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Review
Quantum Topological Photonics
Qiuchen Yan, Xiaoyong Hu,* Yulan Fu,* Cuicui Lu,* Chongxiao Fan, Qihang Liu,
Xilin Feng, Quan Sun, and Qihuang Gong
DOI: 10.1002/adom.202001739
A variety of research has been carried
out on aspects of the physics of quantum
information,[6] including quantum walks,[7]
quantum light sources,[8] boson sam-
pling,[9] quantum integrated devices,[10]
quantum networks,[11] quantum com-
puting,[12] and teleportation of quantum
information.[13] Using quantum optics,
communications can be transmitted
faster and farther and big data can be pro-
cessed accurately and quickly. In addition,
quantum optics has a strong information-
carrying ability and is being developed
for integration onto chips for computing
system applications. It is also gradually
being discovered that some phenomena
that violate the laws of classical physics
can be explained using quantum physics,
which provides a concise physical compre-
hension of these phenomena. Quantum
physics simplifies the complex world and
quantum optics thus oers unparalleled
advantages when compared with classical
optics. Furthermore, on the experimental
side, researchers have focused on realiza-
tion of the N-phonon entangled state and
have had remarkable eects on integrated
optical chips. With the increasing require-
ments of light information processing, quantum optics and
quantum information applications are now in urgent need of
further development. However, several limitations still exist in
quantum systems and one of the greatest problems is with the
decoherence and fidelity involved in systems that contain two
Quantum topological photonics is a new research field with great potential
that is based on developments in both quantum optics and topological pho-
tonics. Topological photonics oers unique properties, including topological
robustness and an anti-backscattering property, and these advantages are
strongly required in quantum optics. Quantum technology, which includes
quantum optics, represents an important direction for future technological
development. However, existing quantum light sources are unstable and
quantum information may easily be lost during transmission. These dis-
advantages have troubled researchers for a long time and no perfect solu-
tion is available thus far. Fortunately, application of topological photonics
to quantum optics can help to generate robust quantum light sources and
protect photons from decoherence during photon propagation. This allows
the correlation and entanglement to be maintained even when photons travel
over long distances. To date, quantum topological photonics has provided
major breakthroughs in certain quantum devices. This Review presents the
basic concepts of quantum topological photonics and summarizes how the
topological protection property works in quantum light sources, quantum
information transmission, and other quantum devices. Finally, an outlook
is provided on the remaining challenges and potential future directions of
quantum topological photonics, which can aid in exploration of additional
new phenomena.
The ORCID identification number(s) for the author(s) of this article
can be found under https://doi.org/10.1002/adom.202001739.
Dr. Q. Yan, Prof. X. Hu, C. Fan, Q. Liu, X. Feng, Prof. Q. Gong
State Key Laboratory for Mesoscopic Physics & Department of Physics
Collaborative Innovation Center of Quantum Matter &
Frontiers Science Center for Nano-Optoelectronics
Beijing Academy of Quantum Information Sciences
Peking University
Beijing 100871, P. R. China
E-mail: xiaoyonghu@pku.edu.cn
Prof. X. Hu, Prof. Q. Gong
Collaborative Innovation Center of Extreme Optics
Shanxi University
Taiyuan, Shanxi 030006, P. R. China
Prof. X. Hu, Prof. Q. Sun, Prof. Q. Gong
Peking University Yangtze Delta Institute of Optoelectronics
Nantong, Jiangsu 226010, P. R. China
Prof. Y. Fu
Institute of Information Photonics Technology and Faculty of Science
Beijing University of Technology
Beijing 100124, P. R. China
E-mail: fuyl@bjut.edu.cn
Prof. C. Lu
Key Laboratory of Advanced Optoelectronic Quantum Architecture and
Measurements of Ministry of Education
Beijing Key Laboratory of Nanophotonics and Ultrafine Optoelectronic
Systems
School of Physics
Beijing Institute of Technology
Beijing 100081, P. R. China
E-mail: cuicuilu@bit.edu.cn
1. Introduction
Quantum topological photonics is a new field in optics that has
seen very rapid growth in recent years[1–5] and can be regarded as
a crossover between quantum optics and topological photonics.
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or more photons. Furthermore, quantum systems are more
complex and are dicult to simulate using current computing
facilities because of the incompatibility of classical systems with
the novel properties of quantum systems. Usually, quantum
networks and quantum computing systems are multibody sys-
tems, which are vulnerable to environmental changes, and the
fidelity of these systems therefore suers. At this bottleneck
in the development of quantum optics, topological photonics
provides a new strategy to solve the problems described above
because of the robustness of the topology provided.
Since the discovery of the integer quantum Hall eect
(IQHE) in 1980, the topological phases of matter, and par-
ticularly topological insulators, have been active subjects of
research and this led to the 2016 Nobel Prize in Physics being
presented to three of the pioneers in the field, David Thouless,
Duncan Haldane, and J. Michael Kosterlitz. Topological pho-
tonics aims to mimic the properties of topological electronic
structures using bosonic degrees of freedom and many novel
system configurations have been proposed that dier from
the topological systems of fermions and are more suitable for
bosons, e.g., robust optical delay lines.[14] Although it is known
that topological physics oers numerous surprising advantages,
including anti-backscattering behavior, unidirectional propaga-
tion and robustness, its most obvious feature is that light prop-
agates along the boundaries of the structures involved. Because
of these unique properties, topological photonics has become
a promising approach to the development of photonic devices
and even to quantum photonics, remedying the shortcomings
of the existing methods. In recent years, 1D, 2D, and even
higher-dimensional topological photonic structures with robust
edge states have been proposed successfully using ingenious
designs. In the 1D structures, multilayer gain dielectric films
are used to fabricate the Su–Schrieer–Heeger (SSH)chain
and other 1D topological structures.[15–19] In 2D structures,
shrinking and expanding photonic crystal lattices are used to
form artificial topological edge states, and large-scale resonant
rings are also used.[20,21] In summary, the topological photonics
field is seeing vigorous development and the properties of topo-
logical photonics can compensate for some of the deficiencies
in quantum optics that were discussed above.
Quantum optics needs to overcome problems in terms
of stability and topological photonics oers the advantage of
robustness; it is thus beneficial to combine the two fields to
produce the field of quantum topological photonics. Quantum
topological photonics focuses on quantum photonic processes
and phenomena based on topological photonic structures and
their applications in integrated quantum photonics, including
quantum walks, quantum simulations, quantum computing,
and quantum communications. The quantum topological tran-
sition was proposed at an early stage of development,[22] and
then quantum walks and boson sampling approaches using
topological photonic structures were also studied.[23–26] These
studies have demonstrated the considerable feasibility of com-
bining quantum photonics and topological photonics. For
example, stable and deterministic quantum sources represent
the basic guarantees that must be met to enable realization of
the follow-up functions of quantum optical chips; regardless of
whether the current sources are the single photon, squeezed
photon state, or entangled photon types, they all lack stability
and certainty under current facility conditions. Moreover, these
sources can be only maintained in very strict environments,
such as the ultralow temperatures required for single photons
from quantum dot materials and the large-scale helical inte-
grated waveguides required for entangled photons. Because of
their high losses and instability, it is inconvenient to integrate
the quantum sources and then detect the photons. In quantum
topological photonics, the topological properties of the struc-
tures protect the quantum light sources from decoherence and
help the correlated photon state to propagate over longer dis-
tances, even with the disorder, which represents a breakthrough
for quantum optics. Similarly, topological photonics has been
applied in other quantum devices, including lasers, quantum
interferometers and beam splitters, quantum amplifiers, and
chiral coupling splitters. In addition, topological photonics is
based on the behavior of group particles and thus is suitable
for quantum network and quantum computing applications.
The topology provides a new degree of freedom and provides
a new research perspective for quantum optics. Furthermore,
researchers have also attempted to simulate topological struc-
tures using atom lattices, which oer a new platform for study
of topological photonics using quantum methods.
In this review, we discuss the recent progress in quantum
topological photonics. Section 1 gives an introduction to the
topic. The concepts of quantum optics and topological photonics
are then introduced systematically in Section2, including quan-
tized photon representation, the novel strategies used to realize
quantum walks, orbital angular momentum (OAM), and 1D,
2D, 3D, and higher-dimensional topological photonics. In Sec-
tion3, we provide a brief presentation of the current methods
used to generate quantum light sources and topological lasers,
along with a detailed discussion of quantum topological light
sources. This involves discussion of correlated and entangled
biphotons, single-photon chiral optics, protected photons trave-
ling along extended dimensional waveguides with specific topo-
logical structures such as the Su–Schrieer–Heeger model,
formation of optical delay lines using coupling resonant rings,
and topological network systems. Section 4 mainly discusses
the relationship between quantum information and topological
photonics. Quantum communication and the associated opera-
tors are also incorporated into this section. Section 5 mainly
introduces quantum topological devices, including quantum
beam splitters, quantum simulators, and quantum amplifiers,
through careful design. In addition, the fundamental principles
of and experimental configurations for quantum topological
photonics are summarized, a brief outlook on the remaining
challenges is provided, and the development directions and
prospects for quantum topological photonics are also reviewed,
thus providing a suitable reference for the future development
of quantum topological photonics.
2. Basic Concepts and Realization of Quantum
Optics and Topological Photonics
In this section, we introduce the basic concepts of quantum
optics and topological photonics. Section 2.1 describes the
development of and basic expressions used in quantum optics.
Section 2.2 then mainly introduces topological photonics,
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including common concepts and the dierent research
dimensions.
2.1. Basic Concepts of Quantum Optics
Quantum optics has seen rapid development in recent years
thanks to the development of appropriate experimental
techniques. The quantum properties with reference to the
electromagnetic field have been revealed and an in-depth
understanding of the interactions between light and matter
has been achieved. The invention of the laser in 1960 opened
up new possibilities for experiments in quantum optics. Inspi-
rational experiments, including the realizations of coherent
states and squeezed states, along with the verification of state
collapse and revival processes, have opened up new and impor-
tant prospects for quantum optics. One of the most important
experiments to be performed in quantum optics is the light
intensity correlation experiment designed by Hanbury Brown
and Twiss (HBT). This experiment expanded the quantum
theory of optical coherence because it focused on correlation
of the intensity fluctuations rather than on phase correlation of
the optical field at two dierent spatiotemporal points. A sche-
matic of the experimental setup used is shown in Figure 1a.
The input beam is split equally into two beams and detected
using two photon detectors, with one of the beams experi-
encing a time delay. The signals from the two photon detectors
are then sent to a correlator to measure the correlation of the
intensity fluctuations. This correlation can be written as shown
in Equation (1)
∆∆ =− −
=−
(,)(,) ((,) )( (,))
(,)(,) (,)(
,)
111222 11 112222
111222 11 1222
IrtIrt IrtIIrtI
IrtIrt IrtIrt
(1)
where Ii(ri,ti) represents the instantaneous field at a point (ri,ti),
I
i is the average intensity, and 〈〉 represents the average value
obtained from repeated measurements. Higher accuracy can be
obtained when compared with that obtained using phase cor-
relation measurements, even under the influence of external
disturbances. In quantum optics, the coherent light field is
defined by the coherent state. The eigenstate of the annihila-
tion operator is a single-mode coherent state and as a result
of vacuum fluctuations, the coherent state contains quantum
noise.
The quantum field is a very important concept in quantum
optics and contributes to the formation of electromagnetic
fields. There are two main categories of quantum states in a
quantum field. When a quantum field can be described using
a state vector |ψ〉, it is called a pure state; however, when the
quantum state is a mixture of dierent pure states |ψ〉 that
have dierent probabilities Pψ, it is then called a mixed state.
A mixed state cannot be described using a state vector; instead,
it is described using a density operator as
∑
ψψρ
=
ψ
ψ
ms P,
where the subscript “ms” represents the mixed states. In fact, a
pure state can also be expressed using a density operator with
the form of ρ=|ψ〉〈ψ|, where P1
∑=
ψ
ψ
in the mixed states.
Furthermore, it should be noted that the distinction between
mixed states and pure states can be seen in superposition form.
Adv. Optical Mater. 2021, 9, 2001739
Figure 1. a) Experimental setup for the standard HBT experiment. Incident light is split into two beams by beam splitters (BSs) and the two beams
are then detected using two photon-detectors (D1 and D2). The signals from these two detectors are sent to a coincidence logic device to measure
the correlation of the intensity fluctuations. b) Schematic of the perturbed topological structure, which is designed for operation in the microwave
measurement band with a gyromagnetic photonic crystal slab and a metal wall. c) Use of the TE and TM modes of the optics to mimic the electron
spin. In the case where the electron spins upward, the modes can be replaced with TE+TM; and when the electron spins downward, the modes can be
replaced with TE−TM. d) Topological phase transition generated by dierences in the lattice constants of photonic crystals. a0 is the lattice constant
and R is the radius of the circles. The left side of the figure shows the trivial photonic crystals, the right side shows the topological case, and the critical
state is shown in the middle. e) An interface between dierent photonic crystals that have dierent lattice constants can have a topological edge state.
The upper structure is the shrunken photonic crystal and the lower structure is the expanded photonic crystal. This robust edge state can maintain
dierent circular polarizations. a) Reproduced under the terms of Creative Commons Attribution License 4.0.[163] Copyright 2019, The Author. b) Repro-
duced with permission.[37] Copyright 2009, Springer Nature. c) Reproduced with permission.[164] Copyright 2013, Springer Nature. d) Reproduced with
permission.[165] Copyright 2015, American Physical Society. e) Reproduced under the terms of Creative Commons Attribution NonCommercial License
4.0.[166] Copyright 2020, The Authors, published by AAAS.
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Comparison of the pure states with the mixed states shows
that the density operators of the mixed states contain diagonal
terms only, while in pure states, in addition to these diagonal
terms, the operators also have o-diagonal terms. The o-diag-
onal terms can cause the interference eect and are thus called
coherent terms. The interactions between quantum systems
and their environments lead to the decay of the density opera-
tors of these quantum systems; the decay of the diagonal terms
is caused by energy dissipation, while the decay of the o-diag-
onal terms is caused by a reduction in the coherence, which is
called decoherence. However, although quantum systems can
easily become decoherent, topological systems can help main-
tain the coherence, and this advantage is reflected particularly
well in quantum light sources. In current quantum devices,
this decoherence invalidates the quantum superposition prin-
ciple. The decoherence thus restricts the number of sequential
gate operations that can be performed while maintaining a
meaningful and coherent quantum state. This case can also be
protected by the topological robustness. More detailed descrip-
tions of the concepts mentioned above and other quantum the-
ories, including quantization of the light field and the quantum
entanglement state, can be found in the literature.[27–30] In
this paper, however, we focus on topological phenomena in
quantum optics, which are described below.
2.2. Topological Photonics
2.2.1. Basic Concepts of Topological Physics
In recent decades, with the development of solid-state theory
and appropriate experimental measurement methods, topo-
logical phenomena in electronics have gradually attracted more
widespread attention.[31] In mathematics, topology concerns
the properties of a geometric object that are preserved under
continuous deformations and the dierent topological shapes
are characterized by topological invariants, which are integers
that remain constant under arbitrary continuous deformations
of the system. Calculation of the Gauss curvature of a closed
surface allows a discrete integer to be obtained that reflects
the number of holes on the surface; this number is called
the Chern number. In electronics, the topology is defined on
the dispersion bands in reciprocal space and the topological
invariant of a 2D dispersion band is the Chern number, which
describes the quantized global feature of this dispersion band.
The most obvious feature of such a topological system is that
the electrons travel along the surface while the inside of the
system is insulated; this is called a topological insulator. Sim-
ilar to the electronic properties in solid-state physics, periodic
optical nanostructures known as photonic crystals aect the
motion of photons in much the same way that ionic lattices
aect the motion of electrons in solids. Therefore, the topo-
logical phenomena of photons can be studied using photons in
nanophotonic structures, in a manner analogous to that of top-
ological electronics. The basic concepts of topological physics
are reviewed in the following.
The discovery of the integer quantum Hall eect[32] in 1980
represents the beginning of topological physics. In 2D electron
systems that were subjected to low temperatures and strong
magnetic fields, it was observed that the Hall conductance
σ undergoes quantum Hall transitions to take on quantized
values of an integral multiple of e2/h. Later, Thouless etal.[33]
and Kohmoto[34] represented the Hall conductance σ using a
topological invariant that was naturally an integer related to
the first Chern number. In solid-state physics, it is known that
the wave function of electrons can be described more clearly in
momentum space over the Brillouin zone. To obtain the Chern
number in a system with an energy bandgap, it is necessary
to know several concepts, including the Berry phase, the Berry
connection, and the Berry curvature of a topological system.
The first and most important of these concepts is the Berry
phase γn. When a state vector moves in a parallel manner in the
parameter space for a period and then returns to the origin, the
Berry phase will accumulate an additional phase
inRt tnR
tt
n
td
dd
0
∫
γ
() ()
() ()
=′′
′′
(2)
The solution for the Berry phase comes from the eigenstate
of the system; as a result, Equation (2) is actually a part of the
original Equation (3)
tERt tinRttnR
tt
t
n
t
() 1dd
dd
00
∫∫
θ
() () ()
() () ()
=′′
−′′
′′
(3)
Here, θ(t) is the phase of the eigenstate of a time-varying
Hamiltonian and is obtained by first solving the eigenvalue
equation of the system and then deriving the phase using cal-
culus. The first term on the right side, ER
tt
n
t
1(())d
0
∫′
, is the
conventional dynamic phase, and the negative second term,
in
Rt tnR
tt
t
∫′′
′′
(()) d
d(()) d
0
, is the Berry phase γn. In Equation (3),
R
is a vector that contains several parameters, including the
magnetic field and the electric field; nR
|(
)
represents the
instantaneous eigenstates at each point on
R
, En is the eigen-
value of the Hamiltonian, ℏ is the reduced Planck constant,
and t represents the dynamic time. The time in the Berry phase
can be removed to focus solely on the dependence of the eigen-
states on the parameters
R
, and the equation then takes the fol-
lowing form
inRn
RR
nR
d
c
∫
γ
() ()
=∇
(4)
In Equation (4), c represents the closed paths. Furthermore,
similar to the imitation of electron transmission in the electro-
magnetic field and for improved use and understanding, the
Berry connection
AR
n
()
,[35] which is a vector function, is also
defined without a time parameter from Equation (4)
AR inRRnR
n
() () ()
=∂
∂
(5)
According to this definition, the Berry connection
AR
n
()
is gauge-dependent. In the closed paths, given that the Berry
phase is gauge-invariant, it can then be derived as follows from
the Stokes theorem
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SnRnR
SnRnRSnR nR
n
iijk j
Im d· ||
Im d|Im d· ||
K
γ
ε
()
()
() ()
() ()
()
() ()
=− ∫∇×∇
=− ∫∇ ∇=−∫ ∇×∇
(6)
In Equation (6), the Berry curvature is defined
as nR nR∇×∇()||
()
, which is the curl of the
Berry connection represented by Fjk and is given by
FnRnRn
Rn
R
jk jk
kj
=∇ ×∇ −∇ ×∇()|| () ()||
()
, where i, j, and k
represent dierent directions in the system. Finally, the Chern
number is the integer value obtained by integrating the Berry
curvature over the entire Brillouin zone based on the concepts
above and is given by
∫
π
=1
2(,)
BZ
2
CdRF RR
nj
kjk
(7)
Interested readers can find more details of the above deriva-
tion in the literature.[35]
2.2.2. 2D Topological Photonics
Many of the works in topological photonics were performed
in 2D photonic structures because it was easier to observe the
anti-scattering robust topological edge states in these 2D sys-
tems by measuring the transmissive spectrum. The photons
experience periodical potentials in photonic crystals that
are analogous to electrons in the crystals, thus giving rise to
band structures for the photons. The photonic analog of the
quantum Hall eect was proposed theoretically in photonic
crystals by Haldane and Raghu, whose paper was published in
2008,[36] and the eect was confirmed experimentally by Wang
et al. in 2009.[37] Well-designed photonic crystals can provide
various band structures and can be fabricated finely following
the development of micromachining technology. Furthermore,
photonic crystals can be realized for dierent working frequen-
cies and polarizations by simply tuning the lattice structure and
size; the required impurities or disorders can be also inserted
into photonic crystals by varying the shape or size of some of
the units individually. Therefore, photonic crystals provide an
eective platform for verification of the theories of topological
physics and application of these theories in practical photonic
devices, and the main emphasis of topological photonics is on
photonic band structure design and realization.
The most important phenomenon observed to date in topo-
logical photonics is that the edge mode transmits unidirection-
ally without backscattering along the interface between two
photonic crystal regions with dierent Chern numbers. Early
experimental realizations were performed successfully in the
microwave band for ease of sample fabrication in the mid-to-
late 2000s;[31,36,38–41] a detailed schematic diagram of such a
topological structure is shown in Figure 1b. Magneto-optical
materials were initially used to mimic the electronic topological
edge states directly; other methods have also been proposed for
use in photonic crystals, including breaking the time-reversal
symmetry to provide an analog of the quantum Hall eect.[42]
As a boson, the photon carries the spin angular momentum
(SAM) ± ℏ; however, the pseudospin ± 1/2ℏ can be imitated by
combining the transverse electric (TE) modes with the trans-
verse magnetic (TM) modes. The mixed TE+TM/TE−TM or
TE/TM[43] states are used to realize degeneracy and to form the
clockwise and counterclockwise states, as shown in Figure1c.
The usual method used to tune the system from a trivial
state into a topology involves use of the symmetry of the
photonic crystal itself. The symmetry has been used to design
the photonic topological edge states and to realize the topolog-
ical phase transition. The duty cycle that occurs between the
dielectric and the air (in most situations) has a major impact
on the energy band structure, while the energy band states
during opening and closing determine the topological state
of a system. As shown in Figure1d, the band structure varies
with the ratio of the radii of the outer and inner circles. It is
clearly shown that the energy band experiences an open state,
a closed state and another open state. Each of these states rep-
resents dierent photonic properties and the phase transition
point is the node of the energy band. After the energy bandgap
opens again, the polarization state of the photons reverses,
and the original low-order model becomes a high-order model
at the same energy level or vice versa; this demonstrates that
a topological transition occurs and it can also be understood
from Figure 1e. In addition to the changes in the lattice,
valley photonics can also achieve the same goals and create
a topological edge state. For example, through a process of
K-valley reversal, the topological phase transition occurs when
inverting the refractive indexes or switching the positions of
the two dierent materials, during which an intersection of the
energy bands will occur. In addition, Hafezi et al.[14] proposed
another concept for construction of an artificial gauge field,
which involves the use of resonant ring arrays. The details of
this structure will be introduced in Section3.3.4. 2D topolog-
ical photonics generally focuses on the creation of structural
symmetries and artificial gauge fields to imitate the properties
of electronic systems.
In addition, besides the long-range order of topological sys-
tems, the short-range order also contributes to the topological
formation. Zhou etal. proposed the photonic amorphous topo-
logical insulator, in which the glass-like matter has short-range
order.[192] In this system, by tuning the disorder strength in the
lattice, the photonic topological edge states can persist into the
amorphous regime prior to the glass-to-liquid transition, which
illustrates the key role of short-range order in the formation
of topological edge states. It reveals that the topology depends
on the orderliness, whether long-range or short-range order.
This will benefit the quantum and classical photonic systems
because the short-range order has strong controllability and
flexibility, which can be used for reconfigurable topological sys-
tems in classical optics and exerting flexible topological protec-
tion in quantum optics.
2.2.3. 1D and Higher-Dimensional Topological Photonics
At almost the same time, studies of the topological photonics
in both lower and higher dimensions were also developed.
Naturally, among these lower and higher dimensions, the lower
dimensions showed more rapid development, particularly
for 1D topological photonics. In the following, we give a brief
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introduction to topological photonics in one dimension and in
higher synthesis dimensions.
The most classical model of a 1D topological structure is the
SSH configuration,[44] which consists of two dierent substance
types arranged in a specific order. Figure 2a shows a chain with
staggered hopping amplitudes that is composed of N unit cells,
where two sites are occupied by a single unit cell.[45] The black
balls are called substance A and the white balls are called sub-
stance B. The SSH model can be understood easily by consid-
ering the interactions between the individual cells, and thus
coupling only exists between the nearest-neighbor particles.
Using the structure shown in Figure2a as an example, the cou-
pling coecients for intra-unit cell hopping and inter-unit cell
hopping are denoted by v and w, respectively, which are con-
sidered to be real numbers. The bulk Hamiltonian of the SSH
configuration is then written as
∑∑
()
()
=+++ +
==
−
ˆ|, ,|h.c. |1,,|h.c.
11
1
Hv
mB mA wmAmB
m
N
m
N
(8)
Here, N represents the total number of structures, m is the
number of each unit cell, |m, B> represents the state of the
chain on unit cell m, and h.c. represents the Hermitian con-
jugation. The energy band structure of the 1D model can be
obtained by solving for the eigenvalues of the Hamiltonian. The
physical meaning of the above is that after the external degrees
of freedom, e.g., the number of unit cells are fixed, and a point
k in the Brillouin zone and the index n of the energy band are
determined, it is possible to solve for the eigenvalues based on
these conditions using Equation (9)
Ek
vewvwv
wk
ik
() || 2cos
22
=± +=±++
−
(9)
From Equation (9), we see that if we take the energy of the
system as the vertical coordinate, v as the horizontal ordinate
and w is fixed, then it is obvious that when v is less than w,
an edge state mode will appear; this is the so-called topological
state of the 1D configuration. Further details about the energy
dispersion can be also found by referring to Figure2a.
The SSH model provides a simple and convenient way to
study the topological properties of photonic systems and dielec-
tric and metal pillars have been used for the localized photonic
mode[46] and the local surface plasmon resonance (LSPR)
mode,[47] respectively. However, the topological edge states in
Adv. Optical Mater. 2021, 9, 2001739
Figure 2. a) Schematic diagram of the most basic SSH model along with its energy spectrum and wave functions of finite size. The parameters v and
w marked in this figure represent the dierent coupling coecients that form the intracoupling and intercoupling between the particles. Topological
states will occur only if the intracellular coupling is smaller than the intercellular coupling, i.e., if v< w. b) Schematic diagram of N dimers in a chain.
The chain is composed of a pair of metal nanoparticles labeled as A and B. Each nanoparticle supports localized surface plasmon resonance. c) In
the microwave band, metal rings are used to constitute an AAH model with dierent gaps as per the model equation, and a magnetic probe is then
used to detect the intensity of each ring when placed above them. d) Harper chain with distance tuning and topological band structure obtained by
solving the vector Green’s matrix of the Harper chain with 500 particles; only some of the bandgap can be viewed. The inset shows an enlargement of
the intersection around the edge state, which is the exact ϕ value of the AAH model. e) Non-Hermitian 1D finite SSH model with arrayed waveguides.
Blue and red colors in the waveguides represent the loss and gain, respectively. The z direction is the waveguide extension direction. a) Reproduced
with permission.[45] Copyright 2016, Springer Nature. b) Reproduced with permission.[167] Copyright 2017, American Physical Society. c) Reproduced
with permission.[131] Copyright 2018, OSA Publishing. d) Reproduced with permission.[54] Copyright 2018, OSA Publishing. e) Reproduced with permis-
sion.[57] Copyright 2019, American Physical Society.
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this case cannot be detected directly through transmission in
the manner of 2D structures, and thus researchers generally
use near-field observation technologies such as the scanning
near-field optical microscope (SNOM)[47,48–50] or the atomic
force microscope (AFM).[51,52] There are other 1D configura-
tions similar to the staggered hopping structure, e.g., the line
arrangement with periodic strong/weak coupling[53] shown
in Figure2b. Furthermore, in addition to periodic structures,
the quasiperiodic model approach is also popular, particularly
the Aubry–André–Harper (AAH) model,[54] which is shown
in Figure2c,d. Details of the AAH model will be provided in
Section5.1.2.
In addition to the basic and classical configurations used
in 1D topological photonics, topological waveguides can be
regarded as “one-plus-1D” topological structures.[55] In all 1D
topological photonic structures, the light field is known to be
localized at the ends of the boundaries, which limits the appli-
cation of these structures. To move the light in these structures,
an extended dimension is then added to the 1D topological
photonic structures. When it has been added, the localization in
the 1D systems can become similar to 2D transmission because
the position of the topological localization will evolve at dierent
times of the light’s existence, which means that the photons will
propagate along the waveguides. Therefore, by extending the
1D structures into a second dimension, topological waveguides
are formed and can thus be conveniently applied to photonic
devices. Figure 2e shows the waveguide structures formed by
extending the SSH model. Because the topological configura-
tion is insensitive to both smooth deformations of the structure
and environmental disturbances, these topological waveguides
can reduce the coupling losses between the waveguides. It has
been found that even if there is a discrepancy of ≈20%[56] caused
by the disturbances, the coupling between waveguides will still
work; furthermore, the device performance can be maintained
and even improved considerably when compared with that of
ordinary waveguides. With regard to 2D topological structures,
such as resonant ring waveguide arrays and photonic crystal
waveguides, the relevant one-plus-1D topological waveguides
are simple to fabricate and have a high fabrication tolerance,
while their simple and small-scale designs are favorable for
quantum on-chip integration. Several dierent ways to realize
these “one-plus-one” waveguides have been reported. For clas-
sical light sources, these topological waveguides will be bene-
ficial for use in integrated photonics and robust photonic sys-
tems, e.g., optical couplers and beam splitters.[57] For quantum
light sources, quantum topological photonics is intended to
realize robust quantum entanglement and prevent decoherence
to allow these structures to be applied in quantum information
processing and complex quantum computing. In particular, in
single-photon sources and correlated or entangled two-photon
sources, because the topological waveguide structures oer
robust protection, they can be applied where the light losses
are great or the quantum properties are fragile. Therefore, the
advantages of 1D topological photonics can be demonstrated
clearly, particularly in on-chip quantum integration applica-
tions, and these applications have in turn encouraged the devel-
opment of quantum topological photonics.
3D topological photonics has developed relatively slowly
when compared with one/2D topological photonics as a result
of diculties in both the theoretical research and realization of
the manufacturing technology required. In a manner similar to
lower-dimensional topological structures, higher-dimensional
topological structures are also studied by exploring the band
structures of photonic crystals. The Weyl point is the 3D linear
degenerate point among the bands[58] and is analogous to the
Dirac point in 2D systems. Research generally focuses on the
process through which the energy bandgaps close and then
open by breaking the time-reversal symmetry, which can gen-
erate the topological edge states. Alternatively, creation of parity-
time (PT) symmetry represents another eective way to achieve
this goal. Initially, almost all realizations of topological systems
in three or higher dimensions were based on electronics,[59] and
these systems were not realized in the optical frequency range
until 2019,[60] when a 3D topological photonic configuration was
demonstrated based on a cleverly designed all-dielectric meta-
material platform. Finally, a recent report about probing of the
4D quantum Hall eect via topological pumping has inspired
researchers to explore the synthetic dimensions in photonics.[61]
3. Realization of Quantum Topological
Light Sources
From this section onward, quantum topological photonics will
be introduced fully based on the quantum optics and topo-
logical photonics concepts discussed above. Light sources are
well known to be among the most important devices in inte-
grated optics. Researchers have proposed numerous methods
to realize integrated light sources, including use of photons in
quantum integration, along with the on-chip lasers used in inte-
grated photonics. Furthermore, the eciency and stability of
the light source, regardless of whether it is a quantum or clas-
sical light source, are vitally important. Quantum light sources
play a major role in current studies of quantum physics, but
these sources are easily lost because of environmental changes.
The topological protection property can help light sources to be
more robust and have low losses, which is necessary for both
quantum and classical light sources. To date, many works on the
generation of classical light sources such as lasers and quantum
light sources such as entangled photon pairs have been realized
experimentally through use of topological protection.
Consequently, we will introduce topological quantum light
sources individually in this section in the following order.
First, in Section3.1, we will introduce the way in which top-
ological photonics works to protect light sources and give an
example based on the classical laser to aid in understanding of
the concept. Section3.2 describes the generation principles of
quantum light sources, including entangled photon pairs and
single-photon sources. In Section3.3, the topological quantum
emitter, the path entanglement of topological quantum sources
and entangled photons in 1D and 2D topological structures will
be presented in detail.
3.1. Topological Lasers
The laser (from the acronym for light amplification by stimu-
lated emission of radiation) is one of the applications of
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atomic energy-level transition technology and has become the
most important light source for use in experimental physics.
Because of the demand for minimization and integration of
optical devices, the development of on-chip lasers has attracted
increasing attention. However, there are still problems to be
solved for on-chip lasers, including their low power charac-
teristics and instability. Topological photonics provides new
approaches to resolve and overcome these shortcomings
because it oers robustness, immunity to backscattering, and
low losses. Recently, there can be found many reviews on topo-
logical lasers that can be used for reference because of the rapid
development of it.[201,202]
3.1.1. 1D Topological Laser
Many 1D configurations have been used to realize topological
lasers. In 2016, Pilozzi and Conti proposed a topological laser
based on use of a 1D AAH chain to form an array of resonant
structures with dierent materials that have dierent dielectric
functions. A schematic of this laser, in which they obtained a
local-topological edge mode to generate laser radiation using
an ultrashort self-induced transparency pulse, is shown in
Figure 3a They found that the laser beams could be emitted
with tunable radiation. In 2017, Jean et al. reported experi-
mental realization of a polarization-selected topological laser in
a 1D lattice with a material system composed of GaAs quantum
wells sandwiched between two Ga0.05Al0.95As/Ga0.8Al0.2As Bragg
mirrors, which is shown in Figure3b. In 2018, Parto etal.[62]
observed laser radiation generated using microring resonators
with gain materials. They studied the lasing properties of SSH
active microring resonator arrays both theoretically and experi-
mentally. In these arrays, the distances between every pair of
microrings were arranged periodically as a long distance and a
short distance in a staggered manner to represent the coupling
dierences in the SSH model. It should be noted that, because
this system has either gain or loss, it is a non-Hermitian system
with PT symmetry, and chiral-time (CT) symmetry is satis-
fied rather than chiral symmetry. The topological structure is
constructed by stacking of InGaAsP quantum wells, which are
enclosed between two barrier layers. The well and barrier layers
are made from the same material, but the material composi-
tions of these layers are dierent. Each 10 nm thick well is com-
posed of Inx = 0.56Ga1−xAsy = 0.93P1−y and each 20 nm thick barrier
layer is composed of Inx = 0.74Ga1−xAsy = 0.57P1−y. There are four
substances in each layer, which are typical quantum well mate-
rials,[19] as illustrated in Figure3c. The total thickness of each
microring resonator is 220 nm and the width of each micro-
ring is 500nm. 16 microring resonators are arranged to form
the SSH model, which can provide sucient energy for lasing.
When the pump power was adjusted to select a gain level in
this system, a strong nonlinearity was observed. To enable
detection of each microring resonator individually, a decou-
pling grating was fabricated for each microring. Each microring
was integrated with an arc waveguide, which was then con-
nected to the decoupling gratings at the two ends. Although
this design is convenient for detection of the laser modes and
the specific circumstances of each microring, the detection
Adv. Optical Mater. 2021, 9, 2001739
Figure 3. a) Schematic of 1D topological laser. Topological edge states are excited using an ultrashort pulse. b) Band structure associated with
polarization-excited light and its electric field distribution. The scanning electron microscope (SEM) image shows the zigzag chain with the stacked
quantum wells, which are embedded in the micropillars between the Bragg reflectors; the topological states occur at the edge of the chain under
nonresonant optical pumping. c) Microrings composed of the InGaAsP multilayer quantum well materials. The rings are arrayed as an SSH model
with dierent coupling coecients. The figure shows an SEM image of the experimental structures with 16 microrings and the insets show the details
of the microrings and the coupling grating. d) Schematic diagram of a topological hybrid silicon micro laser composed of nine microring resonators
with weak and strong coupling arrayed as per the SSH model. The yellow microrings were formed by depositing by 10nm of Cr on top of the rings to
introduce gain and loss. The middle microring is the location of occurrence of the boundary mode. e) Schematic of the topological nanocavity design
concept. Nanocavities with dierent Zak phases form the topological edge state. Dierent unit cell selection methods cause their topological properties
to be dierent and thus constitute a topological state at the interface. a) Reproduced with permission.[168] Copyright 2016, American Physical Society.
b) Reproduced with permission.[19] Copyright 2017, Springer Nature. c) Reproduced with permission.[62] Copyright 2018, American Physical Society.
d) Reproduced under the terms of Creative Commons CC BY License.[63]Copyright 2018, The Authors, published by Springer Nature. e) Reproduced
under the terms of Creative Commons CC BY License.[64] Copyright 2018, The Authors, published by Springer Nature.
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waveguide destroys the topological properties of the structure
to some degree. In other words, rather than use a direct detec-
tion method, an indirect detection method is required. In the
experiment, the existence of the topological edge states was
first confirmed and the robustness of the topological structure
was also verified. The CT symmetry was maintained even when
disorders were introduced into the system, and the gain, loss,
and resonance frequency characteristics barely changed. The
results also demonstrated that gain saturation is very important
for stabilization of the lasing edge mode at dierent pumping
levels, and that the lasing edge modes are strongly aected by
phase transitions with carrier dynamics; only a topological edge
state that is displayed within a certain phase range can generate
laser radiation, which is the core concept of laser generation in
topological photonics. When the topological phase transition
occurs, lasing from a single edge-mode switches to form the
final multimode output. However, no clear theoretical descrip-
tion has been presented for non-Hermitian topological systems
to date. Thanks to the localization properties of the 1D topolog-
ical structures, the localized edge states provide a favorable way
to generate robust lasing, and increasing numbers of works on
this subject have been reported in recent years.[19,63]
In addition to the SSH model, the topological edge state
can also appear at the spliced interfaces of 1D structures with
dierent Chern numbers. Zhao et al.[63]realized lasing using
coupled microrings in a spliced topological state with gain and
loss materials composed of quantum wells and Cr, respectively.
Quantum wells are widely used as laser materials. However, if
other substances are added to the system, losses will then occur.
For example, if Cr is coated evenly on the microrings and these
Cr-coated microrings are then separated by InGaAsP-coated
microrings, the system’s gains and losses will coexist. In the
work of Zhao et al., hybrid silicon was also used as substrate
and structural body to form cavities and the microrings were
coated with InGaAsP and Cr. As a result, the coupling between
the photons in the cavities and the excitons in the quantum
well provided an eective approach for single-mode genera-
tion and laser threshold adjustment. As shown in Figure 3d,
the topological edge state occurs at the interface, which exists
in the form of an isolated microring. In on-chip lasers, fabrica-
tion defects, temperature fluctuations and other environmental
disturbances will aect the local optical potential and cause
instability; all of these problems can be overcome by arranging
the circles into 1D topological models to ensure that the robust
protection works. However, it is dicult to fabricate these stag-
gered microrings using multiple materials with gain or loss
properties and thus new materials and fabrication methods
must be explored.
Some traditional structures for laser generation based on a
1D topological configuration have been reported. Ota et al.[64]
proposed use of two distinct topological crystals spliced
together and fabricated in the form of nanocavities. This repre-
sents a feasible method to create a boundary for the zero-edge
mode, where the zero mode is a so-called localized topological
edge state that can localize the light and achieve a high quality
(Q) value. Figure 3e illustrates the configuration described
above. The blue region is composed of unit cells with a vertical
length of a, while the red region is also composed of unit cells
with a vertical length of a, but these unit cells have dierent
structural shapes. By integrating the Berry curvature within the
first Brillouin zone, the Zak phase was calculated to obtain both
the distinct band structures and the topological properties. The
parameters were selected by maintaining only one mode in the
bandgap. Furthermore, a high Q-factor can also be obtained by
regulating the proportions of d1/(d1+ d2) and thus varying the
lattice of the photonic crystals, which is advantageous for light
confinement. The pump power also plays an important role in
determining the laser gain level, which means that the cavity
mode emission increases greatly with increasing pump power.
In summary, topological laser generation is inseparable
from the gain material used, and the topological configura-
tions guarantee high stability and high power for these lasers.
When compared with higher-dimensional topological photonic
structures, the 1D structures are more convenient for both min-
iaturization and integration. The same applies for the genera-
tion of quantum light sources, which will be introduced in Sec-
tions3.3.2 and3.3.3.
3.1.2. 2D Topological Lasers
The 2D topological photonic laser was realized in early 2017.[65]
Similar to the 1D systems, the edge states in 2D structures
can support light transport. Bahari et al. realized a topolog-
ical system experimentally that was composed of two types of
photonic crystal (PhC) with dierent topological properties.
The PhCs were made from InGaAsP multiple quantum wells
(MQW) and the substrate was made from yttrium iron garnet
(YIG) grown on a gadolinium gallium garnet (GGG) surface.
By applying an external magnetic field, the time-reversal sym-
metry was broken and the nonreciprocal topological states
were then created along the interface between the two types of
PhC. In principle, the geometries of the topological structures
can be arbitrary, although the right-angled corners of the arbi-
trarily shaped cavities showed direct proof of the robustness of
the topological laser, which had not been achieved in previous
structures because of the time-reversal symmetry. The topolog-
ical properties were also proved theoretically by calculating the
Chern number and analyzing the dispersion curve of the edge
mode. The laser reached an insolation ratio of 11.3dB at tele-
communication wavelengths at room temperature. A schematic
of the laser and its photoluminescence spectrum are shown
in Figure 4a. However, one of the shortcomings of this laser
is that it requires an external magnetic field, which makes the
lasing conditions more demanding. In addition, the system has
a topological bandgap of Δλ∼42 pm, which is extremely small
for use in practical applications.
Another topological laser configuration was proposed later
in 2018 that overcame the shortcomings described above. Two
configurations were used for the theoretical analysis[66] and
the experimental measurements,[67] as shown in Figure 4b,c,
respectively. These configurations are based on the Haldane
model, which is a typical model used in topological photonics,
as proposed by Hafezi et al. Each structural unit consists of
four site resonators and four link resonators. The site resona-
tors are also called main resonators, and these eight resonators
form a circle. Every main resonator and link resonator is cou-
pled to each other, thus creating an artificial gauge field, which
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is similar to the electronic spin field. A detailed introduction
to artificial gauge fields is provided in Section 3.3.4. In this
pseudospin field, photons are eectively confined in the topo-
logical boundary and can be transmitted along the edge without
spreading into the bulk. The edge modes are shown clearly in
Figure4d, in which the clockwise and counterclockwise paths
are separated. This property is conducive for wavelength divi-
sion multiplexing (WDM) and chirally based conversion, which
will be explained later in this review. The topological laser is
composed of a ring resonator array in a honeycomb lattice;
however, only the rings on the edges are pumped, thus leaving
the rings in the center to be lossy. Because the excited modes
are the edge modes, no bulk states are excited even if the
internal rings are pumped and because the pump light is in the
bandgap, no actions will occur. The energy flux travels along
the edge with almost uniform intensity. The performances of
lasers with various topological band gaps and their topological
trivial counterparts are analyzed in the presence of disorders.
It is found that the topological laser maintains its slope e-
ciency as the disorder increases, with its performance being
followed by that of the topological small bandgap laser, while
the trivial version provides the worst eciency performance.
The experimental configuration is composed of a square lattice
containing the InGaAsP resonators and input and output ports.
Similar to the theoretical case, the boundary part of the struc-
ture is pumped. Apart from a slight shift in the spectra, both
high slope eciency and high emission intensity are shown in
Figure4c.
Arbitrary cavity shapes can be used in 2D topological lasers
for a much richer range of shapes than 1D topological lasers;
in addition, the transmissive paths of 2D topological lasers
are clear, which means that the transmission spectrum can
be measured directly to enable better study of the wavelength
range, thus greatly simplifying the characterization methods
required when compared with those for 1D topological pho-
tonics. However, complete theoretical explanations for the non-
Hermitian topological systems in both 1D and 2D topological
photonic structures have yet to be provided. Nevertheless,
these structures are also applicable to quantum light sources
for production of entangled photon pairs and single-photon
sources. Besides, the room-temperature topological valley laser
was realized by Smirnova etal. recently.[193] This nanophotonic
topological-cavities laser is generated by the gain medium of
III–V semiconductor quantum wells, and it has the proper-
ties of the narrow spectrum, high coherence, and threshold
behavior. Two interfaced valley-Hall periodic photonic lattices
are used to compose the structures, and the emitted beam
hosts a singularity formed by a triade cavity mode. The mode
is in the bandgap between the two lattices with opposite parity
breaking, which is the specific characteristic of valley topo-
logical photonic crystals. In order to obtain the single narrow-
linewidth lasing, the pump intensity should be above the lasing
threshold. In this case, the spatial distribution of emission
demonstrates that there are intensity strong points in the sin-
gularity shown in Figure 4e. There are also other reports on
valley topological lasers.[79,194,195]
Adv. Optical Mater. 2021, 9, 2001739
Figure 4. a) Schematic of the arbitrarily shaped topological insulator laser and the associated photoluminescence spectrum with dierent external mag-
netic field directions. The laser consists of two types of PhC: square lattice PhCs with star-shaped pillars in the outer part and triangular lattice PhCs with
air holes in the inner part. Light from outside is coupled into the cavity through a defect waveguide. Edge states are created along the interface between
the two types of PhC. The emission power can reach a maximum insolation ratio of 11.3dB. b) Schematic of theoretical structure based on the Haldane
model. The energy flux travels along the edge with almost uniform intensity. The slope eciency versus disorder strength characteristics shows the full
or small topological band gaps and their trivial counterparts. c) Schematic of the experimental structure and the topological edge modes. Experimental
measurement results of the output intensity of the topological laser, where the topological slope eciency is high. Experimental measurement results
of the emission spectra of the topological laser versus that of the trivial laser. d) Schematic of coupled ring resonators in two dimensions to imple-
ment the integer quantum-Hall model. Clockwise and counterclockwise paths of topological robust transmission are shown clearly with one common
input and one common output. e) Schematic of valley topological laser and its spatial distribution of emission. a) Reproduced with permission.[65]
Copyright 2017, AAAS Publishing. b) Reproduced with permission.[66] Copyright 2018, AAAS Publishing. c) Reproduced with permission.[67] Copyright
2017, AAAS Publishing. d) Reproduced with permission.[88] Copyright 2016, OSA Publishing. e) Reproduced under the terms of Creative Commons CC
BY License.[193] Copyright 2020, The Authors, published by Springer Nature.
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3.2. Basic Concepts of Quantum Light Sources
In this section, we focus on the methods for generation of
quantum light sources. Quantum physics research has focused
on the eciency and robustness of both single-photon sources
and correlated photonic pairs for a long time. Therefore, the
quantum integrated light source has become one of the most
important aspects of quantum integrated optics. Spontaneous
four-wave mixing, spontaneous parametric downconversion
(SPDC), and excitation of quantum dot materials are the usual
methods used to generate quantum light sources and will be
introduced in the following.
3.2.1. Entangled Photon-Pair Sources
Photon-pair sources are mostly generated via nonlinear eects
such as spontaneous four-wave mixing (SFWM) and SPDC,
which are illustrated in Figure 5a,b, respectively. Entangled
photon pairs are usually divided into three categories: fre-
quency-entangled pairs, polarization-entangled pairs, and path-
entangled pairs. The first two types of entangled photon pair can
be generated easily via nonlinear waveguides and this method
is widely used in quantum optics. Generation of path entangle-
ment will be introduced in detail in Section3.3.2. Through a
nonlinear process, the generated photon pairs are able to prop-
agate along the waveguides, which is convenient for quantum
integration. For example, frequency-entangled photon pairs can
be large-scale integrated into an array and each photon in the
generated pair can be manipulated individually because they
are strongly identical. However, this approach still has some
shortcomings. Unlike the generation of single-photon sources,
parametric photon-pair generation is reliant on the eciency
of the nonlinear optical eect, and because nonlinear processes
are probabilistic events, the generation rate is only ≈5–10%.[68]
As a result, long nonlinear waveguides, such as the mosquito-
repellent incense structured waveguide shown in Figure 5c,
are always used to guarantee the occurrence of the parametric
process. With improvements in the experimental techniques,
including multiplexing of both time[69] and space,[70] the prob-
ability of entangled photons has improved gradually.
3.2.2. Single-Photon Sources
Quantum dots, which are semiconductor particles with sizes
of a few nanometers, have become a popular material type
for generation of single photons because of their high perfor-
mance.[71] Semiconductor luminescence originates from the
electronic transition that occurs between the valence and con-
duction bands and thus is a deterministic process. However,
because each of the single photons comes from a change in the
semiconductor material’s energy level, it is dicult for them to
be identical single photons. As a result, active demultiplexing
is required to generate correlated photon pairs by tuning each
of the single photons. Furthermore, single-photon sources
based on quantum dot materials are not conducive to optical
integration and light propagation. Quantum dots are some-
times immersed in stacked semiconductor layers, as shown in
Figure5d. When the cylinder of stacked layers is excited, pho-
tons are emitted vertically with respect to the semiconductor
layers, which is unsuitable for in-plane integration. For on-chip
applications, PhC waveguides can be fabricated close to the
quantum dots to support light propagation.
Adv. Optical Mater. 2021, 9, 2001739
Figure 5. a) Spontaneous four-wave mixing (SFWM) process that must obey energy and momentum conservation; ωI, ωP, and ωS represent the idler,
pump and signal frequencies and satisfy the following equation: ωS+ωI= 2ωP. b) Spontaneous parametric downconversion (SPDC), which is similar
to SFWM, where the dierence is that SPDC is a χ(2) process of nonlinear optics, while SFWM is a χ(3) process. The figure shows the polarization-
entangled states (left) of SPDC and the energy-entangled states (right) of SFWM. c) Experimental setup of large-scale integrated optics for multidimen-
sional quantum entanglement. The SFWM source shown in the figure is in the mosquito-repellent incense shape for improved quantum light source
generation. d) Schematic of emitting semiconducting quantum dot with stacked layers. The emitter center is the source of the single photon and can
achieve high strength in the vertical direction. a) Reproduced with permission.[70] Copyright 2013, Springer Nature. b) Reproduced with permission.[169]
Copyright 2016, American Physical Society. c) Reproduced with permission.[10] Copyright 2018, AAAS Publishing. d) Reproduced with permission.[170]
Copyright 2016, American Physical Society.
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3.3. Quantum Topological Light Sources
As mentioned earlier, quantum light sources must be protected
from decoherence. In practical applications such as quantum
sensing, communications, and imaging, many quantum
sources with identical characteristics are usually needed. How-
ever, imperfections and irregularities are unavoidable in nano-
fabrication processes, and disorder limits the scalability of such
quantum sources. Therefore, it is very dicult to improve the
scalability of quantum systems, and the diculty increases
exponentially as the number of qubits increases. Topological
photonics paves the way to the scalability of quantum sources,
owing to its properties of anti-disturbance, robustness, and
low loss, which can help keep the qubits in a coherent state as
the number of qubits increases and prevent them from being
aected by exponentially increased losses.[189,190]
In this section, we focus on how topological photonics adds
robustness to quantum optics. First, the topological quantum
emitter is reviewed. Second, path entanglement with the topo-
logical waveguides is introduced. Finally, we explain generation
of frequency-entangled photon pairs under 1D or 2D topolog-
ical protection.
3.3.1. Topological Quantum Emitters
In Section 3.2, we introduced the quantum light source, and
in this section, we will describe how quantum emitters are
realized in topological photonics. Quantum emitters can be
compared to a “photon gun,” emitting a single photon after
each triggering. For integrated quantum optics applications,
quantum emitters need to be integrated on-chip and the single
photons need to be emitted stably and transmitted eciently.
However, in traditional waveguides, the scattering losses that
occur during propagation can barely be avoided, particularly
when turning corners in the waveguides. Because of their
anti-scattering and robustness properties, topological photonic
structures have been used in on-chip quantum optics to protect
the single photons.
Early in 2018, Barik et al.[72] first proposed fabrication of a
topological quantum light source by combining two photonic
crystals with dierent lattice constants. Because the photonic
bandgaps in the two regions of the photonic crystals are dif-
ferent, the topological edge state appears at the interface
between them. In addition, the authors also proposed the chiral
nature of single-photon emission. When the laser light arrives
at the middle region of the photonic crystals, the emitted single
photons with left-hand and right-hand circular polarizations
are guided in opposite directions, as shown in Figure 6a. The
topological interface waveguide can be designed using arbitrary
shapes and supports single-photon propagation robustly and
with low losses. This novel method for combining topological
photonics with quantum optics provided a good platform for
later research in quantum topological photonics.
More recently, Barik et al.[73] used a valley-Hall topological
photonic crystal waveguide to integrate a single-photon source,
as shown in Figure6b. They also took advantage of a topolog-
ical edge state similar to that of previous work, but realized a
topological ring resonator with a high Q-factor. These works
have paved the way for integrated quantum optics with topo-
logical photonics.[204] Researchers have also attempted to inte-
grate large numbers of quantum light sources on a single chip
to achieve both spatial control and high eciency.[74]
3.3.2. Path Entanglement of Topological Quantum Sources
In addition to the frequency-entangled photon pairs and polar-
ization-entangled photon pairs mentioned above, directional
entanglement is another type of entangled state, i.e., path
entanglement. This type of entanglement relies on photon
interference and most of this interference occurs in waveguides
during propagation of these photons. One interesting path-
entangled state is the NOON state, which is created by superpo-
sition of two items: one item contains N particles in quantum
state a and 0 particles in quantum state b, while the other
item contains 0 particles in quantum state a and N particles
in quantum state b. In two-particle NOON states in particular,
if a photon is detected in a channel, the other photon will be
detected in the same place without any doubt. Traditional wave-
guides have been used to realize path entanglement and topo-
logical photonics can help improve these realizations because
of its enhanced robustness and low loss. Topological protec-
tion is mainly used to avoid decoherence of the qubits in the
quantum information of photonic networks.
In 1D topological photonic structures the light is usually
localized rather than propagating, which limits practical on-
chip applications of these structures. However, the time-varying
properties of topological localization can be studied in these
1D systems. The details of the system were presented in Sec-
tion2.2.3 when 1D topological photonics was introduced. Theo-
retical analysis shows that the topological edge state occurs at
an inflection point in the band structure and at this point, the
second-order diraction term of the path entanglement is zero;
this means that as the wave packet size increases, the amount
of diraction of the wave packets will decrease to zero more
rapidly when compared with the other points in the band struc-
ture. The reflection term should not be neglected in the tradi-
tional model because if the reflection term increases, the path-
entanglement NOON state will then decline. However, in the
topological system, the reflection term becomes zero because
of the topological protection. Although the honeycomb lattice
composed of helical waveguides shown in Figure6c is slightly
dicult to fabricate,[171] this model can help us to improve our
understanding of quantum walks and quantum light sources;
because there are abundant artificial gauge fields in this con-
figuration, it is then easy to contain topological properties and
realize the required topological protection from decoherence.
To simplify the honeycomb-lattice model and to determine
the topological properties, researchers have tried to take advan-
tage of the photonic lattice to support photon propagation,
including single-photon and multiphoton propagation; a sche-
matic diagram of the photonic lattice is shown in Figure6d. It
is vitally important that the waveguides are excited individually
and cannot be excited simultaneously to allow the topological
edge mode to be excited. In the experiments, the edge mode
in the bandgap can be excited using spatial light modulators
or via the emission of specific wavefunctions, and the results
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show that the photon correlation is closely related to the phase
operator for single-particle propagation. We can thus conclude
that topological protection can be used not only to protect the
state of a single photon but also to protect the path entangle-
ment of double photons.
At present, the topological protection of photonic path entan-
glement is somewhat limited by the fabrication and detection
technologies available in the experiments. In theory, the reflec-
tion term of this model can be zero as a consequence of the
topology, but in reality, it is dicult to shape the spatial wave
function to cause the topological edge state to overlap with it;
however, the bulk state should not aect the dynamic process.
Therefore, the reflection term cannot be exactly zero. Further-
more, the experiments must be designed carefully in case the
high losses of long waveguides, especially those caused by
bending, absorption, and scattering, will aect the counting of
the photons. In summary, these simplified quantum topolog-
ical structures need to be improved. However, some researchers
also noted that it was not necessary to protect the photons from
decoherence because the interactions between photons were
weak and the decoherence speed was thus low. From this view-
point, it can only be said that the topological structures can
indeed reduce the losses and protect the photons from decoher-
ence. If possible, the superiority of topological quantum protec-
tion can be reflected using multiphoton systems.
3.3.3. Correlated Photons in 1D Topological Structures
In this section, we present an introduction to biphotons in
topological photonics. Biphotons usually comprise correlated
photon pairs or entangled photon pairs and topological photo-
nics has the exact property that allows light to propagate along
the boundaries. Biphotons in topological photonics use the top-
ological edge state to generate the biphotons and protect them
from decoherence.
In 2018, the topologically protected biphoton state[75] was
realized experimentally, thus demonstrating that a correlated
biphoton can propagate for long distances in 1D topological
photonic waveguides. A schematic of the experimental setup
is shown in Figure 7a. In this setup, the biphotons were gener-
ated through SFWM when the picosecond laser was focused
onto the middle defect of the SSH waveguides and the verifi-
cation was accomplished by interferometry between the signal
and idler photons. With topological protection, the generated
biphotons propagate in the defect topological mode and the
correlation between the photons after the interferometer shows
a good function of the lattice position, regardless of whether
disorders are present or not. Furthermore, the detectors at the
end detect that the biphotons are kept in the defect topological
mode; without topological protection, however, the system is
in a trivial condition and there is no correlation, regardless
of whether there are disorders or not, and the photons are
scattered into other waveguides. The experimental setup was
ingeniously designed using components such as supercon-
ducting single-photon detectors (SSPDs) and time correlation
circuits (TCCs) and this work has successfully verified the
robustness of the correlated photons when using topological
photonics.
In 2019, Michelle et al. used the same detection device to
observe entangled photonic states and verified the topological
protection of the chiral symmetry for photons in the disordered
system. The details of the experimental setup are shown in
Figure7b. Recently increasing numbers of reports on quantum
topological photonics have been published and these reports
indicate that one easy way to realize topological edge states is
Adv. Optical Mater. 2021, 9, 2001739
Figure 6. a) Schematic of the topological quantum optics interface. The blue and yellow lattices have dierent topological properties and their interface
supports two helical edge states with opposite propagation directions and circular polarizations (σ+ and σ−). Two grating couplers on each side collect
the light. The insets show the band structures of the two lattices. b) Topological ring resonator consisting of dierent photonic crystals coupled to a
linear topological waveguide, which is embedded with semiconductor quantum dots. c) Schematic of helical waveguides with honeycomb lattice and
ordinary waveguides. The direction of the arrow indicates a rotation axis and provides spatial transmission. d) Photonic lattice that can support photon
propagation; the two-photon state is injected from the ordered region into the disordered region. a) Reproduced with permission.[72] Copyright 2018,
AAAS Publishing. b) Reproduced with permission.[73] Copyright 2020, American Physical Society. c) Reproduced with permission.[172] Copyright 2018,
Springer Nature. d) Reproduced with permission.[173] Copyright 2016, OSA Publishing.
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to use one-plus-1D topological photonic structures such as the
extended SSH model. Nanowires are arranged in an array, with
strong or weak coupling between adjacent nanowires, as in the
SSH model. Light propagates in these nanowires and is then
coupled to fibers in an array through gratings to measure the
correlation of the photons. In fact, there are two typical features
of this spatial-extension SSH model, i.e., this model is robust
against disturbances and oers chiral symmetry protection.
These features can be deduced based on the propagation con-
stant, which represents the energy term in condensed matter
physics, and the zero amplitude wave function can then be
obtained. In addition to the frequency correlation of the pho-
tons after SFWM, the spatial modes of the photons in a lattice
can also become entangled after formation of frequency-corre-
lated photon pairs, and the phase matching condition must be
satisfied under four-wave mixing conditions as well. For this
lattice, the additional freedom of the photonic system is added,
which means that the topology provides a more optional way
to protect the biphoton. Another aspect that should be men-
tioned is that the dierent topological states can be realized via
a topological pump. In general, the Archimedes screw pump
and the topological pump are the most common pumping
methods within the semiclassical limit and can help form the
synthetic dimensions of the topological defect modes.[76–78] On
this basis, the biphoton should overlap considerably with the
wave function of the topological defect modes. The wave func-
tion can have an important eect on the correlated biphotons,
in that the biphoton correlated state that we expect will only
appear if the wave function is nearly closed or zero.
The 1D topological waveguides are easy to fabricate and con-
tain fewer imperfections; however, if all the protected quantum
topological configurations are based on a single dimension, this
can limit the versatile range of properties of topological pho-
tonics. Topological properties in 2D can have more rich and
varied features; for example, the entangled photon pairs can
propagate in dierent directions and then converge afterward.
The following provides an introduction to entangled photons in
2D topological structures.
3.3.4. Correlated Photons in 2D Topological Structures
The development of topological photonics has shown its poten-
tial in one dimension, two dimensions, three dimensions, and
even synthetic dimensions to date, and the origins of early
topological photonics actually come from 2D structures. In
Section 2.2.2, we briefly introduced 2D topological photonic
structures, with particular emphasis on formation of the top-
ological edge states in photonic crystals. In general, 2D topo-
logical structures rely on bandgap reversion or on use of an
artificial gauge field, and one typical example is the coupled-
resonator optical waveguide (CROW),[14] which was mentioned
in Section3.1.2. The CROW structure uses artificial ring reso-
nators to mimic the quantum spin Hall eect, including the
main rings and the connecting rings, and the main rings play a
major role; these rings are illustrated in Figure 8a. The Hamil-
tonian of the artificial photonic system is written as
Ha
ae aa e
aa aa
xy
xy xy
iy
xy xy
iy
xy xy xy xy
ˆˆ ˆˆ
ˆˆ ˆˆ
0
,,
1, ,
2
,1,
2
,1 ,,1,
††
††
∑
κ
=− +
++
σ
σσπα σσσ πα σ
σσ σσ
+−+
++
(10)
The coupled ring resonators can be also described using
classical theory. The rings are not only waveguides that carry
light, but also are microcavities, and each ring is coupled with
the rings to its left and right to form a unit cell, as shown in
Figure8b. After the light propagates along a total circle com-
posed of four main rings and four connecting rings, an integral
multiple phase of 2p or −2p will be accumulated. Because the
photons at dierent frequencies can propagate along dierent
directions and thus their propagation lengths are dierent, the
dierent phases that contribute to the synthetic gauge field
show dierent topological properties. Experiments have proved
that these CROW structures provide topological protection e-
ciently under disturbance conditions, as illustrated in Figure8c.
Even when a ring resonator is removed, light can still propagate
along the edge of the resonator array. The CROW configura-
tion has been used widely in topological photonics to realize
topological lasers,[79–85] the quadrupole topological state[86] and
corner state, and reconfigurable topological phases.[87]
A robust transformation of correlated photons in the CROW
system was proposed by Hafezi and co-workers[88] in 2016. They
Adv. Optical Mater. 2021, 9, 2001739
Figure 7. a) Experimental setup used to measure biphotons with SSH
waveguides; the center waveguide has a defect with a topological property
that can protect biphotons over long distances. b) Schematic of entan-
gled photonic experimental setup with SSH waveguides. Two paths can
be selected for the photons, similar to a reverse Hong–Ou–Mandel inter-
ferometry process. a) Reproduced with permission.[75] Copyright 2018,
AAAS Publishing. b) Reproduced under the terms of Creative Commons
CC BY License.[174] Copyright 2019, The Authors, published by De Gruyter.
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initially studied the properties of correlated photons under
topological protection theoretically and in 2018, they demon-
strated these properties experimentally in a CROW system.
The correlated photons were generated through SFWM and
were then input into the CROW system. In the CROW system,
the light propagated along the boundaries and the correlated
photons at dierent frequencies propagated in the clockwise
direction and the counterclockwise direction, as illustrated in
Figure8d. The two beams met at the output port and the meas-
ured g(2) demonstrated the robustness of the correlation of the
two beams, meaning that even after the photons propagated
over a long distance or suered from the eects of defects, they
remained correlated.
Later, another 2D configuration was proposed based on 1D
topological structures. The corner states were gradually raised
by extending the SSH model to two dimensions. Gorlach and
Poddubny[89] showed that the topological edge state linked the
interaction of the entangled quantum with the 2D SSH con-
figuration; a schematic diagram of the configuration is shown
in Figure 8e The bound photon pair and the single-photon edge
state can be obtained robustly by choosing dierent 2D SSH
units. Furthermore, within this interacting case of a simple
lattice, there are many richer physical phenomena. They also
used the concept of the edge-state’s existence to determine the
quantum walk graph connectivity further and more clearly.
In general, 2D photonic crystals are widely used to achieve
dierent functions in both integrated photonics and 2D topo-
logical photonic structures. However, to date, very few works
have been reported on propagating entangled and correlated
photons in 2D topological photonic structures and more eort
must be made in topological structure design and the related
experimental measurements. Nonlinear eects and propaga-
tion losses must be considered in these future studies, and
other waveguide configurations such as the mosquito-repel-
lent waveguide shape[90] can also be used to provide longer
propagation lengths. Reconfiguration[91–93] of the tuning of the
quantum topological states in two dimensions is also a new
field at present. There are numerous tuning measures using
dierent properties of materials, including thermal[94,95] and
electro-optic modulators[96–98] and nonlinear index modulation
techniques.[99–101] When tuning is realized, both the path entan-
glement and the frequency entanglement can be freely tuned
as required, which will save considerable time and production
costs and improve the photon-pair generation eciency.
4. Realization of Topological Quantum
Information Process
In the previous section, we introduced topological quantum
light sources in detail. In this section, quantum information,
another important aspect of quantum optics, will be described
exhaustively with reference to topological photonics. At present,
information technology is playing an increasingly important
role in many areas of everyday life and it can also benefit from
integrated quantum optics in the near future, because quantum
information and quantum computing have seen rapid devel-
opment in recent years. Unlike classical digital information,
which has only two states, 0 and 1, that correspond to “o” and
“on,” respectively, a quantum state can be encoded using more
Adv. Optical Mater. 2021, 9, 2001739
Figure 8. a) Ring resonant structures used to form an artificial gauge field, including the main rings and the connecting rings based on the quantum
spin Hall eect (QSHE). b) Coupling between microcavities. Each ring has two nearest neighbors in the form of the left and right microcavity wave-
guides and this represents a basic unit of the topological resonant ring configuration. c) Topological protection against a defect. Even when a resonant
ring is intentionally removed from the completed structure, the edge state can be maintained well because of the robustness. d) Schematic of experi-
mental generation of correlated photons from resonant rings by SFWM. Transmission measurements are shown of two clockwise and counterclockwise
paths with dierent frequencies. The spectra in distinct colors show oset frequencies representing the edge and bulk modes. e) 2D SSH configuration
for two-photon interaction. The dashed lines in the figure denote the unit cell boundaries used to calculate the Zak phase. a) Reproduced with per-
mission.[14] Copyright 2011, Springer Nature. b) Reproduced with permission.[175] Copyright 2013, Springer Nature. c) Reproduced with permission.[176]
Copyright 2018, American Physical Society. d) Reproduced with permission.[177] Copyright 2018, Springer Nature. e) Reproduced with permission.[89]
Copyright 2017, American Physical Society.
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degrees of freedom, including the OAM, polarization, spin and
the spatial modes. Observation and measurement of a multi-
particle system will promote the development of quantum
information systems. Pan etal. realized an 18-qubit entangle-
ment with six photons and three dierent degrees of freedom.
However, the dilemma with this approach is that during infor-
mation transmission and processing, noise will overwhelm the
information. Some of the intrinsic degrees of freedom of pho-
tons oer topological protectivity, which implies that a method
to propel quantum information robustly may be available.
Researchers have reported that the geometrical phase was not
only calculated but also discovered experimentally in quantum
systems.[102] The topological phase has been used recently as
a robust tool in quantum computers. In this section, we will
present several methods to obtain and process quantum infor-
mation using topological photonics. In this section, the basic
concepts of the topological phase and qudits will be introduced
in Section4.1, and quantum information processing and error
correction, which forms an important part of quantum infor-
mation, will be detailed and presented with topological protec-
tion in Section4.2.
4.1. Topological Phase and Qudits
The topological phase can be obtained in a pair of entangled
photons and this phase has been applied in higher-dimensional
scenarios, e.g., in multiphoton entanglement. The value of the
phase is an integer multiplied by 2p/d, where d is the dimen-
sion of a quantum system. Obviously, for the case where d is
higher than 2, fractional phases may be found. All possible
values of this phase have been proved theoretically. A qudit is
defined as a quantum state in the d-dimensional system, fol-
lowing from the definition of the qubit in two dimensions.
The properties of this nontrivial topology are predicted to oer
a possible method for quantum computation and it may over-
come the shortcomings of traditional quantum computers
because of its robustness.
The topological phases of qudit states were measured experi-
mentally by Matoso etal. Two entangled states were prepared
and, after passing through a polarizing beam splitter (PBS) and
being unentangled, one of the states was applied to a unitary
SU(d) operator. An interferometer was set to interfere with one
state and with the operating state. Figure 9a shows the scheme
used to measure this fractional topological phase. Qudits are
encoded using two degrees of freedom: transverse paths and
polarizations. The SU(d) operation causes fractional topo-
logical phases. Higher-dimensional quantum states have been
observed experimentally and will provide a way to realize logic
gates based on the topological phase and robust quantum infor-
mation processing.
4.2. Quantum Information Processing and Error Correction with
Topological Photonics
Quantum theory provides a powerful tool for information pro-
cessing. However, as mentioned above, all quantum systems
need to overcome both noise and errors during information
processing. In this section, we review some research in which
the topological property is used to overcome errors. In classical
information processing systems, the switch is a basic com-
ponent that is used to represent complex information using
binary numbers. In quantum systems, some of the discrete
degrees of freedom are used to encode the data. A topological
photonic switch based on OAM was proposed by Luo etal.[103]
Adv. Optical Mater. 2021, 9, 2001739
Figure 9. a) Schematic of light path used to measure the fractional topological phase of the qudit states. Two entangled photons with orthogonal
polarizations are generated and injected into a polarizing beam splitter (PBS) to split the two photons. Both photons pass through a half-wave plate
(HWP) and a unitary operator is applied to one of the photons. The two photons interfere at the end of the structure and the phase is observed.
b) Experimental schematic image of toric code. m anyons are moving around three loops. a) Reproduced with permission.[178] Copyright 2016, American
Physical Society. b) Reproduced with permission.[109] Copyright 2019, OSA Publishing.
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They used topological pumping to realize a usable photonic
OAM switch. They pumped photons adiabatically to target the
OAM lattice and form a tunable double-well lattice. The topo-
logical property ensured the switching precision, although the
pump cycle was not arranged perfectly. With the exception of
its robustness against disorder, this structure showed eciency
as high as 90% for rapid switching. Photon losses during the
procedure of transportation to the target OAM state were negli-
gible. This result demonstrates the possibility of this approach
being applied in a multiphoton state or a coherent state.
The quantum computer will hopefully provide an upgrade
on classical computers because of its exponential eciency
in calculating huge programs.[104] However, quantum systems
still suer from both decoherence and dephasing. Quantum
error correction is used to reduce the noise in quantum sys-
tems. In Nemoto’s research, they selected a negatively charged
nitrogen-vacancy (NV) center in diamond that had numerous
satisfactory characteristics, such as atomic nucleus spin. Fur-
thermore, the steady memory of this center for information
and photons can be coupled to spin to form interfaces with
other units. This model was then used to construct a computer
architecture. Cavities with these NV centers are linked using
photons in a fiber network. Quantum information is recorded
and processed in the nucleus spin. The entire unit is placed in
a regular 2D lattice with topological structures. The topological
cluster-state error correction process is based on the connec-
tions between the units.[105] When one module is added to two
connected units, the entangled pair and the new module will
form a cluster state. This cluster state can be used to simulate
quantum gates and thus allow parallelizable computation.
Hill etal.[106] also applied this module to construct a quantum
computer using silicon and phosphorus. They then used this
architecture to build a control NOT gate on the surface and the
gate operation provides a method of surface quantum error
correction for silicon quantum computers. Consequently, the
control circuit is simplified and the error lies below the code
threshold. This silicon configuration can be a useful compo-
nent for use in huge-scale quantum computing. Herrera-Marti
et al.[105] proposed another design in which they replaced the
module with a fusion gate. Fusion gates will link a pair of pho-
tons with a third photon, if the fusion is successful. However,
if the fusion fails, the coherence of the qubits will be destroyed.
Fusion gates make the coupling of two photons easier, but also
increase the number of photons involved. Despite this, these
gates are not very tolerant of photon loss because a cluster state
can also be formed with a small possibility, and there are no
ways available to observe whether the entanglement exists to
date. The researchers therefore proposed a two-error model
and analyzed its ability to correct for computational errors and
photon losses during the processing procedure. This structure
forms an error-tolerant optical computer with a single photon
gate. The error rate was approximately equal to their theoretical
estimation.
The superconducting quantum circuit, which is based on
superconductivity, is capable of surface-code error correc-
tion.[107] On a 2D lattice, any gate can only entangle with four
neighboring gates. The fidelity of the processing in this case is
≈99%. Josephson quantum computing has proved to be scal-
able and error-tolerant. Anyons represent a new property in top-
ological quantum information. The anyon is a quasiparticle that
can produce a phase that is intermediate to bosons or fermions.
When using a superconducting circuit, anyonic braiding statis-
tics have a topological property that is error-tolerant. Pan etal.
generated a 7-qubit toric model.[108] The topological robust-
ness of anyons was discovered to be related to the fact that an
anyon’s spatial path is not related to its phase. They soon com-
pleted a 9-qubit planar code based on anyonic braiding
A
()
Ψ=∏+ …|1|0 0
s (11)
The ground state in this circuit was prepared and m anyons
were created, as shown in Figure 9b m anyons represent a type
of anyon. The paths of the anyon pair are loops 1, 2, and 3.
Dierent loops are used to measure the dierent phases after
the anyons are annihilated. Loops 2 and 3 are topologically
equivalent, where the phases of the anyons in these loops are
−1. However, in loop 1, braiding is not achieved and thus the
phase remains +1. These results show robustness if these loops
are applied in a larger web and this work was the first to prove
the topologically path-independent nature of anyonic braiding
in a photonic system. In their experiment, eight photons were
used to form this process, where three pairs of the photons
were entangled and one pair was not.[109] All photons were
injected into a linear network at the ground state. Then,
braiding operations were applied at the ground state because
the ground state created pairs of anyons. These operations,
including the braiding operation, the creation, and the annihi-
lation operation were realized using a combination of single-
qubit gates. They also suppressed the nonlinear photon emis-
sions, which were the main origin of the noise. The noise eect
has been observed closely and reflects the imperfections of the
states.
5. Novel Quantum Topological Photonic Devices
In this section, we will begin to introduce other novel quantum
topological devices. Topological quantum light sources and
topological quantum information, which were introduced in
the previous two sections, are actually parts of the applications
of quantum topological devices. However, because they play
very important roles in quantum optics, we introduced these
two concepts as separate sections. Quantum optical devices
are designed based on quantum eects and are then applied
in quantum information processing and quantum computing.
Usually, the quantum processes are highly sensitive to the
fabrication precision and to environment fluctuations, which
means that even small disturbances and instrument defects
may have a major impact on the performances of these devices.
Fortunately, the development of topological photonics also
brings new strategies for use in quantum devices. However,
although topological phenomena have been realized widely
using classical light, topological eects have rarely been dem-
onstrated in quantum optics until recent years. This section
mainly introduces quantum photonic devices that are topologi-
cally protected, including topological beam splitters, quantum
amplifiers, and quantum simulators. When compared with
their trivial counterparts, these devices show robustness against
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disorders and better performances, thus demonstrating their
broad prospects for applications.
Although quantum topological photonics has been used
for many kinds of devices because of its robustness, there are
fewer reports about the quantum topological detectors. The
detectors generally require high sensitivity as environment
changes, while the topological photonic structures are robust
and anti-disturbance to the external environment. Therefore,
the topological detectors are helpful in detecting the change of
a system, for example, the topological quantum phase transi-
tion, avoiding the impact of environmental disturbance on
detection accuracy.[196–199] We know that the high Q-factor of
optical cavity can improve the sensitivity of the detectors, and
topological optical cavities can have high Q-factor and they have
robustness with small Q-factor fluctuation, they can work in
harsh environments. Furthermore, trivial devices have defect of
high dark current noise, and the quantum channels generated
by topological devices can reduce it. In this review, we will not
give detailed introduction on topological detectors.
Section 5.1 first introduces traditional beam splitters and
interferometers and then quantum topological beam split-
ters and interferometers using the AAH model are presented.
Section 5.2 mainly describes the OAM with respect to topo-
logical photonics and its applications in quantum optics.
Section5.3 introduces the topological quantum simulation and
shows that its realization is sometimes inseparable from OAM.
The topological quantum walk, which is a basic phenomenon
in quantum optics and can be protected successfully using top-
ological photonics, is presented in Section5.4. In Section5.5,
quantum topological amplifiers are demonstrated. In addition
to the devices described above, several other quantum topo-
logical devices have also been reported.[110–119] It is believed that
many more of these devices with novel functions will be pro-
posed in the future.
5.1. Beam Splitters (BS) and Interferometers
BS are widely used optical devices on both the macroscale and
the microscale. BSs are also basic components for quantum
integration. The traditional on-chip BS can be divided into
three types: the directional coupler,[120–122] the Y-junction beam
splitter[123–125] and the multimode interferometer. In this sec-
tion, we will first provide brief introductions to these three types
of BS, and will then introduce a novel BS that takes advantage
of topological photonics.
5.1.1. Traditional Beam Splitters and Interferometers
There are three common BSs: directional couplers, Y-junction
BSs, and multimode interferometers. First, we would like to
introduce directional couplers. A directional coupler consists of
two coupled waveguides that have input and output ports on
their ends. The incident light input from one port is redistrib-
uted proportionally to the two output ports and this proportion
can be tuned by varying the coupling length, the waveguide
widths, the distances between the waveguides and other para-
meters, which can also aect the transmission and coupling
coecients of the waveguides. Figure 10a shows a schematic of
the directional couplers. However, this type of BS is sensitive to
the waveguide size and its process tolerance is poor, i.e., when
fabrication errors or environmental disturbances occur, the
splitting eciency will be very dierent to the expected value.
Another shortcoming is that the dispersion of the waveguide
mode is usually high, and as a result, the splitting ratios of the
same device vary widely at dierent wavelengths. Researchers
have made numerous eorts to solve these problems. For
example, subwavelength gratings have been used to reduce the
eects of dispersion.[126] We will demonstrate later that these
problems can be solved eectively using topological photonics.
The Y-junction BSs are relatively simple. Their symmetrical
structures guarantee a 50/50 splitting ratio and the phases of
both arms are identical. Although two curved arms are usually
used rather than a T-shape, the insertion loss of this type of
structure is still high and is usually ≈2dB.[127] A well-optimized
Y-junction can have an insertion loss of as low as 0.28 dB and
a schematic diagram of such a junction is shown in Figure 10b
The multimode interferometer (MMI) is based on self-imaging
of the multimode waveguide. The propagation constants βv
of the dierent order modes v in a multimode waveguide of
length L can be approximated as
vv
L
v
ββ π
−≈+
π
()
(2)
3
0
(12)
In the multimode waveguide region, the optical field can
be expanded as follows into a superposition of dierent order
modes
∑
ψπ
Ψ=
+
π
=
−
(,)()exp (2)
3
0
1
yL cy jvv
LL
vv
v
m
(13)
In the formula above, when the multimode waveguide length
L is equal to an integer multiple of 3Lp, the phases of the dif-
ferent modes are the same. After a coherent superposition, the
optical field distribution is the same as the initial optical field
distribution, and this is called “self-imaging.” Double and quad-
ruple images can also be obtained at other positions, where the
energy is divided into two or four equal parts. Furthermore,
the MMI can be described as M× N, with dierent num-
bers of input ports M and output ports N. The most common
types of MMI are 1 × 2, 2 × 2, and 1 × 4, which are illustrated
in Figure10c–e, respectively. In the MMI structure, the output
intensities at the ports are usually equal, which means that the
energy at each output port is 1/N of the total intensity.[128–130]
Next, a brief introduction to traditional interferometers
will be given. At present, electro-optical modulators based on
Mach–Zehnder interferometers (MZIs) are among the most
widely used high-speed modulators in on-chip optical sensing
and communications because of their high modulation rates,
low transmission losses, low driving voltages, low modula-
tion frequency chirp characteristics, and small wavelength
dependences.
As an example of the use of traditional interferometers in
quantum optics, the Hong–Ou–Mandel (HOM) eect describes
the phenomenon where two photons interact with a BS. There is
a 50/50 chance that each photon will either pass through the BS
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or be reflected. Therefore, for two interfering photons, there are
the following four possibilities, as illustrated in Figure 11a: the
upper and lower photons are both transmitted; the upper and
lower photons are both reflected; the upper photon is reflected
and the lower photon is transmitted; or the upper photon is
transmitted and the lower photon is reflected. Because photons
are identical, the first and second possibilities cannot be distin-
guished, and because there is a phase dierence of p between
the transmitted photon and the reflected photon, these two pos-
sibilities cancel each other out. In the end, two photons will be
detected on the same side of the beam splitter. The HOM eect
is the most basic optical eect caused by the indistinguishability
between photons and it occupies an extremely important posi-
tion in linear optical quantum computing. In this experiment,
traditional interferometers were used for the two interfering
photons, and topological interferometers can also work with
topological robust protection and lower losses. The following
sections describe these devices in detail.
5.1.2. Quantum Topological Interferometers and Beam Splitters
Losses are always a problem in traditional beam splitters and
interferometers, particularly in quantum optical integration,
and the high losses in quantum networks prevent their prac-
tical application. Topological photonics provides new strategies
to solve the loss problem and researchers have made consider-
able eorts to realize quantum beam splitters and interferom-
eters via topological photonics.
Tambasco et al. proposed a quantum topological interfer-
ometer and beam splitter based on the 1D topological struc-
ture of the AAH model using “one-plus-one” waveguides.[185]
Unlike the SSH model, the AAH chain has an aperiodic struc-
ture. First, let us assume that each atom is a primitive of the
AAH chain, and it is then arranged according to the following
formula
dd
n
n
επ
τϕ
=− +
1cos 2
0
(14)
where dn is the distance between the nth and (n+ 1)th atoms,
d0 is a constant that depends on the original structural parame-
ters, e controls the modulation intensity of the coupling and t is
the golden ratio with the specific value of +
(5
1)
/2
. Because of
the irrational value of t, a quasiperiodic spatial coupling modu-
lation is obtained. Additionally, φ denotes the topological para-
meter, and when φ varies continuously from 0 to 2p, the system
transits from a trivial state into a topological nontrivial state.
In the 1D system, the distance represents a visualization of the
coupling strength of two primitives,[131,54] which means that we
can also describe this adiabatic deformation of the chain using
k in the following equation, where k is the coupling strength.
n
n
κκ
επ
τϕ
=− +
1cos 24
0
(15)
As described above, φ plays the most important role in this
relation. In general, to determine the values of the various
parameters that can support the topological state, the first step
is to find the eigenvalue and then draw the energy band diagram
by solving the Hamiltonian of the system. Then, the value of φ
can be acquired at the intersection point on the basis that the
total number of atoms n has been determined because the dif-
ferent numbers of a chain show dierent φ values. The system’s
topological state is determined by its parameters. The obvious
topological phenomenon of the AAH chain is almost the same
as that of the SSH chain. It is clearly shown in Figure11b that
the light fields are localized at the two ends of both chains.
Adv. Optical Mater. 2021, 9, 2001739
Figure 10. a) Directional coupler with two inputs and two outputs. Light can be transferred gradually from one waveguide to another. b) Schematic
diagram of original Y-junction beam splitter. c) Multimode interferometer with one input and two outputs. d) Multimode interferometer with two inputs
and two outputs. e) Multimode interferometer with one input and four outputs in a single plane. a) Reproduced with permission.[179] Copyright 2015,
OSA Publishing. b) Reproduced with permission.[180] Copyright 2013, OSA Publishing. c) Reproduced under the terms of Creative Commons Attribu-
tion 4.0 License.[181] Copyright 2019, The Authors, published by EDP Sciences. d) Reproduced with permission.[182] Copyright 2008, AIP Publishing.
e) Reproduced with permission.[183] Copyright 2013, OSA Publishing.
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There have been some reports of devices and phenomena
based on the AAH chain, including ring resonators,[131] PT sym-
metry,[132] topology transcending dimensions,[133] and several
others.[134–136] Here, we introduce topological AAH model wave-
guides for beam splitters and interferometers. Application of
the 1D extending-dimensional configuration to light transmis-
sion, which is also important for both topological interferom-
eters and beam splitters, is highly practical.
Topological waveguide structures have been detailed and
demonstrated in Section2.2.3 and these structures can also be
applied to the AAH chain. The original 1D topological struc-
tures show topological zero modes and the optical field is
simply confined at both ends of these structures; time-varying
1D topological structures are produced by extending the atoms
to waveguides, as in the SSH model. However, this case diers
from the SSH chain in that the cross-section of the waveguide
can be considered to be a 1D topological structure composed
of the AAH chain at every moment, while each AAH chain is
dierent at the dierent moments. For better understanding
of this concept, we can imagine that the waveguides consist of
countless small AAH chains. For example, the waveguides in
the SSH chain are arranged to maintain the same para meters
as the foremost SSH model at all times; however, unlike the
simple extension of the SSH waveguide, every parameter con-
figuration of the AAH chain changes all the time for robust
edge-mode propagation. As shown in Figure 11c the wave-
guides are a series of curves rather than straight lines, which
also means that at dierent times during propagation, the AAH
chain parameters will be dierent. In this case, e and φ are
replaced with e(t) and φ(t), respectively, which both vary with
time. Equation (15) then becomes Equation (16)
tt
nt
n
κκ
ε
π
τ
ϕ
=+ +
() 1()cos 2()
0
(16)
By tuning the parameters of e(t) and φ(t), the topological
edge state can transit from one edge to the other; further-
more, the total interferometer and beam splitter designed by
Tambasco et al. is symmetrical above and below, i.e., after the
incident light is input, both classical light and quantum light
sources are fine. Interference occurs during transmission in
the middle of the waveguides because the edge-mode evolu-
tion is complete and the beam splitting function is then real-
ized. It should be noted that when the classical incident light
turns into quantum photons, the detectors on the ends of the
AAH waveguides should measure the output photons with
HOM interference and either visibility or relative visibility. The
measured results also show the good quantum properties and
favorable stability and robustness of these devices that result
from the topological protection, which proves and emphasizes
the quantum eects. Additionally, because of the high visibility
of this novel quantum beam splitter, the system loss is very low,
which provides a strong foundation for development of large-
scale integrated quantum optics. This is a representative aspect
of quantum topological photonics.
5.2. Applications using Orbital Angular Momentum
The OAM is an important degree of freedom of light, but
it did not attract much attention until recently. Following
in-depth study, the OAM has been applied in many fields,
including high-precision control,[137–140] optical communica-
Adv. Optical Mater. 2021, 9, 2001739
Figure 11. a) Schematic of Hong–Ou–Mandel (HOM) eect with four possible situations. From left to right and from top to bottom, they are: the
upper and lower photons are both transmitted; the upper and lower photons are both reflected; the upper photon is reflected and the lower photon is
transmitted; and the upper photon is transmitted and the lower photon is reflected. b) Topological phenomenon of the AAH chain. Two ends of the
chain can localize light after excitation of the center of the chain with a light source. c) Schematic of AAH waveguides using a series of curves (right)
rather than straight lines (left). The time-varied AAH chain has dierent parameters for e and φ at each moment. d) Spatial distributions of spiral waves
in lasers with dierent spiral defects. The lasers contain three, five, and eight defects and the rotation methods of these lasers are dierent because
they rely on the number of topological charges. Vortex lines, which can constitute complex topological characteristics, can be either unbounded (red)
or in loops (white). a) Reproduced with permission.[184] Copyright 2019, IOP Publishing. b) Reproduced with permission.[131] Copyright 2018, OSA
Publishing. c) Reproduced with permission.[185] Copyright 2018, AAAS Publishing. d) Reproduced with permission.[140] Copyright 2011, OSA Publishing.
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tions,[141,142] optical tweezers,[143] and other areas.[144] Rotation
of the light beams is the origin of angular momentum. There
are two important types of angular momentum of light: one
is the OAM and the other is the SAM, which corresponds to
the polarization of the light. Usually, the OAM is much greater
than the SAM. The OAM of light is obtained using spiral phase
plates, which can generate helical phases. The OAM can be
also described using the topology. Complex paths are formed
in three dimensions using optical vortices, i.e., phase singulari-
ties. Additionally, the OAM has complex topological features
with various topological numbers because of the topological
fractal, as shown in Figure11d, which is somewhat similar to
the Chern number. Furthermore, the OAM contains nonlinear
eects and quantum optical properties, e.g., the Kerr eect,
as a result of the parametric downconversion. In addition, the
photon pairs required for quantum entanglement can be gener-
ated through parametric downconversion with the OAM, and
the specific superpositions of the states are obtained from the
diraction gratings, which are displaced slightly from the beam
axis.[138] The OAM of the photons can also be used as a discrete
walker coordinate that corresponds to the helical wavefronts.
Therefore, by applying optical OAM in topological photonics,
this degree of freedom can bring new information to topolog-
ical systems. First of all, light with dierent OAM carries more
information than normal light and can be used to expand the
functions of a single structure, generate multiplexing functions
in topological photonics and improve the scalability of quantum
systems. Secondly, the topological charge from topological sys-
tems can be unbounded, which is in contrast with currently
proposed methods in metasurface that have limited topological
charges. This feature can help design large OAM structures for
multi-purpose in both quantum and classical communications
systems. In addition, the OAM of light provides a novel way
to study topological systems because it carries both classical
and quantized information at the same time, which enlightens
people to realize the switch between classical and quantum sys-
tems under the topological protection with low loss.
The OAM itself can be used to fabricate devices in inte-
grated optics. For example, by controlling the OAM states in
a cavity,[144] topological photonic switches can be obtained. A
schematic of this cavity is shown in Figure 12a, in which the
Hamiltonian can be described as a generalized AAH model.
The main cavity, which is shown in red, is coupled with the
auxiliary cavity, which is shown in blue. There are three steps
involved in switching the OAM states of the photonic signals:
first, the input photon pulse excites the L0 cavity mode; second,
the photon is pumped to the desired OAM state based on the
topological AAH model; third, the photon pulse is released
from the cavity. To obtain high OAM states and also to ensure
a high switching rate, cascaded degenerate-cavity systems can
be used, enabling a change of OAM to be obtained at 10q while
only q degenerate main cavities and at most 5q pumping cycles
are required.
One thing worth mentioning is that the integrated laser
with OAM information can be directly generated by using 2D
topological rings and forming circular boundaries.[191] The gen-
erated multiplexing coherent beams can carry arbitrarily large
OAM under topological protection. Using topology, such limita-
tions of topological charge and integrated device by alternating
concentric circular boundaries between two PhCs of distinct
topologies, an arbitrary number of orthogonal OAM beams
of alternating chiralities can be obtained in a planar manner
using a single aperture. The topological charge can thus be
made arbitrarily large with the radii of rings. By alternating
circular boundaries between PhC1 and PhC2, an arbitrary
number of orthogonal OAM beams of alternating signs can be
Adv. Optical Mater. 2021, 9, 2001739
Figure 12. a) Main cavity consisting of eight mirrors, which are shown in both red and blue. A three-auxiliary-cavity system with parameters in accord-
ance with the AAH model with two beam splitters is shown below. b) Schematic of the 1D cavity array. The first row of images shows each main cavity,
consisting of two beam splitters (BSs) and two spatial light modulators, which can vary the OAM of a portion of light in the cavity by ±1. Each coupling
cavity between two main cavities also contains two BSs. The second and third rows of images show the extra dimensions given by the OAM. The system
can be regarded as a 2D rectangular lattice. a) Reproduced with permission.[186] Copyright 2016, American Physical Society. b) Reproduced under the
terms of Creative Commons CC BY License.[144] Copyright 2015, Springer Nature.
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integrated in a planar manner using a single aperture. This is
an achievable way to improve the scalability of quantum sys-
tems with unbounded topological charge.
5.3. Topological Quantum Simulation
A quantum simulator is a controllable system used to reproduce
the dynamics and the quantum state of the original system.
Classical computers are unable to enumerate quantum states
one at a time. Therefore, classical computers are unable to
simulate quantum processes. In quantum systems, photons are
usually manipulated to simulate the quantum processes. When
compared with other quantum particle systems, e.g., fermions,
photonic quantum simulators oer the following advantages.
First, they do not interact easily because no electricity or other
interaction field is involved, which leads to a decoherence-free
system. Second, their mobility allows these simulators to move
freely in various media, particularly without rigorous condi-
tions such as very low temperatures.[145] Finally, simulations of
various quantum processes and quantum models with a spe-
cific form of Hamiltonian can be realized in these simulators,
including quantum walks, the wave function, the energy level
in a hydrogen molecule, and various topological problems.
Here, we introduce a 1D array of cavities, as illustrated in
Figure 12b, which can simulate the 2D topological physics.[144]
The OAM provides an extra dimension in quantum simula-
tions. In this work, the OAM was brought into quantum simu-
lations for the first time. The simulator has a simple structure
thanks to its low dimensions. The cavities are placed along the
x-direction, where each main cavity contains two beam splitters
and two spatial light modulators. The coupling cavity between the
two main cavities also contains two beam splitters. Because the
OAM brings another extra degree of freedom, the system can be
regarded as a 2D lattice. The change from a 1D array into a 2D
lattice can simulate dierent types of physics problems flexibly,
including photon transmission spectroscopy, edge-state transport,
topological phase transitions, and Chern number measurements.
5.4. Topological Quantum Walk
Quantum walks provide a versatile platform to enable study
of the topological phases of cold atoms,[146–148] photons, and
ions.[149,150] Various new discoveries about quantum walk are
constantly being reported, for example, the quantum-clustered
two-photon walks.[203] Many works have demonstrated simula-
tions of the topological phase,[23,25,151–155] but quantum walks
are periodic quantum phenomena and represent a special case
among these works. This type of phenomenon, in which the
variance of the particle distribution diers with the position
probability σ2(t), can solve problems that are highly complex
in classical systems. The variance of the probability distribu-
tion is a function of time. In classical random walks, σ2(t) is
proportional to the time and the probability distribution at the
long-term limit is a Gaussian distribution; in contrast, in a
quantum random walk, σ2(t) is proportional to the square of
the time, and as a result has a ballistic diusion characteristic
and a faster diusion speed. This provides a theoretical basis
for quantum computation.[156–159] Furthermore, systems of
quantum walks are simulated using a time-independent Hamil-
tonian and contain many novel physical phenomena, including
topological physics. With the development of suitable sources
for light generation, e.g., lasers, and other detection technolo-
gies, e.g., photoemission electron microscopy, study of the
dynamic processes of ultracold atoms, photons, and other par-
ticles becomes feasible. Based on this, we can perform some
direct research on quantum characteristics. Periodically driven
coherent manipulation of the quantum system is a mature
technology and researchers have created synthetic gauge fields
in dierent quantum systems to realize changes between dif-
ferent material forms and the topological energy structure. In
this review, we focus on discrete-time quantum random walks
in photonic systems to study the topological photonics.
As previously reported, the topological properties of a single
photon have been studied directly to a great extent through
photonic quantum walks.[23,24] Quantum walks in photonic sys-
tems represent the quantum analog of classical random walks.
The principle of photonic quantum walks can be understood
with reference to Figure 13a. Each step can be understood as
being the product of a coin-flip operator and a conditional shift
operator. In 2012, Kitagawa etal.[24] observed the discrete-time
quantum walk process of a single photon experimentally. The
experimental scheme used for split-step quantum walks is
shown in Figure13b. The process consists of four steps: first,
find a suitable half-wave plate to realize a single-photon polari-
zation rotation R(θ1); second, move one lattice displacement to
the right using birefringent calcite beam displacers through a
polarization-dependent translator T1 of |H〉; third, rotate twice
by R(θ2); finally, move one lattice displacement to the left using
a polarization-dependent translator T2 of |V〉. The output prob-
ability distribution shows a topologically protected bound state
of the single photon. By varying the polarization, Kitagawa etal.
discovered a topologically protected pair of bound states. It
is clearly shown that every eigenstate of the quantum walk is
interrelated with the quasi-momentum and the superposition
between |H〉 and |V〉. Quantum walks are translationally invar-
iant and this type of quantum walk has chiral symmetry. This
requires the polarization component of any eigenstate to be
confined to a specific circle on the Bloch sphere. On this basis,
topological invariants of the winding number can be defined as
follows: a closed path, which was obtained by varying the quasi-
momentum from −p and p through the Brillouin zone and the
polarization component of the eigenstate contributing it, winds
around the original point, where the total number of winding
operations is called the winding number. Therefore, the
quantum walks of photons represent a tool for measurement
of topological properties and periodic driving. Furthermore, the
topological bound states are not the eigenstates of the quantum
walks when decoherence occurs, but we can still observe their
relative operation. The SSH model was used in this experiment
and it laterally proves the formation of a topological “soliton.”[44]
In 2016, Cardano etal.[23] reported experimental realization
of a discrete quantum walk occurring in the OAM space of
light for both a single photon and two indistinguishable pho-
tons. Unlike the previous work, the quantum walk occurred
in a single light beam rather than in an interferometer. Each
step of the quantum walk consisted of shifts in the polarization
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and the OAM value. The intensity distributions at the dierent
OAM values in each step showed the topological properties of
light. The results could be analyzed using only one output. By
injecting both single-photon sources and two non-entangled
identified photons, the results have shown a topologically pro-
tected state, as predicted theoretically. In addition, by simu-
lating the Gaussian wavepacket dynamics in OAM space, they
investigated the eective band structure of the quantum walk
and found that its topological features are associated with the
spin-orbit coupling. Furthermore, Cardano etal. also reported
the topological quantum transitions occurring in a 1D periodic
system with OAM that was based on the quantum walk pro-
cess. In this work, the probability distribution moments of the
walker’s position after many steps could be used to find the
topological quantum transition point. The topological quantum
transition represents the conversion between the nontrivial top-
ological phase and the trivial topological phase. The topological
phase is connected to the topological winding number, which
was discussed above. The winding number then switches at the
transition point, as shown in Figure13c. This work therefore
oers a new platform to study quantum topological photonics.
Later in 2018, Wang et al.[154] used a similar photonic
quantum walk system to detect the topological invariants and
reveal the topological phase. They detected the winding num-
bers of a 1D quantum walk by measuring the average chiral
displacement and then identified the topological quantum
transition using this displacement in a similar manner to pre-
vious work. Later, they reported another work that detected a
large winding number by introducing a partial measurement
into this system, and the process then became a non-unitary
discrete-time quantum walk. There are also several other
works that focused on simulation of dierent topological phe-
nomena using the discrete-time quantum walk of a single
photon.[25,151, 153,160,161] Figure13c shows the conceptual diagram
for a quantum walk using topological SSH waveguides. From
the works discussed above, we can conclude that not only can
the topological protection work for quantum walks, but also
that quantum walks represent a powerful way to explore new
physical phenomena, e.g., topological photonics.
5.5. Quantum Topological Amplifiers
Amplifiers are usually required in the quantum integration pro-
cess when the input signal is weak or there are heavy losses
in the system. When light travels through the pumped region,
it is amplified through an energy transformation. Similar to
many other microscaled devices, traditional amplifiers require
high fabrication accuracy, and even a small amount of dis-
order may cause a strong disturbance after amplification. The
robustness of the topologically protected edge modes oers a
new approach to solve this problem. However, only a few works
on topologically protected amplifiers have been reported in
recent years. Here, we provide an example of a wave amplifier
based on the topologically protected edge modes. Apart from
the amplification of the vacuum fluctuations from the input
port, this amplifier is robust against disorders and suppresses
both reflection and scattering significantly. The amplification is
induced by the instability of the bosonic Hamiltonian caused by
parametric driving.[162] Under topological protection, the bulk is
hardly perturbed because of the anti-dissipation property, while
the quantum fluctuations along the edge are strongly distorted.
Adv. Optical Mater. 2021, 9, 2001739
Figure 13. a) 1D discrete-time quantum walk with symmetrical beam splitters; each photon after the coin and step operator has two possible outputs,
with one of them to be selected. b) Experimental setup for photonic quantum walk. c) Sketch of SSH model for quantum walks in waveguides. The
phase transition point remains robust, even when a perturbation is introduced intentionally. d) Schematic of the quantum topological amplifier. The
signal from the input port travels along the edge and is amplified, while the signal from the output port will be guided to the sink port. The trans-
mission power gain spectra show a dark thick line for the disordered sample and a light thin line for a clean sample. The reverse gain and the input
reflection with frequency deviation are represented by blue curves and gray curves, respectively. a) Reproduced with permission.[187] Copyright 2014,
American Physical Society. b) Reproduced with permission.[24] Copyright 2012, Springer Nature. c) Reproduced with permission.[188] Copyright 2019,
American Physical Society. d) Reproduced under the terms of Creative Commons Attribution 3.0 License.[162] Copyright 2016, The Authors, published
by American Physical Society.
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A schematic diagram of the amplifier is shown in Figure13d.
The amplifier consists of 30 × 12 nanocavities with one input
port, one output port and one sink port. The edge state inside
the topological bandgap travels chirally along the boundary and
this state will be amplified because the phase is appropriate.
The transmission power gain, the reverse gain, and the input
reflection are also demonstrated in Figure 13d, which shows
the amplifier’s robustness against disorder and its suppression
of reverse propagation.
6. Discussion and Outlook
In this review, we have discussed the principles and applica-
tions of quantum topological photonics that have been proved
experimentally to date. However, there are still several areas
that need to be discussed. First, on the one hand, we have rec-
ognized that the topological photonic structures that are most
popularly used to realize quantum eects are 1D nanopatterns,
because of the ease of their fabrication and simpler principles
when compared with higher-dimensional structures. On the
other hand, however, because of their dimensional constraints,
it is dicult for the light from both classical light sources and
quantum light sources to propagate for long distances, thus
indicating that it is necessary to extend research from the
original 1D topological structures into higher dimensions. The
most common strategy used in this case is the “one-plus-one”
waveguide array, which involves approaches such as extension
of the AAH chain into AAH time-varied waveguides. As a result
of the topological properties, the robustness protects the light
from energy dissipation, even after long-distance transmission.
Based on this idea, reports of experimental quantum walks,
chiral quantum emitters, directional couplers, and quantum
beam splitters are emerging almost endlessly. In addition, the
2D topological photonics in an optical delay line is another pop-
ular artificial structure that is also used for entangled quantum
generation and beam splitters. Second, there are still unsolved
problems in this field. One of these problems is that regard-
less of whether time-varied waveguides or optical delay lines
are used, they all enlarge the device size from the micrometer
scale to the millimeter scale. Because more compact quantum
devices are expected, the sizes of these topological photonic
structures need to be scaled down. Third, for quantum telepor-
tation and communication applications, the robustness in topo-
logical photonics can be used to find an ecient approach to
generate entangled photons while maintaining their coherence.
At present, the research is mainly focused on long-distance
quantum communications and multi-qubit quantum entangle-
ment. However, optical information transmission and photon
entanglement and correlation processes can be implemented in
a better manner using low-loss topological photonic structures.
In conclusion, quantum topological photonics is an attractive
topic that adds topology, a new degree of freedom in optics, to
quantum photons and thus brings us a new world of photonics.
However, although topological photonics has been studied
for years, both the theory and the experimental methodolo-
gies still require further improvement. To provide theory and
methods that can better serve the development of quantum
optics and quantum systems, including the atom array and the
multi-qubit network, and particularly in terms of topological
photonics, it will be necessary to attempt some new approaches
to this work. To date, it has been common to use the charac-
teristics of topological protection to study quantum optical
systems. In contrast, there have been few reports on use of a
quantum system to study topological photonics. In fact, these
two aspects are complementary to each other. If we consider
this from another viewpoint, topological physics is related to
group theory, and the constraints on the spatial structures of
group particles or the responses to the perceived gain or loss
on the time scale also contribute to topological physics. This
indicates that it is appropriate for topological physics to be
used in the study of multiparticle systems. Fortunately, multi-
particles and their symmetries are typical properties of group
theory. As a result, the study of topological physics cannot be
performed without multiparticles, which are the components of
quantum systems. Furthermore, it has been demonstrated that
photons are the best carriers for quantum information. Unlike
electrons, there are no direct interactions between the photons.
Therefore, in turn, the same principle applies to the study of
topological photonics using photon systems, e.g., for the topo-
logical photonics in a quantum walk.
As reviewed above, quantum topological photonics is a
new but promising field, and in recent years, many dierent
enlightening and profound achievements have been reported,
including descriptions of quantum properties with the new
degree of freedom provided by topological photonics. This
includes the realization of quantum walks, robust quantum
light sources, low-loss integrated quantum devices, and
quantum teleportation, which were all accomplished from a
new quantum perspective to realize topological photonics. Fur-
thermore, we have reviewed the theoretical and experimental
studies of these quantum processes and their applications in
topological photonic structures and we have also demonstrated
their advantages when compared with the corresponding pro-
cesses in traditional photonic structures. Generally speaking,
quantum topological photonics is dependent on the basic prin-
ciples of quantum mechanics, the topological phase transition,
and time-reversal symmetry, which is the bridge that connects
quantum optics with topological photonics, thus encouraging
researchers to explore new phenomena and applications in
photonics. Additionally, the topological robustness and unidi-
rectional transition properties are used to provide protection in
both classical and quantum photonic systems in practical appli-
cations. The complexity of quantum walks, the decoherence of
entangled photons and the high losses in integrated quantum
devices can be all remedied using appropriate topological
photonic systems, which will certainly be beneficial in quantum
computing and quantum information applications.
In short, because quantum topological photonics is a new
field, there are still tremendous challenges that must be over-
come. First, the basic theories of the quantum topological net-
work must still be clarified. Second, the available fabrication
precision currently limits the experimental research and appli-
cation of quantum topological photonics in the visible wave-
length region. Therefore, because the topological structures in
the microwave and infrared wavelength regions are too large
to integrate, the materials, structures, and fabrication methods
of the existing topological protection structures still require
Adv. Optical Mater. 2021, 9, 2001739
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© 2021 Wiley-VCH GmbH
2001739 (25 of 29)
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improvement. Thirdly, the output power of current topological
lasers and quantum light sources is still low. Since the output
power of single-mode lasers is generally relatively high, the
existing topological lasers are rarely guaranteed to be single-
mode lasers. In addition, the existing nano-fabrication tech-
nology precision limit the output power. It can be improved by
designing single mode topological laser and improving fabri-
cation accuracy.[200] What’s more, designing 3D structures and
specific topological channel transmission is another method,
for example, adding a mirror in the vertical direction of a 2D
topological structure to prevent laser “leakage.” Finally, higher-
dimensional topological photonic structures could be used to
carry more quantum information because they oer more
degrees of freedom, which may enable ultra-compact devices
to be realized for quantum information applications. Conse-
quently, it is vitally important to develop higher- and synthetic-
dimensional topological photonics to establish multiphoton
and complex quantum networks.
This vital new field not only oers immediate achievements
but also has great potential for future development. In addition
to the development directions that have been mentioned in this
review, there are many other novel paths that can be explored in
quantum topological photonics. For example, for quantum light
sources, further topological configurations and materials such
as electro-optic materials, which are tuned using the applied
voltage, may be able to control when and where the topological
phase transition occurs. Through targeted manipulation, such
approaches can be tried in dierent topological structures, in
addition to the spliced interfaces of photonic crystals, resonant
rings, and the SSH model. Consequently, flexible and precisely
positioned quantum light sources may be obtained. Quantum
information has tremendous development potential for use in
quantum memory and storage applications, in which the infor-
mation can be retained by tuning the gain or loss in the optical
delay line systems to change to the required topological edge
state. Furthermore, the quantum network is a complex system
and topological photonics can only depict it to a small degree.
If the network is combined with some algorithmic networks,
such as quantum topological neural networks, and driven using
the same algorithm, it may realize high eciency and low loss,
which would allow it to provide better integration of quantum
optical systems. Finally, for quantum optical devices, there are
numerous related applications to be addressed, including use
of the strong nonlinear eects of topological photonics to con-
stitute the quantum network, addition of PT symmetry to the
network to obtain an exception point or new quantum phe-
nomena, and fabrication of quantum-entangled chips by vortex
light emission based on the OAM.
Acknowledgements
This work was supported by the National Key Research and
Development Program of China under Grant No. 2018YFB2200403 and
2018YFA0704404, the National Natural Science Foundation of China
under Grant Nos. 61775003, 11734001, 91950204, 11527901, 11704017,
91850117, and 11654003, Beijing Municipal Science & Technology
Commission No. Z191100007219001, the Natural Science Foundation of
Beijing Municipality No. Z180015, and Beijing Institute of Technology
Research Fund Program for Young Scholars.
Conflict of Interest
The authors declare no conflict of interest.
Keywords
integrated photonic devices, quantum optics, topological photonics
Received: October 9, 2020
Revised: March 4, 2021
Published online: April 29, 2021
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Qiuchen Yan is a doctoral candidate student of Prof. Xiaoyong Hu at Peking University. She
received her B.S. degree in Optoelectronic Information Science and Engineering from Nankai
University. Her current research focuses on the study of mesoscopic optics and nanophotonics.
Xiaoyong Hu is the Cheung Kong Professor of Physics at Peking University. He worked as a post-
doctoral fellow with Prof. Qihuang Gong at Peking University from 2004 to 2006. Then he joined
Prof. Gong’s research group. Prof. Hu’s current research interests include photonic crystals and
nonlinear optics.
Qihuang Gong is a member of Chinese Academy of Sciences and the Cheung Kong professor
of physics at Peking University, China, where he is also the founding director of the Institute of
Modern Optics and executive vice dean of Graduate School of Peking University. In addition, he
serves as the director of the State Key Laboratory for Mesoscopic Physics. Prof. Gong’s cur-
rent research interests are ultrafast optics, nonlinear optics, and mesoscopic optical devices for
applications.
Adv. Optical Mater. 2021, 9, 2001739
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