# Proposal for non-cryogenic quantum repeaters with hot hybrid alkali-noble
gases

Jia-Wei Ji<EMAIL_ADDRESS>Institute for Quantum Science and
Technology, and Department of Physics & Astronomy, University of Calgary, 2500
University Drive NW, Calgary, Alberta T2N 1N4, Canada  Faezeh Kimiaee Asadi
Institute for Quantum Science and Technology, and Department of Physics &
Astronomy, University of Calgary, 2500 University Drive NW, Calgary, Alberta
T2N 1N4, Canada  Khabat Heshami  National Research Council of Canada, 100
Sussex Drive, Ottawa, Ontario K1A 0R6, Canada  Department of Physics,
University of Ottawa, Advanced Research Complex, 25 Templeton Street, Ottawa,
Ontario K1N 6N5, Canada  Christoph Simon<EMAIL_ADDRESS>Institute
for Quantum Science and Technology, and Department of Physics & Astronomy,
University of Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4,
Canada

###### Abstract

We propose a quantum repeater architecture that can operate without
cryogenics. Each node in our architecture builds on a cell of hot alkali atoms
and noble-gas spins which offer a storage time as long as a few hours. Such a
cell of hybrid gases is placed in a ring cavity, which allows us to suppress
the detrimental four-wave mixing (FWM) noise in the system. We investigate the
protocol based on a single-photon source made of an ensemble of the same hot
alkali atoms. A single photon emitted from the source is either stored in the
memory or transmitted to the central station to be detected. We quantify the
fidelity and success probability of generating entanglement between two remote
ensembles of noble-gas spins by taking into account finite memory efficiency,
channel loss, and dark counts in detectors. We describe how the entanglement
can be extended to long distances via entanglement swapping operations by
retrieving the stored signal. Moreover, we quantify the performance of this
proposed repeater architecture in terms of repeater rates and overall
entanglement fidelities and compare it to another recently proposed non-
cryogenic quantum repeater architecture based on nitrogen-vacancy (NV) centers
and optomechanical spin-photon interfaces. As the system requires a relatively
simple setup, it is much easier to perform multiplexing, which enables
achieving rates comparable to the rates of repeaters with NV centers and
optomechanics, while the overall entanglement fidelities of the present scheme
are higher than the fidelities of the previous scheme. Our work shows that a
scalable long-distance quantum network made of hot hybrid atomic gases is
within reach of current technological capabilities.

##  I Introduction

The realization of global quantum networks would bring many fascinating
applications to the world, which include secure communication [1], blind
quantum computing [2], private database queries [3], and eventually, a quantum
internet that connects quantum computers and other quantum information
processing devices [4, 5, 6]. In such a quantum network, photons are used as
information carriers for establishing long-distance connections, but they are
adversely affected by transmission loss, which significantly limits the
distance of connecting remote locations. Unlike its classical counterparts,
photon loss cannot be compensated by amplification as unknown quantum states
cannot be perfectly cloned according to the no-cloning theorem [7]. Therefore,
quantum repeaters have been proposed to solve this issue but this requires
stationary quantum memories for storing and processing the quantum information
[8, 9, 5]. Currently, a vast majority of approaches to quantum networks need
either vacuum equipment and optical trapping or cryogenic cooling [10, 11, 12,
13, 14, 8, 15, 16, 17], which makes scaling up such architectures very
difficult. However, there have been some efforts in proposing quantum networks
that operate at room temperature based on solid-state systems [18, 19] but
they require complex setups and have high demands in designing the hardware
for realizing the spin-photon interface. On the other hand, hot alkali vapors
have been actively investigated as quantum memories for the application of
quantum networks [20, 21, 22, 23, 24], and as they require relatively simple
setups, it is easier to scale, which even offers a great potential for being
deployed in space [25].

In spite of the appealing features of hot alkali vapor, there are a few
challenges in the system. The main roadblock towards using this system for
quantum networks is four-wave mixing (FWM) noise as it is quite significant
and ubiquitous in $\Lambda$-type hot atomic ensembles, posing serious
challenges to the single-photon level applications [26, 27]. Proposed
solutions to this issue include blocking FWM channels by polarization
selection rules [28], Raman absorption-enabled suppression in a mixed hot
vapor [29], cavity engineering [30], and by means of coherent destructive
interference of FWM [31]. The advantage of using a cavity to suppress FWM
noise compared to other solutions is that it offers enhanced light storage and
retrieval efficiency while only introducing a cavity. It has been
experimentally verified, reporting a noise floor of around $1.5\times 10^{-2}$
photons per pulse in a Raman-type hot vapor memory [32]. Another significant
challenge in the $\Lambda$-type hot atomic ensembles is short storage time in
the collective spin state, which is mainly affected by the atomic collisions
between the hot vapor and the buffer gas and the collisions in the hot vapor
itself. Due to this detrimental effect, the storage time in hot ensembles is
limited to a microsecond [33], thus restricting its application in quantum
networks. However, there has been some work towards reducing this detrimental
effect either by the motional averaging method [20, 24] or by using a
decoherence-free subspace of spin states [34] with the spin coherence time
extended to a second but even second-long coherence time may not be sufficient
for long-distance quantum networks [8].

Rare isotopes of noble gas have non-zero nuclear spins, which are isolated
from the environment by electronic shells. Thus, they maintain hours-long
coherence time even at room temperature [35]. They can be accessed either via
the collisions with metastable helium atoms or via the collisions with alkali
atoms [35]. A quantum interface between noble-gas spins and alkali atoms has
been proposed based on weak spin-exchange collisions [36]. Using this
interface, the storage time can be significantly enhanced, which has been
experimentally demonstrated with the coherence time of a minute [37] and an
hour [38].

In this work, we propose a quantum repeater architecture without cryogenics,
which is based on hot alkali vapor and noble-gas nuclear spins. In our
proposal, we adopt the cavity engineering method to suppress FWM noise when
the input gets stored as a collective spin excitation in hot vapor via the
off-resonant Raman protocol [30], and then it is mapped to noble-gas spins via
weak spin-exchange collisions [36, 39]. We consider the single-photon-based
protocol [8] where single-photon sources and quantum memories are used for
entanglement generation and swapping. We propose to use the same hot alkali
atomic ensembles for single-photon sources. We quantify and analyze the
entanglement generation efficiency and fidelity between two remote ensembles
of noble-gas spins. Then, we show how entanglement swapping can be done to
extend the entanglement to longer distances. Finally, we compute the repeater
rates and overall fidelities and compare them to quantum repeaters with NV
centers and optomechanics.

This paper is organized as follows. In Sec. II, we introduce the hybrid system
of hot vapor and noble-gas spins in a ring cavity. The single-photon protocol
is presented in Sec. III. Sec. IV discusses the repeater rates and fidelities.
Sec. V gives more details on system implementation. We conclude and provide an
outlook in Sec. VI.

##  II Hybrid atomic gas system

Figure 1: The level diagram of a hybrid system with hot vapor and noble gas.
The $\ket{g}$-$\ket{e}$ transition is coupled by the input signal (Stokes
field) with the strength proportional to $\sqrt{n_{a}}$ where $n_{a}$ is the
number density of alkali atoms in the cell. The control field is coupled to
the $\ket{e}$-$\ket{s}$ transition with a time-dependent Rabi frequency
$\Omega(t)$. Both fields are detuned from $\ket{e}$ by $\Delta_{s}$, and the
control field can also couple the $\ket{g}$-$\ket{e}$ transition with the
detuning $\Delta_{a}$, which generates the anti-Stokes field A. The collective
spin state for noble-gas atoms is denoted by $\ket{k}$. The $\ket{s}-\ket{k}$
transition is coupled to each other via the spin-exchange collision with a
constant strength $J$ when these two states are in resonance with other, i.e.
$\delta_{k}=0$. The collective excited state, alkali spin state, and noble-gas
spin state decohere at the rates of $\gamma_{e}$, $\gamma_{s}$, and
$\gamma_{k}$ respectively. Typically, $\gamma_{k}\ll\gamma_{s}$ as noble-gas
spins have extremely low decoherence rates. Figure 2: (a) Schematic of the
setup of hybrid quantum memories. A cell that contains alkali atoms (red dots)
and noble-gas atoms (blue dots) is placed inside a ring cavity where the green
signal and red control field interact with alkali atoms directly. The FWM
noise is the generated blue anti-Stokes field during the storage and retrieval
processes, which can be largely suppressed via tuning the cavity. The
interface between alkali atoms and noble-gas spins is based on spin-exchange
collisions. (b) Schematic of a two-link repeater with single-photon sources
[40] as an example. There are two elementary links with four nodes where each
one has a single-photon source which is a hot alkali atomic ensemble, a beam
splitter, and a hot alkali-noble gases hybrid system serving as a quantum
memory. The single-photon source emits a photon that either transmits through
the beam splitter or gets reflected to enter the quantum memory. Two steps are
required to establish the entanglement between nodes A and D. The first step
is to generate the entanglement between A and B, C and D. The second step is
to perform the entanglement swapping between B and C to distribute the
entanglement to A and D. (c) Schematic of post-selection. This is the same as
in the DLCZ protocol [10]. The entanglement is established both in links A-D
and A’-D’, and the stored photons are retrieved to be detected, which allows
us to rotate the measurement basis by adjusting the transmission coefficients
and phases of the beam splitters [8].

As shown in Fig. 2(a), the hybrid atomic gas system is composed of a ring
cavity and a cell of two hot atomic gases: alkali atoms and noble-gas atoms.
This cell placed inside the cavity is driven by the control field (red) and
the Stokes field (green). The ring cavity consists of two fully reflective
mirrors and a mirror which serves as input-output coupler with amplitude
reflectivity $r$. The coherent interaction between the noble-gas spins and
alkali atoms is achieved by spin-exchange collisions [36]. As shown in Fig.1,
an ensemble of alkali atoms is modelled as a $\Lambda$-type system with a
collective ground state $\ket{g}$, a collective spin state $\ket{s}$ and an
excited state $\ket{e}$. Each noble-gas atom is modelled as a spin-1/2 system
with up and down states $\ket{\Uparrow}$, $\ket{\Downarrow}$. Here, we denote
the collective noble-gas spin state as $\ket{k}$. The input signal (Stokes
field) $S$ couples the $\ket{g}-\ket{e}$ transition with the strength
proportional to the density of alkali atoms $n_{a}$, and the control field
couples to the $\ket{e}-\ket{s}$ transition with the Rabi frequency
$\Omega(t)$. Both fields are detuned from $\ket{e}$ by $\Delta_{s}$. The
control field can also couple the $\ket{g}$-$\ket{e}$ transition with the
detuning of $\Delta_{a}$, which generates the anti-Stokes field A (FWM noise)
because here all alkali atoms are prepared in one of the ground states that
has higher energy [30]. Due to the effect of spatial diffusion, there could be
many spatial modes for the alkali and noble gases. However, in the light-
dominated regime where the power broadening (proportional to the control field
strength $|\Omega|^{2}$ and the optical depth $d$) in the alkali atoms due to
the control beam dominates over diffusion in the alkali atoms, the collective
spin mode of the alkali gas and the collective spin mode of the noble gas can
be well approximated as single uniform modes by engineering the spatial
profile of the control field [39]. This condition is generally satisfied when
both $|\Omega|^{2}$ and $d$ are large enough to be in the strong coupling
regime, which is the case in this work as discussed in Sec. V.

After polarizing both the alkali and noble gases along the vertical axis using
standard optical pumping and spin-exchange optical pumping (SEOP), the
collective alkali spin state $\ket{s}$ can be coupled to the collective noble-
gas spin state $\ket{k}$ via weak spin-exchange collision with the strength
$J$. The coupling strength $J$ is given by
$J=\zeta\sqrt{(2I+1)p_{a}p_{b}n_{a}n_{b}/4}$ where $\zeta$ is the local
average interaction strength of an alkali-noble atom pair in a single
collision, and $p_{a}$ and $p_{b}$ are the polarization degrees of alkali and
noble gases, and $n_{a}$ and $n_{b}$ are the densities of alkali and noble
gases in the cell. $I$ is the nuclear spin of an alkali atom. Thus, $J$ is the
effective interaction strength in multiple collisions with each collision
averaging over all alkali-noble atom pairs in the ensembles [36]. $\delta_{k}$
is the detuning between these two states, which can be tuned by applying a
magnetic field along the vertical axis. This detuning can be used to decouple
these two species of atoms [36, 38]. $\gamma_{e}$, $\gamma_{s}$, and
$\gamma_{k}$ are the decoherence rates for the collective excited state, spin
state and noble-gas spin state respectively. Moreover, we have
$\gamma_{k}\ll\gamma_{s}$ as noble-gas spins have an extremely low decoherence
rate.

The Maxwell-Bloch equations of this hybrid system with the excited state
$\ket{e}$ being adiabatically eliminated take the form [30, 39]

$\displaystyle(c\partial_{z}+\partial_{t})S$
$\displaystyle=ic\sqrt{\frac{d\gamma_{e}}{L}}\frac{\Omega}{\Gamma_{s}}B-\kappa_{s}S,$
(1) $\displaystyle(c\partial_{z}+\partial_{t})A$
$\displaystyle=ic\sqrt{\frac{d\gamma_{e}}{L}}\frac{\Omega}{\Gamma_{a}}B^{\dagger}-\kappa_{a}A,$
$\displaystyle\partial_{t}B$
$\displaystyle=-i\sqrt{\frac{d\gamma_{e}}{L}}\frac{\Omega^{*}}{\Gamma_{s}}S+i\sqrt{\frac{d\gamma_{e}}{L}}\frac{\Omega}{\Gamma_{a}}A$
$\displaystyle-(\frac{1}{\Gamma_{s}}+\frac{1}{\Gamma^{*}_{a}})|\Omega|^{2}B-\gamma_{s}B-iJK,$
$\displaystyle\partial_{t}K$ $\displaystyle=-(\gamma_{k}+i\delta_{k})K-iJB,$

where $S,A,B$, and $K$ are the annihilation operators for the signal field,
anti-Stokes field, bosonic collective spin wave and collective noble-gas spin
wave (we use the same notation as in [36]).
$\Gamma_{s,a}=\gamma_{e}-i\Delta_{s,a}$ is the complex detuning of the signal
and anti-Stokes fields. $d=g^{2}p_{a}n_{a}L/\gamma_{e}$ is the optical depth
where the alkali density $n_{a}$ is only nonzero in the cell, and $g$ is the
average coupling strength between the Stokes/anti-Stokes fields and the alkali
atoms, which is given by
$g=\sqrt{\frac{1}{N_{a}}\sum^{N_{a}}_{i=1}{|g_{i}(r_{i})|^{2}}}$ with $N_{a}$
being the number of the alkali atoms in the cell. This approximation is valid
when the number of excitations is much smaller than $N_{a}$ [41], which is the
case here. $L$ is the length of a roundtrip in the cavity, and $c$ is the
speed of light. The coordinate $z$ indicates the direction along the optical
path inside the cavity. Moreover, the bosonic operators $B$ and $K$ take the
form as $B=\sum_{j\in\delta z}{\ket{g}_{j}\bra{s}/(\delta
z\sqrt{p_{a}n_{a}})}$ and $K=\sum_{i\in\delta
z}{\ket{\Downarrow}_{i}\bra{\Uparrow}}/(\delta z\sqrt{p_{b}n_{b}})$. As
mentioned before, the spin-exchange coupling rate $J$ is proportional to the
densities of the two atomic gases $n_{a}$ and $n_{b}$, i.e.
$J\propto\sqrt{n_{a}n_{b}}$ [36]. Thus, by increasing the pressure, one can
increase this interaction strength. $\kappa_{s}=cd\gamma_{e}/(L\Gamma_{s})$
and $\kappa_{a}=cd\gamma_{e}/(L\Gamma^{+}_{a})$ stand for the decay rates of
the Stokes field and anti-Stokes field in the ring cavity where
$\Gamma^{+}_{a}=\gamma_{e}-i(\Delta_{a}+\delta_{s})$ with $\delta_{s}$ being
the splitting between the states $\ket{g}$ and $\ket{s}$. Strictly speaking,
Eq. (1) should also have the Langevin noise operators. However, for both the
signal and the anti-Stokes field, the noise is vacuum which is zero in normal
ordering [42]. Now, we have the boundary condition where the intra-cavity
fields at $z=0$: $S_{0}$ and $A_{0}$ can be related to the input fields
$S_{\text{in}}$, $A_{\text{in}}$ and the intra-cavity fields at $z=L$: $S_{L}$
and $A_{L}$ by the input-output coupler. Thus, we obtain the following
relations [30]:

$\displaystyle S_{0}$ $\displaystyle=re^{ik_{s}L}S_{L}+t_{r}S_{\text{in}},$
(2) $\displaystyle A_{0}$
$\displaystyle=re^{ik_{a}L}A_{L}+t_{r}A_{\text{in}},$

where $t_{r}=\sqrt{1-r^{2}}$ is the transmission coefficient of the coupler,
and $k_{s}$ and $k_{a}$ are the wavevectors of the signal and the anti-Stokes
fields respectively. Moreover, $S_{L}$ and $A_{L}$ can be directly related to
$S_{0}$ and $A_{0}$ by Taylor expansion, which gives us:

$\displaystyle S_{L}$ $\displaystyle\approx
e^{-k_{s}\frac{L}{c}}(S_{0}+iL\sqrt{\frac{d\gamma_{e}}{L}}\frac{\Omega}{\Gamma_{s}}B_{0}-\frac{L}{c}\partial_{t}S_{0}),$
(3) $\displaystyle A_{L}$ $\displaystyle\approx
e^{-k_{a}\frac{L}{c}}(A_{0}+iL\sqrt{\frac{d\gamma_{e}}{L}}\frac{\Omega}{\Gamma_{a}}B^{\dagger}_{0}-\frac{L}{c}\partial_{t}A_{0}),$

where $B_{0}$ is the collective alkali spin operator for $z=0$. Combining
these two sets of relations, one can obtain the following Maxwell-Bloch
equations:

$\displaystyle\partial_{t}s$
$\displaystyle=-\tilde{\kappa}_{s}s+i\sqrt{\frac{d\gamma_{e}}{\tau}}\frac{\Omega}{\Gamma_{s}}b+e^{-i\phi_{s}}\frac{t_{r}}{\mu_{s}\sqrt{\tau}}S_{\text{in}},$
(4) $\displaystyle\partial_{t}a$
$\displaystyle=-\tilde{\kappa}_{a}a+i\sqrt{\frac{d\gamma_{e}}{\tau}}\frac{\Omega}{\Gamma_{a}}a+e^{-i\phi_{a}}\frac{t_{r}}{\mu_{a}\sqrt{\tau}}A_{\text{in}},$
$\displaystyle\partial_{t}b$
$\displaystyle=-\gamma_{s}b+i\sqrt{\frac{d\gamma_{e}}{\tau}}(-\frac{\Omega^{*}}{\Gamma_{s}}s+\frac{\Omega}{\Gamma_{a}}a)$
$\displaystyle-(\frac{1}{\Gamma_{s}}+\frac{1}{\Gamma^{*}_{a}})|\Omega|^{2}b-iJk,$
$\displaystyle\partial_{t}k$ $\displaystyle=-(\gamma_{k}+i\delta_{k})k-iJb,$

where $\tau=L/c$ is the cavity roundtrip time.
$s=\sqrt{\tau}S_{0},a=\sqrt{\tau}A_{0},b=\sqrt{L}B_{0}$, and $k=\sqrt{L}K_{0}$
are the intra-cavity amplitudes for the signal, anti-Stokes field, collective
alkali spin state, and collective noble-gas spin state respectively.
$\tilde{\kappa}_{s,a}$ is the resonant and anti-resonant decay rates for the
signal and anti-Stokes field. They are given by [30]:

$\frac{1}{\tilde{\kappa}_{s,a}}=\tau\frac{\mu_{s,a}e^{i\phi_{s,a}}}{1-\mu_{s,a}e^{i\phi_{s,a}}},$
(5)

where $\phi_{s,a}=k_{s,a}L-\Im{\kappa_{s,a}}\tau$ is the accumulated phases in
the cavity roundtrip by the signal and anti-Stokes fields, and
$\mu_{s,a}=re^{-\Re{\kappa_{s,a}}\tau}$ is the cavity roundtrip amplitude
transmission for the fields.

Eq. (4) can be solved in the bad-cavity regime where the signal/anti-Stokes
field evolved at a rate much slower than the corresponding decay rate, i.e.
$|\tilde{\kappa}_{s,a}|\gg|\sqrt{d\gamma_{e}/\tau}\Omega/\Gamma_{s,a}|$ [42].
In this limit, we can set $\partial_{t}a\approx 0$ and $\partial_{t}s\approx
0$. Moreover, as we can decouple the alkali and noble gases by applying a
large magnetic field, we can break the storage into two steps: first consider
the storage in the alkali atoms in the presence of the anti-Stokes field and
then consider the transfer from the alkali to the noble gas. This sequential
storage is optimal when the signal pulse duration $T$ satisfies $T\ll
1/\gamma_{s}$ [39], which is adopted in this work. We discuss how this
sequential storage is achieved in detail and the optimal storage efficiency in
Sec. III.1. The main noise present in the system is FWM, and in order to
achieve the maximum suppression of this noise, we need to tune the ring cavity
to be in resonance with the signal and to be in anti-resonance with the anti-
Stokes field, which means $\phi_{s}=0$ and $\phi_{a}=\pi$. This is crucial in
the first step of storage and retrieval, which is discussed in more detail in
Sec. III.2. Other sources of noise in hot vapor systems include collision-
induced fluorescence noise, the Doppler broadening, and inhomogeneous
broadening for the $\ket{g}-\ket{e}$ transition. However, in this system, we
ignore these effects as it has been demonstrated that fluorescence noise is
negligible for the off-resonant scheme with a short pulse input [43], and this
is also true for the Doppler broadening and inhomogeneous broadening with the
detuning $\Delta_{s}$ much larger than their bandwidth, which is discussed in
Sec. V.

##  III The single-photon repeater

Here, we focus on the single-photon-based protocol [40] for entanglement
generation and entanglement swapping where each node consists of a beam
splitter (BS), a single-photon source (SPS) and a hybrid quantum system as
depicted in Fig. 2(b), where we just show a two-link repeater as an example.
As the noble-gas spins offer ultralong coherence time at room temperature,
they will be used as the memory for storing the signal. There are two steps to
establish the entanglement between two remote locations, and Fig. 2(a) shows
how to achieve this between nodes A and D by cutting this distance into small
pieces of equal length. In Fig. 2(b), it is cut into two equal pieces: A-B and
C-D but it can be more general to have more links. In this example, we need to
first establish the entanglement between A and B, and C and D, which is called
entanglement generation, and then we perform the entanglement swapping between
two local memories B and C to distribute the entanglement to A and D, i.e.
only entangling A and D. Moreover, as a single excitation in the noble-gas
spins is shared between A and D, it is difficult to perform measurements in
other bases than the basis $\\{\ket{k}$, $\ket{0}\\}$. In order to relax this,
we can introduce another entangled link $A^{\prime}-D^{\prime}$ where nodes
$A^{\prime}$ and $D^{\prime}$ are in the same locations as $A$ and $D$
respectively [40], which is depicted in Fig. 2(c). In this way, we can use two
beam splitters and two detectors in each location to read out the stored
photons, which allows measurements in an arbitrary basis by choosing the
transmission coefficients and phases. This step is known as post-selection. In
this section, we show how entanglement generation, entanglement swapping and
post-selection can be achieved in our hybrid system, and we also quantify the
established entanglement generation fidelity and efficiency in the elementary
link.

###  III.1 Entanglement generation

Before we characterize how the entanglement generation can be done, we would
like to first talk about how signal storage can be achieved and give optimal
storage efficiency. Our goal is to store the signal in the quantum memory as a
collective excitation in the noble-gas spins, and this process can be divided
into two steps: storing the signal in the collective spin excitation of alkali
atoms and transferring this excitation to the collective excitation in noble-
gas spins. This sequential storage is optimal when the signal pulse duration
$T$ satisfies $T\ll 1/\gamma_{s}$ [39]. In order to execute the first step, we
make the detuning $\delta_{k}$ between $\ket{s}$ and $\ket{k}$ large enough
such that $\delta_{k}\gg J$, and when this condition is satisfied, the states
$\ket{s}$ and $\ket{k}$ are decoupled from each other [36]. Then, this process
is simply described by the first three equations in Eq. (4) with $J=0$. Given
that the maximum suppression of noise is achieved by tuning the ring cavity to
be in resonance with the signal and to be in anti-resonance with the anti-
Stokes field ($\phi_{s}=0$ and $\phi_{a}=\pi$), it has been shown that the
optimal storage efficiency in the first step is
$\eta_{1}=1-\sqrt{d}\gamma_{e}/(\sqrt{2}\Delta_{s})$ in the strong coupling
regime (more details can be in Sec. III.2) and the far-detuned regime
($\Delta_{s}\gg\gamma_{e}$) without mode mismatch in the cavity [30]. This
efficiency could be achieved when using lossless optical components. The
requirements for all the related parameters can be realized experimentally,
which are discussed in Sec. V. The second step is to transfer the signal
stored in the alkali atoms to the noble-gas spins. Thus, we need to turn off
the control field $\Omega(t)$ and tune $\ket{k}$ on resonance with $\ket{s}$
to make them interact, which can be done using an external magnetic field
[38]. The efficiency of this transfer is maximized when the transfer time is
set to be $\pi/(2J)$ and it is in the strong coherent coupling regime, i.e.
$J\gg\gamma_{s}\gg\gamma_{k}$ [39]. Then, we obtain the optimal transfer
efficiency $\eta_{2}=\text{exp}(-\frac{\pi(\gamma_{s}+\gamma_{k})}{2J})$,
which gives us the total storage efficiency:

$\eta_{s}=\eta_{1}\eta_{2}=(1-\frac{\sqrt{d}\gamma_{e}}{\sqrt{2}\Delta_{s}})\text{exp}(-\frac{\pi(\gamma_{s}+\gamma_{k})}{2J}).$
(6)

Now, we shall see how entanglement can be established in an elementary link.
There are two links illustrated in Fig. 2(b), and here we focus on the first
link for describing how the entanglement generation is achieved. In this link,
for the left node, a single photon emitted from the source after a beam
splitter can be described as $(\alpha a^{\dagger}_{1}+\beta
a^{\dagger}_{2})\ket{0}$ where $\alpha$, $\beta$ are reflection and
transmission amplitudes of a beam splitter, and they satisfy the relation
$|\alpha|^{2}+|\beta|^{2}=1$. The same is true for the right node where the
state of a single photon after a beam splitter is $(\alpha
b^{\dagger}_{1}+\beta b^{\dagger}_{2})\ket{0}$. Thus, the joint state is given
by:

$[\alpha^{2}a^{\dagger}_{1}b^{\dagger}_{1}+\alpha\beta(a^{\dagger}_{1}b^{\dagger}_{2}+a^{\dagger}_{2}b^{\dagger}_{1})+\beta^{2}a^{\dagger}_{2}b^{\dagger}_{2}]\ket{0}.$
(7)

The first term in this state is the case where both single photons are
reflected to be stored in quantum memories, ideally yielding no heralding in
detectors. However, the detector dark counts could potentially lead to
spurious clicks, thus causing infidelity in the desired entangled state. This
probability is given by $\epsilon_{0}(1-\epsilon_{0})\alpha^{4}$ where
$\epsilon_{0}$ is the probability of having no dark counts in detectors. Here,
we take it into account, but later on, we will see that its effect can be
negligible if we choose the detector and detection window time properly. The
second and third terms are the main contributions to single photon heralding
where $a^{\dagger}_{1}$ and $b^{\dagger}_{1}$ are to be stored in quantum
memories. We use noble-gas nuclear spins as quantum memories where the storage
of a single photon is achieved in two steps as described above. As the finite
storage efficiency, $\eta_{s}$ could create vacuum components, we take it into
consideration in this work. The probability of having this contribution is
given by $\epsilon_{0}\alpha^{2}\beta^{2}\eta_{t}\eta_{c}\eta_{d}\eta_{s}$
where $\eta_{t},\eta_{d},\eta_{c}$ are the transmission, detection, and
frequency conversion efficiencies. The last term could also lead to the
single-photon detection event when one of the two photons gets lost in the
transmission, thus creating vacuum components as well. As discussed in Sec.
III.2, although the hindsight from post-selection tells us that the vacuum
components can be eliminated, which seems to have no effect on overall
fidelity, it still could decrease the overall repeater rates. This probability
is given by
$\epsilon_{0}\beta^{4}\eta_{t}\eta_{c}\eta_{d}(1-\eta_{t}\eta_{c})$. Moreover,
we assume that the probability that the single-photon source emits a photon is
$p_{1}$, which depends on the source we use.

Here, we choose to use the same hot alkali gas as a single-photon source,
which can be charged with a single excitation via FWM process used in the DLCZ
protocol [10], and this atomic excitation can then be reverted to emit a
single photon. A few experimental works have been reported for using hot
rubidium atoms to generate bright and indistinguishable photons [44, 45]. In
this way, we do not need to perform frequency conversion to match with the
alkali gas we use in the system, but the frequency conversion is needed for
long-distance communication, i.e. for $a_{2}$ and $b_{2}$. Using atomic
ensembles to generate single photons could lead to multi-photon errors thus
degrading the repeater fidelities. This is discussed in detail in Sec. IV. We
envision using the reverse-proton exchange (RPE) PPLN waveguide technique to
convert a single photon emitted from the source to a telecom photon, which can
operate at room temperature with a conversion efficiency of $23\%$ for the 863
nm signal [46] but it is promising to apply it to the signal of different
wavelengths. Moreover, by choosing the proper waveguide mode filter and fibre
type, one can greatly improve this conversion efficiency to $60\%$ [46], and
we use a higher value of $80\%$ in Sec. IV for rates calculations. Also, we
assume that the relative phase in two optical fibres remains stable.
Practically, this requirement can be achieved by actively stabilizing the
lengths of fibre [8], or through the use of self-compensating Sagnac-type
configurations [47].

After taking all these effects into account, the entanglement generation
fidelity and efficiency of the state created by detecting a single photon in
one of the detectors are given by:

$\displaystyle F_{\text{gen}}$
$\displaystyle=\frac{\alpha^{2}\beta^{2}\eta_{t}\eta_{c}\eta_{d}\eta_{s}}{\beta^{2}\eta_{t}\eta_{c}\eta_{d}+(1-\epsilon_{0})\alpha^{4}-\beta^{4}\eta^{2}_{t}\eta^{2}_{c}\eta_{d}},$
(8) $\displaystyle\eta_{\text{gen}}$
$\displaystyle=2p_{1}(\epsilon_{0}\beta^{2}\eta_{t}\eta_{c}\eta_{d}+\epsilon_{0}(1-\epsilon_{0})\alpha^{4}-\epsilon_{0}\beta^{4}\eta^{2}_{t}\eta^{2}_{c}\eta_{d}),$
(9)

where $\epsilon_{0}=\text{exp}(-\lambda T_{d})$ with $\lambda$ being the dark
count rate, and $T_{d}$ is the detection window time which is set to be the
time duration of the signal, that is $T_{d}=T$. $\eta_{t}$ is the function of
the length of an elementary link $L_{0}$, which takes the following form:
$\eta_{t}=\text{exp}(-L_{0}/2L_{\text{att}})$ with $L_{\text{att}}=22$ km
being the attenuation length for telecom photons. The factor of 2 in the
efficiency expression comes from the fact that the detectors are symmetric,
and the heralding in either of them contributes to the efficiency. We envision
using silicon single-photon avalanche diodes (Si SPADs) [48, 49] and frequency
conversion to detect telecom photons. Si SPADs combined with a monolithic
integrated circuit of active quenching and active reset (AQAR) can enable
detection efficiency as high as $75\%$ with dark count rates below 100 Hz at
785 nm [48]. This type of detector can operate at non-cryogenic temperatures
which only require a thermoelectric cooler. The parameters are taken to be
$\alpha^{2}=0.84$, $\beta^{2}=0.16$, $\eta_{d}=0.6$, $\eta_{c}=0.8$,
$\eta_{s}=0.938$, $T_{d}\sim 12.5$ ns (the signal bandwidth is around 80 MHz,
which is compatible with the hot vapor bandwidth as discussed in Sec. V). In
this regime, the term $(1-\epsilon_{0})\alpha^{4}$ is a few orders of
magnitude smaller than $\beta^{2}\eta_{t}\eta_{c}\eta_{d}$ so Eq. (9) can be
approximately written as $F_{\text{gen}}\approx\alpha^{2}\eta_{s}$, and
$\eta_{\text{gen}}\approx 2p_{1}\beta^{2}\eta_{t}\eta_{c}\eta_{d}$. Moreover,
we can now write the entangled state for each elementary link as

$\alpha^{2}\eta_{s}\ket{\psi_{ab}}\bra{\psi_{ab}}+[\alpha^{2}(1-\eta_{s})+\beta^{2}]\ket{0}\bra{0},$
(10)

where
$\ket{\psi_{ab}}=\frac{1}{\sqrt{2}}(\ket{k_{a}}\ket{0_{b}}+\ket{0_{a}}\ket{k_{b}})$.
The storage inefficiency $1-\eta_{s}$ increases the vacuum component
proportion, and therefore it decreases the repeater rates. Here, we assume the
storage efficiency to be $93.8\%$. Such a high storage efficiency is possible
to achieve, which is discussed in Sec. V. The required input pulse is short as
it satisfies the condition $T\ll 1/\gamma_{s}$, which is also the requirement
for the optimal signal storage in noble-gas spins using the sequential scheme
[39]. Moreover, when we have two elementary links, there is some waiting time
for both links to establish entanglement, and as noble-gas spins offer
ultralong coherence time, the decoherence that happened during the waiting
time is ignored.

###  III.2 Entanglement swapping

After we successfully establish the entanglement in two adjacent elementary
links as shown in Fig. 2(b), we then need to perform entanglement swapping to
propagate the entanglement between A and D. This can be done by recalling the
single photon stored in either quantum memories B or C that are in the same
location, and the heralding at one of the beam splitters informs us of the
success in the swapping process, leading to the entangled state shared between
A and D. At this level, it is well known that the swapping probability takes
the following form [8]:

$P_{1}=\frac{p_{1}F_{\text{gen}}\eta}{2}(2-p_{1}F_{\text{gen}}\eta),$ (11)

where $\eta=\eta_{d}\eta_{r}$ is the product of the detection efficiency and
the retrieval efficiency. Here, the retrieval process happens in two phases as
well. First, we map the excitation in noble-gas spins to the excitation in hot
vapor via the spin-exchange interaction by turning on the magnetic field for
the amount time of $\pi/(2J)$ [36, 39]. Second, we need to read out the signal
from the collective spin state of the hot vapor. In this process, we need to
turn on the control field $\Omega(t)$ and decouple the hot vapor from the
noble gas by applying an external magnetic field to detune $\ket{s}$ from
$\ket{k}$. The efficiency of retrieving the signal from hot vapor is the same
as $\eta_{1}$, and it only holds under the condition that the decoherence of
$\ket{s}$ is negligible during this process, which is true as the decoherence
happens on the time scale much slower than that of memory interactions [39,
30]. Thus, the overall retrieval efficiency is $\eta_{r}=\eta_{s}$, which is
given in Eq. (6).

Now, putting all together, we can further simplify Eq. (11) as
$P_{1}=p_{1}\alpha^{2}\eta_{\text{tot}}(1-\frac{1}{2}p_{1}\alpha^{2}\eta_{\text{tot}})$,
where $\eta_{\text{tot}}=\eta_{s}\eta_{r}\eta_{d}$. If we have more than two
elementary links, the entanglement swapping is nested, which requires higher
levels of swapping. This leads to a more general expression for the success
probability of entanglement swapping at the $i$th level [8]:

$P_{i}=\frac{p_{1}\alpha^{2}\eta_{\text{tot}}}{2}\frac{[2^{i}-(2^{i}-1)p_{1}\alpha^{2}\eta_{\text{tot}}]}{[2^{i-1}-(2^{i-1}-1)p_{1}\alpha^{2}\eta_{\text{tot}}]^{2}}.$
(12)

After the entanglement swapping, a single excitation in noble-gas spins is
shared between two remote locations (in Fig. 2(b), it is between A and D). As
mentioned before, we need to perform post-selection by reading out the stored
photons in each location, which allows us to generate an effective state
$1/\sqrt{2}(\ket{k_{A}k_{D^{\prime}}}+\ket{k_{A^{\prime}}k_{D}})$. Here, the
dark counts are negligible because of the short detection time $T_{d}$ as
mentioned in Sec. III.1. Then, the success probability of performing this
projection is given by [8]:

$P_{ps}=\frac{(p_{1}\alpha^{2}\eta_{\text{tot}})}{2}\frac{1}{[2^{i}-(2^{i}-1)p_{1}\alpha^{2}\eta_{\text{tot}}]^{2}}.$
(13)

This post-selection step enables us to eliminate the vacuum components in Eq.
(10) as it is impossible to detect a single photon on each side if both links
are vacuum. Hence, the overall fidelity is not affected by the vacuum
components in Eq. (10) but as mentioned they have a significant impact on
repeater rates.

In the retrieval process, FWM noise can be strongly suppressed by choosing
$\phi_{s}=0$ (on resonance) and $\phi_{a}=\pi$ (anti-resonance). In the strong
coupling regime, this noise can be quantified by calculating the
$g^{(2)}_{\text{re}}$ function of the retrieved signal, which is equal to
$2|x|^{2}\zeta_{1}|\Gamma_{s}|^{2}/|\Gamma_{a}|^{2}$ [30] when there is no
mode mismatching, and the input signal contains one photon. $x$ is the FWM
noise suppression factor, which is given by

$x\approx\frac{1-\mu_{s}}{2\mu_{s}}=\frac{1-re^{-d(\frac{\gamma_{e}}{\Delta_{s}})^{2}}}{2re^{-d(\frac{\gamma_{e}}{\Delta_{s}})^{2}}}.$
(14)

$\zeta_{1}\gg 1$ is the dimensionless coupling strength between both the
signal and anti-Stokes field and the alkali gas, which is proportional to the
control field strength $|\Omega|^{2}$ and the optical depth $d$ (More details
are discussed in Sec. V). Then, the readout fidelity is given by

$F_{\text{re}}=\frac{1}{1+\text{SNR}^{-1}},$ (15)

where $\text{SNR}^{-1}=g^{(2)}_{\text{re}}/2$ is the signal-to-noise ratio
[30]. Here, we ignore the infidelity that comes from the detector’s dark
counts as the detection window time is assumed to be around $12.5$ ns. Using
the parameters discussed in Sec. V, it is possible to have a readout fidelity
as high as $98.6\%$.

$\mathbf{(a)}$

$\mathbf{(b)}$

Figure 3: (a) Repeater rates as a function of total distance $L$ with
$F_{\text{targ}}=0.9$ for hot hybrid gases-based scheme (referred to as scheme
1), and repeater rates for NV centers and optomechanics-based scheme (referred
to as scheme 2) [19] with corresponding fidelities shown in (b). Here, we plot
4-link case (B), and 8-link case (A) for scheme 2, and we also plot 4-link
case (D), multiplexed 4-link case (C), and 8-link case (E) for scheme 1. The
direct transmission (F) is plotted with a single-photon source of $10$ GHz.
For C, it is multiplexed by a factor of 100. In general, the rates of scheme 2
are much higher than the rates of scheme 1. All these repeaters outperform
direct transmission. We assume $\eta_{c}=0.8$, $\eta_{d}=0.6$,
$\eta_{s}=\eta_{r}=0.938$, $\alpha^{2}=0.84$, $\beta^{2}=0.16$,
$t_{\text{trans}}=1.5$ ms, $t_{\text{ch}}=0.048$ ms and $1.03$ ms for D and E
respectively. The emission probability for single-photon source $p_{1}$ is
assumed to be 0.9 for all repeaters in both schemes. (b) Repeater fidelities
as a function of total distance $L$ for schemes 1 and 2. A and B are 4-link
and 8-link cases in scheme 2 [19]. C stands for a 100-multiplexed 4-link
repeater in scheme 1 with $F_{\text{targ}}=90\%$ and $F_{\text{re}}=98.6\%$.
As a multiplexed 8-link repeater in scheme 1 has fidelities very close to C,
it is not shown here. Scheme 1 yields much higher fidelities than scheme 2,
and they are independent of the total distance.

##  IV Repeater rates and overall fidelities

In our system as the storage and retrieval time are mainly limited by how fast
we can transfer the coherence from hot vapor to noble-gas spins via the spin-
exchange collisions, and these times are given by $t_{\text{trans}}=\pi/2J$,
which is around $1.5$ ms based on the parameters in Sec. V. This transfer time
is on the same order as the two-way communication time $L_{0}/c$ for $L_{0}$
ranging from 50 km to 100 km with $c=2\times 10^{8}$ m/s, which makes the
total length of an 8-link repeater ranging 400 km to 800 km. Furthermore, the
average charging time $t_{\text{ch}}$ in the ensemble also needs to be taken
into account as it is comparable to $t_{\text{trans}}$ both in a four-link
repeater and an eight-link repeater with the final target fidelity
$F_{\text{targ}}=0.9$ as discussed later in this Section. Now, taking
$\eta_{\text{gen}}$, $P_{i}$, and $P_{ps}$ into the standard entanglement
distribution time for the single-photon protocol [8] plus the extra time spent
for retrieving the signal and charging the ensemble, we obtain

$T_{\text{tot}}=\frac{3^{n+1}}{2}(\frac{L_{0}}{c}+t_{\text{trans}}+t_{\text{ch}})\frac{\prod_{i=1}^{n}{(2^{i}-(2^{i}-1)p_{1}\alpha^{2}\eta_{\text{tot}})}}{\eta_{t}\eta_{c}\eta_{d}p^{n+3}_{1}\beta^{2}\alpha^{2n+4}\eta^{n+2}_{\text{tot}}}.$
(16)

where $n$ indicates the number of nesting levels, and the number of links
associated with it is $2^{n}$. Thus, the total length of a repeater is
$L=2^{n}L_{0}$. In Fig. 3(a), we plot 4-link case (D), multiplexed 4-link case
(C), and 8-link case (E) for hot hybrid gases-based scheme (referred to as
scheme 1) with respect to total distance when $F_{\text{targ}}=0.9$ as
discussed below. Also, we plot 4-link case (B), and 8-link case (A) for NV
centers and optomechanics-based scheme (referred to as scheme 2) [19] for
comparison. The direct transmission (F) is plotted with a source of $10$ GHz.
For C, it is multiplexed by a factor of 100, which can be implemented
spatially [50, 8] or spectrally [12] as discussed in Sec. V. All these
repeaters outperform direct transmission at some point but in general, the
rates of scheme 2 are much higher than the rates of scheme 1. The generally
lower rates for scheme 1 are due to the fact that the single-photon protocol
is nested as the entanglement swapping and post-selection are probabilistic as
opposed to the non-nested scheme used in [19], and the interface between
alkali atoms and noble-gas spins is also quite slow, which further degrades
the repeater rates. The other factors that limit the repeater rates in this
proposal are detection efficiency $\eta_{d}$ and frequency conversion
efficiency $\eta_{c}$, which could be improved to further enhance the rates.
We expect an order of magnitude increase in rates when we increase $\eta_{d}$
from 0.6 to 0.9. The slow interface between hot vapor and noble-gas spins also
plays a role in reducing the rates, but the room for improving the speed of
this interface is limited as it is based on weak spin-exchange interactions
[36], which means $J$ cannot be too large. It is worth noticing that it is
much easier to perform multiplexing in hot hybrid gases-based repeaters than
NV centers and optomechanics-based repeaters because the latter requires much
more complex setups than the former [19]. Moreover, there is a trade-off
between the target fidelity $F_{\text{targ}}$ and repeater rates as
$F_{\text{targ}}$ determines $t_{\text{ch}}$. However, the improvement in
rates is not significant when we set a lower target fidelity.

The infidelities in our repeaters mainly come from multiphoton emissions of
the single-photon source and FWM noise in the entanglement swapping and post-
selection. The effect of FWM noise in the signal readout has been estimated in
Sec. III.2 based on the parameters discussed in Sec. V, which gives us a high
readout fidelity of $98.6\%$. In addition, since we use noble-gas spins as
quantum memories, which have hours-long coherence time, thus the infidelity
induced by spin decoherence can be negligible as long as the repeater rate is
above the order of $10^{-3}$ Hz. This is true for multiplexed repeaters and
non-multiplexed 4-link repeaters for $L$ up to 800 km as shown in Fig. 3(a).
Now, the overall fidelity is given by

$F_{\text{tot}}=F_{\text{targ}}\times(F_{\text{re}})^{n+2},$ (17)

where $n+2$ is the number of performing readouts. $F_{\text{targ}}$ is a
target fidelity of repeaters, which we choose to be $90\%$ for all repeaters
in scheme 1 with different nesting levels. This fidelity is determined by
errors due to multiphoton emission in the ensemble-based single-photon source
[8]. The probability of having a two-photon contribution is given by
$p_{2}=2p(1-\eta_{\text{st}})p_{1}$ where $p$ is the probability of emitting
the Stokes photon when charging the ensemble, and $\eta_{\text{st}}$ is the
efficiency of detecting a Stokes photon, assumed to be 0.75 using a silicon
single-photon detector [48, 49]. In order to make $p_{2}$ small enough to have
$F_{\text{targ}}=0.9$, we need to make $p$ sufficiently small. It can be shown
that when we have a four-link repeater, the maximum value that $p_{2}$ can
take is $0.00093$ [8], which leads to $p=0.0021$. This emission probability
results in a charging time given by $t_{\text{ch}}=1/(Rp)=0.048$ ms with the
repetition rate $R=10$ MHz. If we have an eight-link repeater with
$F_{\text{targ}}=0.9$, we obtain $p=9.73\times 10^{-5}$, which leads to
$t_{\text{ch}}=1.03$ ms. Assuming the readout fidelity for both swapping and
post-selection is $98.6\%$, the overall fidelities of a 4-link and 8-link
repeaters in scheme 1 are estimated to be $85.1\%$ and $83.87\%$. In Fig.
3(b), we plot the overall fidelities as a function of total distance $L$ for
scheme 1 and scheme 2. A and B are 4-link and 8-link repeaters in scheme 2
which decrease as total distance increases due to thermal noise present in the
system which are treated as dark counts [19]. C is a 4-link repeater in scheme
1, which is independent of the total distance. In general, scheme 1 yields
fidelities that are significantly higher than the fidelities in scheme 2,
which is mainly due to the fact that the accumulated infidelities induced by
vacuum components are eliminated in the end by post-selection. Overall, these
two schemes have their own advantages and disadvantages. Scheme 1 is much
slower than scheme 2 but has much higher fidelities, and scheme 1 requires
much less complex setups than scheme 2 which also facilitates multiplexing.
Moreover, it is possible to boost the fidelities using entanglement
purification [51], but this comes at the cost of further reducing the rates. A
quantitative discussion of repeaters including purification goes beyond the
scope of the present work.

##  V Implementation

Here, we consider ^39 K atoms as the hot vapor and $\ce{{}^{3}He}$ atoms as
the noble-gas spins in our system, where the optical depth $d$ of the hot
vapor is assumed to be 100, which can be achieved by choosing the length of
the cell given that the temperature of hot vapors is fixed. The linewidth of
the excited state $2\gamma_{e}$ is taken to be $27$ GHz for broadened
$\text{D}_{1}$ line due to collisions with buffer gas, which is much smaller
than the assumed detuning $\Delta_{s}=2700$ GHz so it makes the system in the
far-off resonant regime [30]. Moreover, such a large detuning $\Delta_{s}$
also makes the Doppler broadening negligible which is around 1 GHz at
$230^{\circ}$ C. So far, the experimentally achieved value of $J$ is around
$78$ Hz [38] but if we further increase the pressure to increase the gas
densities, it is possible to have $J=1000$ Hz [36]. In this condition, for
$\gamma_{s}$ and $\gamma_{k}$, they are estimated to be $17.5$ Hz and
$10^{-4}$ Hz respectively, which is dominated by intra-gas and inter-gas
collisional spin-rotation couplings [36, 37, 38]. Thus, this yields an optimal
storage efficiency of around $93.8\%$. Moreover,
$\Delta_{a}=\Delta_{s}+\delta_{s}$ where $\delta_{s}$ is the splitting between
the states $\ket{g}$ and $\ket{s}$, which is around 0.46 GHz in
$\ce{{}^{39}K}$ vapor. In the strong coupling regime ($\zeta_{1}\gg 1$), the
noise suppression factor $x$ is given in Eq. (14), and when the storage and
retrieval efficiencies are optimized, the reflectivity $r$ is given by
$r=(1-\sqrt{1-\alpha^{2}_{s}})/\alpha_{s}$ with
$\alpha_{s}\approx\exp\\{-d(\gamma_{e}/\Delta_{s})^{2}\\}$, which is estimated
to be $93.2\%$. Thus, we obtain the signal readout fidelity $F_{\text{re}}\sim
98.6\%$. The cavity linewidth $\kappa_{c}$ is linked to $r$ and the hyperfine
splitting $\delta_{s}$ as $\kappa_{c}=8\delta_{s}(1-r)/r$, which is estimated
to be $0.27$ GHz. Moreover, in the bad cavity regime, the bandwidth
$\delta_{B}$ of this hybrid quantum memory is upper bounded by the cavity
linewidth as $0.3\kappa_{c}$ [30], which gives $\delta_{B}\sim 80$ MHz. The
size of the ring cavity is given by the length of roundtrip $L=\pi
c/(2\delta_{s})=160$ mm. As for the time-bandwidth product, this hybrid
quantum memory yields an unprecedented value of $8\times 10^{11}$ which is
mainly attributed to the hours-long storage time in the noble gas and the
large bandwidth of the hot vapor. The multiplexing can be implemented either
spatially or spectrally. For spatial multiplexing, we envision having many
hybrid memories in each node [50], which is possible thanks to the mm-scale
system size. The spectral multiplexing also requires many hybrid memories in
one node but the emitted photons need to be converted to different frequencies
fed into a common channel [12, 52]. This can be accomplished using frequency
translation which can be noise-free using waveguide electro-optic modulators
[53]. The feeding to a common channel can be achieved by a tunable ring
resonator filter that enables MHz-level resonance linewidths [54].

##  VI conclusions and outlook

We presented a quantum network architecture based on hot hybrid alkali-noble
gases that can operate without cryogenics. We showed that under realistic
conditions, high-fidelity entanglement can be distributed over long distances
thanks to the ultra-long coherence time of noble-gas spins. We showed that the
rates of our proposed quantum repeaters can outperform direct transmission,
and with realistic multiplexing, the rates can be greatly enhanced, close to
the corresponding rates of NV centers and optomechanics-based repeaters.
Furthermore, our system requires no extra components other than a ring cavity.
This significantly reduces the complexity of the system while offering a great
potential to be scalable, which is more advantageous over the room-temperature
repeaters with NV centers and optomechanics. We hope that this work could
further stimulate the development of high-efficiency silicon single-photon
detectors and even room-temperature detectors that offer both high detection
efficiencies and low dark count rates for telecom photons.

We here have focused on hot atomic gas-based quantum repeaters on the ground,
but this compact hybrid quantum system also offers a good potential for being
used as memory in space [25], which could unlock the possibility of
establishing a truly global quantum network [55, 56, 57] that goes beyond the
limit of terrestrial quantum repeaters, and such a global quantum network
could enable ultra-long distance quantum teleportation, quantum entanglement
and applications in fundamental physics tests [25].

## Acknowledgements

This work was supported by the Natural Sciences and Engineering Research
Council of Canada (NSERC) through its Discovery Grant (DG), CREATE, and
Strategic Project Grant (SPG) programs, and by the National Research Council
(NRC) of Canada through its High-Throughput Secure Networks (HTSN) challenge
program, and by Alberta Innovates Technology Futures (AITF) Graduate Student
Scholarship (GSS) program. In the spirit of respect and reconciliation,
J.-W.J., F.K.A., and C.S. acknowledge that the University of Calgary is
located on the traditional territories of the people of the Treaty 7 region in
Southern Alberta, which includes the Blackfoot Confederacy (comprising the
Siksika, Piikani, and Kainai First Nations), as well as the Tsuut’ina First
Nation, and the Stoney Nakoda (including the Chiniki, Bearspaw, and Wesley
First Nations). The City of Calgary is also home to the Métis Nation of
Alberta (Region 3). K.H. acknowledges that the NRC headquarters is located on
the traditional unceded territory of the Algonquin Anishinaabe and Mohawk
people.

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