of 7
λ
(nm)
875
880
885
Transmission spectrum (a. u.)
0
500
1000
1500
2000
2500
10 μm
a
b
Nd:YSO resonator array
4
F
3/2
-
4
I
9/2
Supplementary Figure 1
.
An array of nano-beam resonators fabricated in Nd:YSO.
(
a
)
Scanning electron microscope image of 6 devices. (
b
) Measured resonance modes (color matched
to the corresponding device) near the 883 nm transition of Nd:YSO (grey line).
t (
μ
s)
0
500
1000
Counts
10
0
10
1
λ
=0 nm
λ
=0.4 nm
Supplementary Figure 2
.
Lifetime measurements for a 0.003% doped Nd:YSO res-
onator.
Lifetime changes from 289
μ
s when the cavity of
Q
=1,500 is detuned by ∆=0.4 nm, to
200
μ
s when the cavity is resonant with the ions.
1
time
τ
echo
π
/2
π
/2
π
/2
τ
T
w
T
w
(
μ
s)
02
04
06
0
Echo decay time 4
τ
(
μ
s)
65
70
75
80
85
90
95
T
w
(
μ
s)
02468
Echo decay time 4
τ
(
μ
s)
2
2.2
2.4
2.6
2.8
3
Γ
h
= 100 kHz
R
= 6.1 kHz
μ
s
-1
R
= 380 Hz
μ
s
-1
Γ
h
= 3.4 kHz
a
b
c
Supplementary Figure 3
.
Spectral diffusion of cavity-coupled ions
(
a
) Three-pulse photon
echo sequence (
π/
2 -
π/
2 -
π/
2). (
b, c
)
T
w
dependent broadening of the effective linewidths of ions
coupled to the 0.2% (
b
) and 0.003% (
c
) doped cavities, measured via three-pulse photon echoes.
Linear fits indicate the spectral diffusion rates
R
for both doping concentrations.
<n
cav
>
10
-6
10
-5
10
-4
Normalized transmission on resonance
0
0.2
0.4
0.6
0.8
1
λ
(nm)
882.8
883
883.2
0
1
Supplementary Figure 4
.
Controlled cavity transmission versus intra-cavity photon
number for probe laser at zero detuning.
The cavity transmission is normalized by empty-
cavity transmission at the zero detuning. Saturation occurs at
n
cav
〉≈
2
×
10
5
. The inset shows
a transmission spectrum at onset of saturation with a normalized transmission at the dip of
54%.
2
Detuning (kHz)
-10
-5
0
5
10
Count (/s)
0
100
200
300
400
500
600
700
800
15 kHz
Supplementary Figure 5
.
Cavity transmission when coupled to a single Nd
+3
ion.
The simulation uses Quantum Optics Toolbox [5] with parameters
Q
=4,400, Γ
h
=3.1 kHz, and
g= 2
π
×
10 MHz. The transmission dip has a full-width at half-maximum (FWHM) of
15 kHz.
Supplementary Note 1.
Fabrication and characterization of arrays of YSO nano-beam resonators.
The
YSO nano-resonators were fabricated in batch with careful focused ion beam (FIB) alignment
and drift compensation. Supplementary Figure 1a shows a SEM image of an array of six
Nd:YSO nano-resonators. All of the devices have resonance modes near the designed 883
nm wavelength, shown in the transmission spectrum in Supplementary Figure 1b. The color
of each spectrum maps to that of the device in Supplementary Figure 1a. Measured quality
factors in this batch range from 1,500 to 4,400. The spread of resonance wavelengths is
about 10 nm, indicating the robustness of this fabrication process.
Supplementary Note 2.
Measurements of inhomogeneous linewidths for cavity-coupled ions.
The inho-
mogeneous linewidth for the ions coupled to the 0.2% nano-cavity was measured to be
inhom
=16.0 GHz from the dipole-induced transparency signal in Fig. 4b. For the 0.003%
cavity, the ion density was too low for a similar measurement of the inhomogeneously broad-
ened distribution. Instead, the linewidth of photoluminescence (PL) from the cavity was
3
measured with a high resolution spectrometer, and ∆
inhom
=5.9 GHz was estimated by de-
convolving the PL signal with the minimally resolvable linewidth of the spectrometer. For
both doping levels, the same inhomogeneous linewidths were measured from the bulk via
absorption spectroscopy. The agreement between the inhomogeneous linewidth of the cavity-
coupled Nd ions and the bulk confirms the excellent spectral stability of REIs when embed-
ded in nanophotonic resonators.
Supplementary Note 3.
Requirement on the inhomogeneous linewidths for scalable QLMIs.
We consider a
network of QLMIs each being a nano-cavity coupled to ensembles of emitters with inhomo-
geneous linewidth ∆
inhom
. Efficient QLMIs require the emitters to emit photons dominantly
into the cavity mode. The cavity photons in a single spatial mode could then be efficiently
coupled to waveguides or fibres for routing to other QLMIs operating at the same frequency.
The probability of an emitter to emit a photon into the cavity mode is
βF/
(1 + (
F
1)
β
),
where
β
is the branching ratio of the dipole transition, and
F
is the Purcell factor in Eq. 2.
Assuming 99% of the dipole emission into the cavity and typical branching ratio of
β
10%,
the required Purcell factor should be
1,000. Considering a photonic crystal nano-cavity
with a small mode volume of 1(
λ/n
)
3
,
F
1
,
000 corresponds to a quality factor
Q
1
×
10
4
and a cavity linewidth
κ
30 GHz (for the 883 nm transition). Thus, for scalability, the
emitters and cavities need to be aligned within
10 GHz (order of magnitude). This limits
the inhomogeneous broadening of the dipole ensembles to be
<
10 GHz for implementing
robust and scalable QLMIs. This condition is satisfied by most REI transitions.
Supplementary Note 4.
Calculation of ensemble averaged Purcell enhancement factor.
The spontaneous
emission rate of a dipole coupled to a nano-resonator is enhanced relative to the bulk medium,
by the factor
1 +
βF
[1], where
β
is the branching ratio of the transition, and
F
is given
by [2],
F
=
F
cav
(
E
(
r
)
·
μ
|
E
max
||
μ
|
)
2
1
1 + 4
Q
2
(
λ/λ
cav
1)
2
(1)
4
where
μ
is the dipole moment,
E
(
r
) is the local electric field at the emitter location r,
λ
cav
is the cavity resonant wavelength,
λ
is the emitter wavelength, and
|
E
max
|
is the maximum
electric field in the resonator. For a dipole that is resonant with the cavity and ideally
positioned and oriented with respect to the maximum cavity field,
F
cav
=
3
4
π
2
(
λ
cav
n
)
3
Q
V
mode
.
(2)
We consider an ensemble of Nd ions uniformly distributed inside the YSO cavity. The
enhancement of the emission from the ensemble can be estimated by averaging
F
cav
(Eq. 1)
over the entire population of Nd ions in the cavity. Based on the 3 dimensional field profile in
Fig. 1b, the mode volume 1.65(
λ/n
)
3
and
Q
=4,400, we numerically calculate this averaged
Purcell factor to be 45 when the cavity is resonant with the transition.
If the emission rate for uncoupled Nd ions is 1
0
= 1
883
+ 1
other
, in the coupled case
the rate becomes 1
c
= (1+
F
)
883
+1
other
, where 1
883
and 1
other
are the spontaneous
emission rates into the 883 nm transition and other 4f-4f transitions, respectively. The
Purcell factor is then experimentally extracted as
F
= (
τ
0
c
1)
, where
β
is the branching
ratio of the 883 nm line. Based on the measured branching ratio
β
= 4.5%, the observed
change in lifetimes leads to an ensemble averaged Purcell factor
42, which matches well
with the calculated value. Furthermore, the averaged value of 42 means the expected Purcell
enhancement for an ideally positioned and oriented Nd dipole is
F
cav
200.
Supplementary Note 5.
Purcell enhancement in 0.003% doped Nd:YSO nano-resonators.
A 0.003% doped
Nd:YSO nano-resonator was fabricated, measuring a resonance mode at 879 nm with quality
factor
Q
=1,500. Spontaneous emission rate enhancement in this cavity was estimated from
lifetime measurements in the same way as for the 0.2% cavity. As shown in Supplementary
Figure 2, a change of lifetime from 289
μs
when the cavity is detuned by ∆ =0.4 nm, to
195
μs
at resonance gives rise to an ensemble averaged Purcell factor
F
10. Note that
a longer
T
1
=290
μs
in the low density Nd:YSO bulk sample (
T
1
= 300
μs
reported in [3])
yields a slightly larger branching ratio
β
5
.
4%. The longer
T
1
is most likely due to weaker
dipole-dipole interactions in 0.003% doped sample.
5
Supplementary Note 6.
Spectral diffusion of Nd
3+
ions coupled to the nano-cavities.
The dynamic coherence
properties of the cavity-coupled Nd ions were investigated by three-pulse photon echoes
((
π/
2 -
π/
2 -
π/
2)) that gives information about the spectral diffusion on time scales up
to
T
1
[4] (Supplementary Figure 3a). The third pulse, delayed by a time
T
w
after second
pulse, is diffracted on the spectral grating from the first two pulses and produces an echo.
Spectral diffusion - frequency shifts of the optical transition due to the fluctuating rare earth
environment - gradually erases the grating during
T
w
, and causes faster echo decays thus
broadening of the effective linewidth Γ
eff
. Linearly increasing Γ
eff
= Γ
h
+
RT
w
at a rate
of
R
= 380 Hz
μ
s
1
was measured for the 0.003% cavity (Supplementary Figure 3b) and
6.1 kHz
μ
s
1
for the 0.2% cavity (Supplementary Figure 3c). Higher spectral diffusion is
expected for higher doping because of stronger dipole-dipole interaction between Nd ions.
Nevertheless, the measured linewidth broadening is much smaller than our Rabi frequency
(
6 MHz in Fig. 3c). This indicates the coupled ions, either singles or ensembles, can
be optical addressed repeatedly up to 10s of
μ
s, which is desirable for optical quantum
information processing.
Supplementary Note 7.
REI-controlled cavity transmission and saturation of the coupled ions.
Supple-
mentary Figure 4 plots the on resonance transmission at zero detuning as a function of the
average photon number in the cavity
n
cav
.
n
cav
was estimated from the input probe laser
power P
in
(measured after the objective), the coupler efficiency
η
, and cavity coupling rate
(
κ/
2) as
n
cav
=
ηP
in
̄
. Black curve is the theoretical calculation using the Quantum
toolbox [5], which shows close agreement with the experiment.
Supplementary Note 8.
Towards detection and control of single REI ions coupled to the nano-cavity.
The
measurement of
N
(∆
λ
) indicate that this system can be used to detect and control a single
ion coupled to the cavity - a key ingredient for realizing quantum networks interconnecting
6
multiple quantum bits encoded in individual REI ions. In the 0.003% low density devices,
we estimated a peak ion density of
N
=0.07 per Γ
h
=1/
πT
2
=3.1 kHz. Correspondingly, the
single ion cooperativity of
η
=1.6 can be attained with the same cavity
Q
=4,400,
V
=
1
.
65(
λ/n
)
3
, and
g
= 2
π
×
10 MHz (typical for REI transitions and we assume the ion is
positioned at maximum cavity field). Simulation using Quantum Optics Toolbox [5] yields
a transmission dip
>
80% due to a single Nd ion, as shown in Supplementary Figure 5. The
main technical challenge to detect single ion in this cavity system is the requirement of a
highly stabilized laser, with linewidth
<
1 kHz and minimal long term drift, for scanning
the single ion spectrum, which should be attainable with state of the art laser spectroscopy
technology.
Supplementary References
[1] Faraon, A., Barclay, P. E., Santori, C., Fu, K. C., & Beausoleil, R. G. Resonant enhancement
of the zero-phonon emission from a colour centre in a diamond cavity.
Nat. Photon.
5,
301-305
(2011).
[2] Purcell, E. M. Spontaneous emission probabilities at radio frequencies.
Phys. Rev.
69,
681
(1946).
[3] Usmani, I, Afzelius, M., de Riedmatten, H., & Gisin, N. Mapping multiple photonic qubits into
and out of one solid-state atomic ensemble.
Nat. Commun.
1,
12 (2010).
[4] Perrot, A.
et al.
Narrow Optical Homogeneous Linewidths in Rare Earth Doped Nanocrystals.
Phys. Rev. Lett.
111,
203601 (2013).
[5] Tan S. M. A computational toolbox for quantum and atomic optics.
J. Opt. B: Quantum
Semiclass. Opt.
1,
424 (1999).
7