††thanks: These two authors contributed equally††thanks: These two authors
contributed equally

# Squeezed vacuum used to accelerate the search for a weak classical signal

M. Malnou<EMAIL_ADDRESS>JILA, National Institute of Standards
and Technology and the University of Colorado, Boulder, Colorado 80309, USA
Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
D. A. Palken<EMAIL_ADDRESS>JILA, National Institute of Standards
and Technology and the University of Colorado, Boulder, Colorado 80309, USA
Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
B. M. Brubaker  JILA, National Institute of Standards and Technology and the
University of Colorado, Boulder, Colorado 80309, USA  Department of Physics,
University of Colorado, Boulder, Colorado 80309, USA  Leila R. Vale  National
Institute of Standards and Technology, Boulder, Colorado 80305, USA  Gene C.
Hilton  National Institute of Standards and Technology, Boulder, Colorado
80305, USA  K. W. Lehnert  JILA, National Institute of Standards and
Technology and the University of Colorado, Boulder, Colorado 80309, USA
Department of Physics, University of Colorado, Boulder, Colorado 80309, USA

###### Abstract

Many experiments that interrogate fundamental theories require detectors whose
sensitivities are limited by the laws of quantum mechanics. In cavity-based
searches for axionic dark matter, vacuum fluctuations in the two quadratures
of the cavity electromagnetic field limit the sensitivity to an axion-induced
field. In an apparatus designed to partially mimic existing axion detectors,
we demonstrate experimentally that such quantum limits can be overcome through
the use of squeezed states. By preparing a microwave cavity in a squeezed
state and measuring just the squeezed quadrature, we enhance the spectral scan
rate by a factor of $2.12\pm 0.08$. This enhancement is in excellent
quantitative agreement with a theoretical model accounting for both imperfect
squeezing and measurement.

##  I INTRODUCTION

In searches for dark matter axions Peccei and Quinn (1977a, b); Preskill  _et
al._ (1983); Abbott and Sikivie (1983); Dine and Fischler (1983), the
quadratures $\hat{X}$ and $\hat{Y}$ of a resonant cavity’s electromagnetic
field carry the imprint of the dark matter signal as a slight excess in their
power spectra. Problematically, the quantum noise Caves (1982) intrinsic to a
measurement of these observables overpowers the signal by roughly three orders
of magnitude, and the signal frequency is a priori unknown Bradley  _et al._
(2003). Even at the quantum limit, scanning the 1–10 GHz frequency band at the
pessimistic benchmark DFSZ Zhitnitsky (1980); Dine  _et al._ (1981) coupling
with present detector technologies Du  _et al._ (2018); Zhong  _et al._ (2018)
will take upwards of 20,000 years of experimental live time. Realizing the
several thousandfold speed-up required to put these detectors on expedient
schedules will require many parallel innovations in detector design and
sensitivity. Among the most promising advances are those which allow access to
a fundamentally distinct parameter regime in which axion searches are no
longer limited by quantum noise.

In detectors measuring the quadratures of a resonant mode, quantum noise can
be circumvented by preparing the mode in a squeezed state Caves (1981).
Squeezing unbalances the uncertainties in the two quadrature observables,
thereby permitting precise knowledge of one at the expense of the other.
Leveraging quantum squeezing to enhance measurement sensitivity has been a
long-standing goal in the optical domain for the sensing of gravitational
waves Caves (1981); Collaboration (2011); Aasi  _et al._ (2013); Kimble  _et
al._ (2001). At microwave frequencies, meanwhile, squeezing has been
demonstrated in principle Murch  _et al._ (2013); Clark  _et al._ (2016);
Bienfait  _et al._ (2017), but it has yet to aid in a search for new physical
phenomena.

But is microwave squeezing beneficial in an axion dark matter search given
current experimental constraints? This question involves both matters of
principle and practice. Specifically, in axion haloscope Sikivie (1983)
searches, the dark matter signal is a persistent tone of unknown frequency and
phase, but more coherent than the resonant cavity itself. As such, a haloscope
uses a tunable cavity to scan a resonance through frequency, with the critical
parameter being the rate at which the cavity can be tuned with some specified
sensitivity to an axion. The suitability of squeezing to this type of search
has not received specific theoretical attention. Furthermore, if squeezing is
in principle beneficial in this case, can microwave losses be made
sufficiently low to yield a practical improvement?

In this article, we show both theoretically and experimentally that squeezing
increases the scan rate in a search for a weak, axion-like signal of unknown
frequency. This improvement exists in spite of the fact that squeezing does
not improve the sensitivity of a haloscope to a tone of known frequency. We
demonstrate this speed-up in an apparatus designed to mimic the behavior of
existing haloscopes. By delivering a maximum of $4.5\pm 0.1$ dB of squeezing
from one Josephson parametric amplifier (JPA) Yamamoto  _et al._ (2008);
Castellanos-Beltran  _et al._ (2008); Zhou  _et al._ (2014) to another in a
squeezed state receiver (SSR) configuration, we perform a realistic
acquisition and processing protocol both with and without squeezing enabled.
We demonstrate that the SSR accelerates the scan rate at constant sensitivity
to axion-like signals by a factor of $2.12\pm 0.08$. Equipping the SSR to a
dark matter detector such as HAYSTAC Brubaker  _et al._ (2017a); Zhong  _et
al._ (2018) or ADMX Du  _et al._ (2018) will mark the transition of searches
for physics beyond the standard model to the sub-quantum limited noise regime.

##  II THEORY OF SQUEEZING-ENHANCED SCAN RATE

Figure 1 shows a representative experimental apparatus in which a resonant
cavity is coupled to an SSR comprising a pair of JPAs. We start by considering
the cavity, whose internal mode (frequency $\omega_{c}$) has Hamiltonian
$\hat{H}=\hbar\omega_{c}(\hat{X}^{2}+\hat{Y}^{2})/2$, where $\hat{X}$ and
$\hat{Y}$ are the noncommuting quadratures of the cavity field, obeying
$[\hat{X},\hat{Y}]=i$. This cavity is modeled as exchanging energy with three
ports. First, a measurement port couples the cavity mode to the propagating
modes of a transmission line with power decay rate $\kappa_{m}$. Along this
line, a microwave circulator spatially separates incoming and outgoing
propagating modes. Second, a loss port, connected to a fictitious transmission
line, models the cavity’s internal energy dissipation at rate $\kappa_{l}$.
Third, the cavity’s coupling to the signal of interest at rate $\kappa_{a}$ is
modeled as occurring through another fictitious transmission line; the signal
itself is modeled as a microwave generator characterized by its frequency
$\omega_{a}$, spectral width $\Delta_{a}$, and amplitude $\mathcal{E}_{a}$. We
assume $\mathcal{E}_{a}\gg 1$, implying that the displacement of the cavity
mode by the signal is classical (i.e., the contribution of the signal to the
cavity’s quantum fluctuations can be neglected in comparison to those coming
from the loss and measurement ports). We also assume a narrow band signal
($\Delta_{a}\ll\\{\kappa_{l},\kappa_{m}\\}$) so weakly coupled
($\mathcal{E}_{a}^{2}\kappa_{a}\ll\kappa_{l}$) that the time required to
resolve the displacement is much longer than the signal’s phase coherence
time. Therefore, on average, this displacement yields a small excess power
above the vacuum fluctuations, isotropic in quadrature space (see bottom panel
in Fig. 1). These inequalities are very well satisfied in the case of the
axion field (see Appendix B).

Figure 1: Schematic of the SSR and cavity. Two JPAs, SQ and AMP, respectively
squeeze and read out a microwave field interacting with a cavity at rate
$\kappa_{m}$. An axion-like field $\mathcal{E}_{a}$, coupled to the cavity at
rate $\kappa_{a}$, displaces the cavity state. Energy leaves the cavity
through internal absorption at a rate $\kappa_{l}$. Bottom panel: quadrature
representation of a vacuum state detuned from the cavity resonant frequency,
traveling in the SSR. At the SQ’s input (a) it is Gaussian, azimuthally
equiprobable in the $(\hat{X},\hat{Y})$ plane (red disk). The state is
squeezed along $\hat{X}$ by the SQ (b), displaced along a random phase within
the cavity (c), and amplified along $\hat{X}$ by the AMP (d). Comparing to
what happens without squeezing (e)–(g), the size of the signal-plus-noise
(green) relative to the noise (red) in this quadrature is larger with (d) than
without (g) squeezing.

The SSR itself comprises the two JPAs shown in Fig. 1, which couple
respectively to incoming and outgoing modes at the cavity’s measurement port.
This configuration exploits the fact that a portion of the vacuum noise
exiting the measurement port arises from vacuum noise incident on that same
port. The first JPA (called SQ) squeezes these input fluctuations along the
$\hat{X}$ quadrature, reducing the observable’s variance below vacuum levels:
$\sigma_{X}^{2}<1/2$ Mallet  _et al._ (2011); Bienfait  _et al._ (2017);
Menzel  _et al._ (2012); Boutin  _et al._ (2017); Malnou  _et al._ (2018). To
satisfy the uncertainty principle, the opposing quadrature’s fluctuations
exceed vacuum, $\sigma_{Y}^{2}>1/2$, but are not measured. The squeezed input
field subsequently enters the cavity, where a small displacement by the signal
would yield a small excess power in both quadratures. At the measurement port
output, the second JPA (called AMP) noiselessly amplifies only the $\hat{X}$
quadrature with sufficient gain to overwhelm the noise added by following
amplifiers and mixers Yamamoto  _et al._ (2008); Zhou  _et al._ (2014);
Pogorzalek  _et al._ (2017). Note that in the absence of squeezing there is
neither a benefit nor a penalty associated with measuring only one quadrature
compared to the usual two-quadrature case (see Appendix C).

The benefit of squeezing can be understood by analyzing the microwave network
formed by the combination of SSR and cavity using input-output theory (see
Appendix A). The signal spectral density at the measurement port output is
equal to the signal spectral density at the measurement port input, weighted
by the susceptibility of the measurement port output to the signal port input.
Similarly, the noise density at the measurement port output is a
susceptibility-weighted sum of squeezed and unsqueezed noise from the
measurement and loss ports, respectively. In the absence of transmission
losses between the JPAs and the cavity, the ratio of output signal spectral
density to total output noise spectral density (hereafter called the signal
visibility) is thus

$\alpha(\omega)\approx\frac{n_{A}\kappa_{a}\kappa_{m}}{\left(n_{T}+\frac{1}{2}\right)\left(\kappa_{l}\kappa_{m}+\frac{\beta(\omega)}{G_{s}}\right)}.$
(1)

Here, $\omega$ is the frequency relative to cavity resonance,
$n_{T}=1/[\exp{(\hbar\omega_{c}/k_{B}T)}-1]$ is the mean thermal photon number
incident on the cavity from the measurement and loss ports,
$n_{A}=\mathcal{E}_{a}^{2}$ is the mean photon number sourced by the
fictitious generator,
$\beta(\omega)=(\kappa_{m}-\kappa_{l})^{2}/4+\omega^{2}$, and $G_{s}$ is the
power gain of the SQ (ideally equal to the reduction of the squeezed state
variance below the vacuum value).

When optimizing $\alpha(\omega)$ in Eq. (1), we assume that $\kappa_{m}$ and
$G_{s}$ can be freely varied, whereas $n_{T}$, $n_{A}$, $\kappa_{l}$, and
$\kappa_{a}$ are fixed by the physics of the signal source or technical
constraints of the detector. On cavity resonance ($\omega=0$), $\alpha$ is
maximized at critical coupling ($\kappa_{m}=\kappa_{l}$). Because $\beta(0)=0$
at critical coupling, $\alpha(0)$ is independent of $G_{s}$ and there is no
benefit from squeezing; physically, the squeezed state injected into the
cavity is completely absorbed in it, while all the unsqueezed noise from the
loss port reaches the AMP. For $\omega\neq 0$, squeezing increases
$\alpha(\omega)$ for any value of $\kappa_{m}$, as $G_{s}$ reduces the amount
of measurement port noise reaching the output. In the limit where
$G_{s}\rightarrow\infty$, the $\beta(\omega)$ term in the denominator can be
neglected, and $\alpha(\omega)$ approaches the critically coupled resonant
value $\alpha(\omega=0,\kappa_{m}=\kappa_{l})$ for any $\kappa_{m}$ and all
$\omega$. This illustrates an important point of principle: squeezing cannot
improve the peak sensitivity of a haloscope, but there is no fundamental limit
to how much it can enhance the detector bandwidth over which this peak
sensitivity is achieved. When $G_{s}$ is finite, overcoupling
($\kappa_{m}>\kappa_{l}$) increases the cavity bandwidth at the cost of
reducing $\alpha(0)$. This can be a favorable trade-off (even in the absence
of squeezing) because the signal’s frequency $\omega_{a}$ is a priori unknown
Irastorza and Redondo (2018); Chaudhuri  _et al._ (2018), and broader
bandwidth enables larger cavity tuning steps. Moreover, squeezing mitigates
the reduction of $\alpha(0)$ from overcoupling, thus enabling faster tuning
without significant degradation of sensitivity.

To quantify this speed-up, we calculate the rate $R$ (in Hz/s) at which we can
tune the cavity resonance through frequency space in search for a signal. This
scan rate is inversely proportional to the measurement time at each tuning
step, which in turn scales with $\alpha^{-2}$ as a consequence of Gaussian
noise statistics (see Appendix A.1). Hence
$R\propto\Delta_{a}\int_{-\infty}^{\infty}\alpha^{2}(\omega)d\omega$; carrying
out the integral we obtain

$R\propto\frac{\Delta_{a}\sqrt{G_{s}}n_{A}^{2}\kappa_{a}^{2}\kappa_{m}^{2}}{\left(n_{T}+\frac{1}{2}\right)^{2}\left[\kappa_{l}\kappa_{m}+\frac{1}{G_{s}}\left(\frac{\kappa_{l}-\kappa_{m}}{2}\right)^{2}\right]^{3/2}}.$
(2)

Without squeezing ($G_{s}=1$), $R$ is maximized when the cavity is twice
overcoupled ($\kappa_{m}=2\kappa_{l}$), and at this optimal coupling the scan
rate scales as $R_{u}^{\mathrm{max}}\propto\kappa_{a}^{2}/\kappa_{l}$, a known
result Kenany  _et al._ (2017). When $G_{s}\gg 1$, the optimal coupling is
$\kappa_{m}=2G_{s}\kappa_{l}$ and the scan rate scales as
$R_{s}^{\mathrm{max}}\propto G_{s}\kappa_{a}^{2}/\kappa_{l}$. Comparing the
two situations, the scan rate is improved by $G_{s}$, which shows that an
ideal SSR greatly accelerates the search for a weak classical signal when
squeezing and overcoupling.

In practice, however, losses in microwave components reduce the SQ-to-AMP
transmission efficiency $\eta$ and hence the benefit of squeezing, because
part of the squeezed state is replaced with unsqueezed vacuum (see Appendix
A.2). Figure 2 compares the theoretical scan rate enhancement
$E_{t}=R_{s}/R_{u}^{\mathrm{max}}$ (a) when $\eta=1$ (perfect transmission),
and (b) when $\eta=0.69$ (efficiency we observe in practice), as a function of
$G_{s}$ and $\kappa_{m}/\kappa_{l}$. In the first case, $E_{t}$ improves
arbitrarily as squeezing and coupling are together increased. In the second
case, it plateaus at $E_{t}^{\mathrm{max}}\approx 2.2$ for $G_{s}>20$ when
optimally coupled. In Secs. III and IV, we present experimental results for
the scan rate enhancement consistent with this theory.

Figure 2: The scan rate enhancement $E_{t}$, calculated as a function of SQ
single-quadrature power gain $G_{s}$ and coupling ratio
$\kappa_{m}/\kappa_{l}$. With perfect efficiency, i.e. $\eta=1$, (a) $E_{t}$
grows steadily with $G_{s}$ and $\kappa_{m}$. When $\eta=0.69$ (b), it
plateaus at $2.2$ for $\kappa_{m}/\kappa_{l}=5$ and $G_{s}>20$. The color
scales for (a) and (b) differ by close to a factor of $10$.

##  III EXPERIMENTALLY IMPROVED SIGNAL RECOVERY WITH SQUEEZING

The scan rate can be degraded not only by transmission losses, but also by
distortions in the squeezed state Malnou  _et al._ (2018); Boutin  _et al._
(2017) and added noise in the amplifier chain. In order to investigate how
accurately Eq. (2) predicts the performance of a squeezing-enhanced haloscope,
we construct an apparatus that mimics its microwave network, but without some
of the cumbersome features. Specifically, in an axion haloscope the
mechanically tunable cavity must reside in a large static magnetic field to
enhance $\kappa_{a}$ and $n_{A}$. In our apparatus, the cavity has a fixed
frequency $\omega_{c}=2\pi\times 7.146$ GHz and there is no applied field.
Consequently our cavity, constructed from superconducting aluminum, has much
lower intrinsic loss than a copper haloscope cavity. We create additional loss
through an explicit port that extracts energy from the cavity at a rate
$\kappa_{l}=2\pi\times\ 100$ kHz, a typical value for a haloscope cavity. We
introduce an axion-like signal into our cavity through a microwave generator
connected to a second port weakly coupled at rate $\kappa_{a}=2\pi\times 100$
Hz. Finally, a third port couples the cavity mode to an SSR with a rate chosen
to be close to the optimum value for the case with ($\kappa_{m}=10\kappa_{l}$)
or without ($\kappa_{m}=1.5\kappa_{l}$) squeezing, creating a physical
realization of the model in Fig. 1.

For the largest increase in scan rate, the SSR should be attached to the
cavity measurement port with as little transmission loss as possible. To
investigate the transmission loss independent of the cavity loss, we use the
fact that the JPAs are narrow band ($\sim 5$ MHz), tunable amplifiers and
detune both the SQ and AMP far off cavity resonance ($\sim 10$ MHz) so that
the squeezed state is promptly reflected from the cavity. Figure 3 illustrates
our ability to efficiently generate, transport, and amplify a squeezed state
in this off-resonance configuration. It shows histograms of the measured
voltage in the AMP’s amplified quadrature $\hat{X}_{\mathrm{out},m}$ as a
function of $\theta$, the phase between the amplified quadrature of the SQ and
the amplified quadrature of the AMP. When $\theta=\pi/2$ or $3\pi/2$, one
quadrature is first squeezed and then amplified. At these points, comparing
the output noise variances $\sigma^{2}_{\mathrm{on}}$ and
$\sigma^{2}_{\mathrm{off}}$ measured with SQ on and off (not pumped),
respectively Malnou  _et al._ (2018), we obtain a squeezing
$S=\sigma^{2}_{\mathrm{on}}/\sigma^{2}_{\mathrm{off}}=-4.5\pm 0.1$ dB. We
emphasize that this is not an inferred squeezing; rather, we directly measure
an overall $4.5$ dB reduction in the noise floor of $\hat{X}_{\mathrm{out},m}$
over the whole bandwidth of the quadrature measurement. This amount of
squeezing (consistent with our estimate of $\eta=0.69\pm 0.01$, see Appendix
D.4) required particular care in reducing the transmission losses between the
two JPAs, which was facilitated by using flux-pumped JPAs Yamamoto  _et al._
(2008); Zhou  _et al._ (2014); Pogorzalek  _et al._ (2017).

Figure 3: Histograms of voltage fluctuations $V_{X}$ (a) and corresponding
vacuum squeezing (b) $S=\sigma_{\mathrm{on}}^{2}/\sigma_{\mathrm{off}}^{2}$
measured along $\hat{X}_{\mathrm{out},m}$, as a function of the SQ-AMP
relative phase $\theta$.

This large amount of delivered squeezing implies that the SSR should improve
our ability to resolve a weakly coupled signal detuned from cavity resonance.
To demonstrate this improvement, we center the SQ and AMP bands on cavity
resonance, overcouple the cavity’s measurement port
($\kappa_{m}=10\kappa_{l}$), and set the phase $\theta$ to $\pi/2$ (see Fig.
3). We complete the single-quadrature measurement by mixing down
$\hat{X}_{\mathrm{out},m}$ with a local oscillator (LO) at the cavity resonant
frequency and computing its power spectral density. As for any such
measurement, the frequency component at $\omega$ in the down-mixed output is a
linear combination of two frequency components at the mixer’s high-frequency
input, $\pm\omega$ detuned from the LO frequency. Figure 4a shows the spectral
density when the tone is $1$ MHz detuned from the cavity’s resonance.
Comparing two situations, SQ on and SQ off, the signal visibility improves by
roughly $4$ dB in the presence of squeezing, with $0.5$-dB degradation from
cavity loss.

Figure 4: Improvement in microwave tone’s signal visibility due to squeezing.
Power spectra normalized to the unsqueezed vacuum power, with (red) and
without (purple) squeezing are shown in (a) for a tone $1$ MHz detuned from
the resonant frequency of the overcoupled cavity ($\kappa_{m}=10\kappa_{l}$).
The $x$-axis has been shifted between the two situations for visual clarity.
In (b), the visibility $\alpha(\omega)$ is measured as a function of the
tone’s detuning from cavity resonance ($\omega=0$) for two cases: no squeezing
while near critical coupling ($\kappa_{m}=1.5\kappa_{l}$, blue circles), and
squeezing while strongly overcoupled ($\kappa_{m}=10\kappa_{l}$, red circles).
In both cases $\alpha(\omega)$ is normalized to the expected maximum value
$\alpha_{\mathrm{max}}=\alpha(0)$ evaluated at $\kappa_{m}=\kappa_{l}$. The
theoretical expectation for each case is superimposed as a black curve,
calculated using $\kappa_{l}=2\pi\times 100$ kHz, $\eta=0.69$ and $G_{s}=13$
dB.

To estimate the associated increase in scan rate from the SSR, we step the
tone across the cavity’s resonance and measure the visibility $\alpha(\omega)$
at each detuning $\omega$. In order to compare the optimal squeezed and
unsqueezed cases, we measure $\alpha(\omega)$ for two cases, displayed in Fig.
4b: for $\kappa_{m}=10\kappa_{l}$ with squeezing, and for
$\kappa_{m}=1.5\kappa_{l}$ without squeezing (see Appendix E for two other
complementary cases). Without squeezing, $\alpha(0)$ is greater but the
bandwidth is poor. With squeezing, $\alpha$ remains relatively high as
$\omega$ increases. We extract
$R\propto\int_{-\infty}^{\infty}\alpha(\omega)^{2}d\omega$ for the two cases,
and obtain the estimated scan rate enhancement $E_{e}=2.05\pm 0.04$. By
independently measuring $\eta$, $\kappa_{m}$, and $G_{s}$, we calculate
expected values for $\alpha(\omega)$, shown as solid lines, in excellent
agreement with the measured values in Fig. 4b. Finally, from the expected
$\alpha(\omega)$ we calculate $E_{t}=2.11\pm 0.07$, also in quantitative
agreement with the data-based estimate $E_{e}$.

##  IV SQUEEZING-ENHANCED SEARCH FOR AN AXION-LIKE SIGNAL

In a real axion search, the aim is to detect a signal many orders of magnitude
smaller than that in Fig. 4. Inferring the presence or absence of an axion at
each frequency requires combining the measured powers from many adjacent
cavity tunings Brubaker  _et al._ (2017a); Du  _et al._ (2018). Furthermore,
over a long integration time the benefit inferred from Fig. 4 is vulnerable to
practical non-idealities. Drifts of either JPA’s gain, drifts of the SQ-AMP
relative phase $\theta$, non-Gaussian noise processes, and interfering rf or
IF tones are of particular concern. In this section, we demonstrate that our
SSR indeed matches the performance presented in the previous section when
searching for a feeble tone over a wide frequency range.

We attempt to detect a fake axion, or “faxion,” sent through the cavity weakly
coupled port. It is synthesized from a randomly modulated microwave tone,
whose power is adjusted such that the faxion spectral density emerging from
the cavity is roughly $1\%$ of vacuum, and whose width is broadened to roughly
$\Delta_{a}\approx 9$ kHz, comparable to expectations for a realistic axion at
frequency $\omega_{a}\approx 2\pi\times 7$ GHz Brubaker  _et al._ (2017b).
Stepping the faxion tone frequency “backwards” past a stationary cavity
simulates a realistic axion search without the hardware demands imposed by a
tunable cavity. The faxion’s initial frequency is chosen randomly within a
$2$-MHz window around the cavity resonance, and is then tuned in discrete
$-10$-kHz steps over a $4$-MHz window. At each tuning step, we record an
output power spectral density as described in Sec. III. These spectra are
mixed down and referred to the mixer’s input by symmetrizing about $\omega=0$.
Spectra are artificially shifted in steps of $+10$ kHz to simulate cavity
tuning. We then rescale the spectra by $\alpha(\omega)$ such that frequency
bins with higher sensitivity to the faxion are weighted more. These rescaled,
shifted spectra are then added into a grand spectrum (see Appendix F and Ref.
Brubaker  _et al._ (2017b)). With this procedure, the faxion’s contributions
in each individual spectrum add at its initial frequency as if we had tuned
the cavity, creating a clear excess of power in the grand spectrum. Figure 5a
presents some symmetrized spectra, obtained while squeezing with an
overcoupled cavity ($\kappa_{m}=10\kappa_{l}$), in which the faxion excess
power is too small to rise above vacuum fluctuations. The spectra were
obtained at different fictitious cavity tunings, normalized to their measured
standard deviations, and vertically offset from one another for visual
clarity. In the resulting grand spectrum (Fig. 5b), a prominent faxion peak
emerges with $6\sigma$ visibility.

Figure 5: Response of the SSR in the presence of a faxion. The acquired power
spectra are symmetrized (a), rescaled and then shifted in frequency to align
all the frequency bins containing the faxion with each other (red dots),
thereby effectively tuning the cavity. The power excess of the processed
spectra is plotted in units of their standard deviation $\sigma_{p}$.
Combining all the spectra into a grand spectrum (b), whose excess power is
plotted in units of its standard deviation $\sigma_{g}$, a large power excess
is observed at the initial frequency of the faxion. Repeating this measurement
allows us to compute distributions of faxion powers (c). When squeezing with
$\kappa_{m}=10\kappa_{l}$, the powers (red points) are drawn from a Gaussian
distribution $\mathcal{N}(\mu_{s},\sigma_{g}^{2})$ (solid line), with
$\mu_{s}=6.05\pm 0.07$. When not squeezing with $\kappa_{m}=1.5\kappa_{l}$,
the powers (blue points) are drawn from $\mathcal{N}(\mu_{u},\sigma_{g}^{2})$
(solid line), with $\mu_{u}=4.15\pm 0.07$. The powers (black points) drawn
from the faxion-less noise distribution $\mathcal{N}(0,\sigma_{g}^{2})$ (solid
line) are also represented for comparison.

In order to extract a scan rate enhancement from such a realistic signal
search, we acquire two distributions of faxion powers, one when squeezing and
one when not squeezing, with near-optimal $\kappa_{m}$ for each case. We
obtain each distribution by repeating the faxion injection and detection
protocol $200$ times. Over the course of the measurement, which takes roughly
nine hours per configuration, we keep the relative phase $\theta$ between SQ
and AMP quadratures at $\pi/2$ via a feedback loop (see Appendix D.3). Figure
5c displays the two faxion power distributions, as well as the noise power
distribution. The no-faxion distribution of mean zero and variance
$\sigma_{g}^{2}$ is guaranteed by the central limit theorem to be Gaussian
distributed, $\mathcal{N}(0,\sigma_{g}^{2})$. The faxion adds a small mean
excess of power $\mu$ insufficient to enlarge the variance. The faxion
distribution is therefore $\mathcal{N}(\mu,\sigma_{g}^{2})$. The signal and
noise distributions separate as the total measurement time squared, and thus
the speed-up due to squeezing is equal to $E_{m}=(\mu_{s}/\mu_{u})^{2}$, where
$\mu_{s}$ $(\mu_{u})$ is the mean of the faxion power distribution obtained
when squeezing (not squeezing). The two distributions of faxion powers are
characterized by $\mu_{s}=6.05\pm 0.07$ and $\mu_{u}=4.15\pm 0.07$, leading to
a measured scan rate enhancement of $E_{m}=2.12\pm 0.08$, in quantitative
agreement with the estimate $E_{e}$ obtained from visibility measurements in
Sec. III.

##  V CONCLUSION

The low signal-to-noise ratios achievable in state-of-the-art axion haloscopes
Du  _et al._ (2018); Zhong  _et al._ (2018); Chung (2016); McAllister  _et
al._ (2017) and similar resonant searches for faint electromagnetic signals
sourced by dark matter axions Ouellet  _et al._ (2019); Silva-Feaver  _et al._
(2017); Majorovits  _et al._ (2017) make the timescales over which they can
sweep out appreciable fractions of parameter space unreasonably long.
Partially replacing the vacuum noise in an axion haloscope with squeezed
vacuum circumvents the standard quantum limit on the noise of the measurement
apparatus, enabling high sensitivity over a broader bandwidth and a more rapid
dark matter search. Squeezing moreover moves haloscopes into a qualitatively
distinct design parameter space: whereas a quantum limited haloscope’s scan
rate plateaus with improving microwave transmission efficiencies $\eta$, a
sub-quantum limited haloscope will benefit almost arbitrarily as microwave
losses are further reduced. Thus, haloscopes can reap larger benefits from
increased efficiency (for example, a tenfold scan rate enhancement at
$\eta=0.91$) as low-loss quantum technologies such as on-chip circulators and
directional amplifiers mature Sliwa  _et al._ (2015); Macklin  _et al._
(2015); Chapman  _et al._ (2017).

## Acknowledgements

We thank Kyle Thatcher for his help in the design and fabrication of the SSR
mechanical parts and Felix Vietmeyer for his help in the design and
fabrication of room temperature electronics. This work was supported by the
National Science Foundation, under Grants No. PHY-1607223 and No. PHY-1734006,
and by the Heising-Simons Foundation under Grant No. 2014-183.

##  Appendix A Theory of the SSR operation

In this appendix, we track the propagating electromagnetic fields through the
SSR and cavity, so as to derive the susceptibility matrix of the entire
system. We then calculate the signal visibility $\alpha(\omega)$ and the scan
rate $R$. We first consider the case where the propagating fields experience
no loss (the cavity mode still decays partially out its loss port), which
nonetheless faithfully illustrates the utility of the SSR, then we treat the
full system in the presence of transmission losses.

###  A.1 Lossless case

We model the energy exchange between the cavity’s ports in its own rotating
frame. The time evolution of the cavity is governed by the Heisenberg-Langevin
equation

$\frac{d\hat{A}}{dt}=-\frac{\kappa_{T}}{2}\hat{A}(t)+\sum_{j}\sqrt{\kappa_{j}}\hat{a}_{\mathrm{in},j}(t),$
(3)

where $\hat{A}$ is the cavity ladder operator,
$\kappa_{T}=\kappa_{m}+\kappa_{l}+\kappa_{a}$, and $\hat{a}_{\mathrm{in},j}$
($j=m,l,a$) are the annihilation field operators of the input modes incident
on ports indexed by $m,$ $l$, and $a$. At the measurement port, we physically
separate input ($\hat{a}_{\mathrm{in},m}$) and output
($\hat{a}_{\mathrm{out},m}$) fields with a circulator. Given the input-output
relations
$\hat{a}_{\mathrm{out},j}(t)=\hat{a}_{\mathrm{in},j}(t)-\sqrt{\kappa_{j}}\hat{A}(t)$,
the input and output field operators are related in the Fourier domain by

$\hat{a}_{\mathrm{out},j}(\omega)=\sum_{k}\chi_{jk}(\omega)\hat{a}_{\mathrm{in},k}(\omega),$
(4)

where

$\chi_{jk}(\omega)=\frac{-\sqrt{\kappa_{j}\kappa_{k}}+(\kappa_{T}/2+i\omega)\delta_{jk}}{\kappa_{T}/2+i\omega}$
(5)

are the elements of a $3\times 3$ susceptibility matrix fully describing the
behavior of the cavity Clerk  _et al._ (2010).

By cascading the input-output relations for each element presented in Fig. 1,
we can calculate the benefit from squeezing. We work in the quadrature basis
and consider the vector of input quadratures
$\vec{x}_{\mathrm{in}}=[\hat{X}_{\mathrm{in},m},\hat{X}_{\mathrm{in},l},\hat{X}_{\mathrm{in},a}]^{\mathrm{T}}$
aligned with our squeezing. We will calculate the SSR/cavity susceptibility
matrix $\bm{\Xi_{X}}$, in terms of which the vector of output quadratures is
$\vec{x}_{\mathrm{out}}=\bm{\Xi_{X}}\vec{x}_{\mathrm{in}}$.

The first element in the system is the SQ, which performs a one-mode squeezing
(OMS) operation on the measurement port’s input quadrature:

$\vec{x}_{s}=\bm{S_{X}}\vec{x}_{\mathrm{in}}=\begin{bmatrix}\frac{1}{\sqrt{G_{s}}}&&0&&0\\\
0&&1&&0\\\ 0&&0&&1\\\ \end{bmatrix}\vec{x}_{\mathrm{in}},$ (6)

where the subscript $s$ refers to the SQ output port and $G_{s}$ is the SQ
single-quadrature power gain. The OMS operation also amplifies the other
quadrature of the measurement port mode, $\hat{Y}_{\mathrm{in},m}$ by
$\sqrt{G_{s}}$ in order to preserve the Heisenberg uncertainty relation. We do
not track the evolution of the $\hat{Y}$ quadrature here, as it is irrelevant
for SSR performance.

Next, the cavity transforms the quadrature operators. In the cavity rotating
frame, the vectors of quadrature operators are obtained from the vectors of
ladder operators by

$\begin{bmatrix}\vec{x}\\\
\vec{y}\end{bmatrix}=\frac{1}{\sqrt{2}}\begin{bmatrix}\bm{I_{3}}&&\bm{I_{3}}\\\
-i\bm{I_{3}}&&i\bm{I_{3}}\end{bmatrix}\begin{bmatrix}\vec{a}(\omega)\\\
\vec{a}^{\dagger}(-\omega)\end{bmatrix},$ (7)

or
$[\vec{x},\vec{y}]^{\mathrm{T}}=\bm{P}[\vec{a}(\omega),\vec{a}^{\dagger}(-\omega)]^{\mathrm{T}}$.
Here $\bm{I_{3}}$ is the $3\times 3$ identity matrix, and the mode ordering in
each vector is $m,l,a$: for example,
$\vec{a}(\omega)=[\hat{a}_{m}(\omega),\hat{a}_{l}(\omega),\hat{a}_{a}(\omega)]^{\mathrm{T}}$.
Thus, the cavity susceptibility matrix in the quadrature basis
$\bm{\tilde{\chi}}$ is:

$\bm{\tilde{\chi}}=\bm{P}\begin{bmatrix}\bm{\chi}(\omega)&&\bm{0_{3}}\\\
\bm{0_{3}}&&\bm{\chi}^{*}(-\omega)\end{bmatrix}\bm{P}^{-1}=\begin{bmatrix}\bm{\chi}(\omega)&&\bm{0_{3}}\\\
\bm{0_{3}}&&\bm{\chi}(\omega)\end{bmatrix},$ (8)

where $\bm{0_{3}}$ is the null matrix. We see that the cavity’s effect on
field and quadrature operators is identical, which leads to

$\vec{x}_{o}=\bm{\chi}(\omega)\vec{x}_{s},$ (9)

where $o$ refers to the cavity output port.

Finally the AMP performs a second OMS operation so as to amplify the
quadrature that the SQ originally squeezed:

$\vec{x}_{\mathrm{out}}=\bm{A_{X}}\vec{x}_{o}=\begin{bmatrix}\sqrt{G_{a}}&&0&&0\\\
0&&1&&0\\\ 0&&0&&1\\\ \end{bmatrix}\vec{x}_{o},$ (10)

where $G_{a}$ is the AMP power gain.

The SSR and cavity thus transform input to output quadratures according to

$\vec{x}_{\mathrm{out}}=\bm{A_{X}}\bm{\chi}(\omega)\bm{S_{X}}\vec{x}_{\mathrm{in}}=\bm{\Xi_{X}}\vec{x}_{\mathrm{in}}.$
(11)

We can then calculate the single-quadrature output spectral density matrix
$\bm{\Sigma_{\mathrm{out,}X}}=\langle[\vec{x}_{\mathrm{out}}]^{\dagger}[\vec{x}_{\mathrm{out}}]^{\mathrm{T}}\rangle/2\pi$,
where the Hermitian conjugation here does not transpose the vector, in order
to determine the total output signal and noise powers. Substituting for
$\vec{x}_{\mathrm{out}}$ yields

$\bm{\Sigma_{\mathrm{out,}X}}=\bm{\Xi_{X}}^{*}\bm{\Sigma_{\mathrm{in,}X}}\bm{\Xi_{X}}^{\mathrm{T}},$
(12)

where
$\bm{\Sigma_{\mathrm{in,}X}}=\langle[\vec{x}_{\mathrm{in}}]^{\dagger}[\vec{x}_{\mathrm{in}}]^{\mathrm{T}}\rangle/2\pi$
is the input noise spectral density matrix. Since the SSR is connected to
three incoherent and uncorrelated modes (see Fig. 1),
$\langle\hat{X}_{\mathrm{in},j}\hat{X}_{\mathrm{in},k}\rangle/2\pi=\delta_{jk}(n_{\mathrm{in,}j}+1/2)$,
where $n_{\mathrm{in,}j}$ is the input mean photon number per unit time per
unit bandwidth (henceforth simply mean photon number) of mode $j$. Thus,

$\bm{\Sigma_{\mathrm{in,}X}}=\begin{bmatrix}n_{T}+\frac{1}{2}&&0&&0\\\
0&&n_{T}+\frac{1}{2}&&0\\\ 0&&0&&n_{A}+\frac{1}{2}\\\ \end{bmatrix}.$ (13)

Taking the first entry in the matrix $\bm{\Sigma_{\mathrm{out,}X}}$ we obtain
the single-quadrature output spectral density at the measurement port:

$\displaystyle\Sigma_{\mathrm{out,}X,m}\approx\frac{G_{a}}{B(\omega)}\bigg{[}$
$\displaystyle n_{A}\kappa_{a}\kappa_{m}$ (14) $\displaystyle+$
$\displaystyle\left(n_{T}+\frac{1}{2}\right)\left(\kappa_{l}\kappa_{m}+\frac{\beta(\omega)}{G_{s}}\right)\bigg{]},$

where $B(\omega)=(\kappa_{m}+\kappa_{l})^{2}/4+\omega^{2}$,
$\beta(\omega)=(\kappa_{m}-\kappa_{l})^{2}/4+\omega^{2}$, and we used
$\kappa_{a}\ll\\{\kappa_{l},\kappa_{m}\\}$ and $n_{A}\gg 1/2$.

The visibility $\alpha(\omega)$, defined as the ratio of signal and noise
spectral densities at the measurement port output, can be directly extracted
from $\Sigma_{\mathrm{out,}X,m}$, since the signal is the term proportional to
$n_{A}$ and the noise is $\Sigma_{\mathrm{out,}X,m}(n_{A}=0)$:

$\alpha(\omega)\approx\frac{n_{A}\kappa_{a}\kappa_{m}}{\left(n_{T}+\frac{1}{2}\right)\left(\kappa_{l}\kappa_{m}+\frac{\beta(\omega)}{G_{s}}\right)},$
(15)

which is identical to Eq. (1).

We now consider a search protocol comprising many measurements of
$\Sigma_{\mathrm{out,}X,m}$, as the cavity frequency is tuned by discrete
steps $\delta_{c}$. The spectral scan rate $R$ is obtained by taking the
limit:

$R=\lim_{\begin{subarray}{c}\delta_{c}\to 0\\\ \tau\to
0\end{subarray}}\frac{\delta_{c}}{\tau},$ (16)

where $\tau$ is the duration of each measurement. In the rest of this section
we derive Eq. (2) for the scan rate. We begin by defining the signal-to-noise
ratio (SNR) at a single tuning step as

$\overline{\alpha}(\omega)=\sqrt{\tau\Delta_{a}}\frac{\alpha(\omega)}{2},$
(17)

which scales as the square root of the number of independent measurements of
the power contained in the bandwidth $\Delta_{a}$, centered on a frequency
detuned from cavity resonance by $\omega/2\pi$ Dicke (1946); Brubaker  _et
al._ (2017a); the factor of 2 is a consequence of our single-quadrature
measurement scheme (see Appendix C).

For $\delta_{c}\lesssim\kappa_{T}/2\pi$, multiple tuning steps will contribute
to the SNR at each frequency. Without loss of generality, we evaluate the net
SNR for some putative signal frequency $\omega_{a}$ which coincides exactly
with the cavity resonance at a particular step. We then consider the
contributions to the SNR of the surrounding $2n+1$ tuning steps, where
$2\pi\delta_{c}=2\kappa_{T}/(2n+1)$. Contributions to the SNR add in
quadrature, so the net squared SNR is

$\overline{\alpha}_{2n+1}^{2}=\frac{\tau\Delta_{a}}{4}\cfrac{1}{\delta_{c}}\sum_{k=-n}^{k=n}\alpha^{2}(k2\pi\delta_{c})\delta_{c}.$
(18)

In the limit $n\to\infty$, $\tau\to 0$, we obtain the integrated squared SNR

$\overline{\alpha}_{I}^{2}=\frac{\Delta_{a}}{4}\frac{1}{R}\int_{-\infty}^{\infty}\alpha^{2}(\omega)\frac{d\omega}{2\pi},$
(19)

where we have extended the limits of integration to $\pm\infty$: the
contributions from tuning steps where the cavity is detuned by more than
$\kappa_{T}$ are negligible.

Note that our arbitrary choice of $\omega_{a}$ appears nowhere explicitly on
the RHS of Eq. (19); thus the integrated SNR will be frequency independent,
insofar as the quantities contributing to $\alpha(\omega)$ remain constant as
we tune the cavity. In practice, we set a certain target value of
$\overline{\alpha}_{I}$ for which a real signal would appear as a sufficiently
prominent peak in the grand spectrum (see e.g., Fig. 5b); this choice
determines the required spectral scan rate $R$. Solving for $R$, we obtain

$\displaystyle R$
$\displaystyle=\frac{\Delta_{a}}{4\overline{\alpha}_{I}^{2}}\int_{-\infty}^{\infty}\alpha^{2}(\omega)\frac{d\omega}{2\pi}$
(20)
$\displaystyle=\frac{\Delta_{a}\sqrt{G_{s}}n_{A}^{2}\kappa_{a}^{2}\kappa_{m}^{2}}{16\overline{\alpha}_{I}^{2}\left(n_{T}+\frac{1}{2}\right)^{2}\left[\kappa_{l}\kappa_{m}+\frac{1}{G_{s}}\left(\frac{\kappa_{l}-\kappa_{m}}{2}\right)^{2}\right]^{3/2}},$

in agreement with Eq. (2). In a comparison of squeezed and unsqueezed scan
rates, the $\Delta_{a}/4\overline{\alpha}_{I}^{2}$ factor drops out; thus, we
need only compare the integral in Eq. (20) for the two cases.

###  A.2 SSR operation in the presence of transmission losses

Here we generalize the calculations of Appendix A.1 to account for imperfect
power transmission efficiency $\eta$ between SQ and AMP. Integrating the
squared SNR over all frequencies, we obtain the scan rate as a function of
$\eta$, used in Fig. 2. For simplicity, we treat the transmission efficiency
$\lambda$ between the SQ and the cavity as identical to that between the
cavity and the AMP, hence $\eta=\lambda^{2}$. Note that, in principle, loss
between SQ and cavity is slightly less harmful than loss between cavity and
AMP, as loss that occurs after the cavity degrades the signal along with the
squeezing.

As in Appendix A.1, we track the vector of input quadratures
$\vec{x}_{\mathrm{in}}$ throughout the SSR and cavity system, neglecting the
orthogonal quadratures. First, the SQ performs the OMS operation defined by
Eq. (6): $\vec{x}_{s}=\bm{S_{X}}\vec{x}_{\mathrm{in}}$. Next we must account
for transmission losses between the SQ and the cavity, modeled as a
beamsplitter interaction between the quadrature operators of the measurement
line at the SQ output $\\{\hat{X}_{s,m},\hat{Y}_{s,m}\\}$, and those of
another, uncontrolled mode $\\{\hat{X}_{s,\lambda},\hat{Y}_{s,\lambda}\\}$. We
could define generalized quadrature vectors $\vec{x}_{s}$ and $\vec{y}_{s}$ to
include $\hat{X}_{s,\lambda}$ and $\hat{Y}_{s,\lambda}$ respectively, but we
do not need to keep track of the electromagnetic field’s evolution in loss
modes. Among the modes we do keep track of, only the measurement port
experiences the loss:

$\vec{x}_{i}=\begin{bmatrix}\sqrt{\lambda}&&0&&0\\\ 0&&1&&0\\\ 0&&0&&1\\\
\end{bmatrix}\vec{x}_{s},$ (21)

or $\vec{x}_{i}=\bm{L_{X}}\vec{x}_{s}$. Here $i$ refers to the cavity’s input,
and $0\leq\lambda\leq 1$ is the single-sided transmission efficiency.

Since we are not tracking the SQ-to-cavity propagating loss mode, the
unsqueezed vacuum that it introduces enters as an added noise term,

$\bm{N_{X}}=\begin{bmatrix}(n_{T}+\frac{1}{2})(1-\lambda)&&0&&0\\\ 0&&0&&0\\\
0&&0&&0\\\ \end{bmatrix}.$ (22)

We thus have the single-quadrature noise spectral density at the cavity input:
$\bm{\Sigma_{i,X}}=[\bm{L_{X}}\bm{S_{X}}]^{*}\bm{\Sigma_{\mathrm{in,}X}}[\bm{L_{X}}\bm{S_{X}}]^{\mathrm{T}}+\bm{N_{X}}$,
where $\bm{\Sigma_{\mathrm{in,}X}}$ is given by Eq. (13). This expression
yields

$\bm{\Sigma_{i,X}}=\begin{bmatrix}\left(n_{T}+\frac{1}{2}\right)\left(\frac{\lambda}{G_{s}}+1-\lambda\right)&&0&&0\\\
0&&n_{T}+\frac{1}{2}&&0\\\ 0&&0&&n_{A}+\frac{1}{2}\\\ \end{bmatrix},$ (23)

where both the partially attenuated squeezed vacuum term
($\propto\lambda/G_{s}$), and the unsqueezed contribution from the loss mode
($\propto 1-\lambda$) appear clearly. Note that in the absence of loss,
$\bm{\Sigma_{i,X}}=\bm{S_{X}}^{*}\bm{\Sigma_{\mathrm{in,}X}}\bm{S_{X}}^{\mathrm{T}}$,
in agreement with Eq. (12).

We can now calculate $\bm{\Sigma_{\mathrm{out,}X}}$, the output spectral
density matrix along $\vec{x}$. Given Eq. (8) for the cavity’s susceptibility
in the quadrature basis, we may write
$\bm{\Sigma_{o,X}}=\bm{\chi}^{*}(\omega)\bm{\Sigma_{i,X}}\bm{\chi}(\omega)^{\mathrm{T}}$.
Then losses between cavity and AMP are accounted for in the same manner as
before, and finally the AMP performs another OMS operation, amplifying the SQ
squeezed quadrature. We thus obtain

$\displaystyle\bm{\Sigma_{\mathrm{out,}X}}$
$\displaystyle=\bm{A_{X}}^{*}[\bm{L_{X}}^{*}\bm{\Sigma_{o,X}}\bm{L_{X}}^{\mathrm{T}}+\bm{N_{X}}]\bm{A_{X}}^{\mathrm{T}}$
(24)
$\displaystyle=\bm{A_{X}}^{*}[\bm{L_{X}}^{*}\bm{\chi}^{*}(\omega)\bm{\Sigma_{i,X}}\bm{\chi}(\omega)^{\mathrm{T}}\bm{L_{X}}^{\mathrm{T}}+\bm{N_{X}}]\bm{A_{X}}^{\mathrm{T}}$

as the analog of Eq. (12) in the presence of loss.

The first entry in the matrix $\bm{\Sigma_{\mathrm{out,}X}}$ is the output
spectral density at the measurement port along $\hat{X}$:

$\displaystyle\Sigma_{\mathrm{out,}X,m}=$
$\displaystyle\left(n_{T}+\frac{1}{2}\right)G_{a}(1-\lambda)$ (25)
$\displaystyle+\frac{G_{a}\lambda}{B(\omega)}\bigg{[}$
$\displaystyle\left(n_{A}+\frac{1}{2}\right)\kappa_{a}\kappa_{m}$
$\displaystyle+$
$\displaystyle\left(n_{T}+\frac{1}{2}\right)\left(\kappa_{l}\kappa_{m}+\left(1-\lambda+\frac{\lambda}{G_{s}}\right)\beta(\omega)\right)\bigg{]},$

where $B(\omega)$ and $\beta(\omega)$ are defined as in Eq. (14), and the same
approximations have been made.

From $\Sigma_{\mathrm{out,}X,m}$ we can extract the signal visibility
$\alpha(\omega)$ in the presence of loss:

$\displaystyle\alpha(\omega)=$ $\displaystyle\frac{\lambda
n_{A}}{\left(n_{T}+\frac{1}{2}\right)}\times$ (26)
$\displaystyle\frac{\kappa_{a}\kappa_{m}}{B(\omega)\left(1-\lambda\right)+\lambda\left[\kappa_{l}\kappa_{m}+\left(1-\lambda+\frac{\lambda}{G_{s}}\right)\beta(\omega)\right]}.$

When $\lambda=1$, Eq. (26) reduces to Eq. (1).

Finally, the scan rate enhancement $E_{t}$ in the presence of loss, presented
in Fig. 2b, is $\int_{-\infty}^{\infty}\alpha^{2}(\omega)d\omega$, normalized
by the same integral with $G_{s}=1$ and $\kappa_{m}=2\kappa_{l}$.

##  Appendix B Model for the axion field

Figure 1 models the axion field as a fictitious generator that drives the
cavity through a weakly coupled port. In this appendix, we relate the
fictitious port coupling $\kappa_{a}$ and the power spectral density $n_{A}$
at the generator output to physical parameters normally found in the haloscope
literature, and show that for representative values, the axion field acts as a
classical force.

The measurable axion-sourced power is obtained from the first term in the
output spectral density, Eq. (14). Referring to the cavity output and
multiplying by the axion linewidth $\Delta_{a}$, the on-resonance
($\omega_{a}=\omega_{c}$) signal power is

$P_{\mathrm{sig}}=4\hbar\omega_{a}n_{A}\Delta_{a}\frac{\kappa_{a}\kappa_{m}}{(\kappa_{m}+\kappa_{l})^{2}}.$
(27)

The two model parameters $\kappa_{a}$ and $n_{A}$ can readily be related to
physical parameters by comparing Eq. (27), to e.g. Eq. (1) of Ref. Kenany  _et
al._ (2017) evaluated on resonance. In our notation this expression takes the
form

$P_{\text{sig}}=\left(g_{a\gamma\gamma}^{2}\frac{\hbar
c^{3}\rho_{a}}{\mu_{0}\omega_{a}^{2}}\right)\times\left(B_{0}^{2}VC_{mn\ell}\frac{\omega_{c}^{2}\kappa_{m}}{(\kappa_{m}+\kappa_{l})^{2}}\right),$
(28)

where, in the first set of parentheses, $g_{a\gamma\gamma}$ parametrizes the
axion field’s coupling to electromagnetism, $\rho_{a}$ is the local dark
matter energy density, $c$ is the speed of light, $\mu_{0}$ is the vacuum
permeability, and we have related the axion rest mass $m_{a}$ to the frequency
of axion-induced photons $\omega_{a}$ as $m_{a}c^{2}=\hbar\omega_{a}$. The
second parenthetical expression contains properties of the haloscope: $B_{0}$
is the static magnetic field, $C_{mnl}$ is the cavity mode-dependent form
factor, and $V$ is the cavity volume. Equating Eqs. (27) and (28), we obtain

$n_{A}\kappa_{a}=\frac{g_{a\gamma\gamma}^{2}\rho_{a}c^{3}}{4\omega_{a}\mu_{0}\Delta_{a}}B_{0}^{2}C_{mnl}V.$
(29)

To derive a second expression relating $\kappa_{a}$ to $n_{A}$, we observe
that the fictitious generator in our haloscope model may equivalently be
represented as a second harmonic oscillator mode with very high occupancy but
very weak coupling to the haloscope cavity. The axion field in any laboratory-
scale volume constitutes such an oscillator mode, with resonant frequency
$\omega_{a}$. Specifically, we model the oscillations of the axion field as a
fictitious cavity occupying the same volume as the real haloscope cavity. The
quanta of this fictitious cavity are axion particles, so its total occupancy
is $N_{A}=V\rho_{a}/\hbar\omega_{a}$; it is coupled to the haloscope cavity
with interaction Hamiltonian $\hat{H}_{\mathrm{int}}=\hbar
g(\hat{A}+\hat{A}^{\dagger})(\hat{B}+\hat{B}^{\dagger})$, where $g$ is the
interaction strength and $\hat{A}$ ($\hat{B}$) is the annihilation operator of
the haloscope (fictitious) cavity. The Hamiltonian of the closed system is
$\hat{H}=\hat{H}_{0}+\hat{H}_{\mathrm{int}}$, where
$\hat{H}_{0}=\hbar\omega_{c}(\hat{A}^{\dagger}\hat{A}+1/2)+\hbar\omega_{a}(\hat{B}^{\dagger}\hat{B}+1/2)$.
Coupling the haloscope cavity to measurement and loss ports at rates
$\kappa_{m}$ and $\kappa_{l}$, respectively, we write down the Heisenberg-
Langevin equations of motion for the open system:

$\displaystyle\frac{d\hat{A}}{dt}$
$\displaystyle=-i\omega_{c}\hat{A}(t)-ig[\hat{B}(t)+\hat{B}^{\dagger}(t)]-\frac{\kappa_{m}+\kappa_{l}}{2}\hat{A}$
(30)
$\displaystyle+\sqrt{\kappa_{m}}\hat{a}_{\mathrm{in},m}(t)+\sqrt{\kappa_{l}}\hat{a}_{\mathrm{in},l}(t)$
$\displaystyle\frac{d\hat{B}}{dt}$
$\displaystyle=-i\omega_{a}\hat{B}(t)-ig[\hat{A}(t)+\hat{A}^{\dagger}(t)],$
(31)

where we describe the input bath associated with the measurement (loss) port
with the annihilation operator $\hat{a}_{\mathrm{in},m}$
($\hat{a}_{\mathrm{in},l}$).

We restrict ourselves to the classical limit of the system, with operators
demoted to complex amplitudes, and the case where the resonances coincide,
$\omega_{a}=\omega_{c}$, with no power entering via the measurement and loss
ports, $a_{\mathrm{in},m}=a_{\mathrm{in},l}=0$. Transforming into the rotating
frame of the haloscope cavity,
$\\{A(t),B(t)\\}\rightarrow\\{A(t)e^{-i\omega_{c}t},B(t)e^{-i\omega_{c}t}\\}$
and making a rotating wave approximation, Eqs. (30) and (31) reduce to

$\displaystyle\frac{dA}{dt}$
$\displaystyle=-igB(t)-\frac{\kappa_{m}+\kappa_{l}}{2}A(t)$ (32)
$\displaystyle\frac{dB}{dt}$ $\displaystyle=-igA(t),$ (33)

where $g$ plays the role of the field exchange rate corresponding to the power
decay rate out of the axion cavity, $g=\kappa_{a}/2$. These equations of
motion describe an exchange of energy between the two cavities and a decay of
that energy out from the haloscope cavity via the measurement and loss ports.
We are interested in the steady-state ($dA/dt=0$) output field
$A_{\mathrm{out},m}=-\sqrt{\kappa_{m}}A$ when the occupancy of the axion
cavity is $|B|^{2}=N_{A}$. We find a steady-state occupancy of the haloscope
cavity $|A|^{2}=[\kappa_{a}/(\kappa_{m}+\kappa_{l})]^{2}N_{A}$, implying an
output signal power of

$P_{\mathrm{sig}}=\hbar\omega_{a}|A_{\mathrm{out},m}|^{2}=\frac{\hbar\omega_{a}N_{A}\kappa_{a}^{2}\kappa_{m}}{(\kappa_{m}+\kappa_{l})^{2}}.$
(34)

Equations (27) and (34) for the output power must agree, implying

$\frac{n_{A}}{\kappa_{a}}=\frac{N_{A}}{4\Delta_{a}}=\frac{V\rho_{a}}{4\hbar\omega_{a}\Delta_{a}}.$
(35)

In terms of physical parameters, Eqs. (29) and (35) yield

$\displaystyle n_{A}$
$\displaystyle=\frac{|g_{a\gamma\gamma}|\rho_{a}B_{0}V}{4\omega_{a}\Delta_{a}}\sqrt{\frac{C_{mnl}c^{3}}{\hbar\mu_{0}}}$
(36) $\displaystyle\kappa_{a}$
$\displaystyle=|g_{a\gamma\gamma}|B_{0}\sqrt{\frac{C_{mnl}\hbar
c^{3}}{\mu_{0}}}.$ (37)

quantity | value
---|---
$\rho_{a}$ | $0.45\ \mathrm{GeV}/\mathrm{cm}^{3}$
$B_{0}$ | $9\ \mathrm{T}$
$g_{a\gamma\gamma}$ | $-7.7\times 10^{-24}\ \mathrm{eV}^{-1}$
$\Delta_{a}$ | $5\ \mathrm{kHz}$
$\omega_{a}/2\pi$ | $5\ \mathrm{GHz}$
$V$ | $1.5\ \mathrm{L}$
$C_{mnl}$ | $0.5$
Table 1: Representative physical values used to estimate our model parameters.

Using representative values for the HAYSTAC experiment Kenany  _et al._ (2017)
in the presence of a 5-GHz KSVZ Kim (1979); Shifman  _et al._ (1980) axion
shown in Table 1, we obtain the values for our model parameters shown in Table
2. We see that the fictitious generator is well into the classical regime,
$n_{A}\gg 1/2$, while its extremely feeble coupling $\kappa_{a}$ nonetheless
makes its presence a challenge to detect.

parameter | value
---|---
$N_{A}$ | $3.3\times 10^{16}$
$\kappa_{a}/2\pi$ | $2.3\ \mathrm{\mu Hz}$
$n_{A}$ | $2.4\times 10^{7}$
Table 2: Model parameter values calculated using the physical values from
Table 1.

##  Appendix C Single vs. double quadrature measurement

Single-mode squeezing can only enhance sensitivity to displacements along a
single quadrature of the cavity field. In a situation such as an axion search,
the signal distributes its excess power equally between the two quadratures.
Thus switching to single-quadrature measurement from double-quadrature
measurement, currently the operational mode of choice of axion haloscopes
Brubaker  _et al._ (2017a); Du  _et al._ (2018), seems detrimental. In this
appendix, we show that in the absence of squeezing, neither measurement scheme
has an advantage over the other.

If we neglect amplifier added noise, the signal visibility $\alpha$ is
independent of whether we measure one quadrature or both. Specifically, an
axion signal characterized by its spectral density $S_{a}$ at the amplifier
input divides itself equally as $S_{a}/2$ between the two quadratures.
Similarly, the vacuum noise spectral density $\hbar\omega_{a}/2$ divides its
power equally as $\hbar\omega_{a}/4$ between the quadratures.

However, we must account for the quantum limits on two-quadrature
measurements. Any linear amplifier that measures both quadratures adds at
least a second half-quantum of input-referred noise, evenly distributed
between the two quadratures Caves (1982). An ideal double-quadrature
measurement thus yields $\alpha_{\mathrm{2Q}}=S_{a}/\hbar\omega_{a}$ in each
quadrature. By comparison, there is no quantum limit on single-quadrature
amplification, so an ideal single-quadrature measurement yields
$\alpha_{\mathrm{1Q}}=2S_{a}/\hbar\omega_{a}=2\alpha_{\mathrm{2Q}}$ in the
amplified quadrature.

To make a fair comparison between the single-quadrature and double-quadrature
cases, we must consider the improvement in the SNR (defined in Appendix A.1).
Because all pertinent differences between the two cases enter when considering
a single tuning step, we neglect tuning in the following discussion. The SNR
$\overline{\alpha}$ is given in terms of the spectral density ratio $\alpha$
by

$\overline{\alpha}=\sqrt{\frac{N}{2}}\frac{\Delta_{a}}{\Delta}\alpha,$ (38)

where $\Delta_{a}/\Delta$ is the ratio of signal to noise bandwidths, and $N$
is the number of independent measurements of the voltage contained in the
noise bandwidth $\Delta$. In considering the appropriate values of $N$ and
$\Delta$ for the two cases of interest, we find two independent effects, each
reducing $\overline{\alpha}_{\mathrm{1Q}}$ by a factor of $\sqrt{2}$ relative
to $\overline{\alpha}_{\mathrm{2Q}}$. Together, these effects cancel the
apparent benefit stemming from the absence of quantum noise limits in the
single-quadrature case.

First, the Nyquist theorem guarantees that there are
$N_{\mathrm{2Q}}=2\tau\Delta_{\mathrm{2Q}}$ independent measurements of the
noise voltage in a double-quadrature measurement of duration $\tau$ and
bandwidth $\Delta_{\mathrm{2Q}}$, where the factor of 2 counts the two
independent quadrature amplitudes for each resolved Fourier component. Thus
there are $N_{\mathrm{1Q}}=\tau\Delta_{\mathrm{1Q}}$ measurements of the noise
voltage in a single-quadrature measurement of bandwidth
$\Delta_{\mathrm{1Q}}$.

Second, noiseless single-quadrature measurement with a parametric amplifier
creates an irreversible ambiguity between the output signal and idler
frequencies, equally spaced about the amplifier band center. This ambiguity
necessitates mapping amplifier outputs at a given detuning from band center to
the input signal and idler frequencies (see Appendix F). The consequent
addition of spectral densities, half of which are guaranteed not to have an
axion-induced excess power, effectively increases the noise bandwidth in a
single-quadrature measurement by a factor of $2$ relative to the signal
bandwidth: $\Delta_{\mathrm{1Q}}=2\Delta_{a}$. In the standard double-
quadrature measurement scheme, the signal and noise bandwidths are equal:
$\Delta_{\mathrm{2Q}}=\Delta_{a}$.

Putting this all together, the two measurement schemes are seen to be
equivalent:

$\displaystyle\overline{\alpha}_{\mathrm{1Q}}$
$\displaystyle=\sqrt{\frac{\tau\Delta_{\mathrm{1Q}}}{2}}\frac{\Delta_{a}}{\Delta_{\mathrm{1Q}}}\alpha_{\mathrm{1Q}}$
(39)
$\displaystyle=\sqrt{\frac{2\tau\Delta_{\mathrm{2Q}}}{2}}\frac{\Delta_{a}}{2\Delta_{\mathrm{2Q}}}2\alpha_{\mathrm{2Q}}$
$\displaystyle=\overline{\alpha}_{\mathrm{2Q}}.$

The first line agrees with Eq. (17) for the single-quadrature SNR.

##  Appendix D Characterization of the experimental apparatus

In this appendix we discuss and characterize key features of the SSR,
including its control electronics. As the performance of the SSR is primarily
limited by its transmission losses, many of our design choices were driven by
loss mitigation. Additionally, we designed the experiment to have a minimum
number of adjustable controls, so that the faxion search could be easily
automated.

###  D.1 Flux-pumped JPAs

The requirement of running the SSR automatically for several days makes
monochromatically current-pumped JPAs a poor choice for SQ and AMP, as the
strong current pump tone remains present at the center of the amplification
band output Castellanos-Beltran  _et al._ (2008); Mallet  _et al._ (2011);
Malnou  _et al._ (2018). If not canceled by a $\pi$-phase shifted tone of
equal amplitude, the SQ pump would reach and saturate the AMP. Furthermore,
with insufficient microwave isolation the AMP pump could also be reflected
from the cavity and perturb the AMP operating point. Keeping both cancellation
tones stable in amplitude and phase, while controlling both SQ and AMP pumps
to obtain good gain and good squeezing is impractical over a long period of
time.

Flux-pumped JPAs do not require cancellation tones, because they are pumped at
twice their operating frequency, far outside the bandwidth of the receiver
chain. In addition, their topology is such that pump and signal propagate
along spatially distinct paths, ensuring good isolation between the two (see
Fig. 6). This also negates the need for a directional coupler in front of the
chip, and therefore reduces the insertion loss along the squeezed state path.
When flux pumping, a JPA can be thought of as a linear, frequency-tunable $LC$
resonator, capacitively coupled to a transmission line. Josephson junctions
arranged to form a series array of superconducting quantum interference
devices (SQUIDs) provide the tunable inductance. The bare resonance
$\omega_{0}$ of the $LC$ resonator is tunable via a dc magnetic flux. The pump
generates an ac magnetic flux which modulates the SQUID inductance at
$\omega_{p}=2\omega_{0}$, generating parametric amplification.

We optimized several constraints to ensure efficient flux pumping. First, we
took care in the design to ensure identical coupling between the pump line and
each of the SQUIDs that collectively constitute the tunable inductor. Uniform
coupling guarantees that the JPA’s dynamics are spatially homogeneous,
resulting in improved power handling. Second, we placed the pump line close to
the SQUID array, in order to minimize the pump power required to drive the
$LC$ resonator. At the same time, the coupling of the $LC$ circuit to the pump
line is kept much lower than the coupling to the line carrying the signal, in
order to avoid losing part of the signal through the pump port: the
transmission between the two ports is kept below $-20\,$dB. Finally, we shaped
the flux line as a U around the SQUID array, as shown in Figs. 6a and b. With
this configuration, the pump couples to the differential mode current
circulating inside each SQUID loop and is isolated from the common-mode
current, unidirectional across the SQUID array.

Figure 6: A flux-pumped JPA, similar to those used in the SSR. It is
fabricated in the standard niobium-aluminum-niobium process in a coplanar
waveguide geometry. A scanning electron microscope image (a) shows, in false
color, the main elements of the JPA: a $550$-fF interdigital capacitor
(purple), an inductance comprising a SQUID array of $6$-µA Josephson junctions
(red), and a flux line, shorted to ground (blue). A closer view (b) shows
several SQUID loops (red), surrounded by the flux line (blue). The pump
current $I(\omega_{p})$ is circulating back and forth along the flux line,
thereby favoring the differential mode current and attenuating the common-mode
current in the SQUID loops. The JPA equivalent circuit (c) represents the two-
port device: an incoming signal on port 1 is reflected and amplified. On port
2, the pump at $\omega_{p}=2\omega_{0}$ modulates the SQUID inductance.

###  D.2 Experimental setup

The full experimental setup is represented in Fig. 7. The SQ-cavity-AMP
ensemble is attached to the bottom plate of a dilution refrigerator. For each
JPA, the pump tone is routed though a $10$-dB directional coupler connected to
the pump port. The placement and configuration of these directional couplers
ensures that the large pump tone required for flux pumping primarily heats up
a $50$-$\ohm$ termination whose Johnson noise propagates back up the pump
line, away from the JPA. A coil around each chip, connected to a dc current
source at room temperature, generates a dc magnetic field, and the chip-coil
ensemble is magnetically shielded with aluminum and cryoperm. Each JPA is
connected to a circulator through superconducting NbTi cables in order to
minimize transmission losses.

An ensemble of four circulators routes the squeezed state and provides
microwave isolation between the SSR elements. Two circulators between the SQ
and the cavity protect the SQ from power reflected back from the cavity. These
circulators provide sufficient isolation when the SQ is operated with $13$ dB
of signal gain. Similarly, three circulators separate the AMP and the cavity,
as the AMP is operated with higher signal gain, around $25$ dB. We
experimentally observed that with only two circulators, undesirable feedback
between the cavity and AMP perturbs the AMP’s gain by effectively changing its
pump’s power.

At the chain’s input, either vacuum noise or a probe tone can be injected via
a $20$-dB directional coupler. This probe tone is useful when characterizing
the JPA gain profiles or when characterizing the cavity. Finally, at the
output, a double-junction isolator protects the AMP from signals reflected
from the next amplifier, a HEMT at $4$ K.

Figure 7: Full schematic of the SSR, with its room temperature control
electronics. The input (black) and output (red) coaxial cables are in bold. At
the $20$-mK stage, NbTi superconducting coaxial cables are shown in green, and
the three adjacent circulators form a triple-junction circulator fabricated by
QuinStar (QCE-070100CM30). The thinner (orange) lines represent dc cables.

At room temperature, a single microwave generator (Keysight E8257D) drives
both JPAs and also serves as LO for the in-phase/quadrature (IQ) mixer in the
readout line. On the path leading to the SQ pump’s input, a voltage-controlled
variable attenuator and phase shifter provide control of both the SQ pump
amplitude and phase. On the path leading to the IQ mixer, a frequency divider
converts $\omega_{p}$ to $\omega_{0}$, the bare resonant frequency of both
JPAs and the cavity. This divider (Pasternack PE88D2000) can input a wide
range of powers (from $-20$ to $5$ dBm) while always outputting the same power
(roughly $-4$ dBm). Thus, the AMP gain can be tuned freely with the microwave
generator’s output power. A second generator injects a tone through the weakly
coupled port of the axion cavity. This tone can be shaped into a $9$-kHz-wide
Lorentzian with an arbitrary waveform generator (AWG).

At the output, a $6$ dB directional coupler routes a fraction of the power to
a vector network analyzer (VNA), which monitors signals either from the SSR
chain’s input or from the cavity’s weakly coupled port. The VNA is used to
measure in situ the couplings $\kappa_{m}$ and $\kappa_{l}+\kappa_{a}$. The
other portion of the output power reaches the IQ mixer’s rf port. After being
mixed down, the in-phase and quadrature signals are amplified and directed
onto $1.9$ MHz anti-aliasing low-pass filters, then finally digitized by an
analog-to-digital converter (ADC).

###  D.3 Phase locking of SQ and AMP pumps

Enhancing the scan rate with squeezing is only beneficial if the AMP amplifies
the SQ squeezed quadrature. To this end, as suggested by Fig. 3, the phase
$\theta$ between SQ and AMP pumps must be maintained at $\pi/2$ (or $3\pi/2$)
to a high precision, given the sharp dependence of the degree of squeezing $S$
on $\theta$. When $S>0$ dB, the output is noisier with the SQ on than off, and
the SSR is therefore detrimental to the axion search. Furthermore $S$ also
depends on the SQ gain $G_{s}$, i.e. on the SQ pump power, and there is an
optimal $G_{s}$ for which $S$ reaches its minimum value $S_{\mathrm{min}}$. In
fact, as the gain increases, $S$ first improves as the squeezed state
elongates in phase space, but then saturates due to distortion effects Malnou
_et al._ (2018); Boutin  _et al._ (2017). For this type of JPA, we
experimentally find $S=S_{\mathrm{min}}$ at $G_{s}\approx 13$ dB.

In order to achieve $S=S_{\mathrm{min}}$ throughout a data run, we implemented
a feedback loop that uses the output variance $\sigma_{\mathrm{on}}^{2}$ as
its control parameter on the voltage-controlled variable attenuator and phase
shifter. When initializing a $9$ hr acquisition, a 2-dimensional sweep of the
variable attenuation $A_{s}$ and phase shift $\theta$ is used to estimate the
global minimum of $\sigma_{\mathrm{on}}^{2}$. Then, periodically throughout
the data run, a fast gradient descent-type algorithm corrects for small drifts
of $\sigma_{\mathrm{on}}^{2}$. Note that this approach is robust to possible
phase shifts due to changes in the variable attenuation and vice versa.
Empirically, we are able to automatically maintain $S$ to $S_{\mathrm{min}}\pm
0.1$ dB, over the course of the entire experiment.

We did not need to adjust the AMP pump power, as it remained stable around
$25$ dB. However, when implementing the SSR in HAYSTAC, it will have to be
adjusted, in particular because the frequency of the JPAs will also be stepped
in time. Note that in a practical haloscope run, sizable fluctuations of the
net receiver gain in a SSR-integrated setup on timescales shorter than the raw
spectrum acquisition time can be detrimental to axion detection. This is true
even if all sources of added noise are overwhelmed. For our raw spectra
acquisition times of $0.32$ s, the receiver gain fluctuations are negligibly
small.

###  D.4 Transmission loss between SQ and AMP

The presence of four circulators and several microwave connectors, including
adapters from SMP (used on the JPAs chips) to SMA standards, inevitably
reduces the transmission efficiency $\eta$ between SQ and AMP. We minimize
$\eta$ through the use of a triple junction circulator, and superconducting
SMA cables between the two JPAs and the circulators. However, $\eta$ still
provides the primary limitation on the efficacy of the SSR in accelerating
axionic dark matter searches.

We estimate $\eta$ by measuring the output power spectral density
$P_{\mathrm{out}}$ of vacuum fluctuations in a single quadrature amplified by
the AMP with the SQ off, then repeating this measurement with the role of the
JPAs interchanged Malnou  _et al._ (2018). Assuming that the gain of the
amplification chain from the HEMT to the ADC stays constant, we can then
deduce the microwave loss between the two JPAs as the difference in the net
gain between the two cases.

More precisely, having only vacuum noise at the chain’s input, we have, in one
quadrature:

$P_{\mathrm{out}}=\frac{1}{4}\hbar\omega
BG_{\mathrm{JPA}}^{c}G_{\mathrm{JPA}},$ (40)

where $\omega$ is the rf angular frequency at the center of integration
bandwidth $B$ for the power spectral density, $G_{\mathrm{JPA}}^{c}$ is the
chain’s gain after the JPA, and $G_{\mathrm{JPA}}$ is the SQ or AMP gain.
Thus, there is a linear relation between $P_{\mathrm{out}}$ and
$G_{\mathrm{JPA}}$, whose slope gives $G_{\mathrm{JPA}}^{c}$. Figure 8 shows
$4P_{\mathrm{out}}/(\hbar\omega B)$, when varying either the SQ or the AMP
gain. We extract $\eta=G_{\mathrm{SQ}}^{c}/G_{\mathrm{AMP}}^{c}=0.69\pm 0.01$.

Figure 8: Measurement of the single-quadrature output power spectral density
$P_{\mathrm{out}}$, normalized, as a function of the JPA gain
$G_{\mathrm{JPA}}$. Two configurations are represented: when operating only
the SQ (red circles) and when operating only the AMP (blue circles). A linear
fit (solid lines) to these linear responses allows us to extract $\eta$.

The efficiency $\eta$ observed in our mock-haloscope setup should not be
significantly degraded by the large magnetic fields required in a real
haloscope experiment such as HAYSTAC Zhong  _et al._ (2018) or ADMX Du  _et
al._ (2018). This is because the primary constraint imposed by the large field
is an increased spatial separation between the axion cavity and the
circulators and amplifiers that process the signal. This $\sim 1$-m distance
is bridged with superconducting coaxial cables, whose attenuation Kurpiers
_et al._ (2017) is far subdominant to other sources of transmission loss, such
as the microwave circulators in our setup, and is robust to large magnetic
fields Brubaker  _et al._ (2017a).

From $\eta$ we can estimate the squeezing $S$ that we should obtain far off
cavity resonance ($\sim 10$ MHz detuned). Considering the ideal case where the
reduction of the squeezed state variance is equal to the SQ single-quadrature
power gain $G_{s}$, we obtain:

$S=\frac{\eta}{G_{s}}+1-\eta,$ (41)

which, for $G_{s}=13$ dB leads to $S=-4.6$ dB, in quantitative agreement with
what we measure in practice.

##  Appendix E Complementary measurements of the microwave tone’s improvement
in resolution

We presented in Fig. 4b the visibility $\alpha(\omega)$ of a microwave tone as
a function of $\omega$, the detuning between the tone’s frequency and the
cavity bare resonance. We compared $\alpha(\omega)$ between the two relevant
cases: squeezing with a strongly overcoupled cavity versus not squeezing with
a near-critically coupled cavity. Figure 9 presents these, along with
measurements of $\alpha(\omega)$ for the two complementary cases: strongly
overcoupling without squeezing, and near-critically coupling with squeezing.
In all four cases, there is excellent agreement with predictions from the
theory, developed in Appendix A.2 and shown in solid lines. The theory curves
are not fits; we used parameter values for $\kappa_{l}$, $\eta$, and $G_{s}$
measured independently. Furthermore, the values are the same across all four
cases.

Squeezing is beneficial, even when not overcoupling the cavity’s measurement
port, as it enhances $\alpha$ off cavity resonance while having no effect on
resonance. Experimentally, we obtain the complementary estimate
$E^{c1}_{e}=1.72\pm 0.03$ when squeezing and near-critically coupling,
compared to the same situation without squeezing, in agreement with the
theoretical prediction, $E^{c1}_{t}=1.77\pm 0.03$. Conversely, the scan rate
is worse when strongly overcoupling without squeezing, compared to near-
critically coupling with squeezing: from the data, we obtain
$E^{c2}_{e}=0.57\pm 0.01$, not far from the theoretical value of
$E^{c2}_{t}=0.52$.

Figure 9: Signal visibility $\alpha(\omega)$, as a function of the signal’s
detuning from cavity resonance. It is normalized by $\alpha_{\mathrm{max}}$,
obtained at zero detuning for $\kappa_{m}=\kappa_{l}$. Four cases have been
measured, with the corresponding theoretical predictions represented as solid
lines: overcoupling ($\kappa_{m}=10\kappa_{l}$) with squeezing (red circles),
near-critically coupling ($\kappa_{m}=1.5\kappa_{l}$) without squeezing (blue
circles), overcoupling without squeezing (purple triangles), and near-
critically coupling with squeezing (green triangles). The theory lines have
been calculated for $\kappa_{l}=100$ kHz, $\eta=0.69$, and $G_{s}=13$ dB.

##  Appendix F Processing of the SSR spectra

In order to detect the small cavity displacement of Sec. IV, we acquire 401
“raw” spectra, which we process and combine into one “grand” spectrum. The
faxion tone that we inject is sufficiently small relative to the size of the
vacuum fluctuations that in any one raw spectrum it typically does not stand
out. However, after processing the spectra, the faxion emerges often well
above the level of vacuum noise, as in Fig. 5b. This appendix provides an
overview of the steps in the processing of the spectra. Our processing
procedure is based closely on the work of Ref. Brubaker  _et al._ (2017b). As
such, we borrow the terminology used for the intermediate processing stages
set forth in that work, and we also omit many details and rationales which are
there covered extensively.

Using the setup of Fig. 1, fluctuations emerging from the cavity, possibly in
the presence of squeezed vacuum noise, are directed into a chain of amplifiers
led by the AMP. The fluctuations are mixed down to dc, further amplified, low-
pass filtered up to $1.9\ \mathrm{MHz}$, and sampled with an Alazar ATS9462
digitizer at $6\ \mathrm{MS/s}$. Each raw spectrum, distinguished as the least
processed data saved to a hard disk, itself actually comprises the frequency-
averaged power spectral densities of $32$ “subspectra,” each acquired over
$10\ \mathrm{ms}$ and fast Fourier transformed to provide a spectral
resolution of $\Delta_{b}=100\ \mathrm{Hz}$. Since $401$ raw spectra are
acquired for each of the $200$ squeezed and $200$ unsqueezed data runs, the
live acquisition time of all the spectra totals just over $14$ hours, not
counting the dead time. In practice it took roughly $18$ hours.

Once the raw spectra are in hand, they are symmetrized. Since the output of a
flux-pumped JPA at a given detuning from the center of its amplification band
is, in the high-gain limit, identical to the output at the opposite detuning
from band center, the best one can do is to infer the spectral density
measured in the homodyne configuration as coming equally from both sides of
the pump.

The symmetrized spectra are then averaged according to their real frequencies
(i.e. not accounting for the fictitious stepping of the cavity used to
simulate a haloscope search) in order to detect excess power in either the
pertinent rf or IF band 111In a real haloscope search, since the local
oscillator being used for homodyne measurement would be stepped along with the
cavity, non-axion-induced power excesses in the rf are trickier to reject than
their IF counterparts Brubaker  _et al._ (2017b).. This mean spectrum is then
high-pass filtered by the profile of a Savitsky-Golay (SG) filter Savitzky and
Golay (1964) applied to the average spectrum. A SG filter is simply a
computationally quick means of applying the $d^{\mathrm{th}}$-degree
polynomial generalization of a moving average within a window of $2W+1$ bins.
If the width $W$ is much larger than the size of interesting spectral
features, those features will be minimally attenuated when the spectrum is
divided by the filter response, whereas features extending over wide
bandwidths will be effectively removed. Following Ref. Brubaker  _et al._
(2017b), we use $d=10$, $W=500$ for this first filtering.

The resulting real-frequency-averaged spectrum is used to detect and remove
excesses of power which do not act like our faxion. Bins that exhibit power
fluctuations more than $4$ standard deviations above the mean at their real
frequencies are discarded along with their close neighbors as contaminated for
all spectra. In practice, we keep over $99\%$ of bins, only eliminating
extreme outliers. Note that since in the real frequency space, the faxion tone
is being stepped, its $400$ tunings do not combine over any one narrow block
of bins, and this bin rejection procedure will therefore not eliminate a
faxion signal. Once contaminated bins are removed, the SG filter is
recalculated, as it will be slightly different without the presence of outlier
bins, and reapplied to each spectrum individually.

This baseline removal leaves a set of nearly flat “normalized” spectra, each
centered near $1$. Residual structure left from fluctuations in the overall
receiver gain or its profile is then removed by dividing out the SG profile
($d=4$, $W=500$) of each individual spectrum. The mean value of $1$ is then
subtracted from each spectrum to form the “processed” spectra, several of
which are shown in Fig. 5a.

Next we must rescale the spectra to account for the varying sensitivity to a
faxion tone as a function of its detuning from cavity resonance in any given
spectrum. To form these “rescaled” spectra, the processed spectra are divided
by the relative visibility profile of the squeezed state receiver: Eq. (26)
with an additional term accounting for the small contribution of the HEMT’s
added noise referred to the input of the AMP, as the weak frequency dependence
of AMP gain over the bandwidth of the cavity in the presence of spectrally
flat HEMT added noise contributes a correspondingly weak frequency dependence
to the noise. The rescaled spectra have the important property that a given
power excess from the weakly coupled port of the cavity produces a constant
expectation value of rescaled spectrum power excess, regardless of detuning
from the center of the cavity. As a consequence, the variance of the power
distribution for each bin grows with detuning from cavity resonance.

Aligning bins along the fictitious frequencies for which the stepped faxion
tone would appear stationary, it is then possible to construct the maximum
likelihood estimate of the power excess in each bin. Doing so yields the
“combined” spectrum, a single spectrum whose bins contain as few as one (at
the extreme edges) and as many as hundreds of contributions from individual
rescaled spectra. The combined spectra bins each have a resolution of
$\Delta_{b}$, far below the linewidth $\Delta_{a}$ of the faxion. The power
within non-overlapping sets of $K^{r}=10\ll\Delta_{a}/\Delta_{b}$ bins are
thus averaged, to form the “rebinned” spectrum, which has $K^{r}\Delta_{b}=1\
\mathrm{kHz}$ spectral resolution.

Finally, overlapping sets of $K^{g}=41\gg\Delta_{a}/K^{r}\Delta_{b}$ rebinned
spectrum bins are combined, accounting for the independently experimentally
determined lineshape of the faxion, in order to produce the maximum likelihood
estimate of the faxion-shaped power centered on each bin. The resulting
spectrum of estimated powers is the grand spectrum, shown renormalized to have
mean power excess 0 and standard deviation 1 in Fig. 5b. In the grand
spectrum, the faxion typically stands out well above the surrounding noise.

The acquisition and processing of sets of 401 raw spectra are repeated 200
times apiece for the optimal squeezed and unsqueezed cases. The two data sets
require slightly different processing, chiefly because the visibility profile
of the unsqueezed case does not include the frequency-dependent contribution
from the squeezing. This results not from finite bandwidth effects of the SQ,
but from the fact that squeezed noise is preferentially absorbed near cavity
resonance by the cavity’s loss port, wherefrom it is replaced with unsqueezed
vacuum.

Each of the 400 total acquisitions provides one measured faxion power. The
powers are histogrammed for the optimal squeezed and unsqueezed cases, along
with the far larger number of grand spectrum powers of bins not containing a
faxion, in Fig. 5c. While the absolute mean values of the squeezed and
unsqueezed faxion distributions carry little meaning, as they scale with our
somewhat arbitrary choice of faxion power, their squared ratio, which would be
preserved for a faxion of any power, gives the scan rate enhancement obtained
from squeezing, as discussed in Sec. IV.

## References

  * Peccei and Quinn (1977a) R. D. Peccei and H. R. Quinn, “$\mathrm{CP}$ conservation in the presence of pseudoparticles,” Phys. Rev. Lett. 38, 1440–1443 (1977a).
  * Peccei and Quinn (1977b) R. D. Peccei and Helen R. Quinn, “Constraints imposed by $\mathrm{CP}$ conservation in the presence of pseudoparticles,” Phys. Rev. D 16, 1791–1797 (1977b).
  * Preskill  _et al._ (1983) John Preskill, Mark B. Wise, and Frank Wilczek, “Cosmology of the invisible axion,” Phys. Lett. B 120, 127–132 (1983).
  * Abbott and Sikivie (1983) L.F. Abbott and P. Sikivie, “A cosmological bound on the invisible axion,” Phys. Lett. B 120, 133–136 (1983).
  * Dine and Fischler (1983) Michael Dine and Willy Fischler, “The not-so-harmless axion,” Phys. Lett. B 120, 137–141 (1983).
  * Caves (1982) Carlton M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
  * Bradley  _et al._ (2003) Richard Bradley, John Clarke, Darin Kinion, Leslie J Rosenberg, Karl van Bibber, Seishi Matsuki, Michael Mück, and Pierre Sikivie, “Microwave cavity searches for dark-matter axions,” Rev. Mod. Phys. 75, 777–817 (2003).
  * Zhitnitsky (1980) A. R. Zhitnitsky, “On possible suppression of the axion hadron interactions,” Sov. J. Nucl. Phys. 31, 260 (1980).
  * Dine  _et al._ (1981) Michael Dine, Willy Fischler, and Mark Srednicki, “A simple solution to the strong CP problem with a harmless axion,” Phys. Lett. B 104, 199 (1981).
  * Du  _et al._ (2018) N. Du, N. Force, R. Khatiwada, E. Lentz, R. Ottens, L. J Rosenberg, G. Rybka, G. Carosi, N. Woollett, D. Bowring, A. S. Chou, A. Sonnenschein, W. Wester, C. Boutan, N. S. Oblath, R. Bradley, E. J. Daw, A. V. Dixit, J. Clarke, S. R. O’Kelley, N. Crisosto, J. R. Gleason, S. Jois, P. Sikivie, I. Stern, N. S. Sullivan, D. B Tanner, and G. C. Hilton (ADMX Collaboration), “Search for invisible axion dark matter with the axion dark matter experiment,” Phys. Rev. Lett. 120, 151301 (2018).
  * Zhong  _et al._ (2018) L. Zhong, S. Al Kenany, K. M. Backes, B. M. Brubaker, S. B. Cahn, G. Carosi, Y. V. Gurevich, W. F. Kindel, S. K. Lamoreaux, K. W. Lehnert, S. M. Lewis, M. Malnou, R. H. Maruyama, D. A. Palken, N. M. Rapidis, J. R. Root, M. Simanovskaia, T. M. Shokair, D. H. Speller, I. Urdinaran, and K. A. van Bibber, “Results from phase 1 of the HAYSTAC microwave cavity axion experiment,” Phys. Rev. D 97, 092001 (2018).
  * Caves (1981) Carlton M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
  * Collaboration (2011) The LIGO Scientific Collaboration, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962 (2011).
  * Aasi  _et al._ (2013) J. Aasi, J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, _et al._ , “Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light,” Nat. Photon. 7, 613 (2013).
  * Kimble  _et al._ (2001) H. J. Kimble, Yuri Levin, Andrey B. Matsko, Kip S. Thorne, and Sergey P. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics,” Phys. Rev. D 65, 022002 (2001).
  * Murch  _et al._ (2013) K. W. Murch, S. J. Weber, K. M. Beck, E. Ginossar, and I. Siddiqi, “Reduction of the radiative decay of atomic coherence in squeezed vacuum,” Nature 499, 62–65 (2013).
  * Clark  _et al._ (2016) Jeremy B Clark, Florent Lecocq, Raymond W Simmonds, José Aumentado, and John D Teufel, “Observation of strong radiation pressure forces from squeezed light on a mechanical oscillator,” Nat. Phys. 12, 683–687 (2016).
  * Bienfait  _et al._ (2017) A. Bienfait, P. Campagne-Ibarcq, A. H. Kiilerich, X. Zhou, S. Probst, J. J. Pla, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, K. Moelmer, and P. Bertet, “Magnetic resonance with squeezed microwaves,” Phys. Rev. X 7, 041011 (2017).
  * Sikivie (1983) P. Sikivie, “Experimental tests of the ”invisible” axion,” Phys. Rev. Lett. 51, 1415–1417 (1983).
  * Yamamoto  _et al._ (2008) T. Yamamoto, K. Inomata, M. Watanabe, K. Matsuba, T. Miyazaki, W. D. Oliver, Y. Nakamura, and J. S. Tsai, “Flux-driven Josephson parametric amplifier,” Appl. Phys. Lett. 93, 042510 (2008).
  * Castellanos-Beltran  _et al._ (2008) M. A. Castellanos-Beltran, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, “Amplification and squeezing of quantum noise with a tunable Josephson metamaterial,” Nat. Phys. 4, 929–931 (2008).
  * Zhou  _et al._ (2014) X. Zhou, V. Schmitt, P. Bertet, D. Vion, W. Wustmann, V. Shumeiko, and D. Esteve, “High-gain weakly nonlinear flux-modulated Josephson parametric amplifier using a SQUID array,” Phys. Rev. B 89, 214517 (2014).
  * Brubaker  _et al._ (2017a) B. M. Brubaker, L. Zhong, Y. V. Gurevich, S. B. Cahn, S. K. Lamoreaux, M. Simanovskaia, J. R. Root, S. M. Lewis, S. Al Kenany, K. M. Backes, I. Urdinaran, N. M. Rapidis, T. M. Shokair, K. A. van Bibber, D. A. Palken, M. Malnou, W. F. Kindel, M. A. Anil, K. W. Lehnert, and G. Carosi, “First results from a microwave cavity axion search at $24\text{ }\text{ }\mu\mathrm{eV}$,” Phys. Rev. Lett. 118, 061302 (2017a).
  * Mallet  _et al._ (2011) F. Mallet, M. A. Castellanos-Beltran, H. S. Ku, S. Glancy, E. Knill, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, “Quantum state tomography of an itinerant squeezed microwave field,” Phys. Rev. Lett. 106, 220502 (2011).
  * Menzel  _et al._ (2012) E. P. Menzel, R. Di Candia, F. Deppe, P. Eder, L. Zhong, M. Ihmig, M. Haeberlein, A. Baust, E. Hoffmann, D. Ballester, K. Inomata, T. Yamamoto, Y. Nakamura, E. Solano, A. Marx, and R. Gross, “Path entanglement of continuous-variable quantum microwaves,” Phys. Rev. Lett. 109, 250502 (2012).
  * Boutin  _et al._ (2017) Samuel Boutin, David M. Toyli, Aditya V. Venkatramani, Andrew W. Eddins, Irfan Siddiqi, and Alexandre Blais, “Effect of higher-order nonlinearities on amplification and squeezing in Josephson parametric amplifiers,” Phys. Rev. Appl. 8, 054030 (2017).
  * Malnou  _et al._ (2018) M. Malnou, D. A. Palken, Leila R. Vale, Gene C. Hilton, and K. W. Lehnert, “Optimal operation of a Josephson parametric amplifier for vacuum squeezing,” Phys. Rev. Appl. 9, 044023 (2018).
  * Pogorzalek  _et al._ (2017) Stefan Pogorzalek, Kirill G. Fedorov, Ling Zhong, Jan Goetz, Friedrich Wulschner, Michael Fischer, Peter Eder, Edwar Xie, Kunihiro Inomata, Tsuyoshi Yamamoto, Yasunobu Nakamura, Achim Marx, Frank Deppe, and Rudolf Gross, “Hysteretic flux response and nondegenerate gain of flux-driven Josephson parametric amplifiers,” Phys. Rev. Appl. 8, 024012 (2017).
  * Irastorza and Redondo (2018) Igor G. Irastorza and Javier Redondo, “New experimental approaches in the search for axion-like particles,” Prog. Part. Nucl. Phys. 102, 89–159 (2018).
  * Chaudhuri  _et al._ (2018) S. Chaudhuri, K. Irwin, P. W. Graham, and J. Mardon, “Fundamental limits of electromagnetic axion and hidden-photon dark matter searches: part I – the quantum limit,” arXiv:1803.01627 (2018).
  * Kenany  _et al._ (2017) S. Al Kenany, M.A. Anil, K.M. Backes, B.M. Brubaker, S.B. Cahn, G. Carosi, Y.V. Gurevich, W.F. Kindel, S.K. Lamoreaux, K.W. Lehnert, S.M. Lewis, M. Malnou, D.A. Palken, N.M. Rapidis, J.R. Root, M. Simanovskaia, T.M. Shokair, I. Urdinaran, K.A. van Bibber, and L. Zhong, “Design and operational experience of a microwave cavity axion detector for the 20-100 $\mu$ev range,” Nucl. Instr. Meth. Phys. Res. A 854, 11–24 (2017).
  * Brubaker  _et al._ (2017b) B. M. Brubaker, L. Zhong, S. K. Lamoreaux, K. W. Lehnert, and K. A. van Bibber, “HAYSTAC axion search analysis procedure,” Phys. Rev. D 96, 123008 (2017b).
  * Chung (2016) Woohyun Chung, “CULTASK, The Coldest Axion Experiment at CAPP/IBS in Korea,” PoS CORFU2015, 047 (2016).
  * McAllister  _et al._ (2017) Ben T. McAllister, Graeme Flower, Eugene N. Ivanov, Maxim Goryachev, Jeremy Bourhill, and Michael E. Tobar, “The ORGAN experiment: an axion haloscope above 15 GHz,” Phys. Dark Univ. 18, 67–72 (2017).
  * Ouellet  _et al._ (2019) Jonathan L. Ouellet, Chiara P. Salemi, Joshua W. Foster, Reyco Henning, Zachary Bogorad, Janet M. Conrad, Joseph A. Formaggio, Yonatan Kahn, Joe Minervini, Alexey Radovinsky, Nicholas L. Rodd, Benjamin R. Safdi, Jesse Thaler, Daniel Winklehner, and Lindley Winslow, “First results from ABRACADABRA-10 cm: A search for Sub-$\mu\mathrm{eV}$ axion dark matter,” Phys. Rev. Lett. 122, 121802 (2019).
  * Silva-Feaver  _et al._ (2017) M. Silva-Feaver, S. Chaudhuri, H. Cho, C. Dawson, P. Graham, K. Irwin, S. Kuenstner, D. Li, J. Mardon, H. Moseley, R. Mule, A. Phipps, S. Rajendran, Z. Steffen, and B. Young, “Design overview of DM Radio pathfinder experiment,” IEEE Trans. Appl. Supercond. 27, 1–4 (2017).
  * Majorovits  _et al._ (2017) B. Majorovits _et al._ (MADMAX interest group), “MADMAX: A new road to axion dark matter detection,” arXiv:1712.01062 (2017).
  * Sliwa  _et al._ (2015) K. M. Sliwa, M. Hatridge, A. Narla, S. Shankar, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, “Reconfigurable Josephson circulator/directional amplifier,” Phys. Rev. X 5, 041020 (2015).
  * Macklin  _et al._ (2015) C. Macklin, K. O’Brien, D. Hover, M. E. Schwartz, V. Bolkhovsky, X. Zhang, W. D. Oliver, and I. Siddiqi, “A near–quantum-limited Josephson traveling-wave parametric amplifier,” Science 350, 307–310 (2015).
  * Chapman  _et al._ (2017) Benjamin J. Chapman, Eric I. Rosenthal, Joseph Kerckhoff, Bradley A. Moores, Leila R. Vale, J. A. B. Mates, Gene C. Hilton, Kevin Lalumière, Alexandre Blais, and K. W. Lehnert, “Widely tunable on-chip microwave circulator for superconducting quantum circuits,” Phys. Rev. X 7, 041043 (2017).
  * Clerk  _et al._ (2010) A. A. Clerk, M. H. Devoret, S. M. Girvin, Florian Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).
  * Dicke (1946) R. H. Dicke, “The measurement of thermal radiation at microwave frequencies,” Rev. Sci. Instrum. 17, 268–275 (1946).
  * Kim (1979) Jihn E. Kim, “Weak-interaction singlet and strong $\mathrm{CP}$ invariance,” Phys. Rev. Lett. 43, 103–107 (1979).
  * Shifman  _et al._ (1980) M.A. Shifman, A.I. Vainshtein, and V.I. Zakharov, “Can confinement ensure natural CP invariance of strong interactions?” Nucl. Phys. B 166, 493–506 (1980).
  * Kurpiers  _et al._ (2017) Philipp Kurpiers, Theodore Walter, Paul Magnard, Yves Salathe, and Andreas Wallraff, “Characterizing the attenuation of coaxial and rectangular microwave-frequency waveguides at cryogenic temperatures,” EPJ Quantum Technol. 4, 8 (2017).
  * Note (1) In a real haloscope search, since the local oscillator being used for homodyne measurement would be stepped along with the cavity, non-axion-induced power excesses in the rf are trickier to reject than their IF counterparts Brubaker  _et al._ (2017b).
  * Savitzky and Golay (1964) Abraham. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures.” Anal. Chem. 36, 1627–1639 (1964).