# A multiplexed quantum repeater based on single-photon interference
with mild stabilization

Daisuke Yoshida<EMAIL_ADDRESS>Yokohama National University, 79-5
Tokiwadai, Hodogaya, Yokohama 240-8501, Japan  LQUOM Inc., 79-5 Tokiwadai,
Hodogaya, Yokohama 240-8501, Japan  Tomoyuki Horikiri  Yokohama National
University, 79-5 Tokiwadai, Hodogaya, Yokohama 240-8501, Japan  LQUOM Inc.,
79-5 Tokiwadai, Hodogaya, Yokohama 240-8501, Japan

###### Abstract

Quantum repeaters are pivotal in the physical layer of the quantum Internet.
For its future development, it is desirable to have quantum repeaters capable
of facilitating robust and high-speed communication. In terms of efficiency,
quantum repeater schemes based on single-photon interference are seen as
promising. However, this method, involving first-order interference with light
sources at distant nodes, requires stringent phase stability in the
components. In this paper, we present a quantum repeater scheme that leverages
single-photon interference, utilizing multimode quantum memories and multimode
two-photon sources. Compared to conventional quantum repeater methods, our
proposed scheme significantly reduces the phase stability requirements by
several orders of magnitude. Additionally, under specific conditions, it is
demonstrated that our scheme achieves a higher coincidence rate between end
nodes compared to existing schemes.

##  I INTRODUCTION

The quantum internet [1, 2, 3], a global network for distributing
entanglement, is anticipated to facilitate technological advancements such as
distributed quantum computation [4, 5, 6, 7],quantum sensor networks [8, 9,
10, 11], and end-to-end quantum key distribution [12, 13, 14]. For the
globalization of entanglement distribution, it is considered essential to have
quantum repeaters that expand quantum entanglement from between elementary
links to between end nodes through entanglement swapping [15].

One of the promising configurations for elementary links with quantum
repeaters involves each node having an absorptive memory and a two-photon
source [16]. A quantum repeater using a built-in type of quantum memory in
which the photon source and memory are integrated is also possible [17, 18,
19, 20, 21, 22, 23, 24]. However, most quantum memories do not emit photons in
the telecom wavelength band. On the other hand, in a configuration using
absorptive memory, a nondegenerate two-photon source can be used to select the
telecom wavelength band as the photon for long-distance transmission. In
addition, time-division multiplexing, which is advantageous for high rates,
has recently been demonstrated by two experimental groups [25, 26]. Although
the configuration is similar in these two demonstrations, the entanglement
generation schemes are different. Ref. [25] demonstrated a single-photon
interference-based single-excitation entanglement generation scheme (SS),
while Ref. [26] demonstrated a two-photon interference-based two-excitation
entanglement generation scheme (TT). SS is more efficient than TT in scenarios
with low photon pair generation rates or high optical transmission losses,
such as long elementary link distances [27]. This is because TT’s entanglement
heralding relies on two-photon coincidence counting at the central station,
whereas SS uses single-photon detection. However, SS requires precise control
or monitoring of photon path sub-wavelengths and the phase of the laser
pumping the two-photon source, presenting significant challenges in phase
control.

In this paper, we propose a single-photon interference-based two-excitation
entanglement generation scheme (ST). This scheme significantly reduces the
phase requirement by over two orders of magnitude compared to traditional
single-photon interference methods, while preserving the high efficiency of
entanglement generation. We demonstrate that ST can outperform the SS in
entanglement generation rates under specific conditions and discuss potential
setups for implementing ST.

Our approach is inspired by the mode-pairing quantum key distribution [28,
29], and a frequency multiplexed quantum repeater [30].

##  II Results

Figure 1: (a) Schematic diagram of the ST scheme’s elementary link. One photon
of each pair emitted by the two-photon source (TPS) is transmitted to a remote
station and inputted into the central Bell state analyzer (CBSA). The other
photon is inputted into a multimode quantum memory (MQM). $L_{\mathrm{A(B)i}}$
denotes the optical path length from Node A(B)’s TPS to the CBSA.
$L_{\mathrm{A(B)s}}$ represents the optical path length from Node A(B)’s TPS
to MQM. (b) Overview of entanglement swapping in the ST scheme. Photons are
regenerated from the MQM to coincide in timing with photons regenerated from
adjacent MQMs. Additionally, Frequency Mode Matching (FMM) is employed to
achieve indistinguishability in the frequency domain of photons inputted into
the Swapping Bell State Analyzer (SBSA). (c) Schematic representation of the
TPS. (d) Schematic diagram of the MQM. (e) Overview of the FMM process.

###  II.1 Components of the proposed scheme

Figure 1(a) presents a schematic diagram of the entanglement generation
process in an elementary link using the ST. Prior to detailing the procedure,
we outline the function of each component as follows.

####  II.1.1 A two-photon source

Figure 1(c) illustrates an overview of the two-photon source (TPS) utilized in
the ST. The TPS generates photon pairs across $M$ frequency modes and $N$
temporal modes in each trial, with a frequency mode interval of
$\mathit{\Delta_{\mathrm{f}}}$ and a temporal mode interval of
$\mathit{\Delta_{\mathrm{t}}}$. The probability of photon pair emission in
each mode is $p_{\mathrm{tps}}$. One photon of each pair is designated as the
signal photon, transmitted to the adjacent multimode quantum memory (MQM), and
the other as the idler photon, sent to the central Bell state analyzer (CBSA)
via a long-distance transmission line. These photon pairs are phase-
correlated, as is typical in spontaneous parametric down-conversion (SPDC)
processes. In SPDC, the sum of the phases of signal and idler photons is
determined by the pump laser’s phase [31, 32]. We assume that the sum of the
phases for signal and idler photons remains constant across all modes within
the same trial at each node in this scheme. The sum of phases for photon pairs
at node $Z$ is denoted as $\mathit{\theta_{Z}}$. The frequencies of idler and
signal photons in the $m$-th frequency mode at node $Z$ are represented as
$f_{Z\mathrm{i},m}$ and $f_{Z\mathrm{s},m}$, respectively. In this scheme,
idler photons from adjacent nodes are assumed to have matching frequencies in
each mode for interference at the CBSA, i.e.
$f_{\mathrm{i},m}=f_{A\mathrm{i},m}=f_{B\mathrm{i},m}$. Additionally, we
assume that the sum of the frequencies of idler and signal photons remains
constant within each node, i.e.,
$f_{\mathrm{i},m}+f_{Z\mathrm{s},m}=f_{\mathrm{i},m^{\prime}}+f_{Z\mathrm{s},m^{\prime}}$.
This assumption aligns with the principles of SPDC, where typically, the
combined frequencies of these photon pairs equal the frequency of the pump
laser. Although SPDC processes can potentially produce multiple photon pairs,
we operate under the assumption that the probability of such occurrences is
negligible [27, 30]. Thus, the state of photon pairs at node Z in one trial is
represented as
$\sum_{n=1}^{N}\sum_{m=1}^{M}(\sqrt{1-p_{\mathrm{tps}}}+e^{i\theta_{Z}}\sqrt{p_{\mathrm{tps}}}z^{\dagger}_{\mathrm{s},mn}z^{\dagger}_{\mathrm{i},mn})\ket{0}$,
where $z_{\mathrm{s(i)},mn}$ signifies the mode for the $m$-th frequency and
$n$-th temporal mode signal (idler) photon at node $Z$, and $\ket{0}$ is the
vacuum state.

####  II.1.2 A multimode quantum memory

Figure 1(d) provides an overview of the Multimode Quantum Memory (MQM). We
assume that the MQM at each node can store signal photons in $M$ frequency
modes and $N$ temporal modes, generated by an adjacent TPS. The MQM is also
assumed to preserve the phase of each input photon across all modes and
regenerate them at any desired time. Essentially, the MQM can transform input
photons into any temporal modes. Within the ST, this capability enables the
conversion of photons from two different modes into temporal modes
$t_{\mathrm{e}}$ and $t_{\mathrm{l}}$, respectively, forming a time-bin qubit
with a time interval $\mathit{\Delta_{\mathrm{t}}}$. As depicted in Figure
1(d), the MQM, for instance, converts photons in temporal modes $n$ and
$n^{\prime}$ into $t_{\mathrm{e}}$ and $t_{\mathrm{l}}$. We also assume that
the MQM’s efficiency, denoted as $\eta_{\mathrm{QM}}$, is uniform across all
modes.

####  II.1.3 A frequency mode mapper

Figure 1(e) shows an overview of the Frequency Mode Mapper (FMM). We assume
that the FMM can convert the frequency of an incoming photon to a pre-
determined frequency, denoted as $f_{\mathrm{fix}}$. As illustrated in Figure
1(e), the FMM, for example, modulates photons in frequency modes $m$ and
$m^{\prime}$ to $f_{\mathrm{fix}}$. We also assume that the modulation
efficiency of the FMM, $\eta_{\mathrm{FM}}$, is consistent across all modes.

####  II.1.4 A Bell state analyzer

In this scheme, two types of BSAs are utilized: a CBSA, positioned between
distant nodes, and a swapping BSA (SBSA) for entanglement swapping. The CBSA
transforms two input modes, $a$ and $b$, into two output modes, $d_{+}$ and
$d_{-}$, with $d_{\pm}=(a\pm b)/\sqrt{2}$ [17]. Each output mode is equipped
with a photon detector capable of identifying the temporal and frequency modes
of detected photons. In the SS, detecting a single photon in either $d_{+}$ or
$d_{-}$ mode in each trial signifies successful entanglement generation
between elementary links. The SBSA is a type-2 fusion gate-based BSA designed
for time-bin qubits with a time interval $\mathit{\Delta_{\mathrm{t}}}$ [33].

###  II.2 Description of the proposed scheme

####  II.2.1 Elementary link

In this section, we describe the entanglement generation procedure of the SS
between nodes A and B, forming an elementary link. The generation of each
photon pair is timed so that photons in the same temporal mode arrive
simultaneously at the CBSA. The phase acquired by the idler (signal) photon of
the $m$-th frequency mode and $n$-th temporal mode at node $Z$, on its path
from the TPS to the CBSA (MQM), is $2\pi
f_{\mathrm{i(\mathit{Z}s)},m}L_{Z\mathrm{i}(Z\mathrm{s}),n}/c$, where
$L_{Z\mathrm{i}(Z\mathrm{s}),n}$ represents the optical path length and $c$ is
the speed of light in the optical path. We assume that during
$N\mathit{\Delta_{\mathrm{t}}}$, the fluctuations in the optical path lengths
are sufficiently small compared to the wavelengths. Consequently, the optical
path length is independent of the temporal mode,
i.e.,$L_{Zs(Zi),n}=L_{Zs(Zi)}$. Detection of a photon in the $m$-th frequency
mode and $n$-th temporal mode at $d_{\pm}$ results in the following states
between the memories:

$\ket{\Psi_{\pm}}_{\mathrm{AB},mn}=\frac{1}{\sqrt{2}}(a^{\dagger}_{\mathrm{s},mn}\pm
e^{i\theta_{\mathrm{AB},m}}b^{\dagger}_{\mathrm{s},mn})\ket{0},$ (1)

where,

$\displaystyle\theta_{\mathrm{AB},m}=\frac{2\pi}{c}(f_{\mathrm{i},m}\mathit{\Delta_{L_{\mathrm{i}}}}+f_{B\mathrm{s},m}L_{B\mathrm{s}}-f_{A\mathrm{s},m}L_{A\mathrm{s}})+\mathit{\Delta_{\theta}},$
(2)

with $\mathit{\Delta_{L_{\mathrm{i}}}}=L_{\mathrm{Bi}}-L_{\mathrm{Ai}}$ and
$\mathit{\Delta_{\theta}}=\theta_{\mathrm{B}}-\theta_{\mathrm{A}}$. The global
phase is omitted for simplicity. The state described in Eq. (1) is also
generated between memories in the SS.

Now, let’s consider the scenario where one photon in the $m$-th frequency mode
and $n$-th temporal mode, and another photon in the $m^{\prime}$-th frequency
mode and $n^{\prime}$-th temporal mode, are detected in $d_{+}$. When
extracting only the case where each of the nodes A and B has one excitation in
the state
$\ket{\Psi_{+}}_{\mathrm{AB},mn}\otimes\ket{\Psi_{+}}_{\mathrm{AB},m^{\prime}n^{\prime}}$,
the resulting state between the memories is given by:

$\begin{split}\ket{\Psi_{++}}_{\mathrm{AB},mnm^{\prime}n^{\prime}}=&\frac{1}{\sqrt{2}}(a^{\dagger}_{\mathrm{s},mn}b^{\dagger}_{\mathrm{s},m^{\prime}n^{\prime}}\\\
&+e^{i\theta_{\mathrm{AB},mm^{\prime}}}a^{\dagger}_{\mathrm{s},m^{\prime}n^{\prime}}b^{\dagger}_{\mathrm{s},mn})\ket{0},\end{split}$
(3)

where,

$\displaystyle\begin{split}\theta_{\mathrm{AB},mm^{\prime}}=\frac{2\pi(m-m^{\prime})\mathit{\Delta}_{\mathrm{f}}}{c}\big{(}\mathit{\Delta_{L_{\mathrm{i}}}}-\mathit{\Delta_{L_{\mathrm{s}}}}\big{)},\end{split}$
(4)

with $\mathit{\Delta_{L_{\mathrm{s}}}}=L_{\mathrm{Bs}}-L_{\mathrm{As}}$ As in
Eq. (1), the global phase is omitted in Eq. (3). The state in Eq. (3)
represents the two-excitation entanglement state created between the memories
of elementary links in the ST scheme.

Comparing the relative phase $\theta_{\mathrm{AB},m}$ (Eq. (2)) in the SS with
$\theta_{\mathrm{AB},mm^{\prime}}$ (Eq. (4)) in the ST scheme highlights two
key points. Firstly, the impact of optical path fluctuations on the phase is
significantly smaller in the ST scheme. In SS, the phase term involves the
product of each photon’s frequency and the optical path, whereas in ST, it
involves the product of the frequency mode spacing and the optical path. Given
that photon frequencies are typically around $\sim 10^{14}~{}\mathrm{Hz}$ and
frequency mode spacing can be much smaller, e.g., $\sim
10^{10}~{}\mathrm{Hz}$, the optical path stability required for equivalent
entanglement fidelity in ST is reduced by approximately four orders of
magnitude compared to SS. Secondly, in ST, the sum of the photon pair initial
phases does not influence the entangled state’s phase. In SS,
$\theta_{\mathrm{AB},m}$ can vary each trial as
$\mathit{\Delta_{\mathrm{\theta}}}$ affects the phase, but in ST,
$\mathit{\Delta_{\mathrm{\theta}}}$ is a global phase and does not impact
$\theta_{\mathrm{AB},mm^{\prime}}$. When SPDC is utilized as the TPS, where
the photon pair’s phases are determined by the pump laser’s phase, the ST only
necessitates a pump laser with a coherence time longer than
$N\mathit{\Delta_{\mathrm{t}}}$. This means phase synchronization or
monitoring the phase difference between remote lasers is not required in ST.

In Eq. (3), the combination of temporal and frequency modes varies with each
trial, posing challenges for quantum swapping and other applications. To
address this, the modes of regenerated photons are adjusted to a predetermined
mode using MQMs and FMMs. By converting the modes of
$\ket{\Psi_{++}}_{\mathrm{AB},mm^{\prime}}$ and applying phase modulation to
achieve $\theta_{\mathrm{AB},mm^{\prime}}=0$, we obtain the following time-bin
entangled state $\ket{\Psi_{++}}_{\mathrm{AB}}$:

$\ket{\Psi_{++}}_{\mathrm{AB}}=\frac{1}{\sqrt{2}}(a^{\dagger}_{\mathrm{s,e}}b^{\dagger}_{\mathrm{s,l}}+a^{\dagger}_{\mathrm{s,l}}b^{\dagger}_{\mathrm{s,e}})\ket{0}.$
(5)

####  II.2.2 Entanglement swapping

”This section describes the entanglement swapping process between the heralded
states $\ket{\Psi_{++}}_{\mathrm{AB}}$ and $\ket{\Psi_{++}}_{\mathrm{CD}}$, as
illustrated in Figure 1(b). Previously, we focused on cases where two memories
in an elementary link are each excited once. Now, we extend our analysis to
include scenarios where one memory is doubly excited, resulting in the
combined state:

$\begin{split}&\ket{\Psi_{++}}_{\mathrm{AB}}\otimes\ket{\Psi_{++}}_{\mathrm{CD}}\\\
&=\frac{1}{4}(a^{\dagger}_{\mathrm{s,e}}b^{\dagger}_{\mathrm{s,l}}+a^{\dagger}_{\mathrm{s,l}}b^{\dagger}_{\mathrm{s,e}}+a^{\dagger}_{\mathrm{s,e}}a^{\dagger}_{\mathrm{s,l}}+b^{\dagger}_{\mathrm{s,e}}b^{\dagger}_{\mathrm{s,l}})\\\
&\otimes(c^{\dagger}_{\mathrm{s,e}}d^{\dagger}_{\mathrm{s,l}}+c^{\dagger}_{\mathrm{s,l}}d^{\dagger}_{\mathrm{s,e}}+c^{\dagger}_{\mathrm{s,e}}c^{\dagger}_{\mathrm{s,l}}+d^{\dagger}_{\mathrm{s,e}}d^{\dagger}_{\mathrm{s,l}})\ket{0}.\end{split}$
(6)

To avoid successful entanglement swapping when one memory is doubly excited,
we use a type-2 fusion gate-based Bell measurement in the SBSA [34, 33].
Ideally, with perfect device efficiencies, successful Bell measurement events
happen only when each node has exactly one excited memory. However, in
practice, due to losses in each device, the resultant state $\rho_{\rm{AD}}$
between nodes A and D after a successful Bell measurement is expressed as:

$\rho_{\rm{AD}}=\alpha_{11}\rho_{11}+\alpha_{1}\rho_{1}+\alpha_{0}\rho_{0},$
(7)

where $\rho_{11}$ is the state with one excitation in both memories,
$\rho_{1}$ represents the state with a single excitation across the two
memories, and $\rho_{0}$ is the vacuum state. The coefficients $\alpha_{11}$,
$\alpha_{1}$, and $\alpha_{0}$ correspond to the probabilities of these
states. (For more details, please refer to Supplementary Note 1).

###  II.3 Performance comparison

Figure 2: (a) Computed coincidence rates for each scheme within the elementary
link. The horizontal axis represents the inter-node distance. Solid lines
depict results with a frequency mode number $M=100$, while dashed lines
represent $M=10$. For both, the temporal mode number is set at $N=10$. The
dotted line considers the original DLCZ scheme, assuming no multiplexing for
both time and frequency modes; hence, no FMM is performed. Other parameters
include detector efficiency $\eta_{\mathrm{det}}=0.9$, quantum memory storage
efficiency $\eta_{\mathrm{QM}}=0.5$, temporal mode interval
$\mathit{\Delta_{\mathrm{t}}}=620\mathrm{ns}$, and an optical path loss of
0.2dB/km. (b) Results from (a) are normalized by the number of frequency
modes.

Next, we compare the coincidence rates of photons emitted from the memory of
end nodes in the SS, ST, and TT. For this comparison, we consider a symmetric
setup where the distance $L/2$ from each node to the CBSM is identical.
Additionally, we assume that the coherence time is sufficiently long to ensure
that the efficiency of the MQM is not affected by storage time. We also
presume that the dark count of the photon detectors is negligible. In
practice, detectors with counts lower than Hz have been achieved [35]. The
detection efficiency of the photon detectors is denoted as
$\eta_{\mathrm{det}}$. To ensure a fair comparison, we suppose that the SS and
TT have the same multiplicity as the ST scheme and that mode conversion is
feasible.

We first analyze the coincidence rate in the elementary link. Let
$K_{X}^{(0)}$ represent the number of heralding events for entanglement in
each scheme, where $X=\mathrm{SS,ST,TT}$. In the SS, $K_{\mathrm{SS}}^{(0)}$
is the number of modes where a single photon detection event occurs in CBSM.
Conversely, in the TT scheme, $K_{\mathrm{TT}}^{(0)}$ equals the number of
modes with a specific two-photon detection event in CBSM [34, 36]. For the ST
scheme, $K_{\mathrm{ST}}^{(0)}=\lceil K_{\mathrm{SS}}^{(0)}/2\rceil$, as it is
established by pairing two single-excitation entanglements. Consequently, the
expected values of $K_{\mathrm{SS}}^{(0)}$, $K_{\mathrm{TT}}^{(0)}$, and
$K_{\mathrm{ST}}^{(0)}$ are calculated as follows:

$\displaystyle E(K_{\mathrm{SS}}^{(0)})$
$\displaystyle=2NMp_{\mathrm{tps}}\eta_{\mathrm{det}}\eta_{\mathrm{att}}\left(\frac{L}{2}\right),$
(8) $\displaystyle E(K_{\mathrm{TT}}^{(0)})$
$\displaystyle=\frac{1}{2}NM\left(p_{\mathrm{tps}}\eta_{\mathrm{det}}\eta_{\mathrm{att}}\left(\frac{L}{2}\right)\right)^{2},$
(9)
$\begin{split}E(K_{\mathrm{ST}}^{(0)})&=\left(\sum_{k=0}^{\frac{NM}{2}-1}k\left(P_{\mathrm{h}}(2k)+P_{\mathrm{h}}(2k+1)\right)\right)\\\
&+\frac{NM}{2}P_{\mathrm{h}}(NM),\end{split}$ (10)

where $NM$ is assumed to be an even number. $P_{\mathrm{h}}(k)$ is defined as
${}_{NM}\mathrm{C}_{k}(2p_{\mathrm{tps}}\eta_{\mathrm{det}}\eta_{\mathrm{att}}(\frac{L}{2}))^{k}(1-2p_{\mathrm{tps}}\eta_{\mathrm{det}}\eta_{\mathrm{att}}(\frac{L}{2}))^{NM-k}$.

From Eq. (8) and (9), it’s evident that $E(K_{\mathrm{SS}}^{(0)})$ is linearly
affected by various efficiencies, whereas $E(K_{\mathrm{TT}}^{(0)})$ is
impacted quadratically. Fig. 2(a) displays the calculated two-photon
coincidence rate for photons re-emitted from both memories in the elementary
link. The derivation methods are detailed in the Methods section “Calculations
of coincidence rate at elementary link”. The parameters used in these
calculations are specified in the caption of Fig. 2. The ST scheme’s rate
decays at a slope similar to the SS scheme in low transmission loss regions,
and closer to the TT scheme in high loss regions. Notably, there are distances
where the SS and ST rates are comparable. Fig. 2(b) then presents the
coincidence rate normalized by the frequency mode number. In the ST scheme,
the distance at which the rate’s slope changes is influenced by the frequency
mode number. Interestingly, even with a relatively small number of modes, such
as M = 1, the normalized rate in the ST exceeds that of the TT by more than an
order of magnitude.

Figure 3:  Diagram of a transmission path composed of $2^{J}$ elementary
links, where Node 1 and Node $2^{(J+1)}$ serve as the end nodes. In the $j$ th
round of swapping, Bell measurements are performed at nodes $2^{j}(2h-1)$ and
$2^{j}(2h-1)+1$, attempting to generate entanglement between these
$2^{j+1}(h-1)+1$ and $2^{j+1}l$ nodes.  Figure 4: The graph illustrates the
coincidence rate at the end nodes of a quantum communication path consisting
of $2^{j}$ elementary links The horizontal axis represents the distance
between end nodes. The rate at the number of links with the best rate at each
end node distance is plotted. The solid line depicts results using ideal
parameters with values $\eta_{\mathrm{det}}=0.95$, $\eta_{\mathrm{QM}}=0.9$,
and $\eta_{\mathrm{FM}}=0.95$. The dashed line represents results using
realistic parameters with values $\eta_{\mathrm{det}}=0.9$,
$\eta_{\mathrm{QM}}=0.5$, and $\eta_{\mathrm{FM}}=0.9$. Other parameters were
the same for both, $p_{\mathrm{tps}}=0.01$, $M=100$, and $N=30$.

Next, we analyze the coincidence rate following entanglement swapping. Let’s
assume that the communication channel consists of $2^{J}$ elementary links
($J\geq 1$) and involves $J$ rounds of swapping, as shown in Fig. 3. In the
$j$-th round of swapping, Bell measurements are conducted on nodes
$2^{j}(2h-1)$ and $2^{j}(2h-1)+1$ ($h\leq 2^{J-j}$). The total time
$T^{(J)}_{\mathrm{tot},X}$, starting from the beginning of the trial and
leading up to the point of achieving a single coincidence event from photons
regenerated at the end node’s memory, is expressed for $X=$ST or TT as
follows:

$\displaystyle
T^{(J)}_{\mathrm{tot},X}=\sum_{j=0}^{J-1}\left\\{t^{(j)}_{X}\frac{1}{p_{X}^{(J)}p_{\mathrm{ps,}X}^{(J)}}\prod_{h=j}^{J-1}\left(n_{\mathrm{ex}}(p_{X}^{(h)})\right)\right\\}.$
(11)

For $X=$SS, where coincidence is attempted at the end node using single
excitation entanglement in two different modes, it is represented as:

$\displaystyle
T^{(J)}_{\mathrm{tot},X}=\sum_{j=0}^{J}\left\\{t^{(j)}_{X}\frac{1}{p_{\mathrm{ps,}X}^{(j)}}\prod_{h=j}^{J}\left(n_{\mathrm{ex}}(p_{X}^{(h)})\right)\right\\}.$
(12)

Here, $t^{(j)}_{X}$ is the time required to inform the necessary node of the
results from the $j$-th round of entanglement swapping ($2^{j-1}L/c$ for
$j\geq 1$) or the time for measurement results from CBSM to reach the adjacent
node after initiating entanglement distribution on the elementary link
($L/c+N\mathit{\Delta}_{\mathrm{t}}$ for $j=0$). $p_{X}^{(h)}$ is the
probability of successful entanglement swapping in the $h$-th round for $h\geq
1$, and the probability of heralding at least one entanglement on the
elementary link for $h=0$. $n_{\mathrm{ex}}(p)$ is the expected number of
trials for two independent events with success probability $p$ to both succeed
at least once, given by $n_{\mathrm{ex}}(p)=\frac{3-2p}{(2-p)p}$.
$p_{\mathrm{ps,}X}^{(J)}$ is the probability of achieving coincidence at the
end node. It is also assumed that even if multiple entanglements are heralded
between memories, only one mode can be swapped. Details on each probability
calculation are provided in Supplementary Material Note 1.

Figure 4 depicts the coincidence rate
$R^{(J)}_{\mathrm{X}}=1/T^{(J)}_{\mathrm{tot},X}$ for $J=1,2,3,4,5$, where the
number of links has been optimized for each distance between end nodes to
maximize the rate. The parameters used in these calculations are listed in
Fig. 4’s caption. In practical scenarios (represented by dashed lines), the
ST’s rate is 1-2 orders of magnitude lower than the SS. This large difference
mainly stems from the different swapping requirements between two-excitation
and single-excitation entanglement: single-excitation entanglement involves
detecting a single photon, while two-excitation entanglement requires
coincidence detection [27]. In the optimistic scenario (solid lines), the
disparity between ST and SS narrows. Interestingly, some end-node distances
show the ST outperforming SS. This can be understood by examining the state
$\rho_{X}^{(J)}$ between end-node memories after a successful $J$-th swapping
operation, represented as follows [27, 34]:

$\displaystyle\rho_{\rm{SS}}^{(J)}=\alpha^{(J)}\rho_{1}+(1-\alpha^{(J)})\rho_{0},$
(13)
$\displaystyle\rho_{\rm{ST}}^{(J)}=\alpha_{11}^{(J)}\rho_{11}+\alpha_{1}^{(J)}\rho_{1}+\alpha_{0}^{(J)}\rho_{0},$
(14) $\displaystyle\rho_{\rm{TT}}^{(J)}=\rho_{11}.$ (15)

In SS, $\rho_{1}$ is the desired state, while $\rho_{0}$ is not.
$\alpha^{(J)}$ indicates the probability of $\rho_{\mathrm{SS}}$ being
$\rho_{1}$. For ST, $\rho_{1}1$ is desired, with $\rho_{1}$ and $\rho_{0}$ as
undesired states. The coefficients for each state, influencing the likelihood
of successful coincidence events, are described as:

$\displaystyle\alpha^{(J)}=\frac{\alpha^{(J-1)}}{2-\alpha^{(J-1)}\eta},$ (16)
$\displaystyle\alpha_{11}^{(J)}=\frac{1}{\left(1+\frac{\alpha_{1}^{(1)}}{2\alpha_{11}^{(1)}}\right)^{2}},$
(17)
$\displaystyle\alpha_{1}^{(J)}=\frac{\alpha_{1}^{(1)}}{\alpha_{11}^{(1)}\left(1+\frac{\alpha_{1}^{(1)}}{2\alpha_{11}^{(1)}}\right)^{2}},$
(18)
$\displaystyle\alpha_{0}^{(J)}=\left(\frac{\alpha_{1}^{(1)}}{2\alpha_{11}^{(1)}+\alpha_{1}^{(1)}}\right)^{2}.$
(19)

Here, $\eta$ is defined as
$\eta=\eta_{\mathrm{QM}}\eta_{\mathrm{FM}}\eta_{\mathrm{det}}$. Given that
$2-\alpha^{(J)}\eta>1$, $\alpha^{(J)}<\alpha^{(J-1)}$ as per Equation (16),
suggesting a decrease in the coefficient for the desired state $\rho_{1}$ in
SS as swapping operations increase. Conversely, in the ST scheme, according to
Equations (17) to (19), the coefficients are constant regardless of the number
of swapping operations. This constancy might explain why the rate of the ST
scheme surpasses that of the SS scheme in certain regions. For cases where
$J\geq 2$, the expressions in Equations (17) to (19) are applicable. For the
specific case of $J=1$, please refer to the expressions provided in
Supplementary Note 1.

##  III Discussion

We consider a potential configuration for ST, possibly involving the use of an
Atomic Frequency Comb (AFC) [37] created within the inhomogeneous broadening
of a Pr^3+:Y2SiO5 crystal (Pr:YSO) as the MQM, cavity-enhanced SPDC (cSPDC) as
the TPS [38], and a phase modulator as the FMM [39]. The AFC offers extensive
multiplexing capabilities. Particularly, the AFC in Pr:YSO, with a frequency
mode interval of 100 MHz and a temporal mode interval of 620 ns, is estimated
to store about 100 frequency modes and 32 temporal modes with an efficiency
exceeding 50% [40]. Already demonstrated is the on-demand storage of 30
temporal modes [40]. Additionally, storing multiple frequency mode photons
emitted from cSPDC has been achieved [41, 42]. Regarding the FMM, employing a
phase modulator for serrodyne modulation enables frequency modulation without
creating surplus modes [39]. The modulation bandwidth of the FMM should be
about $M\mathit{\Delta_{\mathrm{f}}}$, which is around 10 GHz when using a
Pr:YSO-based AFC as MQM. With high-bandwidth signal generators and phase
modulators, serrodyne modulation of several GHz or more is feasible [43]. For
sub-ns pulses, modulation of 20 GHz has been achieved with nearly 100%
efficiency [44].

Now, let’s consider the stabilization of the relative phase
$\theta_{\mathrm{AB},mm^{\prime}}$ (Eq. (4)) in the entangled state produced
by ST. Assuming that the phase fluctuation follows a normal distribution with
a standard deviation $\sigma$, the fidelity F can be expressed as
$(1+e^{-\frac{\sigma^{2}}{2}})/2$ [45]. For a fidelity of 0.99, $\sigma$ is
approximately 0.2 rad, which serves as our reference value. For
$(m-m^{\prime})\mathit{\Delta_{\mathrm{f}}}$ equal to 10 GHz, a displacement
of about 1 mm in $\mathit{\Delta_{L\mathrm{i}}}-\mathit{\Delta_{L\mathrm{s}}}$
results in a phase shift of roughly 0.2 rad.
Here,$\mathit{\Delta_{L\mathrm{s}}}$ denotes the difference in light paths
within a node, which are relatively short and more easily controlled for
environmental factors such as temperature. Conversely,
$\mathit{\Delta_{L\mathrm{i}}}$ represent the difference in longer light
paths, which could be field-deployed fiber, and are thus more susceptible to
uncontrolled environmental conditions. Therefore, it is assumed that the
dominant factor influencing $\sigma$ is the fluctuation in
$\mathit{\Delta_{L\mathrm{i}}}$, with the aim being to stabilize this
fluctuation to within approximately 1 mm. This stabilization could potentially
be achieved by transmitting a reference pulse from each node to the CBSM,
measuring the time difference in arrival with a high-speed detector, and
subsequently providing feedback to a piezoelectric actuator or a stage.
Moreover, employing a wavelength different from that of the idler photons for
the reference light in this stabilization process could allow for nearly 100%
duty cycle in entanglement distribution.

Finally, we emphasize the significance of reducing the requirements for
optical path length stability and phase stability of the pump lasers in TPSs,
which is a key aspect of the ST scheme. Stringent requirements for optical
path stability lead to frequent need for monitoring or stabilizing the phase
difference. This necessitates multiple interventions before an entangled state
is heralded between end nodes, potentially complicating both operational
procedures and the design of the optical system. For instance, in the process
of entanglement swapping between adjacent quantum memories, fluctuations in
the optical path on the order of sub-$\mathrm{\mu}$m can occur while waiting
for entanglement heralding in both memories. In the SS, simultaneously
monitoring or stabilizing the interferometer’s phase difference and storing
photons in quantum memory is not straightforward. However, by sufficiently
relaxing these requirements, such challenges can be mitigated. Additionally,
even with a stable optical path, when using SPDC for photon pairs, the SS
would require monitoring or locking the phase difference of pump lasers
between adjacent nodes. This might require a more complex system involving
optical frequency combs or wavelength conversion for phase stabilization or
monitoring, particularly because the wavelengths of pump lasers are generally
not well-suited for long-distance transmission. In contrast, the ST scheme, as
detailed in the Results section, only requires the pump light to have a
coherence time of about $N\mathit{\Delta_{\mathrm{t}}}$, eliminating the need
to stabilize or monitor phase differences between distant lasers. Therefore,
this relaxation of stability requirements is expected to greatly simplify the
setup.

##  IV Methods

###  IV.1 Calculations of coincidence rate at elementary link

Figure 5 illustrates the process of attempting coincidence detection at the
elementary link for each scheme. The duration of one trial,
$t_{\mathrm{ele}}$, spans from the start of transmitting the photon in the
first temporal mode to the CBSA, until each node receives the result of the
photon observation in the $N$-th temporal mode at CBSA, calculated as
$N\tau+L/c$.

For the SS, the results shown in Fig. 2 were generated using a Monte Carlo
simulation based on the flow in Fig. 5. In SS, a method for obtaining
coincidences through two transmission paths has been proposed [17, 46]. This
method involves heralding single excitation entanglement on both paths,
followed by attempts to achieve coincidences. Let $K^{(0)}_{\mathrm{SS}}$ and
$K^{\prime(0)}_{\mathrm{SS}}$ denote the number of heralded entanglements in
one trial for each path. Coincidence attempts are made
$\min\left(K^{(0)}_{\mathrm{ST}},K^{\prime(0)}_{\mathrm{ST}}\right)$ times
after both $K^{(0)}_{\mathrm{SS}}$ and $K^{\prime(0)}_{\mathrm{SS}}$ reach at
least 1. The probability of a successful coincidence in each trial is
$\eta^{2}/2$, where
$\eta=\eta_{\mathrm{QM}}\eta_{\mathrm{FM}}\eta_{\mathrm{det}}$.

For the ST and TT, $K_{\mathrm{ST}}^{(0)}$ and $K_{\mathrm{TT}}^{(0)}$ not
only represent the number of attempts for coincidence in one trial but also
indicate the number of heralded entanglements in that trial. The probability
of successful coincidence after heralding differs between ST and TT, as ST may
have two excitations on one memory side. The probabilities are $\eta^{2}/2$
and $\eta^{2}$, respectively. Thus, the coincidence rates $R_{ST}^{(0)}$ and
$R_{TT}^{(0)}$ for the elementary link in the ST and TT schemes are given by:

$\displaystyle
R_{\mathrm{ST}}^{(0)}=\frac{\eta^{2}E(K_{\mathrm{ST}}^{(0)})}{2t_{\mathrm{ele}}},$
(20) $\displaystyle
R_{\mathrm{TT}}^{(0)}=\frac{\eta^{2}E(K_{\mathrm{TT}}^{(0)})}{t_{\mathrm{ele}}}.$
(21)

The results for ST and TT in Fig. 2 were plotted using Equations (20) and
(21).

Figure 5: (a) Flowchart of attempted coinsidence in the ST scheme. $k$ and
$k^{\prime}$ represent the number of iterations until one or more
entanglements are heralded in each transmission channel. (b) and (c) are
flowcharts for attempting cointegration in the ST and TT schemes,
respectively.

###### Acknowledgements.

We thank Mayuka Ichihara and Koji Nagano for valuable discussions. This
research was supported by the SECOM Foundation, JSPS KAKENHI (JP20H02652),
Deep-Tech Startups Support Program (New Energy and Industrial Technology
Development Organization, Japan), JST Moonshot R&D (JPMJMS226C), and R&D of
ICT Priority Technology Project (JPMI00316) We also acknowledge the members of
the Quantum Internet Task Force, which is a research consortium aiming to
realize the quantum internet, for comprehensive and interdisciplinary
discussions of the quantum internet.

## Note added.

During the preparation of this work, a work by Li et al. [47] that employs a
similar idea of pairing two single-photon interference events appeared on
arXiv. Our contribution extends this concept by introducing frequency
multiplexing and proposing a specific physical system for implementation.

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## Supplimentary Note1: Calculation
of coincidence rates

###  IV.2  Calculation of coincidence rates
for SS schemes with $2^{j}$ links

In the context of the elementary link in the SS scheme, after a heralding
event occurs in a certain mode, the state between the memories of that mode is
denoted as
$\rho_{\mathrm{SS}}^{(0)}=\alpha^{(0)}\rho_{\mathrm{1}}+(1-\alpha^{(0)})\rho_{\mathrm{0}}$.
Assuming negligible dark counts, $\alpha^{(0)}=1$. Furthermore, the state
between memories after entanglement swapping between
$\rho_{\mathrm{SS}}^{(j-1)}$ (with $j\geq 1$) is represented as
$\rho_{\mathrm{SS}}^{(j)}=\alpha^{(j)}\rho_{\mathrm{1}}+(1-\alpha^{(j)})\rho_{\mathrm{0}}$.
Here, $\rho_{1}$ signifies the desired state with a single excitation
entangled state, while $\rho_{0}$ represents the vacuum state. Additionally,
$\alpha^{(0)}$ and $\alpha^{(j)}$ denote the probabilities that
$\rho_{\mathrm{SS}}^{(0)}$ and $\rho_{\mathrm{SS}}^{(j)}$ are in the desired
state, respectively. Considering entanglement swapping between
$\rho_{\mathrm{SS}}^{(j-1)}$ states, the successful event of this swapping
between $\rho_{\mathrm{SS}}^{(j-1)}$ states is characterized by the detection
of a single photon in SBSA. This event can unfortunately occur when both
states are $\rho_{1}$ ,albeit with losses, or if only one of them is
$\rho_{1}$. These unintended successful events contribute to the vacuum state
of $\rho_{\mathrm{SS}}^{(j)}$. The probabilities of successful events of
entanglement swapping between $\rho_{\mathrm{SS}}^{(j-1)}$ states,
$p_{\mathrm{SS}}^{(j)}$, and $\alpha^{(j)}$ are given as

$\displaystyle
p_{\mathrm{SS}}^{(j)}=\alpha^{(j-1)}\eta\left(1-\left(\frac{\alpha^{(j-1)}\eta}{2}\right)\right),$
(S1) $\displaystyle\alpha^{(j)}=\frac{\alpha^{(j-1)}}{2-\alpha^{(j-1)}\eta}.$
(S2)

In Equation (S2), the parameter $\alpha^{(j)}$ is determined by dividing the
probability of desired successful events in the swapping process, leading to
$\rho_{1}$, by $p_{\mathrm{SS}}^{(j)}$. After a pair of
$\rho_{\mathrm{SS}}^{(j)}$ is heralded at the end nodes of a quantum
communication channel composed of two parallel $2^{j}$-elementary links, the
probability of successfully attempting a coincidence is given by

$\displaystyle p_{\mathrm{ps,SS}}^{(j)}=\left(\alpha^{(j)}\eta\right)^{2}/2.$
(S3)

By substituting the probabilities $p_{\mathrm{SS}}^{(j)}$ and
$p_{\mathrm{ps,SS}}^{(j)}$, as presented earlier, into Equation (12) in the
main text, the coincidence rate in the SS scheme can be calculated.

###  IV.3 Calculation of coincidence rates
for ST schemes with $2^{j}$ links

In the ST scheme, the state heralded between the memories of the elementary
link is represented as
$\rho_{\mathrm{ST}}^{(0)}=\alpha_{\mathrm{11}}^{(0)}\rho_{\mathrm{11}}+\alpha_{\mathrm{20}}^{(0)}\rho_{\mathrm{20}}$.
$\rho_{\mathrm{11}}$ represents the desired state where both memories are
individually excited once, while $\rho_{\mathrm{20}}$ denotes the undesired
state where one memory is excited twice and the other memory is not excited.
Additionally, $\alpha_{\mathrm{11}}^{(0)}$ and $\alpha_{\mathrm{20}}^{(0)}$
represent the probabilities of $\rho_{\mathrm{ST}}^{(0)}$ being in states
$\rho_{\mathrm{11}}$ and $\rho_{\mathrm{20}}$, respectively. Next, we consider
the swapping of $\rho_{\mathrm{ST}}^{(0)}$ states. To reduce the occurrence of
unexpected success events in the swapping process, we employ a BSA based on a
type 2 fusion gate [34, 33]. In this case, the probability of a successful
swapping event is denoted as

$\begin{split}p_{\mathrm{ST}}^{(1)}=\eta^{2}(\alpha_{\mathrm{11}}^{(0)})^{2}/2&+\eta^{2}(1-\eta)\alpha_{\mathrm{11}}^{(0)}\alpha_{\mathrm{20}}^{(0)}\\\
&+\eta^{2}(1-\eta)^{2}(\alpha_{\mathrm{20}}^{(0)})^{2}/2.\end{split}$ (S4)

Furthermore, the state between the memories after a successful swap is given
by
$\rho_{\mathrm{ST}}^{(1)}=\alpha_{\mathrm{11}}^{(1)}\rho_{\mathrm{11}}+\alpha_{\mathrm{1}}^{(1)}\rho_{\mathrm{1}}+\alpha_{\mathrm{0}}^{(1)}\rho_{\mathrm{0}}$
, where
$\alpha_{\mathrm{11}}^{(1)}=\eta^{2}(\alpha_{\mathrm{11}}^{(0)})^{2}/2p_{\mathrm{ST}}^{(1)}$,
$\alpha_{\mathrm{1}}^{(1)}=\eta^{2}(1-\eta)\alpha_{\mathrm{11}}^{(0)}\alpha_{\mathrm{20}}^{(0)}/p_{\mathrm{ST}}^{(1)}$
and
$\alpha_{\mathrm{0}}^{(1)}=\eta^{2}(1-\eta)^{2}(\alpha_{\mathrm{20}}^{(0)})^{2}/2p_{\mathrm{ST}}^{(1)}$.
Moreover, the state between the memories after entanglement swapping between
$\rho_{\mathrm{ST}}^{(j-1)}$ (with $j\geq 1$) is represented as
$\rho_{\mathrm{ST}}^{(j)}=\alpha_{\mathrm{11}}^{(j)}\rho_{\mathrm{11}}+\alpha_{\mathrm{1}}^{(j)}\rho_{\mathrm{1}}+\alpha_{\mathrm{0}}^{(j)}\rho_{\mathrm{0}}$.
Here, for cases where $j\geq 2$, the probability of a successful swapping
event between $\rho_{\mathrm{ST}}^{(j-1)}$ is given by

$\begin{split}p_{\mathrm{ST}}^{(j)}=\eta^{2}(\alpha_{11}^{(j-1)})^{2}/2&+\eta^{2}\alpha_{11}^{(j-1)}\alpha_{1}^{(j-1)}/2\\\
&+\eta^{2}(\alpha_{1}^{(j-1)})^{2}/8,\end{split}$ (S5)

and the coefficients of $\rho$ are
$\alpha_{11}^{(j)}=\eta^{2}(\alpha_{11}^{(j-1)})^{2}/2p_{\mathrm{ps,ST}}^{(j)}$,
$\alpha_{1}^{(j)}=\eta^{2}\alpha_{11}^{(j-1)}\alpha_{1}^{(j-1)}/2p_{\mathrm{ps,ST}}^{(j)}$,
and
$\alpha_{0}^{(j)}=\eta^{2}(\alpha_{1}^{(j-1)})^{2}/8p_{\mathrm{ps,ST}}^{(j)}$.
When these calculations are employed, it becomes evident that for $j\geq 2$,
the values of $\alpha_{11}^{(j)}$, $\alpha_{1}^{(j)}$, and $\alpha_{0}^{(j)}$
do not depend on j. When attempting coincidence on the state
$\rho_{\mathrm{ST}}^{(j-1)}$, the success probability is given as

$\displaystyle p_{\mathrm{ps,ST}}^{(j)}=\eta^{2}(\alpha_{11}^{(j)})^{2}.$ (S6)

By substituting the probabilities $p_{\mathrm{ST}}^{(j)}$ and
$p_{\mathrm{ps,ST}}^{(j)}$, as presented earlier, into Equation (11) in the
main text, the coincidence rate in the ST scheme can be calculated.

###  IV.4 Calculation of coincidence rates
for TT schemes with $2^{j}$ links

In the TT scheme, this study assumes that multiple photon pairs do not occur
in a single mode in the each TPS. Based on this assumption, the state heralded
between the memory links of the elementary link is denoted as
$\rho^{(0)}_{\mathrm{TT}}=\rho_{\mathrm{11}}$. Furthermore, when performing
entanglement swapping between $\rho^{(0)}_{\mathrm{TT}}$, the success
probability $p^{(1)}_{\mathrm{TT}}$ using linear optical elements is
$\eta^{2}/2$. Subsequently, the state remains
$\rho^{(1)}_{\mathrm{TT}}=\rho_{\mathrm{11}}$. Similary, the state after
entanglement swapping between $\rho^{(j-1)}_{\mathrm{TT}}$ (with $j\geq 1$) is
$\rho^{(j)}_{\mathrm{TT}}=\rho_{\mathrm{11}}$ with a success probability of

$\displaystyle p^{(j)}_{\mathrm{TT}}=\eta^{2}/2.$ (S7)

Finally, when attempting coincidence measurements on the
$\rho^{(j)}_{\mathrm{TT}}$, the success probability is given by

$\displaystyle p^{(j)}_{\mathrm{ps,TT}}=\eta^{2}.$ (S8)

By substituting the probabilities $p_{\mathrm{TT}}^{(j)}$ and
$p_{\mathrm{ps,TT}}^{(j)}$, as presented earlier, into Equation (11) in the
main text, the coincidence rate in the TT scheme can be calculated.


# Near-unity third-harmonic circular dichroism driven by quasi-BIC in
asymmetric silicon metasurfaces

Marco Gandolfi<EMAIL_ADDRESS>CNR-INO (National Institute of
Optics), Via Branze 45, Brescia, Italy  Department of Information Engineering,
University of Brescia, Via Branze 38, Brescia, Italy  Andrea Tognazzi  CNR-INO
(National Institute of Optics), Via Branze 45, Brescia, Italy  Department of
Information Engineering, University of Brescia, Via Branze 38, Brescia, Italy
Davide Rocco  CNR-INO (National Institute of Optics), Via Branze 45, Brescia,
Italy  Department of Information Engineering, University of Brescia, Via
Branze 38, Brescia, Italy  Costantino De Angelis  CNR-INO (National Institute
of Optics), Via Branze 45, Brescia, Italy  Department of Information
Engineering, University of Brescia, Via Branze 38, Brescia, Italy  Luca
Carletti<EMAIL_ADDRESS>CNR-INO (National Institute of Optics), Via
Branze 45, Brescia, Italy  Department of Information Engineering, University
of Brescia, Via Branze 38, Brescia, Italy

###### Abstract

We use numerical simulations to demonstrate third-harmonic generation with
near-unity nonlinear circular dichroism (CD) and high conversion efficiency
($10^{-2}\ \text{W}^{-2}$) in asymmetric Si-on-SiO2 metasurfaces. The working
principle relies on the spin-selective excitation of a quasi-bound state in
the continuum, characterized by a very high ($>10^{5}$) quality-factor. By
tuning multi-mode interference with the variation of the metasurface
geometrical parameters, we show the possibility to control both linear CD and
nonlinear CD. Our results pave the way for the development of all-dielectric
metasurfaces for nonlinear chiro-optical devices with high conversion
efficiency.

Bound states in the continuum, asymmetry, third-harmonic generation, circular
dichroism, reflectivity, conversion efficiency, circularly polarized light

Polarization is one of the most fundamental properties of electromagnetic
radiation. The capability to measure, manipulate, and control the polarization
state of a light beam is essential for a huge range of applications such as
classical and quantum optical communications Wang et al. (2012, 2015); Virte
et al. (2013), biological and chemical sensing Barron (2009); Estephan et al.
(2010); Pellegrini et al. (2018), imaging and holography Ni et al. (2013); Lee
et al. (2020), spectropolarimetry and metrology Sterzik et al. (2012); Salvail
et al. (2013). In this context, optical rotation (OR) - rotation of the
polarization axis of linearly polarized waves - and circular dichroism (CD) -
difference in the absorption/reflectivity/transmittivity spectra of left-
circularly polarized (LCP) and right-circularly polarized (RCP) light - are of
paramount importance. Although these chiro-optical phenomena are extremely
weak in natural materials, the recent development of chiral metamaterials and
metasurfaces (MSs) Wang et al. (2016); Valev et al. (2013); Mun et al. (2020);
Hentschel et al. (2017); Keren-Zur et al. (2016) has demonstrated giant OR and
CD enhancement suitable for chiral biosensing and imaging Hentschel et al.
(2017), polarization conversion Zhao et al. (2012), negative refraction Pendry
(2004), spin-controlled wavefront shaping Ye et al. (2016), and chiral
detectors Li et al. (2015).
Most chiral metamaterials demonstrated so far are made of three-dimensional
(3D) metallic nanostructures Wang et al. (2016); Valev et al. (2013); Mun et
al. (2020); Hentschel et al. (2017); Fasold et al. (2018); Tanaka et al.
(2020). However, the complexity of the associated fabrication technology and
optical absorption have hindered their application. Although all-dielectric
chiral metamaterials have been recenlty demonstrated Tanaka et al. (2020),
two-dimensional (2D) or planar metamaterials - i.e. MSs - are highly
attractive due to compatibility with planar fabrication technologies. Despite
a lower chiro-optical response due to inherent reflection symmetry, planar
structures sustaining Fano resonances can exhibit strong non-reciprocal CD due
to modal interference Hopkins et al. (2016). Thus, the recent insurgence of
dielectric nanostructures and MSs enabled the realization of lossless planar
structures with strong chiro-optical effects Kuznetsov et al. (2012); Anthur
et al. (2020); Gómez et al. (2019); Gorkunov et al. (2018); Ullah et al.
(2018); Zhu et al. (2018); Solomon et al. (2018); Hu et al. (2019); Wu et al.
(2014); Ma et al. (2018); Gorkunov et al. (2020). Chiral dielectric MSs are
cornerstone for developing efficient integrated components enabling the
measurement of the polarization state of light which is an essential
functionality for many applications Li et al. (2015); Gorkunov et al. (2020);
Tanaka et al. (2020); Overvig et al. (2021).
Over the past decade, investigations of nonlinear optical effects in chiral
metamaterials and MSs have grown in interest Wang et al. (2016); Li et al.
(2017); Ren et al. (2012); Chen et al. (2016); Tang et al. (2020); Kim and Kim
(2020); Kim et al. (2020) due to the possibility to enhance the chiro-optical
response. Nonlinear CD, i.e. strong difference in second-harmonic generation
(SHG) or third-harmonic generation (THG) conversion efficiency between LCP and
RCP pump, opens to new possible applications such as nonlinear holography,
nonlinear wavefront control, and chiral sensors. Although near-unity nonlinear
CD has been demonstrated Wang et al. (2016); Li et al. (2017); Ren et al.
(2012); Chen et al. (2016); Tang et al. (2020), the conversion efficiency and
pump intensity have been limited by the employment of metallic structures.
Recently, enhanced nonlinear CD with high conversion efficiency has been
demonstrated in an hybrid multiple-quantum-well structure Kim et al. (2020);
Makarov et al. (2017). However, optical absorption still limits the pump power
that these structures can sustain. Albeit mainly plasmonic and hybrid
structures have been investigated so far, dielectric resonators demonstrated
considerably higher SHG and THG conversion efficiencies Pertsch and Kivshar
(2020). In particular, concepts inspired by the bound state in the continuum
(BIC) physics are providing a new framework to enhance nonlinear optical
phenomena at the nanoscale Carletti et al. (2018); Koshelev et al. (2020);
Carletti et al. (2019); Koshelev et al. (2019); Anthur et al. (2020); Liu et
al. (2019); Overvig et al. (2021). Non-plasmonic symmetry-broken structures
sustaining quasi-BIC, which were recently shown to display a very important
linear CD in the transmittivity Gorkunov et al. (2020); Overvig et al. (2021),
are thus promising to enable both high nonlinear CD and high conversion
efficiency Chen et al. (2016); Li et al. (2017); Kim et al. (2020); Kim and
Kim (2020); Frizyuk et al. (2021).
In this work, we use numerical simulations to demonstrate symmetry-broken Si
MSs supporting a quasi-BIC, the latter enhancing THG conversion efficiency up
to $10^{-2}\,\text{W}^{-2}$ and enabling near-unity nonlinear CD ($>99.9\%$).
The high ($>10^{5}$) quality-factor ($Q$-factor) of our engineered quasi-BIC
and the high value of the third-order nonlinear susceptibility ($\chi^{(3)}$)
in silicon in the near-infrared spectrum render this approach promising for an
efficient THG. Our results show that the quasi-BIC is selectively excited only
for one circular polarization enabling near unity nonlinear CD.
The unique opportunity to engineer a high-Q mode in lossless dielectric
metasurfaces provides the unprecedented possibility to enhance simultaneously
the nonlinear response and the nonlinear CD which, so far, have been hindered
by the use of plasmonic metasurfaces.
The designed MS is depicted in Fig. 1. It is composed by periodic unit cells
made by two Si blocks on a silica substrate, which is compatible with the
silicon-on-insulator material system. The MS periodicity is $P$, whereas the
width and the height of each Si block are $W$ and $H$, respectively. The
distance between the two blocks is $t_{gap}$. To induce the symmetry breaking,
the length of one block in the unit cell is reduced by the quantity $\alpha
L$, where $L$ is the length of the unperturbed block and $\alpha$
($0\leq\alpha\leq 1$) is the asymmetry parameter (see Fig. 1).

Figure 1: Sketch of the MS with the relevant geometrical parameters and of the
illumination scheme with LCP and RCP light.

When $\alpha=0$, this MS can sustain symmetry protected BICs. As the symmetry
of the unit cell is broken, the BIC evolves into a quasi-BIC due to the
opening of a radiation channel to free-space Koshelev et al. (2018, 2019). In
this work we fix the following geometrical parameters as: $P=700\,\text{nm}$,
$W=214\,\text{nm}$, $H=575\,\text{nm}$, and $t_{gap}=116\,\text{nm}$, chosen
in order to tune the BIC wavelength in the telecom range.
We develop a full-vectorial numerical model in COMSOL Multiphysics Gandolfi et
al. (2018) for the calculation of the optical eigenmodes in the structure (see
Appendix for further details).

Figure 2: a) Eigenmodes wavelength (horizontal axis) vs asymmetry parameter
(vertical axis). The numbers in the panel indicate the labels assigned to the
bands. The quasi-BIC is highlighted with orange markers. b) $Q$-factor of the
quasi-BIC as a function of the asymmetry parameter. The full orange (dashed
blue) line is obtained from the eigenfrequency analysis (from fit of the
reflectivity spectra with the Fano formula). c) The color map shows the
normalized electric field magnitude inside the two Si blocks for the quasi-
BIC. The electric field vector is shown with black arrows. d) Cross sectional
view (over a plane parallel to the substrate-air interface and passing through
the blocks centers) of the normalized magnetic field amplitude. The latter two
panels refer to the BIC occurring for $\alpha=0$. e) Reflectivity spectra for
three different asymmetry parameters. f) Fano parameter, $q$, (left axis,
black) and width, $\gamma$, (right axis, red) vs asymmetry parameter. The gray
vertical bar in b) and f) highlights the $Q$-factor local maximum. All the
panels are obtained for $L=600$ nm.

We start our analysis from the case $L=600$ nm and we study how the modes in
the MS evolve as the asymmetry parameter increases from $0$ up to $0.35$. Fig.
2a shows the wavelength, $\lambda$, (horizontal axis) of the MS eigenmodes as
a function of the asymmetry parameter (vertical axis). We highlight in orange
the quasi-BIC which is characterized by a much higher $Q$-factor, the latter
defined as

$Q=\frac{1}{2}\frac{Real(\tilde{\nu})}{Imag(\tilde{\nu})},$ (1)

where $\tilde{\nu}$ is the complex eigenfrequency of the mode. The $Q$-factor
of the quasi-BIC as a function of $\alpha$ is shown in Fig. 2b (orange curve).
For $\alpha=0$, the $Q$-factor diverges towards infinity and observation of
the electric and magnetic field profiles shown in Figs. 2c and 2d reveals that
it is a symmetry protected BIC. In particular, each block sustains a magnetic
dipole mode with opposite momentum that results in an odd-symmetric
y-component of the electric field at the MS-air interface. As $\alpha$ is
slightly increased ($0<\alpha\leq 0.1$), the BIC evolves into a quasi-BIC due
to the opening of a radiation channel to free-space and
$Q\propto\alpha^{-2}$Koshelev et al. (2018, 2019).
For further increase of $\alpha$, we notice that the $Q$-factor shows a local
maximum around the critical value $\alpha^{*}=0.25$. The origin of the latter
can be ascribed to accidental BIC formed due to a simultaneous destructive
interference of leakage channels Hsu et al. (2016), as demonstrated at THz
frequencies Han et al. (2021).
The quasi-BIC physics is further investigated by solving the electromagnetic
scattering problem with full-vectorial numerical calculations (see Appendix
for further details). The incident light is linearly polarized with the
electric filed parallel to the block side with length $L$ and it impinges on
the MS from the air side at normal incidence. The reflectivity as a function
of the wavelength is computed for different asymmetry parameters, $\alpha$. In
Fig. 2e, we report the reflectivity for three cases, i.e. $\alpha$ in
proximity of the $Q$-factor local maximum (red curve) and on the two flanks
(black and blue curves). The spectra show a marked Fano resonance in proximity
of the quasi-BIC wavelength. At the critical value $\alpha^{*}$ the resonance
width shrinks, consistently with the modal $Q$-factor enhancement.
We fit the reflectivity (R) spectra with a Fano function Limonov et al.
(2017):

$R(E)=R_{0}+B\frac{\left(q\frac{\gamma}{2}+E-E_{0}\right)^{2}}{\left(\frac{\gamma}{2}\right)^{2}+\left(E-E_{0}\right)^{2}},$
(2)

where $E$ is the photon energy, $E_{0}$ is the resonance energy, $\gamma$ is
the resonance full width at half maximum (FWHM), $q$ is the Fano parameter,
$R_{0}$ is an offset, and $B$ is a multiplicative constant.
In Fig. 2f we report the Fano parameter, $q$, (left axis, black) and $\gamma$
(right axis, red) as function of the asymmetry parameter, $\alpha$. Below
$\alpha=0.10$ the quasi-BIC is very close to another mode with lower
$Q$-factor (see Fig. 2a), thus our fit with a single Fano resonance does not
allow a careful reproduction of the reflectivity curve.
The dashed blue line in Fig. 2b shows the $Q$-factor retrieved from Fano fit
on the reflectivity curves obtained from the full vectorial simulations
(calculated as $Q=\lambda/\gamma$). This is in excellent agreement with the
eigenmode analysis results, meaning that the two methods are consistent and
reliable. We observe that the Fano parameter, $q$, is always negative except
in proximity of $\alpha^{*}$, meaning that in the latter situation the quasi-
BIC and the continuum phases are shifted by almost $\pi/2$ Limonov et al.
(2017).
The quasi-BIC with high $Q$-factor occurring for $\alpha\sim 0.25$ is very
interesting for technological applications. This scenario, corresponding to a
MS with a high asymmetry, supporting a Fano resonance, and exhibiting a strong
electric field enhancement at the same time, is the perfect playground where
nonlinear CD may be observed. Indeed, the exploitation of a quasi-BIC with
high $Q$-factor is beneficial to enhance the electric field within the
structure. Temporal coupled-mode theory indicates that the coupling efficiency
of the incident energy to the resonator scales as the inverse of the
$Q$-factor while the electric field is enhanced due to the resonant response
of the resonator and scales as $Q^{2}$ Maier (2006). As a result, the electric
field enhancement inside the resonator is proportional to $Q$, thus the quasi-
BIC is expected to yield a strong electric field enhancement that is
beneficial to enhance optical nonlinearities Carletti et al. (2018); Koshelev
et al. (2020). For these reasons, we investigate the MS response to circularly
polarized (CP) light at normal incidence when $\alpha=\alpha^{*}$.
We solve the electromagnetic problem for different values of $L$, both for LCP
and RCP light, in order to analyze the CD. We use the following convention to
define the CD of a certain variable $A$ Chen et al. (2016); Tang et al.
(2020):

$CD=\frac{A_{LCP}-A_{RCP}}{A_{LCP}+A_{RCP}}.$ (3)

Figure 3:  a) Average intensity enhancement CD vs block length (vertical axis)
and wavelength (horizontal axis). The computed eigenmodes are reported as
dashed white lines, while the quasi-BIC is highlighted by the full orange
line. Horizontal cyan and red dashed lines indicate $L=540\,\text{nm}$ and
$L=600\,\text{nm}$, respectively. b) Intensity enhancement in the two Si
blocks at the quasi-BIC wavelength for LCP (top row) and RCP (bottom row) pump
and for $L=540\,\text{nm}$ (left column) and $L=600\,\text{nm}$ (right
column). The intensity enhancement of top left, top right and bottom right
panels are multiplied by a factor 4, 20 and 2, respectively. c) $Q$-factor of
the quasi-BIC vs block length. d) Same as a), but for the reflectivity CD. e)
Reflectivity spectra (left axis, black) for LCP (dashed black line) and RCP
(full black line) light. The red line (right axis) shows the reflectivity CD.
The vertical green bar highlights the wavelength of the quasi-BIC. The block
length is 540 nm. f) same as e), but for $L=600$ nm. In all the panels, the
asymmetry parameter is $0.25$.

The first variable for which we calculate the CD is the average intensity
enhancement, defined as:

$<I_{en}>=\frac{1}{V}\int_{V}\frac{I(\mathbf{r})}{I_{0}}dV.$ (4)

where $I(\mathbf{r})$ and $I_{0}$ are the local and the incident light
intensity, and $V$ is the volume of the two Si blocks. Fig. 3a shows the
average intensity enhancement CD as a function of $L$ and wavelength. A high
average intensity enhancement CD can be observed when the wavelength of the
incident light is tuned in proximity of a MS mode. In particular a strong
average intensity enhancement CD up to $\sim 1$ is observed close to the
quasi-BIC wavelength (orange line) for $L=540\,\text{nm}$ and
$L=600\,\text{nm}$ (cyan and magenta dashed lines). The strong average
intensity enhancement CD occurring at the quasi-BIC for $L=$ 540 and 600 nm
can also be observed in Fig. 3b, where the map of
$|\mathbf{E}_{FF}|^{2}/|\mathbf{E}_{0}|^{2}$ inside the Si blocks is reported
($\mathbf{E}_{0}$ and $\mathbf{E}_{FF}$ are the incident and the local
electric field at the fundamental frequency, respectively). We can observe
that the quasi-BIC is selectively excited by RCP incident light and only
weakly excited by LCP light. Furthermore, the high maximum intensity
enhancement (up to 800) is due to the high quasi-BIC $Q$-factor that is shown
as a function of the block length in Fig. 3c.
Scenarios with high average intensity enhancement CD and high intensity
enhancements are promising for the achievement of nonlinear CD in THG with
high conversion efficiency. Indeed, if the intensity enhancement within the Si
blocks is very different upon LCP or RCP excitation, then we expect that the
THG may follow the same trend.
Fig. 3d shows the reflectivity CD ($R_{CD}$) spectra as a function of block
length, $L$. We can observe that $R_{CD}$ around the quasi-BIC is high for
$L=540$ nm, but not for $L=600$ nm. To better appreciate this aspect, we plot
the reflectivity curves (left axes, black) as a function of the wavelength
(horizontal axes) for $L=540$ nm (Fig. 3e) and for $L=600$ nm (Fig. 3f). The
dashed and solid black lines correspond to LCP and RCP light illumination,
respectively. The reflectivity CD is reported as a red line (right axis, red).
As we can see, in proximity of the quasi-BIC (green vertical bar), the
reflectivity CD is much higher in the case of $L=540$ nm ($R_{CD}\approx 0.8$)
than for $L=600$ nm ($R_{CD}<0.25$). The high CD obtained for the MS with
$L=540$ nm is due to interference between the quasi-BIC and the other mode of
the MS in the same wavelength range. Indeed, by analysing the field
distributions of these modes, we can observe that while the quasi-BIC
originates from a magnetic resonance, the second resonance corresponds to an
electric resonance (see Fig. A2 of Appendix). This enhancement of both
electric and magnetic fields provides an enhancement of the electromagnetic
density of chirality Solomon et al. (2018).

Figure 4:  a) $\eta_{RH}^{(0,0)}$ (left axis, black) vs pump wavelength for
LCP (dashed black line) and RCP (full black line) pump. The red line (right
axis) indicates the THG CD. The block length is 540 nm. b) same as a) but with
block length of $600\,\text{nm}$. c)
$|\mathbf{E}_{TH}|^{2}/|\mathbf{E}_{0}|^{2}$ in the two Si blocks at the
quasi-BIC wavelength for LCP (top row) and RCP (bottom row) pump and for
$L=540\,\text{nm}$ (left column) and $L=600\,\text{nm}$ (right column).
$|\mathbf{E}_{TH}|^{2}/|\mathbf{E}_{0}|^{2}$ has been multiplied by a factor
100 (top left), $3\times 10^{4}$ (top right) and 15 (bottom right).

We point out that the linear CD for the transmittivity shows the same trend as
$R_{CD}$.
We analyze the THG of the MS using full-vectorial numerical simulations
assuming an undepleted pump regime Smirnova et al. (2016) (see Appendix for
further details). We tune the pump wavelength in proximity of the quasi-BIC
and evaluate the normalized TH conversion efficiency $\eta=P_{TH}/P_{0}^{3}$,
where $P_{TH}$ and $P_{0}$ are the power at the frequency $3\omega$ emitted by
the MS (in all diffraction modes and towards both air and substrate) and the
incident pump power, respectively. The conversion efficiency is maximum when
the pump is tuned at the quasi-BIC wavelength regardless of the incident
polarization handedness. The $\eta$ peak value is $\sim 10^{-2}$ W-2 (see
Figs. A1 c and d of Appendix). This is comparable with the measured THG
efficiency of hybrid multiple quantum-well and Gammadion-type plasmonic chiral
nanoresonators Kim et al. (2020), and higher than state-of-the-art
demonstration in all-dielectric structures Liu et al. (2019).
The wavelength of the TH occurs in the absorption band of the silicon Green
(2008). As a consequence, resonant modes in that range would be spectrally
broad and are not expected to significantly contribute to the enhancement of
the emitted TH signal. This is also confirmed from the fact that the peak THG
efficiency is obtained for a pump wavelength precisely corresponding to the
quasi-BIC and that, as the pump wavelength is changed close to the quasi-BIC
condition, the THG efficiency follows the quasi-BIC spectral signature. If a
mode at the TH wavelength significantly contributed to the enhancement of the
THG efficiency, the peak of the latter would more likely occur at a slightly
different light wavelength and result in a spectrally broader peak. This is
not the case, confirming that the resonance at the pump wavelength is the
leading mechanism to achieve a high THG conversion efficiency.
The THG emission is distributed over 9 diffraction orders since the TH
wavelength is shorter than the MS period, $P$. We focus our attention on the
THG emitted in the (0,0) diffraction order in air. This light is emitted
normally to the MS plane and can be collected by using low-cost optics with
small numerical aperture. By defining $P_{RH}^{(0,0)}$ and $P_{LH}^{(0,0)}$
the THG emitted powers with right-hand, RH, or left-hand, LH, polarization, we
can define the polarization-dependent TH conversion efficiency through the
(0,0) diffraction order as $\eta_{RH}^{(0,0)}=P_{RH}^{(0,0)}/P_{0}$ for RH
light and $\eta_{LH}^{(0,0)}=P_{LH}^{(0,0)}/P_{0}$ for LH light, where $P_{0}$
is the incident pump power. In Figs. 4a and 4b we report $\eta_{RH}^{(0,0)}$
(left axes, black, log scale) as a function of the incident wavelength for
$L=540$ and $L=600$ nm, respectively. In particular, the full black (dashed
black) lines are obtained for a RCP (LCP) pump.
The strong differences between $\eta_{RH}^{(0,0)}$ obtained with RCP vs LCP
pump imply the presence of an important THG CD. The latter is plotted in Figs.
4a and 4b as red lines (right axes, red). As we can see, in both cases the THG
CD is very high in absolute value, up to 1. Indeed, $\eta_{RH}^{(0,0)}$ may
differ by more than 2 orders of magnitude if the pump is LCP or RCP. This is
due to a selective excitation of the quasi-BIC with RCP incident light with
respect to LCP as observed in Fig. 3b. Consequently, the TH fields in the MS
are mainly focused in the region corresponding to high intensity enhancement
at the quasi-BIC wavelength (see Fig. 3b) and increase considerably depending
on the polarization handedness as it can be observed in Fig. 4c, where the map
of $|\mathbf{E}_{TH}|^{2}/|\mathbf{E}_{0}|^{2}$ within the Si blocks is
reported ($\mathbf{E}_{TH}$ being the electric field at frequency $3\omega$).
We point out that these results are obtained for RCP TH light emitted in (0,0)
diffraction order towards the air. However, identical results in terms of
nonlinear CD are observed for any other diffraction order (including the ones
towards the substrate) and also for LCP THG (in Figs. A1 a and b of Appendix
we report the plot of $\eta_{LH}^{(0,0)}$ and its CD). The same conclusions
extend to the TH conversion efficiency summed over all the diffraction orders
$\eta$ (see Figs. A1 c and d of Appendix).
In conclusion, we have used extensive numerical simulations to characterize
the linear and nonlinear chiro-optical response of a Si MS with a broken-
symmetry supporting a quasi-BIC with high ($>10^{5}$) $Q$-factor. Exploiting
the consequent strong electric field confinement, we demonstrate a THG
conversion efficiency up to $10^{-2}$ W-2. Owing to the chiral broken-symmetry
and the mode interaction in the MS, selective excitation of the quasi-BIC is
achieved for RCP incident light. This results in near-unity nonlinear CD (or
THG CD) with high conversion efficiency that is comparable to state-of-the-art
hybrid multi-quantum-well structures demonstrated in the infrared Kim et al.
(2020). Furthermore, by tuning the length of the blocks we observe that the
linear CD in reflectivity can be enhanced due to multi-mode interference quite
independently from the THG CD. Indeed, for block length $L=540$ nm both the
reflectivity and THD CDs are high ($>0.8$) for incident light wavelength at
quasi-BIC. On the other hand, for $L=600$ nm and at the quasi-BIC wavelength
only the THG CD is high ($\sim 1$), whereas the reflectivity CD does not
exceed 0.25.
These results show the potential of lossless all-dielectric MSs for nonlinear
CD and pave the way for new applications requiring a precise control of the
spin of the light such as telecommunications, quantum optics, holography,
information storage, and biology.

###### Acknowledgements.

This work is partially funded by the European Commission Horizon 2020
H2020-FETOPEN-2018-2020 grant agreement no. 899673 (METAFAST), by the National
Research Council Joint Laboratories program, project SAC.AD002.026 (OMEN), and
by the Italian Ministry of University and Research (MIUR) through the PRIN
project NOMEN (2017MP7F8F).

## Appendix

_Modelling_. The numerical models are developed in COMSOL Multiphysics. The
refractive index for the Si blocks is taken from Ref. Green (2008), while the
refractive index for the silica substrate is $\sim 1.45$ for all the
considered wavelengths Malitson (1965).
The computational cell is a block corresponding to the unit cell of the MS.
Periodic boundary conditions are enforced, in order to mimic the MS. The air
and substrate domains are truncated with perfectly matched layers (PML)
domains to avoid spurious reflections Gandolfi et al. (2018).

_Eigenmode analysis_. An eigenfrequency solver is used to find the MS modes in
the telecom region. The refractive index dispersion of Si is accounted by
using an auxiliary-field formulation Yan et al. (2018). The Si permittivity
from Ref. Green (2008) at wavelengths between $1200\,\text{nm}$ and
$2000\,\text{nm}$ is modeled with a single pole Lorentz-Drude model

$\varepsilon(\omega)=\varepsilon_{\infty}-\varepsilon_{\infty}\frac{\omega_{p}^{2}}{\omega^{2}-\omega_{0}^{2}+i\omega\gamma}$

with $\varepsilon_{\infty}=1$, $\omega_{0}=5.92\times 10^{15}\,\text{Hz}$,
$\omega_{p}=1.93\times 10^{16}\,\text{Hz}$, and $\gamma=0\,\text{Hz}$.

_Linear simulations_. The reflectivity and transmittivity spectra are obtained
using two ports at the air-PML and substrate-PML boundaries. The excitation is
set only on the former port by setting either a linearly or a circularly
polarized plane wave. A frequency domain solver is used to compute the fields
at the same frequency of the incident radiation.

_THG simulations_. The electric fields $\mathbf{E}(\omega)$ calculated with
the linear simulations are used to calculate the nonlinear polarization in the
Si blocks, defined as:

$P_{j}^{NL}(3\omega)=\varepsilon_{0}\left\\{3\chi^{3}_{1122}[\mathbf{E}(\omega)\cdot\mathbf{E}(\omega)]E_{j}(\omega)\right.+$
$\left.+(\chi^{3}_{1111}-3\chi^{3}_{1122})E_{j}^{3}(\omega)\right\\},\ \ \ \
j=1,2,3,$

where $\chi^{3}_{1122}=2.8\times 10^{-19}$ m2/V2 and
$\chi^{3}_{1111}/\chi^{3}_{1122}=2.14$ Moss et al. (1990); Smirnova et al.
(2016).
The nonlinear polarization is used as a source term for the TH calculations
and no other sources are present (i.e. there are no ports). A frequency domain
study is used to compute the fields at frequency $3\omega$. The power,
$P_{TH}$, emitted by the MS at frequency $3\omega$ is:

$P_{TH}=\int_{\sigma}\mathbf{S}_{NL}(3\omega)\cdot\mathbf{N}d\sigma,$

where $\mathbf{S}_{NL}(3\omega)$ is the Poynting vector at frequency
$3\omega$, $\sigma$ is the external surface of the air and substrate domains
(i.e. the surface in contact with the PML domains), and $\mathbf{N}$ is the
unit vector normal to the surface $\sigma$ pointing outside the computational
domain (i.e. in the opposite direction with respect to the MS).

_CD for LH light emitted through the diffraction order (0,0) towards the air._
In Fig. A1 a) and b) we report $\eta_{LH}^{(0,0)}$ and its CD, as functions of
the wavelength, in analogy to Fig. 4.

_Normalized total THG conversion efficiency_. In Fig. A1 c) and d) we report
the total TH conversion efficiency $\eta=P_{TH}/P_{0}^{3}$ vs pump wavelength,
where $P_{TH}$ is the power at the frequency $3\omega$ emitted by the MS in
all diffraction orders, towards both air and substrate and with both LCP and
RCP polarizations. The full black (dashed black) line is obtained for RCP
(LCP) pump. $P_{0}$ is the incident pump power. The corresponding nonlinear CD
is reported as red lines (right axes, red).

Figure A1: a) $\eta_{LH}^{(0,0)}$ and c) $\eta$ (left axis, black) vs pump
wavelength for LCP (dashed line) and RCP (full line) pump. The red line (right
axis) indicates the THG CD. The block length is 540 nm. b) and d) same as a)
and c) respectively, but with block length of $600\,\text{nm}$.

_$Q$ -factor reduction due to fabrication defects_. The results reported in
the manuscript are obtained for an ideal structure, with structures with
vertical and smooth side-walls and infinite, exactly identical unit cells. A
fabricated device, provided that the patterned area is large enough to
accommodate some tens of unit cells, will differ from this ideal structure
mainly in both side-wall quality (verticality and roughness) and lattice
uniformity. These fabrication imperfections and defects will decrease the
photons lifetime in the mode. To account for these phenomena, we can define a
photon lifetime $\tau_{def}$ associated with fabrication features. Hence, the
total photons lifetime, $\tau_{tot}$, may be expressed as

$\frac{1}{\tau_{tot}}=\frac{1}{\tau}+\frac{1}{\tau_{def}}.$

where $\tau$ is the photon lifetime of the ideal structure. Thus, the lifetime
$\tau_{def}$ strongly depends on the specific fabrication procedure used for
the realization of the metasurface.
The $Q$-factor is equal to $Q_{tot}=\omega_{0}\tau_{tot}$ with $\omega_{0}$
the angular frequency of the quasi-BIC. The photons lifetime due to
fabrication related features decreases the $Q$-factor according to

$\frac{1}{Q_{tot}}=\frac{1}{Q}+\frac{1}{Q_{def}}.$

Comparing the measured $Q$-factor with simulations results, an estimation of
$\tau_{def}$ lifetime can be obtained.

_Electric field profiles of the quasi-BIC and the closer mode for the
structure with $L=540$ nm_. In Fig. A2a we report the electric field profile
of the quasi-BIC mode of the structure with $L=540$ nm and $\alpha=0.25$,
occurring at wavelength 1525 nm.

Figure A2: Normalized electric field profiles for the quasi-BIC (a) and closer
mode (b) for a structure with $L=540$ nm and $\alpha=0.25$. The electric field
vector is shown with black arrows.

The mode profile shows a magnetic resonance.
In Fig. A2b we display the electric field profile of the other mode, at
wavelength 1515 nm, falling closer to the quasi-BIC. This second mode
corresponds to an electric resonance.

Marco Gandolfi, Andrea Tognazzi, Davide Rocco, Costantino De Angelis, and Luca
Carletti Phys. Rev. A 104, 023524.
Copyright 2021 by the American Physical Society.

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