of 27
S
upplement
ary
I
nformation:
Microwave
-
to
-
optical transduction with erbium ions coupled to planar
photonic and superconducting resonators
Rochman et al.
Supplementary Note
1.
R
are
-
earth ion
cavity coupling theory
The
coupling between a cavity and ensemble of ions
was simulated
for both microwave and optical
frequencies
us
ing
the model described in Ref
[
1
]
.
A
cavity mode
,
̂
,
with a resonance frequency of
,
0
is
coupled to
N two
-
level atoms
(
)
in
the
weak excitation regime. The Hamiltonian is:
=
0
̂
̂
+
1
2
,
+
(
̂
,
+
,
̂
)
(1)
Here,
describes the coupling strength between
̂
and
,
. With input
-
output theory, the
transmission of the input field is
:
(
)
=
/
2
0
+
푖휅
2
(
)
,
(2)
where the
cavity
-
ensemble coupling term
is given by
(
)
=
2
(
1
,
2
,
)
ion
+
푖훾
2
+
Δ
for a
n ensemble with a
Lorentzian
inhomogeneity
line
shape
that
is
centered at
ion
with
a
FWHM
,
Δ
,
and a mean population
within the two energy levels of
1
,
and
2
,
.
The cavity and atomic decay rates are denoted by
and
,
r
espectively. The poles of the transmission
spectrum
can be found at
:
±
=
,
0
+
ion
2
푖훾
+
푖휅
+
2
Δ
4
±
2
(
1
,
2
,
)
+
(
2
(
푖표푛
0
)
푖훾
2
Δ
+
푖휅
4
)
2
(3)
The real
componen
t
of the pole represents the
frequency
of the
coupled
modes.
In the ensemble strong
coupling regime, the coupled modes are split into two polariton modes.
The imaginary
component
represents the linewidth the
modes
.
Supplementary Note
2.
REI transduction theory
The system we have for the transducer
can be approximated as
an ensemble of three level atoms
coupled to three electromagnetic fields (see Figure
S1
). Two of these fields correspond to an optical and
microwave cavity mode, which a
re the input and output modes of the transducer, while the third field
corresponds to an optical laser field that compensates for the frequency difference between the input
and output fields.
In order to model this atom
-
cavity system, we can use the follow
ing Hamiltonian
[
2
]
:
=
,
̂
̂
+
,
̂
̂
+
,
22
,
+
,
33
,
+
(
,
̂
21
,
+
Ω
o
,
k
σ
32
,
k
+
,
̂
31
,
+
H
.
C
.
)
(4)
w
here
,
is the optical light
-
cavity detuning,
,
is the microwave light
-
cavity detuning,
,
is the
detuning between the k
th
atom and the microwave field,
,
is the detuning between
the k
th
atom and
the
optical
field,
,
is the coupling strength between
the k
th
atom and the microwave cavity,
Ω
o
,
k
is
the optical pump Rabi frequency of
the k
th
atom,
,
is the coupling strength between the k
th
atom and
the optical cavity,
H
.
C
.
is the Hermitian conjugate and N is the number of atoms. We denote the optical
cavity
by the annihilation operator,
̂
, the microwave cavity by the annihilation operator,
̂
, and
푖푗
=
|
|
are the operators of the atoms
.
This Hamiltonian is suited for ground state transduction, but it can
be reformulated in a similar fashion for excited
state transduction.
Figure
S1
:
Energy level diagram of a REI transducer using the
a)
ground
spin
state and
b)
excited
spin
state.
We are interested in determining the transduction efficiency for this system. That is
=
|
out
in
|
2
or
=
|
out
in
|
2
, where
푖푛
and
푖푛
are the cavity input fields and
out
and
out
are the cavity output
fields for cavity mode
s
̂
&
̂
, respect
ively. We can start with the equations of motion of the two cavity
fields in terms of this Hamiltonian:
̂
̇
=
[
̂
,
]
2
,
in
in
(5)
̂
̇
=
[
̂
,
]
2
,
in
in
(6)
w
here
and
are the total microwave and optical cavity decay rates
, respectively,
and
,
in
and
,
in
are the microwave and optical input cavity decay rates
, respectively
.
In general, solving these equations
exactly is challenging due to the large number of atoms
, so we use
d
an approximated form to make the
calculation easier.
Adiabatic
m
odel
In order to
gain some intuition for the transducer efficiency, we can make an adiabatic approximation to
find a relatively simple analytical solution
for the
transducer
Hamiltonian in the steady state
[
2
].
The
transduction efficiency can be determined to be:
=
|
2
2
+
1
|
2
(7)
w
here
=
2
.
is the effective linear coupling strength between the microwave and optical
cavity in the adiabatic limit
,
which can be expressed as:
=
Ω
,
,
,
,
=
훼퐹
Ω
max
2
(8)
w
here
contains the spectroscopic parameters of the ions, F is the mode overlap between the three
fields and
Ω
max
is the maximum optical Rabi frequency of the pump field. We note that
31
32
21
Δ
Δ
,
wh
ere
푖푗
and
푖푗
are the dipole moments of
the
atomic
transition
between level
i
an
d
j
,
N
is the
number of atoms
and
Δ
and
Δ
are
the
optical and microwave transition
inhomogeneous linewidths. In
the adiabatic limit, the detunings will be proportional to the inhomogeneous linewidths.
Linear
m
odel
We can solve the transducer Hamiltonian in a more general case
(i.e. not limited to the adiabatic limit)
using a master
equation
-
based
model and a linear approximation for the cavity fields [
3
]. This
model
allows us to incorporate atomic energy and dephasing loss mechanisms and solve the Hamiltonian
for
choice of
detuning
s
.
The linear approximation
is appropriate for small cavity fields
and
makes the
system computationally easier to solve.
The dynamics of the system can be described by:
푑푡
=
[
,
]
+
(9)
where
is the density matrix of each atom and
describes energy
loss
and dephasing of the atoms.
can be written as the
sum of the different contributions by
=
푖푗
, where
푖푗
describes the energy
loss or dephasing between energy levels
i
and
j
.
We can write the density matrix as a 9x1 vector,
, and
the Hamiltonian interactions and Lindblad decay processes as a 9x9 matrix,
, such that
̇
=
푳흆
[
4
]
.
We can make a linear approximation t
o make this
system of equations
easier to compute
[
3
]. First,
can
be rewritten
without approximation in term
s of its linear dependence on the two cavity fields:
=
+
+
+
+
(10)
Next, we make
an approximation that our density matrix for each atom depends linearly on the cavity
field amplitudes, which should be true for sufficiently small cavity field amplitudes:
=
+
+
+
+
(11)
From here, we can solve for
in the steady state using
0
=
0
,
Tr
(
)
=
1
and
=
for
=
,
,
,
.
We can use this result to find the cavity field amplitudes. Importantly, these equations
result in a linear coupling between the two cavity modes and can be expressed in matrix form:
[
,
in
in
,
in
in
]
=
[
,
+
+
,
31
,
31
,
21
,
+
+
,
21
]
[
]
(12)
w
here
,
푖푗
=
,
,
푖푗
,
(and similar for
,
푖푗
).
We can solve this system of equations analytically and
find the transduction efficiency
equation:
=
|
out
in
|
2
=
|
,
31
,
in
,
in
,
21
,
31
+
(
,
+
+
,
31
)
(
,
+
+
,
21
)
|
2
(1
3
)
Supplementary Note
3.
Resonator
design
and
simulations
The geometric parameters of the optical
resonator
are
summarized in Table
S1
and
the mirror geometry
is shown i
n Figure S
2
. The optical cavity consists of
a
100
μ
m long waveguide between two photonic
crystal mirrors
that are patterned out of amorphous silicon on the
Er
3+
:YVO
4
surface
.
The amorphous
silicon waveguide has a height of 300 nm and a width of 600 nm.
One of the optical mirrors has a
short
er
length (i.e. 2 mirror periods) that we use
d
for coupling, while the second mirror has a long
er
length (i.e. 25 mirror periods) that we us
e
d
for high reflectivity.
The photonic crystal mirror consists of
ellipses with a pitch, a
o
, and radii of
,
0
and
,
0
.
The photonic crystal mirrors mode is tapered to the
waveguide mode by reducing the size of the photonic crystal mirror holes linearly
over 15 periods
on
both sides of the two
mirrors
to a final period of, a
t
, and radii of
,
and
,
, as shown in Figure
S
2
.
Coupling to the
resonator
from free space wa
s done with a grating coupler
and optical waveguide
. Two
grating couplers are pat
terned with the resonator, but for experiments we only couple to the grating
connected to
the low reflectivity mirror and measure the reflected signal.
The magnetic field of the
optical waveguide mode
,
a
s shown in Figure S
3
,
is the relevant field for our e
xperiments because
we use
the magnetic dipole moment for the optical transitions in the transducer.
Figure
S2
:
Photonic crystal geometry.
The
geometric
parameters
that define the photonic crystal mirror
including the mirror period and radii and the
final taper period and radii
.
Table S1: Optical resonator geometric parameters
Optical Resonator
Parameter
Value
Waveguide height
300 nm
Waveguide wi
d
th
600 nm
Waveguide length
100
μ
m
a
o
350 nm
,
0
95 nm
,
0
105 nm
315 nm
,
20
nm
,
20 nm
Mirror periods
2,25
Taper periods
15
Figure
S3
:
Normalized m
agnetic field profile of the amorphous silicon waveguide TM mode.
The
dominant component is B
x
, which is what we use to couple to the erbium optical
transitions.
The geometric parameters of the microwave resonator are included in Table S2 and the mode profile is
shown in Figure S4. The microwave resonator is made from 150 nm thick film of niobium that is
patterned on the surface. It consists of a 100
μ
m long, 1
μ
m wide inductive wire, that we use
d
to confine
the
microwave
magnetic field to the optical resonator, and an interdigitated capacitor to set the
resonance frequency of the cavity. The geometry is similar to other low impedance microwave
resonat
ors used for coupling to spins [
5,6
]. Coupling to the microwave cavity
wa
s done with a co
-
planar
microwave waveguide that is 4
μ
m away from the interdigitated capacitor. The optical cavity
was
patterned
1.2
μ
m away from the microwave cavity when measured e
dge
-
to
-
edge.
Table S2: Microwave resonator geometric parameters
Microwave Resonator
Parameter
Value
Inductive wire length,
ind
100
μ
m
Inductive wire width,
ind
1
μ
m
Capacitor finger length,
cap
485
μ
m
Capacitor finger width,
cap
10
μ
m
Capacitor finger gap,
cap
5
μ
m
Capacitor finger number,
cap
20
Optical gap width,
,
235
μ
m
Optical gap height,
,
55
μ
m
Waveguide coupling gap,
푤푔
,
cou
4
μ
m
Waveguide width,
wg
10
μ
m
Waveguide gap,
wg
4.5
μ
m
Figure
S4
:
Microwave cavity geometry.
The geometric parameters that define the pattern of the
microwave
resonator including the parameters related to the inductive wire, the interdigitated
capacitor, the coupling waveguide an
d the gap for the optical resonator.
Figure
S5
:
The normalized magnetic field distribution of the microwave resonator. a) The
m
agnetic field
distribution in the plane of the niobium
film
.
The dominant component is B
z
at the optical resonator
position
, which is used for coupling to the erbium spin transitions.
b) A closer look at the magnetic field
distribution near the inductive wire. The black line indicates the location of the optical resonator for
reference. c) The
cross
-
section of the microwave resonator magnetic field.
Supplementary Note
4.
Atomic system transduction simulations
We utilize optical transitions between the Z
1
crystal field levels within the
4
I
15
/2
manifold and the Y
1
crystal field levels within th
e
4
I
1
3
/2
for the transducer. Electron spin transitions in the Z
1
doublet are used
for ground state transduction and
transitions in the Y
1
doublet are used for excited state transduction.
Bias
m
agnetic
f
ield
a
ngle
As presented in
R
ef
[
7
],
sigma
-
polarized light can drive all four Z
1
-
Y
1
transitions
when the DC bias
magnetic
field
is
parallel to the
c
-
axis. T
he
spin
-
flipping
transitions
are electric
-
dipole (ED)
transitions
and the
spin
-
preserving
transitions
are magnetic
-
dipole (MD)
transition
s
.
In this magnetic field
configuration, transduction cannot occur when coupling the ions to a standing
-
wave cavity. This is
because
the cavity mode overlap factor [
2
]
:
1
|
(
)
(
)
(
)
3
|
(1
4
)
,
: microwave, optical mode volume
respectively
(
)
,
(
)
,
(
)
: microwave, optical
, and optical pump
modes respectively
will vanish if
the
two optical transitions are ED and MD,
because
the electric field and magnetic field
inside
the
standing
-
wave cavity
are
90
o
out of phase. Therefore,
to avoid a vanishing mode overlap
factor, we appl
ied
the DC magnetic field at an angle between crystal c
-
axis and a
-
axis to mix the
electronic states. Therefore, both ED and MD transitions are allowed for all four transitions in Z
1
-
Y
1
manifold.
A simulati
on was performed to study the state mixing effect at different
magnetic
field angle
s
, as
showed in Fig
ure
S6
, following the methodology described previously [
8
]
.
The figure of merit is
|
31
32
|
2
as the transducer efficiency scale
s
with this factor.
A magnetic field angle of
35
o
from
the
c
-
axis
is
optimal
for mixing the states.
Beyond optimizing the dipole
moment
product, the magnetic field angle determines
the g
-
factors of Z
1
and Y
1
, which need
to be
sufficiently
different
to have isolated optical transitions
.
W
hen the magnetic
field angle from the c
-
axis is great
er
than
45
o
,
the dispersive shift of the microwave cavity
when
the
excited state spins are resonant with the microwave cavity is <10 MH
z
. Therefore, we
set the magnetic
field angle to be
50
o
from the c
-
axis in the experiment to have
sufficient
state mixture
of
the electronic
states
and
enough difference between
the
g
-
factors
of Z
1
and Y
1
.
The spectroscopic parameters of the
Z
1
-
Y
1
transition
at the 50
o
angle are shown in Table
S
3
based on the values from [
7,
9
].
Figure
S6
:
a)
Dipole moment
product for the different transition dipole operators (i.e. electric or
magnetic) and orientations (i
.e. parallel to a
-
axis or c
-
axis) as a function of the applied magnetic field
angle.
b) Optical frequencies of the four different transitions as a function of the magnetic field angle
(
|
B
|
=
80
mT
)
.
Table
S3
:
Spectroscopic parameters for the Z
1
-
Y
1
transition at a magnetic field angle 50
o
from the c
-
axis.
Parameter
Value
dc
,
1
5
.
89
dc
,
1
4.55
,
[
10
32
C
m
]
0.48
,
[
10
32
C
m
]
3.26
,
[
10
32
C
m
]
1.58
,
[
10
32
C
m
]
0.23
,
[
10
32
C
m
]
0.52
,
[
10
32
C
m
]
0.86
Simulated
t
ransduction
e
fficiency
In ord
er to
simulate the transducer efficiency, we follow the master equation model with a linear
approximation as outlined above. Computationally, this requires finding the inverse of N
sim
9x9 matrices
,
where
N
sim
is the number of atoms in the simulation, and
some a
dditional matrix multiplication.
When implementing this model we make a few assumptions. First, we assume the microwave coupling
for the spins within the optical cavity is constant. Second, there are far too many ions to have a unique
density matrix for ea
ch atom. To overcome this, we simulate
N
sim
= 1
0
6
unique ions and assume these
ions are representative of the ensemble in terms of the inhomogeneity of the parameters and scale
their result (i.e.
density matrix) for the rest of the ions. We also ignore all the other level
s
within the
Er
3+
:YVO
4
level structure for simplicity and assume there are no
167
Er isotope ions.
The
parameter
inhomogeneity
that is modelled includes
the spectral
inhomogeneitie
s
, of both the spin
and optical transitions, and the inhomogeneity in the optical coupling
,
,
,
and equivalently in the
optical pump Rabi frequency
,
Ω
,
.
The
distributions of these three parameters
are randomly sampled
over N
sim
number of ions. The
,
distribution is determined from the
optical cavity magnetic field
distribution within the
Er
3+
:YVO
4
material
(see Figure
S3
)
. We assume
Lorentzian
distributions for the
optical and microwave transition inhomogeneities.
With this methodology
,
the device parameters in Table
S4
,
and the system calibration detailed in
Section 6,
we
simulate the transducer and predict
a total device efficiency of
=
1
.
1
10
7
at an
optical pump power of 5
5
0
μ
W
. This
is similar to the measured value of
=
8
10
8
. This difference
can be attributed to several possible factors. First, we are not accounting for the
167
Er hyperfine
transitions in the simulation, which could degrade the efficiency slightly, even though in the experiment
we choose the magnetic field to
minimize their detrimental effect. Also, the optical and spin dephasing
rates used in the simulation (i.e.
2D
=
3D
=
2
10
MHz
) and the excited state spin linewidth are
assumed values
to best match the experimental data
. F
urther spectroscopy is needed to
veri
f
y their
values in the precise experimental
con
figuration
used here.
In order to compare our transducer to other rare
-
earth ion transducers, we can also calculate the
theoretical
effective linear coupling strength,
, of our transducer.
Following the notation in Williamson
et al [
2
], from our spectroscopy, we obtain
=
1
.
54
10
10
s for detunings of
three
inhomogeneous
linewidths. From electromagnetic simulation of the cavities, we obtain a mode overlap
=
9
.
5
10
4
and a maximum optical pu
mp Rabi frequency of
Ω
max
=
2
11
.
5
MHz
(for an
intracavity
photon
number of
~
10
4
)
, which results in
=
2
6
MHz.
Table S4: Parameters of device for simulation
Parameter
Value
Optical lifetime
3.3 ms
*
Optical dephasing rate,
3D
/
2
10 MHz
X
Optical inhomogeneous linewidth,
Δ
300 MHz
*
Spin lifetime
1 ms
*
Spin dephasing rate,
2D
/
2
10 MHz
X
Spin inhomogeneous linewidth,
Δ
65 MHz
*
Optical cavity input decay rate,
,
in
/
2
2.9 GHz
!
Optical cavity total decay rate
,
,
tot
/
2
13.2 GHz
!
Microwave cavity input decay rate,
,
in
/
2
0.85 MHz
!
Microwave cavity total decay rate,
,
tot
/
2
2 MHz
!
Maximum optical pump Rabi
frequency
,
Ω
,
max
/
2
11.5 MHz
+
Maximum optical ion cavity coupling rate,
,
max
/
2
783 kHz
+
Spin cavity coupling (
for
transducer
ions
)
,
,
1
/
2
255 Hz (165 Hz)
+
Total spin coupling
,
,
tot
/
2
120 MHz (91 M
H
z)
+
Ions in optical
cavity
2.1e8
+
Ions in microwave cavity
1.6e15
+
Temperature
100 mK
X
Optical ion
-
cavity detuning (zero field)
1.8 GHz
!
Values in the brackets correspond to excited state parameters, while the nominal value is for the ground
state. The
parameters were determined from spectroscopy (
*
), assumed value (
X
), measurement (
!
) or
simulation (
+
).
Supplementary Note
5.
Set up diagram
Figure
S7
:
Diagram of the full experimental set
-
up. This includes the optical and microwave signal
generation, the dilution fridge setup, and the optical and microwave signal detection. The device setup
in the fridge consists of: 1) gas tuning line, 2) microwave coax
, 3) superconducting magnet, 4) optical
fiber and lens tube, and 5) three
-
axis piezo stack. Further details are in the text.