††thanks: These two authors contributed equally

# Field-deployable Quantum Memory for Quantum Networking

Yang Wang$\dagger$, Corresponding author<EMAIL_ADDRESS>Alexander N.
Craddock  Rourke Sekelsky  Mael Flament  Mehdi Namazi  Qunnect Inc. 141
Flushing Ave, Ste 1110, Brooklyn, NY 11205-1005

###### Abstract

High-performance quantum memories are an essential component for regulating
temporal events in quantum networks. As a component in quantum-repeaters, they
have the potential to support the distribution of entanglement beyond the
physical limitations of fiber loss. This will enable key applications such as
quantum key distribution, network-enhanced quantum sensing, and distributed
quantum computing. Here, we present a quantum memory engineered to meet real-
world deployment and scaling challenges. The memory technology utilizes a warm
rubidium vapor as the storage medium, and operates at room temperature,
without the need for vacuum- and/or cryogenic- support. We demonstrate
performance specifications of high-fidelity retrieval (95%) and low operation
error $(10^{-2})$ at a storage time of 160 $\mu s$ for single-photon level
quantum memory operations. We further show a substantially improved storage
time (with classical-level light) of up to 1ms by suppressing atomic
diffusions. The device is housed in an enclosure with a standard 2U rackmount
form factor, and can robustly operate on a day scale in a noisy environment.
This result marks an important step toward implementing quantum networks in
the field.

##  I Introduction

Quantum information science holds the key to unprecedented information
integration, processing, and distribution capabilities [1, 2, 3]. Similar to
the evolution of the classical internet, where our modern-day applications
were unimaginable in the earliest demonstrations of networking. The “quantum
internet” [4, 5] has the potential to enable revolutionary applications such
as unconditionally-secure secret key exchange [6, 7, 8, 9], distributed
quantum computation [10, 11], enhanced quantum metrology [12, 13], and tests
of fundamental physics [14, 15]. Given the expansive telecommunications fiber
infrastructure, and ease of information processing with photons, fiber
networks are a prime candidate for hosting a global network. However, despite
decades of optimization for digital communication, the transmission loss (>
0.2 dB/km) still presents considerable challenges to quantum networking since
the quantum states cannot be copied or amplified (no-cloning theorem [16,
17]). This constraint fundamentally limits the distances over which remote
parties can be directly networked at a reasonable rate. The concept of a
quantum repeater was theorized [18, 19, 20, 21] to overcome this challenge,
where one divides the long channel into many elementary links and uses pair-
wise entanglement swapping to distribute entanglement between two remote
parties.

For most common quantum repeater schemes, a quantum memory (QM) is a core
enabling device, which allows single photons to be temporarily stored on a
long-lived matter state and retrieved on-demand [22, 23], enabling the storage
of entanglement generated over elementary links. It thus provides an edge in
entanglement swapping (since the elementary entanglement process is
probabilistic) [24, 25], significantly increases the entanglement distribution
rate, serving as a foundation for global-scale quantum networks.

Here, we report the performance of our QM, a self-contained, rackmount
solution optimized towards field deployment. We demonstrate a simple, robust,
and scalable light-matter interface, based on a warm atomic vapor, which is
versatile for a broad range of quantum optics applications. We push the QM
performance while maintaining technical practicality. We package a tabletop
quantum optics experiment with supporting electronic and mechanical systems
into a compact module that fits a standard 2U 19-inch rack system (Fig. 1).
Finally, we demonstrate the robustness of the device in noisy environments
(stray electromagnetic fields, mechanical vibrations, and temperature drifts).

Our device boasts high fidelity retrieval (95%) at 5% storage efficiency and a
long storage time (up to 1 ms). This is consistent with the performance of
well-controlled lab-based setups in a rackmount form factor. The acceptance
bandwidth (typically a few MHz) is readily compatible with other atomic-based
technologies, such as neutral atom quantum computers [26, 27], trapped ions
[28] and NV centers [29]. In addition, the wavelength (795 nm) permits high-
fidelity one-step frequency conversion to telecom bands (e.g., O-/C-) using
nonlinear crystals [30] or atomic media [31]. These features put our QM at an
advantageous position in rapid field deployment and connecting different
quantum nodes at the metropolitan scale ($\sim$ 100 km). We have identified
promising technical routes to upscale the device performances further,
extending the network distance to inter-city and ultimately global level.

Figure 1: Operating principle, implementation and design of the quantum
memory. (a) Energy levels and laser coupling schemes for realizing our QM
based on electromagnetic induced transparency (EIT) with 87Rb. A strong
$\sigma^{+}$-polarized control field (blue) and a weak $\sigma^{-}$-polarized
probe field (orange) couple a $\Lambda$-system composed of F=2 and F=1 atoms.
The single-photon detuning $\Delta$ and two-photon detuning $\delta$ can be
independently controlled. (b) A photo of the device with the memory unit (see
(c)) slid out halfway. An enclosure (cap removed), with a standard 19-inch 2U
rackmount form-factor, hosts the memory unit, the filter unit, and supporting
electronic systems. (c) Schematics of QM’s optical layout, divided into two
physical modules based on the relevant functions. In the memory unit, a photon
carrying quantum information in its polarization degree of freedom is mapped
onto a collective spin state of a warm vapor of 87Rb atoms and later retrieved
using a strong control field. In the filter unit, the recalled photon is
spectrally filtered with two high-finesse etalons to remove excess noise
photons to achieve high fidelity. PBS: polarization beamsplitter; PPR:
polarization plane rotator; GL: Glan-Laser prism; QWP: quarter waveplate; HWP:
half waveplate; BT: beam trap; L: lens; FP: Fabry-Pérot cavity; BD: beam
displacer; FR: Faraday rotator.

The manuscript is structured as follows. In Sec. II we discuss the need and
requirements for a field-deployable quantum memory. Sec. III details our
research and engineering efforts to realize a compact and robust QM with high
performance. The performance of our QM in its final form is benchmarked in
Sec. IV. We conclude in Sec. V and discuss future research directions.

##  II Field-deployable Quantum Memories

QMs are devices capable of storing and retrieving photonic qubits on-demand
[32]. This primary function can be characterized by three parameters: fidelity
$\mathcal{F}$, which evaluates the degradation of the input quantum state of
the photons; storage efficiency $\eta$, defined as the probability of storing
and retrieving photons; and storage time $T$, which assesses the time it takes
for the storage efficiency to decay significantly. For a QM in a quantum
network, these parameters jointly determine the entanglement distribution rate
and scaling over distances [21], and can thus be considered as the “quantum
performance”. However, the real potential and applications of QMs are only
unleashed when they can be successfully integrated into the existing
telecommunication infrastructure, where a further set of metrics apply: fiber-
hub compatibility, which prefers devices low in size, weight, and power
consumption (SWaP); robustness against environmental noises (electromagnetic,
thermal, mechanical); and scalability for mass deployment. We consider these
requirements as the “hardware performance”. A field-deployable quantum memory
serving a large-scale quantum network needs to satisfy both criteria sets
[33].

Different physical systems have been researched to realize a QM, such as
ensembles of atoms [34], single atoms and ions [35, 36], rare-earth-doped
crystals [37, 38], defects in diamonds [39, 40] (and similar structures[41]).
The _quantum performance_ has been brought to high levels in various systems.
For example, a one-hour storage time of classical optical pulses has been
reported in a solid-state system [38] (although such memory has not been
demonstrated to work in the quantum regime). 92 % efficiency can be achieved
in a cold atom ensemble [42]; and >99 % fidelity $\mathcal{F}$ has been
demonstrated [43, 44]. However, from the perspective of _hardware performance_
, the resources and technology required to achieve such high quantum
performance, such as cryogenic cooling, ultra-high vacuum system, and
sophisticated laser cooling and trapping schemes (e.g, optical lattices [34]),
are prohibitive to be made field-deployable and scalable.

Here, we focus on warm atomic-vapor systems, which have shown promising
results in both quantum and hardware performances. Warm vapor systems provide
a simple and robust physical platform that operates at, or above, room
temperature (cryogenic-free and vacuum-free), with access to the rich internal
properties of atoms such as fine/hyperfine structures for useful light-matter
coupling mechanisms. Therefore, warm-vapor systems are prime candidates for
practical quantum optics applications that require robustness, low-SWaP, and
high scalability. Successful products have already been developed with
widespread deployment, such as chip-scale atomic clocks [45], atomic
magnetometers [46], atomic interference gravimeters [47], etc. There have been
a plethora of research activities on warm-vapor based QM, and very promising
memory performances have been demonstrated, such as 87% efficiency [48], 1
second storage time [49] (with classical optical pulses), and close-to-unity
fidelity [50, 51].

Diving into the details of warm-vapor-based QMs, there exist several types of
storage mechanisms: electromagnetic induced transparency (EIT) with near-
resonant fields [52, 53, 54, 55], far-off-resonant Raman transitions [56], and
gradient echo memory (GEM) [57, 48]. Although they each work under very
distinct conditions, the inherent physics and performance can be obtained in a
similar way by modeling a three-level system [58, 25]. While the high quantum
performance of the individual parameters of {$\eta,\mathcal{F},T$} have been
demonstrated in these systems [48, 49, 50, 51], simultaneous high performance
of all three parameters has not been realized. This often-overlooked gap in
the research space is due to the conflicting physical processes which arise
during the optimization of the experimental system. For example, getting high
fidelity requires exquisite filtering of the noise photons due to the strong
control field (an external optical field modifies the optical response of the
atomic medium). This can be achieved by angling the control field in a
$\Lambda$-system [59] or having two fields counter-propagating in a ladder-
system [51, 50]. However, the storage time is severely limited in both of
those approaches, due to either the short spin-wave (SW) wavelength
($\Lambda$-system) or rapid dephasing due to the excited state (ladder-
system). However, since this is often due to the details of a specific
implementation rather than fundamental physics, accessing the complete
{$\eta,\mathcal{F},T$} parameter space is possible.

Here, we will describe a field-deployable QM using EIT for storage in a warm
atomic vapor. We demonstrate optimization of the three critical performance
parameters in one experimental setup, which is then packaged into a rack-
mounted modular system fitting standard fiber hub environments. An innovative
engineering design provides robust performance and stability. The combined
high quantum and hardware performances makes our QM the backbone of a memory-
backed quantum repeater for real-world applications.

##  III technical implementation

In this section, we describe our research and engineering efforts towards a
field-deployable, low-SWaP quantum memory optimized for quantum networking.

###  III.1 Quantum Performance

####  III.1.1 operating principle

Our quantum memory, shown in Fig. 1, is designed to store photonic
polarization qubits that are near-resonant with the $F=1$ to $F^{\prime}=1$
transition of the rubidium D1 line. As is typical for an EIT based quantum
memory[54, 60], a strong control field $\Omega_{c}$, addressing the $F=2$ to
$F^{\prime}=1$ transition of the rubidium D1 line, opens an EIT window under
which a probe photon, $\Omega_{p}$, on two-photon resonance can propagate. By
turning $\Omega_{c}$ off when the probe photon is within the atomic medium, we
coherently map it onto a hyperfine $F=1,2$ SW on Rb atoms. Restoring
$\Omega_{c}$ back on at a later time maps the SW back into a propagating
photon.

In order to store polarization qubits of arbitrary polarization, we separate
the two orthogonal polarization modes ($\ket{H}$ and $\ket{V}$) into two
spatial “rails”. [54, 60] These rails propagate through the Rb vapor in
parallel, each with their own overlapping control field. After exiting the
vapor cell, the rails are re-combined into a single spatial mode to recover
the initial polarization qubit.

####  III.1.2 storage time $T$

Figure 2: Storage time study. (a) We plot $T$ (defined as 1/e decay time of
the storage efficiency) as a function of the probe field beam waist under
different buffer gas (Ne) pressures. The solid lines are from a collective fit
over all data points with two free parameters: Rb atom diffusive constant (D0)
and longitudinal magnetic field gradient. The dashed lines represent the
inferred storage time without a magnetic field gradient. The actual beam waist
in our final QM product is 1.6 mm (vertical dotted line). The experiment is
performed with weak laser pulses as the probe field (< 1 $\mu$W). (b) Semi-log
plot of the normalized memory efficiency $\eta$ as a function of time for an
Rb vapor with 2 Torr Ne buffer gas with different probe beam waists. Solid
lines are exponential fits. (c) Semi-log plot of the normalized memory
efficiency $\eta$ as a function of time for an Rb vapor with 0.65 mm probe
beam waist under different buffer gas (Ne) pressures. Solid lines are
exponential fits.

In a quantum repeater network, the storage time of a QM heavily limits the
total network length [21], and is therefore a critical metric of memory
performance. In our QM, photons are stored in the form of an SW on Rb atoms.
The primary decoherence mechanism of this SW is the atomic free motion and
inhomogeneous magnetic fields [61]. Here we systematically study the
decoherence mechanism and devise corresponding strategies to achieve a $1/e$
storage time of $\sim$ 0.8 ms.

Magnetic fields cause dephasing of the quantum state of atoms by shifting
their Zeeman levels. To suppress the ambient magnetic field, we use a magnetic
shield made of $\mu$-metal material. The Zeeman splitting of Rb atoms connects
the characteristic dephasing time with the magnetic field. A detailed
calculation can be found in Appendix. B. The design of magnetic shielding has
been extensively studied elsewhere [62, 63]. Here, due to space limitations
(height under 2U) and optical access requirements, we design a cylindrical
magnetic shield of two concentric layers with 18-mm diameter center holes on
the end caps for optical access. The ratio between the diameters is optimized
to maximize the shielding effect in the transverse direction. [62] We
construct a Helmholtz coil to measure the shielding performance. However,
within our testing capability, we can only set a lower bound of >30 dB in each
directions, which allows us to achieve > 1 ms storage time (when other
decoherence mechanisms are not the limiting factor).

Since the quantum state of the photon is encoded on the atomic vapor with
finite size, free atomic motion causes decoherence. By co-propagating
$\Omega_{c}$ and $\Omega_{p}$ we achieve a maximum SW wavelength of
$\lambda_{\rm{SW}}$ = 4.3 cm in the longitudinal direction. Transversely, the
SW is defined by the probe field profile. Since the typical beam size (a few
millimeters) is $\ll\lambda_{\rm{SW}}$, the diffusion of atoms in the radial
direction is the primary dephasing mechanism, while the longitudinal,
dephasing is secondary. By adding buffer gas (Ne) to the system, this
diffusion is significantly slowed, with a diffusion constant $D$ determined by
the buffer gas pressure according to $D=D_{0}\frac{P_{0}}{P}$ ($P_{0}$ and
$D_{0}$ defined at 760 Torr). We model the decay of the storage efficiency due
to the diffusion process in Appendix B, which predicts the dependence of $T$
on buffer gas pressure $P_{\rm{Ne}}$ and beam size $w$.

In Fig. 2 (a) we show the measured $T$, defined as $1/e$ decay time of the
retrieved signal, as a function of $P_{\rm{Ne}}$ and $w$, while in (b) and (c)
we show the raw dephasing data with either fixed pressure (2 Torr) or beam
size (0.65 mm). In those cases ((b) and (c)) $T$ scales quadratically with $w$
and linearly with $P_{\rm{Ne}}$, because the radial diffusion is the dominant
dephasing mechanism. In (a) we plot the theoretical prediction of $T$ when
such dephasing is the only source (dashed color lines), which fits the data in
the low $P_{\rm{Ne}}$ and small $w$ regime. However, in the high $P_{\rm{Ne}}$
and large $w$ limit, we see the divergence between the simple theory and the
experimental results, which suggests that the $T$ is now bottlenecked by some
mechanism other than the atomic diffusion. We note that this long-storage time
performance is only achieved when the system is well degaussed, and we measure
slightly different results after each degaussing process.

We attribute this additional dephasing mechanism to a slight magnetic field
gradient in the longitudinal direction. The finite optical access inevitably
compromises the shielding capability along this direction, and the long SW
makes the memory more sensitive to small gradients. This is in contrast to a
cold-atom based approach, where the small size of the atomic cloud (e.g., 100
$\mu$m [42]) makes it possible to actively cancel the residual field precisely
(and gradients). We take this magnetic dephasing into account and plot the
predictions in Fig. 2 (a) as the solid color lines, which matches with
experimental results well. The inferred magnetic field change from the fitting
suggests > 30 dB suppression, which matches with the lower bound of our
measurement.

Despite the imperfect magnetic shielding, we have achieved $T$ = 0.8 ms, which
could support an metropolitan scale quantum network. We anticipate that better
magnetic shielding, particularly in the longitudinal direction, should improve
the storage time to a few ms levels while using modest beam sizes and buffer
gas pressures (high pressure has other negative consequences, explained
later).

####  III.1.3 fidelity $\mathcal{F}$

The memory fidelity $\mathcal{F}$ can be parsed into the operational fidelity
$\mathcal{F}_{\rm{o}}$ and measurement fidelity $\mathcal{F}_{\rm{m}}$ via
$\mathcal{F}=\mathcal{F}_{\rm{o}}\times\mathcal{F}_{\rm{m}}$. Here,
$\mathcal{F}_{\rm{o}}$ refers to the degradation of the quantum state as the
probe field travels through the device without storage, and the measurement
fidelity $\mathcal{F}_{\rm{m}}$ reflects how a finite signal-to-noise ratio
(SNR) affects the quantum state measurement. In our case, the former is an
engineering effort and can be made close to unity (see data in Sec. IV). Thus,
we mainly focus on $\mathcal{F}_{\rm{m}}$, which is directly related to SNR
via $\mathcal{F}_{\rm{m}}=1-\frac{1}{2(1+\rm{SNR})}$ for a single incoming
photon (see Appendix C).

We now focus on the SNR. The signal is directly related to storage efficiency
$\eta$, while the noise is related to the strong control field $\Omega_{c}$.
We observe noise photons due to $\Omega_{c}$ with two distinct origins. The
first is due to $\Omega_{c}$ not involving atomic transitions, and can be
measured when the Rb vapor is removed from the optical path. This noise is
called “technical noise”. The second is due to the interaction between
$\Omega_{c}$ and Rb atoms, and is defined as the excess noise photons when Rb
vapor is put back. We term this “atomic noise”.

In our system, we attribute atomic noise to two atomic processes: spontaneous
Raman scattering (SRS) [64] and four-wave mixing (FWM) [65]. Here, the SRS is
due to control field scattering of atoms in the $F=2$ state. FWM comes as two
consecutive, phase-matched-enhanced SRS due to $F=1$ population, and is
enhanced by the optical depth (OD) and control field strength. We note that
the majority of atomic noise cannot be filtered as they have the same
frequency as the $\Omega_{p}$ field. To reduce (or eliminate) both types of
noise, we leverage a selection rule by using orthogonal circular polarization
for $\Omega_{c}$ and $\Omega_{p}$ field [66, 67], with a near-resonant
$\Omega_{c}$ field to provide effective optical pumping. As a result, we have
observed a lower noise rate compared with the linear polarization scheme [68,
60] under similar conditions.

Technical noise is composed of the strong, narrowband $\Omega_{c}$ photons and
weak, broadband photons due to the processes of generating and delivering the
control light. Here, $\Omega_{c}$ is generated by a diode laser exhibiting
broadband amplified spontaneous emission (ASE) noise, and delivered via
optical fibers in which Raman scattering occurs [69]. In principle, technical
noise can be eliminated without compromising the memory performance.

####  III.1.4 storage efficiency $\eta$

Figure 3: SNR optimization. In the upper panel we plot the storage efficiency
(red circles) and noise (blue circles) under different vapor temperatures with
fixed control power (20 mW). In the lower panel, we show the corresponding SNR
(black circles) normalized to single-photon input and the measured optical
density of the atomic ensemble (green line) at each vapor temperature. The
input probe pulse contains $\bar{n}=2.74(1)$ photons. Error bars represent one
standard deviations in all plots.

For a given EIT-system, there exists an optimal solution of the photon shape
that maximizes the storage efficiency [58]. It has been shown theoretically
and experimentally [70, 71], that this $\eta_{\rm{max}}$ depends on the OD of
the atomic medium, due to the collective enhancement. In a warm-vapor system,
OD can be conveniently adjusted by changing the vapor temperature. Here, in
Fig. 3 we show a typical result of $\eta$ as we increase OD. For each Rb
temperature, we use an iterative approach ([70] and Appendix. A) to find the
input photon shape that gives maximum efficiency. At low OD, the efficiency
increases monotonically. However, at high OD, the efficiency decreases, which
we attribute to the increased dephasing (such as density-dependent Rb
relaxation [72]) and increased absorption [73]. The highest efficiency reaches
7.0(2)%, which is limited by the destructive interference between two
hyperfine excited levels ($F=1,2$) [74].

####  III.1.5 parameter optimization

We now explore the systematic optimization in the {$\eta,T,\mathcal{F}$}
space. The first parameter we consider is the buffer gas pressure, which
affects both $T$ and $\eta$. As shown in Fig. 2, the buffer gas significantly
improves the storage time by suppressing Rb diffusion. However, the collision
with Ne atoms also broadens the transitions to the hyperfine states [75],
which induces stronger destructive interference. This results in a lower
$\eta$. We measure this effect using a $w$ = 1.1 mm probe beam. In Fig. 4, the
storage time and highest storage efficiency are plotted as a function of
buffer gas pressure. $P_{\rm{Ne}}$ = 10 Torr achieves a sufficient balance
between $T$ and $\eta$. Depending on a user’s specific application (e.g.,
short link length and detection efficiency) one can choose the buffer gas
pressure to maximize the overall performance (e.g., higher efficiency but
lower storage time).

Next, we consider $\mathcal{F}$ and $\eta$. The use of orthogonal circular
polarization greatly suppresses the atomic noise by the selection rule [66].
However, as pointed out in Ref [74] it also limits the maximum achievable
$\eta$ due to the destructive interference between two hyperfine excited
levels ($F=1,2$). We have observed a much improved SNR at the cost of a capped
efficiency. One alternative is to use same circular polarization for both
light fields, transforming the hyperfine levels interference from destructive
to constructive, resulting in a high (> 40 %) efficiency [71]. However, in the
parallel polarization scheme used, FWM is no longer turned off by the
selection rule, causing more atomic noise. Moreover, the lack of polarization
filtering mechanism (up to 50 dB) makes it technically costly to remove the
strong control field itself. Consequently, it is challenging to achieve high
fidelity with this scheme.

Finally, $\mathcal{F}$ is a function of $\eta$ and noise, both of which have
similar dependence on OD, see Fig. 3. We measure both parameters as a function
of Rb vapor temperature, and plot the resulted SNR. Since control power
remains fixed, the excess noise is entirely atomic noise (SRS), which scales
with OD, as expected. Below 60 ∘C, both $\eta$ and noise increase with
temperature, but with slightly different rates. Consequently, the highest SNR
(lower panel) is obtained at the cost of sub-optimal $\eta$. We note that in
applications where slightly degraded fidelity is still acceptable, one can
always gain by increasing memory efficiency.

Figure 4: Storage efficiency (red) and storage time (blue) for a $w$= 1.1 mm
$\Omega_{p}$ beam as a function of buffer gas pressure. For each buffer gas
pressure, we optimize experimental conditions (e.g., OD and pulse shaping) to
find the maximum storage efficiency. We note that since $\eta$ is measured at
$t=5$ $\mu$s (instead of $t=0$), it is underestimated for low $P_{\rm{Ne}}$
conditions. The error bars are one standard deviations.

###  III.2 Hardware Performance

Informed by the optimization studies of the quantum performance
{$\eta,\mathcal{F},T$}, we design a QM solution suitable for field deployment,
with an emphasis on achieving high _hardware performance_ without compromising
its quantum performance.

To this end, the QM design follows a _modular_ approach, permitting efficient
multi-parameter optimization and independent upgrades while reducing system
complexity and removing inter-dependence [76]. It is composed of a _memory
unit_ , which hosts the light-matter interface, and a _filter unit_ , which
provides broadband noise suppression. Both units are housed within a 19-inch
2U-rackmount _enclosure_ providing mechanical and electric support.

####  III.2.1 memory unit

The memory unit (Fig.1(b,c)) contains the Rb vapor cell which acts as the
medium for storing and retrieving $\Omega_{p}$ photons. The design objective
is to reproduce the high quantum performance observed in tabletop experiments
within the limited space of the enclosure with commercial off-the-shelf (COTS)
optical components.

A simplified schematic optical layout is shown in Fig.1 (c), with details to
follow. To ensure high quantum performance, we must: 1) recreate the beam
profiles for light-matter interaction; 2) precisely align the probe field to
minimize transmission losses; 3) precisely overlap $\Omega_{c}$ field with
respect to $\Omega_{p}$ to ensure long SW wavelength [59]; and 4) ensure the
polarization filtering can robustly achieved at > 50 dB suppression.
Critically, the memory needs to be polarization agnostic, which is enabled by
using a dual-rail setup (Appendix. A). On a tabletop setup, one can easily
satisfy these conditions. However, on a portable module with limited resources
and spatial constraints, achieving these parameters is non-trivial.

The first step of the design is to determine the beam size, which is closely
related to $T$ as shown in Fig. 2. A compromise is necessary because: 1) COTS
optical components, such as fiber couplers, waveplates, mirrors, crystals, are
limited by preset dimensions and performances (e.g., surface quality), and 2)
there are diminishing returns when using larger beam diameters, due to the
residual magnetic field gradients. Balancing the two, we use $w$ = 1.6 mm (2
mm) for the probe (control) field, at which point the larger beam sizes make
sense only after we improve the magnetic field shielding. Nevertheless, with
$P_{\rm{Ne}}$ = 10 Torr this arrangement ensures a $T$ > 150 $\mu s$ storage
time.

Next, we consider the implementation of the dual-rail system to efficiently
map the two orthogonal polarization components of the $\Omega_{p}$ field onto
two identical spatial rails with the same polarization. COTS beam displacers
are incapable of providing the required performance, where two large-sized
beams need to be created and widely separated to suppress crosstalk. Instead,
we implement a Sagnac-like versatile beam displacement solution (Fig. 1(c))
[77], which creates two rails with large beam sizes and variable separation.
The self-interference pathways ensure perfect parallelness of the rails and no
optical path length difference [77]. These features allow us to efficiently
create and recombine rails, and most importantly, perfectly overlap
$\Omega_{c}$ and $\Omega_{p}$ (which is often an issue using COTS beam
displacers).

Lastly, we finalize the selection of the remaining optical and mechanical
components. By using mirror mounts with vertical drives, we preserve
adjustment capabilities in a minimal footprint. We identify an adjustment-free
polarization plane rotator for applications where a fixed polarization
rotation is needed (such as the dual-rail setup, see Fig. 1). Compared with
half-wave plates, these rotators greatly simplify the optical design. The
polarization filtering requires a high level of spatial uniformity across the
waveplates (recall we have two large-sized, widely separated beams). That is
not a standard specification for COTS components. We screen batches of COTS
components to select for our requirements. With these efforts, we successfully
package the memory unit into a portable module engineered towards long-term
stability. These results are discussed in Sec. IV.

Figure 5: Engineering etalon thermal stability. (a): the thermal response of
an etalon subjecting to a sudden (< 10 minute) and large (6.3 K) environment
temperature change. In each case (one commercial product and three in-house
iterations), we fit the data with an exponential function. (b,c): the
extracted isolation factor $\xi$ and time constant $\tau$ (see text for
details) of the four cases. In each iteration we empirically change the
housing material, mounting scheme, and mechanical assembly to improve the
thermal performance, based on previous iterations feedback.

####  III.2.2 filter unit

High measurement fidelity, $\mathcal{F}_{\rm{m}}$, requires efficient
filtering of excessive noise photons, specifically the technical noise due to
the strong control field $\Omega_{c}$. There exist three types of (technical)
noise photons: the control field $\Omega_{c}$, which is of the order of
$10^{10}$ photon per pulse; the fiber Raman scattering of $\Omega_{c}$; and
the ASE noise from diode lasers. In the memory unit, we use polarization
elements to combine and separate $\Omega_{c}$ and $\Omega_{p}$ with
suppression of $\geq$ 50 dB. The remaining technical noise is further filtered
spectrally in this filter unit.

The filter unit is comprised of two high-finesse monolithic Fabry-Pérot
etalons, which are thermally tuned to be resonant with $\Omega_{p}$ field
(with a coefficient of -2.4 MHz/mK). The temperature of etalons is actively
stabilized via a feedback servo (with 1 mK resolution) by heating the etalon
housing. A stable etalon temperature is essential to its performance in
rejecting noise and transmitting qubits. Such temperature stability can be
achieved with good thermal isolation, such as vacuum [78]. However, due to our
design goal of low-SWaP requirements, we instead engineer the thermal
properties of the etalon housing to achieve high thermal insensitivity in a
vacuum-free, small footprint system.

In an open thermal system (the etalon plus its housing), the environment
(bath) affects the actual temperature of the etalon. The temperature sensor
and the servo create a virtual pivoting point setting a temperature gradient
between the set-point and the environment. A finite-sized object (etalon)
inevitably experiences this gradient shaped by the environment. Therefore, the
_effective temperature_ (which determines the resonance peak locations) of an
etalon changes as the environment changes, even though the thermal servo is
effective. In order to have both short-term and long-term stability, a well-
engineered system should have a weak and slow response to environmental
temperature fluctuations. We characterize this property by probing the step
response of a temperature-regulated etalon subjected to a sudden and large
change in the environment [78]. The effective temperature of an etalon changes
exponentially with a time constant $\tau$, and the relative temperature change
in the steady-state defines an isolation factor $\xi=\Delta
T_{\rm{room}}/\Delta T_{\rm{cavity}}$. A thermal system with large $\xi$ and
long $\tau$ will maintain a stable, effective temperature over fast and long-
term environmental changes.

We tried several iterations of constructing a thermally stable etalon system,
starting from a COTS heated lens tube. In each iteration, we change the
housing material, sensor locations, and heating methods to improve both
parameters, see Appendix A for details. Our results are shown in Fig. 5. To
test the properties in each case, we first bring an etalon to a thermal
equilibrium state. At t = 0, we introduce a large ($\sim$ 6.3 K) and sudden (<
10 minutes) change to the room temperature. The effective temperature of the
etalon is inferred by measuring its transmission peak location. The results
from three in-house iterations and an COTS module are shown in Fig.5 (a),
including exponential fits. The isolation factor $\xi$ and time constants
$\tau$ are shown in (b) and (c). We successfully achieved a high isolation
factor of 58 and a long relaxation time of 63 minutes, effectively creating a
strong low-pass filter, making the etalon system immune to rapid temperature
changes. Considering common HVAC regulated environments (e.g., a fiber hub)
typically exhibit < 1 K change on a 10 $\sim$ 20-minutes time scale, our
etalon system will reliably maintain high transmission (stay on resonance) for
many hours without user intervention, which is critical for field-deployable
quantum applications. A temperature fluctuation induced cavity transmission
variation as low as 1.9% (relative) on a day timescale is achieved with a room
temperature fluctuation amplitude of $\sim$ 0.5 K; see Appendix A for details.
This well-characterized thermal response also warrants the possibility of
further improvement of long-term performance by monitoring the environmental
temperature.

####  III.2.3 enclosure

The stable operation of any quantum optics apparatus relies on a well-
controlled environment (e.g., temperature, pressure, mechanical) and
supporting equipment. Transitioning from a typical lab-environment into the
field, we engineer a general-propose enclosure to provide a stable local
environment and hardware support. This rack-compatible enclosure allows many
modular quantum devices to be integrated into a node for quantum networking.

First, based on the physical size of the memory and filter units, we determine
a form factor for the enclosure to be 19-inch 2U rackmount (Fig. 1). This
comfortably accommodates most free-space optical elements and mounts (up to 1
inch), while maintaining enough vertical space for mechanical assembly. We
then partition the space into quantum optics devices (front) and supporting
electronics (rear). The former is equipped with a passively-dampened internal
rail system bolted to the solid skeleton of the enclosure. The dampening
mechanism isolates vibrations from nearby oscillating equipment like fans
present in rack cabinets at telecommunication hubs, which would otherwise
cause beam misalignment/pointing instability over time. Both memory and filter
units are built on monolithic tempered custom baseplates for mechanical
stability and optical performance. These baseplates can slide on the internal
mounting rails, such that a user can access each unit (e.g., to diagnose)
without opening the enclosure. The units are further shielded from each other
and the environment with internal hoods to regulate the temperature and
airflow across the free-space optical parts.

The supporting electrical components reside in the rear of the enclosure. A
motherboard with interchangeable slots allows a custom circuit board to
control each quantum device (e.g., precision temperature servos) via a Sullins
Edgecard connector under each baseplate. This arrangement is optimized for
easy maintenance and flexible upgrade, which are critical for field deployment
with limited accessibility. This general-purpose electrical system allows the
enclosure to host different quantum devices. The motherboard also provides
Ethernet/USB interfaces with remote access capabilities, which, together with
our software, allows users to monitor, control, and debug devices remotely
through an API. The total power consumption is 15 W with the memory and filter
units installed and in operation.

##  IV Memory Benchmarking

In this section we detail the results of the various characterizations
performed. Details on how each measurement was performed can be found in
Appendix A.

####  IV.0.1 bandwidth engineering

Figure 6: Engineering photon bandwidth. (a) EIT window with different control
field powers. We plot the (normalized) transmissions of a weak probe field (5
$\mu$W) with different $\Omega_{c}$ powers (see labels) as a function of two-
photon detuning $\delta$. The traces are offset vertically for easy
visualisation. (b) The photon temporal profiles with optimal storage
efficiency for different $\Omega_{c}$ powers (see labels). Traces are offset
vertically. (c) We plot the full width at half maximum (FWHM) of the EIT
window (red) as shown in (a), and the FWHM of spectral peak (blue) of the
retrieved photon (after Fourier transforming the temporal traces in (b)) as a
function of the control field power. The EIT window is well captured by theory
(red solid line), and the retrieved photon is limited by the finite switching
time of the control field, which is captured by a simple theory as well (blue
solid line). The experimental data uncertainty is smaller than the symbols.

One benefit of EIT-based QMs is their ability to store photons of varying
bandwidths to support the users’ application. This is realized by adjusting
the EIT window with the control field power, which affects the accepted
spectral components of the atomic medium. In Fig. 6, we show this effect by
measuring the EIT signal and the bandwidths of optimal photons (photon profile
with maximal $\eta$, see Appendix A) for a weak coherent probe field at
various control field powers. The EIT peak width and the optimal photon
bandwidth increase with increasing control power. The linear increase of
0.128(1) MHz/mW in the EIT window width is well explained theoretically as in
the “power-linear” regime [79]. Similarly, we expect the optimal photon
bandwidth to increase linearly with control field power. However, for
technical reasons, $\Omega_{c}$ could not be turned off sufficiently fast,
resulting in the sub-linear dependence shown in the plot, which is explained
by an ad-hoc theory taking this experimental limitation into account (see
Appendix.A). We anticipate that faster optical switching should bridge the
gap. Importantly, we note that changing the photon bandwidth does not affect
other memory performance such as $\mathcal{F},\eta$, and $T$.

####  IV.0.2 stable & high operation fidelity

Figure 7: High operation fidelity. We plot $1-\mathcal{F}_{\rm{o}}$ of both
the memory unit and the filter unit on a day scale under typical experimental
conditions ( <4 K environment temperature changes, 20$\sim$30 % humidity, in a
rack tower). The errors are caused by the transmission performances between
rails, where the amplitude induced part is shown in the upper panel, and the
phase induced part is shown in the lower panel. See main text for details.

Geared towards field deployment, our QM needs to have robust performance
against environmental changes over many hours. Here, we present the results of
the operation fidelity $\mathcal{F}_{\rm{o}}$ for both memory and filter
units. Since our memory stores arbitrary polarization qubits using two spatial
rails within the atomic vapor, $\mathcal{F}_{\rm{o}}$ is related to the
mechanisms affecting the transmission of two rails, which can be broken into
the amplitude and phase.

The fidelity loss due to differential transmission depends on the input state.
Therefore, we consider the “worst case” scenario, in which a specific input
state results in the lowest fidelity. The amplitude can be measured directly,
where we define $T$ as the transmission ratio between two rails ($0\leq T\leq
1$). An analytical solution of fidelity in the worst-case has the form:
$\mathcal{F_{\rm{o}}}=\frac{4\sqrt{T}}{(1+\sqrt{T})^{2}}$, see Appendix. D for
details. To access the phase, we adjust the input state to be an equal
superposition of two modes and measure the output polarization state using a
polarimeter. The parameters of the polarization ellipse (ellipticity and
azimuth angle) are used to calculate this differential phase.

Figure 7 illustrates the measured operation error, defined as
$1-\mathcal{F}_{\rm{o}}$, due to amplitude (a) and phase (b) as a function of
time. These operation errors remain remarkably low: < 0.002 for the memory and
< 0.02 for the filter over many hours. In the memory unit, the dominant source
of operation error is attributed to the creation and recombination of two
widely-separated and large-waist rails. While in the filter unit, it is mostly
due to the birefringence of the etalon optics. We anticipate optical elements
with better specifications and low-stress etalon mounting techniques [80]
should further lower this error in both cases to $\leq 10^{-3}$ level.

####  IV.0.3 $\mathcal{F},\eta$, and $T$

Figure 8: High fidelity memory operation at single-photon level. (a) A
coherent pulse with an arbitrary polarization containing $\bar{n}=2.74(1)$
photons (red peak) is stored and retrieved (blue peak, scaled by a factor of
20). A 200-ns detection window (dashed green lines) is chosen for analyzing
the signal, while a 2-$\mu s$ window far after the retrieval (dashed orange
lines) is chosen to analyze the noise. (b) The storage efficiency $\eta$ and
fidelity $\mathcal{F}$ at single-photon level as a function of the detection
window size.

To measure the single-photon level performance of the memory, we perform
storage and retrieval of a coherent pulse with a random polarization and mean
photon number of order one. Fig. 8 (a) plots a typical result of memory
operation, where the input pulse has an FWHM duration of 218(1) ns (bandwidth:
$2\pi\times 0.770(2)$MHz), and the Rb vapor temperature is 45 ∘C with an
experimentally measured OD of 2.0(1). To evaluate the SNR, we select a
detection window (green dashed lines) of order the input pulse size to analyze
the signal. We then select a larger window (orange dashed lines), taken after
the photon retrieval (where $\Omega_{c}$ remains on), to determine the noise
rate with better photon statistics, which is consistent with the measured
noise under the detection window when the input pulse is blocked, where the
unconditional noise floor is $1.9(1)\times 10^{-3}$ photons per storage trial.
The SNR is then calculated and scaled to obtain the SNR for an input of
exactly one photon. The choice of the detection window size is a trade-off
between the number of successful events and read-out fidelity. We show in Fig.
8 (b) the dependence of efficiency and fidelity on the width of the detection
window. Depending on the details of the applications (e.g., link length,
bandwidth mismatch, detector efficiency/dark count, etc), the user can choose
a suitable window size to optimize specific functions. Using a 200-ns window
that is matched to the input pulse size, we find $\rm{SNR}_{\bar{n}=1}$ =
8.63(44) under polarization-agnostic operation. The corresponding measurement
fidelity is $\mathcal{F}_{\rm{m}}=94.8(2)\%$. For applications with known
input photon polarization state, higher SNR and fidelity can be
straightforwardly obtained by using only one rail, leading to
$\rm{SNR}_{\bar{n}=1}=17.3$. For a generous window size that encloses the
entire pulse, our QM offers > 5% efficiency at $\mathcal{F}$ > 0.9. The above
high fidelity proves our memory works in the quantum regime, even when
compared to a modified threshold for a weak coherent state with non-unity
device efficiency [81, 82]: 85.3% (with mean photon number of one and
including all the optical loss of the device).

We also measure the storage time $T$, see Fig. 9. To demonstrate polarization-
agnostic operation, we plot the (normalized) efficiencies as a function of
storage time for both rails, which are very close to each other. Both rails
exhibit $T>150$ $\mu$s.

##  V conclusion and outlook

Figure 9: Storage time. We plot the normalized storage efficiency $\eta$ as a
function of storage time for the two rails. The colored lines represent
exponential fits, with 1/e decay constants of 180(6) $\mu s$ (left rail) and
157(4) $\mu s$ (right rail), respectively.

To conclude, we have systematically studied and optimized a quantum memory
based on a warm atomic (Rb) vapor fully supported by scalable room-temperature
technologies. We designed and packaged the experimental quantum optics system
into a 2U-rackmount-sized product. The design is robust against
environmentally-induced noises while preserving high quantum performance
($\mathcal{F}\sim 95\%$, $T>160\mu$s, and $\eta>5\%$) for single photons
carrying quantum information of arbitrary polarization. With further diffusion
suppression, the storage time can be increased to 1 ms (demonstrated using
classical-level light).

As the design continues to evolve, we will explore other methods to further
improve {$\eta,\mathcal{F},T$} within warm-vapor-based QMs. First, the storage
time $T$ can be lengthened through several approaches: the longitudinal
magnetic shielding can be improved if the magnetic field direction is
different from the optical path. One way to realize this is to place mirrors
inside of the shielding. The use of an anti-relaxation coated vapor cell could
also help maintain the long storage time with modest-sized beams. However, its
compatibility with the dual-rail configuration remains an open question. After
surpassing the millisecond-level storage time, one will fight higher-order
dephasing mechanisms, where the use of decoherence-free subspace [49] with
Zeeman states provides a promising direction. Second, we consider the storage
efficiency $\eta$. It has been proposed [83] and demonstrated [84, 85, 86, 87]
that an optical cavity outside of an atomic vapor could enhance the light-
matter interaction while, at the same time, suppressing the noise generation,
which yields high $\eta$ without sacrificing $\mathcal{F}$. Third, there are
methods to further suppress the noise to achieve even higher fidelity, such as
the judicious use of “magic detuning” [86].

In parallel to the research agenda, we will also advance the hardware
performance towards building more robust, compact, and cost-effective product,
ready for mass production. We will further explore different approaches to
miniaturization, for example, the use of integrated photonics. Efforts are
also ongoing to design a companion product containing the control field laser
and electronics with a rackmount form factor.

Apart from this product-oriented agenda, we plan to study and optimize the
interfacing of this QM with a Rb vapor-based entangled photon-pair source
[88], currently under development. In this type of experiment, the finite SNR
(e.g., noise floor) of a QM and the heralding efficiency of the photon-pair
source jointly determine the interfacing performance (e.g., nonclassical
correlations). The high SNR (17, single rail) of our QM would require a
heralding efficiency as low as 5.7% to exceed the classical bound (e.g.,
Cauchy-Schwarz parameter $R\geq 1$), which is very doable compared with other
similar experiments [89, 90, 91]. The combination of those two components not
only form the basic building blocks of a quantum repeater, but also patches
the memory to the existing telecom infrastructure for direct use in quantum
networks.

## Acknowledgments

We thank Gabriel Bello Portmann for help with control software development,
Anita Richardson for supporting electronics, and Noel Goddard for manuscript
preparation. This work is supported by the U.S. Department of Energy under
award no. DE-SC0019702.

##  Appendix A Method

General. Our quantum memory is based on electromagnetic induced transparency
(EIT) in a warm 87Rb atom vapor. A strong, $\sigma^{+}$ polarized control
field, $\Omega_{c}$, addresses the ${}^{5}S_{1/2}$ $F=2$ to ${}^{5}P_{1/2}$
$F=1$ transition, while a weak, $\sigma^{-}$ polarized probe field (the
photonic qubit to be stored), $\Omega_{p}$, addresses the ${}^{5}S_{1/2}$
$F=1$ to ${}^{5}P_{1/2}$ $F=1$ transition. We reference the $\Omega_{c}$ field
to the 87Rb transition using saturated absorption spectroscopy. We beatnote
lock the $\Omega_{p}$ field to be detuned from the $\Omega_{c}$ field by
approximately the ground-state hyperfine splitting. For this work we fix the
single photon detuning, $\Delta=-2\pi\times 120\,\mathrm{MHz}$, but vary the
two-photon detuning, $\delta$ according to the specific operation.

The $\Omega_{c}$ field originates from a tapered amplifier (TOPTICA TA pro).
In order to suppress ASE noise from the TA we use a pre-filter comprising of a
narrow-band ($\sim 2\pi$ $\times$ 40 GHz) volume-Bragg-grating (Coherent
ASE-794.98) with up to 70 dB out-of-band suppression, and a flat etalon with a
$\sim 2\pi$ $\times$ 500 MHz bandwidth (LightMachinery OP-7423-6743-2).

The Rb vapor (>99% enriched 87Rb) is contained in a cylindrical glass (quartz)
cell that is 80 mm long and 25.4 mm in diameter. A set of pre-made vapor cells
with different buffer gas (Ne) pressures (2 - 30 Torr) were employed in our
study. These buffer gas atoms homogeneously broaden the Rb resonance lines,
effectively increasing the inverse lifetime of the ${}^{5}P_{1/2},F=1$ state
with a scaling of 9.8 MHz/Torr[75].

We construct a double-layer cylindrical magnetic shields using $\mu$-metal
(Ad-Vance Magnetics, AD-MU-80). Outside the outer layer we use parallel
radiant heat films to regulate the cell temperature while introducing
negligible magnetic field. Outside the heater we have a saturation layer made
of different $\mu$-metal material (Ad-Vance Magnetics, AD-MU-00) meant to
protect the double-layer shields.

The memory unit. For polarization-agnostic operation, we convert two
orthogonal polarization components $|H\rangle$ and $|V\rangle$ into two
spatial modes of the same polarization using a Sagnac-like beam displacing
setup. We then use a polarization plane rotator to rotate the polarization of
one rail by 90∘, making two rails identical. We combine the rails of the
orthogonally polarized control and probe fields using a Glan-Laser (GL) prism.
A quarter waveplate turns the subsequent polarization to $\sigma^{+,-}$ before
these fields interact with a Rb vapor. After the cell a set of waveplates and
a GL are used to filter the control from the probe field, with a suppression
of >50 dB. We then use another polarization rotator on one of the rails and
another Sagnac setup to recombine the probe’s two spatial rails back into its
original polarization form (single rail).

The filter unit. The filter unit is comprised of two monolithic, 0.5-inch
diameter, plano-convex high-finesse Fabry-Pérot etalons (LightMachinery,
custom made) with incommensurate free spectral ranges (FSR). This combination
of high finesse (200 $\sim$ 300), and FSR (13 GHz and 21 GHz), provide >80 dB
suppression at the control field photon and > 40 dB suppression to the
broadband noise. The corresponding linewidths are $2\pi$ $\times$ 40 and
$2\pi$ $\times$ 100 MHz, respectively. With proper mode matching practice we
can routinely achieve >80% transmission for single cavity, and >50% through
the filter unit.

To prevent the back reflection from the second etalon causing interference we
construct a polarization-agnostic optical isolator composed of two calcite
beam displacers with 2.7mm separation, a Faraday rotator and half waveplate.
This provides >40 dB rejection for light propagating backwards.

Memory operation. A typical memory operation starts with 50 $\mu$s optical
pumping with $\Omega_{c}$ field, resulting in a stead-state distribution with
atoms nearly evenly divided among these four states (Fig.1 (c)):
$|2,+2\rangle,|1,+1\rangle,|1,0\rangle,|1,-1\rangle$. Only the last two states
participate in the memory operation.

For storage, a pulse of $\Omega_{p}$ field enters the atomic medium under EIT
created by the $\Omega_{c}$ field. The pulse is “compressed” as it propagates
due to the reduced group velocity inside the Rb vapor. The $\Omega_{c}$ field
is rapidly switched off, mapping this pulse onto a SW. To retrieve,
$\Omega_{c}$ is turned back on, mapping the SW back to the $\Omega_{p}$ field
in a time-reversal manner.

To characterize (diagnose) the QM performance (such as $\eta$ or $T$), we send
a probe pulse that contains a large number of photons >$10^{6}$, and measure
the output using an amplified photodiode (Thorlab APD430A). To obtain the
memory $\mathcal{F}$ at single photon level, we optically attenuate the probe
field so that, on average, each pulse contains of order one photon (a coherent
state). We then use a single photon counting module (EXCELITAS, SPCM-
AQRH-15-FC) with a quantum efficiency of 70% to measure the output. The single
photon events are time-tagged with a time-to-digital converter (qutools GmbH,
quTAU) with 81 ps resolution. The data shown in Fig:8 is accumulated over 5
minutes at a repetition rate of 200 Hz.

Iterative pulse shaping It has been shown theoretically and experimentally
[70, 71] that an optimal photon storage scheme for a three-level system exists
and can be accessed via an iterative approach. We implement this method in our
experiment: we first send in a probe pulse with pre-determined profile; the
retrieved pulse (which may be very different from the pulse we send it) is
measured, which is closer to the optimal shape; and a time-reversed copy of
this pulse is sent to the memory for the next iteration. Three such iterations
typically result in convergence. This iterative process automatically
maximizes $\eta$ since it matches the photon’s spectrum to that of the EIT
medium, which depends on the experimental details such as OD, $\Omega_{c}$,
$\Delta$, and so on.

Deviation between EIT and photon bandwidth The finite turn-off and turn-on
time of control light is responsible for the bandwidth mismatch of retrieved
photon and EIT window Fig. 6. The control light is intensity modulated by a
free space acoustic-optical-modulator (AOM) with a rise time of 100-ns, which
is comparable to the pulse duration. This finite rise time creates an
effective $\Omega_{c}$ that is lower than its peak value, reducing the storage
efficiency of high-frequency spectral components. We model this behavior by
numerically integrating the optical Bloch equation and find satisfying
agreement (blue solid line). We anticipate that with faster switching (e.g.,
using an EOM) of $\Omega_{c}$ the retrieved photon linewidth at high control
field power will increase to eventually overlap with the EIT window, which has
been shown elsewhere [79].

Etalon housing iterations The first iteration is inspired by one COTS solution
(Thorlab, SM1L10HR) we use in our study. It consists of a simple Al-6061 lens
tube embedded in an Al-6061 mount, bolted directly on the baseplate. The lens
tube is temperature stabilized with a ceramic ring-shaped resistive heater at
the end, without any thermal decoupling. Effectively, the heater can affect
the mounting and the environment, or vise-versa. The second iteration adds
isolation between the etalon and the baseplate/assembly by changing the
material of the mount. The same lens tube is used but held in a Delrin block
mount bolted to the baseplate. Because of the mount material change, the
ceramic heater is replaced with a Kapton resistive heating strip. While the
isolation factor improves significantly, the time constant decreases due to
the smaller heat capacity of the thermal bath (Delrin). The third iteration
improves upon both factors by replacing the lens tube with a larger solid
cylinder of Tempered Al-7075 with small apertures. This part is wrapped in
Aerogel and Teflon tape to isolate it from the environment and then held in
place concentrically in a Stainless-Steel 316 armature bolted to the
baseplate. Only the part containing the etalon is temperature regulated. This
provides a more stable bath, better environmental isolation, and solid
mounting to the baseplate with minimal thermal coupling.

Etalon stability In Fig. 10 we show a characteristic etalon operation under a
fluctuating thermal environment. We stabilize the frequency of the probe field
to a Rb reference, and measure the transmission of it through a typical high-
finesse etalon for many hours (upper panel); we also monitor the room
temperature, maintained by an active air conditioning system exhibiting fast
($\sim$ 10 minutes) temperature swings ($\pm$ 250 mK. For a continuous 14-hour
operation the mean cavity transmission is 73.1% with a standard deviation of
1.4%. The well engineered thermal properties of the etalon housing makes the
etalon insensitive to rapid temperature changes.

Figure 10: Etalon thermal stability. Top: cavity transmission. Bottom: room
temperature with fast HVAC fluctuations plus slow drifts. Our cavity is robust
against rapid temperature changes. And a feed-forward style correction based
on the well-characterized cavity response would improve the long-term cavity
performance.

##  Appendix B Spin Wave Dephasing

Multiple processes can cause dephasing, or effective dephasing of the stored
spin wave. In general, the storage efficiency is proportional to the overlap
integral of the initial spin wave, $S(r)$, and that at the retrieval time,
$S^{\prime}(r)$

$\eta\propto\absolutevalue{\int dr\,S^{*}(r)S^{\prime}(r)}^{2}.$ (1)

Here, we consider several dephasing processes and their effect on the storage
efficiency.

###  B.1 Radial Expansion of the Mode Volume

In warm atom systems a major contributor to dephasing is the motion of atoms
after storage. In our experimental geometry the radial size of the SW
(typically a few mm) is significantly smaller than the wavelength of the SW
($\lambda_{\rm{SW}}$ = 4.3 cm). Therefore, radial (as opposed to longitudinal)
motion of the atoms, is the dominant process to consider. Its effect is
calculated here.

When the SW is stored we assume that it has a 2D Gaussian profile identical to
that of the probe beam profile:

$\absolutevalue{S(x,y)}^{2}=\frac{2}{\pi
w^{2}}e^{-2\frac{x^{2}+y^{2}}{w^{2}}},$ (2)

where $w$ is the $1/e^{2}$ radius of the probe field.

After a time, $t$, the transverse spatial profile of the SW is given by

$\displaystyle\left|S^{\prime}(x,y)\right|^{2}=$ $\displaystyle\int\int
dx^{\prime}dx^{\prime\prime}dy^{\prime}dy^{\prime\prime}\left|S(x^{\prime},y^{\prime})\right|^{2}C(x^{\prime\prime})C(y^{\prime\prime})$
(3)
$\displaystyle\delta(x-(x^{\prime}+x^{\prime\prime}))\delta(y-(y^{\prime}+y^{\prime\prime})),$

where

$C(x)=\frac{1}{2\sqrt{\pi D\,t}}e^{-\frac{x^{2}}{4Dt}},$ (4)

is the spatial profile of diffusion from a point source, with a diffusion
constant $D$. Performing the integration, we find

$\displaystyle\left|S^{\prime}(x,y)\right|^{2}$
$\displaystyle=\frac{2}{\pi\left(8Dt+w^{2}\right)}e^{-2\frac{x^{2}+y^{2}}{8Dt+w^{2}}}$
(5) $\displaystyle=\frac{2}{\pi w^{\prime 2}}e^{-2\frac{x^{2}+y^{2}}{w^{\prime
2}}},$

where

$w^{\prime 2}=w^{2}+8Dt.$ (6)

If we assume the expansions imparts no radial phase upon the atoms, then the
storage efficiency is given by

$\displaystyle\eta$ $\displaystyle\propto\frac{4w^{2}w^{\prime
2}}{\left(w^{2}+w^{\prime 2}\right)^{2}}$ (7) $\displaystyle\propto
1-\frac{1}{\left(1+\frac{w^{2}}{4Dt}\right)^{2}}.$

###  B.2 Magnetic Field Gradients

Due to the differing magnetic field sensitivities of the states involved in
the spin wave, magnetic field gradients can cause different parts of the SW to
precess at different rates, and therefore cause dephasing. Here, we calculate
the effect of magnetic field gradients on the SW coherence.

In our system the longitudinal extent of the spin wave is significantly larger
than that in the radial direction. Additionally, the magnetic shielding means
the gradient in the radial direction is significantly smaller than in the
longitudinal direction. Therefore, here we shall only consider a gradient
along the axis of the spin wave. After some storage time we can write the spin
wave as

$S^{\prime}(z)=S(z)e^{-i\Delta(z)t},$ (8)

where $\Delta(z)$ is the position dependent energy shift due to the magnetic
field. If we assume that the magnetic field has a linear gradient

$B(z)=B_{0}+B^{\prime}z,$ (9)

then we can write SW

$S^{\prime}(z)=S(z)e^{-i\left(B_{0}+B^{\prime}z\right)\mathcal{E}_{B}t},$ (10)

where $\mathcal{E}_{B}$ is the shift per unit magnetic field. If we further
assume $S(z)=$constant then the storage efficiency is given by

$\displaystyle\eta$
$\displaystyle\propto\absolutevalue{\int_{0}^{L}\frac{dz}{L}\,e^{-i\left(B_{0}+B^{\prime}z\right)\mathcal{E}_{B}t}}^{2}$
(11)
$\displaystyle\propto\mathrm{sinc}^{2}\left(\frac{B^{\prime}\mathcal{E}_{B}Lt}{2}\right)$

where $L$ is the spatial extent of the spin wave.

##  Appendix C SNR affect on fidelity

The finite signal-to-noise-ratio (SNR) of the memory is a limiting factor in
our quantum memory. Here we calculate how the finite SNR of the memory effects
its fidelity. We assume an noise term of the form

$\hat{\rho}_{n}=\frac{1}{2}\left(\ket{H}\bra{H}+\ket{V}\bra{V}\right).$ (12)

The signal (input photon), $|\psi\rangle=\alpha|H\rangle+\beta|V\rangle$, is a
pure state

$\hat{\rho}_{s}=\ket{\psi}\bra{\psi}$ (13)

the density matrix after the memory would have the form

$\hat{\rho}=A\hat{\rho}_{s}+B\hat{\rho}_{n}.$ (14)

where $A+B=1$, and $A/B$=SNR. This means $A=\frac{SNR}{1+SNR}$, and
$B=\frac{1}{1+SNR}$. The fidelity of the density matrix post memory due to the
SNR is then given by

$\begin{split}\mathcal{F}&=\left(\mathrm{Tr}\sqrt{\sqrt{\hat{\rho}_{s}}\hat{\rho}\sqrt{\hat{\rho}_{s}}}\right)^{2}\\\
&=1-\frac{1}{2(1+SNR)}\end{split}$ (15)

##  Appendix D Differing rail transmission affect on fidelity

Another factor which can affect the fidelity of the memory is the differing
transmission of the two rails. To quantify this effect let us consider an
arbitrary state

$\ket{\psi}=\alpha\ket{H}+\beta\ket{V},$ (16)

where $\alpha$ and $\beta$ are complex numbers normalised such that
$|\alpha|^{2}+|\beta|^{2}=1$. We’ll assume that one polarization experiences
perfect transmission, while the other is imperfect with a real coefficient
$0\leq t\leq 1$, resulting in a state

$\ket{\psi^{\prime}}=\frac{1}{\sqrt{|\alpha|^{2}+|\beta|^{2}t^{2}}}\left(\alpha\ket{H}+\beta
t\ket{V}\right),$ (17)

where we have re-normalized to account for the loss in the $\ket{V}$ mode. The
fidelity, given the loss, is then

$\begin{split}F&=\left|\bra{\psi}\ket{\psi^{\prime}}\right|^{2}\\\
&=\frac{\left(1+\mathcal{B}\left(t-1\right)\right)^{2}}{1+\mathcal{B}\left(t^{2}-1\right)},\end{split}$
(18)

where we’ve made use of the fact that $|\alpha|^{2}+|\beta|^{2}=1$ to
eliminate $\alpha$ and set $\mathcal{B}=|\beta|^{2}$. Differentiating this to
find the extrema

$\frac{\partial
F}{\partial\mathcal{B}}=0\rightarrow\mathcal{B}=\frac{1}{1+t},\frac{1}{1-t}.$
(19)

The second solution is unphysical, while the first solution corresponds to the
state for which the fidelity is a minimum, thus

$F_{\rm{transmission}}\geq\frac{4\sqrt{T}}{\left(1+\sqrt{T}\right)^{2}},$ (20)

where $T=t^{2}$ here is the transmission for the imperfectly transmitted rail.

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