Coherent control and single
-
shot readout of
a
rare
-
earth
ion
embedded in
a
nanophotonic
cavity
Jonathan M. Kindem
1,2
, Andrei Ruskuc
1,2
, John G. Bartholomew
1,2
, Jake Rochman
1,2
, Yan Qi
Huan
1,2
, Andrei Faraon
1,2,
*
Affiliations:
1
Kavli Nanoscience Institute
and Thomas J. Watson, Sr., Laboratory 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.
*Correspon
dence to:
faraon@caltech.edu
Abstract:
Quantum networks based on optically addressable spin qubits promise to enable secure
communication, distributed quantum computing, and tests of fundamental physics. Scaling up
quantum networks based on solid
-
state luminescent centers requires coherent spin
and optical
transitions coupled to photonic resonators. Here we investigate single
Yb
!"!
#
$
ions in yttrium
orthovanadate coupled to a nanophotonic cavity. These ions possess optical and spin transitions
that are first
-
order insensitive to magnetic field f
luctuations, enabling
optical linewidths less than
1 MHz and spin coherence times exceeding 30 ms for cavity
-
coupled ions
. The cavity
-
enhanced
optical emission rate facilitates efficient spin initialization and conditional single
-
shot readout
with
fidelity
greater than 95%. These results showcase a solid
-
state platform based on single
coherent rare
-
earth ions for the future quantum internet.
Main text:
The distribution of entanglement over long distances using optical quantum networks is
an intrigui
ng macr
oscopic quantum phenomenon
with applications in quantum systems for
advanced computing and secure communication
(
1
,
2
)
. Solid
-
state emitters coupled to photonic
resonators
(
3
)
are promising candidates for implementing quantum light
-
matter interfaces
necessary for scalable quantum networks. A variety of systems have been investigated for this
purpose, including quantum dots and defects in diamond or silicon carbide
(
4
–
8
)
. So far, the
ability to scale up these systems has remained
elusive and motivates the development of
alternat
iv
e platforms. A central challenge is identifying emitters that exhibit coherent optical and
spin transitions while coupled to photonic cavities that enhance the optical transitions and
arXiv:1907.12161v1 [quant-ph] 28 Jul 2019
channel emission into
optical fibers. Ensembles of rare
-
earth ions (REIs) in crystals are known to
possess highly coherent 4f
-
4f optical and spin transitions
(
9
,
10
)
, but only recently have single
REIs been isolated
(
11
,
12
)
and coupled to nanocavities
(
13
,
14
)
. The crucial next steps toward
using single REIs for quantum networks are demonstrating long spin coherence and single
-
shot
readout in photonic re
sonators.
Here we demonstrate spin initialization, coherent optical and spin manipulation, and
high
-
fidelity single
-
shot optical readout of the hyperfine spin state of single
Yb
!"!
#
$
ions
coupled to a nanophotonic cavity fabricated in an yttrium orthovanad
ate (YVO) host crystal. The
relevant energy level structure of
Yb
!"!
#
$
in YVO
is shown in Fig. 1A (see Fig. S
2 and Ref.
(
15
)
for additional details).
Yb
!"!
#
$
directly substitutes for
Y
#
$
in a
site
that
has non
-
polar
symmetry (D
2d
), which reduces the sensitivity to electric field fluctuations that can cause optical
decoherence. At zero applied magnetic field
,
the hyperfine interaction partially lifts t
he
degeneracy of the ground state
퐹
(
"
(
)
(
0
)
leading to coupled electron
-
nuclear spin states of the
form
|
0
⟩
/
=
|
↓
⇑
⟩
−
|
↑
⇓
⟩
√
2
,
|
1
⟩
/
=
|
↓
⇑
⟩
+
|
↑
⇓
⟩
√
2
,
and
|
푎푢푥
⟩
/
=
|
↑
⇑
⟩
,
|
↓
⇓
⟩
.
Here we denote the electron spin as
|
↑
⟩
=
B
푆
D
=
!
(
E
,
|
↓
⟩
=
B
푆
D
=
−
!
(
E
and the nuclear spin as
|
⇑
⟩
=
B
퐼
D
=
!
(
E
,
|
⇓
⟩
=
B
퐼
D
=
−
!
(
E
.
We use states
|
0
⟩
/
and
|
1
⟩
/
, which are
separated by ~675 MHz,
to form the spin qubit. The
|
0
⟩
/
and
|
1
⟩
/
states
have zero net magnetic moment and as a result
the
|
0
⟩
/
→
|
1
⟩
/
transition is first
-
order insensitive to magnetic fluctuations that induce
decohe
rence
(
10
)
. The
|
0
⟩
/
→
|
1
⟩
/
transition retains the strength of the electron spin transition,
which enables fast and efficient microwave manipulation.
A typic
al experimental sequence with spin initialization, control, and readout is shown in
Fig. 1B. The
Yb
!"!
#
$
ions are coupled to a photonic crystal cavity with small mode volume
~
1
(
휆
푛
KLM
⁄
)
#
and large quality factor
(
1
×
10
P
)
(Fig 1C, D. See SI
1.1
). This enhanc
es the
emission rate, collection efficiency, and cyclicity of the optical transitions
A
and
E
via the
Purcell effect
(
16
)
. The qubit is initialized into
|
0
⟩
/
by optical and microwave pumping on
F
,
A
,
and
f
e
to empty
|
푎푢푥
⟩
/
and
|
1
⟩
/
, followed by cavity
-
enhanced decay
into
|
0
⟩
/
via
E
(Fig. 1A)
. A
subsequent microwave
휋
pulse applied on
푓
/
optionally
initializes the ion into
|
1
⟩
/
.
The
|
1
⟩
/
state population is read out by exciting on
A
and collecting the resulting ion fluorescence.
Measurements are performed in a cry
ostat at 40 mK unless mentioned otherwise (See SI
6.5
for
discussion of sample temperature). The ions are optically addressed using two frequency
-
stabilized continuous
-
wave lasers, while a microwave coplanar waveguide allows for driving of
the
spin transit
ions (Fig. 1E
).
The
YVO material
used
has a ~20 ppb residual concentration of
171
Yb
that is
distributed
over an optical inhomogeneous linewidth of ~200 MHz in the device due to variations in the
local crystalline environment. This enables frequency isolation of single ions via pulsed resonant
photoluminescence excitation (PLE) spectroscopy on tr
ansition
A
. The PLE scan in Fig
.
2A
shows peaks in fluorescence that are confirmed to originate from single
ions by measuring the
pulse
-
wise second
-
order photon correlation of the resonant emission
(Fig. 2B). For ion X
(marked in Fig. 2A),
푔
(
[
0
]
=
0
.
147
±
0
.
011
. The observed bunching behavior
for
푡
>
0
is
expected for a multi
-
level system wi
th
long
-
lived shelving states (S
ee SI
2.3
). An optical lifetime
of
푇
!
=
2
.
27
휇푠
is measured for ion X (Fig. 2C), which is
a reduction from the bulk lifetime
(
267
휇푠
) by
훽
퐹
`
=
117
(with
훽
=
0
.
35
the branching ratio for emission via
A
) and corresponds
to a single
-
photon coupling rate of
푔
=
2
휋
×
23
MHz
. Similar measureme
nts were performed
on ion
marked
as Y in Fig. 2A (See SI
2
).
The cavity
-
enhanced optical transitions enabl
e coherent optical control and efficient spin
initialization (
Fig. S7
).
Measurements of the
resonant PL with varyi
ng excitation pulse length
show
optical Rabi oscillations (Fig. 2D), enabling calibration of
휋
and
휋
/2
pulses for optical
control and spin readout. An optical Ramsey measurement (Fig. 2E) gives a dephasing time of
푇
(
,
c
∗
=
370
ns
, a factor of 12 shorter than the lifetime
-
limited
푇
(
=
2
푇
!
.
Further optical echo
measurements give
푇
(
,
c
=
4
.
1
휇
푠
(Fig. S8
), whi
ch implies that
푇
(
,
c
∗
is limited by quasi
-
static
fluctuations of the transition frequency. We can extend the
푇
(
,
c
∗
beyond
1
휇
푠
by using post
-
selection to ensure
the ion is on resonance with the readout sequence (Fig. S9). We measure the
long
-
term sta
bility, or equivalently the spectral diffusion, of the ion using PLE readout over 6
hours (Fig. 2F) and observe a narrow integrated linewidth (FWHM) of 1.4 MHz.
We use this optical initialization and detection to demonstrate coherent spin manipulation
by
driving Rabi oscillations on the
|
0
⟩
/
→
|
1
⟩
/
qubit transition (Fig. 3A). We perform a spin
Ramsey measurement to extract a spin dephasing time of
푇
(
,
e
∗
=
8
.
2
휇푠
(Fig. 3B)
.
The spin
coherence is further extended using dynamical decoupling sequences
(
17
)
to suppress quasi
-
static contributions to dephasing. Fig. 3C shows the resulting coherence decay for increasing
numb
ers of
휋
pulses using a Carr
-
Purcell
-
Meiboom
-
Gill (CPMG) sequence (Fig. 3C inset). For a
single
휋
pulse, or spin echo sequence, we observe non
-
exponential behavior characteristic of a
spin coupled to a slowly
-
fluctuating dipolar spin
-
bath
(
18
,
19
)
with
푇
(
,
e
=
43
.
5
휇
푠
. This is
further evidenced by a measurement of the coherence time with
N,
the number of
휋
pulses,
which scales as
푁
h
.
"h
±
h
.
h!
(
Fig. S9).
CPMG scans taken with finer temporal resolution reveal
periodic collapse and revivals of coherence indicative of coupling to nearby nuclear spins (Fig.
S10) that could potentially be used as local quantum registers.
We explore the limits of the
spin coherence time by increasing the number of rephasing
pulses with a fixed pulse separation of 5.74
휇
푠
to avoid unwanted interactions with the nuclear
spin bath
(
20
)
. This enables extension of the CPMG cohere
nce time to 30 ms (Fig 3D). While the
CPMG sequence does not allow for preservation of arbitrary quantum states, we also
demonstrate coherence times longer than 4 ms using an XY
-
8 sequence
(
17
)
(Fig. S14
) suitable
for use in long
-
range quantum networks
(
21
)
. The measured q
ubit lifetime of 54
ms (Fig. S15
)
indicates that the observed coherences are approachin
g the lifetime limit. We repeated these
measurements at cryostat temperatures up to 1.2 K (Fig. 3D) and observed minimal changes in
the spin coherence and lifetime, providing evidence that they are not limi
ted by spin
-
lattice
relaxation (
See
SI
6.5
)
.
To ha
rness this long spin coherence lifetime for quantum networks, it is essential to read
out the qubit state in a single measurement. We achieve this with the scheme shown in Fig. 4A,
which consists of two consecutive optical read periods on
transition
A
sepa
rated by a microwave
휋
pulse to invert the qubit population. This scheme was designed considering that in this device
direct resonant PL readout of
the qubit state
can only be performed using a series of optical
휋
pulses
on transition
A
(
See
SI
2.2
)
.
The Purcell
-
enhanced cyclicity of transition
A
(
훽
∥
>
99
.
6%
,
see Fig. S6) allows for multiple photon emitting cycles before the ion is optically pumped out of
the qubit subspace into
|
푎푢푥
⟩
/
(
|
0
⟩
l
→
|
0
⟩
/
is forbidden at zero
-
field). Fig
.
4B shows the
measure
d photon count distribution in the two readout sequences for the ion initialized in
|
0
⟩
/
(blue) or
|
1
⟩
/
(red). We assign the ion to
|
1
⟩
/
if we measure
≥
1
photons during the first readout
sequence and 0 photons during the second readout sequence, and vice v
ersa for
|
0
⟩
/
. This
discriminates between
|
0
⟩
/
and
|
푎푢푥
⟩
/
to
ensure that the ion was in the qubit subspace during
the measurement. By implementing this scheme, we achieve an average readout fidelity of
95.3% (Fig. 4C).
These measurements showcase single
171
Yb
3+
in YVO as a promising system for solid
state quantum networking technologies. The
measured spin coherence
times correspond to light
propagation for thousands of kilometers in optical fibers, which is necessary for long
-
distance
quantum networks. F
urthermore, the preservation of this coherence lifetime at temperatures of
1.2 K is promising for developing a viable technology with economical
4
He cryogenics. To
generate spin
-
spin entanglement with the current optical dephasing times will require a post
-
selection protocol similar to what has already been developed for other quantum networks
(
22
)
(Fig. S9). While the source of the optical dephasing is still under investigation, it will likely be
improved in higher purity samples (
See
SI
6.5
). While not explored here, the high magnetic field
regime offers the possibility of longer
optical and spin
coherence
times at the expense of weaker
spin transition strengths
(
15
)
. The crucial next steps are increasing the cavity Q/V by an order of
magnitud
e and optimizing the collection efficiency, which should enable high rates of
indistinguishable photon emission. This could be achieved with the current device architecture,
or by using a hybrid platform where cavities are fabricated in a high
-
index materi
al like GaAs
and bonded to the YVO substrate. Multi
-
qubit gates necessary to establish entanglement on
larger scale networks could be performed using the interaction with neighboring vanadium atoms
or other REIs
(
23
)
. The technology demonstrated here with single REI qubits complements other
capabilities that could potentially be realized with
Yb
!"!
#
$
:
YVO
,
including quantum memories
(
24
)
for synchronizing photon traffic and quantum transducers
(
25
)
for coupling to qubits
operating at microwave frequencies, thus pointing to a unified platform for the future qu
antum
internet.
Acknowledgements
:
This work was funded by a National Science Foundation (NSF) Faculty
Early Career Development Program (CAREER) award (1454607), the AFOSR Quantum
Transduction Multidisciplinary University Research Initiative (FA9550
-
15
-
1
-
0
02), NSF
1820790, and the Institute of Quantum Information and Matter, an NSF Physics Frontiers Center
(PHY
-
1733907) with support from the Moore Foundation. The device nanofabrication was
performed in the Kavli Nanoscience Institute at the California Insti
tute of Technology. J.G.B.
acknowledges the support from the American Australian Association’s Northrop Grumman
Fellowship. J.R. acknowledges the support from the Natural Sciences and Engineering Research
Council of Canada (NSERC) [PGSD3
-
502844
-
2017]. Y.Q.
H. acknowledges the support from the
Agency for Science, Technology and Research (A*STAR) and Carl & Shirley Larson as a
Frederick W. Drury Jr. SURF Fellow. The authors would like to thank Dr. Matthew Shaw, Dr.
Sae Woo Nam
,
and Dr. Varun Verma for help with superconducting photon detectors. The
authors would like to thank Alp Sipahigil for useful discussion, Keith Schwab for help with
electronics
,
and
Daniel Riedel for supporting measurements
.
Note:
While completing this wor
k, we beca
me aware of a related publication
on
single erbium
ions
showing results related to those presented in Fig. 4 (
26
).
Figure 1
: Experimental platform. A) Zero
-
field energy level structure of
Yb
!"!
#
$
:
YVO
. States
|
0
⟩
/
and
|
1
⟩
/
form the spi
n qubit. Red transitions
A
and
E
are coupled (co
-
polarized) to the
cavity, while the blue transition
F
is cross
-
polarized to the cavity mode. B) Typical experimental
sequence used to initialize the ion into
|
0
⟩
/
,
manipulate the qubit, and optically read
out the spin
state. C) Scanning electron microscope image of a photonic crystal cavity fabricated in YVO
(Scale bar =
10
휇푚
). D) Reflection spectrum of the cavity. E) Schematic of experimental setup.
The optical transitions are addressed using pulses gen
erated from two frequency
-
stabilized
lasers. The qubit is directly manipulated using microwave (MW) control pulses using a coplanar
waveguide (CPW) next to the photonic crystal (PC) cavity, which is mounted in a dilution
refrigerator. Light collected from
the cavity is detected using a superconducting nanowire single
photon detector (SNSPD).
Figure 2
: Optical detection and coherent optical manipulation of single
171
Yb
3+
ions. A)
Photoluminescence excitation (PLE) spectrum showing resolved peaks co
rresponding to single
Yb
!"!
#
$
ions. B) Pulsed autocorrelation measurement on ion X with
푔
(
[
0
]
=
0
.
147
±
0
.
011
. C)
Normalized photoluminescence emission from ion X coupled to the cavity (red) compared to
typical photoluminescence from ions in bulk crystal (blue)
showing lifetime reduction of ~120.
D) Optical Rabi oscillations on transition
A
after initialization into state
|
1
⟩
/
(top). The resulting
fluorescence is plotted versus pulse length for different average cavity photon number n
cav
. Plots
are offset for cl
arity. E) Optical Ramsey measurement on transition
A
, indicating a dephasing
time
푇
(
∗
=
370
ns
. F) Measurement of
spect
ral diffusion over six hours
in which we repeatedly
measure the transition frequency using
PLE.
Bottom plot shows typical scan (red) and
sum of
counts acquired during all scans (black), which is fit to a Gaussian with FWHM of 1.4 MHz.
Figure 3
: Coherent spin state control of a single
Yb
!"!
#
$
ion. A) Typical Rabi oscillations on
the
|
0
⟩
/
→
|
1
⟩
/
microwave transition. B) Ramsey measurement on qubit transition (inset) that
gives
푇
(
,
e
∗
=
8
.
2
μ
s
. The excitation is detuned by 400 kHz to give rise to oscillations on the free
-
induction decay. C) Measurement of CPMG spin
-
coherence with increasing number
of rephasing
pulses. D) Me
asurement of spin coherence
times up to 30 ms at temperatures up to 1.2 K using
CPMG with fixed pulse separation and increasing numbers of rephasing pulses.
Figure 4
: Single
-
shot readout (SSRO) of single
Yb
!"!
#
$
spin stat
e. A) Scheme for SSRO. The
ion is repeatedly excited using
optical
휋
pulses on transition
A
and the resulting fluorescence is
collected. An ion in state
|
1
⟩
/
(red) will return to
|
1
⟩
/
with branching ratio
훽
∥
before eventually
being pumped to
|
푎푢푥
⟩
/
,
w
hile the ion in
|
0
⟩
/
(blue)
will be largely unaffected by the readout. A
microwave
휋
pulse is then applied to the spin transition to invert the population of
|
0
⟩
/
and
|
1
⟩
/
and
the
ion is optically read
out again. The state of the ion is assigned based on the number of
photons detected in the first and second read sequences. B) Photon
-
count distributions for the
first and second read sequence used in the SSRO protocol with the ion initially prepared in
|
0
⟩
/
(blue) or
|
1
⟩
/
(red). The dashed line shows the state assignment threshold (1 photon). C)
Distribution of assigned states
ij
for dual readout scheme
with the ion initially prepared in
|
0
⟩
/
(blue) or
|
1
⟩
/
(red)
,
where
i(j)
is the state assigned on the fi
rst (second)
read. By conditioning the
ion on the detection of
01 or 10, we obtain an average fidelity of 95.3%.
References
1.
H. J. Kimble, The quantum internet.
Nature
,
453
, 1023
–
1030 (2008).
2.
S. Wehner, D. Elkouss, R. Hanson, Quantum internet: A vision for the road ahead.
Science,
362
(2018)
3.
D. D. Awschalom, R. Hanson, J. Wrachtrup, B. B. Zhou, Quantum technologies with
optically interfaced solid
-
state spins.
Nat. Photonics
.
12
,
516
–
527 (2018).
4.
B. Hensen
et al.
, Loophole
-
free Bell inequality violation using electron spins separated by
1.3 kilometres.
Nature
,
526
, 682
–
686 (2015).
5.
W. F. Koehl, B. B. Buckley, F. J. Heremans, G. Calusine, D. D. Awschalom, Room
temperature cohe
rent control of defect spin qubits in silicon carbide.
Nature
,
479
, 84
(2011).
6.
S. Sun, H. Kim, Z. Luo, G. S. Solomon, E. Waks, A single
-
photon switch and transistor
enabled by a solid
-
state quantum memory.
Science,
361
, 57
–
60 (2018).
7.
A. Sipahigil
et al.
, An integrated diamond nanophotonics platform for quantum
-
optical
networks.
Science,
354
, 847
–
850 (2016).
8.
D. D. Sukachev
et al.
, Silicon
-
Vacancy Spin Qubit in Diamond
: A Quantum Memory
Exceeding 10 ms with Single
-
Shot State Readout.
Phys. Rev.
Lett.
223602
, 1
–
6 (2017).
9.
M. Zhong
et al.
, Optically addressable nuclear spins in a solid with a six
-
hour coherence
time.
Nature
.
517
, 177
–
180 (2015).
10.
A. Ortu
et al.
, Simultaneous coherence enhancement of optical and microwave transitions
in solid
-
state electronic spins.
Nat. Mater.
17
, 671
–
675 (2018).
11.
R. Kolesov
et al.
, Optical detection of a single rare
-
earth ion in a crystal.
Nat. Commun.
3
,
1029 (2012).
12.
T. Utikal
et al.
, Spectroscopic detection and state preparation of a single praseo
dymium
ion in a crystal.
Nat. Commun.
5
, 3627 (2014).
13.
T. Zhong
et al.
, Optically Addressing Single Rare
-
Earth Ions in a Nanophotonic Cavity.
Phys. Rev. Lett.
121
, 183603 (2018).
14.
A. M. Dibos, M. Raha, C. M. Phenicie, J. D. Thompson, Atomic Source
of Single Photons
in the Telecom Band.
Phys. Rev. Lett.
120
, 243601 (2018).
15.
J. M. Kindem
et al.
, Characterization of
171
Yb:YVO
4
for photonic quantum technologies.
Phys. Rev. B
.
80
, 1
–
10 (2018).
16.
E. M. Purcell, Spontaneous emission probabilities at radio frequencies.
Phys. Rev.
69
, 681
(1946).
17.
D. Suter, G. A. Álvarez, Colloquium: Protecting quantum information against
environmental noise.
Rev. Mod. Phys.
88
, 1
–
23 (2016).
18.
G. De Lange, Z. W
ang, S. Riste, V. V. Dobrovitski, R. Hanson, Universal Dynamical
Decoupling of a Single Solid
-
State Spin from a Spin Bath.
Science
.
330
, 60
–
64 (2010).
19.
J. R. Klauder, P. W. Anderson, Spectral Diffusion Decay in Spin Resonance Experiments.
Phys. Rev.
12
5
(1961)
20.
M. H. Abobeih
et al.
, One
-
second coherence for a single electron spin coupled to a multi
-
qubit nuclear
-
spin environment.
Nat. Commun.
9
, 1
–
8 (2018).
21.
P. C. Humphreys
et al.
, Deterministic delivery of remote entanglement on a quantum
netwo
rk.
Nature
.
558
, 268
–
273 (2018).
22.
H. Bernien
et al.
, Heralded entanglement between solid
-
state qubits separated by three
metres.
Nature
.
497
, 86
–
90 (2013).
23.
M. Zhong, R. L. Ahlefeldt, M. J. Sellars, Quantum information p
rocessing using frozen
core
Y
3+
spins in Eu:Y
2
SiO
5
.
New J. Phys.
21
, 033019 (2019).
24.
T. Zhong
et al.
, Nanophotonic rare
-
earth quantum memory with optically controlled
retrieval.
Science,
357
, 1392
–
1395 (2017).
25.
L. A. Williamson, Y.
-
H. Chen, J. J. Longdell, Magneto
-
Optic Modulator with Unit
Quantum Efficiency.
Phys. Rev. Lett.
113
, 203601 (2014).
26.
Mouktik Raha, Songtao Chen, Christopher M. Phenicie, Salim Ourari, Alan M. Dibos, Jeff
D. Thompson
,
Optical qua
ntum nondemolition measurement of a solid
-
state spin without
a cycling transition
.
arXiv
(2019),
1907.09992
Supplementary materials: Coherent control and
single-shot readout of a rare-earth ion embedded in a
nanophotonic cavity
Materials and Methods
1 Experimental setup
S2
1.1 Nanophotonic cavity in YVO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S2
1.2 Detailed experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S2
2 Identifying single
171
Yb
ions
S4
2.1 Energy structure of
171
Yb
:
YVO
. . . . . . . . . . . . . . . . . . . . . . . . . . . S4
2.2 PLE scans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S4
2.3 Verifying single ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S5
3 Purcell enhancement and optical branching ratio
S6
3.1 Predicted Purcell enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . S6
3.2 Modification of branching ratio in cavity . . . . . . . . . . . . . . . . . . . . . . . S7
4 Spin initialization
S7
5 Spectral diffusion and post-selection
S8
6 Additional spin measurements
S9
6.1 Optically-detected magnetic resonance . . . . . . . . . . . . . . . . . . . . . . . . S9
6.2 Calibration of pulses and readout . . . . . . . . . . . . . . . . . . . . . . . . . . . S9
6.3 Magnetic field dependence of spin coherence . . . . . . . . . . . . . . . . . . . . S9
6.4 Noise spectroscopy and dynamical decoupling . . . . . . . . . . . . . . . . . . . . S10
6.5 Spin lifetime and device temperature . . . . . . . . . . . . . . . . . . . . . . . . . S11
7 Single-shot readout fidelities
S11
7.1 Photon count distributions and readout fidelity . . . . . . . . . . . . . . . . . . . . S11
7.2 Conditional readout fidelities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S13
List of Figures
S1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S16
S2 Detailed level structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S17
S3 PLE scans and expected ion distribution . . . . . . . . . . . . . . . . . . . . . . . S18
S1
S4 Identification of Yb-171 ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S18
S5 Additional
g
(
2
)
measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S19
S6 Branching ratio measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S19
S7 Spin initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S20
S8 Optical echo measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S20
S9 Post-selected optical Ramsey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S21
S10 Ground-state ODMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S21
S11 Magnetic field dependence of spin coherence . . . . . . . . . . . . . . . . . . . . S22
S12 Fine CPMG scans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S22
S13 Scaling of CPMG coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S23
S14 Coherence decay using XY-8 sequence. . . . . . . . . . . . . . . . . . . . . . . . S23
S15 Spin lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S24
S16 Single-shot readout fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S24
1 Experimental setup
1.1 Nanophotonic cavity in YVO
Nanophotonic cavities are fabricated directly in a yttrium orthovanadate (YVO) crystal using focused-
ion-beam (FIB) milling (Fig. 1C). Periodic trenches are made in a triangular nanobeam to form a
photonic band gap with the spacing of these cuts tapered in the middle to form the defect required
for the optical cavity mode. The reflectivity of one side of the cavity is lowered by reducing the
number of photonic crystal lattice periods to allow for more efficient coupling into the collection path.
Light is coupled into and out of these devices via total internal reflection using 45 degree couplers
fabricated on both sides of the device. Further details on design and fabrication of nanocavities in
YVO can be found in Ref. (
S1
).
Devices are fabricated in a
c
-cut sample of YVO with the
E
-field of the fundamental TM mode
aligned with the stronger optical dipole of Yb:YVO, which is polarized along the crystal
c
-axis.
The device used here has an energy decay rate of
κ
=
2
π
×
30
.
7
GHz (
Q
∼
1
×
10
4
). The mode
volume extracted from FDTD simulations is
V
=
0
.
095
μ
m
3
≈
1
(
λ
/
n
)
3
, where
n
=
2
.
17
is the
refractive index of YVO for
E
‖
c
. The coupling rate from free-space into the nanobeam waveguide
is determined to be
∼
24%
by direct measurement of the reflection from the device off resonance. The
coupling rate of the input mirror of the cavity,
κ
in
, is extracted from the cavity reflection spectrum
(Fig. 1D) to be
κ
in
/
κ
≈
0
.
14
. The cavity is determined to be undercoupled by measuring the phase
response using a polarization interferometer (
S2
).
The sample used for this work is cut and polished from a boule of YVO grown by Gamdan Optics.
While nominally undoped in the growth process, the crystal contained residual concentrations of
rare-earth ions. From optical absorption measurements in bulk crystals and glow discharge mass
spectrometry (GDMS, EAG laboratories), the total concentration of all Yb isotopes is estimated to be
0.14 ppm. Assuming natural isotopic abundance (
14
.
3%
171
Yb
3
+
), this gives a
171
Yb
3
+
concentration
of
∼
20
ppb, which corresponds to
∼
23
171
Yb
3
+
ions within the cavity mode volume.
1.2 Detailed experimental setup
Fig. S1A shows a schematic of the optical network used in these experiments. The ions in the
cavity are optically addressed using two continuous-wave lasers. The state of the ions is read out on
S2
transition
A
(Fig. 1A) using a Ti:Sapphire laser (M2 Solstis) and optical pumping on transition
F
is
performed using an external-cavity diode laser (ECDL, Toptica DLPro).
A small portion of the Ti:Sapph is picked off to enable locking to a high-finesse Fabry-Perot
cavity (Stable Laser Systems) that serves as a stable long-term frequency reference. A fiber-based
phase-modulator (EOSpace) imposes variable-frequency sidebands onto the light before the reference
cavity and the first-order sideband is locked to the cavity using the standard Pound-Drever-Hall (PDH)
technique. Scanning the frequency of this sideband enables quasi-continuous scans over a 3 GHz
range while locked to a single longitudinal mode of the reference cavity. The wavelength of the laser
is monitored using a wavemeter (Bristol Instruments) to reliably lock to the same longitudinal mode
of the cavity. The ECDL is held at a fixed frequency with respect to the Ti:Sapph by measuring
the beat note between the two lasers on a fast photodiode (Newport) and feeding back to the ECDL
current and piezo control.
Each laser is independently amplitude-modulated using two free-space acousto-optic modulators
(AOMs) in double-pass configuration with total extinction of
≈
120
dBm. The lasers are coupled into
fiber and combined using a fiber-based beamsplitter. A set of variable attenuators and polarization
controllers allow for further adjustment of the amplitude and polarization of the light that passes
through a 99/1 fiber splitter before being directed to the device. Light reflected or emitted from the
device is directed by the 99/1 splitter through an additional free-space AOM shutter before being
detected by a
WSi
2
superconducting nanowire single photon detector (SNSPD) (
S3
). This detector
has high efficiency (
∼
75%
) and low intrinsic dark counts (
<
1
Hz). The AOM shutter serves to
protect the SNSPD from latching during preparation and readout pulses. All fiber connections are
spliced when possible to minimize reflections and additional losses. The total system detection
efficiency (probability of detecting a photon emitted by an ion in the cavity) is
∼
1%
.
A broadband supercontinuum source (Fianiuum WhiteLase Micro) and a spectrometer (Princeton
Instruments SP-2750, PIXIS 2KB eXcelon) are used for measuring the cavity reflection spectrum
when aligning and tuning the device.
Fig. S1B shows a schematic of the experimental setup inside of a Bluefors LD250 dilution
refrigerator. The device is held stationary on a copper sample mount on the mixing chamber (MXC)
plate. Light is coupled into and out of the device from fiber using an aspheric doublet mounted
on an XYZ piezo-stage (Attocube) that allows for optimization of this coupling at dilution fridge
temperatures.
Devices are tuned onto resonance with the ion transition of interest by nitrogen deposition (
S4
).
To implement this in the dilution refrigerator, a gas tuning line is installed from room temperature
down to the mixing chamber (Fig. S1B). This tuning line consists of stainless-steel (SS) tubing from
room temperature to 4 K with the tubing thermalized at each stage and isolated between stages by
a PTFE break. From 4 K to the mixing chamber plate, the tuning line consists of a free-hanging
copper tube that is thermally isolated from the components below 4 K. The output of this line is
directed onto the sample on the mixing chamber plate. A resistive heater attached to the tuning line
near the 4 K stage enables warming of the line such that gas flows through the line without freezing
and is deposited onto the device. Careful adjustment of the heater power allows for fine red-tuning
of the cavity resonance at rates
<
0
.
1
nm/minute. The cavities can be detuned by sublimation of the
frozen nitrogen through optical heating of the device using
∼
100
μ
W
of laser power resonant with
the cavity mode.
Fig. S1C shows the setup for microwave control of single ions. Microwave tones to drive the
ground and excited state transitions are generated using two signal generators (Stanford Research
Systems SG380). The amplitude and phase of the pulses used on the ground state transition are
controlled using IQ modulation driven by a fast function generator (Tektronix AWG5204). Both
sources pass through a set of microwave switches (Minicircuits ZASWA-2-50DR+) that provide
S3
additional extinction before passing through narrow-band filters. The microwave tones are combined,
amplified, and sent to the device in the dilution fridge. To ensure adequate microwave power at the
device for these initial measurements, minimal attenuation is used on the input coaxial lines inside
the fridge with a single 20 dB attenuator on the still plate and 0 dB attenuators on the other plates.
A gold coplanar waveguide is fabricated next to the optical cavity to allow for microwave
manipulation of the ions. The center strip of this waveguide is 60
μ
m
wide with a spacing of 30
μ
m
to the ground plane. The optical device sits inside this 30
μ
m
gap. Launching microwaves through
this waveguide gives rise to an oscillating magnetic field along the crystal
c
-axis, which enables
driving of the desired transitions (
|
0
〉
→
|
1
〉
) at zero-field. The YVO chip sits inside a microwave
launch board (Rogers AD1000, fabricated by Hughes Circuits) with SMP connectors on both input
and output. This launch board is wire-bonded to the chip with as many wirebonds as possible to give
additional cooling through the surface.
Static magnetic fields are applied to the device inside the fridge using a set of homebuilt su-
perconducting magnets made by winding superconducting wire (SC-T48B-M-0.254mm, Supercon
Inc).
With the full experiment loaded, the temperature of the mixing chamber plate is
∼
40
mK.
Experiments are performed with the mixing chamber plate temperature up to 1.2 K to investigate
temperature dependence of spin coherence and lifetime. Further measurements above this tempera-
ture have not been performed at this time as this leads to spurious dark counts and increased latching
on the SNSPDs. Further measurements of the temperature of the device are presented in Section 6.5.
2 Identifying single
171
Yb
ions
2.1 Energy structure of
171
Yb
:
YVO
The
4
f
13
configuration of
Yb
3
+
consists of two electronic multiplets,
2
F
7
/
2
(ground state) and
2
F
5
/
2
(excited state), that are split by the crystal field of YVO into four and three Kramers doublets,
respectively. The optical transition of interest is between the lowest energy doublets of the ground
state and excited state (
2
F
7
/
2
(
0
) →
2
F
5
/
2
(
0
)
), which occurs at approximately 984.5 nm for
Yb
3
+
doped into YVO. At cryogenic temperatures, the Kramers doublets can be treated as spin-
1
/
2
systems and described using an effective spin Hamiltonian (
S5
). For isotopes with non-zero nuclear
spin, the hyperfine interaction adds additional energy structure. The
171
Yb
isotope is unique among
the rare-earth ions as the only stable Kramers ion with nuclear spin
I
=
1
/
2
, which gives rise to the
simplest possible level structure with both electron and nuclear spin. Additional discussion on the
level-structure and effective spin Hamiltonian can be found in (
S6
).
The energy level structure of the lowest crystal field levels of
171
Yb:YVO at zero-field is shown
in Fig. S2. The red (blue) lines correspond to optical transitions allowed for light polarized parallel
(perpendicular) to the
c
-axis of the crystal. The red transitions are the Purcell-enhanced transitions
co-polarized with the cavity mode. The transitions
|
0
〉
g
→
|
0
〉
e
and
|
1
〉
e
→
|
1
〉
g
are forbidden at
zero-field by symmetry.
2.2 PLE scans
Potential single ions are identified with pulsed resonant photoluminescence excitation (PLE) scans.
Fig. S3A shows an extended PLE line scan over a 12 GHz region around the center of the optical
transition. Clusters of peaks in fluorescence correspond to the different isotopes of Yb, which
S4
have the expected transitions shown in Fig. S3B. These PLE scans are taken with Rabi frequencies
>
10
MHz to intentionally power-broaden the optical transitions of the ions and enable coarser and
faster scans. As shown in Fig. S3B, transition
A
of Yb-171 does not spectrally overlap with optical
transitions from the other isotopes, while the other cavity-coupled optical transition from the qubit
subspace (transition
E
) overlaps with the inhomogeneous distribution of the zero-spin isotope. This
makes it difficult to isolate and address single Yb-171 ions using transition
E
without simultaneously
exciting a large number of zero-spin ions. As a result, finer scans are performed around transition
A
(Fig. 2A of the main text) to identify potential Yb-171 ions.
To determine whether an isolated peak corresponds to a Yb-171 ion, we investigate the energy
level structure using optical pumping. The readout laser is tuned on resonance with one of these
peaks and a second laser is scanned across transition
F
. If an observed peak corresponds to the
A
transition of a Yb-171 ion, the pump laser will move population into the qubit subspace as it comes
into resonance with
F
and result in an increase in counts after the readout pulse. Fig. S4 shows
examples of these scans performed on the ions labeled as X and Y in Fig. 2A. A small splitting of
transition
F
is observed that is unexpected for the ion at zero magnetic field. Further investigations
into the behavior of this splitting with applied magnetic field confirm that this is not due to a residual
magnetic field at the ion.
This splitting is attributed to the ions occupying strained or otherwise distorted sites in the crystal.
Here, we isolate single ions by working in the tails of the inhomogeneous distribution arising from
variations in the local environment within the crystal. As a result, we are in practice preferentially
selecting for strained ions. A distortion of the local crystal lattice can reduce the site symmetry of the
ion, which would lead to a breaking of the degeneracy of the lowest energy levels in the ground state
(i.e. break the degeneracy of
|
aux
〉
g
) (
S7
). Further studies are necessary to understand the nature
and cause of this strain and its consequences for the properties of the ion. Depending on the resulting
site symmetry of the ion, these ions could potentially now possess a DC Stark shift (
S8
). This would
have negative implications for long-term optical spectral diffusion due to fluctuating electric fields,
but would also open the door to tuning and stabilization of the optical transition through applied DC
electric fields.
2.3 Verifying single ions
Second-order intensity correlation measurements are performed to verify that an isolated peak
corresponds to emission from a single ion. In Fig. 2B,
g
(
2
)
[
t
]
on ion X is measured by alternating
between a single initialization pulse on the
C
transition and a readout pulse on transition
A
. The
pulsewise correlation is calculated on the counts observed after the excitation pulse on transition
A
. We note that because the detector deadtime is short compared to the excited state lifetime and
photon rate, these measurements were performed using a single detector and by calculating a full
autocorrelation.
The bunching behavior observed for
t
>
0
is expected for a multi-level system with long-lived
shelving states, where the amplitude of the bunching corresponds to the ratio of effective decay rates
into and out of
|
1
〉
g
(
S9
). The single initialization pulse is not sufficient in this case to completely
initialize the ion into
|
1
〉
g
before each readout, but was chosen to enable faster repetition of the
experiment. We expect that this effect would not be observed with full initialization of the ion
before each readout. This bunching could also be attributed to spectral diffusion or blinking (
S10
).
Additional
g
(
2
)
[
t
]
measurements performed on ions identified as zero-spin isotopes at zero-field (i.e.
no shelving levels) do not exhibit this bunching, providing further evidence that this is due to the
multiple long-lived levels of Yb-171 at zero-field.
S5
To show that the observed bunching behavior is related to a population effect, this measurement
was repeated without any initialization into state
|
1
〉
g
as shown in Fig. S5. In this case, the rate of
pumping into
|
1
〉
g
is reduced, while the effective pumping rate out of
|
1
〉
g
is held constant. This
leads to a drastically increased bunching behavior for
t
>
0
as shown.
Measurements performed on ion Y show similar behavior and give
g
(
2
)
[
0
]
=
0
.
30
±
0
.
03
. These
results indicate that we have correctly identified X and Y as single
171
Yb
3
+
ions.
3 Purcell enhancement and optical branching ratio
3.1 Predicted Purcell enhancement
The oscillator strength of the
2
F
7
/
2
(
0
)→
2
F
5
/
2
(
0
)
transition for light polarized along
c
(i.e. transitions
A
,
E
,
I
) was determined to be
f
=
4
.
8
×
10
−
6
from bulk absorption measurements (
S6
). Note that
here we are using the real-cavity local field correction factor between absorption and oscillator
strength (
S11
). The corresponding dipole moment of these transitions is
1
.
06
×
10
−
31
C
·
m
, which
gives an emission rate for
E
‖
c
of
1
/(
763
μ
s
)
. Using the bulk excited state lifetime of
267
μ
s
, this
gives a branching ratio for decay with
E
‖
c
of
β
‖
∼
0
.
35
.
The optical decay rate of the atom in the nanophotonic cavity,
γ
cav
, is enhanced from its free
space value
γ
0
=
1
/(
267
μ
s
)
by
γ
cav
γ
0
=
1
+
4
g
2
κγ
0
=
1
+
η,
(S1)
where we have assumed that the cavity is resonant with the optical transition. Here,
η
is referred to
as the effective Purcell factor to distinguish from the enhancement of the cavity-coupled transition by
F
p
and the resulting overall change in the lifetime determined by the branching ratio (i.e.
η
=
β
‖
F
p
).
The coupling between atom and cavity field is described by the single photon Rabi frequency,
2
g
,
where
g
=
μ
~
√
~
ω
2
0
n
2
V
,
(S2)
μ
is the transition dipole moment,
n
is the refractive index of the medium, and
V
is the optical mode
volume of the cavity. The cavity energy decay rate is
κ
=
2
π
×
30
.
7 GHz
(Fig. 1D). For simplicity, we
assume that the ion is placed at the maximum of the cavity field and optimal polarization alignment
between the cavity mode and the transition dipole.
For the system parameters presented above, the maximal expected coupling is
g
max
=
2
π
×
25
.
5 MHz
. The maximum effective Purcell enhancement in the cavity used here is then
η
max
=
143
,
which corresponds to a cavity lifetime of
1
.
87
μ
s
. This is in reasonable agreement with the
measured lifetime of ion X of
2
.
27
μ
s
, which corresponds to an effective Purcell enhancement
of
β
F
p
=
117
. The resulting cavity QED parameters for this system are
(
g
, κ, γ
)
=
2
π
×
(
22
.
9 MHz
,
30
.
7 GHz
,
596 Hz
)
. Similar measurements on ion Y give a Purcell-enhanced lifetime
of
2
.
3
μ
s
indicating that ion X and ion Y are nearly identically coupled to the cavity.
In addition to enhancing the emission rate, the Purcell effect leads to preferential emission of
photons into the cavity mode from which they can be more readily collected. The fraction of the
ion emission into the cavity mode,
P
cav
, is given by the ratio of emission into the cavity to the total
emission rate:
P
cav
=
β
F
p
1
+
β
F
p
.
(S3)
The measured Purcell enhancement corresponds to
P
cav
=
99
.
1%
.
S6
3.2 Modification of branching ratio in cavity
The Purcell enhancement in this system improves the cyclicity of the optical transitions. In this
context, cyclicity describes the probability that an excited ion will return to its original ground state
upon emission of a photon. High cyclicity is essential for single-shot readout in which the qubit state
is assigned based on the number of photons detected during repeated optical excitation of the ion.
We assume the ion starts in
|
1
〉
g
and is excited to
|
0
〉
e
on transition
A
. Once in the excited state,
the ion can decay via the 984.5 nm transition back to
|
1
〉
g
with rate
γ
‖
or to
|
aux
〉
g
with
γ
⊥
. It can
also decay back to the ground state through the other crystal field levels with rate
γ
other
. The total
excited state decay rate
γ
0
is then
γ
0
=
γ
‖
+
γ
⊥
+
γ
other
.
(S4)
For an ion in the bulk crystal, the overall branching ratio for decay via
A
is
β
‖
=
γ
‖
/
γ
0
≈
0
.
35
(
S6
). The cavity enhances the emission rate for
E
‖
c
by
1
+
F
p
, which results in a cavity-enhanced
branching ratio for this transition:
β
cav
‖
=
(
1
+
F
p
)
β
‖
γ
0
γ
cav
=
(
1
+
F
p
)
β
‖
1
+
F
p
β
‖
(S5)
=
1
−
(
1
−
β
‖
)
T
cav
1
T
bulk
1
.
(S6)
From the observed cavity lifetime of
T
cav
1
=
2
.
3
μ
s
and bulk lifetime of
T
bulk
1
=
267
μ
s
, we
predict a branching ratio in the cavity of
β
cav
‖
≥
0
.
994
. Here we have assumed that decay through
the other crystal field levels will bring the ion to a different ground state to provide a lower bound to
the expected branching ratio in the cavity.
The optical branching ratio is measured directly by initializing the ion into
|
1
〉
g
and measuring
the optical pumping of the population as a function of the number of optical read pulses applied.
Fig. S6 plots the cumulative PL counts,
N
c
, observed as a function of number of read pulses,
N
p
, on
A
for the ion initialized into
|
1
〉
g
. This is fit to the form
N
c
(
N
p
)∝
1
−
β
N
p
e f f
1
−
β
e f f
,
(S7)
where
β
e f f
is the effective branching ratio
β
e f f
=
(
1
−
p
exc
)
+
p
exc
β
par allel
to take into account the
excitation probability,
p
exc
, of the readout pulses. This gives
β
e f f
=
0
.
997
, which is in agreement
with the predicted bound on the branching ratio from the lifetime. using
p
exc
≈
0
.
94
determined
from optical Rabi measurements,
β
‖
=
0
.
9968
.
Further improvements to the cyclicity could be
achieved with larger Purcell enhancement in cavities with higher quality factors.
4 Spin initialization
This section provides additional information on the pulse sequence used to initialize the spin of the
single ions (Fig. 1B in main text).
The single ion is first initialized into the qubit subspace by optical pumping out of
|
aux
〉
on
transition
F
, which consists of two
2
.
5
μ
s
pulses alternating between the two split transitions discussed
earlier (Fig. S4) with a total repetition rate of 100 kHz. Transition
F
is not enhanced by the cavity,
S7
but can be driven using light orthogonal to the cavity mode. Once in the excited state
|
1
〉
e
, the ion
decays by the cavity-enhanced transition
E
with high probability to
|
0
〉
g
. The ion is initialized within
the qubit subspace by optical pumping on transition
A
, which consists of
2
.
5
μ
s
long pulses with a
200 kHz repetition rate. As the optical transition from
|
0
〉
e
→
|
0
〉
g
is not allowed at zero-field, a
microwave pulse is applied simultaneously to the excited state transition
f
e
during optical pumping
on
A
to create a two-photon transition between
|
1
〉
g
and
|
1
〉
e
. Once in
|
1
〉
e
, the ion efficiently decays
to
|
0
〉
g
by transition
E
with branching ratio
β
‖
. This sequence initializes the ion into
|
0
〉
g
. To
initialize into
|
1
〉
g
, a microwave
π
pulse is applied on the ground state transition after the optical
initialization sequence.
To demonstrate and assess the quality of the spin initialization scheme, the population in
|
1
〉
g
is
measured for varying lengths of the preparation sequences. Fig. S7A shows optimization of optical
pumping out of the
|
aux
〉
state by varying the number of pulses on
F
, while keeping the number of
initialization pulses on
A
+
f
e
fixed at 100. From the observed count rate, optical branching ratio and
detection efficiency, the initialization into the qubit subspace is estimated to be
>
95%
. Fig. S7B
shows initialization into
|
1
〉
g
(red) or
|
0
〉
g
(blue) as the number of pulses on
A
+
f
e
is increased
while holding the number of pulses on
F
fixed at 150. Without any subtraction of background
count contributions, a population contrast of
91%
is observed, which corresponds to an initialization
fidelity of
96%
within the qubit subspace. This demonstrates that this pumping scheme allows for
efficient initialization between these two spin states in under
500
μ
s
. Here, the state population in
|
1
〉
g
is measured using PLE with a series of 500
π
pulses on transition
A
. The initialization measured
in this way will be limited by the readout fidelity of this pulse sequence, so represents a lower bound.
5 Spectral diffusion and post-selection
Fig. S8 shows an optical echo measurement on transition
A
, which gives a coherence time of
T
2
,
o
=
4
.
06
μ
s
. The considerably shorter optical Ramsey coherence time (Fig. 2E) indicates that the
measured
T
∗
2
,
o
is limited by quasi-static fluctuations in the frequency of transition
A
. The current
T
∗
2
,
o
will be detrimental to photon indistinguishability, but can be improved, for instance, by using
post-selection to ensure the ion is on resonance with the excitation pulse (
S12
). We demonstrate the
possibility of this approach in this system by post-selecting Ramsey measurements based on number
of photons,
n
c
detected during a subsequent probe sequence consisting of a series of resonant, low
power optical pulses on the transition
A
.
Fig. S9A shows the results of postselected resonant Ramsey measurements, where improvements
in
T
∗
2
,
o
are observed for increasing number of probe photons detected. Fig. S9B shows similar
measurements with the excitation pulses detuned by 1 MHz to give rise to characteristic Ramsey
fringes, verifying that this indeed corresponds to a coherence decay. Post-selecting with
n
c
=
2
leads to a
T
∗
2
,
o
of
1
.
0
±
0
.
1
μ
s
albeit with approximately
84%
of the Ramsey experiments discarded.
Higher collection efficiency in future devices should enable further improvements in
T
∗
2
,
o
using this
or similar post-selection technique.
One possible cause of these quasi-static fluctuations in the optical transition frequency is the
magnetic dipole-dipole, or superhyperfine (SHF), interaction between the Yb electron spin and host
nuclei, specifically vanadium (
I
V
=
7
/
2
) and yttrium (
I
Y
=
1
/
2
). Coupling to the two nearest vana-
dium ions is expected to dominate due to the
1
/
r
3
scaling of the magnetic dipole-dipole interaction
and the relative size of the nuclear g-factors (
g
V
=
1
.
5
and
g
Y
=
−
0
.
27
) (
S13
). Simulations of the
optical transition that take into account the superhyperfine interaction give a broadening on
A
of <
50 kHz (FWHM). This does not fully account for the observed 370 ns
T
∗
2
,
o
time, which corresponds
S8
to an 860 kHz linewidth. Further investigation of the limits of the
T
∗
2
,
o
due to the SHF mechanism
and other factors, such as the second-order DC stark shift of this transition, will be the subject of
future research.
6 Additional spin measurements
6.1 Optically-detected magnetic resonance
Optically detected magnetic resonance (ODMR) measurements were performed on the ground state
spin transition
|
0
〉
↔
|
1
〉
for initial calibration of the spin transition frequencies and to bound the
coherence time. For this measurement, the ion is initialized into state
|
0
〉
, a
160
μ
s
long microwave
pulse is applied, and the population in state
|
1
〉
g
is read out optically. Fig. S10 shows a series of
ODMR frequency scans on ion Y performed at successively lower microwave powers to reduce the
effects of power broadening. At the lowest microwave power used, we measure a linewidth (full-
width-half-maximum) of 48 kHz. This places a lower bound on the spin
T
∗
2
,
s
time of
6
.
6
μ
s
, which
in agreement with the spin
T
∗
2
,
s
measured directly using a Ramsey sequence. From this and similar
measurements on ion X, we extract the qubit transition frequencies of ion X and Y to be 674.48 MHz
and 673.24 MHz respectively (the inhomogeneous linewidth of this transition measured in 100 ppm
171
Yb:YVO is <1 MHz).
The asymmetric profile of the ODMR spectrum is attributed to second-order perturbations of the
qubit transition by magnetic dipole-dipole (superhyperfine) interactions with nearby nuclear spins
(yttrium and vanadium). To verify this this asymmetry is due to the superhyperfine interaction, the
energy level structure of the
|
0
〉
g
→
|
1
〉
g
transition was modeled by introducing magnetic dipole-
dipole coupling of the Yb electron spin to neighboring nuclear spins (
S13
). Fig. S10 plots the
simulated spectrum due to coupling with 3 nearest vanadium and 1 nearest yttrium, which shows
good agreement with the experimental results.
6.2 Calibration of pulses and readout
Additional calibration of the center frequency of the spin transition is accomplished by minimizing
the frequency of Rabi oscillations as a function of microwave drive frequency.
The length of microwave control pulses is extracted from the Rabi oscillations. Finer calibrations
of
π
pulse lengths are performed by initializing the ion into
|
0
〉
g
, applying an even number of
π
pulses, and minimizing the resulting population in
|
1
〉
g
as a function of pulse length.
For the spin coherence measurements presented in the text, the phase of the final
π
/
2
pulse is
chosen to be
180
◦
out of phase with the initial
π
/
2
pulse to map the coherence to a population on the
|
0
〉
g
. This gives rise to the increasing exponential decay observed in Fig. 3.
Unless otherwise specified, optical readout of the spin-state for coherence and lifetime measure-
ments is performed by measuring the fluorescence observed in a single series of readout pulses.
6.3 Magnetic field dependence of spin coherence
As described inthe main text, at zero field the hyperfine interaction gives rise to mixed electron-
nuclear spin states of the form
|
1
〉
=
1
√
2
(|
↑⇓
〉
+
|
↓⇑
〉)
and
|
0
〉
=
1
√
2
(|
↑⇓
〉
−
|
↓⇑
〉)
. These states
have zero net magnetic moment and are thus first-order insensitive to perturbations by the Zeeman
S9