# Optimization and readout-noise analysis of a warm vapor EIT memory on the Cs
D1 line

Luisa Esguerra<EMAIL_ADDRESS>German Aerospace Center (DLR),
Institute of Optical Sensor Systems, Rutherfordstr. 2, 12489 Berlin, Germany
TU Berlin, Institute for Optics and Atomic Physics, Hardenbergstr. 36, 10623
Berlin, Germany  Leon Meßner  German Aerospace Center (DLR), Institute of
Optical Sensor Systems, Rutherfordstr. 2, 12489 Berlin, Germany  Institut für
Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, 12489 Berlin, Germany
Elizabeth Robertson  German Aerospace Center (DLR), Institute of Optical
Sensor Systems, Rutherfordstr. 2, 12489 Berlin, Germany  TU Berlin, Institute
for Optics and Atomic Physics, Hardenbergstr. 36, 10623 Berlin, Germany
Norman Vincenz Ewald  German Aerospace Center (DLR), Institute of Optical
Sensor Systems, Rutherfordstr. 2, 12489 Berlin, Germany  Mustafa Gündoğan
German Aerospace Center (DLR), Institute of Optical Sensor Systems,
Rutherfordstr. 2, 12489 Berlin, Germany  Institut für Physik, Humboldt-
Universität zu Berlin, Newtonstr. 15, 12489 Berlin, Germany  Janik Wolters
German Aerospace Center (DLR), Institute of Optical Sensor Systems,
Rutherfordstr. 2, 12489 Berlin, Germany  TU Berlin, Institute for Optics and
Atomic Physics, Hardenbergstr. 36, 10623 Berlin, Germany

(August 27, 2024)

###### Abstract

Quantum memories promise to enable global quantum repeater networks. For field
applications, alkali metal vapors constitute an exceptional storage platform,
as neither cryogenics, nor strong magnetic fields are required. We demonstrate
a technologically simple, in principle satellite-suited quantum memory based
on electromagnetically induced transparency on the cesium D1 line, and focus
on the trade-off between end-to-end efficiency and signal-to-noise ratio, both
being key parameters in applications. For coherent pulses containing one
photon on average, we achieve storage and retrieval with end-to-end
efficiencies of $\eta_{\text{e2e}}=13(2)\%$, which correspond to internal
memory efficiencies of $\eta_{\text{mem}}=33(1)\%$. Simultaneously, we achieve
a noise level corresponding to $\mu_{1}=0.07(2)$ signal photons. This noise is
dominated by spontaneous Raman scattering, with contributions from
fluorescence. Four wave mixing noise is negligible, allowing for further
minimization of the total noise level.

Quantum key distribution protocols [1, 2] harness superposition and
entanglement to ensure secure information transfer between two parties [3].
However, the longstanding bottleneck of long-distance quantum communication
are photonic losses. The quantum repeater (QR) was proposed in Ref. [4] as a
general solution to this problem by dividing a long-distance link into
segments, and making use of entanglement swapping between the nodes to
generate entanglement over the long-distance link. An essential requirement
for QRs for ground- and satellite-based quantum networks [5, 6, 7] are quantum
memories (QM) [8, 9]. These act as interfaces between flying and stationary
qubits and allow for storage of quantum information, ideally in an efficient
and noise-free manner. QMs are being developed in many different platforms,
ranging from ultracold atoms [10, 11, 12] to solid-state systems [13, 14, 15,
16, 17] and warm atomic vapors [18, 19, 20, 21, 22]. The latter constitute
technologically simple and scalable systems, more suitable for use on
satellites, as neither laser cooling, strong magnetic fields nor ultra-high
vacuum are needed for their implementation. Due to a high optical depth, high
efficiencies are possible. However, these types of memories are often
considered to be prone to noise, especially four-wave-mixing (FWM) [23].

Figure 1:  Overview of the experiment. (a) Three-level $\Lambda$-system with
signal (control) laser frequency red-detuned by $\Delta$ from the respective
hyperfine transitions of the Cs D1 line. A pump laser initially prepares the
$F=3$ ground state ($\ket{g}$). Both ground states $\ket{g}$ and $\ket{s}$ are
naturally long-lived. (b) Schematic memory-experiment setup. EOM: electro-
optic modulator; AFG: arbitrary function generator; ATT: attenuator; SOA:
semiconductor optical amplifier; CP1, CP2: calcite prisms; Cs: vapor cell
inside magnetic shielding; $\lambda/2$, $\lambda/4$: wave plates; APD: single-
photon counting avalanche photodiode; AOM: acousto-optic modulator. (c) Signal
and control pulse sequence (not to scale). The signal pulse enters the memory
shortly after the first control pulse and is written into the memory by it,
while the second control pulse reads out the information 80-200 ns later.
(d) Exemplary arrival time histogram of detected photons in a memory
experiment integrated for $60\,$s for an incoming coherent state with a
Gaussian envelope containing $\absolutevalue{\alpha}^{2}=1.0(1)$ photons on
average (blue), for blocked input signal ($\absolutevalue{\alpha}^{2}=0$)
(red), as well as the resulting noise-corrected signal (yellow). The storage
time is $t_{\text{storage}}=140(1)\,$ns as measured from the maximum of the
input signal to the global maximum of the retrieval pulse. The shaded area
shows the detection window of width $t_{\text{max}}-t_{\text{min}}$ used for
further analysis of the end-to-end efficiency $\eta_{\text{e2e}}$ of the
memory setup and the signal-to-noise ratio SNR. We use
$t_{\text{max}}=155\,$ns (black dashed line), being a good compromise between
$\eta_{\text{e2e}}$ and SNR (Inset).

Building secure global quantum communication networks [24] will make it
necessary to pair such memories with bright and efficient single-photon
sources such as semiconductor quantum dots (QDs) [25, 26, 27, 28, 29],
spontaneous parametric down-conversion (SPDC) sources [30, 31, 32, 33], or
sources based on room temperature atomic ensembles [34, 35]. Essential
requirements on compatible quantum memories for this endeavor are high end-to-
end efficiencies $\eta_{\text{e2e}}$, high signal-to-noise ratios (SNR), and
sufficiently long storage times in the millisecond range. In order to
benchmark the memory performance, we use the parameter $\mu_{1}$, defined as
the mean photon number in the input that results in a SNR of 1 in the output
[36]. In Refs. [5, 6], key rates for specific QR configurations are simulated
with realistic memory parameters, such as memory efficiencies of
$\eta_{\text{mem}}=70-80\%$ and storage times of some milliseconds.
Previously, an internal memory efficiency of $\eta_{\text{mem}}=80\%$, without
considering the filtering system [21], and a low noise figure of
$\mu_{1}=0.20(2)$ [22] have been reported. A $1/e$ storage time of
$t_{\text{store}}=430\,$ms for attenuated coherent pulses was also achieved
[20]. Nevertheless, in these state-of-the-art implementations only one
parameter $(\eta_{\text{e2e}},\,\mu_{1},\,t_{\text{store}})$ was optimized at
a time. Especially regarding applications in quantum communication, it is
imperative to find operating conditions which maximize all relevant parameters
simultaneously.
Our work focuses on the simultaneous optimization of $\eta_{\text{e2e}}$ and
$\mu_{1}$. The used EIT $\Lambda$-configuration with copropagating signal and
control is in principle suitable for long storage times above $1\,$ms, which
will be pursued in follow-up experiments. We achieve
$\eta_{\text{e2e}}=13(2)\%$ and $\mu_{1}=0.07(2)$ simultaneously. The internal
memory efficiency reaches $\eta_{\text{mem}}=33(1)\%$. The origin of the read-
out noise is found to be a combination of spontaneous Raman scattering (SRS)
and fluorescence noise, where the former dominates at larger control pulse
energies. Apart from practical applications in long-distance QKD, these
results are of general interest for the community involved in the exploitation
of laser-induced atomic coherence.
The presented hyperfine ground-state memory scheme is based on EIT [37, 38] on
the Cs D1 transition, following the scheme shown in Fig. 1(a), and is, to our
knowledge, the first realization of a warm single-photon-level memory in
cesium on this transition, although there exist realizations using cold atoms,
e.g. Ref. [39]. Cesium offers a higher possible bandwidth, due to a larger
ground-state hyperfine splitting, as compared to e.g. rubidium. Comparatively,
cesium also shows a larger excited state hyperfine splitting, which reduces
the influence of the additional, unwanted excited state. Further advantages of
cesium include its larger mass and longer D1 wavelength, which result in
slower thermal movement and therefore less Doppler broadening, and in less
absorption in optical fibers compared to the Rb D1 line, respectively. As
depicted in the experimental setup in Fig. 1(b), the orthogonal and linearly
polarized signal ($S$) and control ($C$) lasers (ECDLs from Sacher
Lasertechnik) are red-detuned by $\Delta$ from the
$F\,=3\rightarrow\,F^{\prime}=3$ and $F\,=4\rightarrow\,F^{\prime}=3$
transitions respectively, and are offset-locked to a frequency difference of
$\sim 9.2\,$ GHz. A small deviation from the exact hyperfine splitting is
caused by the AC-Stark shift induced by the control laser. Gaussian signal and
control pulses of varying full width at half maximum (FWHM) are generated by
fiber-based electro-optic modulators controlled by arbitrary function
generators. The control pulses are amplified by a semiconductor optical
amplifier (SOA). The SOAs amplified spontaneous emission is suppressed by a
combination of a narrow-band dielectric filter ($1\,$nm FWHM) and a Fabry-
Pérot etalon (free spectral range FSR $=205.5\,$GHz, finesse
$\mathcal{F}=47$). This results in a control laser peak power of
$P_{C,\text{max}}=12.9\,$mW. For single-photon-level measurements the signal
pulses are attenuated with optical filters. To calibrate the photon number,
$10\%$ of the signal pulse ($N_{\text{mon}}$) is monitored on an avalanche
photodiode (APD). The signal and control beams are overlapped using a calcite
prism and sent through the Cs memory cell. Hereby, we put special care into
making signal and control beams co-propagate through the cylindrical vapor
cell ($7.5\,$cm length, $2\,$cm diameter). The cell contains 133Cs and
$5\,$Torr N2 buffer gas, is enclosed inside a $\mu$-metal magnetic shielding
to nominally reduce the oscillating magnetic field in its interior by a factor
of $1000$, and is kept at $T_{\text{cell}}=60\,$°C. We estimate the wavelength
of the spin wave due to the residual angle to be
$\lambda_{\text{sw}}=2\pi/|\vec{k}_{S}-\vec{k}_{C}|\approx 5\,$mm. The control
and signal beam diameters are $109(5)\,\mu$m and $93(5)\,\mu$m FWHM at their
focus at the center of the memory cell, and their Rayleigh length (as for all
beams) extends until the cell’s end. With a dipole moment of $d=2.7\cdot
10^{-29}\,$C$\cdot$m [40] for the Cs D1 transition, this yields a peak control
Rabi frequency of $\Omega_{C}\approx 2\pi\cdot 540\,$MHz. A pump laser on the
$F\,=4\rightarrow\,F^{\prime}=4$ transition, turned on and off using an
acousto-optic modulator (AOM), is used for state preparation in the $F\,=3$
ground state ($\ket{g}$). Its beam diameter is $540(5)\,\mu$m FWHM at its
focus, with a peak power of $P_{\text{pump}}=12.3\,$mW. The measured pumping
efficiency at this configuration is approximately $n_{F=3}/n_{F=4}=80\%$.
To filter the signal from the control pulses we employ a combination of
polarization- and spectral filtering. The former yields a suppression by 7
orders of magnitude for the control light. The following spectral filtering is
accomplished by two Fabry-Pérot etalons (FSR$\,=25.6\,$GHz, $\mathcal{F}=47$),
each of which further attenuates the control laser by $3$ orders of magnitude
within the signal beam path. In total, this amounts to a control laser
suppression by $13$ orders of magnitude. Meanwhile, the signal light is only
attenuated by a factor of $2.5$. At the output, the signal photons are
detected by a single-photon-counting APD (Laser Components COUNT-50C-FC) and
the resulting signal is analyzed using time-correlated single photon counting.

Figure 2:  End-to-end efficiency ($\eta_{\text{e2e}}$) and signal-to-noise
ratio (SNR). $\eta_{\text{e2e}}$ (blue circles) and SNR (green squares) of the
memory as a function of signal pulse width $\Delta t_{S}$ (a), control pulse
energy $E_{C}$ (b), and laser detuning $\Delta$ (c). The values selected for
the measurements are $\Delta t_{S}=25\,$ns, $280(50)\,\text{pJ}\leq E_{C}\leq
560(50)\,$pJ, and $\Delta=2300(100)\,$MHz red detuning from the atomic
resonance, with only the respective parameter being varied.

A single run of the storage-retrieval experiment begins by switching on the
pump laser for $10\,\mu$s to prepare the state $\ket{g}$. After pumping, the
control and signal pulses enter the memory. Hereby, the signal pulse is
delayed for several ns with respect to the control pulse, as depicted in Fig.
1(c). To retrieve the signal photon from the memory, a second identical
control pulse is sent into the cell after $t=200\,$ns. The storage and
retrieval experiment is repeated at a rate $f_{\text{rep}}=1/(11\mu$s) for an
integration time $t_{\text{int}}$ of $60\,$s.
Figure 1(d) shows a typical measured photon arrival time histogram as used for
all further analysis. Shown are measured data for storage and retrieval of a
coherent state containing
$\absolutevalue{\alpha}^{2}=(N_{\text{mon}}\,\sigma)/(f_{\text{rep}}\,\eta_{\text{APD}})=1.0(1)$
photons on average, a noise measurement resulting from storage of a vacuum
state with $\absolutevalue{\alpha}^{2}=0$, arising from a blocked input, and
the noise-corrected signal. The uncertainty of $\absolutevalue{\alpha}^{2}$
arises from the error in the signal/monitor splitting ratio $\sigma$, measured
with a power-meter (Thorlabs PM160), the uncertainty of the APD efficiency
$\eta_{\text{APD}}=0.33(5)$, and the statistical error in the monitored photon
number $N_{\text{mon}}$. The peak at $t=0\,$ns corresponds to the part of the
signal pulse that is transmitted through the memory (leakage). The peak at
$t=140\,$ns corresponds to the signal read out from the memory. As shown in
Figure 1(d), the temporal profile of the retrieved signal peak is distorted by
the memory and shows two distinct maxima. We define the storage time
$t_{\text{storage}}$ as the difference between the first (and highest) of
these maxima-, and the peak of the incoming signal at $t=0\,$ns. The used
storage times were chosen to be short in order to focus on the
$\eta_{\text{e2e}}$ and SNR, but can be extended into the $\mu$s range with
the present setup, and into the millisecond range by increasing the beam
diameters and changing the $\Lambda$-system configuration by exploiting the
magnetic Zeeman sublevels of the hyperfine ground states [20].
Given the temporal envelope of the retrieved signal, the $\eta_{\text{e2e}}$
of the memory setup, including the filtering system, and the SNR at single
photon level are given as

$\displaystyle\eta_{\text{e2e}}$
$\displaystyle=\frac{N_{\text{signal}}-N_{\text{noise}}}{\absolutevalue{\alpha}^{2}\,\eta_{\text{APD}}\,f_{\text{rep}}\,t_{\text{int}}},$
(1) $\displaystyle\text{SNR}_{\alpha}$
$\displaystyle=\frac{N_{\text{signal},\alpha}-N_{\text{noise}}}{N_{\text{noise}}},$
(2)

where $N_{\text{signal},\alpha}$($N_{\text{noise}}$) correspond to the number
of signal(noise) counts. Due to the long integration time, the statistical
error is small and the uncertainty in $\eta_{\text{e2e}}$ is dominated by the
systematic error in the power measurement and the calibration of the single
photon level. For the SNR, as it is measured at single-photon level, the error
bars are proportional to the uncertainty of $\absolutevalue{\alpha}^{2}$. The
$\eta_{\text{e2e}}$ increases with the length of the retrieval time
integration window $t_{\text{max}}$, while the SNR decreases, as can be taken
from the inset in Fig. 1(d). Hence, a compromising value of
$t_{\text{max}}=155\,$ns was chosen. The lower integration limit is always
taken at the minimum before each retrieval peak.

Figure 3:  Noise analysis. (a) Noise detected on the APD, while scanning one
of the etalons used for spectral filtering over two free spectral ranges (FSR)
(shaded area corresponds to one FSR), while keeping the other etalon on
resonance with the signal frequency. Large peaks at $-16.4\,$GHz and
$9.2\,$GHz detuning correspond to leaked control-laser light, and small peaks
close to the signal frequency (and at $25.6\,$GHz detuning) originate from
atomic noise (fluorescence, SRS and FWM). Vertical dashed lines indicate the
signal (blue) and control (red) frequencies and vertical black solid lines
indicate the limits of one FSR.
(b) Exemplary measurement of total noise counts (the two different datasets
were taken on different days) per retrieval attempt as a function of the
energy in the control pulse $E_{C}$ for $\Delta=334\,$MHz (red detuning), and
corresponding fits of the different noise components: fluorescence,
spontaneous Raman scattering (SRS), and four-wave mixing (FWM). (c) Systematic
study of the measured total noise counts and the corresponding fitted noise
components at retrieval as a function of the control laser detuning $\Delta$.
Negative values of $\Delta$ correspond to blue detuning from resonance. The
noise is mainly dominated by SRS and florescence, while FWM is negligible in
this setting.

Figure 2 shows the dependence of $\eta_{\text{e2e}}$ and SNR on signal pulse
width $\Delta t_{S}$ (a), energy in the control pulses $E_{C}$ (b), and
detuning $\Delta$ (c). $\Delta t_{S}\,$ is related to the bandwidth
$\Delta\omega_{S}$ (FWHM) of the pulses that are stored in the memory as
$\Delta t_{S}\,\Delta\omega_{S}=2\,ln(2)/\pi$, for transform-limited pulses.
For the measurements we took $\Delta t_{S}=25\,$ns and $\Delta=2300(100)\,$MHz
as baseline and varied only the respective parameter. $E_{C}$ was hereby
adjusted in a range $280(50)\,\text{pJ}\leq E_{C}\leq 560(50)\,$pJ. In all
three cases, $\eta_{\text{e2e}}$ shows a saturation behavior, while the SNR
also saturates in Fig. 2(a), but reaches a maximum in Figs. 2(b) and (c), and
then decreases again. Therefore, there exists a trade-off between reaching the
maximally possible efficiency, and adding more noise to the retrieval signal.
To describe $\eta_{\text{e2e}}$, the following models were used: The
efficiency as a function of signal pulse width (Fig. 2(a)) is
$\eta_{\text{e2e}}(\Delta
t_{S})={\eta}_{0,t_{S}}\,/\sqrt{1+(\frac{4\,ln(2)}{\Delta
t_{S}\,\Delta\omega_{\text{mem}}})^{2}}$, where $\eta_{0,t_{S}}$ is the
maximal achievable efficiency and $\Delta\omega_{\text{mem}}$ is the bandwidth
of the memory. This model arises from the Maxwell-Bloch equations and
describes how efficiently a pulse of given width $\Delta t_{S}\,$ is converted
into an atomic spin wave, when limited by the available control Rabi frequency
$\Omega_{C}$, which determines the spectral width of the EIT window, and
therefore the memory bandwidth [41]. It is in good agreement with the measured
data and yields ${\eta}_{0,t_{S}}=0.128(7)$ and
$\Delta\omega_{\text{mem}}=220(30)\,$MHz. In Fig. 2(b) we model
$\eta_{\text{e2e}}(E_{C})=\eta_{0,C}\,e^{-a/E_{C}}$, motivated by the
dependence on the control pulse energy of the former model. For the fit
parameters we find $\eta_{0,C}=0.107(7)$ and $a=1.56(9)\cdot
10^{2}\,\text{pJ}$. Due to non-ideal etalon adjustments, $\eta_{0,C}$ is
slightly reduced compared to $\eta_{\text{e2e}}$. The dependence on the
detuning $\Delta$ shown in Fig. 2(c) is modeled by
$\eta_{\text{e2e}}(\Delta)=\eta_{0,\Delta}\,e^{-\alpha(\Delta)}$, where
$\alpha(\Delta)$ is a Lorentzian, reflecting that the memory efficiency is
limited by collision-broadened absorption from the atomic ensemble. These
models are not to be taken as a result from a microscopic theory, but more as
a hint to the underlying physical mechanisms. A full theoretical model is
outlined in Refs. [42, 43, 44, 45, 46, 47] (see 111We are currently developing
an implementation of such a model and plan on publishing the numerical
calculations in a dedicated paper). The models for the SNR are determined as
the ratio between the efficiency- and the noise models, the latter of which
are described in the next section.
To understand the limiting noise sources, a thorough experimental noise
analysis is performed. When the temporal, spatial, and spectral leakage of the
control light is sufficiently suppressed, the three most relevant noise
sources in warm vapor quantum memories are fluorescence, SRS, and FWM [9].
Figure 3(a) shows the frequency dependence of the measured noise count rate,
obtained by scanning the second etalon of the filtering system over two FSR,
each of $25.6\,$GHz, by temperature tuning, while keeping the first etalon on
resonance with the signal frequency. This measurement is taken with blocked
signal laser. Two large peaks at $-16.4\,$GHz and $9.2\,$GHz detuning appear,
corresponding to leaked control laser light. The transmitted control laser
power is negligible when the etalon is in resonance with the signal, i.e.
during storage experiments. Additionally, two smaller peaks close to the
signal frequency appear approximately at $0$ and $25.6\,$GHz detuning. This
indicates that the only relevant noise sources during memory operation are of
atomic origin. In order to attain a better understanding of these, we measured
the number of noise photons $N_{\text{noise}}$ counted on the APD by varying
$\Delta$, while monitoring the control laser frequency using a wavemeter
(Wavelength Meter WS/7-60). Here, $\Delta=0(60)\,$MHz corresponds to the
control transition. At each point, we determined the noise count dependence on
$E_{C}$, yielding a two dimensional analysis.
Figure 3(b) depicts the noise counts as a function of the control pulse energy
$E_{C}$ at $\Delta=334\,$MHz (red detuning). The data can be modeled by a
second-order polynomial plus a saturating component,
$N_{\text{noise}}=bE_{C}^{2}+cE_{C}+dE_{C}/(e+E_{C})$, where the quadratic and
linear contributions account for FWM and SRS respectively, and the saturating
part corresponds to fluorescence noise. For the shown detuning value we find
$b=0(10^{-8})\,(\text{pJ})^{-2}$, $c=4(2)\cdot 10^{-5}\,(\text{pJ})^{-1}$,
$d=7(2)\cdot 10^{-3}\,(\text{pJ})^{-1}$, and $e=16(16)\,\text{pJ}$, indicating
that both fluorescence and SRS noise have a significant contribution, while
FWM vanishes. Since from Figure 2 we see that memory efficiency increases when
operating far detuned, fluorescence can be disregarded here (see Fig. 3(c).
Figure 3(c) depicts the behavior of the noise for varying detuning $\Delta$ at
maximal $E_{C}$. The error bars of the total noise are of statistical origin.
From fitting the different noise components as a function of the control field
energy at each detuning value (as in Figure 3(b)), the corresponding
components fluorescence, SRS, and FWM are determined at each point. Here, the
error bars arise from the fit error of the $E_{C}$ dependence. The error bars
at $\Delta=-526\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ 774\,$MHz
are comparatively large, since the fit hardly converged and FWM was manually
excluded for these data points. The analysis shows that FWM is negligible in
the experiment, SRS yields a constant contribution to the noise, and the
detuning dependence of the total noise arises mainly from fluorescence. The
latter was modeled as a Voigt distribution with a Gaussian component of
$380\,$MHz width at FWHM due to Doppler broadening, and a Lorentzian component
of $920\,$MHz at FWHM, arising from power broadening by the control light and
pressure broadening due to collisions with the buffer gas, with the only free
parameter being the height. This simple model conveys the general trend, but
does not match the data perfectly. At detunings close to zero, a local minimum
appears in the noise data. We presume this minimum to be caused by re-
absorption of the light due to residual spatial miss-match of the beams along
the whole length of the vapor cell. A more detailed model considering this re-
absorption has too many free parameters to fit the number of existing data
points. More investigation will be given to this fact in further experiments
and theoretical calculations. The resulting total noise model
$N_{\text{noise}}(\Delta)=N_{\text{SRS}}+N_{\text{fl}}V(\Delta)+N_{\text{FWM}}$
is then a sum of the constant SRS contribution and the Voigt distribution
resulting from fluorescence. For completeness, we also add the FWM cotribution
to the model. The resulting parameters are $N_{\text{SRS}}=14(1)\cdot
10^{-3}$, $N_{\text{fl}}=7(2)\cdot 10^{-3}$, and $N_{\text{FWM}}=0(1)\cdot
10^{-3}$. It is thus essential to reduce the former two for future
optimization.
Combining the models for $\eta_{\text{e2e}}$ with the noise models yields the
SNR models shown in Fig. 2(a)-(c). These are in reasonable agreement with the
measured data, and confirm the applicability of the former models.
In summary, we have realized, to the best of our knowledge, the first single-
photon-level storage in a warm hyperfine EIT cesium memory on the Cs D1 line
with an end-to-end efficiency of $\eta_{\text{e2e}}=(13\pm 2)\%$, which
translates to an internal memory efficiency of $\eta_{\text{mem}}=(33\pm
1)\%$. These values approach those needed for practical implementations on QRs
[5, 6]. The simultaneously obtained SNR is $14\pm 2$, which can be converted
to a noise level corresponding to $\mu_{1}=0.07(2)$ signal photons. As the
limiting noise sources, SRS and fluorescence are identified. The present
memory is not yet fully optimized for long storage times, but the geometry
with parallel signal and control beams in principle allows for storage times
not limited by motional dephasing, at least in the ms regime. Nonetheless, our
device can already find a use in quantum computation and sensing applications
where deterministic generation of multiphoton states is required [49]. The
used control powers in this experiment were comparatively low. However, much
higher powers on the order of hundreds of mW will be needed in future
experiments, where larger beam sizes are planned, in order to further extend
the storage times. For minimizing SRS noise, the pumping efficiency should be
increased using higher pump laser powers and larger beams. Further noise
reduction is planned by exploiting the Zeeman-substructure of the cesium
hyperfine states and preparing the atoms in the $m_{F}=+4$ sublevel with
$\sigma^{+}$ polarized light, thus suppressing residual FWM by selection
rules, and reducing possible SRS noise contributions [50, 51, 20]. In this new
configuration, storage times of a few milliseconds are within reach. They will
thus be extended to reach the benchmark $t_{\text{store}}=1\,$ms for practical
applications in QRs, by increasing the beam diameter up to the cell diameter,
hence inhibiting atomic diffusion out of the optical interaction volume. By
use of optimal control pulses as in Ref. [21], we will simultaneously boost
the memory efficiency even further. With this, end-to-end efficiencies above
$70\%$ and simultaneously even higher signal-to-noise ratios, as well as ms-
long storage times are within reach, laying the foundations for memory-
assisted, satellite-suited QRs for long-distance quantum communication.

###### Acknowledgements.

The authors kindly thank Gianni Buser for interesting and fruitful
discussions. We acknowledge the support by the German Space Agency (DLR) with
funds provided by the Federal Ministry of Economics and Technology (BMWi)
under grant number 50RP2090 (QuMSeC) and by the Federal Ministry of Education
and Research (BMBF) under grant number 16KISQ040K (Q-ToRX). M.G. acknowledges
funding from the European Union’s Horizon 2020 research and innovation program
under the Marie Sklodowska-Curie grant agreement No 894590. E.R. acknowledges
funding through the Helmholtz Einstein International Berlin Research School in
Data Science (HEIBRiDS)

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embedding of the post title, (2) the average embedding of all the post’s
comments (3) the post’s score, and (4) the number of comments made on the
post. IMDB-B and IMDB-M [133] are two movie-collaboration datasets. Each graph
includes the actors/actresses and genre information of a movie. In a graph,
nodes indicate actors/actresses, while edges indicate if them if they play the
same movie. In REDDIT-BINARY (RDT-B) and REDDIT-MULTI-5K (RDT-M) [133], each
graph denotes an online discussion thread. In each graph, the nodes are the
users, and the edges are the comments between users. COLLAB [133] is a
scientific-collaboration dataset, where each graph is a ego-network of an
researcher, and the label is the field of this researcher.

###  E.4 Bio-chemical Graphs

Biochemical graphs are related to biology and chemistry domains. The Protein-
Protein Interaction (PPI) [137] dataset contains 24 biological graphs where
nodes are proteins and edges are the interactions between proteins. In MUTAG
[138], each graph indicates a nitro compounds, and its label denotes whether
they are aromatic or heteroaromatic. PROTEINS [139] and D&D [140] are also
protein datasets, where graphs indicate proteins, and labels represent whether
they are enzymes or non-enzymes. The PTC [141] and PTC-MR [142] datasets
contain a series of chemical compounds, while the labels indicate whether they
are carcinogenic for male and female rats. NCI-1 and NCI-109 [143] also
consist of chemical compounds, labeled as to whether they are active to hinder
the growth of human cancer cell lines. The Open Graph Benchmark (OGB) molecule
property prediction benchmark [136] contains 8 molecule graph datasets from
different sources, including BBBP, Tox21, ToxCast, SIDER, ClinTox, MUV, HIB,
and BACE. In these datasets, the graphs indicate different types of molecules,
while the labels express their specific properties. For detailed information
of the domains please refer to [136].

##  Appendix F Open-source Implementations

We collect the implementations of graph SSL approaches reviewed in this survey
if there exists an open-source code for this method. The hyperlinks of the
source codes are provided in Table IX.

##  Appendix G Other Graph SSL Applications

For zero-shot expert linking problem in the field of expert finding, COAD
[152] leverages a same-scale contrastive learning framework to pre-train the
expert encoder model. SLAPS [153] integrates a denoising node feature
generation task into a classification model for graph structure learning. SCRL
[154] applies a prototype-based pretext tasks for few-label graph learning.
SDGE [155] is a community detection approach where the model is trained by a
same-scale contrastive learning objective. Aiming to repair program from
diagnostic feedback, DrRepair [156] pre-trains the model by a repaired line
prediction task which is learned with the automatically corrupted programs
from online dataset. C-SWM [157] introduces a same-scale node-level
contrastive learning strategy to train structured world models that simulate
multi-object systems. Sehanobish et al. [158] use a clustering-based auxiliary
property classification task to train a GAT [2] model and regard the learned
edge weights as the edge feature for downstream tasks learned on biological
datasets including SARS-CoV-2 and COVID-19. To learn representations for
medical images, Sun et al. [159] apply a context-based hybrid contrastive
learning model that maximizes patch- and graph- level agreement on anatomy-
aware graphs extracted from medical images. For federated learning [160, 161],
FedGL [162] introduces an auxiliary property classification task to provide a
global view for the local training in clients. Specifically, the pseudo labels
for each node is acquired by a weighted average fusion of clients’ prediction
results in a server.

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|  Yixin Liu received the B.S. degree and M.S. degree from Beihang University,
Beijing, China, in 2017 and 2020, respectively. He is currently pursuing his
Ph.D. degree in computer science at Monash University, Australia. His research
concentrates on data mining, machine learning, and deep learning on graphs.
---|---
|  Ming Jin received the B.Eng. degree from the Hebei University of
Technology, Tianjin, China, in 2017, and M.Inf.Tech. degree from the
University of Melbourne, Melbourne, Australia, in 2019. He is currently
pursuing his Ph.D. degree in computer science at Monash University, Melbourne,
Australia. His research focuses on graph neural networks (GNNs), time series
analyse, data mining, and machine learning.
---|---
|  Shirui Pan received a Ph.D. in computer science from the University of
Technology Sydney (UTS), Ultimo, NSW, Australia. He is currently an ARC Future
Fellow and Senior Lecturer with the Faculty of Information Technology, Monash
University, Australia. His research interests include data mining and machine
learning. To date, Dr. Pan has published over 150 research papers in top-tier
journals and conferences, including the TPAMI, TKDE, TNNLS, NeurIPS, ICML, and
KDD. He is recognised as one of the AI 2000 AAAI/IJCAI Most Influential
Scholars in Australia (2021).
---|---
|  Chuan Zhou obtained Ph.D. degree from Chinese Academy of Sciences in 2013.
He won the outstanding doctoral dissertation of Chinese Academy of Sciences in
2014, the best paper award of ICCS-14, and the best student paper award of
IJCNN-17. Currently, he is an Associate Professor at the Academy of
Mathematics and Systems Science, Chinese Academy of Sciences. His research
interests include social network analysis and graph mining. To date, he has
published more than 70 papers, including IEEE TKDE, ICDM, AAAI, CIKM, IJCAI
and WWW.
---|---
|  Yu Zheng received the B.S. and M.S. degrees in computer science from
Northwest A&F University, China, in 2008 and 2011, respectively. She is
currently pursuing her Ph.D. degree in computer science at La Trobe
University, Melbourne, Australia. Her research interests include image
classification, data mining, and machine learning.
---|---
|  Feng Xia (M’07-SM’12) received the BSc and PhD degrees from Zhejiang
University, Hangzhou, China. He was Full Professor and Associate Dean
(Research) in School of Software, Dalian University of Technology, China. He
is Associate Professor and former Discipline Leader (IT) in School of
Engineering, IT and Physical Sciences, Federation University Australia. Dr.
Xia has published 2 books and over 300 scientific papers in international
journals and conferences. His research interests include data science,
artificial intelligence, graph learning, and systems engineering. He is a
Senior Member of IEEE and ACM.
---|---
|  Philip S. Yu received the Ph.D. degree in electrical engineering from
Stanford University, Stanford, CA, USA. He is currently a Distinguished
Professor of computer science with the University of Illinois at Chicago,
Chicago, IL, USA, where he is also the Wexler Chair in Information Technology.
He has published more than 830 articles in refereed journals and conferences.
He holds or has applied for more than 300 U.S. patents. His research interests
include big data, data mining, data streams, databases, and privacy. Dr. Yu is
a fellow of the ACM and IEEE.
---|---