www.sciencemag.org/cgi/content/full/scien
ce.
aan5959
/DC1
Supp
lementary
Material
s
for
Nanophotonic
rare
-
earth quantum memory with
optically
controlled retrieval
Tian
Zhong
,
Jonathan M.
Kindem
,
John G.
Bartholomew
,
Jake
Rochman
,
Ioana
Craiciu
,
Evan
Miyazono
,
Marco
Bettinelli
,
Enrico
Cavalli
,
Varun
Verma
,
Sae Woo
Nam
,
Francesco
Marsili
,
Matthew D.
Shaw
,
Andrew D.
Beyer
,
Andrei
Faraon
*
*Corresponding autho
r. Email: faraon@caltech.edu
Published
31 August
20
1
7
on
Science
First Release
DOI:
10.1126/science.
aan5959
This PDF file
includes:
Materials and Methods
Supplementary Text
Fig
s. S1 to S4
Table
S1
References
1
Materials and Methods
Nd:YVO nanocavity design and fabrication
The triangular nanobeam has a width of 690 nm.
31 (11 on one side of the cavity defect mode and 20 on the other side) periodic subwavelength
grooves with widths of 147 nm along the beam axis were milled on top of the nanobeam. The
period of the grooves was modulated quadratically over 14 grooves to form defect modes in
1
the photonic bandgap (
16
). The fundamental TM mode, with top side views shown in Fig. 1
C,
is chosen because it aligns with the strongest Nd dipole of th
e 879.7 nm transition along c
axis of YVO crystals. The theoretical quality factor is 1
×
10
5
for the asymmetric cavity, and
2
×
10
6
for the symmetric cavity (20 grooves on both sides). The left
-most groove interfacing the
waveguide was tapered to minimize scattering. The scatteri
ng due to this taper was negligibly
small from optical microscope measurements. The efficiency
of the coupler was determined to
be 27% by comparing the cavity-reflected power at a off resona
nce wavelength (in the photonic
bandgap) to that from a flat surface of known reflectivity.
The optimization procedure for a one-sided cavity was start
ed by fabricating symmetric two-
sided nanocavities with sufficiently long mirrors (20 groov
es) on each side of the cavity mode.
The measured Q was in the range of 20,000 to 40,000, correspon
ding to an intrinsic loss (likely
due to surface scattering) rate
κ
sc
=2
π
×
9-18 GHz. In subsequent devices, the number of grooves
on one side was decremented from 20 to 11, and the correspondi
ng Q decreased from 20,000 to
3,700 (
κ
= 2
π
×
90 GHz). The final device (with 11 grooves on the input side) wa
s expected to be
dominantly one-sided and over-critically coupled with the
waveguide:
κ
in
/κ
=0.8. In reality, the
measured reflection spectrum in Fig. 2A shows a near critical
coupling to the waveguide with
κ
in
∼
0.5
κ
=2
π
×
45 GHz. The deviation from the design might be due to addition
al fabrication
imperfections when making highly asymmetric groove patter
ns, which could be improved in
future fabrication processes.
Preparation and probing of an atomic frequency comb in the na
nocavity
The comb in Fig.
3D was prepared with 1500 pairs of pulses each containing
∼
100 photons (at the input to the
cavity). The comb was read out as modulated cavity reflection
spectrum using a weak probe
field. The profile of the comb in terms of atomic spectral densi
ty was normalized to an empty
cavity (i.e. 0 ions), and to the full population when no prepa
ration was performed (i.e. 100%
ion density). To convert the reflection spectrum to a normali
zed atomic density, we used the ex-
2
pression
R
(
ω
) = 1
−
T
0
(1 + 4
N
(
ω
)
g
2
/κ
Γ
h
)
−
2
, where
T
0
= 0
.
9
takes into account the residual
on-resonance reflection of a bare cavity (i.e.
N
=0). This expression assumes negligible correla-
tion between ions of different frequencies, which is valid f
or an inhomogeneous ensemble that
is coupled to a cavity below the strong collective coupling r
egime (
15
).
Efficiency of AFC optical quantum memory based on a nanophoto
nic cavity
Taking into
account the cavity QED effects (e.g. Purcell enhancement) p
resent in a nanocavity, we de-
rive the following device storage efficiency from the Tavis-
Cummings Hamiltonian method as
in (
21
).
η
dev
=
(
4
κ
in
Γ
comb
(
κ
+ Γ
comb
+ Γ
bg
)
2
)
2
e
−
7
/F
2
(1)
The storage process - mapping of a input photon onto a dipole e
xcitation in the cavity - has
an efficiency of
4
κ
in
Γ
comb
/
(
κ
+ Γ
comb
+ Γ
bg
)
2
where
Γ
comb
=
n
comb
g
2
/γ
h
,
Γ
bg
=
n
bg
g
2
/γ
h
are Purcell-enhanced atomic absorption rates per bandwidt
h for ions contributing to the comb
and the background, respectively (
n
comb
and
n
bg
are atomic number densities for the comb
and the background). The retrieval process - emission of AFC
photon echo to the waveguide
- is an exact time-reversal of the storage process, therefor
e having the same efficiency. The
additional dephasing due to the comb is given by the term
e
−
7
/F
2
assuming Gaussian-shaped
teeth (
13
). The conditions to approach unity efficiency are
κ
in
/κ
∼
1
,
Γ
bg
= 0
,
F
≫
1
, and
Γ
comb
=
κ
. The last condition is the perfect cavity-ensemble impedan
ce-matching condition
that could almost be met with the current nanocavity devices
.
Measurement of time-bin qubit fidelity
We used attenuated laser pulses to generate test time-
bin qubit states
|
e
i
,
|
l
i
,
|
+
i
,
|−i
. The fidelity
F
e
(
l
)
for input state
|
e
(
l
)
i
is determined experi-
mentally as
F
e
(
l
)
=
C
e
(
l
)
/
(
C
e
(
l
)
+
C
l
(
e
)
)
where
C
e
(
l
)
is the number of counts in the early (late)
time bin. For states
|
+
i
, the recalled photons were sent through a qubit analyzer con
sisting of a
fibre-based unbalanced Mach-Zehnder interferometer with a
long-short path difference
δ
t
=30 ns
that was matched with the time separation between early and l
ate bins. The interferometer was
3
phase-stabilized, and the phase difference between long-s
hort paths (0 or
π
) was set by a piezo-
electric fibre stretcher. The fidelity is determined by
F
+(
−
)
=
C
+(
−
)
/
(
C
+(
−
)
+
C
−
(+)
)
where
C
+(
−
)
is the counts at the interferometer output when the phase is s
et to 0(
π
). The fidelity pa-
rameters
F
e,l,
+
,
−
were measured at two different mean photon numbers
α
1
=0.26,
α
2
=0.58, from
which the lower bounds on the single-photon qubit fidelity
F
(1)
e,l,
+
,
−
were calculated (Supple-
mentary section 7). The average qubit fidelity is then
F
=
1
3
F
(1)
e/l
+
2
3
F
(1)
+
/
−
.
AC Stark pulse generation and spectral compression of AFC
The AC Stark pulses detuned
at
±
1 GHz from the center of the AFC were generated by driving the p
hase electro-optic mod-
ulator (EOM) at a 2.4
V
π
voltage for a duration of 16 ns. The first-order sidebands con
stituted
the Stark pulses, and the extinction of the zero-order power
was about -20 dB. The second and
higher order sidebands also contribute to AC Stark shifts of
AFC, but their effects dropped off
quadratically. The photon numbers in Fig. 4C-E represent on
ly the number in the first-order
sidebands. We estimate the single photon Stark Rabi frequen
cy to be
∼
2
π
×
30 MHz (
22
) in
the nanocavity. The theoretical compression of the AFC comb
spacing
δ
∆
is 48 kHz per Stark
photon in the cavity.
2 Additional details of the experiment apparatus
Single photon detectors
The WSi superconducting nanowire single photon detector (S
NSPD)
has a system detection efficiency of 82
±
2
.
4
% at 880 nm measured at 480 mK. The detector
dark count rate is
<
10 counts per second at a bias current of 5.0
μ
A with the critical current at
6.3
μ
A.
Optical pulse generation and shuttering
The optical pulses were generated by first passing
the CW laser output (M Squared Solstis) through two AOMs (Iso
met 1250c), both in double-
pass configuration, in series, which produced a minimum puls
e width of 50 ns with a 120 dB
extinction ratio. Then the pulses were further trimmed down
to a minimum of 8 ns using an
4
intensity EOM (Jenoptik) with an extinction ratio of 30 dB. F
or fidelity measurements on
|
+
/
−i
qubit states, an additional phase EOM was used in the fibre pat
h to generate a 0 or
π
phase shift
between the two time bins. During the optical pumping sequen
ce, the optical MEMS switch
was off (extinction of 60 dB) to avoid blinding and latching o
f the SNSPD.
System transmission efficiency
The system transmission efficiency including the waveguide
to
fiber coupling (27%), the fiber splicing and connectors loss (
50% transmission in total), circu-
lator loss (49.2% transmission), optical MEMS switch loss (
72.5% transmission), and detector
efficiency is 3.95%.
3 Modelling optical pumping dynamics in Nd:YVO
We consider a
Λ
-system with
ρ
1
,
ρ
2
and
ρ
3
being the population fraction in the two ground
states and in the excited state, respectively. We also intro
duce an effective fourth level
ρ
4
that
takes into account decays from the excited state to other cry
stal field levels (
Z
2
−
5
, e.g.1064 nm
transition of Nd:YVO), and then relaxes quickly to either of
the Zeeman ground states. Opti-
cally pumping populations between ground state spin levels
(through excitation and incoherent
relaxation) can be modelled with a set of rate equations.
d
dt
ρ
1
=
w
21
ρ
2
−
w
12
ρ
1
+ Ω
31
ρ
3
−
Ω
13
ρ
1
+
w
41
ρ
4
(2)
d
dt
ρ
2
=
w
12
ρ
1
−
w
21
ρ
2
+
A
32
ρ
3
+
w
42
ρ
4
(3)
d
dt
ρ
3
= Ω
13
ρ
1
−
Ω
31
ρ
3
−
A
32
ρ
3
−
A
34
ρ
3
(4)
d
dt
ρ
4
=
A
34
ρ
3
−
w
41
ρ
4
−
w
42
ρ
4
.
(5)
where
w
ij
are the spin relaxation rates between the ground states i, j.
w
41
, w
42
represent the
spin decay from higher-lying crystal-field levels, that are
very fast (on the order of ps).
A
3
j
are
the spontaneous emission rates between excited state 3 and g
round state j.
Ω
31
,
Ω
13
≫
A
is the
5
ρ
3
ρ
2
ρ
1
ρ
4
A
32
A
34
ω
21
ω
41
ω
42
ω
12
Ω
Figure S1:
Energy level scheme of Nd:YVO for modelling the optical pump
ing dynamics.
Rabi frequency assuming an intense optical pumping laser on
resonance with the transition 1-3
is driving the transition with equal absorption and emissio
n rates. In the steady state,
dρ/dt
= 0
and
ρ
1
=
ρ
3
. Then the spin polarization, that is the ratio of population
s between the two ground
states, is given by
ρ
2
ρ
1
=
w
12
+
A
32
+
A
34
w
42
/
(
w
42
+
w
41
)
w
21
≈
1 +
2
T
z
(
β
32
+
β
34
w
42
/
(
w
42
+
w
41
))
T
1
(6)
where the second approximation assumes
w
12
=
w
21
= 1
/
2
T
z
which is valid when the thermal
energy
k
B
T
is larger than the Zeeman splitting. The relevant branching
ratios
β
32
, β
34
are the
probability that ions in the excited state decay to the Zeema
n state 2 or the lumped effective
crystal field level 4. To achieve efficient optical pumping an
d strong spin polarization, it is
therefore desirable to have a Zeeman lifetime
T
Z
that is considerably longer than the relevant
spontaneous decay time
1
/A
32
=
T
1
/β
32
or
1
/A
34
. In general, the condition
T
z
≫
T
1
does
not hold for Kramers ions even at temperatures below 4 K. This
factor, combined with poor
branching ratios, generally results in inefficient holebur
ning for Kramer ions including erbium
and neodynium. Here our strategy is to increase
A
32
via Purcell enhancement using an optical
6
nanocavity, creating a situation in which optical pumping c
an be enhanced.
The spontaneous decay rate of a transition coupled to an opti
cal cavity is enhanced by the
Purcell factor
F
+ 1
, where
F
is
F
=
3
4
π
2
(
λ
n
)
3
(
Q
V
)
∫
V
(
E
(r)
·
μ
|
E
max
||
μ
|
)
2
,
(7)
where the last integral over the mode volume of the cavity tak
es into account the local field
intensities experienced by the atoms which are randomly dis
tributed in the cavity. The sponta-
neous decay rate
γ
from the optical excited state will be modified according to
γ
=
(
F
31
+ 1)
β
31
T
1
+
(
F
32
+ 1)
β
32
T
1
+
γ
indirect
(8)
where both 1-3, 2-3 transitions are assumed to be coupled to t
he cavity with the same Purcell
factor because their splitting is much less than the cavity l
inewidth, whereas
γ
indirect
takes into
account decay to other crystal-field levels that are not coup
led to the cavity (i.e. other four
Kramers doublets Z
2
-Z
5
, which include the 1064 nm transition). The Purcell enhance
ment
effect shortens the total excited state lifetime, and also e
ffectively increases the branching ratios
of transitions coupled to the cavity.
For clarity, here we focus on the steady state
ρ
1
that can be calculated from Eq. (4), given
ρ
2
+
ρ
1
= 1
. To model the experimental conditions, a few additional fac
tors need to be taken
into account:
1. The branching ratio
β
32
(
θ
)
is determined from the spin Hamiltonian for any given mag-
netic field orientation
θ
with respect to c-axis, as well as the fact
β
32
+
β
31
=
T
1
/T
spon
=0.273
in a bulk crystal calculated from the spontaneous decay time
T
spon
to the
4
I
9
/
2
level.
2. The percentage of ions belonging to the odd nuclear spin is
otope, which has a different
spin Hamiltonian.
7
3. The finite spin population relaxation between turning off
the optical pumping laser and
the time when the spin population is read out.
The second factor in the list above can be included by separat
ing the ensemble of ions
into subgroups. Odd-isotope ions will in general hole burn v
ery well because the hyperfine
ground states are mixed and they have long nuclear spin lifet
imes (
>
100 ms) for trapping the
population (
33
). The expression to include these physical situations is
ρ
1
(0) =
α
E
1
/
2
1 +
T
z
((
F
32
+1)
β
(E)
32
+
β
34
β
(E)
ID
)
T
1
+
α
O
1
/
2
1 +
T
HF
((
F
32
+1)
β
(O)
32
+
β
34
β
(O)
ID
)
T
1
(9)
(10)
where
α
E
and
α
O
(
α
E
+
α
O
= 1
) represent the percentages of even and odd isotopes. We
know that
α
E
=
(142)
0
.
272 +
(144)
0
.
238 +
(146)
0
.
172 +
(148)
0
.
057 +
(150)
0
.
056 = 0
.
795
, and
α
O
= 0
.
205
. The term
β
ID
=
w
42
/
(
w
42
+
w
41
))
is introduced to track the effective branching
ratio for spin relaxation via the indirect path. The precise
value of this term is unknown, but
we characterize it with the same Nd:YVO crystal sample at 3.5
K (results in the next section).
Generally, excited ions decaying to the ground states via in
termediate levels tend to hole burn
well because spin flips are induced during the time spent in in
termediate states. Furthermore,
the hyperfine states in Nd:YVO have considerably longer life
time, i.e.
T
HF
=
320 ms, that
effectively traps the population away from the ground state
1. Consequently the values for
β
(
O
)
32
and
β
(
O
)
ID
are not critical as the odd isotopes holeburn very efficientl
y and contribute almost none
to the ground state population
ρ
1
during the time scale of consideration (
≤
50 ms). Lastly, we
have
β
34
= 1
−
β
31
−
β
32
=0.727. To include the third factor in the list above, the pop
ulation in
ρ
1
after a time
τ
after the end of the optical pumping pulse is
ρ
1
(
τ
) =
ρ
1
(0) + (
ρ
1
(
∞
)
−
ρ
1
(0)) exp
−
τ/T
Z
,
(11)
8
where
ρ
1
(
τ
)
is the population at time
τ
after the optical pumping laser is off.
Below we discuss the parameters relevant for optical pumpin
g dynamics in two cases.
Case
1: In the photonic nanocavity
•
The ensemble averaged Purcell factor is
F
31
=
F
32
≈
70
, derived considering the short-
ening of
T
1
from 90
μ
s to 4.5
μ
s shown in Fig. 2B in the main text.
•
β
32
(
θ
)
is given by the spin Hamiltonians of the
4
I
9
/
2
and
4
F
3
/
2
levels, and the magnetic
field angle relative to the c-axis (
θ
) (
24
)
•
T
Z
(
θ
)
is the Zeeman lifetime, which is also a function of magnetic fi
eld angle
θ
. With a
field of 350 mT oriented approximately along the c-axis, the l
ifetime at 480 mK is approx-
imately 12.5 ms. Currently, our measurements at 480 mK does n
ot allow changing field
angles away from c axis. However, prior results indicate tha
t the electron spin lifetime at
a 350 mT field is limited by resonant spin-spin interactions (
10
).
•
wait time
τ
=
200
μ
s, which is
∼
40 times the cavity
T
1
.
Case 2: In the bulk crystal
•
F
32
= 1
in the absence of Purcell enhancement
•
β
SH
(
θ
)
is given by the spin Hamiltonians of the
4
I
9
/
2
and
4
F
3
/
2
levels and the magnetic
field angle relative to the c-axis (
θ
) (
24
)
•
T
Z
(
θ
)
is the bulk Zeeman lifetime as a function of magnetic field ang
le
θ
, which is ex-
pected to be limited by the spin-spin interactions as in the p
revious case.
•
wait time
τ
=
2 ms, which is
∼
20 times the bulk
T
1
.
9
4 Extracting Nd:YVO optical pumping properties
The simple model described in the previous section allows th
e optical pumping dynamics to be
predicated from Eq. (8) with the knowledge of
T
z
,
F
32
,
β
32
and
β
ID
for a given experimental
condition. Among these parameters,
T
z
,
F
32
,
β
32
can be directly measured in experiments.
β
ID
cannot be measured but is possible to infer its value from the
remaining ground state population
ρ
1
(
τ
)
after optical pumping. We assume
β
ID
has a dominant functional dependence on the field
angle, and is not sensitive to temperatures (in the range bel
ow 4 K) and field strengths (below 1
T).
We performed a set of optical pumping experiments in a 0.5 mm-
think Nd:YVO bulk sam-
ple that is cut from the same boule (nominally 100 p.p.m.) as t
he one on which nanocavities
were fabricated. The results are summarized in Fig. S2. The s
ample was mounted in a cryo-
station (Montana Instruments) at 3.6 K. The magnetic field ge
nerated from a pair of rare-earth
permanent magnets was 340 mT, and its orientation
θ
with respect to the crystal c-axis was var-
ied manually. The branching ratio
β
32
in Fig. S2 was extracted from the absorption spectra of
the transitions 1-3 and 1-2, which agree with the theoretica
l Spin Hamiltonian calculations (
24
)
(blue curve). The optical pumping laser was polarized along
the c axis, and was on for a suf-
ficiently long time such that a maximal spin polarization was
achieved. Then the spectral hole
(measured as an increase in transmission) was probed by a wea
k pulse at increasing delays
from the optical pumping pulse. From the decay of the spectra
l hole depth, we extracted the
Zeeman lifetime
T
z
at each field orientation. Then based on the remaining ground
state popu-
lation, we estimate the effective
β
ID
values from Eq. (8). There was a clear trend of increasing
β
ID
(increasing spin mixing via other crystal field levels) with
θ
. For the experimental con-
figuration described in the main text, the field angle is nearl
y along the c-axis, thus we expect
β
32
, β
ID
→
0
. This means the optical pumping efficiency is expected to be l
ow (poor spin po-
10
larization) in the absence of cavity enhancement. In the mea
ntime, the enhancement of optical
pumping by the Purcell effect is expected to be most pronounc
ed in such a field configuration.
5 Estimation of the magnetic field orientation
In Fig. S3, we show measurements of the absorption spectrum o
f the 0.5 mm thick bulk
Nd:YVO crystal in which the device was fabricated. The bulk c
rystal experienced the same
field as the ions in the cavity assuming negligible field inhom
ogeneity across the sample thick-
ness. The probe light was polarized perpendicular to c axis,
which caused the branching ratios
of four Zeeman split transitions to be different from the con
figuration with the probe polarized
along c-axis (
24
). The transitions a, c are overlapped, whereas the transiti
ons b, d are clearly
resolved and they are split by 9.92 GHz.
Each ground and excited doublet
Z
1
and
Y
1
is characterized by the principal values g
⊥
(perpendicular to c) and g
k
(parallel to c) of the g-factor. We used the following values
for
Z
1
:
|
g
⊥
|
= 2.361,
|
g
k
|
= 0.915; for
Y
1
:
|
g
⊥
|
= 0.28,
|
g
k
|
=1.13. The relative frequencies of the
transitions a-d can be then calculated from the effective g-
factors at a field angle
θ
with respect to
the c aixs. We measured the field strength at the sample to be 34
0
±
2 mT (from a 1” diameter rod
N52 grade rare-earth permanent magnet). In Fig.S3, we plot t
he expected transition frequencies
(black curves) for B=340 mT and varying angles
θ
. The measured frequency separation between
the transition b and d corresponded to a
θ
=8.2
±
1.5
◦
. The red dotted lines are expected transition
frequencies for
θ
=8.2
◦
.
6 Noise performance of the nanophotonic AFC quantum in-
terface
One of the key metrics for an AFC quantum interface is the nois
e level due to incoherent sponta-
neous emissions from the ensemble in the time window an echo p
hoton is retrieved. Low-noise
11
and thus high fidelity memory is typically achieved in bulk ra
re-earth doped crystals by im-
posing a long wait time (e.g. 20-40 optical
T
1
times) between the input photons and the last
pulses in the optical pumping (AFC preparation) sequence. T
his is to ensure no atoms are in
the excited state and the spontaneous emission is absent. In
a nanocavity, the optical
T
1
is sub-
stantially reduced (e.g.
∼
20 times in Fig. 2B) due to the Purcell effect. Therefore the n
ecessary
wait time required to achieve low-noise performance is also
reduced. In our AFC experiment
in the nanocavity, the wait time was 200
μ
s,
∼
40 times the Purcell enhanced
T
cav
1
but only
∼
2 times the bulk
T
1
. A typical single-pulse input (mean photon number 0.58) and
the echo
signal after a storage time of 75 ns is plotted in Fig. S4A. Wit
hout subtraction of detector dark
counts, the noise level per bin (8 ns) is 17 dB below the peak co
unts in the echo pulse. This
noise level did not decrease when the wait time was increased
, for instance, to 2 ms, which
indicated that the current noise was not caused by the sponta
neous emission from the optical
pumping pulses. The remaining noise is likely to be mostly co
ntributed by the background ions
(imperfect optical pumping as evidenced in Fig. 3D) absorbi
ng the input photon and emitting
spontaneously. Figure S4B plots the AFC echo efficiency vers
us the storage time. The red data
point corresponds to the condition in which the results show
n in Fig. 3D, E of the main text are
measured.
7 Qubit storage fidelity from decoy-state analysis
For characterization of the memory storage fidelity, we used
attenuated laser pulses to mimic
true single photons (i.e. Fock state) as inputs to the memory
. Decoy state strategy is a general
technique that allows extraction of the qubit error rate, af
ter transmission in the context of
quantum key distribution or after storage and retrieval in q
uantum memories, due to the single
photon component of a coherent state signal. Experimentall
y it requires measuring detection
error probabilities at 3 different mean photon numbers of th
e input coherent states:
μ
s
for signal
12
state, a non-zero
μ
d
1
for the first decoy state, and
μ
d
2
=0 (i.e. vacuum) for the second decoy
state. Closely following the treatment in (
35, 25
), here we outline the essential steps to derive
the lower bound on the single photon qubit storage fidelities
from experimentally measured
quantities.
First, we define the error rate as
E
ψ
= 1
−
F
ψ
=
C
ψ
/
(
C
ψ
+
C
φ
⊥
)
, where
F
is the fidelity
defined in the main text. The error
E
is thus the probability of detection in a wrong basis that
is orthogonal to the input state. The error rate
E
(1)
for the single photon component of the
coherent pulses is upper bounded by
E
(1)
U
given by
E
(1)
≤
E
(1)
U
=
E
(
μ
d
1
)
Q
(
μ
d
1
)
e
μ
d
1
E
(
μ
d
2
)
Q
(
μ
d
2
)
e
mu
d
2
(
μ
d
1
−
μ
d
2
)
Y
(1)
L
(12)
=
E
(
μ
d
1
)
Q
(
μ
d
1
)
e
μ
d
1
E
(0)
Y
(0)
μ
d
1
Y
(1)
L
,
where
Y
(0)
and
Y
(1)
L
are the zero-photon yield and the lower bound for the single p
hoton
yield, respectively.
μ
d
1
=0.26 and
μ
d
2
=0 (vaccum) are the mean photon numbers for the two
decoy states used in our experiment, and
E
(
μ
d
1
)
and
E
(
μ
d
2
)
=
E
(0)
are the corresponding error
probabilities obtained experimentally. The gain
Q
(
μ
)
is the detection probability for an echo
photon for each input, which reflects the mean photon in the in
put states, inefficiency of the
memory device, the coupling loss from the device to the fiber,
and all the losses in the optical
path up to and including the detector. Experimentally, the g
ain was measured by the total echo
photon count rate (per second) over the repetition rate of th
e input signal.
The lower bound on
Y
1
L
the single photon yield for the case of
μ
d
2
=0 is given by
Y
(1)
≥
Y
(1)
L
=
μ
s
μ
s
μ
d
1
−
μ
2
d
1
(
Q
(
μ
d
1
)
e
μ
d
1
−
μ
2
d
1
μ
2
s
Q
(
μ
s
)
e
μ
s
−
μ
2
s
−
μ
2
d
1
μ
2
s
Y
(0)
)
(13)
where
μ
s
=0.58 is the mean photon number of the signal state. Once the u
pper bound on the error
probability
E
(1)
U
is obtained from Eq. 11, the lower bound on the qubit fidelity i
s
F
(1)
L
= 1
−
E
(1)
U
,
which represents the fidelity we would have obtained if true s
ingle photons were used to encode
13
qubits at the memory input. Table I lists the measured fidelit
ies for retrieved qubits at different
input photon numbers, from which the single photon qubit fide
lities are calculated.
Input photon number
F
e
/
l
F
+
/
−
0.58
93.63
±
0.64% 94.41
±
0.58%
0.26
94.91
±
0.81% 92.58
±
0.65%
1
98.63
±
0.33% 95.91
±
0.41%
Table S1: Fidelities of the retrieved time-bin qubits.
14
0 10 20 30 40 50 60 70 80 90
0
0.02
0.04
0.06
β
32
0 10 20 30 40 50 60 70 80 90
3
4
5
6
T
z
(ms)
0 10 20 30 40 50 60 70 80 90
θ (degree)
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
ρ
1
after optical pumping
0
0.1
0.2
0.3
0.4
0.5
β
ID
θ
c axis
B
A
C
B
Figure S2:
Characterization of optical pumping properties in a 100 p.p
.m. bulk Nd:YVO
crystal.
(
A
)
β
32
extracted from absorption spectra of the 1-3, 2-3 transitio
ns. The inset shows
the magnetic field angle
θ
, definited with respect to crystal c axis. (
B
) Measured Zeeman life-
times
T
z
as the
1
/e
decay constants of the spectral hole depths. (
C
) Measured remaining ground
state population after optical pumping, from which we estim
ate the effective
β
ID
values at each
field angle.
15
3.40685
3.4069
3.40695
3.407
3.40705
GHz
×
10
5
0
0.1
0.2
0.3
0.4
Optical depth
10
20
30
40
50
60
70
80
90
θ (deg.)
0
a
c
b
d
Figure S3: Absorption spectrum of a bulk Nd:YVO at 480 mK (bot
tom panel) and the expected
transition frequencies as a function of magnetic field angle
θ
with respect to c axis (upper panel).
The dotted red lines indicate the expected frequencies of al
l four optical transitions. From the
intersections of the dotted and the black lines, we estimate
d the magnetic field angle.
16
0
50
100
-50
ns
10
0
10
1
10
2
10
3
10
4
Counts (/300s)
17 dB
echo
input
16 dB
0
100
200
300
storage time (ns)
0
0.9
1.8
2.7
AFC echo efficiency (%)
A
B
Figure S4: (
A
) A representative single-pulse input and AFC echo photon si
gnal measured in the
nanocavity.(
B
) AFC echo efficiency versus storage time. The results shown i
n Fig. 3D, E of the
main text are measured at the condition represented by the re
d data point.
17
References and Notes
1
.
H. J.
Kimble
,
The quantum internet
.
Nature
453
,
1023
–
1030
(
2008
).
doi:10.1038/nature07127
Medline
2
.
C. W.
Chou
,
H.
de Riedmatten
,
D.
Felinto
,
S. V.
Polyakov
,
S. J.
van Enk
,
H. J.
Kimble
,
Measurement
-
induced entanglement for excitation stored
in remote atomic ensembles
.
Nature
438
,
828
–
832
(
2005
).
doi:10.1038/nature04353
Medline
3
.
H.
de Riedmatten
,
M.
Afzelius
,
M. U.
Staudt
,
C.
Simon
,
N.
Gisin
,
A solid
-
state light
-
matter
interface at the single
-
photon level
.
Nature
456
,
773
–
777
(
2008
).
doi:10.1038/nature07607
Medline
4
.
A.
I.
Lvovsky
,
B. C.
Sanders
,
W.
Tittel
,
Optical quantum memory
.
Nat. Photonics
3
,
706
–
714
(
2009
).
doi:10.1038/nphoton.2009.231
5
.
W.
Tittel
,
M.
Afzelius
,
T.
Chaneliére
,
R. L.
Cone
,
S.
Kröll
,
S. A.
Moiseev
,
M.
Sellars
,
Photon
-
echo quantum memory in solid state systems
.
Laser Photonics Rev.
4
,
244
–
267
(
2010
).
doi:10.1002/lpor.200810056
6
.
E.
Saglamyurek
,
N.
Sinclair
,
J.
Jin
,
J. A.
Slater
,
D.
Oblak
,
F.
Bussières
,
M.
George
,
R.
Ricken
,
W.
Sohler
,
W.
Tittel
,
Broadband waveguide quantum memory for entangled photons
.
Nature
469
,
512
–
515
(
2011
).
doi:10.1038/nature09719
Medline
7
.
S.
Marzban
,
J. G.
Bartholomew
,
S.
Madden
,
K.
Vu
,
M. J.
Sellars
,
Observation of photon
echoes from evanescently coupled
rare
-
earth ions in a planar waveguide
.
Phys. Rev. Lett.
115
,
013601
(
2015
).
doi:10.1103/PhysRevLett.115.013601
Medline
8
.
G.
Corrielli
,
A.
Seri
,
M.
Mazzera
,
R.
Osellame
,
H.
de Riedmatten
,
Integrated optical memory
based on laser
-
written
waveguides
.
Phys. Rev. Appl.
5
,
054013
(
2016
).
doi:10.1103/PhysRevApplied.5.054013
9
.
C. W.
Thiel
,
T.
Böttger
,
R. L.
Cone
,
Rare
-
earth
-
doped materials for applications in quantum
information storage and signal processing
.
J. Lumin.
131
,
353
–
361
(
2011
).
doi:10.1016/j.jlumin.2010.12.015
10
.
Y.
Sun
,
C. W.
Thiel
,
R. L.
Cone
,
R. W.
Equall
,
R. L.
Hutcheson
,
Recent progress in
developing new rare earth materials for hole burning and coherent transient applications
.
J. Lumin.
98
,
281
–
287
(
2002
).
doi:10.1016/S0022
-
2313(02)00281
-
8
11
.
J. J.
Longdell
,
E.
Fraval
,
M. J.
Sellars
,
N. B.
Manson
,
Stopped light with storage times
greater than one second using electromagnetically induced transparency in a solid
.
Phys.
Rev. Lett.
95
,
063601
(
2005
).
doi:10.1103/PhysRevLett.95.0636
01
Medline
12
.
M. P.
Hedges
,
J. J.
Longdell
,
Y.
Li
,
M. J.
Sellars
,
Efficien
t quantum memory for light
.
Nature
465
,
1052
–
1056
(
2010
).
doi:10.1038/nature09081
Medline
13
.
M.
Afzelius
,
I.
Usmani
,
A.
Amari
,
B.
Lauritzen
,
A.
Walther
,
C.
Simon
,
N.
Sangouard
,
J.
Minár
,
H.
de Riedmatten
,
N.
Gisin
,
S.
Kröll
,
Demonstration of atomic
frequency comb
memory for light with spin
-
wave storage
.
Phys. Rev. Lett.
104
,
040503
(
2010
).
doi:10.1103/PhysRevLett.104.040503
Medline
14
.
T.
Zhong
,
J. M.
Kindem
,
E.
Miyazono
,
A.
Faraon
,
Nanophotonic coherent light
-
matter
interfaces based on rare
-
earth
-
doped crystals
.
Nat. Commun.
6
,
8206
(
2015
).
doi:10.1038/ncomms9206
Medline
15
.
T.
Zhong
,
J. M.
Kindem
,
J.
Rochman
,
A.
Faraon
,
Interfacing broadband photonic qubits to
on
-
chip cavity
-
protected rare
-
earth ensembles
.
Nat. Commun.
8
,
14107
(
2017
).
doi:10.1038/ncomms14107
Medline
16
.
T.
Zhong
,
J.
Rochman
,
J. M.
Ki
ndem
,
E.
Miyazono
,
A.
Faraon
,
High quality factor
nanophotonic resonators in bulk rare
-
earth doped crystals
.
Opt. Express
24
,
536
–
544
(
2016
).
doi:10.1364/OE.24.000536
Medline
17
. See the supplementary materials.
18
.
F.
Marsili
,
V. B.
Verma
,
J. A.
Stern
,
S.
Harrington
,
A. E.
Lita
,
T.
Gerrits
,
I.
Vayshenker
,
B.
Baek
,
M. D.
Shaw
,
R. P.
Mirin
,
S. W.
Nam
,
Detecting single infrared photons with 93%
system efficiency
.
Nat. Photonics
7
,
210
–
214
(
2013
).
doi:10.1038/nphoton.2013.13
19
.
M.
Sabooni
,
Q.
Li
,
S.
Kröll
,
L.
Rippe
,
Efficient quantum memory using a weakly absorbing
sample
.
Phys. Rev. Lett.
110
,
133604
(
2013
).
doi:10.1103/PhysRevLett.110.133604
Medline
20
.
M.
Afzelius
,
C.
Simon
,
Impedance
-
matched cavity quantum memory
.
Phys. Rev. A
82
,
022310
(
2010
).
doi:10.1103/PhysRevA.82.022310
21
.
S. A.
Moiseev
,
S. N.
Andrianov
,
F. F.
Gubaidullin
,
Efficient multimode quantum memory
based on photon echo in an optimal QED cavity
.
Phys. Rev. A
82
,
022311
(
2010
).
doi:10.1103/PhysRevA.82.022311
22
.
D. L.
McAuslan
,
J. J.
Longdell
,
M. J.
Sellars
,
Cavity QED using rare
-
earth
-
metal
-
ion dopants
in monol
ithic resonators: What you can do with a weak oscillator
.
Phys. Rev. A
80
,
062307
(
2009
).
doi:10.1103/PhysRevA.80.062307
23
.
B.
Lauritzen
,
S. R.
Hastings
-
Simon
,
H.
de Riedmatten
,
M.
Afzelius
,
N.
Gisin
,
State
preparation by optical pumping in erbium
-
doped solids using stimulated emission and
spin mixing
.
Phys. Rev. A
78
,
043402
(
2008
).
doi:10.1103/PhysRevA.78.043402
24
.
M.
Afzelius
,
M. U.
Staudt
,
H.
de Riedmatten
,
N.
Gisin
,
O.
Guillot
-
Noël
,
P.
Goldner
,
R.
Marino
,
P.
Porcher
,
E.
Cavalli
,
M.
Bettinelli
,
Efficient optical pumping of Zeeman spin
levels in Nd
3+
:YVO
4
.
J. Lumin.
130
,
1566
–
1571
(
2010
).
doi:10.1016/j.jlumin.2009.12.026
25
.
N.
Sinclair
,
E.
Saglamyurek
,
H.
Mallahzadeh
,
J. A.
Slater
,
M.
George
,
R.
Ricken
,
M. P.
Hedges
,
D.
Oblak
,
C.
Simon
,
W.
Sohler
,
W.
Tittel
,
Spectral multiplexing for
scalable
quantum photonics using an atomic frequency comb quantum memory and feed
-
forward
control
.
Phys. Rev. Lett.
113
,
053603
(
2014
).
doi:10.1103/PhysRevLett.113.053603
Medline
26
.
Z.
-
Q.
Zhou
,
W.
-
B.
Lin
,
M.
Yang
,
C.
-
F.
Li
,
G.
-
C.
Guo
,
Realiz
ation of reliable solid
-
state
quantum memory for photonic polarization qubit
.
Phys. Rev. Lett.
108
,
190505
(
2012
).
doi:10.1103/PhysRevLett.108.190505
Medline
27
.
N.
Sinclair
,
K.
Heshami
,
C.
Deshmukh
,
D.
Oblak
,
C.
Simon
,
W.
Tittel
,
Proposal and proof
-
of
-
principle demonstration of non
-
destructive detection of photonic qubits using a