using confocal photoluminescence techniques, we

characterized quality factors as high as 2,300 in low-level intrinsic photoluminescence with

calculated mode volumes [1.8, 4.6]× (λn)3. Through a theoretical analysis, we identified TM

and TE polarized modes, and deduced their whispering gallery radial and angular order.

The successful demonstration of SiC microdisk resonators with high Q factors at room

temperature in the visible paves the way for the exploration of cavity-QED effects for room

temperature switching and nonlinear frequency conversion. Some of the approaches involve

doping the 3C-SiC films with emitters such as the divacancy, silicon-vacancy or G-band

emitters, as well as coupling these microresonators to impurities in other nanocrystalline

materials such as vacancies in ZnO or silicon-vacancies in diamond.

33

7 Hybrid Silicon Carbide-Nanodiamond

Cavity Color Center Emission

Enhancement

In this chapter, we develop hybrid silicon carbide-nanodiamond microresonators for en-

hancing SiV− and Cr-related color center emission in diamond [65]. Our approach utilizes

the similarity of the refractive indices between the two materials, allowing for the resonant

mode to be distributed across both components, which is beneficial for achieving Purcell

enhancement. In a diamondoid-seeded chemical vapor deposition step, color center rich

nanodiamonds are grown on top of 3C-SiC microdisks fabricated on a silicon wafer. The

microdisk geometry facilitates preferential positioning of nanodiamonds relative to the whis-

pering gallery mode field distribution. This scalable design has high yield and integrates

diamond color centers with a CMOS-compatible platform.

7.1 Hybrid Microdisk Modeling

We use the Finite Difference Time Domain (FDTD) method to model the electromagnetic

waves inside the hybrid microdisk. The system consists of a 2.3 µm diameter, 200 nm thick

cubic silicon carbide (3C-SiC) disk (n = 2.6), supported by a 500 nm wide cylindrical silicon

pedestal (n = 3.77), and a semi-spherical nanodiamond (n = 2.4) located on the periphery

of the microdisk (Fig. 7.1a). We excite transverse electric (TE) and transverse magnetic

(TM) modes with a 738 nm pulse with magnetic and electric field along the vertical axis,

respectively, and analyze the electric field profile and the resonance quality factor in the

cavity ring-down. The nanodiamonds are positioned at 85% disk radius distance from the

center, which has been determined favorable in a separate analysis. The mode volume

in all considered systems is V ≈ 5.5 (λ/n)3, while the resonant wavelength shift between

the hybrid microdisks with different sizes of nanodiamonds is up to 1 nm. The presence

of the nanodiamond introduces a perturbation to the whispering gallery modes supported

by a bare silicon carbide microdisk. Figure 7.1b shows how different sized nanodiamonds

affect the electric field profile and system parameters on a representative WGM. The quality

factor deteriorates with the larger nanodiamond diameter dND, while the ratio of the field

35

7 Hybrid Silicon Carbide-Nanodiamond Cavity Color Center Emission Enhancement

maxima inside the nanodiamond END to that inside the entire hybrid microdisk Emax

increases. These two terms factor in as linear and quadratic, respectively, in estimating

the maximal Purcell enhancement. For the case when the dipole moment is aligned to the

polarization of the electric field maximum and resonant with the field, the Purcell factor is

Fmax = 34π2 (λ/n)3 Q

V

(ENDEmax

)2. While our model indicates that smaller nanodiamonds and

higher Q-factors could facilitate higher Purcell enhancement (Fig. 7.1b) , the state-of-the-

art quality factors for SiC microdisks [63] are of the order of Q ∼ 103. With this in mind,

we reevaluate Purcell enhancement for Q = 1, 000 and conclude that the trend reverses for

a realistic set of parameters, now favoring larger nanodiamonds with FQ=1,000 = 8.4. Our

analysis assumes that the emitter linewidth is narrower than the cavity linewidth of ∼ 1 nm,

which is fulfilled at cryogenic temperatures. At room temperatures color center emission

broadens to ∼ 5 nm linewidths and the emission lifetime reduction is not expected.

Figure 7.1: (a) Model of the hybrid silicon carbide-nanodiamond microresonator. (b) The analysis of

different sized nanodiamonds influence on a representative whispering gallery mode and

its parameters in the hybrid microresonator, simulated by the FDTD method. c) Growth

and fabrication processes for generating arrays of hybrid silicon carbide-nanodiamond

microresonators with preferential positioning of color centers, as described in the text.

36

7.2 Growth and Fabrication

7.2 Growth and Fabrication

The production of hybrid microresonators is based on the growth of group-IV materials on

a silicon wafer (Fig. 7.1c). First, a 200 nm thick 3C-SiC layer is grown heteroepitaxially on

(001)-Si using a two step Chemical Vapor Deposition (CVD) method [31]. Microdisks with

diameters varying from 1.8 - 2.4 µm are lithographically defined and processed in plasma

and gas phase etchers [63], etching through the silicon carbide and releasing most of the

underlying silicon. In order to generate the oxide layer, the microdisk was exposed to oxygen

plasma for 5 minutes. The surface of microdisk was exposed to oxygen plasma for 5 minutes

at 300 mTorr pressure and 100 W power. The diamondoids of [1(2,3)4] pentamantane as the

seeding were covalently attached on the oxidized surfaces of SiC microdisk via phosphonyl

dichloride bonding with SiOx. Then, the nanodiamond formation is placed in a CVD

chamber for a sequence procedure of diamond growth. The CVD growth process consists

of two steps, one responsible for nanodiamond nucleation and the other determining the

final nanodiamond size. The nucleation step is performed at 400 W power of Ar plasma,

350o C stage temperature and 23 mTorr pressure for 20 minutes. The flow of gases is: 5

sccm H2, 10 sccm CH4 and 90 sccm Ar. The second growth step is varied between 15 and

45 minutes with 300 sccm H2, 0.5 sccm [CH4/SiH4(1%)] at 1,300 W power of hydrogen

plasma and 650o C stage temperature [35]. Due to the charge concentration at the edges

of the structure, the CVD plasma is denser around the ring-edge of microdisk; as a result,

the nanodiamonds that grow on microdisks are preferentially positioned by the outer edge

– in the vicinity of the whispering gallery modes. The seeding diamondoids are relatively

labile and prone to destruction and detachment from the substrate at high temperature and

plasma during nucleation and etching. To overcome that, we modify the CVD growth to a

two-step process, as described in the Methods section. SiV− and Cr-related color centers

are generated as a result of in-situ doping with a controlled flow of silane and the residual

chromium present in the chamber. The duration of the CVD growth defines the size and

density of nanodiamonds on chip, and, consequentially, the number of color centers. The

growth time tG is varied between 15 and 45 minutes. The smallest nanodiamonds are in

the range 60-100 nm, and occurred on 8 out of 100 microdisks i.e. with 8% yield, while

the largest nanodiamonds are 330-380 nm in diameter and had 60% yield, often resulting

in multiple particles per microdisk (Fig. 7.2).

7.3 Optical Characterization

We use a 2D scanning confocal photoluminescence setup (Fig. 7.3a) to characterize the

optical properties of the hybrid microdisks. The sample is excited by a 532 nm continuous

37


Figure 7.2: Three generations of hybrid silicon carbide-nanodiamond microdisk arrays with variable

growth time tG.

38


Figure 7.3: (a) 2D scanning photoluminescence setup; PL: photoluminescence, LP: longpass filter,

BP: bandpass filter, SM: single mode, MM: multimode, CW: continuous wave, NA:

numerical aperture, G: green, B: blue, R: red. Images show SEM image and 2D PL

maps without and with a 740 nm band pass filter. (b) Photoluminescence from hybrid

microresonators with 80 nm, 150 nm and 330 nm nanodiamond size, showing SiV− and

Cr-related color center zero-phonon lines at T = 10 K in hybrid microresonators, as well

as ∼ 1 nm linewidth WGMs in both SiC microdisks and hybrid microresonators.

39


wave (Verdi V-10, Coherent) or 495 nm pulsed (frequency doubled MaiTai tunable ultrafast

laser, Spectra Physics) laser. The laser is reflected off of a two-axis rotating mirror (Newport

FSM-300), whose deflection is mapped onto the sample by the 4-f optical configuration and

a high numerical aperture microscope objective (Zeiss NEOFLUAR, NA = 0.75). Filtered

signal is collected in the red part of the spectrum using a 600 nm longpass filter. A photon

counting module (Perkin Elmer SPCM-AQRH-14-20117) detects the emitted photons in

order to build up a scanned image, while a spectrometer with a liquid nitrogen cooled Si

CCD (Princeton Instruments) is used to spectrally resolve the fluorescence. After collecting

the fluorescence on the photodiode, the time-resolved photon detection is registered on time-

correlated single-photon counting module (PicoHarp 300). A photoluminescence image is

constructed by scanning the excitation laser pixel-by-pixel across the chip and collecting

the filtered signal in the red part of the spectrum. Intrinsic emission from silicon carbide

and bright color center emission from diamond can be distinctly recognized in the image,

and therefore correlated to scanning electron microscope images of the region. We then

collect the spectrum and the time-resolved signal from the points of interest, from which

we determine the origin of the emission and the emitter lifetime.

Figure 7.4: Photoluminescence spectra at 10 K on a hybrid microresonator (shown in the SEM

image) with a 60 nm nanodiamond, taken at the location of the nanodiamond (red) and

away from the nanodiamond (black), exhibiting whispering gallery mode resonances and

two SiV− zero phonon lines at 736 nm and 740 nm.

Cryogenic photoluminescence characterization reveals narrow emission lines originating

40


from both SiV− and Cr-related color center zero-phonon transitions. The number of color

centers varies with the nanodiamond size, from several to more than a hundred zero-phonon

lines (ZPLs) (Fig. 7.3b) . The smallest nanodiamond that exhibited color center emission

was 60 nm in diameter, as shown in (Fig. 7.4). Whispering gallery modes with linewidths ∼1 nm are present in all hybrid microresonator spectra, however they are harder to identify

in structures with largest nanodiamond due to an overwhelming emission from the ZPLs

that form a 5 nm linewidth peak. Color center emission broadens with temperature and

above 100 K individual zero phonon lines become indistinguishable in the broadened peak.

Figure 7.5 shows temperature photoluminescence scans of free diamonds confirming that

no individual ZPL emission can be extracted at temperatures higher than 200 K.

Figure 7.5: Temperature dependent photoluminescence spectra of two free diamonds showing no

distinction between individual emitters at room temperatures.

The room temperature lifetimes at the color center peak emission frequencies are τSiV =

0.9 ns and τCr = 0.6 ns, obtained both from nanodiamonds grown on and off the microdisks

(Fig. 7.6a). Nanodiamonds exhibit spectral variation in lifetime, likely induced by strain

(Fig. 7.6b). We observe high quality factor whispering gallery modes that slightly degrade

with nanodiamond diameter, consistently with the FDTD modeling trends. The highest

quality factor hybrid whispering gallery mode resonance has Q = 2, 700, measured on

41


Figure 7.6: (a) Time resolved measurements of SiV− (left) and Cr-related (right) color centers in

nanodiamonds collected at 1.5 nm band filtered peak wavelengths. Lifetimes obtained

from exponential decay fits are τSiV = 0.9 ns and τCr = 0.6 ns. (b) Spectral variation

in color center lifetime at room temperature.

microdisks with 80 nm nanodiamonds, while for microdisks with 350 nm nanodiamonds

Q = 1, 400. Furthermore, the wavelength spacing of the observed WGMs matches the

expected diameter-dependent free spectral range of 14-20 THz studied in previous work for

sets of TE and TM polarized resonances [63].

We observe enhanced emission signal in hybrid microdisks where the whispering gallery

modes couple to the color center emission at room temperature. Figure 7.7a shows an

example of a hybrid microdisk where SiV− emission is resonantly enhanced at 744 nm by

30% of its intensity, while other (less prominent) peaks noticeable in the spectrum represent

uncoupled WGMs. The quality factors are Q = 960, and 700 < Q < 1, 500, respectively.

The spectrum is compared to the neighboring free nanodiamond whose growth is a result

of same local conditions in terms of the expected color center generation. Figure 7.7b

shows an example of Cr-related color center emission enhancement at 758 nm by 60%. In

this case Q = 550, and Q-factors of the other WGMs in the spectrum range from 530 to

920. Time-resolved photoluminescence measurements do not reveal change in the emission

lifetime, however, the cavity modifies the local density of states and increases the radiative

transition rate, resulting in an enhanced signal from the emitter. As known from literature,

SiV center has a strong non-radiative component at temperatures above 70 K [66, 67]. We

conclude that the non-radiative transitions dominate the color center dynamics at the room

temperatures, and therefore the resonant changes in the radiative decay remain unnoticed

in the lifetime measurements and are present only in the radiative emission enhancement.

The presence of strain causes differences in individual color centers between the nanodia-

monds, as well as within the same nanodiamond. We perform lifetime measurements of SiV

42


Figure 7.7: Hybrid disk and neighboring free nanodiamond photoluminescence at room temperature.

Hybrid disks shows resonant enhancement of a) SiV− and b) Cr-related color center

emission, detail shown inset. Red (yellow) circles correspond to red (black) plot curves.

Uncoupled WGMs are also present in the spectrum at expected free spectral range.

– emission in hybrid microresonators at T = 6.3 K. While we do seem to observe a trend

of lifetime shortening in hybrid microresonators compared to the free nanodiamonds – the

results are inconclusive due to the variability in non-radiative lifetimes.

To further characterize the cavity influence on the color center emission, we take a more

detailed photoluminescence temperature scan in a Montana Cryostation (featuring better

temperature stability and NA = 0.9) by collecting the photoluminescence signal in a con-

focal microscope setup. We first follow a SiV− resonance from room temperature to T = 5

K (Fig. 7.8a) observing change in its wavelength and linewidth, consistent with the ZPL

shift in literature [66]. We observe a similar behavior for a resonant Cr-center (Fig. 7.8b).

The conclusion is that one or several color centers are significantly affected by the whis-

pering gallery mode. The coupling between the emitter and the cavity has spectral and

spatial overlap components which qualifies only a fraction of all color centers for potential

Purcell enhancement. Moreover, the spectral overlap is temperature-dependent, therefore

we analyze closely individual zero-phonon lines in hybrid microresonator spectra shown in

Figure 7.8c. We identify one resonant and two non-resonant SiV− ZPLs whose emission can

be isolated from the spectra though a Lorentzian fit. Figure 7.8d shows the temperature

dependence of the ZPL peak intensities. For comparison, the ZPL emission intensity I in

CVD grown diamond [67] was found to fit the function

I ∼ R

R+ Ce−EkT

,

where R is the probability for radiative transition assumed to be temperature-independent

(R = const.), k is the Boltzmann constant, and energy of the non-radiative transition

E = 70 meV. The non-resonant ZPLs seem to obey this function well, which is not the

43


Figure 7.8: Temperature scans of (a) SiV− and (b) Cr-related center in resonance with WGMs

(spectra are normalized to a mean number of counts for comparison); (c) Hybrid mi-

crodisk emission indicating an on-resonant (red circle) and two off-resonant (dark and

light blue circles) ZPLs; (d) Color-coordinated ZPL peak intensities obtained through

the Lorentzian fit to the emission spectra in (c) fitted with the constant and non-constant

radiative rate intensity dependencies.

44


case for the resonant ZPL that differs from the dependence for temperatures below 70

K, where the emission is almost purely radiative [66]. This signals a resonant effect in

the probability of radiative transition, now R(T ) due to the temperature dependence of the

spectral mismatch. At low temperatures ZPL frequency shift can be approximated as linear

in temperature [66]. Extrapolating from the WGM frequency temperature dependence in

silica microresonators [68] we approximate as linear. Therefore, the Purcell enhanced ZPL

intensity has the following temperature dependence [16]

IT<70K ∼g2

κ

∆ωWGM/2

(ωWGM − ωZPL)2 + (∆ωWGM/2)2∼ A

B(T − T0)2 + C,

due to temperature-independence of the cavity linewidth ∆ωWGM and decay rate κ, as well

as the cavity-emitter coupling g which is constant due to the emitter lifetime saturation

[66, 16] below 70 K. Figure 7.8d shows a good fit of the derived dependence to the resonant

SiV− emission below 70 K and of the non-radiatively dominated dynamics above 70 K. The

comparison between these functions reveals 2.2-fold enhancement of the SiV− emission at

T = 5 K.

Figure 7.9: (a) Photoluminescence spectra taken from two hybrid microdisks and two free nanodia-

monds. (b) Spectrally resolved lifetime measurements, color-coordinated with (a).

To access lifetime measurements at cryogenic temperatures we excite the system with 710

nm pulses from MaiTai tunable ultrafast laser (Spectra Physics). Due to the low photon

counts in the collection direction (out of the disk plane), we focus only on SiV− color

centers in two hybrid microresonators resonant at 741-742 nm (blueshifted from the room

temperature wavelength) and two nearby free nanodiamonds. Spectral resolution of 1.5 nm

is achieved by narrowing the slit between the spectrometer and the Single Photon Counting

Module (SPCM). We observe up to twofold shorter lifetimes in hybrid structure at the

45


resonant wavelength (Fig. 7.9), however, unlike the results presented in Figure 7.8, this

result is not conclusive of Purcell enhancement. Nanodiamond strain, whose presence is

obvious from the ∼ 5 nm ZPL inhomogeneous broadening, potentially affects non-radiative

lifetimes as well, which in turn does not allow us to access precisely the change in the

rates of the radiative transitions. These results may be useful for future studies of strain in

diamond.

7.4 Summary

In summary, we utilized a novel hybrid microcavity coupling mechanism based on the match-

ing refractive index values to enhance diamond SiV− and Cr-related color center emission.

We developed a scalable fabrication approach resulting in preferential color center position-

ing of nanodiamonds on the outer edge of 3C-SiC microdisks, which enables excitation of

high quality factor modes shared across the hybrid structure. Our approach can be extended

to applications pertaining to integration of solid-state quasi-atoms with silicon platforms, or

as an interface between telecom and color center wavelengths that would utilize the second

order optical nonlinearity of silicon carbide.

46

8 Photonic Design for Color Center

Spintronics

Silicon carbide is a promising platform for single photon sources, quantum bits (qubits)

and nanoscale sensors based on individual color centers. Towards this goal, we develop a

scalable array of nanopillars incorporating single silicon vacancy centers in 4H-SiC, readily

available for efficient interfacing with free-space objectives and lensed-fibers [36]. A com-

mercially obtained substrate is irradiated with 2 MeV electron beams to create vacancies.

A subsequent lithographic process forms 800 nm tall nanopillars with 400-1,400 nm diame-

ters. We obtain high collection efficiency, up to 22 kcounts/s optical saturation rates from a

single silicon vacancy center, while preserving the single photon emission and the optically

induced electron-spin polarization properties. Our study demonstrates silicon carbide as

a readily available platform for scalable quantum photonics architecture relying on single

photon sources and qubits.

8.1 The Design

Silicon carbide (SiC) benefits from the decades of utilization in power electronics [69] –

wafer scale substrates are available for purchase, doping and various etching processes [57,

63, 70, 29] are at an engineer’s disposal. This makes SiC advantageous over diamond and

rare-earth doped crystals for industrial applications in quantum computing and sensing, as

well as for integration of point defects with electronics and CMOS devices. Electron spins of

various SiC color center ensembles have been coherently controlled to date [71, 72, 73]. We

focus our attention on individual silicon vacancy (VSi) centers in 4H-SiC, whose electron

spin was recently employed as a room temperature qubit [50], the first one at the single

defect level in SiC. This system features spin coherence times of at least 160 µs, and with a

potential for improvement through isotopic purification. To bring this system closer to the

applications, we address VSi/4H-SiC platform from the points of scalability, device footprint,

optical interface and collection efficiency, while preserving optically induced electron-spin

polarization properties.

Our platform consists of a 2D array of simultaneously fabricated nanopillars in electron

47

8 Photonic Design for Color Center Spintronics

beam irradiated 4H-SiC. The nanopillars feature 0.16-2 µm2 footprint and efficient signal

collection from single color centers into a free space objective of NA = 0.65. Such numerical

aperture is also available among lensed fibers [74], as a potential alternative interface. In

comparison, a previous approach has relied on the non-scalable fabrication of a solid immer-

sion lens (SIL) interface [50]. Here, focused ion-beam milling with heavy ions is used, and

an intense combination of wet and reactive ion etching needs to be performed to remove

radiation damage in the SiC lattice. A footprint of an individual SIL is ∼1,000 µm2. In

another approach, a bulk wafer was irradiated to generate a 2D array of color center sites

[75]. Both of these approaches relied on collection into a high numerical aperture (NA =

1.3) oil immersion lens, in a direct contact with the substrate. Vertically directed wave-

guiding in nanopillars allowed us to observe up to 22 kcounts/s (kcps) collection from single

color centers with half the NA value. This is comparable to the best previous result of 40

kcps achieved in a non-scalable system [50], and a threefold increase over a 7 kcps result

in a scalable array in bulk [75]. The use of NA = 0.95 air objective lens would result in

a few-fold increase in collection efficiency, as will be discussed in Figure 8.3. We also ob-

serve 2-4% relative intensity in optically detected magnetic resonance (ODMR), comparable

to the previous result [50]. Finally, the preservation of optical and spin properties of VSi

centers makes this platform applicable for wafer-scale arrays of single photon sources and

spin-qubits.

8.2 Fabrication

We develop a scalable nanofabrication process, based on electron irradiation and electron

beam lithography, which produces tens of thousands of nanopillars per square millimeter in

a 4H-SiC substrate that incorporates VSi color centers. Figure 8.1a illustrates the process

flow. The initial sample is a commercially available high purity semi-insulating 4H-SiC

crystal, irradiated with 2 MeV electrons at a fluence of 1014 cm−2 at room temperature.

For the purposes of hard etch mask, 300 nm of aluminum was evaporatively deposited

onto the sample. Microposit ma-N negative tone resist was spun over it for patterning of

nanopillars in 100 keV JEOL (JBX-6300-FS) electron beam lithographic system. Aluminum

hard mask was etched in Cl2 and BCl3 plasma, and the pattern was transferred to silicon

carbide in an SF6 plasma process, both performed in the Versaline LL-ICP Metal Etcher.

The mask was stripped in KMG Aluminum Etch 80:3:15 NP.

The pattern consists of 10×10 arrays of nanopillars with 5 µm pitch and diameters varied

between 400 nm and 1.4 µm. The profile features 800 nm tall cylinders whose upper third

has the shape of a cone frustum, reducing the top diameter by 120 nm. Figures 8.1b-c show

the SEM images of the fabricated structures.

48

8.2 Fabrication

Figure 8.1: (a) Lithographic process for fabrication of nanopillars in 4H-SiC. (b) SEM images of

fabricated array of nanopillars with variable diameters. (c) SEM image of a 600 nm

wide, 800 nm tall nanopillar.

49


8.3 Collection Efficiency

We model emission of color centers placed in pillars using the Finite-Difference Time-

Domain method. Index of refraction distribution is made according to the profile of fab-

ricated pillars (nSiC = 2.6, nair = 1) and continuous wave dipole excitation at 918 nm is

positioned within. We consider nanopillars of 400 – 1,400 nm diameter with excitation at

the central axis of the pillar (r = 0, z = 400 nm) with in-plane (Er) and vertically oriented

(Ez) dipole components. The orientation of VSi dipole in our samples is along the main

crystal axis which forms a 4o angle with z-axis, and therefore we add the components to

obtain total collected field (Fig. 8.2a). We evaluate the fraction of collected emission by a

two-step analysis. First, we monitor the power that penetrates the walls of the simulation

space and evaluate the fraction of light that is emitted vertically upward (Fig. 8.2b).

Next, we analyze the far field image (Fig. 8.2c) and obtain the power fraction of vertically

emitted light that is collected within a given numerical aperture (NA). Figure 8.3 shows

how the collection efficiency would change if the objective lens had NA=0.95. Due to the

dominant contribution of the dipole-component along the z-axis, which is also noticeable in

Figure 8.2a, we consider only vertical excitation Ez for additional analyses. In an extensive

search for the optimal dipole-emitter position for each pillar diameter, we learn that color

center positioning closer to the bottom of the pillar is advantageous for collection efficiency

(Fig. 8.4). We also run this analysis for unpatterned silicon carbide, in order to estimate

the expected collection efficiency increase achieved through nanopillar waveguiding, which

yields 1-2 orders of magnitude improvement. The results are presented in Figures 8.2d-e

showing favorable collection efficiency of 600 nm and 1,000 nm diameter nanopillars.

We implement the identical design in the substrate with higher concentration of VSi

centers, as shown in Figure 8.5. We observe three-fold increase in the collection efficiency

from the 1 µm diameter pillars compared to the unpatterned bulk. Moreover, we observe no

change in the linewidth between these two signals, indicating that nanofabrication does not

compromise optical properties of the emitters. This is an important conclusion for silicon

carbide nanophotonics which indicates potential for multi-emitter quantum photonics in

SiC.

8.4 Single Photon Emission

Photoluminescence (PL) measurements obtained in a home-built confocal microscope at

room temperature are shown in Figure 8.6. Light from a single-mode laser diode operating

at 730 nm was sent to an air objective (Zeiss Plan-Achromat, 40x/0.65 NA), in order to

excite the defects. The PL signal was collected with the same objective and sent through

50


Figure 8.2: (a) Collection efficiency from a dipole emitter positioned at the center of 800 nm tall SiC

nanopillar (indicated by inset drawing), with 4o offset from z-axis (red), plotted with

its vertical (yellow) and radial (blue) components. (b) Finite Difference Time Domain

model of electric field propagation from a dipole emitter placed in the center of a 600

nm wide nanopillar. (c) Far field emission profile of (b) with light line (white) and

NA line (red). (d) Color center depth dependence of collection efficiency from a dipole

emitter in bulk SiC, with components marked as in (a); inset drawing illustrates the

direction of increasing depth. (e) Diameter dependence of optimal (red) and centrally

(blue) positioned color center in nanopillar, showing up to two orders of magnitude higher

collection efficiency compared to bulk value of 0.0017.

51


Figure 8.3: Collection efficiency from VSi at the center of the nanopillar (r = 0, h = 400 nm) into

air objective with NA = 0.65 and NA = 0.95.

Figure 8.4: (a) Collection efficiency dependence on vertical position along the central axis (r =

0), where negative values correspond to the top of the nanopillar, and positive ones

to the bottom, plotted for various diameters. (b) Collection efficiency dependence on

horizontal position along the radius at the central heights (h = 400 nm), plotted for

various diameters.

52


Figure 8.5: (a) SEM image of a pattern with nanopillars and unetched bulk regions. (b) Photolu-

minescence measurements at T = 7 K, showing same linewidth for nanopillars as in the

bulk unetched region, as well as the increased collection efficiency.

a dichroic mirror and subsequently spatially filtered by a 50 µm diameter pinhole. The

photons emitted by the defects were spectrally filtered by a 905 nm long pass filter and

finally collected with two single photon counting modules in a Hanburry-Brown-Twiss con-

figuration, or, alternatively, with a spectrometer. The density of color centers is too high

to resolve single defects in the bulk crystals with no fabricated structures, however, com-

bined with the nanopillars, a spatial separation of single defects is possible, simultaneously

providing a moderate yield of defects placed in the pillars, as shown in Figure 8.6a. Ap-

proximately 4 out of 100 pillars host single VSi defects with desirable lateral and vertical

placement, and show an enhanced photon count-rate compared to defects without photonic

structure [75]. For each investigated VSi defect a laser power dependent saturation curve of

the PL was collected as shown in Figure 8.1b. The background was obtained by collecting

the light from an empty pillar nearby and subtracted from the defect’s emission. The influ-

ence of the diameter was experimentally examined by a statistical analysis of the maximum

PL intensity as a function of the diameter. The maximum PL was obtained by fitting the

background-corrected saturation curve with a power law: I(P ) = IS1+α/P , where I is the

PL intensity as a function of the excitation power P , α is excitation power when saturated

and IS denotes the saturation photon count-rate. The result of the fit parameter is plotted

as a function of the diameter in Figure 8.6c, and show a qualitative match to the modeled

collection efficiency trend. The error-bars were obtained by the error from the least-square

fits. The maximum PL from a single defect was found in the pillars with 600 nm diameter,

showing Imax = 22 kcps, which is an increase of factor of 2-3 compared to 5-10 kcps for a

single VSi defect without photonic structures reported in literature [50, 75]. This increase

53


is smaller than expected from the model, which is attributed to the imperfect fabrication

profile. A typical room temperature PL spectrum is shown in Figure 8.6d, while Figure

8.6e shows a typical second order autocorrelation measurement with a clear single photon

emission characteristic.

Figure 8.6: (a) Photoluminescence intensity map of a block of pillars. (b) PL intensity of a single

VSi as a function of the optical power, saturating at 22 kcps. The red line shows a fit to

the data. (c) Diameter dependent collected count rate. (d) Typical room temperature

PL spectrum of a single VSi emitter in a nanopillar. (e) Typical second order correlation

function showing anti-bunching at zero time delay, a clear evidence for single photon

emission. The red line shows a double exponential fit to the data. (f) ODMR signal

from a single VSi in a nanopillar. The red line shows a Lorentzian fit to the data.

8.5 Single Spin Optically Detected Magnetic Resonance at

Room Temperature

In order to address the single spins, a radio-frequency (RF) signal was generated by a Rohde

and Schwarz vector signal generator, subsequently amplified and applied to the VSi in pillars

through a 25 µm copper wire, which was spanned over the sample [50]. Continuous wave

(cw) ODMR is observable under cw laser excitation where spin-dependent transitions over

a metastable state lead to a spin polarization into the magnetic sublevels. At the same

time RF is swept over the spin resonance. When the RF signal is resonant to the Larmor-

frequency, the spin state is changed and the transition rates over the metastable state is

altered, hence a change in PL intensity can be detected. About 10% of the single VSi defects

which show enhanced emission also showed clear spin imprints, as illustrated in a typical

ODMR measurement at a zero magnetic field plotted in Figure 8.6f, which is in agreement

with previous studies [50, 75]. VSi defects in 4H-SiC can originate from two inequivalent

54

8.6 Summary

lattice sites [49]. Among them, the so-called V2 center can be identified by their 70 MHz

zero-field splitting of the ground state spin of S = 3/2. The center position at 70 MHz of

a broadened ODMR spectrum as shown in Figure 8.6f confirm that the found VSi defects

showing ODMR signals are V2 centers. The side-peaks surrounding the center-frequency

originate from a non-zero parasitic magnetic field oriented in an arbitrary direction in the

experimental environment, which is typically about 0.2-0.6 Gauss [76, 77]. The peak at 40

MHz can often be observed, also in other samples [49] but the further investigation of the

origin is not in the scope of this work. The relative intensity of ODMR was calculated as

and it varies between 2-4%, which is in good agreement with previous single spin studies

[50, 75].

8.6 Summary

In summary, we have developed a scalable and efficient platform for collecting single silicon

vacancy color center emission in silicon carbide through a free space objective or a lensed

fiber interface, applicable for single photon generation and coherent control of spin-qubits.

This platform presents an improvement in terms of scalability, device footprint and optical

interface, while providing competitive collection efficiency. The approach can be extended to

incorporate other color centers in bulk 3C-, 4H- and 6H-SiC samples [47, 71, 73, 78, 79]. In

addition, silicon carbide is a biocompatible substrate [80] and can be directly interfaced with

cells for potential magnetic [37, 81] and temperature sensing applications [82], whose proof-

of-principle have recently been reported. Ideas for future development include increasing

collection efficiency by fabricating taller nanopillars, isotopic purification of the sample

for accessing qubits with longer spin coherence times, increased yield through improved

irradiation conditions, and integration with microwave control on chip. Another exciting

direction is the investigation of recently suggested spin-photon interface protocols based

on the VSi in SiC [34], that requires the enhanced photon collection efficiency and the

drastically improved ODMR signal of the VSi defects, which can be accessed at cryogenic

temperatures [83].

55

9 Multi Emitter-Cavity Quantum

Electrodynamics

As discussed in the introduction, single-emitter-cavity solid state systems exhibit highly

nonlinear interaction which enables the development of integrated devices such as ultrafast

optical switches and single-photon sources. To increase the speed and the performance of

these devices, as well as to study new regimes of solid state physics we propose multi-emitter

cavity quantum electrodynamics systems at the level of several emitters. In this chapter,

we explore the necessary conditions for strong coupling of multiple nonidentical emitters,

akin to color center ensembles, to a common cavity [17].

9.1 Cavity Quantum Electrodynamics

Cavity quantum electrodynamics (QED) studies strong interaction of quantum emitters

with electromagnetic field in the cavity. A regime where this interaction is stronger than

the losses to the environment and dominates the dynamics of the system is called the

strong coupling regime. The spectral signature of strong coupling is the splitting of the

cavity transmission spectrum (or emitter emission spectrum) into two polariton peaks, as

opposed to a single Lorentzian peak [17].

9.1.1 Cavity QED Parameters

The relevant cavity QED parameters are:

- Cavity energy decay rate κ, also known as the cavity linewidth because it corresponds

to full-width-at-half-maximum (FWHM) of the transmission spectrum centered at ωa (blue

line in Fig. 9.1). Note, in some references cavity field decay rate is also marked as κ,

which is the half of the energy decay rate and therefore induces 1/2 factor in the relevant

equations.

- Emitter decay rate γ, also known as homogeneous emitter linewidth which corresponds

to FWHM of the single-emitter emission spectrum, a Lorentzian centered at ωe. The lower

bound for γ is the Fourier transform of the emitter lifetime.

57

9 Multi Emitter-Cavity Quantum Electrodynamics

60 40 20 0 20 40 60

(ω−ωc )/2π (GHz)

Tra

nm

issi

on (

a.u

.)

Figure 9.1: Empty cavity (blue) transmission spectrum vs. strongly coupled emitter-cavity system

(red) featuring two polariton peaks. Parameters are {g, κ, γ}/2π = {15, 25, 0.1} GHz.

- Emitter inhomogeneous broadening δ, also known as emitter ensemble linewidth. It

represents a measure of non-identicality in an ensemble of emitters. Due to the changes in

local environment, especially lattice strain, color centers emit at varying frequencies ωe,n

given by normal distribution with the FWHM of δ. The inhomogeneous broadening in

quantum dot ensembles occurs due to their varying size.

- Emitter-cavity interaction rate g (or gn when emitters are not equally coupled), also

known as emitter-cavity coupling. It represents interaction between emitter dipole moment

and resonant electric field of the cavity, and is approximately equal to half the spacing

between the two polariton peaks (red line in Fig. 9.1).

- Collective emitter-cavity coupling rate GN which characterizes interaction strength of

N emitters coupled to a common cavity. We introduced this notation, there is no standard

yet.

- Emitter-cavity detuning ∆ = ωe − ωa. In case of collective coupling each emitter is

characterized by individual emitter-cavity detuning ∆n = ωe,n − ωa.

For convenience, we express all these values divided by 2π and in GHz. The emission

58

9.1 Cavity Quantum Electrodynamics

lines of the color centers of interest are at approximate frequencies ωe/2π ∈ [300, 500] THz.

Strong cavity QED coupling is characterized by

g > κ/4, γ/4,

for individual emitters, and

GN > κ/4, γ/4

for ensembles. In solid-state systems at cryogenic temperatures, cavity linewidth is usually

several orders of magnitude higher than emitter linewidth: κ � γ, so strong coupling

is determined by the ratio between coupling strength and the cavity linewidth. When

emitter(s) is (are) resonant with the cavity, the polariton peak linewidth is |κ+γ|2 ≈ κ

2 , while

their spacing is 2g (2GN ).

9.1.2 Determining Cavity QED Parameters

Cavity linewidth is calculated as

κ =ωaQ,

where Q stands for the cavity quality factor. Table 9.1 shows typical values for cavity

frequency 400 THz.

Q κ/2π [GHz]

100 4,000

400 1,000

1,000 400

4,000 100

10,000 40

40,000 10

100,000 4

Table 9.1: Relationship between quality factor Q and cavity linewidth (energy decay rate) κ for

cavity frequency ωa/2π = 400 THz.

Color center emitter linewidth is measured as the linewidth of the zero-phonon line (ZPL)

of an individual emitter. Despite having similar lifetime at cryogenic and room tempera-

tures, the emitter broadens to several THz as the system heats up to 300 K. Table 9.2

quantifies diamond SiV center linewidth dependence on temperature [66].

Inhomogeneous broadening is taken only from the experiment as the ZPL linewidth of

an ensemble of color centers. For the same color center complex both homogeneous and

59


T [K] γ/2π [GHz]

350 5,500

300 3,000

250 1,800

200 800

100 150

10 0.3

4 0.1

Table 9.2: Temperature dependence of the emitter linewidth γ for SiV center in diamond [66].

inhomogeneous linewidth vary from sample to sample. In good color centers δ/2π ∈ [0.5, 30]

GHz.

Emitter-cavity coupling can be calculated as

g =

√3πc3ρZPL2τω2

an3V

∣∣∣∣EecosφEmax

∣∣∣∣ ,where ρZPL is the ratio of emission that branches into ZPL (typical is 3-70%), τ is emitter

lifetime, c is speed of light, n is index of refraction of the host material (2.4 for diamond, 2.6

for SiC), V is the mode volume of the cavity, Ee is the electric field intensity at the position

of the emitter, Emax is the maximum of the electric field in the cavity mode, and φ is the

angle between the dipole orientation and the local resonant field polarization direction. The

mode volume of the cavity is often expressed though the volume factor v as

V = v

(λ

n

)3

,

so g can be rewritten as

g =

√3cρZPL8πλτv

∣∣∣∣EecosφEmax

∣∣∣∣ .Table 9.3 provides approximate coupling rates for diamond SiV with τ = 1 ns and λ = 738

nm, and perfect dipole alignment at the maximum of the field:

9.2 Multiple Emitters in a Cavity

The nonlinearity in a cavity QED system can be significantly increased through the collective

coupling of multiple emitters (as opposed to a single emitter) to the resonant mode (Fig. 9.2).

In this picture, N emitters are effectively described as a single emitter strongly coupled to

the cavity with an increased interaction rate by a factor of√N . Such coupling is achievable

60

9.3 The Tavis–Cummings Model

v g/2π [GHz] (ρZPL = 1) g/2π [GHz] (ρZPL = 0.7)

0.1 111 93

0.25 70 59

0.5 50 42

0.75 41 34

1 35 29

1.5 29 24

2 25 21

Table 9.3: Relationship between the cavity mode volume factor v and the maximal emitter-cavity

coupling rate g cited for λ = 738 nm, τ = 1 ns and two values of ρZPL.

in systems where the inhomogeneous broadening is comparable to the collective coupling

rate [84]. Collective coupling is possible even when the inhomogeneous broadening exceeds

the cavity linewidth, and the polariton width is dominantly defined by cavity and emitter

linewidths, due to an effect called cavity protection. This phenomenon was previously

investigated with rare-earth ions in a solid-state cavity for a very large ensemble of emitters

approximating a continuum [85]. In contrast, we are interested in a regime of several

emitters coupled to a nano-optical cavity and giving rise to a discretized energy ladder

suitable for demonstrations of advanced photon blockade effects. Similar multi-emitter

cavity QED systems have been demonstrated in atomic [86, 87] and superconducting circuit

systems [88]. With the developments in techniques of substrate growth and processing, the

implementation of these systems is expected with color-centers in solid-state nanocavities

as well, which will have an impact on the development of GHz-speed optical switches and

high-quality integrated sources of single photons.

9.3 The Tavis–Cummings Model

We first discuss the original Tavis–Cummings model for atoms, which have negligible in-

homogeneous broadening, and then extend to the model to include the inhomogeneous

broadening of solid-state quantum emitters. The Tavis–Cummings model was developed to

describe an ensemble of atoms that strongly couple to a cavity mode [89], and gives rise to

the Hamiltonian

HTC = ωaa†a+

N∑n=1

[ωaσ

†nσn + gn

(σ†na+ a†σn

)]. (9.1)

Here, a single cavity mode individually couples to each of the N emitters that otherwise

do not interact with one another. As before, a and ωa represent the cavity operator and

61


�� +Δ�,1 �� +Δ�,�

�

�

�

��1

��

. . .

Figure 9.2: Illustration of N non-identical emitters coupled to a cavity mode.

frequency, while the atoms are characterized by their dipole operator σn and cavity coupling

rate gn. The variables gn account for non-equal positioning of emitters relative to the cavity

field intensity.

Coherent interactions in the system give rise to a new set of eigenfrequencies that form a

dressed ladder of states. The first rung contains N+1 states, corresponding to an additional

excitation of the cavity mode or one of the emitters. The second rung contains (N+1)(N+2)2

states, representing either an excitation of one of the previously non-excited emitters or a

new photon in the cavity mode.

For a system with equally coupled atoms (gn = g), we illustrate these levels in Fig. 9.3. In

particular, the levels in the first rung represent two polaritonic states and N −1 degenerate

states. The degenerate states are at the cavity frequency and their eigenvectors have no

cavity component; therefore, the states do not couple to the environment and are referred

to as the subradiant states. For the first rung, the eigenenergies are given by

E1 = ωa − g√N, (9.2)

E2,...,N = ωa, (9.3)

EN+1 = ωa + g√N. (9.4)

The splitting between the polaritonic states E1 and EN+1 is a result of the collective coupling

with an effective coupling rate of GN = g√N . When the emitters are unequally coupled

(i.e. gn 6= gm), then the collective coupling rate can be calculated as

GN =

√√√√ N∑n=1

g2n. (9.5)

62

9.4 Strong-Coupling Cavity QED with an Ensemble of Color-Centers

N = 1 N = 2 N = 3

0

0

0

+𝑔 2�

−𝑔 2�

+𝑔 6�

−𝑔 6�

0

+𝑔−𝑔

+𝑔 2�

−𝑔 2�

0

0

0

+𝑔 3�

−𝑔 3�

+𝑔 10�

−𝑔 10�

+𝑔

−𝑔

Figure 9.3: Dressed ladder of states for N = 1, 2, 3 identical emitters coupled to a single cavity

mode.

However, the degeneracy between the other N −1 states occurs only for identical emitters

and is lifted with the introduction of any non-identical quantum emitters. Compared to

atoms, color-centers are non-identical quantum emitters, and therefore we need to expand

the original Hamiltonian to include a set of emitter frequencies that capture the inhomoge-

neous broadening in the ensemble, i.e. with

H ′TC = ωaa†a+

N∑n=1

[(ωa + ∆e,n)σ†nσn + gn

(σ†na+ a†σn

)]. (9.6)

Specifically, the inhomogeneous broadening is represented by the emitter detunings ∆e,n =

ωe,n − ωa. In the following sections, we will explore how the inhomogeneous broadening,

resulting in ∆e,n 6= ∆e,m, affects the ability to observe collective oscillations, subradiant

states, and quantum coherent phenomena.

9.4 Strong-Coupling Cavity QED with an Ensemble of

Color-Centers

In our cavity QED model of an ensemble of color-centers in a cavity, the following factors

are considered:

63


1. The loss of photons from the cavity to the environment. The loss is defined by

the quality factor of the mode Q = ωa/κ, and is often practically imposed by fabrication

limitations.

2. The loss of photons from the emitters to the environment. This mechanism is governed

by the ratio of color-center emission that is directed into the cavity mode and is affected

by the density of states. A high density of photonic states results in a low photon loss rate,

which can be encouraged by high quality factor and small mode volume cavities, as well as

by the good positioning of the emitters within the cavity.

3. The distribution of emission frequencies in the ensemble. This factor is usually influ-

enced by the local strain in the lattice and can be especially pronounced in nanoparticles

or heteroepitaxial layers, compared to bulk substrates.

4. The variable positioning of individual emitters in the cavity. Some control over the

depth of the color-centers in the substrates can be gained though selective doping during

substrate growth or by defining the ion energy during irradiation of the pre-grown sample.

Laterally, the use of a focused ion beam or masked apertures can provide a degree of

localization.

5. The ratio of emission into the zero-phonon line (ZPL). This is an intrinsic property

of each color-center, and it is the property that motivates the search for new systems with

high Debye-Waller factor.

We now discuss how these effects are incorporated into our quantum-optical model. The

cavity and emitter losses can be modeled through Liouville’s equation. Now, the super-

operator that describes the system dynamics has the form

LTCρ(t) = i[ρ(t), H ′TC

]+κ

2D[a]ρ(t) +

N∑n=1

Γ

2D[σn]ρ(t), (9.7)

where κ represents the cavity energy decay rate, and Γ represents the individual emitter

linewidth which is assumed to be constant within the ensemble.

The emission frequencies are sampled from a Gaussian distribution centered at ωa + ∆e

with standard deviation of δ2 , where δ is the inhomogeneous linewidth of the ensemble. It

is worth noting that the shape of individual color-center emission is a Lorentzian (homo-

geneous broadening), however, the distribution of central frequencies in the ensemble is a

Gaussian (inhomogeneous broadening).

The coupling strength is limited by the maximal value gmax imposed by the system

parameters: ZPL extraction ratio (Debye-Waller factor) ρZPL, emitter lifetime τ , index of

refraction n, cavity mode volume V and operating frequency ω [90]

gmax =

√3πc3ρZPL

2τω2n3V. (9.8)

64

9.4 Strong-Coupling Cavity QED with an Ensemble of Color-Centers

The coupling rate is reduced by imperfect positioning of emitters rn relative to the resonant

field maximum Emax and the angle between the dipole and field orientation φn

gn = gmax

∣∣∣∣E(rn)

Emaxcos(φn)

∣∣∣∣ . (9.9)

The distribution of the intensity in range [0, gmax] is also specific to the system, mainly to

the spatial positioning of emitters and the resonant field intensity distribution. Here, we

will sample the coupling gn from a uniform distribution.

We choose a realistic set of parameters

{κ/2π,Γ/2π, gmax/2π} = {25 GHz, 0.1 GHz, 10 GHz}

to model the dynamics of the multi-emitter cavity system. In order to numerically inves-

tigate the transmission spectra of the multi-emitter cavity system, we use the incoherent

cavity pumping technique. From these simulations, in Fig. 9.4a we plot the increasing po-

laritonic spacing with an increased number of identical and equally-coupled emitters. The

trend is consistent with the lossless case analysis where the collective coupling is given as

GN = g√N . As discussed previously, non-identicallity of emitters lifts the degeneracy of

the new states in the ladder and makes them visible in the spectrum. With increasing

emitter-emitter detuning, the subradiant states centered between the emitter frequencies

become more visible. This is illustrated with the emergence of an intermediate peak in the

transmission spectrum for N = 2 non-identical emitters in Fig. 9.4b.

Next, we analyze the influence of the inhomogeneous broadening to the transmission spec-

trum and present one of our most important findings, that the inhomogeneous broadening

will not obscure the observation of collective many-body physics in systems comprising

color-centers with small inhomogeneous broadening. First, we investigate this phenomenon

in Fig. 9.5, which shows randomly generated spectra with N = 4 emitters for variable

inhomogeneous broadening with

δ/2π ∈ {1 GHz, 10 GHz, 100 GHz}.

The first two sets of spectra show a small or a moderate perturbation to the system with

identical emitters, featuring two polariton peaks and several subradiant peaks, suggesting

that silicon-vacancy related complexes could be used in multi-emitter cavity QED systems.

In the case of large (100 GHz) inhomogeneous broadening, however, the spectrum is signifi-

cantly perturbed and the identification of the polariton peaks is difficult or impossible (Fig.

9.5c). Therefore, a multi-emitter systems with such broadening may not show the signs of

collective strong coupling.

To further analyze the character of the collective strong coupling, we generate 100 systems

(GN > κ/2) for variable inhomogeneous broadening with

δ/2π ∈ {1 GHz, 10 GHz, 20 GHz}

65


40 30 20 10 0 10 20 30 40(ω−ωa )/2π (GHz)

Tran

smis

sion

(arb

.)

N=1

N=2

N=3

N=4

N=5

(a)

40 30 20 10 0 10 20 30 40(ω−ωa )/2π (GHz)

Tran

smis

sion

(arb

.)

0 GHz

1 GHz

2 GHz

3 GHz

4 GHz

∆e,2/2π:(b)

Figure 9.4: (a) Transmission spectra of N identical and equally coupled emitters. (b) Transmission

spectra showing the emerging subradiant peak as the second of N = 2 emitters becomes

off-resonant.

and plot the spacing between the emitters against the expected collective coupling GN . We

use the same formula as in the case of identical emitters: GN =√∑N

n=1 g2n, so that we

benchmark the results with inhomogeneous broadening against the ideal multi-emitter cav-

ity systems. As seen in Fig. 9.6 the effective coupling rate is fit well by 2GN , implying that

the collective coupling rate of non-identical emitters is comparable to identical emitters for

small inhomogeneous broadening. Crucially, because the numerically investigated polariton

splittings closely match the GN for the cases of diamond SiV (δ ∼ 1 GHz) and SiC silicon

vacancy (δ ∼ 20 GHz) systems, we expect these emitters to exhibit multi-emitter cavity

QED phenomenon in spite of their inhomogeneous broadening.

9.5 Effective Hamiltonian Approach to Multi-Emitter Cavity

QED

The quantum master equation provides a full numerical treatment of the system dynamics

in transmission, however, the diagonalization of the density matrix makes this approach

computationally challenging. This practically limits the number of emitters in the system

to N . 5. The effective Hamiltonian approach introduced can provide a significant speedup.

For example, in the modeling of the transmission spectra, computation is effectively reduced

to the diagonalization of a (N + 1)× (N + 1) matrix, from the initial 2(N+1) × 2(N+1) size.

66

9.5 Effective Hamiltonian Approach to Multi-Emitter Cavity QED

40 20 0 20 40(ω−ωa )/2π (GHz)

Tran

smis

sion

(arb

.)

(a)

40 20 0 20 40(ω−ωa )/2π (GHz)

(b)

40 20 0 20 40(ω−ωa )/2π (GHz)

(c)

Figure 9.5: Randomly generated transmission spectra for N = 4, {κ/2π,Γ/2π, gmax/2π} =

{25GHz, 0.1GHz, 10GHz} and δ/2π = (a) 1 GHz, (b) 10 GHz, and (c) 100 GHz.

10 12 14 16 18 20GN /2π (GHz)

20

25

30

35

40

Pol

arito

n sp

acin

g (G

Hz)

(a)

10 12 14 16 18 20GN /2π (GHz)

20

25

30

35

40(b)

10 12 14 16 18 20GN /2π (GHz)

20

25

30

35

40(c)

Figure 9.6: Polariton spacing of randomly generated strongly coupled systems vs. expected collec-

tive coupling rate for N = 4, {κ/2π,Γ/2π, gmax/2π} = {25GHz, 0.1GHz, 10GHz} and

δ/2π = (a) 1 GHz, (b) 10 GHz and (c) 20 GHz. The red line corresponds to the system

with identical emitters.

67


The effective Hamiltonian for multi-emitter system is

HEFF = H ′TC − iκ

2a†a− i

N∑n=1

Γ

2σ†nσn. (9.10)

Its eigensystem can be represented as a set of eigenenergies and eigenvectors{EEFFn , ψEFF

n

}.

Bearing in mind that Re {EEFFn } gives the frequency of an energy level and 2 Im{EEFF

n } rep-

resent its linewidth, we can reconstruct some of the spectral information obtained through

the quantum master equation approach. We again generate 100 random systems for

δ/2π ∈ {1 GHz, 10 GHz, 20 GHz}

and plot the frequency difference between the two eigenstates with the highest linewidths

against the expected strong coupling rate GN (Fig. 9.7). Here, the eigenstates with the

highest linewidths represent the two states of strongest cavity emission since we operate

in the bad-cavity limit. The correlation between the simulated polariton splitting for non-

identical emitters to the one expected for identical emitters is excellent. The latter can

be analytically calculated by approximating the multi-emitter system as a single-emitter

system with effective coupling GN . The polariton splitting is expressed as

EN+1 − E1 = 2

√G2N −

(κ− Γ

4

)2

. (9.11)

While here we confirm that the non-identical multi-emitter systems behave similarly to

identical ones in the effective Hamiltonian approximation, additional steps are needed to

derive a close approximation to the transmission spectrum. To do this, we first introduce a

driving term to the effective Hamiltonian

Hdrive(ωL) = E(aeiωLt + a†e−iωLt), (9.12)

where ωL is the laser frequency and E is proportional to the laser field intensity. Transform-

ing the Hamiltonian into the rotating-wave frame and diagonalizing the effective Hamil-

tonian, we obtain the eigensystem {EEFFn (ωL), ψEFF

n (ωL)}. Now, we assume that each

energy level emits light with Lorentzian intensity distribution centered at Re {EEFFn (ω)}

and with linewidth of 2Im {EEFFn (ω)}. To combine the individual levels into a trans-

mission spectrum, we weight each of the eigenstates by their cavity occupation terms

〈ψEFFn (ωL)|a†a|ψEFF

n (ωL)〉. From there, we derive our first effective Hamiltonian approx-

imation to the spectrum SEFF(ω) as

SEFFI (ω) =

N∑n=1

〈ψEFFn (ω)|a†a|ψEFF

n (ω)〉L(ω; Re {EEFF

n (ω)}, Im {EEFFn (ω)}

), (9.13)

68


0 5 10 15 20GN /2π (GHz)

0

5

10

15

20

25

30

35

40

Pol

arito

n sp

acin

g (G

Hz)

(a)

0 5 10 15 20GN /2π (GHz)

0

5

10

15

20

25

30

35

40(b)

0 5 10 15 20GN /2π (GHz)

0

5

10

15

20

25

30

35

40(c)

Figure 9.7: Polariton spacing calculated using the effective Hamiltonian diagonalization

vs. expected collective coupling rate for N = 4, {κ/2π,Γ/2π, gmax/2π} =

{25GHz, 0.1GHz, 10GHz} and δ/2π = (a) 1 GHz, (b) 10 GHz and (c) 20 GHz. The red

line corresponds to the system with identical emitters.

where L(ω; ω0, η) represents Lorentzian distribution defined as

L(ω; ω0, η) =1

πη

[1 +

(ω−ω0η

)2] . (9.14)

In Fig. 9.8 we compare the spectra obtained through the effective Hamiltonian approach

to the ones calculated by the quantum master equation. The qualitative agreement is very

good, while the quantitative match between peak locations and intensities is close, but not

complete, because it does not capture the interference effects between the polaritons. To

improve on this, we derive another approximation which models the light field interference

more reliably.

Because we consider the transmission spectra when the emitters are all weakly excited,

they may be considered to primarily emit coherent radiation [91]. Thus, we can ignore their

incoherent portions of emission and use the classical limit

〈a†a〉 ≈ 〈a†〉〈a〉. (9.15)

Now, we can take the fields (∝ 〈a〉) rather than intensities (∝ 〈a†a〉) to combine the trans-

mission profiles in the spectrum, thereby incorporating the interference effects between

the different polaritons and subradiant states. Then, in our second effective Hamiltonian

69


60 40 20 0 20 40 60(ω−ωa )/2π (GHz)

Tran

smis

sion

(arb

.)

(a)SEFFI QME

60 40 20 0 20 40 60(ω−ωa )/2π (GHz)

(b)SEFFI QME

60 40 20 0 20 40 60(ω−ωa )/2π (GHz)

(c)SEFFI QME

Figure 9.8: A comparison between the transmission spectra calculated using quantum master equa-

tion (QME) and the first effective Hamiltonian approximation (SEFFI ) for a N = 4

multi-emitter system {κ/2π,Γ/2π, gmax/2π} = {25GHz, 0.1GHz, 10GHz} and emitter

frequencies sampled for δ/2π = (a) 1 GHz, (b) 10 GHz, and (c) 20 GHz.

approximation, the transmission spectrum is calculated as

SEFFII (ω) =

∣∣∣∣∣N∑n=1

〈ψEFFn (ω)|a|ψEFF

n (ω)〉√L (ω; Re {EEFF

n (ω)}, Im {EEFFn (ω)})

∣∣∣∣∣2

(9.16)

where now the terms inside the summation represent field contributions that are interfered

before the final modulus squared is taken to calculate the intensity. This result is formally

very similar to that obtained by diagonalizing a set of coupled Heisenberg-Langevin equa-

tions and using the approximation that the emitters are all in their ground states [92].

While these two methods arrive at similar answers, we believe the intuitive connection to

the complex eigenstates of the non-Hermitian effective Hamiltonian provides additional in-

sight. Figure 9.9 shows that this approximation provides a close fit to the quantum master

equation results.

With these approximations at hand, we can conclude that the effective Hamiltonian ap-

proach can provide an excellent insight into transmission properties of a cavity QED system.

This is especially valuable for systems with a large number of emitters where the solving

of the quantum master equation would require unrealistic computational resources. We

analyze one such system based on N = 100 emitters for δ/2π ∈ {10 GHz, 40 GHz, 80 GHz}.Figure 9.10 shows the extension of the cavity protection effects to systems with larger inho-

mogeneous broadening, granted by the increase in the collective coupling amounting from

70


60 40 20 0 20 40 60(ω−ωa )/2π (GHz)

Tran

smis

sion

(arb

.)

(a)SEFFII QME

60 40 20 0 20 40 60(ω−ωa )/2π (GHz)

(b)SEFFII QME

60 40 20 0 20 40 60(ω−ωa )/2π (GHz)

(c)SEFFII QME

Figure 9.9: A comparison between the transmission spectra calculated using quantum master equa-

tion (QME) and the second effective Hamiltonian approximation (SEFFII ) for a N = 4

multi-emitter system {κ/2π,Γ/2π, gmax/2π} = {25GHz, 0.1GHz, 10GHz} and emitter

frequencies sampled for δ/2π = (a) 1 GHz, (b) 10 GHz, and (c) 20 GHz.

160 80 0 80 160(ω−ωa )/2π (GHz)

Tran

smis

sion

(arb

.)

(a)

160 80 0 80 160(ω−ωa )/2π (GHz)

(b)

160 80 0 80 160(ω−ωa )/2π (GHz)

(c)

Figure 9.10: Transmission spectra calculated by the second effective Hamiltonian approximation for

a N = 100 multi-emitter system {κ/2π,Γ/2π, gmax/2π} = {25GHz, 0.1GHz, 10GHz}and emitter frequencies sampled for δ/2π = (a) 10 GHz, (b) 40 GHz, and (c) 80 GHz.

71


an increased number of emitters in the system. Despite the large inhomogeneous broad-

ening, collective oscillations as the highest-energy and lowest-energy states are still readily

apparent. These collective excitations manifest as two polaritons, each with approximately

half the cavity linewidth, that are separated by 2GN .

In the next section, we will analyze new opportunities for nonclassical light generation in

multi-emitter cavity QED systems.

72

10 Nonclassical Light Generation in Multi

Emitter Cavity QED Platform

Cavity quantum electrodynamics (cavity QED) studies phenomena, such as photon block-

ade and photon-induced tunneling, that produce nonclassical light in the form of n-photon

emission. Applications of n-photon bundles are wide ranging and include quantum key dis-

tribution [93], quantum metrology [94] and quantum computation [95]. While cavity QED

systems have been extensively studied in atomic platforms [96, 97], a significant advance

in terms of operating rates and on-chip integration took place with their subsequent im-

plementation in the solid-state. Notably, the InAs/GaAs quantum dot platform has been

used to develop devices with GHz speeds, such as the single-photon source [98, 99] and the

low-power all-optical switch [3, 100, 101]. As discussed in the previous chapter, the ability

to use multiple (N > 1) nearly-identical emitters, as opposed to one, would further increase

the interaction rate by a factor of√N via the effects of collective coupling [86, 87]. This

would in turn provide higher speed and quality of on-chip cavity QED devices. However,

the large inhomogeneous broadening among quantum dots is prohibitive of this advance.

Recently, a solid-state cavity QED platform with smaller inhomogeneous broadening—

rare-earth ions in a nanocavity—achieved the strong multi-emitter cavity coupling using

a large (N ∼ 106) ensemble of emitters [85]. In this limit, however, the rungs of the

energy ladder approximate a continuum of modes, and are therefore not suitable for non-

classical light generation. In contrast, coupling several (N & 1) emitters to a cavity would

preserve the desired discrete energy character, but would also require emitters to have larger

dipole moments. Such values have been seen among color centers in diamond and silicon

carbide incorporated into emitter-cavity platforms [102, 65, 103, 44]. With inhomogeneous

broadening often less than 30 GHz [14, 104] these quasi-atoms could meet the requirements

for multi-emitter strong cavity QED coupling if hosted in high quality factor and small

mode volume cavities. This regime has been studied for superconducting cavities [84] and

is characterized by the collective emitter coupling rate GN =√∑N

n=1 g2n that is dominant

over the cavity (κ) and emitter (γ) linewidth-induced loss mechanisms: GN > κ/4, γ/4.

Currently, little is known about coherence properties of such systems and their potential

for nonclassical light generation.

73

10 Nonclassical Light Generation in Multi Emitter Cavity QED Platform

The nonclassical light generation in cavity QED systems finds its origin in the discretized

and anharmonic character of the energy states. The differences in the dressed ladder of

states for an increasing number of emitters are presented in Figure 9.3 and compared to

the Jaynes–Cummings ladder. These more complicated level structures for multi-emitter

systems bring in new opportunities for n-photon emission [105].

10.1 Resonant Photon Blockade

We consider photon blockade under continuous-wave (cw) excitation. Emerging technologies

for photodetection such as superconducting single-photon detectors are beginning to provide

the timing resolution required for the study of cw photon correlations. Because these

correlations do not involve averaging over an entire pulse, they can act as a more sensitive

probe of the underlying system dynamics.

For the cw case, the second-order coherence g(2)(0) is defined as

g(2)(0) =〈a†a†aa〉〈a†a〉2

, (10.1)

and it is related to the statistics of simultaneously emitting two (or more) photons [106].

When the cw statistic g(2)(0) is less than one (ideally zero), the light has a sub-Poissonian

nonclassical character that indicates its potential for single-photon emission. However,

to confirm the single-photon character of the emitted light in the cw case, higher-order

coherences need also to be taken into account [107].

Now, we use this tool to study transmission through the collective polaritonic excitations

of multi-emitter cavity QED systems. Considering the transmission through polaritonic

states, we naively expect that the photon blockade effect should become even more pro-

nounced with a√N increase in coupling strength. However, this seems to be the case only

while GN < κ; we attribute this somewhat unexpected finding to the addition of energy

states in the second-rung (see Fig. 9.3) with an increasing number of emitters. While

a detailed investigation is needed for full understanding of this phenomenon, we suggest

that the photon blockade effects are greatly influenced by the dipolar coupling mechanisms

between the two energy rungs. We present these effects on a system of identical emitters,

resonantly and equally coupled to the cavity. Figure 10.1 compares second-order coherence

g(2)(0) values with an increasing N for three systems whose parameters capture the change

in the trend around GN ≈ κ. As with the single-emitter photon blockade, the points of

best single-photon emission are located around the frequencies of the polariton transmis-

sion peaks. When GN < κ, the photon blockade improves with the increasing N , while

for GN > κ the behavior reverses. Hence, we have identified a potential fundamental limit

in multi-emitter photon blockade under resonant conditions, in that high quality factor

74

10.2 Detuned Photon Blockade in the Multi-Emitter System

40 20 0 20 40(ω−ωa )/2π (GHz)

0.2

0.4

0.6

0.8

1.0

1.2

1.4

g(2)

(0)

(a)N=1

N=2

N=3

Laser

40 20 0 20 40(ω−ωa )/2π (GHz)

0.2

0.4

0.6

0.8

1.0

1.2

1.4(b)N=1

N=2

N=3

Laser

40 20 0 20 40(ω−ωa )/2π (GHz)

0.2

0.4

0.6

0.8

1.0

1.2

1.4(c)N=1

N=2

N=3

Laser

Figure 10.1: Photon blockade trends with an increasing N for parameters {∆e,n/2π,Γ/2π, gn/2π} =

{0GHz, 0.1GHz, 10GHz} and κ/2π = (a) 25 GHz (GN < κ), (b) 15 GHz (GN ∼ κ), and

(c) 8 GHz (GN > κ). Dashed lines illustrate the transmission spectra.

cavities may not necessarily yield the best photon blockade in a multi-emitter cavity QED

system. Notably, this limit is relaxed for detuned photon blockade.

Next, we analyze the effect of unequal coupling on the photon blockade, and compare

the second-order coherence values for systems with variable coupling of three emitters (Fig.

10.2). Starting with a system of identically coupled emitters, we gradually decrease coupling

rates of the second and the third emitter, which corresponds to the decrease of the collective

coupling rate GN . We observe that the quality of the photon blockade decreases [g(2)min(0)

increases] with the lowering of the collective coupling rate.


As in the single emitter case [98], we expect an enhanced photon blockade for a system where

multiple emitters are detuned from the cavity as a result of increased ladder anharmonicity

which helps overcome the system losses. Here, we focus on a two-emitter system and

analyze the minima of g(2)(0) as a function of the emitters’ detuning from the cavity. In

Figure 10.3 we see that the photon blockade improves as the emitters detune from the

cavity. For unequally coupled emitters, the trend becomes less pronounced for the lowered

GN =√g2

1 + g22. Peculiarly, we also see an effect that additionally lowers g

(2)min(0) and

occurs when both emitters are highly detuned from the cavity and a little detuned from

75


40 20 0 20 40(ω−ωa )/2π (GHz)

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

g(2)

(0)

{10,10,10

} GHz{

10,10,8}

GHz{10,8,8

} GHz

{10,8,6

} GHz

Laser

Figure 10.2: Photon blockade trends for N = 3, {κ/2π,Γ/2π,∆e,n/2π} = {25GHz, 0.1GHz, 0GHz}and the variable coupling parameters {g1/2π, g2/2π, g3/2π} noted in the legend.

76


30

20

10

0

10

20

30

∆ω

e,1 (

GH

z)

g2 =g1 g2 =0.9g1

0 0.5 1

g(2)

min(0)

30 20 10 0 10 20 30∆ωe,2 (GHz)

30

20

10

0

10

20

30

∆ω

e,1 (

GH

z)

g2 =0.8g1

30 20 10 0 10 20 30∆ωe,2 (GHz)

g2 =0.7g1

Figure 10.3: Zero-time second-order coherence minima for a detuned system with N = 2 emitters

and {κ/2π,Γ/2π, g1/2π} = {25GHz, 0.1GHz, 10GHz}; g2 is quoted in each panel.

77


one another. This behavior becomes even more pronounced for unequally coupled emitters

providing a lower second-order coherence value, and shifting the g(2)min(0) further away with

the emitters’ detuning from the cavity. The effect is asymmetric and favors systems where

the most strongly coupled emitter is also the highest detuned one.

40 30 20 10 0 10 20 30 40(ω−ωa )/2π (GHz)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

g(2)

(0)

(a)

Laser{0,0}

GHz{10,30

} GHz{

26,20}

GHz

(b)

Figure 10.4: (a) Zero-time second-order coherence for N = 2, {κ/2π,Γ/2π, g1,2/2π} =

{25GHz, 0.1GHz, 10GHz} and the different emitter detunings {∆e,1/2π,∆e,2/2π} noted

in the legend; plotted with transmission spectra (dashed lines). (b) Dressed ladder

of states calculated with our effective Hamiltonian approach for {∆e,1/2π,∆e,2/2π} =

{26GHz, 20GHz}, plotted with their linewidths; black vertical arrows illustrate first and

second excitation at the point of the best photon blockade. Energy spacing between

the rungs is significantly reduced for clarity.

Let us look more closely into what causes this additional second-order coherence reduc-

tion. Figure 10.4a shows the g(2)(0) dependence on the transmission wavelength. We see

that the minima of the function occur around the frequencies of the transmission peaks.

When the emitters are identical there are only two local minima corresponding to transmis-

sion near the polariton peaks. For the non-identical emitters the second-order coherence

has three local minima, with the middle one corresponding to the frequency of the emerging

peak. For large detuning, transmission through this intermediate peak gains an advanta-

geous second-order coherence value, which corresponds exactly to the peculiar minima in

g(2)(0) seen in Figure 10.3. The origin of this effect has been traced back to the specifics of

the dressed ladder of states [105]. Time evolutions of the second-order coherence function

[for details see [105]] show that the dynamics of the single-photon emission from either of

the photon blockade frequencies evolves at the 100 ps scale, which represents a significant

78

10.3 Effective Hamiltonian Approach to n-Photon Generation

speedup in the single-photon emission relative to the expected 10 ns lifetime of individual

color centers. Additionally, the coupling to cavity increases the collection efficiency.

10.3 Effective Hamiltonian Approach to n-Photon

Generation

Here, we consider theoretically the coherence effects in a N = 2 multi-emitter cavity QED

system [Fig. 10.5(a)] and analyze the influence of inhomogeneous broadening to n-photon

generation. Using the quantum master equation in an extended Tavis-Cummings model,

we discover a new interference effect resulting in high quality single-photon generation that

is robust to emitter property variations. We explain the origin of this phenomenon via the

effective Hamiltonian approach.

Figure 10.5: (a) An illustration of the multi-emitter cavity QED system with two nonidentical emit-

ters coupled to a common cavity mode. (b) The ladder of dressed states, frequencies and

unnormalized eigenvectors, for the case of two identical and equally coupled emitters,

∆C = ∆E = 0 and g1 = g2 = g.

We rewrite the interaction Hamiltonian into the form with explicit detunings between

the emitters and the cavity

HI = ωCa†a+ (ωC + ∆C)σ†1σ1 + g1(σ†1a+ a†σ1) + (ωC + ∆C + ∆E)σ†2σ2 + g2(σ†2a+ a†σ2),

where a and ωC represent the annihilation operator and resonant frequency of the cavity

mode; σn and gn are the lowering operator and cavity coupling strength of the n-th out of

N = 2 emitters; ∆C is the first emitter detuning from the cavity and ∆E is the detuning

between the two emitters.

The system is characterized by the cavity energy decay and emitter linewidth {κ, γ}/2π =

{25, 0.1} GHz, corresponding to quality factor Q ≈ 15, 000, emitter lifetime τ = 10 ns, and

79


the individual emitter-cavity coupling rate gn/2π = 10 GHz. Comparing to the state-of-the-

art results with silicon-vacancy centers in a diamond photonic crystal cavity [44], the quoted

values could be achieved by designing nanocavities with doubled quality factor and five-fold

increased coupling rate, which can be achieved by reducing the mode volume several times

and providing higher-precision positioning of the color centers at the field maximum [108].

The three-fold mismatch in emitter linewidth, which is the smallest rate in the system, is

tolerable in the bad cavity regime. The collective coupling of two emitters in the system

gives rise to the strong coupling regime (√g2

1 + g22 > κ/4).

0 0.5 1

(a) Emission

0 1 2

(b) g(2) (0)

0 1 2

(c) g(3) (0) (d) Heff rungs

I II III

20100

102030405060

ω/2π [

GH

z]

(i)

20100

102030405060

ω/2π [

GH

z]

(ii)

0 5 10 15 20∆E /2π [GHz]

20100

102030405060

ω/2π [

GH

z]

(iii)

0 5 10 15 20∆E /2π [GHz]

0 5 10 15 20∆E /2π [GHz]

0 5 10 15 20∆E /2π [GHz]

Figure 10.6: (a) Emission spectrum, (b) second- and (c) third-order coherence of transmitted light

calculated by the quantum master equation. (d) Frequency overlap between the k-th

order rungs of the dressed ladder of states of Heff, presented as ωk/k−ωC with indicated

linewidths; ∆C/2π takes values of (i) 0, (ii) 20 and (iii) 30 GHz.

We gain a more intuitive understanding of the system dynamics by diagonalizing the

effective Hamiltonian, the real and the imaginary part of its eigenvalues represent frequencies

and half-linewidths of the excited energy states, respectively, while the eigenvectors quantify

the cavity-like and the emitter-like character of the excited state. Figure 10.5(b) shows the

80


eigenfrequencies and eigenvectors in the ladder of dressed states for an identical and equally

coupled system. The eigenvector elements are labeled with c for cavity, e1 and e2 for the

first and the second emitter excitation, respectively. In contrast to cavity QED systems with

a single emitter where all excited states contain two levels, here we see that an additional

emitter generates new excited states. The new middle state in the first rung resembles the

wavefunction of a subradiant state (e2− e1)/√

2, known from atomic systems not to couple

to the environment. There are also two new middle states in the second rung, which are

degenerate for this set of detunings. These additional levels ultimately lead to a much richer

set of physics phenomena. Not only do we find an enhanced regime of photon blockade from

the subradiant states, but a newly discovered un-conventional photon blockade regime [107]

can also be achieved in the system. We now explore these effects as a function of the emitter

detuning ∆E .

We present calculation results for cavity detunings ∆C/2π = 0, 20 and 30 GHz which

capture system’s trends and features. The emission spectra are shown in Fig. 10.6(a). In

parallel, we calculate the eigenstates of the first rung of the effective Hamiltonian, shown

as red surfaces in Fig. 10.6(d). The three transmission peaks are in a close agreement

with these eigenfrequencies, which expectedly indicates that the bottom peak is cavity-like,

and the top two peaks emitter-like. To more closely understand the effects that non-

identical emitters bring into cavity QED, we now focus on the emerging subradiant state

for ∆E/2π ≤ 3 GHz. Qualitatively similar spectra have been calculated for superconducting

cavities coupled to an ensemble of spins [109] and experimentally observed in superconduct-

ing circuits [88]. The collective strong coupling rate G2 =√g2

1 + g22 places the two polariton

peak frequencies close to ω∆E=0pol± = ωC + ∆C

2 − iκ+γ

4 ±√

4∆2C+16G2

2+2κγ+4i∆Cκ−4i∆Cγ+iκ2+iγ2

4 .

Here, we describe the subradiant state in more detail by deriving approximations to its

frequency ωsub and state vector vsub for ∆E � G2,κ−γ

2 :

ωsub = ωC + ∆C +g2

1

G22

∆E − i(γ

2+κ− γ8G2

2

∆2E

),

vsub =1√A

−g2∆E

G22

+ iκ−γ8g2

∆2E

G22

−g1g2

8g22+i(κ−γ)∆E

8g21−i(κ−γ)∆E

1

,where A is a normalization factor. For vanishing ∆E the cavity term in vsub becomes zero,

diminishing state’s coupling to the environment. With an increasing ∆E , the frequency of

the subradiant state grows linearly, while the linewidth and the amplitude increase quadrat-

ically, closely matching the trend in the simulated emission spectra for ∆E/2π ≤ 3 GHz.

Next, we quantify the system’s potential for n-photon generation by analyzing the second-

and third-order coherence of light transmitted through the system [Figs. 10.6(b-c)]. Areas

81


0 5 10 15 20∆E /2π [GHz]

0.0

0.2

0.4

0.6

0.8

1.0Sta

te c

hara

cter (a)

|⟨ψ I

1 |c⟩|2

|⟨ψ I

2 |e1

⟩|2

|⟨ψ I

3 |e2

⟩|2

0 5 10 15 20∆E /2π [GHz]

(b)

|⟨ψ II

1 |cc⟩|2

|⟨ψ II

2 |ce1

⟩|2

|⟨ψ II

3 |ce2

⟩|2

|⟨ψ II

4 |e1 e2

⟩|2

Figure 10.7: Prevalent character of the eigenstates of (a) the first and (b) the second rung of Heff

for ∆C = 20 GHz.

of suppressed g(2)(0) statistics (red areas) indicate the potential for single-photon emis-

sion, which is further supported if the g(3)(0) values are simultaneously reduced. We find

such areas close to the frequencies of the emission peaks, especially for higher detuned

emitter-like peaks. This is consistent with the findings in single-emitter cavity QED sys-

tems where the photon blockade at polariton peak frequency improves with emitter-cavity

detuning [98]. However, there is no direct analog for the transmission through the newly

formed subradiant state. We reveal the origin of both effects by analyzing the frequency

overlap Ek/k − ωC between eigenstates of different order (k) rungs of Heff [Fig. 10.6(d)].

We find that the two photon emission is suppressed for the frequencies where single pho-

ton absorption is possible (red areas), but the second photon would not have a spectrally

close state to be absorbed at (no overlapping blue areas). Surprisingly, for large ∆E this

condition disappears for the subradiant peak, but the photon blockade still persists. To

understand this apparent contradiction, we perform an additional analysis of the eigenstate

character for ∆C/2π = 20 GHz, presented in Fig. 10.7. We find that for an increasing

∆E the eigenstates ψI1 , ψI2 , ψ

I3 , ψ

II1 , ψII2 , ψII3 and ψII4 (see Fig. 10.5b), prevalently exhibit

c, e1, e2, cc, ce1, ce2, and e1e2 character, respectively. Therefore, the dipolar coupling be-

tween ψI2 and ψII3 has to be inhibited due to⟨ce2|a†|e1

⟩= 0, which in turn suppresses the

second photon absorption at the higher detunings. With this combination of spectral and

vector component properties obtained from the diagonalization of the effective Hamiltonian

we predict parameters that give rise to enhanced single photon emission. The properties

also hold for nonidentically coupled emitters (g1 6= g2, G2 > κ/4), which has favorable

practical implications for systems with randomly positioned color centers whose coupling

strength can not be imposed uniformly.

In addition to these trends, we also find conditions for realization of a superbunching effect

82


recently dubbed unconventional photon blockade [107], which confirms that the reduction

in two-photon statistics is not a sufficient condition for single-photon emission. This effect

occurs in the system with ∆C/2π = 30 GHz in the region around ω/2π = 25 GHz which

has a low g(2)(0) but high g(3)(0) value, indicating preferential three-photon emission. To

understand the occurrence of this regime in our system, we look into the third rung of the

dressed ladder shown in yellow at Fig. 10.6(d-iii) and reveal that this frequency region

has an overlapping three-photon absorbing state, but no two-photon absorbing state, which

explains the calculated statistics. Thus the multi-emitter cavity QED system will not just

advance the single-photon blockade, but also allow for the exploration of exciting regimes

of multi-photon physics and statistics.

10 0 10 20 30 40 50(ω−ωC )/2π [GHz]

0

0.5

1

1.5

2(a)

g(2) (0)g(3) (0)Emission

0.0 0.5 1.0 1.5 2.0 2.5τ [ns]

0.0

0.5

1.0

1.5

2.0

g(2)

(τ)

(b) 26.4 GHz31.7 GHz39.3 GHz

Figure 10.8: (a) Zero-delay second- and third-order coherences for ∆C ,∆E/2π = 30, 5 GHz system

as a function of laser detuning from the cavity; emission spectrum in dashed lines illus-

trates the relationship between features. (b) g(2)(τ) at frequencies of the corresponding

g(2)(0) minima from (a).

Finally, we analyze the system dynamics in the time domain in terms of interferences

between the excited states. We look into the three g(2)(0) dips for {∆C ,∆E}/2π = {30, 5}GHz [Fig. 10.8(a)]. The second-order coherence evolution at the frequency of the un-

conventional photon blockade [blue line in Fig. 10.8(b)] represents a damped oscillation

and once again confirms that the system can not be a good single photon source at these

frequencies. The dominant oscillation frequency (ω/2π) is 5.2 GHz, while the first 200 ps

also exhibit additional 11 GHz oscillation. The other two second-order coherence traces are

at frequencies of conventional photon blockade, and represent a decay to a coherent state.

The characteristic half-width at half-maximum times are 0.7 ns and 0.2 ns for the transmis-

sion through states with e1 (yellow) and e2 (green) characters, respectively, and represent

a speedup in single photon emission compared to a bare quasi-emitter. Both functions ex-

hibit 44 GHz oscillations with small amplitudes in the first 200 ps, which corresponds to

83


40 30 20 10 0 10 20 30 40(ω−ωC )/2π (GHz)

0.2

0.4

0.6

0.8

1.0

1.2g(

2)(0

)

N=2 :{ωE1, g

},{ωE2, g

}N=1 :

{ωE1, g

}N=1 :

{ωE2, g

}N=1 :

{ωE1, g

√2}

N=1 :{ωE2, g

√2}

Figure 10.9: Comparison between second-order coherence in N = 2 and N = 1 emitter-cavity

systems; emission spectrum in dashed lines illustrates the relationship between fea-

tures. Each emitter is characterized by frequency and coupling rate, {ωE1, ωE2, g}/2π =

{20, 26, 10} GHz.

the oscillation between the polaritonic eigenstates of the first rung of the dressed ladder

and is analogous to the experimentally observed oscillations in single atom-cavity systems

[110]. The function plotted in green also oscillates at 7.7 GHz at longer times, whose origin

we assign to the interference between the upper polaritonic and the subradiant states. We

observe this trend for other sets of parameters as well.

To illustrate the advantages of multi-emitter over single-emitter cavity quantum electro-

dynamics we plot the second-order coherence as a function of laser detuning for comparable

systems (Fig. 10.9). First, we notice that only the two-emitter system (black line) exhibits

the intermediate dip characterized in this paper. Next, its g(2)(0) value is lower than the

one of the individually coupled emitters (magenta and red), and even of the individual emit-

ters coupled with a collective rate for a two-emitter system g√

2 (green and blue). From a

practical point of view, in addition to providing lower g(2)(0) value, transmission through

the subradiant state extends an opportunity for a more robust single photon generation.

The frequency and the second-order coherence values of light transmitted through the state

ψI2 are close to constant for variable emitter detuning. Practically, this implies that for

any pair of nonidentical emitters (3 GHz ≤ ∆E/2π ≤ 20 GHz) the quality of single photon

emission is dictated by the cavity detuning from the first emitter. Experimentally, the cav-

84

10.4 Summary

ity detuning can be controlled by gas tuning techniques without influencing the operating

laser frequency [44]. Finally, the cross-polarized reflectivity method to addressing CQED

systems [111] can be applied to block the pump laser at the output channel.

10.4 Summary

Multi-emitter cavity QED at the level of several emitters is close to its first demonstrations in

a solid-state platform. Recent developments in low-strain material growth, nanofabrication

in bulk substrates, and controlled implantation of color centers are setting the path for

achieving high quality factor, small mode volume cavities with preferentially positioned

nearly-identical quasi-emitters.

We have set forth an extended Tavis–Cummings model that captures the core dynamics

of a multi-emitter cavity QED system. The initial theoretical findings on improved photon

blockade and photon bundle generation are expected to be expanded and unveil rich physics,

even with the inhomogeneous broadening of emerging color centers. Robust regimes of

operation have been predicted, with promising applications in reliable quantum optical

networks. Experimental data will set this theory to a test, and based on the experience

with the single emitter systems, we envision additional inclusion of factors in the quantum

model explicitly, such as dephasing and phonon interaction.

The systems with lossy cavities will require a higher number of emitters to reach the

strong coupling regime. Our new effective Hamiltonian approach to estimating transmission

spectra could provide insights into the dynamics of such systems by finding the best fit to the

transmission spectra, identifying individual emitter frequencies, and establishing regimes of

operation. From there, the same framework can be used to evaluate frequency overlap

between the rungs of excited states and target operating frequencies for nonclassical light

generation.

85

11 Outlook

The findings of this thesis – the development of silicon carbide nanofabrication, the demon-

stration of telecom and visible wavelengths microcavities in this material, the coupling of

the microcavities to color center emission in silicon carbide and diamond, the demonstration

of a scalable array of single photon sources and spin-qubits in silicon carbide, and modeling

of multi-emitter cavity quantum electrodynamics with an ensemble of non-identical emitters

– open up new directions in quantum photonics research.

11.1 Biophotonics

In addition to its excellent optical properties, silicon carbide also exhibit low cytotoxicity

[80] which makes it a compatible substrate to interface cells. Silicon carbide is chemically

inert to acids, alkalis and salts, and does not expand in liquids. Its bioapplications include

skin and connectivity tissue, cardiovascular system, central nervous system, and brain-

machine interfaces. The novel fabrication techniques developed in this thesis can be used

to bring existing devices to the nano-scale or to provide novel functionalities using photonic

mechanisms, such as bio-imaging signal enhancement though the Purcell enhancement of

bio-dye emission in vicinity of a nanoresonator [112], or in-cell drug delivery though the

functionalization of nano-needles [113].

Figure 11.1: SEM images of HeLa cells grown on 4H-SiC substrate patterned with array of

nanopillars.

In collaboration with Moerner group at Stanford University, we have tested viability of

seeding cells on nanofabricated SiC devices. Figure 11.1 shows a colony of HeLa cells that

87

11 Outlook

was planted on a 4H-SiC substrate patterned with an array of nanopillars. The images

confirm that cells grew equally on patterned and non-patterned areas, and their bending

is visible over the nanopillars. Figure 11.2 shows a colony of HeLa cells [114] seeded on a

substrate with an array of 3C-SiC microdisks with silicon pedestals. Here, as well, the cells

grew across the sample, however, a closer look at the microdisk array shows that the cells

grew on top of the disks without touching down on the silicon.

Figure 11.2: SEM images of HeLa cells seeded on an array of 3C-SiC on Si pedestals.

11.2 Nonlinear Optics

Similar to the well known nonlinear material GaAs, cubic polytype of silicon carbide (3C-

SiC) exhibits second order optical nonlinearity (χ(2) > 0) and has the zincblende lattice.

Microdisks than in GaAs have been used for second harmonic generation between infrared

wavelengths [115] by utilizing quasi-phasematching effect granted by the lattice symmetry.

88

11.3 Spin Physics

In zincblende materials, 90o rotation around a cubic axis corresponds to inversion of the

lattice 11.3a. The difference in the phase accumulated by a generated second harmonic wave

and the nonlinear polarization generating it has a discrete sign flip from one quadrant to

the next. If the radius of the path is suitable, this sign flip can be used to quasi-phasematch

the interaction [115]. This mechanism can be applied in 3C-SiC by utilizing microdisk

resonators developed in this thesis 11.3b. Because the bandgap in 3C-SiC is greater in

GaAs, this would allow extending the nonlinear light conversion to visible wavelengths, for

example from 1550 nm to 750 nm. These wavelengths in particular would be of interest for

converting light from telecommunications wavelengths (e.g. traveling through a fiber) to

the wavelength of diamond color centers studied in our hybrid silicon carbide-nanodiamond

resonators.

Figure 11.3: (a) Two views of a zincblende lattice that are simultaneously a mirror image and 90o ro-

tation relative to one another (adjusted from [116]). (b) Quasi-phasematching domains

in a 3C-SiC microdisk for the second harmonic generation process ω2 = 2ω1 illustrated

inset.

11.3 Spin Physics

Electron spins in color centers in diamond and silicon carbide have been used as nanoscale

biophysics magnetometers with unprecedented spatial resolution [41, 51, 76, 77, 81]. The

state of the art detection sensitivity, based on the efficient collection of photons emitted from

the color center, is an order of magnitude away from the fundamental limit of the quantum

projection noise [117]. A promising way of bridging this gap is by modifying emission rates

of the color centers via cavity coupling to microresonators studied in this thesis.

Finally, optical polarization and readout of S = 1 electron spin in diamond nitrogen

vacancy center have contributed to the studies of entanglement and depolarization dynamics

[118]. Color centers in silicon carbide, such as S = 3/2 silicon vacancy center in 4H-SiC

studied in this thesis, provide novel opportunities for studying coherence properties and

collective spin effects. Combining isotopic purification and substrate irradiation techniques

89

11 Outlook

the density of electron spins and their nuclear environment can be controlled for many body

solid-state studies of spin with an optical interface.

90

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