Supplemental material for optically addressing single rar
e-earth ions in a
nanophotonic cavity
Tian Zhong,
1, 2, 3,
∗
Jonathan M. Kindem,
1, 2
John G. Bartholomew,
1, 2
Jake Rochman,
1, 2
Ioana Craiciu,
1, 2
Varun Verma,
4
Sae Woo Nam,
4
Francesco Marsili,
5
Matthew D. Shaw,
5
and Andrei Faraon
1,2,
†
1
Kavli Nanoscience Institute and Thomas J. Watson, Sr., Labo
ratory 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
Institute for Molecular Engineering, University of Chicag
o, Chicago, IL 60637, USA.
4
National Institute of Standards and Technology,
325 Broadway, MC 815.04, Boulder, Colorado 80305, USA
5
Jet Propulsion Laboratory, California Institute of Techno
logy,
4800 Oak Grove Drive, Pasadena, California 91109, USA
(Dated: October 8, 2018)
DETAILS ON THE EXPERIMENTAL SETUP
Figure S1 illustrates the experimental setup with more
details. The Nd
3+
:YVO
4
sample crystal was soldered
with indium onto a copper plate that was mounted on
top of a 3-axis nanopositioner, and was thermally con-
nected to the 20 mK base plate of the dilution refrig-
erator. Cavity tuning was realized by gas condensation
using N
2
gas. The gas tube (brown line in Fig. S1) was
thermally anchored to the 3.8 K stage. When performing
gas tuning, a heater on the gas tube heats it up to
>
30
K to allow gas to flow through. The heater was turned
off after tuning. This configuration was used to minimize
the heat load generated by the tube, allowing the lowest
possible temperature at the sample.
Fiber-waveguide coupling efficiency
The coupling to
the waveguide was realized by a 45
◦
-angled cut into the
YVO
4
substrate [1]. This angled cut allows the verti-
cally focused beam (from a fiber) to be total internally
reflected into the nanobeam waveguide. The aspherical
doublet has an effective focal length of 10.5 mm on the
fiber side and 2.9 mm on the sample side (Fig.S1). The
coupling was optimized by fine scanning the focal spot on
the sample surface using the nanopositioner and looking
for the cavity resonance on a spectrometer. The overall
fiber-waveguide coupling efficiency was characterized by
measuring the reflection of a pulse far off resonance with
the cavity (i.e. in the photonic bandgap). The pulse
propagated from point 1 (marked in Fig.S1) to 2, 3, and
was reflected back to 2, then 4. The transmission effi-
ciency from 1 to 2 (64.1%), and from 2-4 (51.7%) were
directly measured. The only unknown was the coupling
efficiency from 2 to 3. This coupling efficiency could be
uniquely determined from the total pulse reflection, and
was found to be 19%.
Photon collection efficiency
For each photon emitted
by an ion into the cavity, the probability of that pho-
ton to be transmitted to the coupling waveguide was
κ
in
/κ
=45%. The photon then propagated from 3 to 2
(19% waveguide-fiber coupling), and from 2 to 4 (79.9%
transmission through all fiber slices/connectors and
64.7% transmission through the optical circulator), and
was finally detected by the 82%-efficient superconducting
nanowire detector. Thus, the overall collection efficiency
for a cavity photon was 0.45
×
0.19
×
0.80
×
0.65
×
0.82 =
3.6%. To improve this efficiency, further refinement in
fabrication is needed to achieve a highly over-coupled
cavity with
κ
in
/κ >
80%, which should be within reach
since we have achieved symmetric two-sided cavities of
Q as high as 20,000 [2]. Furthermore, the use of ta-
pered fiber-waveguide coupling could improve the fiber-
to-device coupling to 97% [3]. These combined with lower
loss fiber components could increase the overall efficiency
to
∼
70%.
Cavity mean photon number
To obtain the cavity
mean photon number in Fig.3, we first calculate the peak
power of the excitation pulse in the waveguide i.e. P
in
with knowledge of the transmission from 1 to 2 and the
coupling efficiency from 2 to 3. The cavity mean photon
number was ̄
n
= 4
P
in
κ
in
/
~
ω
0
κ
2
, where
κ
in
= 2
π
×
40
GHz was the cavity in-coupling rate,
κ
= 2
π
×
90 GHz
was the total cavity decay rate, and
ω
0
is the photon
frequency.
Optical properties of Nd
3+
in YVO
4
and Purcell
factor
The absorption area
∫
α
(
ν
)
dν
for the
4
I
9
/
2
(Z
1
)
↔
4
F
3
/
2
(Y
1
) optical transition in a 10 ppm
Nd
3+
:YVO
4
sample was measured in zero mag-
netic field with E
k
c by Sun
et al.
[4] to be
3.4 cm
−
1
cm
−
1
= 102 GHz cm
−
1
. Because the
electric dipole transition is heavily
π
-polarized, we can
calculate the transition oscillator strength from the
absorption area
∫
α
(
ν
)
dν
using [5, 6]
f
= 4
πǫ
0
m
e
c
πe
2
1
N
n
χ
L
2
∫
α
(
ν
)
dν ,
(1)
2
AOM
AOM
T = 100 mK
c
SNSPD
c
i
rculat
or
d
evice
s
i
n
gle mod
e fibe
r
Po
l
.
Cont
roller
c
i
rculat
or
Po
a
b
4
OD at
tenu
ato
r
T = 20 mK
3
axis n
ano
posit
i
o
ner
T = 3.8 K
p
erman
ent mag
nets
N
2
gas
FIG. S1: Details of the experimental setup. SNSPD:
superconducting nanowire single photon detector.
where
ǫ
0
is the vacuum permittivity,
m
e
is the mass of
the electron,
e
is the charge on the electron,
c
is the
speed of light,
N
is the number density,
χ
L
is the local
field correction,
n
is the refractive index, and
α
(
ν
) is the
absorption coefficient as a function of frequency
ν
.
For the 10 ppm Nd
3+
:YVO
4
crystal measured by Sun
et al.
[4],
N
= 1
.
24
×
10
23
m
−
3
and
n
= 2
.
1785 for light
polarized along the
c
-axis of the crystal. The local field
correction factor
χ
L
usually takes one of two forms in
the literature, depending on whether the virtual cavity
or real cavity model is used [7–10]:
χ
(
V
)
L
=
n
2
+ 2
3
,
(2)
is the virtual cavity model, and
χ
(
R
)
L
=
3
n
2
2
n
2
+ 1
,
(3)
is the real cavity model correction.
Both the virtual and real cavity models are approxima-
tions to the full local field correction in that they assume
that the field due to the polarization of atoms nearby in
the lattice is zero [10]. The real cavity model has been
shown to be suitable for substitutional ions [11] includ-
ing rare-earth ions in crystalline hosts [7]. In this work,
the predictions based on the real cavity model are more
consistent with our current knowledge of material and
experimental results.
We note that there is inconsistency in the literature
regarding oscillator strengths and dipole moments of 4f -
4f transitions for rare-earth ions in crystals because val-
ues and expressions are not always explicit as to which
local field correction, if any, is assumed. Here we detail
our derivations to make it clear the assumptions we have
made and how that impacts the theoretical predictions.
When assuming the real cavity model for the local field
correction, from Eq. 1 we calculate an oscillator strength
f
= 3
.
7
×
10
−
5
. For the applied field of 390 mT along
the a-axis we expect optical transitions 2 and 3 to be
forbidden. In this case, transitions 1 and 4 each have an
oscillator strength
f
= 3
.
7
×
10
−
5
.
The radiative lifetime
T
rad
is related to the oscillator
strength
f
by [5, 6]
1
T
rad
=
2
πe
2
ǫ
0
m
e
c
(
3
n
2
2
n
2
+ 1
)
2
1
n
n
2
λ
2
f
3
,
(4)
which gives a value of
T
rad
= 237
μ
s. Given the lifetime of
the
4
F
3
/
2
(Y
1
) state measured by fluorescence was
T
1
=
90
μ
s, the branching ratio of emission to the
4
I
9
/
2
(Z
1
)
state is
β
= 0
.
38.
The transition dipole moment
μ
is related to the oscil-
lator strength
f
by [5, 6]
μ
=
√
~
e
2
f
2
m
e
ω
,
(5)
where
ω
= 2
πc/λ
is the frequency
4
I
9
/
2
(Z
1
)
↔
4
F
3
/
2
(Y
1
)
optical transition. Equation 5 differs from the expres-
sions relating
μ
to
f
in [10]. This is because in [10] no
field corrections are assumed in relating
f
to
∫
α
(
ν
)
dν
.
Using Equation 5, we calculate a dipole moment
μ
=
1
.
59
×
10
−
31
C
·
m.
Given
g
0
and the cavity energy decay rate
κ
= 2
π
×
90 GHz, the lifetime of the Nd
3+
ion in the cavity is given
by
T
cav
=
(
4
g
2
0
κ
+
1
−
β
T
1
)
−
1
.
(6)
Therefore, the predicted
T
cav
= 1
.
25
μ
s. The Purcell
enhancement factor of the resonant transition is derived
to be 4
g
2
0
T
rad
/κ
= 189.
PHOTON ECHO MEASUREMENTS
Two pulse photon echo measurements were performed
on an ensemble of ions in the cavity near the center of
inhomogneous distribution (e.g. line 1). The cavity reso-
nance was tuned to different frequencies using a gas con-
densation technique to obtain homogeneous linewidths of
ions at varying Purcell enhancement conditions. Fig.S2
plots the photon echo decays at ensemble-cavity detuning
3
of
δ
∼
22 and
∼
50 GHz. Oscillations in the echo inten-
sities correspond to syperhyperfine interactions between
Nd spins and Y nuclear spins [12] at 740 kHz, which
agree with the beat frequency observed in the Ramsey
interference fringes. Note that the period of the oscilla-
tions appears to be twice long in the Ramsey fringes than
in the echo decays, because the photon echo is emitted
after twice the delay between two pulses. The T
2
were
fitted from the linear section of the decay, which started
after approximately 4
μ
s.
0
2
4
6
8
10
(
s)
4
5
6
7
8
9
10
Log of echo counts
T
2
= 14.9
μ
s,
δ
= 50 GHz
T
2
= 8.0
μ
s,
δ
= 22 GHz
FIG. S2: Photon echoes on sub-ensembles of Nd ions
coupled to the cavity at different cavity-ensemble
detunings. The oscillations evident in the initial echo
intensity decays were due to superhyperfine couplings
between Nd and Y spins. T
2
was fitted from the linear
decay sections (
>
4
μ
s). A difference in T
2
at varying
detunings reflects the change of radiative decay rates
(i.e.T
1
) under different Purcell enhancement factors.
MODELLING SUPERHYPERFINE COUPLINGS
The superhyperfine interaction is between an electronic
spin (Nd in this case) and a neighbouring ligand nu-
clear spin (yttrium or vanadium). This interaction has
been studied in the literature and was typically orders of
magnitude weaker than the rare-earth electronic Zeeman
interactions (the gyromagnetic ratio of Nd
3+
is 3.9-33
GHz/T). As a result, the associated Hamiltonian can be
treated as a perturbation to the electronic Zeeman cou-
pling of Nd
3+
under external applied magnetic field [13].
H
′
=
−
μ
Y
·
(
B
−
μ
0
4
π
[
h
μ
Nd
i
r
3
−
3
(
h
μ
Nd
i ·
r
)
·
r
r
5
))
(7)
where
μ
Y
is the Y
3+
nuclear spin magnetic moment,
B
the externally applied magnetic field,
μ
Nd
the Nd
3+
elec-
tronic spin and
r
the vector connecting two spins. The
superhyperfine coupling at zero field - the terms in Eq. (7)
that are not dependent on
B
- causes each of the Zeeman
levels (both ground and excited) to split into a nuclear
doublet. Such splitting can be readily calculated from
the electronic magnetic dipole moment from the known
anisotropic g-factors of Nd. As the applied field increases,
eventually the term
−
μ
Y
·
B
dominates over other terms
and the splitting becomes approximately linear with
B
.
Thus, in high fields, the superhyperfine splitting strongly
depends on the gyromagnetic ratios of specific ligand nu-
clear spins. For yttrium it is 2.1 MHz/T. For vanadium
it is 11.2 MHz/T. At B = 390 mT along a-axis, the ex-
pected superhyperfine splitting of each of the Zeeman
branches can be estimated based on atomic coordinates
of the yttrium or vanadium ions surrounding the Nd cen-
ter.
For Nd-Y coupling, there are 4 nearest neighbour Y
ions at equal distance of 3.9
̊
A from each Nd ion. The
zero-field superhyperfine splitting is calculated to be
∼
80
kHz and
∼
30 kHz for the optical ground and excited lev-
els, respectively. With an applied field of 390 mT along
a-axis, the total splitting from Eq. (7) gives ∆
g
∼
740 kHz
and ∆
e
∼
790 kHz for ground and excited levels.
For Nd-V coupling, the nearest distance between them
is 3.14
̊
A. Since vanadium has 7/2 spins, there are a
total of 8 superhyperfine sublevels. The splittings be-
tween those levels at 390 mT field ranges from
∼
4 -30
MHz. Given that the optical excitation pulse only has a
bandwidth of 2 MHz, it is unlikely that multiple Nd-V
superhyperfine sublevels were excited.
The envelope modulation in Fig. 3(b) can be modelled
after derivations in [14], which include beatings at fre-
quencies ∆
e
, ∆
g
, (∆
e
−
∆
g
) and (∆
e
+ ∆
g
), with their
relative strengths dependent on the degree of spin mixing.
Due to limited sampling and likely over-simplification of
the model, the experimental data (black) in Fig. 3(b)
cannot be fitted well with a known analytical form. In-
stead, an empirical fit (red) was used to identify the dom-
inant beat frequency. The best fit gives 740 kHz, which is
in general agreement with the ∆
e
, ∆
g
calculated above.
We therefore infer that the observed beating was likely
originated from the Nd-Y superhyperfine interaction but
unlikely from Nd-V couplings. The latter is expected to
give a splitting
>
4 MHz. The Nd-V superhyperfine cou-
pling was indeed reported in [15] in the same Nd
3+
:YVO
4
material to be about 5 MHz at 0.3 T field.
OPTICAL DEPHASING IN Nd
3+
:YVO
4
Possible contributions to the optical dephasing
γ
∗
=9.7
kHz include superhyperfine coupling between Nd spins
and yttrium/vanadium nuclear spins, the Nd spin flip-
flops, direct phonon couplings, and other higher order
processes. Here we discuss contributions from two po-
4
tentially dominant mechanisms.
Superhyperfine interaction
The experimental condi-
tion in the current work closely reassembles that in [4] in
which optical T
2
for a 10 ppm doped Nd
3+
:YVO
4
sam-
ple was measured at varying magnetic field applied along
the a-axis of the crystal. It was found in [4] that with a
field greater than 1.5 T, the T
2
of 27
μ
s became limited
by the Nd-Vanadium superhyperfine interaction. The
corresponding dephasing rate could be calculated from
1/(
π
T
2
) - 1/(2
π
T
1
) = 10.0 kHz, which was very close to
the dephasing rate measured here. We thus expect that
the superhyperfine interactions contribute substantially
to the measured
γ
∗
. Based on the spin interaction mod-
els in [16], we could numerically estimate the broaden-
ings due to Nd-Y and Nd-V interactions to be 34 and 14
kHz, respectively; these values are in order-of-magnitude
agreement with the measurement.
Nd spin flip-flops
Dephasing due to the Nd spin flip-
flops is a function of the Nd doping concentration and
temperature. To better understand this process, we mea-
sured optical T
2
times in both the 50 ppm doped (the
same crystal on which the devices were fabricated) and
a nominally undoped YVO
4
crystal. From the absorp-
tion spectroscopy and secondary ion mass spectroscopy
(SIMS), we estimated the doping concentration of Nd
to be
≈
0.2 ppm in the undoped YVO
4
. Therefore, the
dephasing owing to spin flip-flops is expected to be rela-
tively small in that sample. Both crystals were soldered
to a common sample holder. With the same magnetic
field configuration as in the main text, the ground level
splitting was
μ
B
g
⊥
B
=12.88 GHz where
g
⊥
=2.36 is the
ground state g-factor [17, 18], and
B
= 0.39 T. We then
used the ratio between the absorptions of two Zeeman
transitions to calibrate the crystal temperature. When
both crystals were at
∼
500 mK, we measured a T
doped
2
= 25.4
μ
s and T
undoped
2
= 27.0
μ
s in 50 ppm doped and
undoped YVO
4
crystals, respectively. The difference in
linewidths, which amounts to
<
1 kHz, serves as an upper
bound on dephasing due to Nd spin flip-flops at 500 mK.
Using the models put forth by [16, 19], the optical de-
phasing due to Nd-Nd spin flip-flops can be estimated
from a Lorentzian spectral diffusion model as 1
/πT
M
[19]
where
T
M
=
2Γ
0
Γ
SD
R
(
−
1 +
√
1 +
Γ
SD
R
π
Γ
2
0
)
,
(8)
where Γ
SD
is the Nd magnetic dipolar interaction,
Γ
SD
=
πμ
0
|
g
g
−
g
e
|
g
g
μ
2
B
n
Nd
9
√
3
~
sech
2
(
g
g
μ
B
B
2
kT
) (9)
and
R
= 1
/T
spin
1
. Using
n
Nd
= 6
.
3
×
10
23
m
−
3
as the Nd
concentration at 50 ppm doping, and Nd spin T
spin
1
=98
ms measured from spectral holeburning at 500 mK, we
predict a corresponding optical dephasing to be 30 Hz
[19]. This implies that our measurement is dominated by
other interactions, such as the superhyperfine interaction.
∗
Electronic address:
tzh@uchicago.edu
†
Electronic address:
faraon@caltech.edu
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