# Precision measurement of the
$e^{+}e^{-}~{}\rightarrow~{}\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ cross
section near threshold

M. Ablikim1, M. N. Achasov9,d, S. Ahmed14, M. Albrecht4, M. Alekseev55A,55C,
A. Amoroso55A,55C, F. F. An1, Q. An52,42, J. Z. Bai1, Y. Bai41, O. Bakina26,
R. Baldini Ferroli22A, Y. Ban34, K. Begzsuren24, D. W. Bennett21, J. V.
Bennett5, N. Berger25, M. Bertani22A, D. Bettoni23A, F. Bianchi55A,55C, E.
Boger26,b, I. Boyko26, R. A. Briere5, H. Cai57, X. Cai1,42, O. Cakir45A, A.
Calcaterra22A, G. F. Cao1,46, S. A. Cetin45B, J. Chai55C, J. F. Chang1,42, G.
Chelkov26,b,c, G. Chen1, H. S. Chen1,46, J. C. Chen1, M. L. Chen1,42, P. L.
Chen53, S. J. Chen32, X. R. Chen29, Y. B. Chen1,42, X. K. Chu34, G.
Cibinetto23A, F. Cossio55C, H. L. Dai1,42, J. P. Dai37,h, A. Dbeyssi14, D.
Dedovich26, Z. Y. Deng1, A. Denig25, I. Denysenko26, M. Destefanis55A,55C, F.
De Mori55A,55C, Y. Ding30, C. Dong33, J. Dong1,42, L. Y. Dong1,46, M. Y.
Dong1,42,46, Z. L. Dou32, S. X. Du60, P. F. Duan1, J. Fang1,42, S. S.
Fang1,46, Y. Fang1, R. Farinelli23A,23B, L. Fava55B,55C, S. Fegan25, F.
Feldbauer4, G. Felici22A, C. Q. Feng52,42, E. Fioravanti23A, M. Fritsch4, C.
D. Fu1, Q. Gao1, X. L. Gao52,42, Y. Gao44, Y. G. Gao6, Z. Gao52,42, B.
Garillon25, I. Garzia23A, A. Gilman49, K. Goetzen10, L. Gong33, W. X.
Gong1,42, W. Gradl25, M. Greco55A,55C, M. H. Gu1,42, Y. T. Gu12, A. Q. Guo1,
R. P. Guo1,46, Y. P. Guo25, A. Guskov26, Z. Haddadi28, S. Han57, X. Q. Hao15,
F. A. Harris47, K. L. He1,46, X. Q. He51, F. H. Heinsius4, T. Held4, Y. K.
Heng1,42,46, T. Holtmann4, Z. L. Hou1, H. M. Hu1,46, J. F. Hu37,h, T.
Hu1,42,46, Y. Hu1, G. S. Huang52,42, J. S. Huang15, X. T. Huang36, X. Z.
Huang32, Z. L. Huang30, T. Hussain54, W. Ikegami Andersson56, Q. Ji1, Q. P.
Ji15, X. B. Ji1,46, X. L. Ji1,42, X. S. Jiang1,42,46, X. Y. Jiang33, J. B.
Jiao36, Z. Jiao17, D. P. Jin1,42,46, S. Jin1,46, Y. Jin48, T. Johansson56, A.
Julin49, N. Kalantar-Nayestanaki28, X. S. Kang33, M. Kavatsyuk28, B. C. Ke1,
T. Khan52,42, A. Khoukaz50, P. Kiese25, R. Kliemt10, L. Koch27, O. B.
Kolcu45B,f, B. Kopf4, M. Kornicer47, M. Kuemmel4, M. Kuhlmann4, A. Kupsc56, W.
Kühn27, J. S. Lange27, M. Lara21, P. Larin14, L. Lavezzi55C, H. Leithoff25, C.
Li56, Cheng Li52,42, D. M. Li60, F. Li1,42, F. Y. Li34, G. Li1, H. B. Li1,46,
H. J. Li1,46, J. C. Li1, J. W. Li40, Jin Li35, K. J. Li43, Kang Li13, Ke Li1,
Lei Li3, P. L. Li52,42, P. R. Li46,7, Q. Y. Li36, W. D. Li1,46, W. G. Li1, X.
L. Li36, X. N. Li1,42, X. Q. Li33, Z. B. Li43, H. Liang52,42, Y. F. Liang39,
Y. T. Liang27, G. R. Liao11, J. Libby20, C. X. Lin43, D. X. Lin14, B. Liu37,h,
B. J. Liu1, C. X. Liu1, D. Liu52,42, F. H. Liu38, Fang Liu1, Feng Liu6, H. B.
Liu12, H. L Liu41, H. M. Liu1,46, Huanhuan Liu1, Huihui Liu16, J. B. Liu52,42,
J. Y. Liu1,46, K. Liu44, K. Y. Liu30, Ke Liu6, L. D. Liu34, Q. Liu46, S. B.
Liu52,42, X. Liu29, Y. B. Liu33, Z. A. Liu1,42,46, Zhiqing Liu25, Y. F.
Long34, X. C. Lou1,42,46, H. J. Lu17, J. G. Lu1,42, Y. Lu1, Y. P. Lu1,42, C.
L. Luo31, M. X. Luo59, X. L. Luo1,42, S. Lusso55C, X. R. Lyu46, F. C. Ma30, H.
L. Ma1, L. L. Ma36, M. M. Ma1,46, Q. M. Ma1, T. Ma1, X. N. Ma33, X. Y. Ma1,42,
Y. M. Ma36, F. E. Maas14, M. Maggiora55A,55C, Q. A. Malik54, Y. J. Mao34, Z.
P. Mao1, S. Marcello55A,55C, Z. X. Meng48, J. G. Messchendorp28, G.
Mezzadri23B, J. Min1,42, R. E. Mitchell21, X. H. Mo1,42,46, Y. J. Mo6, C.
Morales Morales14, N. Yu. Muchnoi9,d, H. Muramatsu49, A. Mustafa4, Y.
Nefedov26, F. Nerling10, I. B. Nikolaev9,d, Z. Ning1,42, S. Nisar8, S. L.
Niu1,42, X. Y. Niu1,46, S. L. Olsen35, Q. Ouyang1,42,46, S. Pacetti22B, Y.
Pan52,42, M. Papenbrock56, P. Patteri22A, M. Pelizaeus4, J. Pellegrino55A,55C,
H. P. Peng52,42, Z. Y. Peng12, K. Peters10,g, J. Pettersson56, J. L. Ping31,
R. G. Ping1,46, A. Pitka4, R. Poling49, V. Prasad52,42, H. R. Qi2, M. Qi32, T.
.Y. Qi2, S. Qian1,42, C. F. Qiao46, N. Qin57, X. S. Qin4, Z. H. Qin1,42, J. F.
Qiu1, K. H. Rashid54,i, C. F. Redmer25, M. Richter4, M. Ripka25, M. Rolo55C,
G. Rong1,46, Ch. Rosner14, A. Sarantsev26,e, M. Savrié23B, C. Schnier4, K.
Schoenning56, W. Shan18, X. Y. Shan52,42, M. Shao52,42, C. P. Shen2, P. X.
Shen33, X. Y. Shen1,46, H. Y. Sheng1, X. Shi1,42, J. J. Song36, W. M. Song36,
X. Y. Song1, S. Sosio55A,55C, C. Sowa4, S. Spataro55A,55C, G. X. Sun1, J. F.
Sun15, L. Sun57, S. S. Sun1,46, X. H. Sun1, Y. J. Sun52,42, Y. K Sun52,42, Y.
Z. Sun1, Z. J. Sun1,42, Z. T. Sun21, Y. T Tan52,42, C. J. Tang39, G. Y. Tang1,
X. Tang1, I. Tapan45C, M. Tiemens28, B. Tsednee24, I. Uman45D, G. S. Varner47,
B. Wang1, B. L. Wang46, D. Wang34, D. Y. Wang34, Dan Wang46, K. Wang1,42, L.
L. Wang1, L. S. Wang1, M. Wang36, Meng Wang1,46, P. Wang1, P. L. Wang1, W. P.
Wang52,42, X. F. Wang44, Y. Wang52,42, Y. D. Wang14, Y. F. Wang1,42,46, Y. Q.
Wang25, Z. Wang1,42, Z. G. Wang1,42, Z. Y. Wang1, Zongyuan Wang1,46, T.
Weber4, D. H. Wei11, J. H. Wei33, P. Weidenkaff25, S. P. Wen1, U. Wiedner4, M.
Wolke56, L. H. Wu1, L. J. Wu1,46, Z. Wu1,42, L. Xia52,42, Y. Xia19, D. Xiao1,
Y. J. Xiao1,46, Z. J. Xiao31, Y. G. Xie1,42, Y. H. Xie6, X. A. Xiong1,46, Q.
L. Xiu1,42, G. F. Xu1, J. J. Xu1,46, L. Xu1, Q. J. Xu13, Q. N. Xu46, X. P.
Xu40, F. Yan53, L. Yan55A,55C, W. B. Yan52,42, W. C. Yan2, Y. H. Yan19, H. J.
Yang37,h, H. X. Yang1, L. Yang57, Y. H. Yang32, Y. X. Yang11, Yifan Yang1,46,
M. Ye1,42, M. H. Ye7, J. H. Yin1, Z. Y. You43, B. X. Yu1,42,46, C. X. Yu33, J.
S. Yu29, C. Z. Yuan1,46, Y. Yuan1, A. Yuncu45B,a, A. A. Zafar54, Y. Zeng19, Z.
Zeng52,42, B. X. Zhang1, B. Y. Zhang1,42, C. C. Zhang1, D. H. Zhang1, H. H.
Zhang43, H. Y. Zhang1,42, J. Zhang1,46, J. L. Zhang58, J. Q. Zhang4, J. W.
Zhang1,42,46, J. Y. Zhang1, J. Z. Zhang1,46, K. Zhang1,46, L. Zhang44, S. Q.
Zhang33, X. Y. Zhang36, Y. Zhang52,42, Y. H. Zhang1,42, Y. T. Zhang52,42, Yang
Zhang1, Yao Zhang1, Yu Zhang46, Z. H. Zhang6, Z. P. Zhang52, Z. Y. Zhang57, G.
Zhao1, J. W. Zhao1,42, J. Y. Zhao1,46, J. Z. Zhao1,42, Lei Zhao52,42, Ling
Zhao1, M. G. Zhao33, Q. Zhao1, S. J. Zhao60, T. C. Zhao1, Y. B. Zhao1,42, Z.
G. Zhao52,42, A. Zhemchugov26,b, B. Zheng53,14, J. P. Zheng1,42, Y. H.
Zheng46, B. Zhong31, L. Zhou1,42, Q. Zhou1,46, X. Zhou57, X. K. Zhou52,42, X.
R. Zhou52,42, X. Y. Zhou1, A. N. Zhu1,46, J. Zhu43, K. Zhu1, K. J. Zhu1,42,46,
S. Zhu1, S. H. Zhu51, X. L. Zhu44, Y. C. Zhu52,42, Y. S. Zhu1,46, Z. A.
Zhu1,46, J. Zhuang1,42, B. S. Zou1, J. H. Zou1
(BESIII Collaboration)
1 Institute of High Energy Physics, Beijing 100049, People’s Republic of China
2 Beihang University, Beijing 100191, People’s Republic of China
3 Beijing Institute of Petrochemical Technology, Beijing 102617, People’s
Republic of China
4 Bochum Ruhr-University, D-44780 Bochum, Germany
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
8 COMSATS Institute of Information Technology, Lahore, Defence Road, Off
Raiwind Road, 54000 Lahore, Pakistan
9 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
10 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
11 Guangxi Normal University, Guilin 541004, People’s Republic of China
12 Guangxi University, Nanning 530004, People’s Republic of China
13 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
14 Helmholtz Institute Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz,
Germany
15 Henan Normal University, Xinxiang 453007, People’s Republic of China
16 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
17 Huangshan College, Huangshan 245000, People’s Republic of China
18 Hunan Normal University, Changsha 410081, People’s Republic of China
19 Hunan University, Changsha 410082, People’s Republic of China
20 Indian Institute of Technology Madras, Chennai 600036, India
21 Indiana University, Bloomington, Indiana 47405, USA
22 (A)INFN Laboratori Nazionali di Frascati, I-00044, Frascati, Italy; (B)INFN
and University of Perugia, I-06100, Perugia, Italy
23 (A)INFN Sezione di Ferrara, I-44122, Ferrara, Italy; (B)University of
Ferrara, I-44122, Ferrara, Italy
24 Institute of Physics and Technology, Peace Ave. 54B, Ulaanbaatar 13330,
Mongolia
25 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
26 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
27 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
28 KVI-CART, University of Groningen, NL-9747 AA Groningen, The Netherlands
29 Lanzhou University, Lanzhou 730000, People’s Republic of China
30 Liaoning University, Shenyang 110036, People’s Republic of China
31 Nanjing Normal University, Nanjing 210023, People’s Republic of China
32 Nanjing University, Nanjing 210093, People’s Republic of China
33 Nankai University, Tianjin 300071, People’s Republic of China
34 Peking University, Beijing 100871, People’s Republic of China
35 Seoul National University, Seoul, 151-747 Korea
36 Shandong University, Jinan 250100, People’s Republic of China
37 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
38 Shanxi University, Taiyuan 030006, People’s Republic of China
39 Sichuan University, Chengdu 610064, People’s Republic of China
40 Soochow University, Suzhou 215006, People’s Republic of China
41 Southeast University, Nanjing 211100, People’s Republic of China
42 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
43 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
44 Tsinghua University, Beijing 100084, People’s Republic of China
45 (A)Ankara University, 06100 Tandogan, Ankara, Turkey; (B)Istanbul Bilgi
University, 34060 Eyup, Istanbul, Turkey; (C)Uludag University, 16059 Bursa,
Turkey; (D)Near East University, Nicosia, North Cyprus, Mersin 10, Turkey
46 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
47 University of Hawaii, Honolulu, Hawaii 96822, USA
48 University of Jinan, Jinan 250022, People’s Republic of China
49 University of Minnesota, Minneapolis, Minnesota 55455, USA
50 University of Muenster, Wilhelm-Klemm-Str. 9, 48149 Muenster, Germany
51 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
52 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
53 University of South China, Hengyang 421001, People’s Republic of China
54 University of the Punjab, Lahore-54590, Pakistan
55 (A)University of Turin, I-10125, Turin, Italy; (B)University of Eastern
Piedmont, I-15121, Alessandria, Italy; (C)INFN, I-10125, Turin, Italy
56 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
57 Wuhan University, Wuhan 430072, People’s Republic of China
58 Xinyang Normal University, Xinyang 464000, People’s Republic of China
59 Zhejiang University, Hangzhou 310027, People’s Republic of China
60 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at Bogazici University, 34342 Istanbul, Turkey
b Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia
c Also at the Functional Electronics Laboratory, Tomsk State University,
Tomsk, 634050, Russia
d Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
e Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia
f Also at Istanbul Arel University, 34295 Istanbul, Turkey
g Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
h Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology,
Ministry of Education; Shanghai Key Laboratory for Particle Physics and
Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240,
People’s Republic of China
i Government College Women University, Sialkot - 51310. Punjab, Pakistan.

###### Abstract

The cross section of the
$e^{+}e^{-}~{}\rightarrow~{}\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ process is
measured with unprecedented precision using data collected with the BESIII
detector at $\sqrt{s}=4574.5$, $4580.0$, $4590.0$ and $4599.5$ $\mathrm{MeV}$.
The non-zero cross section near the $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$
production threshold is cleared. At center-of-mass energies $\sqrt{s}=4574.5$
and $4599.5$ $\mathrm{MeV}$, the higher statistics data enable us to measure
the $\Lambda_{c}$ polar angle distributions. From these, the $\Lambda_{c}$
electric over magnetic form factor ratios ($|G_{E}/G_{M}|$) are measured for
the first time. They are found to be $1.14\pm 0.14\pm 0.07$ and $1.23\pm
0.05\pm 0.03$ respectively, where the first uncertainties are statistical and
the second are systematic.

The electromagnetic structure of hadrons, parameterized in terms of
electromagnetic form factors (EMFFs), provides a key to understand quantum
chromodynamics effects in bound states. The nucleon has been studied
rigorously for more than 60 years, but new techniques and the availability of
data with larger statistics from modern facilities have given rise to a
renewed interest in the field, e.g. the proton radius puzzle pradius .
Recently, the access to strange and charm hyperon structure by time-like EMFFs
provides an additional dimension. Assuming that one-photon exchange dominates
the production of spin-1/2 baryons $B$, the cross section of the process
$e^{+}e^{-}$ $\rightarrow$ $B\bar{B}$ can be parameterized in terms of EMFFs,
i.e. $G_{E}$ and $G_{M}$, in the following way OPEXModel :

$\sigma_{B\bar{B}}(s)=\frac{4\pi\alpha^{2}C\beta}{3s}|G_{M}(s)|^{2}\big{[}1+\frac{2m_{B}^{2}c^{4}}{s}\big{|}\frac{G_{E}(s)}{G_{M}(s)}\big{|}^{2}\big{]}.$
(1)

Here, $\alpha$ is the fine-structure constant,
$\beta$=$\sqrt{1-4m_{B}^{2}c^{4}/s}$ the velocity of the baryon, $s$ the
square of the center-of-mass (CM) energy, and $m_{B}$ is the mass of the
baryon. The Coulomb factor $C$ parameterizes the electromagnetic interaction
between the outgoing baryon and antibaryon. For neutral baryons, the Coulomb
factor is unity, while for point-like charged fermions it reads $C=\varepsilon
R$ C1 ; C2 , where $\varepsilon={\pi\alpha}/{\beta}$ is an enhancement factor
resulting in a nonzero cross section at threshold and
$R={\sqrt{1-\beta^{2}}}/(1-e^{-\pi\alpha{\sqrt{1-\beta^{2}}}/\beta})$ is the
Sommerfeld resummation factor C1 . The ratio of EMFFs associated with the
polar angle distribution of the baryon can also parameterize the differential
production cross section of the corresponding baryon OPEXModel .

In the $e^{+}e^{-}\rightarrow p\bar{p}$ process, the BaBar collaboration
observed a rapid rise of the cross section near threshold, followed by a
plateau around 200 MeV above threshold babar2 . The BESIII collaboration also
observed the cross section enhancement ppbarbes . The non-vanishing cross
section near threshold as well as the wide-range plateau have led to various
theoretical interpretations, including i) final-state interactions TheoFSInter
, ii) bound states or meson-like resonances TheoRes and iii) an attractive
Coulomb interaction Resumfct . Recently, the BESIII collaboration has observed
the non-zero cross section near threshold in the process
$e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}$ BAM0139 . Naturally, it is also
interesting to explore the production behavior of $\Lambda_{c}^{+}$, the
lightest baryon containing the charm quark. Previously, the Belle
collaboration measured the cross section of
$e^{+}e^{-}~{}\rightarrow~{}\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ using the
initial-state radiation (ISR) technique Belle , but the results suffer from
significant uncertainties in CM energy and cross section. Therefore, near
$\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ threshold, precise measurements of the
production cross section and EMFF ratios are highly desirable.

In this work, the cross section of the reaction
$e^{+}e^{-}~{}\rightarrow~{}\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ is measured
at four CM energies: $\sqrt{s}=4574.5$, $4580.0$, $4590.0$, and $4599.5$
$\mathrm{MeV}$. At each CM energy, ten Cabibbo-favored hadronic decay modes,
$\Lambda_{c}^{+}\rightarrow pK^{-}\pi^{+}$, $pK_{S}^{0}$, $\Lambda\pi^{+}$,
$pK^{-}\pi^{+}\pi^{0}$, $pK_{S}^{0}\pi^{0}$, $\Lambda\pi^{+}\pi^{0}$,
$pK_{S}^{0}\pi^{+}\pi^{-}$, $\Lambda\pi^{+}\pi^{+}\pi^{-}$,
$\Sigma^{0}\pi^{+}$, and $\Sigma^{+}\pi^{+}\pi^{-}$, as well as the ten
corresponding charge-conjugate modes are independently used to reconstruct
$\Lambda_{c}^{+}$ or $\bar{\Lambda}_{c}^{-}$. Each mode will produce one
measurement of the cross section and the total cross section is obtained from
a weighted average over the 20 individual measurements. In addition, the
higher statistics data samples at $\sqrt{s}=4574.5$ and $4599.5$
$\mathrm{MeV}$ enable the study of the polar angle distribution of
$\Lambda_{c}$ in the CM system. From these distributions, the ratios between
the electric and the magnetic form factors, i.e. $|G_{E}/G_{M}|$, are
extracted for the first time.

The data samples are collected with the BESIII detector BESIII at BEPCII. The
detector has a geometrical acceptance of 93% of the 4$\pi$ solid angle. It
contains a small-celled, helium-based main drift chamber (MDC), a time-of-
flight system (TOF) based on plastic scintillators, an electromagnetic
calorimeter (EMC) made of CsI(Tl) crystals, a muon system (MUC) made of
Resistive Plate Chambers, and a superconducting solenoid magnet.

Monte Carlo (MC) simulations based on geant4 geant4 are performed to determine
detection efficiencies, optimize selection criteria, extract signal shapes and
study backgrounds. The $e^{+}e^{-}$ collisions are simulated by the kkmc
generator KKMC , which takes the beam energy spread and the ISR correction
into account. The distribution of the $\Lambda_{c}$ polar angle is considered
in the generator by parameterizing it with the function
$f(\theta)~{}\propto~{}1~{}+~{}\alpha_{\Lambda_{c}}~{}\cos^{2}\theta$. After
an iterative procedure, the values of $\alpha_{\Lambda_{c}}$ at $\sqrt{s}$ =
$4574.5$ and $4599.5$ $\mathrm{MeV}$ are obtained from real data (see Table 4)
and at the remaining CM energies by a linear interpolation.

Using the branching fractions (BR) measured in Ref. BAM0162 , all tagged
$\Lambda_{c}$ decays are simulated by weighting phase-space events according
to the decay behavior observed in real data. The subsequent decays listed by
the particle data group (PDG) PDG are modeled with evtgen EVTGEN . The
inclusive MC samples include $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ pair
production, $\ell^{+}\ell^{-}$ ($\ell=\textit{e,~{}$\mu$,~{}$\tau$})$ events,
open charm processes DDsMC , ISR-produced low-mass $\psi$ states and the
continuum process $e^{+}e^{-}\rightarrow q\bar{q}~{}(q=\textit{u,~{}d,~{}s})$.

Charged tracks as well as the intermediate states $\pi^{0}$, $K_{S}^{0}$,
$\Lambda$, $\Sigma^{0}$ and $\Sigma^{+}$ are selected and reconstructed with
the same method described in Ref. BAM0162 .

In the final states of decay modes $pK_{S}^{0}\pi^{0}$ and
$pK_{S}^{0}\pi^{+}\pi^{-}$, potential background from $\Lambda\rightarrow
p\pi^{-}$ is eliminated by rejecting events with $M_{p\pi^{-}}$ lying in the
mass window (1100, 1125) $\mathrm{MeV}/c^{2}$, where $M_{p\pi^{-}}$ is the
invariant mass of $p\pi^{-}$ combinations in the final state. For the decay
mode $\Sigma^{+}\pi^{+}\pi^{-}$, the corresponding exclusion window is (1110,
1120) $\mathrm{MeV}/c^{2}$ due to the smaller observed width of the
$M_{p\pi^{-}}$ peak in data. Similarly, background from the intermediate state
$\Sigma^{+}$ is removed from the $pK_{S}^{0}\pi^{0}$ sample by rejecting
events with $M_{p\pi^{0}}$ in the mass window (1170, 1200)
$\mathrm{MeV}/c^{2}$. In modes $\Lambda\pi^{+}\pi^{+}\pi^{-}$ and
$\Sigma^{+}\pi^{+}\pi^{-}$, events with $M_{\pi^{+}\pi^{-}}$ within (490, 510)
$\mathrm{MeV}/c^{2}$ are rejected to suppress $K_{S}^{0}$ background.

\begin{overpic}[width=121.41307pt,height=80.94205pt,angle={0}]{fit_mbc_at_4575_mode1.eps}
\end{overpic}\begin{overpic}[width=121.41307pt,height=80.94205pt,angle={0}]{fit_mbc_at_4580_mode1.eps}
\end{overpic}

\begin{overpic}[width=121.41307pt,height=80.94205pt,angle={0}]{fit_mbc_at_4590_mode1.eps}
\end{overpic}\begin{overpic}[width=121.41307pt,height=80.94205pt,angle={0}]{fit_mbc_at_4600_mode1.eps}
\end{overpic}

Figure 1: Fit results of $M_{\textrm{BC}}$ distribution of
$\Lambda_{c}^{+}\rightarrow pK^{-}\pi^{+}$ in data at
$\sqrt{s}=4574.5~{}\mathrm{MeV}$ (a), $4580.0~{}\mathrm{MeV}$ (b),
$4590.0~{}\mathrm{MeV}$ (c), $4599.5~{}\mathrm{MeV}$ (d). Dots are Poisson
averages of the data in the bins and error bars represent one time of
corresponding standard deviations, the blue solid curves are the sum of fit
functions, the barely visible green dashed lines are the background shapes.
$N$ is the yield with statistical uncertainty of $\Lambda_{c}^{+}$ signal, and
$\varepsilon$ represents the corresponding detection efficiency and
uncertainty.

According to energy and momentum conservation, two discriminating variables,
the energy difference $\Delta E$ and beam-constrained mass $M_{\textrm{BC}}$,
are utilized to identify the $\Lambda_{c}$ signals. The energy difference is
defined as $\Delta E~{}\equiv~{}E~{}-~{}E_{\textrm{beam}}$, where $E$ is the
energy of the $\Lambda_{c}$ candidate and $E_{\textrm{beam}}$ the mean energy
of the two colliding beams. In each tagged mode, the $\Lambda_{c}$ candidates
are formed by all possible combinations of the final state particles, and only
the one with minimum $|\Delta E|$ is stored. In the following analysis, events
are rejected if they fail the $\Delta E$ requirements specified in Ref.
BAM0162 . The beam-constrained mass is defined as
$M_{\textrm{BC}}c^{2}\equiv\sqrt{E_{\textrm{beam}}^{2}-p^{2}c^{2}}$, where $p$
is the momentum of the $\Lambda_{c}$ candidate. Both $\Delta E$ and
$M_{\textrm{BC}}$ are calculated in the initial $e^{+}e^{-}$ CM system. In
Fig. 1, the $M_{\textrm{BC}}$ distributions for $\Lambda_{c}^{+}\rightarrow
pK^{-}\pi^{+}$ at the four CM energies are shown. Clear peaks at the nominal
$\Lambda_{c}^{+}$ mass are observed. Studies of the inclusive MC samples show
that the cross feeds among the ten tagged modes are less than 1.5% and the
background shape can be described by the ARGUS function Argusf .

Performing an unbinned maximum likelihood fit to each $M_{\textrm{BC}}$
distribution gives the corresponding event yields, as partly illustrated in
Fig. 1. The signal shape of the fit is obtained from convolving the
$M_{\textrm{BC}}$ shape of MC simulations with a Gaussian function to
compensate a possible resolution difference between data and MC simulations.
The background is described by an ARGUS function with the high-end truncation
fixed. At $\sqrt{s}=4599.5~{}\mathrm{MeV}$, the parameters of the ARGUS and
the Gaussian functions used in the convolution are obtained from the fit. At
the remaining CM energies, all parameters obtained at the highest energy,
except for the mean of the Gaussian, are used to fix parameters in the new
fits. Yields are extracted from the signal region 2276 $\mathrm{MeV}$
$<M_{\textrm{BC}}c^{2}$ $<$ $E_{\textrm{beam}}$ in each fit. The detection
efficiency of each decay mode is evaluated by MC simulations of the
$e^{+}e^{-}~{}\rightarrow~{}\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ process.
Figure 1 gives the efficiencies of mode $pK^{-}\pi^{+}$ at the four CM
energies.

The cross section of the $i$-th mode is determined using

$\sigma_{i}=\frac{N_{i}}{\varepsilon_{i}\cdot\mathcal{L}_{\textrm{int}}\cdot
f_{\textrm{VP}}\cdot BR_{i}\cdot f_{\textrm{ISR}}},$ (2)

where $N_{i}$ and $\varepsilon_{i}$ represent the yield and corresponding
detection efficiency. The integrated luminosity $\mathcal{L}_{\textrm{int}}$
is taken from Ref. LumXYZ ; LumRscan . The vacuum polarization (VP) correction
factor $f_{\textrm{VP}}$ is calculated to be 1.055 at all four CM energies
VPfactor . The $BR_{i}$ represents the product of branching fractions of the
$i$-th $\Lambda_{c}$ decay mode and its subsequent decay(s). The
$f_{\textrm{ISR}}$ is the ISR correction factor derived in Ref. ISRfactor and
implemented in kkmc. Since the calculation of $f_{\textrm{ISR}}$ requires the
cross section line-shape as input, an iterative procedure has been performed.

Table 1: Summary of the reconstruction related, mode-specific, relative systematic uncertainties of the cross section at $\sqrt{s}$ = $4599.5~{}\mathrm{MeV}$, quoted in %. |  |  |  |  |  | MC |  Signal |  Total |
---|---|---|---|---|---|---|---|---|---
Source | Tracking | PID | $K_{S}^{0}$ | $\Lambda$ | $\pi^{0}$ | statistic |  model |  BR |
$pK^{-}\pi^{+}$ | 3.2 | 4.6 | – | – | – | 0.2 | – | 6.0 |
$pK_{S}^{0}$ | 1.3 | 0.5 | 1.2 | – | – | 0.6 | 0.2 | 5.6 |
$\Lambda\pi^{+}$ | 1.0 | 1.0 | – | 2.5 | – | 0.8 | 0.5 | 6.2 |
$pK^{-}\pi^{+}\pi^{0}$ | 3.0 | 7.6 | – | – | 1.0 | 0.6 | 2.0 | 8.3 |
$pK_{S}^{0}\pi^{0}$ | 1.0 | 1.8 | 1.2 | – | 1.0 | 1.1 | 1.0 | 7.5 |
$\Lambda\pi^{+}\pi^{0}$ | 1.0 | 1.0 | – | 2.5 | 1.0 | 0.6 | 0.6 | 6.0 |
$pK_{S}^{0}\pi^{+}\pi^{-}$ | 2.8 | 5.3 | 1.2 | – | – | 1.0 | 0.5 | 9.3 |
$\Lambda\pi^{+}\pi^{+}\pi^{-}$ | 3.0 | 3.0 | – | 2.5 | – | 0.9 | 0.8 | 7.9 |
$\Sigma^{0}\pi^{+}$ | 1.0 | 1.0 | – | 2.5 | – | 1.1 | 1.7 | 6.7 |
$\Sigma^{+}\pi^{+}\pi^{-}$ | 3.0 | 4.0 | – | – | 1.0 | 0.8 | 0.8 | 7.4 |

The systematic uncertainties of the cross section can be classified into
reconstruction related and general contributions. The reconstruction related
contributions are mode-specific and mainly originate from tracking, PID,
reconstruction of intermediate states and total BRs. The uncertainties of
$\Delta E$ and $M_{\textrm{BC}}$ requirements are negligible after correcting
for the difference in resolution between simulated and real data samples. The
uncertainties from tracking and PID of charged particles are investigated
using control samples from
$e^{+}e^{-}~{}\rightarrow~{}\pi^{+}\pi^{-}\pi^{+}\pi^{-}$,
$K^{+}K^{-}\pi^{+}\pi^{-}$ and $p\bar{p}\pi^{+}\pi^{-}$ collected at
$\sqrt{s}>4.0~{}\mathrm{GeV}$ BAM0216 . The uncertainties are obtained after
weighting according to the momenta of the corresponding final states.
Reconstruction uncertainties of $K_{S}^{0}$, $\Lambda$ and $\pi^{0}$ have been
found to be 1.2%, 2.5% and 1.0% BAM0162 . Statistical uncertainties of
detection efficiencies are considered as systematic uncertainties. The
dependence of the reconstruction efficiency on the MC model for the ten decay
modes also gives a small contribution to the systematic uncertainty BAM0162 .
Uncertainties originating from the total BRs of the tagged modes are quoted
from Refs. BAM0162 ; PDG . A summary of the reconstruction related systematic
uncertainties are given in Table 1. The total uncertainty at each energy has
been calculated assuming that the values given at $\sqrt{s}$ = $4599.5$
$\mathrm{MeV}$ are valid at all CM energies.

Table 2: Summary of the general relative systematic uncertainties of the cross section originating from the factors $f_{\textrm{ISR}}$, $f_{\textrm{VP}}$ and $\mathcal{L}_{\textrm{int}}$, quoted in %. | $f_{\textrm{ISR}}$ |  |
---|---|---|---
Calculation | Line- | CM | Energy |
$\sqrt{s}$ ($\mathrm{MeV}$) | model | shape | energy | Spread | Total |  $f_{\textrm{VP}}$ |  $\mathcal{L}_{\textrm{int}}$
4574.5 | 3.4 | 1.2 | 18.0 | 3.0 | 18.6 |  0.5 | 1.0
4580.0 | 0.7 | 0.6 | – | 0.2 | 0.9 |  0.5 | 0.7
4590.0 | 0.2 | 1.7 | – | – | 1.7 |  0.5 | 0.7
4599.5 | 0.1 | 2.6 | – | – | 2.6 |  0.5 | 1.0

The general contributions to the systematic uncertainty originate from
uncertainties in $f_{\textrm{ISR}}$, $f_{\textrm{VP}}$ and
$\mathcal{L}_{\textrm{int}}$ in Eq. (2) and are the same for all decay modes.
The $f_{\textrm{ISR}}$ is obtained using the kkmc generator which requires a
cross section line-shape as input. The line-shape is in turn obtained by an
iterative fitting procedure of the cross section data using Eq. (1). In the
fit, the $|G_{E}/G_{M}|$ value at an arbitrary CM energy is assigned by linear
interpolation between the two known values listed in Table 4. For simplicity,
$|G_{M}|$ is assumed to be independent of the CM energy. To precisely describe
the data, the $\alpha$ in the Sommerfeld resummation factor is replaced by
$\alpha_{s}(=0.25)$. In the line-shape, the cross section at the CM energy
region $(2m_{\Lambda_{c}}c^{2},~{}4574.5)$ $\mathrm{MeV}$ is obtained from
extrapolating the fit; below threshold it vanishes, as shown by the blue solid
curve in Fig. 2. Four sources of systematic uncertainty from the
$f_{\textrm{ISR}}$ are considered: First, the uncertainty of the calculation
model is studied using a different algorithm mentioned in Ref. ConExc .
Second, the uncertainty associated with the input line-shape is estimated
using different fit functions. Third, the $f_{\textrm{ISR}}$ depends on the CM
energy of the
$e^{+}e^{-}~{}\rightarrow~{}\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ process. The
uncertainty of the CM energy therefore contributes near the threshold. At the
lowest energy point, the CM energy is measured to be $\sqrt{s}=4574.50\pm
0.72$ $\mathrm{MeV}$ XYZECMS . Finally, the beam energy spread, which has been
estimated as $1.55\pm 0.18~{}\mathrm{MeV}$, is important near threshold and
contributes to the $f_{\textrm{ISR}}$ uncertainty. For the other, higher,
energies, the effects from the CM energy uncertainty and the beam energy
spread are less than 0.1% and can be neglected due to the flat line-shape of
the cross section. The uncertainty of $f_{\textrm{VP}}$ is calculated to be
0.5% at all four CM energies VPfactor . The uncertainty from the integrated
luminosity has been found to be 0.7% at $\sqrt{s}$ = $4580.0$ and $4590.0$
$\mathrm{MeV}$ and 1.0% at $\sqrt{s}$ = $4574.5$ and $4599.5~{}\mathrm{MeV}$
LumXYZ ; LumRscan . A summary of the general contributions to the systematic
uncertainties is given in Table 2.

The cross sections obtained in different decay modes are combined using the
method mentioned in Ref. AverageData , in which the cross section is given by:

$\sigma=\sum_{i}w_{i}\sigma_{i}~{}~{}~{}\textmd{with}~{}~{}~{}w_{i}=(1/\Delta\sigma_{i}^{2})\bigg{/}\bigg{(}\sum_{i}1/\Delta\sigma_{i}^{2}\bigg{)}.$
(3)

Here, $w_{i}$ and $\Delta\sigma_{i}$ denote the weight and the total
uncertainty, respectively, of the measured cross section $\sigma_{i}$ of mode
$i$. The sum is performed over all 20 decay modes of $\Lambda_{c}^{+}$ and
$\bar{\Lambda}_{c}^{-}$ IndividualCSs . The combined uncertainty is calculated
by

$\Delta\sigma^{2}=\sum_{i,j}w_{i}(\mathbf{M}_{\sigma})_{ij}w_{j},$ (4)

where $\mathbf{M}_{\sigma}$ represents the covariance matrix of these cross
section measurements, in which the correlations between any two measurements
$\sigma_{i}$ and $\sigma_{j}$ are considered. The resulting cross sections at
the four CM energies are listed in Table 3 and shown in Fig. 2 together with
the Belle data Belle for comparison.

Table 3: The average cross section of $e^{+}e^{-}~{}\rightarrow~{}\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ measured at each CM energy, where the uncertainties are statistical and systematic, respectively. The observed cross section can be obtained by multiplying the $f_{\textrm{ISR}}$ and the $\sigma$. $\sqrt{s}$ ($\mathrm{MeV}$) |  $\mathcal{L}_{\textrm{int}}$ $(\textmd{pb}^{-1})$ |  $f_{\textrm{ISR}}$ |  $\sigma$ (pb)
---|---|---|---
$4574.5$ | 47.67 | 0.45 | $236\pm 11\pm 46$
$4580.0$ | 8.54 | 0.66 | $207\pm 17\pm 13$
$4590.0$ | 8.16 | 0.71 | $245\pm 19\pm 16$
$4599.5$ | 566.93 | 0.74 | $237\pm^{{\color[rgb]{1,1,1}1}}3\pm 15$
Figure 2:  Cross section of
$e^{+}e^{-}~{}\rightarrow~{}\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ obtained by
BESIII (this work) and Belle. The blue solid curve represents the input line-
shape for kkmc when determining the $f_{\textrm{ISR}}$. The dash-dot cyan
curve denotes the prediction of the phase space (PHSP) model, which is
parameterized by Eq. 1 but with $C=1$ and flat $|G_{M}|$ with respect to
$\sqrt{s}$.

\begin{overpic}[width=121.41307pt,height=80.94205pt,angle={0}]{angular_dis_at_4575_draft.eps}
\end{overpic}\begin{overpic}[width=121.41307pt,height=80.94205pt,angle={0}]{angular_dis_at_4600_draft.eps}
\end{overpic}

Figure 3: Angular distribution after efficiency correction and results of the
fit to data at $\sqrt{s}$ = $4574.5$ $\mathrm{MeV}$ (left) and $4599.5$
$\mathrm{MeV}$ (right).

The data sets collected at $\sqrt{s}$ = $4574.5$ and $4599.5$ $\mathrm{MeV}$
are large enough to perform a detailed study in the CM frame of the
$\Lambda_{c}$ polar angle $\theta_{\Lambda_{c}}$, which is defined as the
angle between the $\Lambda_{c}$ momentum and the beam direction. The data
fulfilling all selection criteria are divided into ten bins in
$\cos\theta_{\Lambda_{c}^{+}}$. In each $\cos\theta_{\Lambda_{c}^{+}}$ bin,
the total yield is obtained by summing the yields of all the ten tagged modes.
The one-dimensional bin-by-bin efficiency corrections are applied on these
total yields. The same procedure is performed by tagging
$\bar{\Lambda}_{c}^{-}$ decay channels. The total yields of $\Lambda_{c}^{+}$
and $\bar{\Lambda}_{c}^{-}$ are combined bin-by-bin, and the shape function
$f(\theta)~{}\propto~{}(1+\alpha_{\Lambda_{c}}\cos^{2}\theta)$ is fitted to
the combined data, as shown in Fig. 3. Table 4 lists the resulting
$\alpha_{\Lambda_{c}}$ parameters obtained from the fits, as well as the
$|G_{E}/G_{M}|$ ratios extracted using the equation

$|G_{E}/G_{M}|^{2}(1-\beta^{2})=(1-\alpha_{\Lambda_{c}})/(1+\alpha_{\Lambda_{c}}).$
(5)

Table 4: Shape parameters of the angular distribution and $|G_{E}/G_{M}|$ ratios at $\sqrt{s}$ = $4574.5$ and $4599.5$ $\mathrm{MeV}$. The uncertainties are statistical and systematic, respectively.  $\sqrt{s}$ ($\mathrm{MeV}$) |  $\alpha_{\Lambda_{c}}$ |  $|G_{E}/G_{M}|$
---|---|---
4574.5 | $-0.13\pm 0.12\pm 0.08$ | $1.14\pm 0.14\pm 0.07$
4599.5 | $-0.20\pm 0.04\pm 0.02$ | $1.23\pm 0.05\pm 0.03$

The systematic uncertainties of the $\alpha_{\Lambda_{c}}$ considered here are
the contributions from the fit range and the bin size. A change of the fit
range in $\cos\theta$ from ($-1.0,~{}1.0$) to ($-0.8,~{}0.8$) and in the
number of bins from 10 to 20 are performed, and the differences in the
obtained $\alpha_{\Lambda_{c}}$ are regarded as the systematic uncertainty.
Systematics originating from the model dependencies in the efficiency
correction are found to be negligible compared to the statistical
uncertainties.

In summary, using data collected at $\sqrt{s}=4574.5$, $4580.0$, $4590.0$, and
$4599.5~{}\mathrm{MeV}$ with the BESIII detector, the cross sections of
$e^{+}e^{-}~{}\rightarrow~{}\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ have been
measured with high precision, by reconstructing $\Lambda_{c}^{+}$ and
$\bar{\Lambda}_{c}^{-}$ independently with ten Cabibbo-favored hadronic decay
channels. The most precise cross section measurement is achieved so far at
$\sqrt{s}=4574.5~{}\mathrm{MeV}$, which is only $1.6~{}\mathrm{MeV}$ above the
threshold. The measured value is ($236\pm 11\pm 46$) pb, which highlights the
enhanced cross section near threshold and indicates the complexity of
production behavior of the $\Lambda_{c}$. At $\sqrt{s}=4574.5$ and
$4599.5~{}\mathrm{MeV}$, the data samples are large enough to study polar
angle distributions of $\Lambda_{c}$ and measure the $\Lambda_{c}$ form factor
ratio $|G_{E}/G_{M}|$ for the first time. These results provide important
insights into the production mechanism and structure of the $\Lambda_{c}$
baryons.

The BESIII collaboration thanks the staff of BEPCII, the IHEP computing center
and the supercomputing center of USTC for their strong support. This work is
supported in part by National Key Basic Research Program of China under
Contract No. 2015CB856700; National Natural Science Foundation of China (NSFC)
under Contracts Nos. 11235011, 11335008, 11375205, 11425524, 11625523,
11635010; the Chinese Academy of Sciences (CAS) Large-Scale Scientific
Facility Program; the CAS Center for Excellence in Particle Physics (CCEPP);
Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under
Contracts Nos. U1332201, U1532257, U1532258; CAS under Contracts Nos.
KJCX2-YW-N29, KJCX2-YW-N45, QYZDJ-SSW-SLH003; 100 Talents Program of CAS;
National 1000 Talents Program of China; INPAC and Shanghai Key Laboratory for
Particle Physics and Cosmology; German Research Foundation DFG under Contracts
Nos. Collaborative Research Center CRC 1044, FOR 2359; Istituto Nazionale di
Fisica Nucleare, Italy; Koninklijke Nederlandse Akademie van Wetenschappen
(KNAW) under Contract No. 530-4CDP03; Ministry of Development of Turkey under
Contract No. DPT2006K-120470; National Natural Science Foundation of China
(NSFC) under Contracts Nos. 11505034, 11575077; National Science and
Technology fund; The Swedish Research Council; U. S. Department of Energy
under Contracts Nos. DE-FG02-05ER41374, DE-SC-0010118, DE-SC-0010504, DE-
SC-0012069; University of Groningen (RuG) and the Helmholtzzentrum fuer
Schwerionenforschung GmbH (GSI), Darmstadt; WCU Program of National Research
Foundation of Korea under Contract No. R32-2008-000-10155-0

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# Optimizing the chiral Purcell factor for unidirectional single photon
emitters in topological photonic crystal waveguides using inverse design

Eric Nussbaum<EMAIL_ADDRESS>Centre for Nanophotonics, Department
of Physics, Engineering Physics and Astronomy, Queen’s University, Kingston,
Ontario, Canada, K7L 3N6  Nir Rotenberg<EMAIL_ADDRESS>Centre for
Nanophotonics, Department of Physics, Engineering Physics and Astronomy,
Queen’s University, Kingston, Ontario, Canada, K7L 3N6  Stephen Hughes
<EMAIL_ADDRESS>Centre for Nanophotonics, Department of Physics,
Engineering Physics and Astronomy, Queen’s University, Kingston, Ontario,
Canada, K7L 3N6

###### Abstract

We present an inverse design approach to significantly improve the figures-of-
merit for chiral photon elements and quantum emitters in topological photonic
crystal slab waveguides. Beginning with a topological waveguide mode with a
group index of approximately 10 and a maximum forwards or backwards Purcell
factor at a chiral point of less than 0.5, we perform optimizations of the
directional Purcell factor. We use a fully three dimensional guided-mode
expansion method to efficiently calculate waveguide band dispersion properties
and modes, while automatic differentiation is employed to calculate the
gradient of objective functions. We present two example improved designs: (i)
a topological mode with an accessible group index of approximately 30 and a
maximum unidirectional Purcell factor at a chiral point greater then 4.5
representing a nearly 10-fold improvement to the Purcell factor, and (ii) a
slow light mode, well away from the Brillouin zone edge with a group index
greater then 350 and a maximum unidirectional Purcell factor at a chiral point
greater than 45.

##  I Introduction

Chiral quantum elements play an increasingly important role in quantum optical
experiments and technologies [1] as they enable non-reciprocal devices [2, 3,
4, 5], quantum logic gates [6, 7] and networks [8]. One potential route
towards scalable, on-chip chiral elements are photonic crystal waveguides
(PCWs) integrated with semiconductor quantum dots (QDs) [9, 10, 11]. Photonic
crystal waveguides are known to collect nearly all photons emitted by a
quantum dot [12] while enhancing this emission [13], providing
indistinguishable photons over long time scales [14] and enabling efficient
few photon quantum optical nonlinearities [15]. Chiral interactions arise when
a QD with circular polarized transition dipoles is placed in a region where
the guided mode field is circularly polarized [16, 17], or when both dipole
and field are elliptically polarized [17], with the direction determined by
the matching of the signs of the dipole and field polarizations [18]. This has
been demonstrated in conventional PCWs [2] and more recently in a topological
PCWs [19, 20].

A significant issue for exploiting PCWs in quantum photonic circuits is that
they suffer from disorder-induced backscattering, which is especially severe
in the slow light regime [21, 22, 23, 24, 25, 26, 27] where emission
enhancement occurs. Photon transport through topological PCWs is known to be
robust against certain, symmetry-preserving defects such as 60 degree bends
[28, 29, 19], and can reduce scattering losses in general even while enhancing
chiral light-matter interactions [30]. Despite these known advantages, the
question of how much a topological PC waveguide can enhance chiral
interactions remains. Traditionally, PCW designs have been improved using
brute force searches across a few parameters [31, 32, 33, 34, 35]. This has
allowed the slow-light properties of PCWs [36, 32] and the quality factors of
photonic crystal cavities to be significantly improved. More recently,
nanophotonic inverse design has enabled further optimization, leading to
robust performance compared to devices designed with conventional methods [37,
38, 39, 40]. Instead of relying on the designers intuition or brute force
searches, inverse design optimizes a large number of parameters describing the
device, allowing novel and intuitive designs to be obtained [41].

In this work, we show how to optimize a state of the art topological PCW [19]
to enhance quantum chiral interactions. We use shape optimization [38] to
demonstrate an order of magnitude improvement to the directional Purcell
factor. To achieve this task, we use the efficient guided-mode expansion
method (GME) [42] to calculate the band structure and modes of PCWs, and
automatic differentiation to calculate the gradient of objective functions. In
Sec. II, we outline the GME method, discuss how the directional Purcell factor
is calculated, and how we apply inverse design. In Sec. III, the results of
two optimizations are presented, one operating at a moderate group index
$n_{g}\approx 30$ and the other operating at a larger group index
$n_{g}\approx 350$. In both cases, single mode operation, well away from the
Brillouin zone edge, is maintained. Finally, we conclude in Sec. IV.

##  II General Theory and Optimization Protocol

In this section, we describe how we can efficiently and rapidly optimize
complex, topological PCWs without the need for computationally intensive
methods such as finite element or finite-difference time-domain (FDTD)
techniques. We briefly describe how the GME method allows us to obtain PCW
modes and band structure efficiently, then show how the directional Purcell
factor is calculated, and finally discuss how we apply inverse design to
rapidly improve chiral figures of merit.

###  II.1 Guided-mode expansion method

The GME solves Maxwell’s equations by expanding the magnetic field in the
basis of the modes of the effective homogeneous slab, and then solves the
resulting eigenvalue problem [42]. The GME is a 3D vectorial method and is
significantly faster than alternative methods such as FDTD for calculating the
band structure and modes of PCWs [43]. In this work, we use a GME Python
library Legume of Minkov et al. [38, 44]. The main details of the GME method
are summarized below.

Within a linear dielectric medium, Maxwell’s equations for the electric and
magnetic fields, $\bm{E}(\bm{r},t)$ and $\bm{H}(\bm{r},t)$ respectively, can
be rewritten in the frequency domain to obtain an eigenvalue equation. In
terms of the magnetic field, $\bm{H}(\bm{r},\omega)$, one has:

$\bm{\nabla}\times\left(\frac{1}{\epsilon(\bm{r})}\bm{\nabla}\times\bm{H}(\bm{r},\omega)\right)=\left(\frac{\omega}{c}\right)^{2}\bm{H}(\bm{r},\omega),$
(1)

where $\epsilon(\bm{r})$ is the dielectric constant, with the condition
$\nabla\cdot\bm{H}(\bm{r},\omega)=0$. We assume lossless media and neglect
frequency dispersion in the dielectric constant, which is a good assumption
for the materials of interest for designing and making PCW waveguide modes.
Once the magnetic field is calculated, the electric field is easily obtained
from

$\bm{E}(\bm{r},\omega)=\frac{i}{\omega\epsilon_{0}\epsilon(\bm{r})}{\bm{\nabla}}\times\bm{H}(\bm{r},\omega).$
(2)

To solve equation (1), the GME method expands the magnetic field into an
orthonormal set of basis states as

$\bm{H}(\bm{r},\omega)=\sum_{\mu}c_{\mu}\bm{H}_{\mu}(\bm{r}),$ (3)

and so equation (1) can be written as

$\sum_{\nu}\mathcal{H}_{\mu\nu}c_{\nu}=\frac{\omega^{2}}{c^{2}}c_{\mu},$ (4)

where the elements of the Hermitian matrix $\mathcal{H}_{\mu\nu}$ are defined
as

$\mathcal{H}_{\mu\nu}=\int\cfrac{1}{\epsilon(\bm{r})}\left({\bm{\nabla}}\times\bm{H}^{*}_{\mu}(\bm{r})\right)\cdot\left({\bm{\nabla}}\times\bm{H}_{\nu}(\bm{r})\right)d\bm{r}.$
(5)

To define an appropriate basis set $\bm{H}_{\mu}(\bm{r})$, the GME method uses
the guided modes of the effective homogeneous slab waveguide, with a
dielectric constant taken as the spatial average of the dielectric constant in
the slab layer of the PCW being studied [42]. The guided modes of the
homogeneous slab depend on a wave vector, which can take any value in the slab
plane, while the modes of the PCW depend on $\bm{k}$, which we restrict to the
first Brillouin zone. Thus, for the magnetic field at each wave vector, only
the effective waveguide modes with wave vector $\bm{k}+\bm{G}$, are included
in the basis. The guided mode expansion for the Bloch modes is then

$\bm{H}_{\bm{k}}(\bm{r})=\sum_{\bm{G},\alpha}C_{\bm{k}+\bm{G},\alpha}\bm{H}_{\bm{k}+\bm{G},\alpha}^{\rm
guided}(\bm{r}),$ (6)

where $\bm{H}_{\bm{k}+\bm{G},\alpha}^{\rm guided}(\bm{r})$ is a guided mode
(which is known analytically) of the effective waveguide and $\alpha$ is the
index of the guided mode. Since the dielectric is periodic both the magnetic
and electric fields can be written according to Bloch’s theorem.

The electric and magnetic field Bloch mode can be written as

$\displaystyle\bm{E}_{\bm{k}}(\bm{r})$
$\displaystyle=\bm{e}_{\bm{k}}(\bm{r})e^{i\bm{k}\cdot\bm{r}},$ (7)
$\displaystyle\bm{H}_{\bm{k}}(\bm{r})$
$\displaystyle=\bm{u}_{\bm{k}}(\bm{r})e^{i\bm{k}\cdot\bm{r}},$ (8)

where $\bm{e}_{\bm{k}}(\bm{r})$ and $\bm{u}_{\bm{k}}(\bm{r})$ are the electric
and magnetic field Bloch modes for wave vector $\bm{k}$, which are normalized
according to $\int_{\rm
cell}\epsilon(\bm{r})|\bm{e}_{\bm{k}}(\bm{r})|^{2}d\bm{r}=1$ throughout this
paper, and the integration is carried out over one unit cell. Note, the GME
can also be used to compute intrinsic losses for Bloch modes above the light
line [45, 46].

###  II.2 Purcell factor for a single quantum emitter

The coupling between a point-dipole quantum emitter and a waveguide mode can
be calculated from a Green’s function analysis [9, 18]. The Green’s function
for the waveguide mode, describing the response at $\bm{r}$ to an oscillating
dipole at $\bm{r}_{0}$, is

$\displaystyle\bm{G}_{w}(\bm{r},\bm{r}_{0},\omega)$
$\displaystyle=\bm{G}_{f}(\bm{r}_{0},\bm{r},\omega)+\bm{G}_{b}(\bm{r},\bm{r}_{0},\omega)$
(9)
$\displaystyle=\frac{ia\omega}{2v_{g}}[\Theta(x-x_{0})\bm{e}_{k}(\bm{r})\otimes\bm{e}^{*}_{k}(\bm{r}_{0})e^{ik(x-x_{0})}$
$\displaystyle+\Theta(x_{0}-x)\bm{e}_{k}^{*}(\bm{r})\otimes\bm{e}_{k}(\bm{r}_{0})e^{-ik(x-x_{0})}],$

where $v_{g}$ is the group velocity and $\Theta$ is the Heaviside step
function. The subscripts refer to backward (‘$b$’) and forward (‘$f$’) mode
contributions.

The rate of photon emission, from a QD exciton into the waveguide mode can be
split into forwards and backwards rates. Representing the dipole as a unit
vector $\hat{\bm{\mu}}$ with dipole moment $d_{0}$, the forwards and backwards
emission rates are [18], respectively:

$\displaystyle\Gamma_{w}^{f}(\bm{r}_{0})$ $\displaystyle={\rm
Im}\left[\frac{2d_{0}^{2}\hat{\bm{\mu}}^{\dagger}\cdot\bm{G}_{f}(\bm{r}_{0},\bm{r}_{0},\omega)\cdot\hat{\bm{\mu}}}{\hbar\epsilon_{0}}\right],$
(10) $\displaystyle\Gamma_{w}^{b}(\bm{r}_{0})$ $\displaystyle={\rm
Im}\left[\frac{2d_{0}^{2}\hat{\bm{\mu}}^{\dagger}\cdot\bm{G}_{b}(\bm{r}_{0},\bm{r}_{0},\omega)\cdot\hat{\bm{\mu}}}{\hbar\epsilon_{0}}\right].$
(11)

These expressions are valid for lossless media [47], otherwise a more careful
quantum analysis is needed, where the rate of emission is no longer
proportional to the projected local density of states, and one must include
additional contributions from vacuum fluctuations.

The Purcell factor is defined as $P=\Gamma_{w}/\Gamma_{0}$, where

$\Gamma_{0}=\frac{\omega^{3}d_{0}^{2}\sqrt{\epsilon_{s}}}{3\hbar\pi\epsilon_{0}c^{3}},$
(12)

is the emission rate into the corresponding homogeneous medium with dielectric
constant $\epsilon_{s}$ [9]. Separating the Purcell factor into its forwards
and backwards components,

$\displaystyle P^{f}(\bm{r}_{0})$ $\displaystyle=\frac{3\pi
c^{3}a}{v_{g}\omega^{2}\sqrt{\epsilon_{s}}}\hat{\bm{\mu}}^{\dagger}\cdot\left[\bm{e}_{k}(\bm{r}_{0})\otimes\bm{e}_{k}^{*}(\bm{r}_{0})\right]\cdot\hat{\bm{\mu}}$
(13) $\displaystyle P^{b}(\bm{r}_{0})$ $\displaystyle=\frac{3\pi
c^{3}a}{v_{g}\omega^{2}\sqrt{\epsilon_{s}}}\hat{\bm{\mu}}^{\dagger}\cdot\left[\bm{e}_{k}^{*}(\bm{r}_{0})\otimes\bm{e}_{k}(\bm{r}_{0})\right]\cdot\hat{\bm{\mu}},$

where $v_{g}$ and ${\bm{e}}_{k}$ implicitly depends on frequency.

At a point where the field of the forward propagating mode is right circularly
polarized (RCP), with unit vector
$\hat{\bm{n}}_{R}=2^{-1/2}(\hat{\bm{x}}+i\hat{\bm{y}})$, a RCP dipole
$\bm{\mu}=\bm{\sigma}_{+}$ will only couple to this mode and not to the one
propagating backwards; similarly, a left circularly polarized (LCP) dipole
$\bm{\mu}=\bm{\sigma}_{-}$ placed at the same position, will only couple to
the backwards propagating mode [18].

With a focus on optimizing chiral photon elements, we also define the chiral
directionality,

$C=\frac{P^{f}-P^{b}}{P^{f}+P^{b}},$ (14)

as well as the chiral forwards Purcell factor,

$\tilde{P}^{f}=CP^{f},$ (15)

which will be used below as representative figures of merit.

###  II.3 Inverse design methodology

Inverse design treats the design process as an optimization problem. Instead
of using a heuristic understanding of a specific phenomenon and adjusting a
small number of related parameters, inverse design optimizes a large number of
device parameters. This can lead to unintuitive designs with significantly
better performance then their traditionally designed counter parts [37].

Gradient based optimization algorithms are usually used in inverse design. To
efficiently calculate objective function gradients the adjoint variable method
is widely used [38]. However, implementing the adjoint variable method can be
extremely difficult with a solver which solves an eigenvalue problem, such as
the GME method [38]. Instead, automatic differentiation can be used to
calculate the gradient of an arbitrarily complex objective function. An
automatic differentiation library tracks all sub-function executions, for
which the gradient is known, and then uses the chain rule to compose the
objective functions gradient [38]. In this work, we use the automatic
differentiation Python library Autograd [48]. The GME library we use,
‘Legume’, was developed to be compatible with Autograd making it
straightforward to calculate the gradient of the objective functions based on
a GME calculation [38].

Starting from an initial design, shape optimization modifies the boundaries
between different materials in order to optimize the objective function. In
comparison, topology optimization places fewer constraints on how the material
distribution can evolve, including allowing the number of holes to be changed.
Shape optimization is therefore well suited to optimizing designs where the
general shape is known before hand [49] while topology optimization can
produce remarkably inventive designs.

##  III Results

Beginning with a state-of-the-art topological PCW [19], we will show several
improved designs. The optimizations are performed in two phases: first the
maximum $\tilde{P}^{f}$ is optimized for a particular mode and wave vector,
and second, the dispersion is optimized for a slow light region. One design is
optimized to operate at an accessible group index $n_{g}=c/v_{g}$ of
approximately 30, while the other has a region of very slow light with a group
index of greater then 350 away from the Brillouin zone edge.

###  III.1 Initial design

Figure 1: (a) Schematic of the topological edge-mode waveguide. The direction
of propagation $x$ and relevant geometric parameters are indicated. (a)
Schematic (top-down view) of the crystal. The rhombic unit cell is outlined.
Magnitude of the Berry curvature for the mode bellow the band gap
${\Omega}(\bm{k})$ (units of $[a^{2}/4\pi^{2}]$) for the upper (b) and the
lower (c) crystals shown in (a) for a slab thickness $h=170$ nm, $a=250$ nm,
$r_{1}=28$ nm, $r_{2}=62.5$ nm. The outer dashed black line is the edge of the
first Brillouin zone and the inner dashed line’s vertices are the $K$ and
$K^{\prime}$ points.

The optimization is started from a reference topological PCW in which chiral
coupling of single QDs to the waveguide mode has been experimentally realized
[19], using a promising design. The PCW modes are formed at the interface
between two topologically distinct crystals. The two crystals are inversion
symmetry partners and are formed from a hexagonal lattice of unit cells
containing two circular holes of different sizes, with radii $r_{1}$ and
$r_{2}$, as shown in Fig.1a.

In analogy with topological electronics, the topology of these crystals is
related to their Chern number [50] which due to the time-reversal symmetry of
this photonic system, is equal to zero. However, this need not be the case for
the valley Chern index which we calculate by partitioning the integration of
the Berry curvature

$\bm{\Omega}_{n}(\bm{k})=\bm{\nabla}_{\bm{k}}\times
i\Braket{\bm{u}_{\bm{k},n}}{\bm{\nabla}_{\bm{k}}}{\bm{u}_{\bm{k},n}},$ (16)

where $n$ is the index of the mode, over the Brillouin zone to around only the
$K$ or $K^{\prime}$ valleys [29], as shown in figure 1b,c. For details on how
to numerically calculate the Berry curvature see Ref. [51]. The valley Chern
index is $C_{V}=C_{K}-C_{K^{\prime}}$ where, for example,
$C_{K}=\frac{1}{2\pi}\oiint_{BZ,K}\Omega_{n}(\bm{k})dk^{2}$ is the integral of
the Berry curvature over half the Brillouin zone around the $K$ valleys.
Unlike the Chern number, the valley Chern index is not necessarily quantized
[29, 51]. The two crystals on each side of the interface have the same
$|C_{V}|$ though with opposite sign. This opposing sign gives rise to valley
Hall edge states which are expected, and have been shown to be more robust
against defects that do not couple the counter-propagating states [28, 29].

This initial topological waveguide is designed to operate near a wavelength of
1000 nm in a 170-nm thick GaAs (dielectric constant of 11.6) membrane, and so
has a pitch $a=266$ nm, $r_{1}=28$ nm, and $r_{2}=62.5$ nm [19]. As shown in
figure 2a, the initial design has two guided modes in the frequency range of
interest. We refer to these modes as trivial (T) and non-trivial (NT), and in
what follows focus on the latter. This is because the NT mode has been
experimentally demonstrated to support chiral coupling to single QDs and used
to form triangular ring resonators [19]. Further, as demonstrated in Ref.
[19], when a corner is placed in the waveguide the NT mode propagates around
the corner while the T guided mode is prone to backscattering. We note that,
if we begin with the bearded interface in figure 1(a) and increase the radius
of one of the rows of small holes at the interface to the size of the large
holes such that there is a smooth transition to a zigzag interface, then the
NT mode remains throughout, while the other guided mode disappears into the
bulk bands [19].

For this initial design, we observe that the NT edge state has a single mode
(SM) bandwidth positioned around 296 THz. Within this frequency band, at 296.5
THz, at a spectral separation of 1.6 THz away from the band edge, the mode has
a group index $n_{g}=8.5$, as shown in Fig. 2b by the circular marker.
Although chiral emission into this structure has been observed [19], the
calculated $P^{f}$ map peaks at 0.96 (Fig. 2c), demonstrating that this
emission is in fact suppressed. Worse, the maxima of $P^{f}$ and $C$ do not
overlap for this initial design, and we find a maximal $\tilde{P}^{f}=0.41$
(Fig. 2e). In the region where $C\geq 0.97$, the maximum forwards Purcell
factor is $\max(P^{f})=0.41$.

Before optimizing the design, we shift the pitch to $a=250$ nm to increase the
operational frequency to ensure that the optimized design can function around
$320$ THz, the desired frequency for working with high-quality gallium
arsenide quantum dots [52]. However, because of fabrication limits, we do not
decrease $r_{1}$ bellow our minimum allowed size of $r_{\min}=27.5$ nm [19] so
we use the same hole sizes in $r_{1}=28$ nm and $r_{2}=62.5$ nm with $a=250$
nm. We note that changing the hole radii by this amount does not adversely
affect the concentration of the Berry curvature with opposite signs at the $K$
and $K^{\prime}$ points. After the optimization, the periodicity can be
increased if it is necessary to decrease the operational frequency, but the
periodicity can not in general be further decreased without violating our
fabrication-imposed constraints.

Figure 2:  (a) Band structure for the initial design [19] with $a=266$ nm. The
(NT) guided mode is drawn in solid purple, the second guided mode in dashed
purple, and bulk modes in black. The NT mode’s SM bandwidth is highlighted in
turquoise. Additional guided modes existing at the secondary interface at the
edge of the GME unit cell have been removed in this figure, as well as in
figures 3 and 4 (see Appendix A). (b) The group index for the NT mode is drawn
in purple and the maximum chiral forwards Purcell factor is drawn with a solid
blue line. Frequencies at which the NT mode is above the light line are
highlighted in grey. The following fields are calculated at the circular
marker in (a) and (b) where $n_{g}=8.5$. The $xy$-plane cross sections are at
the center of the slab. (c) Forwards Purcell factor $P^{f}$ for
$\bm{\mu}=\bm{\sigma}_{-}$, (d) the associated chiral directionality $C$, and
(e) the associated chiral forwards Purcell factor $\tilde{P^{f}}$. The red
dashed box in (c) encloses the region shown in (d) and (e).

###  III.2 Optimization

One of our objectives for the optimization is to improve the system’s
performance as a bright, or rapid, chiral light-matter interface. We therefore
demand single mode operation, and only consider positions that are at least 35
nm away from the nearest hole edge, to ensure that the QDs can be electrically
gated [53]. The optimized design should have an accessible group index of less
than 50 with a Purcell factor, in a region where $C\approx 1$, of
approximately 5 which is close to the optimum Purcell factors for enabling
efficient pulse-triggered single photon sources [54, 55, 56]. As discussed
above and shown in figure 2, the maximum chiral Purcell factor $P^{f}$ for the
initial design is 0.41. We are therefore targeting an increase by a factor of
approximately 10. With the same restrictions, apart from allowing for a large
group index greater then 50, we also seek a design with a maximized chiral
forwards Purcell factor.

A shape parameterization is used where the position and radii of the 12 holes
closest to the interface are allowed to be modified. To ensure the final
design can be fabricated extremely small features must be avoided. Therefore
the minimum allowed hole radius is 27.5 nm and the minimum allowed distance
between hole edges is 40 nm. At the start of each optimization, iteration a
projection is applied to the design parameters which transforms designs which
violate theses constraints to a similar design which does not. The “Adam” [57]
optimization algorithm is used throughout.

A two-step optimization is performed. The first optimization phase is focused
on improving $\max(\tilde{P}^{f})$ in a region approximately 35 nm away from
the nearest hole edge. The objective function for the first phase, is

$\mathcal{F}_{1}(\bm{\gamma})={\rm max}[G(\bm{r})\tilde{P^{f}}(\bm{r})]G_{\rm
SM},$ (17)

where $\bm{\gamma}$ is a vector of the device parameters, $G$ is a smooth step
function that is equal to 0 inside a hole and approximately 1 more then 35 nm
from a hole,

$G(\bm{r})=\dfrac{1}{1+e^{-g(r_{m}(\bm{r})-r_{s})}},$ (18)

where $r_{m}(\bm{r})$ is the minimum distance to a hole edge, $r_{s}=30$ nm
and $g=74~{}{\rm nm}^{-1}$ are parameters, $G\tilde{P}^{f}$ are calculated for
the NT mode at a fixed wave vector, and $G_{\rm SM}$ encourages a certain
magnitude and position of SM bandwidth. The objective function for the first
phase is maximized.

The intermediate design from the first phase is further optimized in a second
phase of optimization which is focused on improving the dispersion
characteristics for slow-light operation. Two different objective functions
are used to obtain two different styles of design. The first is designed to
obtain our primary objective, of operation at an accessible group index,

$\mathcal{F}_{2A}(\bm{\gamma})=\int_{k_{x,0}}^{k_{x,1}}dk_{x}(n_{g}-n_{g,d})^{2},$
(19)

where $n_{g}$ is the group index of the NT mode, $n_{g,d}=30$ is the desired
group index, and the integral is performed over $k_{x}a=[0.6,0.9]\pi$. The
second, is used to obtain a very large $n_{g}$ inside the NT mode’s SM
bandwidth,

$\mathcal{F}_{2B}(\bm{\gamma})=\omega_{b}-\omega_{{\rm
SM},\max}+\frac{D}{\max_{\rm SM}(n_{g})}$ (20)

where $\omega_{b}$ is the frequency, under the light line, where $n_{g}$ first
changes sign, $\omega_{{\rm SM},\max}$ is the maximum frequency of the NT
mode’s lowest frequency region of SM bandwidth, $D$ is a weight, and
$\max_{\rm SM}(n_{g})$ is the maximum $n_{g}$ in the NT mode’s lowest region
of SM bandwidth. OF B therefore works to make the NT mode’s bandwidth entirely
SM and to increase the maximum $n_{g}$ in the SM bandwidth. Both
$\mathcal{F}_{2A}(\bm{\gamma})$ and $\mathcal{F}_{2B}(\bm{\gamma})$ are
minimized.

###  III.3 Improved designs

The results of our first optimization routine, designed to operate at a
moderate group index, i.e., optimizing equation (17) and then equation (19),
are summarized in figure 3. The band structure is shown in figure 3(a). The NT
mode has a SM slow light region with group index $n_{g}\approx 30$ deep inside
the bandgap and away from the Brillouin zone edge. At a frequency where
$n_{g}=30$, indicated with a circular marker in figure 3(a) and (b), the NT
mode has $\max(\tilde{P}^{f})=4.5$, with $C=0.91$. In the region at least 35
nm from the nearest hole edge where $C\geq 0.95$, the maximum forwards Purcell
factor is $\max(P^{f})=4.7$. This design satisfies our primary objective, the
maximum Purcell factor at a region of high directionality far from a hole edge
has been improved by a magnitude of order to values around our desired value
of 5 [54, 55, 56].

Figure 3:  Summary of new design A. (a) Band structure. (b) Group index and
the maximum chiral forwards Purcell factor for the NT mode. The following
fields are calculated at the circular marker in (a) and (b) where
$n_{g}=30.2$. (c) Forwards Purcell factor $P^{f}$ for
$\bm{\mu}=\bm{\sigma}_{-}$, (d) the associated chiral directionality $C$, and
(e) the associated chiral forwards Purcell factor $\tilde{P^{f}}$.

Next, we optimize a waveguide to operate in the slow-light regime, well away
from the band edge, using 20 and summarizing the results in figure 4. For this
design, we observe a slow light, SM bandwidth with a maximum $n_{g}>350$ ,
deep inside the bandgap and away from the mode edge (Figs. 4a and b). For the
mode with $n_{g}=360$, we find $\max(\tilde{P}^{f})=48$, with $C=0.98$. The
position of $\max(\tilde{P}^{f})$ is 40 nm away from the nearest hole edge.
This is a two magnitude of order increase to the maximum chiral forwards
Purcell factor from the initial design. In the region at least 35 nm from the
nearest hole edge where $C\geq 0.95$, the maximum forwards Purcell factor is
$\max(P^{f})=50$.

Figure 4:  Summary of new design B. (a) Band structure. (b) Group index and
the maximum chiral forwards Purcell factor for the NT mode. The following
fields are calculated at the circular marker in (a) and (b) where $n_{g}=360$.
(c) Forwards Purcell factor $P^{f}$ for $\bm{\mu}=\bm{\sigma}_{-}$, (d) the
associated chiral directionality $C$, and (e) the associated chiral forwards
Purcell factor $\tilde{P^{f}}$.

##  IV Conclusions

Using an efficient nanophotonic inverse design approach, we have optimized a
state of the art topological PCWs for directional QD-photon interactions. We
present PCWs designed to operate in the single mode regime, well away from the
mode edge, and with moderate (30) and large (360) group indices. The designs
have maximum forwards Purcell factors, at least 35 nm away from the nearest
hole edge with $C>=0.95$, of 4.7 and 50 at group indexes of 30 and 360
respectively.

Our optimized designs, while respecting all fabrication requirements, break
the symmetry of the initial [19] and other previously reported [30, 29, 28,
58, 59] topological PCWs, as the first 3 original crystal unit cells on each
side of interface differ (see Appendix B for details). As a result, the T mode
disappears, ensuring the SM operation of the remaining, NT mode. Moreover,
this broken symmetry leads to an asymmetric (with respect to the direction of
propagation, $\hat{x}$) field (and hence, $P^{f}$) distributions. This is most
noticeable for slow-light modes (cf. Fig. 4c) but can also be observed in the
faster mode (cf. Fig. 3c). In essence, our optimization further increases the
chiral-projected local density of states by borrowing from other regions, in a
way that is not possible with standard, topological PCW designs. Our results
demonstrate the power of efficient inverse optimization methods, providing a
key to unlock the full potential of topological quantum photonics.

###### Acknowledgements.

We thank Nicholas Martin, Mahmoud Jalali Mehrabad, Andrew Foster, and Luke
Wilson for useful discussions. This work was supported by the Natural Sciences
and Engineering Research Council of Canada, the Canadian Foundation for
Innovation, and Queen’s University, Canada.

##  Appendix A Additional edge states existing at the edges of the GME
supercell

The GME method assumes 2D periodicity. As the crystals on each side of the
waveguide are inversion symmetry partners, at the $y$-limits of the GME
waveguide unit cell a second topological waveguide is modeled. This is a
legitimate design, though not the design of interest. As an example the band
structure for the unit cell shown in figure 5(b) is plotted in figure 5(a).
The unit cell contains two interfaces between the topologically distinct
crystals, the first being the waveguide [19] which the optimization was
started from and the second being another topological waveguide which has
previously been described in Ref. [29].

Figure 5: (a) Band structure for the unit cell in (b). The modes existing at
guide 1 are drawn in purple, the modes existing at guide 2 are drawn in green,
and the bulk modes are drawn in black. (b) GME unit cell containing the
waveguide which the optimization was started from [19] and another waveguide
[29].

##  Appendix B Specifications of improved designs

In this appendix the position and radii of holes in the initial and improved
designs are provided. Table 1 lists the position and radii of the 12 holes
closest to the interface, which are the holes whose position and radii were
allowed to be modified during the optimizations, for the initial design,
design A and design B.

Table 1: Location and radius of the 12 holes closest to the interface in the initial design, design A, and design B. The x axis is at the center of the waveguide initial design, see figure 1 for the coordinate system.  Design | x [a] | y [a] | r [a]
---|---|---|---
Initial Design | 0.0000 | -2.4537 | 0.2350
0.0000 | -1.8764 | 0.1053
0.5000 | -1.5877 | 0.2350
0.5000 | -1.0104 | 0.1053
0.0000 | -0.7217 | 0.2350
0.0000 | -0.1443 | 0.1053
0.5000 | 0.1443 | 0.1053
0.5000 | 0.7217 | 0.2350
0.0000 | 1.0104 | 0.1053
0.0000 | 1.5877 | 0.2350
0.5000 | 1.8764 | 0.1053
0.5000 | 2.4537 | 0.2350
Design A | 0.0316 | -2.5706 | 0.2151
-0.1319 | -2.1138 | 0.1100
0.4879 | -1.6942 | 0.2961
0.9557 | -1.2321 | 0.1100
0.0089 | -0.6223 | 0.3422
0.4922 | -0.1223 | 0.1819
-0.0059 | 0.0781 | 0.1100
0.4866 | 0.7157 | 0.3260
0.1667 | 1.2187 | 0.1100
-0.1048 | 1.7316 | 0.3098
0.3516 | 2.1249 | 0.1100
0.5047 | 2.5600 | 0.1909
Design B | 0.0579 | -2.6139 | 0.1884
-0.1001 | -1.9734 | 0.1100
0.4850 | -1.6825 | 0.2367
0.9028 | -1.2260 | 0.1100
0.0849 | -0.6747 | 0.3106
0.4854 | -0.0509 | 0.1347
-0.0244 | -0.0332 | 0.1188
0.4899 | 0.7060 | 0.3301
0.1269 | 1.1884 | 0.1100
-0.1016 | 1.7545 | 0.2626
0.3338 | 2.0794 | 0.1100
0.4764 | 2.5520 | 0.1327

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