118 712021

Photometric and spectroscopic investigations of the Galactic field RRc
candidates V764 Mon and HY Com J. M. Benkő1,2, Á. Sódor1,2 and A. Pál1
1Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, ELKH,
Konkoly Thege Miklós út 15-17, H-1121 Budapest, Hungary
2MTA CSFK Lendület Near-Field Cosmology Research Group
<EMAIL_ADDRESS>
January 29, 2021

By analysing photometric and spectroscopic time series in this paper, we show
that the pulsator V764 Mon, assumed to be the brightest RR Lyrae star in the
sky, is in fact a rapidly rotating $\delta$ Scuti star with an unusually long
dominant period ($P_{1}=0.29$ d). Our spectroscopy confirmed the discovery of
the Gaia satellite about the binarity of V764 Mon. In the case of HY Com, a
‘bona fide’ RRc star, we present its first complete radial velocity curve.
Additionally, we found that the star continues its strong phase variation
reported before. stars: oscillations – stars: variables: RR Lyrae – stars:
variables: $\delta$ Scuti – methods: data analysis – space vehicles

##  1 Introduction

The catalogue of Galactic RR Lyrae stars (Maintz 2005) contains only eight
first-overtone (RRc) pulsators in the sky that are visually brighter than 10
mag at maximum light. The brightest one is V764 Mon (= HD 61176, A5IV,
$V_{\mathrm{max}}=7.13$ mag). The variable nature of the star was discovered
by the Hipparcos satellite (ESA 1997). On the basis of its period ($P=0.290$
d), it was classified as an RRc candidate, as such, not just the brightest RRc
star, but it would even be the brightest of all known RR Lyrae, because it is
also brighter than the brightest fundamental-mode RR Lyrae pulsator, the
namesake of the class, RR Lyr ($V_{\mathrm{max}}=7.45$ mag) itself. It is
therefore surprising how neglected this star is. Up to now no dedicated study
targeted it.

Kervella et al. (2019a,b) enumerate V764 Mon recently as a ‘bona fide’ RR
Lyrae among the binary cepheid and RR Lyrae candidates of the Gaia DR2. On the
basis of its proper motion anomaly value ($\Delta_{G2}=7.1$), which is the
second highest of the Gaia RR Lyrae sample, the binary nature of the star is
certain. Kervella et al. (2019a) estimated the upper limits of the semi-major
axis ($a_{\mathrm{max}}\sim 300$ AU) and the orbital period
($P_{\mathrm{orb}}\sim 5000$ year) of this binary system.

With our present observations, we aimed to construct a spectroscopic time
series of complete phase coverage for this potentially interesting star. Two
secondary targets (T Sex, $V_{\mathrm{max}}=9.91$ mag, and HY Com,
$V_{\mathrm{max}}=10.33$ mag) were also chosen to best utilise our observation
time. The results for T Sex are published elsewhere (Benkő et al. 2021). This
paper presents the results on V764 Mon and HY Com.

rcrrc12.5cmUsed photometric data Star Time $\langle\sigma\rangle$ $n$ Source
dd.mm.year (mmag)

V764 Mon 12.03.1990–03.03.1993 8 97 (1)
07.01.2019–02.02.2019 0.05 1093 (2)
HY Com 29.12.2002–11.07.2009 30 368 (3)
01.04.2012–27.05.2012 72 1411 (4)
09.03.2013–28.11.2018 20 649 (5)
(1) Hipparcos (ESA 1997); (2) TESS (Ricker et al. 2015); (3) ASAS-3 (Pojmański
et al. 2005); (4) Qatar Exoplanet Survey (Bramich et al. 2014); (5) ASAS-SN
Variable Stars II (Jayashinge et al. 2019)

##  2 The case of V764 Mon

###  2.1 Photometric data

We collected and analysed the two space-born photometric time series data of
V764 Mon observed by Hipparcos and TESS satellites. Time length, number of
observed data points and their typical accuracy of these data sets are given
in Table 1. Some large-scale ground-based surveys also observed V764 Mon
(ASAS-3, Pojmański 2002, 2003; ASAS-SN, Shappee et al. 2014, Jayasinghe et al.
2019; KELT, Pepper et al. 2007, 2012; ZTF, Bellm et al. 2019, Masci et al.
2019). However, V764 Mon is well above the saturation limit of all these
surveys. In some cases the image of the saturated stars can still be corrected
and acceptable light curve can be reconstructed (e.g. Kochanek et al. 2017).
Nevertheless, checking V764 Mon data sets, we came to the conclusion that such
correction is not possible for any of the survey data, therefore, these data
sets were not used for analysis.

For bright stars the Hipparcos astrometric satellite provided also some
photometric measurements during its mission (van Leeuwen et al. 1997). The
Hipparcos observation of V764 Mon has been made by a wide non-standard pass-
band $H_{\mathrm{p}}$.

The photometric space telescope TESS (Ricker et al. 2015) is measuring the sky
with four 4k$\times$4k CCD cameras, each of them providing 24$\times 24$ deg
field of view. The mission consists of 27-day-long observing runs in which the
camera positions are fixed and they scan a single 24$\times$90 deg strip,
called a ‘sector’. Thirteen such sectors almost cover an entire ecliptic
hemisphere. The observations are continuous within the sectors, apart from
some technical interruptions. One image is taken every 2 minutes by each
camera. Fifteen consecutive images are combined on-board into so-called full
frame images (FFI)111FFIs are publicly available at the Mikulski Archive for
Space Telescopes: https://archive.stsci.edu/ which results in time-series
observations of $\sim$30 min cadence. The spectral response function of the
TESS CCD cameras (Ricker et al. 2015) defines a broad ($\sim 4000$ Å) non-
standard band, most sensitive at red optical wavelengths, somewhat similar to
the Johnson-Cousins band $I$.

V764 Mon was so far observed by TESS only in Sector 7. We extracted the data
on our target and its surroundings from the FFIs and processed the resulting
small images by using a shell script written by A. Pál (Konkoly Observatory).
The script relies on the tools of the fitsh package (Pál 2012) and beyond the
target selection, it performs a full image subtracting photometry, and also
prepares the light curve for analysis. The detailed description of the process
is given in Plachy et al. (2021). Each observing run of TESS includes two
complete orbits of the satellite around Earth. As a consequence, generally two
different systematic trends can be seen in the raw light curves, one for each
circulation. This is the situation for V764 Mon data as well. For eliminating
these trends, quadratic functions were fitted to the two data parts
separately, then these functions were subtracted from the respective
sections222The used TESS data are available here:
https://konkoly.hu/KIK/data_en.html..

###  2.2 Analysis of the photometric time series

Photometric analysis of V764 Mon. Observed light curves (left panels); phased
light curves with the main period $P_{1}=0.290002$ d (middle panels); and the
Fourier spectra of the corresponding data sets (right panels). First row shows
Hipparcos data while second one shows the TESS measurements.

The time series analysis was performed by using the tools of the Period04
software package (Lenz and Breger 2005). The Fourier spectrum of the Hipparcos
data show only one significant ($S/N=4.5$) peak at the frequency of
$f_{1}=3.44825\pm 0.00002$ d-1 (top right panel in Fig. 1), which is the known
main period $P_{1}=0.290002\pm 0.000002$ d. This period could really be a
pulsation period of an RRc star, but the phased light curve (top middle panel
in Fig. 1) show an order of magnitude lower amplitude ($\sim 0.03$ mag) than
expected for a typical RRc star. Although the Fourier spectrum shows some
structure, after we pre-whiten the data with the main frequency, no further
significant ($S/N>4$, Breger et al. 1993) peak appears in the residual
spectrum.

lrccccrl10.5cmSignificant frequencies of V764 Mon from the TESS data. ID
Frequency $\sigma(f)$ Amplitude Phase $\sigma(\phi)$ $S/N$ ident.
(d-1) (d-1) (mag) (rad) (rad)
$f_{1}$ 3.44757 0.0004 0.0233 2.332 0.016 47.57
$f_{2}$ 11.58466 0.0008 0.0109 3.454 0.035 25.96
$f_{3}$ 4.58517 0.0010 0.0089 0.096 0.043 19.00
$f_{4}$ 0.15345 0.0023 0.0036 1.316 0.101 4.82
$f_{5}$ 0.80614 0.0022 0.0031 1.779 0.097 4.70
$f_{6}$ 9.88850 0.0067 0.0017 5.552 0.299 7.18
$f_{7}$ 14.26846 0.0085 0.0015 2.065 0.377 7.75
$f_{8}$ 7.00563 0.0064 0.0014 5.521 0.284 5.35 $=f_{2}-f_{3}$
$f_{9}$ 20.21424 0.0067 0.0014 4.538 0.295 8.37
$f_{10}$ 7.81177 0.0081 0.0012 0.960 0.357 5.57 $=f_{5}+f_{8}$
$f_{11}$ 18.41987 0.0050 0.0011 5.495 0.221 6.27
$f_{12}$ 6.65780 0.0076 0.0010 3.701 0.335 4.08
The Fourier spectrum of the TESS data shows several highly significant
frequencies. The highest-amplitude peak is at the main frequency
($f_{1}=3.44757$ d-1, bottom right panel in Fig. 1). Folding the light curve
with this period, we obtain a thick phase curve (middle bottom panel in Fig.
1), indicating the multi-periodic nature of the signal. Two high peaks are
situated at $f_{2}=11.58466$ d-1 and at $f_{3}=4.58517$ d-1 frequencies, and
some lower-amplitude peaks are also detectable. Yet, no evident harmonics of
$f_{1}$ can be seen.

Traditionally, the frequency content of multi-periodic stars have been
determined by successive pre-whitening steps starting with the highest-
amplitude frequency, and continuing with the next highest-amplitude frequency
of the residual, and so on. As Balona (2014b) has demonstrated, for many low-
amplitude frequencies, which are common with stars measured by space
photometry, this method could give misleading results. He suggested no or as
few pre-whitening steps as possible. Following this, we pre-whiten the data
with the three highest amplitude frequencies ($f_{1}$, $f_{2}$ and $f_{3}$).
The residual spectrum shows further nine significant frequencies (see Table
2). Three of these ($f_{4}$, $f_{5}$ and $f_{11}$) are significant in the
original spectrum as well. After pre-whitening the original data with all 12
frequencies, no further significant frequencies were found in the residual.
Examining the obtained frequencies, we find two linear combination frequencies
($f_{8}$ and $f_{10}$), that is, ten independent frequencies describe the
light curve within the observed accuracy.

The result of the frequency analysis is presented in Table 2. The frequencies
are arranged in order of decreasing amplitude. After the designation of the
frequency (col. 1) we show the frequency values and their 1$\sigma$ errors
(cols 2–3), the amplitude (col. 4), the phase with their errors (cols 5–6),
signal-to-noise ratio (col. 7) and possible identification (col 8). The
uncertainties of Fourier parameters are estimated by the analytical formulae
of Breger et al. (1999) implemented in the Period04 package. These formulae
uniformly give an error of 0.0004 mag for the amplitudes, which have not been
indicated individually. We mention that the two largest-amplitude non-
significant peaks in the spectrum of the Hipparcos data can be identified with
$f_{10}$ and $f_{7}$, respectively. This suggests that the amplitudes of the
pulsations are changing in time.

Based on the TESS spectrum, the RRc classification can no longer be
maintained. The small-amplitude multi-periodic light-curve variation together
with its spectral classification (A5IV) suggests that V764 Mon is a possible
$\delta$ Scuti star. Or, since its ten independent frequencies are distributed
between 0.15345 and 20.2142 d-1, maybe a hybrid $\delta$ Scuti/$\gamma$ Dor
star. This classification is controversial as to whether it says more than the
$\delta$ Scuti type alone because Balona (2014a) and Balona et al. (2015) have
shown that low-frequencies associated with the g-modes and characteristic of
$\gamma$ Dor stars can be detected in almost all Kepler $\delta$ Scuti stars.
The frequency distribution of V764 Mon, shown in the bottom right panel of
Fig. 1, is similar to the Kepler $\delta$ Scuti stars described by Balona and
Dziembowski (2011), as stars with one dominant mode with a side frequency. The
mode density of V764 Mon (2.0 per d-1) is also typical in this sub-group.
Summarising our photometric analysis, it can be concluded that V764 Mon is a
low-amplitude multi-periodically pulsating star, probably a $\delta$ Scuti or
related variable.

###  2.3 Spectroscopic data

Spectroscopic time series were obtained with the ACE echelle spectrograph
mounted on the 1-m RCC telescope set up in Piszkéstető Mountain Station of the
Konkoly Observatory. The spectra cover the 4150 – 9150 Å wavelength range with
a resolution of $R\approx$20 000. The log of the spectroscopic observations
are given in Table 3. The integration time was set to 30 min which provides
signal-to-noise ratios between 85 and 380, depending on the weather conditions
(see column 4 in Table 3 for averaged nightly $S/N$ values). These S/N values
were estimated by using the tool of iSpec package (Blanco-Cuaresma et al.
2014; Blanco-Cuaresma 2019).

The spectra were processed by using standard IRAF (Tody 1986, 1993) tasks
including bias, dark and flat-field corrections, aperture extraction, and
wavelength calibration. For the latter purpose we used thorium-argon
calibration images which were taken after every three object frames. The
normalisation, cosmic-ray filtering and order merging were performed by our
python scripts. Each spectrum was also corrected to the barycentric frame333
The final normalised spectra are available from the authors upon request..

rccrr12.5cmLog of the spectroscopic observations Star Night JD $\langle
S/N\rangle$ $n$
(yyyy-mm-dd) (-2 400 000)
V764 Mon 2015-03-05 57087 84 3
2015-03-06 57088 163 10
2015-03-09 57091 376 8
HY Com 2015-03-06 57088 18 8
2015-03-08 57090 43 12
2015-03-09 57091 27 6
Columns contain: name of the target star; night of the observation; Julian
day; average signal-to-noise ratio of the single exposures; number of spectra

###  2.4 Spectroscopic results

Radial velocities were calculated by cross-correlating the spectra with a
metallic line mask containing 622 metallic lines between 4800 and 5600 Å from
a synthetic A-type spectrum. We fitted a rotation-broadened line profile to
the time-averaged mean pulsation profile for determining radial velocity
$v_{\mathrm{rad}}$ and $v\sin i$ simultaneously, using an in-house developed
Python application relying on packages of Phoebe 2.0 (Prša et al. 2016). The
fitting process also provided micro- and macro-turbulences, however, the
results on the turbulences were too uncertain and degenerate, while their
values did not affect $v_{\mathrm{rad}}$ at all, and also affected $v\sin i$
only marginally. The uncertainties of the derived turbulences are caused
partially by the limits of our data, but also the non-radial, time- and
location-dependent pulsation motions. The systematic errors resulted in the
data processing and the stability of the wavelength calibration system of the
ACE instrument are better than 0.36 km s-1, based on observations of radial
velocity standards (Derekas et al. 2017). The radial velocities and their
standard deviations are given in Table 4. The average radial velocity is
$v_{\mathrm{rad}}=29\pm 1$ km s-1 while $v\sin i=188\pm 5$ km s-1. The latter
value is an order of magnitude higher than the known values of RRc stars
($\sim 2-15$ km s-1, Preston et al. 2019), while typical for near-main-
sequence A-F stars, including $\delta$ Scuti-type pulsators.

The spectral measurements were optimised with the assumption that V764 Mon is
a bright RRc star. Once this was disproven, the observations were also found
to be sub-optimal for the study of a potential $\delta$ Scuti pulsator. E.g.,
preparing a proper radial velocity curve of a $\delta$ Scuti star needs an
order of higher precision than ours, on one hand because the expected
amplitude of the radial velocity curve is far less than that of an RRc (of
about 5 km s-1, see e.g. Yang and Walker 1986; Mathias and Aerts 1996), and on
the other hand because the uncertainties of the radial-velocity measurements
of fast rotators are significantly higher. Nevertheless, if we fold the radial
velocity data of V764 Mon with its dominant period $P_{1}$ (left panel in Fig.
2), the curve show a definite phase-dependence, but it does not allow a
detailed examination, because of the relatively large errors, and because of
the relatively few number of data points, considering the multi-periodic
nature of the pulsation.

cccccc7cmRadial velocities of V764 Mon BJD $v_{\mathrm{rad}}$
$\sigma(v_{\mathrm{rad}})$ BJD $v_{\mathrm{rad}}$ $\sigma(v_{\mathrm{rad}})$
(d) km s-1 km s-1 (d) km s-1 km s-1
2457087.38387 36.12 1.49 2457088.42919 26.31 0.72
2457087.40482 34.11 1.53 2457088.45018 27.68 0.89
2457087.42577 40.69 2.32 2457091.27534 29.10 0.32
2457088.25168 30.69 0.46 2457091.29628 28.99 0.43
2457088.27263 30.19 0.69 2457091.31723 27.67 0.50
2457088.29362 28.08 0.58 2457091.34924 28.78 0.37
2457088.31748 28.14 0.45 2457091.37021 29.34 0.37
2457088.33847 27.67 0.47 2457091.39116 29.56 0.49
2457088.36434 28.76 0.56 2457091.42026 30.41 0.33
2457088.38530 28.25 0.47 2457091.44123 30.20 0.48
2457088.40628 34.06 0.62

On the left: Phased radial velocities of V764 Mon with its dominant period
$P_{1}=0.29$ d. The scatter in radial velocity is partly attributed to the
multi-periodic nature of the pulsation. On the right: Line profile variation
of V764 Mon in the night of 2015-03-09. Consecutive profiles are plotted from
top to bottom with vertical offsets from each other. The offsets are
proportional to the differences in the observation time of each spectrum. Red-
ward propagating ripples in the profiles can be observed, typical of $l\geq 1$
azimuthal-order, non-radial pulsations.

Investigating individual cross-correlation functions two phenomena can be
seen: (i) The line profiles show red-ward propagating distortions caused by
non-radial pulsation. The time series spectroscopy of $\delta$ Scuti stars is
primarily used to identify their non-radial modes. However, for a successful
mode identification, each of the known methods (see for reviews e.g.
Mantegazza 2000, Zima 2008) requires at least an order of magnitude more
individual spectra, preferably with the resolution of at least 40 000 and
$S/N>200$. In the best night (2015-03-09) the $S/N$ ratio is above this limit
and the line profiles variations can clearly be followed (right panel in Fig.
2).

(ii) By combining all spectra, the line-profile variations caused by the
pulsation are mostly averaged out, while a clearly visible absorption feature
emerges at around 40 km s-1(Fig. 3, left panel). This signal most probably
belongs to the companion star of this binary system. The spectra are
unsuitable for decomposition. This might require better $S/N$ ratio. We can,
however, fit the average radial velocity of the companion (Fig. 3 in the
right) as $v_{\mathrm{rad}}=34.7\pm 2$ km s-1 and the projected rotation
velocity $v\sin{i}=12\pm 3$ km s-1. If the rotation axes are parallel, the
latter would mean slow rotation, but in the case of wide binaries, the
rotation axes are not necessarily aligned. The equivalent width of the cross-
correlated line profile of the companion is $\sim 16$% of that of the V764
Mon, that is, the secondary star is certainly significantly fainter than the
$\delta$ Scuti pulsator primary, but a quantitative estimate is not possible
without knowing the spectral types of both components. Without this
information, we cannot estimate the luminosity ratio. A plausible assumption
is, however, that the companion is a red main-sequence star.

In the left: Average of the weighted cross-correlation function of the
spectroscopic time series of V764 Mon (red line) and the fitted line profile
(blue). The black line shows the averaged profile after we subtracted the
fitted profile from it and shifted for clarity (black). In the right: The fit
of the secondary component’s line profile using the subtracted average (top
curve in the left panel). Line colours have the same meaning as in the left
panel of the figure.

From 60 to 80 percent of the stars in the Galactic field are members of a
binary or multiple system. Although the presently known ratio for $\delta$
Scuti stars is smaller than this, lots of binary systems are known (Liakos and
Niarchos, 2017, Lampens et al. 2018). Therefore, it is not surprising that
V764 Mon is member of a binary system, in contrast with the binary RR Lyrae
scenario suggested by Kervella et al. (2019a, b).

###  2.5 Rotation

Although V764 Mon is not listed in the $\delta$ Scuti catalogue of Rodríguez
et al. (2000), Bush and Hintz (2008) included it in their $\delta$ Scuti
rotational velocity survey and they have determined a projected velocity of
$v\sin{i}=131\pm 8$ km s-1. As we have seen, our projected rotation velocity
is significantly higher than the measurement of Bush and Hintz (2008). The
phase dependent radial velocity due to pulsation could cause some difference
in the $v\sin i$ measurements, but this effect cannot explain the whole
difference. Bush and Hintz (2008) used a standard technique for determining
$v\sin i$ which imply the assumption of a nearly linear relation between the
line width and $v\sin i$. This method could, however, systematically
underestimate the projected equatorial velocities, especially for rigidly
rotating star with radiative envelope, if the gravity darkening is not taken
into account properly (see e.g. Townsend et al. 2004). The rotation study of
A–F stars by Ammler-von Eiff and Reiners (2012) demonstrated well that the
hotter ($>7000$ K) and faster rotating ($v\sin i>50$ km s-1) stars typically
rotate as rigid bodies. V764 Mon fulfils both conditions, it is probably a
rigid rotator and the effect of the gravity darkening explains the difference
between $v\sin i$ measurements. Since a very large fraction of spectroscopic
binaries are members of a triple or multiple system (Tokovinin et al. 2006),
there might be an alternative explanation. If the V764 Mon binary has a third
component on a wide orbit its inclination could change in the time between the
two measurements (2002 – 2003 and 2015) due to an orbital plane precession
caused by the third star.

Correlation between projected velocity $v\sin i$ and $\sin i/P_{\mathrm{rot}}$
for A-F stars determined by Ammler-von Eiff and Reimers (2012). Our linear fit
for the upper envelope is shown with the continuous line. The three vertical
bars mark the positions of the found three potential rotation periods of V764
Mon. The vertical length of the bars result from the allowed $\sin i$ ranges.

We attempted to identify any TESS frequency that might correspond to the
rotation. When we plot the projected velocity $v_{\mathrm{t}}\sin i$ versus
$\sin i/P_{\mathrm{rot}}$ determined by Ammler-von Eiff and Reimers (2012), we
obtain a clear correlation (Fig. 4). It naturally follows from the
relationship between the rotation period $P_{\mathrm{rot}}$ and the tangential
velocity $v_{\mathrm{t}}$:

$v_{\mathrm{t}}=\frac{2\pi R}{P_{\mathrm{rot}}}.$ (1)

If we fit the upper envelope in Fig. 4 as

$\frac{\sin i}{P_{\mathrm{rot}}}=av_{\mathrm{t}}\sin i,$ (2)

we obtain $a=0.0112\pm 0.00027$ which gives a lower limit for the rotation
period: $av\sin i=2.1$, $P_{\mathrm{rot}}>\sin i/2.1$ d.

Using the formulae for the critical rotation speed $v_{\mathrm{crit}}$ (e.g.
Towsend et al. 2004):

$v_{\mathrm{crit}}=\sqrt{G\frac{M}{R_{\mathrm{e}}}}=\sqrt{2G\frac{M}{3R_{\mathrm{p}}}},$
(3)

where $G$ is the gravitational constant, $M$ is the stellar mass,
$R_{\mathrm{e}}$ and $R_{\mathrm{p}}$ are the equatorial and polar radii, with
the parameters of V764 Mon its critical speed is about
$v_{\mathrm{crit}}\sim$308 km s-1. (Here we accepted the equatorial radius of
Gaia DR2: $R_{\mathrm{e}}=4.52$ R⊙). Then the minimum value of the $\sin i$
can be around 0.61, and the lower limit of the rotation period is
$P_{\mathrm{rot}}>0.29$ d.

There are three detected periods in the TESS data above this limit:
$P_{1}=0.290$, $P_{4}=1.240$ and $P_{5}=6.517$ d. According to the above rough
estimation, it is possible that $P_{1}$ is not the main pulsation period but a
rotational variation. If so, the star is very close to the break up limit.
Although it would not be completely unprecedented, since several known
$\delta$ Scuti stars rotate as fast as about 80 – 90% of their break-up limit
(e.g. Suárez et al. 2005; Monnier et al. 2010), but not very likely,
nevertheless (see also Fig. 4). Considering the distribution shown in Fig. 4,
$P_{4}$ is the most probable value of the rotation period in V764 Mon. In this
case, if we estimate the inclination using the formulae (1, 2) and the radius
estimation ($R=4.52$ R⊙), we find that $i$ is very close to 90 degrees.

###  2.6 Physical parameters

rrcccr12.5cmEstimated physical parameters of V764 Mon. Parameter McDonald1
LAMOST2 DR23 EDR34 this work
$T_{\mathrm{eff}}$ (K) 7197$\pm 317$ 7234$\pm$48 7455$\pm{324}$ 8063$\pm 40$
8090$\pm 160$
$\log g$ 3.71$\pm$0.06 3.4986$\pm 0.1$ 3.60$\pm 0.1$
$[$Fe/H$]$ 0.146$\pm$0.1 $-0.015\pm 0.04$ $-0.30\pm 0.3$
$L$ (L⊙) 62.97 26.8$\pm$6.3 56.84$\pm 8.5$ 83$\pm$29 63$\pm$30
$M$ (M⊙) 2.0$\pm$0.1 2.5209 2.26$\pm$0.65
$R$ (R⊙) 5.24 3.3$\pm$0.3 4.52$\pm 0.31$ 4.7$\pm$0.8 4.02$\pm$0.8
(1) McDonald et al. (2012); (2) from the empirical relation of Qian et al.
(2018); (3) Gaia DR2 (Andrae et al. 2018); (4) Gaia EDR3 (Gaia Collaboration
2020). The normal fonts mean literature values while italic fonts are the
calculated parameters.

McDonald et al. (2012) estimated the effective temperature and luminosity of
more than 100 000 stars observed by the Hipparcos satellite comparing their
optical and infrared spectral energy distribution (SED) with model
atmospheres. The obtained values of V764 Mon are shown in column 2 in Table 5.
No infrared excess, i.e. circumstellar material was detected for V764 Mon.

Qian et al (2018) published empirical relations for physical parameters of
$\delta$ Scuti stars based on the LAMOST spectroscopic survey. They isolated a
separate group of multi-periodic stars which are significantly cooler than the
normal $\delta$ Scuti stars. They pulsate with 0.09 and 0.22 d periods,
therefore, V764 Mon cannot be a member of this group. Qian et al. (2018)
consider longer period ($\sim 0.35$ d) stars as ‘normal’ $\delta$ Scuti stars.
Applying their formulas to V764 Mon we obtain the value of column 3 in Table
5. This effective temperature is in good agreement with that determined
McDonald et al. (2012) from their SED fit.

Gaia DR2 (Gaia Collaboration 2016, 2018, Andrae et al. 2018) estimated the
effective temperature, luminosity and radius of stars on the basis of their
three-band photometry ($G$, $G_{\mathrm{BP}}$ and $G_{\mathrm{RP}}$) and
parallax (col. 4 in Table 5). The recently released Gaia EDR3 (Gaia
Collaboration 2020) contains updated and several additional parameters for
V764 Mon ($\log g$, [Fe/H] and $M$) that were not yet included in DR2
catalogue (column 5 in Table 5), even though these values are actually based
on DR2 measurements (Recio-Blaco et al. 2016). These parameters are in good
agreement with those ones we obtained from different previous works. The only
exception is the effective temperature where the former studies support the
lower DR2 value.

Based on our spectra, we give an independent estimate of the atmospheric
parameters of V764 Mon. We prepared a cumulative median spectrum from the six
highest $S/N$ ratio spectra, then we fitted synthetic spectra between 5400 and
5900 Å to this median spectrum by minimizing $\chi^{2}$. The fitting process
was similar to what we described for T Sex in detail (Benkő et al. 2021). The
only difference was that we used Kurucz atmosphere models (Castelli and Kurucz
2003) that better fit our parameters instead of the MARCS GES models used for
T Sex. We applied the synthetic spectral fitting tool of iSpec package
(Blanco-Cuaresma et al. 2014; Blanco-Cuaresma 2019) for this process.
Different initial value combinations of $T_{\mathrm{eff}}$, $\log g$ and
[Fe/H] have chosen from the values in Table 5. The initial micro-turbulent
velocity was the default value of the algorithm ($v_{\mathrm{mic}}=1.05$ km
s-1). For the macro-turbulent velocity, the empirical relation offered by the
algorithm was accepted. During several runs, $v\sin i$ was also a free
parameter, with the initial value 188 km s-1. We obtained 199$\pm$20 km s-1
from the fit. Since this value is consistent with our previous result, we
fixed $v\sin i$ at 188 km s-1 for the further runs. The atmospheric parameters
resulting from the fit and their errors are shown in column 6 of Table 5. As
we have seen these values similar to the former estimations but they have
rather high standard deviation.

We completed Table 5 with several calculated parameters (italic numbers) using
the basic formulae of luminosity, $\log g$ and the calibrated empirical
relation for mass and radius by Torres et al. (2010). For pre-main sequence
stars, the latter relation can not be applied. If we plot our star on the
Hertzsprung-Russell diagram (HRD) using physical parameters from different
publications (Fig. 5), we find that the estimated position we obtained is the
closest to that of EDR3.

The position of V764 Mon in HR diagram from the different estimations listed
in Table 5. Light-grey symbols show normal $\delta$ Scuti stars classified by
the LAMOST spectroscopic survey (Qian et al. 2018). The straight line show the
instability strip of classical $\delta$ Scuti stars (Breger and Pamyatnykh
1998)

An additional physical parameter might be estimated. Searching for regular
patterns in the TESS pulsation frequencies in Table 2. with the SSA method
(Paparó et al. 2016a,b), we found a frequency sequence of 4 elements
consisting of $f_{1}$, $f_{4}$, $f_{6}$ and $f_{12}$. The average spacing
between the frequencies is 3.24$\pm 0.04$ d-1. If we assume this spacing value
to be the large separation ($\Delta\nu$), the value of V764 Mon fits well to
the theoretical relation between $\Delta\nu/\Delta\nu_{\odot}$ and
$\overline{\rho}/\overline{\rho}_{\odot}$ of Suárez et al. (2014). (The solar
values of the large separation and average density was the same as in Suárez
et al. 2014 work.) The average density of V764 Mon determined this way
($\overline{\rho}=0.159$ g / cm-3) is typical for a $\delta$ Scuti star.

###  2.7 Classification

To sum up, V764 Mon is a binary system with a fast rotating main component
showing small-amplitude, multiperiodic light variations originating from non-
radial pulsation. The star is above the main sequence in the HRD and,
depending on the parameter estimations, within or at the edge of the $\delta$
Scuti instability strip (Fig. 5).

This picture is disturbed only by the fact that the dominant period of the
star is unusually long compared to typical $\delta$ Scuti stars. The Kepler
$\delta$ Scuti sample of Balona and Dziembowski (2011) ends at less than 0.2
d. The most up-to-date Galactic $\delta$ Scuti catalogue (Chang et al. 2013)
gives a period of $\sim$0.25 days for the edge of the $\delta$ Scuti group.
Even though this catalogue contains six stars with $P>0.28$ d, by examining
these stars individually, it turns out that they either actually have a
shorter dominant period, or are not $\delta$ Scuti stars at all. However, the
LAMOST spectroscopic survey (Qian et al. 2018) identifies a number of $\delta$
Scuti stars up to a period of $\sim$0.35 days.

The evolutionary state of V764 Mon is unclear. Its position above the HRD
shows that it is a pre- or post-main sequence star. $\delta$ Scuti variation
of such stars are known (see Breger 2007; Zwintz et al. 2014 and references
therein). A distinction between these evolutionary stages based purely on the
position in the HRD is not possible because pre- and post-main sequence
evolutionary tracks intersect (Breger and Pamyatnykh 1998). Evolutionary
identification with detailed asteroseismology would only be possible with
accurate mode identification.

##  3 HY Com

###  3.1 Photometry

The variability of HY Com was discovered by Oja (1981). He determined the
period ($P=0.448614$ d) from $UBV$ photoelectic observations, and classified
the star as an RRc type pulsating variable. Oja detected a phase variation of
the light curve and drew the attention to the unusually long pulsation period
compared to other RRc variables. He re-observed the star in the 1990s (Oja
1995) and found that HY Com shows evident short-term period variation of
unknown nature. Using new photometric data that has become available in the
meantime (NSVS, Woźniak et al. 2004; ASAS-3 data between 2002 and 2006,
Pojmański et al. 2005), Wils (2008) confirmed the considerable phase/period
changes in the star. This work showed that the photoelectric $V$ light curve,
collected between 1994 and 2006, folded with the pulsation period shows only a
small scatter, assuming three ”sudden” period changes in 2001, 2004 and 2005.
The shape of the light curve seemed to be stable within the detection error.

Since then, the ASAS-3 photometric data were extended with three more
observing seasons (2006/2007, 2007/2008 and 2008/2009). For the proper fit of
these new observations to the phase curve of Wils (reproduced in Fig. 6) at
least one additional period change have to be assumed between the last two
seasons (2007/2008 and 2008/2009). The determined periods of the season
2008/2009 is $P_{1}=0.448698\pm 0.000017$ d, the assumed period change is
+11.4 s.

Phase curves of HY Com from the ASAS-3 observations (bottom curve), from the
Qatar Exoplanet Survey data (middle curve) and from ASAS-SN survey (top
curve). The $V$ magnitudes for ASAS and $V-0.7$ mag for ASAS-SN data are
indicated in the left axis, the QES unfiltered magnitudes are shown in the
right vertical axis. Points of ASAS and ASAS-SN data are marked in a different
colour for each observing season.

The Qatar Exoplanet Survey (QES, Alsubai et al. 2013, Bramich et al. 2014)
also provide us a large amount of photometric data (see Table 1). The
instrument of QES consists of five cameras with partially overlapping fields
of view. HY Com was observed by two cameras. The two data sets come from the
two cameras differ systematically, mainly around the maximum. Therefore, we
used only the data set with smaller standard deviation in our analysis. Even
so, we had 775 data points from a 50-day-long interval in 2012.

Analysing the data with the Period04 Fourier fitting tool, we found two
significant frequencies: the main frequency $f_{1}$ and its third harmonic
$3f_{1}$. This gives a pulsation period of $P_{1}=0.448783\pm 0.000014$ d for
2012, $7$ sec longer than that of the last ASAS-3 season (in 2008/2009).

The ASAS-SN survey (Shappee et al. 2014, Jayasinghe et al. 2019) observed HY
Com from its Hawaii station at Haleakala Observatory in standard $V$-band.
Some 650 data points were observed over five years between 2013 and 2018 (see
Table 1). Although the target was saturated, the pipeline handled the issue
relatively well (Kochanek et al. 2017).

We analysed the entire data set and the data sets of each season separately.
The main frequency $f_{1}$ was the only significant frequency in all cases.
Based on the periods of individual seasons, we can detect two further
significant period changes. The period between seasons 2013/14 and 2015/16 is
$P_{1}=0.448540\pm 0.000002$ d which is 21 s shorter than the period of the
QES data, while the last two seasons (between 2016/17 and 2017/18) it is again
longer with 9.8 s ($P_{1}=0.448632\pm 0.000004$ d) than the previous seasons.

Strong and continuous irregular period changes of RRc stars are rather common.
This behaviour is especially frequent amongst longer-period RRc stars (see
e.g. Jurcsik et al. 2001, 2015). Details of these period changes (e.g. how
abrupt are these) were unclear for a long time. Continuous Kepler measurements
have revealed that period changes occur in a few hundred days with continuous
change (Moskalik et al. 2015, Sódor et al. 2017), and not e.g. with a sudden
jump. However, the physical origins of the phenomenon are still unclear.

###  3.2 Spectroscopy

The spectroscopic observations and reduction were made for HY Com with the
same equipment and the same way as we described for V764 Mon in Sect. 2.3. The
log of observations is given in Table 3. As can be seen from Table 3, the
signal-to-noise ratio of the HY Com measurements is significantly lower than
that of the V764 Mon measurements. The most obvious use of spectroscopic data
of this quality is to determine the radial velocity curve. This is especially
important, since no such data has been published before.

The radial velocities were calculated with the same cross-correlating
technique as it was done for V764 Mon. Gaussian fitting functions were used
for determining the radial velocities from the the cross-correlation
functions. The estimated standard errors of the individual radial velocity
values are $\sim$1 – 2 km s-1. The radial velocities and their errors are
given in Table 6. cccccc7cmRadial velocities of HY Com BJD $v_{\mathrm{rad}}$
$\sigma(v_{\mathrm{rad}})$ BJD $v_{\mathrm{rad}}$ $\sigma(v_{\mathrm{rad}})$
(d) km s-1 km s-1 (d) km s-1 km s-1
2457088.50709 $-49.06$ 2.31 2457090.51150 $-26.85$ 1.58
2457088.52806 $-46.04$ 1.44 2457090.53246 $-27.14$ 2.12
2457088.54903 $-42.83$ 1.28 2457090.55341 $-27.53$ 1.89
2457088.57320 $-41.25$ 1.37 2457090.57530 $-29.55$ 2.01
2457088.61515 $-35.69$ 1.30 2457090.59625 $-32.05$ 1.74
2457088.63712 $-32.99$ 1.00 2457090.61720 $-41.75$ 2.68
2457088.65807 $-33.50$ 0.77 2457091.55324 $-51.62$ 1.19
2457090.42245 $-36.12$ 1.57 2457091.57640 $-55.81$ 1.82
2457090.44865 $-31.27$ 2.72 2457091.59736 $-58.43$ 2.46
2457090.46960 $-31.11$ 1.89 2457091.61831 $-52.83$ 1.74
2457090.49055 $-28.74$ 1.77

The radial velocity curve is constructed by folding the radial velocities with
the period of photometry closest to the measurements in time (ASAS-SN data,
Fig. 7). The total amplitude of curve is $\sim$32 km s1. Its shape is
asymmetric: a proper fit requires four harmonic components of the Fourier
solution. The mean velocity of HY Com from this four-element Fourier fit is
$-39.9\pm 0.7$ km s-1. Where different nights are overlapped in phase (around
$\phi=0.3-0.5$) a slight radial velocity shift can be seen which might be
caused by some period change.

Radial velocity curves of HY Com. The error bars show the calculated formal
error (typically 1-2 km s-1). The continuous line shows a four-element Fourier
fit. The initial phase ($\phi=0=1$) was set to the minimum of the radial
velocity curve.

##  4 Summary

On the basis of our spectra and the TESS photometric data we showed that the
brightest RRc candidate V764 Mon, which was also a candidate binary RR Lyrae
star, is, in fact, not an RR Lyrae star but a fast rotating $\delta$ Scuti
variable. Analysing the TESS data, we identified ten independent frequencies
that describe the multi-periodic light curve variation. Our spectra show
characteristic signs (time-dependent line-profile distortions) of non-radial
pulsation, which might explain most of the found frequencies in the
photometry. Using spectroscopic cross-correlation functions, we determined the
projected rotation velocity ($v\sin i=188\pm 5$ km s-1), which shows that V764
Mon is a fast rotating star. We also detected the presence of a companion
star, that is, we spectroscopically confirmed the binary nature of V764 Mon,
discovered by the Gaia satellite by astrometric means. Although the dominant
photometric period of V764 Mon is unusually long (0.29 d), all other
photometric and spectroscopic behaviour and parameters confirm its
classification as a $\delta$ Scuti pulsator.

Investigating the previously unanalyzed photometric data of HY Com, we found
that, in agreement with previous studies, the shape of the light curve is
surprisingly stable, but its phase is not constant over time. We need to
assume at least four phase changes in order to transform the new data into a
common ephemeris with the older ones. We also presented here the first
complete radial velocity curve of HY Com.

This work was supported by the Hungarian National Research, Development and
Innovation Office by the Grant NN-129075 and the Lendület Program of the
Hungarian Academy of Sciences, project No. LP2018-7/2018. JMB thanks to Dr M.
Paparó for her valuable comments.

## References

  * [1] Alsubai et al.201363465
  * [2] Ammler-von Eiff, M. and Reiners, A.2012Å542A116
  * [3] Andrae, R. et al.2018Å616A8
  * [4] Balona, L. A.2014a4371476
  * [5] Balona, L. A.2014b4393453
  * [6] Balona, L. A. and Dziembowski, W. A.2011417591
  * [7] Balona, L. A., Daszyńska-Daszkiewicz, J. and Pamyatnykh, A. A.20154523073
  * [8] Bellm, E. C. et al.2019131018002
  * [9] Benkő, J. M., Sódor, Á. and Pál, A.20215002554
  * [10] Blanco-Cuaresma, S.20194862075
  * [11] Blanco-Cuaresma, S., Soubiran, C., Heiter, U. and Jofré, P.2014Å569A111
  * [12] Bramich, D. M. et al.2014IBVS61061
  * [13] Breger, M.2007CoAst15025
  * [14] Breger, M. and Pamyatnykh, A. A.1998Å332958
  * [15] Breger, M. et al.1993Å271482
  * [16] Breger, M. et al.1999Å349225
  * [17] Bush, T. C. and Hintz, E. G.20081361061
  * [18] Castelli, F. and Kurucz, R. L. G.2003IAU Symp210A20
  * [19] Chang, S.-W., Protopapas, P., Kim, D.-W. and Byun, Y.-I.2013AJ145132
  * [20] Derekas, A. et al.20174641553
  * [21] ESA1997The Hipparcos and Tycho Catalogue1-17ESA SP-1200
  * [22] Gaia Collaboration2016Å595A1
  * [23] Gaia Collaboration2018Å616A1
  * [24] Gaia Collaboration2020Å arXiv:2012.01533
  * [25] Jayasinghe, T. et al.20194861907
  * [26] Jurcsik, J. et al.2001121951
  * [27] Jurcsik, J. et al.2015219id.25
  * [28] Kochanek, C. S. et al.2017129104502
  * [29] Kervella, P. et al.2019aÅ623A116
  * [30] Kervella, P. et al.2019bÅ623A117
  * [31] Lampens, P. et al.2018Å610A17
  * [32] Lenz, P. and Breger, M.2005CoAst.14653
  * [33] van Leeuwen, F. et al.1997Å323L61
  * [34] Liakos, A. and Niarchos, P.20174651181
  * [35] McDonald, I., Zijlstra, A. A. and Boyer, M. L.2012427343
  * [36] Maintz, G.2005Å442381
  * [37] Masci, F. J. et al.2019131018003
  * [38] Mathias, P. and Aerts, C.1996Å312905
  * [39] Mantegazza, L.2000ASP Conf Ser.210p. 138
  * [40] Monnier, J. D. et al.20107251192
  * [41] Moskalik, P. et al.20154472348
  * [42] Oja, T.1981Å103339
  * [43] Oja, T.1995IBVS42761
  * [44] Pál, A.20124211825
  * [45] Paparó, M., Benkő, J. M., Hareter, M. and Guzik, J. A.2016822100
  * [46] Paparó, M., Benkő, J. M., Hareter, M. and Guzik, J. A.201622441
  * [47] Pepper, J. et al.2007119923
  * [48] Pepper, J. et al.2012124230
  * [49] Plachy, E. et al.2021 accepted, arXiv:2012.09709
  * [50] Pojmański, G.200252397
  * [51] Pojmański, G.200353341
  * [52] Pojmański, G., Pilecki, B. and Szczygieł, D.200555275
  * [53] Preston, G. W., Sneden, C., Chadid, M., Thompson, I. B. and Shectman, S. A.2019157153
  * [54] Prša, A. et al.201622729
  * [55] Qian, S.-B., Li, L.-J., He, J.-J., Zhang, J., Zhu, L.-Y. and Han, Z.-T.2018475478
  * [56] Recio-Blanco, A. et al.2016Å585A93
  * [57] Ricker, G. R. et al.2015J. Astron. Tel. Instr. Sys.1id.014003
  * [58] Rodríguez, E., López-González, M. J. and López de Coca, P.2000144469
  * [59] Shappee B. J., et al.2014788id.48
  * [60] Sódor, Á., Skarkra, M., Liška, J. and Bognár, Zs.2017465L1
  * [61] Suárez, J. C., Bruntt, H. and Buzasi, D.2005Å438633
  * [62] Suárez, J. C. et al.2014Å563A7
  * [63] Tody, D.1986Proc. SPIE Instrumentation in AstronomyVIp. 627
  * [64] Tody, D.1993ASP Conference Ser.Vol 52p. 173
  * [65] Tokovinin, A., Thomas, S., Sterzik, M. and Udry, S.2006Å450681
  * [66] Torres, G., Andersen, J. and Giménez, A.2010A&AR1867
  * [67] Townsend, R. H. D., Owocki, S. P. and Howarth, I. D.2004350189
  * [68] Yang, S. and Walker, A. H.1986PASP981156
  * [69] Wils, P.2008Perem. Zvezdy28No. 1
  * [70] Woźniak, P. R. et al.20041272436
  * [71] Zima, W2008J. of Phys: Conf. Ser.118id.012014
  * [72] Zwintz, K. et al.2014Science34539
  * [73]


# Two-photon charging of a quantum battery with a Gaussian pulse envelope

C. A. Downing<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Exeter, Exeter EX4 4QL, United Kingdom  M. S. Ukhtary
Department of Physics and Astronomy, University of Exeter, Exeter EX4 4QL,
United Kingdom  Research Center for Quantum Physics, National Research and
Innovation Agency (BRIN), South Tangerang 15314, Indonesia

###### Abstract

ABSTRACT Quantum energy science is rapidly emerging as a domain interested in
the generation, transfer and storage of energy at the quantum level. In
particular, quantum batteries have the scope to exploit the wonders of quantum
mechanics in order to boost their performance as compared to their
electrochemical equivalents. Here we show how an exponential enhancement in
stored energy can be achieved with a quantum battery thanks to a two-photon
charging protocol. We consider theoretically a quantum battery modelled as a
quantum harmonic oscillator, which when driven by a quadratic field
(manifested by a Gaussian pulse envelope) gives rise to squeezing of the
battery. This quantum squeezing ensures that the population of the battery is
driven exponentially up its bosonic energy ladder. Our results demonstrate a
plausible mechanism for quickly storing a vast amount of energy in a quantum
object defined by continuous variables, which may be explored experimentally
in a variety of quantum optical platforms.

Keywords: quantum battery, quantum technology, energy storage, quantum
harmonic oscillator, quantum optics, squeezing.

##  1 Introduction

Three years ago an influential experiment exploiting ultrafast optical
spectroscopy demonstrated the impressive energy storage capacity of an organic
microcavity Quach2022 . These paradigmatic empirical results kickstarted a
journey towards the creation of a quantum device enabling the temporary
hoarding of energy, commonly known as a quantum battery. This pivotal
experiment itself came in the wake of a pioneering theoretical proposal for a
quantum battery by Alicki and Fannes back in 2013 Alicki2013 , with further
and significant theory developments arising in the years immediately
afterwards Binder2015 ; Campaioli2017 ; Campaioli2023 .

Groundbreaking experiments have recently revealed how one can benefit from
dark states in the charging of a battery based upon a superconducting qutrits
Hu2022 ; Zheng2023 , as well as probing the charging of photonic quantum
batteries using single-photon interferometric networks Qu2023 , and exploring
the impact of entanglement on the charging performance of quantum batteries
within a linear optical platform Huang2023 . In parallel, the charging of
quantum batteries has also been paid its due theoretical attention, including
studies of the influence of various external time-dependent drives on the
efficacy of the battery Zhang2019 ; Chen2020 ; NewCrescente2020 ; Ukhtary2024
; Gemme2024 , as well as specialist setups exploiting directional interactions
Sturges2022 ; Ahmadi2024 . The aim of such research is to propose and realize
a charging protocol which enables the storage of large amounts of energy on a
short timescale, ensuring an impressive charging power of the quantum battery
device Campaioli2023 .

Here we consider an alternative approach to charge a quantum battery, by
exploiting a quadratic driving field Savona2017 ; Savona2019 ; Campinas2023 ;
Downing2023 . In particular, we consider a two-photon drive Leghtas2015 ;
Wang2019 ; Gaikwad2023  carrying a Gaussian pulse envelope as a prototypical
example of our proposed charging protocol. The quadratic nature of the drive
induces quantum squeezing in the system Walls1983 ; Loudon1987 , which acts to
propel the mean population of the battery exponentially up its energy ladder.
In what follows we present the analytic theory underpinning our idea,
including explicit expressions for the enhanced stored energy, charging time
and charging power.

Figure 1: Charging the quantum battery. Panel (a): a cartoon of the quantum
battery model [cf. Eq. (1)], represented by a quantum harmonic oscillator
(associated with the bosonic operator $b$) with the energy level spacing
$\omega_{b}$ [cf. Eq. (2)]. The oscillator is driven quadratically by a pulse
envelope (green line) of dimensionless strength $\zeta$ [cf. Eq. (3)]. In this
drawing, the battery is storing an energy $E_{b}=3\omega_{b}$ (three yellow
bars). Panel (b): the temporal shape of the pulse envelope $f(t)$ is
considered to be Gaussian [cf. Eq. (4)].

##  2 Model

The total Hamiltonian operator $\hat{H}$ of the quantum battery model is
comprised of just two pieces [cf. the sketch of Fig. 1 (a)]

$\hat{H}=\hat{H}_{b}+\hat{H}_{d}.$ (1)

Here the bare Hamiltonian $\hat{H}_{b}$ describes the energy ladder of the
battery (white and yellow bars in the cartoon), which is characterized by its
regular energy level spacing $\omega_{b}$. Energy enters the battery via a
pulse envelope $f(t)$, which has the dimensionality of frequency and which is
carried by some cosinusoidal quadratic driving. The driving Hamiltonian
$\hat{H}_{d}$ is also distinguished by some dimensionless strength $\zeta>0$.
Together these constituent Hamiltonians then read (we set $\hbar=1$
throughout)

$\displaystyle\hat{H}_{b}$ $\displaystyle=\omega_{b}\,b^{\dagger}b,$ (2)
$\displaystyle\hat{H}_{d}$
$\displaystyle=\zeta\cos\left(2\omega_{d}t\right)f(t)\left(b^{\dagger}b^{\dagger}+bb\right),$
(3)

where the bosonic creation and annhilhiation operators $b^{\dagger}$ and $b$
satisfy the commutation rule $[b,b^{\dagger}]=1$, and which act to track the
scaling of excitations up and down the bosonic energy ladder. Notably, quantum
harmonic oscillators are known to be excellent models for quantum optical
battery systems in the real world Qu2023 . The two-photon cosinusoidal carrier
driving is of frequency $2\omega_{d}$, while the pulse envelope $f(t)$ is
considered to be of a Gaussian character [cf. the plot of Fig. 1 (b)]

$f(t)=\frac{\mathrm{e}^{-\frac{t^{2}}{2\tau^{2}}}}{\sqrt{2\pi\tau^{2}}}.$ (4)

Clearly, the pulse has a maximum exactly at the instant in time $t=0$, where
$f(0)=1/\sqrt{2\pi\tau^{2}}$, and the pulse displays a full width at half
maximum of $2\sqrt{2\ln\left(2\right)}\tau\simeq 2.35\tau$, where the temporal
height and width parameter $\tau\geq 0$. The limiting case of $\lim_{\tau\to
0}f(t)=\delta(t)$ recovers a toy model Dirac delta function problem
Christensen1979 ; Downing2024 .

Figure 2: Energy storage in the quantum battery. Panel (a): The energy $E_{b}$
stored in the battery [cf. Eq. (13)] in units of the upper energetic bound
$\omega_{b}\sinh^{2}\left(\zeta\right)$, as a function of time $t$ (in units
of the pulse temporal parameter $\tau$). Dashed grey line: the equivalent
result in the delta pulse limit [cf. Eq. (14)]. Panel (b): the charging time
$t_{\alpha}$ (which has elapsed since the central action of the pulse at
$t=0$) at which the battery holds the fraction $\alpha$ of its energetic
maximum. In panels (a) and (b), we consider several values of the
dimensionless pulse strength $\zeta$ as marked in panel (a). Panel (c): a
logarithmic plot of the generalized quadrature variances
$\sigma_{X_{\theta}}^{2}$ (light green line) and $\sigma_{P_{\theta}}^{2}$
(dark green line) as a function of the quadrature twist angle $\theta$ [cf.
Eq. (23) and Eq. (24)]. Dashed green line: the product of the standard
deviations $\sigma_{X_{\theta}}\sigma_{P_{\theta}}$. Thin grey line: a guide
for the eye at $1/2$, the Robertson–Schrödinger minimum uncertainty. In panel
(c) we take $\zeta=2$ and we consider the behaviour at the snapshot in time
when $t=0$.

The quantum state of the quantum battery may be described by a density matrix
$\rho$, which evolves in time according to the Liouville–von Neumann equation
$\partial_{t}\rho=\mathrm{i}[\rho,\hat{H}]$, where the Hamiltonian operator
$\hat{H}$ is given by Eq. (1). Upon switching to a rotating frame with the aid
of the time-dependent unitary transformation
$U=\exp{(\mathrm{i}\hat{H}_{b}t)}$ [cf. Eq. (2)], the quantum state $\rho$
transforms into the rotated version $\tilde{\rho}=U\rho U^{\dagger}$, whose
dynamics are instead governed by the twisted von Neumann equation

$\partial_{t}\tilde{\rho}=\mathrm{i}\left[\tilde{\rho},\hat{\mathcal{H}}\right],$
(5)

where the rotated Hamiltonian operator
$\hat{\mathcal{H}}=U\hat{H}_{d}U^{\dagger}$. At driving resonance (where
$\omega_{d}=\omega_{b}$) this object simplifies into

$\hat{\mathcal{H}}=\tfrac{\zeta}{2}f(t)\left(b^{\dagger}b^{\dagger}+bb\right),$
(6)

where we have also neglected the quickly rotating terms coming from the cosine
function first appearing in Eq. (3).

We are primarily interested in two figures of merit which appraise the
performance of the proposed quantum battery: the dynamical energy $E_{b}$
stored in the battery and the instantaneous power $P_{b}$ of the device. These
two quantities are defined by Campaioli2023

$\displaystyle E_{b}$ $\displaystyle=\omega_{b}\langle b^{\dagger}b\rangle,$
(7) $\displaystyle P_{b}$ $\displaystyle=\partial_{t}E_{b}.$ (8)

The stored energy $E_{b}$ is counted using the mean population $\langle
b^{\dagger}b\rangle$ of the battery. For example, in the cartoon of Fig. 1 (a)
the three yellow bars represent the storing of an energy $E_{b}=3\omega_{b}$
in the quantum battery. Meanwhile, the instantaneous power $P_{b}$ is a
popular measure of the charging prowess of quantum batteries Salamon2020 ;
Kamin2020 ; Rosa2022 . A third quantity, the so-called ergotropy
$\mathcal{E}$, describes how much work can be extracted from the quantum
battery Alicki2013 ; Allahverdyan2004 . The thermodynamic efficiency of the
proposed model of Eq. (1) is perfect ($E_{b}=\mathcal{E}$) since the quantum
harmonic oscillator is only coherently pumped (it starts in the vacuum state
at $t\to-\infty$ and then it evolves forwards in time according to the
quadratic Hamiltonian of Eq. (1), such that it remains in a coherent state at
all times, with zero passive state energy). This thermodynamic point is
discussed further in Ref. Farina2019  and in the Supplementary Material.

##  3 Results

The mean population of the system $\langle b^{\dagger}b\rangle$ can be readily
computed using the trace propery $\operatorname{Tr}{(O\tilde{\rho})}=\langle
O\rangle$ for an arbitrary operator $O$, where the quantum state
$\tilde{\rho}$ is defined using Eq. (5) and Eq. (6). Carrying out this tracing
process leads to the following equation of motion for the three second moments
of the quantum battery

$\mathrm{i}\partial_{t}\Psi=\mathcal{M}\Psi+\mathcal{P},$ (9)

where the mean values of the operators are housed within $\Psi$, the driving
term is $\mathcal{P}$ and the dynamical matrix is $\mathcal{M}$. These three
objects are together defined by

$\Psi=\begin{pmatrix}\langle b^{\dagger}b\rangle\\\ \langle bb\rangle\\\
\langle b^{\dagger}b^{\dagger}\rangle\\\
\end{pmatrix}\quad\quad\mathcal{P}=\begin{pmatrix}0\\\ \zeta f(t)\\\ -\zeta
f(t)\\\ \end{pmatrix},$ (10)
$\setstackgap{L}{1.1}\fixTABwidth{T}\mathcal{M}=\parenMatrixstack{0&-\zeta
f(t)\zeta f(t)\\\ 2\zeta f(t)00\\\ -2\zeta f(t)00}.$ (11)

The exact solution of Eq. (9) can be found using the variation of parameters
method. By applying the boundary condition of all of the second moments being
zero when the driving pulse is vanishing (that is, when $t\to-\infty$), the
exact solution in terms of a special function read $\langle
b^{\dagger}b\rangle=\sinh^{2}(\zeta[1+\operatorname{erf}{(y)}]/2)$, $\langle
bb\rangle=-\mathrm{i}\sinh(\zeta[1+\operatorname{erf}{(y)}])/2$ and its
complex conjugate $\langle
b^{\dagger}b^{\dagger}\rangle=\mathrm{i}\sinh(\zeta[1+\operatorname{erf}{(y)}])/2$,
with the scaled time variable $y=t/\sqrt{2}\tau$. Here we have employed the
so-called error function $\operatorname{erf}{\left(z\right)}$, as defined by
its integral representation

$\operatorname{erf}{\left(z\right)}=\frac{2}{\sqrt{\pi}}\int_{0}^{z}\mathrm{e}^{-x^{2}}\mathrm{d}x.$
(12)

Hence, using the definition of Eq. (7), the total energy $E_{b}$ stored in the
quantum battery over time is described by

$E_{b}=\omega_{b}\sinh^{2}\left(\tfrac{\zeta}{2}\left[1+\operatorname{erf}{\left(\tfrac{t}{\sqrt{2}\tau}\right)}\right]\right).$
(13)

This short expression features a remarkable exponential enhancement in energy
storage (as compared to batteries driven by linear fields Ukhtary2024 ),
rendering the proposed charging protocol as potentially interesting for future
experimental study. The energetic formula of Eq. (13), scaled by the
($t\to\infty$) asymptotic energetic maximum $\omega_{b}\sinh^{2}(\zeta)$, is
plotted as a function of time in Fig. 2 (a). The coloured curves highlight the
smooth step-like profile of the stored energy $E_{b}$ for several values of
the dimensionless drive strength $\zeta$. In the extremely sharp pulse
($\tau\to 0$) limit, where the envelope profile $f(t)$ of Eq. (4) behaves like
a Dirac delta function, Eq. (13) reduces to the even simpler form

$\lim_{\tau\to
0}E_{b}=\omega_{b}\sinh^{2}\left(\zeta\right)\Theta\left(t\right),$ (14)

where $\Theta\left(t\right)$ is the Heaviside step function. This
straightforward limit is plotted by the dashed grey line in Fig. 2 (a).

Clearly from Fig. 2 (a), the maximum possible stored energy is only achieved
asymptotically (when $t\to\infty$). This fact stems from the unboundedness
(from above) of the energy levels defining the quantum battery, since we
employ the infinite energy ladder of a quantum harmonic oscillator in our
model [cf. Eq. (2)]. In practice, this infinity is not problematic and a
physical realization will have a large but finite number of energy levels. The
charging time $t_{\alpha}$ at which the battery holds the fraction
$0\leq\alpha\leq 1$ of its energetic maximum is given by the inversion of Eq.
(13) as

$t_{\alpha}=\sqrt{2}\tau\operatorname{erf}^{-1}{\left(\frac{2}{\zeta}\operatorname{arcsinh}\left[\sqrt{\alpha\sinh^{2}\left(\zeta\right)}\right]-1\right)}.$
(15)

This charging time $t_{\alpha}$ (measuring the elapsed time since the central
action of the pulse at $t=0$) is plotted in Fig. 2 (b) for several values of
$\zeta$ [those corresponding to panel (a)]. In particular, the limiting
behaviours of Eq. (15) at very small and very large $\zeta$ are encapsulated
by the formulae

$\displaystyle\lim_{\zeta\ll 1}t_{\alpha}$
$\displaystyle=\sqrt{2}\tau\operatorname{erf}^{-1}{\left(2\sqrt{\alpha}-1\right)},$
(16) $\displaystyle\lim_{\zeta\gg 1}t_{\alpha}$
$\displaystyle=\tau\sqrt{\ln\left(\frac{2}{\pi\ln^{2}\left(\alpha\right)}\frac{\zeta^{2}}{\ln\left(\frac{2\zeta^{2}}{\pi\ln^{2}\left(\alpha\right)}\right)}\right)}.$
(17)

When $\zeta\ll 1$ and Eq. (16) holds [cyan line in panel (b)] one notices that
$t_{\alpha}\sim\tau$ for a large range of energetic fractions $\alpha$ (at
least, away from $\alpha=1$). When $\zeta\gg 1$, such that Eq. (17) is a good
approximation for large enough $\alpha$ [yellow line in panel (b)], the
logarithmic character of this charging time $t_{\alpha}$ is most apparent in
the curves of Fig. 2 (b).

Figure 3: Power in the quantum battery. Panel (a): The instantaneous power
$P_{b}$ [cf. Eq. (18)], in units of
$\omega_{b}\zeta\sinh\left(2\zeta\right)/\tau$, as a function of time $t$ (in
units of the pulse temporal parameter $\tau$). We consider several values of
the dimensionless pulse strength $\zeta$ as marked by the plot legend. Panel
(b): the time $t_{P}$ (measured from the central action of the pulse at $t=0$)
at which the power $P_{b}$ is at its maximum as a function of $\zeta$. Dotted
cyan line: a quadratic approximation in the regime $\zeta\ll 1$ [cf. Eq.
(19)]. Dashed pink line: the asymptotic expression when $\zeta\gg 1$ [cf. Eq.
(20)]. Thin yellow line: a logarithmic approximation for $\zeta\gg 1$ [cf. Eq.
(21)]. Panel (c): the maximum power $P(t_{P})$, in units of $\omega_{b}/\tau$,
as a function of $\zeta$. Dashed pink line: the approximate expression of Eq.
(22).

The instantaneous power $P_{b}$ of the quantum battery follows directly from
the definition of Eq. (8) and the result of Eq. (13) as

$P_{b}=\frac{\omega_{b}}{\tau}\frac{\zeta}{\sqrt{2\pi}}\sinh\left(\zeta\left[1+\operatorname{erf}{\left(\tfrac{t}{\sqrt{2}\tau}\right)}\right]\right)\mathrm{e}^{-\frac{t^{2}}{2\tau^{2}}}.$
(18)

We plot the power $P_{b}$, in units of
$\omega_{b}\Omega\sinh\left(2\zeta\right)/\tau$, as a function of time in Fig.
3 (a) for the same spread of dimensionless pulse strengths $\zeta$ as in Fig.
2 (a, b). Naturally, the exponential enhancement in the stored energy is
likewise found in the power $P_{b}$ (please note the
$\sinh\left(2\zeta\right)$ scaling in the unit used in the plot). Larger
values of $\zeta$ act to delay the moment of maximum power as shown in Fig. 3
(a). The turning points of Eq. (18) suggest that the maximum power occurs at
the instant in time $t=t_{P}$, such that $\max_{t}{\\{P_{b}\\}}=P_{b}(t_{P})$.
This key time $t_{P}$ is plotted as function of the dimensionless pulse
strength $\zeta$ with the purple line in Fig. 3 (b). An asymptotic analysis of
the transcendental turning point equation yields the limiting behaviours at
small and large $\zeta$ as follows

$\displaystyle\lim_{\zeta\ll 1}t_{P}$ $\displaystyle\simeq 0.506\tau,$ (19)
$\displaystyle\lim_{\zeta\gg 1}t_{P}$
$\displaystyle=\tau\sqrt{W\left(\tfrac{2\zeta^{2}}{\pi}\right)}.$ (20)

When $\zeta\ll 1$ the peak power occurs at the specific delay time
$t_{P}\simeq 0.506\tau$, and this lower temporal bound grows quadratically in
$\zeta$ from then on, as is sketched with the dotted cyan line in Fig. 3 (b).
Conversely, when $\zeta\gg 1$ the time of peak power nicely follows Eq. (20),
which is given in terms of the Lambert $W$-function (sometimes called the
product logarithm function). This special function is defined as the inverse
function of the equation $f(W)=W\exp{(W)}$. The slowly growing formula of Eq.
(20) is plotted with the dashed pink line in Fig. 3 (b), and it can be
reasonably approximated using de Bruijn’s expansion in terms of simple
logarithms (thin yellow line) as

$\lim_{\zeta\gg
1}t_{P}\simeq\tau\sqrt{\ln{\left(u\right)}-\ln{\left(\ln{u}\right)}+\tfrac{\ln{\left(\ln{u}\right)}}{\ln{\left(u\right)}}},$
(21)

with the dimensionless variable $u=2\zeta^{2}/\pi$. This underlines the rapid
charging response of the proposed quantum battery. The maximum power
$P_{b}(t_{P})$ of the quantum battery is similarly plotted as the purple line
in Fig. 3 (c), highlighting the extremely large charging powers which can be
generated thanks to the exponential enhancement induced in this model. The
result can be nicely approximated by the expression

$P\left(t_{P}\right)\simeq\frac{\omega_{b}}{\tau}\frac{\mathrm{e}^{2\left(\zeta-1/3\right)}}{4}\sqrt{W\left(\tfrac{2\zeta^{2}}{\pi}\right)},$
(22)

which is marked by the dashed pink line in panel (c). The formula of Eq. (22)
matches well the orders of magnitude increase in maximum instantaneous power
with growing dimensionless drive strength $\zeta$.

It is important to charge the device as fast as is practicable, which
necessitates a maximization of the average power $\bar{P}$ (stored energy per
unit time) of the quantum battery Ferraro2018 ; Rosa2024 . This average
charging power $\bar{P}$ can be estimated here by considering the amount of
energy $E_{b}(t)$ deposited during some time interval $t_{-}\leq t\leq t_{+}$
[cf. Eq. (13)]. It is convenient to consider the two times
$t_{\pm}=\pm\sqrt{2\ln(2)}\tau\simeq\pm 1.18\tau$, which correspond to the
temporal boundaries which define the full width at half maximum of the driving
pulse envelope [cf. Fig. 1 (b)]. With this choice of time interval, the
average charging power
$\bar{P}\simeq\omega_{b}\sinh(\zeta)\sinh(\zeta\operatorname{erf}{[\sqrt{\ln(2)}]})/(2\sqrt{2\ln(2)}\tau)$
features a highly desirable exponential enhancement, as is by now familiar due
to the proposed charging protocol. This charging boost is most easily seen
when the drive strength is strong enough ($\zeta\gg 1$), such that the average
power reduces to the explicitly exponential expression
$\bar{P}\simeq\omega_{b}\exp{[\zeta(1+\operatorname{erf}{[\sqrt{\ln(2)}]})]}/(8\sqrt{2\ln(2)}\tau)$.

The quantum squeezing induced by the quadratic nature of the pulse envelope
[cf. Eq. (3)] can be better understood by considering the generalized
quadrature variance
$\sigma_{X_{\theta}}^{2}=\langle\hat{X}_{\theta}^{2}\rangle-\langle\hat{X}_{\theta}\rangle^{2}$
and its partner quantity
$\sigma_{P_{\theta}}^{2}=\langle\hat{P}_{\theta}^{2}\rangle-\langle\hat{P}_{\theta}\rangle^{2}$
Walls1983 ; Loudon1987 . The two operators of generalized position and
momentum are defined by
$\hat{X}_{\theta}=(\exp{(\mathrm{i}\theta/2)}b^{\dagger}+\exp{(-\mathrm{i}\theta/2)}b)/\sqrt{2}$
and
$\hat{P}_{\theta}=\mathrm{i}(\exp{(\mathrm{i}\theta/2)}b^{\dagger}-\exp{(-\mathrm{i}\theta/2)}b)/\sqrt{2}$
respectively, where $\theta$ is an angle allowing for a twist of the
quadratures. Together, these operators satisfy the canonical commutation
relation $[\hat{X}_{\theta},\hat{P}_{\theta}]=\mathrm{i}$. The exact solution
of Eq. (9) then leads (back in the original frame) to the twin expressions

$\displaystyle\sigma_{X_{\theta}}^{2}$
$\displaystyle=\tfrac{1}{2}+\sinh^{2}\left(\tfrac{\zeta\xi}{2}\right)-\tfrac{1}{2}\sin\left(2\omega_{b}t+\theta\right)\sinh\left(\zeta\xi\right),$
(23) $\displaystyle\sigma_{P_{\theta}}^{2}$
$\displaystyle=\tfrac{1}{2}+\sinh^{2}\left(\tfrac{\zeta\xi}{2}\right)+\tfrac{1}{2}\sin\left(2\omega_{b}t+\theta\right)\sinh\left(\zeta\xi\right),$
(24)

where the variable $\xi=1+\operatorname{erf}{(t/\sqrt{2}\tau)}$ also carries a
time dependency. We plot the variances $\sigma_{X_{\theta}}^{2}$ (light green
line) and $\sigma_{P_{\theta}}^{2}$ (dark green line) in Fig. 2 (c) for the
driving strength $\zeta=2$, and at the instant coinciding with the peak of the
pulse ($t=0$). The plot showcases the $\hat{X}_{\theta}$–squeezed nature of
the system, since $\sigma_{X_{\theta}}^{2}$ dips below the
Robertson–Schrödinger minimum uncertainty of $1/2$ (thin grey line). Meanwhile
the dashed green line in Fig. 2 (c) confirms that the product of standard
deviations always obeys the celebrated rule
$\sigma_{X_{\theta}}\sigma_{P_{\theta}}\geq 1/2$. These important (and phase-
dependent) quantities can be readily probed using homodyne detection
experiments Shaked2018 ; Porto2018 . This brief quadrature analysis serves to
confirm the role of the dimensionless pulse strength $\zeta$ as a kind of
squeezing parameter for the proposed quantum battery model [cf. Eq. (23) and
Eq. (24)].

In order to support our idea of exploiting quantum squeezing in order to boost
the performance of quantum batteries, we have built a minimal working theory
by considering a closed quantum system [cf. Eq. (2) and Eq. (3)]. Of course,
the effect of the external environment can also be important for the
functioning of quantum batteries Farina2019 ; Barra2019 ; Carrega2020 ;
Seah2021 ; Smirne2023 , although – quite remarkably – the expected performance
degradation of quantum batteries as open quantum systems can be stabilized in
certain circumstances Gherardini2020 ; Munro2020 ; Mitchison2021 ; Morrone2023
. For our purposes here, we have checked that the exponential enhancement in
stored energy which we have reported is indeed robust against the inclusion of
dissipation, as we show in the Supplementary Material. We have also listed
some auxiliary analytical results for other pulse functions $f(t)$ (including
for hyperbolic secant, Pöschl–Teller, algebraic and Lorentzian shapes) in the
Supplementary Material to complement the Gaussian choice of Eq. (4).

##  4 Conclusion

In conclusion, we have suggested a charging protocol for a quantum battery
which gives rise to an exponential enhancement in both stored energy and
charging power. The main idea is to utilize a quadratic driving field, here
primarily modelled with a Gaussian pulse envelope, in order to induce quantum
squeezing into the system. This squeezing in turn boosts the mean population
of the bosonic battery in an energetically (and ergotropically) favourable
manner. We hope that our theoretical proposal can stimulate further
experimental work in the area of quantum energy storage Hu2022 ; Zheng2023 ;
Qu2023 ; Huang2023 , with an overall view towards advancing the nascent field
of quantum energy science Campaioli2023 ; Metzler2023 .

## Acknowledgements

CAD is supported by the Royal Society via a University Research Fellowship
(URF/R1/201158) and by Royal Society Enhanced Research Expenses which support
MSU. CAD also gratefully acknowledges an Exeter-FAPESP SPRINT grant with the
Universidade Federal de São Carlos, and thanks R. Bachelard, A. Cidrim and C.
J. Villas-Boas for fruitful discussions.

DECLARATION OF COMPETING INTERESTS
The authors have no conflicts to disclose.

DATA AVAILABILITY
No data were generated or analyzed in the presented research.

ORCID.
C. A. Downing: 0000-0002-0058-9746.
M. S. Ukhtary: 0000-0001-5197-7354.

## References

  * (1) J. Q. Quach, K. E. McGhee, L. Ganzer, D. M. Rouse, B. W. Lovett, E. M. Gauger, J. Keeling, G. Cerullo, D. G. Lidzey and T. Virgili, Superabsorption in an organic microcavity: toward a quantum battery, Sci. Adv. 8, eabk3160 (2022).
  * (2) R. Alicki and M. Fannes, Entanglement boost for extractable work from ensembles of quantum batteries, Phys. Rev. E 87, 042123 (2013).
  * (3) F. C. Binder, S. Vinjanampathy, K. Modi and J. Goold, Quantacell: powerful charging of quantum batteries, New J. Phys. 17, 075015 (2015).
  * (4) F. Campaioli, F. A. Pollock, F. C. Binder, L. Céleri, J. Goold, S. Vinjanampathy and K. Modi, Enhancing the charging power of quantum batteries, Phys. Rev. Lett. 118, 150601 (2017).
  * (5) For a review on quantum batteries, see: F. Campaioli, S. Gherardini, J. Q. Quach, M. Polini and G. M. Andolina, Colloquium: quantum batteries, arXiv:2308.02277.
  * (6) C.-K. Hu, J. Qiu, P. J. P. Souza, J. Yuan, Y. Zhou, L. Zhang, J. Chu, X. Pan, L. Hu, J. Li, Y. Xu, Y. Zhong, S. Liu, F. Yan, D. Tan, R. Bachelard, C. J. Villas-Boas, A. C. Santos and D. Yu, Optimal charging of a superconducting quantum battery, Quantum Sci. Technol. 7, 045018 (2022).
  * (7) Y. Ge, X. Yu, W. Xin, Z. Wang, Y. Zhang, W. Zheng, S. Li, D. Lan and Y. Yu, Efficient charging and discharging of a superconducting quantum battery through frequency-modulated driving, Appl. Phys. Lett. 123, 154002 (2023).
  * (8) D. Qu, X. Zhan, H. Lin and P. Xue, Experimental optimization of charging quantum batteries through a catalyst system, Phys. Rev. B 108, L180301 (2023).
  * (9) X. Huang, K. Wang, L. Xiao, L. Gao, H. Lin and P. Xue, Demonstration of the charging progress of quantum batteries, Phys. Rev. A 107, L030201 (2023).
  * (10) Y.-Y. Zhang, T.-R. Yang, L. Fu and X. Wang, Powerful harmonic charging in a quantum battery, Phys. Rev. E 99, 052106 (2019).
  * (11) J. Chen, L. Zhan, L. Shao, X. Zhang, Y. Zhang and X. Wang, Charging quantum batteries with a general harmonic driving field, Ann. Physik 532, 1900487 (2020).
  * (12) A. Crescente, M. Carrega, M. Sassetti and D. Ferraro, Charging and energy fluctuations of a driven quantum battery, New J. Phys. 22, 063057 (2020).
  * (13) C. A. Downing and M. S. Ukhtary, Energetics of a pulsed quantum battery, EPL 146, 10001 (2024).
  * (14) G. Gemme, M. Grossi, S. Vallecorsa, M. Sassetti and D. Ferraro, Qutrit quantum battery: comparing different charging protocols, Phys. Rev. Research 6, 023091 (2024).
  * (15) C. A. Downing and T. J. Sturges, Directionality between driven-dissipative resonators, EPL 140, 35001 (2022).
  * (16) B. Ahmadi, P. Mazurek, P. Horodecki and S. Barzanjeh, Nonreciprocal quantum batteries, Phys. Rev. Lett. 132, 210402 (2024).
  * (17) V. Savona, Spontaneous symmetry breaking in a quadratically driven nonlinear photonic lattice, Phys. Rev. A 96, 033826 (2017).
  * (18) R. Rota, F. Minganti, C. Ciuti and V. Savona, Quantum critical regime in a quadratically driven nonlinear photonic lattice, Phys. Rev. Lett. 122, 110405 (2019).
  * (19) C. A. Downing and A. Vidiella-Barranco, Parametrically driving a quantum oscillator into exceptionality, Sci. Rep. 13, 11004 (2023).
  * (20) C. A. Downing and M. S. Ukhtary, A quantum battery with quadratic driving, Commun. Phys. 6, 322 (2023).
  * (21) Z. Leghtas, S. Touzard, I. M. Pop, A. Kou, B. Vlastakis, A. Petrenko, K. M. Sliwa, A. Narla, S. Shankar, M. J. Hatridge, M. Reagor, L. Frunzio, R. J. Schoelkopf, M. Mirrahimi and M. H. Devoret, Confining the state of light to a quantum manifold by engineered two-photon loss, Science 347, 853 (2015).
  * (22) Z. Wang, M. Pechal, E. A. Wollack, P. Arrangoiz-Arriola, M. Gao, N. R. Lee and A. H. Safavi-Naeini, Quantum dynamics of a few-photon parametric oscillator, Phys. Rev. X 9, 021049 (2019).
  * (23) C. Gaikwad, D. Kowsari, W. Chen and K. W. Murch, Observing parity-time symmetry breaking in a Josephson parametric amplifier, Phys. Rev. Research 5, L042024 (2023).
  * (24) D. F. Walls, Squeezed states of light, Nature 306, 141 (1983).
  * (25) R. Loudon and P. L. Knight, Squeezed light, J. Mod. Opt. 34, 709 (1987).
  * (26) M. J. Christensen and J. Piepenbrink, Quantum dynamics of Hamiltonians perturbed by pulses, J. Math. Phys. 20, 2187 (1979).
  * (27) C. A. Downing and M. S. Ukhtary, Hyperbolic enhancement of a quantum battery, Phys. Rev. A 109, 052206 (2024).
  * (28) S. Julià-Farré, T. Salamon, A. Riera, M. N. Bera and M. Lewenstein, Bounds on the capacity and power of quantum batteries, Phys. Rev. Research 2, 023113 (2020).
  * (29) F. H. Kamin, F. T. Tabesh, S. Salimi and A. C. Santos, Entanglement, coherence, and charging process of quantum batteries, Phys. Rev. E 102, 052109 (2020).
  * (30) J.-Y. Gyhm, D. Šafránek and D. Rosa, Quantum charging advantage cannot be extensive without global operations, Phys. Rev. Lett. 128, 140501 (2022).
  * (31) A. E. Allahverdyan, R. Balian and Th. M. Nieuwenhuizen, Maximal work extraction from finite quantum systems, EPL 67, 565 (2004).
  * (32) D. Ferraro, M. Campisi, G. M. Andolina, V. Pellegrini and M. Polini, High-power collective charging of a solid-state quantum battery, Phys. Rev. Lett. 120, 117702 (2018).
  * (33) J.-Y. Gyhm, D. Rosa and D. Šafránek, Minimal time required to charge a quantum system, Phys. Rev. A 109, 022607 (2024).
  * (34) Y. Shaked, Y. Michael, R. Z. Vered, L. Bello, M. Rosenbluh and A. Pe’er, Lifting the bandwidth limit of optical homodyne measurement with broadband parametric amplification, Nat. Commun. 9, 609 (2018).
  * (35) C. Porto, D. Rusca, S. Cialdi, A. Crespi, R. Osellame, D. Tamascelli, S. Olivares and M. G. A. Paris, Detection of squeezed light with glass-integrated technology embedded into a homodyne detector setup, J. Opt. Soc. Am. B 35, 1596 (2018).
  * (36) D. Farina, G. M. Andolina, A. Mari, M. Polini and V. Giovannetti, Charger-mediated energy transfer for quantum batteries: an open-system approach, Phys. Rev. B 99, 035421 (2019).
  * (37) F. Barra, Dissipative charging of a quantum battery, Phys. Rev. Lett. 122, 210601 (2019).
  * (38) M. Carrega, A. Crescente, D. Ferraro and M. Sassetti, Dissipative dynamics of an open quantum battery, New J. Phys. 22, 083085 (2020).
  * (39) S. Seah, M. Perarnau-Llobet, G. Haack, N. Brunner and S. Nimmrichter, Quantum speed-up in collisional battery charging, Phys. Rev. Lett. 127, 100601 (2021).
  * (40) D. Morrone, M. A. C. Rossi, A. Smirne and M. G. Genoni, Charging a quantum battery in a non-Markovian environment: a collisional model approach, Quantum Sci. Technol. 8, 035007 (2023).
  * (41) S. Gherardini, F. Campaioli, F. Caruso and F. C. Binder, Stabilizing open quantum batteries by sequential measurements, Phys. Rev. Research 2, 013095 (2020).
  * (42) J. Q. Quach and W. J. Munro, Using dark states to charge and stabilize open quantum batteries, Phys. Rev. Research 14, 024092 (2020).
  * (43) M. T. Mitchison, J. Goold and J. Prior, Charging a quantum battery with linear feedback control, Quantum 5, 500 (2021).
  * (44) D. Morrone, M. A. C. Rossi and M. G. Genoni, Daemonic ergotropy in continuously monitored open quantum batteries, Phys. Rev. Applied 20, 044073 (2023).
  * (45) F. Metzler, J. I. Sandoval and N. Galvanetto, The emergence of quantum energy science, J. Phys. Energy 5, 041001 (2023).