# Topological photonics: fundamental concepts, recent developments, and future
directions

Mahmoud Jalali Mehrabad<EMAIL_ADDRESS>Joint Quantum Institute,
NIST/University of Maryland, College Park, Maryland 20742, USA  Quantum
Technology Center, University of Maryland, College Park, Maryland 20742, USA
Sunil Mittal  Department of Electrical and Computer Engineering, Northeastern
University, Boston, Massachusetts 02115, USA  Mohammad Hafezi<EMAIL_ADDRESS>Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland
20742, USA  Quantum Technology Center, University of Maryland, College Park,
Maryland 20742, USA

###### Abstract

Topological photonics is emerging as a new paradigm for the development of
both classical and quantum photonic architectures. What makes topological
photonics remarkably intriguing is the built-in protection as well as
intrinsic unidirectionality of light propagation, which originates from the
robustness of global topological invariants. In this Perspective, we present
an intuitive and concise pedagogical overview of fundamental concepts in
topological photonics. Then, we review the recent developments of the main
activity areas of this field, categorized into linear, nonlinear, and quantum
regimes. For each section, we discuss both current and potential future
directions, as well as remaining challenges and elusive questions regarding
the implementation of topological ideas in photonics systems.

###### Contents

  1. I Introduction
    1. I.1 Key demonstrations in developments of topological photonics
    2. I.2 Scope and aims
  2. II Linear topological photonics
    1. II.1 Concept of Topological Invariants
    2. II.2 Toy model: charged particles in strong magnetic field
      1. II.2.1 Classical picture
      2. II.2.2 Semi-classical picture
      3. II.2.3 Quantum picture: Landau quantization
    3. II.3 Hofstadter Butterfly
    4. II.4 Photonic lattice
    5. II.5 Various topological photonic models and their implementations
    6. II.6 Topological photonic crystals
  3. III Nonlinear Topological Photonics
    1. III.1 Nonlinearity Induced Topological Phase Transitions
    2. III.2 Spatial Solitons
    3. III.3 Dissipative Kerr Temporal Solitons and Frequency Combs
  4. IV Quantum Topological Photonics
    1. IV.1 Topological sources of quantum light
    2. IV.2 Topological robustness for propagating quantum states of light
    3. IV.3 Topological photonic systems coupled to quantum emitters
  5. V Remaining challenges and future directions
    1. V.1 Linear topological photonics
    2. V.2 Nonlinear topological photonics
    3. V.3 Quantum topological photonics
    4. V.4 Strong photon-photon interaction and coupled electron-photon systems

##  I Introduction

###  I.1 Key demonstrations in developments of topological photonics

The entry of topology into physics started with the discovery of the quantum
Hall effect in 1980 [1], in which Hall conductance was demonstrated to be
robustly quantized in a 2D electron gas. Subsequently, it was realized that
such robustness is due to the topological properties of the system energy
bands [2]. The idea of band structure topology was later extended to a wider
class of systems known as topological insulators [3, 4]. Meanwhile, it was
realized that such phenomena are not limited to electronic systems and they
can be also realized in any bosonic system. This was initially considered in
the context of ultracold atoms, both in rotating Bose-Einstein condensates and
optical lattices with synthetic gauge fields [5] and followed up by other
bosonic systems such as photonics [6, 7], acoustics [8], phononics [9],
electronic circuits [10], and mechanics [11, 12]. Specifically, in the
photonic context, an analog of the quantum Hall model was proposed to realize
a one-way edge state for the propagation of electromagnetic field in
gyromagnetic photonic crystals [13, 14], and subsequently demonstrated [15,
16]. However, to break time-reversal symmetry (TRS) this scheme relies on the
presence of external magnetic fields, while the magneto-optical response of
materials is weak.

To address this issue, several theoretical proposals were put forward to
synthesize magnetic fields for photons [17, 18, 19, 20]. This was followed by
two experimental demonstrations of topological edge states in optical systems
without external fields [21, 22]. To bring these ideas to photonic crystals,
the realization of spin [23] and valley [24] quantum Hall models were
theoretically proposed. Subsequently, the spin Hall [25] and valley Hall [26]
topological photonic crystals were experimentally demonstrated.

Figure 1:  A selection of key developments in topological photonics
(references are listed in the main text).

However, topological invariants are not directly accessible in photonic
systems. Specifically, photons are bosons, and quantization of conductance
does not apply in this context. Nevertheless, quantum Hall physics can be
manifested in the form of a spectral flow [27], which was experimentally
observed in 2016 [28]. To expand the field of topological photonics into the
nonlinear regime, several types of topological lasers were demonstrated [29,
30, 31, 32, 33]. Although the nature and degree of robustness of these lasers
are still subject to investigation. Extending to the quantum regime,
topological quantum sources of light were demonstrated [34, 35] around the
same time. An intriguing direction is to explore strong light-matter coupling
to induce strong interaction between photons. To achieve this, integration to
quantum dots [25] and exciton-polariton were demonstrated in micro-cavities
[36] and transition metal dichalcogenides [37]. Remarkably, the Laughlin state
of two photons was realized [38] as a major step towards few-body interacting
topological systems. Some other key developments include topological antennas
[39, 40], 4-dimensional quantum Hall effect [41], higher-order topological
insulators [42, 43, 44, 45, 46], simulation of Landau levels for photons in a
cavity [47], and topological solitons [48, 49]. Finally, three recent
demonstrations showed robust topological funneling for light in a lattice
geometry [50], photonic quantum Hall effect and generation of large orbital
angular momenta [51] and topological beaming of light [52]. Some of these
developments have been summarized in Fig. 1.

In order to have a broader perspective of the above-mentioned developments,
one can classify the observed phenomena based on the involved photon number
and the strength of photon-photon interaction. The classification can be seen
in Fig. 2. In the regime of weak optical nonlinearity, classical photonic
topological phenomena are shown along the vertical axis with increasing photon
number, starting from low photon number cases such as silicon photonic coupled
ring resonators [22, 21], to topological antennas [39, 40], spatial and
temporal topological solitons [48, 49] and lasers [29, 30, 31]. Moving along
the horizontal axis, strong light-matter interaction enables one to induce
photon-photon interaction: Starting from the weak interaction regime
(topological quantum light generation [34], and quantum optics interface
between single emitters and photonic crystals [25]), to the strong interaction
limit, enabling the generation of two-photon Laughlin states [38]. An example
of the intermediate regime of interaction and large photon number is
topological polaritons in micropillar semiconductor systems [36, 53].

Figure 2:  A selection of emerging topological photonic systems categorized
based on photon number and photon-photon-interaction strength, focused on the
optical and infra-red domain [30, 38, 48, 36, 49, 25, 34, 21, 22, 39].

###  I.2 Scope and aims

The scope of this Perspective is to introduce basic concepts and discuss
recent development and potential future directions in the field of topological
photonics. We hope that this Perspective is useful for researchers with no
background in topological physics who are interested to explore this exciting
field.

This Perspective is structured in four sections consisting of linear,
nonlinear, and quantum photonic topological systems. The linear section will
include a concise pedagogical section to introduce the minimum intuitive and
mathematical descriptions of the key concepts required to study topological
photonics. Then, linear photonic implementations such as topological photonic
crystals and passive waveguides and routers are reviewed in this section. The
nonlinear section focuses on nonlinear effects in topological systems such as
lasers, spatial and temporal solitons, and frequency combs. The quantum
section will review the topological quantum sources of light, topological
protection for the propagation of quantum states, chip-integrated quantum
emitters, and systems of strongly interacting electron-photons. The last
section includes detailed remarks on current challenges and more specific
potential future directions as well.

It is not possible to discuss all the developments in the field of topological
photonics in all platforms and frequency domains. Here, the focus of this
Perspective is on the optical and infrared domain. In particular, we do not
focus on the microwaves regime, for which a comprehensive recent review is
available elsewhere [54]. While we provide basic simple ideas behind linear
topological photonics, we do not provide a pedagogical review of nonlinear and
quantum topological photonics. We refer the reader to a comprehensive review
that summarizes the development up to 2020 for nonlinear photonics [55] and
quantum [56, 57]. Moreover, we refer the reader to a review on Non-Hermitian
topological photonics [58], which is another emerging direction. Higher-
dimension and higher-order topological photonics are reviewed in Ref.[59]. A
review on topological lasers can be found here [60]. Synthetic dimension and
other developments can be found in a recent exhaustive roadmap [7].

In this Perspective, we highlight challenges for specific topological
platforms, some of which are fundamental and some are more technical, as
potential directions for future research. These challenges will be separately
discussed in linear, nonlinear, and quantum topological photonics.

Before proceeding, we make a clarifying remark. While the word “topological
insulator” has been extensively used in the literature of topological
photonics, we refrain from its usage in this Perspective to avoid confusion.
Strictly speaking, almost all photonic states studied so far are not
insulating states, due to their bosonic nature. In electronic systems, either
because of Pauli exclusion (Fermionic nature) or interaction, such as band or
Mott insulators, the system can be in an insulating state if it is probed at
the corresponding Fermi levels. In the photonic context, there is no notion of
the Fermi level, since bosonic states can have unlimited occupation in the
absence of interaction. Instead, one can have a photonic bandgap and the
transmission of light can be zero if photons are injected into the system
within the frequency bandwidth of this bandgap.

Moreover, it is important to distinguish between general topological states
and topologically-ordered states. We use the former as a general term for any
state with classical or quantum topological properties, such as vortex states,
Chern band insulators. We reserve the latter term for strongly interacting
systems, where the order is a consequence of interaction and entanglement, and
therefore, can be defined and classified accordingly. This field is an active
area of research, mainly theoretical due to the lack of clean and unambiguous
experimental platforms (see [61] for a recent review). With this definition,
topological states encompass topologically-ordered states. In this
Perspective, we mainly focus on only topological states, primarily in the
single-particle/classical physics regime, and only briefly discuss
topologically-ordered states.

##  II Linear topological photonics

In the following sections, we start with a very broad introduction to the role
of topology in photonic systems, and then we introduce models and relevant
photonic implementation of these concepts.

###  II.1 Concept of Topological Invariants

Topology is a branch of mathematics that studies the general or global
characteristics of a system. For example, when studying a system of
geometrical objects, instead of the specific shapes, topology primarily deals
with how objects are connected. In other words, topology is concerned with the
global geometrical characteristics of a system, rather than the specifics of
its building blocks. As an intuitive example, one can consider the case of
swimming around an island versus a peninsula 111from Latin peninsula, from
paene ‘almost’ + insula ‘island’. Note that regardless of the shape of an
island, we call it an island, but once its topology is changed, we use a
different word (Figure 3). Starting from a point and coming back to the same
location, a swimmer can swim around the island, a process that does not depend
on the shape of the island, or the path taken. However, the number of lapses
around the island, which is an integer number (from $\mathbb{Z}$), is
_topologically_ robust, and that number can be considered as a topological
invariant. We associate the sign of the integer to the clockwise (CW) vs the
counter-clockwise (CCW) orientation of the swimmer. Note that this number is
always zero for a peninsula since a complete round trip around is not
possible. The robustness here means that under small perturbations of the
island’s shape or the swimming path (specific “local” features of the
swimmer’s path and the island’s shape), the integer number (global topological
property of the system) will remain invariant. If one relates a physical
observable (conduction, transmission, resistance, etc.) to such integer
numbers, then that observable will be similarly topologically robust. This is
one of the central motivations for the implementation of topology in physics.

The case of the island and peninsula is a classical example, with no notion of
phase. A classical wave or quantum-mechanical analog such an example can be
realized by considering how electron (or photon) wavefunction winds around a
certain point. An example is a vortex state in two dimensions, where the
wavefunction phase can wind an integer number of times. Considering a polar
coordinate $(r,\theta)$, in the presence of rotational symmetry, the
wavefunction can be of the form $\psi(r,\theta)=\rho(r)e^{im\theta}$, where
the phase winds an integer number of times $m$, and the radial part of the
wavefunction $\rho(r)$ has a singularity at the center. More generally, in the
absence of rotational symmetry, the wavefunction can be described as
$\psi(r,\theta)=\ket{\psi(r,\theta)}e^{i\phi(r,\theta)}$, where
$\phi(r,\theta)$ is the local phase. Therefore, in the context of photons,
these states can be thought of as a solution to the Maxwell equation, where
weak spatial variation in the dielectric constant can deform the spatial form
of this wavefunction but can not change the winding number $m$, i.e.
$\oint{\nabla}\phi(r,\theta)\cdot d\vec{l}=2\pi m$. This is already an example
of the topological robustness of a photonic wavefunction in space.

Figure 3: Number of lapses when starting from a point (marked by the red
cross) and going around an (a) island is an integer number, while it is zero
for (b) a peninsula. Note that this integer number is invariant regarding the
island’s shape, swimming direction, or the taken path.

How can we generalize this idea? Consider a typical periodic photonic system
such as a photonic crystal, e.g., a bipartite lattice as shown in Fig. 4(a).
Solving Maxwell’s equation in such a periodic setting can provide several
informative properties of the system, including the band gaps, group velocity,
dispersion, etc.

Most physical properties are indeed a local function of the band structure
$\varepsilon_{\alpha}(\vec{k})$, where $\varepsilon_{\alpha}$ is the energy
and $\vec{k}$ is the wave vector in the corresponding Brillouin zone.
Remarkably, there are other properties that depend on global properties of the
band structure. For example, let us take a two-band system, with wavefunctions
denoted as $\psi_{\uparrow}(\vec{k})$ and $\psi_{\downarrow}(\vec{k})$, as
shown in Fig. 4(b). In particular, for such a two-band model, the state of the
system can be represented by the unit vector on a Bloch sphere. Then, let us
consider how the wavefunction varies as we move in the Brillouin zone (Fig.
4(c)). If the system is topologically non-trivial, the wavefunction should
accumulate a non-zero phase, when the unit vector is swept around a closed
loop. Loosely speaking, there is an associated integer similar to the island
and peninsula example. The latter case takes place in real space, while the
former does so in momentum space. Nevertheless, the robustness of certain
global properties of the system remains warranted. More specifically, as long
as the spatial variation in the susceptibility and the dielectric function is
weak, this global integer remains invariant. Therefore, the associated
photonic observable, such as a transmission, to this invariant remains robust
to a certain disorder. We briefly clarify this connection in the photonic
context in the following section.

For a survey of band topological models and a step-by-step derivation, the
reader can consult [63]. An introduction to quantum Hall physics can be found
in [64, 65].

Figure 4: (a) A periodic (honeycomb) photonic crystal with sub-lattices A and
B. (b) Band structure of the photonic crystal, with an energy band gap
separating the valence (blue) and conduction (yellow) bands. (c) Eigenvectors
of the system in a unit sphere.

###  II.2 Toy model: charged particles in strong magnetic field

In the following, we study a simple model of a charged particle in two
dimensions, under a uniform magnetic field. While this model describes the
topological physics behind the electronic quantum Hall effect, we later find
it useful to synthesize similar physics in two-dimensional photonic systems.
The key concepts, such as the role of gauge field, topological robustness, and
topological edge state, can be understood using this simple model.

####  II.2.1 Classical picture

In the classical picture, the electrons, with charge $e$ and mass $m$ undergo
a cyclotron motion in the presence of an external magnetic field $\vec{B}$.
Considering the Lorentz force, the dynamics of the velocity $\vec{v}$ in 2D is
given by $m\frac{d\vec{v}}{dt}=-e\vec{v}\times\vec{B}$. Assuming a circular
motion with radius $R$, and angular velocity $\omega$, we have
$m\omega^{2}R=e\omega RB$. One immediately observes that the angular velocity
is radius independent $\omega_{c}=\frac{eB}{m}$ and $\omega_{c}$ is called the
cyclotron frequency. Note that the center and the radius of this orbit are not
constrained.

####  II.2.2 Semi-classical picture

For a semi-classical description, we can use the Bohr-Sommerfield
quantization:$\oint\\!p.\,dq=n\hbar$ to realize such orbits should be
quantized. In fact, the integer $n$ is the phase winding number introduced
earlier in our island analogy. In particular, one can set $n=1$, to find an
orbit with the smallest possible radius.

Alternatively, we can use a simple Heisenberg-limited picture, which gives us
a lower limit on how small the radius of the orbits can be. Specifically,
assuming $\Delta p$ and $\Delta x$ to the uncertainty in momentum and
position, respectively, we have $\triangle x\triangle p\simeq\hbar/2$, where
for a circular motion, $\Delta x\simeq R$ and $\Delta p\simeq m\omega_{c}R$.
Apart from a factor of two, this means the smallest orbit radius is set by

$l_{B}=\sqrt{\frac{\hbar}{m\omega_{c}}}=\sqrt{\frac{\hbar}{eB}},$ (1)

where from now on, we call this the magnetic length. Based on the Pauli
exclusion principle, we can have only a single electron in each state.
Therefore, we evaluate how many of these orbits one can fit in an area of
$A=L_{x}L_{y}$, as shown in Fig. 5a. The total number of orbits that be fit in
the system is

$\begin{split}N_{\phi}=\frac{L_{x}L_{y}}{2\pi
l_{B}^{2}}=\frac{AB}{\Phi_{0}}\end{split}$ (2)

where $\Phi_{0}$ is the quantum of magnetic flux, $AB$ is the total magnetic
flux, and $N_{\phi}$ is the total number of flux quanta. This suggests that
the lowest-energy state of the system is $N_{\phi}$-fold degenerate.

Therefore, this suggests that as long as the number of electrons is less than
$N_{\phi}$, they can be easily fitted into the system, if we ignore the
interaction between them. But if $N_{e}>N_{\phi}$, we have to pay some energy
price $\hbar\omega_{c}$. We can revisit this argument more precisely in the
quantum picture.

Moreover, in the presence of a confining potential at the boundary of the
system, the degeneracy is lifted for the state at the edge of the system.
These edge states are robust against a certain amount of disorder. In other
words, the states form skipping orbits to avoid disorder, instead of reversing
their directions [66]. See below for a quantum description of edge states.

Figure 5: (a) cyclotron motion of electrons with a radius of R in a 2D
electron gas under a perpendicular magnetic field which generates a quantum of
flux due to each electron orbit. (b) energy levels of the system, with
$N_{\phi}$-fold degeneracy. Note that the adjacent levels are separated by
$\hbar\omega_{c}$. (c) Wavefunction concentric orbits. (d) accumulated phase
for a particle looping on the sites of a lattice.

####  II.2.3 Quantum picture: Landau quantization

To write the Hamiltonian of a particle with a charge ($-e$) in the presence of
a classical electromagnetic field, characterized by the vector potential
$\vec{A}=(A_{x},A_{y})$, we should simply replace all momenta by
$\vec{p}\rightarrow\vec{p}+e\vec{A}$:

$\begin{split}\hat{H}&=\frac{1}{2m}(\vec{p}+e\vec{A})^{2}\\\
&=\frac{1}{2m}[(p_{x}+eA_{x})^{2}+(p_{y}+eA_{y})^{2}].\end{split}$ (3)

It is more common to choose the Landau gauge: $\vec{A}=-yB\vec{x}$, and obtain
the $N_{\phi}$ degeneracy by applying the boundary conditions. Instead, we can
choose the symmetric gauge:
$\vec{A}=-\frac{yB}{2}\vec{x}+\frac{xB}{2}\vec{y}$, since the derivation is
more elegant and easily generalizable. Defining mechanical momenta as

$\begin{split}\hat{\Pi}_{x}&=\vec{p}_{x}+e\vec{A}_{x},\\\
\hat{\Pi}_{y}&=\vec{p}_{y}+e\vec{A}_{y},\end{split}$ (4)

for any magnetic field, we have

$\begin{split}[\hat{\Pi}_{x},\hat{\Pi}_{y}]=-ie\hbar B.\end{split}$ (5)

Therefore, if the magnetic field is uniform, by properly re-scaling these
momenta operators, we can consider them as position $\hat{x}$ and momentum
$\hat{p}$ operators. More interestingly, the Hamiltonian is that of a harmonic
oscillator $\hat{H}=\frac{1}{2m}(\hat{\Pi}_{x}^{2}+\hat{\Pi}_{y}^{2})$. The
ladder operators can be defined as:

$\begin{split}\hat{a}=\frac{1}{\sqrt{2e\hbar
B}}(\hat{\Pi}_{x}-i\hat{\Pi}_{y})\end{split}$ (6)

$\begin{split}\hat{a}^{\dagger}=\frac{1}{\sqrt{2e\hbar
B}}(\hat{\Pi}_{x}+i\hat{\Pi}_{y}).\end{split}$ (7)

Consequently, $[\hat{a},\hat{a}^{\dagger}]=1$ and
$\hat{H}=\hbar\omega_{c}(\hat{a}^{\dagger}\hat{a}+\frac{1}{2})$, as shown in
Fig. 5(b). This tells us that the energy levels are evenly spaced by
$\hbar\omega_{c}$, known as Landau Levels. In order to get the Landau level
degeneracy, we can similarly identify another pair of operators that commute
with $\hat{H}$:

$\begin{split}\widetilde{\Pi}_{x}&=\vec{p}_{x}-e\vec{A}_{x},\\\
\widetilde{\Pi}_{y}&=\vec{p}_{y}-e\vec{A}_{y}\end{split}$ (8)

and similarly,

$\begin{split}[\widetilde{\Pi}_{x},\widetilde{\Pi}_{y}]=ie\hbar B\end{split}$
(9)

and all the other commutators are zero if we choose the symmetric gauge:
$[\widetilde{\Pi}_{x},\hat{\Pi}_{x}]=[\widetilde{\Pi}_{y},\hat{\Pi}_{y}],[\widetilde{\Pi}_{x},\hat{\Pi}_{y}],[\widetilde{\Pi}_{y},\hat{\Pi}_{x}]=0$.
This gives us another harmonic oscillator, where ladder operators are
$\hat{b}=\frac{1}{\sqrt{2e\hbar
B}}(\hat{\Pi}_{x}-i\hat{\Pi}_{y}),\hat{b}^{\dagger}=\frac{1}{\sqrt{2e\hbar
B}}(\hat{\Pi}_{x}+i\hat{\Pi}_{y})$, and $[\hat{b},\hat{b}^{\dagger}]=1$. As
shown in Fig.5(b), the eigenstates are simply number states corresponding to
these two harmonic oscillators:

$\begin{split}\ket{n,m}=\frac{\hat{a}^{\dagger 2}\hat{b}^{\dagger
2}}{\sqrt{n!m!}}\ket{0,0}.\end{split}$ (10)

The explicit form of the lowest Landau level wavefunction can be found in [64,
65]. Here we make a few remarks on the properties of these wavefunctions.

  * •

The wavefunctions are concentric orbits with average radius $r=\sqrt{2m}l_{B}$
and a width proportional to $\frac{l_{B}}{\sqrt{m}}$, as shown in Fig. 5(c).
Therefore, as $m$ increases, the orbits become more packed, until the radius
hits the system size. The maximum value of $m$ is
$m_{max}=\frac{1}{2}(\frac{d}{l_{B}})^{2}$, assuming the system to be of a
disk shape with radius $d$. The expression is simply the area in units of the
magnetic length, which is basically the total number of magnetic flux
$N_{\phi}$. Therefore, we recovered the Landau level degeneracy to be
$N_{\phi}$. More precisely, $m$ from zero to $N_{\phi}-1$.

  * •

This physical meaning of $\widetilde{\Pi}_{x}$ and $\widetilde{\Pi}_{y}$, in
the symmetric gauge, is simply the center of orbits. Specifically
$\hat{X}=-\frac{\widetilde{\Pi}_{y}}{eB}$,
$\hat{Y}=\frac{\widetilde{\Pi}_{x}}{eB}$. Using Eq. 9, we find that
$[\hat{X},\hat{Y}]=il_{B}^{2}$, which means we can localize the orbits both in
x and y coordinates. This is essentially the same argument we had in the semi-
classical picture.

  * •

Unfortunately, this model up to here can not explain the integer quantum Hall
effect and one needs to add both a confining potential and disorder to the
model. The introduction of the two ingredients leads to the confinement of the
state in the bulk of the system and the emergence of the edge states at the
system boundary [67, 68, 69]. In fact, this concept is more general and is
known as bulk-boundary correspondence, where the bulk properties dictate the
properties of the edge and vice versa. An intuitive understanding of this
concept is based on gauge invariance, either through Laughlin’s argument [27]
(see [64, 65] for a pedagogical presentation) or the Chern-Simons response
theory [70] (see Ref. [71] supplementary material for a derivation of this
concept in photonic systems).

  * •

The above orbits are also eigenstates of angular momentum:
$\hat{L}_{Z}=i\hbar(x\partial_{y}-y\partial_{x})$, with
$\hat{L}_{Z}{\Psi_{LLL}}^{(m)}=m\hbar{\Psi_{LLL}}^{(m)}$. In other words,
these states have a well-defined non-zero phase winding, similar to our island
analogy. In the presence of weak disorder, the orbits can deform but keep
their phase winding. In the strong disorder limit, the states are completely
washed out.

###  II.3 Hofstadter Butterfly

So far we assumed the considered system is a 2D continuum. Let us now consider
that the charged particles are confined to move on a square lattice, with
lattice spacing $l_{s}$, in the presence of a uniform magnetic field. We then
investigate to see in what regimes these two models are equivalent.

Inspired by the Aharanov-Bohm phenomena, the essence of the magnetic field is
a non-zero $2\pi\alpha$ that the particle acquires on each plaquette.
Formally, one needs to modify each hopping term by the corresponding gauge
field on that link. This is known as Peierls substitution $J\rightarrow
J\exp\left[{\frac{ie}{\hbar}\int_{\rm link}\vec{A}.\vec{dr}}\right]$, where
$J$ is the hopping rate. So naturally, different gauge conventions correspond
to spreading the total phase $2\pi\alpha$ along each link of a plaquette (Fig.
5(d)). Using the Landau gauge: $(A_{x},A_{y})=(-Byl_{s},0)$ where
$(x,y)\in\mathbb{Z}$, and are simply locations on a $(N_{x},Ny)$ square
lattice. The Hamiltonian describing the dynamics is

$\begin{split}\hat{H}&=-J\sum\hat{a}_{x+1,y}^{\dagger}{\hat{a}_{x,y}}^{-2i\pi\alpha
y}+\hat{a}_{x,y}^{\dagger}{\hat{a}_{x+1,y}}^{2i\pi\alpha y}\\\
&+\hat{a}_{x,y+1}^{\dagger}\hat{a}_{x,y}+\hat{a}_{x,y}^{\dagger}\hat{a}_{x,y+1},\end{split}$
(11)

where $\hat{a}_{x,y}$ is the annihilation operator of particle at site
$(x,y)$. So far, we are considering a single particle so the statistics of
particles are not important. But in the following sections, we assume the
operators obey the bosonic commutation relations.

One can verify that the total accumulated phase for a counter-clockwise
propagation on a single plaquette is: $-2\pi\alpha
y+0+2\pi\alpha(y+1)+0=2\pi\alpha$. The number of magnetic flux in each
plaquette is:

$\frac{\Phi}{\Phi_{0}}=\frac{eBl_{s}^{2}}{h}=\alpha.$ (12)

In other words, $\alpha$ is the fraction of a flux in a plaquette. Note that
the essence is the presence of this phase, and the charge $e$, $\hbar$, etc.
drop out. Therefore, one can generalize this model to neutral particles
(atoms/photons) by ”synthesizing” the phase. This is the key insight in Ref.
[19].

The spectrum of this Hamiltonian is periodic when $\alpha\rightarrow\alpha+1$,
and is known as Hofstadter Butterfly [72], with many interesting fractal
properties, as shown in Fig. 6(a). One key point is the presence of band gaps
when the system is considered on periodic boundary conditions, i.e., torus.
Let us recall a tight-binding 2D square lattice ($\alpha=0$ in Eq.11) of the
size $(N_{x},N_{y})$ has a single band with $N_{x}N_{y}$ states with energies:

$E(n,m)=-2J[\cos(k_{x}a)+\cos(k_{y}a)],$ (13)

where $k_{x}l_{s}N_{x}=2\pi n$,$k_{y}l_{s}N_{y}=2\pi m$. In this model, the
translational symmetry is clearly broken, but if $\alpha=\frac{p}{q}$, and we
go around $q$ plaquette, we get $2\pi p$ phase, which is like having no phase
and a zero $\alpha$ 2D tight-binding model. This suggests that at
$\frac{p}{q}$ we have $q$ bands, each containing $\frac{N_{x}N_{y}}{q}$
states, as shown in Fig. 6(a).

When an open boundary condition is considered, in-gap states appear, as shown
in Fig. 6(b). Such states are localized at the boundary and propagate in a
chiral fashion.

Now, we can investigate the effect of the disorder. Imagine a disorder in the
form of an onsite potential $\hat{a}_{x,y}^{\dagger}\hat{a}_{x,y}$. When the
chiral edge state encounters such an obstacle, it is energetically preferred
to go around the disorder site, instead of reversing the propagation path.
Recall that CW and CCW edge states have different energies. Loosely speaking,
this is similar to a “quantum swimmer”, where, in the presence of an obstacle,
the path is modified to make sure the wavefunction remains single-valued,
instead of reversing the path.

Figure 6: (a,b) The spectrum of Eq.(11) on a $10\times 10$ lattice, for a
closed and open boundary condition, respectively. The lower panel illustrates
light intensity on the lattice corresponding to three typical states. Apart
from the localized bulk states in the middle, in the open boundary case, edge
states can form, while propagating in a clockwise (red) and counter-clockwise
(blue) fashion.

###  II.4 Photonic lattice

Can we engineer a 2D array of optical resonators to simulate the previous
Hamiltonian? As we observed above, the essence is the extra phase in hopping.
Let’s start with two coupled resonators:

$\begin{split}\hat{H}=-J{\hat{a}_{L}}^{\dagger}\hat{a}_{R}-J{\hat{a}_{R}}^{\dagger}{\hat{a}_{L}}\end{split}$
(14)

where $\hat{a}_{L},\hat{a}_{R}$ are the annihilation operators of a photon in
left and right resonators, respectively, as shown in Fig. 7(a). $J$ is the
coupling strength and depends on the overlap of electromagnetic modes in the
left and right resonators. The sign of $J$ here depends on the definition of
${\hat{a}_{L,R}}$ modes. Recall that phases usually do not have a meaning
until they are evaluated for a closed loop. More importantly, two coupled
resonators can not have a complex hopping phase, and our goal is to engineer
one (see below Eq.15).

The form is simply that of the coupled mode theory. In fact, in the absence of
nonlinearity, single photon and coherent state dynamics are the same
$(\hat{a}\rightarrow\langle a\rangle$) and we remove the hats going forward.

For example, the Heisenberg picture dynamics is equivalent to two coupled mode
equations of motions: $\dot{a}_{L}=i[{H},{a}_{L}]=iJa_{R}$, and
$\dot{a}_{R}=iJa_{L}$. Now, we want to consider two resonators coupled with a
waveguide in between. One can use the transfer matrix formalism (see
supplementary of [22]). Here we use the quantum input-output formalism [73]
that is shorter and provides insight, following Ref. [22]. For a resonator
mode ($a$) coupled to a waveguide, as shown in Fig. 7(b), decay/input can be
described by: $\dot{a}=-\kappa a-\sqrt{2k}E^{in}$ where $E^{in},E^{out}$ is
the input/output fields, respectively. The boundary condition is written as:
$E^{out}=E^{in}+\sqrt{2\kappa}a$. We want to engineer a situation where:

$\begin{split}H&=-Je^{i\phi}{a^{\dagger}_{L}}a_{R}-Je^{-i\phi}{a^{\dagger}_{R}}a_{L}.\end{split}$
(15)

Figure 7: (a) A pair of coupled ring resonators described by Eq. 14 (b) a ring
resonator coupled to a waveguide that can be described by the input-out
formalism described in the text. (c) two resonators coupled to each other
using another resonator that is anti-resonator with the side resonators. By
positioning the middle resonator and creating a differential optical path one
gets Eq. 17. (d) Clockwise (left) and counter-clockwise (right) propagation
under opposite magnetic fields, as illustrated by the blue arrows.

Note that we can not put arbitrary coefficients in front of the two terms, the
Hamiltonian should be Hermitian. Consider the scheme in Fig. 7)c, where two
resonators are coupled through an “anti-resonant” resonator in an asymmetric
way:

The total optical length of the middle ring is chosen such that photons
resonant with left/right resonators do not interfere constructively in the
middle, and therefore only circulate one on the above/below arm. The total
optical path is $(4m+3)\pi$, where $m$ is a positive integer. Under this
condition, photons spend most of their time on the left or right resonator,
and we can find an effective Hamiltonian with $\hat{a}_{L}$ and $\hat{a}_{R}$,
without the middle ring. The equations of motion take the following form:

$\begin{split}E^{\rm in}_{R}&=E^{\rm out}_{L}e^{2i\pi m+i\frac{3\pi}{2}-2\pi
i\alpha}\\\ &=-iE^{\rm out}_{L}e^{-2\pi i\alpha},\\\ E^{\rm in}_{L}&=-iE^{\rm
out}_{R}e^{+2\pi i\alpha},\\\ E^{\rm
out}_{R,L}&=E^{\rm{in}}_{R,L}+\sqrt{2\kappa}a_{R,L},\\\ \dot{a}_{R,L}&=-\kappa
a_{R,L}-\sqrt{2\kappa}a_{R,L}E^{\rm in}_{R,L}.\end{split}$ (16)

By eliminating $E_{R,L}^{in,out}$, we find the effective Hamiltonian:

$\begin{split}H&=-\kappa{a_{R}}^{\dagger}a_{L}e^{-2\pi
i\alpha}-k{a_{L}}^{\dagger}a_{R}e^{2\pi i\alpha}\end{split}$ (17)

By choosing the length of the resonators accordingly, e.g. by increasing the
$\alpha$ linearly in row number $y$, we can implement the Hofstader
Hamiltonian.

If the system is driven with photons corresponding to the frequencies of the
edge band (see Fig. 6), they circulate around the system either in CW or CCW
way. In other words, photons experience an effective magnetic field $B$, and
orbiting around the system in CW/CCW fashion leads to opposite energies, in
direct analogy to $-\vec{L}\times\vec{B}$, where $\vec{L}$ is the angular
momentum.

We have not applied any external magnetic field that breaks the time-reversal
symmetry (for example this is needed in an optical isolator). However, we have
a magnetic field-like Hamiltonian for a passive system. How is that possible?
In fact, we have two pseudo-spins $\frac{1}{2}$ corresponding to CW/CCW
circulating photons, inside each resonator, each experiencing an opposite
magnetic field. Therefore, a more accurate analogy is spin-orbit interaction,
where each spin orientation experiences an opposite magnetic field
$\vec{S}\times\vec{L}$ where $\vec{S}$ is that photon pseudo-spin of photons.
In other words, the TRS is preserved for the entire system, but we can
selectively drive the system in a “spin-polarized” way, e.g. pumping the CW
mode of the resonator. As long as photons do not get scattered from CW to CCW
mode, each experience an opposite magnetic field.

Since the word chiral is preserved for edge state with broken time-reversal
symmetry, here we use helical edge states.

In any physical realization, such scattering is present, however, if the rate
of such backscattering processes is slower than the hopping rate, then we can
ignore such processes. In the optics language, we need to operate in an
unresolved mode coupling regime.

We treated the left/right rings as single-mode resonators, while the middle
ring was treated as a waveguide. Is that correct? Yes, but this is only valid
for photons close to the resonance of left/right rings. For a rigorous
derivation, one needs to use the transfer matrix theory.

We again emphasize that the Hamiltonian and the second quantization formalism
are not necessary for this part. However, this formalism allows one to
understand its physics without getting lost in the details of transfer matrix
theory.

###  II.5 Various topological photonic models and their implementations

The above model was implemented in arrays of coupled ring resonators
fabricated on $SiO_{2}$ operating at 1550nm wavelength [22]. Generally, it is
crucial to study how the topological invariants are manifested in such systems
and what are the physical observables compared to electronic systems. For
electrons, the system is filled up to the Fermi level, and then the electrical
conductance is measured as the main physical observable. If the system has a
non-zero integer topological invariant, the conduction is quantized,
genetically with the same integer. Filling up the Fermi sea is simply a
consequence of the Pauli exclusion principle that is absent for photons. In
these photonic systems, however, one can probe the system with an incoming
laser field with a given frequency. If the field is resonant with any of the
system’s modes, the photons enter (couple into) the lattice. Otherwise, the
light is completely reflected [22]. This state spectroscopy can be used to
measure topological invariants as a spectral flow when the system is subject
to an extra magnetic flux [74, 28], in an analogy to Laughlin flux insertion
argument [27].

From a more general point of view, during the development of topological
photonics, several models have been developed and subject to intense research,
starting from integer quantum Hall, followed by anomalous quantum Hall (also
known as the Haldane model), and subsequently, spin and valley-Hall effects.
Other topological models have been also implemented in rings, for example,
Su–Schrieffer–Heeger (SSH) model [55], and topological laser arrays [75].
Other models have been also implemented in helical Floquet waveguides [21]. A
broad overview of these models, their characteristics, and implementations can
be found here [55].

It is also important to note highlight here that, different from its
electronic counterpart, topological photonics can offer an opportunity to
harness several unique degrees of freedom which are either in part or
completely unavailable in electronic systems. For example, various
polarization [25] and orbital angular momentum [51] degrees of freedom of
light can offer powerful design flexibility and novel functionalities. For
example, synthetic modal dimensions have been recently implemented to
synthetic hybrid spatial-modal lattice configurations beyond conventional
lattice geometries [76]. A review of the relevant concepts and recent
implementations can be found here [77].

###  II.6 Topological photonic crystals

A simple and useful topological model can be formalized for photonic crystals
based on band inversion and the formation of bound states. A useful way to
think about this is to use continuum models, in particular the ones that led
to the emergence of the topological insulators. The essence of these models is
captured in the Jackiw-Rebbi (JR) model and the concept of band inversion.
Considering a 2D system with the following dispersion:

$\Bigl{[}-i\hbar
v(-\sigma_{x}\partial_{x}+\sigma_{y}\partial_{y})+m\sigma_{z}\Bigr{]}\Psi=E\Psi,$

where $\Psi(x,y)$ is a spinor. $v$ and $m$ are the velocities and the
effective mass, respectively. In contrast to the electron’s spin states, here
the spinor represents a pseudo-spin, e.g., two modes of the electric field. We
assume the mass changes sign at the crossing point $y=0$, specifically,
$m(x,y)=m(y)$, and $m(0)=0$ and $\frac{dm}{dy}<0$. There is a bound state
solution at $y=0$ that propagates along the $x$-axis, described by

$\Psi(x,y)=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
1\end{pmatrix}e^{\frac{1}{\hbar
v}\int_{0}^{y}m(y^{\prime})\,dy^{\prime}}e^{ik_{x}x},$

which is schematically shown in Fig. 8. Note that if TRS is not broken, e.g.,
in the valley and spin-Hall effects, we get two copies of the above
Hamiltonian that are connected by TR. Therefore, we have helical states
(instead of chiral) that propagate in opposite directions with opposite spin
(polarization for photons) [59]. Here, for one of the polarization states,
$m(y)$ goes from a negative to a positive sign, while it changes sign in an
opposite manner for the other polarization. A more detailed investigation of
these concepts can be found here [78].

Figure 8:  Illustration of formation of in-gap edge states at the interface
between two topologically distinct mediums with inverted band structure.
Adapted from [79, 80].

An exciting implementation of the JR model is engineering topologically
distinct photonic crystals (TPCs). Specifically, by changing the photonic
structure, one can engineer a band inversion between two topologically
distinct photonic crystals to form propagating states at the interface, as
first proposed in Ref. [23] and demonstrated in Ref.[25]. These states have
three main characteristics: they are unidirectional (photons with opposite
polarization travel in opposite directions), spatially confined (in $y$ for an
interface along the x-axis)) at the boundary, and robust against certain
disorders. In the spin-Hall TPCs, as described in Ref. [23, 81], the opposite
circular polarisation of the two edge modes can be described by the in-plane
electric field profiles of the TPC’s hexagonal unit cell. The in-plane
electric field is highly circularly polarized, with opposite handedness for
these two Jackiw-Rebbi solutions. The in-plane electric field circular
polarization of ${\sigma}_{\pm}$ can be considered as pseudo-spins for this
topological photonic crystal.

Similarly, one can exploit the valley degree of freedom, and engineer a band
inversion. This leads to valley-Hall TPCs as first proposed in [24] and
demonstrated in [26]. Similar to spin-Hall TPCs, the in-plane electric field
of the TPC’s unit cell has two circular polarization that propagates in
opposite directions, and forms two helical topological edge states.

Topological edge states in TPCs were imaged directly in a lattice of silicon
Mie resonators [82], where the opening of photonic gaps around double
degenerate Dirac cone as well as the formation of topological edge states was
demonstrated using high-resolution optical microscopy. Another development was
recently reported in which valley and spin degrees of freedom were shown to be
presented simultaneously in a topological crystal [78].

Next, we will discuss some of the recent implementations of these types of
TPCs in a variety of passive linear systems.

Examples of experimental demonstrations of linear topological photonic
platforms are shown in 9. These systems include robust photonic waveguides and
ring resonators in both all-pass and add-drop filter configurations. These
devices use valley-Hall edge states. The quantum photonic section will cover
the first demonstration of spin-Hall type photonic crystal waveguide [25].
Recently, it was proposed that adiabatic tuning of the topological bandgap in
valley-Hall type photonic crystal can be utilized to form a topological mode-
taper [83]. Moreover, a similar approach has been implemented to realize
topological rainbow trapping in photonic crystals [84, 85].

Figure 9:  Implementation of topological photonic edge states in the linear
regime. (a) a passive suspended valley-Hall photonic crystal waveguide [26].
(b) a topological photonic mode taper [83]. (c) a valley-Hall topological ring
resonator with access bus waveguide, forming an all-pass filter [86]. (d) A
topological add-drop filter, comprising a valley-Hall resonator and access
waveguides [87].

##  III Nonlinear Topological Photonics

Until very recently photonic systems (as well as other systems like acoustics,
electrical circuits, etc) have been largely used to emulate single-particle
electronic topological Hamiltonians, that is, systems where interactions
between particles are negligible. This includes topological Hamiltonians such
as the SSH model, the integer and anomalous quantum Hall effect, the spin and
valley-Hall physics, higher-order topological insulators, Floquet topological
insulators, and so on. Nevertheless, electronic topological systems also
include effects such as the fractional quantum Hall effect where interactions
between particles lead to a very rich physics. It is, therefore, natural to
ask if one can use photonic topological systems to emulate ”interacting”
topological systems. Though single-photon interactions are very weak, we can
still achieve ”mean-field” nonlinear interactions between photons, at high
enough photon flux, by using a nonlinear medium (whose polarization is a
nonlinear function of the applied electric field). Examples of such nonlinear
interactions include self-phase modulation, cross-phase modulation, sum,
difference and harmonic-frequency generation, optical parametric oscillation,
lasing, etc. Along these lines, one of the research directions explores if
such nonlinear interactions affect the topology of the system: can they induce
topological phase transitions, or are the topological edge states stable in
the presence of such nonlinear interactions? On a more fundamental level, such
nonlinear topological photonic systems have no counterparts in fermionic
systems and can lead to the emergence of topological models that are unique to
photons. In parallel, another research direction explores the applications of
topological phenomena, like edge states, to engineer nonlinear processes in a
medium, for example, for efficient and robust lasers, generation of quantum
states of light, optical frequency conversion, etc. Even more so, one can also
achieve true single-photon-level nonlinearities mediated by atoms or
artificial atoms like quantum dots or superconducting qubits, or excitons in
semiconductors. Such systems can then realize a photonic analog of interacting
topological systems such as the fractional quantum Hall effect. In the
following, we will review advances in these sub-fields of nonlinear
topological photonics. We will limit our discussion to parametric
nonlinearities like the Kerr effect.

###  III.1 Nonlinearity Induced Topological Phase Transitions

Optical nonlinearities, like the Kerr effect, change the refractive index of a
medium as a function of the optical intensity [88]. This refractive index
change can lead to a change in the on-site potential or the coupling strength
between waveguides or resonators and subsequently be used to induce
topological phase transitions. One of the first demonstrations of such a
topological phase transition was carried out in topo-electric circuits that
realized the 1D SSH model [89]. Here the nonlinearity modified the couplings
between alternate lattice sites and a topologically trivial phase at low
intensities transitioned to a topological phase at high enough intensities.

At optical frequencies, a nonlinearity-induced topological phase transition
was proposed by Leykam et al. in a 2D coupled ring resonator system that
implemented the Haldane-like anomalous quantum Hall model in a bipartite
lattice [90]. In this model, a topological phase transition can be introduced
by adding unequal on-site potentials $M$ (mass terms) to the two sets of
lattice sites such that $M>2J$. Leykam et al. considered a system with built-
in (during fabrication) on-site potentials just below the transition
threshold. A broadband high-intensity pump pulse was then injected into the
link rings of the lattice such that their resonance frequency would red shift
compared to the site rings. This relative frequency shift would reduce the
effective coupling between the site rings and thereby, induce the topological
phase transition.

An experimental realization of the nonlinearity-induced topological phase
transition was demonstrated recently by Maczewsky et al. in a 2D coupled
waveguide array implementing anomalous Floquet topological model [91]. The
array is fabricated such that alternating waveguides have a non-zero on-site
potential (introduced by alternating the waveguide width), and the system is
topological only at each coupling region between the waveguides, the power
transfer $t>50\%$. In the linear regime, the presence of on-site potential
ensures that this power transfer ratio is less than $50\%$ and the system is
topologically trivial. Nevertheless, on injecting high enough power into the
waveguide with a thinner core (lower effective refractive index), the Kerr
nonlinearity reduces the on-site potential difference between the waveguides
and increases the coupling ratio such that the system transitions to a
topological phase. Note that this pump is injected only into a single
waveguide and this topological phase transition is local, meaning injecting a
weaker beam elsewhere in the lattice would still experience a topologically
trivial phase.

###  III.2 Spatial Solitons

An optical beam propagating through a medium with optical nonlinearities, like
the Kerr effect, can experience self-focusing wherein the central high-
intensity region of the beam sees a higher refractive index compared to its
low-intensity tails. At specific beam intensity, the self-focusing effect can
exactly balance the diffraction of light, and lead to the formation of spatial
solitons [92, 93, 94, 95]. Optical spatial solitons have been observed in many
platforms, including coupled waveguide arrays - very similar to those used for
the realization of photonic topological insulators [96]. This immediately
leads to the question: can such photonic topological systems host spatial
solitons? Will these solitons live on the edge of the bulk of the system? Are
these solitons robust against disorders?

Very recently, spatial solitons were observed in topological waveguide arrays,
in both bulk and edge states. Specifically, Mukherjee et al. observed bulk
solitons in a 2D anomalous Floquet topological insulator [49], as shown in
Fig. 10. The system consisted of a 2D array of waveguides with periodic/cyclic
couplings to their nearest neighbors. On exiting the bulk waveguides at high
enough input optical power, Mukherjee et al. observed solitons that undergo
cyclotron motion while hopping neighboring waveguides. Because of this
cyclotron motion, the intensity distribution of the soliton would repeat only
after propagating through a complete period (along the waveguide) of the
lattice. Nevertheless, as expected, the soliton would not diffract into the
bulk of the lattice. Furthermore, the quasi-energies of the solitons were
observed to be in the bandgap and the extent of localization of the solitons
was observed to increase (decrease) with the increasing (decreasing)
separation between the quasi-energies of the soliton and the linear band.

Figure 10: (a-c) Schematic of the 2D array of coupled waveguides and variation
of their coupling strengths, used to observe spatial topological bulk
solitons. (e-h) Intensity profile in the array showing soliton behavior. (i)
Schematic of the 1D array of coupled waveguides used for pumping of
topological solitons and the variation of the coupling strength between the
waveguides. (j) Topological pumping in the linear regime. (k) Topological
pumping of solitons at higher pump powers, and (l) trapping of solitons at
very high pump powers.

In another related article, Mukherjee et al. also observed a soliton-like
solution on the edge states of the anomalous Floquet topological waveguide
array [49]. As before, the input light is coupled to a single waveguide, but
now on the edge of the array. Because the input is confined to a single
waveguide, it can excite all the edge modes with different quasi-energies (in
this case quasi-energy is propagation constant/momentum along the waveguides).
The finite curvature of the edge band dispersion can then lead to the
broadening of the edge excitation. Note that, even in linear topological
systems, the excitations on the edge states stay localized to edge states. As
such, the broadening here refers to the increase in the number of waveguides
on the edge that are occupied by the beam as it propagates along the edge of
the array. In the presence of nonlinearity, Mukherjee et al. observed minimal
broadening - an indication of balancing the broadening of the beam against
nonlinearity-induced self-focusing. Nevertheless, these soliton-like features
on the edge were observed to scatter some power into the bulk of the lattice.

Following these observations of bulk and edge spatial solitons, another
fascinating observation by $\text{J}\ddot{\text{u}}\text{rgensen}$ et al. has
been the demonstration of Thouless pumping of solitons [97, 98, 99]. For this
experiment, they used a 1D array of coupled waveguides that simulates the off-
diagonal Aubry-Andre-Harper (AAH) model for photons. In this array, the
coupling strength (off-diagonal elements of the Hamiltonian matrix) between
the waveguides varies periodically as a function of position along the
waveguide length. This 1D model is related to the 2D Chern insulator model and
exhibits an identical band structure. At the input of this array, a soliton,
which is an eigenstate of the nonlinear Hamiltonian, was injected. Then, the
Thouless pumping manifested as a quantized displacement of the soliton to
neighboring waveguides by one unit cell after propagating one period of
coupling-strength modulation. Even more, by choosing a soliton solution that
bifurcated from a different band (with Chern number +2), the authors also
observed quantized displacement by two unit cells in one period of propagation
length. Evidently, the displacement of the soliton corresponded to the Chern
number of the band from which the soliton bifurcated. Using this same platform
the authors have also recently demonstrated nonlinear pumping of solitons by
fractional integers that are quantized [100].

In another very different platform, that of cavity polaritons, Pernet et al.
also observed spatial solitons [101]. Their system consisted of a 1D array of
micropillars, each of which hosts a cavity polariton. The coupling between the
neighboring micropillars was staggered to realize the 1D SSH model. The
nonlinearity in this system originates from the coulomb repulsion between the
excitonic part of the cavity polaritons. When the topological edge state at
the interface between topologically trivial and no-trivial regions was
strongly pumped, a topological gap soliton was observed to be localized at the
same interface state. Evidently, this topological soliton bifurcates from the
mid-gap topological edge state and exhibits a spatial intensity distribution
(localized on a single sublattice) that is similar to the linear case. More
interestingly, when a dimer in the bulk of the array was pumped, with pump
frequency in the topological bandgap, the authors observed the formation of
topological bulk solitons. Furthermore, the pump power threshold for the
formation of topological bulk solitons was found to be robust against defects
(introduced by another laser) only in one sublattice and not in the other
sublattice. Going further, the authors demonstrated that by controlling the
phase of the pump excitation over two micropillars of the dimer, they could
achieve sub-lattice-polarized topological solitons such that the soliton
wavefunction was predominantly localized to only the sublattices, and this
polarization could be controlled by controlling the relative phase of the two
pump beams.

###  III.3 Dissipative Kerr Temporal Solitons and Frequency Combs

The presence of optical Kerr nonlinearity in optical resonators with multiple
free spectral ranges (FSRs) can lead to the fascinating physics of temporal
dissipative Kerr solitons and optical frequency combs [102, 103, 104, 105,
106, 107, 108, 109]. Because of the spontaneous four-wave mixing process
mediated by the Kerr nonlinearity, a continuous-wave pump beam with a
frequency near one of the resonances leads to the generation of new
frequencies in the resonator. The energy and momentum conservation dictates
that the newly generated frequencies are also close to the resonator
frequencies at other FSRs. In the limit of a weak pump, this process is
spontaneous and as we discussed earlier, is used to generate energy-time
entangled photon pairs. With increasing pump power, the newly generated
frequencies beat with the pump and also with other frequencies, ultimately
leading to a stimulated four-wave mixing process. Because the generated
frequencies are almost aligned with the ring resonances, they constitute an
optical frequency comb with frequency separation almost equal to the FSR of
the resonator. However, the comb lines are, in general, not phase-locked and
the frequency comb is chaotic. In this regime, the field profile inside the
resonator is also chaotic

By appropriately designing the dispersion of the resonator, mostly to be in
the anomalous regime, and by tuning the pump frequency and power, it is indeed
possible to get to a regime where the linear dispersion of the ring is exactly
canceled by the dispersion introduced by the Kerr nonlinearity and the
resonator loss is balanced by the four-wave mixing (FWM) gain. This dual
balance leads to the generation of a coherent optical frequency comb where all
the comb lines are phase-locked and precisely equally spaced by the FSR of the
resonator. The corresponding field profile inside the resonator corresponds to
one or more soliton pulses in time, called Dissipative Kerr Solitons, that
propagate without dispersion. Such DKSs have been observed in a variety of
resonator geometries, including whispering gallery mode, bottle, integrated
ring, and Fabry-Perot resonators, and also in a number of material platforms,
including silica, silicon, silicon-nitride, aluminum-nitride, silicon-carbide,
and so on. From an application perspective, coherent optical frequency combs
find a number of applications, for example, in precision time-keeping,
spectroscopy, WDM transceivers, LiDARs, etc.

Recently, there has been growing interest in using coupled-resonator systems
to engineer novel DKS solutions and comb spectra that are not accessible using
single resonator geometries [110, 111, 112]. On a more fundamental level,
these systems also explore the self-synchronization of coupled resonators.
Some of the early demonstrations in this regard used resonators made of fiber
loops or a fiber-loop coupled to an integrated ring resonator [113, 114]. More
recently, the field of frequency combs has seen an influx of ideas from the
field of topological photonics.

Specifically, Mittal et al. theoretically studied the generation of DKSs and
optical frequency combs in two-dimensional ring resonator arrays that, as we
discussed earlier, create a synthetic magnetic field for photons, and thereby,
stimulate the integer or the anomalous quantum Hall physics for photons [115].
Given that this system realizes one copy of the anomalous Hall model near each
of the single-ring resonance frequencies, it is effectively a three-
dimensional system with two real and one synthetic dimension in frequency. For
a linear system, the different copies at different ring resonance frequencies
are uncoupled. However, the introduction of a four-wave mixing process (Kerr
nonlinearity) couples these copies by mediating the hopping of photons between
them. This demonstration is summarised in Fig. 11.

As we discussed earlier, the linear dispersion and the spatial confinement of
topological edge states lead to efficient phase-matching of the spontaneous
four-wave mixing process for the generation of entangled photon pairs. A
similar phenomenon was observed for the generation of optical frequency combs
in topological ring resonator arrays. The comb generation was efficient only
when the pump beam was close to one of the edge mode resonances. This can be
easily understood considering that the topological edge states circulate
around the complete periphery of the lattice, and hence, realize a super-ring
resonator composed of smaller rings. The edge state resonances then simply the
longitudinal modes of this super-ring resonator. So when the pump beam is
close to an edge-state resonance, in addition to the linear dispersion, the
FWM process is resonantly enhanced by the edge-state super-ring resonator.

Figure 11: (a) Schematic of the 2D array of ring resonators used to generate
temporal nested solitons. The resonator array simulates the anomalous quantum
Hall model for photons. The pump laser is coupled to the array using the
input-output waveguide. The nested comb output is collected using the same
waveguide. (b) Phase-locked Turing rolls along the edge of the 2D array. (c)
Phase-locked nested solitons propagating along the edge of the array. Note
that there is a single soliton in each ring resonator that is a part of the
nested soliton. (d) Comb spectrum in the regime of phase-locked Turing rolls
showing oscillation of a single edge mode in each FSR. (e) Comb spectrum in
the regime of nested solitons showing the oscillation of multiple edge modes
in each FSR.

By tuning the pump frequency and pump power, Mittal et al. observed two very
distinctive regimes, namely that of phase-locked Turing rolls and Nested
Solitons. In the regime of Turing rolls, all resonators that lie on the edge
of the lattice show the presence of multiple equidistant peaks in the rings
and only a single edge mode resonance is oscillating. Remarkably, the phase of
the Turing rolls in all the edge rings was locked. At higher pump powers, a
regime of Nested solitons was observed. In this regime, there was a single
pulse in the ring on the edge of the lattice, and also a single super-pulse in
the super-ring resonator formed by the edge states. Once again, the pulse
positions in the rings were phase-locked. This nested-soliton pulse would
circulate around the edge of the lattice, and around defects, without losing
its phase-locking. The comb spectrum in this regime showed oscillation of
multiple edge modes resonances in each FSR (each copy of QAHE), and the
underlying dispersion was canceled by that introduced by the Kerr
nonlinearity. It is worth noting that merely exciting the edge state
resonances of the lattice did not lead to the formation of nested solitons, it
also required tuning the pump frequency around the edge state resonance and
the pump power. As with single-ring resonators, the phase diagram (pump
frequency vs pump power) of the topological frequency comb was largely
dominated by a chaotic regime. Only in very narrow regimes of pump frequency
and power, were these phase-locked patterns observed. It is expected that
similar physics could be explored in similar coupled-resonator systems [116]
or other platforms, for example, topological circuits where it is also
possible to introduce nonlinearities. Nevertheless, an experimental
realization of the topological frequency comb is yet to be realized in any
platform.

##  IV Quantum Topological Photonics

Due to their built-in robustness against decoherence, photonic systems are
poised to play a central role in the development of quantum technologies. In
addition to being the natural choice for quantum communications, photonic
systems also offer a versatile platform for quantum simulations, for example,
of random walks, molecular quantum dynamics, quantum-enhanced sensing, and
full-scale quantum computation using measurement-based computing [117, 118,
119, 120, 121]. This is facilitated by the many photonic degrees of freedom,
for example, polarization, orbital angular momentum, temporal and spectral
modes, etc., onto which quantum states can be encoded, manipulated, and
measured. However, the key challenge in harnessing the full potential of
quantum photonic systems and further diversifying their functionalities is to
achieve a scalable route for quantum engineering of the various photonic
degrees of freedom via large-scale integration of photonic elements on a
single chip. This large-scale photonic integration is mainly hindered by the
unavoidable fabrication disorder that leads to random variations in the
photonic mode structure and manifests as device-to-device variations in
behavior. Following the various demonstrations of topological robustness for
classical photonic systems, it is then natural to investigate if topological
protection could also be used to design robust quantum photonic devices. A
number of recent theoretical and experimental works have explored such quantum
topological photonic systems in various contexts [28, 122, 34, 35, 25, 123,
124, 125, 125, 48, 126]. One broad category of these systems has explored the
extent of topological robustness in the propagation of photons carrying
quantum information, for example, encoded in temporal or spatial entanglement
[28, 122, 123, 125, 125]. Propagation of entangled photons through a
disordered system can, in general, lead to the loss of quantum information. In
contrast, using numerical simulations, Mittal et al. [28] and Rechtsman et al.
[122] proposed that the topological edge states can reliably carry entangled
photons. We note that the quantum information in these systems is generated
outside of the topological device.

The second category of quantum topological photonic systems has explored the
generation of quantum states of light [34, 35]. These systems use the second
or third-order optical nonlinearities of the medium and implement spontaneous
parametric processes that naturally lead to the creation of photon pairs
correlated in energy-time, and space-momentum. The presence of topological
edge states is then exploited as a novel and robust route to engineer the
spectral or spatial correlations in generated photon pairs. The third category
seeks to interface solid-state quantum emitters, for example, quantum dots
with topological photonic systems [[25, 124]. The inherent directionality and
the robustness of topological edge states in these systems lead to chiral
light-matter interactions. In the following, we review some of the
experimental demonstrations of topological robustness in quantum photonic
systems.

###  IV.1 Topological sources of quantum light

Sources of quantum light, in particular, correlated and entangled photon
pairs, have relied on spontaneous processes such as spontaneous parametric
down-conversion (SPDC) and spontaneous four-wave mixing (SFWM), in optical
media with $\chi^{(2)}$ or $\chi^{(3)}$ nonlinearity, respectively [127, 88].
In these processes, one (SPDC) or two (SFWM) photons from a strong, classical
pump beam annihilate and create two daughter photons, called signal and idler
photons. The parametric nature of these processes indicates that no energy or
momentum is transferred between the photons and the nonlinear medium, and
therefore, the pump and the generated photons conserve both energy and
momentum. For example, in SFWM, $2\omega_{p}=\omega_{s}+\omega_{i}$, and
$2\vec{k}_{p}=\vec{k}_{s}+\vec{k}_{i}$, where $\omega$ and $\vec{k}$ are the
frequencies and the momenta of the pump (p), signal (s) or idler (i) photons.
The underlying dispersion relation $\omega$ of the photonic mode structure
couples these two relations together, and eventually, leads to non-classical
energy-time and position-momentum correlations in the generated photon pairs
such that they are described by a two-photon wavefunction.

Implementing SFWM and SPDC on a photonic chip offers a scalable and versatile
platform to generate photon pairs with engineered spectral or spatial
correlations [128, 129, 130, 131, 132]. In particular, on-chip quantum light
sources, using SPDC or SFWM, have now been realized on a variety of material
platforms, such as silicon, silicon-nitride, lithium-niobate, aluminum-
nitride, etc. [133, 132, 134]. A common feature of these sources is the use of
a ring resonator that can resonantly enhance the strength of nonlinear
interactions and lead to higher generation rates [129, 130, 131].

With the aim of further enhancing the generation of photon pairs, and
simultaneously, engineering their spectral and temporal correlations in a
topologically robust way, Mittal et al. [34] used the system of coupled
silicon ring resonators to implement SFWM. As we discussed earlier, this
system realizes a synthetic magnetic field, and thereby, simulates the integer
quantum Hall effect for photons [19, 22, 28]. They chose the synthetic
magnetic field flux $\phi=\pi/2$ such that the transmission spectrum of the
device exhibits two edge bands, with edge states circulating around the
lattice in clockwise and counter-clockwise directions. Using transmission and
delay measurements made over a number of devices, the edge states in this
system have been shown to be quantitatively robust against common fabrication
disorders, for example, a mismatch in the ring resonance frequencies [135].

While the topological robustness of transmission through photonic edge states
has been extensively explored for applications in integrated photonic devices,
Mittal et al. exploited the linear dispersion of the edge states to engineer
the spectral correlations of generated photons. In particular, the spectral
correlations between generated photon pairs, as well as their generation rate
dictated mainly by the phase matching between the pump, the signal, and the
idler photons, that is,
$2\vec{k}_{p}\left(\omega_{p}\right)=\vec{k}_{s}\left(\omega_{s}\right)+\vec{k}_{i}\left(\omega_{i}\right)$.
To understand these spectral correlations, they measured the generation rate
of photons as a function of the input pump frequency and the spectra of
generated signal and idler photons Fig. 12c-f. They showed that the maximum
number of photons is generated when the pump frequency is in the edge band of
the device (highlighted by the white box). Furthermore, this also limits the
spectra of generated photon pairs to the same edge band. This spectrally
confined and enhanced generation of photon pairs is because of the linear
dispersion of the edge states, which naturally satisfies the phase-matching
condition when all the photon fields are in the edge band of the device.
Furthermore, their confinement at the edge of the lattice ensures that they
also have an excellent spatial overlap and enhanced the generation of photon
pairs. In contrast to the edge modes, the bulk modes show a much weaker
generation of photon pairs with no spectral confinement.

To test the robustness of spectral correlations between photons generated by
their topological source, Mittal et al., made measurements over a number of
devices, and also compared their results against a similar source implemented
using a topologically-trivial one-dimensional array of ring resonators (Fig.
12a) [136, 137]. Although these devices were fabricated at state-of-the-art
commercial silicon foundries, they had a significant disorder in the ring
resonance frequencies, hopping strengths, as well as hopping phases, [135].
Nevertheless, as expected, they observed that for topological sources, the
maximum number of photons was always generated when the pump, the signal, and
the idler fields constituted the edge modes of the device, and therefore, the
spectral correlations in the edge band were very similar across different
devices. In contrast, the topologically-trivial 1D sources showed very
significant variations in their correlations, with a much lower similarity
across devices. Using second-order cross- and self-correlation measurements
between generated photons, they confirmed that their source was operating in
the quantum regime. More recently, this scheme has also been extended to
generate path-entangled photon pairs [48, 126]. These results bode well for
the use of topological sources to achieve quantum interference between photons
generated by independent sources.

In another similar experiment, Blanco et al. [35] investigated the generation
of correlated photon pairs in a 1D lattice of coupled waveguides. The coupling
strength between the waveguides is modulated by alternating the gap (small or
large) between the waveguides, such that the lattice simulates the SSH model
[6], and the edge states appear at the physical boundary between the two
topological phases. Furthermore, the edge state wavefunction vanishes at
lattice sites immediately neighboring the edge site and alternating waveguides
thereafter.

The waveguides were fabricated using silicon which allowed for the generation
of correlated photon pairs via SFWM. In particular, the edge state of the
lattice was pumped using a pulsed laser that generated signal and idler photon
pairs as it propagated through the lattice. At the output of the lattice,
Blanco et al. measured the spatial correlations in the generated signal and
idler photons. They observed that similar to the classical (single-particle)
edge state wavefunction in the SSH model, the spatial correlations in the two-
photon wavefunction also showed zero amplitude at alternating waveguides.
Furthermore, they showed that these zeros of the wavefunction were robust
against disorder in the coupling strength between the waveguides.

From a more fundamental perspective, the SFWM process coherently adds or
removes photon pairs from the topological lattice. The number of particles in
the lattice is, therefore, not conserved and it operates in the non-Hermitian
regime, with no analogs in fermionic systems. Furthermore, as theoretically
shown by Peano et al. [138], an increase in the strength of the SFWM
interaction would naturally lead to the generation of squeezed light such that
only the topological edge modes of the lattice are effectively squeezed.

Figure 12: (a), 2D ring resonator array used to realize a topological source
of correlated photon pairs generated via SFWM. Because of the synthetic
magnetic field, photons acquire a non-zero, direction-dependent phase $\phi$
when they circulate around a closed path of four site rings (cyan) and four
links (yellow) rings. The clockwise (CW) and the counter-clockwise (CCW) edge
states are highlighted in color. (b), Measured transmission spectrum showing
edge and bulk bands. (c-f) Measured spectral correlations, that is, the number
of photons generated as a function of the pump and signal frequencies. The
dashed lines indicate the edge band region. The spectral correlations for 2D
topological devices are very similar in the edge band region. (g) SEM image of
a topologically trivial 1D array of coupled ring resonators. (h-k) Measured
spectral correlations for 1D devices. The correlations differ significantly
across devices because of disorder.

###  IV.2 Topological robustness for propagating quantum states of light

While photons do not interact with one another, they do exhibit quantum
interference which forms the basis of many algorithms used in quantum
communications, quantum simulations and quantum computation using photons
[118, 119, 120]. This is best exemplified by the Hong-Ou-Mandel interference
where two indistinguishable photons arriving in two different input ports of a
beam-splitter tend to bunch at either of the output ports [139]. This
interference phenomenon has led to the observation of quantum walks of
correlated photons and the realization of boson sampling in spatial networks
of integrated beam-splitters [121, 140].

However, scaling this multi-photon quantum interference and boson sampling
schemes to a larger number of photons requires a significant reduction in the
variations of the splitting ratio of the on-chip beam-splitters. It is
therefore natural to investigate if topological protection can be used to
design robust beam splitters.

Along these lines, Tambasco et al. [123] realized a beam-splitter using the
topological edge modes. Their system consists of 1D arrays of coupled
waveguides, such that the coupling strength between them is modulated both
along the lattice and along the length of the waveguides. This system
simulates the off-diagonal Harper model and hosts a pair of edge states at its
boundaries, similar to the SSH model. However, modulation of the coupling
strength along the length of the waveguides allows them to adiabatically
delocalize the edge states from the boundary to the bulk of the lattice such
that the photons traveling in the edge states can now interfere. The edge
modes are then again localized at the boundaries of the lattice. This setup
then realizes an integrated beam-splitter for photons but uses edge modes for
guiding photons.

At the input of this topological beam-splitter, Tambasco et al. injected two
indistinguishable photons generated via off-chip SPDC. By tuning the relative
delay between the input photons and using coincidence measurements at the
output, they observed a high visibility HOM interference dip, which confirmed
the intended operation of their topological beam-splitter. However, the
robustness of the beam-splitting ratio of this topological beam-splitter
against fabrication disorder is yet to be studied.

In another experiment using similar 1D arrays of waveguides that simulates the
off-diagonal Harper model, Wang et al. [125] investigated the robustness of
intensity correlations between indistinguishable photon pairs as they
propagate through the lattice. Similar to the experiment of Tambasco et al.,
the correlated photon pairs were generated off-chip using SPDC. Wang et al.
showed that when both the photons are injected in the edge mode of the array,
they maintain the intensity correlations. In contrast, when the photon pairs
propagate through bulk modes, there is a suppression in their correlations.

In a similar context, Mittal et al. [28] numerically studied the propagation
of time-bin entangled photons through their 2D topological system of coupled
ring resonators. Similarly, Rechtsman et al. [122] investigated the
propagation of spatially entangled photon pairs through their Floquet
topological system of coupled helical waveguides. These investigations are
similar in essence to quantum walks of photon pairs through networks of beam-
splitters or coupled waveguides. They observed that propagation through edge
states preserves the temporal and spatial correlation between photon pairs,
respectively, even in the presence of disorder.

###  IV.3 Topological photonic systems coupled to quantum emitters

Coupling light to matter degrees of freedom, such as, quantum dots, can
mediate the interaction between photons and lead to novel quantum states of
light [141]. In turn, the photonic mode structure can significantly alter the
properties of solid-state systems. For example, photonic cavities can be used
to manipulate the emission spectra and the excitation lifetimes in quantum
dots [142, 141]. Coupling quantum dots and solid-state emitters to topological
photonic edge states is, therefore, an exciting avenue to investigate chiral
light-matter interactions that could lead to many-body states [143].

Barik et al. [25] realized such a quantum optics interface between quantum
dots and photonic edge states. Their topological photonic system was designed
using a 2D photonic crystal with triangular holes in a GaAs membrane Fig. 13b.
When the holes are arranged in a honeycomb lattice, the band structure of the
photonic crystal exhibits a Dirac point, very similar to that of graphene [23,
81]. Nevertheless, a deformation of the unit cell of the lattice leads to the
appearance of a bandgap. More specifically, expanding the unit cell of the
lattice, that is, increasing the distance between the holes in the unit cell
while keeping the boundaries of the unit cell constant, resulted in a bandgap
that was topological in nature. In contrast, shrinking the unit cell also
opened a bandgap, but a trivial one. Therefore, an interface between the
shrunken and the expanded domains hosts topological edge states. This model
realizes the quantum spin Hall effect where the in-plane circular polarization
of the electric field constitutes two pseudo-spins of the system. The edge
states corresponding to the two pseudo spins propagate along the interface in
opposite directions.

The GaAs membrane used by [25] was embedded with InAs quantum dots, with their
emission spectra well aligned to the bandgap of the photonic crystal structure
[25]. To couple the quantum dots to the in-plane circularly polarized photonic
edge states, they used an out-of-plane magnetic field that induced a Zeeman
splitting in the excited state energies of the quantum dots (Fig. 13d). In
this configuration, the two Zeeman-split energy levels were selectively
coupled to the two circularly polarized photonic edge states propagating along
opposite directions. In this work, [25], Barik et al. excited a single quantum
dot in the middle (M) of the interface (see Fig. 13b) and measured the spectra
of photons collected from either side (L or R) of the interface, as a function
of the magnetic field strength. Because of the pseudo-spin selective coupling
of the excited states to counter-propagating edge states, they observed that
the lower wavelength (higher energy) photons were primarily guided by the edge
states to the right end of the interface whereas the higher wavelength (lower
energy) photons were guided towards the left end of the interface. To
demonstrate the robustness of this topological quantum-optics interface, they
showed that the chiral propagation of photons is robust against any disorders
that do not flip the two pseudo-spins, for example, bends in the interface.
Furthermore, they used second-order correlation measurements to observe the
anti-bunching of photons which ensured that they were indeed single photons.

In another experiment, Ota et al. [124] explored the coupling of quantum dots
to nanophotonic cavities realized using corner states of light in higher-order
topological systems. Similar to Barik et al., their system comprised a GaAs
photonic crystal membrane, with embedded InAs quantum dots. The photonic
crystal was designed by etching two sets of square holes, with different
lengths of their sides, in the GaAs membrane. This difference in the hole
dimensions opened up a bandgap. More importantly, a $90^{\circ}$ interface
between two photonic crystal regions with swapped hole dimensions led to the
emergence of corner states in the bandgap, physically located at the bend in
the interface. This system is analogous to a two-dimensional SSH model with
alternating coupling strengths in both dimensions. To probe the existence of
corner states in their photonic crystal, Ota et al. used the photoluminescence
from the ensemble of quantum dots as a broadband light source (pumped by a
laser). As an indication of the corner states, they observed a sharp peak in
the photoluminescence spectrum, in the region expected to host corner states.
They confirmed their observation of the corner states by showing that this
peak in the photoluminescence spectrum originated from a narrow spatial region
at the $90^{\circ}$ bend in the interface between topological and trivial
domains of the photonic crystal. We note that unlike Barik et al. who coupled
a single quantum dot to the topological edge states, Ota et al. used an
ensemble of quantum dots.

Figure 13: (a-c) SEM image, and the band structure of the shrunken and
expanded honeycomb lattice used to realize a topological interface between
quantum dots and helical edge states. (c) An applied magnetic field introduces
a Zeeman splitting between the pseudo-spin (right and left-circular) polarized
photons. (d-f) The pseudo-spin polarized edge states propagate along the
interface in opposite directions, where they are collected using grating
couplers. (g,h) The measured second-order correlation function
$g_{2}\left(\tau\right)$ shows the operation of quantum dots as single photon
sources. [144].

##  V Remaining challenges and future directions

Here, we highlight some of the challenges and potential future directions in
topological photonics in linear, nonlinear, and quantum regimes, as well as
coupled electron-photon systems, ranging from fundamental to application
perspectives. For the latter, it is essential to go beyond proof-of-principle
experiments, and a side-by-side comparison of the efficiency and yield of a
topological design compared to trivial counterparts (with the same fabrication
process and material) should be established. Such side-by-side comparisons and
yield estimates remain scarce in the literature [135, 145].

###  V.1 Linear topological photonics

  * •

In spin-Hall PhC waveguides [25] and ring resonators [146] propagation length
and Q factors are low since the edge states are above the light-cone, and
therefore, radiative. Moreover, practical edge state bandwidths in spin-Hall
PhC waveguides in strong perturbation (shrinking and expanding) regimes [25]
are limited. One intriguing improvement can be the realization of broadband
spin-Hall TPC waveguides and high Q photonic cavities with below-light cone
edge dispersion. Another remaining challenge is the efficiency of mode
conversion at the topological-conventional waveguide interfaces [147, 26,
148]. Further optimization of such mode conversion is essential for the
efficient integration of these optical components for scalable photonic
circuitry. One recent approach to address this challenge can be found here
[149]. More promising approaches may be using the recently demonstrated
topological funneling of light and edge mode tapering, although it should be
noted that topological funneling of light in these studies is based on non-
Hermitian physics [50, 83]. Moreover, so far many valley-Hall PhC waveguides
[26, 24, 147] have been proposed and studied. In addition to a recent
investigation of the nature and degree of backscattering against sharp bends
and fabrication imperfections [145], further studies are required to confirm
if there is any protection against real-world defects and classify and
quantify the strength of such protection rigorously. A recent study can be
found here [150].

  * •

While helical topological waveguides have been experimentally realized [122],
their photonic applications have not been explored yet. This is in contrast to
coupled rings used for solitons and lasers, and TPCs used for routing and QDs.
It would be desirable to explore whether similar or other exclusive
applications such as chip-integrated photonic circuits are realizable using
the topological helical waveguides.

  * •

One exciting direction can be the design and realization of reconfigurable
topological devices with phase-changing materials. Such a reconfigurable
platform can be inspired by recently demonstrated reconfigurable non-Hermitian
topological photonic routing [151]. In particular, it is intriguing to be able
to imprint a wide variety of optical components such as waveguides, ring
resonators, and beam splitters all within the same device with a compact
footprint. A recent proposal has explored the possibility of such systems in
spin-Hall TPCs [152].

  * •

Another potential direction in either topological ring resonator arrays or
TPCs can be the realization of topological bandpass and notch filters, enabled
by the robustness propagation of the edge states in larger device sizes. In
particular, the realization of robust topological delay lines has been
proposed in such devices [19]. One can investigate what other devices are
possible to realize that can benefit from topological protection. For example,
the realization of topological photonic taper [83] and application of
topological photonic beaming [52] has not been reported yet. Moreover, the
scalability and application of topological antennas have not been investigated
fully yet [39, 40].

  * •

Spin-flip in topological photonic resonators and photonic crystal waveguides
is an undesirable feature. Using inverse design or more generally machine
learning techniques to address these issues and design practical topological
photonic systems can be an intriguing avenue to improve device functionality
and scalability [153]. In this approach, topology can be hard coded in the
optimization process. For example, a non-zero Berry phase can be encoded as a
loss function.

  * •

Despite decades of theoretical and experimental work on the electronic integer
quantum Hall effect, the plateau transition remains an active area of debate
[154, 155]. It is intriguing to explore whether photonic systems can shed
light on the nature of extended states and transition between plateaus [156].
This might require strong interaction between photons.

  * •

In TPCs, valuable recent work [145] has exhaustively investigated the nature
and the degree of scattering in topological photonic waveguides. More
experimental comparison is direly needed to further deepen the fundamental and
technical limits of topological protection for device engineering.

  * •

With recent remarkable advances in machine learning, it is interesting to
investigate its implication in physics [157], and more specifically in
topological photonic systems. This can be in diagnosing and classifying
topological states and even finding new band structures with topological
phases [158]. In addition to designing topological photonics devices,
exploring the connection between topological data analysis (TDA) and machine
learning and their potential implications in photonic systems may also be
another emerging avenue in this field [159].

  * •

Another novel direction can be the realization of reconfigurable topological
photonic devices using phase-changing materials. In particular, the
demonstration of multi-functional operation within the same photonic device
footprint can be intriguing. A recent example of such a study can be found
here [160].

###  V.2 Nonlinear topological photonics

  * •

Recent theoretical studies suggest a rich set of phenomena to occur in
topological lasers [161, 162]. Experimental exploration of such ideas can
significantly expand the use of topological edge states in real-world
applications in novel lasers. Experimentally, an unambiguous demonstration of
topological robustness in topological lasers and a comparison of their
efficiency compared to trivial one-dimensional counterparts in a side-by-side
comparison remains an active area of research. Another direction includes
Dirac-point lasers in 2D geometries [163, 32, 33]. In particular, it is useful
to optimize the stability and laser emission in broader chip areas without
multi-mode operations in these devices. Moreover, Weyl-points in 3D photonic
crystals are novel candidates for expanding topological lasers to more than 2D
configurations [164, 165]. Polaritons, which are hybrid photon-exciton
particles, and either photon, excitons, or their coupling form can also be
engineered to have topological properties. In these topological exciton-
Polaritons, which were demonstrated recently ([36]), the band gap is very
small, which makes their broadband application and spectrally-resolved
demonstration challenging. It would also be intriguing to explore concepts
such as spin-selective strong light-matter interaction in topological exciton-
polariton systems [166].

  * •

Recently Topological frequency combs and nested temporal solitons have been
theoretically proposed [48]. Nevertheless, the experimental realization of
topological optical frequency combs using coupled ring resonators is expected
to be challenging. In particular, the currently estimated pump power
requirement for topological frequency combs is high, more than 10W. This is
mainly set by the disorder in ring resonance frequencies which, even for
state-of-the-art photonic integration, is of the order of a few tens of GHz.
This sets a lower limit on the coupling strength $J$ between the rings and
limits the loaded quality factor of the rings. Furthermore, lowering pump
power requirements will also reduce the deleterious thermal effects which are
problematic even for single-ring combs. Another area of concern for ring-
resonator-based topological combs will be the mode-mixing between transverse
modes of the ring waveguides. To lower losses, single-ring resonator frequency
combs often employ waveguides that support multiple transverse modes. However,
for coupled ring resonators, mixing between different transverse modes can be
a significant challenge. For proof-of-principle demonstration, many of these
issues could be mitigated by designing rings with lower coupling strengths
(higher loaded quality factors and lower topological edge bandwidth) and
coarse-tuning the ring resonator frequencies (say using heaters) such that the
disorder falls within the reduced topological edge bandwidth. Nevertheless, it
will be interesting to investigate other topological photonic designs that
could lead to lower pump power requirements and make them more appealing for
practical applications. From a theoretical perspective, the topological
frequency combs could host a much more diverse range of nonlinear solutions,
such as breathing solitons, dark solitons, platicons, etc.; these solutions
have not yet been explored. The development of an analytical approach to
describe these multi-resonator systems might be very helpful, but again, it is
expected to be challenging.

  * •

While the nature of the linear and quantum many-body topological states has
been heavily studied and understood, the nature of topological invariants in
the nonlinear topological photonic systems remains elusive. Recent works have
shown that in the nonlinear regime, the topological character of the linear
regime is inherited in some form [167, 168, 99]. It is interesting to see
whether there are genuinely nonlinear topological states. Moreover, is it
possible to have a bulk-edge correspondence for the nonlinear topological
states?

  * •

Inspired by the above examples, are there other photonic phenomena without
electronic counterparts? An example can be using topological confinement for
more efficient lasers [169]. In particular, it is useful to answer whether
there are other implications for using the same confinement, such as optical
sensing. Moreover, further investigation of the topological amplifier which
was recently reported [170] is also another potential future direction.

###  V.3 Quantum topological photonics

  * •

In pair generation in a topological lattice, the key advantage is the robust
phase-matching compared to conventional counterparts [34]. It would be
intriguing to find more applications of this robust phase-matching advantage
for the realization of non-linear quantum optical effects. Moreover, the
device footprint in a topological quantum light generation device is
considerably large since it is comprised of rings with several hundreds of
microns.

  * •

Position-dependence of chiral coupling in QD-coupled topological waveguides is
a significant limitation in these quantum optics interfaces. Moreover,
chirality and high Beta-factor areas are mostly present in the holes rather
than the material [171, 147, 172], which is detrimental for coupling to solid-
state quantum emitters. Also, low Purcell factor (currently only up to less
than 5 is reported [144]) is another limitation in QD-coupled TPCs. In
topological waveguides, the emitter’s coupling efficiency, as well as emission
enhancement, may be improved with either slow-light effect [173] or smaller
mode volume (for example harnessing topological mode tapering [83]). Moreover,
in whispering-galley mode ring resonators, [144, 147], possibilities for
achieving higher Q factors (for example using surface passivation for
suppression of the out-of-plane scattering) can be investigated. Similar to
[174], coupling multiple quantum dots to edge states can be a very interesting
direction to explore, in particular of interest is the collective dynamics
between distance emitters [175], however, this is currently challenging due to
the beta factor being position-dependent in current TPC waveguides. Chiral
coupling is also position dependent in these systems so the directional sub
and superradiance effects from embedded quantum emitters in topological
waveguides are challenging. One avenue to explore can be using inverse design
to address this issue, and in turn, investigate if such a platform can be
utilized to address the spatial inhomogeneity challenge in chip-integrated
solid-state quantum emitters. Finally, the realization of on-chip quantum
interference of single photons in add-drop filter photonic crystal
configurations, in which chiral coupling between emitters and several modes of
the resonator was shown recently [148], would be another intriguing
possibility to explore. In particular, combining recent demonstrations of
broadband slow-light enhancement in a topological photonic ring resonator with
an integrated add-drop filter configuration may be studied [176, 148].

  * •

Another interesting development is the possibility of coupling emitters to
topological photonic structures. This includes coupling emitters to 1D
structures[177, 178, 179], and 2D systems[180, 181] with novel forms light-
matter hybrids.

  * •

An extremely exciting avenue is the realization of Laughlin states with a
higher photon number than two [47], and more broadly, other topologically-
ordered states and potentially braiding them [182, 183]. Specifically, the
excitations above the ground states of such models can have _anyonic
statistics_ this is neither fermionic nor bosonic [61]. The non-Abelian anyons
have been proposed as a robust scheme for topological quantum computation
[184]. Note that one can simulate such exotic statistics even with non-
interacting photons [185, 186, 187]. However, the many-body features such as
the topological robustness for quantum computation are absent in such non-
interacting systems.

###  V.4 Strong photon-photon interaction and coupled electron-photon systems

If the interaction between cavity photons is so strong that a single photon
can prevent the transmission of another one (photon blockade), one can expect
even more exotic topological states, such as the photonic counterpart of
fractional quantum Hall states. In particular, there is a whole class of
topological states, known as topologically ordered states, which are distinct
from the states covered in the introduction. In these states, entanglement and
strong interaction play a central role, for which a brief review can be found
here [61]. In fact, once photons strongly interact with each other, they can
be considered as spin-1/2 particles and therefore many of the topologically
ordered models will be directly applicable. For the case of photonic
fractional quantum Hall states, the essential ingredients are gauge fields
(discussed earlier) and strong photon-photon interaction. The important
parameter in such systems is the magnetic filling factor: the number of
particles divided by the number of magnetic flux (introduced in Sec.II.2.1).
The simplest case for bosons is $\nu=1/2$, where the ground state is Laughlin
state, and it is both unique and gapped. For a pedagogical review of electric
and boson fractional quantum Hall refer to [64, 188], respectively.

Generically in such systems, the ground state on a torus (periodic boundary
condition) has a finite degeneracy. Therefore, a Chern number can not be
associated with a single state, and indeed it is shared among the degenerate
states, and therefore, it can be fractional.

The strong interaction could be achieved in various ways such as Rydberg atoms
or superconducting qubits [189]. Remarkably, in the Rydberg systems,
fractional quantum Hall states (Laughlin states) of a few photons have been
observed [38]. Scaling such a system to a larger number of particles remains a
challenge. In fact, one may ask how small of a system one can call a
topologically ordered matter. In other words, given a wavefunction on finite
system size, is it possible to identify whether a system is topologically
ordered or not? Can one extract the Chern number, without prior knowledge of
the Hamiltonian and application of any field? The answer to these questions
seems to be positive based on recent analytical and numerical works [190, 191,
192], however, there is no experimental demonstration to this date.

So far we considered purely photonic models, whereby by engineering a
Hamiltonian with topological properties, directly for photons, one can observe
various topological phenomena. An interesting direction is to consider light-
matter coupling where either the photonic or matter part has some topological
properties, and therefore, the coupled system inherits those topological
features. In other words, the matter part is not integrated out and the light-
matter interplay is the essence.

We note that the above categorization might sometimes seem artificial since
the underlying microscopic theory for all the cases in this Perspective is
quantum electrodynamic. Specifically, what is purely photonic or matter is a
matter of length scale over which we integrate out microscopic degrees of
freedom to write an effective Hamiltonian for the system. For example, one can
call quantum Hall states coupled to optical cavities also a topopolariton
since the optical excitations in the cavity-quantum Hall system can be
considered excitons that are coupled to the cavities. Below, we highlight
several directions.

  * •

Light-matter interaction in electronic quantum Hall systems: As we mentioned
at the beginning of this Perspective, electronic quantum Hall systems are the
first physical systems to manifest topological properties in transport
measurements. However, from early on optical measurements were also performed
on such systems [193], for example, to probe electronic incompressibility.
More recently, there have been interacting experiments to couple such states
to cavities, either in the THz, [194, 195] or optical domain [196], to probe
and manipulate intra- and interband states, respectively.

Regardless of being in the cavity or free space, it is intriguing to ask
whether light-matter coupling could be exploited to create and manipulate
electronic topological states, and eventually perform braiding.

Moreover, it has been theoretically argued that the light-matter interaction
is dramatically modified in quantum Hall states, since the chirality and
topological robustness of the electronic states may lead to the spatially
large wave functions, which could be comparable to the corresponding optical
transitions [197]. In particular, the dipole approximation can be violated and
the system could be sensitive to the gradient of the electric field and the
phase of an optical vortex beam. The latter is only possible if the electron
is phase coherent around the optical vortex and experiences the phase winding
of the optical beam. It has been proposed that such light-matter coupling
could lead to radial current in quantum Hall systems in the absence of any
electric field bias [198, 199]. Such optical vortex beams could be used to
optically create topological excitation in fractional quantum Hall systems
[200]. A recent experiment demonstrated that photocurrent could be sensitive
to the beam phase winding [201]. In the context of this section, these systems
are particularly interesting because both the electronic and photonic states
have topological properties and such topological interplay is an interesting
direction of future research.

  * •

Topological photonic crystals: In the linear section, we discussed photonic
crystals that can have topological properties, manifested in the presence of
helical waveguides and their coupling to point-like emitters, like QDs [172].
One can also couple extended excitons states, such as the ones in 2D materials
with optical transition, to such helical states. Since layered 2D materials
are essentially one or a few atomic layers thick, they can strongly couple to
the confined electromagnetic modes of the topological photonic crystals.
Particularly interesting are recent studies on the hybridization of
topological photonics states to condensed matter systems, where 2D transition
metal dichalcogenides were shown to be strongly coupled to topological
photonic crystal metasurfaces, forming a polaritonic metasurface [37].

Similar to the case of quantum QDs [172], the chiral light-matter coupling is
sensitive to the location of the emitter with respect to the transverse
position of the waveguide. In fact, one would naively expect that chiral
light-matter coupling for 2D material excitons would be absent in such
systems, because excitons, with the same polarization, are present all along
the transverse direction of the waveguide, half coupled to the left-
propagating modes and half to the right-propagating modes. However,
experimental observation does not agree with this argument, and this subject
remains an active area of research.

###### Acknowledgements.

We wish to gratefully acknowledge Erin Knutson, Daniel Leykam, Mikael
Rechtsman, and Alberto Amo for their insightful comments and discussions
during the preparation of this manuscript. This work was supported by AFOSR
FA9550-20-1-0223, FA9550-19-1-0399, and ONR N00014-20-1-2325, NSF IMOD
DMR-2019444, ARL W911NF1920181, Minta Martin and Simons Foundations.

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