ABSTRACT
Title of dissertation: Chiral Quantum Optics
using Topological Photonics
Sabyasachi Barik
Doctor of Philosophy, 2020
Dissertation directed by: Professor Edo Waks and Professor Mohammad Hafezi
Department of Physics
Topological photonics has opened new avenues to designing photonic devices
along with opening a plethora of applications. Recently, even though there have
been many interesting studies in topological photonics in the classical domain, the
quantum regime has remained largely unexplored. In this thesis, I will demonstrate
a recently developed topological photonic crystal structure for interfacing a single
quantum dot spin with a photon to realize light-matter interaction with topolog-
ical photonic states. Developed on a thin slab of Gallium Arsenide(GaAs) mem-
brane with electron beam lithography, such a device supports two robust counter-
propagating edge states at the boundary of two distinct topological photonic crystals
at near-IR wavelength. I will show the chiral coupling of circularly polarized lights
emitted from a single Indium Arsenide(InAs) quantum dot under a strong magnetic
field into these topological edge modes. Owing to the topological nature of these
guided modes, I will demonstrate this photon routing to be robust against sharp
corners along the waveguide. Additionally, taking it further into the cavity-QED
regime, we will build a topological photonic crystal resonator. This new type of
resonator will be based on valley-Hall topological physics and sustain two counter-
propagating resonator modes. Thanks to the robustness of the topological edge
modes to sharp bends, the newly formed resonators can take various shapes, the
simplest one being a triangular optical resonator. We will study the chiral coupling
of such resonator modes with a single quantum dot emission. Moreover, we will
show an intensity enhancement of a single dot emission when it resonantly couples
with a cavity mode. This new topological photonic crystal platform paves paths for
fault-tolerant complex photonic circuits, secure quantum computation, and explor-
ing unconventional quantum states of light and chiral spin networks.
CHIRAL QUANTUM OPTICS USING TOPOLOGICAL
PHOTONICS
by
Sabyasachi Barik
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2020
Advisory Committee:
Professor Edo Waks, Chair/Advisor
Professor Mohammad Hafezi, Co-Advisor
Professor Thomas Murphy
Professor Trey Porto
Professor Rajarshi Roy
Professor John Cumings
?c Copyright by
Sabyasachi Barik
2020
Dedicated to my loved ones
ii
Acknowledgments
I owe my gratitude to all the people who have made this thesis possible and
who have made my graduate experience one that I will cherish forever.
First and foremost, I?d like to thank my advisor, Professor Edo Waks, for giving
me an invaluable opportunity to work on challenging and extremely interesting
projects over the past six years. He has always made himself available for help and
advice, and there has never been an occasion when I?ve knocked on his door and he
hasn?t given me time. It has been a pleasure to work with and learn from such an
extraordinary individual.
I would also like to thank my co-advisor, Professor Mohammad Hafezi. With-
out his extraordinary theoretical ideas and computational expertise, this thesis
would have been a distant dream. Thanks are due to Professor Thomas Murphy,
Professor Trey Porto and Professor Rajarshi Roy for agreeing to serve on my thesis
committee and for sparing their invaluable time reviewing the manuscript. A special
thanks goes to Professor John Cumings for agreeing to be the dean?s representative
and helping me out tremendously at short notice.
I am indebted to Professor Thomas Murphy for all the insightful discussions.
His clear perception, intellectual contributions and expertise in scientific writing and
presentation have been immensely valuable.
My colleagues at the Quantum Nanophotonics group have enriched my grad-
uate life in many ways and deserve a special mention. I would like to thank Dr.
Hirokazu Miyake for being an excellent mentor in my first two years as a PhD stu-
iii
dent. I learned so many experimental techniques by shadowing him in the lab and
discussing with him. In particular, I would like to thank Dr. Tao Cai and Dr.
Shuo Sun for sharing with me all of their hands-on experiences ranging from optical
alignment to nano-fabrications. They helped me start off by guiding me through
the lab and getting started on the compact optical setups along with help regarding
simulations. I would also express my gratitude to Dr. Hamidreza Chalabi for being
an excellent mentor and collaborator during my PhD research. I learned a lot of
simulation techniques and honed my computational skills while working with him.
Thanks also go to my collaborators including Professor Glenn Solomon, Pro-
fessor Thomas Murphy and Dr. Wade DeGottardi. None of my research work could
be carried out without their sincere support. My interactions with Dr. Shahriar
Aghaeimeibodi, Subhojit Dutta, Aziz Karasahin, Zhouchen Luo, Mustafa Atabey
Bu?yu?kkaya, Chris Flower, Uday Saha and Harjot Singh have always been very fruit-
ful. I would like to acknowledge my past and present lab mates for their support:
Dr. Hyochul Kim, Dr. Chad Ropp, Dr. Shilpi Gupta, Dr. Kangmook Lim, Dr.
Kaushik Roy Choudhury, Dr. Je-hyung Kim, Dr. Jose Algarin, Dr. Zhili Yang,
Peng Zhou, Dr. Youngmin Martin Kim, Dr. Dima Farfurnik, Dr. Chang-Min Lee,
Dr. Robert Pettit, Yu Shi, Chang-Mu Han, Sam Harper, Shantam Ravan, Yuqi
Zhao and Xinyuan Zheng.
I am also grateful to members of Professor Mohammad Hafezi?s group at JQI
including Professor Elizabeth Goldschmidt and Dr. Sunil Mittal, who provided a
cordial atmosphere in the lab. They were always available for any kind of help in
setting up the experiments and discussing theory.
iv
I acknowledge support from technicians in the UMD nanofabrication center:
Tom Loughran, Mark Lecates, Jon Hummel, John Abrahams, and Wen-an Chiou. I
thank them all for maintaining the equipment in good shape and always being there
to help when necessary. I would also like to thank all the staff members in IREAP,
past and present, Bryan Quinn, Don Schmadel, Thomas Weimar, Kathryn Tracey,
Leslie Delabar, and Judi Cohn Gorski . Specially, Nolan?s technical help is highly
appreciated, as is the computer hardware support from Edward Condon, LaTex and
software help from Dorothea Brosius, purchasing help from Nancy Boone and travel
related help from Taylor Prendergast.
I owe my deepest thanks to my family - my mother and father who have always
stood by me and guided me through my career, and have pulled me through against
impossible odds at times. Words cannot express the gratitude I owe them. A special
thanks goes to my younger brother Biplab. His cheerful attitude always gave me
that additional boost. I would also like to thank Sarthak, Subhojit, Tiffany, Aditya,
Swarnav, Tamoghna who are like family members to me. My housemates at my
place of residence have been a crucial factor in my finishing smoothly. I?d like to
express my gratitude to Soubhik and Amit for their friendship and support.
I would like to acknowledge financial support from the Office of Naval Research
(ONR), Air Force Office of Scientific Research (AFOSR), Sloan Foundation and the
Physics Frontier Center (PFC) at the Joint Quantum Institute (JQI) for all the
projects discussed herein.
It is impossible to remember all, and I apologize to those I?ve inadvertently
left out.
v
Lastly, thank you all!
vi
Table of Contents
Preface ii
Acknowledgements iii
Table of Contents vii
List of Tables x
List of Figures xi
List of Abbreviations xiii
List of Publications to Date xiv
Chapter 1: Introduction 1
1.1 Brief Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Chapter 2: Topological Edge States in Photonic Crystals 7
2.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Proposed structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Honeycomb Lattice with Circular Holes . . . . . . . . . . . . . 9
2.2.2 Honeycomb Lattice with Triangular Holes . . . . . . . . . . . 11
2.3 Band structure for TE modes . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Band Folding . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.2 Perturbation to the system parameters . . . . . . . . . . . . . 15
2.3.3 Band inversion . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Topological edge sates . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.1 Non-trivial topology . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.2 One-dimensional band structure . . . . . . . . . . . . . . . . . 18
2.5 Unidirectional propagation . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5.1 Helical nature of edge states . . . . . . . . . . . . . . . . . . . 18
2.5.2 Topological confinement . . . . . . . . . . . . . . . . . . . . . 18
2.6 Topological protection and robustness . . . . . . . . . . . . . . . . . . 20
2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
vii
Chapter 3: Chiral coupling of a single photon inside a Topological Photonic
Crystal Waveguide 22
3.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Fabricated structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.1 Device Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.2 SEM image . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.3 Band structures : FDTD results . . . . . . . . . . . . . . . . . 26
3.3 Waveguiding via edge states . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.1 Transmission of the waveguide . . . . . . . . . . . . . . . . . . 28
3.3.2 Probing the guided modes at the interface . . . . . . . . . . . 28
3.4 Observation of chiral coupling of a single emitter . . . . . . . . . . . . 29
3.4.1 Zeeman effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.2 Polarization of Quantum Dots Under Magnetic Field . . . . . 29
3.4.3 Chiral coupling of quantum dot emission into the waveguide . 30
3.5 Topological robustness . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.5.1 Waveguiding across a bend . . . . . . . . . . . . . . . . . . . . 32
3.5.2 Routing single photon . . . . . . . . . . . . . . . . . . . . . . 34
3.5.3 Robustness to disorder . . . . . . . . . . . . . . . . . . . . . . 34
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Chapter 4: Quantum optics with topological photonic crystal resonator 37
4.1 Review : Chiral quantum optics . . . . . . . . . . . . . . . . . . . . . 37
4.2 Proposed Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2.1 Valley-Hall topological photonic crystal . . . . . . . . . . . . . 39
4.2.2 Fabricated resonator structure . . . . . . . . . . . . . . . . . . 41
4.2.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 Observation of topological resonator modes . . . . . . . . . . . . . . . 42
4.4 Observation of chiral coupling of an emitter inside topological resonator 44
4.4.1 Waveguide-Resonator Hybrid Device : FDTD results . . . . . 44
4.4.2 Chiral coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4.3 Polarization-momentum locking . . . . . . . . . . . . . . . . . 46
4.5 Intensity enhancement of a single emitter coupled to the resonator . . 48
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Chapter 5: Conclusion and Outlook 51
Appendix A: 54
A.1 Honeycomb Lattice with a Six-Site Basis and Band Folding . . . . . . 54
A.2 Tight-Binding Description of the Dispersion . . . . . . . . . . . . . . 55
A.3 Topology and Edge States . . . . . . . . . . . . . . . . . . . . . . . . 62
A.4 Inversion of the Eigenstates . . . . . . . . . . . . . . . . . . . . . . . 65
A.5 Polarization Pseudo-Spin of the Eigenstates . . . . . . . . . . . . . . 67
Appendix B: 69
B.1 Device Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
viii
B.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
B.3 Grating Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
B.4 Coupling Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Appendix C: 75
C.1 The polarization profile of a topological resonator . . . . . . . . . . . 75
C.2 Sample fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
C.3 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
C.4 The quality factor of resonator in hybrid structures . . . . . . . . . . 78
C.5 Resonator designs with Valley-Hall physics . . . . . . . . . . . . . . . 79
Bibliography 81
ix
List of Tables
B.1 Estimation of coupling efficiency . . . . . . . . . . . . . . . . . . . . . 74
x
List of Figures
2.1 Band structure of a honeycomb lattice with circular holes . . . . . . . 10
2.2 Schematic of our proposed honeycomb-lattice-like photonic crystal . . 12
2.3 Band structures show opening and closing of a band gap around the
Dirac point as we perturb the lattice. . . . . . . . . . . . . . . . . . . 14
2.4 Schematic and band structure which gives rise to topological edge
states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Three-dimensional, vertically-confined topological edge states at op-
tical frequencies in an all-dielectric material. . . . . . . . . . . . . . . 19
3.1 Design of honeycomb-like photonic crystal . . . . . . . . . . . . . . . 24
3.2 Fabricated device and band structure . . . . . . . . . . . . . . . . . . 25
3.3 Transmission characteristics of the topological waveguide . . . . . . . 27
3.4 Polarization of quantum dot emission in bulk under magnetic field . . 30
3.5 Chirality in a straight topological waveguide . . . . . . . . . . . . . . 31
3.6 Robust transport in two dimensions along a bend . . . . . . . . . . . 33
3.7 Effect of defects on the transmission of edge modes . . . . . . . . . . 35
4.1 Valley-Hall topological photonic crystal resonator and FDTD simu-
lation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Topological waveguide-resonator system . . . . . . . . . . . . . . . . 43
4.3 Chiral waveguide-resonator-emitter coupling . . . . . . . . . . . . . . 45
4.4 Position dependent spin-momentum locking . . . . . . . . . . . . . . 47
4.5 Purcell enhancement of a quantum emitter coupled to a topological
resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
A.1 Correspondence between the two-site and six-site bases . . . . . . . . 56
A.2 Schematic of our lattice parameters and the labeling of the lattice
sites for our tight-binding model . . . . . . . . . . . . . . . . . . . . . 58
A.3 Physics of band inversion . . . . . . . . . . . . . . . . . . . . . . . . . 66
A.4 Out-of-plane magnetic field and in-plane electric field polarization . . 68
B.1 Mask design for fabrication of triangles . . . . . . . . . . . . . . . . . 70
B.2 Transmission data from left and right gratings . . . . . . . . . . . . . 73
C.1 Simulated left/right circular polarization components of the electric
field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
xi
C.2 Schematic of the experimental setup . . . . . . . . . . . . . . . . . . 77
C.3 Quality factor of cavity in hybrid structure . . . . . . . . . . . . . . . 79
C.4 More resonator ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
xii
List of Abbreviations
QD Quantum Dot
QED quantum electrodynamics
qubit quantum bit
PBS Polarization Beam Splitter
InAs Indium Arsenide
GaAs Gallium Arsenide
InP Indium Phosphide
FDTD finite-difference time-domain
SPCM Single Photon Counting Module
LED Light Emitting Diode
HF Hydro Flouric
xiii
List of Publications to Date
1. S Barik et al. ?Chiral quantum optics using a topological resonator?, Physical
Review B 101 (20), 205303 (2020)
2. H Chalabi, S Barik et al. ? Guiding and confining of light in a two-dimensional
synthetic space using electric fields? Optica 7 (5), 506-513 (2020)
3. S Barik, M Hafezi, ?Robust and compact waveguides?, Nature Nanotechnology
14 (1), 8 (2019)
4. H Chalabi, S Barik et al. ?Synthetic Gauge Field for Two-Dimensional Time-
Multiplexed Quantum Random Walks?, Phys. Rev. Lett. 123, 150503 (2019)
5. S Dutta, EA Goldschmidt, S Barik et al. ?An Integrated Photonic Platform
for Rare-Earth Ions in Thin-Film Lithium Niobate?, Nano letters 20 (1), 741-
747 (2019)
6. S Barik et al. ?A topological quantum optics interface?, Science 359 (6376),
666-668 (2018)
7. S Dutta, T Cai, MA Buyukkaya, S Barik et al. ?Coupling quantum emitters
in WSe 2 monolayers to a metalinsulator-metal waveguide? Appl. Phys. Lett.
113 (19), 191105 (2018)
8. JM Howard, EM Tennyson, S Barik et al. ?Humidity-induced photolumines-
cence hysteresis in variable Cs/Br ratio hybrid perovskites?, The journal of
physical chemistry letters 9 (12), 3463-3469 (2018)
9. Y Xu, EM Tennyson, J Kim, S Barik et al. ?Active Control of Photon Re-
cycling for Tunable Optoelectronic Materials?, Advanced Optical Materials 6
(7), 1701323 (2018)
10. S Barik et al. ?Two-dimensionally confined topological edge states in photonic
crystals?, NJP 18 (11), 113013 (2016)
11. K Lim, C Ropp, S Barik et al. ?Nanostructure-induced distortion in single-
emitter microscopy?, Nano Letters 16 (9), 5415-5419 (2016)
12. K S Alee, S Barik et al. ?Fo?rster energy transfer induced random lasing
at unconventional excitation wavelengths?, Appl. Phys. Lett. 103, 221112
(2013)
1
1The research works published in 1,6 and 10 are similar to the content presented here in the
dissertation. The rest of the publications are unrelated and different.
xiv
Chapter 1: Introduction
1.1 Brief Overview
After its discovery, the topological insulator in condensed matter system [1,2]
gained significant interest for its potential applications. Then the concepts were
translated to optics, giving rise to topological photonics [3, 4]. The conceptual
framework of band topology from condensed matter has successfully been applied
to photonic systems at microwave frequencies in various photonic crystal archi-
tectures [3, 6?8, 60]. Efforts are underway to extend these results into the optical
regime, and there have been notable achievements, for example the realization of
topologically protected edge states with near-infrared light in silicon ring resonator
arrays [9, 10] and visible light in fused silica photonic crystal fibers [11, 12]. One
particularly intriguing phenomena observed with such materials is that they ex-
hibit electronic edge states that can travel along the edge in one direction without
scattering even in the presence of impurities. This is due to the fact that band
topologies are a property of the entire system and not a local property, and so local
perturbations cannot alter the topology of the system.
A major motivation to introduce topologically protected edge states into pho-
tonic systems at optical frequencies is that they could potentially be incorporated
1
into photonic integrated circuits which could form an important element in future
telecommunication technologies and reduce loss in the transmission of optical sig-
nals [13?15]. Looking beyond classical information processing, it is conceivable to
interface topologically protected edge states with quantum emitters to realize quan-
tum information processing [16], or more generally a quantum internet [17], and
to perform quantum simulation with photons [18, 19], for example giving rise to
fractional quantum Hall states [20].
The benefit of being able to interface topological light with quantum emitters
is that this can realize strong interactions between photons with directional con-
trol. Although photons are by themselves essentially non-interacting, it is possible
to introduce effective interactions through intermediaries such as cavities [21, 22],
atoms [23?25] or quantum dots [26?28]. Numerous theoretical studies have explored
ways to realize topological states of light in photonic systems [29,30,32?35], but to
my knowledge none have led to the proposal of a readily realizable system in an
all-dielectric substrate which has topological edge states in three dimensions and
can interface directly with quantum emitters and other nanophotonic elements. For
example, many of the structures considered theoretically use an array of rod-like ele-
ments, but the electromagnetic modes of these systems cannot be confined unless the
ends are capped, for example with metals which are inherently dissipative at optical
frequencies, and so cannot be used as a waveguide. Although recent experimental
results have demonstrated chiral modes in one-dimensional photonic waveguides in-
terfaced to emitters [36?38], these modes are not topologically protected, so it would
be difficult to extend to two dimensions as would be necessary for on-chip routing
2
of light. The photonic crystal architecture we propose allows the integration of edge
states that are topologically protected in two dimensions which can interface with
quantum emitters and is confined with dielectric materials. Another notable aspect
of this system is that it does not require any magnetic fields or magnetic materials
to realize topological edge states [60].
Moreover, controlling light-matter at the single-photon level is at the heart
of quantum optics, and plays a central role in quantum information applications.
One way to control these interactions is by engineering the dielectric environment,
which can modify the emission and absorption of a quantum emitter. In addition,
proper design of the dielectric environment can create chiral modes of light that
lock polarization and momentum. When coupled to quantum emitters, these chiral
modes can control the directionality of spontaneous emission and modify photon-
mediated interactions [96]. This control opens up new opportunities to tailor light-
matter quantum states such as entangled spin states [109] and photonic clusters
states [110].
Nanophotonics provides an alternate platform to engineer chiral light-matter
coupling in a compact and scalable chip-integrated device. While chiral light-matter
interactions have been successfully realized in waveguides [37], nanofibers [36], and
millimeter-scale bottle resonators [111,112], the application of these ideas to optical
resonators that can strongly enhance coupling between emitters and photons, has
remained elusive. This is partly due to the complications in designing resonators
with chiral/helical properties. This is mainly because of the challenges in designing
chiral resonators while maintaining small mode volumes, minimizing bending losses
3
and the susceptibility of the nanophotonic devices to fabrication disorders. However,
A topological resonator could enable strong chiral light-matter interactions in a
compact chip-integrated device and provide robustness against disorder, a major
problem for nanophotonics in general.
Meanwhile, the field of topological photonics has emerged as a new paradigm to
create a photonic structure with robustness against deformation [4,58]. In particular,
in photonic crystals, this approach has been useful in realizing sharp bends and
zigzag structures, without requiring fine-tuning of system parameters [29,100].
To that aim, in this thesis, I examine different nanoscale photonic crystal struc-
tures with triangular holes that sustains topological edge states which can couple to
a quantum emitter in a chiral fashion. Structures like this can be fabricated with
popular dielectric materials like Si or GaAs, which can be incorporated into photonic
devices for reflectionless transport of light. We chose photonic crystals as the plat-
form to realize topological edge states because of the flexibility they offer to engineer
their band structures [39, 40]. This property is critical in realizing non-trivial band
topologies. Moreover, they allow the realization of nanophotonic structures ranging
from optical cavities [41, 42] to waveguides [44, 76] with well understood nanofabri-
cation technologies and techniques in slab materials, where the in-plane confinement
is provided by photonic band gaps and out-of-plane confinement is provided by total
internal reflection at the dielectric-air interface. Furthermore, dielectric nanopho-
tonic structures such as Gallium Arsenide (GaAs) can be readily integrated with
other nanophotonic elements, such as epitaxially grown quantum dots [45, 46, 69]
and cold atom systems [156?159].
4
1.2 Outline of Thesis
In Chap. 2, we present an all-dielectric photonic crystal structure that sup-
ports two-dimensionally confined helical topological edge states. The topological
properties of the system are controlled by the crystal parameters. An interface be-
tween two regions of differing band topologies gives rise to topological edge states
confined in a dielectric slab that propagates around sharp corners without back-
scattering. Three dimensional finite-difference time-domain calculations show these
edges to be confined in the out-of-plane direction by total internal reflection. Such
nanoscale photonic crystal architectures could enable strong interactions between
photonic edge states and quantum emitters.
The application of topology in optics has led to a new paradigm in develop-
ing photonic devices with robust properties against disorder. Although significant
progress on topological phenomena has been achieved in the classical domain, the
realization of strong light-matter coupling in the quantum domain remains unex-
plored. In Chap. 3, we demonstrate a strong interface between single quantum emit-
ters and topological photonic states. We demonstrate light-matter coupling between
topologically protected photonic edge states and a quantum emitter. Our approach
creates robust counter-propagating edge states at the boundary of two distinct topo-
logical photonic crystals. We demonstrate the chiral emission of a quantum emitter
into these modes and establish their robustness against sharp bends. This approach
may enable the development of quantum optics devices with built-in protection,
with potential applications in quantum simulation and sensing.
5
In Chap. 4, we demonstrate a chiral coupling of a quantum emitter inside
a topological photonic crystal resonator. Chiral nanophotonic components, such
as waveguides and resonators coupled to quantum emitters, provide a fundamen-
tally new approach to manipulate light-matter interactions. The recent emergence of
topological photonics has provided a new paradigm to realize helical/chiral nanopho-
tonic structures that are flexible in design and, at the same time, robust against
sharp bends and disorder. Here we demonstrate such a topologically protected chi-
ral nanophotonic resonator that is strongly coupled to a solid-state quantum emitter.
Specifically, we employ the valley-Hall effect in a photonic crystal to achieve topo-
logical edge states at an interface between two topologically distinct regions. Our
helical resonator supports two counter-propagating edge modes with opposite po-
larizations. We first show chiral coupling between the topological resonator and the
quantum emitter such that the emitter emits preferably into one of the counter-
propagating edge modes depending upon its spin. Subsequently, we demonstrate
strong coupling between the resonator and the quantum emitter using resonant Pur-
cell enhancement in the emission intensity by a factor of 3.4. Such chiral resonators
could enable designing complex nanophotonic circuits for quantum information pro-
cessing and studying novel quantum many-body dynamics.
In Chap. 5, I will provide the conclusion to the thesis along with potential
applications of this work in the filed of quantum nanophotonics.
6
Chapter 2: Topological Edge States in Photonic Crystals
2.1 Review
Topology is a ubiquitous concept in physics, ranging from electrons in solid
state [1, 2], quantum degenerate gases [52, 53], and sound [54?57]. A key mani-
festation of topological physics is the presence of edge modes which are robust to
local disorder. The prospect of using topological photonic materials for such robust
propagation of light has attracted a great deal of interest [58, 59].
Topologically-protected edge states have been experimentally demonstrated
in systems at microwave frequencies [60, 61] and optical frequencies, specifically in
ring resonators [9, 10], and in coupled waveguides [11]. Subsequent work measured
the invariants characterizing the topology of two-dimensional photonic systems [62].
Embedding quantum emitters into these optical frequency devices could generate
strong optical non-linearities that exhibit new physical behavior. Theoretical work
has shown that the interplay between emitters and chiral states results in intriguing
phenomena such as many-body position-independent scattering [63], dimerization
of driven emitters [64] and fractional quantum Hall states [20,65,66].
Strong light-matter interactions with optical emitters usually require the con-
centration of light to small mode-volume nanophotonic devices [67]. Two-dimensional
7
photonic crystals are one of the most promising nanophotonic platforms for this ap-
plication because they confine light to less than an optical wavelength [68, 69]. Re-
cently, several works have proposed all-dielectric photonic crystal structures where
deformations open a gap in the Dirac cone dispersion to achieve non-trivial topo-
logical bands [29?31, 70]. However, these proposals consider the flow of light using
two-dimensional numerical simulations with infinite extent in the third dimension
and therefore do not address practical issues associated with propagation and con-
finement of light in a physically realizable crystal. A notable exception is the work
of Wu et al. [29] which does consider structures of finite extent in the out-of-plane
direction, but uses metallic mirrors to achieve confinement in this dimension. In this
case the structure is no longer all-dielectric. Since metallic sheets are highly absorb-
ing at optical frequencies, these mirrors would render the device extremely lossy.
Thus, it would be highly desirable to create an all-dielectric topological photonic
crystal.
In this chapter, we demonstrate that an all-dielectric topological photonic
crystals can exhibit two-dimensionally edge states confined by total internal reflec-
tion in a dielectric slab. This design enables low-loss confinement of light in the
third dimension. This structure addresses the challenge of experimentally realizing
topological photonic crystals and enabling strong interactions with optical emitters.
Our system exhibits spin quantum Hall physics for pseudo-spin photonic polariza-
tions. As a result of time-reversal symmetry, the edge states are helical: edge states
of opposite helicity travel in opposite directions. We utilize a honeycomb periodic
structure with six-fold symmetry based on triangular holes. This structure ensures
8
a complete bandgap for transverse electric modes. Deformations of the unit cell
that preserve its rotational symmetry change the topology of the structure. We
show that interfacing two materials of different band topologies results in robust
two-dimensionally confined edge states that can propagate around sharp bends.
2.2 Proposed structure
2.2.1 Honeycomb Lattice with Circular Holes
One idea is to implement our honeycomb-lattice-like photonic crystal structure
with circular holes instead of triangular holes, as shown in Fig. 2.1(a). Although
the band structure of the transverse-electric modes, as shown in Fig. 2.1(b), does
give rise to a Dirac point, it turns out that it does not give rise to a band gap
in the region of interest. The horizontal white dashed line shows the frequency at
the Dirac point and the white dotted region encloses a range of wavevectors for
which one of the bands crosses the frequency at the Dirac point, thus preventing the
appearance of a band gap across the Brillouin zone even after perturbation, which
is critical for realizing topological edge states. This can be avoided with the use of
equilateral triangular holes, where a band gap is possible after perturbations to the
system. This will be explored in detail next and will be a major point throughout
this project.
9
Figure 2.1: Band structure of a honeycomb lattice with circular holes : (a) Schematic
of a honeycomb lattice made of circular holes, where the parameters a0 is the lattice
constant of the hexagonal clusters (white hexagons) which constitute a triangular
lattice, R is the distance from the center of the cluster to the center of a circular
hole within the cluster (R = a0/3 in this case), and s is the diameter of the circular
hole. (b) Band structure for transverse-electric modes of the structure shown in (a),
showing the appearance of a Dirac cone at 312 THz (indicated by the horizontal
white dashed line). White dotted ellipse shows one of the bands crossing the fre-
quency at the Dirac point, which prevents the appearance of a band gap across the
Brillouin zone after perturbation. Calculations were done with a0 = 350 nm and
s = 140 nm.
10
2.2.2 Honeycomb Lattice with Triangular Holes
Fig. 2.2 shows a schematic of our photonic crystal structure. The reason behind
this geometry will be clear once we study the band structure of this system in the
next section. Unlike the previous case : the lattice with circular holes, in this case
there will be a complete band gap in the reciprocal space of the photonic lattice.
The starting point is a honeycomb lattice made of equilateral triangular holes in a
dielectric material as shown in Fig. 2.2(a). We can view this system as a triangular
lattice with a basis consisting of two triangular holes, as is typically done in studies
of graphene [71]. The black outline shows such a two-hole unit cell. Fig. 2.2(b)
shows the first Brillouin zone (dashed line), which is a hexagon. We denote the
high-symmetry points [161] by ?, M? and K?. Alternatively, we can also view this
structure as a triangular lattice of six-hole unit cells [white dashed hexagons in
Fig. 2.2(a) which we call honeycomb clusters], where the relevant parameters are
the lattice constant of the triangular lattice a0, the distance between the center of
each cluster to the centroid of each triangular hole R, the length of each side of
the equilateral triangular holes s, and the height of the dielectric material h. In
the honeycomb lattice, the relationship R = a0/3 holds. Fig. 2.2(b) shows the first
Brillouin zone as a solid hexagon and ?, M and K indicate the high symmetry points.
Note that the first Brillouin zone for the six-hole unit cell is smaller than for the
two-hole unit cell due to the larger real space unit cell area.
11
Figure 2.2: Schematic of our proposed honeycomb-lattice-like photonic crystal (a)
Baseline structure of equilateral triangular holes arranged in a honeycomb lattice in
a dielectric material. This honeycomb lattice can be viewed as a triangular lattice of
two-hole unit cells (black solid rhombus), or alternatively as a triangular lattice of
six-hole unit cells (white dashed hexagons), which we call honeycomb clusters with
R = a0/3. (b) First Brillouin zone for the six-hole (solid) and two-hole (dashed)
unit cells. The letters indicate high-symmetry points. (c) [and (d)] Same structure
as in (a) except that R < a0/3 (R > a0/3), which we call shrunken (expanded)
clusters.
12
2.3 Band structure for TE modes
We first analyze the band structure of this photonic crystal in the two-hole
unit cell picture using three-dimensional numerical finite-difference time-domain cal-
culations (Lumerical FDTD Solutions). We perform simulations using GaAs as the
dielectric substrate, with index of refraction taken from Ref. [73]. The parameters
we use are a0 = 445 nm, s = 140 nm, and h = 160 nm, which are typical dimensions
for photonic crystal structures [46, 74, 75]. We focus on the transverse-electric-like
modes of the system where the electric field at the symmetric plane of the system
lies in-plane. Fig. 2.3(a) shows the band structure of the honeycomb lattice corre-
sponding to Fig. 2.2(a) along the high-symmetry points of the Brillouin zone. The
gray region indicates the portion of the band structure above the light line where
there are no guided modes confined in the dielectric material of finite thickness [76].
There is a Dirac point at the K? point, indicated by the red arrow in Fig. 2.3(a),
located below the light line. Near this Dirac point, we can modify the topological
properties of the photonic crystal by changing the ratio R/a0 [29]. However, these
perturbations also change the symmetry of the lattice and so we can no longer use
the rhombus-shaped two-hole unit cell to construct the band structure. Instead,
we use the hexagonal six-hole unit cell to construct the band structure without
destroying the rotational symmetry of the system.
13
Figure 2.3: Band structures show opening and closing of a band gap around the
Dirac point as we perturb the lattice. (a) Band structure of the honeycomb lattice
in the two-hole unit cell picture. The gray area represents the region above the light
line, where light can leak out of the plane. A Dirac point exists at the K? point
(red arrow) and is below the light line. (b),(c) and (d) Band structure calculated
with the six-hole unit cell with honeycomb clusters (R = a0/3), shrunken clusters
(R = 0.91 ? a0/3), and expanded clusters (R = 1.09 ? a0/3) respectively. The red
arrow indicates the Dirac point, and the green areas represent the band gap.
14
2.3.1 Band Folding
We obtain the band structure for the six-hole unit cell by appropriate band
folding of the bands obtained from the two-hole unit cell (Appendix A). Although
both Brillouin zones share the same ? point, the K? and K?? points for the two-hole
unit cell [71] become folded over onto the ? point of the six-hole unit cell to form
a doubly-degenerate Dirac point at 319 THz (which corresponds to 940 nm) as
indicated by the red arrow in Fig. 2.3(b).
2.3.2 Perturbation to the system parameters
We perturb this system by varying R with respect to a0 to get clusters that
are shrunken (R < a0/3) or expanded (R > a0/3) as shown in Figs. 2.2(c) and (d)
respectively. Figs. 2.3(c) and (d) show the corresponding band structures specifically
for R = 0.91 ? a0/3 and R = 1.09 ? a0/3 respectively. Increasing or decreasing
the ratio R/a0 about the honeycomb lattice opens a band gap at the Dirac point.
In particular, the band gaps are 13 THz and 25 THz wide for the shrunken and
expanded clusters respectively. By comparing the eigenstates at the ? point for the
expanded and shrunken structures, we see that the eigenstates are inverted between
the two structures, indicating that the band topology changes as we tune the ratio
R/a0 (appendix A).
15
2.3.3 Band inversion
To further confirm the numerically observed band inversion, we also analyti-
cally study the system with a tight-binding model(Appendix A). The Hamiltonian
of our system reduces to the Bernevig-Hughes-Zhang model for the quantum spin
Hall effect [77], where the mass term changes sign when the clusters are shrunken
and expanded around R = a0/3. Consequently, the bands acquire non-zero Chern
numbers that are the direct indication of non-trivial band topology. In this case,
the polarization profile of the in-plane electric field acts as the pseudo-spin.
2.4 Topological edge sates
2.4.1 Non-trivial topology
Non-trivial band topologies manifest themselves most dramatically in the form
of guided topological edge states at the boundary between two gapped regions that
have different band topologies. To confirm this, we perform three-dimensional sim-
ulations of the structure schematically shown in Fig. 2.4(a) using the same values
for the parameters a0, s and h as previously. We examine topological edge states at
an interface between one region composed of unit cells with shrunken clusters (13
clusters wide) and another region of expanded clusters (12 clusters wide). These
two regions share a common band gap in bulk as shown in Figs. 2.3(c) and (d).
16
Figure 2.4: Schematic and band structure which gives rise to topological edge
states. (a) Schematic of two regions with different band topologies. White dotted
line marks the boundary between the two regions. The star (green) indicates the
location where we placed a circularly-polarized electric dipole to excite topological
edge states. (b) Corresponding one-dimensional band structure shows two bands
crossing the band gap in bulk. The opposite group velocities in the crossing region
indicate the existence of counter-propagating directional edge states.
17
2.4.2 One-dimensional band structure
Figure 2.4(b) shows the one-dimensional band structure along the x-direction.
Note that introducing an interface creates two bands crossing the original bandgap of
the individual regions. The two newly formed bands have opposite group velocities,
indicating counter-propagating directional edge states.
2.5 Unidirectional propagation
2.5.1 Helical nature of edge states
The edge states in this system are helical, i.e., the pseudo-spin degree of
freedom controls the direction of propagation [1]. We verify the helicity of the edge
states by exciting the system with a circularly-polarized electric dipole placed at
the location indicated by the green star in Fig. 2.4(a). By choosing the excitation
polarization to be positively (negatively) circularly polarized, we can selectively
excite an edge mode propagating in the ?x (+x) direction (Fig. 2.5(b-i)[(b-ii)]).
The excitation frequency is 320 THz (equivalent to a wavelength of 938 nm).
2.5.2 Topological confinement
Figs. 2.5(c) and (d) show the the electric field intensity distribution of the
three-dimensional, vertically confined edge state[corresponding to Fig. 2.5(b-ii)] in
xz and yz cross-sections respectively. The field is confined within the dielectric slab
due to total internal reflection at the air-dielectric boundary. This proves that one
18
Figure 2.5: (a) Schematic diagram of the three-dimensional photonic crystal where
the colored planes correspond to the cross-sections shown in (b), (c), and (d). (b-
i) and (b-ii) Electric field intensities for a topological edge state excited with a
positively and negatively circularly-polarized electric dipole show directional prop-
agation in the ?x and +x directions respectively. (c) and (d) Cross-section view
along the xz and yz plane of the electric field intensity confirms that total internal
reflection at the air-dielectric boundary prevents light from leaking out of the plane.
(e) Electric field intensity for an edge state with four 90? bends show that light can
propagate around defects without back scattering.
19
can realize topological edge states in three dimensions within dielectric materials at
optical frequencies without significant out-of-plane loss.
2.6 Topological protection and robustness
One of the most distinguishing features of topological edge states is their ro-
bustness against perturbations. To test this robustness, we introduced four 90?
bends to the structure as shown in Fig. 2.5(e). Excitation of the edge mode in
this configuration shows that there is very little back-scattering along the entire
path. Thus our edge states exhibit topological protection against certain types
of disorder and defects, in contrast to chiral, but topologically-trivial, waveguide
structures [36, 37].
2.7 Discussion
We note that the topological protection we obtain in the presence of time-
reversal symmetry differs in an important respect from that of electronic quantum
spin Hall systems. The general classification of topological insulators reveals that
the Z2 topological invariant describing the latter requires that T
2 = ?1, where T
is the time-reversal symmetry operator. The minus sign is a particular feature of
fermionic systems. In contrast, Maxwell?s equations (and other bosonic systems)
obey T 2 = 1. This symmetry taken alone does not afford any topological protection
in two dimensions.
However, we can construct a pseudo? time-reversal symmetry operator based
20
on the (C6v) crystal symmetry of the lattice which obeys T
2 = ?1 [29]. While
this assures that the bulk may be classified according to a Z2 topological invari-
ant, gapless edge modes are not guaranteed since this symmetry is broken at the
boundaries. This symmetry breaking can mix the counter-propagating edge states
and open a mini-gap in the edge mode [78]; in a quantum spin Hall system, this
would be akin to a magnetic impurity at the edge of the system. Apparently, in
our realization this symmetry breaking is weak since we do not observe a gap in
the edge states [Fig 2.4(b)]. We can decouple the pseudo-spin degrees of freedom
up to linear order in k near the ? point. By considering these degrees of freedom
as being completely decoupled, we can characterize the topology of the system by
a stronger Z spin Chern number given by the difference of the Chern numbers for
each pseudo-spin [79].
21
Chapter 3: Chiral coupling of a single photon inside a Topological
Photonic Crystal Waveguide
3.1 Review
The discovery of electronic quantum Hall effects has inspired remarkable de-
velopments of similar topological phenomena in a multitude of platforms ranging
from ultra cold neutral atoms [83, 84] to photonics [58, 85] and mechanical struc-
tures [86?88]. Like their electronic analogs, topological photonic states are unique
in their directional transport and reflectionless propagation along the interface of
two topologically distinct regions. Such robustness has been demonstrated in vari-
ous electromagnetic systems, ranging from microwave domain [60,89] to the optical
domain [90, 91], opening avenues for a plethora of applications, such as robust de-
lay lines, slow-light optical buffers [92], and topological lasers [93?95], to develop
optical devices with built-in protection. While the scope of previous works remains
in the classical electromagnetic regime, a great deal of interesting physics could
emerge by bringing topological photonics to the quantum domain. Specifically, in-
tegrating quantum emitters to topological photonics structures could lead to robust
strong light-matter interaction [96], generation of novel states of light and exotic
22
many-body states [97?99].
In this chapter, we experimentally demonstrate light-matter coupling in a
topological photonic crystal. We utilize an all-dielectric structure [29, 100] to im-
plement topologically robust edge states at the interface between two topologically
distinct photonic materials, where the light is transversely trapped in a small area,
up to half of the wavelength of light. We show that a quantum emitter efficiently
couples to these edge modes and the emitted single photons exhibit robust transport,
even in the presence of a bend.
3.2 Fabricated structure
3.2.1 Device Design
Figure 3.1 shows a schematic of the device design. We begin with a honey-
comb lattice of equilateral triangles exhibiting hexagonal symmetry as our baseline
structure. This lattice is a triangular lattice of cells consisting of six equilateral
triangular holes, indicated by the dashed line. We use a lattice constant of a0 = 445
nm, an edge length of the equilateral triangle of s=140 nm, and a slab thickness of
h = 160 nm. R defines the distance from the center of a cell to the centroid of a
triangle. In this structure a perfect honeycomb lattice corresponds to R = a0/3.
With these parameters we obtain doubly degenerate Dirac cones at 319 THz
(940 nm). We form the two mirrors by concentrically expanding or contracting the
unit cell.
We create topologically distinct regions by deforming the unit cell of the pris-
23
Figure 3.1: Design of honeycomb-like photonic crystal.
tine honeycomb lattice. One one side of the lattice, we concentrically shift the trian-
gular holes by increasing R to 1.05a0/3, thereby shifting all the triangular holes in an
individual cell outward. This deformation results in creation of a bandgap(Fig. 3.2
C). On the other side, we decrease R to 0.94a0/3, which pulls the holes towards
the center resulting in the band structure with similar bandgap but with different
topological properties(Fig. 3.2 D).
3.2.2 SEM image
Fig. 3.2 A shows the fabricated topological photonic crystal structure. The
device is composed of a thin GaAs membrane with epitaxially grown InAs quantum
dots at the center that act as quantum emitters (see Appendix B). The topological
photonic structure is comprised of two deformed honeycomb photonic crystal lat-
24
Figure 3.2: Fabricated device and band structure. (A) Scanning electron microscope
image of the device composed of two regions identified by blue and yellow highlights,
corresponding to two photonic crystals of different topological properties. The in-
terface between the two photonic crystals supports helical edge states with opposite
circular polarization (?+ / ?? ). Grating couplers at each end of the device scatter
light in the out-of-plane direction for collection. (B) Closeup image of the interface.
(C) and (D) show the band structures for the transverse electric modes of the two
photonic crystals.
25
tices made of equilateral triangular air holes on a GaAs membrane [100]. Details
regarding the fabrication and methods are described in Appendix B. Fig. 3.2B shows
a closeup image of the interface, where the black dashed lines identify a single unit
cell of each photonic crystal. In each region, we perturb the unit cell by concen-
trically moving the triangular holes either inward (yellow region) or outward (blue
region).
3.2.3 Band structures : FDTD results
Fig. 3.2C-D shows the corresponding band structures of the two regions. Sim-
ilar to calculations in chapter 2, we use FDTD methods to compute the shown band
structures. The perturbations open two bandgaps exhibiting band inversion at the
? point [29, 100]. Specifically, the region with compressed unit cell, highlighted in
yellow, acquires a topologically trivial band gap, while the expanded region, high-
lighted in blue, takes on a nontrivial one. We design both regions so that their
bandgaps overlap. Photons within the common bandgap cannot propagate into ei-
ther photonic crystal. However, because the crystals have different topological band
properties, the interface between them supports two topological helical edge modes,
travelling in opposite directions, with opposite circular polarizations at the center
of the unit cell.
26
Figure 3.3: Transmission characteristics of the topological waveguide. (A) A
schematic of the excitation scheme identifying the three relevant regions. (B) Sim-
ulated band structure of transverse electromagnetic modes of a straight topological
waveguide. The grey region corresponds to bulk modes of the individual topological
photonic crystals and red lines represent modes within the bandgap corresponding
to topological edge states. The adjacent panel shows the measured spectrum at the
transmitted end of the waveguide. The shaded region identifies the topological edge
band. (C) Transmission spectrum at grating L as a function of the excitation laser
position.
27
3.3 Waveguiding via edge states
3.3.1 Transmission of the waveguide
To show the presence of the guided edge mode, we measure the transmission
spectrum. We illuminate the left grating (L) with a 780 nm continuous-wave laser
using a pump power of 1.3W, and collect the emission from the right grating (R)
(see Fig. 3.3A). At this power the quantum dot ensemble emission becomes a broad
continuum due to power broadening, resulting in an internal white light source that
spans the wavelength range of 900-980 nm. Fig. 3.3B shows the spectrum at the right
grating, presented with the band structure simulation [100]. Light emitted within
the topological band efficiently transmits through the edge mode and propagates to
the other grating coupler, while photons outside of the bandgap dissipate into bulk
modes.
3.3.2 Probing the guided modes at the interface
To confirm that the emission originates from guided modes at the interface
between the two topological materials, we excite the structure in the middle of
the waveguide (M), and collect the emission at the left and right grating coupler,
which we independently calibrate (Appendix B). Fig. 3.3C shows the transmission
spectrum collected from left coupler as a function of the laser spot position as we scan
the laser along the y-axis across the interface indicated by blue arrow (see Fig. 3.3A).
The spectrum attains a maximum transmission within the topological band when
28
the pump excites the center of the structure. When we displace the excitation
beam, by approximately 1.5 microns along the y direction, the transmission vanishes,
indicating that the photons are coming only from the waveguide.
3.4 Observation of chiral coupling of a single emitter
3.4.1 Zeeman effect
A key feature of topological edge modes is the chiral nature of the coupling
between the helical topological edge mode and the quantum emitter. Specifically,
different dipole spins radiatively couple to opposite propagating helical edge states.
To demonstrate this helical light-matter coupling, we apply a magnetic field in the
out-of-plane (Faraday) direction on the entire sample. This field induces a Zeeman
splitting in the quantum dot excited state [101].This results in two non-degenerate
states that emit with opposite circular polarizations, denoted as ?? as described
next (Fig. 3.5A) . While this magnetic field does not play a role in the topological
nature of the waveguide, it enables us to identify the polarization of the dipole by the
frequency of emitted photons. By spectrally resolving the emissions we can identify
the dipole spin and correlate it with the propagation direction of the emitted photon.
3.4.2 Polarization of Quantum Dots Under Magnetic Field
We first measured the photoluminescence from a bare QD in the bulk. With
the application of magnetic field QD emission spectrum splitted into two branches
with circularly polarized emission as shown in Figure 3.4. A and denoted by ??.
29
Figure 3.4: Polarization of quantum dot emission in bulk under magnetic field. (A)
Splitting of single QD emission into ?+ and ?? exciton branches under application
of magnetic field. (B) Verification of circular polarization of excitonic branches with
polarization selective photoluminescence.
At a very high magnetic field of 3T the separation between two branches becomes
0.3 nm. at this stage to verify the selection rules we introduced a quarter wave
plate and a polarizer before collecting the signal. Figure 3.4.B shows recorded
photoluminescence obtained by rotating the polarizer angle. The antiphase relation
between the two branches along with the detection scheme confirms that they are
indeed circularly polarized in bulk under high magnetic field.
3.4.3 Chiral coupling of quantum dot emission into the waveguide
To isolate a single quantum emitter within the topological edge mode, we
reduce the power to 10 nW, which is well below the quantum dot saturation power.
Using the intensities of the collected light at the two ends, we calculate a lower bound
on the coupling efficiency of 68% (Appendix B), defined as the ratio of the photon
30
Figure 3.5: Chirality in a straight topological waveguide. (A) Schematic of quantum
dot level structure in the presence of a magnetic field, and radiative transitions with
opposite circular polarizations. (B) Emission spectrum collected from the excitation
region as a function of magnetic field. (C) and (D) Transmission spectrum to left
and right gratings, respectively.
31
emission rate into the waveguide to the total emission rate. This high efficiency is
due to the tight electromagnetic confinement of the guided modes which enhances
light-matter interactions. Figure 3.5B shows the emission spectrum as a function of
magnetic field, where we collect the emission directly from point M indicated in the
Fig. 3.3A. As the magnetic field increases, the quantum dot resonance splits into two
branches corresponding to the two Zeeman split bright exciton states. We compare
this spectrum to the one collected from left and right gratings (Fig. 3.5 C-D). At
the left grating we observe only the emission from the ?? branch, while at the right
grating we observe only the ?+ branch. These results establish the chiral emission
and spin-momentum locking of the emitted photons, and provide strong evidence
that the emitter is coupling to topological edge states that exhibit unidirectional
transport. Such chiral coupling is in direct analogy to one dimensional systems
[37, 102]. In contrast to one dimensional systems, the waveguided modes of our
structure originate from two dimensional topology. As a result, the topological edge
mode should exhibit robustness to certain deformations, such as bends.
3.5 Topological robustness
3.5.1 Waveguiding across a bend
In order to establish this topological robustness, we analyze the propagation
of emitted photons in the presence of a bend. We introduce a 60 degree bend into
the structure as shown in Fig. 3.6A, and perform measurements similar to those in
Fig. 3.5. Again we observe that emitted photons propagate in opposite direction in a
32
Figure 3.6: Robust transport in two dimensions along a bend. (A) Schematic of
a modified topological waveguide with a bend. (B) and (C) Photoluminescence
collected from position L and position R, respectively, showing only one branch of
the quantum dot. (D) and (E) Second-order correlation measurement (g2(?)) data
obtained from point L and R, respectively, showing anti-bunching.
33
chiral fashion and arrive at the grating associated with their respective polarization
(Fig. 3.6B-C). The preservation of the chiral nature of the emission demonstrates
an absence of back-reflection at the bend, which would result in a strong signal for
both polarizations at the left grating.
3.5.2 Routing single photon
We also confirm that these routed photons are indeed single photons by per-
forming a second order correlation measurement for photons collected from both
ends of the waveguide, which exhibits strong anti-bunching (Fig. 3.6D-E). We note
that the robustness in this system is due to C6v symmetry, and the boundary and
disorder can break this symmetry and lead to backscattering of the edge modes.
3.5.3 Robustness to disorder
Next, we analyze the transmission of the edge mode to certain types of disorder
and show that it maintains a good level of robustness. The full characterization of
robustness, beyond numerical simulations and tight-binding model [103], requires
further study.
The yellow and blue regions in Figure 3.2A represent the topologically dis-
tinct phases discussed above. Topological modes exist at the boundary, and are
protected from any disorder which respects the six-fold crystal symmetry. Disorder
that breaks this symmetry can lead to the backscattering of the edge modes. In
fact, the formation of the interface itself can break this symmetry, albeit weakly.
34
Figure 3.7: Effect of defects on the transmission of edge modes : Simulation showing
a topological waveguide without a defect (A) compared to one with a large defect
(B) that exhibits no backscattering.
Through extensive simulations, we have found that the zigzag interface in our de-
vice adequately preserves the crystalline symmetry, thereby minimizing the coupling
between the counterpropagating edge modes. In Figure 3.7, we show simulations
illustrating the robustness of the edge modes to a certain type of defect. The defect,
an entire missing cell, breaks C6v crystal symmetry. However, this defect does not
adversely affect the transmission in the gapped region. We should note that the
disorder seen in our device is considerably less severe.
3.6 Discussion
Once thought to be limited to the quantum Hall effect, the notion of topological
insulating phases has revolutionized condensed matter physics and is the inspiration
for topological photonic systems. These states are based on the fact that insulating
Hamiltonians which obey certain combinations of symmetries (such as time-reversal)
can be classified according to their topology. The photonic crystal considered here
is described by a Z2 topological invariant, which takes the value 0 (trivial phase) or
1 (topological phase). The physical manifestation of this is that between regions of
35
differing topology, protected edge modes are found. These modes cannot be coupled
since such a term would violate the protected symmetry.
The photonic crystal considered in this work is an analog of a quantum spin
Hall system for photons and is discussed in greater detail in [29]. The quantum
spin Hall system exhibits topological protection that is based on time-reversal sym-
metry. In the context of the photonic crystal, the role of time-reversal symmetry
is played by the six-fold rotational (C6v) crystal symmetry of the hexagonal unit
cell. In the energy range of interest, the band structure of the system is described
by the Dirac equation, where the mass is controlled by the spacing of triangles in
a hexagonal cluster. The topological Z2 index reflects the sign of the mass, and is
positive (negative) for compressed (expanded) regions. The topologically protected
counterpropagating modes exist in the region at which the mass changes sign. In
the context of the Dirac equation, these states are known as Jackiw-Rebbi states
[165].
36
Chapter 4: Quantum optics with topological photonic crystal res-
onator
4.1 Review : Chiral quantum optics
Chiral propagation of light can fundamentally alter the way it interacts with
matter. In particular, chiral light-matter interactions can control the directionality
of spontaneous emission and modify photon-mediated interactions between quan-
tum emitters [96]. These capabilities in-turn enable engineering of novel quantum
states such as entangled spin states [109] and photonic clusters states [110]. Chi-
ral light-matter interactions based on polarization-momentum locking of evanescent
fields have been achieved previously, for example, using optical fibers [102] and
millimeter-scale bottle resonators [111, 112]. Chiral light-matter interactions have
also been explored using purely opto-mechanical interactions [113?116]. Recently,
nanophotonics has emerged as a versatile platform to engineer chiral light-matter
coupling in a compact and scalable fashion. In particular, chiral/helical waveguides
coupled to solid-state quantum emitters have demonstrated directional spontaneous
emission [117, 118]. However, extensions of these ideas to realize strong coupling
between a chiral resonator and a solid-state emitter have remained elusive. This
37
is mainly because of the challenges in designing a nanophotonic resonator while
preserving polarization-momentum locking.
Recently, topology has emerged as a new paradigm to design chiral photonic
structures [4]. In particular, an interface between two topologically distinct regions
hosts edge states exhibit chiral/helical propagation of light where the photon?s mo-
mentum gets locked to a pseudospin degree of freedom, such as polarization. These
edge states have the additional benefit that they are robust to deformations and
disorders. Specifically, they also allow propagation of light around sharp bends
and defects without scattering [90,119?123], which is essential to engineer compact
resonators that exhibit chirality [124,125].
In this chapter, we demonstrate a topological resonator that exhibits strong
light-matter interaction. We realize this resonator by creating an interface between
two valley-Hall topological photonic crystals. By using an inhomogeneously broad-
ened ensemble of quantum emitters as a broad-band light source, we first establish
that our resonator supports edge modes that extend throughout the length of the
resonator. Subsequently, using a single quantum emitter and a chiral waveguide
coupled to the topological resonator, we demonstrate chiral spontaneous emission
where the direction of the emission depends on the polarization of the emitted light.
Having established the chirality of our resonator, we finally proceed to demonstrate
strong coupling between the quantum emitter and the topological resonator. Specif-
ically, we use a magnetic field to tune a quantum emitter into resonance with the
topological resonator and show Purcell enhancement of emission. Our results pave
the way for explorations of strong interactions between multiple solid-state quantum
38
emitters coupled via a chiral nanophotonic resonator.
4.2 Proposed Structure
4.2.1 Valley-Hall topological photonic crystal
Our chiral resonator is based on a valley-Hall topological photonic crystal
[126?131], composed of a honeycomb lattice of triangular holes with a lattice con-
stant of ?a?, as shown in Fig. 4.1a. The two triangular holes in each rhombic unit
cell(dotted rhombus), have different sizes ( 1?.3a and 0?.7a), which leads to the open-
2 3 2 3
ing of topological bandgaps, at the K and K? points. Because of the time-reversal
symmetry, the Berry curvature integrated over the entire Brillouin zone is zero.
However, the Berry curvature at K and K? valleys have opposite signs. Interchang-
ing the two triangular holes in the unit cell (green and violet regions in Fig. 4.1)
flips the sign of the Berry curvature at each valley. Therefore, by interfacing these
two topologically distinct regions, we form a pair of counter-propagating edge states
with opposite helicity at the boundary between the two topologically distinct regions
(shown as an orange band in Fig. 4.1c). These edge states are transverse electric
(TE) modes composed of an in-plane electric field and can be selectively excited
by placing an emitter at specific position along the interface of the waveguide with
suitable circular polarization (see Appendix C)
39
(b) (c) 1100(a) (c)
1050
1000
a 950
900
850
500nm 800
0 0.2 0.4 0.6 0.8 1
wavevector (    )
(d) (e) 6 (f)
15 10 Topological band
10
2 ?m
(g)
5
1?m
0
900 920 940 960 980 1000 2 ?m
wavelength (nm)
Figure 4.1: Valley-Hall topological photonic crystal resonator and FDTD simulation
results: (a) Schematic of the topological interface showing two topologically distinct
regions (violet and green), by interchanging the triangular holes sizes. Black dashed
lines indicate the rhombic unit cells. (b) SEM image of the fabricated topological
interface. (c) Simulated TE-band structure of the interface. The orange band
highlights the edge band. (d) SEM image of a topological resonator shaped in the
form of a super-triangle, with two topologically distinct regions (violet and green).
(e) Simulated longitudinal modes of the resonator. As in (c), the shaded orange
region corresponds to the topological edge band. (f) and (g) The electric field
distribution of a resonator mode and a bulk mode.
40
intensity counts
wavelength (nm)
4.2.2 Fabricated resonator structure
The topological edge states are robust against sharp bends of 60 and 120
degrees, which preserve the symmetry of the structure. This robustness enables
them to form a resonator using a super-triangle (Fig. 4.1d), where the inside/outside
of the super-triangle corresponds to two topologically distinct regions. Analogous to
the one-dimensional edge states, this confined resonator structure hosts two counter-
propagating modes with opposite helicity. We fabricated the resonator structure on
a GaAs slab of thickness 160nm with an embedded layer of InAs quantum dots
as quantum emitters. We patterned the structure using electron beam lithography
followed by inductively coupled plasma reactive ion etching. We chose the lattice
spacing of a =265nm, such that the topological band gap coincides with the emission
spectra of quantum dots. Appendix C provides a detailed description of fabrication
steps.
4.2.3 Simulation results
The 3D Finite-difference Time-domain (FDTD) simulation for a resonator
of length 13?m (50-unit cells) shows multiple longitudinal modes, separated by a
free-spectral range (FSR) of 15nm (Fig. 4.1e). The calculated electric field profile
of the resonator modes exhibits strong transverse confinement at the topological
interface and extend over the super-triangle (Fig. 4.1f). Moreover, we do not observe
any scattering at the three sharp corners of the resonator, indicating topological
robustness. By contrast, the modes outside of the topological bandgap reside in the
41
bulk and are not extended along the resonator perimeter (see Fig. 4.1g). Since these
modes are not protected, their frequency and their field profile are susceptible to
the disorder.
4.3 Observation of topological resonator modes
To show the modes of the topological resonator, we excite the quantum dots
at the tip of the resonator (shown in Fig. 4.2a) using a high-numerical aperture
objective and a continuous wave laser at 780nm. We perform all measurements in a
closed-cycle Helium cryostat, operating at 3K temperature. At this temperature, we
use a high pump power of 100?W such that the quantum dots saturate and broaden
to form a continuous internal white light source that probes the spectrum of the
resonator. We collect the photoluminescence spectra from different points along the
perimeter of the resonator (Fig. 4.2 b-d). We observe three peaks (within shaded
grey regions) within the expected topological band which do not change when we
move our collection point along the length of the resonator, indicating that these
are the extended longitudinal modes of the topological resonator. Moreover, their
free spectral range (? 13nm) matches closely with the simulation. The other peaks
in the PL spectra change with the collection point, and therefore, correspond to the
bulk modes of the photonic crystal structure.
42
Laser
(a)
1 3
2
1?m
(b) FSR~13nm position-1
(c) position-2
(d) position-3
900 910 920 930 940 950 960 970 980 990 1000
wavelength (nm)
Figure 4.2: Topological waveguide-resonator system: (a) Schematic showing the
excitation and collection points on a topological resonator device. Numbered marks
show the position of collection spots. (b,c,d) The experimentally measured spectrum
of a topological resonator from three different collection points along the length of
the resonator. The peaks inside the grey regions correspond to the resonator modes.
43
intensity counts
4.4 Observation of chiral coupling of an emitter inside topological
resonator
4.4.1 Waveguide-Resonator Hybrid Device : FDTD results
To probe the chiral coupling of an emitter to a topological resonator mode, we
coupled the resonator to a helical topological waveguide, which is similarly designed
(Fig. 4.3a). In this experiment, we excited the quantum dots at the point A on
the resonator and collected the emission from the ends of the waveguide through
the grating couplers. In order to analyze the chirality of the resonator mode, we nu-
merically calculate the Poynting vector for the coupled resonator-waveguide device.
Figure 4.3b shows the Poynting vector when the system is excited with a right
circularly polarized dipole (??). We observe that the electric field travels clock-
wise around the super-triangle, and then again chirally couples to the right of the
waveguide. Due to time reversal symmetry, when the system is excited with a left
circularly polarized dipole, we see the electric field travels anti-clockwise along the
super-triangle, before exiting to the left of the waveguide (Fig. 4.3c). Note that
the clockwise/anti-clockwise mode of the resonator couples to the right/left-going
waveguide mode, respectively. This is due to the fact that the region below the
waveguide has the same topology as the inside of the super-triangle, and therefore,
the helicity is preserved at the boundaries. Here, we emphasize that the above men-
tioned polarization-momentum locking is dependent on the spatial position of the
quantum dot with respect to topological interface.
44
(a) Laser (d)
1
A
0.5
L R
2?m
0
926.6 926.65 926.7 926.75 926.8 Max
wavelength (nm)
(b) (e)
1
0.5
0
926.6 926.65 926.7 926.75 926.8
(c) (f ) wavelength (nm)
1
Min
0.5
0
926.6 926.65 926.7 926.75 926.8
wavelength (nm)
Figure 4.3: Chiral waveguide-resonator-emitter coupling: (a) SEM image of the
topological resonator coupled with a topological waveguide, terminated by two grat-
ing couplers (pink shaded half-circles). The yellow line indicates the interface be-
tween two topological regions, highlighted in green and violet. Point A indicates
the point of excitation. (b,c) Simulated Poynting vector profile along the perime-
ter of the resonator-waveguide system, when the system is excited with a right
circularly-polarized dipole and left circularly-polarized dipole respectively. (d,e,f)
The measured PL signal, as a function of the magnetic field strength, collected from
point A, Left grating, Right grating, respectively.
45
magnetic field (tesla) magnetic field (tesla) magnetic field (tesla)
Normalized intensity
4.4.2 Chiral coupling
To experimentally distinguish the chirality of the emitter-resonator-waveguide
coupling, we perform a photoluminescence measurement on device shown in Fig.
4.3a. Specifically, we use two degenerate dipole transitions of the QD which are
selectively coupled to the two helical edge states in the topological resonator. We
apply a magnetic field on QDs, in a Faraday configuration, such that the Zeeman
shift lifts the degeneracy of the two transitions [101]. This allows us to spectrally
resolve the chiral nature of the light-matter interaction. Note that the helical nature
of the resonator and the waveguide remains intact under such a magnetic field. We
excite the sample at point A with a continuous wave 780nm wavelength laser at
power 7?W, such that individual QDs can be spectrally resolved. When the single
is collected at the point A (Fig. 4.3d), we observe both branches of the Zeeman
split QD spectrum, corresponding to two oppositely-circular polarization. However,
when we collect the signal from either of the grating couplers (Fig. 4.3(e)-(f)), we
observe a single branch, as a signature of chiral coupling. We calculate near 89%
directionality between the two gratings in this measurement.
4.4.3 Polarization-momentum locking
The above mentioned spin-momentum locking in this type of resonator system
is dependent on the position of the quantum emitter along the waveguide. To affirm
this fact, we excite a single resonator modes at different spatial points along the
transverse direction with two circularly polarized dipoles and study the intensity
46
1
0.8 Light Circular
0.6 Reft Circular
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
Transverse cross section (?m)
Left grating Right grating
1 1
0.5 0.5
0 0
903.95 904.1 904.15 903.95 904.1 904.15
wavelength (nm) wavelength (nm)
Min Max
Normalized intensity
Figure 4.4: Position dependent spin-momentum locking : (a)Simulated results for
contrast between the two grating with two different types of polarization excitation
along the transverse direction of the waveguide. (b),(c) The measured PL signal, as
a function of the magnetic field strength, collected from the Left grating, and Right
grating, respectively.
47
magnetic field (tesla) Contrast  (IR - IL)/(IR +IL)
magnetic field (tesla)
variations at the two gratings. The results show that if the position of the emitting
dipole is changed then the direction of the out-coupled light changes as shown in Fig.
4.4(a). Moreover, there are points of high chiral points where the magnitude of con-
trast reaches the maximum. Experimentally we further carry out PL measurements
on another device. We find that within one excitation spot of the laser, depending
on the position of the dot, the corresponding circular polarized emission will couple
differently to the right/left propagating modes. This is apparent from Fig. 4.4(b)
and (c) where the emission from a quantum dot at 903.95nm couples differently into
the waveguide compared to the ones at 904.1nm and 904.15nm. Here, we want to
note that even though the spin-momentum locking is similar to that found in ring
resonators [96, 111, 112], these topological resonator modes have topological origin
and arise from the fact that the two photonic crystals on either side of the triangular
resonator have different band topology and therefore robust to bends and certain
imperfection [130].
4.5 Intensity enhancement of a single emitter coupled to the res-
onator
In order to demonstrate that the quantum dots are coupled to the resonator, we
measure their intensity as a function of cavity detuning. By scanning the magnetic
field, one Zeeman branch of the QD spectrum can be tuned into the resonance of the
topological resonator. Figure 4.5a shows the spectrum at two different excitation
power. At high excitation power (100?W), the saturated QDs emission reveals the
48
(a) 6000 (b)
Resonator 100 W
4 Resonator (c) 3000
6000
5000 20 W 2500
x5 3 50004000 2000
4000
3000 2 3000 1500
2000 2000 1000
QD 1
1000 1000 500
0
0
968.5 969 969.5 970 970.5 971 0
QD 0
968.5 969 969.5 970 970.5 -0.5 -0.25 0 0.25 0.5
wavelength (nm) wavelength (nm) detuning (nm)
Figure 4.5: Purcell enhancement of a quantum emitter coupled to a topological
resonator: (a) The measured PL, at zero magnetic field, exhibiting the QD and the
resonator mode at high and low excitation power. (b) Enhancement of emitter?s
PL, when the QD is tuned to the topological resonator, by scanning the magnetic
field. (c) The intensity of the QD?s emission, as a function of detuning from the
resonator mode.
cavity mode, as shown by the green curve. By decreasing the excitation power to
20?W, the spectrum shows multiple individual quantum dot lines near the cavity
resonance. Fig. 4.5b shows the measured spectra as a function of the magnetic
field. As the magnetic field increases the quantum dots Zeeman split, and either the
lower or upper branch crosses the cavity mode (depending on the initial detuning).
The emission from the quantum dots is enhanced as they tune onto resonance with
the cavity mode.
We focus on one particular dot labeled ?QD? in Fig. 4.5a. This dot becomes
resonant with the cavity at a magnetic field of 2.7 Tesla. To quantify the degree
of enhancement, we fit the quantum dot to a Lorentzian function at each magnetic
field and plotted its intensity as a function of detuning from the cavity (Fig. 4.5c).
At zero detuning from the resonator line, we see a nearly threefold increment in the
49
intensity counts
magnetic field (tesla)
intensity counts
count rates. From this emission enhancement, we estimate an intensity enhancement
of 3.4.
Note: we used different measurement schemes for obtaining results depicted
in different figures. We performed these measurements on different types of devices
with different does array. Data presented in Fig. 4.2 4.3 4.5 has been taken from
sample # D240 m3s2b 22 , # D240 m3s2b1d3 22, # D240 m3s2b1 21 respectively.
4.6 Summary
In summary, we demonstrate chiral light-matter interactions in a topological
resonator. We use valley-Hall topological edge states to realize a helical resonator.
Such a helical resonator is created at the interface of two distinct topological regions
that supports two counter-propagating light modes with opposite polarizations. We
show the chiral coupling of the resonator to a quantum emitter. Moreover, we
achieve an intensity enhancement of 3.4 due to resonant coupling. These findings
provide new approaches to study chiral quantum optics with potential applications
in quantum information processing and quantum many-body physics.
50
Chapter 5: Conclusion and Outlook
In conclusion,we have proposed a new all-dielectric photonic crystal design
and presented simulation results showing that three-dimensionally guided topolog-
ical edge states at optical frequencies can be realized. The design parameters are
amenable to implementation with well-established nanofabrication techniques. The
simulations focus on GaAs as the dielectric substrate but the photonic crystal de-
sign principles that give rise to topological edge states are applicable to many other
dielectric materials which can be fabricated with well-established nanofabrication
techniques. The advantage of GaAs is that it can be integrated with quantum
dots for example made of InAs whose charged states in the presence of magnetic
fields have proven to be a promising solid-state qubit [136?138], and whose elec-
tronic transitions can emit circularly polarized light and can be interfaced with
the helical topological edge states. Another potential substrate is InP, which can
also be integrated with InAs quantum dots to realize topological edge states at
telecommunication wavelengths [?, 139]. Aside from epitaxially grown quantum
dots, defects in two-dimensional materials [80] can act as quantum emitters which
emit circularly polarized light at optical frequencies [140?142], and these can be
placed atop for example a silicon nitride photonic crystal [143, 144] nanofabricated
51
into the honeycomb-lattice-like structure described here. Other potential quantum
emitters that could be integrated with topological edge states are defects in mate-
rials such as diamond [145, 146] or silicon carbide [147?149], which can be coupled
to a photonic crystal [150] or even directly fabricated into photonic crystal struc-
tures [151?153]. Finally, cold atom systems could also be integrated with these
photonic crystals [154?159]. Thus the demonstrated design should be applicable to
a wide range of dielectric materials to realize topological photonics at optical fre-
quencies and interfaced with various quantum emitters. With the further prospect of
integration with various quantum emitters ranging from quantum dots [136?138], de-
fects in two-dimensional materials [80] and diamond [145,146], this system promises
to open a new path to research in topological phenomena with optical systems.
Moreover,we also demonstrated coupling between single quantum emitters and
topologically robust photonic edge states. The present approach opens up new
prospects at the interface of quantum optics and topological photonics. In the con-
text of chiral quantum optics, one can explore new regimes of dipole emission in
the vicinity of a topological photonic structures and exploit the robustness of the
electromagnetic modes [96]. Furthermore, in a chiral waveguide, photon-mediated
interactions between emitters are location-independent [104]. This property could
facilitate the coupling of multiple solid-state emitters via photons while overcom-
ing scalability issues associated with random emitter position, enabling large-scale
super-radiant states and spin-squeezing. Ultimately, such an approach could form
a versatile platform to explore many-body quantum physics at a topological edge
[105], create chiral spin networks [104, 106], and realize fractional quantum Hall
52
states of light [107,108].
Additionally, building on the already demonstrated topological platform, next
we studied a topologically robust and chiral interface between a photonic resonator
and a quantum emitter. This platform could provide a robust and scalable pathway
to engineer chiral light-matter interaction between multiple emitters coupled to a
single resonator. It is important to note that topological design does not require
fine tuning of system parameters, they are like LEGO! On the one hand, such
resonators could enable the generation of entangled states of photons, mediated by
chiral coupling of photons to quantum emitters [96], such as superradiant [132,133]
and cluster states [110]. On the other hand, one could conceive generating entangled
states of several solid-state spins, mediated by the helical-circulating photonic modes
[109]. In contrast to conventional waveguides, the mediated interaction strength
between spins does not depend on the distance, since the emitters cannot form a
mirror in a chiral interface. Ultimately, chiral and topological interfaces provide a
new approach to study QED in a new regime [107,108,134,135].
53
Appendix A:
A.1 Honeycomb Lattice with a Six-Site Basis and Band Folding
The photonic crystal we study is a modification of the usual honeycomb lat-
tice. For the special case that the lattice parameters obey R = a0/3, the standard
honeycomb lattice is recovered (see main text for definitions). Typically, the honey-
comb lattice is taken to be a triangular lattice with a two-site basis. For the general
case R 6= a0/3, it is convenient to consider the system as a triangular lattice with a
six-site basis with primitive lattice vectors
(? )
a1 = 3, 0 a, (A.1)(? )
a2 = 3/2, 3/2 a, (A.2)
?
where a = 3R. Figure A.1(a) shows the first Brillouin zones (FBZs) for both the
two-site (dashed hexagon) and six-site (solid hexagon) bases.
The equivalence of these two descriptions can be verified by counting the total
number of states in each case. The two-site basis is described by two bands over the
FBZ of area A, giving a total number of states corresponding to an effective k-space
area of 2A. In the six-site case, each linear dimension of the FBZ is reduced by a
54
?
factor 3 and thus the area is A/3. Since there are six bands, this again gives a
total effective area of 6? A/3 = 2A.
In the case of graphene, it is well-known that the Dirac cones are located at the
edges of the FBZ [labeled by K? in Fig. A.1(a)]. In the six-site basis, these degrees
of freedom now reside at the zone center (?). The K? and ? points are connected
by a reciprocal lattice vector. The bands in Fig. A.1(c) can be obtained by folding
along the vertical dashed lines in Fig. A.1(b) so that the K? is matched to ?. At
this point, the bands formerly at the two inequivalent Dirac points K? and K? ? come
together to form a doubly degenerate Dirac point. We will designate these degrees
of freedom by a pseudospin (?) [29].
A.2 Tight-Binding Description of the Dispersion
The dispersion of our system near ? (k = 0) can be obtained by a tight-
binding model. Following [160], we take a set of basis states for which the magnetic
field profile is concentrated in a particular hole. The time-evolution of the system is
characterized by ?hopping? to adjacent holes in the lattice. Typically, the application
of tight-binding is limited to electronic systems in which electrons hop between
weakly coupled atomic orbitals [163]. In the present context, the tight-binding
method accurately captures the behavior of the band structure near ? due to the
fact that the near ?, the band-structure is tightly constrained by the symmetries of
the lattice. In particular, the tight-binding Hamiltonian H naturally incorporates
the C6v symmetry of the lattice and the triangular holes. For the generic case that
55
Figure A.1: Correspondence between the two-site and six-site bases : (a) Bound-
aries of the first Brillouin zone for the two-site (dashed hexagon) and six-site (solid
hexagon) bases for the honeycomb lattice. (b) The band structure of the honey-
comb lattice considered with a two-site basis. The labels on the horizontal axis
corresponds to the high-symmetry points in k-space as designated in (a) [161]. The
gray area is the area above the light cone where guided modes are not possible. The
red triangles (blue squares) correspond to the red (blue) paths indicated in (a). (c)
The band structure of the honeycomb lattice with a six-site basis as obtained by
folding the band structure in (b) along the dotted vertical lines.
56
a0 6= 3R, the spectrum is gapped. For a0 = 3R corresponding to a honeycomb
lattice, an additional C3v symmetry ensures that the dispersion remain gapless at
the Dirac point.
We describe our system as a triangular lattice with a six-site basis labeled
A,B,C..., F starting with the right-most site and progressing in a counter-clockwise
manner (Fig. A.2). The states of our system |A?, |B?, |C?...,|F ? are the Wannier
functions for the system. For example, the state |C? describes an electromagnetic
field configuration for which the out-of-plane magnetic field is centered on the C
hole in each six-membered ring. In the bandwidth of interest, the magnetic field
configurations can be written as linear combinations
?
??
?
?? ? ?A ???
???
?? ??B?? ?
??
? ?? ?? ?C ?|?(k )? = ? ???, z ? eik?r? ? , (A.3)?? ?? D ??? ???? ? ????? E ?
?F
57
Figure A.2: Schematic of our lattice parameters and the labeling of the lattice sites
for our tight-binding model. A cluster consists of six sites. Then the system is a
triangular lattice of clusters with lattice constant a0. The distance from the centroid
of each hole to the center of its cluster is R. The tunneling amplitudes t1 and t2
correspond to intra- and inter-cluster tunneling between the nearest neighbor holes.
The labels A, B, C, D, E, F denote each lattice site making up the basis.
58
where r = (x, y) and Eq. (A.3) is written in the basis
?? ? ? ? ? ???? 1 ???? ???? 0 ???? ???? 0 ??? ? ? ? ?
????
??? 0 ???? ???? 1 ?? ? ? ?? ?
? 0 ??
? ? ? ??? ???? ?????? 0 ? ? 0 0 ?|A? = ?? ???? , |B? = ???? ??
?
?? , ... |F ? =
????? ?
???? . (A.4)?? 0?? ???? ??
0
?? ???? ???
0
? ????0 0 0
???? ???? ???? ???? ???? ????
0 0 1
The action of the Hamiltonian operator is to evolve the state in time. Roughly,
the matrix elements of H indicate the field configurations which can evolve into each
other on a time scale ? R/c. On this time scale, only states which are localized
to adjacent sites can evolve into each other appreciably and thus we only consider
nearest-neighbor ?hopping?. The Hamiltonian H = H1 +H2 receives contributions
from intra- and inter-cluster couplings, respectively. Intra-cluster hopping is char-
acterized by a parameter t1 and takes the form
? ?
???? 0 1 0 0 0 1? ?
?
??? ?
??
? 1 0 1 0 0 0 ???? ???0 1 0 1 0 0 ?
H1 = ?t1?????? ?
?
? 0 0 1 0 1 0 ?
? . (A.5)
? ?? ?????
??
0 0 0 1 0 1 ?????
1 0 0 0 1 0
59
Inter-cluster coupling is described by
? ?
??? ik?a1? 0 0 0 e 0 0? ??
???
??? 0 0 0 0 eik?a2? 0? ?
?
? ?????? 0 0 0 0 0 e
ik?(a2?a1)
H = ?t ? ?
?
2 2? ? . (A.6)? ?? e?ik?a1 0 0 0 0 0 ???????? ?? 0 e?ik?a2 0 0 0 0 ???? ??
0 0 t2e
?ik?(a2?a1) 0 0 0
We introduce generalized plane wave states
?? ??1
???? ????? z ?
???
?? 2 ????
?
z
| 1?(k ?, z)? = ? ??? ???? ik?r? ? e , (A.7)6 ?? z3 ??
???? ?? z4 ???? ??
z5
where z is a complex number of unit magnitude whose phase is associated with an-
gular momentum around the hexagonal clusters (or pseudospin, in the terminology
of of [29,160]) and k = (kx, ky). Although the full rotational symmetry is broken by
the crystal axis, the states corresponding to z = e?i?/3 possess strong p-like char-
acter, while those with z = e?i2?/3 have d-like character. This can be most easily
seen by noting that the various states |?(z)? are ?sampled? from continuous angular
60
wave functions as follows
|?(z = e?i?/3)? = e?i? ? |p??, (A.8)
|?(z = e?i2?/3)? = e?i2? ? |d??. (A.9)
The ? labels the pseudo-spin degree of freedom. The geometry of the wavefunctions
is clarified through the definitions
|px? ?
1
= (|p+?+ |p??) , (A.10)
2
| ? ?1py = (|p+? ? |p??) (A.11)
i 2
where |px? is odd about the x-axis, etc. Similarly, we have
| ? ?1dx2?y2 = (|d+?+ |d??) , (A.12)
2
| ? ?1dxy = (|d+? ? |d??) , (A.13)
i 2
where |dx2?y2? is a wave function whose maxima coincide with the x- and y-axes as
? = 0? 2?, etc.
We now derive the spectra associated with these 4 states near ? by expanding
Eq. (A.6) to linear order in kx and ky. In this limit, the effective 4? 4 Hamiltonian
is block diagonal, and only states of the same pseudo-spin are coupled. The effective
61
Hamiltonian for the (+)-pseudo-spin is given by
?
3 [ ]H+ = t2a (?kx?x + ky?y) + t2 ? t1 +O(k2x + k2y) ?z, (A.14)2
in the (|p+?, |d+?)T basis. Similarly, in the (|p??, |d??)T basis we find
?
3 [ ]H? = t2a (kx?x + ky?y) + t2 ? t1 +O(k2x + k2y) ?z. (A.15)2
?
In both cases, we have performed a unitary transformation U = ei ?2 z . We note
that in the limit that the various honeycombs are completely decoupled, t2 ? 0 and
Eqs. (A.14) and (A.15) reflect the fact that the p-states have a lower energy than
the d-states. For t1 = t2, H+ and H? are characterized by a Dirac cone spectrum.
For t1 6= t2, the spectrum acquires a gap of size |t1 ? t2|.
A.3 Topology and Edge States
In the previous section, we showed that a honeycomb structure can be de-
scribed by a gapless Dirac Hamiltonian. When we introduce the lattice deforma-
tions, i.e., shrinking/expanding, a gap opens which can be described a mass term
(m?z). Here, we review the concept why the band inversion, i.e., changing the sign
of mass, results in having a topological edge at the boundary.
When the system is gapped, its topology can be characterized by a Chern
62
number for the pseudospins (?). A spin Chern number takes the form
C = C+ ? C?, (A.16)
where C = ?1? sgn(m?), where m? are the masses for the two pseudo-spins [162].2
Thus, we have
C = sgn(t2 ? t1). (A.17)
Topologically-protected edge modes will exist between gapped regions with different
C ?s, i.e., any place that the quantity t2 ? t1 changes sign.
In order to understand the edge state structure, we begin by considering H+
with a spatially varying mass. For concreteness, we consider the situation outlined
in Fig. 4b in the main text As we will see, edge states are localized to domain walls
for which m(x) = t2 ? t1 ? 0. The edge states satisfy the Heisenberg equation
of motion which, for H+ [Eq. (A.14)], is the Dirac equation. The Dirac equation
corresponding to H+ is
[?ih?v (??x?x + ?y?y) +m?z] ? = E?, (A.18)
?
where v = 3t1a0/2 and E is the energy of the eigenstate ?.
Consider the geometry shown in Fig. 4(b) of the main text, which shows an
area of shrunken hexagons above expanded hexagons. The system is described by a
63
mass which depends only on y, i.e., m(x, y) = m(y) and m(0) = 0 with
dm
< 0. (A.19)
dy
In this case, the topologically protected solution
( ? y )1
?(y) = ? exp m(y?)dy? , (A.20)
h?v 0
is an x-independent solution of the Dirac equation with zero energy where ? is a two-
dimensional spinor. This is the celebrated Jackiw-Rebbi solution of the Dirac equa-
tion with a spatially varying mass [165]. The sign in the exponent of ? [Eq. (A.20)]
ensures that the solution is normalizable. The edge state decays exponentially for
both y > 0 and y < 0. The spinor ? obeys
?x? = ?. (A.21)
Thus, ? ?
?1= ??? ?? 1 ???? , (A.22)2
1
in the (|p+?, |d+?)T basis. The full edge mode is described by
??1 ??? 1 ?
?
?? ( 1 ? )?(x, y) = ? ? yexp m(y?)dy? eikxx. (A.23)2 h?v 0
1
64
Again, plugging into the Dirac equation gives an energy dispersion
E(kx) = ?h?vkx. (A.24)
Since the group velocity is given by v = 1 ?E , this represents an edge state travelling
h? ?kx
in the ?x-direction. Indeed, we see that in Fig. 4(b-i) of the main text, the
excitation of the +-pseudospin leads to a left-moving edge state. Similarly, an edge
state derived from the H? channel (opposite pseudo-spin) would travel in the +x-
direction.
A.4 Inversion of the Eigenstates
We examine the out-of-plane magnetic field eigenstates of the system at the
symmetry plane (z = 0) corresponding to the ? point for the shrunken and expanded
clusters. The band structures for the shrunken and expanded cluster systems are
shown in Fig. A.3(a) and (d) and are the same as Fig. 2(c) and (e) in the main text.
The eigenstates corresponding to these band structures show that the eigenstates
are inverted; by that we mean that e.g., the eigenstate px (dxy) shown in Fig. A.3(b)
[A.3(c)] which appeared on the lower (upper) band for the shrunken cluster appears
on the upper (lower) band for the expanded cluster as shown in Fig. A.3(f) [(e)]. This
band inversion indicates that there is a change in the band topology, as discussed
in the previous section on the tight-binding model.
65
Figure A.3: Band inversion : (a) and (d) Band structures for the shrunken and
expanded cluster systems, which are the same as Fig. 2(c) and (d) respectively
in the main text, with a subset of the eigenstates indicated at the ? point. (b)
and (c) [(e) and (f)] Out-of-plane magnetic field eigenstates at the symmetry plane
z = 0 of the lower and upper band for the shrunken (expanded) cluster system
respectively. We see that e.g., the eigenstate px for the lower band in the shrunken
cluster system appears on the upper band for the expanded cluster system, which
indicates a change in the band topology.
66
A.5 Polarization Pseudo-Spin of the Eigenstates
From Maxwell?s equations, at the symmetry plane z = 0, the out-of-plane
magnetic field eigenstates pxz? and pyz? lead to an in-plane electric field given by
i i
E1 = ?? (pxz?) E2 = ?? (pyz?), (A.25)
?0(r) ?0(r)
where Ei = Eixx? + Eiyy? (i = 1, 2), 0 ? 8.854 ? 10?12 Farad/m is the vacuum
permittivity, and (r) is the position-dependent relative permittivity. The out-of-
plane magnetic fields of the px and py eigenstates are shown in Figs. A.4(a) and (b),
respectively. We see that the px and py modes are related by a ?/2 rotation, so we
have at the center of the cluster (r = 0) the relation
?? ? ? ???? E2x ???? ???? ?E1y= ???? (A.26)
E2y E1x
From this we find the relation
i ?? [(px ? ipy)z?] = (E1x ? iE1y)(x?? iy?). (A.27)
?0(0)
This implies that at the center of the clusters the in-plane electric field po-
larization is either ?+- or ??-circularly polarized depending on the out-of-plane
? ?
magnetic field eigenstates p? = (px ? ipy)z?/ 2 where ?? = (x? ? iy?)/ 2. We can
see this directly in Figs. A.4 (c) and (d), which show ?? ? |E |2+ ? |E?|2 where
?
E? = (Ex ? iEy)/ 2, characterizing the degree of circular polarization. In both
67
Figure A.4: Out-of-plane magnetic field and in-plane electric field polarization in a
cluster of six holes outlined by black triangles. (a) and (b) depict the real part of
the out-of-plane magnetic field (Hz) for the eigenstates px and py respectively. The
colors indicate the strength of Hz. (c) and (d) show ?? ? |E |2+ ? |E?|2, indicating
the pseudo-spin nature of the bands excited at the ? point with px+ ipy and px? ipy
modes, respectively.
cases we see that at the center the in-plane electric field is highly circularly polarized,
except with opposite handedness. Thus the out-of-plane magnetic field eigenstates
p? have an associated in-plane electric field circular polarization of ?? which act as
pseudo-spins for this topological photonic crystal.
68
Appendix B:
B.1 Device Fabrication
To fabricate the device, we began with an initial wafer composed of a 160
nm GaAs membrane on top of 1 ?m sacrificial layer of Al0.8Ga0.2As with quantum
dots grown at the center. The quantum dot density was approximately 50 ?m?2
Based on the given quantum density and cross-sectional area of the waveguide, the
probability of finding two dots in the structure with the same resonance is less than
0.7%. Thus, it is extremely unlikely in a given device for a photon emitted by one
dot to be scattered by a second.
We fabricated the topological photonic crystal structure using electron beam
lithography, followed by dry etching and selective wet etching of the sacrificial layer.
We first spin-coated the wafer with ZEP520A e-beam resist, then patterned the
structure using 100 keV acceleration voltage and developed the resist using ZED50
developer. After patterning, we used chlorine-based inductively coupled plasma
etching to transfer the pattern on the GaAs membrane. We finally performed se-
lective wet etching using HF acid to create a suspended structure with air on top
and bottom. The rectangular structures in the periphery are included to facilitate
undercut of the sacrificial layer.
69
Figure B.1: Mask design for fabrication of triangles. (A) Layout of regular mask.
(B) SEM image of rounded triangles resulted from use of regular mask. (C) Layout
of modified mask; triangles are bent from edges to mitigate etching imperfections.
(D) SEM image of sharp triangles fabricated with use of modified mask.
70
Sharp corners with straight side walls are essential to observe the topological
helical edge modes. It is confirmed via simulation that triangles with rounded cor-
ners are detrimental for the device operation. However, even with highly directional
dry etch, creating sharp features like triangles is challenging at such small length
scales. We observed ? by using a regular mask design (as shown in Figure B.1.A)
? that etching causes widening of holes which eventually results in rounded corners
much like a Reuleaux triangle (Figure B.1.B). We used a modified mask design to
overcome this challenge. Triangles with shrunk edges shown in Figure B.1.C are
used as a mask; this results in sharp triangles with edge lengths of 140 nm. Close
up SEM image of final structure is shown in Figure B.1.D.
B.2 Experimental Setup
To perform measurements, we mounted the sample in a closed-cycle cryo-
stat and cooled it down to 3.6 K. A superconducting magnet, contained within the
closed-cycle refrigerator, surrounds the sample and applies a magnetic field of up
to 9.2 T along the out-of-plane (Faraday) direction in order to generate a Zeeman
splitting between the two bright excitons of the quantum dot. We performed all
sample excitation and collection using a confocal microscope with an objective lens
with numerical aperture of 0.8. We collected the emission and focused it onto a
single mode fiber to perform spatial filtering. To perform spectral measurements,
we injected the signal to a grating spectrometer with a spectral resolution of 7
GHz. For autocorrelation measurements, we used a flip mirror to couple the light
71
out of the spectrometer and processed the filtered emission using Hanbury-Brown
Twiss intensity interferometer composed of a 50/50 beamsplitter, two Single Pho-
ton Counting Modules (SPCMs) and a PicoHarp 300 time correlated single photon
counting system.
The quantum dots are less than 20 nm in diameter, while the laser spot size
is approximately 0.4 um. The density of quantum dots are 50 ?m?2 which means
that there are approximately 25 dots within the excitation spots. However, due to
the large inhomogeneous broadening of the ensemble, each of these dots emits at a
different wavelength. We isolate individual quantum dots by spectral filtering using
a grating spectrometer with a resolution of 0.02 nm. The spectrometer selects the
emission from only a single dot, as evidenced by the anti-bunching dip observed in
Fig. 4D-E which dips below 0.5.
B.3 Grating Calibration
Since both left and right grating couplers are fabricated under similar condition
they are identical in terms of coupling efficiency. To test this fact we calibrated them
with respect to. the transmission spectrum of the topological waveguide. Figure
B.2.A shows the different positions on the device. We shine an intense excitation
beam of 780 nm with 1.5uW power at the center of the waveguide (M). At this
high power all the quantum dots are saturated and emit a broadband spectrum
ranging from 900-980nm. We collected the transmitted signal from left (L) and
right grating (R). Figure S B.2.B shows almost equal counts coming from both
72
Figure B.2: Transmission data from left and right gratings. (A) Scheme for excita-
tion and collection. (B) Transmitted signal collected from two gratings.
the gratings with almost overlapping transmission spectrum. Additionally, the area
under the curves give approximately 40 million counts/sec for the gratings thus
indicating equal coupling efficiency.
B.4 Coupling Efficiency
The coupling efficiency of emission from a single quantum emitter into the
topological waveguide is defined by
IL + IR
? = (B.1)
IL + IR + IM
where IL and IR are the integrated photon counts propagating to the left and
right waveguide modes respectively, and IM is the photon counts emitted directly
from the middle of the device into free space. We can estimate these intensities by
measuring the brightness at the three locations denoted in the main text. Table B.1
shows the coupling efficiencies calculated for different dots coupled to our topological
73
device. The table reports integrated count rates for an integration time of 1s at each
point. We determine the average to be 68%.
Coupled QDs IM IL IR ?(%)
1 699 772 740 77.98
2 655 755 735 88.89
3 680 780 780 84.93
4 1300 1400 1900 75.23
5 802 1080 933 81.17
6 739 1021 654 78.85
7 795 1206 645 77.95
8 1090 1061 724 53.50
9 976 934 667 50.00
10 677 1079 807 92.44
11 869 728 819 54.90
12 1531 809 986 37.56
13 884 716 700 39.06
Avg 68.65
Table B.1: Estimation of coupling efficiency.
74
Appendix C:
C.1 The polarization profile of a topological resonator
The electric field for the topological resonator mode is composed of two circularly-
polarized components, where the high-field intensity points appear in different lo-
cations, as shown in Fig. C.1 Although these waveguides exhibit polarization-
momentum locking, these plots indicate that the polarization profile changes in the
transverse direction. If an emitter is located at the peak of right-circularly polarized
light, and the QD is prepared to emit into right/left-circularly polarized light, then
the emitted light travels in a clockwise/anti-clockwise fashion around the resonator.
C.2 Sample fabrication
The initial wafer is composed of 160 nm GaAs membrane with quantum dots
grown at the center of the growth axis. 1 ?m sacrificial layer of Al0.8Ga0.2As be-
neath the active layer is used to undercut and to create suspended structures. The
quantum dot density was 50 ?m?2, which is high enough to find emitters resonant
with the cavity mode. Based on the given quantum dot density and inhomogeneous
broadening, the photon emitted by one quantum dot to be scattered by a second
75
500nm 500nm
Min Max
Figure C.1: Simulated left/right circular polarization components of the electric
field. White dotted line marks the perimeter of the resonator.
is calculated to be extremely unlikely. Samples are first spin-coated with ZEP520A
positive e-beam resist, followed by patterning using 100 keV high-resolution e-beam
system. Exposed regions are developed by using ZED50 developer. After pattern-
ing, chlorine-based directional ICP etching is performed to transfer the patterns
from ZEP520A hard mask to the GaAs layer. Lastly, selective wet etching using HF
acid is performed to create a suspended structure with air on top and bottom.
Due to imperfections in e-beam exposure and directional dry etching, fabrica-
tion of small and sharp corners is challenging. The loss in the resolution is mitigated
by using a modified layout design for triangles. We incorporate rectangular struc-
tures in the periphery in order to facilitate a homogeneous undercut of the sacrificial
layer. Also, we fabricated an array of devices with different e-beam doses.
76
Figure C.2: Schematic of the experimental setup. OL, BS, SMF represent objective
lens, beam splitter, single mode fiber, respectively.
C.3 Measurements
For photoluminescence measurement, we mount the sample on the cold finger
of a cryostat (Attocube-Attodry system) which can be cooled down to a temperature
3K. The cold mount is surrounded by a superconducting magnet, which provided
a variable magnetic field ranging from zero to 6 Tesla. In our setup, the sample is
mounted in a Faraday configuration, i.e., the applied magnetic field is perpendicular
to the sample plane.
We use a confocal microscope setup (Fig. C.2) for both exciting QDs and also
collecting the photoluminescence (PL) signal from them. We use a continuous-wave
780nm diode laser for excitation. In the experiment, the collimated laser beam
is focused on the sample by an objective lens with NA=0.8. We also image the
77
whole sample surface by shining a broadband light. This helps us to locate both the
excitation and collection spots on the sample. The collected light is then focused
on a single mode fiber (SMF) which acts as a spatial filter. For spectrally resolving
individual QD lines, we pass this collected light through a spectrograph fitted with
grating. In our case the spectral resolution is 0.02nm. This resolved light is then
focused onto a nitrogen-cooled charge-coupled device (CCD) camera array capable
of imaging at the single-photon level.
In experiment, due to 0.6 ?m2 spot size of the laser spot, we excite roughly
25 quantum dots at the same time. Now owing to their inhomogeneous broadening
and random placement all over the structure, they emit at different wavelengths
and couple differently into the waveguide mode. We then isolate individual dots
by spectrally filtering them through a grating spectrometer. We then confirm their
coupling into the waveguide by spectrally locating them from emissions from dif-
ferent places of the device e.g. either from different locations on the resonator, or
gratings at the end of the waveguide.
C.4 The quality factor of resonator in hybrid structures
We simulated the coupled waveguide-resonator system for different system
parameters. In particular, we look at the behavior of resonator quality factor (Q)
as we vary the amount of coupling between the waveguide and the resonator. Since
we fix the perimeter of the resonator, the coupling strength only depends on the
distance between the side of the resonator (facing the waveguide) and the topological
78
(a) (b)
15000
10000
5000
2?m 0
5 10 15 20
Figure C.3: Quality factor of cavity in hybrid structure : (a)Schematic of the sim-
ulation setup. Red super triangle indicates the resonator perimeter. The double
arrowed red line indicates the position of the waveguide. (b) The quality factor (Q)
versus distance (?), showing saturation of Q at high values of ?
waveguide. We define the parameter ? as the number of unit cells, characterizing this
coupling distance (Fig. C.3(a)). As shown in Fig. C.3(b) the Q of the resonator
attains a saturation after ?. In other words, at this parameter, the resonator is
weakly coupled into the waveguide. Also depending on the different values of ? one
can create resonators of different Q. In our experiment, we choose ? to be 6.
C.5 Resonator designs with Valley-Hall physics
Apart from a triangular geometry, one can also fabricate other types of res-
onator designs with the valley-hall topological photonic crystal. Since these valley
edge modes are immune to 600 and 1200, one can think of other resonator designs
such as hexagon, rhombus, 3-side equal trapezoid, or more complicated designs, such
79
Q
(a) (b) (c)
wavelength (nm)
Figure C.4: More resonator ideas : (a) Possible resonator shapes using valley-Hall
edge states (b) Simulated longitudinal modes of a hexagonal resonator showing both
bulk modes and resonating modes. (c) The in-plane electric field distribution for
one of the resonator modes.
as a hexagram (Fig. C.4 a).
Here we analyze a hexagonal resonator. As in the case of the triangular res-
onator in the main text, here the inside region of the hexagon is topologically dis-
tinct from the outside zone. Fig. C.4(b) shows the simulated modes in such a
system. Color shaded regions correspond to different regimes on the band struc-
ture. Specifically, the orange shade represents the bulk modes in the system which
are susceptible to disorder thus can provide random localized modes. And the blue
band corresponds to edge band of the band structure. Thus these modes show strong
confinement along the perimeter of the resonator (Fig. C.4(c)).
80
intensity counts
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