Topological quantum photonics

Abstract

Topological quantum photonics explores the interaction of the topology of the dispersion relation of photonic materials with the quantum properties of light. The main focus of this field is to create robust photonic quantum information systems by leveraging topological protection to produce and manipulate quantum states of light that are resilient to fabrication imperfections and other defects. In this perspective, we provide a theoretical background on topological protection of photonic quantum information and highlight the key state-of-the-art experimental demonstrations in the field, categorizing them based on the quantum features they address. An analysis of the key challenges and limitations concerning topological protection of quantum states is presented. Importantly, this paper takes a thorough perspective look into what future research in this area may bring.

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Topological quantum photonics
Cite as: APL Photon. 10, 010903 (2025); doi: 10.1063/5.0239265
Submitted: 18 September 2024 •Accepted: 13 December 2024 •
Published Online: 9 January 2025
Amin Hashemi,1M. Javad Zakeri,1Pawel S. Jung,2,3and Andrea Blanco-Redondo1,a)
AFFILIATIONS
1CREOL, The College of Optics and Photonics, University of Central Florida, Orlando, Florida 32816, USA
2Department of Physics, University of Miami, 1320 Campo Sano Drive, Coral Gables, Florida 33146, USA
3Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland
a)Author to whom correspondence should be addressed: andrea.blancoredondo@ucf.edu
ABSTRACT
Topological quantum photonics explores the interaction of the topology of the dispersion relation of photonic materials with the quantum
properties of light. The main focus of this field is to create robust photonic quantum information systems by leveraging topological protection
to produce and manipulate quantum states of light that are resilient to fabrication imperfections and other defects. In this perspective, we pro-
vide a theoretical background on topological protection of photonic quantum information and highlight the key state-of-the-art experimental
demonstrations in the field, categorizing them based on the quantum features they address. An analysis of the key challenges and limitations
concerning topological protection of quantum states is presented. Importantly, this paper takes a thorough perspective look into what future
research in this area may bring.
©2025 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution-NonCommercial 4.0
International (CC BY-NC) license (https://creativecommons.org/licenses/by-nc/4.0/). https://doi.org/10.1063/5.0239265
I. INTRODUCTION
Quantum technologies aim to exploit the fundamental proper-
ties of quantum mechanics to achieve quantum advantage, famously,
in computing,1,2 but also in other fields such as communications,3
sensing,4and metrology.5Light-based quantum technologies pos-
sess certain benefits with respect to matter-based approaches, given
that photons do not decohere and can be used to transport quantum
information over long distances. In particular, integrated photon-
ics is a powerful platform for generating and processing quantum
states of light with high fidelity and stability.6Nonetheless, quantum
photonic states are subject to degradation due to inevitable nanofab-
rication imperfections, and safeguarding the information stored in
these states becomes crucial for the scalability of quantum integrated
photonic platforms. In the last few years, topology has been raised
as an avenue to enhance the robustness of these platforms against
certain kinds of disorder.7–9
Initial studies of topological protection of quantum states of
light were done with single photons in free space.10,11 However,
quantum information systems rely heavily on multiphoton states,
and understanding the interplay of topology, disorder, and photon
correlations has become an important research avenue. In 2016, we
witnessed the first preliminary experimental results12 and the first
theoretical investigations13,14 of topologically protected correlated
photon states. This marked the beginning of a series of research
efforts in topological quantum photonics, including the demon-
stration of topologically protected chiral emission15 and quantum
interference16,17 of single photons, topological protection of the
spectral18 and spatial19 features of biphoton correlations, and topo-
logical two-photon20–22 and four-photon entangled states.23 In par-
allel, theoretical studies have explored the limits of topological
protection of quantum states24 against degree of disorder and the
trade-off between degree of entanglement and topological protec-
tion.25 A timeline with the key developments of the field since 2012
is depicted in Fig. 1.
This perspective article begins with a description of the theoret-
ical background necessary to understand the topological protection
of quantum states of light. Following, we delve into discussing the
main findings of the experiments in this field, categorizing these
studies into three classes based on the targeted quantum effects.
A critical look into the limitations of these pivotal experimental
results is provided in Sec. IV. This paper concludes with an in-
depth perspective look into the future of the topological quantum
photonics field.
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FIG. 1. Key demonstrations in topological quantum photonics since its inception and potential future directions. Initial experiments with topologically protected bound
states of single photons in quantum walks were presented in 2012.10 In 2016, we witnessed the first measurements with correlated photons in a topological system12
and the first theoretical studies of topologically protected (TP) entanglement.13,14 2018 was a key moment for the field of topological quantum photonics when several
comprehensive experimental studies established that topology can contribute to robust interfaces of quantum emitters with waveguides,15 quantum interference,16 sources
of quantum light,18 and generation of time-energy biphoton correlations.19 Following these key demonstrations from 2019 to 2021, the field produced the first demonstration
of topologically protected path entanglement,20 tunable quantum interference of TP photon pairs,17 and symmetry-induced error filtering.26 From 2022 to date, we have
seen an increasing degree of complexity in TP entangled states,23,27 the use of topology as an additional degree of freedom (DOF) for entanglement,22 and an exciting result
in TP logic gates.28 Given the state of the field and the latest theoretical proposals and demonstrations, we predict the future of topological quantum photonics will continue
moving toward more applied experiments in quantum computing and sensing, as well as the exploration of novel quantum states involving, among others, photon–photon
interactions, the entanglement of modes with different topological orders, and even non-abelian statistics.
II. THEORY
The field of topological quantum photonics combines the study
of two fundamental properties of nature, entanglement and topol-
ogy, and has arisen as a promising avenue to produce quantum
photonic systems robust to disorder and imperfections. However,
photons hardly interact with the environment and, hence, do not
suffer decoherence, as opposed to other quantum particles. There-
fore, the question arises: Is there really a need to protect the quantum
information encoded in photonic states?
In a theory paper from 2016, Rechtsman et al. dealt with this
question.14 The authors show that, despite individual photons being
robust to decoherence, multiphoton states, the building blocks of
quantum information science, are not immune to disorder. The
key here is to understand that, while photons may be scattered
by defects, this does not necessarily lead to loss of information
(decoherence). In contrast, the information in multiphoton states is
embedded in the quantum correlations between the photons of the
state, which will be indeed degraded by uneven scattering of the pho-
tons. In particular, the authors analytically show that upon disorder-
related backscattering, an initial maximally spatially entangled state
will be destroyed due to nonzero amplitude to measure one trans-
mitted and one reflected photon. In this context, the authors show
that photonic topological insulators (PTIs) can transport quantum
information robustly through photonic networks, even in the pres-
ence of disorder. Figure 1(a) shows the honeycomb lattice of helical
waveguides used to create the 2D photonic topological insulator in
this study. This lattice has previously been shown to support edge
states that propagate without backscattering due to their topological
nature.29
Rechtsman et al. study the propagation dynamics of photons
through this system using the equation
i∂za†
n=∑
⟨m⟩
ceiA0(cos Ωz,sin Ωz)⋅rmn a†
m+una†
n≡∑
mHnm(z)a†
m, (1)
where zis the distance of propagation along the waveguide axis,
and rmn is a unit vector representing the direction of displacement
between waveguides mand n.a†
ncreates a photon on waveguide
n;cis the coupling; A0=kRΩais the gauge field strength (arising
due to the helicity); kis the wavenumber; Ris the helix radius; Ωis
the spatial frequency of the helices; ais the lattice constant; unis a
random number lying in the range [−W,W], representing disorder
(random waveguide refractive index); and HF(z)is the z-dependent
Schrödinger picture Hamiltonian.
AN00N state, often denoted as ψN00N=1
√2(N,0+0,N),
is a maximally entangled quantum state where Nphotons are in a
superposition of being entirely in one mode or entirely in another,
but not in both simultaneously. In this paper, a N00N state ψN00N
=1
2(w†
iw†
i+w†
i+lw†
i+l)0, where w†
nis an operator that creates a pho-
ton in the quantum state centered on waveguide n, is injected along
the edge and its dynamics are simulated.
The injected N00N state preserved its initial entanglement
properties after propagating along the edge of the PTI without disor-
der [Fig. 2(b)] and with disorder [Fig. 2(c)]. In contrast, the authors
showed that the entanglement would be destroyed after propagation
through a disordered 1D trivial system, Fig. 2(d). This robustness
against disorder suggests that quantum information encoded in path
entangled photon pairs can be transmitted with high fidelity, even in
practical, imperfect systems protected by topology.
In a contemporaneous paper, Mittal et al. studied the trans-
port of time-bin entangled photon pairs through a two-dimensional
topological photonic system composed of coupled ring resonators,
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FIG. 2. (a) Honeycomb lattice of heli-
cal waveguides forms a photonic Floquet
topological insulator. (b) Correlation map
evolution of the N00N state along the
edge of the PTI. (c) Same as (b) with
added disorder. (d) Correlation map in
a topologically trivial 1D array for N00N
states, shown in two scenarios: without
the disorder (top row) and with the dis-
order (bottom row). Figure adapted from
Ref. 14.
as shown in Fig. 2(a).13 While the system implements the quantum
spin Hall model, the authors selectively excite a single pseudo-spin,
simulating the integer quantum Hall effect with the tight-binding
Hamiltonian,
H=∑
x,yω0ˆ
a†
x,yˆ
ax,y−Jˆ
a†
x+1,yˆ
ax,yeiyϕ+ˆ
a†
x,yˆ
ax+1,ye−iyϕ
+ˆ
a†
x,y+1ˆ
ax,y+ˆ
a†
x,yˆ
ax,y+1,
(2)
where ω0represents the resonance frequency of the rings, Jdenotes
the rate of coupling between the adjacent lattice sites, and ϕsym-
bolizes the artificial magnetic flux permeating each plaquette. The
operators ˆ
a†
x,yand ˆ
ax,ycorrespond to the creation and annihilation of
photons at the site specified by coordinates (x,y), respectively. This
system has been shown to possess two counter-propagating, topo-
logically protected edge modes.30 Mittal et al. simulate the transport
through these edge modes of a maximally entangled Bell state of the
form
Ψ+=1
√2(e1l2+l1e2), (3)
where e1,2 and l1,2 represent the single-photon states in early and
late time bins, respectively.
The results illustrate that the input Bell state shown in Fig. 3(b)
preserves its quantum correlation when propagating through the
counterclockwise edge, as shown in the output correlation func-
tion in Fig. 3(c). This paper shows equally robust propagation
of the Bell state through the clockwise edge. To demonstrate the
non-trivial topological nature of such robustness, the authors sim-
ulate the transport of the same state through the bulk. Unlike edge
transport, the output correlation function for bulk transport varies
with the input excitation frequency within the band and can signifi-
cantly differ from the input, as shown in Fig. 3(d). Interestingly, the
two photons can bunch at the output even if they were anti-bunched
at the input.13
Finally, the authors compare the transport of entangled pho-
tons with that of a separable two-photon state with distinguishable
photons. They find that correlated quantum states of two photons
are more fragile than the separable states, underscoring the complex
nature of quantum transport in disordered systems.13
To finalize our theoretical background discussion, we will now
introduce the relevant theory for topological quantum experiments
in a one-dimensional (1D) array of silicon nanowires comprising
alternating short and long gaps that modulate the coupling strength
between adjacent nanowires, creating a Su–Schrieffer–Heeger (SSH)
lattice with N sites, as presented in Ref. 19. The Hamiltonian
operator for this system is
ˆ
H=ˆ
Hlat +ˆ
Hnl, (4)
where the linear Hermitian Hamiltonian of the SSH lattice, ˆ
Hlat, is
accompanied by the nonlinear operator, ˆ
Hnl, describing the sponta-
neous four-wave mixing (SFWM) process. The SSH lattice supports
topological edge states, which maintain zero-amplitude conditions
in every other waveguide, even under high levels of disorder. In such
a system, a propagating classical optical pump can generate corre-
lated signal and idler photon pairs via SFWM. The dynamics of these
biphotons can be described by a Schrödinger-like equation with the
effective Hamiltonian
ˆ
Heff =ˆ
Hs+ˆ
Hi+ˆ
Hnl, (5)
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FIG. 3. (a) Schematic of a 2D lattice
of coupled ring resonators implementing
the integer quantum-Hall model. Site res-
onators (black) are coupled using link
resonators (gray). The lattice is cou-
pled to input and output waveguides.
Edge state transport is confined along
the lattice boundary, whereas the bulk
states follow different paths through the
bulk of the lattice. A time-bin entan-
gled photon pair is coupled to the lat-
tice at the input, and the output tempo-
ral correlations are examined. (b) Time-
correlation Γ(t1,t2)for Ψ+input state,
with the standard deviation of the Gaus-
sian envelope and the temporal delay
set to ten times and 30 times the
inverse of the coupling rate, respec-
tively. (c) and (d) Simulated correlation
function at the output port of an 8 ×8
lattice for CCW and CW edge states,
respectively. The delay incurred in the
edge states shifts the correlation function
diagonally, but the correlation of the input
state is preserved. The centers of the two
time-bins are marked with dashed yellow
lines.13
where ˆ
Hsand ˆ
Hidescribe the Hermitian SSH Hamiltonian oper-
ators for the signal and idler, respectively, while the nonlinear
Hamiltonian is given by
ˆ
Hnl =γ∑
nA2
na†
n,sa†
n,i+h.c., (6)
where a†
n,sand a†
n,iare the creation operators for the signal and
idler photons at site n, respectively. γrepresents the nonlinearity
coefficient, and Anis the amplitude of the classical pump field.19
The effective evolution of the biphoton amplitudes ψ∈CN×Nalong
the propagation axis zis governed by a discrete Schrödinger-like
equation,
i∂ψ
∂z=ˆ
Hsψ+ψˆ
H∗
i+γψ0diag(A2
1,...,A2
N). (7)
The discussed theoretical model supports and explains the exper-
imental results of topological quantum phenomena in a one-
dimensional SSH lattice. Under chiral symmetry protection,
biphoton states retain their correlation features, enabling high-
fidelity quantum state transport and robust entanglement within
photonic circuits.19,20
III. EXPERIMENTS
A. Multiphoton quantum states and topology
After clarifying what topological protection of quantum infor-
mation in multiphoton states entails and describing a theoretical
background to understand it, let us delve into some of the pioneering
experimental demonstrations.
Interestingly, the first experimental signatures of topologi-
cal protection of multiphoton correlations were presented in June
2016,12 shortly before the theoretical papers described above. In
these initial measurements,12 and in the 2018 follow-up com-
prehensive experimental study and supporting theory work,19 an
array of silicon waveguides featuring alternating long and short
spacings between the waveguides was employed to emulate the
Su–Schrieffer–Heeger (SSH) model,31 characterized by a chain of
two-partite cells. The SSH model can represent different topological
phases depending on the relative strengths of intercell and intracell
couplings. The alternate coupling strengths were implemented here
via alternating gap sizes, and, by introducing a defect consisting of
long–long spacing at the central waveguide of the array, a topolog-
ical phase transition appeared, and a defect mode localized at the
structure’s center was generated, as initially demonstrated with clas-
sical light in Ref. 32. The on-chip generation of the correlated photon
pairs was achieved through spontaneous four-wave mixing (SFWM)
within the silicon waveguides, a Kerr nonlinear process where two
photons at the pump frequency are annihilated, giving rise to two
correlated photons in a process that satisfies energy and momen-
tum conservation. The higher frequency photon in the generated
correlated pair is referred to as the signal, and the lower frequency
photon is referred to as the idler. When injecting an ultrashort pump
at the center waveguide of the array, i.e., at the center of the topolog-
ical defect, as shown in Fig. 4(a), the nonlinearly generated photon
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FIG. 4. (a) Schematic of the experimental setup for biphoton correlation measurements in a waveguide array platform emulating the SSH model, featuring a topological
defect state localized at the central waveguide. (b) Diagram of a non-topological waveguide array consisting of equidistant waveguides. Biphoton correlation measurements
for the topological defect system, both without disorder and with intentionally introduced disorder, are shown in panels (c) and (d), respectively. Panels (e) and (f) display the
biphoton correlation results for the non-topological system, also without and with the disorder, respectively.
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pairs are generated and waveguided predominantly in the topolog-
ical mode due to spatial overlap. The biphoton correlation map
at the output of the array is characterized using superconducting
nanowire single-photon detectors after having spectrally separated
signal and idler photons and adjusting the polarization. The mea-
sured output biphoton correlation exhibits a spatial profile with
strong localization at the central waveguide and with zero amplitude
in every alternate waveguide, as shown in Fig. 4(c). This character-
istic behavior aligns with the distinctive features of the topological
mode predicted by the SSH model. When deliberate disorder is
introduced in the topological lattice by randomizing the position
of the waveguides, the main features of the biphoton correlation
prevail, as shown in Fig. 4(d). To prove the topological nature of
this robustness, Blanco-Redondo et al. also fabricated and measured
topologically trivial lattices of equidistant silicon waveguides with a
defect created by introducing a wider waveguide in the center; see
Fig. 4(b). Although the degree of localization of the biphoton corre-
lation at the output of this trivial lattice, shown in Fig. 4(e), is similar
to that of the topological array, no spatial features are preserved
under the presence of disorder, as depicted in Fig. 4(f).
These measurements are experimental evidence of the protec-
tion that the underlying topology of the structure can provide to the
spatial features of states living in high-dimensional Hilbert spaces.
For two correlated photons in mwaveguides, the states live in an
m2Hilbert space. However, the question remains whether this pro-
tection is just a logical consequence of the topological protection of
classical light in the SSH model, as shown, for instance, in Ref. 32, or
whether the quantum correlations play a crucial role. To answer this
question, the authors introduced a quantity,
ΓΔ,si
i j =ns
inij−ns
inij, (8)
where ns
inijdenotes the signal (s)–idler (i) correlations, which can
be directly measured from single-photon detectors and time cor-
relation electronics, and ns
inijdenotes the product of the signal
and idler intensities in waveguides iand j, respectively, which can
be obtained from the individual detections of signal and idler pho-
tons. The calculation reveals that the value of ΓΔ,si
i j is almost entirely
determined by the correlations, proving that the topological pro-
tection of multiphoton quantum states is a different regime than
the conventional topological protection of classical electromagnetic
states. A natural question that arises is, given that the robustness of
the state is against a kind of disorder that only exists in a coupled
system, what advantage does this structure with biphotons travel-
ing in a single topological mode entail with respect to biphotons
traveling in a single waveguide? The answer is twofold. On one
hand, the rich dynamics in the coupled structure (m2Hilbert space)
open possibilities toward robust multimode entanglement, as we
will discuss below, and a plethora of possibilities in the context of
quantum walks. On the other hand, it has been shown that quan-
tum topological boundary states are robust to kinds of disorders
that also affect individual waveguides, such as environmental noise
decoherence.33
In another pioneering experiment from 2018, Mittal et al.18
demonstrated the protection of spectral features of the biphoton
correlation. In this work, the authors utilize a square lattice of sili-
con ring resonators, similar to the one in Fig. 3(a), resembling the
quantum spin-Hall model, already discussed in Sec. II. The coupling
between ring resonators was precisely tailored so that photons do
not acquire any phase when coupling vertically neighboring rings.
However, a position-dependent phase is acquired as photons tra-
verse horizontally neighboring rings. The cumulative phase acquired
during a closed-loop traversal emulates the Aharonov–Bohm phase
associated with a synthetic uniform magnetic field. For a cumulative
phase of π2, this system provides two edge bands, ensuring the exis-
tence of two edge states, localized at the boundary of the lattice and
propagating clockwise (CW) and anticlockwise (ACW) as shown in
Fig. 3(a).
In their paper, Mittal et al. measured the transmission spec-
trum of an 8 ×8 lattice, which shows the presence of CW edge,
bulk, and ACW edge bands, as shown in Fig. 5(a). Subsequently,
in Fig. 5(b), they showed the measured intensity of the SFWM
generated signal photons (Γ)at frequency ωsas a function of the
pump frequency, ωp, when ωpis swept across the frequencies of the
CW edge, bulk, and ACW bands for a pump injected at the bound-
ary. The figure shows that the intensity of the generated signal
reaches its maximum when the pump aligns with the CW edge band,
which is associated with a longer propagation distance along the
CW edge than along the ACW edge, translating into a higher
photon-pair flux. In addition, Figs. 5(g) and 5(h) show the cor-
responding signal and idler photon intensities within that for a
pump frequency within the CW edge band, clearly illustrating the
FIG. 5. (a) Transmission spectrum of an
8×8 lattice, similar to the one depicted
in Fig. 3(a). (b) Normalized signal inten-
sity as a function of pump and sig-
nal frequencies. (c)–(h) Signal and idler
frequencies corresponding to pump fre-
quencies within the ACW edge band
(green plots), bulk band (blue plots), and
CW edge band (red plots). Adapted from
Ref. 18.
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confinement of the signal and idler photon spectra. On the other
hand, Figs. 5(c) and 5(d) illustrate the signal and idler intensities
when the pump frequency falls within the ACW edge band, where,
as expected, the maximum intensity for both signal and idler occurs
within this band, but the spectral confinement is less pronounced.
Finally, Figs. 5(e) and 5(f) depict the signal and idler intensities for
a pump frequency within the bulk band, where no spectral confine-
ment is observed. The topological characteristics of the edge band
serve multiple purposes: first, they provide spectral confinement for
the correlated photon pairs, and second, they impart robustness
against disorders, as demonstrated in Ref. 18. Third, they enhance
the efficiency of the correlated photon pairs generation. Further elu-
cidating the previous point, it is worth noting that the edge bands, in
contrast to the bulk band, exhibit a linear dispersion relation. This
characteristic facilitates the fulfillment of the phase matching condi-
tion among the pump, signal, and idler photons, thereby enhancing
the generation of photon pairs, as discussed in Ref. 18.
Topological protection of biphoton quantum correlations was
subsequently studied in a waveguide array platform that emulates
the off-diagonal Aubry–Andre–Harper (AAH) model35,36 by Wang
et al.34 In this work, researchers generated biphoton states outside
the chip and, subsequently, coupled them to an edge state with non-
trivial topology, shown in Fig. 6(a), and to a bulk state with trivial
topology, shown in Fig. 6(b), on the chip. The results demonstrate
that the biphoton state not only remains localized in the edge state
FIG. 6. (a) A waveguide array emulating the AAH model, showing the excitation of
an edge state. (b) Same for a bulk state in (b). (a) and (b) Adapted from Ref. 34.
(c) Illustration of the SSH model implemented in a waveguide array, with two long-
long defects highlighted in yellow. (d) Depiction of the SSH model in a waveguide
array, featuring a short-short defect at the center.
but also retains its quantum correlation to a significant degree due to
the topological properties of the edge state. In contrast, biphotons in
a bulk state spread across the lattice, causing their quantum features
to degrade. Two notable features of topological protection become
apparent in this work. First, the edge state preserves the quantum
correlation even when the two photons comprising the biphoton
state have different wavelengths, demonstrating the robustness of
the edge state against wavelength differences. Second, the edge state
continues to protect the quantum correlation in the biphoton state
even when the two photons are indistinguishable.34
Once it was established that topology can protect certain fea-
tures of biphoton correlations, a logical step was to investigate
whether this protection could be extended to entangled states of
light involving more than one topological mode. In a paper pub-
lished in 2019,20 researchers from the University of Sydney and
the Technion experimentally demonstrated topological protection
of photonic path entanglement on a silicon photonics array of cou-
pled waveguides implementing the SSH models with two uncoupled
topological defect modes, as illustrated in Fig. 6(c). By pumping the
center waveguide of each of these topological modes with a common
pump in the weak pumping regime and leveraging SFWM in the sili-
con waveguides, the authors generated a biphoton N00N state of two
topological modes. The N00N state was shown to preserve its most
important characteristics across different lattices and in the presence
of increasing disorder in the couplings, including no mixed terms
(e.g., signal photon in one mode and idler photon in another) and
zero amplitude observed in every other waveguide in the biphoton
correlation map. This behavior highlights the potential of topology
to protect entangled states of light through measurements of the
biphoton correlations.20
Topological protection has proven effective not only for spatial
entanglement but also for polarization entanglement.21 Researchers
in Ref. 21 generated polarization-entangled photon pairs out-
side the chip and injected them into a chip replicating the SSH
model—similar to one of the long-long defects shown in Fig. 6(c).
Their findings demonstrated that topological states can effec-
tively preserve polarization quantum entanglement, even when the
chip material introduces disorder and causes relative polarization
rotation in phase space.
Naturally, these studies lead to the idea that topology can be
utilized as a new degree of freedom for entanglement. This topic
is addressed in Ref. 22. To achieve states with different topolo-
gies, the authors introduced a short–short defect at the center of a
silicon photonics SSH chain, depicted in Fig. 6(d). In contrast to
the long–long defect, which only supports one localized mode of
topological origin, the short–short defect in the SSH exhibits two
modes with trivial topology colocalized with the already familiar
topology edge mode. Pumping the center of the short–short defect
of this waveguide array results in the generation of photon pairs
through SFWM, similar to previously discussed experiments.12,19,20
However, here, the biphoton propagation dynamics are much more
complex: the pump propagation is engineered in a way that induces
beating between the two trivial modes exclusively, but the bipho-
tons that are generated via SFWM as the pump propagates in this
beating pattern can be generated in states involving different super-
positions of the three colocalized modes, as dictated by quantum
interference between all the different mode combinations. This sur-
prising fact can be explained by considering that SFWM can occur
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between any four possible modes, but the possibility of exciting any
given four-mode combination depends on the overlap of the four
mode profiles. This overlap is dramatically higher for the localized
modes, which favors biphoton generation on combinations of the
co-localized topological edge and the two-trivial modes (see the full
formalism for this mode analysis in the original paper22). Conse-
quently, the resulting entangled biphoton state has topological and
trivial modes as components, constituting the first demonstration
of hybrid entanglement between different topologies in photon-
ics.22 Because of this hybridization between trivial and topological
modes, the impact of disorder manifests differently in this entan-
gled state. The measurements show that the characteristics of the
state change in the presence of significant disorder in the position of
the waveguides. The authors explain this behavior through simula-
tions showing that, as the disorder level increases, photon pairs tend
to occupy the topological mode rather than the trivial modes, given
that it is more robust to disorder. Thus, the weight of the different
modes of the entangled state changes as a function of disorder, which
translates into significant differences in the output biphoton corre-
lation map. Despite this lack of robustness in this demonstration,
more generally, this study opens the door to exploiting entangle-
ment between different topologies, which could have applications in
quantum information, not only because it highlights another degree
of freedom for encoding information but also because it presents
avenues for teleportation of quantum information between modes
of different natures.22
Topological systems, beyond their ability to protect quantum
states of light against specific types of disorders, can also filter out
undesired imperfections in quantum states of light. An example of
such an application is demonstrated in Ref. 37 using a photonic
Lieb lattice, represented in Fig. 7(a). The Lieb lattice is a 2D lattice
with a band structure consisting of three energy bands in momen-
tum space. Figure 7(b) plots the energy band of the lattice, showing
two dispersing bands touching the flat middle band via a cone-like
dispersion at Dirac points, leading to the real-space topology. The
FIG. 7. (a) Schematic of the Lieb lattice composed of an array of waveguides. (b)
Energy band diagram of the Lieb lattice. (c) and (d) Experimental transmission
patterns for ∣ψ(ϕ=π)⟩ and ∣ψ(ϕ=0)⟩, respectively. Adapted from Ref. 37.
states of the flatband can be represented on a basis that is fully local-
ized and exhibits a certain resistance to the introduction of defects.38
Researchers in Ref. 37 demonstrated the distinctive transmission
behavior of a flatband state for spatial qubits with different phases.
They generated spatial qubits defined as ψ=(ϕA+eiϕϕB)√2,
where ϕA,Bare the two vectors in the real space basis corresponding
to two different sites Aand B[see Fig. 7(a)] of the lattice, respec-
tively, and ϕdenotes the phase difference between these two vectors.
They showed that the state ψ(ϕ=π) is localized at sites Aand
Bdue to its complete overlap with flatband states, see Fig. 7(c),
whereas the state ψ(ϕ=0), as it is illustrated in Fig. 7(d), is dis-
persive due to its overlap with dispersive band states. Therefore,
by sufficiently propagating a spatial qubit initially localized at sites
Aand B, if its phase is ϕ=π, it will remain stable. Otherwise, if
its phase is different, i.e., ϕ≠π, the qubit will be dispersive and
spread throughout the lattice. Utilizing this feature, they experi-
mentally demonstrated that the implemented photonic Lieb lattice
can be used to spatially filter out qubits with phase deviations from
ϕ=π. In addition, they successfully defined and experimentally
tested a criterion, termed “visibility,” to discriminate the phase of
unknown qubits, determining whether ϕ=0 or ϕ=π. They further
demonstrated that their filtering scheme is robust against ampli-
tude imperfections in the two spatial components of the qubit, even
if they are not balanced. The dual capabilities of the topological
systems to not only protect quantum states of light against spe-
cific types of disorders but also to filter out undesired imperfections
in quantum states underscore the potential of topological pho-
tonic systems to improve the robustness and accuracy of quantum
technologies, where maintaining the integrity of quantum states is
crucial.
B. Quantum emitters and topology
Another intriguing area of investigation pertains to the interac-
tion between quantum emitters and topological modes, as explored
for the first time in Ref. 15. In this study, the authors established a
setup, shown in Fig. 8(a), involving two honeycomb photonic crystal
lattices with distinct topological phases, thereby generating topologi-
cal modes at the interface of the lattices. Epitaxial growth of quantum
dots was performed at that interface. Through the illumination of
the central interface between the two photonic crystals and subse-
quent spectral measurements of the emitted light at both ends of the
interface, the efficient coupling of quantum emitters to the topo-
logical modes was successfully demonstrated. The study reported
a minimum coupling efficiency of 68% for single-photon coupling
to the topological modes. Notably, the authors highlighted a salient
aspect of the coupling mechanism, namely, its chiral nature. This
implies that photons with different polarizations couple to oppo-
site propagating topological modes. Therefore, the interface of the
two photonic crystals serves as a chiral waveguide transporting pho-
tons with different polarizations in opposite directions. Figures 8(b)
and 8(c) show the transmission spectra for the left and right grat-
ings, labeled L and R in Fig. 8(a), as a function of the external
magnetic field. This magnetic field is applied to induce the Zee-
man effect, which splits the energy levels of the emitted photons
from quantum dots based on their polarization, σ±. The results
clearly demonstrate that the left grating exclusively receives photons
with σ−polarization, while the right grating receives photons with
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FIG. 8. (a) Schematic of the honey-
comb photonic crystal lattice featuring
two topologically distinct regions. The
interface between these regions, high-
lighted in light gray, runs along the center
of the lattice in the xdirection. This
platform is designed to couple quantum
emitters to topological modes. Gratings
positioned at the left (L) and right (R) of
the interface are used to collect the emit-
ted photons. (b) and (c) Display the emis-
sion spectra collected from the left and
right gratings, respectively, as a function
of the external magnetic field (B), which
is applied to break the degeneracy of
quantum dot emissions via the Zeeman
effect.
σ+polarization. This indicates that the emitted photons propa-
gate in opposite directions in a chiral manner. This chiral coupling
characteristic plays a crucial role in providing topological robust-
ness against back-reflection for the chiral topological modes, a
phenomenon demonstrated in Ref. 15 through the introduction
of a bend at the interface of two topologically distinct photonic
crystals.
Building on the previously discussed potential for topologi-
cal protection of entangled states of light,20,21 Dai et al. presented
an impressive demonstration of a topologically protected emitter of
Einstein–Podolsky–Rosen (EPR) states and multiphoton entangled
states.23 The researchers emulated an anomalous Floquet insulator
on a silicon photonic chip consisting of a 2D lattice of coupled
microring resonators, shown in Fig. 9(a) for enabling the gener-
ation of pseudospin-entangled multiphoton states. The emulated
anomalous Floquet insulator lattice features two topological edge
states localized at the lattice boundary and propagating in oppo-
site directions. The up pseudospin and down pseudospin corre-
spond to the edge states traveling in counterclockwise (CCW) and
clockwise (CW) directions, respectively. This pseudospin degree of
freedom, provided by topological edge states, is utilized to create
entanglement in the system. The pseudospin-entangled multipho-
ton states, (ns,upni,up0s,down0i,down+eiNϕ0s,up0i,upns,down ni,down)√2
(also known as N00Nstates), are produced through the SFWM
mechanism at the non-trivial topological edge states of the lattice by
coherently and simultaneously exciting the lattice boundary, where
ns,idenote the number of signal/idler photons and N=ns+ni. By
increasing the excitation pump power, entangled states with a higher
number of photons (N)can be generated. The authors reported
the generation of a 2-photon entangled state 2002and a 4-photon
entangled state 4004using their platform. To characterize the on-
chip generated entangled states, the pseudospin degree of freedom
was mapped to the polarization degree of freedom using a 2D grating
coupler. By performing free-space quantum interference and quan-
tum state tomography, they reported high fidelity and purity for the
generated entangled states. Figures 9(b) and 9(c) present the quan-
tum interference pattern and the reconstructed density matrices
FIG. 9. (a) Schematic of the Floquet topological insulator, composed of coupled
microring resonators. Counter-propagating edge states are localized along the
boundaries, highlighted in green and yellow. (b) Quantum interference pattern of
two non-degenerate photons, measured by detecting twofold coincidence counts
for the EPR entangled state (green curve) and the bulk state (black curve). Exper-
imental data points are indicated. (c) Density matrix of the EPR entangled state
obtained through quantum state tomography. Adapted from Ref. 23.
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for the non-degenerate 2002EPR state, respectively. These
results highlight the high visibility [V=0.933(6)] of the interfer-
ence, as well as the excellent fidelity [F=0.968(4)] and purity
[p=0.962(8)] of the EPR state. Although the entanglement in this
platform is not topological, the topological properties of the edge
states enhance the generation and quality of entangled states in sev-
eral ways: (1) The robustness of the edge states against disorders and
defects increases the purity and fidelity of the generated entangled
state. (2) The linear dispersion property of the edge states enables
phase matching in the SFWM process, resulting in a more efficient
generation of signal and idler photons. (3) The edge states in this
platform provide high transmission for a wide range of frequencies,
enabling the generation of broadband photons.
C. Quantum interference and topology
Quantum interference leverages the wave nature of photons.
When a photon, or an ensemble of photons, is in a superposi-
tion of multiple states, these states can interfere constructively or
destructively, or somewhere in between. Because of this, quantum
interference plays a crucial role in quantum science, particularly
in quantum computing, where it can be used to manipulate and
control quantum states and thence perform computational tasks.
One of the most notable quantum interference effects is known
as Hong–Ou–Mandel (HOM),39 where two photons interfere in a
50:50 beam splitter from two separate inputs. If the two photons
are truly indistinguishable, they will emerge from the same output
FIG. 10. (a) Schematic of the topological beam splitter enabling interference of
two topological modes. Adapted from Ref. 16. (b) Sagnac interference performed
by beam splitter BS-1 transforms superposition state ∣ψ1⟩=∣20⟩A,B+∣02⟩A,B
received at the input ports Aand Bof the beam splitter to the state ∣ψ2⟩=∣11⟩C,D,
which deterministically has one photon at each output port Cand Dof the beam
splitter. This state is used in the second beam splitter, BS-2, for HOM interference
performance. Adapted from Ref. 17.
port. Conversely, as they become more distinguishable, the likeli-
hood that they emerge from different output ports increases. Hence,
the coincidence measurements at the outputs of the beam splitter
serve as a means to quantify the indistinguishability of the input
photons. HOM quantum interference involving single photons trav-
eling in topological modes was initially demonstrated by Tambasco
et al. in 2018.16 In this investigation, researchers utilized an array of
borosilicate waveguides implementing the AAH model and capable
of hosting two distinct topological edge states, as shown in Fig. 10(a).
Through meticulous adjustment of the waveguide spacing along the
propagation direction (i.e., modulation of the off-diagonal elements
of the Hamiltonian), they strategically brought these edge states into
close proximity, allowing for their interference before subsequently
separating them once more—a process akin to that of a balanced
beam splitter for topological states on the chip. Conducting an HOM
interference experiment with their devised topological beam split-
ter, they reported a notably high interference visibility of 93.1%,
indicative of the near-ideal performance of the proposed topological
beam splitter. Although this experiment utilized a non-topological
light source, quantum interference between topological modes
was employed to assess the indistinguishability of the topological
states.
In a reverse situation, non-topologically protected quantum
interference has been applied to characterize topological light
sources.17,23 In Ref. 17, Mittal et al. designed a topological source
of indistinguishable photon pairs using a 2D array of microring
resonators that emulated the anomalous quantum Hall model, sim-
ilar to that shown in Fig. 3(a). This platform supports topological
states at the lattice boundary, where photons propagate either clock-
wise or counterclockwise. By employing dual-pump spontaneous
four-wave mixing (SFWM) and the pump mechanism depicted in
Fig. 10(b), they leveraged the topological characteristics of the plat-
form to generate indistinguishable photon pairs. These photons are
indistinguishable in terms of frequency, polarization, and spatial
mode, and they remain localized at the lattice boundary due to their
overlap with the topological modes. For Hong–Ou–Mandel (HOM)
interference performed by using a beam splitter, it is essential for the
two photons to be deterministically separated so that each photon
arrives at a different input port of the beam splitter. As illustrated in
Fig. 10(b), a Sagnac interferometer was employed to convert the state
ψ1=20A,B+02A,B, generated by the topological source, into the
desired state ψ2=11C,Dwith one photon at each of output ports
Cand D, which is ideal for HOM interference. With the pho-
tons now spatially separated, the team successfully conducted
HOM interference to assess the indistinguishability of the proposed
topological source.
IV. LIMITATIONS OF TOPOLOGICAL QUANTUM
PHOTONICS
Our previous discussion highlighted some of the most rep-
resentative results concerning the interaction of topology and the
quantum properties of light. Much of this discussion revolved
around the protection that topology endows to quantum properties
such as correlations and entanglement. To fully understand these
scientific breakthroughs and to be able to leverage them for applica-
tions in quantum information technology, it is important to realize
their limitations.
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Some of the limitations of topological protection of quantum
states are directly inherited from the linear properties of the topo-
logical mode; in other words, they are a direct consequence of the
limitations of topological protection of classical light. The most per-
vasive of these is the fact that true protection against backscattering
is not possible in reciprocal systems.40,41 There are also limitations
that are specific to each topological platform since a given platform
can only be robust to the disorder that does not affect the under-
lying topology. This is, for instance, the case of the topologically
protected modes in the SSH, which are robust to off-diagonal disor-
der that preserves the chiral symmetry (i.e., disorder in the coupling
strengths) but not to on-diagonal disorder (i.e., disorder in the onsite
energies).20
Other constraints of topological protection are, however, spe-
cific to correlated and entangled states (see Chap. 12 of Ref. 9).
Bergamasco and Liscidini theoretically addressed some of these con-
straints in the context of the generation of correlated photon pairs
via spontaneous four-wave mixing in topologically protected edge
modes in the SSH,24 a scenario that directly applies to several of the
experiments described above.12,19,20,22 In Ref. 24, the authors con-
sider a waveguide array composed of N identical silicon photonic
waveguides whose distances are engineered to support a topologi-
cally protected localized mode, which is robust against certain types
of disorder.32 They simulated the nonlinear generation of bipho-
ton states in this structure when a pump is injected in the center
waveguide of the topological defect mode in the undepleted pump
FIG. 11. JPI of the signal and idler photons exiting from the five central waveguides in the case of (a) no disorder, and the average JPI for the cases of (b) σ=14 nm
disorder and (c) σ=43 nm disorder. The path correlations in the photon pair are well preserved compared to the unperturbed state in the case with σ=14 nm, while they
slightly degrade when the disorder is increased to σ=43 nm, as can be seen by the appearance of nonzero terms on the diagonal, where both photons exit from the same
waveguide. The error bars also show a large standard deviation in the terms as the disorder increases, suggesting that very different path states can be generated. (d)
Energy bandgap diagram associated with the modes supported by the structure (e) values of fidelity (top) and Schmidt number (bottom) for increasing levels of disorder.
For each level, we report the average and the error bars corresponding to one standard deviation. The red curves (diamond points) represent the average value obtained
when intramode SFWM occurs only in the topologically protected mode for each figure of merit. Adapted with permission from N. Bergamasco and M. Liscidini, Phys. Rev.
A100, 053827 (2019). Copyright 2019 APS.
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approximation and analyzed the joint path intensity (JPI) of the
resulting output quantum state in the presence of different levels
of disorder. The results are shown in Figs. 11(a)–11(c), which show
the impact of the disorder on the system by analyzing JPI matri-
ces for 300 realizations of random disorder with σ=14 nm and
σ=43 nm of disorder in waveguide displacement. For σ=14 nm,
the results closely resemble the no-disorder case, with small standard
deviations indicating robust path correlations. However, with σ=43
nm, significant variations in the JPI are observed, particularly in the
(2, 2) and (−2,−2)terms, along with unexpected (1, 1) and (−1,−1)
terms. What is surprising is that, at this higher level of disorder,
the linear properties of the topological mode remain protected, as
heralded by the fact that the topological bandgap in the dispersion
relation remains open, as shown in Fig. 11(d). This finding implies
that quantum correlations are less robust than the linear properties
of the topological mode. To explain this observation, Bergamasco
and Liscidini turn their view to the nonlinear generation mecha-
nism for quantum correlations. They consider two scenarios for the
degradation of the generated photon pairs: intramodal SFWM, in
which the residual part of the pump that couples to modes that
are different from the topological mode gives rise to photon pairs
on those same modes, and intermodal SFWM, in which pump
photons that were originally in the topological mode give rise to
photon pairs in non-topological modes. In Fig. 11(e), the authors
show the fidelity (top) and the Schmidt number (bottom) of the
correlated output state as a function of disorder. The blue curves
show the realistic scenario in which both intramodal and inter-
modal SFWM occur, similar to the experimental observations in
Ref. 19, and the red curves show a scenario in which the pump
couples exclusively to the topological mode, and therefore, only
intermodal SFWM can occur. These results illustrate that for large
levels of disorder, intermodal SFWM contributes strongly to the
degradation of the quantum states, explaining why nonlinearly gen-
erated correlated quantum states are more fragile than their classical
counterparts.
Another interesting theoretical study concerning the limita-
tions of topological protection of quantum states was presented by
Tschernig et al.25 In this work, the authors focused on the trade-
off between the degree of entanglement and topological protection
in 2D topological insulators. This appears to be a natural question
to investigate given that, in principle, a high degree of entangle-
ment, which involves the presence of a large number of modes,
seems to strongly contrast with topological protection, which usually
implies localization in one protected mode. Tschernig et al. per-
form a detailed theoretical analysis of the effects of static disorder
on two-photon states injected on periodic topological lattices imple-
menting the Haldane model and aperiodic lattices implementing a
quantum Hall model. They analyze possible mechanisms of dissipa-
tion of two-photon states and the dephasing between the different
components of the entangled state in the presence of disorder in
these two kinds of lattices. The results show that a high degree of
non-separability (entanglement) can lead to rapid degradation in
the presence of disorder. To counteract this and maintain topologi-
cal robustness, the authors propose to prepare the initial entangled
state in a way that the possibilities for it to overlap with edge–bulk
and bulk–bulk modes are minimized. To formalize this idea, they
propose an interesting concept that they call the “topological win-
dow of protection,” which defines a region within the joint spectral
correlation map of two-photon states that ensures topological pro-
tection against disorder. These findings offer practical guidelines
for creating robust entangled states with high Schmidt numbers
that maximize the potential of topological photonic networks for
transferring quantum information.25
V. FUTURE PERSPECTIVES
The progress discussed here evidences a potential for topol-
ogy to protect crucial aspects of quantum states of light. Several
application areas of quantum information science and technol-
ogy could benefit from the introduction of topological concepts
into their designs (see Chap. 12 in Ref. 9), but significant work
remains ahead for the current research efforts to translate into
technology.
Arguably the application that can benefit the most from
advances in this field is the area of quantum computing.1,2 The
currently unavoidable noise on qubits induced by fabrication imper-
fections severely limits the reliability of quantum gates and, in turn,
the scalability of integrated quantum computing platforms. In the
last couple of years, building on earlier fundamental breakthroughs,
several experiments have commenced to address these challenges
directly.23,28 Beyond the initial demonstrations of individual topo-
logical C-NOT and Hadamard photonic gates,28 we foresee large sys-
tem demonstrations being reported in the coming years, potentially
mitigating the requirement for quantum protocols and algorithms
to detect, locate, and correct quantum errors. An important consid-
eration for the introduction of topological approaches into practical
quantum computing platforms is the additional footprint required
to implement topological lattices in contrast to individual wave-
guides or resonators. Therefore, a crucial question to be addressed in
the near future by the topological photonics and quantum comput-
ing communities—ideally together—is what are the crucial quantum
properties to be protected by topology and the degree of enhanced
reliability that needs to be provided in quantum operations to jus-
tify the extra real state required by topological photonic platforms.
Bearing in mind that the more traditional approaches to quantum
photonic computing rely on the use of ancillary qubits to compen-
sate for the lack of reliability, which also entails a significant amount
of accessory space on-chip, a trade-off point must be found. In this
regard, a promising emerging area pursuing resilient, dense quan-
tum information transmission is that of leveraging entanglement
in topological superlattices, supporting complex entangled states of
many modes in a very compact spatial region.27
Taking into account that the largest obstacle to the scalability
of photonic quantum information platforms is photon loss, a key,
although exceedingly hard, area to revisit is that of backscatter-free
topological waveguides. It has been suggested that backscatter-free
propagation could be obtained in systems under the combination of
parity, time-reversal, and duality symmetries.42 However, the most
widely accepted approach against backscattering is that of breaking
the time reversal symmetry,41 which is, of course, extremely difficult
at optical frequencies. The hope is that the recent breakthroughs in
ultra-low-loss waveguides and, consequently, high-Q resonators,43
in combination with the advent of silicon-photonics-compatible
ultrafast modulation technologies, such as thin-film lithium nio-
bate44 and barium titanate,45 may finally enable the realization
of broken time-reversal symmetry topological lattices via dynamic
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phase modulation.46 Acoustic pumping has also been put forward as
an alternative to break time-reversal symmetry47 and could lead to
experimental advances toward backscatter-free quantum devices in
the future.
Photon–photon interactions are also an area to watch for
their potential impact in producing robust quantum information
platforms. This, still largely unexplored area, has shown enticing
demonstrations of systems with trivial single-particle band struc-
tures but topological two-particle bands and doublon edge states.48
These experiments have shown the coherent operation of supercon-
ducting qubit ensembles and paved the way for a plethora of new
interacting models in topological quantum photonics. In relation to
this, models for topologically protected quantum-limited traveling-
wave amplifiers, critical in superconducting qubit experiments, have
been proposed, displaying natural protection against both internal
losses and backscattering.49 Practical implementations of these mod-
els might have a real impact on the scalability of superconducting
quantum computers.
Quantum sensing—sensing beyond the limits usually imposed
by classical mechanics—could also be critically impacted by the
latest developments in topological quantum and non-Hermitian
topological photonics. For instance, it has been theoretically pro-
posed that certain asymmetric non-Hermitian tight-binding models
with symmetries that lead to non-trivial topologies yield an expo-
nential increase in the quantum Fisher information per photon with
the number of sites.50 In addition, a theoretical model for quantum
non-Hermitian topological sensors with high-precision sensing of
observables strongly coupled to the boundary of the device has been
presented.51 The experimental realizations of these ideas, applied to
relevant measurands, are also something we are expecting to see in
the coming years.
Finally, we believe that the recent advent of programmable
topological photonics52–54 will contribute to fast-tracking progress
in all the future avenues predicted above by enabling rapid test-
ing and characterization of new ideas in the topological quantum
photonics field.
ACKNOWLEDGMENTS
A.B.-R. and M.J.Z. are supported by the National Science
Foundation (NSF) (Award No. 2328993).
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
A.H. and M.J.Z. contributed equally to this work.
Amin Hashemi: Investigation (equal); Writing – original draft
(equal). M. Javad Zakeri: Investigation (equal); Writing – orig-
inal draft (equal). Pawel S. Jung: Writing – review & editing
(supporting). Andrea Blanco-Redondo: Conceptualization (lead);
Funding acquisition (lead); Investigation (lead); Project administra-
tion (lead); Resources (lead); Supervision (lead); Validation (lead);
Writing – original draft (lead); Writing – review & editing (lead).
DATA AVAILABILITY
Data sharing is not applicable to this article as no new data were
created or analyzed in this study.
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