Further author information: Alexander Ling
E-mail<EMAIL_ADDRESS>
# Realizing quantum nodes in space for cost-effective, global quantum
communication: in-orbit results and next steps

Chithrabhanu Perumangatt  Centre for Quantum Technologies, National University
of Singapore, 3 Science Drive 2, S117543, Singapore  Tom Vergoossen  Centre
for Quantum Technologies, National University of Singapore, 3 Science Drive 2,
S117543, Singapore  Speqtral Pte. Ltd, 03-06 73 Science Park Drive Science
Park 1, 118254, Singapore  Alexander Lohrmann  Centre for Quantum
Technologies, National University of Singapore, 3 Science Drive 2, S117543,
Singapore  Speqtral Pte. Ltd, 03-06 73 Science Park Drive Science Park 1,
118254, Singapore  Srihari Sivasankaran  Centre for Quantum Technologies,
National University of Singapore, 3 Science Drive 2, S117543, Singapore
Ayesha Reezwana  Centre for Quantum Technologies, National University of
Singapore, 3 Science Drive 2, S117543, Singapore  Ali Anwar  Centre for
Quantum Technologies, National University of Singapore, 3 Science Drive 2,
S117543, Singapore  Subash Sachidananda  Centre for Quantum Technologies,
National University of Singapore, 3 Science Drive 2, S117543, Singapore
Tanvirul Islam  Centre for Quantum Technologies, National University of
Singapore, 3 Science Drive 2, S117543, Singapore  Alexander Ling  Centre for
Quantum Technologies, National University of Singapore, 3 Science Drive 2,
S117543, Singapore  Physics Department, National University of Singapore, 2
Science Drive 3, S117542, Singapore

###### Abstract

Quantum sources and receivers operating on-board satellites are an essential
building block for global quantum networks. SpooQy-1 is a satellite developed
at the Centre for Quantum Technologies, which has successfully demonstrated
the operation of an entangled photon pair source on a resource-constrained
CubeSat platform. This miniaturized and ruggedized photon pair source is being
upgraded to be capable of space-to-ground quantum key distribution and long-
range entanglement distribution. In this paper, we share results from
SpooQy-1, discuss their relevance for the engineering challenges of a small
satellite quantum node, and report on the development of the new light source.

###### keywords:

Entanglement, satellite, quantum key distribution, Bell’s inequality

##  1 INTRODUCTION

Deployment of quantum technology in spacecraft has been receiving greater
attention over the past decade. Satellites that employ quantum systems in
space have already enabled novel experiments to test fundamental laws of
quantum physics[1] [2], and demonstrated fundamental quantum communication
protocols [3, 4]. These fundamental scientific experiments aside, the
immediate application for space based quantum technologies is the global
exchange of encryption keys via quantum key distribution (QKD). Pioneering
experiments in the field have demonstrated the feasibility of QKD using
satellites[5, 6, 7, 8, 9, 10, 11, 12]. By integrating a satellite quantum node
with large ground-based quantum networks, the first long distance integrated
space and ground quantum network was recently presented[13].

Despite these encouraging developments towards global quantum networks, a
challenge that remains is the prohibitive cost of the technology. While high
costs are already a challenge for commercial ground-based fibre quantum
networks, satellite quantum nodes based on large satellites are orders of
magnitude more expensive. For the space segment, a pathway to lower the costs
is the use of smaller satellites. In particular standardized spacecraft such
as CubeSats may play an important role in future quantum networks. Utilisation
of small spacecraft as quantum nodes has progressed from conceptual design
studies[14] to demonstrations of quantum technology [6]. An important
requirement in both satellite and ground quantum networks is photonic
entanglement. Finding cost-effective ways to generate, control and detect
entanglement is imperative to enable practical quantum networks. The immediate
application for generating and manipulating photonic entanglement on a
satellite is Space-to-ground QKD in which the quantum correlations of
entangled photon pairs are used to generate secure encryption keys, for
example via the well-known BBM92 protocol[15]. Many different implementations
of this QKD protocol have been published, some specifically targeted at space-
to-ground QKD [16, 12, 17]. The implementation of this protocol on a small
spacecraft could significantly lower the costs for future satellite quantum
nodes.

Very recently, we reported the successful operation of an entangled photon
source on board the SPOOQY-1 CubeSat[18]. In this paper, we share the latest
experimental performance of SPOOQY-1 and discuss the next iteration of the
quantum system.

##  2 SPOOQY-1 Mission results

The SpooQy-1 satellite demonstrated quantum entanglement correlations through
a violation of the Bell-CHSH (Clauser-Horne-Shimony-Holt) inequality shortly
after its deployment from the International Space Station on the 19th of June
2019[18]. The satellite featured the second iteration of the Small Photon
Entangling Quantum System (SPEQS-2), which improved upon the correlated source
launched on Galassia [6] by the use of a different nonlinear crystal
arrangement, and added optical components. The SPEQS-2 instrument generates
polarization entangled photon pairs using an optical design in which the axes
of the non-linear crystals (Beta Barium Borate : BBO) are aligned in a
parallel configuration [19]. This gives greater overlap for the orthogonally
polarized photon pairs and offers high quality entanglement without the need
for strong spatial filtering [20]. After splitting the entangled photon pairs
into the signal and idler components, the two photon polarization correlation
is characterized using Liquid Crystal Polarization Rotators (LCPRs),
polarizing beam splitters and a pair of Geiger-mode avalanche photon
detectors. Figure 1 shows the optical layout within the SPEQS-2 instrument and
a cut-out model of the satellite. Experiments on board SpooQy-1 continue as of
February 2021 and the satellite is expected to de-orbit within the next 12
months.

Figure 1: (Left) Optical layout of SPEQS-2. (Right) SpooQy-1 spacecraft with
payload subsystems. The bottom two thirds contain the entangled photon source
and the top third of the spacecraft contains the spacecraft avionics.

(a)

(b)

Figure 2: (Left) Polarization measurement of the polarization entangled photon
pairs produced by the payload. Plot reproduced from 18. (Right) CHSH value
recorded within the first 50 days and 500-600 days after the deployment into
orbit.

The output of a typical entanglement test in orbit is shown in Fig. 2. For
this experiment, the LCPR in the idler path is set to fixed polarization
measurement settings - horizontal, vertical, diagonal and antidiagonal
(H/V/D/A). The LCPR in the signal path is scanned and the number of coincident
detection events is recorded. The number of accidental correlations is
estimated and removed. The expected sinusoidal behaviour ($C\propto
cos^{2}(\theta)$ with $C$ denoting the number of coincidences and $\theta$ the
measurement angle) is observed and fitted to the data. The $S$ parameter of
the CHSH inequality is then estimated from the fit. For the measurement set in
Fig. 2(a), we estimate $S=2.53\pm 0.06$ which is in good agreement with the
pre-launch value recorded after payload assembly[18]. The quality of
entanglement as well as the reliability of the measurement setting are
affected by the environment temperature and it is often necessary to heat the
payload to a temperature range between
$15\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$23\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to achieve satisfactory
entanglement quality. The quality of entanglement was monitored over the
lifetime of the satellite and has not been observed to degrade, as shown in
Fig. 2(b).

In addition to the CHSH value, data collected over the past 595 days include
temperature measurements within the payload and the spacecraft sub-systems and
the detector dark counts for the two single photon detectors (SAP500). The
dark counts are of particular interest, as displacement damage within the
active area of the single photon APDs increases the dark count rate steadily
as a function of total accumulated dose.

Figure 3: (a,b) Dark counts recorded over one orbit plotted as a function of
temperature for 25, 188 and 360 days in orbit. The exponential fits are used
to extrapolate a dark count value at a temperature of
$10\text{\,}\mathrm{\SIUnitSymbolCelsius}$. (c) Dark counts for both detectors
normalized to $10\text{\,}\mathrm{\SIUnitSymbolCelsius}$ as a function of time
spent in orbit. However, it is to be noted that the satellite was deployed 30
days after it was launched into a space environment (International Space
Station).

Since the payload temperature and detector temperature is affected by the
illumination time that changes throughout the year, it was not possible to
record detector dark counts at a specific temperature consistently. Instead,
the dark counts were logged throughout an orbit and plotted as a function of
temperature. An exponential function that describes underlying thermal
excitation of deep-level defects was fit to the dark counts. From the fit the
dark count values at different temperatures were extrapolated. The dark count
values at the beginning of the mission, after 188 days in orbit and after 360
days in orbit are shown in Fig. 3(a) and (b) for the two single photon
detectors in the payload. The extrapolated dark count value for
$10\text{\,}\mathrm{\SIUnitSymbolCelsius}$ is shown in Fig. 3(c) as a function
of days since deployment. The dark counts at this temperature increase by
approximately 170 cps per day for the signal detector and 270 cps per day for
the idler detector. Furthermore, the rate of change of dark counts per day is
plotted against the temperature in Figure 4. Future missions will utilise
cooled detectors to reduce the effects of radiation damage.

Figure 4: Change in the dark counts per day as a function of APD temperature.
Each value for the rate are obtained from the fit given in 3(c).

##  3 Next generation quantum payload

Building on the success of the SpooQy-1 mission, a next generation quantum
payload was designed that is suitable for space-to-ground entanglement
distribution. This section discusses the detailed design of the payload. In
the first section, a QKD model is presented that was used to evaluate key
performance parameters and to inform the design of the system. Next, the
system components are introduced with detailed description of the individual
subsystems. An experimental validation of the key components of the system is
presented to confirm the model while validating the subsystem design and
component choices. Finally, the design of a compact payload prototype is
presented.

###  3.1 Performance modelling and design targets

We begin by introducing the QKD performance model, discuss the validation
steps and the most important findings.

Polarization entangled photon pairs are selected to implement the well-known
BBM92 QKD scheme [21]. The polarization degree of freedom is chosen since
photon polarization is readily preserved in free-space channels and it is
simple and efficient to control and detect. In a first step, entangled photon
pairs, said to consist of an idler and a signal photon, are created on the
satellite. One photon is detected locally on-board the satellite while the
other is transmitted to a ground receiver. The polarization of the signal and
idler photons is measured in two orthogonal bases (e.g. horizontal-vertical:
H-V and diagonal-antidiagonal: D-A). The measurement results of coincident
detection events, i.e. detector ‘clicks’that occur within the same time
window, are correlated if they corresponded to signal and idler photons from
the same pair. Because of these non-classical correlations, once the
spacecraft and the ground station have publicly compared which detector basis
responded at a given time, they share common knowledge of the measurement
results, from which a private encryption key can be extracted. Maximising the
secret key means maximising the number of coincident detection events while
minimizing the number of erroneous coincident detections. These erroneous
event occur due to inherent imperfect entanglement quality, multiple photon
pair emissions within the same coincidence time window, dark counts and
background noise.

We developed a detailed model for satellite-based QKD [22] to estimate
instantaneous/averaged asymptotic/finite key rates and Quantum Bit Error Rates
for different scenarios. The inputs for the model as well as typical values
for our design are given in Table 2. Using the model a number of system trade
off studies were performed.

First, a trade off of brightness (rate of photon pairs generated per second)
and coincidence time window (see Figure 5 (left)) shows that the optimum
achievable asymptotic key rate (for the given input parameters) lies around
70M pairs per second at a coincidence time window of around 1-1.2ns. The
product of the brightness and the coincidence window can be expressed as a
mean photon pair number per coincidence window which in this case lies around
0.07. This is in good agreement with results obtained in long-distance free-
space demonstrations for entanglement based QKD [17].

This optimum exists because accidental coincidences (detection of photons from
two uncorrelated pairs within the same coincidence window) increase
exponentially with the pair production rate rate, while coincident detection
events only scale linearly, a fundamental limitation to the useful brightness
of spontaneously generated entangled photon sources. At a 25% pair to singles
ratio, 70 M pairs per second would lead to 17.5 M photons detected per second
in the space detection setup (4.375 M per detector). This exceeds the count
rates of our time-stamping electronics currently qualified for space, which is
around 8 M/s. However, a 1 ns coincidence time window is achievable given a
detector jitter of around 300 ps for common single photon avalanche diodes.
This implies that time transfer between the space and ground detection setups
must be performed at better than 1 ns and the accuracy with which each
detection event is measured should be at least 0.5 ns or lower.

Second, a trade off of detector dark counts in Figure 5 (right) illustrates a
relative insensitivity of asymptotic key rate to detector dark counts in space
and a strong sensitivity to dark counts on the ground, due to the low chance
of correlating a noise event on the satellite with a detection event on the
ground station, but the high chance of correlating a noise count on the ground
with one on the spacecraft. The latter conclusion also applies to limiting
background light, which can be done through spectral filtering (limited by
SPDC bandwidth), temporal filtering (coincidence time window limited by
detector jitter) and spatial filtering (reduction of field of view of
receiving telescope is limited by pointing performance of the telescope and
tracking mount). This becomes a key performance requirement for the optical
ground station. On the spacecraft dark counts should be limited such that they
do not impact the performance of the system, including time-stamping capacity.
Radiation damage will increase dark counts, probably more severely than on
SPOOQY-1 due to the higher altitude, near-polar orbit of useful QKD
satellites. Mitigation techniques include cooling of the detectors and thermal
annealing of the active area of the detectors. Both options are planned for
the next spacecraft.

Another consideration is the system Quantum Bit Error Rate (QBER), which
comprises contributions from the polarization visibility of the source, multi-
photon pair events, dark counts, noise counts, polarization reference frame
error and polarization state mixing introduced by elements in the optical
link. From an engineering perspective, it is useful to manage this through a
QBER budget, where individual contributions must be allocated. To set a QBER
limit, instead of considering the asymptotic key rate, we can evaluate the
effect of finite raw key sizes on the final secret key. The model calculates
the finite key size for the total raw key accumulated during a pass following
a recent security bound for the BBM92 protocol [16]. First, the size of the
raw key (i.e. sifted coincidences, before error correction and privacy
amplification) that can be created during one ideal pass is shown in Figure 6
(left). Achievable raw key sizes are between 100 and 300 k bits per pass for a
brightness between 25 and 75 M pairs per second at a coincidence time window
of 1 ns. Figure 6 (right) shows that the total QBER should be limited to below
about 8% to get a positive finite key and to between 1-2% to lose only 50%
during finite key extraction. From an operational perspective there is always
a possibility of accumulating raw key over multiple passes, which can mitigate
higher average QBERs. A tentative average QBER target can be set at 5%,
although during operations the average QBER can possibly be optimized by
discarding sequences with high QBER. The finite key size, as well as
instantaneous asymptotic key rate and QBER that are achievable within one pass
are shown in Figure 7. A QBER budget with targets for the individual
contributions is shown in Table 1.

Figure 5: (Left): Maximum asymptotic secret key rate for brightness and
coincidence time window during a satellite pass straight over an optical
ground station in good weather conditions at nighttime. (Right): Maximum
asymptotic secret key rate for detector dark counts in space and on the ground
during a satellite pass straight over an optical ground station in good
weather conditions at nighttime. The brightness was set at 25MCps. Because of
the high singles rates on the spacecraft the system is not sensitive to dark
counts in space. On the ground, however, detector dark counts need to be
limited. Input parameters as in Table 2.

Figure 6: (Left) Raw key created during one ideal pass. (Right): Finite key
size for a given raw key size and average QBER as calculated from Appendix A.2
in [16]. The security parameter $\epsilon_{cor}$ (the probability that Alice
and Bob’s key are non-identical) was set as $10^{-10}$ and $\epsilon_{sec}$
(the probability that the adversary’s total quantum information is
statistically distinguishable from the ideal output state) was also set as
$10^{-10}$. Other parameters as in Table 2.

Figure 7: (Left) Finite key size for one pass (same parameters as Figure 6), but the instantaneous QBER is limited to 5% (so the average QBER is actually slightly below this). (Right) QBER and asymptotic key rate during one pass for a brightness of 25Mcps and a coincidence time window of 1ns. Other parameters as in Table 2. Table 1: QBER budget for a space-to-ground QKD system. QBER contribution | Effect on QBER | Target
---|---|---
Source visibility: QBER=(1-VIS)/2 | Constant | 1%
Detector dark counts (space) | Varies with coincidence rate | Limit dark counts to <100kcps
Detector dark counts (ground) | Varies with coincidence rate | Limit dark counts to <1kcps
Background light | Varies with coincidence rate | Limit background light to <1000cps
| Total noise counts
---
(dark counts and background counts)
Varies with coincidence rate | <1% at closest approach
| Multi-photon pairs in one
---
coincidence time window
leading to an accidental coincidence
Constant |  | <1.5%, but optimize for this
---
by choosing ideal brightness
| Polarization reference frame rotation
---
correction error
Constant | 0.1%
Polarization state mixing | Constant | 0.1%
Total QBER |  $\sim$3.5% at closest approach, $\sim$5% on average

###  3.2 Description of sub-systems

The quantum payload consists out of an entangled photon source, a detection
unit (Alice), control electronics and software, an optical interface and
conventional spacecraft interfaces. This is illustrated in Figure 8. The
source is based on parametric down conversion producing non-degenerate pairs
of photons, signal and idler, entangled in the polarization degree of freedom.
Figure 9 describes the optical layout of the quantum payload. Within the
quantum payload the signal (785 nm) and idler (837 nm) photons, are separated
using a dichroic mirror. The signal is routed to the telescope payload using
an interface mirror which transmits it to the ground station while the idler
photons are measured using a 4-channel polarization detection unit that
projects incoming photons at random into the H/V or A/D bases. To monitor the
efficiency of the coupling of entangled photons, as well as the quality of the
generated entangled state, we employ a “health check detector” analysing the
polarization state of a small portion of the signal photons. Along with the
entangled photons we are transmitting a timing beacon laser that will be used
for time synchronization.

Figure 8: Physical architecture of the quantum payload. Figure 9: Schematic of
the quantum payload. LD- laser diode, pp - prism pair, LCPR - liquid crysat
polarization rotator, PBS - polarizating beam splitter, DM -B beacon dicroic
mirror, L- lens, B- beacon laser diode, QWP- quarter wave plate, HWP- half
wave plate, LB- beacon lens, $YVO_{4}$ \- yttrium orthovanadate , BBO - beta
barium borate,DM- dichroic mirror, LP- long pass filter, LEPD- lateral effect
photodiode, LC - coupling lens, SMF- single mode fibre,LOUT- output lens, BS-
beam splitter, ACHWP- achromatic half wave plate and H,V,D,A, $\theta$ \-
APDs.

####  3.2.1 Source

Entanglement in polarization is achieved by superposing orthogonally polarized
SPDC photons generated using two pump decay paths [23]. Most designs for high
performing entangled sources make use of a Sagnac interferometer. In this
case, the phase stability of the entangled state generated is related to the
alignment stability of the mirrors used in the interferometer with counter
propagating pump beams. Moreover, Sagnac designs suffer from a relatively
large footprint that is required for the interferometer. Therefore, instead of
the common Sagnac source, we use a liner displacement interferometer based
design in which two displaced pump beams co-propagate inside a second order
non-linear crystal. Such linear designs can be more compact and are
particularly easy to align [24].

The source concept is briefly reiterated here. The pump beam of diagonal
polarization is split into a horizontal and vertical beam using spatial walk
off from a birefringent crystal. The horizontally polarized pump beam is
converted to vertical polarization using a half wave plate. A periodically
poled potassium titanyl phosphate crystal kept at a constant temperature
inside an oven is used as the second order non-linear crystal for SPDC. The
crystal is oriented such that the vertically polarized pump photon with 405 nm
wavelength is down converted to two vertically polarized daughter photons with
wavelengths 785 nm (signal) and 837 nm (idler). The polarization of one of the
pump decay path is converted to horizontal and the two decay paths are
superposed using another birefringent crystal of appropriate length.

Propagation of SPDC photons through the birefringent crystal can a introduce
wavelength dependent phase between the orthogonal states. This can reduce the
entanglement visibility which in turn increases the quantum bit error rate. To
compensate for the wavelength dependent phase a Yittrium Orthovanadate (YVO4)
crystal is introduced. We also introduce a pre-compensator crystal in the pump
beam path, with which the phase introduced by different pump wavelength is
always constant. This is helpful to ensure that the generated state is
resilient to the changes in the pump laser wavelength that are generally
associated with volume holographic grating laser diodes. A liquid crystal
retarder (LCR) is used to control the relative phase between horizontal and
vertical pairs.

The pump beam is removed using a dichroic mirror and a long pass filter. The
SPDC photons are coupled into a single mode fiber. Coupling the entangled
photons into a single mode fiber is the most challenging part of the payload
development. To minimize the footprint of the instrument, as well as to avoid
mirror mounts with spring mechanisms, we use lens translation to couple the
beam into the fibre. L2 is translated along the beam path (z direction) while
L1 and LC1 are translated perpendicular to the beam propagation direction (X
and Y). The distance between L1 and L2 changes the pump focus at the crystal.
The pump and collection focus are optimized in order to optimize the
efficiency of collection as well as the brightness [25], with pump waist sizes
around $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and a collection waist of
$55\text{\,}\mathrm{\SIUnitSymbolMicro m}$. We use a single mode fiber of
short length (100 mm) with collimators attached to both ends (LC2 and LOUT).

The source also includes the timing beacon that needs to be transmitted along
with the quantum signal. This is a 785 nm pulsed laser with frequency between
1 kHz to 50 kHz (tunable) and a pulse width of 5 ns. A portion of this laser
is transmitted through the dichroic mirror (DM2) causing a 4-fold coincidence
on the on-board detection system. Most of the beacon is transmitted along with
the signal photons to the ground station. The 4-fold coincident clicks
corresponding to the beacon can be extracted directly from the timestamps and
are used to correct any time drifts (Doppler or clock drifts) between the
satellite and the ground stations by cross correlating beacon events. This
correction is fast and robust up to clock drifts of several
$100\text{\,}\mathrm{\SIUnitSymbolMicro s}$.

####  3.2.2  Detection unit

One photon of the entangled pair is detected within the satellite. A Dichroic
mirror (DM2) transmits the signal and reflects the idler photons. The
detection setup is a standard 4-channel polarization measurement device. A
50:50 beam splitter (BS) selects the measurement basis (H/V or D/A) at random
while an achromatic wave plate (ACHWP) and polarizing beam splitter (PBS)
project the photons into two orthogonal polarization states for each basis
choice. A lens (L2) is used to focus the idler photons to the active area of
the detectors. Free space detectors with an active area of 180 $\mu m$ with
in-built thermo-electric cooler are used. The detectors are actively quenched
using home-built fast quenching circuits with a dead time of less than 100 ns.
The photon arrival time is recorded by a time stamp with resolution better
than 0.5 ns.

A part of the signal that is reflected from DM2 is sent to the health check
detector. A 90:10 beam splitter is used for transmitting a part of the signal
to the health-check detector. The coincidence counts between this detectors
and the other 4 detectors measure the pair to singles-ratio and source
visibility (QBER). The error signal produced from the coincidences is used to
maintain an appropriate phase of the quantum state by applying a corresponding
voltage to the LCR. Additionally, the coincidence rates are used to control
the laser power and the temperature set points of the thermal control loops of
the laser and the nonlinear crystal.

####  3.2.3 Optical interface

The quantum payload is designed such that the payload will be mounted back to
back with the optical terminal payload. This free-space optical interface
between the payloads will involve an interface mirror reflecting the signal
and the beacon from the quantum payload to the optical terminal payload. The
1/$e^{2}$ diameter beam diameter of the quantum and synchronization beams is
3.5 mm.

###  3.3 Validation of key rate model and design

Figure 10:  Table top verification of the link model. The filled squares are
the experimentally obtained data and the lines are from the model. Variation
of (a) heralding efficiency for signal and idler photons (b) Detected
coincidence (c) variation of quantum bit error rate and (d) estimated secret
key rate are plotted against the introduced channel loss.  Figure 11:  A
simulated satellite pass and the performance of the table top system. The
black and blue lines are the measured and predicted sifted key. Similarly, red
and green curves are the measured and predicted QBER. Initial parameters:-
source QBER: 0.9%, average efficiency : 22%, generated pair rate: 34 Mcps,
Coincidence window: 1.5 ns.

We implement the above mentioned experimental setup for QKD on an optical
table. The key rate calculation as well as the optical design were
experimentally validated using this table-top QKD system comprising a
simulated satellite, including the source, the synchronization beacon, the
self-check and the measurement system, and a receiver unit. Between the
simulated satellite and the ground station, a variable attenuation is
introduced to simulate link losses. We regulate the dark count levels,
brightness of the source and coincidence window (software) in order match
realistic satellite operating conditions. While most of the optical components
are representative of the final payload, we use off the shelf components for
control, detection and timing electronics. One critical difference with the
payload is the use of a long single mode fiber which introduces polarization
rotation that needs to be corrected using additional wave plates. The
performances of the detection systems are individually verified using light
from a laser (785 nm and 850 nm) coupled from the source side to the receiver
systems. The total transmission through the receiver unit from the fibre until
the detectors is approximately 83%. The main limitation is the reflection on
the dichroic mirror with a reflection efficiency of 92%. The other optical
efficiencies exceed 98% for all components.

Initially, we measure the source performance, such as the brightness, QBER
heralding efficiency etc, without introducing any losses. With these
measurement results as initial conditions the QKD performance model is used to
predict the behaviour of these parameter and the final secret key rate for
different link losses. For each channel loss introduced by the variable
optical attenuator, timestamps are obtained at the simulated satellite and the
receiver station. The timestamps are then fed into the QKD software stack in
order to get the secret keys along with the other parameters. Figure 10
compares the observed parameters with the prediction from the model. Behaviour
of heralding efficiency, coincidence, quantum bit error rate and secret key
rate with the introduced channel loss are given. It can be observed that even
after 40 dB of loss, the QBER is well below the classical limit and it can
produce secret keys.

After the validation of the performance of the system for a given link loss we
have simulated a satellite pass in which the loss varies dynamically. Figure
11 shows the performance of the QKD system over a simulated satellite pass. At
time 0, the satellite is at its closest to the ground station and this will
contribute to lower QBER and higher sifted key rate. After the error
correction and privacy amplification, we obtain 26 Kbits (101 256-bit AES
keys) of secret keys over a single pass. With the finite key assumption this
corresponds to 13.6 Kbits (53 256-bit AES keys) of secret keys for a single
pass.

###  3.4 Development strategy

Spacecraft instrumentation and spacecraft in general are typically developed
as a series of models, each used to verify part or all of the requirements
posed on the system, culminating in the Flight Model (FM) that is launched
into Space. We follow a similar path, starting from a general proof-of-concept
table-top demonstration iterated in the previous section. The first
approximately to-scale model of the opto-mechanical sub-system of the payload,
called the Functional Model (FM), aimed to demonstrate full performance levels
in a miniaturized form factor and is intended to be used in end-to-end tests
in the lab or using a free-space link. A subsequent model, called the
Engineering Model (EM), should also demonstrate functionality after initial
environmental tests and confirm the final iteration of the optical layout. A
full space qualification campaign is planned for the Qualification Model (QM),
which will be used to qualify the design through extensive functional and
environmental testing, prior to the Flight Model build and launch. Here, we
discuss first results from the functional model build and test and progress on
the next models.

####  3.4.1 Functional model

The design of the functional model is given in Figure 12. The main difference
between the functional model and the planned final design is that it uses
mirrors to steer the SPDC light beam to couple it into a single mode fibre.
The SMF is a short piece of fiber kept in an ‘S’shape in order to avoid the
propagation of cladding modes. In this model we use a single lens before the
fibre. L2 and Lc are moved by external translation stages in the direction of
the beam propagation ($\pm z$) to focus the pump for collection. The beacon
laser mount (B) is moved in z direction while the beacon lens (LB) is moved in
X and Y directions (perpendicular to the beam path) to couple the beacon into
the fibre. All the lenses and mounts are secured via epoxy after obtaining the
desired performance.

After the integration of the source we observed a detected brightness of 600k
counts/s/mw with signal and idler heralding efficiencies of 23.5% and 28%,
respectively. This corresponds to a source brightness of 9M counts/s/mW. With
40 mW of maximum achievable pump power we can easily meet the required
brightness of 25-70 MCPs. The polarization correlation is given in Figure 13.
The average visibility of the curves is 98.6% which corresponds to a source
QBER of only 0.6%. Further integration of the full unit and functionality
tests are currently being undertaken.

Figure 12: Design of the functional model  Figure 13: Polarization correlation
of the entangled photons generated in the functional model

####  3.4.2 Thermal and vibration tests

The satellite payload needs to be resilient against random vibrations, thermal
gradients and thermal cycles. As alignment tolerances for coupling the light
into a single mode fibre are tight, a critical concern is misalignment of the
system. We are currently developing a test unit representative of the fiber
coupling system of the payload in order to test under different conditions.
Once the unit has passed the vibration and thermal tests we will be moving to
the final engineering model design.

####  3.4.3 Engineering model

With the input from the thermal and vibration tests, an engineering model will
be developed. This will be a fully functional quantum payload with optical,
electronic and software systems. The engineering model will undergo thermal-
vacuum testing as well as vibration testing.

##  4 Conclusions

SpooQy-1 demonstrated the feasibility of operating quantum entanglement
sources on board CubeSats and after 600 days of operation, the satellite
continuous to produce and detect high quality entangled photons. The effect of
the radiation environment on the dark counts of the avalanche photo diodes is
measured which is a useful insight for designing next generation quantum
payloads. For satellite to ground QKD, requirements are derived from a
detailed link model which computes the performance of the system for a given
entangled photon source, detectors and timestamping hardware. The model is
validated through a table top experiment for QKD. Early development models of
the next generation quantum payload have demonstrated performance levels
suitable for space-to-ground quantum key distribution and long range
entanglement distribution. The next development models will be built and
tested this year, with a realistic launch window of 2022-2023. The small
volume of 2-4L allows for cost-effective deployment on a wide range of
spacecraft, enabling nodes in a quantum network to be deployed in Space.

###### Acknowledgements.

This research is supported by the National Research Foundation, Singapore
under its Central Gap Fund (NRF2018NRF-CG001-001)

##  Appendix A Key rate model inputs

Table 2: Inputs for space-to-ground key rate calculations | Payload
---
subsystem
Model inputs | Unit | Value | Reference
| Entangled photon
---
source
Pair production rate | Mcps | 1-100 | this is a variable input
| Visibility | % | 98 | 99% achieved in the lab
| Wavelength | nm | 780/842 |  | typical SPDC
---
emission wavelength
| Bandwidth | nm | 10nm | 5nm achieved in lab
Detection setups | Efficiency | % | 0.25 |  | Total detection efficiency
---
or pair-to-singles ratio,
25% achieved in the lab
| Dark counts | cps | 100000/500 |  | SPOOQY-1 results /
---
lab-based numbers (per detector)
| Dead time | ns | 50 | <50ns achieved in lab
| Jitter | ps | 320 |  | Measured for single photon
---
detectors on SPOOQY-1
|  | After-pulsing
---
probability
% | <5 |  | 1% achieved in lab,
---
depends on temperature
Optical link | Transmitter aperture | m | 0.09 |  | Realistic aperture size
---
for a nanosatellite
| Receiver aperture | m | 0.6 |  | Optimum aperture to
---
cost ratio for
PlaneWave telescopes
| Beam quality | M2 | 1.6 |  | Fundamental limit is 1.4
---
due to diffraction
|  | Atmospheric
---
attenuation
dB | 3 |  | 3 at zenith from [bourgoin],
---
scaled with path length
for lower elevation angles
| Pointing jitter | microrad | 5 |  | 1.2 microrad demonstrated [9]
---
on (larger) Micius satellite
| Efficiency | % | 50 |  | Estimated based on
---
reflectivity and number
of optical surfaces
| Background counts | cps | 1300 |  | Measured with representative
---
setup in Singapore
Protocol |  | Error correction
---
efficiency
- | 1.1 | This depends on the protocol
|  | Basis reconciliation
---
factor
- | 0.5 |  | Un-biased basis splitting
---
in detection setup
|  | Coincidence time
---
window
ns | 1 |  | Lowest achievable
---
with given detector jitter
Orbit | Altitude | km | 500 |  | Sun-synchronous orbit
---
at 500km means daily passes [9]
|  | Maximum elevation
---
angle during pass
deg | 90 | pass directly over the GS

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# Uncommon magnetic ordering in the quantum magnet Yb3Ga5O12

S. Raymond111corresponding author<EMAIL_ADDRESS>Univ. Grenoble Alpes, CEA,
IRIG, MEM, MDN, 38000 Grenoble, France  E. Lhotel  Institut Néel, CNRS & Univ.
Grenoble Alpes, 38000 Grenoble, France  E. Riordan  Institut Néel, CNRS &
Univ. Grenoble Alpes, 38000 Grenoble, France  E. Ressouche  Univ. Grenoble
Alpes, CEA, IRIG, MEM, MDN, 38000 Grenoble, France  K. Beauvois  Univ.
Grenoble Alpes, CEA, IRIG, MEM, MDN, 38000 Grenoble, France  C. Marin  Univ.
Grenoble Alpes, Grenoble INP, CEA, IRIG, PHELIQS, 38000 Grenoble, France  M.E.
Zhitomirsky  Univ. Grenoble Alpes, Grenoble INP, CEA, IRIG, PHELIQS, 38000
Grenoble, France

###### Abstract

The antiferromagnetic structure of Yb3Ga5O12 is identified by neutron
diffraction experiments below the previously-known transition at
$T_{\lambda}=54$ mK. The magnetic propagation vector is found to be ${\bf
k}=(1/2,1/2,0)$, an unusual wave-vector in the garnet structure. The
associated complex magnetic structure highlights the role of exchange
interactions in a nearly isotropic system dominated by dipolar interactions
and finds echos with exotic structures theoretically proposed. Reduced values
of the ordered moments may indicate significant quantum fluctuations in this
effective spin-1/2 geometrically frustrated magnet.

Research on geometrically frustrated magnetic materials has been extremely
fruitful for discovering new states of condensed matter, namely exotic spin
liquid, spin ice and spin fragmented states Balents ; Jaubert ; Lhotel1 . The
hallmark geometries associated to these findings are the three dimensional
pyrochlore and two dimensional kagome lattices. Recently interest has been
reborn in the less studied hyperkagome lattice and in particular in the family
of rare-earth gallium garnet systems $R_{3}$Ga5O12. In these cubic crystals
(space group Ia$\bar{3}$d), the rare-earth ions $R^{3+}$, form two three
dimensional interpenetrating lattices of corner sharing triangles called
hyperkagome lattices. Depending on the rare earth element, anisotropy leads to
contrasted behaviors from the realization of multi-axis antiferromagnetic
order in Ising systems ($R=$Nd, Tb, Dy, Ho, Er) Hamman1 ; Hamman2 ; Hamman3 ;
Cai ; Wav ; Petit ; Kibalin ; Hamman4  to exotic spin correlations in nearly
isotropic systems $R=$Gd Paddison  and $R=$Yb Sandberg ; Lhotel2 .

In the present work, we focus on the magnetic state of of the effective
spin-1/2 hyperkagome material Yb3Ga5O12. It has been known for a long time
that the heat capacity exhibits a lambda-type anomaly at $T_{\lambda}$
$\approx$ 54 mK and a broad hump at about 200 mK Filippi . While the latter
feature is associated with magnetic correlations recently studied by neutron
scattering Sandberg ; Lhotel2 , microscopic insight into the nature of the
transition and the magnetic ground state has been missing for more than forty
years. In particular, the Mössbauer spectroscopy measurements did not observe
any hyperfine splitting associated with a possible magnetic ordering below
$T_{\lambda}$ Hodges . The $\mu$SR experiments also did not detect any sign of
long-range order as well pointing instead to the reinforcement of the
dynamical correlations below $T_{\lambda}$ Dalmas03 . In this letter, we show
through neutron diffraction experiments that a static long-range magnetic
order does settle below $T_{\lambda}$. Our measurements reveal that it arises
with a propagation vector ${\bf k}=(1/2,1/2,0)$ that has never been reported,
to our knowledge, at zero field for the garnet systems, where the magnetic
ordering occurs in almost all cases within the crystallographic unit cell,
i.e. with a propagation vector $\textbf{k}=0$ ($\Gamma$ point of the Brillouin
zone) garnet  and a few cases with k= (0, 0, 1) ($H$ point of the Brillouin
zone) Plumier .

Diffraction measurements were performed at the Institut Laue Langevin,
Grenoble on the two-axis thermal-neutron diffractometer D23 equipped with a
lifting detector. A copper monochromator provides an unpolarized beam with the
wavelength of 1.276 Å. The sample is the same as the one used for
magnetization measurements in Ref. Lhotel2, . In order to insure a good
thermalization, the sample was inserted inside a sealed copper box filled with
helium and the box was attached to the mixing chamber of a dilution stick. The
based temperature was 30 mK on the thermometer mounted below the mixing
chamber.

Based on common experimental observations for materials from the $R_{3}$Ga5O12
family as well as by theoretical predictions Capel ; Kibalin , an extensive
search was performed for the magnetic ordering wave-vector ${\bf k}={\bf 0}$
complemented by a few checks at ${\bf k}=(0,0,1)$. No signal was found by
subtraction of the peak intensity collected at 35 mK and 500 mK over 21
representative reflections (10 independent ones). In contrast by expanding the
search along a $\Gamma$-$\Gamma$ line, a magnetic signal has been detected.
Figures 1a and b show scans along the momentum transfer ${\bf
Q}=(Q_{H},Q_{H},0)$ and $(1/2,1/2,Q_{L})$ respectively at 35 mK and 500 mK,
where the coordinates of the momentum transfer $\bf{Q}$ are given in units of
$2\pi/a$ with the lattice parameter $a$ = 12.196 Å (r.l.u. stands for
reciprocal lattice units in the Figure 1). They reveal a peak at $Q_{H}=1/2$
and $Q_{L}=0$ respectively, so that the propagation vector is ${\bf
k}=(1/2,1/2,0)$ (It is recalled the relation between momentum transfer and
propagation vector: ${\bf Q}={\bf G}\pm{\bf k}$ where $\bf{G}$ lies at a
$\Gamma$ point of the reciprocal lattice). Figure 1c shows the intensity
difference obtained between the data collected at 35 and 500 mK along ${\bf
Q}=(Q_{H},Q_{H},0)$ over an extended range in $Q_{H}$. The diffuse scattering
intensity measured above $T_{\lambda}$ on D7-ILL from Ref. Lhotel2,  is also
shown as a dashed line. This highlights the fact that the correlations at
$(1/2,1/2,0)$ evidenced above $T_{\lambda}$ in our previous study transform
into a long range magnetic order below $T_{\lambda}$.

Figure 1:  Momentum transfer cuts at 35 mK (full blue circles) and 500 mK
(open red circles) along (a) ${\bf Q}=(Q_{H},Q_{H},0)$ and (b) along ${\bf
Q}=(1/2,1/2,Q_{L})$. (c) Subtraction of the neutron intensity obtained at 35
mK from the one obtained at 500 mK along ${\bf Q}=(Q_{H},Q_{H},0)$ in an
extended range up to $Q_{H}=2$. The red dashed line indicates the diffuse
scattering data obtained above $T_{\lambda}$ in Ref. Lhotel2, ; the intensity
is scaled to the new data. In all the panels, full lines are guides to the
eye.

Figure 2 shows the temperature dependence of the Bragg peak intensity at ${\bf
Q}=(1/2,1/2,0)$ on warming and on cooling. The temperature was varied by 2 mK
steps, and after a 4 min waiting time, the intensity was counted during 15 min
for each point. The obtained transition temperatures are 62 mK on cooling and
66 mK on warming, which may suggest a first order transition. However, after
cooling the initial low temperature intensity is not recovered. In order to
check whether it is intrinsic or related to the thermalization of the sample,
a point at 50 mK has been measured with a longer stabilization time. The
outcome is shown in Figure 2 where the black diamond points indicate the data
obtained at 50 mK after 1 hour waiting time on warming from 30 mK or 1 hour
waiting time on cooling from 90 mK. The same intensity is recovered
(overlapping diamonds) meaning that there is no intrinsic temperature
hysteresis in the system. Due to the available neutron beamtime, we could not
determine the equilibrium temperature dependence of the magnetic intensity
with this stabilization time protocole for each point. From this analysis, we
estimate that the real Néel temperature is $T_{N}$= 64 $\pm$ 2 mK, a somewhat
higher value than the reported $T_{\lambda}$=54 mK in specific heat.

Figure 2:  Temperature dependence of the neutron intensity measured at ${\bf
Q}=(1/2,1/2,0)$ on warming from 30 to 90 mK (red dots) and on cooling from 90
to 30 mK (blue circles). The black diamond points correspond to data obtained
by longer stabilisation time of 1 h (see text).

In the space group Ia$\bar{3}$d, the star of ${\bf k}=(1/2,1/2,0)$ contains
six equivalent wave-vectors, producing six antiferromagnetic configuration
domains. As expected in the absence of symmetry breaking perturbation, the
integrated peak intensities measured for the six corresponding Bragg
reflections ${\bf Q}={\bf k}$ are found to be very similar. In order to
address the magnetic structure, a collection of magnetic peaks belonging to
the single configuration domain with ${\bf k}=(1/2,1/2,0)$ has been carried
out through rocking curve measurements (“Omega scans”). Representatives of
such scans are shown in Figure 3a, the strongest peak, which corresponds to
${\bf Q}={\bf k}$ being shown in the top panel. Due to the weak signal, long
counting times were necessary (9 min per point) and 21 independent magnetic
Bragg peaks could be collected. Owing to the limited set of data and given the
complexity of the lattice (12 magnetic sites without the centring I), the full
structure could not be solved. The discrimination between the possible
structures was made by least-square fitting of the data, while screening the
combinations of basis vectors given by the magnetic representation analysis
(Details are given in the Supplemental Material Supplement ).

In the space group Ia$\bar{3}$d with the propagation vector ${\bf
k}=(1/2,1/2,0)$, there exist only two representations of order two,
$\Gamma_{1}$ and $\Gamma_{2}$, and the sites are split into 1a sites (8
moments), 1b sites (2 moments) and 1c sites (2 moments) Supplement . The
remaining 12 sites are deduced by the centring symmetry I, where the moments
are reversed given the propagation vector $\bf{k}$. The best fit is obtained
using the $\Gamma_{2}$ representation and by putting the 1a site moments along
(0, 0, $M_{Z}$) (the components ($M_{X}$, $M_{Y}$, $M_{Z}$) are given in the
global coordinate frame ($X$, $Y$, $Z$) associated to the crystallographic
$a$, $b$ and $c$ axes). As far as the 1b site is concerned, the fits show that
a $M_{Z}$ component must exist but they are not conclusive about the symmetry
allowed ($M_{X}$, -$M_{X}$, 0) component. For the 1c site, the data are
compatible with any possibility from zero magnetic moment to magnetic moments
in the direction ($M_{X}$, -$M_{X}$, $M_{Z}$).

Figure 3:  a) Representative rocking curves (red points) performed at 35 mK for ${\bf Q}=(0.5,0.5,0)$, ${\bf Q}=(1.5,2.5,1)$ and ${\bf Q}=(0.5,0.5,2)$ and the associated fitted intensity using a Gaussian function (black lines). b) Magnetic structure in the cubic crystallographic cell (1a sites in black, 1b sites in green, 1c sites in orange). Table 1:  Spin components for the proposed structure of Yb3Ga5O12. The magnetic moment components are given in the global coordinate frame (($M_{X}$, $M_{Y}$, $M_{Z}$) notation) and in the local coordinate frame (($m_{x}$, $m_{y}$, $m_{z}$) notation) at each site indicated by its crystallographic coordinates in units of $a$. The relations between the local and global coordinate frames are given in the Supplemental Material Supplement . Site | coordinates |  $M$ global frame |  $m$ local frame
---|---|---|---
1a | $(1/8,0,1/4)$ | $(0,0,M_{Z})$ | $(-m_{x},m_{x},0)$
| $(3/8,0,3/4)$ | $(0,0,M_{Z})$ | $(-m_{x},m_{x},0)$
| $(1/4,1/8,0)$ | $(0,0,M_{Z})$ | $(m_{x},m_{x},0)$
| $(3/4,3/8,0)$ | $(0,0,-M_{Z})$ | $(m_{x},m_{x},0)$
| $(7/8,0,3/4)$ | $(0,0,M_{Z})$ | $(-m_{x},m_{x},0)$
| $(5/8,0,1/4)$ | $(0,0,M_{Z})$ | $(-m_{x},m_{x},0)$
| $(3/4,7/8,0)$ | $(0,0,-M_{Z})$ | $(-m_{x},-m_{x},0)$
| $(1/4,5/8,0)$ | $(0,0,M_{Z})$ | $(-m_{x},-m_{x},0)$
1b | $(0,1/4,1/8)$ | $(0,0,M_{Z})$ | $(0,0,m_{z})$
| $(0,3/4,7/8)$ | $(0,0,M_{Z})$ | $(0,0,m_{z})$
1c | $(0,3/4,3/8)$ | $(-M_{X},M_{X},0)$ | $(0,m_{y},0)$
| $(0,1/4,5/8)$ | $(-M_{X},M_{X},0)$ | $(0,m_{y},0)$

In order to go further it is thus necessary to consider the different terms of
the Hamiltonian of the system, i.e. the anisotropy and the interactions that
are at play. We have recently shown that a dipolar model reproduces very well
two experimental results in Yb3Ga5O12, the positive Curie-Weiss temperature
$\theta_{p}=97$ mK and the weakly dispersive magnetic excitation spectrum in
the field saturated paramagnetic phase Lhotel2 . The obtained magnetic
structure should thus minimize the energy associated to dipolar interactions.
Concerning the anisotropy, the crystal electric field (CEF) leads to the
ground-state doublet, well separated from the first excited state (the
splitting is about 67 meV) Carson ; Buchanan ; Pearson . As a result, at low
temperatures the excited crystal-field levels can be ignored and Yb3+ ions are
faithfully represented by effective spins $S=1/2$. In the cubic unit cell, 24
ytterbium sites are split into six groups according to the local crystal-field
axes ($x$, $y$, $z$), depending on the orientation of the oxygen octahedron
around each site (the relations between the local and global coordinate frames
are given in the Supplemental Material Supplement ). The $g$-tensor for the
ground-state doublet is described by values $g_{x}=3.74$, $g_{y}=3.60$ and
$g_{z}=2.85$ in the local coordinate frame conventions . Hence, the doublet is
fairly isotropic with a weak easy-plane anisotropy.

The combined effect of dipolar interactions, exchange interactions and
anisotropy has been theoretically addressed in the early studies of gallium
garnets Capel  and reexamined in a recent investigation Kibalin . Such studies
are nevertheless limited to ${\bf k}={\bf 0}$ corresponding to an intra-cell
magnetic ordering of the 24 magnetic atoms. For Ising systems $g_{z}\gg
g_{x},\ g_{y}$, antiferromagnetic ordering occurs in a multi-axial structure
with moments along the local axis $z$ (so-called ”AFA states”). For a given
triangle, the local $z$-axis alternates between the three $X$, $Y$ and $Z$
global axes. This solution is given by a single basis vector in magnetic
representations analysis (There are two such representations, odd or even with
respect to the inversion). These structures occur for $R=$Nd, Tb, Dy, Ho, Er
Hamman1 ; Hamman2 ; Hamman3 ; Cai ; Wav ; Petit ; Kibalin ; Hamman4 . In the
case of nearly isotropic systems such as Yb garnet, a more complex
ferrimagnetic structure was proposed, the so-called ”FC state”, arising from a
three dimensional representation with nine basis vectors Capel . Note that
this ferrimagnetic FC state is associated with an odd representation with
respect to the inversion and an even representation also exists leading to a
complex antiferromagnetic state (the odd/even classification is only valid for
$\bf{k=0}$).

Interestingly, Yb3Ga5O12 exhibits magnetic order at ${\bf k}=(1/2,1/2,0)$,
which has not been anticipated so far in theoretical studies. Based on the
results obtained for ${\bf k}={\bf 0}$, we propose the magnetic structure
detailed in Table 1: as explained above, to be consistent with the measured
intensities, the magnetic moments 1a have to be along (0, 0, $M_{Z}$), which
actually lies in the easy local anisotropy planes of these ions, i.e with
component $(m_{x},m_{y},0)$. Such a collinear state has similarities with one
possible FC state introduced in Ref. Kibalin, . For 1c magnetic moments, the
refinement only imposes the $(-M_{X},M_{X},M_{Z})$ constraint, and it thus
appears logical to put these moments in their easy-plane. This amounts to
cancel the $Z$ component, giving a $(0,m_{y},0)$ moment in the local frame.
Note that this orientation cancels the nearest neighbor Heisenberg exchange
interaction since the nearest neighboring 1a sites have moments along $Z$.
However an absence of ordered magnetic moment on the 1c site cannot be ruled
out with our data. Theoretical studies on Ising spins on the hyperkagome
lattice in the presence of long-range interactions could stabilize such
magnetic configurations with paramagnetic sites ($\bf{k}$=0 calculation)
Yoshioka , but we can expect that the less constrained anisotropy in Yb3Ga5O12
will favor the presence of an ordered moment as supposed here. Finally, for
the 1b moments, a non-zero $M_{Z}$ component (which corresponds to a $m_{z}$
component in the local frame), which opposes to the local anisotropy, is
obtained from the refinement. We propose that the other components (whose
values do not impact the quality of the refinement) are zero, so that, in
analogy with the FC state introduced above, the 1b moments are along
$(0,0,M_{Z})$ and collinear to the 1a ones.

The final magnetic structure is shown in Figure 3b. To better visualize it,
fragments of the structure are displayed in Figure 4. This view indeed points
out the fact that the 1c sites separate lines / blocks of spin up and spin
down magnetic moments from 1a and 1b sites: the 1c site is shared by two
triangles, one with its two remaining spins along $(0,0,M_{Z})$ and the other
with its spins along the opposite directions $(0,0,-M_{Z})$. Such arrays of
parallel spins could be favoured by dipolar interactions, that are expected to
be dominant in Yb3Ga5O12 Lhotel2 .

Figure 4:  Fragments of the magnetic structure highlighting the underlying
organization: the spins of the 1c sites (red arrows) separate lines / blocks
of up and down spins (black arrows). The shown cut corresponds to
$-0.25a~{}\leq~{}X~{}\leq~{}0.4a$, $0~{}\leq~{}Y\leq~{}2.3b$ and
$-0.5c~{}\leq~{}Z\leq~{}0.5c$.

The obtained ordered moment associated to this magnetic structure is 0.61 (1)
$\mu_{\rm B}$. We have constrained its magnitude to be equal on each site and
taken into account the 6 configuration domains while assuming equal population
of the 8 orientation domains. The ordered moments at $T$= 30 mK account for
only about 1/3 of the saturated moments obtained in the magnetization
measurements Lhotel2 . Normally, one would expect to find $\approx$ 3/4 of the
full moments at a temperature roughly twice smaller than $T_{N}$. It is to be
noted that this value of the ordered magnetic moment is robust when assuming
zero magnetic moment on the 1c site. Two physical mechanisms behind the moment
reduction in Yb3Ga5O12 can be considered. First, geometric frustration on the
hyperkagome lattice promotes significant thermal fluctuations at $T<T_{N}$ as
was discussed previously for the Ising antiferromagnet Ho3Ga5O12 Zhou .
Second, an even more interesting possibility is that strong quantum
fluctuations for nearly isotropic spin-1/2 Yb moments reduce magnetic order
even at $T$ = 0 K. For comparison with other three-dimensional frustrated spin
models, note that 30% moment reduction was found for the nearest-neighbor
Heisenberg spin-1/2 antiferromagnet on a fcc lattice Schick22 , whereas even
larger quantum effects are expected for anisotropic spin-1/2 pyrochlore
antiferromagnets in the proximity to classical boundaries Zhitomirsky12 ;
Yan17 . Inelastic neutron scattering at first sight appears as a promising
technique to investigate the fluctuations in the ordered phase, but it is
worth noting that the experimental difficulties in thermalizing Yb3Ga5O12
samples below 60 mK, while keeping reasonable background, would make these
measurements on large single crystals below the transition very challenging.

We now address the puzzling result of the propagation vector ${\bf k}$=(1/2,
1/2, 0), which has never been reported to our knowledge for the magnetic
structure of rare-earth garnets in zero field, where ordering with ${\bf k=0}$
is the paradigm. From our analysis, dipolar interactions and weak anisotropy
appear to be the most important ingredients, but they are expected to give
${\bf k}={\bf 0}$ ordering. Non zero sub-leading antiferromagnetic exchange
interactions are thus needed to stabilize the unusual propagation vector
evidenced in the present neutron diffraction experiments. It is worth noting
that the role of exchange interactions in Yb3Ga5O12 remains debated from
previous studies above the transition temperature Sandberg ; Lhotel2  and new
inelastic neutron scattering studies have been recently performed in order to
gain insight into this point Deen24 .

It is interesting to compare Yb3Ga5O12 with the extensively studied sister
material Gd3Ga5O12. Gadolinium ions have large semiclassical spins $S$=7/2
with a small easy-plane anisotropy in Gd3Ga5O12 Paddison ; Bonville . No
magnetic transition is known for this compound, though complex spin-spin
correlations based on ten-site loops were reported that give rise to a so-
called director state Paddison . A partial incommensurate ordering was also
evidenced Petrenko98 , as well as spin-glass like freezing Schiffer . These
properties are the consequences of competing small energy scales. It was shown
that two types of next-nearest neighbor interactions (inside the hyperkagome
lattice and between the two hyperkagome lattices) have to be included in the
Hamiltonian to understand the observed properties even if they are much
smaller than the dominant antiferromagnetic nearest neighbor interactions and
dipolar interactions Yavorskii . In Gd3Ga5O12, for a magnetic field ${\bf H}$
applied parallel to $[1,\bar{1},0]$, a ${\bf k}=(1/2,1/2,0)$ ordering vector
has been reported in a narrow field range, included in a larger region of the
phase diagram with four incommensurate wave-vectors Petrenko , which
highlights the fragility of such a phase. Calculations show that, specifically
for this field direction, soft modes leading to long range order are possible
along the line $(h,1-h,0)$, which includes (1/2, 1/2, 0) Ancliff . This
comparison with Gd3Ga5O12 suggests that in Yb3Ga5O12, the ${\bf k}$=(1/2, 1/2,
0) ordered phase results from a delicate balance between dipolar, nearest
neighbor and next nearest neighbor exchange interactions.

In the present work, we have considered a single-${\bf k}$ multi-domain
structure but a multi-${\bf k}$ structure cannot be ruled out. Multi-${\bf k}$
structures are often difficult to distinguish from single-${\bf k}$ ones by
neutron diffraction experiments. Remarkably, in the pyrochlore lattice system
Gd2Ti2O7, a 2-$\bf{k}$ magnetic structure is now established Paddison2  with a
${\bf k}=(1/2,1/2,1/2)$ propagation vector, while similarly to garnets, most
magnetically ordered pyrochlores have a k=0 propagation vector. Other
multi-${\bf k}$ structures occurring for magnetic ordering at half integer
reciprocal lattice vector were also theoretically shown to be stabilized in
cubic spinels Zhito . Since the proposed single-${\bf k}$ structure is almost
collinear, further inelastic neutron scattering experiments aiming to
determine the polarization of the lowest energy excitations would indicate if
a more complex multi-${\bf k}$ structure should be considered Jensen .

To conclude, more than forty years after the discovery of the lambda anomaly
at 54 mK in Yb3Ga5O12, long range static antiferromagnetic ordering is
revealed with the unusual propagation vector ${\bf k}=(1/2,1/2,0)$ for the
garnet lattice. The associated magnetic structure has similarities with
complex spin arrangements already theoretically proposed for $\bf{k}$=0 and
never experimentally observed so far, even for this latter propagation vector.
This emphasizes the delicate balance between the competing energy scales at
play in these systems in order to realize the unique magnetic ordering pattern
revealed in Yb3Ga5O12. Additionally, the reduced ordered moment points to
significant fluctuations, whose origin is to be addressed by further
experimental and theoretical investigations. Together with the different
ground state achieved in the notorious system Gd3Ga5O12, these results
highlight the rich possibilities brought by the hyperkagome lattice in the
limit of small anisotropy and the importance of this geometry in the overall
progress in undestanding frustrated magnetism.

We acknowledge B. Vettard, P. Fouilloux, M.-H Baurand, V. Joyet, E. Bourgeat
Lami and X. Tonon for technical support. We acknowledge S. Petit and F. Damay
for fruitful discussions. The structure was determined using Mag2POL and
checked with Fullprof softwares. We acknowledge help from N. Qureshi for
Mag2POL. E. L., C. M., and M. E. Z. acknowledge financial support from Agence
Nationale de la Recherche, France, Grant No. ANR-18-CE05-0023. The neutron
scattering data collected at the ILL for the present work are available at
Ref. DOI1 .

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