# Bright quantum photon sources from a topological Floquet resonance

Shirin Afzal  Institute for Quantum Science and Technology, and Department of
Physics and Astronomy University of Calgary, 2500 University Drive NW,
Calgary, Alberta T2N 1N4, Canada  Tyler J. Zimmerling  Department of
Electrical and Computer Engineering, University of Alberta, 9211 116 Street
NW, Edmonton, Alberta, Canada T6G 1H9  Mahdi Rizvandi  Institute for Quantum
Science and Technology, and Department of Physics and Astronomy University of
Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada  Majid
Taghavi  Institute for Quantum Science and Technology, and Department of
Physics and Astronomy University of Calgary, 2500 University Drive NW,
Calgary, Alberta T2N 1N4, Canada  Taras Hrushevskyi  Institute for Quantum
Science and Technology, and Department of Physics and Astronomy University of
Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada  Manpreet
Kaur  Institute for Quantum Science and Technology, and Department of Physics
and Astronomy University of Calgary, 2500 University Drive NW, Calgary,
Alberta T2N 1N4, Canada  Vien Van  Department of Electrical and Computer
Engineering, University of Alberta, 9211 116 Street NW, Edmonton, Alberta,
Canada T6G 1H9  Shabir Barzanjeh  Corresponding Author:
<EMAIL_ADDRESS>Institute for Quantum Science and Technology, and
Department of Physics and Astronomy University of Calgary, 2500 University
Drive NW, Calgary, Alberta T2N 1N4, Canada

###### Abstract

Entanglement, a fundamental concept in quantum mechanics, plays a crucial role
as a valuable resource in quantum technologies. The practical implementation
of entangled photon sources encounters obstacles arising from imperfections
and defects inherent in physical systems and microchips, resulting in a loss
or degradation of entanglement. The topological photonic insulators, however,
have emerged as promising candidates, demonstrating an exceptional capability
to resist defect-induced scattering, thus enabling the development of robust
entangled sources. Despite their inherent advantages, building bright and
programmable topologically protected entangled sources remains challenging due
to intricate device designs and weak material nonlinearity. Here we present an
advancement in entanglement generation achieved through a non-magnetic and
tunable resonance-based anomalous Floquet insulator, utilizing an optical
spontaneous four-wave mixing process. Our experiment demonstrates a
substantial enhancement in entangled photon pair generation compared to
devices reliant solely on topological edge states and outperforming trivial
photonic devices in spectral resilience. This work marks a step forward in the
pursuit of defect-robust and bright entangled sources that can open avenues
for the exploration of cascaded quantum devices and the engineering of quantum
states. Our result could lead to the development of resilient quantum sources
with potential applications in quantum technologies.

Figure 1: Schematic representation and simulation of a Floquet topological
entanglement source. a, A 2D square microring lattice consisting of individual
ring resonators arranged in identical unit cells. Each unit cell (inset)
comprises three closely coupled microrings arranged in a square formation with
the coupling strength $g$ and center-to-center distance $a$. By unrolling each
microring from the points shown by black arrows, an equivalent coupled
waveguide array can be obtained (inset). Four periodic coupling sequences, in
the microring unit cell and in its equivalent waveguide array, are shown by
light blue, light pink, light green, and dark pink arrows/segments. Gray color
shows uncoupled part of microrings/waveguides. An FDMR with resonance
frequency $\omega_{\mathrm{FDMR}}$ can be created by utilizing a thermo-optics
effect to introduce a phase shift $\Delta\phi$ in one of the rings along the
path of the edge state. The lattice’s periodicity leads to the appearance of
multiple equally spaced modes, separated by the frequency of
$\Delta_{\mathrm{FDMR}}$. By applying a pump to the system via the edge state
and utilizing the SFWM process within the FDMR, we can produce idler-signal
entangled photons at various $\Delta_{\mathrm{FDMR}}$ within the system, as
illustrated in the lower inset. b, The computed band structure for the
photonic anomalous Floquet topological insulator in one Floquet–Brillouin zone
with wavenumbers $K_{x}$ and $K_{y}$, comprising one unit cell periodic in
both $x$ and $y$ directions. The transmission bands of Floquet states, with
Chern number $C=0$, are separated by three topological bandgaps, labeled by
(I, II, II), with Winding number $W=1$. c, Projected band diagrams of a semi-
infinite lattice limited in the $y$-direction (consisting of 10 unit cells)
and extending infinitely in the $x$-direction. The red bands represent the
flat bands of Floquet bulk, obtained using a semi-infinite $5\times 5$
supercell, which are distinct from the transmission band manifolds due to a
phase detune. d, Simulated transmission spectra of an ideal topological
photonic insulator lattice of identical microring resonators with power
coupling of $98\%$, considering intrinsic loss, in the presence (pink line)
and absence (blue line) of the phase shift $\Delta\phi$.

The unique properties of entanglement and topology hold tremendous potential
for advancing quantum technology. Entanglement has emerged as a crucial
resource in quantum information [1, 2]. Its applications span diverse
disciplines, including quantum sensing [3, 4], computing [5, 6], and
communication [7, 8]. Significant progress has been made in the optical
domain, especially concerning the generation of entangled states through
integrated devices [9, 10], opening up avenues for practical real-world
applications [11, 12, 13, 14]. However, preserving and effectively harnessing
entanglement face challenges due to inevitable fabrication imperfections and
defects in the microchips [15, 16]. These disorders can lead to loss,
scattering, and spectral randomness, ultimately affecting the stability and
purity of the entanglement [17, 18].

Similarly, topological photonic insulators (TPIs) have attracted considerable
attention due to their ability to guide light through topologically-protected
edge states [19, 20, 21]. These states have found applications in various
domains such as lasing [22, 23, 24], soliton generation [25, 26] and frequency
comb [27]. The most intriguing aspect of TPIs is their resilience to
scattering caused by defects, making them highly desirable for robust device
development, capable of withstanding fabrication imperfections encountered
during photon transport [28, 29, 30]. Therefore, utilizing TPI systems for
generating nonclassical emissions offers unprecedented possibilities for
advancing quantum technology.

Recently, TPIs have been employed for generating robust entangled photons
emission using coupled waveguides [31] and microring resonators [32, 33], as
well as through Floquet-based TPIs [34]. Nevertheless, the practical
implementation of nonclassical sources based on edge states has faced
challenges due to the weak nonlinear properties of the materials. Ensuring
sufficient pair generation requires either long lattice boundaries, leading to
increased material loss, or utilizing high pump powers that introduce noise
and reduce the purity of the entanglement. These limitations have constrained
the potential applications of TPIs in quantum technologies, especially in
scenarios demanding bright entangled emissions, such as long-distance quantum
communications. Despite demonstrating resonance enhancement of photon pair
generation through spontaneous four-wave mixing (SFWM) in conventional
resonators [35, 36, 37, 38, 39, 40], the experimental realization of a bright
Floquet TPI entanglement source has remained elusive and unexplored until now.

Here for the first time, we explore entanglement generation in a novel
resonance effect called Floquet Defect Mode Resonance (FDMR) in the bulk of a
Floquet TPI [41]. We present compelling experimental evidence that
demonstrates a substantial enhancement in entangled photon pair generation by
harnessing this compact topological cavityless resonance effect. By exciting
the wavelength-tunable FDMR and utilizing SFWM in a silicon microring lattice
coupled to a topologically-protected edge state, we successfully create a
bright quantum photon source. This source demonstrates a second-order cross-
correlation of photon pairs approximately 3300 times higher compared to using
edge mode without resonance. These results demonstrate the potential of FDMR
in significantly improving the efficiency of entangled photon pair generation
in TPI-based systems, thereby removing the necessity for high power or
extended chip boundaries. Our topological quantum device combines the
advantages of integrated topological photonics, tunable localized optical
modes, and robustness to disorder, representing a novel approach to generating
entangled photon pairs. Such an approach holds great promise for various
applications, including the advancement of ultrabright photon-pair generation
for quantum information processing and facilitating the development of TPI
high-dimensional entangled quantum states [42] as well as quantum logic [43].

## Anomalous Floquet resonance mode

Fig. 1a presents the schematic of our Floquet topological photonic insulator,
created using two-dimensional (2D) directly-coupled microring resonators [44].
Each unit cell, see the inset of Fig. 1 a, comprises three strongly coupled
identical microrings arranged in a square shape with the coupling coefficient
$g$. As light propagates around each microring, it evanescently couples to the
neighboring rings in a periodic sequence, see Fig. 1(a), with the period equal
to the microring circumference $L$. The system thus emulates a periodically
driven system with the evolution along the direction of light propagation $z$
rather than time [45]. By varying the coupling between microrings, the
topological phase of the lattice can be tuned, leading to the appearance of
Chern or anomalous Floquet topological insulators in the weak or strong
coupling regimes, respectively as shown in Refs [46, 45, 34]. Our 2D microring
lattice follows a similar framework as the Floquet systems and satisfies the
following eigenvalue equation for the wavefunction
$|\psi_{n}(\textbf{k})\rangle$

$U_{F}(\textbf{k})|\psi_{n}(\textbf{k})\rangle=e^{i\epsilon_{n}(\textbf{k})L}|\psi_{n}(\textbf{k})\rangle,$
(1)

where $\epsilon_{n}(\textbf{k})$ is the quasi-energy band of the lattice with
the periodicity of $2\pi/L$. The Floquet operator
$U_{F}(\textbf{k})=\mathcal{T}e^{i\int_{0}^{L}H(\textbf{k},z^{\prime})dz^{\prime}}$,
where $\mathcal{T}$ represents the time-order operator, depends on the
Floquet-Bloch Hamiltonian $H(\textbf{k},z)=H_{\mathrm{FB}}(\textbf{k},z)$ that
exhibits periodicity along the $z$ direction
$H(\textbf{k},z)=H(\textbf{k},z+L)$, with a period of $L$. This characteristic
mimics the behavior of a periodically driven Hamiltonian, where the variable
$z$ plays the role of time $z\rightarrow{t}$. We can obtain an expression for
$H(\textbf{k},z)$ by transforming the microring lattice into an equivalent
coupled waveguide array, see the inset of Fig. 1a. In the strong coupling
regime, our 2D lattice exhibits three bandgaps with non-zero Winding numbers.
All bands possess trivial Chern numbers $C=0$, making it an anomalous Floquet
insulator, as shown in the band structure of one Floquet-Brillouin zone of the
unit cell in Fig. 1b. To verify the presence of topological edge states, we
impose boundaries along the $y$-axis while assuming the lattice extends
infinitely along the $x$-axis. From the projected quasi-energy band diagram,
we observe the existence of two pseudospin topological edge states in each
bandgap, as shown in Fig. 1c.

By exploiting the natural hopping sequence of our 2D lattice and leveraging
the edge state’s resilience against local defects, we can achieve the
confinement of light within a closed loop, creating a cavityless local
resonator. This confinement is attained by introducing a small perturbation in
the driving sequence, in the form of a phase shift ($\Delta\phi$), in one of
the ring resonators along the edge state’s trajectory, leading to the
formation of a flat-band Floquet mode within the 2D lattice. As a result, the
light becomes effectively trapped within the loop, giving rise to a locally
confined mode referred to as FDMR. This perturbation subsequently modifies the
total Hamiltonian of the system $H=H_{\mathrm{FB}}+H_{\mathrm{FDMR}}$ where
$H_{\mathrm{FDMR}}=\hbar\omega_{\mathrm{FDMR}}a^{\dagger}a$ describes the FMDR
with a resonance frequency $\omega_{\mathrm{FDMR}}$ and annihilation operator
$a$. The associated quasienergy of this mode experiences a shift directly
proportional to the magnitude of the induced phase, as depicted in Fig. 1c
(see Supplementary Materials). This FDMR mode can achieve a large Q-factor,
approaching $10^{5}$, compared to other 2D topological resonators [47, 48],
mainly because it lacks physical boundaries that would otherwise confine light
within the defect resonator. The resonance pattern in FDMR follows the
trajectory of the topologically nontrivial Floquet bulk mode with a total
circumference of $3L$. Fig. 1d illustrates the simulation of the transmission
spectrum of our design in the presence and absence of a phase shift,
demonstrating the appearance of the FDMR. Note that, this mode is coupled to
the edge state existing within the same bandgap, implying that we can control
or access the FDMR via the edge state.

To generate correlated photon pairs through the SFWM process within the
bandgap of the topological 2D lattice, we utilize the third-order nonlinearity
$\chi^{3}$ in silicon. The Hamiltonian that describes this process is given by

$H_{\mathrm{non}}=\hbar
g_{\text{nl}}\big{(}a_{p}a_{p}a_{s}^{\dagger}a_{i}^{\dagger}+h.c\big{)},$ (2)

where $g_{\text{nl}}$ is the strength of the SFWM while $a_{p}$ and $a_{s(i)}$
refer to the annihilation operators of the pump and signal (idler) modes,
respectively. The Hamiltonian (2) represents a four-photon mixing process in
which two photons from the pump are annihilated, resulting in the creation of
photons in the idler and signal modes, which were initially in a vacuum state.
With the presence of the FDMR, the generation of the photon pairs is
significantly enhanced, surpassing the pair generation rate achieved solely
with the edge state

$N_{\mathrm{FDMR}}\propto N_{\mathrm{edge}}\times Q^{3},$ (3)

where $N_{\mathrm{FDMR}}$ and $N_{\mathrm{edge}}$ are the pair generation
rates of the FDMR and edge state, respectively and $Q$ is the quality factor
of the FDMR, see the Supplementary Materials.

Figure 2: Experimental realization. a, A microscopic image of the anomalous
Floquet insulator lattice, featuring a $10\times 10$ unit cells arrangement
across a $595.32\times 595.32$ $\mu$m chip. The insets provide detailed
scanning electron microscopy (SEM) views of a unit cell and the structure of
the heater functioning as a thermo-optics phase shifter. Each microring in our
design has a square shape, with side lengths of 29.64 $\mu$m, and rounded
corners with a radius of 5 $\mu$m to minimize scattering losses. Additionally,
the figure includes a diagram of the experimental setup, which involves a pump
laser passing through a fiber polarization controller (FPC) and being injected
into the sample via a lensed fiber. The transmitted light is measured using a
photon detector and a power meter, two narrow bandpass filters, two
Superconducting Nanowire Single-Photon Detectors (SNSPDs), and a Time
Controller (TC). b, Near Infrared images of the edge state and FDMR in real-
time and comparison with numerical simulations. c, The normalized transmission
(T) of the lattice as a function of the pump wavelength. The solid line
represents the fitted theoretical model, as detailed in the Supplementary
Materials. d Zoomed-in transmission of the FDMR with respect to the wavelength
of the pump.  Figure 3: Enhancement of the photon pairs generation. a and b,
show the measured second-order cross-correlation $g_{si}^{(2)}(t)$ and the
second-order autocorrelation function $g_{ss/ii}^{(2)}(t)$, respectively, for
the power in edge state of 0.29 mW. c, Obtained nonclassical parameter,
$\Gamma$, for both FDMR and edge state at different pump powers. Increasing
the pump power pushes the system to a nonlinear regime such as the two-photon
absorption (TPA). d, Illustrates the transmission and $g_{si}^{(2)}(0)$ versus
pump wavelengths around the resonance frequency of the FDMR for the pump power
of 0.29 mW. e, Shows the measured $g_{si}^{(2)}(0)$ at different
$\Delta_{\mathrm{FDMR}}$ at the same powert. f, Display the obtained CAR for
FDMR and edge state versus pump power in the edge state. For all panels, the
error bars are derived from three independent measurements. For some points,
the size of the dots is larger than the error bars. Fitted solid lines in
panels a and b utilize the Poissonian distribution while fitting for panels c,
d, and f is based on the comprehensive theory outlined in the Supplementary
Materials.

## Results

We experimentally demonstrate the generation and enhancement of entangled
photons within our 2D Floquet lattice. The lattice is structured with a
$10\times 10$ unit cell arrangement and fabricated on a Silicon-on-Insulator
(SOI) substrate, incorporating a total of 300 individual ring resonators, see
Fig. 2a. Each microring in our design is square-shaped and is designed to
achieve efficient power coupling, enabling around $g^{2}=98\%$ of the power
transfer from one microring to the nearest microring. This design is
specifically chosen to achieve the desired anomalous Floquet insulator
behavior around the wavelength of 1545 nm.

Fig. 2b presents the real-time field distribution for the edge state and FDMR,
as measured via a Near-Infrared camera. These experimental results demonstrate
excellent agreement with the numerical simulations obtained by solving the
Schrödinger equation. In particular, the edge state exhibits distinct
nontrivial light propagation along the boundaries of the lattice. By applying
a phase shift in a microring located on the bottom edge of the sample, we
observe the emergence of the FDMR mode, characterized by a strongly localized
field distribution in a loop pattern.

The chip’s transmission spectrum is measured by injecting a laser into the
lattice at the bottom left edge and detecting the output light at the bottom
right edge using a powermeter, as displayed in Fig. 2a. During the
measurement, the laser’s wavelength is swept from 1530 nm to 1557 nm,
equivalent to 13 $\Delta_{\mathrm{FDMR}}$. Note that we define
$\Delta_{\mathrm{FDMR}}$ as the frequency spacing of the FDMR which is
approximately one-third of the FSR of our TPI, measuring 1.72 nm. The
normalized transmission spectrum, shown in blue in Fig. 2c, allows us to
identify topologically nontrivial bandgaps, characterized by regions of high
and flat transmission as labeled by (I, II, and III). Upon activation of the
heater (phase shift), a Floquet bulk mode is lifted into the bandgap,
resulting in a flattened energy band and a resonant mode spatially localized
in a loop pattern, as demonstrated at the bottom of Fig. 2a (see the inset).
The transmission spectrum in the output waveguide, with a phase detune of
$\Delta\phi=2.37\pi$, is represented by the pink trace in Fig. 2c. The
presence of distinct and adjustable resonance dips within each bandgap of each
FSR of the device indicates the excitation of a resonance mode that is coupled
to the edge state. In Fig. 2d, a close-up view of the transmission and the
theoretical fit for a selected mode is shown. This mode will be utilized for
resonance-enhanced entanglement generation.

To generate entangled photon pairs using FDMR, we employ a continuous-wave
pump beam with a frequency that corresponds to one of the FDMRs with a
$Q=4.843\times 10^{4}$, as indicated by an arrow in Fig. 2c. Maintaining
energy conservation via the SFWM process requires the generation of photon
pairs for idler and signal modes at different $\Delta_{\mathrm{FDMR}}$
symmetrically distributed around the pump frequency. To measure photon pairs
that fulfill this condition and suppress the emission from FDMR modes in
different $\Delta_{\mathrm{FDMR}}$, we utilize two narrowband tunable filters.
These filters have center wavelengths placed at $6\Delta_{\mathrm{FDMR}}$,
both below and above the pump wavelength, see Fig. 2c. This setup enables
efficient selection of the desired entangled photon pairs while minimizing
interference from unwanted FDMR modes at different $\Delta_{\mathrm{FDMR}}$.
We verify the non-classical nature of the photon pairs generated from FDMR by
measuring the second-order cross-correlation function defined by
$g_{si}^{(2)}(t)=\frac{\langle\hat{n}_{s}\hat{n}_{i}\rangle}{\langle\hat{n}_{s}\rangle.\langle\hat{n}_{i}\rangle}$,
where $\hat{n}_{s/i}$ is the photon number of the signal/idler modes. This
function determines the normalized probability of detecting signal and idler
photons at a specific time separation $t$ and can be measured using
coincidence rates of the signal-idler pairs and individual photon counts, see
Methods.

Fig. 3a shows the measured second-order cross-correlation function, with a
maximum value of $g_{si}^{(2)}(t)\approx 1450\pm 50$ observed at $t=0$ and at
a fixed pump power of $0.29$ mW, compelling evidence of time-correlated photon
pairs at the FDMR. At the same power, we compare these results with the
second-order autocorrelation functions, $g_{ii}^{(2)}(t)\approx 1$ and
$g_{ss}^{(2)}(t)\approx 1$, for the idler and signal modes, shown in Fig. 3b.
This comparative analysis allows us to examine a formal proof of a non-
classical light source i.e. the violation of the Cauchy-Schwarz inequality,
$\big{[}g_{si}^{(2)}(0)\big{]}^{2}\leq\big{[}g_{ss}^{(2)}(0)\cdot
g_{ii}^{(2)}(0)\big{]}$. Alternatively, we define and measure the
nonclassicality parameter

$\Gamma:=\frac{\big{[}g_{si}^{(2)}(0)\big{]}^{2}}{\big{[}g_{ss}^{(2)}(0)\cdot
g_{ii}^{(2)}(0)\big{]}},$ (4)

for which $\Gamma>1$ indicates a nonclassical source. Fig. 3c illustrates
measurements of the parameter $\Gamma$ as a function of the pump power and
includes a comparison with the photons generated from the edge state when the
FDMR mode is absent. The plot shows the violation of the Cauchy-Schwarz
inequality ($\Gamma>1$), indicating a profound enhancement of nonclassical
properties in the emitted photons due to the presence of the FDMR and its
coupling to the edge state, which effectively increases the cross-correlation.
The experimental result is in good agreement with the theoretical model we
developed in the Supplemental Materials except at high powers region where the
nonlinear effects, such as two-photon absorption (TPA) and pump noise change
the single and coincidence count rates. We note that the small size of our
chip limits the edge state’s ability to generate significant photon pairs
while propagating at the shorter arm of the sample. Hence, unlike Ref. [32]
the measured emission from the edge state is primarily composed of pump
leakage and lacks considerable nonclassical properties i.e. $\Delta\approx 1$.

Table 1:  Comparing the result of entangled photon pairs generation around the $1.55\,\mu$m wavelength, achieved through optically-driven SFWM. Micro-ring resonator ($\mu$-ring), coupled resonator optical waveguide (CROW), topologically protected (TP), pair generation rate (PGR), coincidence-to-accidental rate (CAR), cavityless or not physical boundary.  Integrated photonic sources | On-chip power ($\mu$W) | CAR | PGR | Dimension | TP | Cavityless | Coupling to edge state
---|---|---|---|---|---|---|---
$\mu$-ring [39] | 7.4 | 12100 | 16 kHz | 1D | No | No | No
Coupled $\mu$-rings [40] | 12 | 1100 | 200 Hz | 1D | No | No | No
CROW [35] | 200 | 80 | 1.64kHz | 1D | No | No | No
Topological based/edge state [32] | 1400 | 42 | - | 2D | Yes | - | -
This work | 86 | 2331 | 30kHz | 2D | Yes | Yes | Yes

To investigate the resonance-enhancement of the FDMR, at the pump power of
0.29 mW, the wavelength of the pump was swept across one resonance, allowing
for $g_{si}^{(2)}(0)$ to be compared to the transmission of the FDMR, as shown
in Fig. 3d. The peak of cross-correlation, pink trace, coincides with the
deepest point of the resonance, blue trace, demonstrating successful resonance
enhancement of the generation of entangled photon pairs, with 2nd-order cross-
correlation up to $\approx$1450\. We also measured the $g_{si}^{(2)}(0)$ at
different $\Delta_{\mathrm{FDMR}}$ in Fig. 3e.

Another important parameter for evaluating the nonclassical properties of the
generated photon pairs is the Coincidence to Accidental Rate (CAR). It serves
as a reliable indicator of the signal-to-noise ratio of the entangled photons
and is obtained by integrating $g_{si}^{(2)}(t)$ over the peak at $t=0$. The
CAR is measured as $\mathrm{CAR}\propto\frac{N_{c}}{N_{s}N_{i}}$, where
$N_{c}$ represents the coincident count rate, and $N_{s/i}$ denotes the single
count rates of the signal/idler. Note that, all three count rates $N_{j}$
($j=c,s,i$) scale quadratically with the pump power (see Methods), resulting
in the observed inverse dependence of CAR with the square of power in
agreement with our theoretical model, see Fig. 3f. This figure also provides a
comparison of the CAR between the FDMR and the edge state (heater off) under
various pump powers. Our source achieves an exceptional CAR value of $2300$,
significantly surpassing the $\mathrm{CAR}\approx 1.01$ obtained with edge
states in the absence of the FDMR. Moreover, this performance exceeds what is
typically achieved with other similar sources utilizing edge states in larger
chips and topologically trivial single and cascaded ring resonators. The
result highlights the outstanding performance of the FDMR in enhancing the
nonclassical characteristics of the generated photons. In Table 1, we compare
our Floquet entangled source and the current state-of-the-art photon-pair
sources in silicon. Unlike basic microring resonators [35, 39, 40], FDMR has
the capacity to be turned on and off in situ while also having the capability
to couple with edge states. Multiple FDMRs can be activated and coupled on a
single chip. These characteristics distinguish our system from the traditional
approach of entanglement generation using microring resonators.

## Discussion

In summary, we have demonstrated the resonance-enhanced generation of
entangled photon pairs within anomalous Floquet TPI, operating at room
temperature. This achievement is made possible by employing an FDMR coupled to
a topological edge mode. To verify the nonclassicality of the generated
entangled photons, we conducted measurements of the second-order cross-
correlation, tested the nonclassical characteristic of the emission, and
provided a comparison with the comprehensive theoretical model that we have
developed.

We conclude our discussion by exploring the topological protected properties
of our system. Unlike previous approaches that depend on propagating fields in
the edge state to generate photon pairs, our approach distinctively employs a
confined mode embedded within the bandgap region co-existing with the edge
state. Therefore, the FDMR operates as a frequency-tunable source that
generates entangled pairs, while the edge state acts as a topologically
protected waveguide that delivers the generated photons to the output ports.
Consequently, our system acquires the same topologically protected features as
those inherently present within the edge state. This unique capability sets
the current system apart from other sources of entanglement based on edge
state [32, 33, 34], as well as from the topologically trivial photonic devices
[35, 36, 37, 38, 39, 40]. We note that defects and disorders at the physical
boundaries, where the FDMR forms, have the potential to interfere with the
loop, reduce the mode’s quality factor or shift the mode. However, the
frequency tunability of the FDMR can partially mitigate these effects. In the
Methods, we expand further this discussion and explore various defects along
with their impact on the FDMR.

An essential aspect of our sample is its capability to generate entanglement
within a relatively small chip due to the compact size of the FDMR. This
feature becomes especially crucial when the material loss of the sample
affects the nonclassical property of the photon pairs. The compact nature of
the FDMR allows for efficient entanglement generation and on-chip
distribution, making it well-suited for scenarios where larger devices might
encounter challenges related to propagation loss. The exceptional attributes
of the FDMR, including its high quality factor, compact size, tunability (on
and off), and coupling with the edge state play a critical role in
facilitating the generation of bright entangled photon pairs. It opens up
possibilities for exploring novel techniques of on-chip entanglement
distribution, field-matter interaction, and advancing integrated quantum
electrodynamics. In particular, the FDMR can efficiently interact with
localized atoms/ions in the substrate, facilitating their integration into a
network of cascaded atom-cavity chains connected with edge states. This
platform holds promise for the development of on-chip and topologically-
protected quantum computing and processing.
Acknowledgments We thank Mohammad Hafezi for helpful comments and discussions
and Tyler Zegray for preparing the figures. S.B. acknowledges funding by the
Natural Sciences and Engineering Research Council of Canada (NSERC) through
its Discovery Grant, funding and advisory support provided by Alberta
Innovates (AI) through the Accelerating Innovations into CarE (AICE) –
Concepts Program, support from Alberta Innovates and NSERC through Advance
Grant project, and Alliance Quantum Consortium. V.V. and T.J. acknowledge
funding from NSERC and AI. This project is funded [in part] by the Government
of Canada. Ce projet est financé [en partie] par le gouvernement du Canada.
Contributions S.B. and V.V. conceived the ideas. S.B, S.A, T.Z. T.H. and V.V.
developed and built the experimental setup. S.B., S.A, M.R, and M.T. analyzed
the data. S.B., S.A. T.Z, and M.R. developed the theoretical model and
performed the measurements. V.V., T.Z., and S.A. simulated and fabricated the
sample. S.A. designed the chip. S.A. and M.K. developed the measurement codes.
All the authors contributed in preparing the manuscript.
Disclosures The authors declare no conflicts of interest.
Data Availability Statement Data underlying the results presented in this
paper are available upon proper request.

## Methods

### Device design and Fabrication

We conducted device simulations using Lumerical software with the Mode
Solution and Finite-Difference Time-Domain solver. Our simulations involved
introducing a TE-polarized mode into waveguide couplers, where the coupling
lengths matched the side length of the squared ring. From the simulation
results, we calculated the coupling angle $\theta$ and power coupling
coefficient $g^{2}=\text{sin}(\theta)^{2}$. Subsequently, we designed the
sample and fabricated the device on an SOI chip. The silicon waveguides within
our TPI have dimensions of 450 nm in width and 220 nm in height, while the
cladding consists of $2\mu$m of top SiO2 and $2.2\mu$m of bottom SiO2. Our
design includes square-shaped microrings with side lengths of 29.64 $\mu$m,
and we integrated rounded corners with a radius of 5 $\mu$m to minimize
scattering losses. To achieve efficient power coupling, we precisely set the
gaps between the microrings at 180 nm, enabling approximately $98\%$ of power
transfer from one microring to the next-nearest microring. This particular
design was chosen to attain the desired anomalous Floquet insulator behavior
at a wavelength of 1545 nm.

### Activation of the FDMR

We initiate the excitation of an FDMR and achieve precise tuning of its
resonant frequency by manipulating the phase of one of the ring resonators
situated on the lower boundary of the lattice. To facilitate this tuning
process, we fabricate a titanium-tungsten (TiW) heater, which covers the
rectangular perimeter, and apply current to the heater. As we perform this
manipulation, we observe that the resonant wavelength shift $\delta\lambda$ of
the FDMR exhibits an approximately linear relationship with the applied heater
power $p_{\text{heat}}$, see Supplementary Methods.

### Measurement setup

Fig. 2a illustrates our experimental setup designed to test the correlation
between the generated photon pairs and evaluate their quantum properties. The
setup involves the following components: TE-polarized pump photons generated
by an FPC and a tunable continuous-wave laser (Santec TSL-550) at telecom
wavelengths. The laser output power ranges from 0.086 mW to 1.73 mW, delivered
via a lensed-tip fiber that is butt-coupled to the chip’s facet. The pump and
generated wavelengths are collected at the output fiber, with $1\%$ of the
light sent to a photon detector and a power meter to measure the device’s
transmission spectrum. The remaining light is split into two separate paths
using a 50/50 coupler. The signal and idler photons then pass through bandpass
filters (Optoplex C-Band 50GHz) to suppress the input pump light by 40dB, as
well as other generated photons at different frequencies. Finally, two SNSPDs
”ID Quantique - ID281” detect isolated photons, providing electrical signals
to a Time Controller (ID900) with a timing resolution of 100ps.

In our experiments, we initially inject 1mW of pump power from the laser into
the lattice at the bottom left edge. We measure the output light from the
bottom right edge while sweeping the laser wavelength from 1530nm to 1557nm (5
FSR of the TPI). The total loss from the laser to the power meter is measured
as -10.5 dB, accounting for input and output coupling losses (from fiber to
chip and vice versa), FPC, and loss in the chip. The losses in the FPC and the
chip are measured as 0.24 dB and 2.6 dB/cm, respectively. As a result, for 1mW
laser power, the power in the waveguide is estimated as 0.29 mW. The
transmission spectrum before and after applying phase detuning is shown in
Fig. 2c, respectively by blue and pink traces.

### Single and coincidence counts measurement

Taking into account the losses in both the signal and idler channels, the
measured coincidence rate can be expressed as
$N_{\text{tot},c}=\eta_{i}\eta_{s}N_{c}$, where $\eta_{s/i}$ represents the
transmission efficiency from the chip’s output to the SNSPDs, and $N_{c}$
denotes the inferred coincidence rate excluding measurement losses and
efficiencies (see Supplementary Materials). In addition to system losses, for
the measured signal and idler pair rate, we must consider the noises and
residues related to the sidebands of the pump and pump leakage, which exhibit
a linear relationship with pump power ($N_{s/i,p}\propto P$). Furthermore, we
account for the extra counts associated with the dark counts $D_{s/i}\approx
50$ Hz of the SNSPDs. Consequently, the measured single/idler counts rate can
be defined as follows

$N_{\text{tot},s/i}=\eta_{s/i}N_{s/i}+N_{s/i,p}+D_{s/i}$ (5)

where $N_{s/i}$ signifies the inferred signal/idler single count rates. In
Fig. 4a, we plot the inferred $N_{c}$ as a function of the pump power in the
edge state. As anticipated, the coincidence rate exhibits a rise with the
increasing pump power.

### Second-order correlation function and coincidence-to-accidental rate

Figure 4: a Coincidence rate $N_{c}$ versus the pump power in the edge state.
b Dependency of CAR and coincidence rate $N_{c}$ to the measurement
coincidence window.

To characterize the correlation of the generated photon pairs, we use the
second-order cross-correlation function $g_{si}^{(2)}(t)$ by measuring single
and coincidence counting methods employing the SNSPDs and the Time Controller
(TC). This function allows us to determine the normalized probability of
detecting signal and idler photons with a time separation of $t$ [49]:

$g_{si}^{(2)}(t)=\frac{N_{\text{tot},c}}{N_{\text{tot},s}N_{\text{tot},i}}\times\frac{1}{T_{\mathrm{coin}}}$
(6)

where $T_{\mathrm{coin}}$ is the duration of arrival time called the
coincidence window. We analyzed the second-order cross-correlation function
$g_{si}^{(2)}(t)$, within the smallest measurable coincidence window, 100 ps,
of our Time Controller.

The determination of the coincidence-to-accidental rate (CAR) involved using
the coincidence histogram. We achieved this by calculating the ratio of the
total coincidence counts, within a coincidence window centered around the peak
values, to the accidental counts recorded during the same coincidence window
[50]

$\text{CAR}=\frac{\int_{-t/2}^{t/2}g^{2}_{si}(t)dt}{\int_{\infty-t/2}^{\infty+t/2}g^{2}_{si}(t)dt}.$
(7)

To obtain the proper coincident window, we first fitted the coincidence counts
using a Gaussian function to calculate the variance, $\delta=99.7$ ps, and the
full-width half maximum (FWHM) of the coincidence counts as
$\text{FWHM}=2\sqrt{2\text{ln}2}\delta=234.8$ ps. Fig. 4 b shows the
integration of coincidence count rates, $N_{c}$, over various coincidence
windows and their corresponding CARs. As shown in this figure, there is a
triad-off between the coincidence rate and CAR at different time windows.
Since the coincidence rate gets saturated at coincidence windows greater than
0.5 ns, leading to dropping CAR, we consider the coincidence
window$=3\delta\approx 1.2\text{FWHM}=0.3$ ns to calculate CAR at different
pump power shown in Fig. 3f.

### Resiliency against defect

We emphasize that the nature of FDMR mode comes from the topologically
nontrivial behavior of the lattice, in that, the existence of circulating
loops in the bulk of the lattice [41, 26] is required to mimic the quantum
hall effect that pushes the light to propagate only at the edge of the lattice
unidirectionally. In fact, the topology of the lattice makes FDMR loops to be
robust to fabrication imperfections as long as the defect doesn’t completely
alter the hopping sequence, for instance, by removing one ring in the loop
where the FDMR is excited.

To illustrate the resilience of FDMRs, we conducted simulations of the
lattice’s transmission spectrum, considering deviations of $10\%$ and $20\%$
in coupling coefficients between microrings and variations in the roundtrip
phase of each microring. Our simulations encompassed 50 lattices,
incorporating random variations centered around the FDMR loop. The most
significant impacts on the transmission spectrum are presented in Fig. 5,
shown by the dark green and light blue lines for coupling coefficient and
roundtrip phase variations, respectively. This illustration confirms that
specific types of defects, whether within the FDMR loop or adjacent
microrings, manifest as loss channels. While these defects may alter the
quality factor of the FDMR resonance or cause a resonance shift, they are
unable to destroy the FDMR loop.

Figure 5: Robustness of FDMR vs fabrication imperfections. a and b, Simulation
of respectively $10\%$ and $20\%$ variations in coupling coefficient between
microrings and roundtrip phase of the microrings located around the FDMR
resonance. Pink lines represent the lattice without defects. Light blue and
dark green lines show the maximum changes in FDMR due to coupling defects and
roundtrip phase defects, respectively. The maximum change,in terms of both
frequency shift and resonance linewidth, are chosen from 50 simulations with
random defects in the coupling coefficient representing the variations in the
gaps between microrings and roundtrip phase defects which are due to
variations in the size of the microrings. As illustrated in this figure the
defects in the roundtrip phases cause shifts in resonance and defects in $g$
lead to changes in the Q-factor of the FDMR.

As discussed above, various forms of defects and disorders can appear within
our devices. This section aims to elaborate further on distinct scenarios that
could potentially impact the efficiency of our system.

1\. Defect affecting the edge state: A possible scenario involves the
emergence of defects/disorders along the trajectory of the edge state, far
away from the region where the FDMR forms. In such an instance, the system
demonstrates resilience in the face of defects. The spectral characteristics
of the system remain largely unaffected by the presence of such a defect [34].
It is worth noting that this resilience pertains specifically to certain types
of defects, as the edge state remains robust against select anomalies. For
instance, consider the case of sidewall roughness in the waveguide, which has
the potential to induce back-reflection within the device.

2\. Impact on neighboring unit cells of the FDMR: In this scenario, the
presence of defects surrounds the FDMR mode, yet they do not directly
intersect the FDMR loop itself. Instead, the adjacent unit cells function as
loss channels, taking energy and photons away from the FDMR mode.
Consequently, the introduction of these neighboring defects results in
increased intrinsic losses for the mode. Such imperfections in fabrication can
manifest in various ways, such as by fluctuations in the spacing between rings
composing the FDMR and those situated within neighboring unit cells. These
variations exert a direct influence on the overall quality factor of the mode.
It’s important to highlight that the mode remains resilient against these
defects, provided they do not exceed a certain magnitude (e.g., the absence of
a ring within the FDMR). This assertion finds support in our simulation
results, depicted in Fig. 5.

3\. Defects impacting the FDMR loop: The third category encompasses defects
that directly affecting on the rings forming the FDMR. This type of defect
yields significant importance. It shares similarities with the second type,
potentially inducing frequency shifts due to disordered configurations and
resulting in phase alterations. Moreover, these defects possess the capacity
to influence the quality factor of the FDMR mode by introducing surface
roughness or causing partial light scattering toward adjacent rings. Another
consequence involves the potential disruption of extrinsic coupling with the
edge state. Such disruptions can yield variations in extrinsic quality
factors, either increasing or reducing them. These defects, although
impactful, do not possess the capability to destroy the entire mode, as long
as the defect’s size remains proportionally smaller than that of a single ring
within the loop. Fig. 5 shows simulations depicting these specific defects.
Furthermore, we note that the mode’s inherent frequency tunability plays a
crucial role in mitigating the effects of defects, particularly those
resulting in phase shifts within the loop.

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Supplementary Materials

###  .1 Topological Photonic Insulators

The Floquet-Bloch Hamiltonian of our 2D microring lattice can be derived using
a similar approach as for the general 2D microring lattice in [45], except
here each unit cell contains three microrings instead of four. We first
transform the microring lattice into an equivalent 2D coupled waveguide array
by ”cutting” each microring in a unit cell and unrolling it into a straight
waveguide with length $L$ equal to the circumference of the microring. Thus,
each roundtrip of light circulation in a microring is equivalent to light
propagation along the coupled waveguide array of length $L$, which can be
divided into a sequence of 4 coupling steps of equal length $L/4$. Denoting
the fields in microrings A, B, and C in unit cell $(m,n)$ in the lattice as
$\psi_{m,n}^{A},\psi_{m,n}^{B}$ and $\psi_{m,n}^{C}$, respectively, we can
write the equations for the evolution of light in the waveguide array along
the direction of propagation ($z$-axis) over each period in terms of the
coupled mode equations:

Figure 6: Mapping the coupling sequence of a 2D microring unit cell in the
direction of propagation ($z$). The coupled waveguide array is obtained by
unrolling the microrings in a unit cell from the points shown by black arrows.

$\displaystyle-i\frac{\partial{\psi_{m,n}^{A}}}{\partial{z}}=\beta\psi_{m,n}^{A}+k_{a}(1)\psi_{m,n}^{B}+k_{a}(2)\psi_{m,n}^{C}+k_{a}(3)\psi_{m-1,n}^{B}+k_{a}(4)\psi_{m,n-1}^{C}$
(8)
$-i\frac{\partial{\psi_{m,n}^{B}}}{\partial{z}}=\beta\psi_{m,n}^{B}+k_{a}(1)\psi_{m,n}^{A}+k_{a}(3)\psi_{m+1,n}^{A}$
(9)
$-i\frac{\partial{\psi_{m,n}^{C}}}{\partial{z}}=\beta\psi_{m,n}^{C}+k_{a}(2)\psi_{m,n}^{A}+k_{a}(4)\psi_{m,n+1}^{A}$
(10)

where $\beta$ is the propagation constant of the microring waveguide mode and
$k_{a}(j)$ specifies the coupling between adjacent waveguides in step $j$ of
each evolution period, with $k_{a}(j)=4\theta_{a}/L$ in step $j$ and $0$
otherwise. By making use of the Bloch’s boundary conditions
$\psi_{m+1,n}=\psi_{m,n}e^{ik_{x}\Lambda}$,
$\psi_{m,n+1}=\psi_{m,n}e^{ik_{y}\Lambda}$, where $\Lambda$ is the lattice
constant, for each field in the microrings, we can express the Coupled Mode
Equations in the form

$-i\frac{\partial}{\partial z}|\psi(\textbf{k},z)\rangle=[\beta
I+H_{FB}(\textbf{k},z)]|\psi(\textbf{k},z)\rangle$ (11)

where $|\psi\rangle=[\psi_{m,n}^{A},\psi_{m,n}^{B},\psi_{m,n}^{C}]^{T}$,
$\textbf{k}=(k_{x},k_{y})$ is the crystal momentum vector, and $I$ is the
identity matrix. The Floquet-Bloch Hamiltonian $H_{FB}$ consists of a sequence
of 4 Hamiltonians $H_{FB}=\sum_{j}\Theta(z-L_{j})H(j)$ corresponding to
coupling steps $j=\\{1,2,3,4\\}$, which have the explicit forms for $j={1,3}$

$\displaystyle H(j)=$
$\displaystyle\Bigg{(}\begin{matrix}[c]0&k_{a}e^{-i\delta_{3j}k_{x}\Lambda}&0\\\
k_{a}e^{i\delta_{3j}k_{x}\Lambda}&0&0\\\ 0&0&0\end{matrix}\ \Bigg{)}$

and for $j={2,4}$

$\displaystyle H(j)=$ $\displaystyle\Bigg{(}\
\begin{matrix}[c]0&0&k_{a}e^{-i\delta_{4j}k_{y}\Lambda}\\\ 0&0&0\\\
k_{a}e^{i\delta_{4j}k_{y}\Lambda}&0&0\\\ \end{matrix}\Bigg{)}$

In the above expressions, $\delta_{ij}=1$ if $i=j$ and 0 otherwise. The
Floquet-Bloch Hamiltonian is periodic in $z$ with a periodicity equal to the
microring circumference $L$, $H_{FB}(\textbf{k},z)=H_{FB}(\textbf{k},z+L)$.
The quasienergy spectrum of the lattice consists of three bands in each
Floquet-Brillouin zone, with a flat band state at quasienergies $2m\pi,\
(m\in\mathbb{Z})$. For $\theta_{a}\gtrsim\pi/\sqrt{8}$, all three bandgaps
exhibit Anomalous Floquet Insulator behavior with nontrivial winding numbers
[45].

The FDMR is created by introducing a periodic perturbation to the Hamiltonian
in the form of a phase detune of a microring during a step $j$ in each
evolution period. Thus, if a phase detune of $\Delta\phi$ is applied to
microring B in a unit cell during step $j$, its equation of motion is modified
as

$-i\frac{\partial{\psi_{m,n}^{B}}}{\partial{z}}=(\beta+\Delta\beta(j))\psi_{m,n}^{B}+k_{a}(1)\psi_{m,n}^{A}+k_{a}(3)\psi_{m+1,n}^{A}$
(12)

where $\Delta\beta(j)=4\Delta\phi/L$ in step $j$ and zero otherwise. We
previously showed in [41] that the spatial field localization pattern of the
defect mode depends on which segments of the microring are detuned. To create
an FDMR loop, one needs to apply the phase detune to segments $j=1$ and $4$ of
microring B (corresponding to the bottom half of the target defect resonator).

We can first construct the Hamiltonian matrix $H_{j}(k_{x})$ for each coupling
step $j={1,2,3,4}$ for the $1\times N_{y}$ lattice with Floquet boundary
condition on $x$ and zero Dirichlet boundary condition on $y$ (detailed matrix
construction is given in [45]). The quasienergy bands over one Floquet-
Brillouin zone can be obtained by computing eigenvalues of the Floquet
operator $U_{F}(k_{x})=U_{4}(k_{x})U_{3}(k_{x})U_{2}(k_{x})U_{1}(k_{x})$,
where $U_{j}(k_{x})=e^{-iH_{j}(k_{x})L/4}$. To compute the quasienergy band
for the FDMR at a specific phase detune $\Delta\phi$, we can construct the
Hamiltonian matrices $H_{j}(k_{x},k_{y})$ for a supercell consisting of
$(N_{x}=10\times N_{y}=10)$ cells with Floquet boundary conditions on all
boundaries (detailed matrix construction is given in the Supplementary
Information Section S1 of [41]). With the phase detune applied to coupling
steps $j=4,1$ of ring B, we can obtain the quasienergy bands of the FDMR modes
over one Floquet-Brillouin zone by computing the eigenvalues of the Floquet
operator $U_{F}$ of the supercell.

To obtain the intensity distributions of the edge mode and FDMR, shown in Fig.
2(b) of the main text, we can simulate a $(N_{x}=10\times N_{y}=10)$ lattice
with zero Dirichlet boundary conditions on all boundaries. With an input
waveguide coupled to the bottom left ring A and an output waveguide was
coupled to the bottom right ring B, a continuous-wave light signal applied to
the upper port of the input waveguide allows for the computation of the field
amplitudes in all the microrings in the lattice using the field coupling
method presented in [51].

As fully discussed in the supplementary material of Ref. [41], by simulating
the field distribution of a microring lattice where both normal bandgap and
Floquet topological bandgap exist, it’s been shown that the FDMR defined by
its specific loop pattern, Fig.2b of the main text, only exists at
topologically nontrivial bandgaps, whereas detuning the phase of a microring
in a trivial bandgap trap the light in the defect microring. It is worth
mentioning that the nontopological resonance mode in a normal bandgap can not
be experimentally excited due to the lack of any traveling mode that could
reach and couple to this resonance mode. On the other hand, the FDMR can be
experimentally excited using an edge state that coexists in the same
topological bandgap.

###  .2 Theoretical modeling of the pair generation

Considering the FDMR as a local resonator with a resonance frequency
$\omega_{\mathrm{FDMR}}$ and the edge state as a waveguide connected to the
resonator with a coupling rate $\kappa_{e}$, we can simplify the
representation of the FDMR-edge state system as a resonator-waveguide
platform, as shown in Fig. 7a. This simplification provides a straightforward
approach to effectively characterize and model the system’s dynamics and
describe the generation of nonclassical radiation from the FDMR. Note that,
the validity of this model is confined to the edge state passband, as
illustrated in Fig. 7b. The phenomenological Hamiltonian governing the SFWM of
the system in the effective representation is

$H=\sum_{j=p,s,i}\hbar\omega_{j}a_{j}^{\dagger}a_{j}-\hbar\big{(}g_{\text{nl}}a_{p}^{2}a_{s}^{\dagger}a_{i}^{\dagger}+h.c),$
(13)

where $a_{j}$ with $j=p,s,i$ are the annihilation operators of pump ($p$),
signal ($s$), and idler ($i$). The first term in the Hamiltonian (13)
corresponds to the free energy of each mode, while the last term describes the
Hamiltonian of SFWM with a nonlinear coupling parameter
$g_{\text{nl}}=\frac{3\hbar\omega_{p}^{2}\chi^{(3)}}{4\epsilon_{0}\bar{n}^{4}V_{\text{FDMR}}}$.
In this expression, $\chi^{(3)}$ denotes the nonlinear susceptibility,
$V_{\text{FMR}}$ represents the mode volume of the FDMR, $\omega_{p}$ is the
pump frequency, $\bar{n}$ stands for the average index of refraction of the
FDMR, and $\epsilon_{0}$ corresponds to the permittivity of free space.

The dynamics of the system can be described by the quantum Langevin equations
of motion [52, 53, 54]

$\displaystyle\dot{a_{p}}$ $\displaystyle=$
$\displaystyle-(i\omega_{p}+\frac{\kappa_{p}}{2})a_{p}+2ig_{\text{nl}}a_{p}^{\dagger}a_{s}a_{i}+\sqrt{\kappa_{p}^{\text{ext}}}a_{p}^{\text{ext}}+\sqrt{\kappa_{p}^{\text{int}}}a_{p}^{\text{int}},$
$\displaystyle\dot{a_{s}}$ $\displaystyle=$
$\displaystyle-(i\omega_{s}+\frac{\kappa_{s}}{2})a_{s}+ig_{\text{nl}}a_{p}^{2}a_{i}^{\dagger}+\sqrt{\kappa_{s}^{\text{ext}}}a_{s}^{\text{ext}}+\sqrt{\kappa_{s}^{\text{int}}}a_{s}^{\text{int}},$
(14) $\displaystyle\dot{a_{i}}$ $\displaystyle=$
$\displaystyle-(i\omega_{i}+\frac{\kappa_{i}}{2})a_{i}+ig_{\text{nl}}a_{p}^{2}a_{s}^{\dagger}+\sqrt{\kappa_{i}^{\text{ext}}}a_{i}^{\text{ext}}+\sqrt{\kappa_{i}^{\text{int}}}a_{i}^{\text{int}},$

where the total damping rate
$\kappa_{j}=\kappa_{j}^{\text{ext}}+\kappa_{j}^{\text{int}}$ of each mode
$(j=p,s,i)$ is the sum of intrinsic $\kappa_{j}^{\text{int}}$ and extrinsic
$\kappa_{j}^{\text{ext}}$ damping rates. Furthermore, the term
$a_{j}^{\text{ext}}(a_{j}^{\text{int}})$ accounts for the noise operator
arising from the coupling to the edge state (inaccessible loss channels). When
a strong laser pumps the system at the resonance frequency $\omega_{p}$, we
can simplify the first equation in (.2) by considering only the steady-state
part of the pump and disregarding field fluctuations. In this scenario, the
pump can be treated as a classical field with the amplitude

$\alpha_{p}(\omega)=|\langle
a_{p}\rangle|\approx\sqrt{\frac{\kappa_{p}^{\text{ext}}}{(\omega-\omega_{p})^{2}+\kappa_{p}^{2}/4}}\alpha_{in},$
(15)

where $\alpha_{in}=\sqrt{\frac{P}{\hbar\omega_{p}}}$ and $P$ is the power of
the laser in the edge state. Note that, in the derivation of the above
equation, we neglected the nonlinear term $2ig_{\text{nl}}\langle
a_{p}^{\dagger}a_{s}a_{i}\rangle$ under the assumption of the non-depletion
regime [55]. Considering the pump as a classical field, we can solve the
equations of motion for the idler and signal in the frequency domain. By
taking the Fourier transformation of the second and third equations in (.2),
we obtain

Figure 7: a The coupling between the FDMR and edge state can be represented as
a single ring resonator with a radius of R coupled to a trivial waveguide. In
this representation, the FDMR functions as a single-mode resonator with a
resonance frequency $\omega_{\text{FDMR}}$, an annihilation operator
$a_{\text{res}}$, and a Free-Spectral-Range (FSR). To initiate SFWM in this
configuration, the system is pumped at mode $a_{p}$, resulting in the
generation of photon pairs at the idler $a_{i}$ and signal $a_{s}$ modes,
distributed above and below the pump mode. The input and output light undergo
reflection and transmission at the facets of the waveguide with parameters
$t_{i}$ and $r_{i}$. b The output of the FDMR indicates the regions where the
effective resonator-waveguide model remains valid.

$\displaystyle-i\omega a_{s}$ $\displaystyle=$
$\displaystyle-(i\omega_{s}+\frac{\kappa_{s}}{2})a_{s}+iGa_{i}^{\dagger}+\sqrt{\kappa_{s}^{\text{ext}}}a_{s}^{\text{ext}}+\sqrt{\kappa_{s}^{\text{int}}}a_{s}^{\text{int}},$
(16) $\displaystyle-i\omega a_{i}$ $\displaystyle=$
$\displaystyle-(i\omega_{i}+\frac{\kappa_{i}}{2})a_{i}+iGa_{s}^{\dagger}+\sqrt{\kappa_{i}^{\text{ext}}}a_{i}^{\text{ext}}+\sqrt{\kappa_{i}^{\text{int}}}a_{i}^{\text{int}},$
(17)

where $G=g_{\text{nl}}|\alpha_{p}|^{2}$ describes the effective nonlinear
coupling between idler and signal. These coupled equations can be solved

$\displaystyle\textbf{a}=\textbf{M}.\textbf{a}_{\textbf{in}}^{\text{ext}}+\textbf{N}.\textbf{a}_{\textbf{in}}^{\text{int}},$
(18)

where $\textbf{a}=[a_{s},a_{i}^{\dagger}]^{\text{T}}$,
$\textbf{a}_{\textbf{in}}^{\text{ext/int}}=[a_{s}^{\text{ext/int}},(a_{i}^{\text{ext/int}})^{\dagger}]^{\text{T}}$
with T stands for the transpose of the matrix, and

$\textbf{M}(\omega)=\begin{pmatrix}\frac{\sqrt{\kappa_{s}^{\text{ext}}}\chi_{i}^{*}}{\chi_{s}\chi_{i}^{*}-G^{2}}&\frac{iG\sqrt{\kappa_{i}^{\text{ext}}}}{\chi_{s}\chi_{i}^{*}-G^{2}}\\\
-\frac{iG\sqrt{\kappa_{s}^{\text{ext}}}}{\chi_{s}\chi_{i}^{*}-G^{2}}&\frac{\sqrt{\kappa_{i}^{\text{ext}}}\chi_{s}}{\chi_{s}\chi_{i}^{*}-G^{2}}\end{pmatrix},\\\
\textbf{N}(\omega)=\begin{pmatrix}\frac{\sqrt{\kappa_{s}^{\text{int}}}\chi_{i}^{*}}{\chi_{s}\chi_{i}^{*}-G^{2}}&\frac{iG\sqrt{\kappa_{i}^{\text{int}}}}{\chi_{s}\chi_{i}^{*}-G^{2}}\\\
-\frac{iG\sqrt{\kappa_{s}^{\text{int}}}}{\chi_{s}\chi_{i}^{*}-G^{2}}&\frac{\sqrt{\kappa_{i}^{\text{int}}}\chi_{s}}{\chi_{s}\chi_{i}^{*}-G^{2}}\end{pmatrix},$
(19)

where $\chi_{j}(\omega)=\big{[}-i(\omega-\omega_{j})+\kappa_{j}/2\big{]}$ is
the inverse of the field susceptibility with
$\chi_{j}^{*}=\chi_{j}^{*}(-\omega)$. Since we are interested in the state of
the output fields in the current system, we can rewrite the Eq. (18) in terms
of the output fields using the input-output theory
$a_{j,\text{in}}^{\text{ext/int}}=a_{j,\text{out}}^{\text{ext/int}}+\sqrt{\kappa_{j}^{\text{ext/int}}}a_{j}$
with $j=s,i$, resulting in

$\displaystyle\textbf{a}=\textbf{M}.\textbf{a}_{\textbf{out}}^{\text{ext}}+\textbf{N}.\textbf{a}_{\textbf{out}}^{\text{int}}+\textbf{B}.\textbf{a},$
(20)

where

$\textbf{B}=\begin{pmatrix}\frac{\kappa_{s}\chi_{i}^{*}}{\chi_{s}\chi_{i}^{*}-G^{2}}&\frac{iG\kappa_{i}^{\text{ext}}}{\chi_{s}\chi_{i}^{*}-G^{2}}\\\
-\frac{iG\kappa_{s}^{\text{ext}}}{\chi_{s}\chi_{i}^{*}-G^{2}}&\frac{\kappa_{i}^{\text{ext}}\chi_{s}}{\chi_{s}\chi_{i}^{*}-G^{2}}\end{pmatrix},$
(21)

Note that the final term in this equation introduces reliance on the higher
order of the output field operators
$O([\textbf{a}_{\textbf{out}}^{\text{ext/int}}]^{n})$. However, for our
analysis, we will disregard this term as our focus is solely on the generation
of photon pairs.

The state of photons within the FDMR loop can be determined using the time-
dependent Hamiltonian given by
$|\psi\rangle=\mathcal{T}e^{-i/\hbar\int_{0}^{t}dtH(t)}|\psi(0)\rangle$, where
$\mathcal{T}$ denotes the time order operator and
$|\psi(0)\rangle=|0\rangle_{s,i}|0\rangle_{\mathrm{env}}$ represents the
initial state of the idler, signal, and environment. Here, all modes are
initially in the vacuum state, and hence, we have
$a^{\text{ext}}_{s/i}|0\rangle_{s/i}=a^{\text{int}}_{\text{s/i}}|0\rangle_{env}=0$.
For weak driving fields, the two-photon state can be simplified

$\displaystyle|\psi\rangle\simeq\Big{[}1-\frac{i}{\hbar}\int_{0}^{t}dtH(t)\Big{]}|\psi(0)\rangle=\Big{[}1-i\int_{0}^{t}dt\big{(}g_{\text{nl}}a_{p}^{2}a_{s}^{\dagger}a_{i}^{\dagger}+h.c)\Big{]}|\psi(0)\rangle,$
(22)

where we only considered the nonlinear part of the Hamiltonian. Utilizing the
frequency domain representation of the operators

$O(t)=\frac{1}{\sqrt{2\pi}}\int d\omega e^{-i\omega t}O(\omega),$ (23)

we get

$\displaystyle|\psi\rangle=\Big{[}1-\frac{ig_{\text{nl}}}{(2\pi)^{3/2}}\int_{0}^{t}dt\int
d\omega
d\omega^{\prime}\omega^{\prime\prime}\Big{(}a_{p}(\omega)^{2}a_{s}^{\dagger}(\omega^{\prime})a_{i}^{\dagger}(\omega^{\prime\prime})e^{-i(2\omega-\omega^{\prime}-\omega^{\prime\prime})t}+h.c\Big{)}\Big{]}|\psi(0)\rangle,$
(24)

Replacing the pump drive with its classical form
$a_{p}(\omega)\rightarrow|\alpha_{p}(\omega)|\delta(\omega_{p})$ and perform
the time integral, results in

$\displaystyle|\psi\rangle$
$\displaystyle=\Big{[}1-\frac{ig_{\text{nl}}}{2\pi}\int
d\omega^{\prime}d\omega^{\prime\prime}\Big{(}|\alpha_{p}(\omega_{p})|^{2}a_{s}^{\dagger}(\omega^{\prime})a_{i}^{\dagger}(\omega^{\prime\prime})+h.c\Big{)}\delta(2\omega_{p}-\omega^{\prime}-\omega^{\prime\prime})\Big{]}|\psi(0)\rangle$
(25) $\displaystyle=\Big{[}1-\frac{iG}{2\pi}\int
d\omega\Big{(}a_{s}^{\dagger}(\omega)a_{i}^{\dagger}(2\omega_{p}-\omega)+h.c\Big{)}\Big{]}|\psi(0)\rangle,$
(26)

where the term $\delta(2\omega_{p}-\omega^{\prime}-\omega^{\prime\prime})$
fullfiles the phase matching condition $2\omega_{p}=\omega_{s}+\omega_{i}$
required for SFWM. The output state $|\psi\rangle_{\mathrm{out}}$ can be
obtained by substituting $a_{s/i}$ with $a_{s/i}^{\text{out}}$ using Eq. (18)

$\displaystyle|\psi\rangle_{\text{out}}$
$\displaystyle=\int\frac{d\omega}{2\pi}\Big{[}\zeta_{00}(\omega)|0\rangle_{s,i}|0\rangle_{\text{env}}-\zeta_{11}(\omega)|1_{\omega},1_{2\omega_{p}-\omega}\rangle_{s,i}|0\rangle_{\text{env}}-\zeta_{10}(\omega)|1_{\omega},0\rangle_{s,i}|0,1_{2\omega_{p}-\omega}\rangle_{\text{env}}$
$\displaystyle-\zeta_{01}(\omega)|0,1_{2\omega_{p}-\omega}\rangle_{s,i}|1_{\omega},0\rangle_{\text{env}}-\eta(\omega)|0\rangle_{s,i}|1_{\omega},1_{2\omega_{p}-\omega}\rangle_{\text{env}}\Big{]},$

with

$\displaystyle\zeta_{00}(\omega)$ $\displaystyle=$ $\displaystyle
2\pi\delta(\omega)-iG\big{[}M_{12}^{*}(\omega)M_{22}(2\omega_{p}-\omega)+N_{12}^{*}(\omega)N_{22}(2\omega_{p}-\omega)$
$\displaystyle+$ $\displaystyle
M_{11}(\omega)M_{21}^{*}(2\omega_{p}-\omega)+N_{11}(\omega)N_{21}^{*}(2\omega_{p}-\omega)\big{]},$
$\displaystyle\zeta_{11}(\omega)$ $\displaystyle=$ $\displaystyle
iG\big{[}M_{11}^{*}(\omega)M_{22}(2\omega_{p}-\omega)+M_{12}(\omega)M_{21}^{*}(2\omega_{p}-\omega)\big{]},$
$\displaystyle\zeta_{10}(\omega)$ $\displaystyle=$ $\displaystyle
iG\big{[}M_{11}^{*}(\omega)N_{22}(2\omega_{p}-\omega)+N_{12}(\omega)M_{21}^{*}(2\omega_{p}-\omega)\big{]},$
(28) $\displaystyle\zeta_{01}(\omega)$ $\displaystyle=$ $\displaystyle
iG\big{[}N_{11}^{*}(\omega)M_{22}(2\omega_{p}-\omega)+M_{12}(\omega)N_{21}^{*}(2\omega_{p}-\omega)\big{]},$
$\displaystyle\eta(\omega)$ $\displaystyle=$ $\displaystyle
iG\big{[}N_{11}^{*}(\omega)N_{22}(2\omega_{p}-\omega)+N_{12}(\omega)N_{21}^{*}(2\omega_{p}-\omega)\big{]},$

In Eq. (.2), the initial term represents the presence of signal and idler in a
vacuum state. The subsequent component characterizes the idler-signal two-
photon states, giving rise to coincidences between the idler and signal modes.
The third and fourth terms correspond to situations where one photon is
present in either the idler or signal mode. This occurrence arises due to the
loss of an idler or signal photon, and these terms contribute to the single
count rates. Finally, the last term illustrates the loss of the generated
photons to the surrounding environment. The parameters
$|\zeta_{ij}(\omega)|^{2}$ and $|\eta(\omega)|^{2}$ quantify the probability
of each of these described events. Based on this, the photon-pair generation
rate is expressed as [56, 55, 57]

$N_{c}=\int_{-\infty}^{\infty}d\omega|\zeta_{11}(\omega)|^{2}=G^{2}\int_{-\infty}^{\infty}d\omega\Big{|}M_{11}^{*}(\omega)M_{22}(2\omega_{p}-\omega)+M_{12}(\omega)M_{21}^{*}(2\omega_{p}-\omega)\Big{|}^{2}.$
(29)

The second term of the above equation $M_{12}M_{21}^{*}\propto G^{2}$
represents a higher order pump-dependent correction of the coincidence. In the
weak pumping regime $G^{2}\ll\kappa_{s}\kappa_{i}$ we can simplify the above
equation

$N_{c}\approx
G^{2}\int_{-\infty}^{\infty}d\omega|M_{11}(\omega)|^{2}|M_{22}(2\omega_{p}-\omega)|^{2}\approx
G^{2}\int_{-\infty}^{\infty}d\omega\Big{[}\frac{\kappa_{s}^{\text{ext}}}{(\omega-\omega_{s})^{2}+\kappa_{s}^{2}/4}\Big{]}\times\Big{[}\frac{\kappa_{i}^{\text{ext}}}{(2\omega_{p}-\omega-\omega_{i})^{2}+\kappa_{i}^{2}/4}\Big{]}.$
(30)

We now can perform the frequency integral and find an analytical expression
for the coincidence rate

$N_{c}\approx\frac{8\pi
G^{2}(\kappa_{s}^{\text{ext}}\kappa_{i}^{\text{ext}})(\kappa_{s}+\kappa_{i})}{\kappa_{s}\kappa_{i}\Big{[}(\kappa_{s}+\kappa_{i})^{2}+(2\omega_{p}-\omega_{s}-\omega_{i})\Big{]}},$
(31)

Considering the phase matching condition $2\omega_{p}=\omega_{s}+\omega_{i}$
we find

$N_{c}=\frac{8\pi
G^{2}(\kappa_{s}^{\text{ext}}\kappa_{i}^{\text{ext}})}{\kappa_{s}\kappa_{i}(\kappa_{s}+\kappa_{i})}=\frac{8\pi
g_{\text{nl}}^{2}P^{2}}{(\hbar\omega_{p})^{2}(\kappa_{s}+\kappa_{i})}\Big{(}\frac{\kappa_{s}^{\text{ext}}}{\kappa_{s}}\Big{)}\Big{(}\frac{\kappa_{i}^{\text{ext}}}{\kappa_{i}}\Big{)}\Big{(}\frac{\kappa_{p}^{\text{ext}}}{\kappa_{p}^{2}}\Big{)}^{2},$
(32)

In the symmetric coupling rates $\kappa=\kappa_{s}=\kappa_{i}=\kappa_{p}$ and
critical coupling regime $\kappa_{j}^{\text{ext}}/\kappa_{j}=1/2$ the above
equation reduces to

$N_{c}=\frac{\pi g_{\text{nl}}^{2}P^{2}}{4(\hbar\omega_{p})^{2}\kappa^{3}},$
(33)

as reported in the main text. In reality, the measured coincidence rate
deviates from this equation and is influenced by the detection efficiency and
losses in the measurement chain, leading to the modified expression
$N_{\text{tot},c}=\eta_{i}\eta_{s}N_{c}$.

The single-count rates for the idler and signal can be expressed as follows

$\displaystyle N_{s}=\int_{-\infty}^{\infty}d\omega|\zeta_{10}(\omega)|^{2}$
$\displaystyle\approx\frac{8\pi
g_{\text{nl}}^{2}P^{2}}{(\hbar\omega_{p})^{2}(\kappa_{s}+\kappa_{i})}\Big{(}\frac{\kappa_{s}^{\text{ext}}}{\kappa_{s}}\Big{)}\Big{(}\frac{\kappa_{i}^{\text{int}}}{\kappa_{i}}\Big{)}\Big{(}\frac{\kappa_{p}^{\text{ext}}}{\kappa_{p}^{2}}\Big{)}^{2},$
(34) $\displaystyle
N_{i}=\int_{-\infty}^{\infty}d\omega|\zeta_{01}(\omega)|^{2}$
$\displaystyle\approx\frac{8\pi
g_{\text{nl}}^{2}P^{2}}{(\hbar\omega_{p})^{2}(\kappa_{s}+\kappa_{i})}\Big{(}\frac{\kappa_{s}^{\text{int}}}{\kappa_{s}}\Big{)}\Big{(}\frac{\kappa_{i}^{\text{ext}}}{\kappa_{i}}\Big{)}\Big{(}\frac{\kappa_{p}^{\text{ext}}}{\kappa_{p}^{2}}\Big{)}^{2},$
(35)

Note that, the leakage/residue of the pump, the efficiency of the measurement
chain $\eta_{j}$, and the dark counts $D_{j}$ of the photodetectors add extra
terms to the single-count rates

$\displaystyle N_{\text{tot},j}=\eta_{j}N_{j}+N_{j,p}+D_{j},$ (36)

with $j=s,i$ and as described in the Methods. Note that, the additional noise
photons arising from the pump leakage $N_{j,p}=\beta_{j}P$ shows a linear
dependency to the pump power $P$ [58, 59]. The coefficient $\beta_{j}$ is
affected by imperfections in the bandpass filter and the presence of laser
sidebands. These imperfections enable pump photons to reach the
photodetectors, leading to an overall increase in the total single counts
integrated over the measurement time.

###  .3 Phase detuning calibration

The spectral width of each bandgap is 1.28nm on average in the wavelength
region of interest plotted in Fig. 2b. The FDMR are, on average, detuned
0.92nm from the short-wavelength edge of their respective bandgaps. This
equates to 73.4% of the way across each bandgap, or a phase detune of
approximately 1.47$\pi$.

###  .4 Power transmission spectral

In this section, we analyze the transmission spectrum of the FDMR, considering
the single-mode Kerr nonlinearity, as depicted in the effective resonator-
waveguide representation shown in Fig. 7a. To start, we set
$a_{\text{res}}=a_{p}=a_{s}=a_{i}$ in the Hamiltonian (13). This allows us to
derive the Hamiltonian that describes the single-mode Kerr nonlinearity and
its role in the transmission behavior of the nonlinear FDMR

$H=\hbar\omega_{\text{FDMR}}a_{\text{res}}^{\dagger}a_{\text{res}}-\hbar
g_{\text{nl}}a_{\text{res}}^{\dagger 2}a_{\text{res}}^{2},$ (37)

The equation of motion describing the dynamics of the system then is given by

$\dot{a}_{\text{res}}=-\Big{(}i\big{[}\omega_{\text{FDMR}}-2g_{\text{nl}}a_{\text{res}}^{\dagger}a_{\text{res}}\big{]}+\frac{\kappa}{2}\Big{)}a_{\text{res}}+\sqrt{\kappa^{\text{ext}}}a^{\text{ext}}+\sqrt{\kappa^{\text{int}}}a^{\text{int}},$
(38)

where $\kappa=\kappa^{\text{ext}}+\kappa^{\text{int}}$, with
$\kappa^{\text{ext}}$ and $\kappa^{\text{int}}$ represent the extrinsic and
intrinsic coupling rates, respectively. Furthermore, $a^{\text{ext}}$ and
$a^{\text{int}}$ are the annihilation operators of the input field from the
edge state and loss channels, respectively. In the frequency domain, we can
solve the coherent response of Eq. (38), leading to

$\langle a_{\text{res}}(\omega)\rangle=\frac{\sqrt{\kappa^{\text{ext}}}\langle
a^{\text{ext}}(\omega)\rangle}{\kappa/2-i\Big{(}\omega-\omega_{\text{FDMR}}+2g_{\text{nl}}\langle
a_{\text{res}}^{\dagger}a_{\text{res}}\rangle\Big{)}},$ (39)

here we consider $\langle a^{\text{int}}\rangle=0$. Using the input-output
theory
$a_{\text{out}}=a^{\text{ext}}-\sqrt{\kappa^{\text{ext}}}a_{\text{res}}$ we
can find the coherent transmission of the system

$T_{0}(\omega)=\frac{\langle a_{\text{out}}(\omega)\rangle}{\langle
a_{\text{ext}}(\omega)\rangle}=1-\frac{\kappa^{\text{ext}}}{\kappa/2-i\Big{(}\omega-\omega_{\text{FDMR}}+2g_{\text{nl}}n_{\text{res}}\Big{)}}$
(40)

which has a nonlinear dependency to the photons inside the cavity
$n_{\text{res}}=\langle a_{\text{res}}^{\dagger}a_{\text{res}}\rangle$, as
expected for a cavity-based Kerr nonlinear platforms. In the low-power regime,
the nonlinear term $2g_{\text{nl}}$ in the equation becomes negligible, and
the transmission function $T_{0}$ follows the standard Lorentzian function of
a cavity.

The Total transmission of the system is affected by several factors, which
include the reflection and transmission at the input ($r_{1},t_{1}$) and
output ($r_{2},t_{2}$) facets of the waveguide, as well as the coupling to and
from the edge state. By taking these reflection and transmission parameters
into account, we can determine the normalized power transmission at the output
of the lensed coupled waveguide facet [60]

$T(\omega)=\frac{|t_{1}t_{2}|^{2}|T_{0}(\omega)|^{2}}{\Big{|}1-r_{1}r_{2}T_{0}(\omega)^{2}e^{-i\phi}\Big{|}^{2}},$
(41)

where $\phi$ represents the propagation phase of light in the coupling
waveguide and edge state, which depends on the resonance frequency and
propagation length. To fit the transmission spectrum of the FDMR, as shown in
Fig. 2d of the main text, we used Eq. (41).

Idler | Signal
---|---
FSR | $Q_{t}$ | $Q_{e}$ | $Q_{i}$ | $Q_{t}$ | $Q_{e}$ | $Q_{i}$
1 | 62918 | 95050 | 186120 | 25471 | 30966 | 143542
2 | 30806 | 122399 | 41167 | 47116 | 124749 | 75712
3 | 47184 | 82361 | 110472 | 52778 | 83187 | 144382
4 | 62374 | 95585 | 179523 | 21536 | 108772 | 26852
5 | 33550 | 50157 | 101326 | 33174 | 51111 | 94531
6 | 54625 | 88776 | 142001 | 51767 | 78703 | 151259
7 | 75958 | 227300 | 114081 | 20122 | 113791 | 24445
8 | 36358 | 60597 | 90891 | 28535 | 43627 | 82483
9 | 46185 | 74315 | 122014 | 39249 | 61741 | 107736
Table 2: The extracted values for total $Q_{t}$, extrinsic $Q_{e}$ and
intrinsic $Q_{i}$ quality factors for nine FDMR modes.

###  .5 Second-order correlation function for FDMR

Figure 8 reports the raw data of the second-order cross-correlation function
$g^{(2)}_{si}$ and auto-correlation functions of the idler and signal
$g^{(2)}_{ss}$ and $g^{(2)}_{ii}$ for different pump powers at the presence of
the FDMR. The result indicates the violation of the Cauchy-Schwarz inequality
$[g^{(2)}_{si}(t)]^{2}\leq g^{(2)}_{ss}(t)\times g^{(2)}_{ii}(t)$ at time
$t=0$.

Figure 8: Comparing time-dependent the second-order cross-correlation function
$g^{(2)}_{si}(t)$ and auto-correlation functions of the idler and signal
$g^{(2)}_{ss}(t)$ and $g^{(2)}_{ii}(t)$ for different pump powers at the
presence of the FDMR.

###  .6 Coincidence rate: FDMR versus edge state

In this section, we present the raw data of the measured coincidence counts of
the FDMR and edge state, including the detection efficiency and setup loss
without calibration i.e. $N_{\text{tot},c}=\eta_{i}\eta_{s}N_{c}$, as
described in the Methods. Due to the small size of our sample and the low
efficiency of photon conversion in SFWM, the edge state cannot generate
significant photon pairs in the system. This limitation is attributed to the
short interaction time available for generating pair-correlated photons in the
edge state, which is considerably smaller compared to the resonance condition.

As we increase the power of the laser, noise and pump leakage to the
photodetector become prominent. Therefore, no substantial coincidences have
been observed in the edge state for various laser powers. In contrast, when
the FDMR is activated, a significant level of coincidence has been detected.
Figure 9 illustrates that the noise level increases as the laser power is
raised.

Figure 9: The measured (uncalibrated) coincidence rate of the edge state and
the FDMR as a function of arrival time at different pump powers.

###  .7 Phase detune calibration

To calibrate the heating power used in our experiment and to find its relation
with the applied phase detune, $\Delta\phi$, we tuned the heating power from 7
to 27 mW and we measured the wavelength of the excited FDMR shown in Fig. 10a.
In addition, we simulated the transmission spectrum of the lattice and we
plotted the wavelength of the FDMR for the phase detuning of
$\Delta\phi=1.5\pi$ to $2.65\pi$ applied to an entire micoring at the boundary
of the lattice, Fig. 10b. The simulation and experimental results show that
the wavelength of FDMR is linearly shifted by increasing the phase detune and
heating power. By comparing the simulation and experiment, we obtained the
$\Delta\phi=2.37\pi$ for excitation of the FDMR at the wavelength of 1545.265
nm.

Figure 10: a and b,) Phase detune calibration using plots FDMR wavelength
versus heating power and $\Delta\phi$.


\frac{i\nu(-k_{0})}{{\beta^{r}}(\xi)}e^{-\frac{3\pi}{4}(\nu(-k_{0})+i)}D_{i\nu(-k_{0})-1}(e^{-\frac{3\pi
i}{4}}\zeta)&e^{\frac{\pi}{4}\nu(-k_{0})}D_{-i\nu(-k_{0})}(e^{-\frac{\pi}{4}i}\zeta)\end{array}\right).$
(B.46)

when $\zeta\in\mathbb{C}^{-}$,

$\displaystyle
m_{0}(\zeta)=\left(\begin{array}[]{cc}e^{\frac{\pi\nu(-k_{0})}{4}}D_{i\nu(-k_{0})}(e^{\frac{\pi}{4}i}\zeta)&-\frac{i\nu(-k_{0})}{{\gamma^{r}}(\xi)}e^{-\frac{3\pi}{4}(\nu(-k_{0})-i)}D_{-i\nu(-k_{0})-1}(e^{\frac{3\pi
i}{4}}\zeta)\\\
\frac{i\nu(-k_{0})}{{\beta^{r}}(\xi)}e^{\frac{\pi}{4}(\nu(-k_{0})+i)}D_{i\nu(-k_{0})-1}(e^{\frac{\pi
i}{4}}\zeta)&e^{-\frac{3\pi}{4}\nu(-k_{0})}D_{-i\nu(-k_{0})}(e^{\frac{3\pi}{4}i}\zeta)\end{array}\right),$
(B.49)

Which is derived in [23].

From (B.30), we know that $(m_{0,-})^{-1}m_{0,+}=J_{0}$ and

$\displaystyle q^{r}_{1}(-k_{0})$
$\displaystyle=m_{0,-,11}m_{0,+,21}-m_{0,-,21}m_{0,+,11}$
$\displaystyle=e^{\frac{\pi}{4}\nu(-k_{0})}D_{i\nu(-k_{0})}(e^{\frac{\pi}{4}i}\zeta)\cdot\frac{e^{-\frac{3\pi\nu(-k_{0})}{4}}}{\beta^{r}(\xi)}\left[\partial_{\zeta}(D_{i\nu(-k_{0})}(e^{-\frac{3\pi
i}{4}}\zeta))+\frac{i\zeta}{2}D_{i\nu(-k_{0})}(e^{-\frac{3\pi
i}{4}}\zeta)\right]$ $\displaystyle\hskip
10.00002pt-e^{-\frac{3\pi}{4}\nu(-k_{0})}D_{i\nu(-k_{0})}(e^{-\frac{3\pi}{4}i}\zeta)\cdot\frac{e^{\frac{\pi\nu(-k_{0})}{4}}}{{\beta^{r}}(\xi)}\left[\partial_{\zeta}(D_{i\nu(-k_{0})}(e^{\frac{\pi
i}{4}}\zeta))+\frac{i\zeta}{2}D_{i\nu(-k_{0})}(e^{\frac{\pi
i}{4}}\zeta)\right]$
$\displaystyle=\frac{e^{-\frac{\pi}{2}\nu(-k_{0})}}{{\beta^{r}}(\xi)}{\rm
Wr}\left(D_{i\nu(-k_{0})}(e^{\frac{\pi}{4}i}\zeta),D_{i\nu(-k_{0})}(e^{-\frac{3\pi}{4}i}\zeta)\right)$
$\displaystyle=\frac{e^{-\frac{\pi}{2}\nu(-k_{0})}}{{\beta^{r}}(\xi)}\cdot\frac{\sqrt{2\pi}e^{\frac{\pi}{4}i}}{\Gamma(-i\nu(-k_{0}))}.$
(B.50)

And

$\displaystyle{\beta^{r}}(\xi)=\frac{\sqrt{2\pi}e^{\frac{i\pi}{4}}e^{-\frac{\pi\nu(-k_{0})}{2}}}{{q}^{r}_{1}(-k_{0})\Gamma(-i\nu(-k_{0}))},\quad{\gamma^{r}}(\xi)=\frac{\sqrt{2\pi}e^{-\frac{i\pi}{4}}e^{-\frac{\pi\nu(-k_{0})}{2}}}{{q}^{r}_{2}(-k_{0})\Gamma(i\nu(-k_{0}))}$
(B.51)

Finally we have

$m^{pc}_{-k_{0}}=I+\frac{i}{\zeta}\left(\begin{array}[]{cc}0&-{\beta^{r}}(\xi)\\\
{\gamma^{r}}(\xi)&0\end{array}\right)+\mathcal{O}(\zeta^{-2}).$ (B.52)

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