# Opto-RF transduction in Er3+:CaWO4

Thierry Chanelière<EMAIL_ADDRESS>Rémi Dardaillon  Pierre
Lemonde  Jérémie J. Viennot  Univ. Grenoble Alpes, CNRS, Grenoble INP,
Institut Néel, 38000 Grenoble, France  Emmanuel Flurin  Patrice Bertet
Université Paris-Saclay, CEA, CNRS, SPEC, 91191 Gif-sur-Yvette Cedex, France
Diana Serrano  Philippe Goldner  Chimie ParisTech, PSL University, CNRS,
Institut de Recherche de Chimie Paris, 75005 Paris, France

###### Abstract

We use an erbium doped CaWO4 crystal as a resonant transducer between the RF
and optical domains at 12 GHz and 1532 nm respectively. We employ a RF
resonator to enhance the spin coupling but keep a single-pass (non-resonant)
setup in optics. The overall efficiency is low but we carefully characterize
the transduction process and show that the performance can be described by two
different metrics that we define and distinguish: the electro-optics and the
quantum efficiencies. We reach an electro-optics efficiency of -84 dB for 15.7
dBm RF power. The corresponding quantum efficiency is -142 dB for 0.4 dBm
optical power. We develop the Schrödinger-Maxwell formalism, well-known to
describe light-matter interactions in atomic systems, in order to model the
conversion process. We explicitly make the connection with the cavity quantum
electrodynamics (cavity QED) approach that are generally used to describe
quantum transduction.

###### Contents

  1. 1 Introduction and basic principle
  2. 2 Experimental setup
    1. 2.1 Optical and RF excitations
    2. 2.2 Transduction detection
    3. 2.3 Transduction efficiency calibration
      1. 2.3.1 Electro-optics efficiency
      2. 2.3.2 Quantum number efficiency
  3. 3 Transduction
    1. 3.1 Cooperativities
      1. 3.1.1 RF cooperativity
      2. 3.1.2 Optical absorption
    2. 3.2 Transduction characterization
      1. 3.2.1 Spin detuning
      2. 3.2.2 Cavity detuning
      3. 3.2.3 Wavelength dependency
  4. 4 Theoretical modeling
    1. 4.1 Input-output solution
    2. 4.2 Electro-optics efficiency
      1. 4.2.1 Definition
      2. 4.2.2 Finite temperature and population distribution
    3. 4.3 Quantum number efficiency
      1. 4.3.1 Definition
      2. 4.3.2 Choice of quantization volume
      3. 4.3.3 Expression as function of the cooperativities
    4. 4.4 Discussion
  5. 5 Analysis
  6. 6 Conclusion
  7. A Schrödinger-Maxwell model
    1. A.1 Schrödinger equation for three-level atoms in the perturbative regime
    2. A.2 Maxwell propagation equation
    3. A.3 Input-output solution
  8. B Intracavity RF magnetic field
  9. C Estimation of the spin cooperativity
  10. D Choice of the heterodyne beating frequency

##  1 Introduction and basic principle

In the bestiary of quantum technologies, the need has recently arisen for an
optical to microwave transducer to bridge the gap between two frequency
domains: quantum optical telecommunications traveling through fibre networks
on the one hand and spin or superconducting qubits confined to very low
temperature cryostats on the other, with no real possibility of
interconnection. From a fundamental point of view, the design of such a device
raises questions about the interaction at the quantum scale between an radio-
frequency (RF) field, which can be electric, magnetic or acoustic, and the
optical field in the same material system [1, 2, 3, 4, 5, 6, 7, 8, 9]. These
questions and a unified presentation are well covered by several review
articles [10, 11, 12]. Among the many materials considered, rare earth ions
doped crystals have a strong historical character, since they have been
studied during decades in parallel by optical spectroscopy and electron spin
resonance (ESR), enabling a detailed understanding of properties for
applications in magnetism or laser development. Reconsidering these materials
for microwave to optical transduction is quite natural in that sense [13, 14,
15].

The key parameters that drive the efficiency of the transduction process are
the optical and spin cooperativities, and more precisely the product of the
two quantities. In practice, lightly doped samples as the ones historically
considered [16, 17], few tens of ppm in our case, already offer a significant
optical cooperativity without cavity. In other words, the single-pass
absorption also known as the optical depth approaches one in many crystal.
This experimental fact stimulated the development of optical quantum memories
in solid-state atomic ensembles [18]. Concerning the RF cooperativity, a
resonator is necessary to enhance the spin-microwave interaction and probe
diluted samples. Starting from standard ESR apparatus with simple rectangular
cavities in historical experiments [19, 20, 21], the sensitivity of the most
recent ESR studies has now been pushed to the quantum limits using
superconducting circuits [22].

That being said, the dual integration of RF and photonic circuits is very
beneficial for enhancing the interactions. This approach has already
demonstrated its value in the development of original and demanding devices
such as [23, 24]. These developments are particularly timely, as they can be
carried out in synergy with current efforts to integrate quantum memories, by
sharing common platforms [25]. Despite the tour de force that these
achievements represent, they remain difficult to characterize, precisely
because of their integration. For example, in a doubly resonant optical and RF
scheme, it remains difficult to extract the ion couplings and the passive
resonator characteristics independently.

Based on this observation, we have then decided to operate a simplified setup
to observe opto-RF transduction in Er3+:CaWO4, employing a rectangular RF
resonator to drive the spins and a single-pass laser beam to excite the
optical transition. This kind of setup has been widely employed for the so-
called Raman heterodyne detection, even recently [26]. The goal is to reveal
the spin resonances without quantitative analysis of the emitted signal as it
was the case in the early days of the technique [27, 28]. Our work can
alternatively be seen as the quantitative extension of this method since we
carefully model the intensity of the beating signal. It should be also noted
that this is the first time the mixing process is observed in Er3+:CaWO4.

CaWO4 is an interesting and historically successful material, which can be
produced by a variety of growth techniques [29, 30], because of its relatively
low melting temperature compared to the iconic Y2SiO5, which continues to be
the talk of the town in rare-earth ions based quantum technologies [31]. The
CaWO4 matrix has a low nuclear spin density [32], this allowed the observation
of remarkably long spin coherence time [33, 34]. This renewed effort has
enabled ultimate demonstrations of both optical [35] and RF single-ion
detections [36, 37].

Before detailing the experimental setup, we first remind the basic principle
of the transduction using resonant interactions of the spin and optical
transitions in a so-called $\Lambda$-scheme as summarized in Fig.1 (left). A
more realistic structure of Er3+ in CaWO4 under magnetic field is represented
by a four-level structure in Fig.1 (right).

Figure 1: Left: Ideal $\Lambda$-scheme to implement resonant transduction. A
low level spin transition, at few GHz between $\ket{g}$ and $\ket{s}$, is
excited by the microwave field at $\omega_{\mathrm{rf}}$. The laser excites
the $\ket{s}-\ket{e}$ transition in the optical domain at
$\omega_{\mathrm{o}}$. This interaction are resonant or nearly resonant
(detuned by few linewidths). The transduction signal (green arrow) is
generated at $\omega_{\mathrm{o}}+\omega_{\mathrm{rf}}$. The field are
characterized by their Rabi frequencies $\mu$, $\Omega$ and $\mathcal{E}$ for
the optical Raman, microwave and transduction signal fields respectively.
$\Delta$ and $\delta$ are the field detunings from the optical and spin
transitions respectively. Right: Realistic level structure of Er3+:CaWO4. The
ground and optically excited states are actually spin Kramers doublets that
split under magnetic field into $\ket{g_{-}}$,$\ket{g_{+}}$ and
$\ket{e_{-}}$,$\ket{e_{+}}$ respectively. Additionally, in our temperature
range ($\sim 2-3$K), the spins are only partially polarized so both
$\ket{g_{-}}$ and $\ket{g_{+}}$ are populated. As a consequence, the
transduction is possible on four transitions between the two doublets. This
will be discussed in more details in 3.1.2 and 3.2.3

After having presented the experimental setup in 2, we will then characterize
the cooperativities of the optical and spin transitions independently in 3.1.
Finally, we present the results of the transduction process and its variation
as a function of the accessible experimental parameters in 3.2. We continue
with a detailed quantitative modeling of the transduction signal intensity in
4, and compare these theoretical results with the experiment. In the final
section, we will discuss the discrepancy between the prediction and the
experiment, and suggest some avenues for further studies in 5.

##  2 Experimental setup

###  2.1 Optical and RF excitations

We place the Er3+:CaWO4 crystal ($5\times 4\times 3$ mm) in a rectangular RF
copper resonator (dimensions $a\times b\times d=15\times 10\times 20$ mm
parallel to our reference frame $\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}$
respectively ) inside the inner bore of a superconducting coil providing the
bias magnetic field $\boldsymbol{B_{0}}$ of typically 100 mT along
$\boldsymbol{z}$ in our case (see Fig.2). The 4-mm dimension of the crystal
along $\boldsymbol{y}$ correspond to the crystalline c-axis. The nominal
concentration of erbium is 50 ppm. We excite the spins with a vector network
analyzer (VNA) at $\sim$12 GHz (Anritsu MS46522B) through a single coaxial
line and collect the reflexion $S_{11}$ coefficient. This allows to measure
the RF cooperativity. The exact position of the crystal in the resonator and
the intracavity field profile are detailed in B.

Following [38, eq.(6.40)], we expect the lowest resonant frequency to appear
at $12.49$ GHz. In practice, we observe a narrow resonance near $12.29$ GHz
with a quality factor of few thousands depending on the operating conditions
(temperature, position of the crystal and length of the antenna). This
resonance will be used for the spin excitation in the following. The length
and position of the antenna (see Fig.2) are adjusted at room temperature to
approach the critical coupling to the feed line so $S_{11}$ goes to zero in
order to maximally excite the spins for a given input power.

The laser beam from a Velocity® External Cavity Diode Laser around $1532$ nm
enters and leaves the copper box through two holes (1mm diameter), crossing
the crystal in a single-pass along its c-axis (4-mm dimension).

###  2.2 Transduction detection

The transduction signal should appear as a $\omega_{\mathrm{rf}}=2\pi\times
12.29$ GHz modulation of the transmitted laser beam at $\omega_{\mathrm{o}}$.
This represents a relatively large frequency for optoelectronics devices so we
decided to perform an heterodyne detection by generating a local oscillator
(LO) $\omega_{\mathrm{lo}}=2\pi\times(12.29$ GHz$+44$ MHz) from the same laser
using a standard lithium niobate electro-optic frequency modulator (EOM, see
Fig.2). A low bandwidth photodiode ($<50$ MHz of a Thorlabs PDA255) can then
record the beating between $\omega_{\mathrm{o}}+\omega_{\mathrm{rf}}$ and
$\omega_{\mathrm{o}}+\omega_{\mathrm{lo}}$ at $44$ MHz in our case. The choice
of this beatnote frequency is justified in appendix D.

Figure 2: Transduction measurement setup. Half of the laser power, resonant
with the optical transition passes through the crystal (single-pass along
$\boldsymbol{y}$ parallel the crystalline c-axis) by a 1-mm diameter aperture
in the cooper box (upper arm). The crystal is placed in a rectangular copper
resonator in the inner bore of a bias coil cooled down at cryogenic
temperature ($\sim 2.5$ K). The RF excitation is provided by the pin of a SMA
connector (antenna). $\boldsymbol{B_{0}}$ is the bias magnetic field along
$\boldsymbol{z}$. The heterodyne detection beats at
$\omega_{\mathrm{rf}}-\omega_{\mathrm{lo}}$ between the opto-RF signal (upper
arm) and the local oscillator generated by a fibered EOM (lower arm). The
transduction signal appears as a sideband at
$\omega_{\mathrm{o}}+\omega_{\mathrm{rf}}$ (see upper insert) after passing
through the crystal. This beats on the photodiode with the
$\omega_{\mathrm{o}}+\omega_{\mathrm{lo}}$ component of the local oscillator
generated by the EOM (see lower insert). We adjust the beam polarization after
the crystal (before recombination with the LO) to maximize the interference
contrast on the photodiode. Both frequencies $\omega_{\mathrm{rf}}$ and
$\omega_{\mathrm{lo}}$ are delivered by a dual channel VNA.

###  2.3 Transduction efficiency calibration

####  2.3.1 Electro-optics efficiency

Before considering the quantum perspectives of the transducer, we propose to
use a classical metric to evaluate the opto-RF conversion efficiency. We here
define the electro-optics efficiency as the normalized intensity of the
sideband shifted by $\omega_{\mathrm{rf}}$ with respect to the carrier
(relative intensity). This is a simple classical characterization of the
transduction process that is directly related to experimentally measurable
parameters.

The raw amplitude of the beatnote at
$\omega_{\mathrm{rf}}-\omega_{\mathrm{lo}}$ depends on the relative laser
power between the LO and the crystal transducer arms that form a Mach-Zender
interferometer after recombination in Fig.2. On the LO side, the fibered EOM
efficiency written $\eta_{\mathrm{lo}}$ (as the normalized intensity of the
$\omega_{\mathrm{lo}}$ sideband) scales the amplitude of the recorded beatnote
at $\omega_{\mathrm{rf}}-\omega_{\mathrm{lo}}$. We then measure
$\eta_{\mathrm{lo}}$ independently by comparing the amplitude at
$\omega_{\mathrm{lo}}$ with respect to the carrier through a commercial Fabry-
Perot interferometer. At our working frequency $\sim$12 GHz, the fibered EOM
efficiency is only $\eta_{\mathrm{lo}}\sim 7$% at 18.4 dBm input power
(maximum power delivered by the VNA) because we exceed the nominal modulation
bandwidth of the EOM and have a limited input driving power. Still, this is
sufficient to observe and characterize the transduction signal.

Finally, there is no need to measure the power on both arms of the Mach-Zender
interferometer because the transduction signal is a weak modulation of the
carrier. Instead, we simply observe the Mach-Zender interferometer output DC
signal on the photodiode. This fluctuates because the phase of the few meters
long interferometer is not stable in time. There is no need to stabilize it
for an AC measurement anyway. We record the amplitude of these slow temporal
fluctuations between a maximum and a minimum as the voltage
$V_{\mathrm{fringes}}$ that actually reveals the beating contrast (typically
80% in our case). This latter may be imperfect because of a small polarization
mismatch, the spatial mode overlap being ensured by the single mode fiber
recombiner. At the end, if the beatnote amplitude voltage recorded by the
spectrum analyzer is $V_{\mathrm{sa}}$ then the electro-optics transduction
efficiency $\eta_{\mathrm{eo}}$ is

$\eta_{\mathrm{eo}}=\frac{1}{\eta_{\mathrm{lo}}}\left(\frac{V_{\mathrm{sa}}}{V_{\mathrm{fringes}}}\right)^{2}$
(1)

This formula actually compares the measured power in the transduction sideband
$\displaystyle\left(V_{\mathrm{sa}}\right)^{2}$ and the one in the local
oscillator sideband
$\displaystyle\eta_{\mathrm{lo}}\left(V_{\mathrm{fringes}}\right)^{2}$ that
actually interfere to produce the beating signal as recorded independently.

We will use the electro-optics efficiency $\eta_{\mathrm{eo}}$ (expressed in
dB typically) as a metric to analyze the experimental results. Nevertheless,
it is important to carefully define the quantum number efficiency in the
recent stimulating context.

####  2.3.2 Quantum number efficiency

By definition, the quantum number efficiency is the main metric for the
quantum conversion process. As opposed to the electro-optics efficiency, we do
not compare the intensity of the sideband at $\omega_{\mathrm{rf}}$ with
respect to the optical carrier. Instead we here compare the photon number flux
in the optical sideband at $\omega_{\mathrm{o}}+\omega_{\mathrm{rf}}$ and the
RF photon flux. The powers are then $\eta_{\mathrm{eo}}P_{\mathrm{o}}$ for the
optical sideband and $P_{\mathrm{rf}}$ for the RF respectively, where
$P_{\mathrm{o}}$ is the laser power in the crystal and $P_{\mathrm{rf}}$ the
RF power at the resonator input. So the quantum efficiency $\eta_{\mathrm{Q}}$
reads as

$\eta_{\mathrm{Q}}=\frac{\left(\eta_{\mathrm{eo}}P_{\mathrm{o}}\right)/\omega_{\mathrm{o}}}{P_{\mathrm{rf}}/\omega_{\mathrm{rf}}}=\eta_{\mathrm{eo}}\frac{P_{\mathrm{o}}\omega_{\mathrm{rf}}}{P_{\mathrm{rf}}\omega_{\mathrm{o}}}$
(2)

(1) and (2) connect the two definitions of the efficiencies.

The powers $P_{\mathrm{o}}$ and $P_{\mathrm{rf}}$ are experimental parameters.
They are usually taken at the device inputs to focus on the conversion process
and compensate for parasitic losses than can be avoided by technical
improvements. Anticipating the description in 3.2.1, at the maximum RF source
power, we have $P_{\mathrm{rf}}=37$ mW. The laser power inside the cryostat
(corrected from losses though cryostat windows and crystal interfaces) is
$P_{\mathrm{o}}=1.1$ mW. The power ratio (maximum RF) is then
$\displaystyle\frac{P_{\mathrm{o}}}{P_{\mathrm{rf}}}=2.8\times 10^{-2}$ or
-15.5 dB in log scale. Even if the powers have different orders of magnitude,
we haven’t observed any saturation of both transitions confirming that the
efficiencies are constant and well-defined in our range of parameters.

The frequency ratio is fixed by the level structure, in our case we have
$\displaystyle\frac{\omega_{\mathrm{rf}}}{\omega_{\mathrm{o}}}=6.3\times
10^{-5}$ or -42.0 dB in log scale.

In typical experimental conditions (maximum power), we have
$\eta_{\mathrm{Q}}=\eta_{\mathrm{eo}}-57.5$ dB when expressed in log scale. In
conclusion, the efficiencies are proportional, they both characterize the
transduction process that will be detailed in the next section.

##  3 Transduction

As discussed in the introduction, the RF and optical cooperativities are the
key parameters to evaluate the transduction efficiency. They will be first
characterized.

The reference method to measure the spin cooperativity consists in monitoring
the $S_{11}$ from the VNA as the transition is tuned on resonance by sweeping
the magnetic field. The shift induced by the spin scales as the RF
cooperativity as we will show in 3.1.1.

Concerning the optical cooperativity, since we employ a non-resonant setup,
the single-pass optical transmission can be directly recorded by a photodiode.
As shown in 3.1.2, the cooperativity, equal to the optical depth in free
space, can be deduced from the absorption spectrum.

We will then present the results of the transduction experiment in 3.2.

###  3.1 Cooperativities

####  3.1.1 RF cooperativity

The RF cooperativity can be deduced from the $S_{11}$ spectra for a varying
magnetic field when the spin resonance is swept across as shown in Fig. 3.

Figure 3: Absolute value of the $S_{11}$ coefficient. Top: Map (log. scale) of
$|S_{11}|$ as a function of RF frequency (Y-axis) for a varying magnetic field
(X-axis). Bottom: $|S_{11}|$ spectra for different characteristic values of
the magnetic field. The experimental results are on the left and corresponding
fit on the right (see the text for details). We observe a cavity resonance
shift when the spin resonance is swept across.

The cavity response is described by the well-established input-output theory
[39, 40] [41, Eq.(3.50)]

$S_{11}(\omega_{\mathrm{rf}})=1-\frac{i\kappa_{c}}{\omega_{\mathrm{rf}}-\omega_{c}+i\frac{\kappa_{t}}{2}-W(\omega_{\mathrm{rf}})}$
(3)

where $\kappa_{c}$ is the cavity coupling rate and $\kappa_{t}$ the damping
rate ($\omega_{c}$ is the cavity resonant frequency). The spin interaction is
contained in $W(\omega_{\mathrm{rf}})$ [41, Eq.(3.53)]

$W(\omega_{\mathrm{rf}})=C_{\mu}\frac{i\kappa_{t}/2}{2i\frac{\omega_{\mathrm{rf}}-\omega_{s}}{\gamma}-1}$
(4)

where $C_{\mu}$ is RF cooperativity [40], $\gamma$ the spin linewidth (FWHM)
and $\omega_{s}$ the spin resonant frequency that varies with the magnetic
field $B_{0}$ as $\omega_{s}=g\mu_{B}B_{0}/\hbar$ ($g$ is the erbium g-factor
along $\boldsymbol{B_{0}}$ and $\mu_{B}$ the Bohr magneton).

We fit the map of $|S_{11}|$ in Fig. 3 to the model (3) & (4). So we obtain
the different parameters: $\kappa_{t}=2\pi\times 8.57$ MHz corresponding to a
quality factor of 1430, $\kappa_{c}=2\pi\times 4.52$ MHz demonstrating a close
to critical coupling condition with $\kappa_{t}\approx 2\kappa_{c}$. The spin
linewidth is measured to be $\gamma=2\pi\times 219.14$ MHz limited by the
magnetic field inhomogeneity along the crystal dimensions. The fit predicts
lower values for $|S_{11}|$ reaching -45 dB as compared to the minimum -30 dB
that we observe. An ad-hoc introduction of extra losses (typically -15 dB)
would certainly improve the fit accuracy, but this is difficult to justify a
priori. So we stick to the original model and still expect a correct
prediction of the parameters since these extra losses are almost invisible in
the linear scale that we choose for the fitting procedure.

In any case, this leads to a RF cooperativity of $C_{\mu}=0.14$ in correct
agreement with spin concentration and sample size (see C for details). This
also gives an erbium g-factor of $g=8.20$ in good agreement with the reported
values using this field orientation for $g_{\perp}$ [42, 33]. For comparison
with magnetic field sweep, the spin linewidth $\gamma$ corresponds to $1.9$ mT
or 1.7% bias field inhomogeneity.

We additionally extract a total attenuation coefficient of $-5.4$ dB mostly
attributed to the coaxial line inside the cryostat.

The parameters obtained from this measurement allows to accurately
characterize the excitation of spins coupled to the RF cavity.

####  3.1.2 Optical absorption

An equivalent analysis is conducted in the optical domain by finely sweeping
our Velocity® External Cavity Diode Laser around
$\lambda_{\mathrm{o}}=1532.636$ nm (absorption line position at zero magnetic
field) and monitor the crystal transmission with a photodiode in Fig.4. The
polarization is along $\boldsymbol{z}$ (bias magnetic field) and is
perpendicular to the crystalline c-axis.

Figure 4: Absorption spectrum under a 105mT magnetic field (solid line). Four
lines can be observed corresponding to level structure presented in Fig.1
(right). The optical wavelength is finely swept around the central value
$\lambda_{\mathrm{o}}=1532.636$nm. The strongest lines correspond to the
direct transitions $\ket{g_{-}}\rightarrow\ket{e_{-}}$,
$\ket{g_{+}}\rightarrow\ket{e_{+}}$ (solid lines in Fig.1, right). The weakest
to the crossed transitions $\ket{g_{-}}\rightarrow\ket{e_{+}}$,
$\ket{g_{+}}\rightarrow\ket{e_{-}}$ (dashed lines in Fig.1, right). We fit
data (dashed line) to extract the optical depth of the transitions (see text
for details).

We observe four lines corresponding to the so-called direct and crossed
transitions in Fig.1 (right). The two central lines are the direct ones (less
than 1GHz detuning) and the crossed ones are positioned at roughly $\pm 12$
GHz. The former are stronger that the latter, without generality because this
depends on the magnetic field orientation. The imbalance between the direct
transitions, namely $\ket{g_{-}}\rightarrow\ket{e_{-}}$ and
$\ket{g_{+}}\rightarrow\ket{e_{+}}$, is due to the finite temperature that
partially polarize the spins into the $\ket{g_{-}}$ state. This is also
present but much less visible on the crossed ones because of their weakness.
So we model the transmission as the sum of four gaussian profiles and
attribute two different absorption coefficients to the direct and crossed
transitions. We also include the finite temperature to account for the
Boltzmann thermal populations. We fit data and obtain the different
parameters. The fit is visibly imperfect because of the numbers of parameters
characterizing the transitions that must be completed by auxiliary parameters
to describe the distorted absorption baseline. The latter is not flat because
of the cryostat transmission path going through many insulating uncoated
windows (vacuum ports, radiation shields and cold windows). We nevertheless
keep the fitted parameters after convergence which give a realistic order of
magnitude that can feed the model later on. We obtain an optical depth of
$0.44$ and $0.016$ for the direct and crossed transitions respectively along
the 4mm propagation direction of the crystal. The optical depths are
normalized to a total population of one, so the finite temperature
measurements exhibit a smaller absorption because the population is shared
between $\ket{g_{-}}$ and $\ket{g_{+}}$. In other words, for an expected fully
polarized spins ensemble, the direct and crossed transitions from
$\ket{g_{-}}$ would have an optical depth of $0.44$ and $0.016$ respectively.

The fitted optical linewdith is $\Gamma=2\pi\times 1750$ MHz (FWHM), a typical
value for low concentration rare-earth samples. The value is compatible but
still larger than the previously reported inhomogeneous linewidth in
Er3+:CaWO4, even if the exact lineshape and measurement conditions cannot be
compared at this level [43].

This concludes the characterization of the spin and optical transitions
independently in terms of cooperativity. The parameters extracted from both
fits will be used to model the transduction efficiency in 4. The corresponding
experimental results will be detailed now.

###  3.2 Transduction characterization

Using the detection setup sketched in Fig.2 and described in 2, the opto-RF
transduction signal appears as a $44$ MHz beating between the spin signal
frequency $\omega_{\mathrm{o}}+\omega_{\mathrm{rf}}$ and the local oscillator
$\omega_{\mathrm{o}}+\omega_{\mathrm{lo}}$. The photodiode is directly
connected to a spectrum analyzer (Agilent E4402B, centered at 44 MHz, 50 kHz
span, 1 kHz resolution). The amplitude of the beatnote peaks (raw data) varies
from -75 dBm (maximum signal) to -95 dBm (noise floor) corresponding to the
measured voltage $V_{\mathrm{sa}}$ as defined in 2.3. The data will be
displayed in terms of electro-optics efficiency $\eta_{\mathrm{eo}}$ (in dB,
left plot axis) using (1) and quantum number efficiency efficiency (in dB,
right plot axis) since both are simply related by (2).

We will first characterize the transduction signal as a function of the
magnetic field (spin detuning) in 3.2.1 and $\omega_{\mathrm{rf}}$ (cavity
detuning) in 3.2.2 for a given laser wavelength of
$\lambda_{\mathrm{o}}=1532.727$ nm corresponding to the
$\ket{g_{+}}\rightarrow\ket{e_{-}}$ transition. We will finally vary the
optical wavelength in 3.2.3 to compare the different absorption lines.

####  3.2.1 Spin detuning

We keep the RF and laser frequencies constant as
$\omega_{\mathrm{rf}}=2\pi\times 12.29$ GHz and
$\lambda_{\mathrm{o}}=1532.727$ nm and we sweep the magnetic field to tune the
spins on resonance (see Fig. 5).

The input RF power is 18.4 dBm, the maximum power of our VNA as directly
measured at low frequency ($\sim$1 GHz). The RF power at the cavity input can
be inferred from the -5.4 dB coaxial line attenuation (extracted for the fit
described in 3.1.1) so we assume the input RF power to be reduced by -2.7 dB
at the cavity level namely $P_{\mathrm{rf}}=37$ mW or 15.7 dBm.

Figure 5: Transduction efficiency as a function of the spin detuning. The
latter is adjusted by scanning the magnetic field across the resonance as in
Fig.3. The colored circles correspond to the values $B_{0}=107.0$ mT and
$109.9$ mT that will be used for a RF frequency scanning in Fig.6. The left
axis represents the electro-optics efficiency and the right one the quantum
efficiency. Both are related by (2). The solid line is the theoretical
prediction affected by a rescaling factor as will be discussed in 4.2

The electro-optics efficiency is globally weak with a maximum of -84 dB
approximately, using (1) to convert the measured beatnote into a transduction
performance. Our noise detection floor is -105 dB when expressed in electro-
optics efficiency. A quantitative analysis will be developed in 4.

Concerning the linewidth of the beatnote, we essentially retrieve the spin
linewidth $\gamma$ as measured in 3.1.1. For the experimental curve, the
presence of dip at center and the slight shift from the spin resonance is not
reliable from shot to shot. This is attributed to instability of the magnetic
field during and between the sweeps of the magnet (hysteresis). Apart from the
linewidth, that is well predicted, the model is down scaled by -11 dB to
retrieve the experimental maximum as will be discussed in 5.

####  3.2.2 Cavity detuning

Under the same experimental conditions ($\lambda_{\mathrm{o}}=1532.727$ nm and
18.4 dBm RF power), we now vary the RF frequency for two values of the
magnetic field ($B_{0}=107$ mT and $109.9$ mT) as plotted in Fig.6.

Figure 6: Transduction efficiency as a function of the cavity detuning. We
vary the $\omega_{\mathrm{rf}}$ around the cavity resonance at $12.290$ GHz
spanning $50$ MHz. $B_{0}=107$ mT and $109.9$ mT correspond respectively to
spins on resonance or slightly detuned from the cavity (see Fig. 5). The
dashed lines are a lorentzian fit to the data. The left axis represents
$\eta_{\mathrm{eo}}$ and the right one $\eta_{\mathrm{Q}}$. The dashed line is
a lorentzian fit to the data. The solid lines are the expectation from the
model detailed in 4.2

One roughly retrieves the cavity linewidth on the electro-optics efficiency.
The curves are well approximated by a lorentzian fit (dashed lines in Fig.6)
with 8 MHz and 4.3 MHz linewidths for $B_{0}=107$ mT and $109.9$ mT
respectively. The first value (8 MHz at $B_{0}=107$ mT) should be compared to
the $\kappa_{t}=2\pi\times 8.57$ MHz cavity linewidth obtained from Fig.3. The
second value (4.3 MHz at $109.9$ mT) is narrower than $\kappa_{t}$ but is
visibly limited by our noise detection floor of -105 dB which limit the
experimental peak amplitude and in turn gives an artificially low linewidth. A
quantitative comparison of the efficiency will be detailed in 5.

####  3.2.3 Wavelength dependency

Keeping the spin RF excitation at the previously described optimal conditions,
we now vary the laser excitation wavelength. The laser is finely tuned step by
step ($\sim 0.3$ GHz steps). After each step, we record the laser wavelength
(Burleigh WA-1000 Wavelength Meter) and acquire the transduction signal in
Fig.7 as detailed in 2.3.

Figure 7: Transduction efficiency as a function of the optical wavelength for
$B_{0}=107.0$ mT. Left: broad spectrum covering the different absorption lines
observed in Fig.4. The coloured circle correspond to a wavelength of
$\lambda_{\mathrm{o}}=1532.727$ nm at which other measurements have been
performed. Right: Narrow scan around $\lambda_{\mathrm{o}}$ compared to the
prediction of the model (solid line) that will be presented in 4.2. The dashed
is a lorentzian fit to the data (linewidth 191 MHz). As before, the left axis
represents $\eta_{\mathrm{eo}}$ and the right one $\eta_{\mathrm{Q}}$ as in
Figs.5 and 6.

The transduction spectrum in Fig.7 should be compared to the optical
absorption in Fig.4. One retrieve the position of the different absorption
lines corresponding to the level structure of Fig.1 (right). The efficiency
tends to decrease from one peak to the other when the optical frequency
increases. The tendency is not reproducible from one data set to the other and
may be explained by the narrowness of the lines compared to the frequency
steps of $\sim 0.3$ GHz steps which makes the measurement quite sensitive to
the experimental instabilities.

In any case, the transduction efficiency does not follow the strength of the
transitions. The weakest and the strongest transitions in Fig.4 roughly give
the same transduction signal. This is actually not surprising because the
appearance of the transduction signal in a $\Lambda$-system connects the
strength of both optical transitions involved in the non-linear process
(namely $\ket{s}-\ket{e}$ and $\ket{g}-\ket{e}$, Fig.1, left), more precisely
the product of the direct and the crossed transitions strengths. The latter is
generally constant within $\Lambda$-systems formed by Kramers doublets in the
ground and excited states. This qualitative explanation will be discussed in
the light of the theoretical model developed in the next section 4.

Finally, in Fig.7 (right), we zoom on one peaks and compare the measurement to
the model. This latter predicts a transduction optical linewidth of
$\Gamma=2\pi\times 1750$ MHz equals to the optical absorption linewidth as in
Fig.4. The transduction linewidth is narrower by an order of magnitude (191
MHz FWHM). This is a unexpected feature that will discussed in 5.

##  4 Theoretical modeling

The transduction process in a $\Lambda$-scheme as sketched in Fig.1 can be
described by Schrödinger-Maxwell model that we detail in A. We present the
result in the form of a input-output solution.

###  4.1 Input-output solution

The outgoing signal field after propagation through the crystal (length $z$)
reads as an integral solution for $\mathcal{E}(z)$ after a propagation $z$
(36):

$\mathcal{E}(z)=\frac{1}{2\delta}\left[\exp\left(\frac{-2i\delta\Gamma}{4\delta\Delta-\Omega^{2}}\alpha_{\mathcal{E}}\,z\right)-1\right]\Omega\times\mu$
(5)

The different parameters are schematically represented in Fig.1(left). The
input-output solution (5) relates the different fields represented by their
Rabi frequencies: $\mu$ for RF field, $\Omega$ for the optical Raman field and
$\mathcal{E}$ for the transduction signal. The experimental parameters
$\Gamma$ and $\gamma$ are the optical and spin linewidths (FWHM) respectively.
$\Delta$ and $\delta$ are the detunings of optical and spin transitions
respectively. $\alpha_{\mathcal{E}}$ is the absorption coefficient on the
$\ket{g}-\ket{e}$ transition, or in other words, $\alpha_{\mathcal{E}}\,z$ is
the optical cooperativity.

In order to define the electro-optics efficiency, one has to compare the
intensity of the optical field $\Omega$ and the generated transduction signal
$\mathcal{E}$ at the output of the medium. We choose to express the fields in
units of Rabi frequencies which is a natural choice to write the light-matter
interaction Hamiltonian (29). Nevertheless, the comparison of the different
fields intensities is then indirect and requires the introduction of the
transition dipole moments. This will be discussed specifically in 4.2. One may
be also surprised by the predominance of the optical cooperativity on the
$\ket{g}-\ket{e}$ transition $\alpha_{\mathcal{E}}\,z$, because one expects a
symmetric dependency with respect to the fields involved in the non-linear
mixing process $\mu$ , $\Omega$ and $\mathcal{E}$. This contradiction is only
apparent because of the choice of units as will made explicit in defining the
electro-optics and quantum efficiencies in 4.2 and 4.3.

In the meantime, one can already discuss and simplify the expression (5). The
right-hand side term $\Omega\times\mu$ represent the lowest order frequency
mixing process that generate the transduction signal from the optical and RF
fields. Strictly speaking, the transduction can be seen as a sum-frequency
generation, namely a second order nonlinear optical-RF process as already
noted [10]. The prefactor
$\displaystyle\frac{2\delta\Gamma}{4\delta\Delta-\Omega^{2}}$ potentially
introduces high-order contributions from $\Omega^{2}$, $\Omega^{4}$ … if
written as a Taylor expansion so this is not a second order nonlinear process
anymore. It should be nevertheless noted that the intensity $\Omega^{2}$ is in
practice smaller than $\delta\Delta$. $\Delta$ is indeed of the order of the
optical inhomogeneous linewidth, experimentally much larger than $\Omega$ [44,
45]. This further simplify the analysis.

Additionally, one can consider the low absorption limit
$\alpha_{\mathcal{E}}\,z\ll 1$, keeping only the first order term in the
expansion of $\left[1-\exp\left(\cdots\alpha_{\mathcal{E}}\,z\right)\right]$

$\mathcal{E}(z)=\left[\frac{-i\Gamma}{4\delta\Delta}\alpha_{\mathcal{E}}\,z\right]\Omega\times\mu$
(6)

This simplified expression will be used for the rest of the paper, and can be
used to write the electro-optics and quantum efficiencies in 4.2 and 4.3.

###  4.2 Electro-optics efficiency

####  4.2.1 Definition

As previously discussed, we compare the intensity of the input optical field
$\Omega$ and the generated signal $\mathcal{E}$ to define electro-optics
efficiency. The Rabi frequencies $\Omega$ and $\mathcal{E}$ are related to the
electric field amplitudes $E_{\Omega}$ and $E_{\mathcal{E}}$ by
$\Omega=\displaystyle\frac{\mathcal{P}_{\Omega}E_{\Omega}}{\hbar}$ and
$\mathcal{E}=\displaystyle\frac{\mathcal{P}_{\mathcal{E}}E_{\mathcal{E}}}{\hbar}$
respectively where $\mathcal{P}_{\Omega}$ and $\mathcal{P}_{\mathcal{E}}$ are
the transition dipoles of the optical transitions for the Raman and signal
fields respectively.

The electro-optics efficiency is then
$\eta_{\mathrm{eo}}=\displaystyle\frac{\left|E_{\mathcal{E}}\right|^{2}}{\left|E_{\Omega}\right|^{2}}$.
This can be obtained directly from (6) by introducing the ratio of the
transitions dipole moments (sometimes called the branching ratio) or
equivalently
$\displaystyle\sqrt{\frac{\alpha_{\mathcal{E}}}{\alpha_{\Omega}}}$ introducing
the absorption coefficients of both transitions, leading to

$\eta_{\mathrm{eo}}=\left|\frac{\Gamma}{4\delta\Delta}\right|^{2}\alpha_{\mathcal{E}}z\times\alpha_{\Omega}z\times\left|\mu\right|^{2}$
(7)

The formula involves spectroscopic parameters (absorption coefficients and
linewidths) deduced from the experimental measurements on one side (see 3.1.1
and 3.1.2) and the Rabi frequency of RF field $\mu$ on the other side. The
expression is more symmetric since it involves equally the cooperativities
$\alpha_{\mathcal{E}}z$ and $\alpha_{\Omega}z$ on the both optical
transitions.

When defining the electro-optics efficiency, essentially comparing $\Omega$
and $\mathcal{E}$, without surprise, the RF power $\left|\mu\right|^{2}$
appears as a parameter.

In practice, the relative strengths of the optical transitions need to be
deduced from absorption measurements because the optical electric-dipole
excitation is forbidden to the first order relying on the so-called forced
electric-dipole and/or allowed magnetic-dipole (as for erbium) transitions
[45].

(7) involve the product $\alpha_{\mathcal{E}}\times\alpha_{\Omega}$ so the
transduction efficiency do not depend on which transition is pumped with
$\Omega$ (direct or crossed in Fig.1). In other words, if a weak transition is
pumped with $\Omega$ then a strong transition will generate $\mathcal{E}$ or
the other way around, then the resulting efficiency will be the same for both
cases since the product of strengths will be involved. That is the reason why
in Fig.7 the efficiency does not follow the relative strength of the excited
transitions.

The RF Rabi frequency $\mu$ can be directly deduced a priori from the
g-factors (magnetic dipole of the spin) and the value of the AC magnetic field
inside the resonator (intracavity RF power). In other words, the spin
transition is magnetic-dipole allowed to pursue the analogy, so
$\left|\mu\right|^{2}$ can be deduced from the RF cavity parameters and the
input power (directly proportional). The well-known calculation of the AC
magnetic field distribution in a rectangular resonator is reminded in B.

The RF Rabi frequency follows the resonance of the cavity given by the
relection coefficient (3). As detailed in B, we finally obtain (44) in the low
spin cooperativity limit:

$\mu=\frac{4g_{x}}{\hbar}\sqrt{\mu_{0}}\frac{a}{\sqrt{d^{2}+a^{2}}}\sqrt{\frac{P_{\mathrm{rf}}}{V}}\sqrt{\frac{2\kappa_{c}}{4\left(\omega_{\mathrm{rf}}-\omega_{c}\right)^{2}+\kappa_{t}^{2}}}$
(8)

where $g_{x}=g_{\perp}$ is $g$-factor along the $x$ direction of the RF field.
$a$, $d$ and $V$ are the resonator dimensions and volume, $P_{\mathrm{rf}}$ is
the input resonator power, as defined in B.

####  4.2.2 Finite temperature and population distribution

Moving away from the ideal structure discussed in Fig.1 (left), namely all the
atomic population is in the level $\ket{g}$ (zero temperature), it is
important to consider the imperfect polarization of the spin states as in our
case at a temperature of few kelvins. The population is distributed between
$\ket{g}$ and $\ket{s}$, written $\rho_{\mathrm{gg}}$ and $\rho_{\mathrm{ss}}$
respectively. There is no population in the excited state because of the huge
energy gap.

Rigorously accounting for the atomic population requires the use of the
density matrix formulation, we won’t detail this calculation here, as it’s
lengthy, can be found in a number of textbooks [46, 47] and ultimately gives
an intuitive result. The imperfect polarization can be introduced by by-hand
in the simplified formula (6). The population difference
$\rho_{\mathrm{gg}}-\rho_{\mathrm{ss}}$ actually scales the RF spin coupling
term, or in other words, for a perfect population balance there is no
interaction with the spin. This annihilation would correspond to a perfect
balance between the absorption and stimulated emission on the
$\ket{g}-\ket{s}$ transition. Starting for the structure in Fig.1 (left), one
could alternatively consider the reverse situation with $\ket{s}$ fully
populated and show that both schemes would interfere destructively leading to
the annihilation of the opto-RF process when both levels are populated. This
interpretation can be formally justified using the density matrix, but for the
present analysis, we can stay at the intuitive level and rescale
$\mu\rightarrow\left(\rho_{\mathrm{gg}}-\rho_{\mathrm{ss}}\right)\mu$. This
readily affects the efficiency as

$\eta_{\mathrm{eo}}=\left|\frac{\Gamma}{4\delta\Delta}\right|^{2}\alpha_{\mathcal{E}}z\times\alpha_{\Omega}z\times\left|\mu\right|^{2}\left(\rho_{\mathrm{gg}}-\rho_{\mathrm{ss}}\right)^{2}$
(9)

Beyond the rescaling of the efficiency, the absorption terms
$\alpha_{\mathcal{E}}$ and $\alpha_{\Omega}$ should be handled with
precaution: $\alpha_{\mathcal{E}}$ ($\alpha_{\Omega}$ resp.) is the absorption
coefficient on the $\ket{g}-\ket{e}$ transition ($\ket{g}-\ket{s}$ resp.) if
$\ket{g}$ is fully populated ($\ket{s}$ resp.).

The absorption spectrum as in Fig.4 actually measures $\alpha_{\mathcal{E}}$
and $\alpha_{\Omega}$ and already accounts for the current population at a
given temperature in the states $\ket{g}$ and $\ket{s}$ by assuming a thermal
population proportional to
$\rho_{\mathrm{gg}}=\displaystyle\frac{1}{1+\exp\left(-\hbar\omega_{s}/kT\right)}$
and
$\rho_{\mathrm{ss}}=\displaystyle\frac{\exp\left(-\hbar\omega_{s}/kT\right)}{1+\exp\left(-\hbar\omega_{s}/kT\right)}$.
From the optical spectrum fit, the population imbalance is inferred to
$\displaystyle\left(\rho_{\mathrm{gg}}-\rho_{\mathrm{ss}}\right)=0.12$ that
can feed (9).

In conclusion, the electro-optics efficiency (9) which includes the RF Rabi
frequency (8) only contains already measured parameters that can be used to
compare the experimental results with the previously derived model as we will
discuss in 4.4.

###  4.3 Quantum number efficiency

####  4.3.1 Definition

As compared to the electro-optics efficiency defined in 4.2, it is important
to properly describe the quantum efficiency. The latter is the main figure of
merit for a quantum transducer. Again, (6) characterizes the mixing process of
the optical pump $\Omega$ with the RF field $\mu$ (and vice-versa) to generate
the optical signal $\mathcal{E}$. As previously explained, the electro-optics
efficiency essentially compares the intensities of the signal $\mathcal{E}$
and the pump $\Omega$. The quantum efficiency compares different quantities,
namely the number of photons in the optical signal $\mathcal{E}$ and the RF
field $\mu$. From this point of view, the optical pump $\Omega$ is just a
classical parameter.

The Rabi frequencies of the optical field $\mathcal{E}$ and the RF $\mu$ are
related to the quantized fields, called $\mathcal{A}$ and $\mathcal{B}$ in the
following, by the coupling constant $g_{\mathcal{E}}$ and $g_{\mu}$
respectively in the sense of the Jaynes–Cummings model. So we have
$\displaystyle\mathcal{E}=g_{\mathcal{E}}\,\mathcal{A}$ and
$\displaystyle\mu=g_{\mu}\,\mathcal{B}$ respectively.

In (6), the term $\displaystyle\frac{\Gamma}{4\delta\Delta}\Omega$ can be
taken as a parameter, but it should also be noted that $\alpha_{\mathcal{E}}$
also contains a contribution from the coupling rate $g_{\mathcal{E}}$ between
light and atoms. Both are related by the following standard expressions, for
the coupling constant

$g_{\mathcal{E}}=d_{\mathcal{E}}\sqrt{\frac{\omega_{o}}{2\hbar\epsilon_{0}V_{\mathcal{E}}}}$
(10)

where $d_{\mathcal{E}}$ is the transition dipole and $V_{\mathcal{E}}$ the
quantization volume whose choice will be discussed extensively in 4.3.2, and
for the absorption coefficient

$\alpha_{\mathcal{E}}=\frac{\mathcal{N}\omega_{o}{d_{\mathcal{E}}}^{2}}{\hbar
c\epsilon_{0}\Gamma}$ (11)

where $\mathcal{N}$ is the atomic density. So $\alpha_{\mathcal{E}}$ can be
alternatively expressed as a function of $g_{\mathcal{E}}$ with (10)

$\alpha_{\mathcal{E}}=\frac{2\mathcal{N}V_{\mathcal{E}}{g_{\mathcal{E}}}^{2}}{c\Gamma}$
(12)

This allows to express (6) in terms of quantized fields $\mathcal{A}$ and
$\mathcal{B}$ as

$\mathcal{A}=\left[\frac{-i\Omega}{4\delta\Delta}\right]\,\frac{2\mathcal{N}V_{\mathcal{E}}z}{c}\,g_{\mathcal{E}}g_{\mu}\cdot\mathcal{B}$
(13)

Since the optical and RF fields play an equivalent role, the expression can be
made more symmetric by introducing the cooperativities that we used as key
parameters to characterize the experimental data. In cavity quantum
electrodynamics (cavity QED), the spin cooperativity is defined as

$C_{\mu}=\frac{g_{\mu}^{2}N_{\mu}}{\gamma\kappa_{c}}$ (14)

where $N_{\mu}=\mathcal{N}V_{\mathrm{crystal}}$ is the number of excited spins
($V_{\mathrm{crystal}}$ is the crystal volume) [40]. Usual cavity QED
expressions can be used without ambiguity for the RF spin coupling, but this
is not the case for the optical excitation because there is no optical
resonator in our approach. The quantization of the optical field in free-space
and the definition of the corresponding cooperativity should then be handled
with precaution as we will see now.

####  4.3.2 Choice of quantization volume

Our experimental choice, namely a resonant RF cavity but free-space optical
pass, may be a source of confusion when considering the quantum efficiency by
comparing the number of photons. As we have seen, the efficiency definition
introduces a quantization volume for the optical field that deserves a
discussion. Indeed the quantization volume of the free-space optical pass
should be adequately chosen to illustrate the transduction process. The
transverse dimension of the optical photon are essentially imposed by the beam
size (cross-section $S_{\mathcal{E}}$) but the spatial extension (pulse
length) actually depends on the pulse duration. Surprisingly, the latter
should depend on the RF pulse length. In other words, optical and RF photons
should have the same duration. As a consequence, we impose for the optical
cooperatitiy to read as

$C_{\mathcal{E}}=\frac{g_{\mathcal{E}}^{2}N_{\mathcal{E}}}{\Gamma\kappa_{c}}$
(15)

$N_{\mathcal{E}}=\mathcal{N}S_{\mathcal{E}}z$ is the number of atoms in the
optical path (cross-section $S_{\mathcal{E}}$ and length $z$). That is to say,
we impose the optical quantization volume to depend on the RF cavity linewidth
as

$V_{\mathcal{E}}=S_{\mathcal{E}}\frac{c}{2\kappa_{c}}$ (16)

where $\displaystyle\frac{c}{2\kappa_{c}}$ is the photon spatial extension in
free-space. This choice may appear arbitrary and surprising because
$N_{\mathcal{E}}\neq\mathcal{N}V_{\mathcal{E}}$ for example. But this is
driven by the physics of the conversion process and has the advantage to give
consistent results at the end, somehow extending the cavity QED model to free-
space approaches, that can be explored experimentally.

With this choice, one can alternatively express the optical cooperativity
$C_{\mathcal{E}}$ defined by (15) using the absorption coefficient (11) and
the coupling constant (10) as

$C_{\mathcal{E}}=\alpha_{\mathcal{E}}z$ (17)

Without surprise, cooperativity and optical depth are the same in free-space,
as noted early in the context of quantum memories [48]. This result is quite
intuitive, thus supporting our choice of quantization volume.

####  4.3.3 Expression as function of the cooperativities

(13) can now be simplified by introducing the cooperativities defined by (14)
and (15).

$\mathcal{A}=\left[\frac{-i\Omega\sqrt{\Gamma\gamma}}{4\delta\Delta}\right]\,\sqrt{\frac{S_{\mathcal{E}}z}{V_{\mathrm{crystal}}}}\sqrt{C_{\mathcal{E}}}\sqrt{C_{\mu}}\cdot\mathcal{B}$
(18)

The quantum number efficiency $\eta_{\mathrm{Q}}$ compares the photon numbers
in the quantized fields $\mathcal{A}$ and $\mathcal{B}$ leading to the
expression

$\eta_{\mathrm{Q}}=\left|\frac{\Omega\sqrt{\Gamma\gamma}}{4\delta\Delta}\right|^{2}\,{C_{\mathcal{E}}}{C_{\mu}}\,\frac{S_{\mathcal{E}}z}{V_{\mathrm{crystal}}}$
(19)

(19) should be compared to its equivalent electro-optics efficiency expression
(7). In a same manner, if we account for an incomplete polarization in the
ground state, the efficiency is rescaled by the factor
$\left(\rho_{\mathrm{gg}}-\rho_{\mathrm{ss}}\right)^{2}$ as in (9).

###  4.4 Discussion

(9) and (19) describe the same physical process with different points of view.
The quantum efficiency (19) shows the symmetric role of the laser and RF
fields cooperativities as opposed to (9) where the spin collective coupling is
somehow hidden in the definition of the Rabi frequency. Our derivation of (19)
leads to the same expression of the cavity QED approach, when both fields are
enhanced in resonators [10]. In 4.3, we simply illustrate the experimental
comparison between $\eta_{\mathrm{eo}}$ and $\eta_{\mathrm{Q}}$ than can be
done directly with (2). Nevertheless, we have shown that the well-established
Schrödinger-Maxwell description leads to the same expression that can be
derived from the cavity QED formalism. This is important in our case since the
optical field is propagating in free-space (single-pass). In any case, (19)
shows the prevalence of the cooperativities and the intensity of the pump
field $\Omega$ as key parameters that should drive the process understanding
and the experimental design.

As discussed in 4.2, (9) allows to predict the efficiency by evaluating the
experimental parameters. This is sufficient to compare the data with the
model. Even if the expression (19) involves different parameters (that could
be measured independently). it derives from the same expression. So for a
given set, (9) and (19) are proportional and related by (2). In Figs. 5, 6 and
7, we use two efficiency axes, left and right, for (9) and (19) respectively.
They simply differ by 57.5 dB.

The discrepancy between the experimental data and the model can now be
discussed specifically.

##  5 Analysis

In Figs. 5, 6 and 7, the model only qualitatively reproduces the data. Indeed,
to roughly reproduce the observed efficiencies, we systematically downscale
the model by 11 dBm (in the three Figs. 5, 6 and 7). More precisely, the
experimental efficiency $\eta_{\mathrm{eo}}$ peaks typically at -84 dBm in the
optimal conditions (on-resonance fields) where the model predicts -73.4 dBm.

Concerning the RF excitation, we note that the linewidths, when we vary the
spin detuning in Fig.5 or the cavity detuning in Fig.6, are properly
reproduced when the measured signal largely exceed our noise level. Only the
11 dBm discrepancy remains.

Concerning the optical linewidth of the transduction signal in Fig.4, we find
a FWHM of 191 MHz, that is much narrower than the absorption peaks
$\Gamma=2\pi\times 1750$ MHz in Fig.7, by an order of magnitude. The observed
value of 191 MHz is surprisingly close to the spin linewidth
$\gamma=2\pi\times 219.14$ MHz measured in 3.1.1. This points to a possible
correlation between the spin and optical transitions, the effect of which has
still to be assessed. In any case, the global discrepancy cannot be simply
explained within our framework and deserves further investigations.

The fact that we obtain a transduction feature narrower than the inhomogeneous
absorption profile may also indicate that the mixing process actually resolves
the inhomogeneous linewidth, comparable to the spectral hole burning
mechanism, without reaching the homogeneous linewidth though ($<$MHz). This
may partially explain the discrepancy with the model. Indeed, spectral hole
burning and more generally optical pumping dynamics is not included in our
model because we assume in A.1 that the population stays in the ground state
(so-called perturbative regime), or at minimum that the population
distribution is steady (see 4.2.2). This would require further theoretical
modeling and would for sure introduce another level of non-linearity with the
optical pumping intensity (noted $\Omega^{2}$) acting not only on the
coherence as in (30) and (31), but also on the population as an optical
pumping mechanism.

We conclude with a last remark to further explain the limitations of the
theoretical model and the discrepancy with the experiment. We have briefly
discussed the nature of optical transitions in 4.2 driven by $\Omega$ and
$\mathcal{E}$, forced electric or magnetic dipole, both of which are possible
in the case of erbium. In terms of absorption, there is no distinction. This
is not the case for the transduction process. The branches of the
$\Lambda$-system must in fact be of the same nature $\Omega$ and $\mathcal{E}$
without imposing any special constraint on the spin transition driven by
$\mu$, as judiciously noted in [49, 23]. A fictitious pathological case, where
one branch of the $\Lambda$-system would be purely magnetic dipole and the
other purely electrical dipole, prohibits the non-linear mixing process. The
interaction Hamiltonian introduced in the section A here assumes couplings of
the same nature for the transitions, generally electric dipole for the sake
simplicity, but an equivalent model would give the same result for a magnetic
dipole coupling [50]. A measurement of absorption solely cannot determine the
nature of the transitions, and can only roughly predict the efficiency of the
transduction process. This uncertainty should be minimized in our case, since
the magnetic field is perpendicular to the crystal c-axis, which should avoid
a complete disjunction in the nature of the $\Lambda$-system transitions [51].
However, a rigorous evaluation can only be made through a detailed analysis of
the crystal field and the nature of the wave functions in the levels involved
[49].

##  6 Conclusion

We observed and carefully characterized the opto-RF transduction process in
Er3+:CaWO4 using the resonant excitations of the spin and optical transitions
in a $\Lambda$-system. Our work can alternatively be seen as the quantitative
extension of the Raman heterodyne detection technique, since we carefully
model the intensity of the beating signal. We have decided to operate a
simplified setup, employing a rectangular RF resonator to enhance the spin
driving and a single-pass laser beam to excite the optical transition. This
allows to extract the different experimental parameters independently that can
feed our Schrödinger-Maxwell model.

We reach an electro-optics efficiency of -84 dB for 15.7 dBm RF power
corresponding to a quantum efficiency of -142 dB for 0.4 dBm optical power. We
carefully define both quantities that describe the same process from different
points of view. The model convincingly predicts the different spin-related
linewidths but fails to reproduce the optical linewidth and overestimates the
efficiencies. We bring out a systematic discrepancy and downscale the model by
11 dBm to fit data. The origin of this discrepancy is not understood. In the
context of quantum transduction, where efficiency matters, we hope our
analysis will stimulate further studies, both theoretical and experimental,
involving other materials.

To conclude, it is important to return to the main context motivating our
study. How can we increase the quantum efficiency close to unity ? The
question here is to gain orders of magnitude compared to our proof of
principle, which is limited to -142 dB. The configuration we have chosen
(optical single-pass) can be optimized by using Er3+:Y2SiO5 for example, whose
bare cooperativity (absorption) remains better and the transitions narrower,
with a direct impact on the efficiency (19). A higher optical power can then
be used to achieve typically -90 dB [26]. Another order of magnitude can be
achieved by fully polarizing the spins at reduced temperature (10-20 mK),
which will in any case be necessary for the interconnection with
superconducting qubits for example. There’s still a long way to reach a unit
efficiency.

The use of an optical cavity then offers a typical gain of four orders of
magnitude compared with single-pass configuration, as the cooperativity
follows proportionally the finesse of the optical resonator [26]. Finally, the
obvious way to perfect the job is to use fully concentrated materials [15].
Little studied in optics, they are particularly attractive compared with
diluted samples (10-100 ppm), since the concentration and therefore the
cooperativity increase by 4 or 5 orders of magnitude. At very low
temperatures, they can even exhibit magnetic ordering, further narrowing the
magnon resonances, whose sharpness increases the efficiency. Their study under
coupled optical and RF excitations clearly opens up a new field for rare-earth
doped crystals.

## Acknowledgements

The authors acknowledge support from the French National Research Agency (ANR)
through the projects MIRESPIN (ANR-19-CE47-0011) and MARS (ANR-20-CE92-0041).

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##  Appendix A Schrödinger-Maxwell model

The dynamics of three-level atoms under field excitation (optical or RF) can
be found in many textbooks. The interest for the coherent excitation of
$\Lambda$-scheme has been renewed by the observation of electromagnetically
induced transparency opening a wide spectrum of quantum memory protocols [18].
This represents a solid background for the theoretical description of the
opto-RF transduction process even if this connection has not been made
explicit so far. A modal description as a generalization of cavity quantum
electrodynamics with different fields is usually preferred without specific
assumption on the physical system [11, 10]. We choose to stick to the
Schrödinger equation for three-level atoms as depicted in Fig.8 even if both
approaches are equivalent to some extend.

Figure 8: $\Lambda$-scheme to implement resonant transduction similar to Fig.1
with the notations for the model in 4. $\mu$ is the Rabi frequency of RF field
on the spin transition $\ket{g}-\ket{s}$ (frequency $\omega_{\mathrm{rf}}$
detuned by $\delta$). $\Omega$ is the Rabi frequency of optical field on the
transition $\ket{s}-\ket{e}$ (frequency $\omega_{\mathrm{o}}$ detuned by
$\Delta$). The transduction signal is generated at
$\omega_{\mathrm{o}}+\omega_{\mathrm{rf}}$ with a Rabi frequency $\mathcal{E}$
on the $\ket{g}-\ket{e}$ transition. We first assume that the level $\ket{g}$
is solely populated (zero temperature). This assumption will be correct by-
hand to account for the presence of population in $\ket{s}$ as well (finite
temperature in 4.2.2).

###  A.1 Schrödinger equation for three-level atoms in the perturbative regime

For three-level atoms, labeled $|g\rangle$, $|e\rangle$ and $|s\rangle$ for
the ground, excited and spin states depicted in Fig.8, the rotating-wave
probability amplitudes $C_{g}$, $C_{e}$ and $C_{s}$ respectively are governed
by the time-dependent Schrödinger equation [52, eq. (13.29)]:

$\displaystyle i\partial_{t}\left[\begin{array}[]{c}C_{g}\\\ C_{e}\\\ C_{s}\\\
\end{array}\right]=\left[\begin{array}[]{ccc}0&\displaystyle\frac{\mathcal{E}^{*}}{2}&\displaystyle\frac{\mu^{*}}{2}\\\
\displaystyle\frac{\mathcal{E}}{2}&-\Delta&\displaystyle\frac{\Omega}{2}\\\
\displaystyle\frac{\mu}{2}&\displaystyle\frac{\Omega^{*}}{2}&-\delta\\\
\end{array}\right]\left[\begin{array}[]{c}C_{g}\\\ C_{e}\\\ C_{s}\\\
\end{array}\right]$ (29)

where $\Omega(t)$ and $\mu(t)$ are the complex slowly varying envelopes of the
RF and the optical (sometime called Raman) field respectively. The
transduction signal $\mathcal{E}(z,t)$ potentially contains a spatial
dependency in the single-pass configuration (along $z$) accounting for
absorption and/or amplification though propagation. $\mathcal{E}(z,t)$ is
emitted $\omega_{\mathrm{o}}+\omega_{\mathrm{rf}}$. If the spin level
$|s\rangle$ is empty, the Raman field $\Omega(t)$ is not attenuated
(independent of $z$). The RF field $\mu(t)$ do not depend on $z$ because the
single-pass absorption of the spin is extremely weak so it is uniform in the
medium.

The atomic variables $C_{g}$, $C_{e}$ and $C_{s}$ depend on $z$ and $t$ for
given detunings $\Delta$ and $\delta$. Decay terms $\Gamma$ and $\gamma$ for
the optical and spin transitions can be added by-hand by introducing complex
detunings for $\Delta\rightarrow\Delta-i\Gamma/2$ and
$\delta\rightarrow\delta-i\gamma/2$ where $\Gamma$ and $\gamma$ are the
optical and spin linewidths (FWHM) respectively.

The symmetry of the Hamiltonian in (29) reveals the equivalency between the
optical and RF fields and the bidirectional character of the opto-RF
conversion, from optics to RF and vice versa.

Going on the step further, the so-called perturbative regime assumes that the
atoms stay in the ground, $C_{g}\simeq 1$ to the first order, because the
signal is weak. The atomic evolution (eq.29) is now only given by $C_{e}$ and
$C_{s}$ that we write with $\mathcal{P}\simeq C_{e}$ and $\mathcal{S}\simeq
C_{s}$ to describe the optical (polarization $\mathcal{P}$) and spin
($\mathcal{S}$) excitations [18, and references therein]. The atoms dynamics
from (29) becomes:

$\displaystyle\partial_{t}\mathcal{P}$
$\displaystyle=i\Delta\mathcal{P}-i\frac{\Omega}{2}\mathcal{S}-i\frac{\mathcal{E}}{2}$
(30) $\displaystyle\partial_{t}\mathcal{S}$
$\displaystyle=-i\frac{\Omega^{*}}{2}\mathcal{P}+i\delta\mathcal{S}-i\frac{\mu}{2}$
(31)

###  A.2 Maxwell propagation equation

The propagation of the signal $\mathcal{E}(z,t)$ is described by the Maxwell
equation that can be simplified in the slowly varying envelope approximation
[52, eq. (21.15)]. This reads for an homogeneous ensemble whose linewidth is
given by the decay term $\Gamma$:

$\partial_{z}\mathcal{E}(z,t)+\frac{1}{c}\partial_{t}\mathcal{E}(z,t)=-\displaystyle{i\alpha_{\mathcal{E}}}\,\Gamma
C_{g}^{*}C_{e}$ (32)

The term $C_{g}^{*}C_{e}$ is the atomic coherence on the
$|g\rangle\rightarrow|e\rangle$ transition directly proportional to the atomic
polarization. The light coupling constant is included in the absorption
coefficient $\alpha_{\mathcal{E}}$ on the $\ket{g}-\ket{e}$ transition
(inverse of a length unit), thus the right hand side represents the
macroscopic atomic polarization $\mathcal{P}$ and can be written as

$\partial_{z}\mathcal{E}(z,t)+\frac{1}{c}\partial_{t}\mathcal{E}(z,t)=-\displaystyle{i\alpha_{\mathcal{E}}}\,\Gamma\mathcal{P}(t)$
(33)

The set of coupled equations (30)&(31) and the propagation (33) describe the
generation of the transduction signal in the single optical pass configuration
that we explore experimentally. We now use this model to predict the electro-
optics efficiency as described in 2.3.

###  A.3 Input-output solution

In the continuous pumping regime, the efficiency is given by the stationary
solution of (30)&(31), more precisely

$\mathcal{P}=\frac{2\delta}{4\delta\Delta-\Omega^{2}}\mathcal{E}+\frac{\Omega}{4\delta\Delta-\Omega^{2}}\mu$
(34)

The variables are now time independent. We additionally assume that the fields
are real thus setting a fix phase relation between them, in other words the
fields are not frequency swept. We remind that the decay rates are hidden in
the terms $\Delta\rightarrow\Delta-i\Gamma/2$ and
$\delta\rightarrow\delta-i\gamma/2$ allowing a compact notation. The reader
familiar with the previously mentioned electromagnetically induced
transparency will recognized in the prefactor
$\displaystyle\frac{2\delta}{4\delta\Delta-\Omega^{2}}$ the optical
susceptibility under Raman excitation which vanished (complete transparency)
at $\delta=0$. The other term in $\mu$ represent the optical excitation driven
by the RF field $\mu$ and the Raman field $\Omega$ (mixing process). This is
literally the source of the transduction process.

$\partial_{z}\mathcal{E}(z)=-i\alpha_{\mathcal{E}}\,\frac{2\delta\Gamma}{4\delta\Delta-\Omega^{2}}\mathcal{E}-i\alpha_{\mathcal{E}}\,\frac{\Omega\Gamma}{4\delta\Delta-\Omega^{2}}\mu$
(35)

whose integral solution formally reads as

$\mathcal{E}(z)=\frac{1}{2\delta}\left[\exp\left(\frac{-2i\delta\Gamma}{4\delta\Delta-\Omega^{2}}\alpha_{\mathcal{E}}\,z\right)-1\right]\Omega\times\mu$
(36)

This expression in discussed in 4 and reminded in the text as (5).

##  Appendix B Intracavity RF magnetic field

The electric and magnetic field distributions are well-known is the simple
rectangular $15\times 10\times 20$ mm geometry [38, eq.(6.42)]. The electric
field $E_{y}$ lies along $y$, the small dimension $b=10$ mm. The magnetic
field circulates in the $x,z$ plane with $B_{x}$ and $B_{z}$ components (see
Fig.9 for details). The amplitudes are:

$\displaystyle E_{y}$ $\displaystyle=E_{0}\sin\left(\frac{\pi
x}{a}\right)\sin\left(\frac{\pi z}{d}\right)$ (37) $\displaystyle B_{x}$
$\displaystyle=-j\frac{E_{0}}{c}\frac{a}{\sqrt{d^{2}+a^{2}}}\sin\left(\frac{\pi
x}{a}\right)\cos\left(\frac{\pi z}{d}\right)$ (38) $\displaystyle B_{z}$
$\displaystyle=j\frac{E_{0}}{c}\frac{d}{\sqrt{d^{2}+a^{2}}}\cos\left(\frac{\pi
x}{a}\right)\sin\left(\frac{\pi z}{d}\right)$ (39)

The amplitude of the electric field $E_{0}$ scales the fields distributions
and depends on the total energy stored in the resonator
$\mathbb{E}_{\mathrm{cav}}$ as given by [38, eq.(6.43)]:

$\mathbb{E}_{\mathrm{cav}}=\frac{\epsilon_{0}V}{8}E_{0}^{2}$ (40)

where $V=abd$ is the cavity volume.

Figure 9: Magnetic field lines in the rectangular RF resonator with dimensions
$a\times b\times d$=$15\times 10\times 20$mm along x,y,z respectively. The
crystal (red square) sits at the bottom of the cavity and is crossed by the
laser beam (magenta dot) propagating along y

At the position of the laser schematically represented in Fig.9 ($x=a/2$ and
$z=2$ mm), the magnetic field is uniform (small beam side compared the cavity
dimensions) and along $x$ with

$B_{x}(z_{0})=-j\frac{E_{0}}{c}\frac{a}{\sqrt{d^{2}+a^{2}}}\cos\left(\frac{\pi
z_{0}}{d}\right)$ (41)

with $\displaystyle\cos\left(\frac{\pi z_{0}}{d}\right)\simeq 0.95$

The amplitude of the magnetic field can now be deduced from the incoming power
$P_{\mathrm{rf}}$ and the resonator linewidth [41, Eq.(3.16)] and follows the
resonant character of the energy stored. For an empty cavity (neglecting the
spin cooperativity), we obtain

$\mathbb{E}_{\mathrm{cav}}=\frac{\epsilon_{0}V}{8}E_{0}^{2}=\frac{4\kappa_{c}}{4\left(\omega_{\mathrm{rf}}-\omega_{c}\right)^{2}+\kappa_{t}^{2}}P_{\mathrm{rf}}$
(42)

The spin interaction can also be introduced as in (3) where
$W(\omega_{\mathrm{rf}})$ appears as an extra loss coefficient (spin
absorption) or similarly as a rescaling of $\kappa_{t}$.

So the magnetic field amplitude in the crystal is

$|B_{x}(z_{0})|=4\sqrt{\mu_{0}}\frac{a}{\sqrt{d^{2}+a^{2}}}\sqrt{\frac{P_{\mathrm{rf}}}{V}}\sqrt{\frac{2\kappa_{c}}{4\left(\omega_{\mathrm{rf}}-\omega_{c}\right)^{2}+\kappa_{t}^{2}}}$
(43)

We have dropped the term $\displaystyle\cos\left(\frac{\pi z_{0}}{d}\right)$
for simplicity leading to a small 5% error. The expression can be further
simplified, if needed, under critical coupling condition with
$\kappa_{t}\approx 2\kappa_{c}$. Any way, the different parameters to
determine to local value of the oscillating magnetic field are known for a
given cavity geometry or measured from RF measurements detailed in 3.1.1.

The corresponding Rabi frequency, used to evaluate the electro-optics
efficiency (6), finally reads as

$\mu=\frac{g_{x}\mu_{B}|B_{x}(z_{0})|}{\hbar}=\frac{4g_{x}\mu_{B}}{\hbar}\sqrt{\mu_{0}}\frac{a}{\sqrt{d^{2}+a^{2}}}\sqrt{\frac{P_{\mathrm{rf}}}{V}}\sqrt{\frac{2\kappa_{c}}{4\left(\omega_{\mathrm{rf}}-\omega_{c}\right)^{2}+\kappa_{t}^{2}}}$
(44)

where $g_{x}$ is $g$-factor along the $x$ direction of the RF field. In our
case, the crystalline c-axis lies along $y$ (laser propagation) so
$g_{x}=g_{\perp}$, the perpendicular value of the $g$-tensor.

##  Appendix C Estimation of the spin cooperativity

As introduced in section 3.1.1, the spin cooperativity can be deduced from the
cavity parameters $\gamma$, $\kappa_{c}$ and spin coupling strength as

$C_{\mu}=\frac{g_{\mu}^{2}N_{\mu}}{\gamma\kappa_{c}}$ (45)

where $g_{\mu}$ is the coupling constant and $N_{\mu}$ the total number of
spins. $g_{\mu}$ can be deduced for the single-photon Rabi frequency when
$\mathbb{E}_{\mathrm{cav}}=\hbar\omega$.

The zero-point magnetic field fluctuations using the notation of section B
then reads as

$B_{\mathrm{vac}}=2\frac{a}{\sqrt{d^{2}+a^{2}}}\sqrt{\frac{2\hbar\omega\mu_{0}}{V}}$
(46)

leading to the coupling strength $g_{\mu}=g_{x}\mu_{B}B_{\mathrm{vac}}/\hbar$

$g_{\mu}=g_{x}\mu_{B}\frac{a}{\sqrt{d^{2}+a^{2}}}\sqrt{\frac{2\omega\mu_{0}}{\hbar
V}}$ (47)

As already noted in 4.2.2, because of the finite temperature and the imperfect
spin polarization, we have to account for the population difference
$\rho_{\mathrm{gg}}-\rho_{\mathrm{ss}}$ that scales the spin cooperativity as
well, effectively modifying the number of spins
$N_{\mu}\rightarrow\left(\rho_{\mathrm{gg}}-\rho_{\mathrm{ss}}\right)N_{\mu}$.
At our temperature, we measure
$\displaystyle\left(\rho_{\mathrm{gg}}-\rho_{\mathrm{ss}}\right)=0.12$ (see
4.2.2).

The cooperativity can now be evaluated for a 50 ppm doped sample, by knowing
the Ca concentration ($1.3\times 10^{22}$ at/cm3) and noting that 77% of the
erbium ions (without nuclear spin) are probed by our setup, we obtain
$C_{\mu}=0.12$, in satisfying agreement the experimentally inferred value of
0.14 in section 3.1.1.

##  Appendix D Choice of the heterodyne beating frequency

One may be surprised by our choice of the heterodyne beating frequency that
can be in principle arbitrarily taken within the detector bandwidth.

Indeed, when we record the noise spectrum on the photodiode, we interestingly
observe a minimum close to 44 MHz that we choose as a beatnote heterodyne
detection frequency as discussed in 2.2.

Figure 10: Noise spectrum on the photodiode (500 kHz resolution). We mark with
a vertical dashed line the minimum close to 44 MHz that we choose as a
beatnote heterodyne detection frequency for the transduction signal.

This minimum of the laser intensity noise that our partially fibered optical
setup forms a Mach-Zender interferometer for the optical carrier (see Fig.2).
The interferometer can be seen as a filter in the frequency domain, that
exhibits minima for frequencies $\displaystyle\frac{c}{2L}$,
$\displaystyle\frac{3c}{2L}$, $\displaystyle\frac{5c}{2L}$ … where $L$ is the
optical path length difference between the two arms. This latter is
approximately $3$m in our case, corresponding to the observed $44$ MHz
minimum.