# SpinSpotter: An Automated Algorithm for Identifying Stellar Rotation Periods
With Autocorrelation Analysis

Rae J. Holcomb Department of Physics & Astronomy, University of California
Irvine, Irvine, CA 92697, USA  Paul Robertson Department of Physics &
Astronomy, University of California Irvine, Irvine, CA 92697, USA  Patrick
Hartigan Physics and Astronomy Department, Rice University, Houston, TX 77005,
USA  Ryan J. Oelkers Charles R. and Judith G. Munnerlyn Astronomical
Laboratory, Texas A&M University, College Station, TX 77832, USA  George P.
and Cynthia W. Mitchell Institute for Fundamental Physics and Astronomy,
Department of Physics and Astronomy, Texas A&M University, College Station, TX
77843, USA  Caleb Robinson  Physics and Astronomy Department, Rice University,
Houston, TX 77005, USA

###### Abstract

SpinSpotter is a robust and automated algorithm designed to extract stellar
rotation periods from large photometric datasets with minimal supervision. Our
approach uses the autocorrelation function (ACF) to identify stellar rotation
periods up to one-third the observational baseline of the data. Our algorithm
also provides a suite of diagnostics that describe the features in the ACF,
which allows the user to fine-tune the tolerance with which to accept a period
detection. We apply it to approximately 130,000 main-sequence stars observed
by the Transiting Exoplanet Survey Satellite (TESS) at 2-minute cadence during
Sectors 1-26, and identify rotation periods for 13,504 stars ranging from 0.4
to 14 days. We demonstrate good agreement between our sample and known values
from the literature and note key differences between our population of
rotators and those previously identified in the Kepler field, most notably a
large population of fast-rotating M dwarfs. Our sample of rotating stars
provides a data set with coverage of nearly the entire sky that can be used as
a basis for future gyrochronological studies, and, when combined with proper
motions and distances from Gaia, to search for regions with high densities of
young stars, thus identifying areas of recent star formation and undiscovered
moving group members. Our algorithm is publicly available for download and use
on GitHub.

Stellar rotation — Stellar activity — Light curves

##  1 Introduction

Understanding the rotation of stars has long been foundational to
characterizing stellar formation, magnetism, and evolution. In the past
decade, the study of stellar rotation has been revolutionized by large-scale
photometric surveys, particularly the Kepler and K2 missions, and provided the
first statistical samples of photometrically measured stellar rotation
periods. In 2020, the Transiting Exoplanet Survey Satellite (_TESS_) completed
its primary mission of photometrically surveying bright stars across more than
90% of the night sky, allowing the first all-sky statistical search for bright
rotating stars among hundreds of thousands of targets. To ingest such a wealth
of data, we present an automated algorithm that uses autocorrelation analysis
to identify the rotation periods of stars with high cadence photometric data.
We also present a catalog of stellar rotation periods with $P_{rot}<14$ days
recovered from the sample of 2-minute cadence _TESS_ targets.

Historically, early rotation measurements were performed through spectroscopic
measurements of rotational broadening or through long term ground-based
photometric surveys (e.g. Baliunas et al., 1996). The basis of the photometric
approach is to detect the brightness variations from spot complexes on the
photosphere. Active regions on a star’s surface typically appear darker than
the surrounding surface; thus, as a star rotates this creates quasi-periodic
modulations in the star’s brightness as the spots move in and out of the
observer’s line of sight.

The advent of wide field space-based photometric surveys, such as the Kepler
mission (Borucki et al., 2010), ushered in a new era of rotation studies.
Large scale, consistent photometric monitoring of thousands of targets allowed
the first statistical-scale rotation surveys based on measuring periodic
brightness modulations caused by spots on the surface of rotating stars (e.g.
Affer et al., 2012; Harrison et al., 2012; Aigrain et al., 2015). In
particular, the work of McQuillan et al. (2013) demonstrated the power of
using the autocorrelation function (ACF) to detect rotation periods, and soon
after McQuillan et al. (2014) published the largest sample of measured stellar
rotation periods ever, reporting 34,030 rotation periods from a sample of
133,030 Kepler targets. They demonstrated the advantages of the ACF approach
over Fourier-based methods for recovering periodic brightness modulations that
may vary in shape and amplitude over time as spots form, evolve, and dissipate
on the photosphere of a star. Since then, other papers have further refined
the ACF method. In 2015, Aigrain et al. (2015) performed a comparative
analysis of several different methods for recovering stellar rotation from
Kepler light curves, including those based on Lomb-Scargle periodograms,
autocorrelation, and wavelet decomposition, and concluded that among the
methods investigated, ACF-based approaches were most conducive to automated
analysis. Thereafter, a variant of the ACF method was developed by Saylor et
al. (2018) that was focused on recovery specifically of fast rotators in the
Kepler sample.

The _TESS_ satellite launched in 2018 with the primary mission of surveying
more than 80% of the sky across two years to search for transiting exoplanets
(Ricker et al., 2015). It observed over 200,000 of the brightest stars at a
high 2-minute cadence, and millions more at a slower 30-minute cadence. While
its primary mission is to discover nearby transiting exoplanets, this survey
also provides ideal data to search for fast-rotating stars across the majority
of the night sky. Work on this has already begun, such as Canto Martins et al.
(2020) investigating the rotational properties of _TESS_ stars hosting
candidate transiting planets, Ramsay et al. (2020) searching specifically for
ultra-fast rotating M dwarfs in the southern _TESS_ field, and Claytor et al.
(2021) applying new deep learning techniques to the _TESS_ sample.

Stellar rotation is of particular interest for studying the evolution of stars
during their early lifetimes and evolution onto the main sequence. Solar-type
stars have long been thought to converge to a tight age-period relationship
once they evolve onto the main sequence (Skumanich, 1972), giving rise to the
field of gyrochronology, which sought to establish well described
relationships between stellar age and rotation. Considerable work has been put
into showing the ages of young stellar associations can be derived from
rotation periods of their members (e.g. Agüeros et al., 2018; Curtis et al.,
2020; Spada & Lanzafame, 2020), though age determination of field stars
through gyrochronology remains elusive. Additionally, the period-age
relationship for young late-type dwarfs remains particularly poorly
constrained, as these stars will maintain their rapid rotation for far longer
than their Solar-like counterparts and then undergo a rapid transition onto
the slow rotator track, though the mechanism driving this remains an open
question (Newton et al., 2016). According to gyrochronological models, fast
rotation is often invoked as a signature of youth. However, analysis of fast
rotation in Kepler field stars with Gaia parallaxes has thrown doubt on this
assumption, suggesting that many fast rotating field stars may be tidally
synchronized binaries (Simonian et al., 2019). Expanding our collection of
known rapid rotators to encompass the whole sky will allow new investigations
into these topics.

In this paper, we present two products. First we describe SpinSpotter, a
Python-based algorithm that searches for evidence of stellar rotation in
densely-sampled photometric time series, such as those provided by the _TESS_
and Kepler missions. Second, we applied SpinSpotter to all main sequence stars
observed with a 2-minute cadence from the _TESS_ primary mission (Sectors
1-26), both to demonstrate the effectiveness of our algorithm and to create a
catalogue of rotating stars with rotation periods of two weeks or less that
may be used as a basis for future studies of gyrochronology, stellar activity,
and more. Section 2 describes the workings of our algorithm, Section 3
explains how we applied it to the _TESS_ sample, Section 4 discusses the
significance of our results, and Section 5 presents our conclusions.

##  2 Algorithm Description

This section presents the methodology of the SpinSpotter code package, which
is available for download and use at https://github.com/rae-
holcomb/SpinSpotter. The algorithm builds on the techniques developed in
McQuillan et al. (2013) and Saylor et al. (2018) and aims to provide a tool
that can be flexibly applied to photometric survey data in order to identify
rotating stars from a large sample in a fast, automated way. As input, it
takes photometric light curves (LCs) with densely-sampled data, such as those
provided by the _TESS_ or Kepler missions. As output, it provides the stellar
rotation period if one is identified, as well as a collection of descriptive
parameters which characterize various features of the light curve and its ACF,
and can be used to tune the sensitivity of the period selection.

Figure 1: Examples of LCs and ACFs for illustrative cases of a rotating star
(A), a rotating star with spots in opposite hemispheres thus creating alias
peaks at half values of the rotation period (B), an eclipsing binary system
(C), and a star with no clear evidence of rotation (D). For the case with
spots in opposite hemispheres (B), note the alternating low-high pattern of
peaks in the ACF. The even numbered peaks, which are taller, correspond to the
true rotation period. Figure 2: An example of the _TESS_ light curve (top) and
ACF (bottom) of a star that displays clear evidence of rotation at 5.8 days.
The periodicity causes peaks in the ACF to occur at integer multiples of the
lag associated with the rotation period. We fitted parabolas around each of
these peaks (red) and used the height of the peak vertex (A, pink), the width
of the peak at the zero crossing (B, green), and the $R^{2}$ statistic for the
parabolic model (red) as parameters to accept or reject the rotation period
associated with the location of the peak vertex ($P_{rot}$, yellow).

SpinSpotter performs the following steps. For each star, steps 1-6 are
performed on each sector of data individually and also on the combined
multisector light curve.

  1. 1.

Normalize each sector of data and combine into a single multisector light
curve.

  2. 2.

Calculate the ACF.

  3. 3.

Select the highest peak in the fast Fourier transform (FFT) of the ACF to
identify an initial estimate for rotation period.

  4. 4.

Fit parabolas to the peaks in regions of the ACF corresponding to integer
multiples of the initial estimate of the rotation period.

  5. 5.

From the fit parameters, calculate the descriptive parameters $A_{k}$,
$B_{k}$, $R^{2}_{k}$, and $P_{k}$ for each $k$th peak in the ACF. At this
stage, stars may be flagged as showing a factor of 2 degeneracy in the
rotation period identified.

  6. 6.

Accept or reject the rotation period found for the candidate based on whether
the descriptive parameters pass the selection criteria, and how many sectors
the rotation period was identified in.

_1\. Normalize and stitch the LC._ We begin by normalizing and preparing the
data for analysis. We divide the light curve by its median value and subtract
1 so that the data are centered around zero. If the time series is observed
across multiple segments of time, such as _TESS_ sectors or Kepler quarters,
this normalization is performed separately on each segment before combination.
Next, we stitch the segments together and bin the time series to the desired
cadence. After stitching, we subtract the mean of the entire light curve to
negate any offsets introduced by the stitching or binning process. Finally, we
fill large gaps, such as those introduced by _TESS_ data downlinks and sectors
when target is unobserved, with zeros. The process of filling these gaps with
zero may affect the ACF for lags longer than the baseline of a single segment
of data; however, we only consider lags on timescales shorter than a single
segment and thus the effect is negligible.

_2\. Calculate the ACF._ We calculate the ACF according to its standard form

$r_{k}=\frac{\sum_{i=1}^{N-k}(x_{i}-\bar{x})(x_{i+k}-\bar{x})}{\sum_{i=1}^{N-k}(x_{i}-\bar{x})^{2}}$
(1)

where $r_{k}$ is the autocorrelation coefficient at lag $k$, for each point in
the time series $x_{i}$ ($i=1,…,N$). Each lag $k$ corresponds to
$\tau_{k}=k\Delta t$, where $\Delta t$ is the cadence (Shumway et al., 2000).

_3\. Take the FFT of the ACF to identify the dominant periodicity in the ACF._
Stellar rotation appears in the ACF as multiple peaks occurring at lags equal
to integer multiples of the rotation period, as pictured in Figures 1 and 2.
The number of peaks in the ACF corresponds to the number of cycles of the
rotation period captured in a single segment of the data. This means that the
ACF of a rotating star shows strong periodic behavior at the same period as
the stellar rotation. Thus, to calculate an initial estimate of the rotation
period, we take a fast Fourier transform (FFT) of the ACF and use the location
of highest FFT peak to guide our period search. This approach works better
than simply taking the FFT of the light curve since the shape of the periodic
signal tends to be more consistent across multiple cycles than in the original
light curve. However, this value by itself cannot be used as an accurate
measure of period, since aliases of the rotation signal at late lags in the
ACF may drift from integer multiples of the period and thus influence the
location of the peak in the FFT, as described in Saylor et al. (2018). Thus,
additional steps must be taken to identify the final value of the rotation
period.

_4\. Fit parabolas to the peaks in the ACF._ We refine the period estimate
further by searching for local maxima in the ACF near the value identified
from the FFT. We separate the ACF into sections centered around integer
multiples of the initial period estimate and fit parabolas to them in order to
identify the local maxima in each section, as show in Figure 2. Parabolas are
chosen to provide a simple model for the shape of a smooth peak, whose
coefficients can be used to describe the characteristic shape of each peak. We
use non-linear least square regression to perform these fits in windows that
have width equal to half the initial period estimation, centered on integer
multiples of the candidate rotation period. For the $k$th peak, this takes the
form

$y_{k}=a_{k}x^{2}+b_{k}x+c_{k}$ (2)

where $a_{k}$, $b_{k}$, and $c_{k}$ are fitted parameters.

_5\. Calculate the descriptive parameters $P_{rot,k}$, $A_{k}$, $B_{k}$,
$R_{k}^{2}$._ From the fitted parameters of the parabolas, we derive several
useful diagnostics that describe the shape and location of the parabola fitted
to the peak.

  * $P_{rot,k}=-\frac{b_{k}}{2a_{k}}$ (3)

  * $A_{k}=c_{k}-\frac{b_{k}^{2}}{2a_{k}}$ (4)

  * $B_{k}=-\frac{\sqrt{b_{k}^{2}-4a_{k}c_{k}}}{a_{k}}*\frac{1}{P_{rot}}$ (5)

  * $R_{k}^{2}=\textrm{coefficient of determination for the parabola fit}$ (6)

where $P_{rot,k}$ is the location of the parabola peak along the x-axis,
$A_{k}$ is the height of the parabola vertex above the x-axis, and $B_{k}$ is
the width of the parabola at its zero crossing. These expressions correspond
to geometric features of the fitted parabola. The coefficient of determination
$R_{k}^{2}$ is used to assess the goodness of fit of the parabola to the ACF
peaks, according to

$R_{k}^{2}=1-\frac{\sum_{i=1}(y_{i}-\hat{y}_{i})^{2}}{\sum_{i=1}(y_{i}-\bar{y})^{2}}$
(7)

where $y_{i}$ is the value of the ACF at position $i$, $\hat{y}_{i}$ is the
value of the fitted parabola at position $i$, and $\bar{y}$ is the mean of the
ACF in the relevant window. The values of the $P_{rot,k}$, $A_{k}$, $B_{k}$,
$R_{k}^{2}$ are then averaged across each peak in the ACF to produce
$P_{rot}$, $A_{avg}$, $B_{avg}$, $R_{avg}^{2}$.

In cases where a star is strongly spotted on opposite hemispheres, the light
curve will show strong dimming twice per cycle, thus resulting in intermediary
peaks in the ACF near the half-period values and creating an alternating high-
low pattern in the peaks. Figure 1 shows an example of such a star. To account
for these scenarios, we check the height $A_{k}$ of the first three peaks
appearing at the shortest lag times in the ACF. If the second peak is higher
than its neighbors by more than 10%, then we take this as an indication that
the star is displaying spots in opposite hemispheres, and thus calculate the
rotation period as the average of the vertex location of the even numbered
peaks only. The pipeline flags stars where this was applied, and we caution
that stars flagged in this way should be checked individually for confirmation
that the correct period was selected.

We calculate the period uncertainty with one of two methods. In cases where at
least three peaks are identified in the ACF, we take the period uncertainty to
be the standard deviation of $P_{rot,k}$, representing the vertex location of
each peak in the ACF. When fewer than three peaks are identified, we take the
uncertainty to be the average of the half-width at half maximum of the the
peaks in the ACF. It should be noted that the latter approach systematically
returns larger uncertainties; however, this is appropriate since the rotation
period of a star cannot be well known if multiple cycles of the rotation
period are not captured in the the light curve.

_6\. Apply selection criteria to accept or reject the rotation period._ At
this stage, the pipeline has identified a rotation period, period uncertainty,
and three additional parameters ($A_{avg}$, $B_{avg}$, and $R_{avg}^{2}$,
describing the peak height, width, and goodness of fit respectively) that
describe the ACF peak shape for each star. From these, the user may determine
cut off criteria to decide on whether to accept or reject the rotation period
found for each star. The best thresholds to use for this may vary depending on
the science needs and the input data. In 3.3, we describe how we applied this
algorithm to the _TESS_ 2-minute cadence data and selected thresholds for our
acceptance criteria to identify an all-sky sample of rotating stars with
periods of two weeks or less.

##  3 Identifying Rotators in the _TESS_ Sample

###  3.1 Data Products

We used data from Sectors 1-26 of the _TESS_ mission, which is publicly
available from the the Mikulski Archive for Space Telescopes (MAST). Each
sector has an observational baseline of $\sim 27$ days, during which stars are
observed at a 2-minute cadence, with a 16 hour gap halfway through each sector
due to data downlinks. We used the the PDCSAP_FLUX light curves provided by
the standard _TESS_ pipeline, which corrects for most instrumental and
systematic effects (Jenkins et al., 2016).

###  3.2 Sample Selection

In order to show photometric modulation from photospheric spot complexes, a
star must have a convective outer envelope; that is, it must be on the main
sequence. To ensure that our targets satisfied this requirement, we applied
$T_{\mathit{eff}}-\log g$ cuts recommended by Ciardi et al. (2011), where a
star is considered to be a dwarf if its $T_{\mathit{eff}}$ and $\log g$ fall
within:

$\log(\textit{g})\geq\left\\{\begin{aligned} &3.5&\textrm{if
}T_{\mathit{eff}}\geq 6000\\\ &4.0&\textrm{if }T_{\mathit{eff}}\leq 4250\\\
&5.2-(2.8\times 10^{-4}T_{\mathit{eff}})&\textrm{if
}4250<T_{\mathit{eff}}<6000\end{aligned}\right.$

We used $T_{\mathit{eff}}$ and $\log g$ as provided by the _TESS_ Input
Catalog v7 (Stassun et al., 2018). Any targets with no measured
$T_{\mathit{eff}}$ or $\log g$ values were also excluded. This removed 79,087
stars from our initial sample of 215,955 stars. _TESS_ Objects of Interest
(TOIs), which are stars that have had planetary transit events identified in
their light curves, were included in this analysis, with extra steps taken to
mask out potential transit events. Repeated transit events can create narrow
peaks in the ACF occurring at integer multiples of the transit period, making
it more difficult to isolate the rotation period. Thus, we removed possible
transit events by replacing points that occured during a transit (as reported
in the TOI list maintained in Guerrero et al. (2021)) with 0 after the light
curve is been normalized, as described in Section 2. Since our light curves
were normalized to have a mean value of 0, this means that points occurring
during these putative transit events did not contribute to the $r_{k}$
coefficient when calculating the ACF according to Equation 1. The TOI list
used in this analysis was up to date as of February 4, 2022.

Eclipsing binary systems and pulsating variables can also create periodic
signals not associated with stellar rotation, thereby creating the potential
for false positive detections in our pipeline. At the time of this writing,
there were no published catalogs of eclipsing binaries (EBs) or pulsating
variables (PVs) for the entirety of the _TESS_ field, so additional steps to
remove these objects were applied after the primary analysis based on features
of the light curve itself. We defined a parameter, $X_{var}$, which is
calculated by taking the midpoint between the value at the 95th percentile
point and the value at the 5th percentile point of the normalized light curve.
The value of $X_{var}$ gives a measure of how well the most extreme
variability in the light curve is centered around the mean of the light curve.
Since we first normalized the lightcurves to center around 0, a value of
$X_{var}$ much larger or smaller than 0 indicated that the most extreme points
are highly asymmetric about the mean, which is a feature commonly exhibited in
the photometry of eclipsing binaries. We removed any stars with
$|X_{var}|>0.01$ from our analysis, due to their potential binary nature. Most
PVs were excluded from the sample by the $T_{\mathit{eff}}-\log g$ cuts that
removed evolved stars. To help identify any remaining PVs, we defined a
parameter $Y_{var}$ to be the distance between the 5th and 95th percentile
points of the normalized light curve, to serve as a measurement of the total
variability of the time series. We did not exclude any stars based on the
$Y_{var}$ value, though targets with particularly high $Y_{var}$ values,
especially those with hotter $T_{\mathit{eff}}$, may warrant additional
inspection for evidence of whether their periodic variability arises from
rotation or pulsation. After all cuts were applied, the resulting sample
contained 136,868 stars with observational baselines ranging from a single
sector to 13 sectors if the star was in the continuous viewing zone.

###  3.3 Period Detection of TESS Rotators

After applying all cuts, we ran our targets through the SpinSpotter algorithm
as described in Section 2. We chose to bin the light curves from the 2-minute
cadence provided by _TESS_ to a 30 minute cadence, which offered two
advantages. First, it reduced the contribution of variability on timescales of
less than 30 minutes, resulting in smoother and more easily identifiable peaks
in the ACF on timescales of hours to days, which is the region where we most
expect to find stellar rotation. Second, it matched our data to the cadence of
both the _TESS_ full frame images and the Kepler mission, thus allowing for
easier comparisons between our results and other studies. It should be noted
that this choice of binning imposed a limit on our sensitivity to ultra-fast
rotators, and we did not recover rotation periods of less than 0.4 days in
this study, though we have found it to be possible when using shorter binning
timescales. _TESS_ full-frame image (FFI) targets were not included in this
analysis, as at this time they have been not run through the _TESS_ data
reduction pipeline, though they may be in the future. Once this is done, these
targets would make an excellent sample to analyze with the SpinSpotter
algorithm in future works.

###  3.4 Development of Training Data & Selection Criteria

SpinSpotter provides descriptive statistics about the peak height, width, and
goodness of fit in the ACF, but what values of these statistics indicate
credible identification of the rotation period? In order to define acceptance
criteria for our sample, we visually inspected 3285 LCs and their
corresponding ACFs selected randomly from our sample and assigned them one of
the following categorizations:

  * •

Periodic (379 stars) \- periodicity clearly visible in the LC, with matching
peaks in the ACF.

  * •

Possibly Periodic (222 stars) \- some evidence of periodicity in the LC and
ACF, but less confident than in the Periodic category.

  * •

Eclipsing Binary (32 stars) \- LC indicative of being an eclipsing binary
system.

  * •

Non-Periodic (2652 stars) \- no clear evidence of periodicity.

We randomly assigned 70% of these tagged stars to be used for tuning the
values of our selection criteria, and the remaining 30% to be used to verify
how well they perform, resulting in a tuning set and a verification set of
2300 and 985 stars respectively. Using the former set, we chose the following
selection criteria to accept a period detection as valid:

  * $A_{avg}/B_{avg}>0.25$

  * $.4<B_{avg}<0.6$

  * $R_{avg}^{2}>0.9$

For stars with multiple sectors of data, we also required that a valid period
be detected in a minimum number of sectors, as well as in the combined
multisector light curve. We defined this minimum to be half the total number
of sectors for which the star was observed, rounded up. _TESS_ sectors are
short and often non-consecutive, such that spot patterns that appeared in one
sector of data may no longer be present in later sectors, and the noise level
between different sectors of the same target can also vary dramatically (Dalba
et al., 2020).

Figure 3: Stars from our testing and verification sets, plotted to show our
selection criteria. $A_{avg}$ is the ACF peak height, $B_{avg}$ is the
fractional peak width, and $R^{2}_{avg}$ is the goodness of fit parameter of
the parabolas. Solid black lines show the values of our selection criteria,
dashed lines show suggested values of relaxed criteria that would provide a
higher recovery rate of rotators at the expense of a higher false positive
rate.

In the tuning set of 2300 stars, these criteria identified 203 stars as
rotators, with a false positive rate of 4.93%. We recovered 62.5% of stars
tagged as rotators and 18.5% of those marked as possible rotators. We found
that the majority of our false positives were identified as having periods
longer than 9 days, which matched one third the baseline of a single _TESS_
sector. When the sample was restricted to only accept stars with $P_{rot}<9$
days, our false positive rate fell to 3.3% and we recovered 70.1% of tagged
rotators and 29.8% of tagged possible rotators. Thus, we concluded that our
algorithm has limited ability to recover rotation periods that exceed a third
of the baseline of one _TESS_ sector, the reasons for which we discuss further
in Section 4.

Applying these same conditions to the verification set of 710 stars, we found
that we identified 84 stars as rotators, with a false positive rate of 6.17%.
We recovered 53.1% of stars tagged as rotators and 21.1% of those marked as
possible rotators. Thus, we concluded that the criteria derived from our
tuning set were generalizable to the entirety of our sample of _TESS_
rotators. More potential rotators could be recovered by relaxing these
requirements. For example, lowering the criteria to $A_{avg}/B_{avg}>0.15$,
resulted in the identification of 236 rotating stars in the tuning set and 93
rotating stars in the verification set, thus recovering 71.3% of tagged
rotators and 36.4% of possible rotators, though the false positive rate
increased to 11.8%. Visually, this can be seen in Figure 3, which shows the
distribution of our tagged stars across the descriptive parameters
$A_{avg}/B_{avg}$, $B_{avg}$, and $R_{avg}^{2}$. Our selection criteria,
visible as black lines on the plot, effectively separate rotators from non-
rotators.

##  4 Discussion

Figure 4: _TESS_ rotators identified by our algorithm (blue) alongside those
found by (McQuillan et al., 2014) (pink) from the Kepler sample. Top: A
scatter plot, where each point represents an individual rotating star.
Horizontal lines at 9 days, 13.5 days, and 45 days, represent one third and
one half of the baseline of a _TESS_ sector, and half the baseline of a Kepler
quarter, respectively. Yellow lines show age isochrones from (Mamajek &
Hillenbrand, 2008) at 500 Myr (dotted), 1 Gyr (dash-dotted), and 4 Gyr
(solid). Note the large drop off in survey sensitivity for stars with periods
longer than 9 days; this reflects our limited sensitivity to longer period
rotators due to our short baseline of observations. The increase in rotators
with $10<P_{rot}<14$ days is dominated by stars flagged as showing half period
aliases of the rotation periods. Bottom: Histograms showing the distribution
of stars in the SpinSpotter (bottom left, blue) and McQuillan et al. (2014)
(bottom right, pink) respectively, intended to more clearly show the features
of each distribution. Period bins are equal in log space.

When applied to the whole sample, we found rotation periods for 13,504 stars
out of the 136,868 stars that passed our initial cuts in the _TESS_ 2-minute
cadence sample. Figure 4 shows the period and temperature distribution of our
sample of rotators compared to those found by McQuillan et al. (2014). While
the two samples show broad agreement, particularly for earlier-type MS stars,
there are several salient features of this plot that merit further
examination. These are the absence of rotators recovered at longer periods,
the anomalously large number of stars identified as rotators in our sample
with periods of 10-14 days, and the high number of fast rotating M dwarfs
identified in the _TESS_ sample but absent in previous rotation studies of
Kepler stars.

First, we found few stars with rotation periods longer than 9 days. Only 11%
of stars found to have rotation have periods $>9$ days, which corresponds to
approximately one third of the 27-day baseline of a _TESS_ sector. Two effects
make these longer period targets difficult to detect. First, in order to
identify a rotation period from the ACF, there must be multiple cycles of the
rotation period visible, meaning that the star must have been observed for
multiple cycles of its rotation period. Secondly, the PDCSAP_FLUX from _TESS_
is heavily detrended on long time scales, which effectively removes signals on
time scales comparable to a sector or longer. This means that even for stars
with multiple sectors of data, evidence of rotation may have been removed by
the post-processing pipeline. We surmise that it would be possible to recover
rotation periods for some of these stars with our algorithm from the _TESS_
sample by building a custom pipeline that extracted light curves from _TESS_
FFIs that preserved such long period signals, but this is beyond the scope of
this paper. Additionally, the number of false positive detections among our
training data increases drastically above this cut off. Because of this, we
caution investigators to treat rotation periods that exceed one third of the
baseline of the data with skepticism and recommend additional checks either by
eye or with other methods before using them.

The second feature of interest is the apparent prevalence of stars with
$P_{rot}$ between 11 and 14 days. This is not reflective of the underlying
stellar population. Our algorithm flags stars where the even numbered peaks in
the ACF exceed the height of the odd numbered peaks, since this is often
indicative of a star with spots in opposite hemispheres, leading to additional
peaks at 1/2 aliases of the rotation period. Stars with this configuration
have two peaks in the ACF per rotation period, thus creating more repeated
features compared to the typical case which has one peak per rotation period
in the ACF, thus making it easier for our algorithm to identify long rotators
that have this half period alias. Stars with this configuration make up 74% of
rotators that we identified with periods $>9$ days. However, many are single
sector objects, which makes it difficult to discriminate between the half
period and full period case, since other than ultra-fast rotators, most stars
will only detect a handful of cycles of the rotation. We flagged stars where
the discrepancy in peak height between odd and even peaks is less than 10% and
recommend that these cases be treated on a case by case basis. It is likely
that some of the 11-14 day stars have been incorrectly identified as the half-
period case and are in fact stars with periods of 5-7 days, which may
contribute to the overdensity between 11-14 days in Figure 4.

We also see a slight overdensity of stars with rotation periods near 5 days.
We attribute this to two main effects. For stars near and above Solar
temperatures, we take this overdensity to be representative of the underlying
population, as it tracks well with the increased number of rotators found in
that region in McQuillan et al. (2014) and other Kepler rotation studies.
However, the overdensity persists at cooler temperatures, diverging from the
McQuillan et al. (2014) distribution. Based on individual inspection of a
random selection of these stars, we suspect that this may be due to
uncorrected systematics associated with the PDCSAP_FLUX. Future works will
explore this further, both by applying SpinSpotter to datasets other than the
PDCSAP_FLUX to see if this feature persists when a different lightcurve
reduction pipeline is used, and comparison to the period distribution of
_TESS_ stars as measured by other period-finding methods.

Finally, our algorithm identifies a large population of fast rotating M dwarfs
that is not present in Kepler and K2 rotation surveys such as McQuillan et al.
(2014) and Reinhold & Hekker (2020). We attribute this to differences in the
survey structure of the two missions. The _TESS_ 2-minute cadence sample
prioritizes bright, nearby stars, thus leading to a greater inclusion of late
M dwarfs in the sample. In contrast, the Kepler mission prioritized Sunlike
stars, leading to an exclusion of many M dwarfs from its target list. This
phenomenon has also been documented in other papers that investigate the
_TESS_ population, such as Ramsay et al. (2020).

The $\sim$9 day limit on our detection sensitivity with _TESS_ 2-minute
cadence data has interesting implications for the science cases to which this
sample is relevant, particularly when contrasted with other large-scale
rotation studies. Figure 4 shows the temperature-period distribution of our
sample of _TESS_ rotators overlaid with those identified in the Kepler sample
by (McQuillan et al., 2014). For Sunlike stars, our population looks similar
to the McQuillan et al. (2014) sample, but two key features stand out. First,
note the appearance of our detection limit at $\sim$9 days, above which we
identify very few rotators. Because of this, our sample of rotators will be
most useful in studying very young stars that have not yet spun down to lower
rotation speeds. This makes our sample of high interest to those studying
young stellar objects and open clusters.

###  4.1 Comparison to Previous Rotation Studies

We compared the performance of our algorithm to previous surveys of stellar
rotation. Numerous large scale rotation period studies have been performed on
the Kepler sample; however, the Kepler field covers only a small portion of
the sky compared to _TESS_ , and targeted many stars dimmer than those
prioritized in the _TESS_ 2-minute cadence sample, leading to a limited amount
of crossover between the two populations. To expand coverage of the sky, we
also compared our results to studies of young stellar objects (YSOs) and open
cluster members.

Many of these studies had observational baselines significantly longer than
that of a _TESS_ quarter, and thus were sensitive to longer-period rotators.
We restricted our comparison only to stars with literature values of
$P_{rot}<14$ days, or half of a single _TESS_ sector, which represents the
theoretical limit of the period sensitivity of our data.

The methods of each paper are summarized below.

####  4.1.1 McQuillan et al. (2014)

McQuillan et al. (2014) examined $\sim$150,000 main sequence stars in Kepler
Q3-Q14, excluding known EBs, Kepler Objects of Interest (KOI), and stars with
less than 8 quarters of data. They identified periods by calculating the ACF,
identifying local maxima, and taking the period to be the slope of these local
maxima as a function of peak number. They reported periods for 34,030 stars
ranging from 0.5-60 days. Of these, 29 appear in our sample and have
$P_{rot}<14$ days. This low overlap stems from the overall dimness of stars in
the Kepler field compared to those targeted by high cadence _TESS_
observations, as well as the much longer baseline of Kepler quarters, which
allowed (McQuillan et al., 2014) to identify stars with much longer rotation
periods.

####  4.1.2 Saylor et al. (2018)

Saylor et al. (2018) used K2 light curves to search for fast-rotating
($P_{rot}<4$ days) GKM dwarfs from the SUPERBLINK high proper motion survey
(Lepine, 2018). Their method was the most similar to the one we elected to use
in the analysis of this paper. Saylor et al. (2018) calculated the ACF of the
light curve, then took the FFT of the ACF to find an initial estimate of the
period, which they then refined by fitting parabolas to the peaks appearing at
aliases of that initial estimate. They report 481 rotation periods, of which
19 appear in our sample and have $P_{rot}<14$ days.

####  4.1.3 Reinhold & Hekker (2020)

Reinhold & Hekker (2020) examined K2 campaigns 0-18 with a variety of analysis
techniques. They applied the LS periodogram, wavelet power spectrum, and ACF,
then required good agreement between the periods found by each method. They
restricted their period searches to $P_{rot}>1$ day, and typically resolved
periods of 1-44 days. They reported periods for 32,387 stars, of which 88
appear in our sample and have $P_{rot}<14$ days. The low number of stars
appearing in both samples can be attributed to similar reasons as for
McQuillan et al. (2014).

####  4.1.4 Messina et al. (2010) & Messina et al. (2011)

Messina et al. (2010) investigated young stellar associations within 100 pc,
including TW Hydrae, $\beta$ Pictoris, Tucana/Horologium, Columba, Carina, and
AB Doradus, supplemented with rotation periods for alpha Persei and the
Pleiades. They applied the LS periodogram to time series from the All-Sky
Automated Survey (ASAS; Pojmanski, 1997, 2002) and reported periods of $<$10
days for 144 stars, which were a combination of newly found rotation periods,
updates to known periods, and rotation periods from literature. Of these, 76
appear in our sample. Messina et al. (2011) extended the analysis of Messina
et al. (2010) to include objects in Octans, Argus, $\eta$ Chamaeleontis, and
IC 2391 clusters. Similarly to the preceding paper, they applied LS
periodograms to photometric time series from ASAS and the Wide Angle Search
for Planets (SuperWASP; Pollacco et al., 2006; Butters et al., 2010) archives
and reported rotation periods for 132 stars, which included both newly found
rotation periods and values from the literature. 49 of these stars appear in
our sample.

####  4.1.5 Canto Martins et al. (2020)

Canto Martins et al. (2020) searched for rotation periods among the 1000
_TESS_ Objects of Interest (TOI) from Sectors 1-22 using a combination of fast
Fourier Transform, LS periodograms, wavelet techniques, and visual inspection.
They found rotation periods for 163 stars ranging from 1 to 13.5 days, of
which they designated 131 as “unambigous” periods and 32 as “dubious” periods.
96 of these stars appear in our sample.

Paper | Sample | Method | # of xmatched Targets | # with $P_{rot}$ Agreement within 10% | # with $P_{rot}$ Agreement outside 10% | # with No Rotation Found
---|---|---|---|---|---|---
McQuillan et al. (2014) | Kepler Main Sequence (MS) | ACF | 29 | 14 | 4 | 11
Saylor et al. (2018) | K2 MS | Autocorrelation Function (ACF) | 19 | 12 | 2 | 5
Reinhold & Hekker (2020) | K2 MS | Lomb-Scargle (LS), ACF, Wavelet | 75 | 26 | 10 | 39
Messina et al. (2010) | YSOs within 100 pc | LS | 76 | 44 | 15 | 17
Messina et al. (2011) | YSOs beyond 100 pc | LS | 49 | 30 | 7 | 9
Canto Martins et al. (2020) |  _TESS_ Objects of Interest | Fourier Transform, LS, Wavelet | 96 | 46 | 8 | 42

Table 1: A summary of targets that appear both in our sample and in previous
studies of stellar rotation. We list the literature sources, the population of
the original studies, the primary method of identifying the period used in the
original study, the number of stars that appear in both our analysis and the
original study with $P_{rot}<14$ days, and finally the number of those stars
that agreed with our found rotation period to within 10%, disagreed by more
than 10%, and for which we did not find a rotation period.

####  4.1.6 Results of Comparison to Previous Studies

The results of our comparison are given in Table 1 and Figure 5. Overall, we
found good agreement between our rotation periods and those that appear in the
literature. Over 75% of rotation periods identified by our algorithm fell
within 10% of those reported in the literature. Of those that did not agree,
we found that a significant portion instead are assigned a value either half
or twice that of the literature value. This is visible in Figure 5, as
clusters of points around the lines $y=2x$ and $y=x/2$. This suggests that
both our and previous analyses are identifying signals from the same physical
phenomena, but there remains some ambiguity between the true rotation period
and the half period aliases that must be addressed individually. Additionally,
disagreement is more common among stars with $P_{rot}>8$ days, once again
demonstrating the difficulty of extracting periods that exceed one third of
the baseline of the data.

Among the targets of Messina et al. (2010), there are five stars (TIC
383615666, TIC 385012516, TIC 401820823, TIC 425976940, TIC 363782810) with
anomalous disagreements between our rotation periods and those reported in the
previous study. Upon visual inspection, we find that the period identified by
our analysis is consistent with the features of the ACF generated from the
_TESS_ time series used in this paper, and the value reported previously is
not. Messina et al. (2010) reported the periods for four of these five stars
with caveats regarding their disagreement with other analyses, thus we
conclude with a preference for the rotation periods reported by our analysis.

Our rate of recovering rotators varies greatly from survey to survey, with our
highest yields of $>75\%$ among the YSO surveys by Messina et al. (2010) and
(Messina et al., 2011), and the lowest for Canto Martins et al. (2020) (56%)
and Reinhold & Hekker (2020) (48%). We attribute the “missed” stars to two
main effects. First, all these surveys except for Canto 2020 are performed on
data that are not concurrent with the _TESS_ mission, meaning that the spots
and activity generating the quasi-periodic signals in the light curve may not
be present in both periods of observation. Additionally, many of our “missed”
stars are observed only for one or two sectors, and this short baseline makes
it even more likely that we may have observed the stars during low-activity
periods. This is consistent with visual inspection, which reveals that many of
the “missed” stars show weak, if any, evidence of their rotation period in
their _TESS_ light curve and ACF.

This explanation does not apply to Canto Martins et al. (2020), who
investigated TOIs in _TESS_ sectors 1-22. However, TOIs often show large and
frequent transits in the light curve. This can cause additional peaks not
associated with the rotation period to appear in the ACF or change the shape
of the ACF peaks, thus leading to missed detections. To avoid this, we
recommend that time series be treated to mask possible transit events before
applying the SpinSpotter algorithm.

Finally, in a small number of cases, our algorithm identified a period
consistent with the literature value, but rejected it due to being slightly
below the selection threshold for one or two parameters. In cases like these,
adopting less strict selection criteria would lead to recovery of more of
these borderline cases.

Figure 5: A comparison of rotation values as derived by our algorithm (y-axis)
and literature values (x-axis). Stars with in agreement with the literature
appear on the line y=x (solid black lines). Stars where the period value
disagrees between our algorithm and the literature appear near the lines
$y=2x$ and $y=x/2$ (grey dashed line). Note the increase in scatter about
$y=x$ for stars with $P_{rot}>9$ days, this reflects our lack of sensitivity
to stars periods greater than one third of the data baseline, since multiple
cycles of the rotation period must be present in order to form the
characteristic peaks in the ACF that are indicative of stellar rotation.

###  4.2 Applications

In this section we discuss the utility of both the sample of rotating stars we
identified in the _TESS_ sample and of the SpinSpotter algorithm more broadly.

The rotating stars we identified in this paper provide a large collection of
fast rotators that will be useful as a basis for future statistical
investigations of stellar evolution, young stellar associations, and stellar
activity. Our catalog is dominated by rapid rotators due to the upper limit on
period sensitivity around 9 days imposed by the short baseline of a _TESS_
sector. As a result, our sample is best suited for science cases involving
young stars that have not yet spun down, or stars for which the Skumanich-
style age-period relationship is interrupted by some other process that delays
the spin down process. In the former case, our sample provides a unique
opportunity to develop a holistic view of young, nearby stellar populations
across the entirety of the night sky, thanks in large part to the excellent
sky coverage of _TESS_. Gyrochronology models, such as those in Mamajek &
Hillenbrand (2008), indicate that our sample covers much of the period space
for stars younger than a billion years. Combined with parallaxes and proper
motions from the Gaia mission, this sample may be used to create a fully-
realized phase space map of young objects in the night sky, which will help us
identify and confirm new members of moving groups and young clusters. This in
turn will support studies of gyrochronology in young clusters, which are a
cornerstone of constraining gyrchronological relationships since their ages
can be determined with a greater degree of certainty than for single stars.
Additional members of these groups may be identifiable by their fast rotation
periods first and then confirmed with Gaia proper motions and parallaxes.

For older stars that maintain short rotation periods, recent studies have
shown that M dwarfs may experience a delayed spin-down process, though the
mechanism for it is not understood (Agüeros et al., 2018). However, the
mechanism driving it is thought to take place quickly, as most of these stars
are found either on the rapid rotator or slow rotator track and with few
objects in between (Newton et al., 2016). Curtis et al. (2020) specifically
calls for more M dwarf rotators, particularly ones that are members of young
clusters, to be found to use as benchmarks for investigating this phenomenon
further. Our sample of _TESS_ rotators contains many more fast rotating M
dwarfs than those found by Kepler surveys (e.g. McQuillan et al., 2014;
Reinhold & Hekker, 2020), and therefore provides a ready made sample for
future investigations of this topic.

The second product we present in this paper, the SpinSpotter algorithm itself,
also has potential for applicability beyond just the _TESS_ 2-minute cadence
sample. SpinSpotter is designed to be flexible and usable on any densely-
sampled photometric light curve to detect periodicities of up to one third the
baseline of the time series. We posit that many thousands more rotators can be
found by applying SpinSpotter to _TESS_ 30-minute cadence targets, though
recovery rates may be lower on these targets as the signal to noise on dimmer
targets will make detecting periodicities more difficult. SpinSpotter can also
be used on Kepler light curves, future photometric missions, and even densely-
sampled ground based monitoring of specific stars.

Since SpinSpotter is fast and easily implementable, it is an ideal tool for
searching large surveys of stars, such as the whole _TESS_ sample, for targets
with specific attributes. By adjusting the thresholds for selection criteria,
investigators can quickly identify targets with their desired rotational
properties, making it an ideal tool for sifting through large data sets for
targets that match a desired science case. Additionally, since SpinSpotter
returns a variety of descriptive parameters about the LC, its ACF, and
individual peaks in the ACF, investigators may tune their selection criteria
to suit a variety of inquiries beyond merely the identification of rotation
periods. For example, Basri et al. (2021) investigated spot lifetimes from
Kepler LCs by using the decay in peak heights in the ACF. This sort of
analysis is replicable using SpinSpotter using the $A_{k}$ parameter, which
represents the vertex height of the parabolas fitted to the $k$th peak in the
ACF. Similarly, the other diagnostics of features in the ACF can be calculated
from the parameters $a_{k}$, $b_{k}$, and $c_{k}$ of the fitted parameter,
and, if they can be shown to correspond to physical properties of the star,
used to investigate other aspects of the stellar surface. Another interesting
application would be to investigate differential rotation, particularly for
stars observed during both the _TESS_ primary and extended missions. With a
separation of two years between observation periods, most stars will have
formed entirely new spot complexes which may occupy different latitudes
between the two observation periods. If different rotation periods are
identified with high confidence for these two time periods, this may indicate
that differential rotation is present and that the star is a good target for a
more in depth analysis on the subject.

It is also important to note the limitations of the SpinSpotter algorithm and
of this sample of _TESS_ rotators. We caution investigators against using
exclusion from this sample as indication of a lack of stellar rotation.
Evidence of rotation may be absent due to being overwhelmed by other sources
of variability, whether from more complex stellar activity or contamination in
the _TESS_ aperture, or the star may simply have no large spot complexes on
its photosphere during the relevant observation period. Additionally, stars
with rotation periods exceeding one third of the observation baseline have a
low chance of recovery, and those that are recovered at such periods have a
higher chance of being false positive detections, as described in Section 4.

##  5 Conclusion

We have developed SpinSpotter, an autocorrelation-based algorithm to recover
stellar rotation periods from densely sampled photometric data in a robust,
automated way. SpinSpotter searches for the dominant periodic signal in the
ACF of the light curve using the FFT, then fits parabolas to peaks occurring
at integer multiples of that period value to characterize the shape of these
features in the ACF associated with that period. It uses the average of the
vertex locations of the peaks to determine the value of $P_{rot}$, and also
requires that the peaks display a characteristic height, width, and goodness
of fit to the fitted parabolas. For stars with multiple segments of data, such
as _TESS_ sectors or Kepler quarters, we treat both the combined light curve
and each segment individually, then require the rotation period be found in a
minimum number of segments in order to be accepted. We also perform a check of
whether the half-value alias of the rotation period was found instead, and
report flags for stars where this may be the case.

Our algorithm is fast, flexible, and needs minimal supervision. It is
sensitive to periods up to one third of the baseline of the data, though it
may recover periods up to one half the baseline of the data with a higher
false positive rate. The characteristic peak parameters it reports allow
investigators to tune their selection criteria for finding periods to best
match their science case, and also opens the door to perform other types of
photometric activity analysis beyond just identifying the value of the
rotation period, such as searching for the presence of differential rotation
or the decay times of spot complexes.

We applied our algorithm to $\sim$130,000 main sequence stars observed by
_TESS_ at 2-minute cadence and found rotation periods for 13,504 stars ranging
from 0.4 to 14 days. We compared our results to surveys of known rotators from
the Kepler sample, the _TESS_ TOI list, and members of young clusters and
found good agreement with the literature. Because we are limited to fast
rotators by the short 27-day baseline of a _TESS_ sector, our sample is most
useful for studying young stars, particularly those less than $\sim$1 billion
years old, or those that undergo a delayed spin down process. The latter case
is particularly interesting, since we identify a large population of fast-
rotating M dwarfs not present in previous Kepler rotation studies. These stars
may make an excellent basis for future studies into the mechanisms that govern
the evolution of stellar spin down in late type stars. We also see utility for
this data set in searching for previously unidentified members of nearby young
stellar associations. Rapid rotation, combined with proper motion and parallax
data from the Gaia mission, may provide strong evidence for membership of
individual stars in these groups.

Though not without its challenges, the _TESS_ mission gives us an
unprecedented opportunity to investigate patterns of stellar rotation across
nearly the entirety of the night sky. Through the SpinSpotter algorithm and
our sample of _TESS_ rotators, we aim to provide tools for future
investigations into stellar rotation and evolution.

##  6 Acknowledgements

This paper includes data collected by the TESS mission. Funding for the TESS
mission is provided by the NASA’s Science Mission Directorate.

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††thanks<EMAIL_ADDRESS>
# Optomechanically Induced Transparency/Absorption in a 3D Microwave Cavity
Architecture at Ambient Temperature

Sumit Kumar,1, Matthew Kenworthy1, Henry Ginn1, Xavier Rojas${}^{1,^{*}}$
1Department of Physics, Royal Holloway University of London, Egham, Surrey,
TW20 0EX, U.K.

###### Abstract

Leveraging advancements in cavity optomechanics, we explore Optomechanically
Induced Transparency/Absorption (OMIT/OMIA) in the microwave domain at ambient
temperature. Contrary to previous works employing cryogenic temperatures, this
work exploits a 3D microwave cavity architecture to observe these effects at
ambient temperature, broadening the scope of possible applications. The work
successfully enhances the optomechanical coupling strength, enabling
observable and robust OMIT/OMIA effects, and demonstrating up to 25 dB in
signal amplification and 20 dB in attenuation. Operating in the unresolved
sideband regime enables tunability across a wider frequency range, enhancing
the system’s applicability in signal processing and sensing. The findings
herein highlight the potential of optomechanical systems, presenting a
simplified, cost-effective, and more feasible approach for applications at
ambient temperature.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

Cavity optomechanics, harnessing the interaction between light and mechanical
motion, has seen rapid advancements over the last two decades [1]. A
particular phenomenon of interest in this domain is Optomechanically Induced
Transparency (OMIT) and its counterpart, Optomechanically Induced Absorption
(OMIA). These effects arise due to the strong coupling between
optical/microwave and mechanical modes, leading to a modification of the
cavity response [2, 3, 4, 5, 6].

Traditional microwave optomechanics setups operate at cryogenic temperatures
to suppress thermal noise, which could otherwise mask the weak optomechanical
effects [7, 8, 9, 10, 11, 12]. However, there has been a burgeoning interest
in realizing these effects at ambient temperature [13, 14, 15, 16, 17].
Operating at ambient temperature offers several advantages such as simplifying
experimental setups, cost reductions, and enhanced feasibility for various
applications, especially in the realm of signal processing and sensing.

Microwave cavity optomechanics at ambient temperature presents a unique set of
challenges and opportunities, especially when cooling to cryogenic
temperatures is not feasible or practical. The present work leverages a 3D
microwave cavity architecture to overcome some of these challenges and
demonstrate the potential of OMIT/OMIA for applications in signal processing
and sensing. The emphasis on ambient temperature operations facilitates
experiments with a high photon number, enabling robust and clear observations
of the cavity optomechanical effects.

In this work, we enhanced the optomechanical coupling strength up to
$g=g_{0}\sqrt{n_{\text{d}}}\simeq 2\pi\times 95$ kHz, corresponding to
cooperativity
$C=4g_{0}^{2}n_{\text{d}}/\kappa_{\text{t}}\Gamma_{\text{m}}\simeq 30$, an
important factor in achieving pronounced OMIT/OMIA effects. This coupling
strength enhancement opens avenues for a range of signal processing and
sensing applications. This allowed us to demonstrate up to 25 dB in signal
amplification, 20 dB in signal attenuation, as well as mass sensing
improvement by a factor of 5 compared to the direct measurement of mechanical
resonance frequency. Our cavity optomechanical system operates in the
unresolved sideband regime (i.e. bad cavity limit) allowing fine frequency
tuning across the entire cavity bandwidth ($\kappa_{\rm t}/2\pi=28.5$ MHz). In
the regime, where the mechanical frequency ($\Omega_{\text{m}}$) of our system
is less than the $\kappa_{\rm t}$, the system is able to function effectively
across a variety of frequencies within this bandwidth. This capability is
significant because it allows for a range of operational frequencies, rather
than being fixed to a single resonant frequency. Additionally, by adjusting
the resonance frequency of our cavity, we can achieve finer tuning across a
wider spectrum. This is achieved by altering the gap between the re-entrant
post and the Si3N4 membrane possible by modifying the cavity design, as
detailed in the studies by Floch et al. [18] and Carvalho et al. [19]. This
feature significantly widens the range of potential applications for our
design.

Figure 1: (a) Schematic of the microwave re-entrant cavity showing the chip
window for locating the Si3N4 membrane chip which is mechanically clamped
using a teflon screw, and the pin holes for the couplers. (b) Schematic
showing the re-entrant pillar, seamlessly integrated with the Si3N4 membrane
chip. (c) Circuit diagram showing the methodology for measuring the
transmission $S_{21}(\omega)$ through the microwave cavity.

##  II Experimental apparatus

The core of our experiment, as detailed in Fig. 1, is a pre-stressed silicon
nitride (Si3N4) membrane chip commercially purchased [20]. These membranes
have low mechanical loss, making them particularly suitable for our
experimental setup. With dimensions of $1\text{ mm}\times 1\text{ mm}\times
50\text{ nm}$, the Si3N4 membrane, is further plated with a 55 nm thick layer
of gold deposited by thermal evaporation. The fundamental mode of this
mechanical membrane has a resonance frequency
$\Omega_{\text{m}}/{2\pi}=160\text{ kHz}$ and a dissipation rate of
$\Gamma_{\text{m}}/2\pi=45$ Hz characterized by measuring the membrane’s
thermomechanical motion [12].

Integrated into our system, the membrane interacts capacitively with a 3D
microwave re-entrant cavity defined by a re-entrant post localised in the
center of the cavity, in close proximity of the membrane. This design ensures
a minimal gap, spanning $\sim$ 6 $\mu$m between the membrane and the post. The
gap was estimated by characterizing the 3D cavity dimensions and using Finite
Element Modelling (FEM) simulation. (see supplementary information). In
previous work, a similar design was developed by Pate et al. with smaller gaps
in the few microns range, leading to the observation of a Casimir spring and
dilution effects in microwave optomechanics [21]. In the re-entrant cavity
geometry, the electrical field of the fundamental mode is predominantly
concentrated between the membrane and the post’s top, making it possible to
model this region as an effective parallel plate capacitor. The effective
capacitance undergoes modulations, in response to the membrane’s displacement,
which can be measured through the microwave cavity transmission. The cavity
resonance can be measured using the measurement setup shown Fig. 1.

A parallel LCR circuit models the fundamental mode of resonance of the
microwave cavity. The total capacitance of this circuit $C$ has two
contribution. The capacitance $C_{\rm g}$ represents the geometric capacitance
between the Si3N4 membrane and the pillar’s top, and the capacitance $C_{0}$
represents the effective capacitance of the rest of the microwave cavity. The
microwave cavity is coupled to the input and output transmission lines of the
measurement circuit via pin antenna couplers represented in the circuit by the
capacitances $C_{1}$ and $C_{2}$. The cavity is driven by a strong pump at
$\omega_{\text{d}}=\omega_{\text{c}}+\Delta$ from a signal source (Rohde &
Schwarz SMA100B), where $\Delta$ represents the cavity detuning. The resonant
cavity is then probed by measuring the complex transmission, corresponding to
the scattering parameter $S_{21}$, using a weak probe signal at
$\omega_{\text{p}}$ generated and detected by a vector network analyzer (Rohde
& Schwarz ZNA67).

Comprehensive insights into the design of our cavity are detailed in a
previous study [17]. The experiments were conducted at ambient temperature
(293 K), within a vacuum chamber that consistently maintained a pressure below
$5\times 10^{-6}\text{ mbar}$.

##  III Experimental results

The cavity exhibits a resonance frequency of $\omega_{\text{c}}/2\pi=$ 4.055
GHz and a total loss rate of $\kappa_{\text{t}}/2\pi=$ 28.5 MHz. The external
feedlines are coupled to the cavity via port 1 and port 2 with loss rates of
$\kappa_{1}$ and $\kappa_{2}$ respectively, such that
$\kappa_{\text{ext}}=\kappa_{1}+\kappa_{2}=2\pi\times 16.9$ MHz. Our cavity is
therefore slightly overcoupled as
$\eta=\kappa_{\text{ext}}/\kappa_{\text{t}}=0.59$.

Figure 2: The cavity is driven on resonance where
$\omega_{\text{d}}=\omega_{\text{c}}$, and probed with a weaker tone at
$\omega_{\text{p}}$. The membrane’s coherent drive, when
$|\omega_{\text{d}}-\omega_{\text{p}}|\approx\Omega_{\text{m}}$, gives rise to
sidebands at $\omega_{\text{d}}\pm\Omega_{\text{m}}$ with opposite phases.
These sidebands interfere constructively (amplification) or destructively
(absorption) with the probe signal, at
$\omega_{p}=\omega_{\text{d}}\pm\Omega_{\text{m}}$. The inset magnifies a
typical amplification (orange curve) and absorption (blue curve) for an
arbitrary pump power.

The cavity is driven by a strong pump at
$\omega_{\text{d}}=\omega_{\text{c}}+\Delta$ and probed using a weaker probe
tone at $\omega_{\text{p}}$. The beating of $\omega_{\text{d}}$ and
$\omega_{\text{p}}$ results in a radiation pressure force on the membrane at
the difference frequency $\Omega=\omega_{\text{p}}-\omega_{\text{d}}$. This
leads to coherent drive of the membrane when
$||\Omega|-\Omega_{\text{m}}|<<\Gamma_{\text{m}}$, where $\Gamma_{\text{m}}$
is the mechanical linewidth. The signal at $\omega_{\text{d}}$ is modulated
due to driven mechanics, creating sidebands at
$\omega_{\text{d}}\pm\Omega_{\text{m}}$ as shown in Fig. 2. In this study, we
performed the measurements at zero detuning ($\Delta=0$) that is
$\omega_{\text{d}}=\omega_{\text{c}}$, for which we do not expect dynamical
backaction effects (cooling/heating) on the mechanical system.

The red sideband at $\omega_{\text{c}}-\Omega_{\text{m}}$ interferes
constructively with the probe signal, leading to amplification of the output
signal. In this regime the system can be used as an amplifier for the probe
signal. The amplification gain is tunable with the pump power. We measured
amplification gain up to 25 dB, which was limited by the signal generator
maximum output power. This effect is shown in Fig. 3.

Below an input pump power threshold of 16 dBm, corresponding to a cavity
photons $n_{d}\sim 4.5\times 10^{13}$ photons in this system, the red-detuned
sideband at $\omega_{\text{c}}+\Omega_{\text{m}}$ is in opposite phase with
the probe signal, and interferes destructively with it, leading to signal
absorption. In this regime, the system can be used as a notch filter for the
probe signal. At 14 dBm of input power corresponding to cavity photons
$n_{d}\sim 2.8\times 10^{13}$, we obtain a signal reduction of up to 20 dB on
a narrow frequency range of $\Delta\omega/2\pi=8$ Hz, which is 5 times smaller
than the mechanical linewidth ($\Gamma_{\rm m}/2\pi=45$ Hz). This effect is
shown in Fig. 3. This power threshold is associated with a cooperativity
approaching unity ($C\sim 1$). In this regime, the cavity optomechanical
system can be used as a notch filter of high selectivity and tunability in the
microwave frequency range. This narrow notch filter could also be used for
sensing purposes, allowing, for instance, a sensitive mass detection of 50
attogram deposited on the mechanical membrane. This corresponds to being able
to detect a mechanical frequency shift of the order of half the notch filter
linewidth ($\Delta\omega/2=2\pi\times 4$ Hz), which is 5 times smaller than
the half bare mechanical linewidth. As demonstrated, leveraging cavity
optomechanical absorption for mass detection can substantially enhance
sensitivity compared to solely relying on the mechanical resonator detection
alone. Theoretically, as the cooperativity, $C$, approaches 1, one can achieve
unlimited signal absorption over an infinitely narrow linewidth. In principle,
this would enable the detection of incredibly minute masses. However, the
ultimate limit to this detection capability is set by frequency fluctuations
of the mechanical resonator. Provided that one could reach such limit of $C$
approaching 1, this system would be an ideal probe to study frequency
fluctuations in mechanical resonators, which are not yet totally understood
[22].

Above the input power threshold of 16 dBm, the blue sideband at
$\omega_{\text{c}}+\Omega_{\text{m}}$ interfere constructively with the probe
signal leading to signal amplification as for the blue-detuned sideband.

We measured the transmission of the probe signal through the cavity $S_{21}$
in the vicinity of $\omega_{\text{c}}\pm\Omega_{\text{m}}$ for different pump
powers (-5 dBm to 27 dBm) applied at the signal generator output, hence
spanning cavity photon numbers over three orders of magnitude ($n_{\rm
d}\simeq 10^{12}$ to $10^{14}$ photons). Fig. 3 shows a colormap of $S_{21}$,
highlighting the amplification and absorption of the probe signal after
interfering with the motional sidebands. The blue-detuned sideband at
$\omega_{\text{c}}-\Omega_{\text{m}}$ is continuously amplified with
increasing pump power, while absorption was observed at the red-detuned
sideband $\omega_{\text{c}}+\Omega_{\text{m}}$ for low powers. At higher pump
powers (16 dBm and above), however, we observed amplification at the red-
detuned sideband $\omega_{\text{c}}+\Omega_{\text{m}}$. Similar results have
been obtained in previous microwave optomechanical setups operated at
cryogenic temperatures [6, 23]. This work shows that these effects can also be
observed at ambient temperature. Fig. 3 also shows line cuts of $S_{21}$ along
the colormap for different powers corresponding to different cavity photons
numbers
$n_{\text{d}}=2\kappa_{\text{ext}}P_{\text{in}}/\hbar\omega_{\text{c}}\kappa_{\text{t}}^{2}$
where $P_{\text{in}}$ is the power applied at the cavity input, taking into
account 3.3 dB of attenuation from the cables. Theoretically, the cavity
transmission $S_{21}$ is given as,

$S_{21}=\dfrac{\kappa_{\text{ext}}[1+i\chi]}{\kappa_{\text{t}}-2i(\omega_{\text{p}}-\omega_{\text{c}})}{\text{,}}$
(1)

where,

$\chi=\dfrac{4\Omega_{\text{m}}g_{0}^{2}n_{\text{d}}}{[\Omega_{\text{m}}^{2}-(\omega_{\text{p}}-\omega_{\text{c}})^{2}-i\Gamma_{\text{m}}(\omega_{\text{p}}-\omega_{\text{c}})][\kappa_{\text{t}}-2i(\omega_{\text{p}}-\omega_{\text{c}})]}{\text{,}}$
(2)

and $g_{0}$ is the single photon optomechanical coupling strength. See
supplementary information for a derivation of the transmission. Using the
above equations, we fitted our data at different powers and extracted the
single photon optomechanical coupling strength $g_{0}/2\pi=$ 4 mHz. This
represents an improvement of an order of magnitude compare to our previous
architecture [17]. The other fitting parameters were the cavity linewidth
$\kappa_{\text{t}}$ and the mechanical linewidth $\Gamma_{\text{m}}$, which
have a slight dependence on the photon number $n_{\text{d}}$ inside the
cavity. The ratio $\kappa_{\text{t}}/\Omega_{\text{m}}=178$ indicates that our
system is in the unresolved sideband regime (or bad cavity limit).

Figure 3:  (left) Colormap showing $S_{21}$ for different pump powers (-5 dBm
to 27 dBm) at $\omega_{\text{c}}-\Omega_{\text{m}}$. We see continuous
amplification with increasing pump power. The colored linecuts correspond to
$S_{21}$ at different pump powers. The y-axis is offset for different curves
for clarity. The fits are shown in dashed grey. (right) Colormap showing
$S_{21}$ for different pump powers (-5 dBm to 27 dBm) at
$\omega_{\text{c}}+\Omega_{\text{m}}$. We see absorption from -5 dBm to 15 dBm
and then amplification for powers above 16 dBm. The colored linecuts
corresponds to $S_{21}$ at different pump powers. The y-axis is offset for
different curves for clarity. The fits are shown in dashed grey.

The microwave resonance frequency $\omega_{\text{c}}$ drifts over time with
changes in the ambient temperature due to temperature dependent cavity
parameters. Consequently, $\omega_{\text{c}}$ was monitored to adjust the
frequency of the pump signal maintaining the zero detuning condition
($\Delta=0$). A small power-dependent cavity damping was also observed, which
suggests that non-linear effects come into play at higher driving powers.
Since we obtained an excellent agreement fitting the cavity response at each
power with a standard harmonic oscillator equation, we simply defined a power-
dependent linewidth $\kappa_{\rm t}$.

At high power, we also observed a frequency shift of the mechanical resonance
frequency, which can be attributed to changes in the membrane’s tension caused
by thermal expansion. This spring softening of the membrane at the highest
power is due to classical heating of the membrane, which was described in our
previous work [17]. We would also like to mention that the frequency shift is
not due to This drift of the mechanical resonance frequency, is also monitored
to adjust the probe signal frequency.

The maximum gain/attenuation for each power used in this experiment is plotted
on Fig. 4. Eq. 1 fits well at most power, indicative of a good agreement with
the theoretical description of OMIT/OMIA.

Figure 4:  Plot showing the gain/attenuation originating from the cavity
optomechanical system and its generated sidebands interfering with the probe
signal. The left sideband at $\omega_{\text{c}}-\Omega_{\text{m}}$ (orange
datapoints), and the right sideband at $\omega_{\text{c}}+\Omega_{\text{m}}$
(blue datapoints).

The noise due to thermal fluctuations can significantly impact the signal at
cavity resonance. The noise stems from thermal occupancy of both the cavity
and mechanical resonator at ambient temperature. The thermal fluctuations of
the Si3N4 membrane introduce more noise compared to the cavity, primarily
because cavity resonance frequency is much larger than mechanical resonance
frequency i.e. $\omega_{\text{c}}>>\Omega_{\text{m}}$. This disparity in
frequencies accentuates the influence of the membrane’s thermal fluctuations,
making them a dominant source of noise in the system. The thermal fluctuations
experienced by the membrane are influenced by its effective temperature, which
is influenced by a several factors including the temperature of the
surrounding bath, the phase noise of the pump signal, and classical heating
resulting from microwave absorption. To fully characterise the performances of
these devices as amplifiers/filters, further studies of the noise should be
carried.

##  IV Conclusion

In this work, we have detailed the experimental setup and results for studying
optomechanical interactions in a microwave re-entrant cavity with a Si3N4
membrane. Our results clearly demonstrate OMIT and OMIA effects of large
amplitude, with a good agreement between the experimental observations and
theoretical predictions. The precise control and measurement of these effects
highlight the potential of our system for further optomechanical studies and
applications in signal processing and sensing.

Despite the excellent agreement between experiment and theory, our
observations at high power levels indicate a lack of temperature stability of
our cavity optomechanical system. Improving temperature control in the system
and increasing the strength of coupling would help reduce the impact of
parasitic effects. Also, the operating bandwidth of our system could be
broadened by tuning $\omega_{\text{c}}$, achieved by adjusting the gap between
the membrane and the top of the conical post. This adjustment allows for
enhanced flexibility and a wider range of detectable frequencies. Further
research in these directions could provide additional insights and enable the
enhancement of optomechanical interactions for advanced applications.

## Data availability

The data that support the findings of this study are openly available in
zenodo at https://doi.org/10.5281/zenodo.10560931, Ref. [24]

## Supplementary material

The supplementary material presents our estimation of the reentrant cavity gap
based on FEM simulations, and the derivation of the cavity transmission
$S_{21}$ obtained from the equations of motion, when the cavity is driven by a
control and a probe signal.

## acknowledgment

We thank A. Armour for helpful discussions, and C. Guisca for his help on the
surface characterization of our 3D cavities. This research was supported by
the Royal Society (UF150140, RGF\EA\180099, RGF\R1\180059, RGF\EA\201047,
RPG\2016\186, URF\R\211009, and RF \ERE\210198), the EPSRC (EP/R04533X/1), and
the STFC (ST/T005998/1). Measurements were made at the London Low Temperature
Laboratory, supported by technical staff, in particular R. Elsom, I. Higgs, P.
Bamford, and H. Sandhu. For the purpose of open access, the author has applied
a Creative Commons Attribution (CC BY) license (where permitted by UKRI, “Open
Government Licence” or “Creative Commons Attribution No-derivatives (CC BY-ND)
license” may be stated instead) to any Author Accepted Manuscript version
arising.

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## Supplementary 1: Estimating the gap between re-entrant post and the
membrane

The dimensions of the top of the re-entrant post are determined using optical
interferometery. The estimated effective radius of the top of the re-entrant
post is $\rm R_{t}=0.15$ mm. The resonance frequency of the 3D re-entrant
cavity is predominantly influenced by the separation between the top of the
re-entrant post and the membrane. We conducted Finite Element Method (FEM)
simulations using COMSOL, depicted in Fig. 5, to determine the microwave
resonance frequency. A resonance frequency of 4.055 GHz, obtained from the
simulation, corresponds to the measured resonance frequency given a 6 $\mu$m
gap between the post top and the silicon nitride membrane.

Figure 5:  (top) Figure showing the surface profile of the top of the re-
entrant post, obtained using an optical interferometer, with the dimensions of
the re-entrant cavity shown below. (bottom) Electric field profile of the
simulated re-entrant cavity mode at 4.055 GHz obtained using an FEM simulation
(COMSOL).

## Supplementary 2: Derivation of the equations of motion

We’ll begin by formulating the classic optomechanical equation of motion for
both the mechanical resonator and the optical cavity given by,

${}\ddot{x}=-\Omega_{\text{m}}^{2}x-\Gamma_{\text{m}}\dot{x}+\dfrac{(F_{\text{BA}}+F_{\text{th}})}{m_{\text{eff}}},$
(3)

and

${}\dot{\alpha}=[-\frac{\kappa_{\text{t}}}{2}+i\Delta]\alpha+iGx\alpha+\sum_{\text{k=1,2}}\sqrt{\kappa_{\text{k}}}\alpha_{\text{in,k}}.$
(4)

The motion of the mechanical resonator can be described by Eq. 3 where
$m_{\text{eff}}$ represents the effective mass, $x$, $\dot{x}$, and $\ddot{x}$
are respectively the membrane’s displacement, velocity and acceleration,
$F_{\text{BA}}=\hbar G|\alpha|^{2}$ is the backaction force, and
$F_{\text{th}}$ is the thermal force. $\alpha$ is the intracavity field and
$G=\partial\omega_{\text{c}}/\partial{x}$ is the frequency pull factor. The
equation of motion for the intracavity field $\alpha$ in the frame rotating
with pump frequency $\omega_{\text{d}}$ is expressed by Eq. 4 where
$\kappa_{\text{t}}$ is the total cavity decay rate, $\kappa_{\text{k=1,2}}$ is
the coupling rate of cavity to left (1) and right (2) ports respectively,
$\alpha_{\text{in,k}}$ is the input field, and
$\Delta=\omega_{\text{d}}-\omega_{\text{c}}$. In the context of OMIT/OMIA,
random noise inputs to the mechanical resonator and the cavity are neglected.
Eq. 3 and Eq. 4 are reframed as,

${}\ddot{x}=-\Omega_{\text{m}}^{2}x-\Gamma_{\text{m}}\dot{x}+\dfrac{\hbar
G|\alpha|^{2}}{m_{\text{eff}}},$ (5)

and

${}\dot{\alpha}=[-\frac{\kappa_{\text{t}}}{2}+i\Delta]\alpha+iGx\alpha+\sqrt{\kappa_{\text{1}}}\alpha_{\text{in}}+\sqrt{\kappa_{\text{1}}}s_{\text{p}},$
(6)

where, $s_{\text{p}}=s_{0}e^{-i\Omega t}$ is the probe field,
$\Omega=\omega_{\text{p}}-\omega_{\text{d}}$, and $\alpha_{\text{in}}$ is the
input pump field. We will solve Eq. 5 and Eq. 6 by linearizing them around
their steady-state solutions, $\bar{\alpha}$ and $\bar{x}$, where
$x=\bar{x}+\delta x$ and $\alpha=\bar{\alpha}+\delta\alpha$. The equations of
motion, once linearized, can be expressed as follows,

${}\delta\ddot{x}=-\Omega_{\text{m}}^{2}\delta
x-\Gamma_{\text{m}}\delta\dot{x}+\dfrac{\hbar
G\bar{\alpha}(\delta\alpha+\delta\alpha^{*})}{m_{\text{eff}}},$ (7)

and

${}\delta\dot{\alpha}=[-\frac{\kappa_{\text{t}}}{2}+i\Delta]\delta\alpha+iG\bar{\alpha}\delta
x+\sqrt{\kappa_{\text{1}}}s_{\text{p}}.$ (8)

To solve Eq. 7 and Eq. 8 we use the following ansatz,

${}\delta\alpha=A_{1}e^{-i\Omega t}+A_{2}e^{i\Omega t},$ (9)
${}\delta\alpha^{*}=A_{1}^{*}e^{i\Omega t}+A_{2}^{*}e^{-i\Omega t},$ (10)
${}\delta x=Xe^{-i\Omega t}+X^{*}e^{i\Omega t}.$ (11)

Using above equations and only taking terms with $e^{-i\Omega t}$
coefficients, Eq. 7 simplifies to

${}X=\left[\dfrac{\hbar
G\bar{\alpha}(A_{1}+A_{2}^{*})}{m_{\text{eff}}}\right]\chi_{\text{m}}(\Omega),$
(12)

where

${}\chi_{\text{m}}(\Omega)=\dfrac{1}{m_{\text{eff}}\left[\Omega_{\text{m}}^{2}-\Omega^{2}-i\Gamma_{\text{m}}\Omega\right]}.$
(13)

Similarly separating the terms with $e^{-i\Omega t}$ and $e^{i\Omega t}$
coefficients from Eq. 8 we get,

${}A_{1}=\left[iG\bar{\alpha}X+\sqrt{\kappa_{\text{1}}}s_{\text{p}}\right]\chi_{\text{a}}(\Omega),$
(14)

and

${}A_{2}=\left[iG\bar{\alpha}X^{*}\right]\chi_{\text{b}}(\Omega),$ (15)

where,

${}\chi_{\text{a}}(\Omega)=\dfrac{1}{\frac{\kappa_{\rm
t}}{2}-i(\Delta+\Omega)},\chi_{\text{b}}(\Omega)=\dfrac{1}{\frac{\kappa_{\rm
t}}{2}+i(\Omega-\Delta)},$ (16)

and

${}\chi_{\text{c}}(\Omega)=\chi^{*}_{\text{b}}(\Omega)=\dfrac{1}{\frac{\kappa_{\rm
t}}{2}-i(-\Delta+\Omega)}.$ (17)

Now putting the expression of $A_{2}^{*}$ in Eq. 12 we get,

${}X=\dfrac{A_{1}}{\dfrac{1}{\chi_{\text{m}}(\Omega)\hbar
G\bar{\alpha}}+iG\bar{\alpha}\chi_{\text{c}}(\Omega)}.$ (18)

We now continue by putting the above expression in Eq. 14 and solve for
$A_{1}$ giving us,

${}A_{1}=\sqrt{\kappa_{\text{l}}}s_{\text{p}}\left[\dfrac{1+i\chi}{\dfrac{1}{\chi_{\text{a}}(\Omega)}+i\chi[\dfrac{1}{\chi_{\text{a}}(\Omega)}-\dfrac{1}{\chi_{\text{c}}(\Omega)}]}\right],$
(19)

where,

${}\chi=\hbar
G^{2}\bar{\alpha}^{2}\chi_{\text{c}}(\Omega)\chi_{\text{m}}(\Omega).$ (20)

In our case, we have $\omega_{\text{d}}=\omega_{\text{c}}$, Eq. 20 and thus
Eq. 19 simplifies to

${}\chi=\dfrac{4\Omega_{\text{m}}g_{0}^{2}n_{\text{d}}}{[\Omega_{\text{m}}^{2}-(\omega_{\text{p}}-\omega_{\text{c}})^{2}-i\Gamma_{\text{m}}(\omega_{\text{p}}-\omega_{\text{c}})][\kappa_{\text{t}}-2i(\omega_{\text{p}}-\omega_{\text{c}})]}{\text{,}}$
(21)

and

${}A_{1}=\dfrac{2\sqrt{\kappa_{\text{1}}}s_{\text{p}}[1+i\chi]}{\kappa_{\text{t}}-2i(\omega_{\text{p}}-\omega_{\text{c}})}.$
(22)

The scattering parameter can further be written as,

${}S_{21}=\dfrac{\sqrt{\kappa_{\text{2}}}}{s_{\text{p}}}\dfrac{2\sqrt{\kappa_{\text{l}}}s_{\text{p}}[1+i\chi]}{\kappa_{\text{t}}-2i(\omega_{\text{p}}-\omega_{\text{c}})}.$
(23)

Since $\kappa_{\text{1}}=\kappa_{\text{2}}=\kappa_{\text{ext}}/2$, above
equation simplifies to

${}S_{21}=\dfrac{\sqrt{\kappa_{\text{ext}}}[1+i\chi]}{\kappa_{\text{t}}-2i(\omega_{\text{p}}-\omega_{\text{c}})}.$
(24)