11institutetext:  ASTRON, Netherlands Institute for Radio Astronomy, Oude
Hoogeveensedijk 4, Dwingeloo, 7991 PD, The Netherlands 22institutetext:
Department of Physics and Astronomy, University of New Mexico, Albuquerque,
USA 33institutetext: Kapteyn Astronomical Institute, University of Groningen,
PO Box 72, 97200 AB, Groningen, The Netherlands 44institutetext: Leiden
Observatory, Leiden University, PO Box 9513, 2300 RA, Leiden, The Netherlands
55institutetext: School of Physics, Trinity College Dublin, the University of
Dublin, Dublin-2, Ireland 66institutetext: LESIA, CNRS – Observatoire de
Paris, PSL 92190, Meudon, France 77institutetext: Dublin Institute for
Advanced Studies, Ireland 88institutetext: Thüringer Landessternwarte,
Sternwarte 5, 07778 Tautenburg, Germany

# Large closed-field corona of WX UMa evidenced from radio observations

I. Davis 1122 H. K. Vedantham 1133 J. R. Callingham 1144 T. W. Shimwell 1144
A. A. Vidotto 55 P. Zarka 66 T. P. Ray 77 A. Drabent 88

(Received XXX; accepted YYY)

The space-weather conditions that result from stellar winds significantly
impact the habitability of exoplanets. The conditions can be calculated from
first principles if the necessary boundary conditions– namely on the plasma
density in the outer corona and the radial distance at which the plasma forces
the closed magnetic field into an open geometry– are specified. Low frequency
radio observations ($\nu\lesssim 200$ MHz) of plasma and cyclotron emission
from stars probe these magneto-ionic conditions. Here we report the detection
of low-frequency ($120-167\,{\rm MHz}$) radio emission associated with the
dMe6 star WX UMa. If the emission originates in WX UMa’s corona, we show that
the closed field regions extends to at least $\approx 10$ stellar radii, that
is about a factor of a few larger than the solar value, and possibly to
$\gtrsim 20$ stellar radii. Our results suggest that the magnetic-field
structure of M dwarfs is in between Sun-like and planet-like configurations,
where compact over-dense coronal loops with X-ray emitting plasma co-exist
with a large-scale magnetosphere with lower plasma density and closed magnetic
geometry.

###### Key Words.:

Stars: coronae – Radio continuum: stars – Plasmas – Radiation mechanisms: non-
thermal

##  1 Introduction

M-dwarfs often host multiple terrestrial temperate exoplanets (Dressing &
Charbonneau, 2015; Gillon et al., 2017). Since M-dwarfs display heightened
levels of magnetic activity over gigayear timescales, their high energy photon
and plasma ejection could adversely affect the habitability of exoplanets
(Kopparapu et al., 2013).

Magnetohydrodynamical (MHD) simulations of exoplanet space-weather have been
used to model the interplanetary plasma density and magnetic field strength
around low-mass stars (Vidotto, 2021; Vidotto et al., 2013; Cohen et al.,
2014; Garraffo et al., 2017). These models must necessarily ascribe boundary
conditions on the plasma density and velocity, as well as the magnetic field
strength and topology on some closed surface. The large-scale magnetic field
at the stellar surface can be well constrained by Zeeman Doppler imaging maps
(Morin et al., 2010) but the radial evolution of the field, and the radius at
which it transitions from a closed topology to radially distended open
topology, cannot be determined without explicit knowledge of the plasma
density. The plasma density in these simulations is usually assumed ad-hoc or
estimated from the X-ray luminosity of the star. However, the X-ray
measurements are likely biased towards over-dense gas that is held confined in
short coronal loops (smaller than the stellar radius typically) since X-ray
emissivity depends on the square of the electron density. Instead, it is the
lower density plasma that occupies the bulk of the coronal volume that expands
as the stellar wind. The density contrast between plasma in short coronal
loops and the bulk of the corona could, in principle, be orders of magnitude
on M-dwarfs because their intense magnetic fields can support such a pressure
difference (see Pneuman, 1968, for a theoretical analysis of a generic open-
closed field boundary region). As a result, existing MHD simulations have been
hampered by a lack of empirically determined boundary conditions for the
coronal gas properties.

Polarised lower frequency radio emission, due to the plasma or cyclotron maser
mechanism, is expected to originate at much higher coronal layers (many
stellar radii as we demonstrate here) instead of from short coronal loops
close to the stellar surface where the densest X-ray emitting gas is thought
to be confined. Such radio observations can therefore provide a crucial
constraint on the density of the outflowing plasma and significantly improve
the reliability of exoplanet space-weather simulations.

The vast majority of stellar radio emission has only been studied at
frequencies greater than 1 GHz (Güdel, 2002) owing mainly to sensitivity
constraints. The advent of sensitive radio telescopes at metre wavelengths has
opened up the low-frequency window to stars and brown dwarfs (Lynch et al.,
2017; Villadsen & Hallinan, 2019; Vedantham et al., 2020c; Callingham et al.,
2021a; Vedantham et al., 2020a). As part of our ongoing effort to constrain
coronal parameters using low-frequency emission, here we focus on WX UMa— a
system blindly detected (Callingham et al., 2021b) in the ongoing Low-
Frequency Array (LOFAR) Two-metre Sky Survey (LoTSS) (Shimwell et al., 2017,
2019) with the LOFAR telescope (van Haarlem et al., 2013). We have chosen WX
UMa from our LoTSS detected stars because its magnetic field strength and
topology are well known (Morin et al., 2010). Here we use the LOFAR radio data
to show that, if the emission originates in the corona of WX UMa, then the low
frequency of emission, despite the strong magnetic field of WX UMa, requires
the star to have a closed-field configuration out to a radial distance of
$\gtrsim 10$ stellar radii and possibly $\gtrsim 20$ stellar radii. This is
substantially larger than radial distances to the open-closed field boundary
of $3-5$ stellar radii that has been derived from existing MHD simulations of
M-dwarfs stellar winds (Vidotto et al., 2013, 2014; Kavanagh et al., 2021),
but comparable to the size of ‘slingshot prominences’ (Collier Cameron &
Robinson, 1989) proposed to exist on fast-rotating cool stars (Jardine et al.,
2020).

##  2 Observations & Results

###  2.1 Low-frequency radio detection

Figure 1: Stokes-I (left) and Stokes-V (right) images of the area surrounding
WX UMa. The images are formed from the entire 8 hour observation duration and
entire 44 MHz bandwidth ($120-167\,{\rm MHz}$). The cross-hairs and circles
show the proper-motion corrected position of WX UMa (GJ412 B) and GJ412 A,
respectively, using the astrometric data from Gaia DR2 (Gaia Collaboration et
al., 2018). The flux densities of WX UMa and uncertainties are collated in
Table 1.

WX UMa was discovered as a radio source in the 120-168 MHz LoTSS survey
(Callingham et al., 2021b) as part of our on-going blind survey for stellar,
brown dwarf and planetary radio emission (Vedantham et al., 2020c, b;
Callingham et al., 2019, 2021a). LoTSS surveys the sky with a tessellation of
partially overlapping pointings (Shimwell et al., 2017, 2019). WX UMa was
observed in four such pointings (see Table 1 and Fig. 1). However, two of the
four pointings were observed simultaneously using LOFAR’s multi-beaming
capability and therefore have redundant information between them. The source
is detected in all three independent pointings and has a consistently high
($\gtrsim 70\%$) circular polarisation fraction when averaged over the full
$\approx 8$ h of each observation. Its companion star, GJ 412 A that is
$\approx 30^{{}^{\prime\prime}}$ away, is not detected in any pointing.

LoTSS field | Obs. date | Dist. | Obs. id | Stokes-I (mJy) | Stokes-V (mJy) | Pol. fraction
---|---|---|---|---|---|---
P5Hetdex | 2014-09-07 | $1^{\circ}.81$ | 233996 | $1.8(2)$ | $-1.4(2)$ | $0.8(1)$
P167+42 | 2016-03-22 | $1^{\circ}.54$ | 441476 | $0.6(1)$ | $-0.58(6)$ | $1.0(1)$
P163+45∗ | 2015-11-11 | $2^{\circ}.32$ | 403966 | $1.1(1)$ | $-0.8(1)$ | $0.7(1)$
P163+42∗ | 2015-11-11 | $2^{\circ}.31$ | 403966 | $0.9(1)$ | $-0.6(1)$ | $0.7(1)$
Table 1: Details of LOFAR observations of WX UMa. Quantities in brackets
denote $1\sigma$ root mean square (rms) uncertainty on the least significant
digit. WX UMa is seen through different gains in the primary beam which leads
to differing noise levels. The polarised fraction error is computed by Taylor
expanding the ratio to first order. The two exposures with an ∗ superscript
were taken concurrently using LOFAR’s multi-beam capability.

###  2.2 Spectro-temporal properties

The low frequency radio emission from WX UMa is broad-band and persistent. The
emission persists for most of the 8 hr observation block in each exposure, as
seen in Fig. 2. The spectrum, shown in Fig. 3 is particularly interesting, as
it has a consistent shape across the observations: it is bright at the low-end
of the observational band, and rapidly declines within a few MHz around
$\approx 150\,{\rm MHz}$ to a level below our detection threshold. For
instance, in the 2016-03-22 exposure (P167+42), for which we have the best
Stokes-V sensitivity, the emission changes from $\approx-1500$ $\mu$Jy at 124
MHz to being undetectable (below 3 $\sigma$) in the three 7.8-MHz wide
channels between $\approx 148$ to $\approx 170\,$MHz, which have an rms noise
of $\approx$ 140 $\mu$Jy. This corresponds to a factor of $\approx 4$ decline
over a fractional bandwidth $\Delta\nu/\nu\approx$ 0.15, where $\nu$ is the
observing frequency. The decline in other epochs appears to be even more
abrupt. However, the available signal-to-noise does not allow us to trace the
spectral decline in finer channels. The small number of exposures (three) also
do not allow a meaningful search for patterns in the location of the spectral
turnover with observation epoch.

We also averaged the images in the three 7.8-MHz channels of the P167+42 that
yielded non detections at the $3\,\sigma$ level. The averaged image (not shown
here) revealed a faint $3.5\,\sigma$ source at the location of WX UMa. This
suggests that the emission may settle into a low plateau as opposed to a total
cut-off. Hence, despite the rapid decline, the emission does not appear to
‘turn-off’ entirely at the high frequency end of our band.

Further details of the radio data processing and extraction of dynamic spectra
are given in Appendix A.

Figure 2: Time resolved Stokes-I flux density of WX UMa averaged over the full
120-167 MHz band on the three independent exposures (see Table 1). The
emission persists of most of the 8 hr survey pointing duration. Figure 3:
Stokes-V spectrum of WX UMa in the three independent exposures from the LoTSS
survey observations. The three sets of points show the spectrum for the three
independent exposures. Upper limits correspond to $1\sigma$. The grayscale
images are made from the P167+42 exposure (see Table 1) over 8 hr of synthesis
and $\approx 7.8\,{\rm MHz}$ of bandwidth. The red cross-hair mark the
position of WX UMa. The source consistently shows a drop in flux density
magnitude above $\approx 150\,{\rm MHz}$ in all epochs suggesting that this is
an enduring property of the emitter. We made a 3.5-$\sigma$ detection of WX
UMa when averaging over the last three bands in the P167+42 observation,
suggesting the spectral evolution at these higher frequencies is a low plateau
rather than a cutoff.

##  3 Discussion

Parameter | Value | Reference/Comments
---|---|---
Spectral class | dM6e | Alonso-Floriano et al. (2015)
Mass ($M_{\odot}$) | 0.095 | Newton et al. (2017)
Radius ($R_{\odot}$) | 0.12 | Morin et al. (2010)
Distance (pc) | 4.9 | Gaia Collaboration et al. (2018)
ROSAT X-ray luminosity (ergs s-1) | $3.62\times 10^{27}$ | Callingham et al. (2021b)
H$\alpha$ equivalent width (Å) | 6.85 $\pm$ 0.09 | Newton et al. (2017)
Surface polar magnetic field∗ (kG) | 4.53 | Morin et al. (2010)
Stellar rotation period (days) | 0.78 | Morin et al. (2010)
Average coronal temperature | $5.8\times 10^{6}\,{\rm K}$ | Based on Johnstone & Güdel (2015)
Density scale height | $0.63R_{\ast}$ | Neglecting centrifugal force
Table 2: Known properties of WX UMa from literature. ∗Morin et al. (2010) also
found that the magnetic field is well described by a dipole:
${\bm{B}}=(B_{0}/2)R^{-3}(2\cos\theta\bm{\hat{r}}+\sin\theta\bm{\hat{\theta}})$,
where $B_{0}$ is the polar surface field strength, $R$ is the radial distance
in units of the stellar radius, $\theta$ is the polar angle, and
$\bm{\hat{r}}$ and $\bm{\hat{\theta}}$ are the unit vectors along the radial
and polar-angular directions.

The known properties of WX UMa which will aid in our interpretation of the
radio emission are summarised in Table 2. Because the emission is persistent,
we only consider cases where the emitting plasma is trapped within a closed
magnetic geometry. This is because plasma that unbinds from the star is
expected to expand on a timescale that is much shorter than the duration of
our observations. For example, solar radio emission from plasma flowing along
open field lines typically lasts seconds to minutes as seen in solar Type-II
and Type-III bursts (Wild & Smerd, 1972).

Magnetically trapped plasma in stellar coronae can emit via different
mechanisms depending on the energy of the emitting electrons and the ambient
magneto-ionic parameters (Dulk, 1985). The high circular polarisation fraction
rules out all known incoherent emission mechanisms. Coherent emission
mechanisms that can generate up to $100\%$ circularly polarised emission are
fundamental plasma emission and fundamental and second harmonic cyclotron
emission. While cyclotron emission occurs at the harmonics of the electron
cyclotron frequency (Treumann, 2006; Melrose & Dulk, 1982), plasma emission in
a magnetic trap is thought to occur at the harmonics of the upper hybrid
frequency (Zaitsev & Stepanov, 1983; Stepanov et al., 2001) which is the
quadrature sum of the plasma and electron cyclotron frequencies. Both of these
mechanisms can account for the long duration of the observed emission. Plasma
emission will persist for as long as the turbulent Langmuir wave spectrum that
powers the emission is maintained. CMI from ultracool dwarfs is commonly seen
to be pulsed at the rotation period due to beaming, but for conducive
geometries, it can be visible at all rotation phases. CMI’s beaming geometry
is discussed in more detail in §3.6 and Appendix C.

###  3.1 Emission site

The magnetic field of WX UMa is largely dipolar with a polar surface magnetic
field strength of $\approx 4.5\,{\rm kG}$ (Morin et al., 2010). This yields a
maximum cyclotron frequency at the surface of $\approx 12.6\,{\rm GHz}$.
Because a dipolar field evolves with radial distance $R$ as $R^{-3}$,
fundamental cyclotron emission in the observed frequency band can only
originate at a radial distance of $\approx 4.7R_{\ast}$, where $R_{\ast}$ is
the stellar radius.

If the emission is due to the plasma mechanism, the emission site cannot be
accurately pinpointed because the plasma density profile in WX UMa is not
known apriori. However, a lower bound on the radial distance of the emission
site can be placed based on the requirement for radiation escape. Escaping
radiation in the presence of cyclotron damping is obtained when the ambient
cyclotron frequency is sufficiently low: $\nu_{c}<0.26\nu_{p}$ (see Appendix
B). Observable plasma emission at $120\,{\rm MHz}$ therefore requires the
ambient magnetic field to be below $\approx 11\,$G. For a dipolar field that
evolves with radial distance as $R^{-3}$, this value can only be reached on WX
UMa for $R>7.4\,R_{\ast}$. We note that although there is a narrow window of
low cyclotron opacity at small angles to the magnetic field for
$\nu_{c}>0.26\nu_{p}$ (see Fig. A1 of Vedantham, 2020), the bulk of the
radiation will still be absorbed since spontaneous plasma emission is quasi-
isotropic.

###  3.2 Polarisation parity

The Zeeman doppler imaging (ZDI) maps of WX UMa show the star to have an
intermediate inclination with a small magnetic obliquity and the south
magnetic pole pointed towards us (i.e. field lines pointed towards the star;
Morin et al., 2010). The observed radio emission is right-hand circularly
polarised (see Appendix A.1 for details of sign convention). It is therefore
in the $o$-mode if it originates in the polar regions (where the field points
away from us) and $x$ mode if it originates in the equatorial regions (where
the field points towards us). Fundamental plasma emission is polarised in the
$o$-mode, whereas cyclotron maser emission can be polarised in the $x$-mode or
$o$-mode depending on the ambient plasma density (Dulk, 1985; Vedantham,
2020). Because WX UMa has a mostly axisymmetric dipolar field (Morin et al.,
2010), we use cyclotron maser emission from Jupiter as a benchmark for our
discussion for the possibility of cyclotron maser emission from WX UMa (Zarka,
1998). Following this, the unstable electron distribution in a closed field
region necessary to drive the cyclotron maser is expected to be situated close
to the magnetic poles, and therefore, we only consider $o$-mode cyclotron
emission. The $o$-mode cyclotron maser for a loss-cone distribution is
expected to dominate for $0.1\lesssim\nu_{c}/\nu_{p}\ll 1$, with the $x$ mode
dominating at lower plasma densities ($\nu_{c}/\nu_{p}\lesssim 0.1$) (Hewitt
et al., 1982; Melrose & Dulk, 1982). In summary, the observed polarity
requires the emission to originate in the polar regions and be driven by
either fundamental plasma emission, or $o$-mode cyclotron maser emission in
which case $\nu_{c}/\nu_{p}\gtrsim 0.1$.

###  3.3 Extent of the closed field region

To recap, the two possibilities for the emission mechanism are: (a) plasma
emission at a radial distance of $>7.4R_{\ast}$ with ambient density of
$\approx 2\times 10^{8}\,{\rm cm}^{-3}$, and (b) cyclotron maser at a radial
distance of $4.7R_{\ast}$ with ambient density of $\gtrsim 2\times 10^{6}{\rm
cm}^{-3}$. Detailed MHD simulations are necessary to use these constraints to
determine the size of the closed field region accurately, but we can place
preliminary constraints with simplifying assumptions. Let us assume that the
density has a hydrostatic profile with the inclusion of the centrifugal force
(see eqn. 3 of Havnes & Goertz, 1984). We then use the dipolar structure of
the magnetic field and compute the equatorial distance at which the gas
pressure ($nk_{B}T$ where $T$ is the temperature, $n$ is the number density
and $k_{B}$ is Boltzmann’s constant), is equal to the magnetic pressure.
Beyond this radial distance, we assume that the gas forces open the field
lines. So we identify this radial distance as the radius of the closed field
region. Using this procedure and values relevant for WX UMa from table 2 we
find that the closed-open boundary occurs at radii of $\approx 10R_{\ast}$ and
$22R_{\ast}$ for the case of fundamental plasma emission and cyclotron
emission respectively. Finally, the co-rotation radius, $R_{\rm
C}=(GM/\Omega^{2})^{1/3}$ ($G$ is the gravitational constant, $\Omega$ is the
angular speed, $M$ is the stellar mass) is about $13.6R_{\ast}$ for WX UMa. It
is noteworthy that the corotation radius is comparable to the approximate
open-closed boundary for the case of plasma emission.

###  3.4 Source of emitting electrons

With these considerations, it is still presently unclear what the source of
the emitting electrons is in WX UMa, but we can draw analogies with two known
systems. We first consider the solar paradigm for the case of plasma emission.
In this case, the ultimate energy source is thought to be magnetic
reconnection in coronal loops close to the surface (Priest & Forbes, 2002).
However, the brightness temperature arguments in Sec. 3.5 show that the solar
paradigm can only apply to WX UMa if its entire coronal volume out to a radial
distance of $\gtrsim 7.4R_{\ast}$ is filled with flaring loops which is
unprecedented. We may alternatively adopt the Jovian paradigm, where cyclotron
maser emission is emitted by suprathermal electrons (Dulk, 1985) that are
accelerated by breakdown of co-rotation between the magnetic field and the
magnetospheric plasma far from the planet’s surface, or by unipolar induction
by its moons (Goldreich & Lynden-Bell, 1969). A similar acceleration mechanism
sited in the equatorial plasma sheet at few tens of stellar radii may be
responsible for WX UMa’s radio emission. Such an acceleration mechanism has
also been suggested for magnetic chemically peculiar stars (Linsky et al.,
1992). It it also noteworthy that both WX UMa and magnetic chemically peculiar
stars have strong large scale magnetic fields, hot X-ray coronae and emit
coherent circularly polarised emission (e.g. CU Vir, Trigilio et al., 2000).

###  3.5 Brightness temperature

The brightness temperature of the radio emitter assuming isotropic emission is
$T_{b}\approx 10^{12.5}x^{-2}\,{\rm K}$ where $x$ is the radius of the emitter
in units of the stellar radius. Theory allows the brightness temperature of
cyclotron maser emission to be many orders of magnitude higher than this value
(Treumann, 2006; Melrose & Dulk, 1982), and hence, the observed value can be
readily accommodated. Fundamental plasma emission on the other hand cannot
attain brightness temperatures $\gg 10^{12}\,{\rm K}$, especially at such low
frequencies (Vedantham, 2020). As shown in §3.1, the projected source radius
for plasma emission is $x\approx 7.4$, giving an observed brightness
temperature of $\sim 10^{11}\,{\rm K}$. Although we note there is no
precedence for plasma emission to be simultaneously occurring at this
brightness throughout the corona (at $x=7.4$ in this case), such a brightness
temperature can be reached by plasma emission based on canonical theory if the
plasma is highly turbulent (see Appendix A.1). Therefore, brightness
temperature arguments do not provide a clear discriminant between the two
plausible emission mechanisms.

###  3.6 Origin of the spectral evolution

Coherent stellar and planetary radio emission are known to display narrowband
spectral features but their frequency centroid generally varies stochastically
with time (see Litvinenko et al., 2009; Villadsen & Hallinan, 2019, for
examples).111A possible exception is AD Leo whose bursts observed by Villadsen
& Hallinan (2019) show an apparent cut-off near-consistently around 1.5 GHz.
As such, the rapid spectral evolution from WX UMa is not unusual, but the
persistent location of the spectral turn-over is noteworthy. The persistence
suggests that it encodes meaningful information on the nature and geometry of
emission.

The expressions for the emissivity and absorption coefficients of various
emission processes themselves vary smoothly, to polynomial order, with
frequency (Zheleznyakov, 1996). We must therefore seek an explanation for the
rapid spectral evolution in terms of beaming, resonance conditions or abrupt
boundaries over which the plasma properties can change substantially. With
this in mind, we consider three plausible explanations.

  * •

Beaming: If the emission is driven by the cyclotron maser mechanism, then a
natural explanation for the cut-off would be due to beaming of the emission.
Cyclotron maser emission is beamed into a hollow cone with a large opening
angle, $\theta\lesssim\pi/2$, small thickness, $\Delta\theta\ll\theta$ and
whose axis is parallel to the ambient magnetic field. Radiation cones axes on
a given field line make smaller angles with the magnetic axis at larger
frequencies as illustrated in Fig. 4. Hence it is possible to construct
geometries where the emission goes out of sight at higher frequencies. For
example, consider emission originating on field lines that intersect the
equatorial plane at some radius $L$. If the magnetic field line makes a polar
angle $\alpha(\nu)$ at the point where emission at frequency $\nu$ originates,
then the emission will be visible to all observers within a magnetic co-
latitude ‘visibility band’ that lies between $\theta-\alpha(\nu)-\Delta\theta$
and $\theta+\alpha(\nu)+\Delta\theta$. If $\alpha(\nu)$ evolves sufficiently
rapidly with $\nu$, then a spectral cut off can be explained by beaming. Could
a spectral evolution be observed due to the visibility-band edge moving over
the line of sight to the Earth within the observed frequency range? The
function $\alpha(\nu)$ can be computed for a given $L$, surface magnetic field
strength and dipolar geometry. For the case of LOFAR-band emission from WX
UMa, the slope $\partial\alpha(\nu)/\partial\nu$ varies between
$0^{\circ}.05\,{\rm MHz}^{-1}$ and $0^{\circ}.12\,{\rm MHz}^{-1}$ as $L$ is
varied between $20R_{\ast}$ and $6R_{\ast}$ (further details in Appendix).
This corresponds to a shift in the visibility band by $1^{\circ}$ and
$2^{\circ}.4$ within a 20 MHz band within the observed frequency range. The
angular shift is significantly smaller than the characteristic width of the
emission cone’s surface, $\Delta\theta$ (Zarka, 1998; Melrose & Dulk, 1982) as
well as other geometric parameters such as magnetic obliquity. Hence, an
apparent cutoff due to beaming is possible but with a somewhat contrived
geometric setup. Nevertheless, the beaming hypothesis can be tested by a
search for a characteristic rotation and spectral modulation of the radio
light curves at the rotation rate of WX UMa, which is an expected result of
any magnetic obliquity in the system.

Figure 4: A not-to-scale sketch showing the geometry of cyclotron maser
emission on a single dipolar field line that lies in the plane of the page.
The higher frequency emission cone (blue) is oriented at a smaller polar
angle, $\alpha$, when compared to the emission cone at a lower frequency
(red). Here, $L$ is the distance to the field line for a polar angle of
$0^{\circ}$, $R$ is the distance from the star’s centre to the emission cone
along the field line, $\theta$ is the angle between the direction of emission
and the interior wall of the cone, and $\Delta\theta$ is the angular width of
the emission cone’s wall. The effects of $\alpha$ and $\theta$ on emission
frequency has been exaggerated here and is not to scale for the LOFAR
frequency range.

  * •

Resonant cyclotron absorption: If the emission is due to fundamental plasma
emission, then the cut-off could be due to resonant cyclotron absorption of
higher frequency emission that originates deeper in the corona. This is
possible because the cyclotron frequency evolves with radial distance as
$R^{-3}$ whereas the plasma frequency can evolve much more gradually with
radial distance, particularly if the density scale height is large. In such a
situation, it is possible for the radiation escape condition to be satisfied
at larger radii where lower frequency emission originates, but not at lower
radii, where higher frequency emission originates, leading to a cut-off. Fig.
5 shows plausible hydrostatic density profiles along different field lines,
each with a different base density that leads to catastrophic cyclotron
absorption within the LOFAR band. The density profiles all require that the
base coronal density is restricted to a narrow range of values (about a 10%
range) around $10^{9.1}\,{\rm cm}^{-3}$ which is comparable to the solar
value. If this is the correct mechanism, then similar emission (long duration,
highly polarised) must not be visible from WX UMa at frequencies above the
LOFAR band and long term monitoring should not show clear signs of rotation
modulation in the LOFAR-band intensity. We note that WX UMa is detected
between 2 and 4 GHz in the ongoing VLASS survey (Lacy et al., 2020) at a flux
density of $1.3(1)$ mJy.222Based on the VLASS ‘quick-look’ images. See
https://science.nrao.edu/science/surveys/vlass However, the VLASS survey
observes with 3 minute exposures which is not long enough to associate the
VLASS-detected emission with the long duration LOFAR-detected emission as
opposed to a shorter duration burst.

Figure 5: Plot of the radial profile of the cyclotron frequency (solid black
line) and the upper hybrid plasma frequency (coloured lines) for different
base plasma density values given in units of $cm^{-3}$. The plasma has been
assumed to follow a hydrostatic density structure along any given field-line
while taking into account centrifugal force (Havnes & Goertz, 1984, their
equation 3). The broken lines denote regions where cyclotron absorption will
preclude radiation escape. The yellow-band shows the observed frequency range.

  * •

Surface cut-off: The rapid spectral evolution can be explained if the bulk of
the emission does not originate in WX UMa’s corona but instead in the
magnetosphere of a closeby (unresolved) sub-stellar companion and the cut-off
frequency corresponds to a cyclotron frequency consistent with a surface
magnetic field strength of $\sim 40$ G, much in the same way Jovian cyclotron
emission cuts-off at around 40 MHz. This would be an unprecedented conclusion
if true, as it would amount to a radio discovery of a sub-stellar companion
(exoplanet or brown dwarf) around WX UMa. A prediction of this interpretation
is periodic rotational modulation of the radio emission at a rate that does
not match the known rotation rate of WX UMa. However, this property cannot be
inferred from the current data since the emission has been detected in all
three independent pointings.

In summary, we have identified three possible mechanisms to explain the rapid
spectral evolution all of which can be tested with monitoring the LOFAR-band
emission over several rotation cycles and with gigahertz frequency
observations.

##  4 Summary and outlook

We have detected persistent highly circularly polarised ($\gtrsim 70\%$) radio
emission around 144 MHz from WX UMa. If the emission originates from WX UMa’s
corona, there are two plausible emission mechanisms that can account for the
radio emission: (a) plasma emission at the fundamental at a radial distance of
$\gtrsim 7.4R_{\ast}$ where the ambient plasma density is $\approx
10^{8.5}\,{\rm cm}^{-3}$, and (b) $o$-mode cyclotron maser emission in which
case the emission originates at a radial distance of $\approx 4.7R_{\ast}$
where the ambient coronal density is $10^{6.5}\,{\rm cm}^{-3}$.

We expect our plasma density constraints to provide guidance to MHD
simulations of the stellar wind of WX UMa. A simple balance between gas and
magnetic pressure yield the radial extant of the closed field regions to be
$\approx 10R_{\ast}$ (plasma emission) and $\approx 22R_{\ast}$ (cyclotron
emission) respectively. Such radii are substantially larger than those that
have been inferred from several MHD wind simulations of M-dwarfs (see for e.g.
Vidotto et al., 2014), but are comparable to recent simulation of M-dwarf
winds with low mass-loss rates (Kavanagh et al., 2021). Interestingly, the
closed$-$open field boundary in the equatorial plane for the plasma emission
model is close to the co-rotation radius for WX UMa which is the anticipated
extent of slingshot prominences (Jardine et al., 2020).

WX UMa’s radio spectrum shows a rapid decline within the observing band which
could be a result of beaming of cyclotron emission or due to cyclotron
absorption of plasma emission. Alternatively, the radio emission we observe
might have originated not on WX UMa but on a sub-stellar companion. All three
possibilities can be tested with longer term multi-frequency monitoring of WX
UMa’s radio emission.

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###### Acknowledgements.

ID completed this work as part of the ASTRON summer student programme. JRC
thanks the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) for
support via the Talent Programme Veni grant. AAV and TPR acknowledge funding
from the European Research Council (ERC) under the European Union’s Horizon
2020 Research and Innovation Programme (grant agreements No 817540, ASTROFLOW
and No 743029 EASY respectively). AD acknowledges support by the BMBF
Verbundforschung under the grant 05A20STA. This paper is based on data
obtained with the International LOFAR Telescope (obs. IDs 233996, 441476,
403966 and 403968) as part of the LoTSS survey. LOFAR is the Low Frequency
Array designed and constructed by ASTRON. It has observing, data processing,
and data storage facilities in several countries, that are owned by various
parties (each with their own funding sources), and that are collectively
operated by the ILT foundation under a joint scientific policy. The ILT
resources have benefitted from the following recent major funding sources:
CNRS-INSU, Observatoire de Paris and Université d’Orléans, France; BMBF, MIWF-
NRW, MPG, Germany; Science Foundation Ireland (SFI), Department of Business,
Enterprise and Innovation (DBEI), Ireland; NWO, The Netherlands; The Science
and Technology Facilities Council, UK. This research made use of the Dutch
national e-infrastructure with support of the SURF Cooperative (e-infra
180169) and the LOFAR e-infra group. The Jülich LOFAR Long Term Archive and
the German LOFAR network are both coordinated and operated by the Jülich
Supercomputing Centre (JSC), and computing resources on the supercomputer
JUWELS at JSC were provided by the Gauss Centre for supercomputing e.V. (grant
CHTB00) through the John von Neumann Institute for Computing (NIC). This
research made use of the University of Hertfordshire high-performance
computing facility and the LOFAR-UK computing facility located at the
University of Hertfordshire and supported by STFC [ST/P000096/1], and of the
Italian LOFAR IT computing infrastructure supported and operated by INAF, and
by the Physics Department of Turin university (under an agreement with
Consorzio Interuniversitario per la Fisica Spaziale) at the C3S Supercomputing
Centre, Italy. The Jülich LOFAR Long Term Archive and the German LOFAR network
are both coordinated and operated by the Jülich Supercomputing Centre (JSC),
and computing resources on the supercomputer JUWELS at JSC were provided by
the Gauss Centre for Supercomputing e.V. (grant CHTB00) through the John von
Neumann Institute for Computing (NIC).

##  Appendix A Radio data processing

The data were processed with the usual LoTSS pipeline333The LoTSS pipeline can
be found at https://github.com/mhardcastle/ddf-pipeline (Shimwell et al. 2017,
2019) which is described in Tasse et al. (2020). We then performed an
additional round of self-calibration in the direction of WX UMa with a
procedure described in van Weeren et al. (2020) and also employed in
(Vedantham et al. 2020c). After calibration, the measurement set was further
processed to make source extraction easier for producing light curves and
spectra. WX UMa was removed from the mask image which was then used to produce
a model file using WSClean. WX UMa was also removed from these model images so
that it would not be removed during the subtraction of other sources. The
subtracted model image was then used with WSClean again to predict
visibilities which were recorded in a new modelled data column in the
measurement set. The modelled data was subtracted from the real data column
and recorded in a new column - this served to remove all other bright sources
from the region besides WX UMa. Finally, DPPP was used to phaseshift the
measurement sets to be centred on WX UMa, using the subtracted data as the new
real data.

After extra sources were removed from the field and the set was phaseshifted,
frequency and time slices for Stokes-I and V were produced via WSClean.
Because the signal to noise ratio of the observation was not large enough to
produce a full dynamic spectrum, the frequency slices were averaged over all
time and the time slices were averaged over the first 40 channels (120 to to
135.83 MHz, $\approx$16 MHz bandwidth) where WX UMa is the brightest. The
Background and Noise Estimation (BANE) tool in Aegean (Hancock et al. 2012)
was used with each image to produce rms and background images. Aegean was run
with a 4$\sigma$ seedclip to avoid identifying extraneous sources. The flux
density for WX UMa for a particular image was assigned from the peak flux
found by Aegean if a source was found in the expected location of WX UMa, and
the rms noise was retrieved from the rms image produced by BANE. For images
where no 4$\,\sigma$ source was found at the star’s coordinates, a priorized
fitting was applied using the catalogue from the image with the most
significant detection in order to extract $\approx 3\,\sigma$ detections at
the expected position of WX UMa. The images were also inspected by eye to make
sure the results of Aegean were consistent with the images. The temporal
evolution of WX UMa’s radio flux is shown in Fig. 2 and its spectral evolution
is shown in Fig. 3.

We note that the some the images in the light curve estimation have non-
Gaussian noise due to residual sidelobe noise (from poorer $uv$ coverage) and
low-level calibration errors.

###  A.1 Stokes-V sign convention

There are two commonly used Stokes-V sign conventions in literature. The
pulsar literature typically follows the convention ${\rm V}={\rm LCP}-{\rm
RCP}$ whereas the IAU convention is $V={\rm RCP}-{\rm LCP}$ (van Straten et
al. 2010). To check the convention applicable to the LoTSS images, we compared
the sign of Stokes-V of several pulsar detections to that in the literature
and found that the LoTSS survey data products follow the same convention as
the one in pulsar literature (Callingham et al. 2021b). Hence, WX UMa is
radiating ${\rm RCP}$ radiation, which means that the electric field vector
rotates counter-clockwise as a function of time to an observer looking at
incoming radiation.

##  Appendix B Emission mechanism

###  B.1 Brightness temperature of plasma emission

In Fig. 6, we computed the brightness temperature of plasma emission using the
expressions of Stepanov et al. (2001, speficially, their eqn. 15). In doing
so, we assumed a peak turbulence level of $w=10^{-5}$ and to compute the
plasma temperature, we used the empirical relationship of Johnstone & Güdel
(2015) and the observed X-ray luminosity (Table 2).

Figure 6: Brightness temperature of plasma emission computed for parameters
relevant for WX UMa (annotated in the figure). The black bar is the measured
value based on the radiation escape conditions and the assumption that the
entire projected coronal surface is radiating.

###  B.2 Escape of plasma emission

Fundamental plasma emission is excited at the upper-hybrid frequency:
$\nu_{\rm uh}\approx\sqrt{\nu_{p}^{2}+\nu_{c}^{2}}$ where $\nu_{p}$ and
$\nu_{c}$ are the plasma and cyclotron frequencies respectively. Note that the
upper-hybrid frequency is always larger than the cyclotron frequency. The
cyclotron absorption optical depth for $o$-mode radiation propagating at an
angle $\theta$ to the magnetic field may be approximated by

$\tau=\left(\frac{\pi}{2}\right)^{5/2}\frac{2}{c}\frac{\nu_{p}^{2}}{\nu}\frac{s^{2s}}{s!}\left(\frac{k_{B}T\sin^{2}\theta}{2m_{e}c^{2}}\right)^{s-1}(1+|\cos\theta|)^{2}L_{B}$
(1)

where integer $s=\nu/\nu_{c}>1$ is the cyclotron harmonic number, $T$ is the
coronal temperature, $m_{e}$ is the electron mass, $c$ is the speed of light,
$k_{B}$ is Boltzmann’s constant and $L_{B}$ is the magnetic scale length which
we take to be $R_{\ast}$. For the parameters of interest: $T\sim
10^{6.7}\,{\rm K}$, $L_{B}\sim 10^{10}\,{\rm cm}$, the optical depth is safely
below unity only for $s\geq 4$. Hence, the radiation escape condition is
$\sqrt{\nu_{p}^{2}+\nu_{c}^{2}}>4\nu_{c}$, which gives $\nu_{c}<0.26\nu_{p}$.

##  Appendix C Spectral evolution of the emission-cone

Consider a single dipolar field line which intersects the equatorial plane at
radial distance $L$. The radial distance and polar angle, $\theta^{\prime}$
along the field line are related via $L=R\sin^{2}\theta^{\prime}$. The polar
inclination angle of the field lines, $\alpha$ is given by
$\cos\alpha=(3\cos^{2}\theta^{\prime}-1)(1+3\cos^{2}\theta^{\prime})^{-1/2}$.
The magnetic field strength (and cyclotron frequency) vary continuously along
the field line according to
$\nu(R)=\nu_{0}R^{3}_{\ast}/(2R^{3})(4-3R/L)^{1/2}$, where $\nu_{0}$ is the
peak cyclotron frequency at $\theta^{\prime}=0,\,R=R_{\ast}$. With the above
relationships, we predicted the inclination of the emission cone axis for
various $L$-shell at a range of cyclotron frequencies (see Fig. 7). The
numerically computed slope of the $\alpha$ versus $\nu$ lines at $140\,{\rm
MHz}$ vary between $\partial\alpha/\partial\nu\approx 0.2^{\circ}/{\rm MHz}$
and $\partial\alpha/\partial\nu\approx 0.8^{\circ}/{\rm MHz}$ as $L$ is varied
between $6$ and $20$. The magnitude of the slope decreases for larger $L$ and
must asymptote to zero as $L$ tends to infinity.

Figure 7: Polar angle of the magnetic field at the site of emission (y-axis)
versus frequency of emission (x-axis) for different field lines specified by
the radial distance at which they intersect the equatorial plane.


# Effect on spectral purity due to on-chip temporal manipulation of the pump

Lukas Rodda  Quantum Engineering Technology Labs, University of Bristol,
Nanoscience and Quantum Information Building, Tyndall Avenue, Bristol BS8 1FD,
United Kingdom  Ben M. Burridge  Quantum Engineering Technology Labs,
University of Bristol, Nanoscience and Quantum Information Building, Tyndall
Avenue, Bristol BS8 1FD, United Kingdom  Quantum Engineering Centre for
Doctoral Training, Centre for Nanoscience & Quantum Information, University of
Bristol, Bristol, United Kingdom  Jorge Barreto  Quantum Engineering
Technology Labs, University of Bristol, Nanoscience and Quantum Information
Building, Tyndall Avenue, Bristol BS8 1FD, United Kingdom  Imad I. Faruque
Quantum Engineering Technology Labs, University of Bristol, Nanoscience and
Quantum Information Building, Tyndall Avenue, Bristol BS8 1FD, United Kingdom
<EMAIL_ADDRESS>
###### Abstract

Photonic Integrated Circuits (PIC)s are a promising contender for quantum
information technologies. The spectral purity of photons is one of the key
attributes of PIC photon-pair sources. The dual-pulse pump manipulation
technique [1] showed $>$99% purity in ring-resonator photon-pair sources.
Here, we have developed a PIC to shape a pulse into dual, triple and quadruple
pulses and investigated the effect of these pump pulse configurations on the
purity. Our results show that more complex configurations over dual-pulse do
not result in comparatively higher purity but allow accurate control over
choosing arbitrary values of the purity.

##  1 Introduction

Quantum information is the foundation for many strategic technologies such as
quantum computing and communication [2, 3, 4, 5]. Integrated photonics has
shown to be a promising platform to scale-up and more stable quantum
experiments while benefiting from well developed fabrication techniques [6].
Using photons as quantum bits (qubits) requires a source of single photons,
such as a heralded single-photon source. Recently published results present
various methods to improve the spectral purity of the photon sources, which is
crucial for high-fidelity gate operation [7]. For example, complex resonator
designs [8], or temporal manipulations of the pulsed pumps [1, 9] are two of
the ways to achieve near unity purity, an essential characteristic of photon
pair sources for heralding..

Nonlinear light-matter interaction mechanisms such as spontaneous four-wave
mixing (SFWM) are often used in photonic devices in integrated photonics
platforms (e.g., silicon-on-insulator platform) to generate photon-pairs,
where two pump photons are converted into a signal and idler photon. A ring-
resonator is one such structure that enhances the production of photon-pairs
with high brightness and high purity [7]. The spectral purity of this source
is often quantified using joint spectral amplitude (JSA) of emitted signal and
idler photons. This is given by [10]

$\Phi(\omega_{s},\omega_{i})=\int d\omega_{p}d\omega_{p^{\prime}}\hskip
4.0pt\alpha(\omega_{p})\alpha(\omega_{p^{\prime}})\phi(\omega_{p},\omega_{p^{\prime}},\omega_{s},\omega_{i})L(\omega_{p})L(\omega_{p^{\prime}})L(\omega_{s})L(\omega_{i})$
(1)

where $\alpha(\omega)$ is the pump envelope, $\phi$ is the phase matching
term, $L(\omega)$ is the Lorentzian field enhancement terms arising from the
ring resonances, with pump, signal and idler frequencies: $\omega_{p}$,
$\omega_{s}$ and $\omega_{i}$. The pump is spectrally confined by the ring
resonance at the pump frequencies, which causes correlations in the photon
pairs. Theoretically the purity of a photon pair state generated in a ring is
limited to 0.92 for a single pulse pump [10]. However, by manipulating the
phase and temporal spectrum of the pump, the effect of the resonance at the
pump frequencies can be removed, leading to a separable pair state and
arbitrarily high purity [1].

$\alpha_{target}\propto L(\omega_{p})^{-1}\times\alpha_{single}$ (2)

This ideal target pump can be approximated using a dual pulse (Fig. 1a), of
two $\pi$ phase-shifted pulses, which has been theoretically shown to improve
purity past 0.99 [1], and was demonstrated experimentally using fibre-optics
[9], however pump configurations were only stable for approximately 1h and
tuning ranges of the parameters were limited.

Figure 1: Target pulse for completely pure JSA, compared to dual and triple
pulse pumps. The dual pulse shown has a splitting ratio 0.55 and a delay of
10ps, the best performing case. The triple pulse has splitting ratios of 0.8
and 0.8, and a 10ps delay between each pulse

.

Figure 2: a) Experimental set-up used to collect purity measurements for
different pulse pump shapes. Pulses are created in the first chip using three
delay lines (DL), then combined with a stimulating signal field from a
continuous wave (CW) laser, to generate photon pairs in a ring resonator
source on the second chip. An optical spectrum analyser (OSA) then measures
the intensity of generated idler photons. Also used are: polarisation
controllers (PC), beam splitters (BS), powermeters (PM), an erbium-doped fibre
amplifier (EDFA), a variable optical attenuator (VOA) and dense-wavelength
division multiplexers (DWDM). b) Single, dual and triple pulses, from
experimental and simulated data, where experimental data was collected by the
OSA after setting the parameters on-chip, and simulated data from models using
the same parameters. For the dual pulse $\eta=0.8$ and $\Delta\tau=20ps$
(modelled with equation 4). For the triple pulse $\eta_{1},\eta_{2}=0.8,0.2$
and $\Delta\tau_{1}=20ps$, $\Delta\tau_{2}=40ps$ (modelled with equation 5).
An inverse Fourier transform was used to plot the simulated pulses in time
domain.

Here we present simulated and experimental results which explore alternate
pump profiles, such as a pump containing three pulses, to therefore match,
improve or provide more control over the purity of a dual pulse pump. We used
a PIC for pulse pump manipulation. This allowed multiple pump pulse
configurations compared to the literature [9, 1], and also allowed for
accurate control over parameters while guaranteeing the phase stability of the
composite pump pulse as parameters are changed. The pump can then be used with
any conventional resonator source, allowing for easy integration into new or
existing photonic experiments.

##  2 Methods

The set up shown in Fig. 2a was used to create various pulse pump
configurations and record the spectral purity of our photon pair source. Our
experiment uses two 220 nm thickness silicon-on-insulator chips. We used a
pulsed laser (PriTEL FFL) which produces $sech^{2}$ shaped single pulses at
1550nm wavelength with 41 GHz Full-Width Half Maximum (FWHM) and 50 MHz
repetition rate. Such a pulse pump $\alpha_{single}$ is described using the
following equation

$\alpha_{single}=sech^{2}((\omega-\omega_{0})/\sigma)$ (3)

where, $\omega$ denotes angular frequency, $\omega_{0}$ denotes central
frequency and $\sigma$ denotes FWHM of the pulse. These pulses were coupled to
the first chip, denoted as the pulse shaping chip in the figure, through a
polarisation controller to optimise the coupling into the chip using a
vertical grating coupler, as well as a beam splitter as a 10% tap to monitor
the input power in the chip. Each single pulse is split into two by a Mach-
Zehnder interferometer (MZI). The MZI is controlled by a thermo-optic phase
shifter to have the desired splitting ratio $\eta_{1}$. The two pulses are
then temporally separated using a fixed delay ($\Delta\tau_{1}$) line of 20 ps
($\Delta\tau_{1}=$ 20 ps) and then recombined using another MZI. The relative
phase difference ($\phi_{1}$) between the two pulses can also be set using
another thermo-optic phase shifter. At this point in the circuit, the
resulting pulse configuration can be expressed by Eq. 4 which can be denoted
as a dual pulse.

$\alpha_{dual}=[\sqrt{\eta_{1}}+\sqrt{1-\eta_{1}}\exp{(-i\Delta\tau_{1}(\omega-\omega_{0})+i\phi_{1})}]\times\alpha_{single}$
(4)

The same process continues in series twice more before the resulting pulse is
output from the chip, allowing a pulse configuration of up to four pulses to
be created with arbitrary splitting ratios and phases, and with fixed 20 ps
time delays between them. Thus, a triple pulse, with another splitting ratio
($\eta_{2}$), time delay ($\Delta\tau_{2}$) and phase ($\phi_{2}$), can be
expressed as

$\alpha_{triple}=[\sqrt{\eta_{1}}+\sqrt{(1-\eta_{1})\eta_{2}}\exp((-i\Delta\tau_{1}(\omega-\omega_{0})+i\phi_{1})\\\
+\sqrt{(1-\eta_{1})(1-\eta_{2})}\exp(-i\Delta\tau_{2}(\omega-\omega_{0})+i\phi_{2})]\times\alpha_{single}$
(5)

Where all delays and phases are relative to the first pulse maximum.

A few configurations are plotted in Fig. 2b, where the single pulse Eq. 3 was
numerically fit to the corresponding output of the chip, and the dual and
triple pulse Eqs. 4 and 5 were fit using the same values for $\eta$,
$\Delta\tau$ and $\phi$ as were set on chip. Once the desired pulse
configuration was acquired, it was amplified using an EDFA to be used as a
strong laser pump for stimulated emission tomography (SET) in the second chip
[11].

The second chip contains a racetrack resonator source which can produce
photon-pairs through a nonlinear light-matter interaction through SFWM. To
perform SET, we have combined our pump laser with a continuous wave (CW) laser
(Yenista Tunics) as a seed field using a DWDM (Fig. 2(a)). The pump and the
seed field produce a stimulated idler field in the resonator. A variable
optical attenuator was used to bring the power to an arbitrary level, and an
extra DWDM was used to filter the noise after the EDFA. This was then coupled
into the second chip, with a 1% tap to monitor the combined power and two PCs
to optimise the coupling.

The racetrack resonator is followed by an asymmetric-MZI that removes the pump
field from the signals. After the chip, the pump was monitored using another
tap. The pump and the stimulated idler fields were combined with a final DWDM
which filtered out the seed field. The intensity of the generated idler
photons was measured using a (Finisar WaveAnalyser) optical spectrum analyser
(OSA). A single pulse JSI was collected using SET first by configuring the
pulse shaper circuit so that the original pulse wasn’t split. Then a triple
pulse was created by splitting a single pulse twice with a ratio of
$\eta_{1}=\eta_{2}=0.8$, and the JSI was collected using SET.

The JSA is approximated using experimentally measured $\sqrt{JSI}$.
Alternatively we simulate the JSA numerically in MATLAB with Eq. 1 and the
corresponding pulse shape ($\alpha$). We then calculate the purity for each
pulse pump configuration through a Schmidt decomposition of the JSA [10]. We
have also simulated the in-resonator field spectral distribution given by
$A_{p}(\omega)=L(\omega-\omega_{p})\times\alpha_{pump}((\omega-\omega_{p}))$,
and the corresponding temporal distribution using Fourier transformations.

##  3 Results and discussion

The pulse shaper circuit was characterised by creating single, dual and triple
pulse pumps with varying splitting ratios, phases and delays. These matched up
well with our model (Fig. 2b), showing the chip is capable of a fine level of
control. The single, dual and triple pulse pumps were modelled with Eqs. 3, 4
and 5.

An $n$ pulse pump was modelled to investigate the purity of a pump with an
arbitrary number of pulses. A train of pulses was constructed by adding $n$
single pulses together, with a constant time delay between them and all $\pi$
out of phase with respect to the first pulse. Then two different
configurations, in terms of the pulse amplitudes, were used. First, the pulse
train was set so that after an initial pulse, each next pulse had a constant,
lower amplitude as shown in Fig. 3a. Then a pulse train was set where the
amplitude of each subsequent $n$ pulse was reduced using a splitting ratio
$\eta/2^{n}$ as shown in Fig. 3c, similar to how the pulse shaper circuit
operates.

Figure 3: a) An example of a pulse train used in the $n$ pulse pump
simulations. Here the amplitude is constant after an initial stronger pulse,
with a time delay $\Delta\tau$ between each pulse b) Purity from simulated
JSAs as the number of pulses increases, for the constant pulse train. c) Pulse
train where amplitude decreases according to a splitting ratio $\eta$, for
each pulse $n$, as shown by the envelope. d) Purity as the number of pulses
increases, with different delays ($\Delta\tau$) between them. Here the pulse
train amplitudes decreased according to a splitting ratio $\eta=0.55$, as
shown in c).

Both of these configurations showed similar results (Figs. 3b, d). No
improvement in purity over the dual pulse case was seen as the number of
pulses increased. Furthermore, only a few cases exceed the 0.92 purity limit,
with the first configuration only plateauing at the limit as the pulse train
approximated a single pulse. Since these configurations have fixed time delays
and phases, to find higher purity solutions, we focus on the triple pulse
configuration (Eq. 5) where the splitting ratios and phases can be varied
arbitrarily to explore a larger parameter space to illustrate any benefits
over dual pulse case.

First the two splitting ratios ($\eta_{1,2}$) were varied while keeping the
delays ($\Delta\tau_{1,2}$) and phases ($\phi_{1,2}$) constant at 20ps and
$\pi$ respectively. Then $\eta_{1}$ and $\eta_{2}$ were both fixed to 0.8, and
the phases were varied, keeping the delays the same. The parameters were
scanned, simulating the JSA for each configuration, creating two heatmaps
shown in Fig. 4.

Figure 4: a) Purity as splitting ratios $\eta_{1}$ and $\eta_{2}$ are varied,
for a triple pulse with $\Delta\tau_{1}$ and $\Delta\tau_{2}$ set to 20ps and
40ps respectively, and $\phi_{1}$, $\phi_{2}$ both set to $\pi$. b) Purity as
the phase of the second and third pulse is varied relative to the first. Here
$\eta_{1}$ and $\eta_{2}$ are both 0.8, with the same delays as previously.

The results show (Fig. 3) that the purity does not reach higher than the best
dual pulse case, and only a few configurations perform better than the single
pulse limit. For simulations of the dual pulse (Eq. 4), the best performing
configuration reaches 0.999 purity, with a large set of the parameter space
going above 0.99 [1]. On the other hand, almost all of the space for the
triple pulse gives purity at or below the 0.92 limit, with only a small area
in the phase simulation reaching higher than 0.95. However, this shows that a
wide range of purity values can be selected simply by changing the phase and
splitting ratios, with high tolerance. This suggests more control over the
dual pulse, where changing the time delay is required for similar control,
which is more complicated to achieve experimentally and can only be achieved
in limited discrete time steps on a photonic chip.

Figure 5: Experimentally collected JSIs and calculated Purity P, for a single
pulse (a) and triple pulse (b), with $\eta_{1}=\eta_{2}=0.8$ and
$\Delta\tau_{1}=20ps$,$\Delta\tau_{2}=40ps$. c) and d) show simulated JSIs for
the same single and triple pulse configurations modelled using Eqs.3 and 5.

Fig. 5 presents simulated and experimental data collected for a chosen triple
pulse case to show the reduction in purity when a third pulse is added. To
establish a baseline with our experimental setup we first recorded JSI for a
single pulse as shown in Fig. 5a which agrees with the simulated purity (Fig.
5c). As from the simulation, any triple pulse should show a 0.92 or lower
purity when both phases are set to $\pi$ out of phase. Here, $\eta_{1}$ and
$\eta_{2}$ were set to 0.8 to improve the signal-to-noise ratio of the
measured JSI. The result shows an estimated purity of 0.845 which is very
close to the simulated purity of 0.828.

Figure 6: Simulated in-resonator pump fields for single, dual and triple pulse
pump, in a) frequency and b) time domain, with dual pulse configuration of
$\eta=0.55$ and $\Delta\tau=10ps$ and triple pulse configuration of
$\eta_{1}=\eta_{2}=0.8$ and $\Delta\tau_{1}=20ps$,$\Delta\tau_{2}=40ps$.

Fig. 6 depicts the spectral and temporal pump distribution inside the
resonator source ($A_{p}$), simulated to understand the reduction of purity
when more than two pulses are used. This figure shows that the right
configuration of the dual pulse matches best the target spectral shape (Fig.
1) while a triple-pulse adds extra lobes. Adding the second pulse in the dual-
pulse configuration significantly reduces temporal pump energy inside the
resonator as denoted by the zero-crossing point, thus effectively broadening
the pump resonance and improving purity. A third pulse with the same phase as
the second adds significantly more energy to the residual energy from the
second pulse, hence the effective reduction in purity. While this can be
alleviated somewhat by adjusting the phases, the dual pulse still has a
quicker energy reduction. The triple pulse pumps investigated can get close to
the desired target pump, but the right dual pulse can fit better, as shown in
Fig.1, meaning a dual pulse results in higher purity than any of the tested
triple pulses.

##  4 Conclusion

This work investigated the temporal manipulation of a pulsed pump with greater
than two pulses, both through simulation and by using a photonic integrated
circuit experiment, on the spectral purity of a ring resonator photon-pairs
source. Our results indicate that there was no gain in purity for any pump
with more than two pulses for any of the pump profiles investigated. A dual
pulse pump provides the best intrinsic match for a specific target pump shape
which would result in theoretically maximum purity. An advantage found with
the triple pulse is an accurate level of control: with more parameters that
are easily adjustable experimentally, the purity can be selected within a
large range to any desired value. However, here the parameters investigated
were chosen based on results from the existing literature [9, 1]. In future, a
larger parameter space could be more rigorously explored, either corroborating
these results or finding outliers with higher purity with comparatively high
brightness.

### Funding

B.M.B. acknowledges the support of the EPSRC training grant EP/LO15730/1. The
authors acknowledge the support of the EPSRC Quantum Communications Hub
(EP/T001011/1) and Quantum Photonic Integrated Circuits (QuPIC)
(EP/N015126/1). The authors also include the use of a paid MPW service
CORNERSTONE 2 (EP/T019697/1).

### Acknowledgments

L.R would like to thank Hugh Barrett for assisting with the experimental set-
up and preliminary investigations.

### Disclosures

The authors declare no conflicts of interest.

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