11institutetext: Leibniz-Institut für Astrophysik Potsdam (AIP), An der
Sternwarte 16, 14482 Potsdam, Germany
11email<EMAIL_ADDRESS>22institutetext: Department of Astronomy & Space Sciences,
Faculty of Science, University of Ege, 35100 Bornova, Izmir, Turkey
33institutetext: South African Astronomical Observatory, PO Box 9, Observatory
Road, Observatory 7935, Cape Town, South Africa 44institutetext: Department of
Astronomy, University of Cape Town, Private Bag X3, Rondebosch 7701, South
Africa 55institutetext: Department of Physics, University of the Free State,
PO Box 339, Bloemfontein 9300, South Africa

# Serendipitous discovery of the magnetic cataclysmic variable SRGE
J075818-612027

Samet Ok 1122 Georg Lamer 11 Axel Schwope 11 David A. H. Buckley 334455 Jaco
Brink 3344 Jan Kurpas 11 Dusán Tubín 11 Iris Traulsen 11

(Received …; Accepted …)

We report the discovery of SRGE J075818-612027, a deep stream-eclipsing
magnetic cataclysmic variable found serendipitously in SRG/eROSITA CalPV
observations of the open cluster NGC 2516 as an unrelated X-ray source. An
X-ray timing and spectral analysis of the eROSITA data is presented and
supplemented by an analysis of TESS photometry and SALT spectroscopy. X-ray
photometry reveals two pronounced dips repeating with a period of $106.144(1)$
min. The 14-month TESS data reveal the same unique period. A low-resolution
identification spectrum obtained with SALT displays hydrogen Balmer emission
lines on a fairly blue continuum. The spectrum and the stability of the
photometric signal led to the classification of the new object as a polar type
cataclysmic variable. In this picture the dips in the X-ray light curve are
explained by absorption in the intervening accretion stream and by a self-
eclipse of the main accretion region. The object displays large magnitude
differences on long (months) timescales both at optical and X-ray wavelengths,
being interpreted as high and low states and thus supporting the
identification as a polar. The bright phase X-ray spectrum can be reflected
with single temperature thermal emission with 9.7 keV and bolometric X-ray
luminosity $L_{\rm X}\simeq 8\times 10^{32}$ erg s-1 at a distance of about
2.7 kpc. It lacks the pronounced soft X-ray emission component prominently
found in ROSAT-discovered polars.

###### Key Words.:

cataclysmic variable stars – binary stars – x-rays – individual: SRGE
J075818-612027 – stars: fundamental parameters

##  1 Introduction

AM Her systems or polars are interacting close binaries that consist of an
accreting white dwarf (WD, called primary) and a Roche-lobe filling donor
star, typically on the main sequence, often referred to as the secondary. They
are magnetic cataclysmic binaries or cataclysmic variables (CVs). Hence,
interaction takes place via Roche lobe overflow to the WD. The WDs in these
systems have strong magnetic fields, $B>10$ MG, that keeps both stars in
synchronous rotation, suppresses an accretion disk, guides accreted matter to
the polar regions of the WD, where its potential energy finally is released
through radiation from the cooling shocked plasma at X-ray wavelengths and via
cyclotron radiation at optical and neighboring wavelength regimes (Warner,
1995).

Polars were found as soft X-ray emitters with the Einstein satellite and with
Röntgensatellit (ROSAT). Soft X-ray emission of reprocessed origin or from
deeply buried filamentary accretion shocks (Lamb & Masters, 1979; Frank et
al., 1988) was regarded one of the observational hallmarks of the class.

Interestingly, the polar survey conducted by Ramsay & Cropper (2004) with the
X-ray Multi-Mirror Mission (XMM-Newton) and all serendipitous discoveries made
with XMM-Newton since then (Vogel et al., 2008; Ramsay et al., 2009; Webb et
al., 2018; Schwope et al., 2020) did not reveal any further evidence for soft
X-ray emission from a large number of magnetic CVs. Also, the first eROSITA-
discovered polar did not show pronounced soft X-ray emission (Schwope et al.,
2022). Meanwhile about 150 magnetic CVs are known that belong to the polar
subclass of the CV population (Ritter & Kolb, 2003).

Polars are in the last steps of the binary star evolution and typically have
short orbital periods (Porb¡ 2 hours) (Ritter & Kolb, 2003; Knigge, 2006). The
magnetic torques keep the spin of the WDs synchronized with the orbit so that
for most polars $P_{\rm orbit}=P_{\rm spin}$.

Accordingly, accretion-related emission is modulated on this period. Polars
may show alternating high and low accretion states due to cessation of the
mass flow from the donor to the WD. Such accretion rate changes thought to be
related to the magnetic activity of the secondary star (Livio & Pringle,
1994). The optical brightness of the prototype polar, AM Herculis, which has
more than a century of optical observations, differs by 2.5 mag between high
and low states (Hessman et al., 2000; Wu & Kiss, 2008; Schwope et al., 2020).

The extended Roentgen Survey with an Imaging Telescope Array (eROSITA)
instrument (Predehl et al., 2021) on board the Spektrum-Roentgen-Gamma
spacecraft (SRG; Sunyaev et al., 2021) was launched in July 2019 and brought
into a wide halo orbit around the Sun-Earth $L_{2}$ point. Its main aim is to
perform several X-ray all-sky surveys (called eRASS), each lasting six month,
in the energy range of $0.2-10$ keV.

Prior to the main survey a comprehensive CalPV (Calibration and Performance
Verification) phase of the mission took place, one of the targets was the
galactic open cluster NGC 2516 (also known as Caldwell 96), which was observed
to calibrate the boresight and plate scale. The analysis of the data on the
cluster member stars will be reported in (Fritzewski et al., in prep.). Here
we report on the serendipitous discovery of a new magnetic cataclysmic
variable in the field of NGC 2516, SRGE J075818-612027 or hereafter SRGE0758.
The object could be identified initially based on its regular variability
pattern which became obvious by a visual inspection of the data cube made of
the distribution of events in the space spanned by the $x,y,$ and time
coordinates. The object was also reported as variable in Gaia DR3 catalogue
with a variability flag that indicates long-term photometric variations (see
Eyer et al., 2022, and references therein).

We detail the X-ray observations obtained by eROSITA with the comprehensive
photometric Transiting Exoplanet Survey Satellite (TESS) data and descriptive
spectral observation in Sec. 2. We present the results of the basic features
of the system by considering the X-ray and optical data together in Sec. 3.

Figure 1: X-ray image of the NGC 2516 obtained by eROSITA. The white dashed
circle marks the position of SRGE0758. False colors of the image were made
using photon energies between $0.2-1.1$ keV, $1.1-2.3$ keV and $2.3-5$ keV for
the red, green, and blue channels, respectively. The image displays an area of
1$\times$1 degree.

Figure 2: DSS image of the sky region around SRGE0758. The optical counterpart
is marked with red lines.

##  2 Observations

###  2.1 Discovery with eROSITA

eROSITA carries seven identical cameras (TM1-TM7) each covering the same FoV
with $\sim$ 1 deg diameter.

The X-ray observations of NGC 2516 were on October 5, 2019 (OBS ID 700018,
60.6 ksec good time) and October 31, 2019 (OBS ID 700019, 80 ksec good time)
in pointed mode.

Observation 700018 was carried out with only three cameras (TM5, TM6 and TM7),
while 700019 was carried out with all seven cameras.

The data were reduced with the eROSITA Science Analysis Software System
(eSASS) version eSASSusers_211214  (Brunner et al., 2022). In order to
determine remaining astrometric offsets between the two data sets, both
observations were astrometrically calibrated. Preliminary eSASS source lists
were generated for both observations and corresponding Gaia counterparts were
assigned within a 12 arcsec matching radius. Finding the optimal 3-D rotation
matrix $R_{\mathrm{opt}}$ needed to minimize the differences between X-ray
positions and Gaia reference positions resembles ”Wahba’s problem” (Wahba,
1965). Here we applied a method described in Markley & Crassidis (2014) using
the singular value decomposition (SVD) to find $R_{\mathrm{opt}}$.

For each of the two observations the matrix $R_{\mathrm{opt}}$ defines an
offset in the equatorial coordinates $\alpha$, $\delta$ and in the roll angle
$\phi$. Neglecting the small roll angle correction, we applied the following
corrections to each event:

$\alpha_{\mathrm{corr}}=\alpha_{\mathrm{uncorr}}+\Delta\alpha/\cos{\delta}$
$\delta_{\mathrm{corr}}=\delta_{\mathrm{uncorr}}+\Delta\delta$

with $\Delta\alpha=-2.087$ arcsec , $\Delta\delta=+0.052$ arcsec for
observation 700018 and with $\Delta\alpha=-2.835$ arcsec,
$\Delta\delta=+4.685$ arcsec for observation 700019.

The corrected event lists were then merged and a stacked image in the energy
band 0.2-5.0 keV was created (see Fig. 1). The new object we are reporting
here, SRGE0758, was one of the brightest X-ray sources in the field of NGC
2516 (see Fig. 1).

###  2.2 Gaia observations

In Gaia DR3 (Gaia Collaboration et al., 2021) SRGE0758 is quoted by ID
5290647986316685824 with the sky coordinates of DEC2000=119.57648825065 deg
and RA2000=-61.34105844684 deg. The mean brightness of the object is 19.73$\pm
0.03$, 19.90$\pm 0.13$, and 18.96$\pm 0.09$ in the G, $G_{BP}$, and $G_{RP}$
passbands, respectively. Gaia measured the parallax of SRGE0758 as
0.648$\pm$0.35 mas. We used the geometric distance ($r_{\rm geo}$) to the
system of $2733^{+4147}_{-1225}$ pc, as determined by Bailer-Jones et al.
(2021b). The photometric distance ($r_{\rm pgeo}$) is unreliable at G¿ 19 mag
according to Creevey et al. (2022) and Fouesneau et al. (2022). The sky
position of the source is shown in Fig. 2.

###  2.3 TESS observations

Transiting Exoplanet Survey Satellite (TESS) is providing high time-precision
light curves for time-domain astrophysics (Ricker et al., 2014). The satellite
is equipped with four identical refractive cameras and observes the sky in
sectors with 24 x 96 degrees. Each sector is observed for two orbits of the
satellite around the Earth, or about 27 days on average. Each of the 2k x 2k
CCDs on the satellite has a scale of 21 arc seconds per pixel.

SRGE0758 has observations in seven sectors, that Obs. IDs of 27, 28, 31, 34,
35, 36 and 37 which are reachable from the Mikulski Archive for Space
Telescopes111https://mast.stsci.edu. The observations were made between July
5, 2020, and April 02, 2021, about one year after the eROSITA observations and
were carried out with a time resolution of 475 sec.

We extract the photometry from the TESS full-frame images using eleanor
(Feinstein et al., 2019), which to obtain a cut out of 31x31 pixels postcard
of the calibrated full-frame images centered on target source. eleanor removes
the background, corrects for systematic errors, and derives a light curve for
different apertures.

All sectors were reduced into 11x11 pixel subgrids. We used the default pixel
source extraction area that produced by eleanor for the source to obtain time
series. For each sector we plotted over Gaia DR3 measurements in the field and
check the correct extraction location for the source.

During the extraction process we encountered two revelant challenges. (1) At
$G\simeq 19.7$ the source is very faint, almost at background level. This
indeed led in some sectors to background-subtracted light curves with negative
fluxes. (2) Gaia data have shown that a neighbouring source falls into the
same TESS pixel. This object is $\sim$8 arcsec away from our target with
coordinates RA1=119.57869730126448 and DEC1=-61.34270664883291 (source ID
5290647917596586112, $G=19.71\pm 0.02$ mag). We checked the variability of
this neighbouring source from Gaia DR3. The new Gaia DR3 catalog released with
a variability column (Eyer et al., 2022). In this column, it is noted whether
the system is a variable from long-term photometry. There was no indication
about brightness variability of the contaminating source.

All this means that the already weak signal from our target is further diluted
by the signal from the neighbouring star, so that neither absolute brightness
values nor absolute variability amplitudes can be determined. Giving those
extra challenges we perform variability analysis on normalized light curves.

To investigate possible periodic variations in all sectors, we applied a
Savitzky–Golay smoothing filter algorithm with flatten task in lightkurve
(Lightkurve Collaboration et al., 2018). The data obtained in one of the
sectors (here sector #35) are shown in Fig. 3 (top: original, bottom:
normalized).

Figure 3: TESS light curve of SRGE0758 obtained for Sector 35. (a) is showing
the background corrected TESS light curve in original time series. (b) shows
flattened/normalised light curve for Sector 35. (c) displays folded light
curve of sector 35 according to Eq. 1 in normalised flux.

Figure 4: Lomb-Scargle periodogram of the TESS and eROSITA observations. Main
figure shows the power spectra of the object which obtained from TESS and
eROSITA that are over plotted together with an inset detailing the main peak
of the TESS observations. Dashed black line shows a gaussian line fit to the
main peak. The red dashed line in the main frame gives the 99% significance
level of the eROSITA power spectrum. The main period and its harmonics are
indicated with the $\Omega$ symbol . The horizontal black dashed line within
the inset indicates 99% significance for the TESS spectrum.

###  2.4 Follow-up spectroscopy with SALT

We carried out optical spectroscopic observations of the object with the 11
meter Southern African Large Telescope (SALT) at Sutherland Observatory
(Buckley et al., 2006). The 1200s exposure was performed with the Robert
Stobie Spectrograph (RSS, Burgh et al., 2003) on May 24, 2022. The grism
(pg0300) covers 4000-8000 $\AA$ with a small gap at 4900-5100 $\AA$. The used
grism together with the used slit width of 1$\aas@@fstack{\prime\prime}$5
resulted in a spectral resolution of 17 Å (FWHM). The spectrophotometric
standard star Feige 110 (Oke, 1990), observed with SALT on 2022 May 15, was
used for the flux calibration. The calibration was done in IRAF (Tody, 1986)
by fitting a spline3 function to the observed spectrum of the standard to
determine the sensitivity function, which was then applied to the spectrum of
SRGE0758. We could only obtain relative flux calibrations, from observing a
spectrophotometric standard in twilight, due to the nature of the SALT design
and the moving entrance pupil (Buckley et al., 2018).

Figure 5: The SALT spectrum of the SRGE0758. The wavelength ranges which are
strongly contaminated by sky lines or cosmic rays are indicated in grey.

##  3 Analysis and results

###  3.1 Optical photometry from TESS

To find any periodic changes in the optical light curve, we used the Lomb &
Scargle periodogram which is useful in non-sinusoidal time series (Lomb, 1976;
Scargle, 1982; VanderPlas, 2018). We searched for significant peaks in the
frequency interval from 0 $d^{-1}$ up to the Nyquist frequency of TESS data
(182 $d^{-1}$). The power spectrum shows a main peak at $P=106$ min. Following
Baluev (2008) we computed the false alarm probability (FAP) to check the
significance of the peak. The FAP was zero, confirming that the main peak
belongs to a real periodic variation. We measured the uncertainty of this
period by a Gaussian fit applied to the periodogram peak. At 95% confidence
the uncertainty is $0\aas@@fstack{d}0000009$ so that the final TESS-period is
$0\aas@@fstack{d}0737114(9)$.

We applied the same method to the eROSITA X-ray light curve obtained in the
$0.2-2.3$ keV band. The resulting periodogram also shows a peak at
$P=0.0738(5)$ (106.2(7)min), fully compatible with the TESS-period, so that we
find coherent periodic variations of both the optical and X-ray light curves
at 106 min. The power spectrum of the both light-curves is displayed in the
Fig. 4. No other periodicity was found in the power spectrum except for the
harmonics of the main peak.

In the lower panels of Fig. 3 we show folded TESS light curves. The original
data were initially phase-folded with an arbitrary time zero. The resulting
signal showed roughly a sinusoidal brightness variation in all sectors (around
$\sim$5000 photometric cycles). The (arbitrary) phase of maximum brightness
was determined with a Gaussian fit and the zero time corrected accordingly so
that our final ephemeris for folding of data and the determination of the TESS
maximum brightness reads as

$BJD(T_{0,max})=2459036.3484(4)+0.0737114(9)\times E,$ (1)

where the numbers in parentheses give the uncertainties in the last digits. In
the following, all phases refer to the ephemeris given in Eq. 1.

In all sectors, TESS light curves always show the same sinusoidal brightness
variation but the amplitude of the sine curve changes considerably. We are
working on limits of the instrument due to the relatively faint nature of the
object. Therefore, the variable amplitude is due to the subtraction of
different background counts/levels sector to sector. It is not possible to
make an inference on the long-term variation of the object with TESS, but in
short-term we obtained very strong signal that gives the same photometric
period as the X-ray data in all sectors and this is quite a consistent result.

###  3.2 SALT spectroscopy

In Fig. 5 the SALT spectrum is shown. The object displays several hydrogen
Balmer emission lines superposed on a flat continuum. Other lines were
tentatively detected (HeI5875, OI6300 and HeII4686) but closer inspection of
the raw and sky subtracted 2D-spectra showed that these were possibly mimicked
by cosmic ray hits on the CCD. We consider the information obtained from these
lines unreliable and ignore them for further analysis. They are shaded grey in
Fig. 5.

To determine the parameters of the hydrogen Balmer lines the continuum was
approximated with a low-order polynomial and the spectrum normalized to this
continuum. Line parameters were determined by single Gaussians fits to the
data. They are listed in Table 1. The signal-noise ratio is low at the blue
end of the spectrum and the measurement of the H$\delta$ line in particular
rather uncertain. It has a double-peaked shape, different from the other
lines, which is regarded being apparent only. Its measurements are regarded
controversial and indicated by a colon in Table 1.

From flat continuum at 6250 Å the mean spectral flux density is about
6.5$\times$10-18 erg cm-2 s-1 Å-1, which corresponds to a Gaia G-band
magnitude of 21.5 (Vega). Hence, at the time when the spectrum was obtained
the source appeared considerably fainter than found previously with Gaia,
$G\sim 19.7$ mag). The spectral exposure was started at JD 2459724.2284838,
the spectrum thus covered the phase interval $0.07-0.26$, the decreasing
branch of the TESS light curve.

Strong emission lines of hydrogen and helium, both neutral and ionized, are
main indicators to identify a source as a CV (Szkody, 1998). We have
discovered with certainly only hydrogen Balmer emission lines. This and the
low brightness at the time of the SALT observation is indicative of a low
accretion state of the object during our observations. Low accretion states
are regularly found in strongly magnetic CVs, the polars. In these states
typically only hydrogen Balmer lines (in particular H$\alpha$) are visible
(Schwope et al., 2002). The SALT spectroscopy thus suggests the identification
of SRGE0758 as a polar in a low accretion state.

###  3.3 eROSITA observations

####  3.3.1 X-ray photometry

eROSITA X-ray light curves of both observations of SRGE0758 are shown in
original time sequence in Fig. 6. Only data in the interval $0.2-2.3$ keV and
time bins of 100 s were used. The object showed pronounced brightness
variations with one bright hump and two dip or eclipse-like brightness minima.
While the main minimum can always clearly recognized, the second one following
in time is quite shallow. This overall variability pattern is visible during
all 22 cycles that were covered during the X-ray observations. The main
minimum has sometimes zero count rate but remains finite at other instancies.
Hence, it is not interpreted as a true eclipse, were the count rate would
always be consistent with zero.

We generated energy-resolved and phase-folded X-ray light curves in two energy
bands, a soft band between 0.2 - 0.8 keV and a hard band between 0.8 - 2.3
keV. These are shown in Fig. 7(b) and Fig. 7(c). The first dip is located at
phase 0.40 and has a FWHM of 0.07 phase units. The flux is almost zero in the
soft band ($0.2-0.8$ keV), but stays always positive in the hard band
($0.8-2.3$ keV). The second dip shows a contrasting behaviour. The dip is
centered on phase 0.69, i.e. 0.29 phase units after the first one. It is
shallow in soft X-rays and more pronounced and deeper in the hard band.

Table 1: Spectral parameters of Balmer emission lines in the SALT spectrum.
Flux units are 10-16 erg cm-2 s-1. A colon indicates an uncertain measurement.

Lines | Observed ($\AA$) | Shift ($\AA$) | FWHM ($\AA$) | EW($\AA$) | Flux
---|---|---|---|---|---
H$\alpha(6563\lambda)$ | 6560.7$\pm 1.2$ | 3.0 | 23.7 | -66.6 | 3.7
H$\beta(4861\lambda)$ | 4863.4$\pm 0.9$ | 2.4 | 14.7 | -42.2 | 2.5
H$\gamma(4340\lambda)$ | 4338.5$\pm 1.0$ | -1.5 | 19.3 | -63.3 | 3.4
H$\delta(4101\lambda)$ | 4092.9$\pm 1.8$: | -8.1: | 20.1: | -55.4: | 3.7:

Figure 6: eROSITA $0.2-2.3$ keV X-ray light curve of SRGE0758 with time bins
of 100 s. The top row shows the data from ObsID 700018 the lower row that of
ObsID 700019.

Figure 7: TESS, energy resolved eROSITA light curves, Hardness Ratio and
$N_{\rm H}$ variations of SRGE0758 for ObsID 700019. (a) TESS light curve
obtained from sectors 31 and 37. (b) and (c): the X-ray data in the two
indicated energy bands were originally binned using 50 sec time bins, and
eventually averaged using a phase bin size of 0.005 phase units. (d) hardness
ratio obtained from 0.2 - 0.8 keV and 0.8 - 2.3 keV energy bands. (e) column
density variation. All light curves folded according to photometric period and
T0,max in Eq. 1.

To gain a better understanding of these variations, we calculated the Hardness
Ratio (HR) between the two chosen bands. HR is defined as $HR=(H-S)/(H+S)$ and
thus can vary between $-1$ and $+1$ for supersoft and superhard sources (see
Fig. 7(d)). Outside the dips the HR shows little variability and scatters
around $\sim$0.5.

The dips (or eclipse-like features) could be related to photoelectric
absorption or to geometric effects like self-eclipses and/or geometric
foreshortening of the corresponding emission regions.

The observed spectral hardening during the first dip implies that it is likely
caused by photo-electric absorption. The absorber is the magnetically guided
stream or curtain out of the orbital plane like those studied by e.g. Watson
et al. (1989) and also described by Kuulkers et al. (2010). The energy-
independent variation during the second dip suggests that it is more likely
caused by foreshortening/self-eclipse of the accretion/emission region as the
white dwarf rotates. The self-eclipse is not total but partial only, which can
be explained if the emission region is extended and not completely vanishing
behind the limb of the WD as it rotates.

####  3.3.2 X-ray spectroscopy

Spectral analysis was performed with the XSPEC package (version 12.12.0
Arnaud, 1996; Dorman & Arnaud, 2001). Only photons between $0.2-5$ keV were
included in the spectral analysis. Higher energies were completely dominated
by noise. We started out by selecting photons from the bright phase between
$\phi=0.8-1.2$ only (see Fig 7). Photons were grouped with a minimum of 20
counts per bin and we utilised chi statistics for optimizing the fit.

Initially we used just a single thermal emission model (Mewe et al., 1985;
Liedahl et al., 1995) with solar abundances (Wilms et al., 2000) absorbed by
cold interstellar matter of some column density $N_{\rm H}$, so that our
spectral model reads as TBABS * MEKAL in XSPEC terms. The object is relatively
distant, the inverse parallax is $\sim$1500 pc while the estimate by Bailer-
Jones et al. (2021a) points to an even larger distance of $\sim$2733 pc.
Accordingly, we may expect that the X-ray spectrum ’sees’ a large fraction of
the whole galactic column in this direction which is $N_{\rm H,gal}$ =
$1.26\times 10^{21}$ cm-2.

The simple model already gave a good fit to the data with $\chi^{2}=171$ for
168 degree of freedom, $\chi^{2}_{\nu}=1.02$ (see Fig. 8).

Many ROSAT-discovered polars show a soft, blackbody-like, radiation component
from the bottom of the accretion column and its surroundings. In contrast, we
could not find evidence of such a blackbody component in the spectrum.

We calculated the unabsorbed fluxes with the cflux task in XSPEC. After ideal
spectral fitting, we calculated the bolometric X-ray flux with XSPEC using the
best-fit spectral parameters and a dummy response which covers the broad
energy range between 10-6 and $10^{4}$ keV. The bolometric correction factor
is 1.28. Errors were calculated with the error command in XSPEC (90%
significance). All the fit results and uncertainties are listed in Table 2.

We then wanted to better constrain the spectral parameters as a function of
the spin-phase, in particular the change in the column density through the
main absorption dip. We therefore generated phase-resolved spectra using 10
phase bins of equal length. The photons per phase bin were grouped again with
20 counts per spectral bin. The temperature of the MEKAL model was fixed at
the bright-phase temperature so that only the column density and the model
normalization were allowed to vary. In the bottom panel of Fig. 7 the phase-
dependent behaviour of the column density is shown. It was $N_{\rm
H}=0.5\times 10^{21}$ cm-2 for most of the phase bins as for the mean
spectrum. During the first dip a marked increase by almost a factor 10
(although with large error bar) was observed which was even in excess of the
galactic column density. This accretion-related high column density is very
similar or comparable with other well known polar, HU Aqr during its high
accretion state (factor of 10) (Schwarz et al., 2009).

We may derive a rough estimate of the mass accretion rate limit via the X-ray
luminosity. If one assumes that the WD in SRGE0758 is a typical one, its
likely mass is 0.8 $M_{\odot}$ (Pala et al., 2021). We estimated $R_{\rm wd}$
for this mass using the Nauenberg (1972) mass–radius relation. If one further
assumes that the X-ray luminosity is a good proxy for the accretion
luminosity, $L_{\rm acc}\simeq L_{\rm X}$, one gets $\dot{M}=L_{\rm X}R_{\rm
0.8}/M_{\rm 0.8}G=8.3(7)\times 10^{-11}$ $M_{\odot}$ yr-1 (only flux error
considered here). The accretion rate was calculated for the distance of 2733
pc (Bailer-Jones et al., 2021b). These authors also give a rather large
confidence interval between 1225 pc and 4147 pc. Taking the uncertainty in the
distance into account, the likely accretion rate lies in the interval
$\dot{M}\approx 2-19\times 10^{-11}$ $M_{\odot}$ yr-1

Figure 8: Phase resolved X-ray spectra of the SRGE0758. The black data points
represent the bright phase spectrum and were extracted for the phase interval
$0.8-1.2$. The black line shows the model fit with a reduced chi-square value
of 1.02 ($\chi=171$ for 168 dof). The spectrum of the first dip (shown with
red data points) was obtained from phase at $0.4-0.5$, the fit reveled a
reduced $\chi^{2}$ of 1.51 ($\chi=30$ for 20 dof).

####  3.3.3 eROSITA all sky surveys

In Fig. 10 X-ray images for the first four eROSITA All-Sky Survey (eRASS)
scans of SRGE0758 are shown. The new object was detected in the first three
scans but not in the last (eRASS4). The mean count rates and observation times
are listed in Table 3. Because of the high ecliptic latitude of the object,
every survey scans the position of SRGE0758 quite often every four hours thus
building up a farther deep survey exposure (Freyberg et al., 2020).

We calculate the count rate upper limit at the position of the source in
eRASS4 based on X-ray aperture photometry following the Bayesian approach
described by Kraft et al. (1991). We collect observed counts, background
counts, and exposure time within a circular aperture with a radius of $r_{\rm
ape}=30.5$ arcseconds at the 0.2 - 2.3 keV energy band. The aperture
corresponds with a PSF encircled energy fraction of $\rm EEF=0.75$, which
measures the fraction of total photons within the aperture.

The X-ray flux decrease in eRASS1 and non-detection in eRASS4 suggest overall
X-ray emission cut in the object which indicate different mass accretion
state.

Table 2: Spectral parameters from bright phase interval ($\phi$=0.8 - 1.2):
spectral fit parameters, their uncertainties, fit statistics, and model
bolometric fluxes.

Model: TBABS*(MEKAL)
---
Parameters |
$N_{\rm H}$($10^{22}$cm2) | $0.05^{+0.01}_{-0.01}$
$kT_{\rm mekal}$(keV) | $9.7^{+7.7}_{-2.6}$
$\chi^{2}$(d.o.f) | 1.02(171/168)
Unabsorbed Fluxes $(10^{-13}\mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1})$ |
$F_{0.5-2.5}$ | $3.94^{+0.02}_{-0.02}$
$F_{2.5-10}$ | $7.2^{+0.8}_{-1}$
$F_{\rm bol}$ | $17.7^{+1.6}_{-1.6}$
$L_{x}$(erg s-1)$(10^{32})$ | $7.9(7)$

###  3.4 Accretion geometry

The X-ray light curve of the object does not contains an eclipse of a WD.
Therefore we do not know the true orbital phase zero. The X-ray observations
only reveal one bright hump with two consecutive dips. Conformably, optical
variability obtained with TESS revealed the same photometric period. The
optical light curve shows just one hump. Under these circumstances, it is not
possible to specify the location of the accretion column precisely. But we can
nevertheless derive a few constraints.

Figure 9: Images of the CV during the 4 eROSITA All Sky Surveys in the 0.2–2.3
keV range. During eRASS4,the object is in a low state. Table 3: Observation
log of the eROSITA All Sky Surveys and pointed observations covering the
location of SRGE0758. The given times (MJD) refer to the middle of the
concerned observation.

Survey | Count Rates (s-1) | Time (MJD) | Total Scans | Total Exposure (s)
---|---|---|---|---
eRASS1 | 0.11$\pm 0.02$ | 59002.265 | 22 | 634
eRASS2 | 0.27$\pm 0.03$ | 59188.520 | 24 | 736
eRASS3 | 0.27$\pm 0.03$ | 59371.505 | 26 | 810
eRASS4 |  $<0.038$ (99.87%) |  | 25 | 764
700018(CalPV) | 0.244$\pm 0.005$ | 58762.302 | - | 60.6k
700019(CalPV) | 0.277$\pm 0.006$ | 58788.207 | - | 80k

The increased absorption at the phase of the deep dip suggests an origin due
to photo-electric absorption in the accretion stream. The second dip is likely
caused by a self-eclipse of the accretion column by WD itself. During the
second dip the accretion region is partially hidden behind the limb of the WD.
Accordingly, this accretion region which is visible for most of the orbital
cycle is located in the upper hemisphere (with respect to the orbital plane,
$\beta$¡90∘ with $\beta$ being the co-latitude of the accretion region). The
object does not show an eclipse, hence the inclination is constrained to
$i<75\degr$.

The stream eclipse at 0.4 phase can only occur if the orbital inclination is
larger than the co-latitude of the column ($i>\beta$). If our interpretation
of the second dip as due to a self-eclipse is correct, then $i+\beta\simeq
90\degr$. The mid points of the two dips are separated by $\Delta\phi\sim$0.29
phase. Best visibility of the accretion region is then expected at half a
cycle later, i.e. about 0.2 phase units before the first dip. One would expect
then to observe a clear maximum in the light curve due to the given best
visibility of the accretion region which is not clearly observed. Instead
there is an extended bright phase without a clear center or maximum.

The typical location of a stream dip in polars is between phases 0.8 and 0.9
(Schwope et al., 2001, e.g.). In polars with a pronounced bright-phase hump in
the light curve the dip is often located in the second half of the bright
phase, i.e. it occurs at bright phase center of shortly thereafter. If our
tentative geometry is correct, the situation in SRGE0758 is rather different
with the dip trailing the bright-phase by 0.2 phase units. Such a geometry
seems to be unusual but actually there are other polars which display similar
photometric features, namely QQ Vul (Belle et al., 2000) and V834 Cen (Schwope
et al., 2008). The true orbital phase zero of these systems are known. In both
objects true phase zero was located between the two consecutive dips.

Figure 10: Long-term X-ray light curve of SRGE0758 which corresponds to
eRASS1-4 and pointed eROSITA observations. Black arrow displays the upper flux
limit of the eRASS4 survey for given sky position of the SRGE0758.

##  4 Discussion and conclusion

We have analysed extended eROSITA and TESS observations of a serendipitously
discovered X-ray source in the direction of the open cluster NGC 2516, that
clearly does not belong to the cluster. It is among the X-ray brightest
objects in the eROSITA CalPV observation of the field but is an accretion-
powered background object at a distance of about 1225 to 4147 pc. The cluster
itself is at a distance of 400 pc (Terndrup et al., 2002). We also obtained
SALT spectroscopic follow-up and time-resolved photometry from the SAAO.
eROSITA X-ray and optical TESS observations reveal a periodic variability of
the source at 106 min. This is the only found period in the system. The
identification spectrum was obtained when the object was $\sim$2 mag fainter
compared to its Gaia measurements. The spectrum reveals hydrogen Balmer
emission superposed on a flat or slightly blue continuum. All these features
clearly identify the new object as a magnetic cataclysmic variable. It likely
belongs to the AM Herculis or polar subclass although a nature as an
Intermediate Polars (IP) could not ruled out completely. This identification
is based on the occurrence of just one period in the system, the spin period
of the white dwarf, and the accretion rate changes which are typically
observed in polars. Also, the low state spectrum is similar to other polars.

The X-ray light curves are modulated by 100% on the 106 min period. The most
structuer in the light is an eclipse-like broad dip at phase 0.40 which we
interpret as stream absorption. Stream eclipses are common among the polars,
they are e.g. observed in EF Eri (Beuermann et al., 1991), EK UMa (Clayton &
Osborne, 1994), QQ Vul (Beardmore et al., 1995), HU Aqr (Schwope et al., 2001)
and UZ For (Warren et al., 1995), and require that the magnetically guided
accretion stream lifted out of the orbital plane crosses the line of sight.
The second dip is clearly revealed in the phase-folded X-ray light curve, not
so much in the individual cycles, and is tentatively identified as a possible
self-eclipse of the emission region, because no change of the hardness ratio
or the column density was observed at this phase. The self-eclipse could be
grazing only, the X-ray flux stays finite at this phase. This interpretation
is based on the observation that neither the hardness ratio nor the column
density in the spectral fits show significant variability at this phase.

Due to the lack of a primary star eclipse or other information from e.g.
spectral line variability the true phase zero is unknown. But the fact that
the 106-minute photometric period is the same in both TESS and eROSITA
observations suggesting a synchronous spin/orbital rotation, a typical
property of polars (Warner, 1995). The large portion of the polar population
has orbital periods less than 2 hours (Kuulkers et al., 2010) as our new
object, and also we did not encounter any change implying the white dwarf spin
seen as in Intermediate polars. In IPs, the accretion can be occur in two way,
stream-fed or disc-fed. In both accretion scenario, the matter changes poles
due to the WD spin and this breeds as itself a beat period in periodograms
obtained from X-ray and optical light curves (Norton et al., 1997).

The mean optical light curve from TESS is not as complex as the X-ray light
curve. The variation is very smooth and almost sinusoidal. The variability
amplitude is likely small but cannot properly quantified given the relative
faintness of the star and the contamination due to a second object in the same
pixel. The maxima of the TESS and eROSITA light curves are roughly coincident.
We could not detect any sharp and strong accretion stream or curtain related
absorption in the TESS light curves. The TESS minimum lies between the two
dips in X-rays but closer to the first. Optical variability in polars may be
caused by cyclotron beaming, geometric foreshortening of a heated accretion
region and obscuration of the light sources. Given the lack of detailed light
curve information, we do not speculate about the origin of TESS variability in
SRGE0758.

The X-ray spectrum is compatible with single temperature thermal plasma
emission at around 10 keV, well in line what is typically found in polars
(Kuulkers et al., 2010). Polars have lower X-ray temperatures than IPs (Mukai,
2017). The new object lacks a soft black body component in its X-ray spectrum
which was often prominent in the X-ray spectra of ROSAT-discovered polars
Beuermann & Schwope (1994). This is similar to other polars discovered with
XMM-Newton (Vogel et al., 2008; Ramsay et al., 2009; Webb et al., 2018) and
also the first eROSITA-discovered polar seem to lack this component Schwope et
al. (2022). The strength of the soft X-ray excess in polars and has been a
hotly debated topic (e.g. Vogel et al., 2008; Ramsay et al., 2009). It seems
that the effective energy range of XMM-Newton, which cannot extend to EUV
wavelengths, makes it difficult to detect this component of re-processed
emission from the WD in some systems. Presumably, we are facing with the same
issue in eROSITA which has a similar sensitivity range.

The optical identification spectrum has a flat continuum and contains broad
hydrogen emission lines which is a good indicator for magnetic accretors. The
continuum neither reveals the white dwarf not cyclotron harmonic emission nor
the donor star. A detection of the donor cannot be expected given large
distance and the short orbital period, hence late spectral type (Knigge et
al., 2011, predict a spectral type later than M5.3) and thus faintness of the
star. Another specific feature needs to be argued here is the large scale
brightness change between Gaia and SALT spectrum ($\sim$1.8 mag). While in
polars orbital phase dependent variability of up to 4 magnitudes may occur,
the brightness drop by about 2 mag between Gaia and SALT is more likely due a
reduced mass accretion rate. The lack of clearly detected helium lines is
suggestive of a reduced accretion state. The non-detection in eRASS4, about 5
month prior to the SALT observations support this view. Accretion state
transitions occur randomly in polars. The time they spend in these states is
also randomly changing and unpredictable. The triggering mechanism is thought
to be predominantly chromospheric activity in the donor star (Kafka &
Honeycutt, 2005). On the other hand, accretion state variations are not a
unique indication of a polar. Also the intermediate polars are meanwhile
frequently being discovered in low states, the first perhaps being FO Aqr
(e.g. Littlefield et al., 2020; Kennedy et al., 2020) followed by several
others(e.g. Covington et al., 2022; Hill et al., 2022).

The position of the first dip with respect to the main X-ray hump in the light
curve implies that the object has a very interesting accretion geometry. The
column is not in the direction of the secondary star. The appearance of the
stream dip after the main hump implies that the accretion spot is in a rather
perpendicular position to the system. The accretion to the distant/main pole
is more likely occur at high accretion states (Ferrario et al., 1989) in
polars.

Previously, ROSAT and Swift observatories observed the NGC 2516 but SRGE0758
could not be detected by either mission. Probably, the object was in a low
accretion state during these observations. The sensitivity of eROSITA together
with its optimized scanning strategy led to the discovery of this object
despite its large distance and the switch between high and low states. It may
be expected the eROSITA will uncover many more magnetic CVs through systematic
follow-up observations of all point-like X-rays sources found in the eRASS. In
particular there is a great chance to find those that escaped its detection in
the ROSAT all-sky survey while being in a low state at that time, because the
duty of polars might be as short as 50% (Hessman et al., 2000).

###### Acknowledgements.

Samet Ok is supported by TUBITAK 2219-International Postdoctoral Research
Fellowship Program for Turkish Citizens. Samet Ok acknowledge deep gratitude
to the Leibniz Institute for Astrophysics Potsdam (AIP) and members of the
X-ray astronomy group for the warm hospitality during his stay. This work was
supported by the Deutsches Zentrum für Luft- und Raumfahrt (DLR) under grants
50 QR 2104, 50 OX 1901, and 50 OR 2203. Support by the Deutsche
Forschungsgemeinschaft under grant Schw536/37-2 is gratefully acknowledged.
This research has made use of data, software and/or web tools obtained from
the High Energy Astrophysics Science Archive Research Center (HEASARC), a
service of the Astrophysics Science Division at NASA/GSFC and of the
Smithsonian Astrophysical Observatory’s High Energy Astrophysics Division.
This paper includes data collected by the TESS mission. Funding for the TESS
mission is provided by the NASA’s Science Mission Directorate. This work is
based on data from eROSITA, the soft X-ray instrument aboard SRG, a joint
Russian-German science mission supported by the Russian Space Agency
(Roskosmos), in the interests of the Russian Academy of Sciences represented
by its Space Research Institute (IKI), and the Deutsches Zentrum für Luft- und
Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Association (NPOL)
and its subcontractors, and is operated by NPOL with support from the Max
Planck Institute for Extraterrestrial Physics (MPE). The development and
construction of the eROSITA X-ray instrument was led by MPE, with
contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP (FAU
Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz
Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and
Astrophysics of the University of Tübingen, with the support of DLR and the
Max Planck Society. The Argelander Institute for Astronomy of the University
of Bonn and the Ludwig Maximilians Universität Munich also participated in the
science preparation for eROSITA. The eROSITA data shown here were processed
using the eSASS software system developed by the German eROSITA consortium. We
thank the anonymous referee for useful comments and suggestions which helped
to improve the paper.

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# Practical Scheme for Realization of a Quantum Battery

Maryam Hadipour  0000-0002-6573-9960 Faculty of Physics, Urmia University of
Technology, Urmia, Iran  Soroush Haseli  0000-0003-1031-4815
<EMAIL_ADDRESS>Faculty of Physics, Urmia University of Technology,
Urmia, Iran  School of Physics, Institute for Research in Fundamental Sciences
(IPM), P.O. Box 19395-5531, Tehran, Iran  Dong Wang  0000-0002-0545-6205
<EMAIL_ADDRESS>School of Physics and Optoelectronics Engineering, Anhui
University, Hefei, 230601, People’s Republic of China  Saeed Haddadi
0000-0002-1596-0763<EMAIL_ADDRESS>Faculty of Physics, Semnan
University, P.O. Box 35195-363, Semnan, Iran

###### Abstract

We propose a practical scheme for a quantum battery consisting of an atom-
cavity interacting system under a structured reservoir in the non-Markovian
regime. We investigate a multi-parameter regime for the cavity-reservoir
coupling and reveal how these parameters affect the performance of the quantum
battery. Our proposed scheme is simple and may be achievable for practical
realization and implementation.

Quantum battery; Atom-cavity system; Non-Markovianity

##  I INTRODUCTION

Traditional batteries that are still in use, such as lithium-ion, alkaline,
and lead-acid batteries, operate based on electrochemical reactions that
involve the motion of ions between two electrodes through an electrolyte. The
performance of these batteries strongly depends on various factors such as the
electrolyte composition, the materials used in electrodes, and the overall
design. Quantum batteries (QBs) [1], on the other hand, are a novel concept
that probes the potential of quantum mechanics to enhance energy storage.
These batteries, at least in theory, can use quantum superposition and
entanglement to store and recover energy more efficiently than traditional
batteries [2, 3, 4, 5, 6]. However, QBs are still in the early stages of
development and face notable challenges in terms of stability, scalability,
and practical implementation [7, 8, 9].

A typical model of the QB comprises two components, the battery charger and
the battery holder. The latter, designed to prevent energy loss, is
essentially isolated from the surrounding environment and is treated as a
dissipation-less subsystem. To obtain energy, the battery holder must be
coupled to the battery charger. After a limited charging period, the battery
holder detaches from the battery charger, allowing energy storage and
ultimately extraction of the required energy [10]. Interestingly, this basic
bipartite model and other models of QB, along with its theoretical
explorations in various forms, have been the subject of intensive QB research
in recent years [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28].

QBs illustrate an area of research at the intersection of quantum physics and
energy storage, however, as mentioned above, their development has some
potential challenges and limitations. For example, QBs rely on preserving
quantum coherence [29, 30, 31, 32, 33], which is the ability of quantum
systems to exist in a superposition of states. Practically speaking,
preserving quantum coherence for a sufficient period of time can be difficult
due to some sources of decoherence [34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46]. In other words, a significant challenge is constructing a QB that
can resist external effects and maintain quantum coherence in a real-world
environment. Therefore, addressing environmental effects on the performance of
QBs is necessary for their practical implementation and development.

In this paper, the presence of a mediated cavity can have several effects. It
may serve as a means to protect QB from external disturbance, helping to
preserve the delicate quantum state of QB. Actually, the mediated cavity can
assist in maintaining coherence and reducing decoherence, which are crucial
factors in quantum systems. So, this mediation can lead to more controlled and
efficient energy transfer processes, potentially optimizing the charging
performance of QB. Moreover, the mediated cavity helps to manage the
interactions, allowing for better manipulation and utilization of quantum
properties during the charging process.

Motivated by this, we propose a theoretical scheme for a QB consisting of an
atom-cavity interacting system under a structured (bosonic) reservoir in the
non-Markovian regime [47, 48, 49, 50]. We scrutinize a multi-parameter regime
for the strength of cavity-reservoir coupling and demonstrate how these
parameters affect the performance of the proposed QB. This scheme is simple
yet complex enough for our goal, however, it is feasible for practical
implementations.

##  II cavity-mediated charging performance of QB

The general system we are interested in studying includes a two-level system
(qubit) considered a QB, a single-mode cavity, and a structured bosonic
environment. A visual representation of our proposed scheme is illustrated in
Fig. 1. The cavity acts as a mediator between the QB and the environment. To
find the conditions under which the QB can be charged with better efficiency
[14], two different scenarios for the environment will be considered–one is an
environment with memory and the other is a memory-less environment. The
Hamiltonian of the composite system in the rotating wave approximation can be
written as [51, 52]

$H=H_{0}+f(t)H_{I}.$ (1)

In the above equation, the function $f(t)$ is the regulator of QB charging
time. Its outcome equals $1$ for $t\in[0,\tau)$ with $\tau$ denoting the
charging time of QB and is $0$ for other values of $t$. In other words, this
function plays the role of an on-and-off switch for the interaction of
subsystems. It is assumed that at time $t=0$, the QB will be connected to the
cavity as well as the cavity to the environment. In the considered scenario,
there exists no interaction between QB and the environment. So, energy
transfer from the environment to QB during the charging period $[0,\tau)$
takes place through the cavity, which plays the role of a mediator. At the end
of the charging process, i.e. at time $\tau$, QB and cavity are disconnected
from cavity and environment respectively.

In Eq. (1), $H_{0}$ is the free Hamiltonian of the total system, which can be
written as

$H_{0}=\omega_{0}\sigma_{+}\sigma_{-}+\omega_{c}a^{\dagger}a+\sum_{k=0}^{\infty}\omega_{k}b_{k}^{\dagger}b_{k},$
(2)

where $\sigma_{+}=|e\rangle\langle g|$ and $\sigma_{-}=|g\rangle\langle e|$
are the raising and lowering operators for qubit with $|e\rangle$ and
$|g\rangle$, which are excited and ground state of the qubit respectively.
Besides, $\omega_{0}$ is the transition frequency of qubit. The second term in
Eq. (2) represents the free Hamiltonian of the cavity with $a$ and
$a^{{\dagger}}$ being the annihilation and creation operators of the cavity
and $\omega_{c}$ is the transition frequency of the cavity. Herein, the
resonance condition is considered $\omega_{c}=\omega_{0}$. The third term of
the mentioned equation shows the free Hamiltonian of the environment in which
$b_{k}$ and $b_{k}^{{\dagger}}$ are the annihilation and creation operators
for $k$th mode of the environment with frequency $\omega_{k}$.

In the last term of Eq. (1), $H_{I}$ denotes the interaction term of the
Hamiltonian that describes the qubit-cavity and cavity-environment interaction
given by

$H_{I}=\Omega\left(\sigma^{+}a+\sigma^{-}a^{\dagger}\right)+\sum_{k=0}^{\infty}g_{k}\left(ab_{k}^{\dagger}+a^{\dagger}b_{k}\right),$
(3)

where $\Omega$ is the QB-cavity coupling strength and $g_{k}$ is the coupling
strength between the cavity and $k$th mode of the environment.

Here, the time evolution of the total system will be calculated with a single
excitation, given that the reservoir is in a vacuum state. So, the initial
state of the total system can be considered as

$|\psi(0)\rangle=\left[c_{1}(0)|g_{B}\rangle|1_{c}\rangle+c_{2}(0)|e_{B}\rangle|0_{c}\rangle\right]\otimes|0_{k}\rangle_{\mathcal{E}},$
(4)

where $|0_{c}\rangle$ and $|1_{c}\rangle$ are the vacuum and single-mode
excitation states of the cavity respectively, and
$|0_{k}\rangle_{\mathcal{E}}$ represents the vacuum state of the bosonic
environment. Moreover, $c_{1}(0)$ and $c_{2}(0)$ are the probability
amplitudes. So, the time-evolved state spanned by a single excitation basis
state can be obtained as (see appendix A)

$\displaystyle|\psi(t)\rangle=$
$\displaystyle[c_{1}(t)|g_{B},1_{c}\rangle+c_{2}(t)|e_{B},0_{c}\rangle]\otimes|0_{k}\rangle_{\mathcal{E}}$
(5)
$\displaystyle+\sum_{k}c_{k}(t)|g_{B},0\rangle\otimes|1_{k}\rangle_{\mathcal{E}},$

where $|1_{k}\rangle_{\mathcal{E}}$ is the state of the bosonic environment
with excitation in $k$th mode.

Figure 1: A visual representation of the considered set-up for cavity-mediated
charging process of QB.

##  III Structured with-memory environment

Let the reservoir be assumed to have a Lorentzian spectrum as
$J(\omega)=\frac{\gamma}{2\pi}\frac{\lambda^{2}}{(\omega_{0}-\omega)^{2}+\gamma^{2}}$,
where $\gamma$ is the effective coupling strength and $\lambda$ is the width
of the spectrum which has the inverse relation with memory time of environment
$\tau_{\mathcal{E}}$, i.e. $\lambda^{-1}=\tau_{\mathcal{E}}$. In this work, we
study two different situations. One when the environment has memory
$\tau_{\mathcal{E}}\neq 0$ and the other when it is memory-less
$\tau_{\mathcal{E}}=0$.

Let us assume that the QB is initially empty. Hence, we set $c_{1}(0)=1$ and
$c_{2}(0)=0$ in Eq. (4), implying that the initial state of the whole system
is $|g_{B},1_{c}\rangle\otimes|0_{k}\rangle_{\mathcal{E}}$. If the partial
trace is taken over the cavity and environment, then the reduced density
operator of the QB at time $\tau$ can be written as

$\rho_{B}(\tau)=|c_{2}(\tau)|^{2}|e\rangle\langle
e|+\left[1-|c_{2}(\tau)|^{2}\right]|g\rangle\langle g|,$ (6)

where $c_{2}(\tau)=\kappa(\tau)c_{2}(0)$, with

$\kappa(\tau)=\mathcal{L}^{-1}\left(\frac{-i\Omega}{s^{2}+\Omega^{2}+\frac{\lambda\gamma
s}{2(s+\lambda)}}\right),$ (7)

in which $\mathcal{L}^{-1}$ is the inverse Laplace transformation.

In order to study the charging process of the QB, we first analyze the
dynamics in the context of Markovian (without memory) and non-Markovian (with
memory) aspects in the presence of the environment with memory
$\tau_{\mathcal{E}}\neq 0$. Markovian dynamics are those in which information
continuously flows from a system to its environment during the evolution,
while non-Markovian dynamics are those in which information backflow from the
environment to the system [53, 54]. The Breuer–Laine–Piilo (BLP) measure is
used here to quantify the degree of non-Markovianity of the quantum evolution
[54]. Based on the trace distance between two distinct quantum states
$\rho_{1}(t)$ and $\rho_{2}(t)$, namely
$D(\rho_{1}(t),\rho_{2}(t))=\frac{1}{2}\textmd{tr}|\rho_{1}(t)-\rho_{2}(t)|$,
the BLP measure quantifies the amount of non-Markovianity $\mathcal{N}$ as

$\mathcal{N}=\max_{\rho_{1}(0),\rho_{2}(0)}\int_{\sigma>0}dt~{}\sigma\left(t,\rho_{1}(0),\rho_{2}(0)\right),$
(8)

where
$\sigma(t,\rho_{1}(0),\rho_{2}(0))=\frac{d}{dt}D(\rho_{1}(t),\rho_{2}(t))$ is
the rate of change of the trace distance.

Using a large sample set of pairs of initial states and strong numerical
evidence, Ref. [55] determines that Eq. (8) has a maximum value for states
$\rho_{1}(0)=|e\rangle\langle e|$ and $\rho_{2}(0)=|g\rangle\langle g|$. Based
on these two initial states, we obtain
$\sigma\left(t,\rho_{1}(0),\rho_{2}(0)\right)=\frac{d}{dt}\kappa(t)$. If
$\sigma\left(t,\rho_{1}(0),\rho_{2}(0)\right)>0$, then we have
$\mathcal{N}\neq 0$.

Figure 2: Non-Markovianity $\mathcal{N}$ as functions of $\gamma/\Omega$ and
$\lambda/\Omega$.

In Fig. 2, the non-Markovianity is plotted as functions of $\gamma/\Omega$ and
$\lambda/\Omega$. From this figure, it is clear that these two parameters have
significant effects on the evolution from the Markovian and non-Markovian
aspects, one is the ratio of the width of the spectrum $\lambda$ to the QB-
cavity coupling strength $\Omega$, and the other is the ratio of cavity-
environment coupling strength $\gamma$ to QB-cavity coupling strength
$\Omega$. On one hand, we observe that the strong coupling between the QB and
the cavity $\Omega$ leads to a greater non-Markovian feature of the evolution.
On the other hand, from the dependence of the width of the spectrum $\lambda$
to the memory time $\tau_{\mathcal{E}}$ of the structured environment as
$\lambda^{-1}=\tau_{\mathcal{E}}$, we see that the longer memory time of the
environment increases the amount of non-Markovianity. We also find that with
the increase in the cavity-environment coupling strength $\gamma$, the amount
of non-Markovianity decreases. It has been shown that QBs in the non-Markovian
regime have the best efficiency in the charging process [37], which is why we
have first studied the evolution from both Markovian and non-Markovian
perspectives.

Let us return to the main issue, which is the cavity-mediated charging
performance of QBs. It is straightforward to calculate the amount of energy
that the battery obtains during the charging process as $\Delta
E_{B}(\tau)=E_{B}(\tau)-E_{B}(0)$, where
$E_{B}(\tau)=\textmd{tr}[\rho_{B}(\tau)H_{B}]$. Since the battery is initially
empty, we have $\Delta E_{B}(\tau)=\omega_{0}|c_{2}(\tau)|^{2}$.

Figure 3: (a) Dynamics of stored energy in QB during charging process $\Delta
E_{B}(\tau)$ and (b) ergotropy $\mathcal{W}(\tau)$ as a function of
dimensionless parameter $\Omega\tau$ in non-Markovian regime with
$\lambda/\Omega=0.1$ for different values of $\gamma/\Omega$.

In Fig. 3(a), we have plotted the non-Markovian dynamics of $\Delta
E_{B}(\tau)$ in terms of dimensionless parameter $\Omega\tau$ with
$\lambda=0.1\Omega$ in the presence of structured environment with memory. It
is observed that, during the charging process, the energy obtained by the QB
increases with the enhancement in the QB-cavity coupling strength $\Omega$,
while it decreases with the strengthening of the cavity-environment coupling
$\gamma$.

The ergotropy describes the maximum amount of energy that can be extracted
from the QB under cyclic unitary operation after it has been charged for a
certain period of time $\tau$ [56], defined by (see appendix B)

$\mathcal{W}=\textmd{tr}(\rho_{B}(\tau)H_{B})-\textmd{tr}(\sigma_{\rho_{B}}H_{B}),$
(9)

where $\sigma_{\rho_{B}}$ is called the passive state of $\rho_{B}$ and it is
a state from which no work can be extracted under cyclic unitary operations.
From Eq. (9), the ergotropy takes the following form for the cavity-mediated
scenario

$\mathcal{W}(\tau)=\omega_{0}(2|c_{2}(\tau)|^{2}-1)~{}\Theta(|c_{2}(\tau)|^{2}-\frac{1}{2}),$
(10)

where $\Theta(x-x_{0})$ indicates the Heaviside function.

The time-evolution of ergotropy is plotted in Fig. 3(b) as a function of
$\Omega\tau$ in a non-Markovian regime with memory $\lambda=0.1\Omega$. We
find that the most energy can be extracted from the QB when the QB-cavity
coupling strength $\Omega$ is greater than the cavity-environment coupling
strength $\gamma$.

The maximum stored energy in QB $\Delta E_{B}^{max}$ and the maximum ergotropy
$\mathcal{W}_{max}$ can then be used to evaluate the charging performance of
the cavity-mediated scenario of QBs. They are given by

$\Delta E_{B}^{max}=\max_{\tau}[\Delta
E_{B}(\tau)],\quad\mathcal{W}_{max}=\max_{\tau}[\mathcal{W}(\tau)],$ (11)

where the optimization is done over the charging time $\tau$. Note that
greater $\Delta E_{B}^{max}$ and $\mathcal{W}_{max}$ are required for optimal
charging of the QB.

In Fig. 4(a), the maximum energy stored in QB $\Delta E_{B}^{max}$ is plotted
in terms of $\gamma/\Omega$ and $\lambda/\Omega$. As can be seen, the maximum
stored energy in QB reaches its highest value for stronger QB-cavity coupling
$\Omega$. It can also be said that for larger memory time $\tau_{\mathcal{E}}$
of the structured environment (small value of $\lambda$), $\Delta E_{B}^{max}$
has the highest value. Moreover, Fig. 4(b) illustrates $\mathcal{W}_{max}$ as
functions of $\gamma/\Omega$ and $\lambda/\Omega$. We find that the highest
value of $\mathcal{W}_{max}$ is obtained for smaller values of $\lambda$
(longer memory time of environment $\tau_{\mathcal{E}}$). It is also observed
that $\mathcal{W}_{max}$ increases by amplifying the QB-cavity coupling
$\Omega$ and decreases with increasing the effective cavity-environment
coupling $\gamma$.

Figure 4: (a) Maximum stored energy in QB, $\Delta E_{B}^{max}$, and (b)
Maximum ergotropy, $\mathcal{W}_{max}$, as functions of $\gamma/\Omega$ and
$\lambda/\Omega$.

##  IV Structured memory-less environment

Let us consider the situation in which the memory time of the environment is
zero. For the environment to be memory-less ($\tau_{\mathcal{E}}=0$), it is
necessary to set $\lambda\rightarrow\infty$. So, in this section, we follow
all the steps that we derived in the previous sections, but we just consider
the limit $\lambda\rightarrow\infty$ in all of the calculations. Hence, we can
rewrite Eq. (7) as

$\kappa(\tau)=-4i\exp[-\frac{\mathcal{R}}{4}\tau]\sinh[\frac{\mathcal{R}}{4}\tau],$
(12)

where $\mathcal{R}=\sqrt{\gamma^{2}-16\Omega^{2}}$. From the BLP measure for
non-Markovianity, it is observed that the evolution is non-Markovian for
$\gamma/\Omega<4$. Since the charging performance of QBs is more optimal in
the non-Markovian regime, therefore, we will study here the charging
performance in the non-Markovian regime.

Figure 5: (a) Dynamics of stored energy in QB during charging process $\Delta
E_{B}(\tau)$ and (b) ergotropy $\mathcal{W}(\tau)$ as a function of
$\Omega\tau$ in non-Markovian regime with memory-less environment
$\lambda\rightarrow\infty$ for different values of $\gamma/\Omega$.

Figure 6: (a) Dynamics of stored energy in QB during charging process $\Delta
E_{B}(\tau)$ and (b) ergotropy $\mathcal{W}(\tau)$ as a function of
$\Omega\tau$ in non-Markovian regime for different types of environments with
$\gamma=0.1\Omega$.

Figure 7:  Comparative plots of maximum stored energy $\Delta E_{B}^{max}$ and
maximum ergotropy $\mathcal{W}_{max}$ from the point of view of memory-less
($\lambda\rightarrow\infty$) and with-memory environments as a function of (a)
$\lambda/\Omega$ (with $\gamma=0.1\Omega$) and (b) $\gamma/\Omega$ (with
$\lambda=0.1\Omega$) in non-Markovian regime.

In Fig. 5(a), the stored energy in QB is plotted as a function of
dimensionless parameter $\Omega\tau$ in the memory-less environment for
different values of $\gamma/\Omega$. As can be seen, $\Delta E_{B}(\tau)$
decreases by increasing effective coupling between cavity and environment
$\gamma$. It can also be noticed that the stored energy increases with raising
the QB-cavity coupling strength $\Omega$. Meanwhile, the ergotropy
$\mathcal{W}$ is plotted in terms of $\Omega\tau$ for different values of
$\gamma/\Omega$ in Fig. 5(b). The ergotropy decreases with increasing $\gamma$
and finally tends to zero for larger values of $\gamma$. Notice that the
ergotropy achieves its highest value for greater values of QB-cavity coupling
strength $\Omega$.

##  V Comparison between with-memory and memory-less environments

Here, we examine which of the environments with memory or memory-less is more
efficient in the charging process of QB. To compare the stored energy and
ergotropy in these two different environments, we consider the same effective
coupling $\gamma$ between the environment and the cavity.

Fig. 6(a) shows the evolution of stored energy $\Delta E_{B}(\tau)$ in QB for
different types of environments for the same cavity-environment effective
coupling $\gamma=0.1\Omega$. Since the stored energy $\Delta E_{B}(\tau)$ in
the presence of the memory-less environment (blue dot-dashed line) is less
than the stored energy in the presence of the with-memory environment (black
solid line), the optimal charging performance of QB happens for the with-
memory environment. In Fig. 6(b), the ergotropy $\mathcal{W}(\tau)$ is plotted
as a function of $\Omega\tau$ for different types of environments with
$\gamma=0.1\Omega$. In agreement with the result of Fig. 6(a), the ergotropy
in the with-memory environment is greater than in the memory-less environment.

To make it clear that the charging performance is more optimal in a with-
memory environment, we plot $\Delta E_{B}^{max}$ and $\mathcal{W}_{max}$ in
terms of $\lambda/\Omega$ [see Fig. 7(a)] and $\gamma/\Omega$ [see Fig. 7(b)].
From Fig. 7(a), it is observed that for the memory-less environment
$\lambda\rightarrow\infty$ with $\gamma=0.1\Omega$, the value of maximum
stored energy is equal to $0.925$ (the upper thin dotted line in this plot
indicates this value). A comparison of this value with the maximum stored
energy in the QB reveals that the value of $\Delta E_{B}^{max}$ in the with-
memory environment (black solid line) is higher than that in the memory-less
environment. Moreover, one can see that the maximum value of ergotropy for the
memory-less environment $\lambda\rightarrow\infty$ is equal to $0.851$ (the
lower thin dotted line in this plot indicates this value). Also, we find that
the maximum value of the ergotropy $\mathcal{W}_{max}$ in the presence of an
environment with memory (blue dashed line) is greater than that in a memory-
less environment.

In Fig. 7(b), we sketch $\Delta E_{B}^{max}$ and $\mathcal{W}_{max}$ versus
$\gamma/\Omega$. From this figure, we notice that the maximum values of the
stored energy and ergotropy for the with-memory environment
($\lambda=0.1\Omega$) are always greater than those in the memory-less
environment $\lambda\rightarrow\infty$.

##  VI Conclusion and outlook

In this work, we have studied the cavity-mediated charging performance of QB
in the presence of a structured bosonic environment. This study considered a
practical model in which a cavity acts as a mediator between the QB and the
structured environment. In other words, there exists no direct interaction
between QB and structured bosonic environment. It was observed that, with
increasing the coupling strength between the cavity and environment $\gamma$,
the QB will not charge optimally and the amount of stored energy in the QB
will decrease. Furthermore, increasing the coupling strength between the QB
and the cavity $\Omega$ optimizes the QB charging process and increases the
amount of work that can be extracted from the QB.

We have also studied the effects of two different types of environments from
the memory point of view on the charging process of QB individually. The
environment has memory when its correlation time has a non-zero value, while
the environment is memory-less when its correlation time is equal to zero. We
found that for the with-memory environment, the stored energy and extractable
work are greater than those in which the cavity interacts with a memory-less
environment. Therefore, it can be concluded that it is possible to extract
work from QB in a more efficient manner in with-memory environments.

##  Appendix A Analytical solution for the system dynamics

Here, we examine the model used for the operational investigation of QBs in
more detail. The interaction Hamiltonian in the interaction picture can be
written as

$\tilde{H}_{I}=\Omega(\sigma_{+}a+\sigma_{-}a^{{\dagger}})+\sum_{k=0}^{\infty}g_{k}(ab_{k}^{{\dagger}}e^{i\delta_{k}t}+a^{{\dagger}}b_{k}e^{-i\delta_{k}t}).$
(13)

By considering the above Hamiltonian and inserting Eq. (5) into the
Schrödinger equation, the following coupled differential equations are derived
as

$\displaystyle\frac{d}{dt}c_{1}(t)$ $\displaystyle=-i\Omega
c_{2}(t)-i\sum_{k}g_{k}e^{-i\delta_{k}t}c_{k}(t),$ (14)
$\displaystyle\frac{d}{dt}c_{2}(t)$ $\displaystyle=-i\Omega c_{1}(t),$
$\displaystyle\frac{d}{dt}c_{k}(t)$
$\displaystyle=-ig_{k}e^{i\delta_{k}t}c_{1}(t),$

where the resonance condition is considered $\omega_{c}=\omega_{0}$ and
$\delta_{k}=\omega_{k}-\omega_{0}$. Considering the correlation function
$f(t,t^{\prime})=\sum_{k}|g_{k}|^{2}e^{-i\delta_{k}(t-t^{\prime})}$, the above
coupled differential equations can be solved using the Laplace transformation
method as

$\displaystyle c_{1}(t)$
$\displaystyle=\mathcal{L}^{-1}\left(\frac{c_{1}(0)-i\frac{\Omega}{s}c_{2}(0)}{s+\frac{\Omega^{2}}{s}+\frac{\lambda\gamma}{2(s+\lambda)}}\right)$
(15)

and

$\displaystyle c_{2}(t)$
$\displaystyle=\mathcal{L}^{-1}\left(\frac{c_{2}(0)}{s}-i\frac{\Omega}{s}\frac{c_{1}(0)-i\frac{\Omega}{s}c_{2}(0)}{s+\frac{\Omega^{2}}{s}+\frac{\lambda\gamma}{2(s+\lambda)}}\right).$
(16)

Note that the probability amplitude $c_{k}(t)$ can be obtained through the
normalization condition.

##  Appendix B The ergotropy

The QB as a quantum system can be described by the density matrix $\rho_{B}$
and the Hamiltonian $H_{B}$, which their spectral decompositions are given by

$\rho_{B}=\sum_{n}r_{n}|r_{n}\rangle\langle r_{n}|,\quad r_{1}\geq
r_{2}\geq...\geq r_{n}$ (17)

and

$H_{B}=\sum_{n}\varepsilon_{n}|\varepsilon_{n}\rangle\langle\varepsilon_{n}|,\quad\varepsilon_{1}\leq\varepsilon_{2}\leq...\leq\varepsilon_{n},$
(18)

where the sets $\\{|r_{n}\rangle\\}$ and $\\{|\varepsilon_{n}\rangle\\}$ are
eigenvectors of $\rho_{B}$ and $H_{B}$ respectively. Moreover, $r_{n}$’s and
$\varepsilon_{n}$’s are eigenvalues of $\rho_{B}$ and $H_{B}$ respectively.
The passive state $\sigma_{B}$ of $\rho_{B}$ is the state that can not extract
work from it and is given by

$\sigma_{B}=\sum_{n}r_{n}|\varepsilon_{n}\rangle\langle\varepsilon_{n}|.$ (19)

Thereby, the ergotropy can be formulated as

$\mathcal{W}=\sum_{n,m}r_{n}\varepsilon_{m}\left(|\langle
r_{n}|\varepsilon_{m}\rangle|^{2}-\delta_{m,n}\right),$ (20)

where $\delta_{m,n}$ is the Kronecker delta function.

## Acknowledgments

D. Wang was supported by the National Natural Science Foundation of China
(Grant No. 12075001), and Anhui Provincial Key Research and Development Plan
(Grant No. 2022b13020004). S. Haddadi was supported by Semnan University under
Contract No. 21270.

## Disclosures

The authors declare that they have no known competing financial interests.

## Data availability

No datasets were generated or analyzed during the current study.

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