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PHYSICAL REVIEW APPLIED
12,
024062 (2019)
Nanophotonic Quantum Storage at Telecommunication Wavelength
Ioana Craiciu
,
1,2
Mi Lei,
1,2
Jake Rochman,
1,2
Jonathan M. Kindem,
1,2
John G. Bartholomew,
1,2
Evan Miyazono,
1,2
Tian Zhong,
1,2,
Neil Sinclair,
3,4
and Andrei Faraon
1,2,
*
1
Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratories of Applied Physics, California Institute of
Technology, Pasadena, California 91125, USA
2
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena,
California 91125, USA
3
Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, California
91125, USA
4
Alliance for Quantum Technologies, California Institute of Technology, Pasadena, California 91125, USA
(Received 15 January 2019; revised manuscript received 24 June 2019; published 30 August 2019)
Quantum memories for light are important components for future long-distance quantum networks. We
present on-chip quantum storage of telecommunication-band light at the single-photon level in an ensem-
ble of erbium-167 ions in an yttrium orthosilicate photonic crystal nanobeam resonator. Storage times of
up to 10
μ
s are demonstrated with an all-optical atomic-frequency-comb protocol in a dilution refrigerator
under a magnetic field of 380 mT. We show this quantum-storage platform to have high bandwidth, high
fidelity, and multimode capacity, and we outline a path toward an efficient erbium-167 quantum memory
for light.
DOI:
10.1103/PhysRevApplied.12.024062
I. INTRODUCTION
Optical quantum memories can aid processes involv-
ing the transfer of quantum information via photons, with
applications in long-distance quantum communication and
quantum-information processing [
1
5
]. Rare-earth ions in
crystals are a promising solid-state platform for optical
quantum memories due to their long-lived optical and
spin transitions that are highly coherent at cryogenic tem-
peratures [
6
,
7
]. Among rare-earth ions, only erbium has
been shown to possess highly coherent optical transitions
in the telecommunication C band, which allows integra-
tion of memory systems with low-loss optical fibers and
integrated silicon photonics [
8
].
Fixed-delay quantum storage for less than 50 ns at
telecommunication wavelengths has been demonstrated
in erbium-doped fibers [
9
], and lithium niobate waveg-
uides [
10
] at efficiencies approaching 1%. The protocol
used in both cases, the atomic frequency comb (AFC),
requires spectrally selective optical pumping [
11
]. The
storage efficiencies in these studies were limited in part
by the lack of suitable long-lived shelving states in the
erbium ions in these hosts. Moving to isotopically puri-
fied erbium-167 in an yttrium orthosilicate host (YSO)
offers the prospect of long-lived shelving states in the
*
faraon@caltech.edu
Present address: Institute of Molecular Engineering, Univer-
sity of Chicago, Chicago, Illinois 60637, USA.
form of hyperfine levels [
12
]. While optical storage has
been realized in erbium-doped YSO [
13
15
], including
efficiencies approaching 50% for storage times of 16
μ
s
(revival-of-silenced-echo protocol [
15
]), quantum storage
has yet to be demonstrated in this material.
In this work, we demonstrate on-chip quantum stor-
age of telecommunication light at the single-photon level.
We use a nanophotonic crystal cavity milled directly in
167
Er
3
+
-doped YSO (
167
Er
3
+
:YSO) for coupling to an
ensemble of erbium ions and realize quantum storage using
the AFC protocol [
11
]. The cavity increases the absorption
of light by the ion ensemble, allowing on-chip implementa-
tion of the memory protocol [
16
]. By working in a dilution
refrigerator and using permanent magnets to apply a field
of 380 mT, we access a regime in which the ions have
optical coherence times of approximately 150
μ
s and long-
lived spin states to allow spectral tailoring. For a storage
time of 165 ns, we achieve an efficiency of 0.2%, with
lower efficiencies for longer storage times, up to 10
μ
s.
We demonstrate storage of multiple temporal modes and
measure a high fidelity of storage, exceeding the classical
limit. Lastly, we identify the limits on the storage efficiency
and propose ways to overcome them to achieve an efficient
167
Er
3
+
:YSO quantum memory for light.
II. OVERVIEW OF THE SYSTEM
Memories using spectral tailoring such as the AFC
protocol require a long-lived level within the optical-
2331-7019/19/12(2)/024062(10)
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© 2019 American Physical Society
IOANA CRAICIU
et al.
PHYS. REV. APPLIED
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ground-state manifold, where the population can be
shelved. Hyperfine levels in the optical ground state in
167
Er
3
+
:YSO have been shown to have long lifetimes at
1.4 K and a magnetic field of 7 T [
12
]. In general, these
levels can be long-lived when the erbium electron spin is
frozen, which occurs when

ω
e

k
B
T
, where
ω
e
is the
electron Zeeman splitting [
12
]. In this work, we satisfy this
inequality by using a moderate magnetic field of 380 mT
parallel to the
D
1
axis of the crystal
e
=
2
π
×
80 GHz)
and a nanobeam temperature of approximately 400 mK
(see Appendix
A
). The hyperfine lifetime is measured to
be 29 min in the bulk crystal under the same magnetic field
and cooling conditions.
Figures
1(a)
and
1(b)
show the nanoresonator used in
this experiment. A triangular nanobeam photonic crys-
tal cavity [
17
] is milled in a YSO crystal doped with
isotopically purified
167
Er
3
+
(92% purity) at a nominal
concentration of 50 ppm. The nanobeam is 1.5
μ
m wide
and approximately 20
μ
m long. The slots in the nanobeam
create a photonic crystal band gap, and the periodic pat-
tern (lattice constant 590 nm, groove width 450 nm) is
modified quadratically in the center to create a cavity
mode.
Figure
1(c)
shows a schematic of the optical testing
setup. A fiber-coupled tunable external-cavity diode laser
is used to probe the nanobeam device and implement the
AFC storage protocol. One percent of the laser light is
directed to a wavemeter for measurement. Another 1%
of the light is picked off and sent to a locking setup,
in which the laser frequency is stabilized by locking to
a homebuilt fiber cavity by means of the Pound-Drever-
Hall technique [
18
]. The remaining light is directed to the
sample through two acousto-optic modulators for pulse
shaping, as well as an electro-optic phase modulator to
control the phase of the light and to add strong side-
bands for hyperfine initialization. Neutral-density filters
and polarization paddles provide attenuation and polariza-
tion control, respectively. A circulator directs light to the
167
Er
3
+
:YSO crystal located inside a dilution refrigerator,
thermally linked to the 25-mK stage. An aspheric lens pair
focuses light from an optical fiber onto the angled cou-
pler of the resonator. A stack of
x
-
y
-
z
nanopositioners is
used to optimize free-space coupling. Light from the res-
onator is directed by the circulator onto a superconducting-
nanowire single-photon detector at approximately 100 mK.
Strong initialization pulses are prevented from reaching
laser
D
S
P
C
TT
(a)
(c)
(d)(e)
(b)
Input and
output
FIG. 1. (a) Finite-element-analysis simulation of the TM cavity mode in the triangular nanobeam resonator. The red-blue color
gradient indicates the electric field component normal to the surface,
E
z
; the black outline indicates the YSO-air interface; a yel-
low arrow indicates coupling. (b) Scanning electron micrograph of the resonator, showing input and output coupling through a 45
slot coupler. (c) Experimental setup (details are given in the main text). (d) Energy diagram of
167
Er
3
+
:YSO, showing the
4
I
15
/
2
4
I
13
/
2
optical transition for crystallographic site 2. (e) Reflection spectrum of the cavity when tuned on resonance to the 1539-nm
167
Er
3
+
:YSO transition. Detuning is measured from 194 816
±
2 GHz. The inset shows a close up of ion coupling before (black) and
after (red) partial hyperfine initialization. The circles are data points, and solid black and dashed red lines are fits to theory (see the
main text for details). AOM, acousto-optic modulator; EOM, electro-optic phase modulator; MEMS, microelectromechanical switch;
ND, neutual density; SNSPD, superconducting-nanowire single-photon detector.
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the superconducting-nanowire single-photon detector by
a microelectromechanical switch. A magnetic field
B
=
380 mT is applied to the sample with use of two cylindrical
permanent magnets.
Figure
1(d)
shows an energy diagram of an
167
Er
3
+
ion
in YSO. The ground-state and excited-state manifolds both
have 16 hyperfine states (
I
=
7
/
2 nuclear spin,
S
=
1
/
2
effective electron spin), split into two sets of eight by
electron Zeeman coupling to the applied magnetic field.
The optical transition used in this work couples the lower
eight hyperfine levels of the ground-state and excited-state
manifolds. The coherence time of this optical transition is
measured by the two-pulse photon-echo method [
19
]tobe
149
±
4
μ
s in the nanobeam. This provides an upper bound
on the all-optical storage time. In the bulk crystal, the opti-
cal coherence time of this transition under similar cooling
conditions is measured to be 759
±
41
μ
s. The reduction
in the coherence time as measured in the nanobeam is
likely caused by a combination of higher temperature in
the nanobeam during measurement and the impact of the
fabrication process. At 1 K, the coherence time of ions
in the bulk crystal is found to be 136
±
9
μ
s. In similar
nanobeam devices [
20
,
21
], focused-ion-beam milling does
not significantly impact the coherence properties of ions.
However, the longer bulk coherence times measured in the
current work allows a higher sensitivity probe of the ions’
environment. The optical coherence time does not limit the
storage time achieved in this work.
Figure
1(e)
shows the reflection spectrum of the
nanobeam cavity, which has a measured loaded quality fac-
tor of 7
×
10
3
. The cavity is tuned onto resonance with
the 1539-nm transition of the
167
Er
3
+
ions by our freezing
nitrogen gas onto the nanobeam at cryogenic tempera-
tures [
22
]. The coupling of the ensemble of ions to the
cavity is seen as a peak in the cavity reflection dip. The
inset shows a close up of the ion-cavity coupling (in black).
The ensemble cooperativity is estimated from a fit to this
curve to be 0.1 (see Appendix
B
).
For high-efficiency storage using ions coupled to a
cavity, the ensemble cooperativity should be 1 [
16
,
23
].
An increased ensemble cooperativity of 0.3 [inset in
Fig.
1(e)
, in red], is obtained with a partial-hyperfine-
initialization procedure. The width of the 1539-nm tran-
sition in
167
Er
3
+
:YSO is approximately 1.5 GHz. This is
broader than the true inhomogeneous linewidth of approx-
imately 150 MHz because of the numerous closely spaced
optical transitions arising from hyperfine splitting in both
the ground-state manifold and the excited-state manifold
[see Fig.
1(d)
]. Because the ion population is distributed
among the hyperfine ground states, the optical depth in
the center of the 1539-nm line is lower than for an
I
=
0
isotope. Therefore, to increase the ensemble cooperativity,
the ion population is first initialized into a small number
of hyperfine ground states with optical transitions at the
center of the inhomogeneous line. This is achieved by our
sweeping the laser frequency between 350 and 820 MHz
on both sides of the inhomogeneous line. At 7 T, the entire
optical-ground-state population can be initialized into one
hyperfine state with an efficiency of 95% by pumping on
all

m
=+
1 transitions or all

m
=−
1 transitions [
12
].
At 380 mT parallel to the
D
1
axis, only a partial initializa-
tion can be performed because the

m
1 transitions
are not fully spectrally resolved from the

m
=
0 transi-
tions. The partial-initialization procedure can be improved
by using larger magnetic fields or by changing the angle of
the applied field.
III. ATOMIC FREQUENCY COMB STORAGE
The nanobeam device is used to demonstrate quantum
optical storage using the AFC protocol [
11
]. In this proto-
col, a pulse of light that is absorbed by an atomic frequency
comb with an intertooth spacing of

is stored for
t
=
1
/
.
Frequency-selective optical pumping is used to create a
0
100
200
300
400
Time (ns)
0
20
40
Counts/10
3
×100
–50
–25
0
25
50
Detuning (MHz)
50
55
60
Counts/10
3
n
pump
n
input
amp.
freq.
t
t
(a)
(b)
(c)
FIG. 2. AFC experiment in the nanobeam cavity. (a) A section
of the resonator reflection spectrum, showing an atomic fre-
quency comb in the center of the inhomogeneously broad-
ened
167
Er
3
+
transition. Detuning is measured from 194 814.2
±
0.1 GHz. The apparent slope of the comb is due to its center
frequency not being precisely aligned to the cavity resonance,
leading to a dispersive shape. (b) AFC pulse sequence show-
ing amplitude (yellow) and frequency (purple) modulation of the
laser (pulses not to scale, see the main text for detail). (c) AFC
storage: the input pulse (dashed red line) is partially absorbed
by the comb; an output pulse is emitted at time
t
=
1
/
=
165 ns (black line,
×
100). The black line also shows the par-
tially reflected input pulse (
t
=
0) and a smaller second output
pulse at
t
=
330 ns.
024062-3
IOANA CRAICIU
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comb within the inhomogeneous linewidth, as shown in
Fig.
2(a)
. Figure
2(b)
shows a schematic of the protocol.
First, a long pulse with strong frequency-modulated side-
bands is used for partial hyperfine initialization. The next
15 pulses, repeated
n
pump
=
20 times, create the comb: the
laser frequency is swept through 15 values, separated by

=
6.1 MHz, to optically pump away ions and create 15
spectral transparencies. The following
n
input
=
60 pulses
are zero-detuning weak coherent states that are stored
in the frequency comb. The full experiment is repeated
approximately 10
4
times. As shown in Fig.
2(c)
, 60-ns-
wide pulses with an average photon number
̄
n
of 0.60
±
0.09 are stored for 165 ns with an efficiency of 0.2%.
Despite the partial initialization, the storage efficiency is
limited by the ensemble cooperativity of the device (see
Appendix
D
).
Coherent pulses can be stored in the device for up to
10
μ
s, although with a lower efficiency of 10
5
, as shown
in Fig.
3
. Here, as for all storage times longer than 165 ns,
we use an accumulated-AFC method [
24
] to create the
comb. As shown in the inset in Fig.
3
, weak pairs of
pulses separated by
t
storage
=
10
μ
s are repeatedly sent into
the cavity. The Fourier transform of each pulse pair is a
frequency comb, which imprints onto the
167
Er
3
+
inho-
mogeneous line to create the AFC. This procedure uses
laser-frequency stabilization during comb creation, which
enables the creation of fine-toothed AFCs required for
longer storage. For the 165-ns storage, where a coarser
AFC is practical, the procedure shown in Fig.
2(b)
with no
laser-frequency stabilization leads to higher efficiencies by
creating a more-consistent comb over the entire bandwidth.
This is because the accumulated AFC has a sinc-function
n
pump
n
input
...
× 20 000
t
FIG. 3.
AFC storage for 10
μ
s in the nanobeam resonator. The
dashed red line shows the input pulse. The black line shows the
partially reflected input pulse and the output pulse
(
×
20 000
)
.
The reflected input pulse appears small due to detector saturation.
The inset shows a schematic of the pulse sequence following
hyperfine initialization. Pairs of comb-preparation pulses 10
μ
s
apart are repeated
n
pump
=
10 000 times, followed by input pulses
20 ns wide, repeated
n
input
=
10 times.
envelope. The storage efficiency at 10
μ
s is limited by
residual laser-frequency jitter and by superhyperfine cou-
pling to the yttrium ions in YSO. Superhyperfine cou-
pling limits the narrowest spectral feature to approximately
1MHz [
25
,
26
]. Since this exceeds the period of the comb
needed for this storage time (

=
1
/
t
storage
=
0.1 MHz),
the resulting AFC will have a lower contrast, leading to
lower storage efficiency.
The AFC protocol is capable of storing multiple tem-
poral modes [
11
]. Ten coherent pulses are stored in this
device, as shown in Fig.
4(a)
. The AFC comb in Fig.
2(a)
has a bandwidth of approximately 90 MHz, which can
accommodate storage in multiple frequency modes [
27
].
An inhomogeneous linewidth of 150 MHz limits the band-
width of storage in this system. Although there are methods
to increase this linewidth [
28
], the bandwidth cannot be
increased much further before being limited by overlap-
ping optical transitions from other hyperfine levels.
In quantum storage protocols, the phase of the stored
state must be preserved. A double AFC is used as an
interferometer to characterize the coherence of the storage
process [
24
]. Two overlapping AFCs with tooth spacing
–0.5
0
0.5
1
1.5
2
Detuning (MHz)
0
50
100
150
200
250
Counts
(a)
(b)
FIG. 4.
Multimode and coherent storage in the nanobeam res-
onator. (a) Storage of multiple temporal modes: ten 20-ns-wide
input pulses (reflection off cavity shown) and the correspond-
ing ten output pulses
(
×
1000
)
from a 500-kHz AFC. (b) Vis-
bility curve acquired in a double-comb experiment, with

1
=
5.2 MHz,

2
=
3.4 MHz, and
δ
1
=
0 MHz. The detuning
of the second comb is swept from
δ
2
=−
0.2 MHz to
δ
2
=
2.2
MHz, and the intensity of the two central overlapping output
pulses is measured. Black circles show the sum of counts in the
overlapping pulse region with
N
counts
uncertainty bars. The red
line shows a least-squares fit to a sinusoid. The inset shows the
four output pulses (middle two overlapping) in the case of maxi-
mally constructive (dashed black line) and maximally destructive
(solid red line) interference.
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1
and

2
and with frequency detuning
δ
1
and
δ
2
are cre-
ated so that each input pulse is mapped to two output pulses
at times 1
/
1
and 1
/
2
and with a relative phase
φ
rel
=
2
π
(
δ
2
/
2
δ
1
/
1
)
[
11
]. An input state encoded into
two pulses,
|
ψ
in
=
(
1
/
2
)
(
|
early
+|
late

)
, is therefore
mapped to a total of four output pulses. By our appro-
priately selecting the time interval between the early and
late input pulses, two of the four output pulses are made
to overlap and either constructively or destructively inter-
fere, depending on
φ
rel
[see the inset in Fig.
4(b)
]. Using an
input state with mean photon number
̄
n
=
0.6
±
0.09 and
sweeping
φ
rel
via the detuning
δ
2
, we obtain the interfer-
ence fringe shown in Fig.
4(b)
(see the caption for details).
The measured visibility of
(
91.2
±
3.4
)
% demonstrates the
high degree of coherence of this on-chip storage process.
The visibility is limited by the 12 counts in the total-
destructive-interference case
(
δ
2
=

2
/
2
φ
rel
=
π
)
.A
dark count rate of 18.5 Hz accounts for seven of these.
The remaining counts arise from imperfect cancellation of
the two overlapping output pulses due to slightly different
efficiencies of storage in the two AFCs. The dark-count-
subtracted visibility is
(
97.0
±
3.6
)
%.
The double-comb method is also used to estimate a
lower bound for the fidelity of storing single-photon time-
bin states,
F
(
n
=
1
)
. The fidelity of storage is measured for
four input states—
|
early

,
|
late

,
(
|
early
+|
late

)/
2, and
(
|
early
−|
late

)/
2—with two mean photon numbers
̄
n
=
0.30 and
̄
n
=
0.60. With these values, the decoy-state
method [
27
,
29
] is used to calculate a bound on the fidelity
for storing single-photon states
F
(
n
=
1
)
(
93.7
±
2.4
)
%,
which exceeds the classical limit of
F
=
2
/
3 (details are
given in Appendix
C
). Similarly to the visibility case
discussed above, the measured fidelity is limited in part
by dark counts and in part by the double-comb proto-
col being an imperfect interferometer. The dark-count-
limited-fidelity bound is estimated to be approximately
96.5%. Although the pulses stored in this experiment
are weak coherent pulses, single-photon sources [
30
35
]
and entangled-photon sources [
36
,
37
] are available at
telecommunication wavelengths, and storing photons of
any frequency is enabled by quantum frequency conver-
sion [
38
]. For efficient storage, the bandwidth of compati-
ble photons is bounded by the periodicity of approximately
1 MHz and the bandwidth of approximately 100 MHz of
the AFC.
IV. DISCUSSION
While the storage presented here is limited in efficiency,
a nanophotonic cavity coupled to
167
Er
3
+
ions in YSO
promises to be an efficient quantum storage system. The
main limitations to the storage efficiency in this work
are a low ensemble cooperativity of 0.3 and loss from
the optical nanobeam cavity. The cooperativity can be
increased using higher
167
Er
3
+
doping and better hyperfine
initialization, which would require the applied magnetic
field to be increased [
12
] or its angle to be changed.
A higher-intrinsic-quality-factor resonator would serve to
both increase cooperativity and decrease cavity loss. For
example, with use of a YSO crystal with 200-ppm
167
Er
3
+
doping, optimal hyperfine initialization, and a resonator
with an intrinsic quality factor of 2
×
10
6
, the theoreti-
cal efficiency of the AFC quantum storage is 90% (see
Appendix
D
for analysis). Mature silicon nanofabrication
technology can be leveraged to achieve this goal by use of
a silicon resonator evanescently coupled to
167
Er
3
+
ions
in YSO [
8
,
33
]. With this efficiency level and a storage
time of 10
μ
s, the device would outperform a delay line
composed of standard telecommunication fiber [
39
], an
important benchmark on the way to achieving a quantum
memory suitable for scalable quantum networks.
With the optical AFC protocol alone, it will be difficult
to achieve efficient storage for this duration due to superhy-
perfine coupling. However, the AFC spin-wave protocol,
where the stored information is reversibly transferred from
the optical manifold to the hyperfine manifold [
11
], would
enable storage longer than 10
μ
s without the same require-
ments for narrow spectral features, and would also enable
on-demand recall. The availability of hyperfine states with
coherence times exceeding 1 s [
12
] makes
167
Er
3
+
:YSO a
promising system for spin-wave storage.
V. CONCLUSION
In conclusion, we demonstrate on-chip quantum stor-
age of telecommunication-band light at the single-photon
level. The storage has a bandwidth of approximately
90 MHz, and a storage fidelity for single-photon states
of at least
(
93.7
±
2.4
)
%.
167
Er
3
+
:YSO at temperatures of
approximately 400 mK and a moderate magnetic field is
shown to be a promising material for AFC quantum mem-
ories. A clear path exists for creating a high-efficiency
quantum memory with this material and a nanoscale
resonator.
ACKNOWLEDGMENTS
This work was supported by Air Force Office of Sci-
entific Research Young Investigator Award No. FA9550-
15-1-0252, Air Force Office of Scientific Research Grant
No. FA9550-18-1-0374, and the National Science Foun-
dation (Grant No. EFRI 1741707). I.C. and J.R. acknowl-
edge support from the Natural Sciences and Engineering
Research Council of Canada (Grants No. PGSD2-502755-
2017 and No. PGSD3-502844-2017). J.G.B. acknowl-
edges support of the American Australian Associa-
tion’s Northrop Grumman Fellowship. N.S. acknowl-
edges funding by the Alliance for Quantum Technologies’
Intelligent Quantum Networks and Technologies research
program.
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IOANA CRAICIU
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APPENDIX A: DEVICE TEMPERATURE
Because of poor thermal conduction at low temperatures
in insulating materials such as YSO, and the small cross
section of the nanobeam, the device is warmer than its
approximately-25-mK surroundings when optical pulses
are coupled in. The device temperature is estimated via the
167
Er
3
+
electron spin temperature [
40
], which is computed
from the ratio between the lower and upper electron spin
populations in the optical-ground-state manifold with use
of
N
|↑
/
N
|↓
=
e

ω/
k
B
T
.
–4
–2
0
2
Detuning (GHz)
0
0.5
1
1.5
Normalized PL
0
0.2
0.4
0.6
0.8
1
Refrigerator temperature (K)
0
0.2
0.4
0.6
0.8
1
Electron spin temperature (K)
0.03
0.04
0.2
0.4
0.6
(a)
(b)
FIG. 5.
(a) Photoluminescence (PL) from the nanobeam device
as a function of detuning at three refrigerator temperatures:
720 mK (gray squares), 385 mK (black diamonds), and 37 mK
(red circles). Detuning is measured from 194 810
±
0.1 GHz. PL
is collected after a 500-
μ
s resonant pulse at 0.3 pW (estimated
power in the nanobeam). Background counts are subtracted, and
each curve is normalized and offset for clarity. Solid lines are fits
to a sum of two Gaussians with equal widths and center frequen-
cies 3.2 GHz apart. The
|↓
transition is at the higher frequency.
(b) Electron spin temperatures (EST) computed from the PL data
in (a) as a function of refrigerator temperature. The dashed gray
line indicates where the two temperatures are equal. The inset
shows an enlargement of the EST measurement at 37 mK (black
circle) and the EST estimated during the
T
2
measurement (green
diamond) and the 165-ns storage experiment (blue square). To
estimate the latter two temperatures, the same pattern of laser
pulses as in the actual experiments is sent to the nanobeam,
at 0.3 and 0.02 nW, respectively, and PL is collected after the
pulses. Error bars are propagated standard deviations from pho-
ton counting (
N
counts
). In all measurements the laser frequency
is slowly modulated within each transition to prevent hyperfine
hole burning.
Under an applied field of 380 mT, the electron spin in the
optical ground state is frozen for any temperature under
approximately 500 mK, enabling the long hyperfine life-
times required for AFC storage. To be sensitive to lower
temperatures, measurements of the electron spin popula-
tion are performed with a lower magnetic field of 110 mT
(parallel to the
D
1
axis of the crystal), leading to an electron
Zeeman splitting of
ω
=
2
π
×
23 GHz, where the upper
electron spin state has a detectable population down to
approximately 250 mK. Because the Zeeman splitting is
still considerably greater than the hyperfine splitting, one
can consider two electron spin states,
|↓
and
|↑
, each split
into eight by the hyperfine interaction. The population in
the two electron spin states is measured via the electron-
spin-preserving optical transitions from each level. The
nanobeam is tuned such that these transitions are both res-
onant with the cavity, and photoluminescence is collected
as a function of frequency, as shown in Fig.
5(a)
.
N
|↑
/
N
|↓
is extracted from the area ratio of the two transitions.
Figure
5(b)
shows the electron spin temperatures com-
puted from these ratios for different dilution-refrigerator
temperatures. The inset in Fig.
5(b)
shows the electron
spin temperature measured under input power conditions
identical to two experiments: 317
±
49 mK for the
T
2
measurement in the nanobeam and 413
±
24 mK for the
165-ns storage experiment in Fig.
2
. Assuming the elec-
tron spin is in thermal equilibrium with the device, we
estimate the temperature of our device during experiments
to be approximately 400 mK.
APPENDIX B: ENSEMBLE COOPERATIVITY
To extract the cooperativity of coupling between the
nanobeam resonator and the ensemble of
167
Er
3
+
ions,
each cavity reflection spectrum shown in the inset in
Fig.
1(e)
is fit with
R
=
α
1




(
1
α
f
)
+
α
f
e
i
θ
f
i
κ
in
ω
ω
cavity
+
i
(κ/
2
)
+
W
,
g
total
,

ions
,
ω
ions
)




2
+
α
2
,(B1)
where
α
1
and
α
2
are amplitude and background fit param-
eters,
α
f
e
i
θ
f
accounts for Fano interference (both
α
f
and
θ
f
are fit parameters),
κ
is the total cavity energy decay
rate,
κ
in
is the coupling rate through the input port, and
ω
cavity
is the cavity resonance frequency.
κ
=
27.3 GHz
and
κ
in
=
0.21 are measured from reflection curves
where the cavity is detuned from the
167
Er
3
+
transition.
W
,
g
total
,

ions
,
ω
ions
)
is the rate of absorption of the cav-
ity field by the ensemble of ions,
W

i
[
g
2
i
/(ω
ω
i
)
],
where
g
i
is the coupling between one ion and the
cavity [
16
,
41
]. We approximate the irregular shape of
024062-6
NANOPHOTONIC QUANTUM STORAGE...
PHYS. REV. APPLIED
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024062 (2019)
the inhomogeneously broadened and hyperfine-broadened
optical transition as a Gaussian, and use the expression for
W
from Ref. [
41
]:
W
=
i

π
log2
g
2
total

ions
/
2

1
erf

i

log2
ω
ions
)

ions
/
2

×
exp

log2
ω
ω
ions

ions
/
2
2

,(B2)
where

ions
is the linewidth of the ensemble transi-
tion,
ω
ions
its the center, and
g
2
total
=

i
g
2
i
. Finally, the
ensemble cooperativity is computed with
C
=|
W
=
ω
ions
)
|
/(κ/
2
)
=
4

π
log2
g
2
total
/(κ
ions
)
[
8
].
For the case with no initialization, the fit yields
ω
cavity
ω
ions
=
2
π
×
2.5 GHz,
g
total
=
2
π
×
0.79 GHz,

ions
=
2
π
×
1.4 GHz, and
C
=
0.1.
For the case with initialization, the fit yields
ω
cavity
ω
ions
=
2
π
×
1.5 GHz,
g
total
=
2
π
×
0.70 GHz,

ions
=
2
π
×
0.36 GHz, and
C
=
0.3.
APPENDIX C: FIDELITY
In the absence of a single-photon source, a lower bound
on the storage fidelity of a single-photon input state can be
found by the decoy-state-analysis method [
27
,
29
]. In this
method, a time-bin state
ψ
with a mean photon number
̄
n
is stored with use of the AFC protocol, and the fidelity
F
(
̄
n
)
ψ
of storage is measured as
F
(
̄
n
)
ψ
=
N
ψ
N
ψ
+
N
φ
ψ
,(C1)
where
N
ψ
(
N
φ
ψ
)
is the number of photons measured in
the output time bin corresponding to
ψ(φ
ψ)
, where
φ
ψ
denotes the state orthogonal to
ψ
. The gain of the
output,
Q
(
̄
n
)
ψ
, is estimated as follows:
Q
(
̄
n
)
ψ
=
N
ψ
+
N
φ
ψ
.(C2)
F
(
̄
n
)
ψ
and
Q
(
̄
n
)
ψ
are measured for mean photon numbers
̄
n
1
and
̄
n
2
, where
̄
n
1
<
̄
n
2
,and
̄
n
2
<
1.
The lower bound on the fidelity of storing a one-photon
input state
F
(
n
=
1,
L
)
ψ
is then computed as follows:
F
(
n
=
1,
L
)
ψ
=
1
E
(
̄
n
1
)
ψ
Q
(
̄
n
1
)
exp
̄
n
1
E
(
0
)
Y
(
0
)
Y
(
n
=
1,
L
)
̄
n
1
,(C3)
where
E
(
̄
n
)
ψ
=
1
F
(
̄
n
)
ψ
(C4)
is the error rate of storing a state
ψ
with mean photon
number
̄
n
,and
Y
(
n
=
1,
L
)
=
max
Y
(
0
)
,
̄
n
2
̄
n
2
̄
n
1
− ̄
n
2
1
×
Q
(
̄
n
1
)
e
̄
n
1
Q
(
̄
n
2
)
e
̄
n
2
̄
n
2
1
̄
n
2
2
̄
n
2
2
− ̄
n
2
1
̄
n
2
2
Y
(
0
)
(C5)
is the lower bound on the detection yield for the storage of
a single-photon state (see Ref. [
29
]).
Y
(
0
)
=
Q
(
n
=
0
)
is the
yield when the input state is the vacuum state, and is equal
to the dark counts in both output time bins. The super-
scripts denote the photon number and whether the value is
a lower bound
(
L
)
.
E
(
0
)
=
E
(
n
=
0
)
is the vacuum error rate,
which is 0.5 by definition [
29
].
To obtain an average fidelity bound for all possi-
ble time-bin states, the fidelities for storing the time-
bin states
|
early

,
|
late

,
|+ =
(
|
early
+|
late

)/
2, and
|− =
(
|
early
−|
late

)/
2 are measured for input pho-
ton numbers
̄
n
1
=
0.30 and
̄
n
2
=
0.60. The input pulses
defining the
|
early

and the
|
late

basis are 60 ns wide
and 90 ns apart. A double AFC is used for measurements
of all states, with the memory times associated with the
two combs being
t
1
=
210 ns and
t
2
=
300 ns, such that
t
2
t
1
=
90 ns. Of the three output time bins [see the inset
in Fig.
4(b)
], the first and third are used to measure
F
early
and
F
late
, while the second time bin is used to measure
F
+
and
F
.
Following Eq.
(C3)
,
F
(
n
=
1,
L
)
+
is computed as follows:
F
(
n
=
1,
L
)
+
=
1
E
(
̄
n
1
)
+
Q
(
̄
n
1
)
+
/
exp
̄
n
1
E
(
0
)
Y
(
0
)
+
/
Y
(
n
=
1,
L
)
+
/
̄
n
1
,(C6)
with similar equations for the other three states.
Q
(
̄
n
1
)
+
/
,
Y
(
0
)
+
/
,and
Y
(
n
=
1,
L
)
+
/
are averaged over the
|+
and
|−
fidelity measurements.
The lower bound on the fidelity of storing an arbi-
trary single-photon state,
F
(
n
=
1,
L
)
average
=
(
93.7
±
2.4
)
%, is then
computed as follows:
F
(
n
=
1,
L
)
average
=
1
3

F
(
n
=
1,
L
)
early
+
F
(
n
=
1,
L
)
late
2

+
2
3

F
(
n
=
1,
L
)
+
+
F
(
n
=
1,
L
)
2

.(C7)
Table
I
summarizes the measured fidelity values for stor-
ing weak coherent states. The uncertainties are calculated
using

N
photon
standard deviation on all
N
ψ
values due
to Poissonian statistics of photon counting and the uncer-
tainty, estimated to be 15%, of the mean input photon
numbers,
̄
n
.
024062-7
IOANA CRAICIU
et al.
PHYS. REV. APPLIED
12,
024062 (2019)
TABLE I.
Measured storage fidelities in the nanobeam device.
Input photon number
1
2
F
(
̄
n
)
early
+
F
(
̄
n
)
late

1
2
F
(
̄
n
)
+
+
F
(
̄
n
)

̄
n
=
0.60
±
0.09
(
89.04
±
1.34
)
%
(
91.90
±
1.32
)
%
̄
n
=
0.30
±
0.05
(
82.59
±
1.80
)
%
(
90.75
±
1.84
)
%
n
=
0
50%
50%
APPENDIX D: STORAGE EFFICIENCY
The efficiency of AFC storage in a cavity is given by [
16
,
21
,
23
]
η
AFC
=
4
κ
in
comb
total
+
comb
+
bg
)
2
2
η
d
,
(D1)
where
κ
in
is the rate of cavity coupling through the input
port,
κ
total
is the total energy decay rate of the cavity,
η
d
=
exp

π
2
/(
2 log 2
(/γ )
2
)

accounts for dephasing due to
the finite width of the comb teeth [
11
], and
comb
and
bg
are the rates of absorption of the cavity field by the ensem-
ble of ions in the comb and the background, respectively.
Background ions are those ions remaining after optical
pumping, with transition frequencies where transparency is
desired (i.e., between the teeth of the comb). Nonzero
bg
results from limitations in spectral hole burning. With use
of
η
spectral
, the fractional optical depth of a spectral hole,

,
the intertooth spacing, and
γ
, the width of one comb tooth,
comb
and
bg
can be estimated as follows:
comb
η
spectral
γ

ions
,
(D2)
bg
=
(
1
η
spectral
)
ions
,
(D3)
where
ions
=|
W
=
ω
ions
)
|=

π
log2
g
2
total
/(
ions
/
2
)
is
the rate of absorption of the cavity field by the ensemble of
ions before comb preparation (see Appendix
B
).
We can define an effective AFC cooperativity,
C
:
C
=
comb
+
bg
κ
total
/
2
=

η
spectral
γ

+
(
1
η
spectral
)

C
.
(D4)
Rewriting Eq.
(D1)
gives rise to the following expression
for the AFC storage efficiency:
η
AFC
=
1
(/γ )(
1
η
spectral
1
)
+
1
κ
in
κ
total
4
C
(
1
+
C
)
2
2
η
d
,
(D5)
The efficiency is maximized when
C
1.
The predicted efficiency of storage in the nanobeam is
η
AFC
=
0.17%, which is found with
C
=
0.3,
κ
in
total
=
0.21, and a measured finesse of
/γ
=
2.1 and by our
assuming a perfect comb,
η
spectral
=
1. This is similar to
the measured value of 0.20% for a storage time of 165 ns.
To increase the memory efficiency, the ensemble coop-
erativity and the ratio
κ
in
total
must be increased. We first
consider increasing the concentration and improvement of
initialization of
167
Er
3
+
ions, keeping the nanophotonic
resonator the same. By increasing the ion concentration
to 200 ppm (assuming no significant increase in inho-
mogeneous linewidth), and using the level of initializa-
tion achieved in the work reported in Ref. [
12
] (95%),
we expect a cooperativity of
C
=
3. With the measured
κ
in
total
=
0.21 and a comb finesse of
/γ
=
3,
(
C
=
0.73
)
, the predicted efficiency is 3%. This memory effi-
ciency is mainly limited by the loss from the cavity, the
term
in
total
)
2
in Eq.
(D5)
. The total cavity loss rate
is
κ
total
=
κ
in
+
κ
i
, where
κ
i
is the intrinsic loss, includ-
ing losses from absorption, scattering, and the imperfect
reflectivity of the second mirror.
A memory efficiency greater than 90% can be achieved
if the intrinsic quality factor of the resonator is increased
from its current value of
Q
i
9000 to 2
×
10
6
and
κ
in
total
is increased to 0.97. The latter can be accom-
plished with a resonator with one mirror having relatively
low reflectivity, and with minimal losses through other
channels (transmission through the second mirror, scat-
tering, absorption). This calculation assumes the same
material as described above (200 ppm, ideal initializa-
tion into one hyperfine state) and a comb finesse of 13.
It also accounts for a decrease by a factor of approxi-
mately 3 in the cooperativity that results from switching
to a hybrid silicon-YSO platform where the evanescent
cavity-ion coupling is weaker. With this storage efficiency,
the memory would match the performance of an optical
fiber (0.15 dB/km) at
t
storage
=
10
μ
s[
39
].
For a finesse of
f
=
13 and a storage time of 10
μ
s,
the corresponding comb tooth width is
γ
=
/
f
=
8 kHz,
which is too narrow for
167
Er
3
+
:YSO, where effective
linewidths are limited to 1 MHz by superhyperfine cou-
pling. However, memory times longer than 10
μ
s can be
achieved with a nanoresonator in this material by use of
the spin-wave AFC [
11
].
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