# Referee: Towards reference-free cross-speaker style transfer with low-
quality data for expressive speech synthesis

###### Abstract

Cross-speaker style transfer (CSST) in text-to-speech (TTS) synthesis aims at
transferring a speaking style to the synthesised speech in a target speaker’s
voice. Most previous CSST approaches rely on expensive high-quality data
carrying desired speaking style during training and require a reference
utterance to obtain speaking style descriptors as conditioning on the
generation of a new sentence. This work presents Referee, a robust reference-
free CSST approach for expressive TTS, which fully leverages low-quality data
to learn speaking styles from text. Referee is built by cascading a text-to-
style (T2S) model with a style-to-wave (S2W) model. Phonetic PosteriorGram
(PPG), phoneme-level pitch and energy contours are adopted as fine-grained
speaking style descriptors, which are predicted from text using the T2S model.
A novel pretrain-refinement method is adopted to learn a robust T2S model by
only using readily accessible low-quality data. The S2W model is trained with
high-quality target data, which is adopted to effectively aggregate style
descriptors and generate high-fidelity speech in the target speaker’s voice.
Experimental results are presented, showing that Referee outperforms a global-
style-token (GST)-based baseline approach in CSST.

Index Terms—  style transfer, low-quality data, neural speech synthesis

##  1 Introduction

Neural text-to-speech (TTS) synthesis has been greatly improved in terms of
quality and robustness of synthesized speech in recent years [1, 2, 3, 4].
Nowadays, how to improve the expressiveness of TTS systems for an even better
listening experience has attracted more attention and research effort. Cross-
speaker style transfer (CSST) is a promising technology for expressive speech
synthesis, which aims at transferring a speaking style from a source speaker
to the synthesized speech in a target speaker’s voice.

Most previous CSST approaches require a reference utterance conveying the
desired speaking style to obtained style descriptors, either in utterance
level [5, 6, 7] or in more fine-grained levels [8, 9, 10], during the run-time
inference phase. This hinders their practical use since a reference utterance
is not universally applicable for different textual input and should be
carefully selected for the desired speaking style. Hence, reference-free CSST
approach for expressive TTS is more applicable in real-world scenarios.

Reference-free CSST presents several challenges. First, representing speaking
style in a computable form is challenging, since a speaking style is the
confluence of several factors including pronunciation patterns, intonation,
intensity, etc. Due to the fact that all kinds of speech information are
entangled within an acoustic signal, for robust CSST it is of great importance
to decompose style representations from speaker identity. Second, it is
difficult to predict style descriptors from the text. Sufficient amount of
training data conveying the desired speaking style is required to learn a
robust text-to-style model. Furthermore, it is challenging for a synthesis
module to effectively and efficiently aggregate style descriptors and generate
high-fidelity speech in a desired target speaker’s voice.

Few work focuses on reference-free CSST approach. In [11], an auto-regressive
multi-speaker and multi-style Transformer-based TTS model is augmented with a
phoneme-level prosody module, and is able to conduct reference-free CSST.
However, a sufficient amount of high-quality style data from non-target
speakers are required during the training phase.

Fig. 1: Training and inference procedures of the proposed CSST approach for
TTS.

This work attempts to address the aforementioned challenges, and proposes
Referee, a robust reference-free CSST approach. An overview of the proposed
approach is illustrated in Fig. 1. We regard pronunciation, duration, pitch,
and intensity as basic information characterizing a speaking style. Phonetic
PosteriorGram (PPG) has been widely used to represent pronunciation in speech
synthesis tasks due to its speaker-independent property, such as cross-lingual
TTS [12], code-switched TTS [13], accent conversion [14], one-shot voice
conversion [15], just to name a few. A PPG is obtained from the acoustic model
in a speaker-independent automatic speech recognition (SI-ASR) system, by
stacking a sequence of phonetic posterior probabilistic vectors. Since PPG
features have the same frame-rate as their corresponding acoustic features,
they contain not only pronunciation information but also duration information
of a speech signal. Moreover, PPGs computed from an SI-ASR system, which is
trained on large-scale speech corpus containing various levels environmental
noise, reverberation and any other kinds of acoustic distortions, are robust
representations for pronunciation and duration information. Therefore, this
work uses PPGs as pronunciation and duration descriptors. Phoneme-level pitch
and energy contours are utilized to represent pitch and intensity of a speech
signal, respectively. In Referee, a style extractor is adopted to obtain these
features from wave signals, as shown in Fig. 1.

For reference-free CSST, this work trains a text-to-style (T2S) model to
predict PPGs, phoneme-level pitch, and energy contours from the text. Large-
scale high-quality dataset of a desired speaking style is expensive if not
impossible to collect. In the meantime, we observe that a large number of low-
quality speech data with rich speaking styles can be easily obtained in the
era of data explosion. Based on this, we first pretrain a base T2S model with
a multi-style low-quality dataset, and then refine it with the low-quality
samples carrying the desired style we want to transfer with a novel meta-
learning mechanism, as illustrated by the green arrows in Fig. 1. Finally, we
train a style-to-wave (S2W) model with high-quality data from the target
speaker to aggregate PPGs, pitch and energy (see purple arrows in Fig. 1) and
use it to generate speech carrying desired speaking style in the target
speaker’s voice.

The contributions of this work are summarized as follows:

  * •

We achieve robust reference-free CSST for expressive speech synthesis by
cascading a T2S model with a powerful S2W model.

  * •

We present a novel pretrain-refinement method to fully exploit readily
accessible low-quality data.

The rest of this paper is organized as follows: Details of the proposed
approach are presented in Section 2. Experimental results are shown in Section
3 and Section 4 concludes this paper.

##  2 Proposed approach

The proposed approach consists of a style extractor, a T2S model and an S2W
model, as illustrated in Fig. 1. The style extractor is used to obtain PPGs,
phoneme-level pitch and energy contours from wave signals. The T2S model is
trained with low-quality data and adopted to predict PPGs, pitch and energy
from text, while the S2W is trained with high-quality data from the target
speaker and adopted to generate waveform in the target speaker’s voice from
PPG, pitch and energy.

###  2.1 Style Extractor

The style extractor contains a PPG model, a pitch extractor and an energy
extractor. A Deep-FSMN with Self-Attention (DFSMN-SAN)-based ASR acoustic
model [16] is trained as the PPG model with large-scale (about 20k hours)
forced-aligned audio-text speech data, which contains diverse levels of
pronunciation variations, environmental noises, reverberations, and other
acoustic distortions. We train the ASR model with a frame-wise cross-entropy
loss, where the ground-truth labels are Chinese phonemes (i.e., initial and
final with tone). The pitch extractor and energy extractor compute phoneme-
level F0 values and energy values according to text-audio alignment
information obtained from a hidden-Markov-model (HMM)-based forced aligner.
Ground-truth frame-level F0 and energy contours are averaged by durations to
get phoneme-level pitch and energy.

###  2.2 Text-to-Style Model

Fig. 2: Text-to-Style (T2S) model. “LUT” is abbreviation of “look-up table”.
“CLN” is abbreviation of “conditional layer normalization”.

The network architecture of the T2S model is illustrated in Fig. 2, which is
adapted from FastSpeech 2 [17] and further augmented with conditional layer
normalization (CLN) introduced in AdaSpeech [18]. The T2S model takes phoneme
sequence as input and predicts style descriptors (i.e., PPGs, phoneme-level
pitch and energy contours) conditioned on style IDs. Learning to predict style
descriptors purely from textual input requires considerable amount of training
data. To ease the inconvenience of collecting high-quality speech data, we
fully exploit low-quality data, and design a novel pretrain-refinement scheme
to learn a robust T2S model for a speaking style we want to transfer. The
details will be presented in following subsections.

####  2.2.1 T2S pretraining

We use a low-quality multi-style audio-book corpus (each speaker has only one
style) to train a T2S model as the base model. We incorporating style
information into both the phoneme encoder and the PPG decoder via the
operations of CLN, since we hypothesize that the phoneme encoder output
contains not only content information but also pronunciation patterns.
Specifically, style IDs are first transformed to embedding vectors by a style
look-up table (LUT). The style embedding vectors are then consumed by layer-
dependent linear layers to generate scale and bias vectors, which are used to
modulate the hidden features at the layer normalization layers after each
self-attention and feed-forward network in the phoneme encoder and PPG
decoder. During pretraining, ground-truth phoneme-level pitch and energy
features are added to the phoneme encoder output. Besides, ground-truth
duration information is adopted to expand the encoder output along the
temporal axis. Mean squared error (MSE) criterion is adopted to compute PPG,
pitch, energy and duration (in log-scale) losses.

####  2.2.2 T2S refinement

After pretraining, we refine the base T2S model to a target style with a novel
refinement procedure using a meta-learning mechanism. One challenge of CSST is
that when conducting style transfer for out-of-domain (OD) textual input, the
transfer performance degrades to some degree compared with the cases where the
textual input is in-domain. This is because text and style are somewhat
entangled due to data sparsity in the training set. To mitigate this issue,
during the refinement stage, we make the T2S model simulate the scenario of
conducting style transfer for OD texts using episodic meta-learning, using a
similar but different procedure as shown in Meta-StyleSpeech [19].

Refinement process of the S2T model is illustrated in Fig. 3. For convenient
presentation, we denote the tuple of paired ground-truth (PPG, pitch, energy)
as $S$ and the predicted by the T2S model as $\hat{S}$. We assume that only a
single text sample for a speaking style is available; thus, simulating one-
shot learning for new OD text input. In each episodic, we regard the target
style as the support style ID $z_{s}$ and sample a support text $T_{s}$ in the
target style. In addition, we sample a query text $T_{q}$ from other styles
(i.e., non-target styles). Since there is no ground-truth $S$ for the query
text in the target style, we set a style discriminator $D_{s}$ and a phoneme
discriminator $D_{t}$ as shown in Fig. 3 to give learning signals to the T2S
model. Due to the fact that the Transformer-base T2S model is parameter-
intensive, we only update the parameters of the style embedding, CLN linear
layers, pitch predictor, energy predictor and duration predictor; and freeze
the remaining parameters in the T2S model during refinement. This helps
mitigate the overfitting issue. The discriminators $D_{s}$ and $D_{t}$ are set
in the least squares GAN (LS-GAN) [20] setting. Their loss function is
computed as follows:

$\mathcal{L}_{D_{s}}=\mathbb{E}[(D_{s}(S)-1)^{2}+D_{s}(T2S(T_{q},z_{s}))]$ (1)
$\mathcal{L}_{D_{t}}=\mathbb{E}[(D_{t}(S,E_{s})-1)^{2}+D_{t}(T2S(T_{q},z_{s}),E_{q})]$
(2)

The adversarial loss of the T2S model is the sum of the adversarial losses
from $D_{s}$ and $D_{t}$, as follows:

$\begin{split}&\mathcal{L}_{adv}=\mathcal{L}_{s-adv}+\mathcal{L}_{t-adv}\\\
&=\mathbb{E}[(D_{s}(T2S(T_{q},z_{s}))-1)^{2}]+\mathbb{E}[(D_{t}(T2S(T_{q},z_{s}),E_{q})-1)^{2}]\end{split}$
(3)

To stabilize the refinement process, we also use the reconstruction loss
(e.g., MSE loss) for the T2S model as:

$\mathcal{L}_{recon}=\mathbb{E}[||T2S(T_{s},z_{s})-S||_{2}^{2}]$ (4)

Therefore, the ultimate losses of the T2S model and the discriminators are as
follows:

$\mathcal{L}_{T2S}=\alpha\mathcal{L}_{recon}+\mathcal{L}_{adv}$ (5)
$\mathcal{L}_{D}=\mathcal{L}_{D_{s}}+\mathcal{L}_{D_{t}}$ (6)

We empirically set $\alpha=10$ to balance scale of each loss, and
alternatively update the T2S model and the discriminators.

Fig. 3: Refinement process of the S2T model. The reconstruction loss is not
shown for simplicity. Fig. 4: Training procedure of the style-to-wave (S2W)
model.

###  2.3 Style-to-Wave Model

The S2W model aggregates PPG, pitch, energy features and directly generate
speech waveform in the target speaker’s voice. We adapt the powerful Glow-
WaveGAN model [21] as the S2W model, inspired by its success in high-fidelity
TTS. We make several changes from the original structure to fit our need in
this work: (1) The text encoder is replaced with a PPG encoder; (2) We drop
the monotonic alignment search (MAS) module since PPGs are automatically
aligned frame-wisely with the wave encoder output $Z_{\mu}$ and $Z_{\sigma}$,
as well as the sampled hidden variable $Z$; and (3) we locally condition the
Flow-based decoder with frame-wise pitch and energy features, obtained by
expanding their phoneme-level counterparts according to phoneme durations. The
training process of the S2W model consists of two stages, as illustrated in
Fig. 4. We summarize them as follows, for more detailed description, we refer
readers to [21]: At training stage 1, a generative adversarial network
(GAN)-based variational autoencoder (VAE) is trained on the raw waveform, with
an auxiliary task of predicting F0 from the latent variable $Z$; At training
stage 2, the wave encoder is frozen and used as a hidden variable sampler to
train the PPG encoder and Flow decoder.

###  2.4 Inference

The inference procedure for CSST is illustrated in the right part of Fig. 1.
Text with arbitrary content is fed into the T2S model, which is refined on the
target style, to predict PPG, phoneme-level pitch, energy and duration. The
phoneme-level pitch and energy are then expanded to frame-level according to
the predicted duration. Finally, the S2W model takes the predicted PPG and
frame-level pitch and energy as input and generate waveform in the target
speaker’s voice conveying the target style; thus accomplishes reference-free
CSST.

##  3 Experiments

###  3.1 Setups

Two internal Chinese corpora are used in the experiments. The low-quality
multi-style audio-book corpus contains 38 speakers (each has only one speaking
style) and in total has 28.8 hours speech data (0.76 hours/speaker). Sampling
rate of some audio samples is 16 KHz. We resample all audio to 24 KHz. We
randomly partition the low-quality corpus into training, validation and test
sets according to an 86%-7%-7% scheme. In the experiments, we choose a strong
northeast Chinese accent as our target style, whose corresponding speaker is a
male speaker. The high-quality corpus is recorded for developing TTS systems
by a female target speaker in reading style, which contains 14 hours of speech
data. Audio is sampled at 24 KHz. PPG feature has dimensionality of 218 with a
10ms frame-shift.

The T2S model in the proposed approach uses 4 feed-forward Transformer (FFT)
blocks in both the phoneme encoder and the PPG decoder. The hidden size,
number of attention heads, kernel size and filer size of the 1D convolution in
the FFT block are set as 256, 2, 9 and 1024, respectively. The dimention of
the style LUT is 128. The style discriminator and phoneme discriminator adopt
the same network hyper-parameter settings as in [19]. The S2W model follows
the basic setting as that used in [21]. The frame-level pitch and energy
features are incorporated into the Flow decoder by transforming with 1D
convolutional layers with kernel size 3 and adding elementwisely to the hidden
features in each affine coupling layer as local conditioning. The Adam
optimizer with $\beta_{1}=0.9$, $\beta_{1}=0.98$ and $\epsilon=10^{-9}$ is
used to train all the models. The T2S model is pretrained on the multi-style
low-quality data with a batch size of 64 for 50K steps. Then we meta-refine
the base T2S model on the target style for 5K steps using a fixed learning
rate of $10^{-5}$ and batch size of 16.

###  3.2 Comparisons

Four approaches are compared in the experiments:
GST-DurIAN (baseline) This model is base on the DurIAN model [3], which is a
text-to-mel-spectrogram model augmented with a global-style-token (GST)-based
style encoder [6]. We use style vector from the GST encoder and speaker IDs as
auxiliary input to both the acoustic model and duration model. To disentangle
style and speaker, we adopt the domain adversarial training mechanism [22].
GST-DurIAN is first pre-trained on the combination of both the low-quality
multi-style corpus and the high-quality target speaker corpus, and then
refined to the low-quality target style data and the high-quality target
speaker corpus. During cross-speaker style transfer at inference, a reference
audio conveying the target style is used as the input to the GST encoder. A
HifiGAN vocoder [23] is used to generate waveform from mel-spectrogram.
Style-GWG (topline) This model (Style-Glow-WaveGAN) is the topline of the
proposed approach. We directly feed ground-truth PPG, pitch and energy
obtained from a reference utterance to the S2W model.
Prop-1 (proposed) This model is the proposed approach with a different
refinement procedure. During the refinement, we use only the target style data
to refine the T2S model. To avoid overfitting, only parameters of the style
embedding, CLN linear layers, pitch predictor, energy predictor and duration
predictor are updated.
Referee (proposed) The proposed approach introduced in Section 2.

###  3.3 Evaluations

We conduct the mean opinion score (MOS) tests and similarity MOS (SMOS) tests
to evaluate the naturalness and voice similarity of the generated audio
samples. To evaluate the style transfer performance, ABX style perceptual
preference tests are conducted between compared approaches. 10 samples are
generated from hold-out texts each compared approach. We invite 10 native
Mandarin Chinese speakers as the raters, each presented with 370 stimuli on
average. Audio samples can be found online111[Online]
https://liusongxiang.github.io/Referee/.

Table 1: Mean opinion score (MOS) and similarity MOS (SMOS) results with 95%
confidence interval.

Setting | In-domain text | Out-of-domain text
---|---|---
MOS ($\uparrow$) | SMOS($\uparrow$) | MOS ($\uparrow$) | SMOS($\uparrow$)
Recordings | 4.60$\pm$0.09 | - | - | -
GST-DurIAN | 3.38$\pm$0.14 | 3.53$\pm$0.19 | 3.27$\pm$0.21 | 3.25$\pm$0.18
Style-GWG | 4.04$\pm$0.17 | 3.67$\pm$0.22 | - | -
Prop-1 | 3.85$\pm$0.17 | 3.55$\pm$0.20 | 3.33$\pm$0.20 | 3.23$\pm$0.19
Referee | 3.86$\pm$0.19 | 3.53$\pm$0.21 | 3.35$\pm$0.21 | 3.33$\pm$0.22

Fig. 5: In-domain Style similarity ABX test results. Fig. 6: Out-domain Style
similarity ABX test results.

####  3.3.1 CSST for in-domain text

For the in-domain evaluation, texts from the test set in target style are
used. The MOS and SMOS results are presented in the left part of Table 1. We
can see that the topline approach Style-GWG achieves the best MOS and SMOS
results among the four compared approaches. Prop-1 and Referee have similar
performance in naturalness and voice similarity of the generated samples, both
outperforming the baseline approach GST-DurIAN in naturalness. Since low-
quality data is incorporated in the training process of the GST-DurIAN model,
the style vector computed from a low-quality utterance by the GST encoder also
encodes noise and channel information, leading to quality degradation in
generated speech. The ABX test results for evaluating style transfer
performance are shown in Fig. 5. We can see that Referee is significantly
better than Prop-1 ($p$-value $\approx$ 0.016) and the baseline GST-DurIAN
($p$-value $=2.0e^{-17}$) in terms of style transfer performance. The topline
Style-GWG beats Referee with a large margin ($p$-value = 0.00023). Aggregating
the observations mentioned above, we can safely say that Referee achieves
significantly better performance than GST-DurIAN in terms of naturalness and
cross-speaker style transfer; and on-par performance in terms of voice
similarity. Adopting the pretrain-refinement scheme introduced in section
2.2.2 boosts Referee in naturalness and style transfer, compared with Prop-1.
However, there is still a gap between Referee and the topline Style-GWG. One
possible reason is that there exists mismatch between the generated style
features by the T2S model and those directly extracted from waveform.
Finetuning the S2W model with generated style features should be the first
attempt to address this issue.

####  3.3.2 CSST for out-of-domain text

This subsection presents the out-domain evaluation, where texts for generating
samples are from an unseen reading-style corpus. Since Style-GWG cannot
conduct style transfer when there is no speech sample accessible for the
target style, evaluations on it are not conducted. The MOS and SMOS results
are shown in the right part of Table 1. The results have a significant drop
compared to their in-domain counterparts, showing that the models overfit the
text in the training set in some degree. Enlarging the training set size could
be a good solution to this issue. Referee achieves the best results in
naturalness and voice similarity. The ABX test results are shown in Fig. 6,
where we see that Referee is significantly better than GST-DurIAN ($p$-value
$=3.7e^{-33}$). Adopting the pretrain-refinement scheme introduced in section
2.2.2 helps Referee achieve slightly better style transfer performance
compared with Prop-1 ($p$-value $=0.068$).

##  4 Conclusion

This paper has introduced Referee, which is a robust CSST approach and does
not require a reference stylish utterance at inference. PPG, phoneme-level
pitch and energy are used as style descriptors and are predicted from text
using a T2S model. Referee adopts a novel pretrain-refinement scheme to fully
exploit easily accessible low-quality data. Our experiments show that Referee
outperforms a GST-based baseline system in CSST performance for both in-domain
text and out-of-domain text. Future work includes extending Referee for
supporting robust many-to-many cross-speaker style transfer.

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# Multiplexed telecom-band quantum networking with atom arrays in optical
cavities

William Huie  Department of Physics, The University of Illinois at Urbana-
Champaign, Urbana, IL 61801, USA  Shankar G. Menon  Pritzker School of
Molecular Engineering, University of Chicago, Chicago, IL 60637, USA  Hannes
Bernien<EMAIL_ADDRESS>Pritzker School of Molecular Engineering,
University of Chicago, Chicago, IL 60637, USA  Jacob P. Covey
<EMAIL_ADDRESS>Department of Physics, The University of Illinois at
Urbana-Champaign, Urbana, IL 61801, USA

###### Abstract

The realization of a quantum network node of matter-based qubits compatible
with telecom-band operation and large-scale quantum information processing is
an outstanding challenge that has limited the potential of elementary quantum
networks. We propose a platform for interfacing quantum processors comprising
neutral atom arrays with telecom-band photons in a multiplexed network
architecture. The use of a large atom array instead of a single atom mitigates
the deleterious effects of two-way communication and improves the entanglement
rate between two nodes by nearly two orders of magnitude. Further, this system
simultaneously provides the ability to perform high-fidelity deterministic
gates and readout within each node, opening the door to quantum repeater and
purification protocols to enhance the length and fidelity of the network,
respectively. Using intermediate nodes as quantum repeaters, we demonstrate
the feasibility of entanglement distribution over $\approx 1500\,\text{km}$
based on realistic assumptions, providing a blueprint for a transcontinental
network. Finally, we demonstrate that our platform can distribute $\gtrsim 25$
Bell pairs over metropolitan distances, which could serve as the backbone of a
distributed fault-tolerant quantum computer.

##  I Introduction

The development of a robust quantum network Cirac1997 ; Wehner2018 ;
Kimble2008  will usher in an era of cryptographically-secured communication
Pirandola2019 , distributed and blind quantum computing Jiang2007 , and sensor
and clock networks operating with precision at the fundamental limit Komar2014
. Almost all of these applications require network nodes that are capable of
storing, processing, and distributing quantum information and entanglement
over large distances Kimble2008 . Nodes based on neutral atoms have the
potential to combine highly desirable features including minute-scale
coherence and memory times Young2020 , scalability to hundreds of qubits per
node Ebadi2020 , multi-qubit processing capabilities Saffman2010 ; Levine2019
; Graham2019 , and efficient light-matter interfaces at telecom wavelengths
Uphoff2016 ; Covey2019b ; Menon2020  based on optical cavities Kimble2008 ;
Reiserer2015 .

FIG. 1:  Overview of the network architecture. (a) Nodes based on arrays of
atoms (green circles) in optical cavities generate a Bell pair over each link
(blue dashes) to distribute entanglement between end-users Alice and Bob. We
employ “heralded” entanglement generation based on photon interference on a
50:50 beamsplitter (BS). Fiber-optic switches (FOS) connect adjacent nodes at
will by routing the photons from each cavity. (b) The near-concentric optical
cavities have a mirror spacing of $\approx 1\,\text{cm}$ while the atom array
spans a length of only $\approx 200$ $\mu$m. (c) The time signature of the
photons on the detectors (PD) informs which atoms at each node are in a Bell
state (green check marks). Subsequent, deterministic gates can be achieved by
moving these atoms (dashed red arrows) and performing Rydberg entangling
operations (purple circle). (d) A standing wave in the cavity traps atoms in a
one-dimensional array (blue) to overlap with the highest field strength of the
telecom mode (red). Atoms are positioned with auxiliary optical tweezers
(yellow) that also move the atoms.

Despite recent work establishing neutral atom-based nodes Ritter2012 ;
Hofmann2012 ; Samutpraphoot2020 ; Langenfeld2021 ; Daiss2021 ; Dordevic2021 ,
a major bottleneck for the development of such a network is the exponential
attenuation and long transit time associated with sending single photons – the
quantum bus that distributes entanglement – throughout the network
Reiserer2015 . Since the success probability per entanglement generation
attempt is low and success must be “heralded” via two-way communication
Duan2001 ; Pfaff2013 , there is intense interest in developing architectures
that can “multiplex” many signals in parallel on each attempt Graham2013 ;
Sinclair2014 ; Kaneda2015 ; Zhong2017 ; Wengerowsky2018 . Multiplexing is
necessary to construct networks much larger than the attenuation length in
optical fiber ($\approx$20 km in the telecom band Corning2020 ), but it not
sufficient. Intermediate “repeater” nodes are required to swap the
entanglement and teleport quantum information Duan2001 ; Pirandola2019 .
Additionally, entanglement “purification” protocols Dur2003 ; Bennett1996a ;
Kalb2017  are often needed to improve the fidelity of the distributed quantum
states.

FIG. 2:  Multiplexed remote entanglement generation over $L=100\,\text{km}$
with atom arrays. (a) With only a single atom at each node, the low success
probability necessitates an average of $\approx 5800$ entanglement attempts.
The solid gray bar shows this process with time moving to the right. The zoom
shows a single attempt, in which the cooling and initialization of the atom
(blue) and signal transmission over the link (purple) dominate the duration of
each attempt. The duty cycle of entanglement-producing operations – qubit
rotations (green) and atom-photon entanglement (orange) [see Fig. 3 for
details] – is only $\approx 4$%. (b) With an array of $N=100$ atoms at each
node, the success probability necessitates an average of only $\approx 60$
multiplexed attempts; hence the overall time it takes to create entanglement
(gray bar) is much shorter. Each attempt takes longer and has a much greater
duty cycle of $\approx 50$% for entanglement-producing operations. (c) A
cartoon of the operations required for each attempt. The two qubit rotations
and atom-photon entanglement operations follow the standard protocol for time-
bin entanglement generation Barrett2005 ; Bernien2013 .

Here, we propose a quantum network and repeater node architecture that is
capable of high-rate, multiplexed entanglement generation, deterministic
inter-node quantum gates and Bell-state measurements for purification and
distribution of many-body states, while at the same time operating at telecom
wavelengths where low-loss optical fibers permit long-distance entanglement
distribution. Our architecture is based on arrays Endres2016 ; Barredo2016  of
individual neutral ytterbium (Yb) atoms, an alkaline earth-like species
Cooper2018 ; Norcia2018b ; Saskin2019 , in large ($\approx 1\,\text{cm}$),
near-concentric optical cavities Casabone2013 ; Nguyen2018 ; Davis2019 ;
Zeiher2020  (see Fig. 1). We consider a time-bin entanglement generation
protocol Bernien2013  that combines a strong, $1.48$ $\mu$m-wavelength
transition Covey2019b ; Covey2019c  and long-lived nuclear spin-1/2 qubit
states of ${}^{171}\text{Yb}$ with temporal multiplexing along the array of
atoms.

Based on recent progress with alkaline-earth atomic arrays Cooper2018 ;
Norcia2018b ; Saskin2019 ; Covey2019 ; Norcia2019 ; Madjarov2019 ;
Madjarov2020 ; Wilson2019  and realistic assumptions regarding the operation
of these nodes, we show that our multiplexing protocol can generate Bell pairs
over $>1000$ kilometers within the coherence time of the qubits, and is
compatible with entanglement purification protocols Dur2003 ; Bennett1996a ;
Kalb2017  as well as the distribution of many-body states Komar2014 ;
Polzik2016 ; Bernien2017 ; Choi2021 . Our work lays the foundation for a
versatile metropolitan or transcontinental network through a novel
architecture that combines the use of Rydberg atom arrays Saffman2010 ;
Browaeys2020 , cavity QED with strong atom-photon coupling McKeever2003 ;
Birnbaum2005 ; Tiecke2014 ; Reiserer2015 , and atom-array optical clocks
Madjarov2019 ; Norcia2019 ; Young2020  in one platform for the first time.

##  II Multiplexed remote entanglement generation

To motivate the proposed architecture, we begin with an overview of our
multiplexed time-bin networking protocol. Specifically, we consider the
example of a network link of length $L=100\,\text{km}$. The associated two-way
signal transmission time per attempt is $\tau=2L/c$, where $c=c_{0}/n$ is the
speed of light in optical fiber ($n=1.4$) that includes both the quantum
signal and classical heralding signal; $\tau\approx 1\,\text{ms}$ for this
distance. Per the methods described below, we estimate that $\approx 5800$
entanglement attempts will be required if there is only a single qubit (atom)
at each node, resulting in a $\approx 0.16\,\text{Hz}$ entanglement generation
rate. Figure 2(a) shows the full process of successful entanglement generation
with a zoomed view of each attempt. The attempt time is dominated by signal
transmission (see Appendix D for full timing details) such that the duty cycle
of entanglement-producing operations is only $\approx 4\%$.

If instead we have $N=100$ qubits at each node and multiplex their signals as
described below, we can drastically decrease the number of required attempts
to only $\approx 60$ resulting in a $\approx 50$-fold increase in the
entanglement rate to $8\,\text{Hz}$ at $L=100\,\text{km}$. Figure 2(b) shows
the full process of successful entanglement generation for $N=100$ atoms with
a zoomed view of each multiplexed attempt. In this case the duty cycle for
entanglement-producing operations is $\approx 50\%$. Although the time
required per attempt is longer when multiplexing across a large number of
atoms, the favorable scaling in success probability per attempt over long
network links leads to substantially improved entanglement generation rates
compared to the case of a single atom.

##  III Description of the network architecture

Before summarizing these results in more detail in section IV and V, we
provide an overview of the atom array platform and the atom-photon
entanglement scheme. Further details on these topics can be found in
Appendices B and C, respectively.

###  III.1 Atom arrays in near-concentric optical cavities

There has been intense interest in coupling neutral atoms to optical cavities
with small mode volumes such as nanophotonic Samutpraphoot2020 ; Dordevic2021
; Menon2020  and fiber-gap Fabry-Pérot Hunger2010 ; Haas2014 ; Brekenfeld2020
systems to enhance the atom-photon coupling. However, these systems are not
readily compatible with large atom arrays (and single-atom control therein)
due to their limited optical accessibility. Additionally, the proximity of
dielectric surfaces to the atoms makes the prospect for robust, high-fidelity
Rydberg-mediated gates uncertain as stray electric fields limit the coherence
of Rydberg transitions Sedlacek2016 ; Thiele2015 .

Meanwhile, near-concentric cavities with large mirror spacings ($\ell\gtrsim
1\,\text{cm}$) have recently been used with great success in myriad cavity QED
research directions Casabone2013 ; Nguyen2018 ; Davis2019 ; Zeiher2020 , and
offer enough optical access to enable single-atom control in cavity-coupled
atom arrays. Crucially, the mirror spacing is similar to the size of glass
cells used in many recent high-fidelity Rydberg entanglement studies
Levine2019 ; Graham2019 ; Madjarov2020 ; Wilson2019 . Further, near-concentric
cavities are widely used in trapped ion systems Casabone2013  that are also
sensitive to transient electric fields from dielectric surfaces Teller2021 .
Therefore it is reasonable to expect that these cavities are compatible with
deterministic Rydberg-mediated gates and Bell state measurements needed in a
quantum repeater and purification architecture.

We focus on a near-concentric system with $\ell=0.975\,\text{cm}$ and radius
of curvature $R=5\,\text{mm}$ for which the cavity stability parameter
$\mathcal{G}=1-\ell/R=-0.95$ Nguyen2018 ; Kawasaki2019 . We choose a single-
sided cavity, where the reflectivity of one mirror is much greater than the
other to allow photon passage, with a finesse of $50,000$. We couple this
cavity to the ${{}^{3}\text{P}_{1}\leftrightarrow{}^{3}\text{D}_{2}}$
transition with wavelength $\lambda_{\text{net}}=1480\,\text{nm}$ and decay
rate $\Gamma=2\pi\times 318\,\text{kHz}$. Based on these parameters, the
coupling strength to the cavity is $g_{34}\approx 2\pi\times 1.53\,\text{MHz}$
and the single-atom cooperativity is $C\approx 16$ (for a detailed derivation
see Appendix B).

We trap the atoms in a standing wave at
$\lambda_{\text{trap}}=\lambda_{\text{net}}/2=740\,\text{nm}$ to ensure
maximal coupling with the telecom field (at $\lambda_{\text{net}}$) in the
cavity [see Fig. 1(d)]. The standing wave at $\lambda_{\text{trap}}$ is
fortuitously close to the ‘magic’ wavelength for the optical clock transition
(${}^{1}\text{S}_{0}\leftrightarrow{}^{3}\text{P}_{0}$) where the two states
have equal polarizability Ye2008 : $\lambda_{\text{m}}=760\,\text{nm}$. The
expected $1/e^{2}$ waist radius for this standing wave is
$w_{\text{trap}}\approx 14$ $\mu$m; the trap depth (and frequency) are free
parameters. Optical tweezers are employed to create an atom array from the
magneto-optical trap (MOT) before the standing wave is turned on, and the
tweezers are positioned to overlap the desired anti-nodes of the standing wave
[see Fig. 1(d)]. The standing wave provides strong axial confinement with
$\lambda_{\text{trap}}/2$ spacing between the anti-nodes and guaranteed
maximal overlap with the anti-nodes of telecom cavity mode at
$\lambda_{\text{net}}$, and the tweezers provide strong transverse
confinement.

FIG. 3:  Multiplexed remote entanglement via a four-pulse excitation scheme.
(a) A minimal diagram of the ${}^{171}\text{Yb}$ level structure showing two
hyperfine Zeeman states in the ${}^{3}\text{P}_{1}$ manifold as
intermediaries. (b) Local application of $\Omega_{12}$ and $\Omega_{41}$ on an
atom-by-atom level is the primary mechanism for our time-based multiplexing
scheme. (c) Analysis of the pulses and internal dynamics during the process as
well as the temporal shape of the extracted photon that is entangled with the
nuclear qubit in the ground state. The black arrow highlights that the
relative timing of the two pulses $\Omega_{12}$ and $\Omega_{41}$ is a free
parameter. The maximum Rabi frequencies of these pulses are
$\\{\Omega_{12}^{\text{max}},\Omega_{34}^{\text{max}}\\}=\\{13.2\gamma,23.0\gamma\\}$
and $\gamma=2\pi\times 180\,\text{kHz}$ is the decay rate of
${}^{3}\text{P}_{1}$. (d) The resulting atom-photon entanglement fidelity and
success probability vs the relative timing $\delta$ of $\Omega_{12}$ and
$\Omega_{41}$ in units of $\gamma$. [(c) corresponds to $\delta=0$.] We choose
$\delta$ as shown in the black dashed line for the remainder of this work.

###  III.2 Atom-photon entanglement via four-wave mixing

We entangle the nuclear spin-$1/2$ qubit in the ground state of
${}^{171}\text{Yb}$ with a $1480\,\text{nm}$-photon on the
${{}^{3}\text{P}_{1}\leftrightarrow{}^{3}\text{D}_{2}}$ transition via a four-
pulse scheme that uses two Zeeman states within the ${}^{3}\text{P}_{1}$
manifold as intermediaries [see Fig. 3(a)]. The target state of our protocol
is the atom-photon Bell state ${|\psi\rangle}_{\text{atom-
photon}}=({|0\rangle}_{a}{|\text{early}\rangle}_{p}+{|1\rangle}_{a}{|\text{late}\rangle}_{p})/\sqrt{2}$,
in which the atomic qubit states {${|0\rangle}_{a},{|1\rangle}_{a}$} are
entangled with the photon occupation in an early and late emission time bin
{${|\text{early}\rangle}_{p},{|\text{late}\rangle}_{p}$} Barrett2005 ;
Bernien2013 . Such time-bin encoded states are ideally suited for long-
distance entanglement distribution via optical fibers as they are robust
against birefringence in fibers that would adversely affect other encodings
such as polarization encoded states. To create ${|\psi\rangle}_{\text{atom-
photon}}$ we start by preparing a superposition of the atomic qubit states
$({|0\rangle}_{a}+{|1\rangle}_{a})/\sqrt{2}$. Then a coherent atomic pulse
sequence results in the emission of a photon into the cavity mode only if the
atom is in ${|1\rangle}$. The proposed four-level system that allows such a
state selective emission is shown in Fig. 3(a), and was inspired by similar
sequences that have recently been considered for alkali species Menon2020 .
After the emission in the early time bin, a $\pi-$pulse on the qubit states
flips ${|0\rangle}_{a}$ and ${|1\rangle}_{a}$, and second optical pulse
sequence causes emission in the late time bin. This completes the protocol and
leaves the system in the target state ${|\psi\rangle}_{\text{atom-photon}}$.

We leverage the $F=3/2$ and $F=1/2$ hyperfine structure of the
${}^{3}\text{P}_{1}$ manifold to provide the well-separated intermediate
states ${|2\rangle}$ and ${|4\rangle}$, and we assume a magnetic field of
$B\gtrsim 100$ G although this is not strictly necessary. We apply Gaussian
pulses $\Omega_{12}$ and $\Omega_{41}$ on a per-atom basis within the array
[Fig. 3(b)] as the primary mechanism for our time-based multiplexing scheme.
$\Omega_{23}$ and $g_{34}$ couple to all atoms globally, but are distantly
off-resonant with negligible differential effect on the qubit
${|0\rangle}-{|1\rangle}$ when $\Omega_{12}$ and $\Omega_{41}$ are not applied
to the atom. Hence, we raster the tightly-focused $\Omega_{12}$ and
$\Omega_{41}$ beams across the atoms such that the position of the atom in the
array is mapped to the time-stamp of the photon emitted into the cavity.

We describe the optimization and analysis of the pulse design in Appendix C
and summarize our findings in Fig. 3(c). We leave $\Omega_{23}$ at a constant
value for the entire duration of the four-wave mixing (FWM) protocol. We then
transfer population from ${|1\rangle}$ to ${|2\rangle}$ with $\Omega_{12}$.
These two fields populate ${|3\rangle}$, which is transferred to ${|4\rangle}$
by the coherent cavity coupling $g_{34}$. Note that other schemes for
transferring population from ${|1\rangle}$ to ${|3\rangle}$, such as a two-
photon $\pi$-pulse detuned from the intermediate state ${|2\rangle}$, are
expected to further suppress double-excitation due to decay during the first
half of the FWM protocol to below 1%. We then perform $\Omega_{41}$ to
coherently transfer the atomic population back to ${|1\rangle}$. The relative
timing of the $\Omega_{12}$ and $\Omega_{41}$ pulses introduces a trade-off
between process fidelity and success probability [Fig. 3(d)]. Essentially, the
process is limited by spontaneous emission from ${|4\rangle}$ which occurs at
a rate $\Gamma_{41}\approx 2\pi\times 180\,\text{kHz}\approx g_{34}/8.5$.
Moving the $\Omega_{41}$ pulse earlier mitigates the decay but reduces the
probability of success. Note that the remote entanglement scheme is heralded,
so events that do not produce photons only affect success rates, while events
that produce photons but leave the atom in the wrong state are classified as
successful and lead to infidelity. We choose the values shown in Fig. 3(d) for
which the fidelity (success probability) of producing
${|\psi\rangle}_{\text{atom-photon}}$ with the photon in the fiber is $\approx
0.98$ ($\approx 0.39$).

##  IV Entanglement distribution across a single link

We now return to the discussion of entanglement distribution rates and we
begin by considering a single link between two nodes. Details of the analysis
are described in Appendix D. Figure 4 shows the mean entanglement rate in our
multiplexed scheme versus the distance between the nodes for different atom
numbers $N$. For distances larger than $\approx 20\,\text{km}$, we find a
drastic improvement of the entanglement rate as more atoms per node are used.
At a distance of $100\,\text{km}$ we see a $\approx 50$-fold faster rate when
using $100$ atoms per node compared to the single-atom case (see also Fig. 2).
We find that the entanglement rate sees diminishing returns for $N\gtrsim 200$
due to two main factors. First, the probability of successfully creating a
Bell pair asymptotically saturates at $1$ such that larger numbers of atoms
are not needed for suitably large rates. Second, the time per entanglement
attempt becomes dominated by the total time required to perform the four-wave-
mixing protocol for all the atoms at each node, rather than the classical
signal transmission time between them (see Fig. 2 and Appendix D). This second
effect is clearly visible at short distances below $\approx 25$ km.

FIG. 4:  Analysis of a single link. (a) Multiplexed entanglement generation
between two nodes, each containing an array of atoms in an optical cavity. The
mean entanglement distribution rate versus the length of the link for various
numbers of atoms $N$ is shown as an opacity scale for
$N=\\{10,20,50,100,150,200\\}$ with $N=200$ being fully opaque. This scale is
used in subsequent figures. The dotted line shows the entanglement
distribution rate for $N=1$. The horizontal dashed line shows a conservative
estimate for the anticipated decoherence rate of the atoms. Here we focus on
the rates associated with successfully generating a single Bell pair with one
atom at each node (inset).

We compare the entanglement rate to the coherence time of the qubits in the
nodes. We assume a conservative lower bound of $T_{2}=1\,\text{s}$ for our
nuclear qubits, but note that it could approach the minute scale Young2020 .
Hence, we consider distribution rates above $\Gamma_{\text{coherence}}=1/(2\pi
T_{2})=0.16$ Hz to have a sufficiently high link efficiency Hucul2015  for
useful entanglement. This criterion suggests that our platform will enable the
generation of entanglement over $\approx 180\,\text{km}$ using $N=200$ atoms,
which is well within the reach of current technology Ebadi2020 . For context,
the current record for matter-based qubits is 1.3 km Hansen2015 .

##  V Entanglement distribution using quantum repeater nodes

We now turn to the use of intermediate repeater nodes to extend the range of
entanglement generation to greater distances. The goal is to connect these
intermediate links into a larger chain which we refer to as the “network-
level” architecture. We break the length $L$ between end-users Alice and Bob
into $2^{m}$ segments with length $L_{m}=L/2^{m}$, where $m$ is a non-negative
integer we call the “nesting level” of the network.

###  V.1 Overview of the protocol

We divide the intermediate links into two groups in alternation such that
adjacent links are not in the same group [see Fig. 5(a)]. Our protocol is
based on the generation of Bell pairs across all Group 1 links in parallel
followed by all Group 2 links in parallel. Naively, the mean time required to
generate Bell pairs across all links is approximately twice the mean time
required for a single link. However, the number of attempts required to
successfully create entanglement follows a geometric distribution and both
groups must wait for the success of all constituent links. Hence, we
stochastically sample the distribution of attempts for each link in both
groups in order to estimate mean entanglement generation rates at the network
level (see Appendix D for details). Note that if $N$ atoms are employed in the
multiplexed entanglement generation in Group 1, $N-1$ atoms are available for
generating entanglement in Group 2.

After the Bell pairs have been generated on Group 1 links, the constituent
atom at each node in these Bell pairs – recognized by its time stamp – must be
isolated and preserved from the subsequent operations on the Group 2 links.
Our protocol is based on transferring those qubits from the nuclear spin-$1/2$
ground state (${}^{1}\text{S}_{0}$) to an auxiliary computational basis
Gorshkov2009  of the nuclear spin-$1/2$ metastable clock state
(${}^{3}\text{P}_{0}$) that has a lifetime of $\approx 20\,\text{s}$.
Accordingly, we leverage the (nearly-)clock-magic wavelength of the cavity
standing wave-optical tweezer trap system. The metastable clock state is
transparent with respect to the four-wave mixing sequence and a negligible
relative phase is anticipated on this auxiliary qubit. We expect that
transferring the qubit to the auxiliary basis will occur at a rate much faster
than the entanglement generation rates over distances of interest and
therefore have a negligible effect on the total rate. Rates of
$\Omega_{\text{clock}}\approx 2\pi\times 100\,\text{kHz}$ and a transfer
fidelity of $\gtrsim 0.99$ are anticipated with ${}^{171}\text{Yb}$ Young2020
. Alternatively, the atom(s) could be moved away from the array and the laser
fields to preserve coherence during Group 2 operations.

FIG. 5:  Network-level entanglement generation. (a) In order to distribute
entanglement to end-users Alice and Bob over greater distances, $2^{m}-1$
intermediate repeater nodes are used, where $m$ is the nesting level. Bell
pairs are generated in parallel within Group 1 (blue) and Group 2 (red).
Intermediate nodes have two atoms involved in Bell pairs. (b) Simulated
entanglement distribution rates over the full network versus the network
length $L$ for nesting levels $m=2$ (blue), $3$ (orange), and $4$ (yellow)
with number of atoms per node $N$ shown as the same opacity scale as in Fig.
4. The dashed lines again show conservative estimates of the coherence of the
qubits at each nesting level. Note that the number of qubits depends on $m$,
so the estimated coherence is $2^{m}/(2\pi T_{2})$. The black dotted line
shows for comparison the direct entanglement distribution rate by sending
entangled photon pairs at a rate of $10\,\text{GHz}$ Pirandola2017 .

With Bell pairs across all neighboring links, we now complete the end-to-end
entanglement protocol by entangling atomic pairs and performing deterministic
Bell-state measurements at each node to effectively reduce the nesting level
of the network by 1. Bell pairs between increasingly distant nodes are traced
out of the system through this process (see Appendix A) until end-users Alice
and Bob directly share a Bell pair. We couple to highly-excited Rydberg states
to perform the required local deterministic entanglement operations Levine2019
; Graham2019 ; Madjarov2020 , inspired by a recent approach with alkaline-
earth atoms coupling from the clock state to Rydberg states in the
${}^{3}\text{S}_{1}$ series Madjarov2020 . However, this interaction occurs
only over short distances, requiring the atomic pairs to be re-positioned [see
Fig. 1(c)]. The optical tweezers will remove the atoms from the cavity
standing wave and translate them to within several microns of each other prior
to Rydberg excitation. Tweezer-mediated coherent translation of atomic qubits
over such distances is routinely performed on the $\sim\text{ms}$ timescale
with minimal decoherence Beugnon2007 ; Lengwenus2010 ; Schymik2020 ;
Dordevic2021  and Rydberg-mediated gates are on the $\lesssim\mu$s timescale
Levine2019 ; Graham2019 ; Madjarov2020 . These steps are again much faster
than the entanglement distribution rates and are only performed when remote
Bell pairs have been successfully created, so we can neglect their effect on
the total rate. The expected near-term fidelity of Rydberg-mediated gates and
local measurement is $\gtrsim 0.99$ Covey2019 ; Madjarov2020 , which is high
compared to the fidelity of generating Bell pairs: $\approx 0.98^{2}=0.96$
[see Fig. 3(d)]. A detailed network fidelity budget is outside the scope of
this work.

###  V.2 Summary of the results

We consider the network-level entanglement distribution rate based on this
protocol for varied network length $L$, nesting level $m$, and atom number per
node $N$. We compare this rate against a conservative estimate of the
coherence of all qubits in the system. Naturally, this depends on the nesting
level, and hence the network level coherence estimate is
$\Gamma_{\text{coherence}}^{m}=2^{m}/(2\pi T_{2})=0.16\times
2^{m}\,\text{Hz}$. Figure 5(b) shows the network level generation rate versus
the network length for nesting levels $m=2,\,3,\,4$ with various atom numbers
per node $N$ shown as an opacity scale. We also compare against direct
communication (without intermediate nodes) based on entangled photon pairs at
a wavelength of $1550\,\text{nm}$ with a repetition rate of $10\,\text{GHz}$
Pirandola2017 . The direct communication rate falls sharply, passing below our
coherence time estimates at a distance of $\approx 600\,\text{km}$. We find
that the achievable network length increases for higher nesting level and
saturates for $N\approx 200$ atoms. In particular, for $m=4$ our system
enables a network of $L\approx 1500\,\text{km}$.

##  VI Multiple Bell pairs and entanglement purification

We now consider the generation of multiple Bell pairs with our system, which
are needed for more advanced protocols such as purification and logical
encoding. Entanglement purification (also known as distillation) Dur2003 ;
Bennett1996a ; Kalb2017  is based on taking two (or more) Bell pairs and
consuming them to generate a single Bell pair with higher fidelity (See
Appendix A). Purification requires entanglement operations between the local
qubits in the pairs combined with single-qubit readout within each node. The
former will again be accomplished with Rydberg-mediated gates Levine2019 ;
Graham2019 ; Madjarov2020  while the latter will leverage the auxiliary qubit
basis in the metastable clock state to perform single-atom readout by
scattering photons from the
${}^{1}\text{S}_{0}\leftrightarrow{}^{3}\text{P}_{1}$ transition, to which the
${}^{3}\text{P}_{0}$ clock state is transparent Monz2016 ; Erhard2021 .

FIG. 6:  Multiple Bell pairs at the network and single link levels. (a) The
simulated entanglement distribution rate for $m=4$ versus network distance for
one Bell pair (dot-dashed), two Bell pairs (dot-dot-dashed), and five Bell
pairs (dot5-dashed) with $N=\\{100,150,200\\}$ as an opacity scale. The black
dotted line is again direct communication at $10\,\text{GHz}$. The horizontal
dashed lines are the expected coherence associated with the total number of
qubits. For multiple Bell pairs $B>1$, this is
$\Gamma_{\text{coherence}}^{m,B}=B\cdot 2^{m}/(2\pi T_{2})$. The maximum
distance falls from $\approx 1500$ to $\approx 1100\,\text{km}$ when
increasing the number of Bell pair from one to two. (b) The entanglement
distribution rate of a single link with distance $L=50\,\text{km}$ to
represent a metropolitan-scale network. The rate is plotted versus the number
of Bell pairs with $N$ from 10 to 200 as an opacity scale. This shows a
favorable scaling with $B$, and that $B=26$ Bell pairs can be generated with
$N=200$, where the dashed line is again the expected coherence associated with
the total number of qubits. Entanglement of $B>N$ pairs is impossible; hence
rates for these data points are omitted.

To this end, we study the network-level entanglement generation rate versus
network length $L$ with $m=4$ for various numbers of Bell pairs. We find that
rate associated with generating $B$ Bell pairs in a given attempt decays
exponentially with $B$; hence, we instead use a “ladder” scheme analogous to
the network-level analysis. Specifically, we create $B$ Bell pairs one at a
time on each link [see Fig. 6(a)], and still divide the links into two groups.
Here again, we must sample the distribution of attempts before the successful
generation of each Bell pair on each link, and both Group 1 and 2 are limited
by the time for each constituent link to generate $B$ pairs.

We find that the simulated mean entanglement generation rate for $B=2$ exceeds
the decoherence of the $B\cdot 2^{m}$ Bell pairs for distances up to $L\approx
1100\,\text{km}$. These findings indicate that our platform may be compatible
with the development of a transcontinental terrestrial quantum network with
sufficiently high fidelity – based on entanglement purification – for
subsequent nontrivial operations. Interestingly, we find a favorable scaling
with $B$ and include $B=5$ in Fig. 6(a), showing rates exceeding decoherence
for distances up to $L\approx 500\,\text{km}$.

Finally, we consider the possibility of generating many Bell pairs over a
metropolitan-scale link with $L=50$ km for advanced error correction protocols
or for the distribution of many-body states such as logically-encoded qubits
Fowler2012 ; Albert2020 ; Erhard2021 , atomic cluster or graph states Choi2019
, spin-squeezed states Polzik2016 ; Pezze2018 ; Pedrozo2021  or Greenberger-
Horne-Zeilinger (GHZ) states Komar2014 ; Omran2019 . We analyze the
entanglement generation rate versus the number of Bell pairs per link for
various $N$ in Fig. 6(b). Crucially, we find that 26 Bell pairs can be
generated for $N=200$ – comparable with the largest GHZ states created locally
to date Omran2019 ; Song2019 ; Marciniak2021  – offering new opportunities for
distributed computing and error-corrected networking.

##  VII Outlook and conclusion

We have proposed a platform that combines the strengths of neutral atoms –
efficient light-matter interfaces McKeever2003 ; Birnbaum2005 ; Tiecke2014 ;
Reiserer2015  with telecom operation Uphoff2016 ; Covey2019b ; Menon2020 ,
high-fidelity qubit operations and measurement Levine2019 ; Graham2019 ;
Covey2019 ; Madjarov2020 , scalability to many qubits Bernien2017 ;
Browaeys2020 ; Ebadi2020 , and long coherence times in state-independent
optical traps Madjarov2019 ; Norcia2019 ; Young2020  – for the first time to
enable new directions in quantum communication and distributed quantum
computing. Moreover, we have demonstrated how this platform can offer dramatic
improvements in entanglement generation rates over long distances by time-
multiplexing across an array of atoms within each entanglement generation
attempt.

We show that entanglement generation rates with $N\approx 100$ atoms across
$\gtrsim 100\,\text{km}$-links compare favorably with conservative estimates
of the atoms’ coherence time. We further demonstrate that multiplexed
repeater-based networks with $2^{(m=4)}$ links and $N\approx 100$ atoms at
each node can generate entanglement over $\approx 1500\,\text{km}$.
Additionally, we show that our system is well-suited for entanglement
purification Dur2003 ; Bennett1996a ; Kalb2017  and can achieve a purified
network range to $\approx 1100\,\text{km}$, providing a promising architecture
for a transcontinental quantum network. This network architecture is also
compatible with heterogeneous hardware, and may be combined with microwave-to-
optical transduction Covey2019c ; Lauk2020  to provide a robust network
between superconducting quantum processors Arute2019 . Finally, we consider
the prospects for generating larger numbers of Bell pairs for more advanced
protocols such as distributing logically-encoded or other many-body states
relevant for quantum computing and metrology. We find that $26$ Bell pairs can
be generated over a metropolitan link of $50\,\text{km}$.

More generally, the confluence of the associated research thrusts – Rydberg
atoms arrays Saffman2010 ; Browaeys2020 , cavity QED with strong atom-photon
coupling McKeever2003 ; Birnbaum2005 ; Tiecke2014 , and atom-array optical
clocks Madjarov2019 ; Norcia2019 ; Young2020  – into one platform will enable
new methods to engineer, measure, and distribute many-body entangled states
with single-qubit control. For example, the optical cavity can mediate non-
demolition measurements Boca2004 ; Kalb2015  that could augment the Rydberg-
based quantum computing platform. Conversely, Rydberg-mediated interactions
and single-atom control may help to enhance and distribute spin-squeezed
states of optical clock qubits generated via the cavity Polzik2016 ; Pezze2018
; Pedrozo2021 . Finally, the marriage of short-ranged (Rydberg-mediated) and
infinite-ranged (cavity-mediated) interactions combined with the possibility
of atom-selective control and readout will enable new opportunities for the
study of quantum many-body phenomena such as the simulation of magnetism
Davis2019  and chaotic dynamics Choi2021  in regimes not readily accessible to
classical computers.

## Acknowledgments

We thank Michael Bishof and Liang Jiang for stimulating discussions and
Johannes Borregaard for carefully reading this manuscript. We acknowledge
funding from the NSF QLCI for Hybrid Quantum Architectures and Networks (NSF
award 2016136).

##  APPENDIX A Quantum repeater and purification protocols

The repeater protocol Duan2001 ; Pirandola2019  is based on creating two Bell
pairs, where end-users Alice and Bob each have half of one pair and the
intermediate node has half of both. Then, the combination of a deterministic
two-qubit controlled-NOT gate (CNOT), a single-qubit Hadamard gate, and the
qubit measurements swaps the entanglement out of the two qubits at the
intermediate node and leaves Alice and Bob’s halves entangled in a Bell pair
with no quantum information remaining at the intermediate node [see Fig.
S1(a)].

The purification protocol Dur2003 ; Bennett1996a ; Kalb2017  is based on
creating two Bell pairs, where end-users Alice and Bob each have half of both
pairs. A CNOT gate and a single-qubit measurement at both nodes leaves only
one Bell pair between Alice and Bob that has higher fidelity than either
initial pair. No quantum information remains in the other qubit pair [see Fig.
S1(b)]. This protocol could be extended to the case of intermediate nodes and
could be combined with the repeater protocol. We note that all necessary
inner-node single- and two-qubit operations, and measurements for these
protocols have been demonstrated in atomic arrays Levine2019 ; Graham2019 ;
Madjarov2020 .

##  APPENDIX B Atomic and cavity QED parameters

###  1 Notes on the Yb telecom-band transitions

We begin this section by compiling a list of references for the Yb
${{}^{3}\text{P}_{J}\leftrightarrow{}^{3}\text{D}_{J^{\prime}}}$ transitions
Bowers1996 ; Porsev1999 ; Loftus2002 ; Dzuba2010 ; Guo2010 ; Cho2012 ;
Beloy2012 ; Lee2015 ; Antypas2019 . In the literature, there appears to be
universal agreement that the decay rate from ${}^{3}\text{D}_{2}$ to
${}^{3}\text{P}_{1}$ (the transition of interest in this work) is
$\Gamma_{{{}^{3}\text{D}_{2}\rightarrow{}^{3}\text{P}_{1}}}=2\times
10^{6}\,\text{s}^{-1}$ and the decay rate from ${}^{3}\text{D}_{2}$ to
${}^{3}\text{P}_{2}$ is
$\Gamma_{{{}^{3}\text{D}_{2}\rightarrow{}^{3}\text{P}_{2}}}=2\times
10^{5}\,\text{s}^{-1}$. This corresponds to a branching ratio of the desired
decay path of $0.87$.

FIG. S1:  Schematic overview of quantum repeater and purification protocols.
See text for details.

However, there is disagreement about the decay rates of ${}^{3}\text{D}_{1}$
to ${}^{3}\text{P}_{J}$. In particular, the literature is split between
$\\{\Gamma_{{}^{3}\text{D}_{1}\rightarrow{}^{3}\text{P}_{0}},\Gamma_{{}^{3}\text{D}_{1}\rightarrow{}^{3}\text{P}_{1}},\Gamma_{{}^{3}\text{D}_{1}\rightarrow{}^{3}\text{P}_{2}}\\}=\\{200,100,3\\}\times
10^{4}$ Bowers1996 ; Porsev1999  and $\\{200,10,3\\}\times
10^{4}\,\text{s}^{-1}$ Loftus2002 . We believe that Ref. Loftus2002  – which
came after Refs. Bowers1996 ; Porsev1999  – introduced an error that has since
propagated in the community. References Cho2012 ; Lee2015 ; Covey2019b ;
Covey2019c  have propagated this error, though it has not affected their
arguments or conclusions, while Refs. Antypas2019  and others use the correct
values.

###  2 Cavity QED parameters

We consider a cavity characterized by two parameters: the radius of curvature
$R=5\,\text{mm}$ of its two mirrors and the length $\ell=9.75\,\text{mm}$
between them. For these parameters, the cavity is near-concentric, and
satisfies the stability condition $0\leq\mathcal{G}^{2}\leq 1$, where
$\mathcal{G}=1-\ell/R$ is the cavity stability parameter. We also characterize
the principal mode of the cavity by its waist $w_{0}$, Rayleigh range $z_{0}$,
and volume $V_{m}$ using

$\displaystyle w_{0}$
$\displaystyle=\left[\left(\frac{\lambda_{{}^{3}\text{D}_{2}\rightarrow{}^{3}\text{P}_{1}}\ell}{2\pi}\right)^{2}\left(\frac{1+\mathcal{G}}{1-\mathcal{G}}\right)\right]^{1/4},$
(1) $\displaystyle z_{0}$ $\displaystyle=\frac{\pi
w_{0}^{2}}{\lambda_{{}^{3}\text{D}_{2}\rightarrow{}^{3}\text{P}_{1}}},$ (2)
$\displaystyle V_{m}$ $\displaystyle=\frac{\pi}{4}w_{0}^{2}\ell,$ (3)

where $\lambda_{{}^{3}\text{D}_{2}\rightarrow{}^{3}\text{P}_{1}}$ is the
wavelength of the targeted telecom transition, $1480\,\text{nm}$. We also
assume the cavity to have intrinsic finesse $F_{\text{int}}=10^{5}$,
transmission linewidth $\kappa_{\text{int}}=2\pi\times c/2\ell
F_{\text{int}}\approx 2\pi\times 154\,\text{kHz}$, and free spectral range
$\text{FSR}=c/2\ell\approx 15.4\,\text{GHz}$. For a chosen extrinsic finesse
$F_{\text{ext}}=5\times 10^{4}$ ($\kappa_{\text{ext}}\approx 2\pi\times
307\,\text{kHz}$), this gives a photon collection efficiency of
$\eta_{\text{coll}}=1-F_{\text{ext}}/F_{\text{int}}=0.5$. Now we consider the
atom-cavity interaction parameters essential to the proposed scheme. The
electric dipole matrix element $D$ for our chosen transition is

$\begin{split}D=\Bigg{[}&\frac{3\pi\varepsilon_{0}\hbar
c^{3}}{\omega_{{}^{3}\text{D}_{2}\rightarrow{}^{3}\text{P}_{1}}^{3}}\Gamma_{{}^{3}\text{D}_{2}\rightarrow{}^{3}\text{P}_{1}}\\\
&\phantom{\times}\begin{Bmatrix}J&J^{\prime}&1\\\
F^{\prime}&F&I\end{Bmatrix}(2F^{\prime}+1)(2J^{\prime}+1)\Bigg{]}^{1/2},\end{split}$
(4)

where $\omega_{{}^{3}\text{D}_{2}\rightarrow{}^{3}\text{P}_{1}}\approx
2\pi\times 202\,\text{THz}$,
$\Gamma_{{}^{3}\text{D}_{2}\rightarrow{}^{3}\text{P}_{1}}=2\pi\times
318\,\text{kHz}$ and the term in braces is the Wigner 6-$j$ symbol, giving
$D\approx 1.96\times 10^{-29}\,\text{C\,m}$. Using this, the coherent coupling
to the cavity mode is

$g_{34}=\frac{D}{\hbar}\left[\frac{\hbar\omega_{{}^{3}\text{D}_{2}\rightarrow{}^{3}\text{P}_{1}}}{2\varepsilon_{0}V_{m}}\right]^{1/2}\approx
2\pi\times 1.53\,\text{MHz},$ (5)

which gives cooperativity
$C=g_{34}^{2}/\kappa\Gamma_{{}^{3}\text{D}_{2}\rightarrow{}^{3}\text{P}_{1}}\approx
16.0$ with $\kappa=\kappa_{\text{int}}+\kappa_{\text{ext}}$. Then the
probability of emitting a telecom photon into the cavity mode is
$P_{\text{cavity}}=C/(C+1)\approx 0.941$, and hence the probability of
extracting this photon for use in our scheme is
$\eta_{\text{extract}}=P_{\text{cavity}}\eta_{\text{coll}}\approx 0.471$.

##  APPENDIX C Four-wave mixing

FIG. S2:  Relevant levels of four-wave mixing. States ${|0\rangle}$ to
${|4\rangle}$ were considered for the simulation to estimate the success
probability and fidelity. During the excitation cycle, the population in the
levels ${|2\rangle}$ and ${|4\rangle}$ decay to state ${}^{1}\text{S}_{0}$
with a decay rate $\gamma_{{}^{1}\text{S}_{0}}=2\pi\times 182\,\text{kHz}$,
and population in state ${|3\rangle}$ decays to state ${}^{3}\text{P}_{1}$ and
${}^{3}\text{P}_{2}$ with decay rate $\gamma_{{}^{3}\text{P}_{1}}=2\pi\times
318\,\text{kHz}$ and $\gamma_{{}^{3}\text{P}_{2}}=2\pi\times 48\,\text{kHz}$,
respectively. For simulation purposes, all decays are assumed to accumulate in
a dump level that does not contribute to the coherent evolution.

###  1 Numerical model and results

The atom-telecom photon entanglement generation protocol is similar to the
four-level scheme previously shown for rubidium and cesium atoms coupled to
nanophotonic cavities Menon2020 . The protocol starts with initializing atom
in the superposition state $({|0\rangle}_{a}+{|1\rangle}_{a})/\sqrt{2}$. This
is followed by a pulse sequence that takes the atom through states
${|1\rangle}\rightarrow{|4\rangle}$ before returning back to the initial state
${|1\rangle}$. First, pulse $\Omega_{12}$ transfers population from state
${|1\rangle}$ to ${|2\rangle}$. Then the population is excited to state
${|3\rangle}$ by light field $\Omega_{23}$, which is always on. The population
that reaches the state ${|3\rangle}$ is preferentially transferred to state
${|4\rangle}$ via the emission of a telecom photon into the cavity, which is
resonant with the ${|3\rangle}\leftrightarrow{|4\rangle}$ transition. A second
pulse, $\Omega_{41}$, then transfers the population in the state ${|4\rangle}$
back to state ${|1\rangle}$. The spontaneous decay from excited states (see
Fig. S2) limit the coherent completion of this cycle and leads to
infidelities. Here we define the fidelity as the probability of finding the
atom in the qubit state after the round-trip through states
${|1\rangle}\rightarrow{|4\rangle}$, given the heralding of the telecom
photon.

The requirement of heralding makes this scheme robust to any atomic decays
preceding the photon emission into the cavity and limits the infidelities to
decays from the state ${|4\rangle}$. The optimum parameters for the given
pulse sequence are extracted using a two-step optimization process Menon2020 .
The first step optimizes the Rabi frequencies $\Omega_{12},\,\Omega_{23}$ and
the pulse width of $\Omega_{12}$ to maximize the population transfer to the
state ${|4\rangle}$ and the second step optimizes the timing, pulse width, and
Rabi frequency of $\Omega_{41}$. In both the schemes below, the success
probability accounts for the probability $P_{|1\rangle}$ for the initial
population in ${|1\rangle}$ to emit a telecom photon and return to
${|1\rangle}$, as well as the probability for the emitted photon to couple to
the external coupling mode of the cavity; i.e.

$\text{Success
probability}=\frac{\kappa_{\text{ext}}}{\kappa_{\text{int}}}P_{|1\rangle}$ (6)

####  1.1 Resonant case

In the first case, which we call the “resonant case,” we have the cavity on
resonant with the ${|3\rangle}\leftrightarrow{|4\rangle}$ transition. In this
case the corresponding Hamiltonian in an appropriately chosen rotating frame
is

$\begin{split}\hat{H}&=\Omega_{12}(t){|1\rangle}{\langle
2|}+\Omega_{12}{|2\rangle}{\langle 3|}+g_{34}\hat{a}{|3\rangle}{\langle 4|}\\\
&\phantom{=}+\Omega_{41}(t){|4\rangle}{\langle 1|}+\text{H.c.}\end{split}$ (7)

In this resonant excitation scheme, the population transfer to ${|4\rangle}$
occurs over a time scale that is inversely proportional to atom-cavity
coupling $g_{34}$, and for efficient completion, the second pulse has to be
timed to match. The earlier coherent transfer spend a longer time in
${|4\rangle}$ leading to spontaneous decay. To minimize the contribution to
infidelity, we transfer the population from ${|4\rangle}$ at earlier times,
trading fidelity gains for reduced efficiency, due to incomplete population
transfer. Here we achieve this by applying $\Omega_{41}$ earlier than what is
optimal for the complete population transfer shown in Fig. 3(c). The increase
in fidelity and corresponding reduction in the efficiency are shown in
Fig.3(d). Fixed Gaussian pulses with full widths at half maximum
$116\,\text{ns}$ and $58\,\text{ns}$ were used for $\Omega_{12}$ and
$\Omega_{41}$, respectively. Here, the achieved fidelities were conditioned on
heralding entanglement using the photons that were emitted until the coherent
transfer back to the initial qubit state by $\Omega_{41}$. Detection of
photons emitted from the cavity after the completion of $\Omega_{41}$ leads to
additional infidelities.

FIG. S3:  Detuned four-wave mixing scheme. (a) Time evolution of the coherent
population in the states ${|1\rangle}$ to ${|4\rangle}$ and the photon emitted
from the cavity through external coupling parameter $\kappa_{\text{ext}}$. The
top half of the plot shows pulses $\Omega_{12}$ and $\Omega_{41}$. In the
simulation, the first pulse is assumed to have a constant length of
$65\,\text{ns}$ and the length of the second pulse is optimized for each value
of the detuning. Here, both pulses have a $20\,\text{ns}$ ramp time. (b)
Scaling of fidelity and success probability with detuning. Larger detuning
leads to lower occupation and decay from state ${|4\rangle}$ at the cost of
lower coherent population transfer back to state ${|1\rangle}$.

####  1.2 Detuned case

High-fidelity atom-telecom photon entanglement can also be obtained by using
an off-resonant scheme, where the population transfer to ${|4\rangle}$ is
minimized, since decay from this state is the dominant error in the heralding
protocol. In this case the Hamiltonian considered is

$\begin{split}\hat{H}&=\Omega_{12}(t){|1\rangle}{\langle
2|}+\Omega_{23}{|2\rangle}{\langle 3|}+g_{34}\hat{a}{|3\rangle}{\langle
4|}+\Omega_{41}(t){|4\rangle}{\langle 1|}\\\
&\phantom{=}+\delta{|4\rangle}{\langle 4|}+\text{H.c.}\end{split}$ (8)

In this scheme, the optimal fidelities were also found by a two-step
optimization procedure. For a given detuning, the first step maximized the
population transfer to ${|3\rangle}$ by optimizing the Rabi frequencies
$\Omega_{12},\,\Omega_{23}$ and the pulse width of $\Omega_{12}$, and the
second step optimizes the duration and Rabi frequency of $\Omega_{41}$ to
maximize the population transfer from ${|3\rangle}$ to ${|1\rangle}$ through
the two-photon process. Here, we fix the pulse length of $\Omega_{12}$ to
$65\,\text{ns}$ including a linear ramp time of $20\,\text{ns}$. The length of
$\Omega_{41}$ varies from $300\,\text{ns}$ to $500\,\text{ns}$ according to
the varied detuning. Similar to the resonant case we again find that higher
fidelities can be obtained at the cost of lower success probabilities [see
Fig. S3(b)]. Incomplete population transfer in both schemes will lead to some
residual population left behind in the states that are coupled to the cavity,
which can lead to photon emission even after the end of the pulse sequence.
Detection of these photons will add to infidelity. Overall success
probabilities were found to be greater for the resonant scheme that is used in
the main text for our calculations.

###  2 Phase matching considerations

We consider the importance of phase matching and momentum conservation of the
four light fields that have overlapping amplitude during our four-wave scheme.
We perform a qualitative estimate based on classical four-wave mixing analysis
in which an outgoing wave is produced by the interaction of three incoming
waves with a nonlinear medium Ender1982 . The outgoing field intensity is
proportional to a phase-matching factor whose argument is $\xi=\Delta k\times
L$, where $\Delta k=|\vec{k}_{12}+\vec{k}_{23}-\vec{k}_{34}-\vec{k}_{41}|$ and
$L$ is the effective overlap length of the four fields which in practice is
determined by their size or the size of the medium (whichever is smaller). The
phase matching factor is equal to one when $\xi=0$ and decreases for $\xi\gg
0$.

For the beam configuration shown in Fig. 3(b) assuming a $180^{\circ}$ angle
between $\vec{k}_{12}$ and $\vec{k}_{41}$ and a $45^{\circ}$ angle between
$\vec{k}_{34}$ and $\vec{k}_{23}$, we estimate that $\Delta k\approx
2\pi/(1500\sqrt{2})\,\text{nm}$. The relevant length scale of the single-atom
case should be the size of the atomic wavefunction in the optical trap, which
we assume is $L_{\text{single}}\approx 50\,\text{nm}$. Hence, we estimate that
$\xi<1$ for the case we consider here, so phase matching of the four light
fields is not crucial. We therefore neglect it in our analysis, but choose a
beam geometry to minimize $\Delta k$.

For an atomic ensemble or a solid-state spin ensemble, this factor would be
much higher. Assuming $L=10\leavevmode\nobreak\ \mu$m with the same beam
geometry, $\xi\gg 10$. Hence, phase matching is often crucial in ensemble and
crystalline environments.

##  APPENDIX D Entanglement distribution calculations

We start by considering the rate $\Gamma_{\text{link}}$ at which entanglement
between two adjacent network nodes can be attempted. This rate comprises all
components shown in Fig. 2. The time to cool and initialize all atoms at the
nodes (shown in blue in Fig. 2), performed globally and in parallel over the
arrays of atoms at both nodes, is $1/\Gamma_{\text{init}}=1/10\,\text{kHz}$.
This is based on the maximum scattering rate from the 3P1
($\Gamma_{{}^{3}\text{P}_{1}}\approx 90$ kHz) and an assumption about the
number of photons required for cooling and optical pumping. The total qubit
pulse time comprising globally applied $\pi/2$\- and $\pi$-pulses (all time
windows shown in green in Fig. 2) is
$1/\Gamma_{\pi/2}+1/\Gamma_{\pi}=3/\Gamma_{\pi/2}=3/100\,\text{kHz}$, based on
an assumed Rabi frequency of 50 kHz via stimulated Raman pulses between the
nuclear spin states. The total four-wave-mixing time (shown in orange in Fig.
2) is $2N/\Gamma_{\text{FWM}}=2N/200\,\text{kHz}$ for all $N$ atoms. The four-
wave-mixing rate is determined by the time between when the sequence begins
and when the photon leaves the cavity with high probability [see Fig. 3(c)].
Finally, the time to transmit classical signals through fibers and to herald
entanglement (shown in purple in Fig. 2) is
$1/\Gamma_{\text{comm}}(L)=c/2L\approx 10^{8}\,\text{m\,s${}^{-1}$}/L$.

Table S1:  Rates for constituent steps in the single-link multiplexing protocol. Exact values for rates composing the total rate at which entanglement between two adjacent network nodes $\Gamma_{\text{link}}$ can be attempted (Eq. 12) in terms of the per-node atom number $N$ and distance in fiber between nodes $L$. The listed steps correspond chronologically to the colored time windows shown in Fig. 2(a) and (b).  Step | Symbol | Description | Global? | Rate [kHz]
---|---|---|---|---
1 | $\Gamma_{\text{init}}$ | Optical pumping and cooling | Global | $10$
2 | $\Gamma_{\pi/2}$ |  $\pi/2$-pulse | Global | $100$
3 | $\Gamma_{\text{FWM}}$ | FWM protocol | One-by-one | $200/N$
4 | $\Gamma_{\pi}$ |  $\pi$-pulse | Global | $50$
5 | $\Gamma_{\text{FWM}}$ | FWM protocol | Global (same order) | $200/N$
6 | $\Gamma_{\text{comm}}(L)$ | Heralded entanglement | Global (atom-unique time stamp) | $c/2L$

Using these quantities, $\Gamma_{\text{link}}$ is

$\Gamma_{\text{link}}(L,N)=\left[\frac{1}{\Gamma_{\text{init}}}+\frac{3}{\Gamma_{\pi/2}}+\frac{2N}{\Gamma_{\text{FWM}}}+\frac{1}{\Gamma_{\text{comm}}(L)}\right]^{-1}.$
(9)

The exact values used for these rates are summarized in Table S1.

The probability $p$ of successfully creating a single Bell pair between any
given two atoms at adjacent nodes similarly comprises several components;

$p(L)=\left(\frac{1}{2}\right)^{2}\eta_{\text{FWM}}^{2}\,\eta_{\text{fiber}}^{2}\,\eta_{\text{det}}^{2}\,\eta_{\text{att}}(L).$
(10)

Here, $\eta_{\text{FWM}}\approx 0.364$ is the total success probability of the
four-wave-mixing scheme under the condition shown in Fig. 3(d);
$\eta_{\text{fiber}}=\eta_{\text{det}}=0.9$ are the efficiencies at which
photons may be collected by their respective fibers and subsequently detected;
and $\eta_{\text{att}}(L)=\exp(-L/\lambda_{\text{att}})$ is the attenuation of
the telecom photons ($\lambda_{\text{att}}=20.7\,\text{km}$ at
$1480\,\text{nm}$ Corning2020 ). The two leading factors of $1/2$ are due to
the overlap between the Bell-state and computational bases and an assumed
complete loss of photon polarization in the long-distance fibers. It follows
that the total probability $P_{\text{mux}}$ of creating at least $B$ Bell
pairs between adjacent nodes through multiplexing is

$P_{\text{mux}}(L,N,B)=\sum\limits_{k=B}^{N}\binom{N}{k}p^{k}(1-p)^{N-k}$ (11)

for $N\geq B$ and zero otherwise.

To calculate the rate $\Gamma_{\text{mux}}$ at which these $B$ or more Bell
pairs can be formed between atoms at adjacent network nodes, we consider a
total number of times $M$ that the entire procedure is attempted. While $M$ is
in principle unbounded, it is realistic to choose $M$ such that the mean
number of successful attempts $MP_{\text{mux}}(L,N,B)$ to create $\geq B$ Bell
pairs is one, and hence the average success rate is that at which these $M$
attempts can be performed,

$\Gamma_{\text{mux}}(L,N,B)=\Gamma_{\text{link}}(L,N)\times
P_{\text{mux}}(L,N,B).$ (12)

FIG. S4:  Comparison of simulation with Eq. 11. The simulated mean number of
attempts $M$ required for the formation of one Bell pair across a single link
for various $N$ at $L=100\,\text{km}$ (green circles), averaged over $10,000$
trials, are compared in the main plot with the expected analytically derived
result $M=1/P_{\text{mux}}$ (black dashed line). The error bars show the one-
sided RMS deviation from the mean. The inset shows the distribution over
values of $M$ associated with the $N=10$ data point along with the mean of the
distribution (gray dotted line) and analytical result (black dashed line).

Generalizing to the network-level procedure, the two-group structure [see Sec.
V] requires an extra consideration. The proposed protocol requires that
entanglements in Group 1 complete before those in Group 2 can be attempted,
which precludes the derivation of an analytical formula to describe the
expected rates; hence we turn to numerical simulation to calculate the results
shown in Figs. 5 and 6. We use a simple simulation scheme based on
stochastically sampling the probability distribution over $M$ attempts
required for the formation of Bell pairs across each network link in
accordance with Eq. 11, which gives $M=1/P_{\text{mux}}$ for one success
across a link for given $N$, $L$, and $B$. For our simulations, $M$ – which
follows a geometric distribution – is sampled by counting the number of random
events required for a single success, which occurs with probability
$P_{\text{mux}}$. The resulting averages of these counts over $10,000$ trials
are in good agreement with the expected value obtained using Eq. 11, shown in
Fig. S4. The time taken for each linking attempt is then
$M/\Gamma_{\text{link}}$, and the mean of a set of $Q$ such trials can be
inverted to find the average entanglement rate. It was found that this scheme
could be used for as few as $Q=1000$ trials to faithfully reproduce Fig. 4.

At the network level, the results shown in Figs. 5 and 6(a) were calculated
following the two-group protocol as described. The single-link linking time
$M/\Gamma_{\text{link}}$ was singly sampled for each of the $2^{m-1}$ network
links in Group 1, from which the maximum was selected. This sampling was
repeated for $2^{m-1}$ links in Group 2 (for $N-1$ atoms at each node), and
the two maxima were added to find the total time required for the network. The
mean of $Q=5000$ such trials was then inverted to calculate the average rate
for each value of $m$, $N$, and $L$ shown. For the multi-Bell case shown in
Fig. 6(b), the single single-link procedure described in the previous
paragraph was repeated for $B$ Bell pairs following the “ladder” scheme [see
Sec. VI], the total time for $B$ linking attempts averaged over $Q=5000$
trials, and the average inverted for each value of $N$ and $B$ shown.

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