# Hybrid Quantum Repeaters with Ensemble-based Quantum Memories and Single-
spin Photon Transducers

Fenglei Gu  QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ
Delft, The Netherlands  Shankar G Menon  Pritzker School of Molecular
Engineering, University of Chicago, Chicago, IL, 60637, USA  David Maier
QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The
Netherlands  Antariksha Das  QuTech, Delft University of Technology,
Lorentzweg 1, 2628 CJ Delft, The Netherlands  Tanmoy Chakraborty  QuTech,
Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
Wolfgang Tittel  QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ
Delft, The Netherlands  Department of Applied Physics, University of Geneva,
1211 Geneva 4, Switzerland  Constructor University, 28759 Bremen, Germany
Hannes Bernien  Pritzker School of Molecular Engineering, University of
Chicago, Chicago, IL, 60637, USA  Johannes Borregaard  QuTech, Delft
University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

###### Abstract

Reliable quantum communication over hundreds of kilometers is a daunting yet
necessary requirement for a quantum internet. To overcome photon loss, the
deployment of quantum repeater stations between distant network nodes is
necessary. A plethora of different quantum hardware is being developed for
this purpose, each platform with its own opportunities and challenges. Here,
we propose to combine two promising hardware platforms in a hybrid quantum
repeater architecture to lower the cost and boost the performance of long-
distance quantum communication. We outline how ensemble-based quantum memories
combined with single-spin photon transducers, which are devices that can
generate, convert, and modulate photons with single spins, can facilitate
massive multiplexing, efficient photon generation, and quantum logic for
amplifying communication rates. As a specific example, we describe how a
single Rubidium (Rb) atom coupled to nanophotonic resonators can function as a
high-rate, telecom-visible entangled photon source with the visible photon
being compatible with storage in a Thulium-doped crystal memory (Tm-memory)
and the telecom photon being compatible with low loss fiber propagation. We
experimentally verify that Tm and Rb transitions are in resonance with each
other. Our analysis shows that by employing up to 16 repeater stations, each
equipped with two Tm-memories capable of holding up to 350 storage modes,
along with four single Rb atoms, one can reach a quantum communication rate
exceeding hundreds of qubits per second across distances of up to 1000 km.

##  I Introduction

The ability to transmit quantum information reliably between distant parties
is a prerequisite for any useful application of a quantum internet Kimble
(2008); Wehner  _et al._ (2018). The primary challenge to achieve this is the
exponential attenuation of optical signals in fiber-based networks. To
overcome this challenge, quantum repeaters have been proposed where the
distance is divided into shorter segments over which entanglement can be
established in a heralded fashion. Once entanglement has been successfully
established over the segments, entanglement swapping can extend the
entanglement over the total distance Munro  _et al._ (2015); Azuma  _et al._
(2023).

Different quantum hardware such as solid-state defect centers Hermans  _et
al._ (2022); Knaut  _et al._ (2023), atomic ensembles Heller  _et al._ (2020);
Pu  _et al._ (2021); Askarani  _et al._ (2021), trapped ions Bock  _et al._
(2018); Krutyanskiy  _et al._ (2023) and quantum dots Neuwirth  _et al._
(2021) are currently being developed to enable a functional quantum repeater.
There exist numerous theoretical proposals for repeater architectures tailored
to the specific features of each hardware Sangouard  _et al._ (2011); Nemoto
_et al._ (2016); Borregaard  _et al._ (2020); Sharman  _et al._ (2021).

Quantum repeaters with ensemble-based quantum memories, pioneered by the Duan-
Lukin-Cirac-Zoller (DLCZ) protocol Duan  _et al._ (2001), have been pursued
heavily experimentally due to their technological simplicity and multiplexing
capabilities Lago-Rivera  _et al._ (2021); Li  _et al._ (2021); Businger  _et
al._ (2022). However, using ensembles makes it difficult to perform quantum
logic on the stored information. Repeater protocols thus resort to
probabilistic entanglement swapping schemes based on linear optics, severely
limiting the performance. In addition, the probabilistic generation of
approximate pairs of entangled photons from either the ensembles themselves or
external Spontaneous Parametric Down conversion (SPDC) sources leads to a
fundamental trade-off between the rate and fidelity of the communication Krovi
_et al._ (2016); Yoshida and Horikiri (2024) severely limiting the usefulness
of repeaters.

Quantum repeaters based on individual atoms or atom-like defects represent an
alternative route Hermans  _et al._ (2022); Knaut  _et al._ (2023); Langenfeld
_et al._ (2021). Near-deterministic single-spin photon transducers, which are
devices that can generate, convert, and modulate photons with single spins,
can be achieved by coupling the atomic system to optical resonators enabling
efficient single-photon generation Morin  _et al._ (2019); Knall  _et al._
(2022); Raha  _et al._ (2020). The ability to manipulate the hyperfine states
of single atoms allows for quantum logic enabling deterministic entanglement
swapping and purification techniques Kalb  _et al._ (2017). However, repeater
protocols based on merely individually trapped atoms make large multiplexing a
formidable challenge given current technology.

Here, we propose to combine ensemble-based quantum memories with single-spin
photon transducers to enable a near-term hybrid quantum repeater with massive
multiplexing, efficient photon generation, and near-deterministic entanglement
swapping. In our scheme, a single-spin photon transducer is used for high-rate
generation of entangled photon pairs where, for each pair, one photon is to be
stored in a multi-mode ensemble-based memory in a repeater node and the other
to be transmitted through a fiber to generate distant entanglement over the
elementary segment between the nodes. Successfully entangled stored photons
are read out from the memories and near-deterministic entanglement swapping
can be accomplished with the aid of extra single-spin photon transducers,
thereby extending entanglement over neighboring elementary segments.

Furthermore, we outline a specific implementation with cavity-coupled single
Rb atoms and Tm-doped crystal memories. We show how a single Rb atom coupled
to two nanophotonic cavities with visible and telecom resonance frequencies,
respectively, can function as a robust photon-pair source producing entangled
visible and telecom photons. The telecom photon can propagate in standard
optical fibers with minimum loss Tanzilli  _et al._ (2005) and we
experimentally verify that the visible photon is compatible with the resonance
of the Tm-doped crystal. Thus no frequency conversion is required for the
repeater. We simulate the performance of the repeater for quantum key
distribution and show that rates of hundreds of secret bits per second over
distances of up to 1000 km can be achieved with up to 16 repeater stations,
each containing only two ensemble-based memories and four single Rb atoms.
Further increase of the rate is possible through additional multiplexing.

The article is organized as follows: Section II introduces the overall
protocol and the mechanism of the entanglement swapping. Section III discusses
the mechanism by which Rb atoms emit both telecom and visible photons. Section
IV presents an experimental study demonstrating the compatibility between the
Rb photon source and quantum memory based on Tm:LiNbO3. In Section V, we show
the simulation results of the repeater chain. Finally, Section VI offers a
general discussion and an outlook.

##  II Structure of the Repeater Chain Protocol

Figure 1: Structure of the quantum repeater chain architecture. The whole
system contains two end nodes, Alice and Bob, connected by a series of
repeater nodes. This figure depicts an example with two repeater nodes
showcasing the key components involved. The yellow blocks represent the
repeater nodes with the local devices depicted within. The protocol involves
three main steps: entanglement generation, entanglement transfer, and
entanglement swapping, as circled with dashed lines in different colors in the
figure. The whole protocol harnesses two kinds of devices as the key
components – the single-spin photon transducers and the ensemble-based quantum
memories. The others are common devices including 50-50 optical beam
splitters, photon detectors, and optical switches, as sketched with
conventional symbols in the figure. The solid-line arrows represent the
optical fibers and the dashed-line arrows the classical channels. Both the end
nodes and the repeater nodes have symmetric layouts.

The general structure of the quantum repeater, composed of the single-spin
photon transducers and ensemble-based quantum memories, is depicted in Fig. 1.
The protocol is designed to distribute entanglement between two distant end
nodes, Alice and Bob. It divides the total distance between Alice and Bob into
multiple segments. In each segment, we repeatedly emit entangled photon pairs
on each side, storing one photon from each pair in the ensemble-based memories
and attempting heralded entanglement generation over the segment with the
other photons. Upon success, the now entangled photons are retrieved from the
memories on each side and mapped into single-atom systems in a heralded
fashion. This is followed by entanglement swapping using local Bell
measurements to generate entanglement between Alice and Bob. The protocol thus
consists of three main steps: entanglement generation, entanglement transfer,
and entanglement swapping as circled separately in Fig. 1.

For the initial entanglement generation step, as shown in the dashed red
circle in Fig. 1, two identical single-spin photon transducers are employed to
produce entangled photon pairs continuously. For each photon pair, one of the
photons is to be stored in the multi-mode ensemble-based quantum memory while
the other is sent via optical fiber to the middle station. At the middle
station, a linear optics Bell measurement is performed to entangle the photons
stored in the quantum memories. While the success of this operation is
probabilistic and subjected to photon loss, information about which photon
pairs were successfully entangled is sent back from the middle station to the
nodes where quantum memories are located. The successfully entangled photons
will be read out from the memories for further processing while the failed
ones will be discarded.

The read-out photon pairs are transferred to the single-spin photon transducer
systems. This step is heralded by the detection of the photons after
entangling with the spins in the photon transducers. Since both the storage of
the photons in the quantum memories and the transferring procedure are subject
to loss, heralding signals are sent between the nodes to confirm successful
transfer. This is illustrated in Fig. 1 as circled with the dashed blue line.

Lastly, for the entanglement swapping between neighboring sections, one has to
implement a Bell state measurement between the two adjacent spins in one
repeater node. In this case, one can let one of the spins re-emit a photon,
guide it to interact with the other spin, and finally, detect the photon and
both spins in certain bases, as sketched in Fig. 1 in the dashed orange
rounded rectangle. This operation is deterministic up to local losses.

Below, we outline a concrete implementation of this protocol with trapped
single neutral Rb atoms and Tm-doped crystal quantum memory (Tm-memory) and
assess its performance.

##  III Rb entangled-Photon emitter and entanglement generation protocol

Figure 2:  Rb emitter design and mechanism. (a) Parallel dual cavity design.
The wide and narrow gray strips placed along the X direction represent the
nanophotonic cavities resonant with the telecom and visible photons emitted
from the Rb atom, respectively. They are of a TE mode telecom wavelength
cavity (1470 nm) and a TM mode visible wavelength cavity (795 nm ) both with a
refraction index of 2.6 and a thickness of 300 nm. The light-red cone denotes
the combination of the incident and reflected trapping lights, forming
antinodes of high light intensity marked as dark red. The Rb atom is trapped
in the nearest node to the cavities. (b) The slice of the simulated relative
electric field intensity of the trapping light of the Z-Y plane centered
around the trapping light position. It is normalized to the incident tweezer
beam intensity. The two solid gray rectangles indicate the two nanophotonic
cavities viewed from the X direction. (c) The intended driving path, from
$\ket{1}$ to $\ket{5}$, in the Rb atom. The first two couplings are laser-
induced and the later two are cavity-induced. There is a detuning
$\delta=1.73$ GHz respecting the $|1\rangle-|2\rangle$ energy difference in
the first laser driving. (d) Temporal profile of the driving pulse of the
first laser (orange) and emitted telecom and visible photons (blue).

The Rb entangled-photon emitter consists of a single neutral Rb atom trapped
with optical tweezers and coupled to two nanophotonic cavities, one with
resonance wavelength at 1470 nm and the other at 795 nm. There have been
previous designs for two-mode cavity coupling of emitters with crossed
cavities or using both TE and TM modes in a waveguide Burgess  _et al._
(2009); Vučković  _et al._ (2011). It is however challenging for such setups
to maintain the required cavity quality factor while increasing the frequency
separation of the two modes, which, in our case, corresponds to a wavelength
difference of 675 nm.

Instead, we propose a parallel-cavity arrangement with the Rb atom located on
the top of the two cavities, as shown in Fig. 2(a). Such an architecture
enhances the independence of the two cavities and is feasible by integrating
atoms with on-chip nanophotonic cavities Menon  _et al._ (2023); Kim  _et al._
(2019). However, there are still many issues that need to be considered for
choosing suitable system parameters. On the one hand, we need both cavities to
be close enough to each other to reflect the tweezer light to form a stable
trap for the Rb atoms Thompson  _et al._ (2013a); Samutpraphoot  _et al._
(2020). In addition, placing the cavities closer to each other also provides
larger atom-cavity coupling strengths, resulting in Purcell-enhanced emissions
into the desired cavity mode. On the other hand, as the cavities of 1470 nm
and 795 nm get closer, photonic modes can leak into each other, reducing the
achievable quality factors.

We address these challenges by carefully selecting the material with the
optimum refractive index, cavity separation, and cavity thickness. In our
design, we chose silicon-enriched silicon nitride, SiN, as the cavity material
with a refractive index of 2.6. This selection enables better mode confinement
compared to stoichiometric ${\rm Si_{3}N_{4}}$, allowing for minimal
separations between the cavities while maintaining high cavity quality
factors. For practical fabrication considerations, both cavities are assumed
to have a thickness of 300 nm. We perform simulations with a 1060 nm tweezer
and 400 nm separation between the cavities and present the results in Fig.
2(b). Our findings demonstrate that this design enables the 1470 nm telecom-
photon cavity to achieve a quality factor of $1.4\times 10^{5}$ with an
average cooperativity of 34 with the Rb atom, while the 795 nm visible-photon
cavity reaches a quality factor of $3.8\times 10^{5}$ with an average
cooperativity of 11. More details on the cavity design are provided in
Appendix D.

We will now describe in detail how this Rb-dual-cavity setup can function as a
source of entangled photon pairs with the two photons at telecom and visible
wavelengths, respectively. To achieve this, we harness five specific
electronic orbital states from the ${}^{87}{\rm Rb}$ atom. They are:
$|1\rangle$: $5^{2}\text{S}_{1/2}|F=2,m_{F}=2\rangle$, $|2\rangle\text{:}\
5^{2}\text{P}_{3/2}|F=3,m_{F}=3\rangle$, $|3\rangle\text{:}\
4^{2}\text{D}_{3/2},|F=3,m_{F}=3\rangle$, $|4\rangle\text{:}\
5^{2}\text{P}_{1/2},|F=2,m_{F}=2\rangle$, and $|5\rangle\text{:}\
5^{2}\text{S}_{1/2},|F=1,m_{F}=1\rangle$. Fig. 2(c) shows the coupling and
driving of these five levels. In each photon emission cycle, we initialize the
Rb atom in state $|1\rangle$ and drive it first to the $|2\rangle$ state with
the first laser and then to the $|3\rangle$ state with the second laser. From
the $|3\rangle$ state, the Rb atom will decay initially to the $|4\rangle$
state emitting a photon into the telecom cavity mode (1470 nm), followed by
another decay to the $|5\rangle$ state emitting a second photon into the
visible cavity mode (795 nm).

The second laser is continuously driving the transition
$|2\rangle\leftrightarrow|3\rangle$ at a Rabi frequency of the same order as
the cavity couplings, however, the Rabi frequency of the
$\ket{1}\leftrightarrow\ket{2}$ drive is chosen to be about an order of
magnitude lower. Besides, the pulse of the $\ket{1}\leftrightarrow\ket{2}$
drive is modulated with a pause in the middle of the driving process as shown
by the orange curve in Fig. 2(d). The paused driving is calibrated such that
the telecom and visible photons have equal probabilities of being both
generated in the early time-bin (E) and the late time-bin (L), as shown by the
blue curve in Fig. 2(d). Ideally, this results in an entangled state of the
form:

$\psi_{{\rm
vt}}=\frac{1}{\sqrt{2}}\left(|\text{E}\rangle_{\text{v}}|\text{E}\rangle_{\text{t}}+|\text{L}\rangle_{\text{v}}|\text{L}\rangle_{\text{t}}\right),$
(1)

where the subscripts refer to the visible (v) and telecom (t) photons. We note
that the Rb atom is not entangled with the final photonic state and can be
used to repeatedly emit states of the form above. We simulated this procedure
of emission with the comprehensive Hamiltonian and parameters shown in
Appendix A.

In each elementary segment, both end nodes are equipped with Rb emitters. The
visible photons generated by the two Rb emitters are directly stored in
quantum memories within the nodes while the telecom photons are to be sent via
optical fiber to a middle station. At the middle station, a beam splitter is
used for erasing the which-way information, and photon detectors after the
beam splitter are used to tell in which time-bins the two photons arrive,
respectively. In this way, with a maximum probability of 50% for the photon
detectors to detect both an early and a late photon, it ideally projects the
two visible photons into a maximally entangled state

$\psi_{{\rm
vv}}=\frac{1}{\sqrt{2}}\left(|\text{E}\rangle_{\text{vl}}|\text{L}\rangle_{\text{vr}}\pm|\text{L}\rangle_{\text{vl}}|\text{E}\rangle_{\text{vr}}\right),$
(2)

where the subscripts "l" and "r" denote the left and right ends and the $\pm$
sign is determined by whether the same detector (+) or different detectors (-)
measure a photon in the early and late time bins. In this way, one realizes
the heralded entanglement generation in each segment in a multiplexed way.

In actual implementations, there will be imperfections and noisy processes in
the above procedure. To investigate the effect of these, we develop a detailed
quantum optical model that includes the many-level structure of the Rb-atoms,
faulty laser polarization, finite and fluctuating cavity coupling, and various
loss mechanisms. We refer to Appendix A for details of this model and discuss
only some of the key insights here.

Ideally, the first driving laser should only drive $\sigma_{+}$ transitions.
However, due to the device geometry, good polarization-maintained driving can
only be obtained by driving with linear polarization. This results in an equal
driving of $\sigma_{+}$ and $\sigma_{-}$ transitions. The $\sigma_{-}$
components leak population to many other hyperfine levels as shown in Appendix
A. This problem can, however, be mitigated by tuning the frequency of the
first laser to match a specific resonance of a dressed state resulting from
the continuous driving of the excited states from the second laser. The
details of this method are presented in Appendix B.

Besides, the atoms will oscillate in the trapping potential due to finite
temperature, which results in fluctuating cavity couplings. While this can in
principle be mitigated by cooling the atomic motion Thompson  _et al._
(2013b); Kaufman  _et al._ (2012), it is important to assess the robustness of
our scheme to this.

Fluctuating cavity couplings mean that photons from different emitters will
have varying amplitudes of the photonic time-bins due to the fluctuations.
This, in turn, will degrade the quality of the generated entanglement. While
this cannot be completely circumvented, we identify certain ‘sweet spots’ of
the frequency of the first driving laser where the effect of these
fluctuations can be efficiently suppressed. This can be understood from the
dressed states of the subspace $\\{\ket{2},\ket{3},\ket{4}\\}$ coupled by the
telecom cavity field and the second laser. By tuning the frequency of the
first driving laser, we can target one of the dressed states such that the
effect from the fluctuation of $\ket{3}-\ket{4}$ coupling will be counteracted
by the effective $\ket{1}-\ket{2}$ coupling. This procedure is described in
more detail in Appendix C.

Adopting the error-suppression techniques described above, we simulate that
entangled states with fidelity of 0.98 with respect to the target state in Eq.
(2) can be generated for fluctuating cooperativities of $34\pm 5.0$ (telecom-
photon cavity) and $11\pm 2.2$ (visible-photon cavity) and polarization errors
of 83% (telecom-photon cavity) and 67% (visible-photon cavity) and near-unity
polarization fidelity in the lasers. For these parameters, the emission time
is $\sim 60$ ns. Therefore, we assume the repetition rate of the Rb entangled
photon source to be dominated by the state initialization, which we assume
limits the repetition rate to 1 MHz.

##  IV Thulium-doped Crystal Quantum Memory

In our proposal, multimode quantum memories (QMs) require an ensemble-based
approach Lvovsky  _et al._ (2009); Jing and Bao (2023). Several possible
methods exist, including electromagnetically induced transparency (EIT) Ma
_et al._ (2017), and photon-echo-related approaches Bussières  _et al._
(2013), including the atomic frequency comb (AFC) protocol Afzelius  _et al._
(2009); Jobez  _et al._ (2016). Relevant hardware include alkaline atoms
Bashkansky  _et al._ (2012); Vernaz-Gris  _et al._ (2018); Cho  _et al._
(2016); Hosseini  _et al._ (2011); Guo  _et al._ (2019) and rare-earth-ion-
doped crystals (REID crystals) Tittel  _et al._ (2010); Thiel  _et al._
(2011); Zhong and Goldner (2019); Liu and Jacquier (2006). While the
transition of Rb vapor or laser-cooled Rb is naturally resonant with the
emission wavelengths of photons from a single Rb atom, here we focus on the
use of REID crystals. The reason is two-fold. Firstly, the multimode capacity
of the AFC quantum memory protocol, which is widely used for quantum state
storage in REID crystals Saglamyurek  _et al._ (2016); Businger  _et al._
(2022), does not depend on the optical depth of the material, unlike in the
case of EIT Ma  _et al._ (2017); Nunn  _et al._ (2008). Secondly, some Tm-
doped crystals have matching resonance lines with 87Rb, as shown in Fig. 3
with the example of Tm-doped LiNbO3(Tm:LiNbO3): We performed Doppler-free
saturation spectroscopy Preston (1996) of atomic 87Rb vapor using an
experimental setup schematically shown in a simplified form in Fig. 3(a). A
pump beam propagates through the 87Rb vapor cell and turns into a probe beam
after reflecting from a mirror passing through the 87Rb vapor cell again. The
intensities of both beams are detected by two photon detectors, respectively.
This gives the transmission of the 87Rb vapor as a function of the wavelength
of the laser, i.e. the spectrum of the 87Rb atom as shown in Fig. 3(b). On the
other hand, the transmission spectrum of a cryogenically cooled ($\approx$ 600
mK) Tm:LiNbO3 crystal is depicted in Fig. 3(c), showing spectral overlap with
the 87Rb spectrum. Specifically, we show an enlarged view of the [52S1/2,
$F=1$] $\leftrightarrow$ [52P1/2, $F=2$] resonance line in the inset of Fig.
3(b), which is used in our Rb single-spin photon transducer.

Tm:LiNbO3 has been used in several implementations of QM for light Sinclair
_et al._ (2014); Puigibert  _et al._ (2020). Ignoring spectral diffusion,
which will be discussed below Böttger  _et al._ (2006), the storage efficiency
for photons at 795 nm wavelength is determined by the optical coherence time
$T_{2}$ of its ${}^{3}H_{6}\leftrightarrow$ ${}^{3}H_{4}$ transition. More
precisely, the normalized retrieval rate decreases as $R_{\rm
rtr}=\exp\left(-4t/T_{2}\right)$, where $t$ is the storage time, yielding
$t_{\rm 1/e}=T_{2}/4$. The coherence time itself is upper bounded by
$T_{2}^{\rm max}=2T_{1}$ where $T_{1}$ is the lifetime of the excited state –
in the case of Tm:LiNbO3 around 100 $\mu$s. Experimentally, a $T_{2}$ time of
117 $\mu$s has been reported at a temperature of 810 mK Sinclair  _et al._
(2017), which limits $t_{\rm 1/e}$ to around 30$\mu$s. We note that other Tm-
doped crystals with much longer coherence times are known. In particular,
these include Tm:Y3Ga5O12 (Tm:YGG), for which $T_{2}$=1.1 ms has been measured
at 500 mK Askarani  _et al._ (2021). We expect that the coherence time will
approach the theoretical maximum of 2$T_{1}$=2.6 ms under optimized
experimental conditions, in particular at lower temperatures and optimized
magnetic fields.

In addition to the coherence time, spectral diffusion is another factor that
can limit the possible storage time Sun  _et al._ (2012); Sinclair  _et al._
(2021); Saglamyurek  _et al._ (2012); Sinclair  _et al._ (2017); Askarani  _et
al._ (2021); Böttger  _et al._ (2006). However, there are several ways to
mitigate the effect of spectral diffusion, including so-called zero first-
order Zeeman (ZEFOZ) transitions McAuslan  _et al._ (2012); Davidson  _et al._
(2021), a reduction of the temperature Sinclair  _et al._ (2017, 2021), or co-
doping with other ions Thiel  _et al._ (2012); Böttger  _et al._ (2008). For
the simulations in this paper, we, therefore, assume that the storage
efficiency is only determined by the coherence time, which we furthermore
assume to be given by its upper limit of 2$T_{1}$. In the case of Tm:YGG, we
thus find $R_{\rm rtr}=\exp\left(-4t/2.6{\rm ms}\right)$.

As shown in Fig. 3, the ${}^{3}H_{6}\leftrightarrow$ ${}^{3}H_{4}$ transition
line of Tm:LiNbO3 spectrally overlaps with the [52S1/2, $F=1$]
$\leftrightarrow$ [52P1/2, $F=2$] line in 87Rb. However, the same line in
Tm:YGG is slightly off-resonant (the Rb transition is centered at 794.97 nm
and features a linewidth of 2$\pi\cdot$5.7 MHz while the line of a 1% doped
Tm:YGG crystal is centered at 795.32 nm with 56 GHz linewidth Thiel  _et al._
(2014)). However, crystal engineering, e.g. by co-doping Thiel  _et al._
(2012); Böttger  _et al._ (2008), may solve this problem. Experiments show
that a linear increase of the inhomogeneous linewidth in Tm:Y3Al5O12(Tm:YAG)
can be induced, for example, by 24 GHz/Scandium$\%$ Ferrier  _et al._ (2018)
and by 3.6 GHz/Europium$\%$ Zhang  _et al._ (2020). It is reasonable to assume
that these methods work for Tm:YGG as well. While more work remains to be
done, it therefore appears feasible to match the Rb line with the absorption
line of a Tm-doped crystal, allowing the creation of quantum memory for light
with the desired specifications.

Figure 3:  (a) A simplified schematic of the Doppler-free saturation
spectroscopy setup used to measure the absorption spectrum of rubidium. We use
a laser with continuously sweeping frequency along with two light detectors
(D1 to normalize the pump light, D2 to detect the probe light) to measure the
absorption spectrum of the 87Rb vapor. (b) Absorption spectrum of 87Rb vapor.
The inset shows an absorption peak corresponding to the [52S1/2, $F=1$]
$\leftrightarrow$ [52P1/2, $F=2$] transition which is used in our single-spin
photon transducer. (c) The absorption spectrum of Tm:LiNbO3. The orange bar
indicates the wavelength range in panel (b), identifying the spectral overlap
between the Tm-memory absorption and Rb photon transducer emission.

##  V Simulated performance of the repeater chain

Figure 4:  Simulated Rb-Tm repeater chain performance. (a) Average secrete key
rate $R_{\rm SK}$ over the varied total distances $L$ for different $N_{\rm
tRb}$, the number of Rb atoms at each side of the elementary segment for
entanglement transfer. The dashed black line presents the secret key rate with
no repeaters but only a direct link between the two end notes. It declines
rapidly due to the 0.2 dB/km fiber attenuation. (b) Optimized number of
quantum repeaters $N_{\rm rep}$. The optimization is by maximizing $R_{\rm
SK}/N_{\rm seg}$ with $N_{\rm seg}$ being the total number of divided segments
in the chain. (c) The $N_{\rm mode}$: Number of modes need to be stored in
each Tm-memory to enable continuous operation. The solid lines are fitting
curves for the data with the corresponding color. The fitting excludes the
data points at $L=100$ km where the direct link is the optimal.

To assess the performance of the quantum repeater chain, we conduct an in-
depth simulation to determine the secret key rate achievable when applied to
quantum key distribution (QKD). We assume that the two end nodes can perfectly
store the successfully entangled pairs or that they directly measure the pairs
without needing storage, depending on the QKD protocols. Thus we do not
consider imperfect storage or limited memory capabilities at the end nodes but
we do include this in all repeater nodes.

Through a Monte-Carlo simulation, we track the successful entanglement
generation, transfer, and swapping events throughout the repeater chain,
allowing for accurate modeling of finite memory time effects on the quality of
the entanglement. In the simulation, the repeater is operated continuously,
and by recording the time, $T$, it takes to obtain 1000 successful end-to-end
entangled pairs, we calculate the average success rate as $R_{{\rm
suc}}=1000/T$.

We trace the evolution of the density matrices of each entangled pair
including the decoherence due to imperfect storage in the repeater nodes. As a
result, we obtain the 1000 density matrices of the end-to-end entangled pair
from the 1000 successful events. From these, we derive the average qubit error
rates (QBERs) of Z and X type errors, $Q_{{\rm Z}}$ and $Q_{{\rm X}}$, from
which, based on the BB84 protocol Bennett and Brassard (2014), we estimate the
raw secrete key rate as

$R_{{\rm SK}}=\max\left(0,\ 1-H\left(Q_{{\rm X}}\right)-H\left(Q_{{\rm
Z}}\right)\right)\times R_{{\rm suc}},$ (3)

where $H(\cdot)$ is the binary entropy function. Please note that, in the
original BB84 protocol, there was an overall factor of 1/2 from the random
basis choice. This can, however, be circumvented without loss of security in
the asymptotic limit of long keys Lo  _et al._ (2005); Scarani  _et al._
(2009) that we consider here such that the factor is omitted. In the end, we
repeat the above procedure 100 times and take the average $R_{{\rm SK}}$.

As mentioned in section III, we assume a repetition rate of the Rb emitter of
$\nu_{\rm Rb}=$ 1 MHz. The fiber attenuation rate for the telecom photon is
0.2 dB/km and the fiber’s light speed is $\rm{c}=2\times 10^{5}$ km/s. In
addition, we assume single photon detection efficiencies of 0.99. As mentioned
in section IV, the retrieving rate of the Tm-memory as a function of storage
time $t$ is $R_{\rm rtr}=\exp\left(-4t/2.6{\rm ms}\right)$. The entanglement
transfer is assumed to have an overall success rate of 0.95 including optical
circuit loss and imperfect photon detection efficiency. The coherence time of
the hyperfine levels of the individually trapped Rb atoms is assumed to be 100
ms. The local entanglement swap has an operation time of 200 ns and a success
rate of 0.92 which is a combination of 0.97 efficiency of emitting a photon
from one of the Rb-atoms (assuming a cooperativity of 30) and a total
efficiency of 0.95 for entangling with the other atom in accordance the
entanglement transfer process.

In addition to simulating the case with only one Rb photon transducer for
photon transfer at each end of the segment as shown in Fig. 1, we also
simulate cases with multiple ($N_{\rm tRb}$) Rb photon transducers for photon
transfer. This second level of multiplexing addresses the inefficiencies of
the entanglement transfer step and the entanglement swapping.

Fig. 4(a) shows the simulated average secret key rate $R_{{\rm SK}}$ as a
function of the total distance $L$ for various $N_{\rm tRb}$. We have
optimized the number of repeater nodes to achieve the highest secret key rate
per elementary segment i.e. $R_{\rm SK}/N_{\rm seg}$ with $N_{\rm seg}$ being
the total number segments in the repeater chain. The resulting number of
repeater nodes $N_{\rm rep}=N_{\rm seg}-1$ is shown in Fig. 4(b) and it
determines the needed multiplexing capacity of the Tm-memories for ensuring
continued operation as $N_{\rm mode}=\nu_{\rm Rb}L/N_{\rm seg}/\rm{c}$, which
is shown in Fig. 4(c).

Our simulation shows that a secret key rate above 100 bit/s can be achieved
over 1000 km with a total number of repeaters ranging from 5 to 16 and the
number of modes needed in each Tm-memory ranging from 150 to 350. In essence,
our setup directs a large amount of multiplexing to overcome the fiber loss in
the elementary links to the Tm-memories. Meanwhile, additional multiplexing
for combating the imperfections in the entanglement transfer and swapping can
be achieved with orders of magnitude fewer single Rb atoms. Besides, more
multiplexing could further increase the rate if, e.g., more Rb-atoms were used
in the entanglement generation step to increase the repetition rate for
generating the entangled photon pairs.

##  VI Discussion

In summary, we have proposed a hybrid quantum repeater employing single-spin
photon transducers and ensemble-based photonic memories to achieve high-rate
entanglement distribution over large distances. The hybrid architecture
directs the massive multiplexing necessary for battling transmission loss in
the optical fibers to the ensemble-based memories, while efficient and near-
ideal entanglement swapping and photon pair generation are targeted by the
single-spin photon transducers. Furthermore, we provided a specific example
utilizing Rb atoms coupled to nanophotonic cavities as single-spin photon
transducers and Tm-doped crystal as ensemble-based memories for massive
multiplexing.

The overall framework presented here is relevant to other combinations of
hardware besides the specific example analyzed in this work. Efficient single-
spin photon transducers can be realized with diamond defect centers Knaut  _et
al._ (2023) and quantum dot systems Huber  _et al._ (2018), which can be
matched with other ensemble-based memories based on AFC Afzelius  _et al._
(2009), Raman Heller  _et al._ (2022) or EIT Ma  _et al._ (2017) storage using
impurity-doped crystals Guo  _et al._ (2023); Liu and Jacquier (2006) or
atoms, either laser-cooled or at room temperature. We note that different
hardware combinations may require frequency conversion to be compatible with
each other, which can be achieved through standard techniques based on non-
linear waveguides Tanzilli  _et al._ (2005); Tchebotareva  _et al._ (2019);
Iuliano  _et al._ (2023); Knaut  _et al._ (2023).

While combining different hardware is arguably more complex than a single
hardware repeater, the overall requirements for reaching high-rate
entanglement distribution may be substantially relaxed as demonstrated in this
work. We thus believe that further investigations of hybrid repeater
architectures are a promising direction for future work. In particular,
employing more complex operations such as entanglement purification techniques
Kalb  _et al._ (2017) to boost the fidelity could be envisioned through the
quantum logic enabled by the single-spin photon transducers.

###### Acknowledgements.

J.B. and F.G. acknowledge funding from the NWO Gravitation Program Quantum
Software Consortium (Project QSC No. 024.003.037). J.B. acknowledges support
from The AWS Quantum Discovery Fund at the Harvard Quantum Initiative. H.B.
and S.G.M. gratefully acknowledge funding from the NSF QLCI for Hybrid Quantum
Architectures and Networks (NSF award 2016136), and the NSF Quantum
Interconnects Challenge for Transformational Advances in Quantum Systems (NSF
award 2138068). W.T. acknowledges funding through the Netherlands Organization
for Scientific Research, and the European Union’s Horizon 2020 Research and
Innovation Program under Grant Agreement No. 820445 and Project Name Quantum
Internet Alliance.

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##  Appendix A Rb emitter Hamiltonian

Figure 1: Atomic-level diagram illustrating all the relevant Rb atomic energy
levels participating in the light emission procedure. States $|1\rangle$ to
$|5\rangle$ are the intended used levels. They are connected with thick blue
arrows as their couplings. The first two couplings are laser-induced and the
later two are cavity-induced. The thin red arrows indicate the unintended
drivings and coupling caused by the used laser and cavities. The choices of
polarization ($\pi$, $\sigma^{+}$, $\sigma^{-}$) and wavelengths of light are
noted alongside each coupling arrow. There is a detuning $\delta=1.73$ GHz
respecting the $|1\rangle-|2\rangle$ energy difference in the first laser
driving. The thin red arrows represent the notable faulty couplings induced by
the same lasers and cavity modes. These faulty couplings involve other states
denoted as $|6\rangle$ to $|16\rangle$.

Fig. 1 depicts all the couplings including the intended ones (thick blue
arrows) and the unintended ones (thin red arrows) from the lasers and cavity
modes in the 87Rb atomic system with the levels involved labeled as
$\ket{1}\sim\ket{16}$. From this, the Hamiltonian of the whole system, in a
suitable rotating frame, can be written as

$H={\rm h}\left[\delta|1\rangle\langle 1|+\Omega_{1}\left(|2\rangle\langle
1|+\frac{1}{c_{2,1}}\sum_{i=6}^{8}c_{i,1}|i\rangle\langle
1|+\mathrm{H.c.}\right)+H_{\text{rig}}+H_{\text{flt}}\right]$ (1)

where h is the Plank constant; $\delta$ is the detuning of the first laser
from the transition between the $|1\rangle$ and $|2\rangle$ states;
$\Omega_{1}$ is the Rabi frequency of the first laser driving; $c_{j,i}$ is
the Clebsch-Gordan coefficient corresponding to the $|i\rangle-|j\rangle$
coupling which determines the relative coupling strengths of the couplings
induced by the same laser or cavity coupling;

$H_{\text{rig}}=\Omega_{2}|3\rangle\langle 2|+g_{\rm t}|4\rangle\langle
3|\hat{a}_{\rm t}^{\dagger}+g_{\rm o}|5\rangle\langle 4|\hat{a}_{\rm
o}^{\dagger}+\text{H.c.}\;;$ (2)

and

$\begin{split}H_{\text{flt}}=&-\Delta_{7}|7\rangle\langle
7|-\left(\Delta_{7}+\Delta_{8}\right)|8\rangle\langle
8|-\Delta_{10}|10\rangle\langle
10|-\left(\Delta_{10}+\Delta_{11}\right)|11\rangle\langle 11|-\\\
&\Delta_{13}|13\rangle\langle 13|+\Delta_{15}\left(|15\rangle\langle
15|+|16\rangle\langle 16|\right)+\\\
&\left\\{\frac{\Omega_{2}}{c_{3,2}}\sum_{j=9}^{11}\sum_{i=6}^{8}c_{j,i}\left|j\rangle\langle
i\right|+\frac{g_{\rm
t}}{c_{4,3}}\left[\sum_{i=9}^{11}\left(c_{i,4}\left|4\rangle\langle
i\right|+c_{12,i}\left|12\rangle\langle
i\right|+c_{13,i}\left|13\rangle\langle i\right|\right)\right]\hat{a}_{\rm
t}^{\mathrm{f}\dagger}+\right.\\\ &\left.\frac{g_{\rm
o}}{c_{5,4}}\left[\sum_{i=12}^{13}\left(c_{5,i}\left|5\rangle\langle
i\right|+c_{14,i}\left|14\rangle\langle
i\right|+c_{15,i}\left|15\rangle\langle
i\right|+c_{16,i}\left|16\rangle\langle i\right|\right)\right]\hat{a}_{\rm
t}^{\mathrm{f}\dagger}+\mathrm{H.c.}\right\\},\end{split}$ (3)

where $\Delta_{i}$ are the Rb atomic energy splittings as shown in Fig. 1, and
$\Omega_{2}$, $g_{\rm t}$, and $g_{\rm o}$ are the Rabi frequencies of the
second laser driving, telecom-photon cavity coupling and visible-photon cavity
coupling, respectively, and $\hat{a}_{\rm t}^{\dagger}$, $\hat{a}_{\rm
o}^{\dagger}$ are the creation operators of the right telecom and visible
photon, respectively. The creation operators for faulty ones are $\hat{a}_{\rm
t}^{\mathrm{f}\dagger}$ and $\hat{a}_{\rm o}^{\mathrm{f}\dagger}$ assuming all
the telecom or visible faulty photons are the same.

The atomic decay and the cavity leakage are characterized by the Lindblad
operators below:

$\displaystyle L_{1}=\sqrt{\gamma_{2}}|1\rangle\langle 2|,\quad
L_{2}=\sqrt{\gamma_{3a}}|2\rangle\langle 3|,\quad
L_{3}=\sqrt{\gamma_{3b}}|4\rangle\langle 3|,\quad
L_{4}=\sqrt{\gamma_{4}}|5\rangle\langle 4|,\quad$ (4) $\displaystyle
L_{5}=\sqrt{\kappa_{\rm t}}\hat{a}_{\rm t},\quad L_{6}=\sqrt{\kappa_{\rm
o}}\hat{a}_{\rm o},\quad L_{7}=\sqrt{\kappa_{\rm t}}\hat{a}_{\rm
t}^{\mathrm{f}},\quad L_{8}=\sqrt{\kappa_{\rm o}}\hat{a}_{\rm
o}^{\mathrm{f}}.\quad$ (5)

where $\gamma_{i}$ is the atomic decay rate of the $|i\rangle$ level,
especially, $\gamma_{3a}$ is the decay rate from the $|3\rangle$ state to the
$|2\rangle$ state and $\gamma_{3b}$ to the $|4\rangle$ state. The decay rates
$\kappa_{\rm t}$ and $\kappa_{\rm o}$ characterize the rate of the telecom-
and visible-photon cavity photons going into the fibers, respectively. The
cavity loss is not considered here because it is integrated into the fiber
loss and/or considered in the entanglement swap section. The decay from the
faulty levels has been neglected because the population in the faulty manifold
is small.

Regarding the parameters used in the simulation, the Rb atom has energy
differences $\Delta_{7}=157$MHz, $\Delta_{8}=267$MHz, $\Delta_{10}=50.2$MHz,
$\Delta_{11}=75.3$MHz, $\Delta_{13}=817$MHz, $\Delta_{15}=6.83$GHz and
linewidths $\gamma_{2}=2\pi\times 6.1$MHz, $\gamma_{3a}=2\pi\times 0.19$MHz,
$\gamma_{3b}=2\pi\times 1.5$MHz, $\gamma_{4}=2\pi\times 5.7$MHz. Besides, from
the cavity simulation, we get the intended light proportions to be
$p_{1}=0.98$, $p_{2}=0.99$, $p_{3}=0.83$, and $p_{4}=0.67$, and the cavity
loss rates to be $\kappa_{\rm t}=2\pi\times 1.5$GHz and $\kappa_{\rm
o}=2\pi\times 1.0$GHz. Besides, we also get the cooperativities for the
telecom- and visible-photon cavities $C_{\rm t}=\frac{g_{\rm
t}^{2}}{\kappa_{\rm t}\left(\gamma_{3a}+\gamma_{3b}\right)}=34.4\pm 5.0$ and
$C_{\rm o}=\frac{g_{\rm o}^{2}}{\kappa_{\rm o}\gamma_{4}}=11.2\pm 2.2$ and
those determine $g_{\rm t}$ and $g_{\rm o}$. Moreover, based on the Rb emitter
simulation, the optimized $\Omega_{2}=2.0$GHz, $\delta=1.73$GHz, and
$\Omega_{1}$ is at the order of 100 MHz and time-dependent with the temporal
profile shown in Fig. 2(d) in the main text.

With the parameters above, we numerically calculate the system evolution based
on the master equation

$\frac{\mathrm{d}\rho}{\mathrm{d}t}=\frac{\text{i}}{\rm
h}[H,\>\rho]+\sum_{k}\left(\hat{L}_{k}\rho\hat{L}_{k}^{\dagger}-\frac{1}{2}\left\\{\hat{L}_{k}^{\dagger}\hat{L}_{k},\>\rho\right\\}\right)\;.$
(6)

and get the temporal profiles of the emitted photons as shown in Fig. 2(d) in
the main text. Based on the temporal profiles, we model the entanglement
generation protocol with two Rb emitters. We sample the situations with varied
cavity coupling and different moments within each time bin when the photons
hit the photon detectors. As a result, leaving the fiber loss during distant
transmission and inefficient detection aside, we get a success rate of 0.49
for generating the entangled visible photon pairs with an entangled state

$\rho_{0}=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0.5&0.48&0\\\
0&0.48&0.5&0\\\ 0&0&0&0\end{array}\right).$ (7)

##  Appendix B Suppression of faulty-polarization errors

Figure 1: (a)Energy level diagram of the Rb-cavity system. Each quantum state
is a direct product of the atomic state (first ket) and the photonic state
(second ket). The atomic states are named with numbers in the decimal system.
The two digits in the ket of the photonic state are either 0 or 1, meaning the
vacuum or one-photon states in the telecom- (first digit) and visible- (second
digit) photon cavities. The $\tilde{1}$ with a tilde means the one-photon
state of faulty photons. The diagram is arranged in the way that the energy
levels in the left dashed blue rounded rectangle belong to the intended
subsystem and the ones in the right dashed red round rectangle consist of the
faulty subsystem. (b) The spectrum, or the real parts of the eigenvalues of
the effective Hamiltonians of the intended (left) and faulty (right)
subsystems, respectively. The gray lines in the faulty subsystem have
neglectful coupling to the $|1\rangle|00\rangle$ state. The double arrows
point to the targeted states used in our simulations. (c) Right photon
emission proportion over the sweep of the first laser detuning $\delta$.
Vertical lines denote the eigenenergies of the right subsystem (solid blue)
and the faulty subsystem (dashed red and gray).

For better analysis of the dynamic of the whole Rb-cavity system, we derive
the energy-level diagram of the system incorporating the photonic states shown
in Fig 1(a) where we have separated the intended path and the faulty paths
into two sides. Since the first laser driving $\Omega_{1}$ is about one order
weaker than the driving from the second laser and cavity couplings, it is
suitable to first neglect the couplings induced by it and separate the excited
systems into two parts: Specifically, the intended subsystem driven by the
$\sigma_{+}$ polarized component of the first laser light and the faulty
subsystem driven by the $\sigma_{-}$ polarized component, as circled in with
blue and red dash lines in Fig. 1(a). These two subsystems are governed by the
two Hamiltonians $H_{\text{rig}}$ and $H_{\text{flt}}$, respectively, which
are defined as

$H_{\text{rig}}^{\text{eff}}=H_{\text{rig}}+\frac{h}{2\mathrm{i}}\sum_{k=1}^{6}L_{k}^{\dagger}L_{k}$
(1)

and

$H_{\text{flt}}^{\text{eff}}=H_{\text{flt}}+\frac{h}{2\mathrm{i}}\left(\sum_{k=7}^{8}L_{k}^{\dagger}L_{k}+L_{4}^{\dagger}L_{4}\right)\;,$
(2)

Diagonalizing these two effective Hamiltonians and taking the real parts of
the eigenvalues, one can get the spectrum of two subsystems as shown in Fig.
1(b), while the imaginary parts of the eigenvalues correspond to the effective
decay rates of the eigenstates or the dressed states. From the spectrum, we
find it possible to couple strongly to specific dressed states of the right
subspace while weakly coupling to the ones from the faulty subsystem. Thus,
having the right detuning of the first laser can strongly suppress any driving
from the $\sigma_{-}$ component. Fig. 1(c) shows the right photon emission
rate sweeping over the detuning $\delta$. One can see that the right photon
emission rate peaks at regions where one of the dressed states of the right
subsystem is resonantly driven while the coupling to the dressed states of the
faulty subsystem is weak.

##  Appendix C Stabilization of Population in time-bins

Figure 1: The fluctuation of the possibility that photons are emitted in the
early time bin caused by the fluctuation in the atom-telecom-cavity coupling.
The $t_{\rm h}(\delta)$ is the calibrated time for each $\delta$ making sure
that, when the telecom-cavity coupling $g_{\rm t}$ is at its averaged value,
the photon emission possibility in the early time-bin is always 0.49 (the
yellow horizontal line). Different lines show different population emission
possibilities in the early time bin corresponding to different telecom-cavity
coupling strengths. The blue vertical line marks the eigenenergy of the
targeted state of the intended subsystem. The black vertical line marks the
sweet spot where the emission possibilities are efficiently stabilized.

Our simulation suggests that the Rb-telecom-cavity coupling has an averaged
cooperativity of 34 with a standard deviation of 5.0. This results in roughly
10% standard deviation in the Rb-telecom-cavity coupling strength $g_{\rm t}$,
which is the main cause of the fluctuation in the time-bin population of the
photons. However, by sweeping the first laser frequency, we find sweet spots
where this fluctuation is efficiently suppressed as shown in Fig. 1.

These sweet spots can be understood from the dressed states of the subspace
spanned with $\ket{2}\ket{00}$, $\ket{3}\ket{00}$, $\ket{4}\ket{10}$ states.
Coupled by the second laser and the telecom cavity field, These three states
form a typical $\Lambda$ system whose eigenstates are close to the ones in our
system. The corresponding eigenstate to the one we are targeting reads

$\ket{\psi_{\rm tar}}=\frac{1}{\sqrt{2}}\left(\ket{3}\ket{00}+\frac{g_{\rm
t}}{\sqrt{g_{\rm
t}^{2}+\Omega_{2}^{2}}}\ket{4}\ket{10}+\frac{\Omega_{2}}{\sqrt{g_{\rm
t}^{2}+\Omega_{2}^{2}}}\ket{2}\ket{00}\right).$ (1)

In this picture, one can view the atomic driving procedure as that the
population is driven from the $\ket{1}\ket{00}$ state to the $\ket{\psi_{\rm
tar}}$ with the first laser and then the $\ket{\psi_{\rm tar}}$ experiences
cavity decay leaking the telecom photon into the fiber. In this case, the
effective first laser driving becomes:

$\Omega_{1}^{\rm eff}=\frac{\Omega_{2}\Omega_{1}}{\sqrt{2}\sqrt{g_{\rm
t}^{2}+\Omega_{2}^{2}}},$ (2)

and the effective cavity decay rate turns into:

$\kappa_{\rm t}^{\rm eff}=\frac{g_{\rm t}^{2}\kappa_{t}}{2\left(g_{\rm
t}^{2}+\Omega_{2}^{2}\right)}.$ (3)

Therefore, the fluctuating $g_{\rm t}$ affects positively on $\kappa_{\rm
t}^{\rm eff}$ but negatively on $\Omega_{1}^{\rm eff}$. By introducing a
detuning $\delta^{\prime}$ respecting to the $\ket{1}\ket{00}-\ket{\psi_{\rm
tar}}$ transition, one can tune the relative weights of these two effects and
let them maximally compensate each other. As a result, the transition rate is
stabilized over the fluctuation of $g_{\rm t}$.

##  Appendix D Trapping and cavity designs

To couple an atom with two cavity fields, we design two nanophotonic cavities
at the target frequencies fabricated parallel to each other, combined with a
trap geometry that enables trapping an atom between the cavities. The
individual cavities are designed by quadratic tapering of the filling
fractions on a nanobeam waveguideQuan and Loncar (2003). For practical
fabrication considerations, both cavities are designed with the same
thickness. Due to its larger wavelength, the TE cavity at 1470 nm, requires a
larger thickness for mode confinement and subsequently a high quality-factor.
However, for TE mode cavities for 795 nm, these thicknesses result in low
modal overlap in the atom trapping region due to large mode confinement. To
overcome this issue, we make use of the TM mode cavity design for 795 nm
cavity, which requires a larger thickness while maintaining decent modal
overlap in the trapping region.

Quality factors of both the 780 nm and 1470 nm cavities are individually
optimized and brought close to each other to form a stable trap geometry.
Atoms can be trapped near similar nanophotonic devices by bringing the tweezer
adiabatically on top of these devices and trapping the atom in lattices formed
by the incident and reflected tweezerThompson  _et al._ (2013a); Menon  _et
al._ (2023); Goban  _et al._ (2014). In the case of two parallel cavities,
trapping on top of any one of the two cavities results in minimal coupling to
the other cavity due to minimal cavity mode at the trapping region. To have a
significant coupling strength to both cavities, we propose trapping in between
the two devices as an alternate implementation. If the separation between the
parallel cavities is larger than the wavelength, and the diffraction-limited
spot size of the trapping tweezer, atoms can be trapped in the normal tweezer.
However, the large rayleigh range of a normal tweezer implies an atomic wave
function larger than the device’s thickness. This results in a large variation
in the cavity coupling strength experienced by the atom from shot to shot. To
overcome this, we restrict the separation between the two parallel cavities to
be significantly less than the trapping tweezer wavelength. In this case, the
incident tweezer beam is reflected by the combined device structure to form a
lattice-like potential similar to a single device case. For a reasonable
trapping potential of atoms, the separation between the devices should be
smaller than half the trapping tweezer wavelength.

While the cavities were designed to have high quality factors individually,
when they are kept next to each other, the mode from one cavity can leak into
the other. These additional extrinsic losses $\kappa_{\rm ext}$ through the
second waveguide result in lower overall quality factors
($Q=\frac{\omega}{\kappa_{\rm ext}+\kappa_{\rm int}}$; $\omega$ is the
resonant frequency; $\kappa_{\rm int}$ is the intrinsic losses not coupled
into any waveguides; $\kappa_{\rm ext}$ includes losses into the intended
waveguide mode and the additional losses to the nearby waveguide). This loss
is a function of separation between the two cavities, where the loss increases
with reduced separation between the cavities.

We iterate between device separation, thickness, and refractive index of the
material to find a deep trap potential while maintaining high quality factors
required for the results presented in the paper. To iterate over the
refractive index, we assume tuning of the silicon enrichment ratio in the
silicon nitride. However, the ratio of silicon in the silicon nitride also
modifies the bandgap of the material. As the nitride content is reduced, the
refractive index and the bandgap approaches that of bare siliconPhilipp
(1973). To avoid the above bandgap excitation using 780 nm laser involved in
the protocol, we assume the refractive index of 2.6, assumed in the paper,
results in a bandgap larger than 1.6 eV.

Atoms can be trapped in regions where cavity fields have predominantly linear
polarization direction on top of nanophotonic devices Menon  _et al._ (2020).
However, moving the trap to the edge of the devices results in cavity field
polarization to have components in more than one dimension in the trapping
region, without specific control over the phase between the polarizations.
This results in varying proportions of contributions from $\sigma_{+}$,
$\sigma_{-}$, and $\pi$ in the trapping region. The purity of the intended
sigma polarization is marked in Fig. 1. To overcome this issue, we make a
careful selection of the states involved in the scheme. The level scheme
involved in the protocol is chosen in a way that under correct excitation,
only $\sigma_{-}$ polarization of the cavity fields couple between atomic
states. The corrections coming from the faulty excitation are labeled in the
figure and are accounted for.