1
On
-chip
coherent
microwave
-to-optical
transduction
mediated by ytterbium
in YVO
4
John G. Bartholomew
1,2
,†
, Jake Rochman
1,2
, Tian Xie
1,2
,
Jonathan M. Kindem
1,2
,‡
, Andrei Ruskuc
1,2
, Ioana Craiciu
1,2
, Mi Lei
1,2
, Andrei Faraon
1,2,*
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
*Corresponding author: faraon@caltech.edu
†Current address: School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia
‡Current address: JILA, University of Colorado and NIST, Boulder, CO, USA
;
Department of Physics, University of Colorado, Boulder, CO, USA
National Institute of Standards and Technology (NIST), Boulder, CO, USA
2
Optical networks that
distribute entanglement among quantum technologies will
form a powerful
backbone for quantum science
1
but are yet to interface with
leading
quantum
hardware
such as
superconducting qubit
s. Consequently, these
systems
remain isolated because microwave
links
at
room temperature
are noisy and lossy
. Build
ing
connectivity requires i
nterface
s that
map quantum
information between microwave and optical fields
. While preliminary mic
rowave
-to-optical (M2O)
transduc
ers
have been realized
2,3
, developing efficient
, low
-noise
device
s that match superconducting
qubit frequencies (gigahertz) and bandwidths (10 kHz –
1 MHz) remains a challenge.
Here we
demonstrate a proof
-of-concept on
-chip
M2O
transducer
us
ing
171
Yb
3+
-ions in yttrium orthovanadate
(YVO) coupled to a nanophotonic waveguide and a microwave transmission line
. The
device’s
miniaturiz
ation
, material
4
, and zero
-magnetic
-field
operation
are
important
advance
s for
rare
-earth
ion magneto
-optical devices
5,6
. Further integration with high quality factor microwave and optical
resonators will en
able
efficient
transduction
and
creat
e opportunities
toward
multi
-platform
quantum networks.
Rare
-earth ion (REI) ensembles simultaneously coupled to optical and microwave resonators
have been
proposed for M2O transducers that could a
chieve
an efficiency and bandwidth to challenge other
leading protocols
5,6
. A further advantage of
the REI
platform compared to electro
-optical
7,8
, electro
-
optomechanic
al
9,10
, piezo
-optomechanical
11,12
, and other magneto
-optical
13
scheme
s is the existing
REI
infrastructure for building complex qu
antum
-optical networks including sources
14
–16
and
memories
17–
19
for
quantum states of light
. While
REIs provide promise for future networks
, transducer demonstrations
have been limited to macros
copic devices
20,21
. These millimeter-
scale
transducers
currently
req
uire high
optical pump powers that will be challenging to integrate with cryogenic cooling systems and light
-
sensitive superconducting
circuits
20
. In contrast, o
n-chip
REI
techn
ologies
provide
strong optical mode
confinement
to reduc
e the required pump power by several orders of magnitude
, and miniaturiz
ation
expedites integration of multiple devices
for powerful control of photons at the quantum level.
To
3
achieve further integration with superconducting
qubit platforms, it is also
highly beneficial to extend
REI schemes
5,6
to
zero magnetic field operation
22
. Toward this end
,
171
Yb
3+
is appealing
because it
exhibits the
simplest spin
-state structure with gigahertz
-frequency hyperfine transitions
4,23
.
We report a
magneto
-optic modulator based on
171
Yb
3+
that
allow
s on
-chip
gigahertz
-frequency
M2O
transduction
at near
-zero and
zero
magnetic field
. The concept for the proof
-of -pri
nciple device is
shown in Figure 1
(a
– c). A REI crystal is cooled and simultaneously coupled to optical and microwave
excitation
s. The coherence
generated
on the spin transition
from excit
ation at frequency f
M
is mapped
to an optical coherence at
frequency
f
t
through a
n optical pump
field
at frequency f
o
. We measure the
transduced signal at
f
t
using
optical heterodyne detection
with a strong
local oscillator at
frequency
(f
o
−
280
) MHz
.
A 30
μm-long
nanophotonic waveguide
was
fabricated
in one of the gaps between the ground and signal
lines of a gold
microwave
coplanar waveguide
(Figure 1(d)).
A photonic crystal mirror fabricated on one
end
of the waveguide
allow
ed
optical fields to be laun
ched and collected from
a single 45
° coupler
on
the opposite end of the device.
We
used waveguides to enable operation over a wide frequency range
and t
o test
multiple optical and microwave transitions but using
resonant cavities
rather than
waveguides
will dramatically increase the efficiency of the transduction process
20,21
. The device was
thermally contacted to a dilution refrigerator with a
base temperature of
30 mK (see Supplementary
Information
§I for
details on device temperature).
To achieve efficient
M2O
transduction
using REI
-doped crystals
, it is critical to have an ensemble with
low inhomogeneity and a collective cooperativity greater than unity for their optical and microwave
transitions
5
.
171
Yb
3+
:YVO
satisfies both requirements
4
. Significantly
, the
171
Yb
3+
optical transition
near
984.5 nm
exhibits a
narrow inhomogeneous linewidth
(
Γ
푖푖ℎ
,
표표
≈
200
MHz
at
a doping concentration of
approximately 100 ppm)
, and a
large
4
f
-4
f
oscillator strength (
f =
5. 3 ×
10
−6
), r
esult
ing
in a magneto
-
4
optic nonlinear coefficient 10
0x larger than
other REI
-doped crystals
considered
for transduction
(see
Supplementary Info
rmation
§D)
.
Figure 2(a) illustrates the zero
-field
energy level
s of
171
Yb
3+
in YVO
. For light polarized parallel to the
crystal
c
axis
, only the spin preserving transitions (A, E, and I) are allowed. The relatively large hyperfine
interaction means that
the
three
optical transitions are easily
resolved
in
a waveguide transmission
spectrum at zero magnetic field
(Figure 2(b)
). Figure
2 highlights that for this polarization there are no V
-
or
Λ
- systems available for transduction
with
magnetic field
B = 0
1
. We pursue two strategies to mediate
transduction. First, we create s
uitable three
-level systems by applying small magnetic
fields along the
c
axis, which introduce
spin
-state mixing through the linear Zeeman interaction. Second, we demonstrate
a four-
level scheme that overcomes the need for applied magnetic fields. In both cases
we transduce
microwave photons coupled to
the spin transition in the optical excited state, which will allow future
transducers to benefit from decreased parasitic loss
and dephasing due to coupling with spectator
-ion
ensembles
24
.
Figure 2(c) shows the normalized o
ptical absorption of the ions in the waveguide
as a function of
magnetic field compared to the predictions of the
171
Yb
spin
Hamiltonian
model
4
. Transitions B and D
become allowed
for non
-zero magnetic fields
, which can be used to form two V
-systems
and two
Λ
-
systems.
We transduce classical microwave signals using
the
V-systems containing the
|1
⟩
푒푒
↔
|2
⟩
푒푒
transition
: f
M
=
3.369 GHz
at
B
≈
0 (Figure
2(a)).
Figure
3(a) show
s example
M2O
transduction signals using the three
-level strategy as function of laser
excitation frequency
for increasing magnetic field
. When B
≠
0 and the ions are optically driven on
transition
A (B)
at an offset frequency
∆
opti
cal
around 0
GHz (
0.675
GHz)
, microwave tones resonant with
the excited state transition are transduced to optical photons emitted on the D (E) transition. Without
1
This statements holds for all polarization
s, which is explored in the Supplementary Information
§G and §H
.
5
cavity enhancement the transduction signal is strongest for input
fields resonant with the ion
transitions, whereas cavity coupling
would
allow
high efficiency off resonance
5,25
. As the
magnetic
field
increases
, the transduced signal intensity varies as the dipole moments
and inhomogeneity
of the
optical an
d spin transition
s change. Figure 3
(b) shows a double resonance scan showing the transduced
signal intensity as a function of the pump frequency and the applied microwave frequency
for
B = 5.1
mT
.
The high signal
-to-noise ratio data was
enabled by the opt
ical heterodyne detection
, which
overcomes
the low device photon
-number
efficiency
η
= 1.2
x 10
-13
(see Supplementary Information
§C
). Given the
characterization of our material, temperature,
and driving rates we expect to increase the e
fficiency
by a
factor
≥
3 x 10
12
by targeting optimized microwave and optical cavity coupling (see Supplementary
Information
§D). That is, the same
ensemble of
171
Yb
3+
ions coupled to
one
-sided
microwave and optical
resonators, each with a quality factor of 2 x 10
4
, could perform at the
휂휂
>
0. 3
level.
The dramatic
increase in efficiency is possible because
η
scales
quadratically with the photon
-ion coupling strength
for
휂휂 ≪
1
20,21
.
To characterize the transducer’s bandwidth
, we perform
ed
pulsed M2O
transduction
measurements
(shown in Figure 3(
c)). The decrease in signal for pulse lengths less than 10
μs suggest
s a bandwidth
limited by the spin t
ransition inhomogeneity
Γ
푖푖ℎ
,
푠푠
≈
100
kHz
, which was confirmed by ensemble Rabi
flopping measurements (
Γ
푖푖ℎ
,
푠푠
=
130
kHz
- Supplementary Information
§E
). The
current
bandwidth
is
similar to leading
electro
-optomechanic
al
10
transducers
but
lower than the megahertz
-bandwidths
demonstrated in other schemes
3
including REI demonstrations
20,21
. Bandwidth increases could be
achieved
by intentionally broadening
Γ
푖푖ℎ
,
푠푠
through
increased dopant concentration or
strain.
Performing transduction in atomic systems
enables
quantum memories to be incorporated direc
tly
into
the transduction protocol
6
to enable
synchronization of network links. The coherence lifetime of the
6
spin transition T
2 (Spin)
sets an upper bound on the potential storage time. Using two
-pulse Hah
n echoes
we measure T
2 (Spin)
= 35
μs as
B
→
0 (see Supplementary Information
§E
), which
is sufficiently long to
enable useful storage relative to the timescales of typical microwave qubit operations (10
– 100 ns)
.
Using a coherent three
-level atomic system is a conceptually simple route toward transduction between
the microwave and optical domains. There are, however, disadvantages to this scheme.
Given a fixed
pump field
, the strength of the
optical photon
-ion coupling is reduced by at least a factor of 4 when
using
a V-
or Λ
-system
. This is because the total oscillator strength of the optical t
ransition must be
divided between the two optical branches.
Also
, operating with a small bias magnetic field is not ideal
as
it will require shielding for integration with superconducting qubits
. We present an
alternate
transduction
strategy
us
ing
a four-
level system driven by an optical and a microwave pump as shown in
Figure
4(a).
The ideal implementation of this method harnesses
the full optical oscillator strength of the
ions and for
171
Yb:YVO
the four
-level scheme enables
transduction at zero
magnetic field. The tradeoff
for moving to the four
-level scheme is the need for an additio
nal microwave drive tone resulting in
more
stringent device criteria to operate at high efficiency
(see Supplementary Information
§F).
Figure
4(b)
show
s a
double r
esonance spectr
um
for
the two microwave inputs
, with the optical pump
field fixed at the frequency of maximum transduction
(
∆
optical
= 0)
. In
our
waveguide device the four
-level
scheme is less efficient than the three
-level scheme
and thus, requir
es increased laser power to
measure the signal. The resultant increase in
device temperature
broadens the spin inhomogeneous
linewidth, which in turn decreases the efficiency further.
The signal m
odulation
near
resonance
for both
microwave field
s is most lik
ely produced by coherent destructive interference at specific population
difference
s between the four levels
25
.
This waveguide device illustrates the appeal
of
miniaturized REI devices
for quantum photonic
application
s. We have demonstrated coherent M2O
transduction
, present
ed
a strateg
y to improve the
7
efficiency
to greater than 3
0%
, and extended the protocol to zero
-magnetic
-field operation
. The
enabling high spectral density of the
171
Yb
3+
transitions can also be applied to realize other quantum
photonic
interface
s such as sources and memories
. Future work will
target
high
ly efficient
transducers
that will allow a
detailed noise analysis of the protocol, and ultimately the
ir integration
wit
h photonic
quantum memories
23
and
171
Yb
3+
-ion
single photon sources
16
to create the interfaces for
hybrid quantum
networks.
Methods
Device
A 5 nm thick layer of chromium and a 115 nm thick layer of gold was deposited on a 3 x 3 x 0.5mm (a x a
x c)
86 ppm
171
Yb
3+
-doped
YVO
4
crystal
(Gamdan Optics) using electron beam evaporation (
CHA
Industries Mark 40
). A coplanar waveguide was fabricated from the gold layer using electron beam
lithography (
Raith EBPG 5000+
) followed by wet
-etching in gold etchant.
The photonic structures
were milled within the coplanar waveguide gaps using a Ga
+
focused ion beam
(FEI Nova 600 Nanolab).
The underlying structure for
the nanophotonic waveguide is a suspended beam
with an equilateral triangular cross section with each side equal to approximately
1 μm.
A d
ist
ributed
Bragg reflecting mirror was then milled into the waveguide
, using
similar cuts used to define photonic
crys
tal resonators in
our
previous
work
26
.
Experimental setup
The device chip was bonded
to an oxygen free
high thermal conductivity
(OFHC)
copper sample holder
using a thin layer of silver paint (Pelco)
. The gold coplanar waveguide was wire bonded to a PCB board
from Montana Instruments fitted with SMP type coaxial connectors. The sample hol
der was
incorporated into a home
-built,
OFHC
copper apparatus attached to the mixing chamber of a BlueFors
8
dilution refrigerator. The apparatus incorporates a homebuilt
superconducting solenoid (field coefficient
= 77.3
mT per Amp) and a fiber coupled
-lens
pair mounted onto a three
-axis nanopositioner (Attocube).
Continuous
-wave
transduction
measurements were made using
a Field Fox N9115A spectrum analyzer.
Optical signals from the device were combined
with a strong optical local oscillator on a
50:50 fiber
beam splitter. The output from the beam splitter
wa
s detected by
an
InGaAs fiber coupled
photodetector
with a 5 GHz bandwidth
(DET08CFC
– Thorlabs)
. The output from the detector was
filtered using a bias
-tee (
ZFBT4B2GW+
- Minicircuits
) and the strong beat signal at the local oscillator
offset frequency (280 MHz) was suppressed using a band
-block filter (BSF
-280M
– RF Bay). The signal
was then amplified (PE15A1010
– Pasternack) before being detected by the Field Fox receiver.
For the time domain
measurements, the amplified signal was further amplified by two
Minicircuits
ZX60
-
3800LN
-S+
amplifiers and mixed down
(Minicircuits
ZX05
-30W
-S+
) to a frequency of 21.4 MHz using a
local oscillator signal at approximately 3.6704
GHz (TPI
-1002
-A). The lower fre
quency signal was then
filtered (Minicircuits BBP
-21.4), amplified (SR445
), and detected on a TDS7104 oscilloscope.
To gate the
microwave input to the device we used a Minicircuits
ZASWA-
2-50DR+
TTL controlled switch.
The optical excitation was provided
by a cw titanium sapphire
laser (either
M
2
SolsTiS
or Coherent MBR).
For higher precision measurements, the
SolsTiS
was locked to an ultra
-low expansion reference cavity
(Stable Laser systems) with a controllable offset frequency provided by an electro
-optic modulator (IX
Blue). The laser light was fiber coupled and sent through a free space polarization controller.
The
polarized light was then split into two paths, one acting as the sample pump beam, and the other as the
optical local oscillator. The pump beam was frequency shifted and gated through a fiber acousto
-optic
modulator
(AOM
- Brimrose) and input into
the fridge using a circulator.
Absorption measurements were performed using a home
-bu
ilt external cavity diode laser.
In this case,
the transmitted light was detected by a switchable gain InGaAs photodetector
(PDA10,
Thorlabs) or a
9
Perkin Elmer APD. In the case
of photon counting experiments, time tagging was performed by Sensl or
Picoquant data acquisition electronics.
For pulsed all-
optical measurements, the input light was gated using two double
-pass
AOMs
(Intraction)
and the
signal gated by a third single
-pas
s A
OM before detection on the APD.
For further details refer to Supplementary Information §
A and Figure S1.
Acknowledgements
This work was funded by Office of Naval Research Young Investigator Award No. N00014
-16
-1-2676,
Office of Naval Research Award No. N00014
-19
-1-2182
, Air Force Office of Scientific Research grant
number FA9550
-18
-1-0374,
and Northrop Grumman. The device
nanofabrication was performed in the
Kavli Nanoscience Institute at the California Institute of
Technology. J.G.B. acknowledges the support of
the American Australian Association’s Northrop Grumman Fellowship. I.C. and J.R. acknowledge support
from the Natural Sciences and Engineering Research Council of Canada (Grants No. PGSD2
-502755
- 2017
and No. PGSD3
-502844
-2017). The authors would like to acknowledge Jevon Longdell, Yu
-Hui Chen, Tian
Zhong,
and Mike Fitelson for useful discussions.
Author contributions
J.G.B, J.R., T.X., and A.F. designed the experiments. All authors contributed to the construct
ion of the
experimental apparatus. J.R. fabricated the device, and J.G.B. and T.X. performed the experiments, with
support from all other authors. J.G.B, J.R., and T.X. conducted the data analysis and modelling. J. G. B
and A. F. wrote the manuscript with input from all authors.
Competing financial interests
The authors declare no competing financial interests.
10
Figures
Figure 1 (a)
Conceptual s
chematic of the REI magneto
-optic modulator.
A microwave field B
ac
is
transduced to an optical field (dotted red line) using a REI ensemble in a crystal.
The crystal is coupled to
a microwave transmission line (MW coil) and pumped by a laser field (solid red line).
Magnetic
field
coils
provide
control
of the external dc
field
B. The transduced signal is combined with a local oscillator on a
photodiode to provide high signal
-to-noise
ratio
heterodyne detection. (b) E
xample
3-level
energy
structures proposed for REI magneto
-optic transducers with the input microwave (
f
M
), o
ptical pump (
f
o
),
and
transduced
optical output (
f
t
). (c) Example 4
-level energy structure for transduction in zero
magnetic
field with an additional microwave pump (
f
MG
) (d) False color s
canning electron microscope image of the
planar, on
-chip realization
of the
device in (a).
A 30
휇휇
m-long waveguide with a photonic crystal mirror
defined for the TM mode
(see inset)
. Light (red lines) is coupled to and collected from the device using
the coupler formed from a
45-
degree
cut at
one
end of the waveguide. The gold coplanar waveguide
provides
a microwave frequency oscillating magnetic field
aligned with the crystal
c
axis, while a home
-
built superconducting solenoid (not shown) provided an external
dc
field, also aligned with the cryst
al
c
axis.
11
Figure 2. (a) E
nergy level structure for
171
Yb
3+
:YVO
4
. Transitions A
(304501.0 GHz
≈
984.54 nm
), E, and I
are the allowed, spin
-preserving transitions
at zero magnetic field
, whereas transitions B and D only
become allowed for B
≠
0. (b) Transmission spectrum of the Yb
3+
:YVO
4
nanophotonic waveguide (total
length
≈
60
μm) at B
= 0 with
171
Yb
3+
transitions shaded blue and the impurity
even
Yb
3+
transition shaded
orange
. (c) Comparison of the magnetic field dependent absorption of the
waveguide device compared
to the spin Hamiltonian theory
12
Figure 3. (a) Transduction at approximately 3.37 GHz mediated by optically driving transitions A
(
∆
optical
=
0 GHz)
and B (
∆
optical
= 0.675 GHz) in their respective V
-systems. (
b) A double resonance scan showing the
transduced signal as a function of both optical and microwave frequency
(Detection bandwidth = 3 kHz,
optical pump power in the waveguide
= 2 μ
W, Rabi frequency
Ω
o
≈
6 MHz
, and microwave power of -
5.3
dBm in the coplanar waveguid
e,
Rabi frequency
Ω
m
≈
1 MHz
). White
curves
show the transduced signal
(log scale) as a function of f
M
at the middle of the optical inhomogeneous line.
(c) Pulsed transduction
signals
(offset for clarity)
generated at f
t
(blue) at the maximum efficiency po
int in (b). The yellow pulse
indicates excitation at f
o
only, whereas during the red pulses the ensemble is excited with both f
o
and
f
M
generating
the
transduced
field.
13
Figure
4. (a) The energy levels used for a four
-level
magneto
-optical
transduction
scheme at zero
magnetic field using
171
Yb:YVO
. (b) Transduc
ed signal at the frequency of the optical transition E as a
function of the two microwave input signals
with
the detuning of the optical pump
∆
optical
= 0.
(Detection
bandwidth = 30 Hz, optical pump power in the waveguide
= 25 μ
W,
Rabi frequency
Ω
o
≈
20 MHz, a
nd
microwave power of 3.7 dBm in the coplanar waveguide, Rabi frequency
Ω
ME
≈
3 MHz,
Ω
MG
≈
10 MHz
).
14
1.
Wehner, S., Elkouss, D. & Hanson, R. Quantum internet: A vision for the road ahead.
Science
362
,
eaam9288 (2018).
2.
Lambert, N. J., Rueda, A., Sedlmeir, F. & Schwefel, H. G. L. Coherent conversion between
microwave and optical photons --
an overview of physical implementations.
Arxiv preprint
arXiv:1906.10255
(2019).
3.
Lauk, N.
et al.
Perspectives on quantum transducti
on.
Arxiv preprint arXiv:1910.04821
(2019).
4.
Kindem, J. M.
et al.
Characterization of
171
Yb
3+
:YVO
4
for photonic quantum technologies.
Physical
Review B
98
, 024404 (2018).
5.
Williamson, L. A., Chen, Y.
-H. & Longdell, J. J. Magneto
-optic modulator with un
it quantum
efficiency.
Physical Review Letters
113
, 1
–5 (2014).
6.
O’Brien, C., Lauk, N., Blum, S., Morigi, G. & Fleischhauer, M. Interfacing superconducting qubits
and telecom photons via a rare
-earth
-doped crystal.
Physical Review Letters
113
, 1
–5 (2014)
.
7.
Rueda, A.
et al.
Efficient microwave to optical photon conversion: an electro
-optical realization.
Optica
3
, 597 (2016).
8.
Fan, L.
et al.
Superconducting cavity electro
-optics: A platform for coherent photon conversion
between superconducting and pho
tonic circuits.
Science Advances
4
, eaar4994 (2018).
9.
Andrews, R. W.
et al.
Bidirectional and efficient conversion between microwave and optical light.
Nature Physics
10
, 321
–326 (2014).
10.
Higginbotham, A. P.
et al.
Harnessing electro
-optic correlations in an efficient mechanical
converter.
Nature Physics
14
, 1038
–1042 (2018).
11.
Vainsencher, A., Satzinger, K. J., Peairs, G. A. & Cleland, A. N. Bi-
directional conversion between
15
microwave and optical frequencies in a
piezoelectric optomechanical device.
Applied Physics
Letters
109
, 033107 (2016).
12.
Dahmani, Y. D., Sarabalis, C. J., Jiang, W., Mayor, F. M. & Safavi-
Naeini, A. H. Piezoelectric
transduction of a wavelength
-scale mechanical waveguide.
Arxiv preprint arX
iv:1907.13058
(2019).
13.
Hisatomi, R.
et al.
Bidirectional conversion between microwave and light via ferromagnetic
magnons.
Physical Review B
93
, 174427 (2016).
14.
Ledingham, P. M., Naylor, W. R. & Longdell, J. J. Experimental Realization of Light with Time
-
Separated Correlations by Rephasing Amplified Spontaneous Emission.
Physical Review Letters
109
, 093602 (2012).
15.
Dibos, A. M., Raha, M., Phenicie, C. M. & Thompson, J. D. Atomic Source of Single Photons in the
Telecom Band.
Physical Review Letters
120
, 243601 (2018).
16.
Kindem, J. M.
et al.
Coherent control and single
-shot readout of a rare
-earth ion embedded in a
nanophotonic cavity.
Arxiv preprint arXiv:1907.12161
(2019).
17.
Hedges, M. P., Longdell, J. J., Li, Y. & Sellars, M. J. Efficient quant
um memory for light.
Nature
465
, 1052–
1056 (2010).
18.
Gündo
ğ
an, M., Ledingham, P. M., Kutluer, K., Mazzera, M. & de Riedmatten, H. Solid State Spin
-
Wave Quantum Memory for Time
-Bin Qubits.
Physical Review Letters
114
, 230501 (2015).
19.
Jobez, P.
et al.
Coherent Spin Control at the Quantum Level in an Ensemble
-Based Optical
Memory.
Physical Review Letters
114
, 230502 (2015).
20.
Fernandez-
Gonzalvo, X., Chen, Y.
-H., Yin, C., Rogge, S. & Longdell, J. J. Coherent frequency up
-
16
conversion of microwaves to the
optical telecommunications band in an Er:YSO crystal.
Physical
Review A
92
, 062313 (2015).
21.
Fernandez-
Gonzalvo, X., Horvath, S. P., Chen, Y.
-H. & Longdell, J. J. Cavity
-enhanced Raman
heterodyne spectroscopy in Er
3+
:Y
2
SiO
5
.
Physical Review A
100
, 03380
7 (2019).
22.
Chen, Y.
-H., Fernandez-
Gonzalvo, X. & Longdell, J. J. Coupling erbium spins to a three
-
dimensional superconducting cavity at zero magnetic field.
Physical Review B -
Condensed Matter
and Materials Physics
94
, 1
–5 (2016).
23.
Ortu, A.
et al.
Simultaneous coherence enhancement of optical and microwave transitions in
solid
-state electronic spins.
Nature Materials
17
, 671
–675 (2018).
24.
Welinski, S.
et al.
Electron Spin Coherences in Rare
-Earth Optically Excited States for Microwave
to Optical Qu
antum Transducers.
Physical Review Letters
122
, 247401 (2018).
25.
Fernandez-
Gonzalvo, X. Coherent Frequency Conversion from Microwave to Optical Fields in an
Erbium Doped Y
2
SiO
5
Crystal: Towards the Single Photon Regime. (University of Otago, 2017).
26.
Zhong, T., Rochman, J., Kindem, J. M., Miyazono, E. & Faraon, A. High quality factor nanophotonic
resonators in bulk rare
-earth doped crystals.
Optics Express
24
, 536 (2016).
1
Supplementary Information:
On
-chip coherent microwave
-to-optical transduction mediated by ytterbium in YVO
4
John G. Bartholomew
1,2,
†
, Jake Rochman
1,2
, Tian Xie
1,2
,
Jonathan M. Kindem
1,2,
‡
, Andrei Ruskuc
1,2
, Ioana Craiciu
1,2
, Mi Lei
1,2
, Andrei Faraon
1,2,*
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 Info
rmation and Matter, California Institute of Technology, Pasadena, California 91125, USA
*Corresponding author: faraon@caltech.edu
†Current address: School of Physics, The University of Sydney, Sydney, New South Wa
les 2006, Australia
‡Current address: JILA, University of Colorado and NIST, Boulder, CO, USA;
Department of Physics, University of Colorado, Boulder, CO, USA
National Institute of Standards and Technology (NIST), Boulder, CO, USA
A. Ex
perimental setu
p
Here we include a detailed
schematic of the experiment (Figure S1.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 co
nductivity
(OFHC) copper sample mount (Figure
S1.b) using silver paint. The gold coplanar waveguide fabricated on the YVO crystal surface was wire bonded to the PCB
board as shown in Figure S1.c. SMP connectors were
contacted
to the underside of the PCB bo
ard, which allowed the
coaxial cables shown in Figure S1.a to
connect to
the device.
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 polarizatio
n controlled (POL), and was intensity modulated using a
fiber acousto
-optic modulator (AOM –
Brimrose
, centered at 280 MHz
). Output light from the device was routed
through a 90:10 fiber splitter,
and a fiber isolator. For intensity detection, the light ro
uted to
an AOM
-gated avalanche
photodiode (APD –
Perkin Elmer) or InGaAs photodiode (PD -
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 Figure S1.
e.
2
Figure S1: S
chematic of the experimental setup.
a) Overall apparatus for the experiments showing the optical path (
red
),
the
microwave
path (bl
ue
), and the dilution refrige
rator
mounting.
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.
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 VNA 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 Figure S1.d used one of two lasers. A homebuilt external cavity diode laser (ECDL -
built u
sing 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 a
n M
2
SolsTiS offset
-locked to an
3
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 Figure S1.e was u
sed for continuous wave transduction measurements. The detector was a FieldFox
N9115A spectrum analyser (SA), or a Copper Mountain C1209 vector network analy
zer (VNA) for phase sensitive
measurements. DS
2
was used for pulsed transduction measurements. The electronic signal from Figure S1.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 a 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 copl
anar waveguide. The 65
μm wide conductor was centered between the two ground planes with
the edge-
to-edge distance from con
ductor to ground plane equal to approximately
50
μm. The resist was then removed
with R
emover
PG
.
A further 50 nm of chromium was the
n 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
wa
s a suspended
beam with an equilateral triangular cross section
, with each sid
e equal to approximately 1
μm.
A d
istributed Bragg
reflecting mirror w
as 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).
B.
Crystal structure
, site symmetry
, and energy levels
Figure S2: Unit
cell of YVO
4
, where a 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 Figure S2 t
he
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
.
4
Figure S3: Energy level diagram of
171
Yb
3+
-ions in YVO. The transitions labelled in blue are polarized along the crystal
c
axis and the transitions labelled in red are 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
.
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 Figure S3 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 degen
eracy at
zero field. We transduce microwave photons using the
|1
⟩
푒푒
↔
|2
⟩
푒푒
or
|3
⟩
푔푔
↔
|4
⟩
푔푔
spin transitions, w
hich have large
transition strengths (the
dipole moment is of the order of electron
spin
s)
for ac
-magnetic fields applied along the crystal
c
axis. This is despite the states involved being hybridized electron spin
-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.
C. Efficiency measurement
To determine the 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
휂휂
output
=
0. 09
. This encompasses the coupling efficiency b
etween the free space lens doublet and waveguide
(
휂휂
coupling
=
0. 22
) and losses in fiber connections, the optical isolator, and fiber beam splitters (
휂휂
optical
path
=
0. 4
).
The microwave input coupling efficiency was dependent on frequency with
휂휂
input
(3. 369
GHz
) =
0. 15
and
휂휂
input
(0. 674
GHz
) =
0. 45
. This was made up of the efficiency launching from coaxial cables into the waveguide
{
휂휂
launch
(3.369
GHz
) =
0. 74
,
휂휂
launch
(0. 674
GHz
) =
0. 88
} and other system losses {
휂휂
mw
path
(3. 369
GHz
) =
0. 20
,
휂휂
mw
path
(0. 674
GHz
) =
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
6
photons/s at the output fre
quency.
Given the microwave input power of 3 dBm at a frequency of 3.369 GHz (
8. 9
×
10
20
photons/s), and the efficiencies
휂휂
output
and
휂휂
input
, the photon number efficiency of the transduction process
휂휂
=
1. 2
×
10
−13
.
5
D. Increase in efficiency by using cavit
ies
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
휂휂
=
4
푅푅
2
(
푅푅
2
+
1
)
2
, for
푅푅
=
2
푆푆
�
휅휅
표표
휅휅
푚푚
.
The parameter
푆푆
is the coupling strength between the microwave and optical c
avities provided by the magneto
-optic
nonlinearity of the rare
-earth ion ensemble.
휅휅
표표
and
휅휅
푚푚
are the decay rates of the optical and microwave cavities,
respectively.
푅푅
is the ratio of the coupling strength to the impedance-
matched coupling strength, s
uch that
휂휂
=
1
when
푅푅
=
1
.
푅푅
can be rewritten as
3
푅푅
=
Ω훼훼훼훼
�
푄푄
표표
푄푄
푚푚
,
where
Ω
is the Rabi frequency of the optical pump,
훼훼
describes the density and spectroscopic properties of the ion
ensemble
(magneto
-optical nonlinear coefficient)
,
훼훼
is an effective filling factor describing the mode overlap of the three
fields, and
푄푄
표표
and
푄푄
푚푚
are the quality factors of the one sided optical and microwave resonators, respectively. Although
this theory is derived for the three fields being detuned f
rom 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 transiti
ons
4
.
For
푅푅 ≤
0. 1
,
휂휂
is approximately equal to
4
푅푅
2
: 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
훼훼
given that
3
훼훼
=
�
휇휇
0
ℏ
2
휖휖
0
휇휇
31
휇휇
21
휌휌 �
퐷퐷
푚푚
(
훿훿
푚푚
)
훿훿
푚푚
∞
휖휖
푚푚
푑푑훿훿
푚푚
�
퐷퐷
표표
(
훿훿
표표
)
훿훿
표표
∞
휖휖
표표
푑푑훿훿
표표
,
where
휌휌
is the ion number density
. The other parameters are denoted as follows:
휇휇
0
is the vacuum permeability,
ℏ
is the
reduced Planck constant,
휖휖
0
is the vacuum permittivity,
휇휇
31
is the
electric
dipole moment for
the
output optical
transition,
휇휇
21
is the magnetic dipole moment for the
input microwave transition, and
퐷퐷
푚푚
(
훿훿
푚푚
)
and
퐷퐷
표표
(
훿훿
표표
)
are
the
functions describing the inhomogeneous broadening
of the spin
and optical transitions, respectively, which
are
assume
d
to be Gaussian
.
For all the continuous wave transduction signals, the number of ions contributing to the signal was a factor of
≤
0.25 of
the available population. This is due to the thermal distribution of ions at the elevated temperatures (estimated to be
approximately 1 K) caused by the continuous optical pump and input microwave field. Initializing the population into a
non
-thermal distribution through optical pumping will allow
휌휌
to
increase by a factor of 4. This is a conservative estimate
for the current experiment given that optical pumping can contribute to population being trapped in spin states outside
the V or
Λ system.
The use of an optical cavity will reduce the power required to maintain the same optical pump Rabi frequency
Ω
, which
will decrease the heat load on the device. If required, further decreases in device temperature can be achieved by
running the transducer in a pulsed mode.
Replacing each waveguide with a cavity, will increase
R
by
�
ℱ
표표
ℱ
푚푚
, where
ℱ
is the cavity finesse. Our group has already
demonstrated the ability to fabricate photonic crystal mirrors to create cavities with
푄푄
표표
>
2 ×
10
4
2
[2]
. For a 30
μm long
optical cavity with a resonant frequency around 984.5 nm and
푄푄
표표
= 10
4
, the
cavity FWHM
∆푓푓
=
30
.45
GHz
, and the
finesse
ℱ
표표
=
75
.5
.
6
High quality factor microwave resonators can be fabricated on rare
-earth ion host crystals using superconducting metals
or alloys such as Nb or NbN
5
[5]
. For such cavities
ℱ
푚푚
≈푄푄
푚푚
, allowing
ℱ
푚푚
≤
2. 5 ×
10
4
without limiting the transduction
bandwidth
to less than
the current inhomogeneity of the excited state transition (130 kHz).
For
푄푄
표표
=
푄푄
푚푚
=
2 ×
10
4
, the predicted increase in
R
is approximately
1. 7 ×
10
3
. A
llowing for the more ambitious values
푄푄
표표
=
푄푄
푚푚
=
10
5
, the feasible increase
in
R
approaches
8
. 7 ×
10
3
, albeit accompanied by a reduction in the bandwidth
to
34 kHz.
Another key factor in increasing
R
in the current transducer is increasing the effective filling factor
훼훼
. Given the mode
profiles for the two waveguides and the much larger size of the microwave waveguide compared to the optical
waveguide,
훼훼
can be approximated as
훼훼 ∝�
푉푉
표표
푉푉
푚푚
,
where
푉푉
표표
is the volume of the optical mode in YVO and
푉푉
푚푚
is the volume of the microwave mode on the chip. Several
improvements can be made in
훼훼
. First, the optical structure can be lengthened by a factor of 10 to 300
μm in length.
Second, the coplanar microwave
structure can be compacted by decreasing the conductor and gap widths by a factor of
20. A further order of magnitude improvement in
훼훼
is possible by using a lumped element microwave cavity that
concentrates the magnetic field in the mode volume of the o
ptical cavity. This provides an increase in
R
by a factor of
around 600.
We note that to maintain the adiabatic condition
Ω
2
<
훿훿
표표
훿훿
푚푚
[3] the current
Ω≈
6
MHz w
ould
have to be decreased by
approximately a factor of 2
, reducing
R
by the same factor.
By optimizing
훼훼
and
훼훼
, and using microwave and optical cavity coupling
the increase in
푅푅
is predicted to reach
a factor
of
2 ×
10
6
. The corresponding increase
in the
device transduction efficiency would yield
휂휂 ≈
0. 4
. Such an efficiency is
not fundamentall
y limited
. Further increases to
the
cavity quality factors
(to 10
5
) and
171
Yb
concentration
are
both
feasible strategies for pushing toward unit efficiency.
The potential efficiency of a
171
Yb
3+
:YVO
-based transducer (using the
|1
⟩
푒푒
↔
|2
⟩
푒푒
transition) can also be considered by
comparing its spectroscopic properties to Site 1 Er
3+
:Y
2
SiO
5
(Er:YSO), the material used in other
transduction work
3,6,7
.
Parameter
171
Yb:YVO (86 ppm)
even
Er:YSO (10
ppm) (Site 1)
휌휌
푚푚푚푚푚푚
(assuming spin polarization)
1
.
08
×
10
24
m
-3
9
.
35
×
10
22
m
-3
Optical oscillator strength
푓푓
31
5
.
3
×
10
−
6
(
퐸퐸
|
|
푐푐
)
2
×
10
−
7
(
퐸퐸
|
|
퐷퐷
2
)
Optical dipole moment
휇휇
31
5
.
7
×
10
−
32
C m
2
.
13
×
10
−
32
C m
Spin dipole moment
휇휇
12
17.6
GHz/T (
퐵퐵
|
|
푐푐
,
퐵퐵
푚푚푎푎
|
|
푐푐
)
35.5
GHz/T (
퐵퐵
⊥
푏푏
&
29
o
to
퐷퐷
1
,
퐵퐵
푚푚푎푎
|
|
푐푐
)
Γ
푖푖
ℎ
(
optical
)
200 MHz
500 MHz
Γ
푖푖
ℎ
(
spin
)
0.13 MHz
1 MHz
훼훼
(Calculated
as detailed in [3]
)
1
.
4
×
10
−8
s
4
.
8
×
10
−
11
s
(
1
.
4
×
10
−10
s for
휇휇
12
= 15
휇휇
B
/2 as
presented in [3]
)
Table S2: Comparison of material
spectroscopic properties.
7
As shown in Table S2,
171
Yb:YVO
has a value of
훼훼
at least 100x greater than
even
Er:YSO.
Therefore, for
171
Yb:YVO
the value
of
Ω훼훼
�
푄푄
o
푄푄
m
required for efficient transduction is correspondingly lower than for Er:YSO.
Our modelling of the filling
factor
훼훼
for on
-chip transducer geometries suggests that
훼훼
can approach the value achieved in the loop gap resonator
geometry proposed in [3]
3
and used in [7]
7
(
훼훼
=
0. 0084
). The work in [3]
3
, predicts unit efficiency is possible for
even
Er:YSO with
푄푄
표표
푄푄
푚푚
=
10
10
. Based on Table S2, efficient, on
-chip
171
Yb:YVO
transducers are feasible with cavity
quality factors around
10
4
. This
correlates well with the analysis based on the current device.
E. Pulsed
measurements
of transducer bandwidth
and coherence lifetime
Figure 3(c) of the main text shows pulsed transduction signals as a function of pulse length. In a pulsed regime the
device temperature will be significantly lower than for the CW transduction studies (Section G
), resulting in narrower
spin inhomogeneous l
inewidth
Γ
푖푖ℎ
.
To
measure
Γ
푖푖ℎ
we perform ensemble Rabi flopping using the pulse sequence shown in Figure S
4.a(i). The external field
was 1.9 mT applied parallel to the
c
axis
, with the laser excitation power = 2
휇휇
W in the waveguide and the microwave
power = -
5.3 dBm in the CPW. Population is initialised into
|1
⟩
푒푒
through the application of an optical pulse applied
resonantly with transition B. The spin ensemble was then driven with a resonant microwave field near 3
.369 GHz
resulting in ensemble Rabi oscillations. In the regime where the microwave Rabi frequency
is less than
the optical pump
Rabi frequency
, the transduced field at frequency
휔휔
푡푡
is approximately proportional to the population difference
between states
|2
⟩
푒푒
and
|1
⟩
푒푒
. Thus, the amplitude of the transduced field produced by a combined optical
-microwave
readout pulse, measures the excited state spin inversion. The results are shown in Figure S
4.b. The damping rate of the
Rabi oscillations due to the ense
mble inhomogeneity is consistent with
Γ
푖푖ℎ
=
130
kHz.
Figure S
4: (a) Pulse sequences for optical detection of (i) excited state spin
-ensemble Rabi flopping, and (ii) excited state
spin transition Hahn echoes. (b) Excited state spin
-ensemble Rabi flopping
showing the narrow inhomogeneity of the
transition and an effective spin Rabi frequency of 1 MHz. (
c) Excited state spin transition Hahn echo decays as a function
of sequence delay t. The coherence lifetime T
2
increases with decreasing field as the
171
Yb
3+
-ions approach a clock
transition at B = 0.
8
We extended the pulse sequence as indicated in Figure S4
.a(ii) to measure the excited state spin coherence lifetime. In
this case, a Hahn echo sequence was inserted between the initialization and readout steps
of the sequence. The results
are shown in Figure S4
.c. For an applied field of B = 1.9 mT parallel to the
c
axis
the coherence lifetime was measured to
be 14
휇휇
s. As the field was decreased to B = 1.1 mT (B = 0.6 mT) the coherence lifetime increased to 22
휇휇
s (35
휇휇
s). The
increase in coherence as the
171
Yb
3+
ions approach their zero
-field clock transition indicates that the decoherence is
dominated by magnetic noise from their environment. The experiments were limited to a minimum field of 0.6 mT
because
below this field the B transition becomes prohibitively weak and the transduction signal falls below our
detection circuit noise.
F.
4-level scheme
The 4
-level transduction scheme presented in the paper is analogous to the 4
-level transduction scheme prop
osed for
cold gas atoms
8
. Therefore, the optical non
-linearity, or coupling strength between the microwave and optical cavity is
given by
푆푆
=
√
푁푁
Ω
12
Ω
23
푔푔
푀푀
√
푁푁
푔푔
표표
훿훿
2
훿훿
3
훿훿
4
훼훼
,
where
Ω
12
is the microwave pump Rabi frequency on the
|
3
⟩
푔푔
↔
|4
⟩
푔푔
transition
(
휇휇
=
42
GHz
/T )
,
Ω
23
is the optical
pump Rabi frequency on the
|4
⟩
푔푔
↔
|1
⟩
푒푒
transition (transition A),
푔푔
푀푀
is the single ion coherent coupling rate to the
microwave field on transition
|1
⟩
푒푒
↔
|2
⟩
푒푒
(
휇휇
=
17
GHz
/T)
,
푔푔
표표
is the single ion coherent coupling rate to
the optical
field on transition
|
2
⟩
푒푒
↔
|3
⟩
푔푔
(transition E)
, and the
훿훿
are the detunings relative to the upper three energy levels. We
note that equivalently to the 4
-level scheme proposed in [8]
8
, there is no collective enhancement on the microwave
transition targeted for transduction. Alternatively, a 4
-level scheme in
171
Yb
3+
:YVO could target the ground state spin
transition for transduction, which would be collectively enhanced. The microwave pump would then be applied on the
excited state spin transition.
The 4
-level scheme for transduction is advantageous in situations whe
re the transducer is required to operate at zero
field, or where the optical pump field needs to be minimized. In comparison to the 3
-level scheme
3,6,9
, the 4
-level
coupling strength
푆푆
is modified by a factor of
Ω
12
훿훿
2
<
1
and a reduction due to the filling factor
훼훼
that accounts for the
over
lap of a fourth field. Therefore, to achieve impedance matching for highly efficient transduction using the 4
-level
scheme, the factor
휌휌
�
푄푄
o
푄푄
m
will have to increase to compensate, without adversely impacting other device parameters.
G. Double resonance
scans
Figure S5 shows the transduction signal for the excited state microwave transition
|
1
⟩
푒푒
↔
|2
⟩
푒푒
with the optical pump
polarization parallel to the crystal
c
axis
. In part (a) the signal is shown as a function of the applied microwave frequency
f
M
, the optical pump frequency offset from transition A
∆
optical
and applied dc
magnetic field along the crystal
c
axis
. At
zero field, there is no transduction signal because transitions B and D are forbidden (top left
-hand plot). For non
-zero
fields, trans
duction is observed for two values of
∆
optical
corresponding to two V
-systems: one containing transition A
and the other containing transition B. At each magnetic field t
he signal is strongest when f
M
and
∆
optical
correspond to
center of the spin and optic
al inhomogeneous distributions, respectively.
In part (b) the overlay of 22 two
-dimensional data sets illustrates the simultaneous evolution of the
|1
⟩
푒푒
↔
|2
⟩
푒푒
microwave transition, and optical transitions A and B with the magnetic field. T
he spin Hamilton
ian model developed in
[10]
10
accurately predicts the frequency evolution (red
-dashed curves).