# Zero-Added-Loss Entangled Photon Multiplexing
for Ground- and Space-Based Quantum Networks

Kevin C. Chen  Research Laboratory of Electronics, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  Department of Electrical Engineering and
Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139,
USA  Prajit Dhara  Wyant College of Optical Sciences, The University of
Arizona, Tucson, AZ 85721  NSF-ERC Center for Quantum Networks, The University
of Arizona, Tucson, AZ 85721  Mikkel Heuck  Department of Electrical and
Photonics Engineering, Technical University of Denmark, 2800 Kgs. Lyngby,
Denmark  Yuan Lee  Department of Electrical Engineering and Computer Science,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Wenhan Dai
Department of Electrical Engineering and Computer Science, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA  College of Information and
Computer Science, University of Massachusetts, Amherst, Massachusetts 01003,
USA  Saikat Guha  Wyant College of Optical Sciences, The University of
Arizona, Tucson, AZ 85721  NSF-ERC Center for Quantum Networks, The University
of Arizona, Tucson, AZ 85721  Dirk Englund<EMAIL_ADDRESS>Research
Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge,
MA 02139, USA  Department of Electrical Engineering and Computer Science,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Brookhaven
National Laboratory, Upton, NY 11973, USA

###### Abstract

We propose a scheme for optical entanglement distribution in quantum networks
based on a quasi-deterministic entangled photon pair source. By combining
heralded photonic Bell pair generation with spectral mode conversion to
interface with quantum memories, the scheme eliminates switching losses due to
multiplexing in the source. We analyze this ‘zero-added-loss multiplexing’
(ZALM) Bell pair source for the particularly challenging problem of long-
baseline entanglement distribution via satellites and ground-based memories,
where it unlocks additional advantages: (i) the substantially higher channel
efficiency $\eta$ of downlinks vs. uplinks with realistic adaptive optics, and
(ii) photon loss occurring before interaction with the quantum memory – i.e.,
Alice and Bob receiving rather than transmitting – improve entanglement
generation rate scaling by $\mathcal{O}(\sqrt{\eta})$. Based on numerical
analyses, we estimate our protocol to achieve $>10~{}$ebit/s at memory
multiplexing of $10^{2}$ spin qubits for ground distance $>10^{2}~{}$km, with
the spin-spin Bell state fidelity exceeding 99$\%$. Our architecture presents
a blueprint for realizing global-scale quantum networks in the near-term.

##  I Introduction

Entanglement distribution across distant nodes is fundamental to constructing
quantum networks [1]. However, despite recent progress via optical fiber links
[2, 3, 4], scaling quantum networks to global reach remains a formidable
challenge. One approach to increasing the entanglement rate over low
efficiency $(\eta\ll 1)$ elementary links is to use a deterministic Bell-state
source at Charlie (C) midway between quantum repeaters (QRs) Alice (A) and Bob
(B) in lieu of an entanglement swap, i.e. the midpoint source architecture. If
A (B) post-selects on events where a photon passed at least path length
$\overline{\text{AC}}$ ($\overline{\text{BC}}$), in the regime where the QRs
are memory-limited, the average entanglement rate $\bar{\Gamma}$ improves to
${\propto\sqrt{\eta}}$ compared to ${\propto\eta}$ in typical protocols where
C performs local Bell state measurements (BSM) [5]. Despite the increase in QR
complexity, this “midpoint source” scheme enables an increase in
$\bar{\Gamma}$ by $\mathcal{O}(1/\sqrt{\eta})$ and has motivated research
efforts to produce the required entangled photon pair sources, which should be
near-deterministic for the advantage to persist. Thus far, leading efforts are
based on cascaded atomic sources [6, 7] or spontaneous parametric down
conversion (SPDC) sources. While the latter has demonstrated heralded
production of Bell pairs out of a single [8, 9, 10, 11] or a pair of SPDC
sources [12, 13], the existing approaches still suffer in either efficiency or
fidelity required for near-term quantum networks.

Figure 1: Protocol overview — A global-scale quantum network composed of
flying satellites as the quantum transmitters (qTX) and terrestrial stations
as the quantum receivers (qRX). The qTX (C) emits heralded photonic Bell pairs
$|S_{j}^{(+)}\rangle_{AB}$ to A and B, followed by subsequent spectral mode
conversion. Filled (open) circles represent successfully heralded (lost) Bell
pairs. The two bunched photons signify multi-photon emission due to BPS’s
imperfection. $|S_{j}^{(+)}\rangle_{AB}$ is accompanied by a classical signal
$j_{A,B}$ containing frequency information. A 1x$N$ switch then routes it to
an arbitrary spin memory composed of electron-nuclear spins. The qTX
containing quasi-deterministic heralded BPS can also be implemented in ground-
only quantum networks, as shown by node $D$ connected to other network
stations, separated by ground distance $L_{\text{GG}}$ (considered in Appendix
H).

In this Article, we propose a quasi-deterministic Bell pair source (BPS) that
eliminates compounding switching loss of previously proposed spatially
multiplexed [13] or purely temporally multiplexed [12] heralded BPS. Namely,
this “zero-added-loss multiplexing” (ZALM) BPS leverages the large SPDC phase-
matching bandwidth to achieve high transmission rate via spectral
multiplexing.

A high bandwidth BPS is crucial to general long-distance entanglement
distribution to compensate for channel losses. In this proposal, we consider
one specific application of utilizing space-to-ground optical links to
establish ground-to-ground entanglement, though such a quasi-deterministic BPS
would benefit any general quantum network setting. Reaching the global scale,
however, calls for space-based configurations, one example of which is shown
in Fig. 1, where the midpoint source is a satellite sending entangled photons
to two distant ground stations. This satellite-mediated entanglement
distribution scheme has to contend with extreme channel conditions. Here we
consider three primary factors that contribute to our particular setup: large
transmission loss, channel instability, and heralding latency.

Two canonical choices for satellite-based optical transmission links are the
_downlink_ and _uplink_ configurations. Despite the relative ease of just
having an interferometric system for BSM in space, the uplink suffers from
pointing instability due to the “shower-curtain effect” [14], requiring two-
way adaptive optics yet to be demonstrated, making the downlink configuration
more efficient [15]. Therefore, we limit our investigation to a two-photon
downlink architecture.

A single-rail (heralding on one photon) [2, 16] encoded protocol could
increase $\bar{\Gamma}\propto\eta$ over a dual-rail (heralding on two detected
photons) [17] scheme with $\bar{\Gamma}\propto\eta^{2}$. However, it is
presently unclear whether the optical carrier-level phase tracking required in
the single-photon scheme is compatible with space-to-ground links.

Specifically, we assume: (1) the availability of memory-multiplexed
($k$-qubits) QRs on the ground – but not on the satellite; (2) satellite-to-
ground (SG) downlinks; (3) SG transmission in the telecommunications C-band
via frequency conversion for compatibility with space-qualified photonics
[18]; and (4) two-photon-heralding [2, 16].

This Article is organized as follows. Section II presents an overview of the
satellite-assisted entanglement distribution architecture, one specific
example of global-scale quantum network utilizing the ZALM BPS. Section III
discusses the quantum state description of a heralded entangled state and
evaluates the boosted BPS emission rate with spectral multiplexing. Section IV
then describes the two critical components in the ground-based quantum
receiver: (1) a mode converter for temporal-spectral conversion via a sum-
frequency generation process; (2) a cavity-based spin-photon interface based
on a hybrid photonic integrated circuit. We then evaluate the ground-to-ground
spin-spin entanglement state fidelity and efficiency in Sections V-VI,
accounting for imperfections in the ZALM BPS and the ground-based quantum
receivers. Additionally, we discuss the trade-off between entanglement
fidelity and generation rate due to unheralded photon loss in the channels.
Finally, Section VII concludes the paper and offers thoughts on alternative
approaches for implementing the spectrally multiplexed BPS. Importantly, it
addresses engineering challenges that still need to be overcome in order to
realize the proposed architecture.

##  II Entanglement Distribution Architecture Overview

Improving efficiency and fidelity of quantum links for scalable quantum
networks beyond recent satellite-based demonstrations [19, 20] likely requires
heralded entanglement of quantum memories. Here, we propose an architecture
integrating the merits of satellite-based channels and spin-photon interfaces
containing diamond color centers. On the satellite, C as a multiplexed quantum
transmitter (qTX) emits heralded polarization-encoded photonic Bell states
$|S_{j}^{(+)}\rangle_{AB}=(\ket{H_{A}V_{B}}+\ket{V_{A}H_{B}})/\sqrt{2}=(\ket{1,0;0,1}+\ket{0,1;1,0})/\sqrt{2}$
(polarization-Fock representation [13]), accompanied by a classical heralding
message encoding its frequency information, $j_{A,B}$. Idler photons (of same
polarization unknown to us) of a pair of pulse-pumped SPDC sources are
interfered and detected in wavelength demultiplexed channels to herald Bell
pairs in the signal photons. The detail will be covered in Section IV.

Each photon of the Bell pair described by $|S_{j}^{(+)}\rangle_{AB}$ then
travels to its respective terrestrial quantum receiver (qRX), A and B
(detailed in Section IV). At each receiver, a mode-converter (MC) [21, 22]
converts both the frequency and the spectral bandwidth of the photon to match
those of the quantum memories, crucial for efficient cavity-based spin-photon
interaction [23]. Finally, successful photon detection at both A and B
completes teleportation, i.e. A and B sharing spin-spin entanglement upon
heralding on the same Bell state’s photons.

The analysis and discussion of our proposal in the rest of manuscript is
tailored towards studying the performance and challenges of a single quantum
link; however the utility of a ZALM link is applicable to terrestrial fiber-
based quantum links. Multi-hop linear chain of quantum repeaters (where the
channel between A and B is subdivided into smaller segments) with ZALM
architecture based links for generation of entanglement and local entanglement
swapping between memories (in a repeater station) is a natural extension of
our study, as is a regular grid network (on a well defined ‘lattice’) of
repeaters which can support multiple entanglement generation paths. We reserve
the analysis of these advanced network geometries for future work.

##  III Quantum transmitter

###  III.1 Photonic sources of high-quality dual-rail entangled qubits

Generation of high-quality photonic entangled pairs is an open research
challenge, with multiple possible approaches, each having their own set of
merits and demerits. We consider SPDC sources pumped with a mode-locked laser,
combining previous proposals for quasi-deterministic sources [8, 9, 24, 13,
10, 11, 12, 25, 26, 27]. More specifically, our BPS builds on the proposal of
Ref. [13] of interfering photons of a pair of SPDC sources [28, 29, 30] to
herald Bell pair production. Prior proposals necessitate spatial multiplexing
to boost the Bell pair generation rate. However, this requires optical
switches in a tree-configuration whose number of layers grows exponentially
with the number of spatial multiplexing modes. Given finite loss per switch,
the compounded optical loss through the switch array quickly renders the Bell
pair generation inefficient. For example, the multiplexing requirements are
very demanding ($\sim 10^{7}$ sources running in parallel) to achieve quasi-
deterministic state generation. Considering 0.5 dB loss per switch, $\sim
10^{7}$ would lead to $10^{-0.5/10\cdot\log_{2}(10^{7})}\sim 7\%$ efficiency.

In our current proposal, we leverage the large phase-matching bandwidth
$\sigma_{\text{pm}}=10~{}$THz for spectral multiplexing. After the beam
splitter interaction of the intermediate BSM, we demultiplex the output beams
into dense wavelength division multiplexing channels (DWDM) and perform single
photon detection on each channel separately (Bell state analyzer shown in Fig.
2). Each DWDM channel is spaced apart by $\sigma_{\text{ch}}=12.5~{}$GHz over
$\sigma_{\text{pm}}$ in the C-band. If detection (with the correct click
patterns; see Appendix A) occurs in the same frequency channels for the two
DWDMs, then the BPS has heralded production of a photonic Bell state.

First, we consider the quantum state of a single down-conversion process in a
$\chi^{(2)}$ medium [31, 32, 25],

$\displaystyle\ket{\psi}$ $\displaystyle=c_{0}\ket{\textbf{0}}+c_{1}\int
J(\omega_{S},\omega_{I})\hat{a}_{S}^{\dagger}(\omega_{S})\hat{a}_{I}^{\dagger}(\omega_{I})\ket{\textbf{0}}$
$\displaystyle\quad+c_{2}\int
J(\omega_{S},\omega_{I})\hat{a}_{S}^{\dagger}(\omega_{S})\hat{a}_{S}^{\dagger}(\omega_{I})$
$\displaystyle\quad\quad\quad\;\cdot
J(\omega^{\prime}_{S},\omega^{\prime}_{I})\hat{a}_{S}^{\dagger}(\omega^{\prime}_{S})\hat{a}_{I}^{\dagger}(\omega^{\prime}_{I})\ket{\textbf{0}}$
(1)

up to two-photon contributions, where $\ket{\textbf{0}}$ is signal-idler mode
in the vacuum state, and $\hat{a}^{\dagger}({\omega_{k}})$ is the creation
operator at frequency $\omega_{k}$ for the signal ($k=S$) or idler $(I).$.
$J(\omega_{S},\omega_{I})$ represents the joint spectral amplitude function
(Appendix A).

The constituent terms of Eq.(III.1) are the vacuum ($\ket{0,0}$), single
photon entanglement (with $J(\omega_{S},\omega_{I})$ and terms with a second
order contribution. The latter are detrimental to entanglement distribution
protocols as they lie outside the dual-rail photonic qubit Hilbert space. With
photon loss, these terms yield false click patterns and limit the fidelity of
the distributed entangled pairs [13, 33]. Each SPDC source in Fig. 2 comprises
two down-conversion processes 111Although separate SPDC sources have been
shown here, the same could be achieved by a single source and a spectrally
demultiplexed Franson interferometer for time-basis entanglement swap.. The
qTX performs BSM in each DWDM channel by interfering the idler photons from
both SPDC sources. We take $J(\omega_{S},\omega_{I})$ to exhibit frequency
anti-correlation between signal-idler photons, such that a demultiplexed
detection heralds ‘which-frequency’ information about the Bell pair. Based on
the analysis of Ref. [13] and its extension to the current proposal (see
Appendix A for a detailed derivation), the spectrally-synchronized BSM that
yields one of the desirable photon click patterns (say on channel $j$) heralds
an entangled state with a spectral description given as,

$\displaystyle\ket{S^{(\pm)}_{j}}\propto$
$\displaystyle\int_{\vec{\Omega}_{j}}d\bar{\omega}\Biggl{[}\biggl{(}J(\omega_{A_{1}},\omega_{A^{\prime}_{1}})\,J(\omega_{B_{2}^{\prime}},\omega_{B_{2}})\,\hat{a}^{\dagger}_{A_{1}}(\omega_{A_{1}})\,\hat{a}^{\dagger}_{B_{2}}(\omega_{B_{2}})+(-1)^{m_{1}}J(\omega_{A_{2}},\omega_{A_{2}^{\prime}})\,J(\omega_{B_{1}^{\prime}},\omega_{B_{1}})\,\hat{a}^{\dagger}_{A_{2}}(\omega_{A_{2}})\,\hat{a}^{\dagger}_{B_{1}}(\omega_{B_{1}})\biggr{)}$
$\displaystyle+(-1)^{m_{2}}\biggl{(}J(\omega_{A_{1}},\omega_{A_{1}^{\prime}})\,J(\omega_{A_{2}},\omega_{A_{2}^{\prime}})\,\hat{a}^{\dagger}_{A_{1}}(\omega_{A_{1}})\,\hat{a}^{\dagger}_{A_{2}}(\omega_{A_{2}})+(-1)^{m_{1}}J(\omega_{B_{2}^{\prime}},\omega_{B_{2}})\,J(\omega_{B_{1}^{\prime}},\omega_{B_{1}})\,\hat{a}^{\dagger}_{B_{1}}(\omega_{B_{1}})\,\hat{a}^{\dagger}_{B_{2}}(\omega_{B_{2}})\biggr{)}\Biggr{]}$
$\displaystyle\times\ket{0}_{A_{1}}\ket{0}_{A_{2}}\ket{0}_{B_{1}}\ket{0}_{B_{2}}.$
(2)

Here, the modes $A_{1},A_{2}\,(B_{1},B_{2})$ correspond to the qubits
transmitted to A (B), and $\int_{\vec{\Omega}_{j}}d\bar{\omega}$ represents
integration over the $j$-th detection channel’s spectral bandwidth
$\vec{\Omega}_{j}$. Subscript 1 (2) represents the polarization mode $H$
($V$). Parity bits $m_{1}$ and $m_{2}$ depend on the detection pattern and
determine the distributed entangled state (Appendix A).

Figure 2:  Implementation of qTX. – A multiplexed BPS comprises a pair of
pulse-pumped SPDC sources. A spectral demultiplexed BSM heralds the Bell pair
creation and sends out classical messages encoding its frequency information.

The first two terms in Eq. (2) represent having one photon each in A’s and B’s
frequency correlated channels, corresponding to the quantum state equivalent
to
$(\ket{1,0}_{A}\ket{0,1}_{B}\pm\ket{0,1}_{A}\ket{1,0}_{B})/\sqrt{2}=(\ket{H}_{A}\ket{V}_{B}\pm\ket{V}_{A}\ket{H}_{B})/\sqrt{2}\equiv\ket{\Psi^{\pm}}$
in the spectral-mode basis. The remaining terms of Eq. (2) represent a state
in which A (B) receives both photons and B (A) receives none, i.e. it is
equivalent to a term of the form,
$(\ket{1,1}_{A}\ket{0,0}_{B}\pm\ket{0,0}_{A}\ket{1,1}_{B})/\sqrt{2}$ in the
polarization-Fock basis, contributing to the heralded state infidelity. We
note that leakage photons stemming from the second-order term in Eq. III.1
with non-degenerate frequencies (i.e. $\omega_{s}\neq\omega_{s}^{\prime}$)
would not introduce additional errors in spin-spin entanglement, as discussed
later in Section IV.1.

###  III.2 Average Bell pair generation rate

In calculating the average emission rate of the ZALM BPS, we abstract all the
parameters of the transmission and collection optics into a single channel
loss parameter $\sqrt{\eta}$ for each satellite-ground link; assuming suitable
levels of timing synchronicity, adaptive optics, pointing and tracking,
Doppler compensation and beam forming for the architecture design.

The essence of the heralded BPS relies on leveraging the entirety of the phase
matching bandwidth of the SPDC, which we assume to be
$\sigma_{\text{pm}}=10~{}$THz. To compute the BPS emission rate requires
knowing the density matrix of the heralded BPS’ output state
$\rho_{\mathrm{BPS}}$, which depends on the mean photon number per mode
$N_{s}$ that is effectively dictated by the power of the pump field. Given
$\rho_{\mathrm{BPS}}$ and accounting for imperfections in the quantum
receiver, we can evaluate the resultant spin-spin entangled state and its
fidelity to the ideal Bell state, $\mathcal{F}$, after heralded teleportation
of the photonic qubits. More details on calculations of $\mathcal{F}$ will be
covered in Section IV. Figure 3 shows $\mathcal{F}$ as a function of $N_{s}$
for both the heralded BPS and the non-heralded free-running narrowband SPDC
source. For the former, as $N_{s}$ increases, the effect of loss is suppressed
and hence the fidelity increases. For the latter, however, fidelity increases
from having higher Bell pair contribution relative to the other order terms.
After reaching an optimum, the fidelity begins to drop with increasing $N_{s}$
as a result of multi-photon events degrading the photonic Bell state fidelity.
Note that in general the generation rate can be improved by increasing
$N_{s}$, at the cost of lowering the heralded photonic Bell state fidelity
(thereby the spin-spin Bell state fidelity). For an instance, for the ZALM
BPS, increasing $N_{s}$ to $8\times 10^{-2}$ still maintains a fidelity
$\mathcal{F}\geq 0.995$. On the other hand, with a free-running SPDC, the
fidelity drops off much more quickly due to higher order terms that cannot be
eliminated via heralding.

Figure 3: The spin-spin Bell state fidelity $\mathcal{F}$ as a function of the
mean photon number $N_{s}$ for both the heralded BPS (blue solid) and the non-
heralded free-running narrowband BPS (orange solid). The horizontal black
solid line indicates the fixed fidelity $\mathcal{F}=0.998$, which is chosen
based on particular parameter values considered (see main text). Both the blue
and orange dashed lines indicate their intersections at the respective $N_{s}$
values for the heralded and non-heralded BPS.

To fairly compare the performance of the two, we fix the spin-spin Bell state
fidelity still attainable by the non-heralded BPS at $\mathcal{F}=0.998$ and
derive a corresponding a mean photon number of $N_{s}=2.94\times 10^{-2}$ for
the heralded ZALM BPS. Given $N_{s}$, we numerically compute the density
matrix of the heralded photonic state in the Fock basis for each spectral
mode, and consequently calculate the probability of generating a heralded Bell
pair $p_{\mathrm{gen}}$ by taking a partial trace over the Bell basis states.
We find $p_{\mathrm{gen}}=3.6\times 10^{-4}$ per spectral mode. Now,
accounting for all the spectral modes, we can greatly boost the generation
probability. Specifically, assuming a DWDM channel bandwidth of 12.5 GHz, the
number of spectral-multiplexed modes is simply
$N_{\text{modes}}=\sigma_{\text{pm}}/\gamma=10~{}$THz$/$12.5 GHz$=800$. The
probability of successfully heralding a photonic Bell pair per pulse across
the entirety of the phase-matching bandwidth $\sigma_{\text{pm}}$ is
$p_{\text{ZALM}}=1-(1-p_{\text{gen}})^{N_{\text{modes}}}=2.5\times 10^{-1}$.
If we consider a state of the art mode-locked laser with a pulse generation
repetition rate of $\sigma_{\text{rep}}=30~{}$GHz [35], the average Bell pair
emission rate is $p_{\text{ZALM}}\times\sigma_{\text{rep}}\,\approx 7.45$ GHz.

In contrast, for a narrowband-filtered free-running SPDC source (non-heralded
BPS), we find the required mean photon number mode of
$N_{s,\text{SPDC}}=1.27\times 10^{-3}$, to suppress higher order terms and
match the fidelity of the heralded spin-spin state ($\mathcal{F}=0.998$) with
the heralded ZALM BPS. We again compute its density matrix and find the
probability of generating a Bell pair to be $p_{\text{gen,SPDC}}=2.5\times
10^{-3}$. Since we assume a narrowband-filtered source, the absence of
spectral multiplexing means the success probability is the generation
probability itself, i.e. $p_{\text{gen,SPDC}}=p_{\text{SPDC}}$. Assuming the
same pulsed laser repetition rate of 30 GHz, the average generation rate for
the photonic Bell pair is $p_{\text{SPDC}}\times\sigma_{\text{rep}}=75~{}$MHz.

###  III.3 Requirements on the photon detectors for entanglement swap

Here we consider the requirements on the single photon detectors in qTX.
First, the detector reset time should be short enough such that ideally there
are no missed heralding events. Given $N_{s}$, the average production rate for
the signal-idler pair would be $N_{s}\sigma_{\text{rep}}$. The number of
incident photons per channel within the detector’s reset time $\tau_{r}$ would
be $\mu=\tau_{r}N_{s}\sigma_{\text{rep}}/(4\times N_{\text{modes}})$, where
the factor of 4 stems from the need of four detectors for BSM [13].
Consequently, the probability of detecting $\geq 2$ photons in one channel
within the detector’s reset time is ${p=1-\exp(-\mu)-\mu\exp(-\mu)}$ following
the Poisson distribution.

We take a detector reset time $\tau_{r}=1$ ns [36, 37], resulting in
$\mu=1~{}\text{ns}\times 7.45~{}\text{MHz}/(4\times 800)=2.3\times 10^{-3}$.
The probability of detecting $\geq 2$ photons is then $\sim 2.7\times
10^{-6}\ll 1$, which is sufficiently small to neglect missed detection events
due to multiple-photon incidence on the same detector within its reset time.

Second, the detector jitter must be sufficiently small to avoid projecting the
output of the ZALM BPS to a mixed state. Otherwise, this would lead to sub-
optimal BSM due to spectral distinguishability and an overall reduction in the
mode conversion efficiency at the ground stations. To avoid uncertainty in
projection within the 12.5 GHz spectral window given by the DWDM channel
bandwidth, we need the jitter time to be much less than
$1/12.5~{}$GHz$~{}=80~{}$ps. Based on Ref. [38], we assume the detector jitter
time to be close to state of the art at 1 ps. In Appendix B, we also evaluate
the impact of having jitter time much larger than $1/12.5~{}$GHz$\approx
80~{}$ps on the heralded photonic Bell pair. Namely, having an imperfect
measurement would lead to a reduced interference visibility in the BSM,
consequently degrading the fidelity of the photonic Bell state. However, with
an assumed Gaussian spectral profile per DWDM channel, the JSI almost recovers
perfect purity and results in a theoretical maximum visibility of $v=0.996$
(corresponding to a photonic Bell state measurement infidelity of $2\times
10^{-3}$). Further complications concerning mode conversion and spin-photon
quantum teleportation with increased jitter time are beyond the scope of this
Article and thus tabled for future studies.

##  IV Quantum Receiver

Figure 4:  Implementation of qRX. – The qRX contains a mode converter (MC) and
a $1\times N$ switching array that routes the photon to arbitrary channels in
the memory bank, each containing a diamond nanocavity coupled with an
optically active electron spin (red).

###  IV.1 Mode conversion for frequency and bandwidth matching

For efficient cavity-based spin-photon interaction, the photon must be 1)
resonant with the spin’s optical transition and, 2) narrow in bandwidth
relative to the emitter’s optical linewidth [23]. Hence, we consider a “mode-
converter” (MC) that frequency up-converts from telecom C-band to visible
wavelengths and pulse shapes to reduce the photon’s spectral bandwidth, i.e.
increasing the initial temporal width of the photon $\Delta t_{i}$ to a final
temporal width $\Delta t_{f}\gg\Delta t_{i}$.

As shown in Fig. 4, upon arrival from the satellite, the photon with carrier
frequency $\omega_{a}$ enters a high-Q ring cavity [39] made from a
$\chi^{\scriptscriptstyle(2)}$-material, supporting three resonance
frequencies: $\omega_{a},\omega_{b}$ (target frequency matching the emitter’s
transition at 737 nm, see Section IV.2) and $\omega_{c}$ (strong pump’s
frequency) that satisfy energy conservation:
$\omega_{a}+\omega_{c}=\omega_{b}$ [40]. Upon receiving the classical message
$j_{A,B}$ encoding $|S_{j}^{(+)}\rangle_{AB}$’s frequency information, the
pump field is optimally shaped to convert photons in mode $a$ to mode $b$ [22,
21]. Since each DWDM channel corresponds to a unique frequency, each ground
receiver contains an array of ring resonators whose resonance frequencies are
spaced 12.5 GHz apart. To reduce the intensive requirement of having many ring
resonators, each resonator may include phase shifters to address multiple DWDM
channel frequencies. For example, if each resonator has up to 1 THz of tuning
range [41, 42, 43, 44], corresponding to 80 DWDM channels, the number of ring
resonators needed reduces from $N_{\text{modes}}=800$ to 10.

A subtle feature of the MC is that it acts as an additional spectral filter,
which removes spurious spectral modes from interacting with the spin memories
in the qRX. For example, second-order terms with two photon-pairs per SPDC
source as indicated in Eq. (III.1) may lead to interference between spectrally
degenerate photons (for e.g. single idler photons at $\omega_{I}$ from the
constituent SPDC sources) with an accompanying non-degenerate photon (i.e. one
of idler modes actually generates a two-pair term with
$\omega_{I}\neq\omega^{\prime}_{I}$ ); henceforth we shall term these as
“leakage photons”. With perfect detection efficiency, these events would be
immediately flagged based on the detection pattern (since the pattern would
deviate from the ideal spectral mode-synchronized detection). However, in the
practical case of non-unity detection efficiency, these events may not be
discernible from a scenario in which only first-order photon-pairs from the
two SPDC sources interfere (i.e. idler photon at $\omega^{\prime}_{I}$ is
lost). Given a particular BSM with the correct click pattern, the qTX sends
the known information $j_{A,B}$ about the heralded Bell pair’s frequency.
Regardless of the presence of leakage photons, say at $\omega_{S}^{\prime}$,
the MC would only up-convert $\omega_{S}$ photons specified by $j_{A,B}$. In
this sense, the final spin-spin Bell state is partially post-selected
spectrally. We note that the edge case of having second-order terms with
spectrally degenerate idler photons (and therefore signal by frequency
correlation) is indeed a non-correctable error, but is accounted for in the
fidelity calculations presented in Section V later.

We use the method developed in Ref. [21] to realize a beam splitter
Hamiltonian (in the frequency domain) between modes $a$ and $b$ via sum-
frequency-generation (SFG) between the pump and mode $a$, though other
proposed methods are equally viable [45, 46, 47, 48, 49]. The Hamiltonian
describing the cavity modes is

$\displaystyle\hat{H}=\hbar\chi_{\scriptscriptstyle{\rm{SFG}}}\big{(}\hat{a}\hat{b}^{\dagger}\hat{c}+\hat{a}^{\dagger}\hat{b}\hat{c}^{\dagger}\big{)}+\sum_{q}i\hbar\sqrt{\kappa_{q,\rm{w}}}\big{(}\hat{q}^{\dagger}\hat{w}_{q}-\hat{w}_{q}^{\dagger}\hat{q}\big{)},$
(3)

where $q=\\{a,b,c\\}$, $\hat{w}_{q}$ is the annihilation operator of the input
(i.e. waveguide) mode interacting with cavity mode $q$, and
$\kappa_{q,\rm{w}}$ is the cavity-waveguide coupling rate.
$\chi_{\scriptscriptstyle{\rm{SFG}}}$ is the SFG coefficient. If mode $c$
contains the strong pump mode, it may be treated classically, and the first
term in the Hamiltonian has the beam splitter form [21]

$\displaystyle\hat{H}_{\scriptscriptstyle{\rm{BS}}}=\hbar\Big{(}\Lambda^{\\!*}(t)\hat{a}^{\dagger}\hat{b}+\Lambda(t)\hat{a}\hat{b}^{\dagger}\Big{)},$
(4)

where
$\Lambda(t)\\!=\\!\chi_{\scriptscriptstyle{\rm{SFG}}}\langle\hat{c}(t)\rangle\\!=\\!\chi_{\scriptscriptstyle{\rm{SFG}}}\sqrt{n_{c}(t)}$
with $n_{c}$ being the number of pump photons in the control mode, $c$. Taking
the quantum state of the cavity to be

$\displaystyle\ket{\Psi_{\rm{cav}}}\equiv[\psi_{a}(t)\hat{a}^{\dagger}+\psi_{b}(t)\hat{b}^{\dagger}]\ket{0}_{a}\ket{0}_{b},$
(5)

we can derive the equations of motion from the Schrodinger equations, which
are detailed in Appendix C. The input state may be expressed in a time-bin
basis as [21]

$\displaystyle\ket{\Psi_{\rm{in}}}\equiv\int
dt\xi_{a,\rm{i}}\hat{w}_{a}^{\dagger}(t)\ket{\bm{0}}_{t},$ (6)

with $\ket{\bm{0}}_{t}$ representing the temporal multi-mode vacuum state
$\hat{w}_{a}^{\dagger}(t)$ populates the time-bin indexed by $t$ with one
photon. The output state is

$\displaystyle\ket{\Psi_{\rm{out}}}\equiv\int
dt\xi_{a,\rm{o}}(t)\hat{w}_{a}^{\dagger}(t)\ket{\bm{0}}_{t}+\int
dt\xi_{b,\rm{o}}(t)\hat{w}_{b}^{\dagger}(t)\ket{\bm{0}}_{t},$ (7)

where $\xi_{a,\rm{o}},\xi_{b,\rm{o}}$ are two Schrodinger coefficients
solvable by the equations of motion (see Appendix C). We assume that a maximum
conversion efficiency is achieved when $\xi_{a,\rm{o}}(t)\\!=\\!0$ by solving
for an optimally shaped control pulse described by $\Lambda(t)$.

As an example, let us consider Gaussian input pulses

$\displaystyle\xi_{a,\rm{i}}(t)$
$\displaystyle=\\!\sqrt{\frac{2}{\tau_{\\!\scriptscriptstyle\mathcal{G}}}}\\!\left(\frac{\text{ln}(2)}{\pi}\right)^{\\!\\!\frac{1}{4}}\\!\\!\exp\\!\left(\\!-2\text{ln}(2)\frac{(t-T_{\rm{in}})^{2}}{\tau_{\\!\scriptscriptstyle\mathcal{G}}^{2}}\right),$
(8)

where $|\xi_{a,\rm{i}}(t)|^{2}$ has a full temporal width at half maximum
(FWHM) of $\tau_{\\!\scriptscriptstyle\mathcal{G}}$, spectral width of
$\Omega_{\scriptscriptstyle\mathcal{G}}\\!=\\!4\text{ln}(2)/\tau_{\\!\scriptscriptstyle\mathcal{G}}$,
and integrates to 1 (over the infinite interval from $-\infty$ to $\infty$).

Figure 5: Simulation results for frequency conversion and bandwidth narrowing.
$\xi_{a,\rm{i}},\xi_{a,\rm{o}},\xi_{b,\rm{o}}$ represent the input pulse in
frequency mode $a$, the output pulse in mode $a$, and the target output pulse
in mode $b$. $\Lambda$ is the solved control pulse optimized for the mode
conversion process.

Specifically, we use an input pulse centered at 1550 nm with a temporal width
of $\tau_{\\!\scriptscriptstyle\mathcal{G}}\\!=\\!80\,$ps and
$\kappa_{a,\rm{w}}\\!=\\!4\Omega_{\scriptscriptstyle\mathcal{G}}$, which
ensures efficient absorption into cavity mode $a$. The output pulse has a
target wavelength of 737 nm corresponding to the optical transition of the
spin memory of choice (Section IV.2). Figure 5 plots the input and output wave
packets whose bandwidth is $\kappa_{b,\rm{w}}\\!=\\!2\pi\times 200\,$MHz to
match the bandwidth of the output pulse to the cavity containing the spin
qubit. The small amplitude of the blue line in Fig. 5 indicates a very small
fraction of the incident wave packet leaking through the cavity without being
absorbed. This fraction decreases towards zero very rapidly as
$\kappa_{a,\rm{w}}$ is increased relative to
$\Omega_{\scriptscriptstyle\mathcal{G}}$ [21].

###  IV.2 Spin-photon interface for quantum memory storage

Illustrated in Fig. 4, after passing through the MC, the photons at A and B
each arrive at a bank of spin memories, which we take to be SiV- centers in
diamond coupled to nanocavities as described in Ref. [50] for memory
multiplexing. Each SiV-’s electron spin can map onto a neighboring nuclear
spin for long memory storage via hyperfine interaction [51] (Appendix E),
minimizing decoherence while the repeater waits for subsequent photons. For
calculating the spin-spin Bell state fidelity, we assume a sufficiently long
nuclear spin coherence time $\gg 1~{}$s and neglect dephasing errors.

The spin memories can be optically interfaced through a $1\times N$ switching
array that directly routes from 1 input channel to $N$ output channels. For an
example, beam-steering devices such as a spatial light modulator may be used
to route incoming photons to an arbitrary cavity in a free-space
configuration. For the calculations covered in the remainder of the Article,
we assumed using a sufficiently fast optical switch array with speed
$\sim\sqrt{\eta}p_{\text{ZALM}}\sigma_{\text{rep}}\approx 100~{}$kHz. In
Appendix D, however, we also consider spin memories heterogeneously integrated
into a LiNbO3-based [39, 52] PIC for high-bandwidth operations. For the
latter, the entanglement generation rate decreases with increasing memory
multiplexing due to a $\log_{2}N$ depth Mach-Zehnder interferometer. One
strategu to minimize the required dimension of the switching array is to
assume the presence of multiple memories inside each cavity. Since the
memories (at a given site) would be pre-characterized, their emission
frequencies are known beforehand. The MC can then be used to spectrally select
an emitter via upconversion.

Importantly, prior to the switching array, the modes of polarization-encoded
photonic state are split into two physical paths (on the PIC) by a
polarization beam splitter. Only polarization mode $\ket{H}$ enters the
$1\times N$ optical interposer to an array of $k$ nanocavities. The photon
then reflects off the cavity and acquires a spin-state-dependent phase [23].
Mode $\ket{V}$, on the other hand, acquires a constant phase by reflecting off
a mirror in a path length-matched to that traveled by the $\ket{H}$ component.
Together, the polarization-encoded qubit undergoes a controlled-phase gate
that effectively entangles the photonic qubit with the spin memory. After the
two polarization modes re-interfere at a beam splitter, they are subsequently
detected in the diagonal basis to herald quantum teleportation, i.e. mapping a
photonic qubit $\ket{\psi}_{P}=\alpha\ket{H}+\beta\ket{V}$ onto the spin qubit
$\ket{\psi}_{S}=\alpha\ket{\downarrow}+\beta\ket{\uparrow}$.

Crucially, the controlled-phase gate fidelity depends on the atom-cavity
cooperativity, $C$. In the limit of high cooperativity, the cavity reflection
coefficient (for the $H$-polarization mode) would be

$\displaystyle r\xrightarrow{C\gg 1}\frac{C-1}{C+1}$ (9)

More details on the derivation can be found in Appendix E. For the
entanglement fidelity calculations presented in Section V, we assume a
cooperativity of $C=100$ [3]. Therefore, the mode-converted photon’s bandwidth
of 200 MHz is still much narrower than the spin’s Purcell-broadened linewidth,
a requirement which is paramount to the cavity reflection protocol [23, 53].

##  V Entanglement fidelity

We now analyze the spin-spin Bell state fidelity by accounting for
imperfections in the qTX and qRX. First, for each spectral mode (i.e. spanning
a single DWDM channel bandwidth), we compute the density matrix of the
heralded photonic state $\rho_{\mathrm{ZALM}}$ based on Eq. 2.
$\rho_{\mathrm{ZALM}}$, in the polarization-Fock basis
$\\{\ket{H_{A},V_{A};H_{B},V_{B}}\\}$, contains up to two-photon contributions
for $A$ and $B$. As for the spin qubits each initialized in a superposition
state $\ket{\psi}_{S,i}=\ket{\downarrow}+\ket{\uparrow})/\sqrt{2}$, they form
a product state

$\displaystyle\rho_{S,i}=\left(\ket{\psi}\\!\\!\bra{\psi}_{S,i}\right)^{\otimes
2}$ (10)

Hence, the collective initial state for both photonic and spin qubits is
$\rho_{i}=\rho_{\mathrm{ZALM}}\otimes\rho_{S,i}$.

By considering finite cooperativity and waveguide-cavity coupling strengths
based on the parameters used in Ref. [50], we evaluate the resultant
controlled-phase gate operation acting only on the Bell basis states,
$\hat{U_{\mathrm{CZ}}}$. For non-vacuum basis states outside of the Bell
basis, we treat these contributions as erroneous photonic states that project
the spins into maximally mixed states. Assuming perfect single-qubit gate for
the Hadamard and Pauli-correction operations post-heralding, the teleported
state’s fidelity $\mathcal{F}_{\mathrm{tele}}$ shared between the two spin
memories at A and B is calculated by taking the overlap with an ideal spin-
spin Bell state $\ket{\Phi^{+}}$ (considering here only one of the four Bell
states):

$\displaystyle\mathcal{F}_{\mathrm{tele}}$
$\displaystyle=w_{\mathrm{Bell}}\cdot\langle\Phi^{+}|\widehat{U}_{\mathrm{CZ}}\rho_{i}\widehat{U}_{\mathrm{CZ}}^{\dagger}|\Phi^{+}\rangle+\frac{w_{\mathrm{not-
Bell}}}{4}$ (11)

where $w_{\mathrm{Bell}}$ and $w_{\mathrm{not-Bell}}$ represent the partial
traces over the Bell and non-Bell basis states. The latter term is again due
to terms outside of the Bell basis projecting the spin memories into maximally
mixed states. Accounting for dark counts (later calculations assume
$10^{2}~{}$Hz based on Ref. [20]), we further modify the teleported state’s
fidelity to be,

$\displaystyle\mathcal{F}_{\mathrm{tele}}$
$\displaystyle\rightarrow\left(w_{\mathrm{Bell}}\cdot\langle\Phi^{+}|\widehat{U}_{\mathrm{CZ}}\rho_{i}\widehat{U}_{\mathrm{CZ}}^{\dagger}|\Phi^{+}\rangle+\frac{w_{\mathrm{not-
Bell}}}{4}\right)$
$\displaystyle\quad\times(1-p_{\mathrm{dark}})^{2}+\frac{2(1-p_{\mathrm{dark}})p_{\mathrm{dark}}+p_{\mathrm{dark}}^{2}}{4}.$
(12)

Next, we focus on infidelity caused by photon loss in the channel. The issue
arises from a competition between efficiency and fidelity. Since initializing
spins is time-consuming [3], time can be saved by not re-initializing the
spins after every attempt (where photons transmit through atmosphere and
arrived at qRX). A complication arises if a photon is lost after reflecting
off the cavity. This qubit loss error projects the spin into a maximally mixed
state; however since the loss is unheralded, the spins may not be re-
initialized for the subsequent interacting photons. As a result, the average
spin-spin Bell state fidelity decreases as the number of attempt increases. By
setting a maximum number of attempts $N_{\text{max}}$ before needing to re-
initialize the spins, we can then minimize this photon loss infidelity,
$\epsilon$.

To calculate the dependence of $\epsilon$ on $N_{\text{max}}$, we consider
three scenarios when a photon arrives at the qRX. It can be (1) detected with
probability $\sqrt{\eta}$ (assume identical downlink channels), (2) lost
before reaching the spin with probability $p_{\mathrm{lost}}$, or (3) lost
after reaching the spin with probability $p_{\mathrm{e}}$. Since generally
$p_{\mathrm{lost}}\gg\\{\sqrt{\eta},p_{\mathrm{e}}\\}$, we save time by not
re-initializing the spins after every channel use.

However, as mentioned previously, skipping spin re-initialization after
detector click potentially causes an unheralded error. Thus, we optimize the
spin-spin entanglement fidelity by constraining the number of loading attempts
before we re-initialize the spins. For simplicity of analysis, we fix an
average qTX transmission rate at
$1/\tau_{0}=p_{\mathrm{ZALM}}\sigma_{\mathrm{rep}}\approx 7.45$ GHz (Section
III.2). The probability of at least one unheralded error occurring in the
first $m-1$ time bins conditioned on detector clicks on the $m\mathrm{th}$ bin
is

$\displaystyle P_{\mathrm{error}}(m,m)$ $\displaystyle=1-\left(\xi
p_{\mathrm{lost}}/(1-\sqrt{\eta})\right)^{2(m-1)},$ (13)

where $\xi$ is the probability of both A and B receiving photons, computed
from taking a partial trace of $\rho_{\mathrm{ZALM}}$ over basis states with
$\geq 1$ photons going to A and B. Then, the probability of error over $N$
bins is

$\displaystyle{P}_{\mathrm{error}}$
$\displaystyle=\sum_{m=1}^{N}{P}_{\mathrm{error}}(m,m)(1-\eta)^{m-1}\eta.$
(14)

Finally, the average state fidelity is

$\displaystyle\mathcal{F}$
$\displaystyle=(1-P_{\mathrm{error}})\mathcal{F}_{\mathrm{tele}}+\frac{P_{\mathrm{error}}}{4}.$
(15)

The maximum fidelity $\mathcal{F}_{\mathrm{max}}$ occurs when
$P_{\mathrm{error}}=0$. We can then define
$\epsilon=\mathcal{F}_{\mathrm{max}}-\mathcal{F}$ as the average infidelity
arising from the unheralded photon loss error. Constraining $\epsilon$
therefore limits the maximum number of bins $N_{\text{max}}$ before needing to
re-initialize the spin memories.

##  VI Average entanglement generation rate

Figure 6: Entanglement generation rate $\bar{\Gamma}$ as a function of (a) the
total channel efficiency $\eta$ and (b) the number of spins $k$. (a) In the
memory-limited regime ($k=10^{1}$), the normalized rate
$\bar{\Gamma}\tau_{\mathrm{idle}}/(k-1)$ scales $\propto\sqrt{\eta}$, whereas
it scales $\propto\eta$ in the source-limited regime ($k=10^{7}$). (b)
$\bar{\Gamma}$ vs $k$ for both ZALM and SPDC with varying
${\epsilon\in\\{10^{-2},10^{-3}\\}}$ and total downlink atmospheric
attenuation ${\alpha_{\mathrm{atm}}=\\{40,50\\}~{}\mathrm{dB}}$. We
additionally consider $\overline{AB}=10^{2}~{}$km (purple). These calculations
assume a nuclear spin coherence time $\gg 1~{}$s.

Lastly, we compute entanglement generation rate $\bar{\Gamma}$ by considering
$k$ spins in the qRX. For simplicity, we let only a single initialized spin to
accept photons at a time. Upon successful spin-photon mapping, A communicates
with B to determine if B’s corresponding photon was successfully detected. If
both photons were detected, A then transfers the electron spin to the nuclear
spin for memory storage (see Appendix E). Otherwise, A re-initializes the spin
qubit and awaits subsequent successful detection(s). This communicate-and-
reset sequence takes a time $\tau_{\mathrm{idle}}$, given by the sum of
communication time $\tau_{\mathrm{comm}}$ and the spin reset time
$\tau_{\mathrm{reset}}$.

Since the remaining $(k-1)$ spins are inactive, each spin is “on-duty” for
time $\tau_{\mathrm{idle}}/(k-1)$. The qTX generates
$N_{k}\equiv\tau_{\mathrm{idle}}/[(k-1)\tau_{0}]$ attempts during a single
spin’s on-duty time. Furthermore, the target fidelity
$\mathcal{F}=\mathcal{F}_{\mathrm{max}}-\epsilon(N_{\text{max}})$ limits
$N_{\text{max}}$, as mentioned in Section V. Hence, for a fixed $k$, the on-
duty spin would actually be active for $N=\min(N_{k},N_{\text{max}})$
attempts.

In one clock cycle over time $\tau_{\mathrm{idle}}k/(k-1)$, each spin would
re-initialize after every $N$ attempts. The probability both stations detect
photons is then

$\displaystyle p_{\mathrm{success}}$
$\displaystyle=\eta\left(\frac{1-(1-\sqrt{\eta})^{2N}}{1-(1-\sqrt{\eta})^{2}}\right),$
(16)

yielding an average of $kp_{\mathrm{success}}$ successfully detected pairs.
Finally, $\bar{\Gamma}$ is the ratio between the number of successfully
detected pairs and clock cycle:

$\displaystyle\bar{\Gamma}$
$\displaystyle=\frac{p_{\mathrm{success}}\cdot(k-1)}{\tau_{\mathrm{idle}}}$
(17)

Fig. 6(a) compares the normalized rate
$\bar{\Gamma}\tau_{\mathrm{idle}}/(k-1)$ at $k\in\\{10^{1},10^{7}\\}$. In the
memory-limited regime ($k=10^{1}$), $\bar{\Gamma}$ scales as $\sqrt{\eta}$
(see Appendix F for derivations), highlighting the advantage of the “midpoint
source” [5] architecture. In contrast, in the source-limited regime
($k=10^{8}$), we recover the $\bar{\Gamma}\propto\eta$ scaling.

In Fig. 6(b), we compare the entanglement generation rate $\bar{\Gamma}$
between using the ZALM BPS and using a free-running SPDC source (with a
narrowband 200 MHz filter) as the qTX in both memory-limited and source-
limited regimes. Furthremore, we compare their $\bar{\Gamma}$ between total
downlink atmospheric attenuation $\alpha_{\mathrm{atm}}=40~{}$dB and 50 dB,
corresponding to channel losses $\sqrt{\eta}\approx 0.2\%$ and $0.07\%$,
respectively. The total loss accounts for attenuation in the downlink channel
itself, 3.57 dB from adaptive optics [54, 55], 3 dB from mode conversion
inefficiency (Appendix C), 2.68 dB from the diamond nanocavity (with
cooperativity $C=100$), $\sim$0.8 dB insertion loss from the switching array
(assumed a single-layer low-loss interposer), and 0.044 dB from detector
inefficiency [3].

We also consider infidelity arising from unheralded photon loss error
$\epsilon=\\{10^{-2},10^{-3}\\}$, which effectively dictate the frequency of
spin re-initialization (the consequential rate-fidelity trade-off is
investigated in Appendix G). Lastly, we evaluate two ground-to-ground
distances $\overline{AB}$ at $10^{4}$ km and $10^{2}$ km, with corresponding
communication times $\tau_{\mathrm{comm}}=60$ ms and
$\tau_{\mathrm{comm}}=0.5$ ms, respectively. The spin re-initialization time
is assumed to be $\tau_{\mathrm{reset}}=30~{}\mu$s [3].

For $\overline{AB}=10^{4}$ km (blue, orange, and yellow curves in Fig. 6(b)),
at low $k$, ZALM and SPDC have comparable $\bar{\Gamma}$ since the rate is
limited by a quickly saturated bank of memories, i.e. photons are arriving at
a rate faster than the spins are able to be freed up. However, as $k$
increases, the advantage of the ZALM BPS starts manifesting as its rate
performance surpasses that of a non-heralded SPDC source.

Figure 7: Entanglement generation rate $\bar{\Gamma}$ and the probability of
success for the ZALM BPS $p_{\text{ZALM}}$ as a function of spectrally
multiplexed modes in the qTX, $N_{\text{modes}}$. $\bar{\Gamma}$ is evaluated
for $k=\\{10^{4},10^{5},10^{6}\\}$. Calculations assume $\eta=40~{}$dB,
$\epsilon=10^{-2}$, and $\overline{AB}=10^{4}$ km.

The point of divergence between the ZALM BPS and SPDC depends on $\epsilon$
and $\tau_{\mathrm{comm}}$. To maintain a small $\epsilon=10^{-3}$, A and B
need to re-initialize the spins more frequently , i.e. reducing
$N_{\text{max}}$ regardless of $k$. Hence, ZALM and SPDC achieve similar
$\bar{\Gamma}$. Instead, if $\tau_{\mathrm{comm}}$ is reduced, e.g. smaller
$\overline{AB}=10^{2}~{}$km, the effect of memory saturation is suppressed. As
a result, ZALM would greatly outperform SPDC even with small $k$, shown by the
purple curves in Fig. 6(b). Lastly, $\bar{\Gamma}$ plateaus as $k$ increases
indefinitely, at which point $\bar{\Gamma}$ is solely determined by $\eta$ and
$\tau_{0}$,

$\displaystyle\bar{\Gamma}\rightarrow\eta/(1-(1-\sqrt{\eta})^{2})\cdot
2/\tau_{0}\cdot\log\left(1/(1-\sqrt{\eta})\right).$ (18)

The upper bounds, indicated by the black dashed lines, are $7.3\times
10^{2}~{}$Hz for $\alpha_{\mathrm{atm}}=40~{}$dB and $7.3\times 10^{1}$ Hz
$\alpha_{\mathrm{atm}}=50~{}$dB. Similar calculations for a ground-only
quantum repeater network are presented in Appendix H, where we consider a
midpoint source equidistant from A and B separated by $2L_{\text{GG}}=10^{2}$
km. Expectedly, a qTX based on the ZALM BPS shows a rate advantage by more
than an order of magnitude than one based on a non-heralded SPDC source.

As shown in Fig. 7, we further compute the entanglement generation rate
$\bar{\Gamma}$ and the probability of success $p_{\text{ZALM}}$ as a function
of the number of spectrally multiplexed modes $N_{\text{modes}}$ for
$k=\\{10^{4},10^{5},10^{6}\\}$. The calculations here assume $\eta=40~{}$dB,
$\epsilon=10^{-2}$, and $\overline{AB}=10^{4}$ km. Expectedly,
$p_{\text{ZALM}}$ approaches unity as $N_{\text{modes}}$ increases. However,
despite the increase in the qTX’s transmission rate $1/\tau_{0}$,
$\bar{\Gamma}$ quickly saturates at $N_{\text{modes}}<10^{2}$ for $k=10^{4}$,
a manifestation of the memory-saturation effect. As $k$ increases to $10^{6}$,
the rate at which spin memories is freed up at the qRX becomes comparable to
$1/\tau_{0}$, giving rise to an increase in $\bar{\Gamma}$ monotonic with
$N_{\text{modes}}$. Given the saturation behavior, the number of spin memories
available at the qRX effectively sets the upper bound on $N_{\text{modes}}$
used in ZALM, past which point the rate advantage of the quasi-deterministic
BPS nullifies.

##  VII Discussion and outlook

In this Article, we proposed a midpoint source architecture relying on a
quasi-deterministic ‘zero-added-loss multiplexed’ BPS for quantum networks. It
leverages the large phase-matching bandwidth intrinsic to SPDC sources and
utilizes spectral multiplexing via commercially available DWDMs. Moreover, our
proposed mode converter crucially upconverts the heralded photons to spectral
modes with matching frequency and bandwidth to those of solid-state spin
qubits, a step which is essential for memory-based repeater networks. Our
calculations show that the ZALM BPS greatly increases the entanglement
generation rate $\bar{\Gamma}$, especially in low-transmission, memory-limited
links. In our scheme, upon receiving heralded photonic Bell pairs from the
quantum transmitter on the satellite, the ground-based quantum receivers
perform photon-to-spin mapping and herald spin-spin entanglement between two
remote terrestrial stations. Our calculations show that in the memory-limited
regime, our “midpoint source” scheme has a favorable $\sqrt{\eta}$ rate
scaling. As memory multiplexing increases to $k\gtrapprox 10^{2}$, for ground-
to-ground distance $10^{2}$ km, the ZALM BPS enables $\bar{\Gamma}>10$ Hz with
theoretical spin-spin Bell state fidelity $\mathcal{F}>0.99$. This
significantly outperforms the case where a single SPDC (non-heralded) source
is used, in which $\bar{\Gamma}$ is lower by $\sim 2$ orders of magnitude. We
stress that such a high bandwidth BPS should prove valuable to any two-way
quantum repeater network configurations. For example, the advantage is
apparent in a ground-only quantum network as shown in Appendix H.
Additionally, the ZALM BPS may also benefit applications that must rely on
distribution of optical entanglement, such as optical quantum computing [56,
57, 58], precision measurement [59], and all-optical quantum repeaters [60,
61].

Currently, the upper bound of $\bar{\Gamma}$ is limited by the low probability
of detecting the same Bell pair, which is a function of both channel loss
$\eta$ and the BPS emission rate $1/\tau_{0}$. While $\alpha$ (e.g.
atmospheric attenuation) is often a fixed parameter, $\eta$ can be further
improved by additional engineering: reducing the DWDM channel bandwidth to
increase the number of spectrally multiplexed modes, minimizing loss in
optical transmission (for e.g. increasing adaptive optics transmission
efficiency, using larger optical apertures etc.), minimizing insertion loss
for the $1\times N$ switching array, and increasing the cavity reflection
efficiency (dictated by both design and fabrication capability currently
[62]). As for boosting the ZALM BPS’ emission rate, using higher pump powers
does increase the mean photon numbers $N_{s}$ (and consequently the heralding
probability) at the cost of degraded state fidelity. Having higher single
photon detectioon efficiency on the satellite could also improve $1/\tau_{0}$.
We stress that for ZALM’s advantage to persist, the detection efficiency must
be high in qTX entanglement swap and heralding. Missed photon detection
translates to infidelity in the heralded state, countered only by lowering
$N_{s}$, which further limits the heralding and entanglement distribution
rate.

The implementation of the qTX in the proposal thus far demands having a large
number of detectors in the midpoint source. Alternative to DWDM and having
detectors for each spectral channel, one may also use time-of-flight
measurements via dispersive optical elements already experimentally
demonstrated in Ref. [63]. With a detector reset time $\tau_{r}$ and a pump
whose repetition rate of $\ll 1/\tau_{r}$, a single detector would suffice,
therefore significantly reducing the detector requirement for the BPS.
However, we note that having a high detection efficiency is critical to
correctly heralding Bell state creation in the qTX [13]. However, we note that
there exists a large attenuation loss in the dispersive element ($\sim 10$ dB
based on Ref. [63]). Further technological improvements in constructing high-
efficiency and highly dispersive optics are warranted to realize a practical
qTX based on time-of-flight measurements.

In the qRX, the mode converter is imperative to enabling $\ll 1$ ns long
photons to interact with spin memories with $<$ns temporal width at
frequencies 100s THz away. As opposed to sum-frequency generation in high-$Q$
ring resonators, time-lensing effect already realized in electro-optical
modulating platforms (e.g. LiNbO3) could also be viable alternatives [64, 49].

Given current demonstration of PIC-integrated $k\sim 10^{2}$ SiV- [65], the
achievable $\bar{\Gamma}$ is $\sim 1~{}$ebit/s. It remains another engineering
challenge to realize large-scale integration of solid-state spin qubits in
photonic structures. Potential avenues may include implanting multiple
emitters within each cavity via focused-ion beam [66, 65] and spectrally
select a frequency-unique spin.

Lastly, instead of using a cavity-reflection spin-photon mapping protocol, BSM
based on atomic emission interfering with photonics Bell pairs can also
entangle remote spins [5]. The spin memories suitable for the proposed
architecture extends to other matter qubits such as defects in Si [67] and SiC
[68], rare-earth ions [69, 70], atomic vapors [71], cold atoms, and trapped
ions, may be suitable memory qubits as well. Irrespective of the entanglement
scheme and platform choice, our proposal remains viable as an optical
entanglement distribution protocol for midpoint source based quantum links.

## Acknowledgements

We thank Zheshen Zhang (Univ. of Michigan), Paul G. Kwiat (UIUC), Eric Bersin
(MIT-LL), Franco N. C. Wong (MIT), Babak N. Saif (GSFC, NASA), and Madison
Sutula (MIT/Harvard) for discussions. We also would like to thank Thaddeus
Ladd (HRL), Danny Kim (HRL), Sofiane Merkouche (HRL), Shanying Cui (HRL), and
Shuoqin Wang (HRL) for invaluable feedback. K.C.C. acknowledges funding
support by the National Science Foundation RAISE-TAQS (Grant No.1839155) and
the MITRE Corporation Moonshot program. D.E. acknowledges partial support from
the NSF RAISE TAQS program (Grant No. 1839155). P.D. acknowledges funding
support from the Nicolaas Bloembergen Graduate Student Scholarship in Optical
Sciences. M. H. acknowledges support from the Villum Foundation (QNET-NODES
Grant No. 37417). W.D. is supported by the National Science Foundation to the
Computing Research Association for the CIFellows 2020 Program. P.D., S.G., and
D.E. acknowledge the National Science Foundation (NSF) Engineering Research
Center for Quantum Networks (CQN), awarded under cooperative agreement number
1941583, for supporting this research. D.E. also acknowledges funding from the
NSF C-ACCEL program.

During the preparation of our manuscript, we became aware of a similar work on
heralded frequency-multiplexed BPSs [63].

## Conflicts of Interest

S.G. has outside interests in Xanadu Quantum Technologies, Quantum Network
Technologies, and SensorQ Technologies. These interests have been disclosed to
the University of Arizona and reviewed in accordance with its conflict of
interest policies. Any resulting conflicts of interest from these interests
will be managed by The University of Arizona in accordance with its policies.
D.E. holds shares in Quantum Network Technologies; these interests have been
disclosed to MIT and reviewed in accordance with its conflict of interest
policies, with any conflicts of interest to be managed accordingly.

##  Appendix A Analysis of the Down Conversion Process

We consider a spontaneous parametric down-conversion (SPDC) Hamiltonian [31,
32] in terms of the interacting field operators as

$\displaystyle\hat{H}_{\text{int}}(t)=\int_{V}d^{3}r\chi^{(2)}\hat{E}_{P}^{(+)}(\mathbf{r},t)\hat{E}_{S}^{(-)}(\mathbf{r},t)\hat{E}_{I}^{(-)}(\mathbf{r},t)+\text{
H.c. }.$ (A19)

The consituent terms in this expression are as follows

  * –

$\hat{E}_{j}(\mathbf{r},t)$
$=\hat{E}_{j}^{(+)}(\mathbf{r},t)+\hat{E}_{j}^{(-)}(\mathbf{r},t)$ are the
three interacting fields, with the mode label index $j=P,S,I$ identifying the
pump, signal and idler fields respectively.

  * –

The crystal’s nonlinearity is characterized by the second order nonlinear
coefficient $\chi^{(2)}$. This coefficient is assumed to be equal over the
frequency range of interest.

  * –

$V$ is the volume of interacting field regions in the nonlinear crystal.

To simplify the analysis, the down-converted beams are constrained to be co-
linear with the pump beam. The volume integral in Eq. (A19) then becomes an
integral over only one direction, which we choose to be $z$. The positive
frequency part of the field operator $\hat{E}_{j}(z,t)$, is described by

$\displaystyle\hat{E}_{j}^{(+)}(z,t)=\int
d\omega_{j}A\left(\omega_{j}\right)\hat{a}_{j}\left(\omega_{j}\right)e^{i\left[k_{j}\left(\omega_{j}\right)z-\omega_{j}t\right]},$
(A20)

where $\hat{a}_{j}\left(\omega_{j}\right)$ is the photon annihilation operator
for the mode defined by frequency $\omega_{j}$, the $z$ direction, and the
polarization associated with the index j. The term $A\left(\omega_{j}\right)$
is a slowly varying function of frequency,

$\displaystyle
A\left(\omega_{j}\right)=i\sqrt{\frac{\hbar\omega_{j}}{2\epsilon_{0}n^{2}\left(\omega_{j}\right)\times
V_{j}}}.$ (A21)

This term varies slowly w.r.t. the field frequencies, and hence may be taken
outside the integral. In Eq. (A21), $n(\omega_{j})$ is the frequency dependent
refractive index and $V_{j}$ is the mode volume. In case of a free-space
implementation, this is calculated as the product of the crystal face area and
the propagation length of the photons generated by the down-conversion
process; the crystal face area is replaced by the waveguide mode area are in
case of an on-chip implementation. Since SPDC is a very inefficient process,
the pump field must be relatively large. Accordingly, the electric-field
operator $\hat{E}_{p}^{(+)}(\mathbf{r},t)$ may be replaced by the classical
field
$E_{p}(\mathbf{r},t)=\widetilde{\alpha}(t)e^{ik_{P}\left(\omega_{P}\right)z}$.
The interaction Hamiltonian may now be expressed as

$\displaystyle\hat{H}_{\text{int}}(t)=$ $\displaystyle A\int_{0}^{L}dz\int
d\omega_{I}\int
d\omega_{S}\,\hat{a}_{I}^{\dagger}\left(\omega_{I}\right)\hat{a}_{S}^{\dagger}\left(\omega_{S}\right)\widetilde{\alpha}(t)\times
e^{-i\left\\{\left[k_{I}\left(\omega_{I}\right)+k_{S}\left(\omega_{S}\right)-k_{P}\left(\omega_{P}\right)\right]z-\left[\omega_{I}+\omega_{S}\right]t\right\\}}+\mathrm{H}.\mathrm{c}.,$
(A22)

where $L$ is the length of the crystal and $A\left(\omega_{j}\right)$ has been
grouped into a single parameter $A$, along with several constants defined by

$\displaystyle A=A(\omega_{S})A(\omega_{I})\chi^{(2)}\approx
A(\omega_{P}/2)^{2}\chi^{(2)}$ (A23)

For a continuous wave (CW) pump, the nonlinear interaction prescribed by Eq.
(A22) is a continuous process. We can argue that the interaction has therefore
started long before the emission of signal and idler photons that we expect to
arise as result. Thus we can extend the interaction time
$t_{0}\rightarrow-\infty$ and $t\rightarrow\infty$.

With the revised limits of integration, the integral is somewhat easier to
handle if the pump field is represented as its frequency components. This is
in general true for any pump spectral shape [72]. We shall examine two cases
in our analysis, (1) a CW pump of finite line width and (2) a pulsed mode-
locked pump whose spectral characteristics are given as

  1. 1.

Continuous wave (CW) pump: The pump field for a CW laser may be expressed as
$\tilde{\alpha}(t)=\int d\omega_{P}\,\alpha(\omega_{P})\,e^{-i\omega_{P}t}$
with spectral mode shape function $\alpha(\omega_{P})$. We consider a Gaussian
spectral profile model for $\alpha(\omega_{P})$ of the form,

$\displaystyle\alpha(\omega_{P})=\exp(-(\omega_{P}-\omega_{P0})^{2}/2\sigma_{P}^{2})$
(A24)

where $\omega_{P,0}$ is the pump center frequency and $\sigma_{P}$ is the pump
linewidth.

  2. 2.

Mode-locked pump laser: For the present analysis the mode locked laser is
modelled as a sum of CW-like laser lines separated by preset frequency
separations given by

$\displaystyle\begin{split}\alpha(\omega_{P})&=\sum_{n=-N}^{N}\mathcal{S}(\omega_{P})\exp\left(\frac{-(\omega_{P}-\omega_{P,0}-n\Delta\omega_{P})^{2}}{2\sigma_{P}^{2}}\right),\end{split}$
(A25)

where $\mathcal{S}(\omega_{P})$ is the overall mode envelope function for the
laser. For a comb source with a Gaussian gain envelope, we choose
$\mathcal{S}(\omega)=exp(-(\omega-\omega_{P0})^{2}/2\sigma_{P,BW}^{2})$ to be
the standard functional form, where $\sigma_{P,BW}$ is the comb bandwidth.

Irrespective of the pump shape, the integral in Eq. (A22) then becomes

$\displaystyle\int_{t_{0}}^{t}dt^{\prime}\hat{H}_{\text{int}}\left(t^{\prime}\right)=$
$\displaystyle A\int_{-\infty}^{\infty}dt^{\prime}\int_{0}^{L}dz\int
d\omega_{I}\int
d\omega_{S}\,\hat{a}_{I}^{\dagger}\left(\omega_{I}\right)\hat{a}_{S}^{\dagger}\left(\omega_{S}\right)\int
d\omega_{P}\,\alpha\left(\omega_{P}\right)$ $\displaystyle\times
e^{-i\left\\{\left[k_{I}\left(\omega_{I}\right)+k_{S}\left(\omega_{S}\right)-k_{P}\left(\omega_{P}\right)\right]z-\left[\omega_{I}+\omega_{S}-\omega_{P}\right]t\right\\}}+\text{
H.c. }$ (A26)

The time integral is performed first, yielding a
$2\pi\delta\left(\omega_{I}+\omega_{S}-\omega_{P}\right)$ term. Subsequently
integration over the length of the crystal yields,

$\displaystyle\int_{t_{0}}^{t}dt^{\prime}\hat{H}_{\text{int}}\left(t^{\prime}\right)=$
$\displaystyle 2\pi A\int d\omega_{I}\int
d\omega_{S}\,\hat{a}_{I}^{\dagger}\left(\omega_{I}\right)\hat{a}_{S}^{\dagger}\left(\omega_{S}\right)\times\alpha\left(\omega_{S}+\omega_{I}\right)\Phi\left(\omega_{S},\omega_{I}\right)+\text{
H.c. }$ (A27)

where $\Phi(\omega_{S},\omega_{I})$ is the phase matching function (PMF) given
by,

$\displaystyle\Phi(\omega_{S},\omega_{I})=\frac{\sin\left\\{\left[k_{S}(\omega_{S})+k_{I}(\omega_{I})-k_{P}(\omega_{S}+\omega_{I})\right]L\right\\}}{\left[k_{S}(\omega_{S})+k_{I}(\omega_{I})-k_{P}(\omega_{S}+\omega_{I})\right]L}$
(A28)

Using the concepts developed in the study of time-dependent perturbation
theory, we may now work out a full Dyson series based expansion of the
interaction to obtain the final state. For a generic interaction Hamiltonian
$H_{\text{int}}(t)$ acting on the initial state $\ket{\psi(t_{0})}$ from
$t_{0}$ to $t$, the final state (upto a second order Dyson series expansion)
can be expressed as,

$\displaystyle\ket{\psi(t)}=\left[1+\frac{1}{i\hbar}\int_{t_{0}}^{t}dt^{\prime}\,\hat{H}_{\text{int}}(t^{\prime})+\frac{1}{(i\hbar)^{2}}\int_{t_{0}}^{t}dt^{\prime}\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}\hat{H}_{\text{int}}(t^{\prime})\hat{H}_{\text{int}}(t^{\prime\prime})\right]\ket{\psi(t_{0})}.$
(A29)

Such an expansion yields the following characteristic quantum state after the
necessary integrals are evaluated.

Zeroth Order Term — The zeroth order approximation describes the regime
without any nonlinear field interaction. Hence, this simply yields the
broadband vacuum term $\ket{0,0}$.

First Order Term — The first order Dyson expansion yields the term in which a
single pair of signal and idler photons are created. Their spectral
characteristics are governed by the joint spectral amplitude (JSA),
$J(\omega_{S},\omega_{I})=\alpha(\omega_{S}+\omega_{I})\Phi(\omega_{S},\omega_{I})$
i.e. a product of the pump spectral function and the PMF. The state may then
be expanded as

$\displaystyle\ket{\psi^{(1)}}$ $\displaystyle=\frac{2\pi A}{i\hbar}\int
d\omega_{I}\int
d\omega_{S}\,\alpha(\omega_{S}+\omega_{I})\,\Phi(\omega_{S},\omega_{I})\,\hat{a}_{S}^{\dagger}(\omega_{S})\
\hat{a}_{I}^{\dagger}(\omega_{I})\ket{0,0}$ (A30a) $\displaystyle\equiv
g^{(1)}\int\int
d\omega_{I}\,d\omega_{S}\;J(\omega_{S},\omega_{I})\,\hat{a}_{S}^{\dagger}(\omega_{S})\
\hat{a}_{I}^{\dagger}(\omega_{I})\ket{0,0}.$ (A30b)

Here, we make the simplification $g^{(1)}=2\pi A/i\hbar$. It is important to
note here that the JSA for the mode locked pump is a summation of JSA terms
for the CW pump. Further, the mode spacing ($\Delta\omega_{P}$) is typically
much smaller than the overall envelope’s bandwidth, $\sigma_{P,BW}$; thus the
overall JSA may be modelled by that of the pump envelope, rather the
individual pump spectral lines.

Second Order Term — The second order term must account for time-ordering
effects of subsequent down-conversion steps. Ref. [73] has shown, that due to
destructive interference, nontrivial terms in this expansion are eliminated.
This leads us to calculate the second order terms as

$\displaystyle\ket{\psi^{(2)}}$ $\displaystyle=\frac{1}{2}\left(\frac{2\pi
A}{i\hbar}\right)^{2}\left(g^{\prime}_{0}\ket{0,0}+\\!\\!\int d\omega_{I}\int
d\omega_{S}\int d\omega^{\prime}_{I}\int
d\omega^{\prime}_{S}\,J(\omega_{S},\omega_{I})\hat{a}_{S}^{\dagger}(\omega_{S})\
\hat{a}_{I}^{\dagger}(\omega_{I})\ket{0,0}J(\omega^{\prime}_{S},\omega^{\prime}_{I})\hat{a}_{S}^{\dagger}(\omega^{\prime}_{S})\
\hat{a}_{I}^{\dagger}(\omega^{\prime}_{I})\ket{0,0}\right)$ (A31a)
$\displaystyle\equiv g^{(2)}\left(g^{\prime}_{0}\ket{0,0}+\int_{\text{all
}\omega}J(\omega_{S},\omega_{I})J(\omega^{\prime}_{S},\omega^{\prime}_{I})\hat{a}_{S}^{\dagger}(\omega_{S})\
\hat{a}_{I}^{\dagger}(\omega_{I})\ket{0,0}\hat{a}_{S}^{\dagger}(\omega^{\prime}_{S})\
\hat{a}_{I}^{\dagger}(\omega^{\prime}_{I})\ket{0,0}\right)$ (A31b)

where,

$\displaystyle g^{\prime}_{0}=\int d\omega_{I}\int
d\omega_{S}\,|\alpha(\omega_{S}+\omega_{I})\Phi(\omega_{S},\omega_{I})|^{2}$
(A32a) $\displaystyle g^{(2)}=2(\pi A/i\hbar)^{2}=\left(g^{(1)}\right)^{2}/2$
(A32b)

Hence upto the considered second order expansion, the emitted state is
expressed as,

$\displaystyle\ket{\Psi}_{\mathrm{SPDC}}$
$\displaystyle=(1+g^{(2)}g^{\prime}_{0})\ket{\textbf{0}}+g^{(1)}\int
J(\omega_{S},\omega_{I})\,\hat{a}_{S}^{\dagger}(\omega_{S})\,\hat{a}_{I}^{\dagger}(\omega_{I})\ket{\textbf{0}}$
$\displaystyle\quad+g^{(2)}\int
J(\omega_{S},\omega_{I})\,\hat{a}_{S}^{\dagger}(\omega_{S})\,\hat{a}_{I}^{\dagger}(\omega_{I})\cdot\,J(\omega_{S}^{\prime},\omega_{I}^{\prime})\hat{a}_{S}^{\dagger}(\omega_{S}^{\prime})\,\hat{a}_{I}^{\dagger}(\omega_{I}^{\prime})\ket{\textbf{0}}$
(A33)

where $\ket{\textbf{0}}$ represents the vacuum state. Note that this state is
unnormalized and hence contains information about the total emission rate of
the source.

###  1 Post-Measurement State and Detection Statistics of Zero-Added Loss
Multiplexed Source

The output state for a single entangled pair source (comprised of two down
conversion processes) is

$\displaystyle\ket{\Gamma}=$ $\displaystyle\ket{\Psi}_{\mathrm{SPDC}}^{\otimes
2}=\left(\ket{0,0}+\ket{\psi^{(1)}}+\ket{\psi^{(2)}}\right)^{\otimes 2}$ (A34)
$\displaystyle=$
$\displaystyle\ket{0,0}^{(1)}\ket{0,0}^{(2)}+\ket{\psi^{(1)}}^{(1)}\ket{0,0}^{(2)}+\ket{0,0}^{(1)}\ket{\psi^{(1)}}^{(2)}\text{(vacuum
+ Bell pair )}$
$\displaystyle+\ket{\psi^{(2)}}^{(1)}\ket{0}^{(2)}+\ket{0}^{(1)}\ket{\psi^{(2)}}^{(2)}+\ket{\psi^{(1)}}^{(1)}\ket{\psi^{(1)}}^{(2)}\quad\text{(two
photon terms + vacuum contri.)}$
$\displaystyle+\ldots+\ket{\psi^{(2)}}^{(1)}\ket{\psi^{(2)}}^{(2)}\quad\text{(higher
order terms)}$ (A35)

We clarify on the mode labelling convention used in the generalized analysis
of this Appendix. Each SPDC process generates a signal-idler pair; an
entangled pair source is modelled to comprise of two SPDC processes. Let us
label the signal and idler as $S$ and $I$ respectively; additionally let us
associate the subscript (say $k$) to the entangled pair source. The
superscript is used to label the signal-idler pair (or equivalently, which
constituent SPDC process). So the mode label $S^{(2)}_{\mathfrak{a}}$
signifies the signal of the second SPDC process for the entangled pair source
labelled $\mathfrak{a}$.

Since the idler beams are swapped, we can assume the mode ordering to be
$S^{(1)}_{k},I^{(2)}_{k},S^{(2)}_{k},I^{(1)}_{k}$ to get the standard state
expansion, where $k=\mathfrak{a},\mathfrak{b}$ determines which entangled pair
source (of two) we are referring to. Additionally, we may simplify the mode
notation by adopting the abbreviations proposed in Fig. S1.

Figure S1: Schematic of the individual SPDC entangled pair source using a CW
pump excitation. Figure S2: Schematic of the Zero added loss Multiplexed
(ZALM) Bell pair source.

The spectrally demultiplexed Bell state analyzer in Fig. S2 governs the mode
interactions; we shall analyze a subset of them to determine the final quantum
state. As per our abbreviated notation, the modes labelled by
${\mathfrak{a}4}$ and $\mathfrak{a}3$ interact with modes $\mathfrak{b}1$ and
$\mathfrak{b}2$, respectively. Under the action of the balanced beamsplitters
preceeding the detection, the creation operators are transformed as per the
following rule,

$\displaystyle\hat{a}^{\dagger}_{D_{1}}:=\frac{1}{\sqrt{2}}\left(\hat{a}^{\dagger}_{\mathfrak{a}4}+\hat{a}^{\dagger}_{\mathfrak{b}1}\right),\hat{a}^{\dagger}_{D_{2}}:=\frac{1}{\sqrt{2}}\left(\hat{a}^{\dagger}_{\mathfrak{b}1}-\hat{a}^{\dagger}_{\mathfrak{a}4}\right),$
(A36a)
$\displaystyle\hat{a}^{\dagger}_{D_{3}}:=\frac{1}{\sqrt{2}}\left(\hat{a}^{\dagger}_{\mathfrak{a}3}+\hat{a}^{\dagger}_{\mathfrak{b}2}\right),\hat{a}^{\dagger}_{D_{4}}:=\frac{1}{\sqrt{2}}\left(\hat{a}^{\dagger}_{\mathfrak{b}2}-\hat{a}^{\dagger}_{\mathfrak{a}3}\right),$
(A36b)

where $\hat{a}^{\dagger}_{D_{i}}$ is the creation operator for corresponding
detector bank. Subsequently, we may analyze the modes pairwise to determine
the quantum state emitted in a complementary pair of undetected modes. As an
example, consider the $\mathfrak{a}4\Leftrightarrow\mathfrak{b}1$ interaction;
the corresponding complementary modes are $\mathfrak{a}2$ and $\mathfrak{b}3$
respectively. Detection of a $1,0$ click pattern on the pair of modes i.e.
detectors $D_{1}$ and $D_{2}$ on one of the uDWDM channels imposes a spectral
window on the detected photon. The detection jitter (which is a detector
parameter) in conjunction with the spectral window, govern the temporo-
spectral shape of the heralded photon pair. Given a spectral filter of width
$\Delta\Omega$ Hz and a detector with a detection jitter of $\delta\tau$ s,
the temporal extent of the un-detected photons is given by
$\max(\delta\tau,1/\Delta\omega)$ s.

We consider detection of photons in modes $\mathfrak{a}4$ and $\mathfrak{b}1$
overall a spectral range
$\omega_{\mathfrak{a}4},\omega_{\mathfrak{b}1}\in(\Omega,\Omega^{\prime})$.
This heralds the quantum state $\ket{\varphi}$ on the undetected modes

$\displaystyle\ket{\varphi}\propto\biggl{(}$ $\displaystyle\int
d\omega_{\mathfrak{a}2}\int_{\Omega}^{\Omega^{\prime}}d\omega_{\mathfrak{a}4}\,\alpha\left(\omega_{\mathfrak{a}2}+\omega_{\mathfrak{a}4}\right)\Phi\left(\omega_{\mathfrak{a}4},\omega_{\mathfrak{a}2}\right)\hat{a}^{\dagger}_{\mathfrak{a}2}(\omega_{\mathfrak{a}2})+\int
d\omega_{\mathfrak{b}3}\int_{\Omega}^{\Omega^{\prime}}d\omega_{\mathfrak{b}1}\,\alpha\left(\omega_{\mathfrak{b}1}+\omega_{\mathfrak{b}3}\right)\Phi\left(\omega_{\mathfrak{b}1},\omega_{\mathfrak{b}3}\right)\hat{a}^{\dagger}_{\mathfrak{b}3}(\omega_{\mathfrak{b}3})\biggr{)}\ket{0}_{\mathfrak{a}2}\ket{0}_{\mathfrak{b}3}.$
(A37)

We note that modes $\mathfrak{a}2$ and $\mathfrak{b}3$ span a complementary
frequency range i.e.
$\omega_{\mathfrak{a}2},\omega_{\mathfrak{b}3}\in(\omega_{P}-\Omega^{\prime},\omega_{P}-\Omega)$
ensures energy conservation and phase-matching of the down-converted beams.
Similarly, the detection of a $1,0$ pattern on the detector banks $D_{3}$ and
$D_{4}$ for a the detection channel
$\omega_{\mathfrak{a}3},\omega_{\mathfrak{b}2}\in(\Omega,\Omega^{\prime})$
heralds a quantum state $\ket{\varphi^{\prime}}$ similar to Eq. (A37) above,
and is described by

$\displaystyle\ket{\varphi^{\prime}}\propto$ $\displaystyle\biggl{(}\int
d\omega_{\mathfrak{a}1}\int_{\Omega}^{\Omega^{\prime}}d\omega_{\mathfrak{a}3}\,\alpha\left(\omega_{\mathfrak{a}1}+\omega_{\mathfrak{a}3}\right)\Phi\left(\omega_{\mathfrak{a}3},\omega_{\mathfrak{a}1}\right)\hat{a}^{\dagger}_{\mathfrak{a}1}(\omega_{\mathfrak{a}1})+\int
d\omega_{\mathfrak{b}4}\int_{\Omega}^{\Omega^{\prime}}d\omega_{\mathfrak{b}2}\,\alpha\left(\omega_{\mathfrak{b}2}+\omega_{\mathfrak{b}4}\right)\Phi\left(\omega_{\mathfrak{b}2},\omega_{\mathfrak{b}4}\right)\hat{a}^{\dagger}_{\mathfrak{b}4}(\omega_{\mathfrak{b}4})\biggr{)}\ket{0}_{\mathfrak{a}1}\ket{0}_{\mathfrak{b}4}.$
(A38)

Hence, the complete detection pattern of $1,0,1,0$ on detectors
$D_{1}$—$D_{4}$, yields the final state
$\ket{\varphi}\otimes\ket{\varphi^{\prime}}$, which we may express compactly
as $\ket{\varphi}\otimes\ket{\varphi^{\prime}}\equiv\ket{\varPsi}+\ket{\Xi}$
with the constituent terms,

$\displaystyle\ket{\varPsi}=\int
d\bar{\omega}\left(J(\omega_{\mathfrak{a}1},\omega_{\mathfrak{a}3})\cdot
J(\omega_{\mathfrak{b}1},\omega_{\mathfrak{b}3})\,\hat{a}^{\dagger}_{\mathfrak{a}1}(\omega_{\mathfrak{a}1})\,\hat{a}^{\dagger}_{\mathfrak{b}3}(\omega_{\mathfrak{b}3})+J(\omega_{\mathfrak{a}2},\omega_{\mathfrak{a}4})\cdot
J(\omega_{\mathfrak{b}2},\omega_{\mathfrak{b}4})\,\hat{a}^{\dagger}_{\mathfrak{a}2}(\omega_{\mathfrak{a}2})\,\hat{a}^{\dagger}_{\mathfrak{b}4}(\omega_{\mathfrak{b}4})\right)\ket{0}_{\mathfrak{a}1}\ket{0}_{\mathfrak{a}2}\ket{0}_{\mathfrak{b}3}\ket{0}_{\mathfrak{b}4}$
(A39a) $\displaystyle\ket{\Xi}=\int
d\bar{\omega}\left(J(\omega_{\mathfrak{a}1},\omega_{\mathfrak{a}3})\cdot
J(\omega_{\mathfrak{a}2},\omega_{\mathfrak{a}4})\,\hat{a}^{\dagger}_{\mathfrak{a}1}(\omega_{\mathfrak{a}1})\,\hat{a}^{\dagger}_{\mathfrak{a}2}(\omega_{\mathfrak{a}2})+J(\omega_{\mathfrak{b}1},\omega_{\mathfrak{b}3})\cdot
J(\omega_{\mathfrak{b}2},\omega_{\mathfrak{b}4})\,\hat{a}^{\dagger}_{\mathfrak{b}3}(\omega_{\mathfrak{b}3})\,\hat{a}^{\dagger}_{\mathfrak{b}4}(\omega_{\mathfrak{b}4})\right)\ket{0}_{\mathfrak{a}1}\ket{0}_{\mathfrak{a}2}\ket{0}_{\mathfrak{b}3}\ket{0}_{\mathfrak{b}4}.$
(A39b)

Here,
$J(\omega_{i},\omega_{j})=\alpha(\omega_{i}+\omega_{j})\Phi(\omega_{i},\omega_{j})$,
represents the joint spectral amplitude function for the signal-idler
frequency pair $\omega_{i},\omega_{j}$. The $\ket{\varPsi}$ component is the
spectral equivalent of a broadband Bell pair of the form
$(\ket{1,0}_{A}\ket{0,1}_{B}+\ket{0,1}_{A}\ket{1,0}_{B})/\sqrt{2}$. In
contrast, $\ket{\Xi}$ is a state in which either Alice or Bob receive both
photons; this is equivalent to a broadband
$(\ket{1,1}_{A}\ket{0,0}_{B}+\ket{0,0}_{A}\ket{1,1}_{B})/\sqrt{2}$ state. Note
that the latter component is a spurious term which limits the fidelity of the
generated entangled state. The overall general state may be expressed as

$\displaystyle\ket{S}\propto\int d\bar{\omega}\Biggl{[}$
$\displaystyle\biggl{(}J(\omega_{\mathfrak{a}1},\omega_{\mathfrak{a}3})\cdot
J(\omega_{\mathfrak{b}1},\omega_{\mathfrak{b}3})\,\hat{a}^{\dagger}_{\mathfrak{a}1}(\omega_{\mathfrak{a}1})\,\hat{a}^{\dagger}_{\mathfrak{b}3}(\omega_{\mathfrak{a}3})+(-1)^{m_{1}}J(\omega_{\mathfrak{a}2},\omega_{\mathfrak{a}4})\cdot
J(\omega_{\mathfrak{b}2},\omega_{\mathfrak{b}4})\,\hat{a}^{\dagger}_{\mathfrak{a}2}(\omega_{\mathfrak{a}2})\,\hat{a}^{\dagger}_{\mathfrak{b}4}(\omega_{\mathfrak{b}4})\biggr{)}$
$\displaystyle+(-1)^{m_{2}}\biggl{(}J(\omega_{\mathfrak{a}1},\omega_{\mathfrak{a}3})\cdot
J(\omega_{\mathfrak{a}2},\omega_{\mathfrak{a}4})\,\hat{a}^{\dagger}_{\mathfrak{a}1}(\omega_{\mathfrak{a}1})\,\hat{a}^{\dagger}_{\mathfrak{a}2}(\omega_{\mathfrak{a}2})+(-1)^{m_{1}}J(\omega_{\mathfrak{b}1},\omega_{\mathfrak{b}3})\cdot
J(\omega_{\mathfrak{b}2},\omega_{\mathfrak{b}4})\,\hat{a}^{\dagger}_{\mathfrak{b}3}(\omega_{\mathfrak{b}3})\,\hat{a}^{\dagger}_{\mathfrak{b}4}(\omega_{\mathfrak{b}4})\biggr{)}\Biggr{]}$
$\displaystyle\times\ket{0}_{\mathfrak{a}1}\ket{0}_{\mathfrak{a}2}\ket{0}_{\mathfrak{b}3}\ket{0}_{\mathfrak{b}4}$
(A40)

where the compressed integral $\int d\bar{\omega}$ denotes the limited
frequency integrals (for the corresponding spectral channels)

$\displaystyle\int d\bar{\omega}\equiv\int d\omega_{\mathfrak{a}1}\int
d\omega_{\mathfrak{a}2}\int_{\Omega}^{\Omega^{\prime}}d\omega_{\mathfrak{a}3}\int_{\Omega}^{\Omega^{\prime}}d\omega_{\mathfrak{a}4}\int_{\Omega}^{\Omega^{\prime}}d\omega_{\mathfrak{b}1}\int_{\Omega}^{\Omega^{\prime}}d\omega_{\mathfrak{b}2}\int
d\omega_{\mathfrak{b}3}\int d\omega_{\mathfrak{b}4}$ (A41)

The frequency extent of the undetected modes
$(\mathfrak{a}1,\mathfrak{a}2,\mathfrak{b}3,\mathfrak{b}4)$ are restricted to
a $(\omega_{P}-\Omega^{\prime},\omega_{P}-\Omega)$ by the detection windows
(neglcting the pump linewidth). In Eq.(A40), $m_{1}$ and $m_{2}$ are parity
bits given by Table 1.

Click Pattern | Parity Bits
---|---
$D_{1}$ | $D_{2}$ | $D_{3}$ | $D_{4}$ | $m_{1}$ | $m_{2}$
1 | 0 | 1 | 0 | 0 | 0
0 | 1 | 0 | 1 | 0 | 1
0 | 1 | 1 | 0 | 1 | 1
1 | 0 | 0 | 0 | 1 | 0
Table 1: Table of parity bits $(m_{1},m_{2})$ for a given click pattern in
detectors $D_{1}-D_{4}$

For the purposes of the current protocol, we translate to a qubit notation for
the modes that make up the quantum state transmitted. We choose the equivalent
naming convention where
$\mathfrak{a}1,\mathfrak{a}2,\mathfrak{a}3,\mathfrak{a}4\equiv
A1,A2,A1^{\prime},A2^{\prime}$ respectively and
$\mathfrak{b}1,\mathfrak{b}2,\mathfrak{b}3,\mathfrak{b}4\equiv
B2^{\prime},B1^{\prime},B2,B1$. The ordering for the $\mathfrak{b}$ modes are
reversed to match the mode ordering in Fig. S2 where $\mathfrak{b}3$ and
$\mathfrak{b}4$ represent modes transmitted to Bob. With this mode
relabelling, Eq. (A40) becomes,

$\displaystyle\ket{S}\propto\int$ $\displaystyle
d\bar{\omega}\Biggl{[}\biggl{(}J(\omega_{A1},\omega_{A1^{\prime}})\,J(\omega_{B2^{\prime}},\omega_{B2})\,\hat{a}^{\dagger}_{A1}(\omega_{A1})\,\hat{a}^{\dagger}_{B2}(\omega_{B2})+(-1)^{m_{1}}J(\omega_{A2},\omega_{A2^{\prime}})\,J(\omega_{B1^{\prime}},\omega_{B1})\,\hat{a}^{\dagger}_{A2}(\omega_{A2})\,\hat{a}^{\dagger}_{B1}(\omega_{B1})\biggr{)}$
$\displaystyle+(-1)^{m_{2}}\biggl{(}J(\omega_{A1},\omega_{A1^{\prime}})\,J(\omega_{A2},\omega_{A2^{\prime}})\,\hat{a}^{\dagger}_{A1}(\omega_{A1})\,\hat{a}^{\dagger}_{A2}(\omega_{A2})+(-1)^{m_{1}}J(\omega_{B2^{\prime}},\omega_{B2})\,J(\omega_{B1^{\prime}},\omega_{B1})\,\hat{a}^{\dagger}_{B1}(\omega_{B1})\,\hat{a}^{\dagger}_{B2}(\omega_{B2})\biggr{)}\Biggr{]}$
$\displaystyle\times\ket{0}_{A1}\ket{0}_{A2}\ket{0}_{B1}\ket{0}_{B2}$ (A42)

We use the equivalent notation $\ket{S}\equiv\ket{S(\omega_{P0},\sigma_{P})}$
to denote that the state specified in Eq. (A42) arises from a CW pump field of
the form Eq. (A24). Hence given our choice of the mode-locked pump in Eq.
(A25), the multiplexed heralded state can be succinctly described by

$\displaystyle\ket{\mathbf{S}}=\sum_{n=-N}^{N}\ket{S(\omega_{P0}+n\Delta\omega_{P},\sigma_{P})}.$
(A43)

This state poses additional changes; since there are multiple ‘center
frequencies’, the signal and idler photons have an added degree of
uncertainity in their origin. Considering a single SPDC source (say modes
labelled by $\mathfrak{a}$) a signal-idler pair (say at frequencies
$\omega_{a1}$ and $\omega_{a3}$) could be generated by each pump line
(centered at $\omega_{P}=\omega_{P0}+n\Delta\omega_{P}$). The contribution of
each pump line is determined by the spectral envelope
($\mathcal{S}(\omega_{P})$) of the pump and more crucially, by the joint
spectral amplitude $J(\omega_{\mathfrak{a}_{1}},\omega_{\mathfrak{a}_{3}})$
function.

##  Appendix B Heralded photonic Bell state

The heralded photonic Bell state fidelity may suffer from timing uncertainty
in the detection and imperfect mode conversion. Here we address the
aforementioned issues through an example of calculated joint spectral
amplitude (JSA) function $J$, which is a product of the pump envelope
$\alpha(\omega_{P})$ and the phase matching function
$\Phi(\omega_{S},\omega_{I})$ (Appendix A). We first assume a repetition rate
of 1 GHz for the mode-locked pump laser, with each comb width being 12.5 GHz
(matching that of the DWDM channel as explained later). We also take the SPDC
material to be magnesium-oxide doped LiNbO3. Using the Sellmeier equations for
both the signal (extraordinary) and the idler (ordinary) photons, and applying
a Gaussian approximation [32], we can compute the phase matching function
$\Phi(\omega_{S},\omega_{I})$. Multiplying $\alpha(\omega_{P})$ with
$\Phi(\omega_{S},\omega_{I})$, we then obtain the JSA function. Figure S3
shows the three functions respectively.

Figure S3: Numerically calculated (a) pump envelope function with a frequency
comb source, (b) phase matching function using the Gaussian approximation, and
the resultant (c) joint spectral intensity (JSI) function (modulus square of
the JSA function). The plot axes for (a) and (b) spans across a $10\times
12.5~{}$GHz, with the center wavelength at $2\times 2\pi
c/\omega_{p}=1560~{}$nm. For (c), the axes span $12.5~{}$GHz corresponding to
the spectral width of each DWDM channel.

We further assume a Gaussian spectral profile for the DWDM channel that shapes
the JSA, as illustrated in Fig. S4. To quantify the level of separability, we
perform Schmidt decomposition on the JSA to extract its eigenvalues
$\lambda_{m}$ with orthonormal basis vectors
$\\{u_{m}(\omega)\\},\\{v_{m}(\omega)\\}$ [32]:

$\displaystyle J(\omega_{s},\omega_{i})$
$\displaystyle=\sum_{m}\sqrt{\lambda_{m}}u_{m}(\omega_{s})v_{m}(\omega_{i})$
(B44)

where

$\displaystyle\int
K_{1}(\omega,\omega^{\prime})u_{m}(\omega^{\prime})d\omega^{\prime}$
$\displaystyle=\lambda_{m}u_{m}(\omega)$ (B45) $\displaystyle\int
K_{2}(\omega,\omega^{\prime})v_{m}(\omega^{\prime})d\omega^{\prime}$
$\displaystyle=\lambda_{m}v_{m}(\omega)$ (B46) $\displaystyle
K_{1}(\omega,\omega^{\prime})$ $\displaystyle=\int
J(\omega,\omega_{2})(J(\omega^{\prime},\omega_{2}))^{*}d\omega_{2}$ (B47)
$\displaystyle K_{2}(\omega,\omega^{\prime})$ $\displaystyle=\int
J(\omega_{1},\omega)(J(\omega_{1},\omega^{\prime}))^{*}d\omega_{1}$ (B48)

The entropy of entanglement, which approaches zero as the JSA becomes more
separable, is calculated based on:

$\displaystyle S$
$\displaystyle=-\sum_{k=0}^{\infty}\lambda_{k}\log_{2}\lambda_{k}$ (B49)

Figure S4: The JSI function before and after applying spectral filtering via a
Gaussian-shaped DWDM channel profile. The calculated entropy based on Schmidt
decomposition is $S=0.446$.

If the inverse of the detector jitter time is much greater than the DWDM
channel bandwidth of 12.5 GHz, then the infidelity of the photonic Bell state
stemming from projection into a mixed state is negligible. For calculations
presented in the main text, we assumed a timing uncertainty of 1 ps [38]
corresponding to 1 THz, which greatly exceeds 12.5 GHz. However, even with
ideal detection, the heralded photonic Bell state composed by the signal
photons from two SPDC sources may still have a complex spectral shape, which
may lead to infidelity caused by the mode conversion step. This is largely a
consequence of using a frequency comb source with a repetition rate comparable
to the DWDM channel width.

Two avenues to circumvent this issue are (i) injecting a complex pump pulse
shape in the MC corresponding to the spectral shape in the heralded Bell state
or (ii) engineering the JSA function [32]. For the calculated example shown in
Fig. S3 and Fig. S4, we take the latter approach by matching the pump
linewidth to the DWDM channel bandwidth and increasing the effective crystal
length. As a result, the DWDM-filtered JSI shows a near Gaussian profile, as
assumed in Appendix C.

Lastly, we consider the case of having non-ideal detectors, with the inverse
of the jitter time being smaller than 12.5 GHz. In this scenario, the two-
photon interference visibility is determined by the purity of the DWDM-
filtered state.

We follow Ref. [74]’s formalism to compute the two-photon interference
visibility $v$ with gating (heralding):

$\displaystyle v$ $\displaystyle=\frac{\mathcal{E}}{\mathcal{A}},$ (B50)

where

$\displaystyle\mathcal{E}$ $\displaystyle=(g^{(1)})^{2}\int
d\omega_{1}d\omega_{1}^{\prime}d\omega_{2}d\omega_{2}^{\prime}J(\omega_{1},\omega_{2})J(\omega_{1}^{\prime},\omega_{2}^{\prime})J^{*}(\omega_{1},\omega_{2}^{\prime})J^{*}(\omega_{1}^{\prime},\omega_{2})$
(B51) $\displaystyle\mathcal{A}$ $\displaystyle=(g^{(1)})^{2}\int
d\omega_{1}d\omega_{1}^{\prime}d\omega_{2}d\omega_{2}^{\prime}|J(\omega_{1},\omega_{2})J(\omega_{1}^{\prime},\omega_{2}^{\prime})|^{2}$
(B52)

We find the theoretical maximum visibility to be $v=0.996$.

Finally, we estimate the entanglement swap fidelity
$\mathcal{F}_{\text{swap}}=(1+v)/2$ to be 0.998, which still exceeds 0.99,
highlighting the importance of JSA engineering in the case of imperfect
detection. However, we note that the spectral profile of the heralded photonic
Bell state would be convolved with the detector’s instrument response
function, likely demanding further shaping on the MC’s pump pulse. For this
reason, we assume a sufficiently small detector jitter and leave the effect of
imperfect detection for future studies.

##  Appendix C Frequency and Bandwidth Conversion

Recall that the quantum state of the cavity is

$\displaystyle\ket{\Psi_{\rm{cav}}}\equiv\psi_{a}(t)\ket{10}+\psi_{b}(t)\ket{01},$
(C53)

where $\ket{10}$ corresponds to the photon being in mode $a$ while $\ket{01}$
corresponds to the photon being in mode $b$. The equations of motion for the
Schrödinger coefficients are

$\displaystyle\dot{\psi}_{a}(t)$
$\displaystyle=-\Big{(}i\delta_{a}+\frac{\kappa_{a}}{2}\Big{)}\psi_{a}(t)-i|\Lambda(t)|e^{-i\phi(t)}\psi_{b}(t)+\sqrt{\kappa_{a,\rm{w}}}\xi_{a,\rm{i}}(t)$
(C54a) $\displaystyle\dot{\psi}_{b}(t)$
$\displaystyle=-\Big{(}i\delta_{b}+\frac{\kappa_{b}}{2}\Big{)}\psi_{b}(t)-i|\Lambda(t)|e^{i\phi(t)}\psi_{a}(t)$
(C54b) $\displaystyle\xi_{a,\rm{o}}(t)$
$\displaystyle=\xi_{a,\rm{i}}(t)-\sqrt{\kappa_{a,\rm{w}}}\psi_{a}(t)$ (C54c)
$\displaystyle\xi_{b,\rm{o}}(t)$
$\displaystyle=-\sqrt{\kappa_{b,\rm{w}}}\psi_{b}(t),$ (C54d)

where the photon wave packets are described by the functions
$\xi_{q,\rm{i}}(t)$ and $\xi_{q,\rm{o}}(t)$ with $q\\!=\\!\\{a,b\\}$. The
input state may be expressed in a time-bin basis as [21]

$\displaystyle\ket{\Psi_{\rm{in}}}\equiv\int
dt\xi_{a,\rm{i}}\hat{w}_{a}^{\dagger}(t)\ket{\bm{0}}_{t},$ (C55)

with $\ket{\bm{0}}_{t}$ representing the product state where all time-bins are
empty (i.e. multi temporal-mode vacuum state) and $\hat{w}_{a}^{\dagger}(t)$
populates the time-bin indexed by $t$ with one photon. The output state is

$\displaystyle\ket{\Psi_{\rm{out}}}\equiv\int
dt\xi_{a,\rm{o}}(t)\hat{w}_{a}^{\dagger}(t)\ket{\bm{0}}_{t}+\int
dt\xi_{b,\rm{o}}(t)\hat{w}_{b}^{\dagger}(t)\ket{\bm{0}}_{t}.$ (C56)

Note that the control field was divided into amplitude, $|\Lambda(t)|$ , and
phase, $\exp[i\phi(t)]$, in Eq.( C54a)-( C54b) and the total decay rates are
defined by $\kappa_{q}\\!=\\!\kappa_{q,\rm{w}}+\kappa_{q,\rm{l}}$ with
$\kappa_{q,\rm{l}}$ being the intrinsic loss rates of the cavity modes. We
assume that a maximum conversion efficiency is achieved when
$\xi_{a,\rm{o}}(t)\\!=\\!0$, so the task is to determine $|\Lambda(t)|$ and
$\phi(t)$ such that this is true. Solving Eq. (C54c) yields
$\psi_{a}(t)\\!=\\!\xi_{a,\rm{i}}(t)/\sqrt{\kappa_{a,\rm{w}}}$, which we
substitute into Eq. (C54b) and re-arrange terms

$\displaystyle\frac{d}{dt}\Big{(}\psi_{b}(t)e^{-Q(t)}\Big{)}e^{Q(t)}=\frac{-i}{\sqrt{\kappa_{a,\rm{w}}}}|\Lambda(t)|e^{i\phi(t)}\xi(t)~{}\Rightarrow~{}~{}\psi_{b}(t)=\frac{-i}{\sqrt{\kappa_{a,\rm{w}}}}e^{Q(t)}\int_{t_{0}}^{t}\\!e^{-Q(s)}|\Lambda(s)|e^{i\phi(s)}\xi(s)ds.$
(C57)

We defined the function $Q(t)=-(i\delta_{b}+\kappa_{b}/2)t$ and replaced
$\xi_{a,\rm{i}}$ with $\xi$ for brevity in Eq. (C57). Substituting
$\psi_{a}\\!=\\!\xi/\sqrt{\kappa_{a,\rm{w}}}$ into Eq. (C54a) yields

$\displaystyle\frac{(\kappa_{a,\rm{w}}-\kappa_{a,\rm{l}})}{2}\xi(t)-\dot{\xi}(t)-i\delta_{a}\xi(t)$
$\displaystyle=i|\Lambda(t)|e^{-i\phi(t)}\sqrt{\gamma}\psi_{b}(t).$ (C58)

Multiplying Eq. (C58) by $\xi^{*}(t)\\!\exp(\kappa_{b}t)$ and defining real
functions $f_{i}$ and $g_{i}$, we find

$\displaystyle f_{i}(t)+ig_{i}(t)$
$\displaystyle=|\Lambda(t)|e^{-i\phi(t)}\xi^{*}(t)e^{(-i\delta_{b}+\frac{\kappa_{b}}{2})t}\int_{\\!t_{0}}^{t}\\!e^{(i\delta_{b}+\frac{\kappa_{b}}{2})s}|\Lambda(s)|e^{i\phi(s)}\xi(s)ds,$
(C59)

with

$\displaystyle f_{i}(t)$
$\displaystyle=\Big{(}\frac{\kappa_{a,\rm{w}}-\kappa_{a,\rm{l}}}{2}\xi(t)-\dot{\xi}(t)\Big{)}\xi^{*}(t)e^{\kappa_{b}t}$
(C60a) $\displaystyle g_{i}(t)$
$\displaystyle=-\delta_{a}|\xi(t)|^{2}e^{\kappa_{b}t}.$ (C60b)

Note that Eq. (C60a) assumes an input wavepacket without chirp,
$\frac{d}{dt}[\arg{\xi(t)}]\\!=\\!0$. The right hand side of Eq. (C59) can be
written as

$\displaystyle\big{[}x(t)-iy(t)\big{]}\int_{t_{0}}^{t}\\!\big{[}x(s)+iy(s)\big{]}ds=x(t)\int_{t_{0}}^{t}\\!x(s)ds+y(t)\int_{t_{0}}^{t}\\!y(s)ds+i\Big{(}x(t)\int_{t_{0}}^{t}\\!y(s)ds-y(t)\int_{t_{0}}^{t}\\!x(s)ds\Big{)},$
(C61)

where

$\displaystyle x(t)$
$\displaystyle=|\Lambda(t)||\xi(t)|\exp(\kappa_{b}t/2)\cos\\!\big{[}\phi(t)+\delta_{b}t+\arg(\xi)\big{]}$
(C62a) $\displaystyle y(t)$
$\displaystyle=|\Lambda(t)||\xi(t)|\exp(\kappa_{b}t/2)\sin\\!\big{[}\phi(t)+\delta_{b}t+\arg(\xi)\big{]}.$
(C62b)

By defining the functions

$\displaystyle
X(t)=\int_{t_{0}}^{t}\\!x(s)ds=R(t)\cos\\!\big{[}\theta(t)\big{]},~{}Y(t)=\int_{t_{0}}^{t}\\!y(s)ds=R(t)\sin\\!\big{[}\theta(t)\big{]},$
(C63)

Eq. (C59) can be split into real and imaginary parts

$\displaystyle f_{i}=\dot{X}X+\dot{Y}Y,\qquad g_{i}=\dot{X}Y-\dot{Y}X.$ (C64)

Using the definition in Eq. (C63), we have

$f_{i}=\dot{X}X+\dot{Y}Y=\big{[}\dot{R}\cos(\theta)-R\sin(\theta)\dot{\theta}\big{]}R\cos(\theta)+\big{[}\dot{R}\sin(\theta)+R\cos(\theta)\dot{\theta}\big{]}R\sin(\theta)=\dot{R}R=\frac{1}{2}\frac{d}{dt}\Big{(}R^{2}\Big{)},$
(C65)

which has the solution

$\displaystyle R(t)=\sqrt{2\int_{t_{0}}^{t}\\!f_{i}(s)ds}.$ (C66)

Similarly,

$\displaystyle
g_{i}=\dot{X}Y-\dot{Y}X=\big{[}\dot{R}\cos(\theta)-R\sin(\theta)\dot{\theta}\big{]}R\sin(\theta)-\big{[}\dot{R}\sin(\theta)+R\cos(\theta)\dot{\theta}\big{]}R\cos(\theta)=-R^{2}\dot{\theta}.$
(C67)

Using the result in Eq. (C66), the solution for $\theta$ is

$\displaystyle\theta(t)=-\frac{1}{2}\int_{t_{0}}^{t}\\!\\!\frac{g_{i}(s)}{\int_{t_{0}}^{s}\\!f_{i}(z)dz}ds.$
(C68)

To find the solution for $|\Lambda(t)|$ we evaluate
$x^{2}+y^{2}\\!=\\!|\Lambda|^{2}|\xi|^{2}\exp(\kappa_{b}t)$ using the results
above

$|\Lambda|^{2}|\xi|^{2}e^{\gamma_{L}t}=\dot{X}^{2}+\dot{Y}^{2}=\big{[}\dot{R}\cos(\theta)-R\sin(\theta)\dot{\theta}\big{]}^{2}+\big{[}\dot{R}\sin(\theta)+R\cos(\theta)\dot{\theta}\big{]}^{2}=\dot{R}^{2}+R^{2}\dot{\theta}^{2}=\frac{1}{2\int\\!f_{i}}\big{(}g_{i}^{2}+f_{i}^{2}\big{)}.$
(C69)

Inserting the definition of $g_{i}$ from Eq. (C60b) yields

$\displaystyle|\Lambda|^{2}|\xi^{2}|\exp(\kappa_{b}t)=\frac{1}{2\mathcal{F}_{i}}\Big{(}\delta_{a}^{2}\exp(2\kappa_{b}t)|\xi|^{4}+f_{i}^{2}\Big{)}~{}\Rightarrow~{}|\Lambda(t)|^{2}=\frac{\delta_{a}^{2}|\xi|^{2}e^{\kappa_{b}t}}{2\mathcal{F}_{i}}+\frac{f_{i}^{2}e^{-\kappa_{b}t}}{2|\xi(t)|^{2}\mathcal{F}_{i}},$
(C70)

where $\mathcal{F}_{i}(t)$ is the anti-derivative of $f_{i}(t)$. If
$\delta_{a}\\!=\\!0$, the solution is

$\displaystyle|\Lambda(t)|=\frac{|f_{i}(t)|e^{-\frac{\kappa_{b}}{2}t}}{\sqrt{2}|\xi(t)|}\frac{1}{\sqrt{\displaystyle\int_{t_{0}}^{t}f_{i}(s)ds}}.$
(C71)

Knowing $|\Lambda(t)|$ means $g_{i}$ is a known function and $x$ and $y$ may
be evaluated using $\theta$ from Eq. (C68). Then, the phase $\phi$ is

$\displaystyle\phi(t)$
$\displaystyle=-\delta_{b}t-\arg(\xi)+\tan^{-1}\\!\bigg{(}\frac{y(t)}{x(t)}\bigg{)}.$
(C72)

To obtain $x$ and $y$, note that

$\displaystyle x$
$\displaystyle=\dot{X}=\dot{R}\cos(\theta)-R\sin(\theta)\dot{\theta}=\frac{f_{i}\cos(\theta)+g_{i}\sin(\theta)}{\sqrt{2\int\\!f_{i}}}$
(C73) $\displaystyle y$
$\displaystyle=\dot{Y}=\dot{R}\sin(\theta)+R\cos(\theta)\dot{\theta}=\frac{f_{i}\sin(\theta)-g_{i}\cos(\theta)}{\sqrt{2\int\\!f_{i}}}.$
(C74)

When $\delta_{a}\\!=\\!0$, we have $g_{i}\\!=\\!0$ and Eq. (C72) simplifies to

$\displaystyle\phi(t)$ $\displaystyle=-\delta_{b}t-\arg(\xi).$ (C75)

Using the same example as the one presented in the main text, let us consider
Gaussian input pulses

$\displaystyle\xi_{a,\rm{i}}(t)$
$\displaystyle=\\!\sqrt{\frac{2}{\tau_{\\!\scriptscriptstyle\mathcal{G}}}}\\!\left(\frac{\text{ln}(2)}{\pi}\right)^{\\!\\!\frac{1}{4}}\\!\\!\exp\\!\left(\\!-2\text{ln}(2)\frac{(t-T_{\rm{in}})^{2}}{\tau_{\\!\scriptscriptstyle\mathcal{G}}^{2}}\right),$
(C76)

where $|\xi_{a,\rm{i}}(t)|^{2}$ has a full temporal width at half maximum
(FWHM) of $\tau_{\\!\scriptscriptstyle\mathcal{G}}$, spectral width of
$\Omega_{\scriptscriptstyle\mathcal{G}}\\!=\\!4\text{ln}(2)/\tau_{\\!\scriptscriptstyle\mathcal{G}}$,
and integrates to 1 (over the infinite interval from $-\infty$ to $\infty$).
We also introduce the offset $\Delta\tau$ to investigate the influence of
timing-offset between the incident photon and the control pulse. The control
pulse, $\Lambda(t)$, is then calculated using $\xi(t+\Delta\tau)$ instead of
$\xi(t)$. Solving the equations of motion in Eq. (C54) with the control field
calculated from Eq. (C71)–(C75), we calculate the probability of converting
the photon to $\omega_{b}$ as

$\displaystyle P_{b,\rm{o}}$ $\displaystyle=\int|\xi_{b,\rm{o}}(t)|^{2}dt.$
(C77)

The results are shown in Fig. S5.

Figure S5: Simulation results for frequency conversion and bandwidth
narrowing. (a) Example of the solutions to (C54). (b) Conversion efficiency as
a function of the timing-offset between the incident photon and the control
field. (c) Conversion efficiency as a function of the intrinsic loss rate of
the cavity modes. Parameters:
$\tau_{\\!\scriptscriptstyle\mathcal{G}}\\!=\\!80\,$ps,
$\kappa_{a,\rm{w}}\\!=\\!4\Omega_{\scriptscriptstyle\mathcal{G}}$,
$\kappa_{b,\rm{w}}\\!=\\!2\pi\\!\times\\!100\,$MHz, and
$\kappa_{a,\rm{l}}\\!=\\!\kappa_{b,\rm{l}}\\!=\\!\kappa_{\rm{l}}$. In (a) and
(b) we used $\kappa_{\rm{l}}\\!=\\!0$ and in (a) and (c) we used
$\Delta\tau\\!=\\!0$.

We use an input pulse centered at 1550 nm with a temporal width of
$\tau_{\\!\scriptscriptstyle\mathcal{G}}\\!=\\!80\,$ps and
$\kappa_{a,\rm{w}}\\!=\\!4\Omega_{\scriptscriptstyle\mathcal{G}}$, which
ensures efficient absorption into cavity mode $a$. Fig. S5(a) plots the input
and output wave packets for an example with
$\kappa_{a,\rm{l}}\\!=\\!\kappa_{b,\rm{l}}\\!=\\!\kappa_{\rm{l}}\\!=\\!0$,
$\Delta\tau\\!=\\!0$, and $\kappa_{b,\rm{w}}\\!=\\!2\pi\\!\times\\!100\,$MHz
to match the bandwidth of the output pulse to the cavity containing the spin
qubit. The small amplitude of the blue line in Fig. S5(a) shows that a very
small fraction of the incident wave packet passes by the cavity without being
absorbed. This fraction decreases towards zero very rapidly as
$\kappa_{a,\rm{w}}$ is increased relative to
$\Omega_{\scriptscriptstyle\mathcal{G}}$ [21]. The effect of timing mismatch
is investigated in Fig. S5(b), which shows that the conversion efficiency
stays above 70% if $|\Delta\tau|\leq\tau_{\\!\scriptscriptstyle\mathcal{G}}$.
Note that more robustness could be obtained by optimizing the control pulse
while taking the timing-offset into account. Fig. S5(c) shows the reduction in
conversion efficiency in the case of a finite intrinsic decay rate of the
cavity modes (assumed to be equal for modes $a$ and $b$). The required narrow
bandwidth of the output pulse requires the loaded $Q$ to be larger than
$Q_{b,\rm{w}}\\!=\\!4\\!\times\\!10^{6}$. When
$Q_{\rm{l}}\\!=\\!Q_{b,\rm{w}}$, the spectrum of the output pulse is roughly
Lorentzian with a FWHM bandwidth of 200 MHz and the conversion efficiency is
50% as illustrated using the vertical dased black line in Fig. S5(c).

##  Appendix D Switching array in the receiver

Figure S6: An illustration of a MZI tree network as the interposer to the
memories. Each layer introduces transmission loss from the MZI.

Here we consider using a MZI tree network in a photonic integrated circuit
(PIC) as a fast switching array, as shown in Fig. S6, albeit with finite
transmission loss that scales exponentially with the number of layers. For
LiNbO3-based PIC, propagation loss has been shown to be negligible even for
device length exceeding $100~{}\mu$m [39]. For simplicity, we neglect metal
absorption loss stemming from nearby electrodes used to drive the electro-
optic phase shifters. Rather, we assume the main loss mechanism to come from
imperfection in the directional couplers. We take a state-of-the-art value of
$\sim$0.2 dB per MZI demonstrated on SOI [75] to illustrate the effect of
switching array loss on the entanglement generation rate. Fig. S7 shows
comparable rates as the non-MZI-based scheme shown in the main text, up until
$k\approx 10^{4}$. After which point, the $k$-dependent switching array loss
begins reducing the entanglement generation rate.

Figure S7: Entanglement generation rate $\bar{\Gamma}$ vs $k$ for both ZALM
and SPDC with varying $\epsilon\in\\{10^{-2},10^{-3}\\}$ and total downlink
atmospheric attenuation $\alpha=\\{40,50\\}~{}$dB. We additionally consider
$\overline{AB}=10^{2}~{}$km (purple). The rates are based on a scheme that
employs a MZI tree network that has $k$-dependent transmission loss.

This particular qRX setup requires heterogeneous integration of diamond color
centers into PICs, a feat which has already been demonstrated by Ref. [65] and
is conducive to scaling up a multiplexed quantum repeater network. Despite
solid-state emitters such as SiV- manifesting spectral inhomogeneity, we argue
post-selecting candidates within a narrowed inhomogeneous distribution and
performing subsequent in-situ tuning enabled by an active PIC platform could
still ensure maximal spin-cavity coupling. For example, SiV- can be strain-
tuned [76] to shift its optical transition frequency, while the nanophotonic
cavity’s resonance can be gas-tuned [77] (i.e. index shifting).

##  Appendix E Spin-photon system

###  1 Silicon-vacancy center

The proposed architecture considers diamond’s negatively-charged silicon-
vacancy center as the atomic memory. With an applied magnetic field and
accounting for only Zeeman splitting, the energy ground state splits into two
electron spin states $\ket{\downarrow}$ and $\ket{\uparrow}$. One of the two
optical transitions with the excited state $\ket{e}$,
$\ket{\downarrow}\longleftrightarrow\ket{e}$, is coupled with the cavity mode.
With a nearby nuclear spin, hyperfine splitting further divides the two
electronic spin states to a total of four levels:
$\ket{\downarrow_{e}\downarrow_{n}},\ket{\downarrow_{e}\uparrow_{n}},\ket{\uparrow_{e}\uparrow_{n}},\ket{\uparrow_{e}\downarrow_{n}}$.
Effectively, the electron spin acts as a broker qubit that interfaces with the
photon, and subsequently transfers the qubit state to the nuclear spin that
serves as a long-lived atomic memory [51].

Figure S8: Level-structure of a SiV- center in diamond. The electron spin
(red) contains two Zeeman-split states
$\\{\ket{\downarrow},\ket{\uparrow}\\}$, which further split into four states
$\\{\ket{\downarrow_{e}\downarrow_{n}},\ket{\downarrow_{e}\uparrow_{n}},\ket{\uparrow_{e}\uparrow_{n}},\ket{\uparrow_{e}\downarrow_{n}}\\}$
due to hyperfine coupling with the nuclear spin (blue).

###  2 Spin-dependent cavity reflection

The reflection coefficient of a single-sided cavity coupled with a quantum
emitter is:

$\displaystyle r(\omega)$
$\displaystyle=1-\frac{\kappa_{\text{wg}}\left(i\Delta_{a}+\frac{\gamma}{2}\right)}{\left(i\Delta_{c}+\frac{\kappa}{2}\right)\left(i\Delta_{c}+\frac{\gamma}{2}\right)+g^{2}}$
(E78)

where $g$ is the atom-cavity coupling strength, $\gamma$ is the emitter’s
spontaneous emission rate, $\kappa$ is the cavity’s total decay rate,
$\kappa_{\text{wg}}$ is the waveguide-cavity coupling rate, and
$\Delta_{a}=\omega_{a}-\omega$ and $\Delta_{c}=\omega_{c}-\omega$ are the
atomic and cavity detuning from the probe photon, respectively. In the large
cooperativity $C=4g^{2}/\kappa\gamma\gg 1$ limit and on resonance
$\Delta_{a}=\Delta_{c}=0$, the reflection coefficient of a perfectly over-
coupled cavity simplifies to

$\displaystyle r(\omega)\xrightarrow{C\gg 1}\frac{C-1}{C+1}$ (E79)

Therefore, $r$ approaches $+1$ when $C$ increases, whereas a emitter decoupled
from the cavity mode would yield $r\rightarrow-1$. We consider a spin qubit
whose basis states are $\ket{\downarrow}$ and $\ket{\uparrow}$. When the
emitter is in the $\ket{\downarrow}$ state that is coupled to the cavity mode,
$r=+1$. On the other hand, if it is in the $\ket{\uparrow}$ state that is
decoupled with the cavity mode, $r=-1$. As a result of this state-dependent
phase difference, the probe photon is entangled with the spin via cavity
reflection.

###  3 Photon-to-spin mapping

In the Schrodinger picture, we present an example of how an arbitrary photonic
qubit encoded in the polarization basis $\\{\ket{H},\ket{V}\\}$,
${\ket{\psi}_{P}=\alpha\ket{H}+\beta\ket{V}}$
($\equiv\alpha\ket{0,1}+\beta\ket{1,0}$ in the dual-rail Fock basis), can be
teleported to a spin qubit
${\ket{\psi}_{S,\text{final}}=\alpha\ket{\downarrow}+\beta\ket{\uparrow}}$. We
initialize the spin to be in an equal superposition state:
${\ket{\psi}_{S,\text{init}}=\left(\ket{\downarrow}+\ket{\uparrow}\right)/\sqrt{2}}$.
Their joint state is then:
$\ket{\Psi}=\ket{\psi}_{P}\otimes\ket{\psi}_{S,\text{init}}={\alpha\ket{H,\downarrow}+\alpha\ket{H,\uparrow}+\beta\ket{V,\downarrow}+\beta\ket{V,\uparrow}}$.
Upon entering the receiver node, we envision using a polarization-splitter
rotator to convert the polarization basis to the spatial basis
$\\{a_{H},a_{V}\\}$. Importantly, we re-write the joint state as
${\ket{\Psi}=\alpha\ket{a_{H},\downarrow}+\alpha\ket{a_{H},\uparrow}+\beta\ket{a_{V},\downarrow}+\beta\ket{a_{V},\uparrow}}$.
Subsequently, mode $a_{H}$ acquires a spin-dependent phase upon cavity
reflection, whereas mode $a_{V}$ acquires a constant $-1$ phase from
reflection off a mirror. The resultant state is

$\displaystyle\ket{\Psi}$
$\displaystyle=\alpha\ket{a,\downarrow}-\alpha\ket{a,\uparrow}-\beta\ket{b,\downarrow}-\beta\ket{b,\uparrow}$
(E80)

We note describe the evolution of each of the two spatial modes separately.
After cavity interaction, $\\{\ket{a_{H}},\ket{a_{V}}\\}$ enter a 50:50 beam
splitter whose output modes are
$\\{\ket{A}=t\ket{a_{H}}+r\ket{a_{V}},\ket{B}=r\ket{a_{H}}+t\ket{a_{V}}\\}$,
where $r=i$ and $t=1$ are the reflection and transmission coefficients,
respectively. The two output modes then become

$\displaystyle\ket{A}$
$\displaystyle=\alpha\left(\ket{\downarrow}-\ket{\uparrow}\right)-i\beta\left(\ket{\downarrow}+\ket{\uparrow}\right)$
(E81) $\displaystyle\ket{B}$
$\displaystyle=i\alpha\left(\ket{\downarrow}-\ket{\uparrow}\right)-\beta\left(\ket{\downarrow}+\ket{\uparrow}\right)$
(E82)

The spin undergoes a Hadamard rotation, transforming the states to

$\displaystyle\ket{A}$
$\displaystyle=\alpha\ket{\uparrow}-i\beta\ket{\downarrow}$ (E83)
$\displaystyle\ket{B}$
$\displaystyle=i\alpha\ket{\uparrow}-\beta\ket{\downarrow}$ (E84)

Upon detection on either of the $\ket{A}$ or $\ket{B}$ port, appropriate Pauli
operations can be applied to the spin qubit to obtain the target state
${\ket{\psi}_{S,\text{final}}=\alpha\ket{\downarrow}+\beta\ket{\uparrow}}$.

However, with imperfections such as finite cooperativity and non-unity
coupling to the waveguide mode ($\kappa_{\text{wg}}/\kappa<1$) in the spin-
cavity system, the reflection coefficients would have non-unity amplitudes. We
consider now the photon-to-spin mapping process with generalized reflection
coefficients, $r_{\text{on}},r_{\text{off}}$, for the on- and off-resonance
cases respectively. We still treat the mirror as a lossless component such
that mode $b$ still acquires a constant -1 phase.

Upon reflection, the spin-photon state is

$\displaystyle\ket{\Psi}$
$\displaystyle=r_{\text{on}}\alpha\ket{a_{H},\downarrow}+r_{\text{off}}\alpha\ket{a_{H},\uparrow}-\beta\ket{a_{V},\downarrow}-\beta\ket{a_{V},\uparrow}$
(E85)

After the 50:50 beam splitter, the two output modes are

$\displaystyle\ket{A}$
$\displaystyle=\alpha\left(r_{\text{on}}\ket{\downarrow}+r_{\text{off}}\ket{\uparrow}\right)-i\beta\left(\ket{\downarrow}+\ket{\uparrow}\right)$
(E86) $\displaystyle\ket{B}$
$\displaystyle=i\alpha\left(r_{\text{on}}\ket{\downarrow}+r_{\text{off}}\ket{\uparrow}\right)-\beta\left(\ket{\downarrow}+\ket{\uparrow}\right)$
(E87)

##  Appendix F Dependence of $\bar{\Gamma}$ on $\eta$

###  1 Memory multiplexing dependent rate scaling behavior

From the main text, the entanglement generation rate is defined to be

$\displaystyle\bar{\Gamma}$
$\displaystyle=\frac{p_{\text{success}}\cdot(k-1)}{\tau_{\text{idle}}}$ (F88)

where

$\displaystyle p_{\text{success}}$
$\displaystyle=\eta\left(\frac{1-(1-\sqrt{\eta})^{2N}}{1-(1-\sqrt{\eta})^{2}}\right)$
(F89)

For simplicity, let us neglect the contribution of $\epsilon$ and define
$N=N_{k}=\tau_{\text{idle}}/((k-1)\tau_{0})$. In the memory-limited regime,
$\tau_{\text{idle}}/\tau_{0}\gg(k-1)$, thus $N_{k}\gg 1$. We can re-write
$p_{\text{success}}$ as

$\displaystyle p_{\text{success}}$
$\displaystyle\approx\eta\left(\frac{1-e^{-2N_{k}\sqrt{\eta}}}{2\sqrt{\eta}}\right)$
(F90) $\displaystyle\approx\frac{1}{2}\sqrt{\eta}$ (F91)

where we used the approximations $(1+x)^{\alpha}\approx e^{\alpha x}$ for
large $|\alpha x|\gg 1$ and $(1+x)^{\alpha}\approx 1+\alpha x$ for small
$|\alpha x|\ll 1$. From above, we see that $\bar{\Gamma}$ scales as
$\sqrt{\eta}$ in the memory-limited regime.

On the other hand, with sufficiently high memory multiplexing such that
$N_{k}\ll 1$,

$\displaystyle p_{\text{success}}$
$\displaystyle\approx\eta\left(\frac{2N_{k}\sqrt{\eta}}{2\sqrt{\eta}}\right)$
(F92) $\displaystyle=N_{k}\eta$ (F93)

which recovers the typical $\bar{\Gamma}\propto\eta$ scaling.

##  Appendix G Spin-spin Bell state generation rate fidelity trade-off

Figure S9: Rate-fidelity trade-off at ${k\in\\{10^{2},5\times
10^{2},10^{3}\\}}$. Setting $N_{s}=2.94\times 10^{-2}$ and $C=100$,
$\mathcal{F}$ upper-bounds at 0.998 due to imperfections in the qTX and qRX.
We take $\overline{AB}=10^{2}$ km and $\alpha_{\text{atm}}=40$ dB.

Figure S9 demonstrates the rate-fidelity trade-off at different
${k\in\\{10^{2},5\times 10^{2},10^{3}\\}}$. $\bar{\Gamma}$ approaches zero as
$\mathcal{F}$ increases. Similar to what is shown in Fig. 6 with
$\overline{AB}=10^{2}$ km and $\alpha_{\text{atm}}=40$ dB, $\bar{\Gamma}$
increases monotonically with the number of spins.

##  Appendix H ZALM for ground-only quantum networks

Figure S10: Entanglement generation rate $\bar{\Gamma}$ as a function of the
number of spins $k$. $\bar{\Gamma}$ vs $k$ for both ZALM and SPDC with varying
channel losses $\alpha=\\{50,60,70,80\\}~{}$dB. These calculations assume
$\overline{AB}=10^{2}$ km with corresponding $\tau_{\text{comm}}=30$ ms and
$\epsilon=10^{-3}$.

The quasi-deterministic ZALM BPS is useful for general two-way quantum
repeater networks regardless of their configurations. Although the main text
provides a specific example for a satellite-assisted architecture for global
scale networks, we show here that the same qTX can be equally beneficial for
ground-only quantum networks. Figure S10 shows the rate of generating
entanglement between A and B, with a midpoint source C equidistant from the
two qRX’s. We assume $\overline{AB}=2L_{\text{GG}}=10^{2}$ km with
corresponding classical communication time of 30 ms and $\epsilon=10^{-3}$.
Again, we compare the rate performance between ZALM and a free-running
narrowband-filtered SPDC for channel losses $\alpha=\\{50,60,70,80\\}~{}$dB
accounting for the attenuation loss in optical fibers. Additionally, we assume
the qRX containing a fast-switching PIC whose MZI tree array causes compounded
loss dependent on the number of tree layers. In the memory-limited regime with
$k\geq 10$, ZALM already outperforms SPDC by at least an order of magnitude in
$\bar{\Gamma}$. The gap in rate widens further with greater channel loss. For
example, the difference in $\bar{\Gamma}$ at $k=1$ is much larger for
$\alpha=80$ dB than for $\alpha=50$ dB. Nevertheless, with increasing memory
multiplexing, the rate advantage in using the ZALM BPS immediately manifests.

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