# Measuring a population of spin waves from the electrical noise of an
inductively coupled antenna

T. Devolder  Université Paris-Saclay, CNRS, Centre de Nanosciences et de
Nanotechnologies, 91120, Palaiseau, France<EMAIL_ADDRESS>S.-M.
Ngom  A. Mouhoub  J. Létang  J.-V. Kim  P. Crozat  J.-P. Adam  Université
Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 91120,
Palaiseau, France  A. Solignac  SPEC, CEA, CNRS, Université Paris-Saclay,
91191 Gif-sur-Yvette, France  C. Chappert  Université Paris-Saclay, CNRS,
Centre de Nanosciences et de Nanotechnologies, 91120, Palaiseau, France

###### Abstract

We study how a population of spin waves can be characterized from the analysis
of the electrical microwave noise delivered by an inductive antenna placed in
its vicinity. The measurements are conducted on a synthetic antiferromagnetic
thin stripe covered by a micron-sized antenna that feeds a spectrum analyser
after amplification. The antenna noise contains two contributions. The
population of incoherent spin waves generates a fluctuating field that is
sensed by the antenna: this is the ”magnon noise”. The antenna noise also
contains the contribution of the electronic fluctuations: the Johnson-Nyquist
noise. The latter depends on all impedances within the measurement circuit,
which includes the antenna self-inductance. As a result, the electronic noise
contains information about the magnetic susceptibility of the stripe, though
it does not inform on the absolute amplitude of the magnetic fluctuations. For
micrometer-sized systems at thermal equilibrium, the electronic noise
dominates and the pure magnon noise cannot be determined. If in contrast the
spinwave bath is not at thermal equilibrium with the measurement circuit, and
if the spin wave population can be changed then one could measure a mode-
resolved effective magnon temperature provided specific precautions are
implemented.

##  I Introduction

Spin waves (SW) possess specific properties that make them well-suited for
microwave applications [1, 2]. This includes anisotropic and non-reciprocal
dispersion relations as well as tunability. Moreover, there is a growing use
of the non-linear interactions of spin waves for advanced computing proposals
[3, 4, 5]; these applications rely on magnetic bodies where large populations
of coherently pumped spin waves share a common space with their less coherent
biproducts [6, 7] and the thermal populations of spin waves [8, 9, 10]. It is
thus of interest to develop experimental techniques able of measuring spin
waves in broad frequency intervals with a large dynamic range, ideally from
the floor of the thermal population of spin waves up to the regime of large
amplitudes of magnetization precession. There is also a fundamental interest
in the development of such experimental techniques since observing the
fluctuations of a system is a way [11] to get direct insight into its
intrinsic internal dynamics.

Brillouin Light Scattering (BLS) [12] and its space-resolved variant [13, 14]
are the techniques of choice for SW measurements: they feature the proper
sensitivity and dynamic range, but they involve a somewhat heavy
infrastructure for a lateral resolution that is limited by diffraction. A
better lateral resolution –essentially limited by nanofabrication [15]– is
provided by the electrical methods such as propagating spin wave spectroscopy
(PSWS [16, 17]) or cantilever-based techniques like magnetic resonance force
microscopy [18]. Unfortunately most of these techniques rely on the homodyne
detection of coherent SW that must be pumped in an ad-hoc manner, with no
ability to measure spin waves out-of-sync from the pump. Finally, broadband
magneto-resistive noise spectroscopy is able to measure SW thermal populations
even in sub-100 nm magnet [19], however it requires to apply invasive currents
and can be implemented only on patterned systems that display magneto-
resistance. A versatile electrical method capable of measuring SW populations
over a large dynamic range in any material is therefore still missing.

Here we study whether the noise of a lithographically-defined inductive
antenna can be used to measure a population of incoherent SW placed in its
vicinity. We illustrate the method on a synthetic antiferromagnet (SAF) in
which both the SW frequencies and the SW group velocities can be adjusted
conveniently over large intervals using external fields. This method can
actually be implemented on any type of magnet, not just SAFs. We show that the
method is versatile, broadband but requires the implementation of impedance
matching precautions.

Our samples are described in section II. Section III depicts two experimental
configurations, their operation, and provides the results on the electrical
power spectra delivered by the antenna. The results indicate that the
magnetization-sensitive part of the signal delivered by the antenna arises
from two contributions. We conjecture that the two origins are uncorrelated so
that their noise powers are additive. Section IV evaluates how the field- and
frequency-dependent impedance of the antenna modulates the overall electronic
noise power that it delivers, independently from the actual population of spin
waves that would be present in the sample in the absence of the antenna.
Section V accounts for the power collected by the inductive antenna thanks to
the magnetic fluctuations underneath, assumed to be related to the population
of incoherent SW. We then compare the amplitudes of these two sources of
signal. We finally discuss the conditions on the magnetic material, geometry
and circuit impedance that must be met in order to be able to measure
quantitatively a thermal or non-thermal population of incoherent spin waves
from the total noise delivered by the antenna.

##  II Samples

###  II.1 Materials, geometry and uniform resonances

We use in-plane magnetized Synthetic AntiFerromagnetic (SAF) films of
composition Ta (6 nm) / Co40Fe40B20 ($t$) /Ru (7 Å) / Co40Fe40B20 (t) / Ru
(0.4 nm)/ Ta (6 nm) with thicknesses $t=17~{}\textrm{nm}$ [Fig. 1(a)]. The
magnetization and the damping of the CoFeB material were determined on single
films to be respectively $M_{s}=1.35~{}\textrm{MA/m}$ and $\alpha\approx
0.004$. A small uniaxial anisotropy field of 3 mT exists within the sample
plane; the spin-flop field of the SAF is thus finite ($\approx$0-15 mT) [20,
21] except when the field is applied exactly along the hard axis. In this
paper, the hard axis is taken conventionally along $x$ [Fig. 1(a)], although
the anisotropy field will be neglected in the equations.

Figure 1: Properties of Synthetic Antiferromagnet (SAF) films. (a): Sketch of
the stack. (b): Sketch of the device and main notations. (c): Field derivative
of the imaginary part of the permeability of the film measured by VNA-FMR
prior to patterning using an external stripline of width 0.05 mm and an rf
power of 0 dBm. The optical and acoustical spin wave resonances at zero
wavevector have frequencies corresponding to the red/blue frontiers. (d): Spin
wave thermal spectra measured by BLS microscopy at the middle of a 5 $\mu$m
wide SAF stripe, on a film with thinner (hence more transparent) Ta cap. The
red vertical bars illustrate the frequency range in which thermal spin waves
are detected. Inset: hysteresis loops along the easy and hard axes.

Vector Network Analyser ferromagnetic resonance (VNA-FMR [22]) was used to
check the properties of the complete SAF stack on an unpatterned film [Fig.
1(c)]. The parallel pumping configuration was chosen to allow detecting both
the acoustical and the optical eigenmodes [23], all at zero wavevector $k$. At
remanence, the uniform (i.e. $k=0$) optical mode is at
$\frac{\omega_{\textrm{op}}}{2\pi}=11.3~{}\textrm{GHz}$. The interlayer
exchange field $\mu_{0}H_{j}=-\frac{2J}{{M_{s}}{t}}$ (with $J$ the interlayer
exchange energy) can be estimated from the 2-macrospin approximation [21]. The
formulas are detailed in Table I. A first estimate stems from the optical
frequency $\omega_{\textrm{op}}\big{|}_{H_{x}=0}=\gamma_{0}\sqrt{M_{s}H_{j}}$;
it yields $J\approx-1~{}\textrm{mJ/m}^{2}$. An alternative estimate can be
deduced from a linear fit of the frequency of the uniform (i.e. $k=0$)
acoustical mode $\frac{\omega_{\textrm{ac}}}{2\pi}$ versus $H_{x}$ at low
field (see appendix I). This alternative estimation yields a slightly higher
value of $J\approx-1.2~{}\textrm{mJ/m}^{2}$. We noticed that this value is
very sensitive to the details of material deposition, thermal treatment and
aging.

For this study the SAF films were patterned into stripes of width
$w_{\textrm{mag}}=20~{}\mu\textrm{m}$ along $(x)$ and lengths
$\ell_{\textrm{mag}}=75~{}\mu\textrm{m}$ along $(y)$ [Fig. 1(b)]. We then
deposit a $s=150$ nm insulation layer of Si3N4. We finally make single-wire
antennas of composition Ti(10 nm)/Au(150 nm), width
$\mathcal{A}=1.8~{}\mu\textrm{m}$. The antenna dc impedance amounts to
$\tilde{Z}_{\textrm{out}}\approx(10+0i)\Omega$. . Although our present purpose
–the inductive measurement of the spin wave populations– requires only one
antenna we have deposited a second one at a distance $r=5~{}\mu\textrm{m}$ to
enable to check of the SW main properties using propagating spin wave
spectroscopy (not shown).

Figure 2: Noise sources within the sample: (a) Variance of the Johnson-Nyquist
e.m.f due to the Brownian motion of the electrons within the measurement
conductive and dissipative path, and (b) inductive e.m.f. arising from the
fluctuations within the spin wave bath underneath the antenna.

###  II.2 Identification of spin wave branches using microfocused BLS

To characterize the spin waves in our material, we measured their thermal
spectra by BLS with a laser focused at the diffraction limit (350 nm) at the
surface of a SAF [Fig. 1(d)]. We used a dedicated sample in which the
thickness of the Ta cap layer was reduced to 5 nm to allow optical access to
the topmost CoFeB layer. Besides, the stripe width was 5 $\mu$m. In this
focused optical configuration, the sensitivity is maximal at ${k}={0}$. The
similarity between the BLS and the VNA-FMR spectra [Fig. 1(c) vs (d)] confirms
the identification of the modes even if the mode frequencies of the BLS-
dedicated sample and the fully capped samples are slightly different. We
believe that this difference is due to the large sensitivity of the interlayer
exchange coupling to the structural properties of the material.

Interestingly, this focused BLS measurement is also sensitive to the SW with
finite wavenumbers, typically up to $k_{\textrm{max}}=18~{}\textrm{rad}/\mu$m.
For spin waves with wavevector orientation $\theta=\\{\vec{k},\vec{H_{x}}\\}$,
the frequency band in which the SW can be optically detected has a width
$\frac{1}{2\pi}{v_{g}(\theta)k_{\textrm{max}}}$, where
$v_{g}(\theta)=\frac{\partial\omega}{\partial|k|}$ is the (positive or
negative) group velocity in the $\theta$ direction. The optical SW of a SAF
are always forward waves such that the thermal population of optical SW is
seen as a halo extending only above the frequency of the $k=0$ optical mode
[Fig. 1(d)]. This contrasts with the acoustical SW for which the dispersion
relation includes both forward and backward waves as $\theta$ is varied, such
that the halo extends both below and above the frequency of the $k=0$
acoustical mode.

We emphasize that in the remainder of this study we will use a linear antenna
instead of the circular laser spot. As a result, we shall be sensitive only to
spin waves with either vanishing wavevectors, or with a wavevector
perpendicular to the antenna (the inductive contributions of other SW average
out over the antenna length). In this $\vec{k}\perp\vec{H_{x}}$ configuration,
both the optical and the acoustical SWs have a forward character (see
appendix); their frequencies will always be above the $k=0$ cases. This is a
major difference from the BLS case.

##  III Measurement of the electrical noise power delivered by the inductive
antenna

###  III.1 Qualitative description of the sample impedance and fluctuators

Our goal is now to determine the different contributions to the rf electrical
noise power that the antenna can deliver. For qualitative understanding, it is
convenient to view the {antenna + SAF} system as the noisy electromotive force
(e.m.f) of a Thevenin generator connected in series with a noise-free output
impedance $\tilde{Z}_{\textrm{out}}$ (Fig. 2). The impedance
$\tilde{Z}_{\textrm{out}}$ comprises a (non-stochastic) resistive part and a
(non-stochastic) self-inductance part which increases linearly with the
magnetic susceptibility of the SAF. As a result, the impedance matching
between the sample and the measurement circuit depends on the applied field
and on the frequency.

Besides, the sample contains two a priori independent noise sources that
contribute to its e.m.f.: the electronic fluctuations within the conductors
[the ”Johnson-Nyquist noise”, Fig. 2(a)] and the magnetic fluctuations
inductively collected by the antenna [the ”magnon noise”, Fig. 2(b)].

###  III.2 Experimental configuration n°1: the conventional apparatus

In a first step, the set-up chosen to spectrally analyze this e.m.f is
inspired from the ones widely used for the characterization of spin-torque
oscillators [24] or hard-disk read-heads [25]: the antenna feeds a spectrum
analyzer (SA) using an using an rf probe and a coaxial cable of characteristic
impedance $Z_{0}=50~{}\Omega$, followed by a broadband amplifier [cf. Fig.
3(a)] . The amplifier gain ($G_{\textrm{ampli}}=52~{}\textrm{dB}$) is chosen
large enough to bring the sample noise above the one internally generated at
the front end of the SA (typically 24 dB above thermal noise). The amplifier
noise figure is chosen as low as possible (here: 2 dB).

Figure 3: Experimental set-ups for the measurement of the spectral density of
the electrical power delivered by the antenna. (a): The first ”conventional”
configuration (red color) allows the electrical signal to flow back (red
arrows) and forth (blue arrows) between the antenna and the amplifier which
both partially reflect at the planes $\Pi_{1}$ and $\Pi_{2}$, thereby forming
a cavity hosting standing microwaves. (b) The second ”modified” configuration
(green color) includes an isolator that prevents the back flow (hatched green
arrow) of electromagnetic energy from the amplifier towards the antenna. The
standing waves are largely mitigated in the operating range of the isolator.

Note that in broadband amplifier technologies, a compromise is done between
minimal noise figure and perfect impedance matching: the amplifier input
impedance is $\tilde{Z}_{\textrm{in}}(\omega)\neq Z_{0}$; this mismatch leads
to a partial reflection at the amplifier’s input (here: $-17\textrm{~{}dB}\leq
S_{11}^{\textrm{amp}}\leq-10~{}\textrm{dB}$ in the used frequency band). As a
result, there are inevitably two mismatch planes [labeled $\Pi_{1}$ and
$\Pi_{2}$ in Fig. 3] in this conventional measurement circuit. The
voltage/current electromagnetic waves can travel back and forth in the cavity
formed between these two planes: standing microwaves form within the coaxial
cable. The interferences are constructive along a frequency comb of spacing
$v_{\varphi}/\xi$, where $\xi$ is the length of the cable and
$v_{\varphi}\approx 2.1\times 10^{8}~{}\textrm{m/s}$ is the phase velocity of
the electromagnetic waves in the coaxial cable. The voltage standing wave
ratio depends on the reflection parameters at the two ends of the cavity; in
particular, it depends on the applied field and on the frequency through
$\tilde{Z}_{\textrm{out}}$, as illustrated in the inset of Fig. 4(a) for two
different applied fields.

####  III.2.1 Measurement procedure

Our measurement procedure is illustrated in Fig. 4. We first use a single port
VNA to determine $\tilde{Z}_{\textrm{out}}(\omega,\vec{H})$ and
$\tilde{Z}_{\textrm{in}}(\omega)$ separately. We then mount the measurement
chain of Fig. 3(a). The resolution bandwidth (RBW) of the spectrum analyser is
set to its largest possible value (here $\textrm{RBW}=50~{}\textrm{MHz}$) to
collect maximal energy and thereby speed up the acquisition. The magnitude of
the noise power spectral density arriving at the SA is then typically of the
order of the Johnson-Nyquist noise floor:
$10~{}\textrm{Log}_{10}(k_{B}T_{e}\times\textrm{RBW})+30+G_{\textrm{ampli}}\approx-44~{}\textrm{dBm}$
per investigated frequency point, where $T_{e}$ is the circuit temperature and
the term ’30’ accounts for the mW to Watt conversion. This power spectral
density [PSD, Fig. 4(b)] displays a slow frequency dependence essentially set
by the roll-off of $G_{\textrm{ampli}}(f)$, on top of which faster variations
can be seen as a ripple. This ripple is a manifestation of the presence of
standing electrical waves within the measurement circuit.

We then record the variation of the PSD with the applied field $H_{x}$ and
perform the field-to-field subtraction
$\textrm{PSD}(H_{x})-\textrm{PSD}(H_{\textrm{ref}})$ to reveal the tiny
(typically a few 0.01 dB) variations of the PSD with the magnetic field,
hereafter referred to as the ’excess noise’ [Fig. 4(c) and (d)].
$H_{\textrm{ref}}$ is a arbitrarily chosen reference field; here we have
chosen $H_{\textrm{ref}}=0$. We strongly emphasize that subtracting
$\textrm{PSD}(H_{\textrm{ref}})$ is just a convenient way to reveal the
dependence of the PSD with the applied field. As we will see later, this
”(field-to-field) excess noise” resulting from this field-to-field subtraction
is equivocal and has generaly no clear physical meaning. A quick look at Fig.
4(d) indicates however immediately that the field-to-field excess noise when
measured with this conventional apparatus contains magnetic information.

####  III.2.2 Magnetic information within the field-to-field excess noise for
the conventional experimental configuration

The field-to-field excess noise is found to be flat and featureless at the
frequencies much below the uniform acoustical mode (10.3 GHz at 100 mT in Fig.
4). However near and above that frequency, the excess noise becomes markedly
oscillatory [Fig. 4(c)]. Importantly, striking anticrossings are detected
[Fig. 4(d)] between the branch of uniform acoustical spin waves and the
frequency comb of the $\Pi_{1}-\Pi_{2}$ microwave cavity. This is in stark
contrast with the BLS result [Fig. 1(d)]. These anticrossings evidence the
existence of a significant coupling occurring at $\Pi_{2}$ (i.e.
$\tilde{Z}_{\textrm{out}}$) between the quasiparticles of the measurement
system (the microwave photons confined in the $\Pi_{1}-\Pi_{2}$ cavity) and
the magnons within the magnetic film. We wish to emphasize that these
anticrossings were not present in the BLS measurement.

This leads us to question whether the conventional electrical experimental
configuration actually measures the magnetic noise and/or whether it does so
in a non-invasive manner. We have thus implemented the following modified
measurement configuration meant to change the interaction between the spin
wave bath and the measurement circuit.

###  III.3 Experimental configuration n°2:
for a reduced voltage standing wave ratio

The connection between the antenna and the amplifier can be done in a
different manner. In the configuration n°1 the microwaves could travel back
and forth between the antenna and the amplifier’s input. In the modified
configuration [green in Fig. 3(c)], we insert an isolator before the
amplifier; this non-reciprocal element
($||S_{12}^{\textrm{iso}}||\leq-20~{}\textrm{dB}$ while
$||S_{21}^{\textrm{iso}}||\approx-1~{}\textrm{dB}$) is meant to ensure that
the electromagnetic energy flows only forward from the antenna to the
remainder of the circuit. The isolator is way better matched than the
amplifier and reflects only a tiny part of the electromagnetic energy (here
$||S_{11}^{\textrm{iso}}||\leq-23~{}\textrm{dB}$ while the amplifier had
$-17\textrm{~{}dB}\leq S_{11}^{\textrm{amp}}\leq-10~{}\textrm{dB}$). The goal
is to suppress the standing waves within the operation range of the isolator
(here 8-12 GHz). As a result, the ripple of the signal in this frequency band
is strongly mitigated (compare Fig. 4(b) and 5(a) in the range of the ”wavy”
and ”flatter” labels). This modified measurement configuration gives
strikingly different results. Inserting the isolator seems to fully remove any
magnetic information from the field-to-field excess noise, even when the field
is chosen such that the SW acoustical frequency is within the passing band of
the isolator [compare Fig. 4(c) and 5(b)]. Apart from a non-reproducible drift
of the baseline, the spectra of excess noise become featureless for all
applied fields, including in the passing band of the isolator (8-12 GHz), see
Fig. 5(c).

One generally expects the signal arising from the magnetic fluctuations to
flow from the antenna towards the amplifier. The apparent suppression of
magnetic information in the field-to-field excess noise in the modified
experimental configuration may consequently seem counter-intuitive. Let us
thus model the antenna noise.

Figure 4: Procedure, results and modeling for the first experimental
configuration. (a) Reflection coefficient of the antenna for two different
applied fields and zoom thereof (inset). (b): Spectral density of the noise
power at 100 mT. (c): Excess noise with respect to the $H_{\textrm{ref}}=0$
reference. The black vertical lines in (a-c) are at a frequency
$f_{\textrm{ac}}$ corresponding to the uniform acoustical mode at 100 mT. The
curly brace with the ”wavy” label in (b) is meant for comparison with Fig.
5(a). (d) Field dependence of the experimental excess noise. The horizontal
feature near 1 GHz is an artefact caused by ambient wireless devices. (e)
Expected excess Johnson-Nyquist noise of the antenna calculated from Eq. 2 and
the measured field dependence of the antenna impedance. (d) and (e) were
decremented with the $H_{\textrm{ref}}=0$ reference data. Figure 5: Results
for the second experimental configuration. (a) Spectral density of the noise
power. The passing band of the isolator appears as the grey zone. (b): Excess
noise with respect to the $H_{\textrm{ref}}=0$ reference. The black vertical
line recall the frequency $f_{\textrm{ac}}$ of the acoustical mode at $k=0$.
The curly braces in (a) illustrate the part of the spectrum in which the
presence of the isolator strongly attenuates the ripple, in contrast with Fig.
3(b). Panel (c): field dependence of the excess noise, with scales comparable
to these of Fig. 4(d).

##  IV Magnetic modulation of the Johnson-Nyquist electrical noise power
delivered by the measurement circuit

Let us first evaluate the contribution of the electronic noise to the spectrum
recorded by the spectrum analyser in the experimental configuration n°1. As
already noted, the microwave elements placed between the antenna and the
amplifier form a microwave cavity bounded by the impedance mismatch planes
$\Pi_{1}$ and $\Pi_{2}$ [Fig. 3(c)]. As shown in Fig. 4(a), the reflection
parameter $\tilde{S}_{11}$ at the antenna boundary of this cavity is field-
and frequency-dependent, likely because of the antenna self-inductance. This
changes the Johnson-Nyquist noise present in the $\Pi_{1}-\Pi_{2}$ cavity.

This change of the electric noise present in the cavity contains magnetic
information; however it has nothing to do with the amplitude of the magnetic
fluctuations (here: the thermal amplitude). It is related only with the
magnetic susceptibility which is included in the inductance of the antenna:
this noise contribution scales with the (electronic) temperature $T_{e}$ of
the measurement system. As it is a cavity effect, the fast modulation of the
excess noise is present only in the first measurement configuration. Let us
estimate the Johnson-Nyquist noise arriving at the amplifier’s input. A
prerequisite is to evaluate the equivalent impedance presented at the front
end of the amplifier.

###  IV.1 Impedance shifting within the mismatched cavity

Since the antenna is seen through a $\xi\approx 20~{}\textrm{cm}$-long coaxial
cable, the equivalent impedance presented at the front end of the amplifier
$\tilde{Z}_{\textrm{out}}(\xi)$ is distinct from the antenna impedance
$\tilde{Z}_{\textrm{out}}$. If the cable has a characteristic impedance
$Z_{0}$ and carries the microwaves with a phase velocity $v_{\varphi}$, we can
shift the reference plane from $\Pi_{2}$ to $\Pi_{1}$ and consider that the
amplifier is fed by the {antenna+cable} system describable as a lumped element
of $\xi$-shifted impedance being:

$
\tilde{Z}_{\textrm{out}}(\xi)=Z_{0}\frac{\tilde{Z}_{\textrm{out}}+iZ_{0}\tan(\xi\omega/v_{\varphi})}{Z_{0}+i\tilde{Z}_{\textrm{out}}\tan(\xi\omega/v_{\varphi})}$
(1)

This {antenna+cable} system supplies a Johnson-Nyquist noise of e.m.f.
spectral density $4k_{B}T_{e}\mathcal{R}e(\tilde{Z}_{\textrm{out}}(\xi))$ to
the amplifier of input impedance $\tilde{Z}_{\textrm{in}}(f)$. Assuming a
perfect (noiseless) amplifier, the spectral density of the active power
delivered to the spectrum analyzer is obtained after passing in the voltage
divider configuration, which yields:

$\textrm{PSD}_{\textrm{electronic}}(f,\xi,H)[\textrm{in~{}W/Hz}]=4k_{B}T_{e}~{}\mathcal{R}e\left(\tilde{Z}_{\textrm{out}}(\xi,H,f)\right)\\\
~{}.~{}G_{\textrm{ampli}}(f)~{}.~{}\frac{1}{2}\frac{\mathcal{R}e(\tilde{Z}_{\textrm{in}}(f))}{|\tilde{Z}_{\textrm{out}}(\xi,H,f)+\tilde{Z}_{\textrm{in}}(f)|^{2}}$
(2)

where $T_{e}$ is the temperature of the {antenna+cable} system. All terms can
be obtained from single-port VNA measurements of the circuit elements.

###  IV.2 Comparison of the expected electronic noise and the experimental
field-to-field excess noise

The electronic noise according to Eq. 2 in excess to the $H_{\textrm{ref}}=0$
case is evaluated in Fig. 4(e). The electronic noise depends on the frequency
in an inherently oscillatory manner because of the
$\tan(\xi\omega/v_{\varphi})$ terms in Eq. 1. The plotting of
$\textrm{PSD}_{\textrm{electronic}}(H_{\textrm{x}})-\textrm{PSD}_{\textrm{electronic}}(H_{\textrm{ref}})$
never succeeds in suppressing this oscillatory contribution, whatever the
choice of $H_{\textrm{ref}}$. Indeed the output impedances of the antenna at
reflection coefficient at $H_{x}$ and at $H_{\textrm{ref}}$ are unequal,
especially near the SW resonances. The circuit-dependent oscillation within
the electronic noise must persist near SW resonances whatever field-to-field
subtraction is attempted.

The comparison between Figs. 4(d) and (e) shows that the main features of the
experimental field-to-field excess noise in the conventional measurement
configuration are reproduced by the calculated excess electronic noise. The
red (positive noise) halo at low fields and low frequencies is reproduced.
Also, since the coupling between the microwave photons of the circuit and the
magnons below the antenna is included in the experimental value of
$\tilde{Z}_{\textrm{out}}$, the anticrossings between the branches of the
microwave photon cavity modes and the branches of the acoustical SWs that were
evidenced in the direct noise measurement [Figs. 4(d)] are well reproduced in
the modeled electronic noise [Figs. 4(e)].

From this similarity between the predicted electronic noise and the
experimental excess noise, we can infer that the contribution of the magnon
noise to the total noise is at most a minor part of it. The next section is
dedicated to the evaluation of the magnon noise.

##  V Spectral density of the inductive e.m.f. generated at the antenna by a
given population of incoherent spin waves

Let’s estimate the variance of the e.m.f. present at the antenna because of
the fluctuations of the magnetization in the SAF underneath. The corresponding
signal should reach the amplifiers in both experimental configurations. We
assume that the population of SWs can be described by an effective temperature
$T_{m}$. $T_{m}$ can be equal to $T_{e}$ if the spin wave population is at
equilibrium like in our present experiments, or it can be larger than $T_{e}$
if an overpopulation of spin wave is intentionally generated by a way that
keeps $T_{e}$ unchanged. The calculation of the spectral density of this
e.m.f. requires to first identify how each SW contributes to the flux picked
by the antenna and then to sum over them using the SW density of states.

We will make simplifying assumptions meant to keep an analytical formalism for
the description of the spin waves properties, as detailed in the Appendix. We
for instance disregard the lateral confinement of the spin waves and assume
that they are identical to their counterparts in unbounded SAF films, and that
the magnetizations are uniform across the thicknesses of each magnetic film.

###  V.1 Relevant inductive signal

The inductive e.m.f. is the time derivative of the flux intersecting any
surface encompassed by the contour defined by the measurement circuit. The
relevant parameters are described in Fig. 6(a). We can choose the surface $S$,
defined to extend vertically from the antenna to $z=+\infty$. We define
$\bar{S}$ as the symmetric of $S$, and $S_{0}$ as the cross section of the SAF
stripe. Since the stray field emanating from the magnetic sample decays fast
in space, we can assume with marginal error that the antennas are infinitely
extended in the $x$ direction. We also assume that the film thickness and film
to antenna spacing can be considered as vanishingly small quantities.

Within our assumptions the magnetization precession associated with the spin
waves occurs essentially in the $xy$ plane, with a large ellipticity (Table
1); the inductive contribution of the out-of-plane components of the
precessing magnetizations is of secondary importance compared to the
contribution of the in-plane components, and it shall be neglected. The fluxes
$\Phi_{S}$ and $\Phi_{\bar{S}}$ are then equal. Applying Gauss’s law
($\vec{\nabla}.\vec{B}=0$) on the closed surface ${S_{0}}\cup{S}\cup{\bar{S}}$
[see Fig. 6(a)], we get that $\Phi_{S}=-\frac{1}{2}\Phi_{S0}$. Applying Lenz’s
law, the e.m.f. is thus:

$\textrm{e.m.f.}_{m}=\frac{1}{2}\dot{\Phi}_{\textrm{S0}}=\frac{1}{2}\iint_{\textrm{S0}}\dot{B_{y}}(x,z)dxdz$
(3)

i.e. it is the time derivative (expressed as the dot symbol) of the half flux
of the $B_{y}$ field in the cross section of the magnet right below the
antenna.

Figure 6: Principle of the derivation of the power spectral density of the
inductive e.m.f. generated by a population of incoherent spin waves (i.e. the
magnon noise). (a) Inductive geometry: the antenna collects the flux of the
magnetic field lines (green) intercepting the surface $S$ above the magnetic
film (blue) of cross section $S_{0}$, in the presence of a spin wave (green
arrows) of wavevector $\vec{k}_{y}$ perpendicular to the antenna. (b) Sketch
of the density of states of the spin waves propagating perpendicularly to the
antenna at 50 mT for the acoustic branch. The grey part is when the linear
Taylor expansion of the dispersion relation breaks down. (c) Corresponding
positive and negative wavevectors. (d) Corresponding antenna efficiency
functions $h(k)^{2}$.

###  V.2 Amplitude of the fluctuation of the total moment for a single spin
wave mode

The standard deviation of $\Phi_{\textrm{S0}}$ stems from the standard
deviation of the $y$-component of the total dynamic moment of the two
eigenmodes, which is calculated in Annex I. We write the ground state of the
two films (labelled 1 and 2) as
$\vec{M}_{\textrm{dc}}=\\{\vec{M}_{1}^{\textrm{dc}},\vec{M}_{2}^{\textrm{dc}}\\}$
and the eigenvectors of their $k=0$ spin waves as
$\vec{M}_{\textrm{dyn}}=\\{\vec{M}_{1}^{\textrm{dyn}},\vec{M}_{2}^{\textrm{dyn}}\\}$,
with the acoustical and optical mode frequencies being
$\omega_{\textrm{ac}}/(2\pi)$ and $\omega_{\textrm{op}}/(2\pi)$ (details in
Table 1). We also assume that the population of spin waves can be described by
an effective temperature $T_{m}$.

As a representative example, the $k=0$ acoustical mode in the scissors state
has a fluctuation amplitude for its in-plane component that amounts to:

$\sqrt{\left(M_{1}^{\textrm{dyn}}+M_{2}^{\textrm{dyn}}\right)_{\textrm{RMS}}^{2}}=M_{s}\sqrt{\frac{2k_{B}T_{m}}{\mu_{0}M_{s}H_{j}V}},$
(4)

where $V=t\ell_{\textrm{mag}}w_{\textrm{mag}}$ is the volume of one of the
layers. Eq. 4 follows from the ellipticities of the modes which do not depend
much on $k$ when $kt\ll 1$; We shall thus consider that Eq. 4 also applies to
all the spin waves considered in our system.

Summing the inductive contributions over all SWs requires the knowledge of the
SW density of states (DOS). Since only the SWs with wavevectors perpendicular
to the antenna [red waves in Fig. 3(b)] have a net contribution to the
induction flux picked by the antenna, we shall restrict the analysis to the
sole longitudinal wavevectors.

###  V.3 Density of states within the spin wave manifold

In a 1D stripe of length ${\ell_{\textrm{mag}}}$, the longitudinal wavevectors
$k\equiv k_{y}$ are quantized at values $\frac{2\pi}{\ell_{\textrm{mag}}}n$
with $n\in\mathbb{N}$, and frequency spacings of
$\frac{v_{g}}{\ell_{\textrm{mag}}}$. The SW DOS per unit frequency [Fig. 6(b)]
is the classical result for a 1D system with linear dispersion:

$\textrm{DOS}(f)=2\frac{\ell_{\textrm{mag}}}{v_{g}}\mathcal{H}(f-f_{\textrm{k=0}})~{},$
(5)

where $\mathcal{H}$ is the Heaviside function, $v_{g}=\frac{d\omega}{d|k|}$ is
the group velocity (positive here, see Table 1) of the SW branch under study.
The factor of 2 recalls the two possible signs of $k_{y}$.

Note that since the optical branch has a vanishing group velocity for
$\vec{k}\perp\vec{H}$ within our approximations (see Table 1), the
corresponding DOS is simply a Dirac peak at $f_{\textrm{op}}$. This cannot be
perceived with the RBW settings of the spectrum analyzer. We thus restrict the
remainder of this section to the sole acoustical branch.

Table 1: Main properties of the spin waves of wavevector $k$ propagating in
the direction perpendicular to the applied field in in-plane magnetized
isotropic Synthetic AntiFerromagnets. The SAF is assumed to be fully
symmetrical and the magnetizations are assumed uniform across the thickness of
each of the two magnetic films. The dc field $H_{x}$ is applied in the
$\\{1,0,0\\}$ direction. The dagger symbol recalls that the ground state is
unstable against an arbitrary in-plane global rotation when at remanence. The
group velocities are defined as the derivative of the angular frequency in the
direction of wavevector $\frac{\partial\omega}{\partial|k|}$, i.e. according
to the common practice. The interlayer exchange field is defined as
$\mu_{0}H_{j}=-\frac{2J}{{M_{s}}{t}}$.
Field | $H_{x}=0$ | $H_{0}<H_{j}$ | $H_{0}>H_{j}$
---|---|---|---
Ground state |  $\dagger$ Antiparallel | Scissors | Parallel
$M_{1}^{\textrm{dc}}/M_{s}$ | $\\{0,1,0\\}$ | $\\{\frac{H_{x}}{H_{j}},\sqrt{1-\frac{H_{x}^{2}}{H_{j}^{2}}},0\\}$ | $\\{1,0,0\\}$
$M_{2}^{\textrm{dc}}/M_{s}$ | $\\{0,-1,0\\}$ | $\\{\frac{H_{x}}{H_{j}},-\sqrt{1-\frac{H_{x}^{2}}{H_{j}^{2}}},0\\}$ | $\\{1,0,0\\}$
Acoustical mode: |  |  |
$\omega_{\textrm{ac}}$ at $k_{y}=0$ |  $\dagger$ 0 | $\gamma_{0}H_{x}\sqrt{\frac{M_{s}+H_{j}}{H_{j}}}$ | $\gamma_{0}\sqrt{H_{x}(H_{x}+M_{s})}$
$\Delta\omega_{\textrm{ac}}$ (FWHM) |  $\dagger$ $\alpha\gamma_{0}(M_{s}+H_{j})$ | $\alpha\gamma_{0}(M_{s}+\frac{H_{j}^{2}+H_{x}^{2}}{H_{j}})$ | $\alpha\gamma_{0}(M_{s}+2H_{x})$
Eigenmode: $\vec{M}_{1}^{\textrm{dyn}}$ part |  $\dagger$ $\\{-i,0,0\\}$ | $\left\\{-i\sqrt{\frac{M_{s}+H_{j}}{H_{j}}}\frac{\sqrt{H_{j}^{2}-H_{x}^{2}}}{H_{x}},+i\sqrt{\frac{M_{s}+H_{j}}{H_{j}}},1\right\\}$ | $\left\\{0,i\sqrt{\frac{M_{s}+H_{x}}{H_{x}}},1\right\\}$
Eigenmode: $\vec{M}_{2}^{\textrm{dyn}}$ part |  $\dagger$ $\\{+i,0,0\\}$ | $\left\\{+i\sqrt{\frac{M_{s}+H_{j}}{H_{j}}}\frac{\sqrt{H_{j}^{2}-H_{x}^{2}}}{H_{x}},+i\sqrt{\frac{M_{s}+H_{j}}{H_{j}}},1\right\\}$ | $\left\\{0,i\sqrt{\frac{M_{s}+H_{x}}{H_{x}}},1\right\\}$
$\vec{M}_{1}^{\textrm{dyn}}+\vec{M}_{2}^{\textrm{dyn}}$ |  $\dagger$ $\\{0,0,0\\}$ |  $\left\\{0,2i\sqrt{\frac{M_{s}+H_{j}}{H_{j}}},2\right\\}$ i.e. $\perp$ to dc field | 2$\left\\{0,i\sqrt{\frac{M_{s}+H_{x}}{H_{x}}},1\right\\}$
$\frac{1}{M_{s}}(M_{1}^{\textrm{dyn}}+M_{2}^{\textrm{dyn}})_{\textrm{RMS}}$ | along y: $\sqrt{\frac{2k_{B}T}{\mu_{0}M_{s}H_{j}V}}$ | along y: $\sqrt{\frac{2k_{B}T}{\mu_{0}M_{s}H_{j}V}}$ | along y: $\sqrt{\frac{2k_{B}T}{\mu_{0}M_{s}H_{x}V}}$
$v_{g}^{\textrm{ac}}$, $\vec{H}_{x}\perp\vec{k}_{y}$, $k>0$ | 0 | \+ $\frac{1}{2}\gamma_{0}M_{s}t$ $\frac{H_{0}}{H_{j}}\frac{M_{s}}{\sqrt{H_{j}(H_{j}+M_{s})}}$ | \+ $\frac{1}{2}\gamma_{0}M_{s}t$ ${\frac{M_{s}}{\sqrt{H_{x}(H_{x}+M_{s})}}}$
Optical mode: |  |  |
$\omega_{\textrm{op}}$ at $k=0$ | $\gamma_{0}\sqrt{M_{s}H_{j}}$ | $\gamma_{0}\sqrt{\frac{M_{s}}{H_{j}}}\sqrt{H_{j}^{2}-H_{x}^{2}}$ | $\gamma_{0}\sqrt{(H_{x}-H_{j})(H_{x}+M_{s}-H_{j})}$
$\Delta\omega_{\textrm{op}}$ (FWHM) | $\alpha\gamma_{0}(M_{s}+H_{j})$ | $\alpha\gamma_{0}(M_{s}+\frac{H_{j}^{2}-H_{x}^{2}}{H_{j}})$ | $\alpha\gamma_{0}(M_{s}+2H_{x}-2H_{j})$
$M_{1}^{\textrm{op}}$ eigenstate | $\left\\{+i\sqrt{\frac{M_{s}}{H_{j}}},0,-1\right\\}$ | $\left\\{+i\sqrt{\frac{M_{s}}{H_{j}}},-i\sqrt{\frac{M_{s}}{H_{j}}}\frac{H_{x}}{\sqrt{H_{j}^{2}-H_{x}^{2}}},-1\right\\}$ | $\\{0,\sqrt{\frac{H_{x}+M_{s}-H_{j}}{H_{x}-H_{j}}},-1\\}$
$M_{2}^{\textrm{op}}$ eigenstate | $\left\\{+i\sqrt{\frac{M_{s}}{H_{j}}},0,1\right\\}$ | $\left\\{+i\sqrt{\frac{M_{s}}{H_{j}}},+i\sqrt{\frac{M_{s}}{H_{j}}}\frac{H_{x}}{\sqrt{H_{j}^{2}-H_{x}^{2}}},+1\right\\}$ | $\\{0,-\sqrt{\frac{H_{x}+M_{s}-H_{j}}{H_{x}-H_{j}}},1\\}$
$M_{1}^{\textrm{op}}+M_{2}^{\textrm{op}}$ eigenstate | $\\{2i\sqrt{\frac{M_{s}}{H_{j}}},0,0\\}$ |  $\\{2i\sqrt{\frac{M_{s}}{H_{j}}},0,0\\}$ i.e. $\parallel$ to dc field | $\\{0,0,0\\}$
$\frac{1}{M_{s}}(M_{1}^{\textrm{op}}+M_{2}^{\textrm{op}})_{\textrm{RMS}}$ | along x: $\sqrt{\frac{2k_{B}T}{\mu_{0}M_{s}H_{j}V}}$ | along x: $\sqrt{\frac{2k_{B}T}{\mu_{0}M_{s}H_{j}V}}$ | along x: 0
$v_{g}^{\textrm{op}}$, $\vec{H_{x}}\perp\vec{k_{y}}$, $k>0$ | 0 | 0 | 0

###  V.4 Power spectral density of the e.m.f. due to magnetic fluctuations

To finally get the power spectrum of the $\textrm{e.m.f.}_{m}$, we still need
to account for the variable part of the flux picked-up by the antenna. This is
done by writing:

$\frac{dB_{y}}{dt}=i\omega B_{y}^{\textrm{dyn}}$ (6)

To first order in $k$, the difference between the flux density and the dynamic
magnetization within the magnetic sample is simply
$B_{y}^{\textrm{dyn}}=\mu_{0}M_{y}^{\textrm{dyn}}(1-\frac{kt}{2})$. This flux
is partly transduced to an antenna voltage with the dimensionless antenna
efficiency function [26] that only depends on the antenna width $\mathcal{A}$
and spacing between the mid-plane of the antenna and the mid-plane of the
magnetic film $s_{\textrm{eff}}\approx 250~{}\textrm{nm}$:

$h(k)=\frac{\sin(k\mathcal{A}/2)}{k\mathcal{A}/2}e^{-|k|s_{\textrm{eff}}}$ (7)

Using Eq. 3, 5, 6, this yields the spectral density of the inductive ’noise’
delivered by the population of incoherent magnons for a given branch:

$\frac{d}{df}\textrm{e.m.f.}_{m}^{2}~{}[\textrm{in~{}units~{}of~{}}\textrm{V}^{2}/\textrm{Hz}]=\\\
\frac{2\omega^{2}}{v_{g}}tw_{\textrm{mag}}\mu_{0}^{2}\left[V\left(M_{1y}^{\textrm{dyn}}+M_{2y}^{\textrm{dyn}}\right)_{\textrm{RMS}}^{2}\right]h(k)^{2}\left(1-\frac{kt}{2}\right)^{2}$
(8)

Note that the term enclosed between square brackets is in fact independent
from the sample geometry and volume $V$, as can be seen from Eq. 4. Note also
that $k$ is implicitly defined from the frequency by the dispersion relation,
i.e. to first order: $k=\frac{1}{v_{g}}(\omega-\omega_{k=0})$ [Fig. 6(c)].

This e.m.f. is delivered in the $\Pi_{2}$ plane where the amplifier is seen as
a lumped element of $\xi$-shifted input impedance
$\tilde{Z}_{\textrm{in}}(\xi,f)$ which is:

$\tilde{Z}_{\textrm{in}}(\xi,f)=Z_{0}\frac{\tilde{Z}_{\textrm{in}}+iZ_{0}\tan(\xi\omega/v_{\varphi})}{Z_{0}+i\tilde{Z}_{\textrm{in}}\tan(\xi\omega/v_{\varphi})}$
(9)

such that the active power delivered to the spectrum analyser by the
acoustical branch of the spin waves when in the scissors state is:

$\textrm{PSD}_{\textrm{mag.fluct.}}(f,\xi,H)[\textrm{in~{}W/Hz}]=4k_{B}T_{m}~{}2\mu_{0}w_{\textrm{mag}}\omega_{\textrm{ac}}\\\
~{}\mathcal{H}(f-f_{\textrm{ac}})\times\left[h(k)^{2}\left(1-\frac{kt}{2}\right)^{2}\right]\frac{M_{s}+H_{j}}{M_{s}}\\\
~{}.~{}G_{\textrm{ampli}}(f)~{}.~{}\frac{1}{2}\frac{\mathcal{R}e(\tilde{Z}_{\textrm{in}}(\xi,f))}{|\tilde{Z}_{\textrm{in}}(\xi,f)+\tilde{Z}_{\textrm{out}}(f,H)|^{2}}$
(10)

###  V.5 Color and amplitude of the inductive magnon noise

Several point are worth noticing in Eq. 10.

(i) The inductive magnetic noise is colored and narrow-band, essentially
thanks to the finite k-vector efficiency of the antenna: the noise should
suddenly shoot up at the acoustical frequency and then extend up to the first
zero of the $h(k)$ function occurring at
$f_{\textrm{ac}}+\frac{v_{g}}{\mathcal{A}}$. This contrasts with the focused
BLS configuration for which the group velocity in all directions has to be
taken into account and the antenna width $\mathcal{A}$ has to be replaced by
the laser spot size.

(ii) Neither $M_{s}$ nor $t$ appear explicitly in the inductive magnetic noise
(Eqs. 10). This could seem in contradiction with the intuition that an
inductive signal should scale with the total magnetic moment, as it does for
each value of $k$. Lifting this apparent contradiction requires to notice that
we have also $v_{g}\propto t$: the signal for each $k$ value is spread over a
proportionally large frequency interval, yielding a plateau of
$\textrm{PSD}_{\textrm{mag.fluct.}}$ that is independent from the magnetic
thickness $t$ for $f\approx f_{\textrm{ac}}$. However the frequency interval
in which the magnon noise is substantial is typically
$f_{\textrm{max}}-f_{\textrm{ac}} =\frac{v_{g}^{\textrm{ac}}}{\mathcal{A}}$
which scales proportionally to $\gamma_{0}M_{s}t$ (see Table 1): in brief, the
plateau of the magnon noise is independent from $t$ but its integral is
proportional to $M_{s}t$, in line with the qualitative expectation.

(iii) If the amplifier was hypothetically perfectly matched with
$\tilde{Z}_{\textrm{in}}=Z_{0}$ (or equivalently if a perfectly matched
isolator is placed at the front end of the amplifier), then the antenna would
simply see the lumped impedance $\tilde{Z}_{\textrm{in}}(\xi)=Z_{0}$. The
$\textrm{PSD}_{\textrm{mag.fluct.}}$ would not oscillate with the frequency;
the color of this noise would reflect the product of the SW DOS and the
antenna efficiency function. In the realistic case when the amplifier is not
perfectly impedance matched (i.e. configuration n°1),
$\textrm{PSD}_{\textrm{mag.fluct.}}$ exhibits a ripple along a frequency comb
of spacing $v_{\varphi}/\xi$, i.e. in a manner very similar to the electronic
noise so that they can’t be easily distinguished. The most important question
is thus the respective amplitudes of the magnetic noise (Eq. 10) and the
formerly modeled electronic noise (Eq. 2).

##  VI Discussion on the inductive measurement of spin wave populations

###  VI.1 Respective contributions of the spin wave population-induced noise
and the electronic noise at thermal equilibrium

It is primordial to notice that the term $2\mu_{0}w_{\textrm{mag}}$ in Eq. 10
has the dimension of an inductance, so the term
$2\mu_{0}w_{\textrm{mag}}\omega_{\textrm{ac}}$ has the dimension of an
impedance. For a stripe width $w_{\textrm{mag}}=20~{}\mu\textrm{m}$, this so-
called ”magnon” impedance amounts typically to $0.6-6~{}\Omega$ in the 2-20
GHz interval considered here. This impedance should be compared with the much
larger impedance of the other noise sources present in the measurement chain
(Eq. 2) with our initially chosen geometry.

This comparison of impedances evidences that if the circuit temperature
$T_{e}$ and that of the spin wave bath $T_{m}$ are equal (like in our
experiments) the noise power inductively supplied by the spin waves in a
$w_{\textrm{mag}}=20~{}\mu\textrm{m}$ stripe cannot be expected to be more
than a tiny fraction of the total noise supplied by the circuit. This explains
the finding that the modified experimental configuration collects much less
magnetic signal than the conventional experimental configuration.

Rendering the magnon noise perceivable within the total noise would require
Eq. 10 to exceed Eq. 2, which occurs when the magnon impedance of the antenna
exceeds the input impedance of the amplifier, i.e. typically when:

$T_{m}>T_{e}\frac{50~{}\Omega}{2\mu_{0}w_{\textrm{mag}}\omega_{\textrm{ac}}}$
(11)

If the spin wave bath is at equilibrium with the circuit (i.e. if
$T_{m}=T_{e}$) then the previous condition is not met in our system. One could
obviously increase the width $w_{\textrm{mag}}$ of the magnetic stripe to
collect magnon noise from a longer region; meanwhile, it would be beneficial
to adjust the length and thickness of the antennas to get a better matching
between $\tilde{Z}_{\textrm{out}}$ and $Z_{0}$ and thereby to minimize the
frequency ripple of the electronic noise that is difficult to correct. However
the rigorous subtraction of the electronic noise to get the sole magnon noise
would still be impossible from the sole field-dependence of the noise
delivered by the antenna.

###  VI.2 Quantifying spin wave populations using a two-magnetic-temperature
measurement

There is fortunately another possibility/situation to quantify inductively the
population of spin waves, notably when the latter is not in equilibrium with
the circuit. Such a situation is for instance encountered in a spin-torque
oscillator, and one could think of coupling the latter to an inductive
antenna. When a spin-torque current $I_{\textrm{STT}}$ is applied in a sub-
threshold (non-oscillating) regime, overpopulations of incoherent spin waves
build up far beyond the thermal expectation values [27] within the free layer
of the oscillator. The metric $\textrm{PSD}(I_{\textrm{STT}}\neq
0)-\textrm{PSD}(I_{\textrm{STT}}=0)$ can be used to estimate the spin wave
population $T_{m}$, although the common practice is to express this metric as
a precession angle after having assumed that a single spin wave mode exists in
the system. Another equally relevant situation relies in various non-linear
FMR experiments, when the pumped coherent populations of spin waves coexist
with incoherent populations resulting from multimagnon processes. This happens
for instance in the transient route towards equilibrium when cascades of four-
magnon scattering events have partitioned the excess energy initially injected
by parametric pumping to a single pair of modes [28].

The method to count a spin wave overpopulation can be directly inspired from
the 2-temperature methods implemented to measure the noise figures of rf
receivers: Instead of performing a field-to-field subtraction of some
reference noise spectrum, one can perform a hot-to-cold comparison of the
noise spectra. Indeed, since the electronic noise (Eq. 2) is independent from
the magnon noise (Eq. 10), it can be subtracted if two measurements are
performed at the same circuit temperature $T_{e}$ and the same matching
conditions (hence same applied field) but with two different levels of spin
populations that we shall describe with their effective temperatures $T_{m1}$
and $T_{m2}$.

One of the temperatures must be known (for instance the reference measurement
is $T_{m1}=T_{e}=300~{}\textrm{K}$). One needs also to assume that the spin
wave populations do not affect their resonance frequencies (i.e. neither blue
nor red shift). In this case one can use Eq. 10 to deduce the spin wave
overpopulation $(T_{m2}-T_{m1})$ in a rigorous manner. Once again, adjusting
the electrical parameters of the antenna to get the best achievable matching
between $\tilde{Z}_{\textrm{out}}$, $Z_{0}$ and $\tilde{Z}_{\textrm{in}}$ and
thereby minimizing the frequency ripple can only help to minimize the errors
when evaluating the electrical terms within Eq. 10.

##  VII Summary and concluding remarks

We have studied the noise present in the microwave signal collected by an
inductive antenna on top of a thin magnetic stripe and its spin wave bath. The
spin waves are –by definition– (inductively) coupled to the microwave photons
of the antenna, such that one may question whether it is possible to measure
the populations of spin waves in a non-invasive manner while collecting the
microwave photons of the circuit.

We have found that the experimental noise power delivered by the antenna to
the measurement circuit is clearly correlated with the spin wave frequencies.
Our modeling indicates that this noise power contains two contributions: the
magnon noise, i.e. the contribution of the population of spin waves coupled to
a noise-free measurement circuit (Eq. 10), and that of the electronic thermal
fluctuations coupled to a noise-free (but with a finite susceptibility)
magnetic body (Eq. 2). This Johnson-Nyquist noise is modulated by the antenna
reflection coefficient in a way that depends on the magnetic susceptibility
and on the electrical parameters of the microwave cavity formed by the
measurement circuit if not perfectly impedance-matched.

Since the electronic noise also contains magnetic information, the two sources
of noise cannot be separated by solely field-dependent measurements.
Measurements performed at different levels of spin wave populations, i.e. at
different spin wave effective temperatures, can circumvent this difficulty.
This opens the way for the inductive quantitative measurement of spin wave
populations that are out-of-equilibrium in nanoscale systems. This could be
applied to a great advantage to the situations when the spin wave
overpopulations result from non-linear processes like for instance parametric
pumping [9].

The authors acknowledge financial support from the French National Research
Agency (ANR) under Contract No. ANR-20-CE24-0025 (MAXSAW). M.S.N. and J.L.
acknowledge in addition the FETOPEN-01-2016-2017 [FET-Open research and
innovation actions (CHIRON project: Grant Agreement ID: 801055)]. We thank C.
Dupuis, D. Bouville, A. Harouri and N. Bardou for technical assistance for the
sample fabrication.

## Appendix: analytics of the spin waves in synthetic antiferromagnetics

This appendix gives further details on analytical description of the SW
properties in SAFs. The algebra is done for a SAF but the results listed in
Table I can be easily transposed to a single layer film magnetized along the
in-plane applied field. One should simply consider that the interlayer
exchange coupling field $H_{j}$ vanishes while not forgetting that for a
correct accounting of the dipole-dipole interaction, the single layer film
must be described with a thickness equal to $2t$.

###  VII.1 Simplifying assumptions and methods

We assume that the SAF can be described by two films (labelled 1 and 2) that
are both uniformly magnetized across their thickness. Keeping the same
notations as in the main part of this paper, we write the ground state as
$\vec{M}_{\textrm{dc}}=\\{\vec{M}_{1}^{\textrm{dc}},\vec{M}_{2}^{\textrm{dc}}\\}$.
For an applied field $0<H_{x}<H_{j}$, the SAF is in the scissors state [29,
20, 21]. The dynamical matrix of the system can be calculated following the
standard methodology [30, 31, 23, 32, 33, 34]. The real and imaginary parts of
its eigenvalues are the frequencies of the SWs and their half linewidths. The
complex-valued eigenvectors $\vec{M}_{\textrm{dyn}}$ are the dynamic
magnetization components
$\vec{M}_{\textrm{dyn}}=\\{\vec{M}_{1}^{\textrm{dyn}},\vec{M}_{2}^{\textrm{dyn}}\\}$,
with the dephasing of the components being coded in their complex argument.

The results of the eigenvalues and eigenvectors calculations are summarized in
Table 1 and illustrated in Fig. 6(b). The first mode is ’acoustical’ [29]: the
two magnetizations precess while keeping constant the angle between their in-
plane projections [when seen in top view, this is a rigid rocking of the
scissor state at a frequency $\omega_{\textrm{ac}}/(2\pi)$]. The second mode
is ’optical’: the scissoring angle breathes at a frequency
$\omega_{\textrm{op}}/(2\pi)$.

The group velocity of the acoustical and the optical branches are gathered in
Table 1. The corresponding DOS and dispersion relations are illustrated in
Fig. 6(b, c). The group velocities were found with the procedure of ref. [31]
while augmenting the dynamical matrix with the contribution of finite
wavevectors to the self- and mutual demagnetizing effects of the two magnetic
layers according to Eq. 25 and 26 of ref. [34]. The eigenvalues of the
augmented dynamical matrix were Taylor-expanded to first order in $kt$ at
$k=0$ to get the $v_{g}$’s of the acoustical and optical spin wave branches in
the small $k$ limit. In our specific case of $\vec{k}_{y}\perp\vec{H_{x}}$,
the dispersion relations of the acoustical and optical spin waves are both
reciprocal.

###  VII.2 Amplitude of the magnetization fluctuations

The eigenvectors $\vec{M}_{1}^{\textrm{dyn}}$ and $\vec{M}_{2}^{\textrm{dyn}}$
at $k=0$ can be used to evaluate the amplitude of the magnetic fluctuations in
the uniform modes. For both acoustical and optical modes, the dynamical
component $\vec{M}_{z}^{\textrm{dyn}}$ is in quadrature with the in-plane
component $\vec{M}_{xy}^{\textrm{dyn}}$: the magnetizations recover an in-
plane orientation in a synchronous manner twice per period. Noticing that
$k_{B}T_{m}\gg\hbar\omega_{\textrm{ac}},~{}\hbar\omega_{\textrm{opt}}$, we can
use the equipartition theorem to link the amplitude of magnetic fluctuations
with the effective temperature $T_{m}$ of the magnon bath. The standard
deviations of the in-plane magnetization fluctuations related to a given
eigenmode follow from the average energy of that degree of freedom, i.e.:

$
E(\vec{M}_{\textrm{dc}}+\epsilon\vec{M}^{xy}_{\textrm{dyn}})-E(\vec{M}_{\textrm{dc}})=\frac{1}{2}k_{B}T_{m}~{},$
(12)

where $\epsilon$ is a dimensionless number whose square is proportional to the
number of magnons of the considered type. Solving for $\epsilon\ll 1$ to
second order in $\epsilon$ yields the in-plane amplitude of the fluctuations.
The derivation of the e.m.f. (Eq. 3) requires to vector-sum over the two
layers. Table 1 gathers the
$\frac{1}{M_{s}}(M_{1}^{\textrm{dyn}}+M_{2}^{\textrm{dyn}})_{\textrm{RMS}}$,
i.e. the standard deviations of the $y$-component of this vector sum for the
two eigenmodes.

###  VII.3 Validity of the expressions listed in Table I

The results listed in the last column of Table 1 would strictly hold only in
the ultrathin limit at $t=0$. At finite applied fields, vertical gradients of
the magnetization orientation develop within the magnetic layers unless their
thickness $t$ is much smaller than the two characteristic lengths $A/J$ and
$\sqrt{2A_{\textrm{ex}}/(\mu_{0}M^{2}_{s})}$, where $A_{\textrm{ex}}$ is the
exchange stiffness. In our samples, we estimate these two lengths to be
respectively 12 nm and 4 nm, i.e. thinner than the $t=17$ nm of our two CoFeB
layers. In this situation, the magnetizations of the magnetic regions directly
touching the Ru spacer stay more antiparallel that the other regions, such the
2-macrospin description of the ground state gets inaccurate at finite applied
fields. Micromagnetic calculations (not shown) lead us to conclude that the
acoustical mode, which involves less the interlayer exchange interaction than
the optical mode, is still reasonnably quantitatively described by the
expressions listed Table I. This is not the case of the optical mode,
especially when $H_{x}$ is comparable to $H_{j}$. A collateral consequence is
that the characteristic field $H_{j}$ does not correspond to the saturation of
the magnetization in any part the stack, the saturation being gradual within
the thickness of the films. For our specific film thickness, the expressions
for the optical mode listed in Table I should thus be considered as
qualitative only.

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# Electromagnetically induced transparency in inhomogeneously broadened solid
media

H. Q. Fan  United States Army Research Laboratory, Adelphi, Maryland 20783,
USA  Joint Quantum Institute, University of Maryland, College Park, Maryland
20742, USA  K. H. Kagalwala  Joint Quantum Institute, University of Maryland,
College Park, Maryland 20742, USA  S. V. Polyakov  National Institute of
Standards and Technology, Gaithersburg, Maryland 20899, USA  A. L. Migdall
Joint Quantum Institute, University of Maryland, College Park, Maryland 20742,
USA  National Institute of Standards and Technology, Gaithersburg, Maryland
20899, USA  E. A. Goldschmidt<EMAIL_ADDRESS>United
States Army Research Laboratory, Adelphi, Maryland 20783, USA

###### Abstract

We study, theoretically and experimentally, electromagnetically induced
transparency (EIT) in two different solid-state systems. Unlike many
implementations in homogeneously broadened media, these systems exhibit
inhomogeneous broadening of their optical and spin transitions typical of
solid-state materials. We observe EIT lineshapes typical of atomic gases,
including a crossover into the regime of Autler-Townes splitting, but with the
substitution of the inhomogeneous widths for the homogeneous values. We obtain
quantitative agreement between experiment and theory for the width of the
transparency feature over a range of optical powers and inhomogeneous
linewidths. We discuss regimes over which analytical and numerical treatments
capture the behavior. As solid-state systems become increasingly important for
scalable and integratable quantum optical and photonic devices, it is vital to
understand the effects of the inhomogeneous broadening that is ubiquitous in
these systems. The treatment presented here can be applied to a variety of
systems, as exemplified by the common scaling of experimental results from two
different systems.

##  I Introduction

Coherent processes in atomic ensembles are the basis for many implementations
of quantum memory, coherent control of atomic populations, and mediation of
interactions between optical fields Fleischhauer et al. (2005).
Electromagnetically induced transparency (EIT) is a canonical example of such
a process with applications including slow and stopped light Turukhin et al.
(2001); Schraft et al. (2016); Longdell et al. (2005), atomic-based field
sensing Fleischhauer et al. (2000); Katsoprinakis et al. (2006), lasing
without inversion Wu et al. (2008); Bohnet et al. (2014). Most studies of EIT
and other coherent processes in atomic ensembles have been conducted in
gaseous media over a range of temperatures from ultracold quantum gases to
heated vapor cells Fleischhauer et al. (2005); Novikova et al. (2012), while a
relatively smaller effort has been made in solid-state Turukhin et al. (2001);
Ham et al. (1997a); Schraft et al. (2016); Akhmedzhanov et al. (2018). In
fact, it was originally thought EIT would be impossible in solids Harris
(1997). However, atom-like emitters in solids offer benefits for quantum
optical processes including higher densities, freedom from motional dephasing,
and the possibility of integrated photonics approaches Zhong et al. (2017).
The density of emitters in a solid can be as large as
$\mathrm{10^{22}~{}cm^{-3}}$ while retaining atom-like optical properties
Ahlefeldt et al. (2016). Solid-state ensembles of rare-earth atoms, in
particular, are a promising platform for quantum memory and other applications
due to their long spin coherence times Zhong et al. (2015).

A major difference between solid-state systems and atomic gas systems is the
static inhomogeneity of both the optical and spin transitions common in solid-
state ensembles due to variations in the local electromagnetic field
environment at each emitter location. Some of this variation is from strain
due to material defects and imperfections, but even with high-purity
materials, the inhomogeneous linewidth of the optical transition for an
ensemble of solid-state emitters is often orders of magnitude larger than the
homogeneous linewidth of each emitter. This inhomogeneity has potential
benefits, particularly for the possibility of spectral multiplexing Nunn et
al. (2008); Heinze et al. (2012), but it also complicates coherent processes
like EIT. To date, most studies of the effect of inhomogeneity on EIT and
other quantum optical processes have focused on Doppler broadened gases Javan
et al. (2002); Ye and Zibrov (2002); Figueroa et al. (2006). But
inhomogeneously broadened solids present a different situation where motional
effects are not present and the coherence time is not limited by transit time
broadening Kuznetsova et al. (2002).

Figure 1: (a) Energy level diagram for a $\Lambda$-system. The weak probe
field (thin arrow) is detuned from the $|1\rangle\rightarrow|3\rangle$
transition by $\Delta$. The strong coupling field (thick arrow) on the
$|2\rangle\rightarrow|3\rangle$ transition has Rabi frequency $\Omega$ and the
difference between the probe and coupling detunings, the two-photon detuning,
is $\delta$. The optical and spin transitions have inhomogeneously broadened
widths $\sigma_{\mathrm{opt}}$ and $\sigma_{\mathrm{spin}}$, respectively. (b)
Energy level diagrams for Eu:YSO and Pr:YSO. The three states that make up the
$\Lambda$-system in each case are labeled, as is the transition (dashed arrow)
used for repumping during spectral hole-burning.

We report results of $\mathrm{\Lambda}$-type EIT in two rare-earth doped
solids, yttrium orthosilicate doped with europium (Eu:YSO) and with
praseodymium (Pr:YSO) Liu and Jacquier (2006). We use spectral hole-burning
techniques to control the optical inhomogeneous linewidth Pryde et al. (2000);
Nilsson et al. (2004). We note that while the homogeneous optical linewidth is
much larger in Pr:YSO than Eu:YSO, we observe a large parameter range for both
systems over which the EIT width depends only on the control Rabi frequency
$\Omega$ and the optical inhomogeneous width $\sigma_{\mathrm{opt}}$. These
results quantitatively agree with a theoretical treatment of the system,
suggesting that a large class of inhomogeneously broadened systems exhibit
behavior that does not depend on the single-atom properties of the individual
emitters, but only on the properties of the ensemble as a whole.

##  II Simple theoretical treatment

Consider the $\Lambda$-type energy level scheme depicted in Fig. 1(a). The
transitions from two long-lived ground states, $|1\rangle$ and $|2\rangle$, to
a single excited state, $|3\rangle$, are addressed optically with a weak probe
field and a strong coupling field, respectively. In many solid-state systems,
variations in the local electric field environment cause different emitters to
have slightly shifted transition energies. In rare-earth-doped solids, both
the optical electronic transition and the ground hyperfine transition are
inhomogeneously broadened by this effect.

We are interested in the transmission of the probe field as a function of
detuning for various values of the coupling Rabi frequency $\Omega$ and the
inhomogeneous widths of the optical and spin transitions
$\sigma_{\mathrm{opt}}$ and $\sigma_{\mathrm{spin}}$, respectively. In a
typical homogeneously broadened system (where decay rate, including dephasing,
on the $|i\rangle\rightarrow|j\rangle$ transition is denoted $\gamma_{ij}$),
we can follow the treatment in Fleischhauer et al. (2005) to obtain the shape
of the linear susceptibility $\chi$ of a weak probe field in the perturbative
regime:

$\begin{split}\chi&\propto\frac{2i\gamma_{21}+4\delta}{\Omega^{2}+(\gamma_{21}-2i\delta)(\gamma_{31}-2i\Delta)}\end{split}$
(1)

where parameter definitions can be found in Fig 1(a) and its caption. For a
resonant control field ($\delta=\Delta$), this expression for the
susceptibility exhibits two important features. First, the imaginary part
(which is proportional to the absorption of the probe field) has a
transparency window around the two-photon resonance ($\delta=0$) whose width
can be smaller than the natural linewidth of the probe transition
($\gamma_{31}$). Second, the real part (which describes the dispersion of the
medium) has a sharp slope around the two-photon resonance that leads to
significantly reduced probe group velocity. The appearance of a narrow
transparency window and slowing of light are the hallmarks of EIT Fleischhauer
et al. (2000).

We now consider an inhomogeneously broadened ensemble, which can be thought of
as a collection of homogeneously broadened ensembles, each with some shift of
its transition energy. We have parameterized the detunings relative to state
$\left|1\right\rangle$ such that a shift of the spin transition energy,
$\delta_{s}$, affects the two-photon detuning, $\delta$, but not the probe
detuning and the shift of the optical transition energy, $\delta_{o}$, affects
the probe detuning, $\Delta$, but not the two-photon detuning. Thus,
$\delta\rightarrow\delta-\delta_{s}$ and $\Delta\rightarrow\delta-\delta_{o}$
(where we assume the control field is centered on the optical inhomogeneous
line). We have assumed uncorrelated shifts of the optical and spin transitions
for each atom in the ensemble, which is the case for the rare-earth solids
studied here Ahlefeldt et al. (2016). The susceptibility of the inhomogeneous
system (denoted $\tilde{\chi}$) is the homogeneous susceptibility integrated
over the inhomogeneous profiles $P_{o}(\delta_{o})$ and $P_{s}(\delta_{s})$.

$\begin{split}\chi&\propto\frac{2i\gamma_{21}+4(\delta-\delta_{s})}{\Omega^{2}+(\gamma_{21}-2i(\delta-\delta_{s}))(\gamma_{31}-2i(\delta-\delta_{o}))}\\\
\tilde{\chi}(\delta)&\propto\iint
P_{o}(\delta_{o})P_{s}(\delta_{s})\chi(\delta,\delta_{o},\delta_{s})d\delta_{o}d\delta_{s}\\\
\end{split}$ (2)

By inspection, we see that the expression for $\tilde{\chi}$ can be integrated
analytically if we assume Lorentzian inhomogeneous profiles with full widths
at half maximum (FWHM) $\sigma_{\mathrm{opt}}$ and $\sigma_{\mathrm{spin}}$.
This results in a familiar expression for the probe susceptibility in the
inhomogeneously broadened system:

$\begin{split}\tilde{\chi}(\delta)&\propto\frac{2i(\gamma_{21}+\sigma_{\mathrm{spin}})+4\delta}{\Omega^{2}+((\gamma_{21}+\sigma_{\mathrm{spin}})-2i\delta)((\gamma_{31}+\sigma_{\mathrm{opt}})-2i\delta)}.\end{split}$
(3)

This is the same expression as for the susceptibility of the homogeneously
broadened system with the replacements
$\gamma_{21}\rightarrow\gamma_{21}+\sigma_{\mathrm{spin}}$ and
$\gamma_{31}\rightarrow\gamma_{31}+\sigma_{\mathrm{opt}}$. In the limit of
inhomogeneous linewidths much larger than their homogeneous counterparts, we
have simply replaced the homogeneous values with the inhomogeneous values.
This means that we can use all of our intuition and understanding of EIT in
homogeneously broadened systems, including scaling of the bandwidth, group
velocity, and visibility, and the crossover from an EIT-like regime where the
transparency window is narrower than the optical linewidth, to an Autler-
Townes-like regime where the absorption feature is split into two features
separated by more than their widths Anisimov et al. (2011). We discuss later
the effect on the susceptibility of deviation from a Lorentzian inhomogeneous
profile.

To compare with experiment, we extract expressions for the FWHM and visibility
of the EIT feature, $\Gamma_{\mathrm{EIT}}$ and $V_{\mathrm{EIT}}$. The
visibility is defined such that $1-V_{\mathrm{EIT}}$ is the residual probe
absorption as a fraction of the absorption without the coupling field. We find
that these quantities depend only on $\Omega$, $\sigma_{\mathrm{opt}}$, and
$\sigma_{\mathrm{spin}}$ in the limit of large inhomogeneous widths
($\sigma\gg\gamma$). Furthermore, we expand to first order in
$\sigma_{\mathrm{spin}}$ for the width, as it is smaller than the other
relevant quantities in our systems.

$\begin{split}V_{\mathrm{EIT}}&=\frac{\Omega^{2}}{\Omega^{2}+\sigma_{\mathrm{opt}}\sigma_{\mathrm{spin}}}\\\
\Gamma_{\mathrm{EIT}}&=\frac{\sqrt{\sigma_{\mathrm{opt}}^{2}+4\Omega^{2}}-\sigma_{\mathrm{opt}}}{2}\left(1+\frac{\sigma_{\mathrm{spin}}(\sigma_{\mathrm{opt}}^{2}-\Omega^{2})}{\Omega^{2}\sqrt{\sigma_{\mathrm{opt}}^{2}+4\Omega^{2}}}\right)\\\
\Gamma_{\mathrm{EIT}}&\approx\frac{\Omega^{2}}{\sigma_{\mathrm{opt}}}+\sigma_{\mathrm{spin}}\
\ \ \mathrm{for}\ \Omega\ll\sigma_{\mathrm{opt}}\\\
\Gamma_{\mathrm{EIT}}&\approx\Omega-\frac{\sigma_{\mathrm{opt}}+\sigma_{\mathrm{spin}}}{2}\
\ \ \mathrm{for}\ \Omega\gg\sigma_{\mathrm{opt}}\end{split}$ (4)

We recover the well-known narrowing of EIT in the presence of inhomogeneous
broadening Kuznetsova et al. (2002); Scherman et al. (2012). We further
recover two distinct regimes where the width scales as the square of the Rabi
frequency ($\Omega\ll\sigma_{\mathrm{opt}}$, EIT regime) and linearly with the
Rabi frequency ($\Omega\gg\sigma_{\mathrm{opt}}$, Autler-Townes regime)
Anisimov et al. (2011). Consider a value of $\Omega$ in the Autler-Townes
regime for a homogeneously broadened system ($\Omega\gg\gamma_{31}$), but far
from such a regime in the inhomogeneously broadened system
($\Omega\ll\sigma_{\mathrm{opt}}$). Rather than two absorption peaks split by
$\approx\Omega$, the inhomogeneous system exhibits a transparency window that
resembles a homogeneously broadened system in the EIT regime with linewidth
$\sigma_{\mathrm{opt}}$ (and is thus narrower than the naively expected width
of $\Omega$ by a factor of $\approx\Omega/\sigma_{\mathrm{opt}}\ll 1$).
Reaching the regime with two well-separated absorption peaks requires
$\Omega\gg\sigma_{\mathrm{opt}}$. This limit is difficult to reach in many
Doppler-broadened gases, but we see clear Autler-Townes behavior in our rare-
earth doped crystals with controllable inhomogeneous broadening (see Fig. 3
(b)). Similarly, we see in the visibility the expected behavior that the EIT
disappears for $\Omega^{2}\ll\sigma_{\mathrm{opt}}\sigma_{\mathrm{spin}}$.

##  III Experimental setup

We investigate EIT in two different cryogenically cooled rare-earth-doped
solids. These are a 0.01% doped $\mathrm{Eu^{3+}:Y_{2}SiO_{5}}$ crystal
(Eu:YSO) and a 0.05% doped $\mathrm{Pr^{3+}:Y_{2}SiO_{5}}$ crystal (Pr:YSO),
each held at $\approx 4$ K in a closed-cycle cryostat and addressed by its own
frequency doubled diode laser on the ${}^{7}F_{0}\rightarrow{{}^{5}D_{0}}$
transition at 580 nm for Eu:YSO and the ${}^{3}H_{4}\rightarrow{{}^{1}D_{2}}$
transition at 606 nm for Pr:YSO (Fig. 1(b)). Each diode laser is frequency
stabilized to a reference cavity and the laser linewidths are
$<4~{}\mathrm{kHz}$ and $<1~{}\mathrm{kHz}$ for the Eu:YSO and Pr:YSO
transitions, respectively. We note that non-zero laser linewidth has the same
effect on EIT as spin inhomogeneity, so $\sigma_{\mathrm{spin}}$ is the
quadrature sum of the laser linewidth and the intrinsic spin inhomogeneous
width Lü et al. (1997). For each rare-earth-doped crystal, the probe and
coupling fields are derived from the same laser and given a relative frequency
shift with acousto-optic modulators in a double-pass configuration. The probe
and coupling fields intersect in the crystal at an angle of $<$2∘ that allows
the $\approx 0.5$ mm diameter fields to overlap for the entire 10 mm length of
the crystal. The probe field frequency is scanned over the two-photon
resonance and the transmitted intensity is recorded as a function of time,
which is then converted to frequency. The scan speed is kept below a rate
equal to the spin inhomogeneous width divided by the optical coherence time to
avoid coherent oscillations Meinert et al. (2012).

The two systems differ primarily in the strength of the optical transition,
which is substantially weaker and longer-lived in Eu:YSO compared to Pr:YSO.
As a result, the parameter regimes we can access allow extremely narrow EIT in
Eu:YSO and clear Autler-Townes-type behavior in Pr:YSO. Furthermore, coherent
transients in the EIT are more apparent in Eu:YSO, which may be useful for EIT
based-sensing or other applications Meinert et al. (2012).

The Rabi frequency of the control field is calibrated from the frequency of
the observed optical nutation when the medium is suddenly illuminated by the
control field Sun et al. (2000). The optical inhomogeneous width is determined
from the absorption of the weak probe as a function of detuning in the absence
of the coupling field (see below for more details on controlling the optical
inhomogeneity). The spin inhomogeneous width is an uncontrolled, fixed value
that is due to disorder in the crystal.

##  IV Spectral hole burning

To study the role of optical inhomogeneous broadening in EIT we work in a
regime where $\sigma_{\mathrm{opt}}$ and $\Omega$ are similar in value and
larger than the uncontrolled $\sigma_{\mathrm{spin}}$ because the EIT
visibility drops precipitously for
$\Omega^{2}<\sigma_{\mathrm{opt}}\sigma_{\mathrm{spin}}$ Kuznetsova et al.
(2002). The optical inhomogeneous linewidth of the entire ensemble is $\approx
10^{7}$ times larger than the homogeneous linewidth of the atoms. This
broadening is also larger than the hyperfine splittings, which means that some
selection of a subensemble of atoms is required to observe coherent effects
involving the spin states. We use spectral hole burning techniques to select a
subensemble of atoms within a much narrower spectral region and pump atoms
nearby in the inhomogeneous profile to the non-participating ground hyperfine
level Pryde et al. (2000); Nilsson et al. (2004). This technique allows
inhomogeneous widths larger than the laser linewidth (integrated over the
$\approx\mathrm{ms}$ optical pumping time) and smaller than the hyperfine
splittings. In practice, we generate subensembles with spectral widths
$10^{2}$ to $10^{4}$ times the homogeneous linewidth of the atoms.

We use a multi-step spectral hole-burning sequence to prepare a sub-ensemble
of atoms in a particular frequency class and ground hyperfine state, while
pumping atoms in other nearby frequency classes to ground states such that
they are far from resonance with the probe and control fields Pryde et al.
(2000); Nilsson et al. (2004). An example hole-burned absorption profile is
shown in Fig. 2.

Figure 2: Transmission of a weak probe (with no coupling field) as a function
of frequency near zero detuning following state preparation. Energy level
structures for chosen frequency class plus three other classes shown with
probe (thin arrow) and coupling (thick arrow) fields for reference.

Both Pr:YSO and Eu:YSO have electron spin-singlet ground and excited states
that are split into three doubly degenerate nuclear hyperfine states in zero
magnetic field with splittings in the range of $\approx$5 MHz to 100 MHz.
Praseodymium has a single naturally occurring isotope, while europium has two
isotopes that occur naturally in approximately the same abundance and have
different hyperfine structures. All ground to excited transitions are allowed
with varying transition strengths in both systems and all fields are linearly
polarized along the crystallographic axis that maximizes the light-matter
interaction Liu and Jacquier (2006). The existence of a third metastable
ground state is important as it acts as an auxiliary state where unwanted
population can be shelved to allow coherent processes on the other two states.
In both Eu:YSO and Pr:YSO, we use the upper two ground states for EIT and the
lowest as the auxiliary state.

The large inhomogeneous broadening of the full ensemble means that at any
optical frequency within the inhomogeneous bandwidth there are atoms in nine
different frequency classes resonant on each of the nine different
transitions. (In the natural abundance europium used here, there are an
additional nine frequency classes of the other isotope resonant on its nine
transitions). The first step in the hole burning procedure is selecting a
single frequency class of interest by applying three fields at frequencies
such that the chosen frequency class is resonant with all three fields on
transitions from each of the ground states. All other frequency classes can be
resonant with at most two of the fields and will be optically pumped out of
the ground state(s) with a resonance that matches a resonance of the chosen
frequency class. We sweep these fields over a range much larger than the
ultimate desired subensemble to prepare a spectral region with increased
transparency. We then empty out the two ground states that make up the
$\Lambda$ system by turning off the third field that is at neither the probe
nor control frequency. Finally, we repopulate a narrow spectral region in
$|1\rangle$ with a single frequency repump field, while keeping the field at
the control frequency on to prevent population build up in $|2\rangle$. An
example trace of the transmission of a weak probe measuring the final
absorption profile is shown in Fig. 2.

We generate the absorbing feature by illuminating the sample with a repump
field for a variable amount of time. The width of the feature depends on the
power broadening of the repump light, the spectral diffusion of the atoms
during the repump, and the laser linewidth averaged over the repump time. In
practice, the laser noise during the $>$100 ms repump sets the width of the
absorbing feature, and by controlling the repump time we can vary the final
width from $\lesssim 300~{}\mathrm{kHz}$ to $\gtrsim 2~{}\mathrm{MHz}$ in both
systems. We do not generate wider absorbing features to ensure that the probe
transmission is dominated by atoms in the absorbing feature rather than atoms
outside the transparency window burned around the probe frequency. In Eu:YSO,
the transparency window is limited to $\approx 5~{}\mathrm{MHz}$ by the level
structure and in Pr:YSO we create a single transparency window that covers
both the probe and control frequencies, which leaves the absorbing feature
$\approx 3~{}\mathrm{MHz}$ from the edge of the window.

##  V Experimental results

We measure the EIT over a range of both optical inhomogeneous widths
($\sigma_{\mathrm{opt}}$) and control Rabi frequencies ($\Omega$) in both the
Eu:YSO and Pr:YSO systems. With the control field centered on the hole-burned
feature, the probe field is scanned in frequency across the two-photon
resonance. The probe transmission is recorded as a function of time and
converted to transmission as a function of frequency. We note that coherent
effects on the probe transmission limit the speed of the frequency scan,
particularly in Eu:YSO, where the optical transition has a coherence time
$>1~{}\mathrm{ms}$.

Figure 3: Example measured EIT spectra (black lines) in (a) Eu:YSO and (b)
Pr:YSO. The shaded area is the hole-burned feature. The optical inhomogeneous
width and measured EIT linewidth are noted on each figure.

The available laser power for each system allows us to study EIT for Eu:YSO
only for $\Omega\ll\sigma_{\mathrm{opt}}$, while in Pr:YSO we can reach
$\Omega\gtrsim\sigma_{\mathrm{opt}}$. For $\Omega\ll\sigma_{\mathrm{opt}}$,
the shape of the EIT transmission peak is approximately Lorentzian and its
width can be extracted by fitting. Outside this regime, the overall absorption
appears as two separated peaks and we extract the FWHM without fitting any
particular shape to the feature. The optical depth of the Eu:YSO system is
sufficiently small ($<2$ for all prepared features) that the visibility can be
directly extracted from the probe transmission as the ratio of the
transmission at the EIT peak to the transmission away from the hole-burned
feature. The larger optical depth in the Pr:YSO ($\gtrsim 6$ for the widest
hole-burned features) necessitates fitting a saturated absorption function to
the hole-burned features in order to extract the visibility. Figure 3 shows
typical EIT features in two different parameter regimes. For
$\Omega\ll\sigma_{\mathrm{opt}}$ the EIT window is at the center of the hole-
burned absorbing feature, while for $\Omega\sim\sigma_{\mathrm{opt}}$ the
absorbing feature appears split by the control field as typical of Autler-
Townes splitting. We note that at the largest EIT widths we observe a small
absorption peak at zero detuning (not shown). This can be attributed to atoms
in frequency classes outside the spectral hole burned region. We confirm this
effect by performing the integration in Eq. LABEL:Eqn:int numerically, with
$P_{o}(\delta_{o})$ that includes an absorbing feature at the center of a
hole-burned trench with a broad inhomogeneous ensemble outside.

The measured width of the transparency window is plotted against the expected
value
$\frac{1}{2}(\sqrt{\sigma_{\mathrm{opt}}^{2}+4\Omega^{2}}-\sigma_{\mathrm{opt}})$
in Fig. 4. We have ignored the term in the expected value that depends on the
spin inhomogeneity (Eq. 4), which matters only for the smallest Rabi
frequencies studied and accounts for the deviation of the data from the unit
slope line at small values. The value of the spin inhomogeneity for each
system is noted in Fig. 4 with dashed horizontal lines. The Eu:YSO value of
$4~{}\mathrm{kHz}$ is similar to the measured laser linewidth, suggesting that
the intrinsic spin inhomogeneity is smaller than a previously measured value
in a similar sample Timoney et al. (2013). The Pr:YSO value of
$40~{}\mathrm{kHz}$ is inferred to be the intrinsic spin inhomogeneous width,
as it is much larger than the measured laser linewidth and consistent with
previously measured values Ham et al. (1997b).

Figure 4:  Measured vs. theoretical EIT width for Eu:YSO (solid markers) and
Pr:YSO (hollow markers) at different Rabi frequencies and optical
inhomogeneous widths. Optical inhomogeneous widths are as indicated. Solid
line is the unit slope. The spin inhomogeneity in each system (horizontal
dashed lines).

The visibility for several different configurations and both species similarly
collapse as a function of
$\Omega^{2}/(\sigma_{\mathrm{opt}}\sigma_{\mathrm{spin}})$, as seen in Fig. 5.
We see the expected behavior that the visibility rises from zero to near one,
crossing 0.5 near the expected value of
$\Omega^{2}\approx(\sigma_{\mathrm{opt}}\sigma_{\mathrm{spin}})$ for both
species. The saturation of the visibility at large Rabi frequency is likely
due to residual absorption of far off resonant atoms. This collapse offers
strong support for our model because of the order of magnitude difference in
the value of $\sigma_{\mathrm{spin}}$ for europium and praseodymium.

Figure 5:  Measured EIT width for Eu:YSO (solid markers) and Pr:YSO (hollow
markers) at different optical inhomogeneous widths. Optical inhomogeneous
widths are as indicated.

We observe that the data from the two different systems follows the same
scaling law that depends only on the control field Rabi frequency and optical
inhomogeneous width. The single atom properties of each system, namely the
homogeneous linewidth, does not affect the EIT linewidth. Thus, properties
like the optical lifetime and coherence time are independent of the bulk
ensemble response to the probe and coupling fields. Both the EIT width and
visibility exhibit same scaling with $\Omega$, $\sigma_{\mathrm{opt}}$, and
$\sigma_{\mathrm{spin}}$ for the two different rare-earth species studied over
two orders of magnitude. This occurs despite the different values of the
single atom (homogeneous) parameters for each system.

##  VI Non-Lorentzian inhomogeneous profiles

Most real systems do not exhibit the Lorentzian inhomogeneous profile we
assumed in section II. In Doppler broadened gases the optical inhomogeneity is
Gaussian, and the spectral hole-burned features here have a range of shapes
depending on the specific implementation. In this work, we extract the FWHM of
the hole-burned spectral features without assuming any particular shape. Thus,
it is important to consider the validity of our theoretical treatment for non-
Lorentzian lineshapes.

The integral in Eq. LABEL:Eqn:int cannot in general be calculated analytically
for inhomogeneous profiles with non-Lorentzian distributions. In order to gain
an understanding of the role of the distribution shape, we perform numeric
integration over Gaussian and flat-top profiles for a range of parameters to
obtain $\tilde{\chi}(\delta)$. We then extract the FWHM of the EIT feature and
the EIT visibility (defined as the difference of the maximum and minimum
values of the imaginary part of $\tilde{\chi}$ divided by their sum).

Figure 6: (a) EIT width vs $\Omega/\sigma_{\mathrm{opt}}$ and (b) EIT
visibility vs $\Omega^{2}/(\sigma_{\mathrm{opt}}\sigma_{\mathrm{spin}})$ are
both extracted from Eq. 4 and numerical integration over Gaussian and flat-
topped inhomogeneous profiles. In (a) different colors correspond to different
values of the spin inhomogeneity,
$\sigma_{\mathrm{spin}}/\sigma_{\mathrm{opt}}$, (the values of which are
denoted with horizontal dashed lines) and different curves correspond to
different inhomogeneous shapes (analytical results for Lorentzian shapes are
thick solid lines). Different curves are nearly indistinguishable and
correspond to different spin and optical inhomogeneous shapes showing the
insensitivity to those shapes. In (b), the leftmost results are integrated
over flat-topped (dashed line) and Gaussian (thin solid line) spin
inhomogeneous shapes. The nearly indistinguishable set of curves to the right
are all Lorentzian spin inhomogeneous broadening with different optical
inhomogeneous shapes.

The results of these numerical integrations are shown in Fig. 6 along with the
analytical results for different values of spin inhomogeneity and
inhomogeneity shapes. For the width (see fig. 6a), we see the linear
dependence at large $\Omega$, quadratic dependence at small $\Omega$ and
saturation at the value of the spin inhomogeneity at very small $\Omega$.
Similarly for the visibility of the EIT transmission peak (see fig. 6b), we
see a transition from low transmission when width is saturated, to high
transmission when the spin inhomogeneity is negligible compared to the other
quantities. Neither the linewidth of the EIT nor the visibility depends on the
shape of the optical inhomogeneity. The only dependence suggested by the
numerical results is increased EIT transmission at smaller $\Omega$ for spin
inhomogeneous broadening that falls off more quickly than Lorentzian. Thus,
the replacement of the homogeneous linewidth with its inhomogeneous
counterpart is thus a reasonable technique for considering EIT in a wide range
of inhomogeneous broadened ensembles.

##  VII Conclusion

In conclusion, we have studied EIT in two different inhomogeneously broadened
rare-earth doped solids. As opposed to ensembles of identical or near-
identical cold atoms (homogeneously broadened ensembles), most solids exhibit
large inhomogeneous broadening. As solid-state systems become more common for
quantum optics and quantum information applications due to their advantages in
terms of of motional dephasing, reduced experimental overhead, and
integratability into scalable photonic systems, it is vital to explore and
understand the impact of inhomogeneous broadening. Here we observe good
agreement with a theoretical treatment covering two orders of magnitude in the
coupling Rabi frequency and inhomogeneous linewidth. In addition, a simple
theoretical treatment of inhomogeneous broadening predicts EIT lineshapes
similar to those seen in homogeneously broadened systems, with a direct
replacement of the homogeneous linewidths with their inhomogeneous
counterparts. We further discuss the effect of the shape of the inhomogeneous
profile on the EIT properties, and see that the properties of interest are
largely insensitive to the shape. This work provides important groundwork for
implementing coherent quantum optical processes in solids where inhomogeneous
broadening often plays a major role, thus paving the way to exploiting this
class of materials for quantum information applications.

## Acknowledgments

We acknowledge partial support from the NSF Physics Frontier Center at the
Joint Quantum Institute. The authors thank Paul Lett and Ivan Burenkov for
helpful discussions.

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