1
Supplementary
Information:
On
‐
chip
coherent
microwave
‐
to
‐
optical
transduction
mediated
by
ytterbium
in
YVO
4
Bartholomew
et
al.
2
Supplementary
Note
1
‐
Experimental
setup
Here
we
include
a
detailed
schematic
of
the
experiment
(Supplementary
Figure
1.a)
and
its
different
configurations,
along
with
further
information
to
supplement
the
Methods
section.
The
device
chip
was
thermally
lagged
to
an
oxygen
‐
free,
high
thermal
conductivity
(OFHC)
copper
sample
mount
(Supplementary
Figure
1.b)
using
silver
paint.
The
gold
coplanar
waveguide
fabricated
on
the
YVO
crystal
surface
was
wire
bonded
to
the
PCB
board
as
shown
in
Supplementary
Figure
1.c.
SMP
connectors
were
contacted
to
the
underside
of
the
PCB
board,
which
allowed
the
coaxial
cables
shown
in
Supplementary
Figure
1.a
to
connect
to
the
device.
Supplementary
Figure
1:
Schematic
of
the
experimental
setup.
a
Apparatus
for
the
experiments
showing
the
optical
path
(red),
the
microwave
path
(blue),
and
the
dilution
refrigerator
mounting.
WFM
m
‐
microwave
waveform,
WFM
o
–
optical
waveform,
AOM
–
acousto
‐
optic
modulator,
ISO
–
fiber
isolator,
APD
–
avalanched
photodiode,
PD
–
photodiode,
POL
–
polarisation
control,
LS
–
laser
system,
DS
–
detection
system,
SA
–
spectrum
analyser,
VNA
–
vector
network
analyser,
EOM
–
electro
‐
optic
modulator,
CAV
–
reference
cavity,
WFM
lo
–
local
oscillator
waveform,
DSO
–
digital
oscilloscope.
b
Cross
‐
sectional
view
of
the
dilution
refrigerator
mounting.
c
View
of
the
sample
mount
along
the
axis
of
the
lens
tube.
d
The
two
laser
systems
used
in
this
work.
e
The
two
heterodyne
detection
systems
used
in
this
work.
3
Light
from
the
laser
system
(LS)
was
coupled
to
the
on
‐
chip
devices
using
a
lens
doublet
mounted
on
an
XYZ
nanopositioner
(Attocube).
The
excitation
light
was
polarization
controlled
(POL),
and
was
intensity
modulated
using
a
fiber
acousto
‐
optic
modulator
(AOM
from
Brimrose,
centered
at
280
MHz).
Output
light
from
the
device
was
routed
through
a
90:10
fiber
splitter,
and
a
fiber
isolator
(ISO).
For
intensity
detection,
the
light
was
routed
to
an
AOM
‐
gated
avalanche
photodiode
(APD
from
Perkin
Elmer)
or
InGaAs
photodiode
(PD
from
Thorlabs).
For
heterodyne
detection,
the
light
was
routed
to
a
high
bandwidth
photodiode
after
mixing
with
a
strong
local
oscillator
(LO)
in
a
fiber
beam
splitter.
The
electronic
signal
from
the
heterodyne
PD
was
filtered
using
a
bias
‐
tee
and
a
band
‐
block
filter
(attenuating
the
strong
signal
at
280
MHz
produced
by
the
LO
interfering
with
reflected
pump
light).
The
signal
was
then
amplified
and
sent
to
one
of
the
detection
systems
(DS)
detailed
in
Supplementary
Figure
1.e.
The
optical
waveform
(WFM
o
)
was
generated
by
an
HP8656
B
signal
generator
that
was
gated
using
a
TTL
controlled
switch
(Minicircuits
ZASWA
‐
2
‐
50DR+)
and
amplified
(Minicircuits
ZHL
‐
1
‐
2W+).
The
microwave
waveform
(WFM
m
)
consisted
of
the
amplified
output
signal
from
a
spectrum
analyser
or
vector
network
analyzer
that
was
gated
with
a
TTL
switch.
To
maintain
the
fixed
frequency
and
phase
relationship
of
the
electronic
signals,
all
function
generators
were
locked
to
the
reference
clock
of
the
FieldFox
N9115A.
The
total
phase
stability
of
the
setup
was
limited
to
a
few
seconds
because
of
temperature
and
position
drift
in
the
optical
fibers.
LS
1
shown
in
Supplementary
Figure
1.d
used
one
of
two
lasers.
A
homebuilt
external
cavity
diode
laser
(ECDL
‐
built
using
the
design
outlined
in
[1]
1
)
was
used
for
the
inhomogeneous
linescans.
For
transduction
experiments
we
used
a
cw
titanium
‐
sapphire
laser
(Coherent
MBR)
locked
to
its
own
internal
reference
cavity.
LS
2
used
an
M
2
SolsTiS
offset
‐
locked
to
an
ultra
‐
low
expansion
reference
cavity
using
two
electro
‐
optic
modulators.
The
light
could
be
gated
using
two
double
‐
pass
AOM
setups
or
routed
directly
to
the
experiment.
DS
1
shown
in
Supplementary
Figure
1.e
was
used
for
continuous
wave
transduction
measurements.
The
detector
was
a
FieldFox
N9115A
spectrum
analyser
(SA),
or
a
Copper
Mountain
C1209
vector
network
analyzer
(VNA)
for
phase
sensitive
measurements.
DS
2
was
used
for
pulsed
transduction
measurements.
The
electronic
signal
from
Supplementary
Figure
1.a
was
amplified,
mixed
down
to
21.4
MHz
using
a
local
oscillator,
filtered,
and
further
amplified
before
detection
on
a
digital
oscilloscope.
Device
details
The
sample
used
in
this
work
was
cut
from
an
yttrium
orthovanadate
boule
doped
with
isotopically
enriched
(95%)
171
Yb
3+
(Gamdan
Optics).
The
171
Yb
3+
concentration
was
determined
to
be
86
ppm
relative
to
the
host
yttrium
using
glow
discharge
mass
spectrometry
(GDMS
‐
EAG
Laboratories).
The
3
x
3
x
0.5mm
(a
x
a
x
c)
sample
was
cut
and
polished
by
Brand
Optics.
Following
the
chromium
and
gold
deposition,
a
ZEP
mask
was
defined
by
electron
beam
lithography
(Raith
EBPG
5000+).
The
samples
were
then
wet
‐
etched
in
gold
etchant
to
form
the
coplanar
waveguide.
The
65
μm
wide
conductor
was
centered
between
the
two
ground
planes
with
the
edge
‐
to
‐
edge
distance
from
conductor
to
ground
plane
approximately
50
μm.
The
resist
was
then
removed
with
Remover
PG.
A
further
50
nm
of
chromium
was
then
evaporated
onto
the
sample
as
a
hard
mask.
The
sample
was
milled
using
a
Ga+
focused
ion
beam
(FEI
Nova
600
Nanolab).
The
underlying
structure
for
the
nanophotonic
waveguide
was
a
suspended
beam
with
an
equilateral
triangular
cross
section,
with
each
side
equal
to
approximately
1
μm.
A
distributed
Bragg
reflecting
mirror
was
then
milled
into
one
end
of
the
waveguide
2
,
along
with
the
45
o
couplers.
The
chromium
layer
was
then
removed
using
chrome
etchant
(CR
‐
7).
4
Supplementary
Note
2
‐
Crystal
structure,
site
symmetry,
and
energy
levels
Supplementary
Figure
2:
Unit
cell
of
YVO
4
.
An
Yb
3+
‐
ion
has
substituted
for
the
central
Y
3+
‐
ion
in
the
cell.
Yttrium
orthovanadate
(YVO)
is
a
uniaxial
crystal
in
which
the
Y
3+
‐
ion
sits
at
a
site
of
D
2d
symmetry.
In
Supplementary
Figure
2
the
two
orthogonal
trapezoids
that
connect
the
nearest
8
O
2
‐
‐
ions
give
a
guide
to
the
eye
for
visualizing
D
2d
symmetry.
Importantly,
the
space
point
group
D
2d
is
non
‐
polar,
which
means
that
substitutional
Yb
3+
‐
ions
in
this
site
have
zero
first
order
sensitivity
to
electric
fields.
Supplementary
Figure
3:
Energy
level
diagrams
of
171
Yb
3+
‐
ions
in
YVO.
a
Zero
magnetic
field
energy
levels
and
transitions
polarized
along
the
crystal
c
axis.
b
Non
‐
zero
field
energy
levels
and
transitions
polarized
perpendicular
to
the
c
axis.
Transitions
A,
E,
and
I
are
allowed
at
zero
field,
as
are
transitions
C
1,2
,
F
1,2
,
G
3,4
,
and
H
3,4
.
c
Repeated
data
from
Figure
3(c)
of
the
paper,
where
the
predictions
of
the
Hamiltonian
model
are
superimposed
(in
red)
on
the
experimental
data.
Yb
3+
has
13
electrons
in
the
4f
shell,
which
yields
a
relatively
simple
electronic
energy
level
structure
(it
is
effectively
a
one
‐
hole
system).
The
degeneracy
of
the
two
spin
‐
orbit
manifolds
2
F
7/2
(ground)
and
2
F
5/2
(excited)
is
lifted
by
the
D
2d
–
symmetric
crystal
field
interaction.
We
focus
on
the
lowest
lying
levels
of
both
multiplets,
denoted
in
Supplementary
Figure
3
as
2
F
7/2
(0)
and
2
F
5/2
(0).
Only
2
F
7/2
(0)
is
thermally
populated
at
liquid
helium
temperatures
because
it
is
separated
from
the
next
crystal
field
levels
by
>200
cm
‐
1
(>
6
THz).
171
Yb
3+
has
a
nuclear
spin
of
½,
which
interacts
with
the
ion
electron
spin
to
partially
lift
the
remaining
degeneracy
at
zero
field.
We
transduce
microwave
photons
using
the
|1
⟩
↔
|2
⟩
or
|3
⟩
↔
|4
⟩
spin
transitions,
which
have
large
transition
strengths
(the
dipole
moment
is
of
the
order
of
electron
spins)
for
ac
‐
magnetic
fields
applied
along
the
crystal
c
axis.
This
is
despite
the
states
involved
being
hybridized
electron
‐
nuclear
spin
states.
In
a
magnetic
field,
the
remaining
degeneracy
is
lifted,
and
transitions
B
and
D
become
allowed
because
of
the
mixing
between
the
hyperfine
states.
5
Supplementary
Note
3
‐
Device
Efficiency
measurement
To
determine
the
device
efficiency
of
the
transducer
we
performed
a
calibration
of
the
optical
output
losses,
the
microwave
input
losses,
and
the
sensitivity
of
the
heterodyne
detection
system.
The
output
efficiency
with
which
a
transduced
photon
from
the
waveguide
reaches
the
photodiode
was
휂
୭୳୲୮୳୲
ൌ
0.09
.
This
encompasses
the
coupling
efficiency
between
the
free
space
lens
doublet
and
waveguide
(
휂
ୡ୭୳୮୪୧୬
ൌ
0.22
)
and
losses
in
fiber
connections,
the
optical
isolator,
and
fiber
beam
splitters
(
휂
୭୮୲୧ୡୟ୪
୮ୟ୲୦
ൌ
0.4
).
The
microwave
input
coupling
efficiency
was
dependent
on
frequency
with
휂
୧୬୮୳୲
ሺ
3.369 GHz
ሻൌ
0.15
and
휂
୧୬୮୳୲
ሺ
0.674 GHz
ሻൌ
0.45
.
This
was
made
up
of
the
efficiency
launching
from
coaxial
cables
into
the
waveguide
{
휂
୪ୟ୳୬ୡ୦
ሺ
3.369 GHz
ሻൌ
0.74
,
휂
୪ୟ୳୬ୡ୦
ሺ
0.674 GHz
ሻൌ
0.88
}
and
other
system
losses
{
휂
୫୵
୮ୟ୲୦
ሺ
3.369 GHz
ሻൌ
0.20
,
휂
୫୵
୮ୟ୲୦
ሺ
0.674 GHz
ሻൌ
0.51
}.
The
heterodyne
detection
system
was
calibrated
by
measuring
the
beat
note
of
two
lasers
(M
2
SolsTiS,
and
home
built
ECDL)
locked
at
a
frequency
offset
of
3.65
GHz.
Using
the
measured
detector
responsivity
(0.18
A/W)
and
the
overall
gain
of
the
bias
tee,
filter,
and
amplifier
(39.3
dB),
the
optical
signal
intensity
producing
the
maximum
electrical
signal
observed
in
the
experiments
‐
71.62
dBm
(3
kHz
BW)
was
calculated
to
be
280
fW
.
This
corresponds
to
1.4
ൈ
10
photons/s
at
the
output
frequency.
Given
the
microwave
input
power
of
3
dBm
at
a
frequency
of
3.369
GHz
(
8.9
ൈ
10
ଶ
photons/s),
and
the
efficiencies
휂
୭୳୲୮୳୲
and
휂
୧୬୮୳୲
,
the
photon
number
device
efficiency
of
the
transduction
process
휂ൌ
1.2
ൈ
10
ିଵଷ
.
Supplementary
Note
4
‐
Increase
in
device
efficiency
by
using
cavities
In
this
section
we
detail
the
expected
efficiency
gains
from
several
modifications
to
the
dual
waveguide
device
including
the
use
of
high
quality
‐
factor
cavities
rather
than
broadband
waveguides.
Using
the
model
developed
in
Williamson
et
al.
3
[3]
the
efficiency
of
the
three
‐
level
transduction
process
where
the
ion
ensemble
is
coupled
to
both
a
microwave
and
optical
cavity
is
given
by
Supplementary
Equation
1
휂ൌ
4
푅
ଶ
ሺ
푅
ଶ
1
ሻ
ଶ
,for
푅ൌ
2
푆
ඥ
휅
୭
휅
୫
.
ሺ
1
ሻ
The
parameter
푆
is
the
coupling
strength
between
the
microwave
and
optical
cavities
provided
by
the
magneto
‐
optic
nonlinearity
of
the
rare
‐
earth
ion
ensemble.
휅
୭
and
휅
୫
are
the
decay
rates
of
the
optical
and
microwave
cavities,
respectively.
푅
is
the
ratio
of
the
coupling
strength
to
the
impedance
‐
matched
coupling
strength,
such
that
휂ൌ
1
when
푅ൌ
1
.
푅
can
be
rewritten
as
3
푅ൌΩ훼퐹
ඥ
푄
୭
푄
୫
,
ሺ
2
ሻ
where
Ω
is
the
Rabi
frequency
of
the
optical
pump,
훼
describes
the
density
and
spectroscopic
properties
of
the
ion
ensemble
(magneto
‐
optical
nonlinear
coefficient),
퐹
is
an
effective
filling
factor
describing
the
mode
overlap
of
the
three
fields,
and
푄
୭
and
푄
୫
are
the
quality
factors
of
the
one
sided
optical
and
microwave
resonators,
respectively.
Although
this
theory
is
derived
for
the
three
fields
being
detuned
from
the
relevant
ion
resonances
(but
in
three
photon
resonance)
3
,
the
formulation
extends
to
the
single
pass
regime
4
.
Without
cavities,
the
highest
efficiency
is
achieved
when
the
fields
are
resonant
with
the
ion
transitions
4
.
For
푅
0.1
,
휂
is
approximately
equal
to
4
푅
ଶ
:
improvements
to
the
current
device
coupling
strength
will
contribute
quadratically
to
the
efficiency.
A
significant
gain
in
R
can
be
made
by
using
all
the
ions
available.
This
increases
훼
given
that
3
훼ൌ
ඨ
휇
ℏ
ଶ
휖
휇
ଷଵ
휇
ଶଵ
휌
න
퐷
୫
ሺ
훿
୫
ሻ
훿
୫
ஶ
ఢ
ౣ
푑훿
୫
න
퐷
୭
ሺ
훿
୭
ሻ
훿
୭
ஶ
ఢ
푑훿
୭
,
ሺ
3
ሻ