11institutetext: Institute for Solar Physics, Dept. of Astronomy, Stockholm
University, Albanova University Center, 10691 Stockholm, Sweden
22institutetext: Instituto de Astrofísica de Canarias, C/Vía Láctea s/n,
E-38205 La Laguna, Tenerife, Spain 33institutetext: Universidad de La Laguna,
Dept. Astrofísica. E-38206 La Laguna, Tenerife, Spain 44institutetext:
National Solar Observatory, 3665 Discovery Drive, Boulder, CO 80303, USA

# Transverse oscillations in 3D along Ca II K bright fibrils in the Solar
chromosphere

Sepideh Kianfar 11 Jorrit Leenaarts 11 Sara Esteban Pozuelo 112233 João M. da
Silva Santos 44 Jaime de la Cruz Rodríguez 11 Sanja Danilovic 11
<EMAIL_ADDRESS>
(Received Month dd, yyyy; accepted Month dd, yyyy)

###### Abstract

Context. Fibrils in the solar chromosphere carry transverse oscillations as
determined from non-spectroscopic imaging data. They are estimated to carry an
energy flux of several kW m-2, which is a significant fraction of the average
chromospheric radiative energy losses.

Aims. We aim to determine oscillation properties of fibrils not only in the
plane-of-the-sky (horizontal) direction, but also along the line-of-sight
(vertical) direction.

Methods. We obtained imaging-spectroscopy data in Fe I 6173 Å, Ca II 8542 Å,
and Ca II K obtained with the Swedish 1-m Solar Telescope. We created a sample
of 120 bright Ca II K fibrils and measured their horizontal motions. Their
vertical motion was determined through non-LTE inversion of the observed
spectra. We determined the periods and velocity amplitudes of the fibril
oscillations, as well as phase differences between vertical and horizontal
oscillations in the fibrils.

Results. The bright Ca II K fibrils carry transverse waves with a mean period
of $2.2\times 10^{2}$ s, and a horizontal velocity amplitude of 2 km s-1,
consistent with earlier results. The mean vertical velocity amplitude is 1 km
s-1. We find that 118 out of the 120 fibrils carry waves in both the vertical
and horizontal directions, and 55 of those have identical periods. For those
55, we find that all phase differences between $0$ and $2\pi$ occur, with a
mild but significant preference for linearly polarized waves (phase difference
of $0$ or $\pi$).

Conclusions. The results are consistent with the scenario where transverse
waves are excited by granular buffeting at the photospheric footpoints of the
fibrils. Estimates of transverse wave flux based on imaging data only are too
low because they ignore the contribution of the vertical velocity.

###### Key Words.:

Sun: atmosphere – Sun: chromosphere – Sun: oscillations – Methods:
observational

††offprints: Sepideh Kianfar

##  1 Introduction

The solar chromosphere is pervaded by waves and oscillations that are excited
by the convection and p-modes in the photosphere (e.g., Jess et al., 2015, and
references therein). These waves are considered a prime candidate to transport
the energy needed to sustain chromospheric radiative losses from the
photosphere into the chromosphere.

In areas where the gas pressure is stronger than the magnetic pressure (plasma
$\beta>1$) acoustic-gravity waves dominate. They are well-understood, both
observationally and theoretically (e.g. Carlsson & Stein, 1997; Wedemeyer et
al., 2004), but the size of their contribution to the required energy input
remains under debate (Carlsson et al., 2019). In areas where the magnetic
pressure is stronger ($\beta<1$), the situation is more complex. In a
homogeneous plasma there are now three waves, the slow and fast magneto-
acoustic wave and the Alfvén wave. The chromosphere is obviously not
homogeneous, and a large literature exists about waves in homogenous magnetic
cylinders (referred to as flux tubes in the literature) with the field aligned
to the axis of the cylinder, taken as an approximation of magnetic field
bundles (e.g. Edwin & Roberts, 1983; Zaqarashvili & Erdélyi, 2009; Goossens et
al., 2009). In this geometry there is a complex set of wave modes, including
torsional waves, oscillations of the tube diameter (sausage modes), and a
swaying of the whole tube (kink modes). The most frequently identified
observationally is the kink mode, as it manifests itself as clearly swaying
structures in imagery, especially of spicules that protrude above the limb of
the Sun (e.g. De Pontieu et al., 2007; Suematsu et al., 2008).

Propagating transverse oscillations of fibrils and mottles, elongated
chromospheric structures that emanate from photospheric magnetic elements,
have been observed using imaging data in chromospheric spectral lines, e.g.,
Pietarila et al. (2011); Kuridze et al. (2012); Morton et al. (2014);
Jafarzadeh et al. (2017a). These oscillations have typically been interpreted
as kink waves. Observed velocity amplitudes are on the order of 5 km s-1,
periods in the range of 16 – 600 s, and phase speeds of 50 – 300 km s-1.
Shetye et al. (2021) studied high frequency waves in spicule-like event using
imaging spectroscopy on H$\alpha$. In addition to plane-of-the-sky (POS)
velocities, they use Doppler shift information to estimate line of sight (LOS)
velocity of the spicules. They find periods in the range of 40 – 200 s, and
velocity amplitudes around 5–20 km s-1.

However, when radial and longitudinal inhomogeneities are introduced in
magnetic concentrations, the separation between the different wave types
becomes more difficult (Srivastava et al., 2021). Radiation-MHD models of the
chromosphere show proxies of different wave types often appear in the same
location and become indistinguishable (Danilovic, 2022).

Leenaarts et al. (2015) investigated the relation between oscillations as seen
in H$\alpha$ imagery and transverse oscillations propagating along magnetic
field lines in a radiation-MHD simulation of network. They warned that
observed fibrils might not always trace out single field lines, so that
interpretation of observations in terms of kink waves along flux tubes must be
done with caution. However, radiative transfer computations by Bjørgen et al.
(2019) based on a radiation-MHD simulation of an active region, indicate that
strong fibrils seen in H$\alpha$, Mg II k, and Ca II K trace field lines to a
much larger degree than in the network simulations.

Several observations of transverse oscillations in off-limb spicules using
both POS swaying and LOS motions through measuring Doppler shifts of line
profiles exist (e.g., Gadzhiev & Nikolskii, 1982; De Pontieu et al., 2012;
Antolin et al., 2018) ,but only one where on-limb fibrils are considered
(Shetye et al., 2021). In the latter study, Doppler shifts in H$\alpha$ are
used to trace the LOS variations in velocity which significantly limits the
accuracy.

In this paper we exploit imaging spectroscopy data obtained at the Swedish 1-m
Solar telescope in multiple spectral lines combined with non-LTE inversion to
trace the LOS components of transverse oscillations. This ensures that the LOS
velocity is better resolved. We focus on oscillations in fibrils that appear
bright in Ca II H&K data (Jafarzadeh et al., 2017a; Kianfar et al., 2020). The
physical properties of these bright fibrils have been studied by Kianfar et
al. (2020). We report here on both POS and LOS velocity and displacement
amplitudes, wave periods, and correlations between the POS and LOS properties.

##  2 Observations

###  2.1 Data preparation

We analyse observations of a fibrillar area centered at
$(X,Y)=[70\aas@@fstack{\prime\prime}4,153\aas@@fstack{\prime\prime}2]$, i.e.
$\mu=0.98$, taken at the Swedish 1-m Solar Telescope (SST; Scharmer et al.,
2003). The observed field of view (FoV) was located at the edge of a plage
region in the AR12716. The observations were acquired on 2018-07-22, from
08:23:58 to 08:53:25 UT using the CRisp Imaging Spectro-Polarimeter (CRISP;
Scharmer et al., 2008) and CHROMospheric Imaging Spectrometer (CHROMIS
Scharmer, 2017) instruments. Seeing conditions were excellent throughout the
whole observing time.

The Fe I 6173 Å line was sampled by CRISP in 13 equidistant wavelength
positions in a range of $\pm 180$ mÅ around the line centre. CRISP also
observed the Ca II 8542 Å line. It was sampled at 21 equidistant wavelength
positions between $\pm 550$ mÅ as well as two extra points at $\pm 880$ Å from
the line core. Both lines were observed with full Stokes polarimetry and with
a total cadence of 21 s.

CHROMIS observed the Ca II K line in 27 equidistant wavelength positions in
the range $\pm 1.5$ Å around the line core. In addition, it observed an
additional wavelength point at 4000 Å (continuum). The CHROMIS observations
have a cadence of 8 s.

The observed data were reconstructed by applying the CRISPRED reduction
pipeline (de la Cruz Rodríguez et al., 2015) to the CRISP data and SSTRED
(Löfdahl et al., 2021) to the CHROMIS observations. The image restoration used
in the reconstruction pipelines is the Multi-Object Multi-Frame Blind
Deconvolution (MOMFBD; van Noort et al., 2005) method. After reconstruction,
the CRISP and CHROMIS images were co-aligned with an accuracy of about 0.1
pixel. The CRISP data were resampled from a pixel size of
$0\aas@@fstack{\prime\prime}059$ to the CHROMIS pixel size of
0$\aas@@fstack{\prime\prime}$038\. It was also resampled in time to the
CHROMIS cadence using nearest-neighbour interpolation. The data were
calibrated to the absolute intensity by comparing the average intensity
profile in a quiet region in the field of view to a solar atlas profile
(Neckel & Labs, 1984).

###  2.2 Overview of the FoV

Figure 1: Overview of the observations taken on 2018-07-22 at 08:31:49 UT. Top
panels show intensity images of the continuum at 4000 Å (a) and the
wavelength-integrated Ca II K line intensity (b). The white dashed box marks
the region that we analyse in detail. The red box marks the RoI analysed in
Fig 3. An zoom in of the area enclosed by the white dashed box is shown in the
bottom panels. c: Fe I 6173 Å wavelength-averaged linear polarisation; d: Fe I
6173 Å wavelength-averaged circular polarisation; e: Ca II 8542 Å line core
intensity; f: Ca II K line core intensity. The red solid and white dashed
lines in panel f show the cuts across the Ca II K bright fibrils that we used
(see Sec. 3.1). An analysis of the cut along the white dashed line is
presented in Figs 7 and 8. The white dot marks the zero point on the $y$-axis
in Fig. 7. An animated version of this figure showing the entire time sequence
is available online.

Figure 1 and the accompanying animation show an overview of the observations.
The upper-left corner of the FoV contains a plage region with a concentration
of unipolar small-scale magnetic features as seen in the linear and circular
polarization maps (Fig. 1c and 1d). The area surrounding the plage region is
covered with a forest of elongated bright fibrils in the chromosphere (Fig.
1b). The Ca II 8542 Å line core image in Fig. 1e shows a fibrillar pattern
similar to the one in Ca II K (cf. Fig. 1f, Kianfar et al., 2020). The right
half of the CHROMIS FoV is quiet and therefore devoid of bright fibrils as
shown in the Ca II K line-core image in Fig. 1a. We used this part of the FoV
for the absolute intensity calibration of our observations (see Sect. 2.1).

##  3 Analysis methods

###  3.1 Measuring POS oscillations

Figure 2: Time evolution of the cut across the bright fibrils shown with white
dashed line Fig. 1-f. Top panel: intensity at the nominal line centre of Ca II
K. Bottom panel: contrast-enhanced space-time image of the intensity shown in
the top panel. The oscillation trajectories of the bright fibrils in the POS
are shown as dashed lines. The smoothed trajectories are plotted with solid
curves. The oscillations properties of the curves marked with numbers are
shown in Fig. 8. The zero point of the cut length ($y$-axis) is marked with a
white dot in Fig. 1-f

.

In order to analyse transverse oscillations of bright fibrils visible in Ca II
K we selected a region in our data that was observed by both CRISP and CHROMIS
and contained fibrils (indicated by the white-dashed box in Fig. 1b). Then we
defined 12 cuts of $3\arcsec-12\arcsec$ length perpendicular to the local
orientation of the fibrils in this region. In space-time plots of the Ca II K
line core intensity along these cuts, the fibrils appear as sinusoidal curves
with a higher brightness than their surroundings (the top panel of Fig. 2
shows the space-time plot along the white dashed cut in Fig. 1f). We used the
CRisp SPectral EXplorer computer program (CRISPEX; Vissers & Rouppe van der
Voort, 2012; Löfdahl et al., 2021) software to define and extract the cuts.

We defined the trajectory of the POS oscillations in the space-time plots
using a combination of manual and automatic approaches: First, we visually
defined a preliminary trail of an oscillation by choosing a sparse set of
$(x,t)$ points along a sinusoidal curve. Second, we interpolated in time
between these points to get a continuous curve. Third, for each time step, we
defined the location of the fibril as the point with the highest intensity
within $\pm 0\aas@@fstack{\prime\prime}15$ around the interpolated trajectory.
Finally, we smoothed the oscillation trajectories by applying a 1D boxcar
average in time with a width of 64 seconds, to bring out the wave patterns and
cancel out the small-scale fluctuations.

Using the above procedure enabled us to create a sample of 120 POS
oscillations of Ca II K bright fibrils. The bottom panel of Fig. 2 shows the
interpolated and smoothed trajectories of POS oscillations in the example cut
marked in Fig. 1f. We calibrated the overall axis of the oscillations by
subtracting a linear trend from the trajectories so that the oscillation axis
becomes parallel to the time axis (Leenaarts et al., 2015). Then we computed
the velocity of the POS oscillations by calculating the time-derivative of the
amplitudes.

###  3.2 Inversion and measurement of LOS oscillations

We used the MPI-parallel STockholm inversion Code (STiC; de la Cruz Rodríguez
et al., 2019) to derive the physical properties of the atmosphere from our
observations. We used the inferred atmospheric velocity to estimate the LOS
velocity in our fibril sample.

To solve the polarised radiative transfer equation, STiC uses the radiative
transfer code RH (Uitenbroek, 2001) with Bezier solvers (de la Cruz Rodríguez
& Piskunov, 2013). STiC uses the equation-of-state that is acquired from the
SME package functions (Piskunov & Valenti, 2017), and includes partial
redistribution (PRD) effects in the radiative transfer using the fast
approximation method introduced in Leenaarts et al. (2012b). STiC fits
multiple spectral lines simultaneously for each individual pixel by assuming
the 1.5D approximation (i.e. plane-parallel atmosphere). We ran STiC on the
observations of the three Fe I 6173 Å, Ca II 8542 Å, and Ca II K lines to
retrieve the physical properties in both photosphere and chromosphere. In our
inversion runs, we assumed the statistical equilibrium and non-LTE condition
for Ca II, including partial redistribution for the Ca II H&K lines. For the
Fe I 6173 Å line we assumed LTE.

We did not aim to determine the magnetic field, and therefore only fitted
Stokes $I$, even though our observations included all four Stokes parameters
for Fe I 6173 Å and Ca II 8542 Å. Because Zeeman broadening affects the width
and shape of the line profiles, in particular for Fe I 6173 Å, we included the
longitudinal magnetic field as a quantity in our atmospheric model with two
nodes, and the transverse field with one node. Otherwise we used the same set-
up and nodes to run the inversions as used in Kianfar et al. (2020), i.e.,
nine nodes in temperature, and four nodes for both $v_{\mathrm{LOS}}$ and
$v_{\mathrm{turb}}$, located non-equidistantly in the optical depth span of
${\log(\tau_{500\,\mathrm{nm}})}=[-7,1]$. The inversion results are presented
in Section 4.1.

##  4 Results

###  4.1 Inversion results

####  4.1.1 Region of interest

Figure 3: Time evolution of the physical properties of the RoI marked with a
red box in Fig. 1. Left panels: intensity variation over time (from top to
bottom) at the 4000 Å continuum, and in the line cores of Ca II 8542 Å and Ca
II K. Middle and right panels: time evolution of the temperature and
$v_{\rm{LOS}}$ at a photospheric depth (${\log(\tau_{500\,\mathrm{nm}})}=0.1$)
and two different chromospheric depths, ${\log(\tau_{500\,\mathrm{nm}})}=-3.9$
and $-4.6$. One specific fibril in the RoI is marked in the 22 minute panel of
the Ca II K intensity. Three cuts across the head, middle and the tail of this
fibril are marked and numbered and their properties as function of time are
displayed in Fig. 4. The red and grey dots indicate where the three cuts cross
this specific fibril at different times of its evolution.

To study the time-evolution of the fibrils, we first ran inversions on the
time-series of a 2D region of interest (RoI) marked by the red box in Fig. 1b.
The results are shown in Fig. 3. The RoI covers an area of
$3\aas@@fstack{\prime\prime}1\times 2\aas@@fstack{\prime\prime}2$ centering on
a long-lived fibril (Fig. 3, Ca II K image at 22 minutes) with a clear
transverse oscillation. The temperature and $v_{\rm{LOS}}$ images in the
photosphere (i.e. ${\log(\tau_{500\,\mathrm{nm}})}=0$) show a convection
pattern consistent with the granulation pattern in the intensity images; there
is higher temperature and upflows in the granules, and lower temperature and
downflows in the intergranular lanes. The temperature in
${\log(\tau_{500\,\mathrm{nm}})}=-3.9$ at $t=10$ min and $t=18$ min almost
follows the fibrillar patterns of the intensity images at Ca II 8542 Å line-
core. The fibrillar patterns in the temperature become more significant at
${\log(\tau_{500\,\mathrm{nm}})}=-4.6$ in the chromosphere where the bright
fibrils appear as temperature enhancements in agreement with the Ca II K
intensity image (Kianfar et al., 2020). The upflow and downflow structures in
the $v_{\rm{LOS}}$ images in the chromosphere do not particularly follow the
fibrillar patterns, though they have the same average orientation as the
fibrils (Kianfar et al., 2020).

We chose three perpendicular cuts (Fig. 3, $t=22$ min) across the central
fibril in the RoI. Figure 4 shows space-time plots of the intensity and
inversion result along these cuts. The optical depth in which the inversion
results are extracted is ${\log(\tau_{500\,\mathrm{nm}})}=-4.3$ since the Ca
II K line is most sensitive to the perturbations in the chromosphere at this
optical depth (Kianfar et al., 2020). The smoothed interpolated trajectories
of the fibrillar POS oscillations in the cross-cuts are defined by the method
explained in Sect. 3.1 (overplotted curves in Fig. 4). They exhibit
oscillations with a period $P$ of about 8 minutes that last for two periods.
The period stays constant along the fibril. We calculated the average phase
speed for the oscillation along this fibril as

$\overline{v}_{\phi}=\frac{\Delta r}{\Delta t_{\phi_{c}}},$ (1)

where $\Delta r$ is the average distance between the cuts and $\Delta
t_{\phi_{c}}$ is the average time difference between the two points with the
same phase in the oscillations. Accordingly the wave along this fibril
propagates with a speed of $\sim$13 km s-1 from cut(1) to cut(3). The
amplitude of the oscillations is highest ($A_{1}\approx
0\aas@@fstack{\prime\prime}5$ ) at the head of the fibril (i.e. close to the
magnetic field concentration, top panels of Fig. 4) and it decreases to
$A_{3}\approx 0\aas@@fstack{\prime\prime}2$ in the tail.

The temperature (middle column of Fig. 4) almost follows the oscillatory
patterns in intensity with enhancements of 40–60 K. There are no obvious wave
patterns in $v_{\rm{LOS}}$.

Figure 4: Physical properties of the cuts perpendicular to the bright Ca II K
fibril marked in Fig. 3. Left column shows the intensity-over-time variations
of the cuts at Ca II K nominal line centre. The oscillation in the POS is
visually tracked and marked by red curves. Middle and right columns
demonstrate the temperature and $v_{\rm{LOS}}$ variations of the cuts as
function of time in the chromosphere
(${\log(\tau_{500\,\mathrm{nm}})}=-4.31$). The POS oscillation is overplotted
on these images as well. Length zero on the $y$-axis is where the cuts are
marked by numbers in Fig. 3.The $|$ symbols, marked on the curves in the left
column, show the times of the fibril’s evolution that are displayed in Fig. 3.

We extracted the temperature and $v_{\rm{LOS}}$ along the POS oscillations in
the cuts of Fig. 4 in the atmosphere between
${\log(\tau_{500\,\mathrm{nm}})}=[-6,-2]$. This is roughly the height range
where Ca II K line responds to perturbations of the physical properties in the
atmosphere (Kianfar et al., 2020). The results are shown in Fig. 5. The
oscillatory pattern in $v_{\rm{LOS}}$ (right panel) starts around
${\log(\tau_{500\,\mathrm{nm}})}\sim-3$ and gets larger at larger heights and
peaks between ${\log(\tau_{500\,\mathrm{nm}})}=[-3,-6]$. The temperature
profiles (left panel) do not show any particular wave patterns.

However, there are peaks forming at $t=11$ and 19 minutes, i.e, where the POS
amplitude has extrema (see Fig. 4). These temperature peaks start appearing
around ${\log(\tau_{500\,\mathrm{nm}})}=-3.5$ and get stronger up to
${\log(\tau_{500\,\mathrm{nm}})}=-5$. Higher up they get damped. This height
range is where bright Ca II K fibrils show temperature enhancements compared
to their surroundings (Kianfar et al., 2020).

While the peaks in the temperature profiles get smaller as we move towards the
tail of the fibril (i.e., from cut 1 to cut 3), the amplitude of
$v_{\rm{LOS}}$ oscillations increases. Figure 6 shows a comparison of the time
evolution of $T$ and $v_{\rm{LOS}}$ at ${\log(\tau_{500\,\mathrm{nm}})}=-4.31$
to $v_{\rm{POS}}$. It shows that both the temperature variation and its
average decrease as we move from the head to the tail of the fibril. The
amplitude of $v_{\rm{POS}}$ behaves the same. The $v_{\rm{LOS}}$ curves show
different behaviour, with the highest amplitude peak at about $t=17$ min in
cut (3), i.e., close to the tail of the fibril. We further discuss the
$v_{\rm{POS}}$ and $v_{\rm{LOS}}$ oscillations of our sample and their wave
properties in Sect. 4.2.

Figure 5: Temperature and $v_{\rm{LOS}}$ variations of the three cuts marked
in Fig. 3 over time. Left and right column show the temperature and
$v_{\rm{LOS}}$ variations along the POS oscillation trajectory (marked in Fig.
4) for a depth range where Ca II K is sensitive to the perturbations in the
atmosphere. Solid curves are the smoothed plots of the actual values that are
shown with dots. Red-colored plots mark the depth where Ca II K has the
highest sensitivity to atmospheric perturbations. These curves are plotted
individually in Fig. 6. Figure 6: Variations of the temperature (left column)
and $v_{\rm{LOS}}$ (right column) of the perpendicular cuts across the fibril
shown in Fig. 3. They are plotted based on their values extracted along the
oscillation trajectory in POS marked in Fig. 4 at the depth
${\log(\tau_{500\,\mathrm{nm}})}=-4.31$ (i.e. where the Ca II K line is most
sensitive to atmospheric perturbations). Oscillation of the $v_{\rm{POS}}$ is
plotted in gray for comparison on the panels in the right column.

####  4.1.2 Cross-cuts

Figure 7: Time evolution of the physical properties of the example cross-cut
shown in Fig. 2. Left column shows the inversion results at lower chromosphere
and the right column represents the results in higher chromosphere. Top panels
show the temperature of the cut in the chromosphere. Bottom panels are space-
time images of line-of-sight velocity. The POS oscillations are overplotted on
the right panels. Zero point of the cut length is marked with a white dot in
Fig. 1.

Figure 7 shows the temperature and $v_{\rm{LOS}}$ of the example cut shown
Fig. 2 at two chromospheric heights. The temperature in the lower chromosphere
($\log\tau_{500}=-3.5$) barely shows fibrillar structures. Higher up, at
$\log\tau_{500}=-4.31$, the temperature image shows a similar pattern to the
intensity space-time image (Fig. 2) with enhanced temperature at the location
of the bright fibrils (Kianfar et al., 2020). The $v_{\rm{LOS}}$ space-time
images do not demonstrate a clear correlation to the fibrillar oscillation
trajectories in the POS and appears similar at both heights, with only a
slight difference in the range of the velocity values.

###  4.2 3D oscillations

Figure 8: Comparison of the velocity oscillations of the examples marked with
numbers in Fig. 2. These examples represents four different groups of
oscillations in our sample of Ca II K fibrillar oscillations. Top plot of each
panel, shows the oscillation of velocity in the plane-of-sky (red curve),
compared to the velocity oscillation along the line-of-sight direction (grey
curve). Middle plots of the panels show the results of the auto-correlation
calculation on both POS and LOS velocity oscillations. Normalised products of
auto-correlations, i.e. normalised cross-correlation ($\hat{C}$ function) of
the POS and LOS velocity oscillations are shown in the bottom plots of the
panels. For the interpretation of the above plots and more details on the
different fibrillar groups see Sect. 4.2.1 and 4.2.2. An animation of this
figure is available online that shows all 118 fibrils in the sample.

From the POS oscillation trajectories (Sect. 3.1) and space-time inversion
results of the cuts across the fibrils (Sect. 4.1.2), we determined the LOS
velocity along the fibrils; we extracted the $v_{\rm{LOS}}$ values along the
smoothed trajectory of the POS oscillations (Sect. 3.1) at
${\log(\tau_{500\,\mathrm{nm}})}=-4.31$, i.e. the height where temperature and
velocity at fibrillar locations are most sensitive to the perturbations in the
chromosphere (Kianfar et al., 2020). We compute the velocity in the POS by
taking the time derivative of the POS displacement. The top panels of Fig. 8
show $v_{\rm{LOS}}$ an $v_{\rm{POS}}$ along the four example fibrils marked
with numbers in Fig. 2.

####  4.2.1 Determination of the oscillation properties

We derived the wave properties in our oscillation sample through a combination
of automatic and manual approaches. First, we computed the autocorrelation of
each $v_{\rm{LOS}}$ and $v_{\rm{POS}}$ curve. Besides the peak at a time lag
$\Delta t=0$ s, a curve with an oscillation shows a secondary peak at a time
lag roughly equal to the dominant period of the oscillation. Because the
velocity curves contain noise, and might contain partial oscillations,
interference patterns or oscillation with changing period, the secondary peak
can have a much lower amplitude. Combined with visual inspection, we selected
118 fibrils that have a clear oscillation in both $v_{\rm{LOS}}$ and
$v_{\rm{POS}}$. The middle panels of Fig. 8 show the autocorrelations for the
four example fibrils marked with numbers in Fig. 2.

We then derived periods for this subsample by measuring the time of each
extremum, and compute ”partial periods” as the time difference between two
consecutive minima or maxima. As the periods are not constant throughout one
oscillation, we assigned an average and a dominant period ($\bar{P}$,
$P_{\mathrm{dom}}$) as the mean and maximum of the partial periods for
$v_{\rm{LOS}}$ and $v_{\rm{POS}}$ in each fibril. We also measured the
amplitude of each extremum, and used that to assign an average amplitude
$\bar{A}$ for $v_{\rm{LOS}}$ and $v_{\rm{POS}}$ in each fibril.

To quantify the correlation between $v_{\rm{LOS}}$ and $v_{\rm{POS}}$, we
calculated the normalised cross-correlation function, $\hat{C}(t)$, for each
fibril. Example cross-correlation curves are shown in the bottom panels of
Fig. 8. We considered fibrils with
$max\bigl{\\{}|\hat{C}(t)|\bigl{\\}}\leavevmode\nobreak\ >0.5$ as potentially
exhibiting correlation motion. Then we visually inspected those fibrils to
confirm that $v_{\rm{LOS}}$ and $v_{\rm{POS}}$ indeed had similar periods and
oscillation patterns. The differences in period between the two velocity
components is smaller than 22%. We found that 55 fibrils out of the 118 showed
correlated oscillations. For this subset we computed the phase difference
between the oscillations as

$\Delta\phi=\pi\frac{\Delta t}{P_{M}},\quad-\pi\leq\Delta\phi\leq\pi,$ (2)

where $P_{M}=(\bar{P}_{\mathrm{LOS}}+\bar{P}_{\mathrm{POS}})/2$. The online
animation accompanying Fig. 8 shows all fibrils in our sample.

####  4.2.2 Statistical analysis

Figure 9: Scatter plots and histograms of the average period $\bar{P}$ and average amplitude $\bar{A}$ of the 118 fibrils in our sample that show POS and LOS oscillations. The panels show scatter plots of the quantities indicated in the axis labels. The histogram distribution for each quantity is shown in the side-plots corresponding to their axis. The Pearson correlation coefficient $r$ is annotated in the corner of the left-hand panels. Table 1: LOS and POS fibril oscillation properties | min | max | mean | std. dev.
---|---|---|---|---
$\overline{P}_{\rm{POS}}$ (s) | 118 | 607 | 218 | 87
$\overline{P}_{\rm{LOS}}$ (s) | 91 | 635 | 195 | 78
$\overline{A}_{\rm{POS}}$ (km s-1) | 0.25 | 7.9 | 1.9 | 1.0
$\overline{A}_{\rm{LOS}}$ (km s-1) | 0.17 | 2.7 | 1.1 | 0.45

Figure 9 shows the distribution of and correlations between the measured
periods and amplitudes. Table 1 shows their minimum maximum, average, and
standard deviation.

The top-left panel of Fig. 9 shows a weak but positive correlation between the
$\overline{P}_{\rm{POS}}$ and $\overline{P}_{\rm{LOS}}$ especially between
periods of 150 to 250 s. In contrast, the amplitudes (lower-left panel) do not
appear to have a correlation between the two POS and LOS directions. In
general the $v_{\rm{LOS}}$ oscillations tend to have smaller amplitudes
compared to $v_{\rm{POS}}$, instead. This is also confirmed by the mean values
in Table. 1. There appears to be no correlation between the amplitude and
period in the POS direction (upper right), and likewise for the LOS direction
(lower right).

Figure 10: Distribution of the phase difference $\Delta\phi$ and
$|\Delta\phi|$ for the 55 fibrils in our sample that show velocity
oscillations in the POS and LOS direction with similar periods.

We show the distribution of the phase difference between the $v_{\rm{POS}}$
and $v_{\rm{LOS}}$ oscillations for the 55 fibrils in our sample that show
correlated velocity oscillations in the POS and LOS directions in Figure 10.
The observed phase difference span the whole range of [$-\pi,\pi$]. This
implies that the total velocity vectors
$\mathbf{v}_{tot}=({v}_{\rm{POS}},v_{\rm{LOS}})$ can range from fully linearly
polarized ($\Delta\phi=0$, $-\pi$, or $\pi$) to fully circularly polarized
($\Delta\phi=\pm\pi/2$).

We also show the distribution of $|\Delta\phi|$ in Fig. 10. Because our sample
size is limited and we are only interested in absolute phase differences, this
lowers the statistical noise in our sample. There is a clear peak in the
distribution for linearly polarized waves, and there is a hint of a secondary
peak for fully circularly polarized waves in the $|\Delta\phi|$ distribution.

In order to visualize the motion of the spicule axes we integrated the LOS
velocity to obtain the LOS displacement as a function of time. Together with
the measured POS amplitude we can then trace the motion of a particular point
along a fibril as a function of time. We display these curves for the four
example fibrils shown in Figs. 2 and 8 in Fig. 11.

Figure 11: Time evolution of the motion of a point along the spicule axis
$\mathbf{A}(t)$ of the four example oscillations shown in Figs. 2 and 8. The
oscillations in POS and LOS are shown in the side planes and the projections
of the total motions are plotted in the plane perpendicular to the POS and
LOS. The small arrow shows the direction of the motion in the bottom plane. An
animated version of this figure is available online.

The phase differences between the velocity oscillations of the examples (1)
and (2) are 0 and $\pi$ as shown in Fig. 8, suggesting a linear polarized
wave. This linear behaviour is seen roughly in the total motion of the example
(2) in Fig. 11 but not so clearly in example (1). Example (3) has a
$\Delta\phi\approx\pi/2$, and the projection of the amplitude vector traces
out an elliptical path. Example (4) has velocities that appeared to have low-
ampitude short-period velocities with different periods in the LOS and POS
directions. However, the projection of the amplitude vector reveals it carried
a long-period linear oscillation as well as the short period oscillations.

Figure 12: Spatial distribution of measured fibril oscillation periods and
LOS-POS phase differences. Each symbol marks the coordinates of the
oscillation point. The $\blacktriangle$ symbols mark the points where an
oscillation is detected in both $v_{\rm{POS}}$ and $v_{\rm{LOS}}$. The size of
the $\blacktriangle$ symbols is proportional to their average period. The
color-coding of yellow, orange and red represents the absolute values of the
phase difference between the $v_{\rm{LOS}}$ and $v_{\rm{POS}}$ oscillations
where their period is correlated. The coordinates of the oscillations that do
not have a clear correlation between their $\overline{P}_{\rm{POS}}$ and
$\overline{P}_{\rm{LOS}}$ are shown with gray triangles. The orientation of
the $\blacktriangle$ symbols show the perpendicular direction to the cuts
(Fig. 1-f) across the bright fibrils. The cross symbols mark the points where
an oscillation is detected in the POS direction but not in the LOS direction.
The white filled contour lines indicate areas in the photosphere with magnetic
concentrations, where the footpoints of the fibrils are located.

Finally, we investigated the spatial distribution of the periods and phase
differences. Our results are shown in Fig. 12. Longer period oscillations
($>200$ s) tend to occur preferentially closer to the fibril footpoints in the
network patch. Further away from the network we detect mainly short-period
oscillations ($<200$ s). The phase differences do not show an obvious trend.
The cases where there are no oscillations in the LOS direction (shown by
$\times$ symbols) are very few and all far from the magnetic concentrations.

##### Summary of results

We summarise our main findings: we selected a number of cross cuts through Ca
II K bright fibrils, and identified a sample of 120 fibrils along these cross-
cuts. We determined the velocity amplitudes and periods of the fibril
oscillations for both the LOS and POS directions. The distribution of periods
in both directions is very similar, with a mean period of around $2.2\times
10^{2}$ s. The mean velocity amplitude in the POS direction is however almost
twice as large (1.9 km s-1) as in the LOS direction (1.1 km s-1).

Because the sample was selected based on POS oscillations, all of them exhibit
a POS oscillation. Almost all of them (118) also show a clear LOS oscillation
pattern. Of those, 55 fibrils showed oscillations with similar periods in both
the LOS and POS directions. For these, we measured the phase difference
between the two oscillation directions. All phase differences occur, but there
is a preference for zero phase difference, and possibly a slight preference
for a 90-degree phase difference.

##  5 Discussion

The periods and POS velocity amplitudes that we measure here fall in the range
of transverse oscillations observed in H$\alpha$ fibrils as reported by Morton
et al. (2013). The oscillations in H$\alpha$ spicule-like events reported by
Shetye et al. (2021) have smaller periods and larger amplitudes.

The mean period that we observe is more than twice as large as the values
reported in Jafarzadeh et al. (2017b) for Ca II H fibrils observed with a 0.11
nm filter. Table 2 of Jafarzadeh et al. (2017b) list observed periods and
amplitudes of transverse oscillations in fibrils, mottles, and spicules
observed in H$\alpha$ and Ca II H in different studies using a variety of
instruments. The observed periods range between 16 s and 600 s, roughly
consistent for all studies. However, the differences in periods and amplitudes
might be caused by differences in observed target, observing cadence,
different methods to define and detect spicules, and different data
processing. A limit of our study is that we smooth oscillations in time with a
64 s boxcar average, with effectively blocks us from detecting periods smaller
than $\sim 120$ s, which might explain the larger value for the mean period
that we found compared to those in Jafarzadeh et al. (2017b) and Shetye et al.
(2021).

The range of velocity amplitudes varies, with observations of fibrils in
H$\alpha$ and Type II spicules having median values of 5 km s-1 to 10 km s-1.
The Ca II K fibrils in this study, as well as those observed by Jafarzadeh et
al. (2017b) have a POS velocity amplitude of roughly 2 km s-1. This difference
could be explained by the differences in formation height together with
decreasing mass density with height: for constant wave flux (and phase speed),
lower density gas implies larger velocity amplitude.

For fibrils that appear both in Ca II K and H$\alpha$, the segment of the
fibril that appears bright in Ca II K is likely located lower in the
atmosphere than the segment visible in H$\alpha$. Kianfar et al. (2020) showed
that Ca II K shows fibrils located around $\log\tau_{500}\approx-4.3$, while
the H$\alpha$ line core forms at larger heights ($\log\tau_{500}\approx-5.7$
in the FALC model atmosphere). Type II spicules can protrude far above the
bulk chromosphere and are expected to form at even lower densities.

The presence of oscillation with comparable periods in the LOS and POS
direction for 55 out of 118 fibrils in our sample strongly indicates that both
oscillation directions are excited by the same process. Theoretical
calculations show that curved flux tubes have nearly the same wave speeds in
the vertical and horizontal directions (van Doorsselaere et al., 2009).

We interpret the presence of fibrils with identical periods and zero phase
difference in our sample as evidence of convective buffeting (Spruit, 1981)
with randomly oriented kicks as a driver because there is no obvious
correlation between the LOS and POS velocity amplitudes (Fig. 9). Simulations
of photospheric magnetoconvection at high resolution (e.g. Rempel, 2014) show
however substantially more complex motions of photospheric magnetic elements,
which could explain the whole range of phase differences that we observe.

We note that we detect oscillations with periods longer than 200 s
preferentially close to the network patch that makes up the footpoint region
of our fibril sample (Fig. 12). Further away we find mainly shorter periods.
This hints at an alternative driving mechanism: buffeting of the fibrils from
below by quiet-Sun 3-min slow-mode oscillations. Idealized simulations by
Rosenthal et al. (2002), as well as more complex simulations by Nutto et al.
(2012) indicate that such waves can convert into the fast mode with a
component of its velocity amplitude transverse to the magnetic field. We might
well detect such oscillations in our fibrils far from the network.

We measure an average velocity amplitude in the POS direction of 1.9 km s-1,
almost twice as large as the 1.1 km s-1 measured in the LOS direction. This
difference could be due to the difference in measurement technique: we
determine POS amplitudes from measuring fibril tracks in images, while we
infer the LOS amplitude through non-LTE inversions. The inversions assume a
smooth velocity profile as a function of height, with only four nodes. In
addition, the inversion includes microturbulence, which is of the order of 2.5
km s-1 in fibrils (Kianfar et al., 2020). There is some degeneracy between the
velocity and the microturbulence, which might lower the inferred velocities,
especially given our relatively coarse velocity resolution.

Similarly, Shetye et al. (2021) find smaller LOS amplitudes than POS
amplitudes in H$\alpha$ spicules. They define the LOS velocity as the first
moment of the line profile, which can yield different values than the actual
LOS gas velocity (see Figs. 5 and 6 in Danilovic et al., 2022). Measurement of
fibril oscillations in H$\alpha$ using the Doppler shift of the line profile
minimum as a LOS velocity measurement (Leenaarts et al., 2012a) could provide
a more reliable measurement of the relative sizes of the LOS and POS velocity
amplitudes.

We find a hint of a preference for phase differences of $\pi/2$, indicative of
fully circularly polarized waves, in the context of flux tubes known as
helical kink waves (Zaqarashvili & Skhirtladze, 2008). This leads to circular
motion of the tube axis in the plane perpendicular to the tube. Such motions
have been detected before in both off-limb spicules (e.g., Gadzhiev &
Nikolskii, 1982) and on-disk spicules (Shetye et al., 2021). Similar motions
have also been detected in the chromosphere above photospheric magnetic
elements using feature tracking (Stangalini et al., 2017).

Finally, we note that estimates of the wave flux carried by transverse waves
in chromospheric fibrils based on POS motions only are an underestimate
because they do not include the significant LOS motions. As we show here,
imaging spectroscopy together with non-LTE inversions is a powerful tool to
study chromospheric oscillations in all three dimensions.

###### Acknowledgements.

SK and JL were supported through the CHROMATIC project (2016.0019) funded by
the Knut and Alice Wallenberg foundation. SEP was supported by the Knut and
Alice Wallenberg Foundation and also acknowledges the funding received from
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program (ERC Advanced Grant agreement No.742265). SD
has received funding from Swedish Research Council (2021-05613) and Swedish
National Space Agency (2021-00116). The Swedish 1-m Solar Telescope is
operated on the island of La Palma by the Institute for Solar Physics of
Stockholm University in the Spanish Observatorio del Roque de los Muchachos of
the Instituto de Astrofísica de Canarias. The Institute for Solar Physics is
supported by a grant for research infrastructures of national importance from
the Swedish Research Council (registration number 2021-00169). This project
has received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation program (SUNMAG, grant
agreement 759548). The NSO is operated by the Association of Universities for
Research in Astronomy, Inc., under cooperative agreement with the National
Science Foundation.

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#  Making Noisy Quantum Channels Transparent

Rajeev Gangwar  Department of Physical Sciences, Indian Institute of Science
Education and Research (IISER), Mohali, Punjab 140306, India  Mohit Lal Bera
ICFO – Institut de Ciències Fotòniques, The Barcelona Institute of Science and
Technology, ES-08860 Castelldefels, Spain  G. P. Teja  Department of Physical
Sciences, Indian Institute of Science Education and Research (IISER), Mohali,
Punjab 140306, India  Sandeep K. Goyal  Department of Physical Sciences,
Indian Institute of Science Education and Research (IISER), Mohali, Punjab
140306, India  Manabendra Nath Bera<EMAIL_ADDRESS>Department of Physical
Sciences, Indian Institute of Science Education and Research (IISER), Mohali,
Punjab 140306, India

###### Abstract

One of the major obstacles faced by quantum-enabled technology is the
environmental noise that causes decoherence in the quantum system, thereby
destroying much of its quantum aspects and introduces errors while the system
undergoes quantum operations and processing. A number of techniques have been
invented to mitigate the environmental effects; however, much of these
techniques are specific to the environment and the quantum tasks at hand,
limiting their applicability. Here we propose a protocol that makes arbitrary
environments effectively noise-free or transparent. The protocol exploits non-
local superposition in evolution as a quantum resource. Consequently, it
enables full protection of quantum information and entanglement from
decoherence, and perfect quantum communications across arbitrary noisy
channels for any finite-dimensional quantum system. We also propose
experimental schemes to implement this protocol on linear optical and atomic
systems.

Introduction – A quantum system cannot be fully isolated from its environment,
and the latter induces decoherence in the system, thereby destroying much of
its quantum aspects, and introduces uncontrolled errors [1, 2, 3, 4]. Taking
over environmental noise is thus one of the major challenges in quantum-
enabled technologies today. Some of the well-known techniques to eliminate
environmental effects include dynamical decoupling [5, 6, 7, 8], and coherent
feedback control [9] to suppress the decoherence, quantum error-correction
[10, 11, 12, 13, 14, 4], and error-mitigating methods [15, 16, 17, 18, 19, 20]
for quantum computation and simulations. In the context of long-distance
quantum communication, protocols based on quantum switches and indefinite
causal-order have been proposed to improve channel capacities [21, 22, 23].
However, much of these techniques have limited applicability and specific to
the nature of environments and the quantum tasks under consideration.

Here we introduce a generic protocol that makes arbitrary noisy environment
noise-free or transparent. The protocol finds many potential applications in
quantum information technology. For example, it enables full protection of
quantum information and entanglement from decoherence and environmental noise,
and perfect quantum communications across arbitrary channels, to mention a
few. We also propose two experimentally realizable schemes to implement the
protocol enabling perfect quantum communication and perfect protection of
quantum information in optical and atomic systems.

Unlike other protocols, we exploit the non-local superposition in the
evolution as a quantum resource to achieve transparency of an arbitrary
channel acting on the system. We term our protocol “quantum channel
transparency” (QCT). For simplicity, we consider qubit channels ($d=2$);
however, the protocol can be extended to an arbitrary $d$-dimensional system.
The action of an arbitrary channel $\Lambda^{S}$ on a state $\rho^{S}$ of a
qubit system $S$ reads

$\displaystyle\Lambda^{S}(\rho^{S})=\sum_{m}F_{m}^{S}\ \rho^{S}\
F_{m}^{S{\dagger}},$ (1)

where $F_{m}^{S}$s are the Kraus operators satisfying
$\sum_{m}F_{m}^{S{\dagger}}F_{m}^{S}=\mathds{1}$. Here, $\mathds{1}$ is the
$2\times 2$ identity matrix. The Kraus operators can be expressed in terms of
superposition in the orthonormal Hilbert-Schmidt bases, as
$F_{m}^{S}=\sum_{i=0,x,y,z}c_{mi}\ \sigma^{S}_{i}$, where
$c_{mi}\in\mathds{C}$, $\sigma_{0}=\mathds{1}$, and
$\sigma^{S}_{x},\sigma^{S}_{y}$, and $\sigma^{S}_{z}$ are the Pauli $x,y$, and
$z$ matrices, respectively. The QCT protocol implements the transformation
$\Lambda^{S}\to\mathds{1}^{S}$ on the level of operation, where
$\mathds{1}^{S}$ represents the identity channel.

Figure 1: Quantum circuit for QCT: The QCT protocol uses two ancilla $A$ and
$B$ in the initial state $\ket{++}^{AB}$. In the pre-processing phase, the
non-local unitary $U_{ABS}$ is applied on the system $S$ and the ancilla $AB$,
which couples the three qubits before they are exposed to the environments.
Due to the environment, the system $S$ undergoes an arbitrary noisy operation
$\Lambda^{S}$, while the ancilla $AB$ may be exposed to a wide class of noise
given by the map $\Phi^{AB}$ in Eq. (5). In the post-processing phase, we undo
the operation $U_{ABS}$, i.e., apply $U_{ABS}^{\dagger}$, followed by the non-
local unitary operation $V_{ABS}$, which neutralizes the effect of
environmental noise on $S$.

QCT protocol – The step-by-step description of the QCT protocol is given
below. The protocol requires additional two qubits $A$ and $B$ as ancillary
systems and global operations between the system $S$ and the ancilla $AB$. The
system is in an arbitrary state $\rho^{S}$. A quantum circuit scheme to
realize the protocol is given in figure 1.

  1. Step 1:

The ancilla $AB$ is initiated in the state $\ket{++}$ where
$\ket{+}^{A/B}=(\ket{0}^{A/B}+\ket{1}^{A/B})/\sqrt{2}$.

  2. Step 2:

The composite $ABS$ is evolved by a global unitary operation $U_{ABS}$, given
by

$\displaystyle U_{ABS}=$ $\displaystyle\ |00\rangle\\!\langle
00|^{AB}\otimes\mathds{1}^{S}+|01\rangle\\!\langle
01|^{AB}\otimes\sigma_{z}^{S}$ $\displaystyle+|10\rangle\\!\langle
10|^{AB}\otimes\sigma_{x}^{S}-i|11\rangle\\!\langle
11|^{AB}\otimes\sigma^{S}_{y}.$ (2)

  3. Step 3:

The system $S$ undergoes an arbitrary (possibly unknown) qubit channel
$\Lambda^{S}$, as a result of its interaction with its environment. Similarly,
the ancilla may be allowed to experience environmental noise resulting in a
specific class of channels $\Phi^{AB}$.

  4. Step 4:

The composite $ABS$ is evolved with the unitary $U_{ABS}^{\dagger}$ followed
by another unitary $V_{ABS}$ given by

$\displaystyle V_{BAS}=$ $\displaystyle\
|++\rangle\\!\langle++|^{AB}\otimes\mathds{1}^{S}+|+-\rangle\\!\langle+-|^{AB}\otimes\sigma_{x}^{S}$
$\displaystyle+|--\rangle\\!\langle--|^{AB}\otimes\sigma_{y}^{S}+|-+\rangle\\!\langle-+|^{AB}\otimes\sigma_{z}^{S}.$
(3)

On the level of channel, the Steps 1-4 implement a transformation that leads
to

$\displaystyle\Phi^{AB}\otimes\Lambda^{S}\rightarrow\Psi^{AB}\otimes\mathds{1}^{S},$
(4)

for an arbitrary noisy channel $\Lambda^{S}$, and
$\Phi^{AB}=\sum_{\mu}E^{AB}_{\mu}\rho_{AB}(E^{AB}_{\mu})^{\dagger}$ with the
Kraus operators

$\displaystyle E^{AB}_{\mu}=q_{\mu 0}\mathds{1}\otimes\mathds{1}+q_{\mu
1}\sigma_{x}\otimes\mathds{1}+q_{\mu 2}\sigma_{y}\otimes\sigma_{x}+q_{\mu
3}\sigma_{z}\otimes\sigma_{x},$ (5)

where $q_{\mu i}\in\mathds{C}$. An example of the operation $\Phi^{AB}$ is the
identity operation.

The unitaries $U_{ABS}$ and $V_{ABS}$ are acausal in nature as they possess
(non-local) superposition in the (local) evolution bases
$\\{\outerproduct{mr}{ns}^{AB}\otimes\sigma_{i}^{S}\\}$ and
$i\in\\{0,x,y,z\\}$, and cannot be expressed as the product of two local
unitaries [24, 25]. This acausal nature of these unitaries results in the
following transformation of the Kraus operators $F^{S}_{m}$ of the system

$\displaystyle
F^{S}_{m}\to\sum_{i}\sigma_{i}F^{S}_{m}\sigma_{i}\propto\mathds{1},$ (6)

which results in the transparency of the channel. The acausal nature of the
$U_{ABS}$ and $V_{ABS}$ is responsible for transferring the environmental
effects from the system to the ancilla. Note that, in place of applying the
unitary $V_{ABS}$ in Step 4, we may also preform a non-unitary operation with
the Kraus operators $\\{|++\rangle\\!\langle++|^{AB}\otimes\mathds{1}^{S},\
|+-\rangle\\!\langle+-|^{AB}\otimes\sigma_{x}^{S},\
|--\rangle\\!\langle--|^{AB}\otimes\sigma_{y}^{S},\
|-+\rangle\\!\langle-+|^{AB}\otimes\sigma_{z}^{S}\\}$ to make the channel
$\Lambda^{S}$ transparent.

QCT protocol for qudit channels ($d\geqslant 3$) – The QCT protocol for a
$d$-dimensional system ($S$) follows similar to the qubit systems, where an
arbitrary qudit channel $\Lambda_{d}^{S}$ is made transparent with the help of
two $d$-dimensional ancillary systems $A$ and $B$. In Step 1, the ancilla
state is prepared in $\ket{\psi_{0}}^{A}\otimes\ket{\psi_{0}}^{B}$ where
$\ket{\psi_{0}}=\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}\ket{k}$. In Step 2, the
tri-partite unitary operation $U^{(d)}_{ABS}=(\mathds{1}_{B}\otimes
C^{AS}_{X})(\mathds{1}_{A}\otimes C^{BS}_{Z})$ is applied. Here the control
operations $C^{AS}_{X}$ and $C^{BS}_{Z}$ are defined as

$\displaystyle C^{BS}_{Z}=\sum_{k=0}^{d-1}|k\rangle\\!\langle k|^{B}\otimes
Z^{k},\ \ C^{AS}_{X}=\sum_{k=0}^{d-1}|k\rangle\\!\langle k|^{A}\otimes X^{k},$
(7)

where $Z=\sum_{k=0}^{d-1}e^{(i2k\pi/d)}|k\rangle\\!\langle k|$ and
$X=\sum_{k=0}^{d-1}\ket{(k+1)\ \mbox{mod}\ d}\bra{k}$. In Step 3, the system
is exposed to an arbitrary environment and undergoes a noisy operation
$\Lambda_{d}^{S}$. The ancilla $AB$ may also undergo noisy operation
$\Phi_{d}^{AB}$ with the corresponding Kraus operators

$\displaystyle E^{AB}_{\mu}=\sum_{i,j}c_{\mu ij}Z^{i}X^{j}\otimes X^{d-i}.$
(8)

In Step 4, the $ABS$ composite is evolved with the unitary
$U^{(d){\dagger}}_{ABS}$ and followed by $V^{(d)}_{ABS}$, where

$\displaystyle
V^{(d)}_{ABS}=\sum_{m,n}|\psi_{m}\rangle\\!\langle\psi_{m}|^{A}\otimes|\psi_{n}\rangle\\!\langle\psi_{n}|^{B}\otimes(Z^{m}X^{n})^{\dagger},$
(9)

with $\ket{\psi_{m}}^{A}=Z^{m}\ket{\psi_{0}}^{A}$ and
$\ket{\psi_{n}}^{B}=Z^{[n(d-1)\mod d]}\ket{\psi_{0}}^{B}$. The Steps 1-4
result in the overall transformation on the level of channel as

$\displaystyle\Phi_{d}^{AB}\otimes\Lambda_{d}^{S}\to\Psi_{d}^{AB}\otimes\mathds{1}^{S},$
(10)

where the local channel on the system $S$ becomes transparent.

For potential applications, the protocol can protect quantum information
against arbitrary noise and enables perfect quantum communication. Not only
that, for a multipartite system, it can protect entanglement while the
subsystems are exposed to local environments. In the following, we discuss
implementation schemes for QCT protocol on optical and atomic systems.

Perfect quantum communication using optical system – How efficiently and
accurately one is able to transmit quantum information through noisy
environments without affecting the message is a crucial question in long-
distance quantum communications. In general, for major non-unitary channels,
the quantum communication capacity remains below the maximum possible value,
where the maximum value represents a perfect communication channel. For
example, for a qubit channel, the maximum value is unity. The proposed QCT
protocol, however, can maximize the channel capacity of an arbitrary noisy
channel by making it transparent.

Figure 2: An all optical setup implementing QCT. Initially, the system qubit
is in an arbitrary state $\ket{\psi}=\alpha\ket{1}+\beta\ket{-1}$, where
$\alpha,\beta\in\mathds{C};\leavevmode\nobreak\ |\alpha|^{2}+|\beta|^{2}=1$.
The initial state of the ancilla will be
$\ket{00}^{AB}\equiv\ket{h,a}\equiv\ket{h}\otimes\ket{a}$. The state
$\ket{++}^{AB}\equiv\frac{1}{2}(\ket{h,a}+\ket{h,b}+\ket{v,a}+\ket{v,b})$ of
ancilla $AB$ is prepared by the action of a HWP in the path $\ket{a}$ rotated
by an angle $\pi/8$ followed by a balanced BS. In order to apply the $U_{ABS}$
operation, we use PSDP in path $\ket{a}$ and two DPs followed by a PSDP in
path $\ket{b}$. In the similar fashion we implement $U^{\dagger}_{ABS}$. We
use HWPs, balanced BSs, and cylindrical lenses ($\pi$-converters) to implement
Hadamard operations in order to convert $U_{ABS}$ to $V_{ABS}$.

An experimentally realizable scheme for perfect quantum communication using
QCT is presented in Figure 2. The scheme uses internal degrees of freedom of
light such as polarization modes, the orbital angular momentum (OAM) [26]
modes and the spatial degrees of freedom. The system qubit ($S$) consists of
the OAM states in the basis $\\{\ket{\pm 1}\\}$, which is also used to carry
quantum information and is prone to arbitrary environmental interactions.
Ancilla qubits $A$ and $B$ are the polarization and spatial modes of light,
respectively. The spaces of $A$ and $B$ are spanned by the basis
$\\{\ket{h},\ket{v}\\}$ and $\\{\ket{a},\ket{b}\\}$, respectively.

The most non-trivial operations in the QCT protocol are the non-local unitary
operations $U_{ABS}$ and $V_{ABS}$. The operation $U_{ABS}$ can decomposed as

$\displaystyle U_{ABS}=\outerproduct{a}{a}^{B}\otimes
C^{AS}_{x}+\outerproduct{b}{b}^{B}\otimes
C^{AS}_{x}(\mathds{1}^{A}\otimes\sigma_{z}^{S}),$ (11)

where $C^{AS}_{x}$ is the C-NOT operation on the $AS$ subsystem, where the
control lies with the ancilla $A$. From this decomposition of $U_{ABS}$, it is
clear that we need to perform C-NOT operation jointly on the polarization and
OAM system, and $\sigma_{z}$ operation of the OAM qubit.

The $\sigma_{z}$ operation on the OAM states $\ket{\pm 1}$ can be performed
using a Dove prism (DP) [27]. The action of a DP rotated about the $z$-axis by
angle $\vartheta$ cause the following transformation on the OAM state
$\ket{\ell}$:

$\displaystyle\ket{\ell}\to\exp(2\text{i}\ell\vartheta)\ket{-\ell}.$ (12)

Therefore, two DPs in series, first one at angle $\vartheta=\pi/4$ and second
one at $\vartheta=0$ will result in the transformation matrix

$\displaystyle i\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$ (13)

The phase $i$ can be compensated using a phase plate. Hence, two colinear DPs
can act as $\sigma_{z}$ operation on the OAM states $\ket{\pm 1}$.

The C-NOT operation between the OAM and the polarization qubits can be
implemented using the polarization selective DP (PSDP) [28], which acts like a
DP for some polarization and an identity channel for the orthogonal
polarization. Therefore, the action of the PSDP is given by:

$\displaystyle\ket{h,\pm 1}$ $\displaystyle\to\ket{h,\pm 1};\ \ket{v,\pm
1}\to\ket{v,\mp 1}.$

This is exactly the C-NOT operation we desire.

Note that $V_{ABS}=(H_{A}\otimes H_{B}\otimes H_{S})U_{ABS}(H_{A}\otimes
H_{B}\otimes H_{S})$, where $H_{\eta}$ represents Hadamard operation on the
system $\eta$. The Hadamard operation on the polarization and the spatial
modes can be implemented using half wave-plates (HWPs) and the beam splitters
(BSs), respectively. On the other hand, the Hadamard on the OAM degree of
freedom requires $\pi$-converter [29, 30].

Therefore, the QCT protocol requires only linear optical elements such as DP,
PSDP, BS, HWP, and $\pi$-converters, see Figure 2. Thereby making the channel
implemented on OAM transparent, the protocol enables perfect quantum
communication using OAM modes.

Protecting quantum information in atomic systems – In quantum information
processing tasks, it is often necessary to protect coherence in the state of a
quantum system for a long enough duration. But, the inevitable interaction
with the environment makes the state to decay. The environmental effects can
be nullified using the QCT protocol introduced above hence protect the state
and the quantum information indefinitely. To demonstrate that, we present an
experimental scheme in Figure 3, where the state of an atom is protected for
an indefinite time.

In this scheme, we consider a three-level atom with a two-fold degenerate
ground state and an excited state trapped inside an optical cavity. The system
qubit $(S)$ consists of the state space spanned by the low-energy states
$\ket{\pm 1}$ of the atom, whereas the ancilla qubits ($A$ and $B$) consist of
two single-photons.

To implement the operation $U_{ABS}$ we first notice that this operator can be
decomposed as a product of two controlled two-qubit operators as follows:

$\displaystyle U_{ABS}=\big{(}\mathds{1}_{B}\otimes
C_{x}^{AS}\big{)}\big{(}\mathds{1}_{A}\otimes C_{z}^{BS}\big{)},$ (14)

where
$C_{x}^{AS}=\outerproduct{0}{0}^{A}\otimes\mathds{1}^{S}+\outerproduct{1}{1}^{A}\otimes\sigma_{x}^{S}$
and
$C_{z}^{BS}=\outerproduct{0}{0}^{B}\otimes\mathds{1}^{S}+\outerproduct{1}{1}^{B}\otimes\sigma_{z}^{S}$
are respectively the control-NOT and control-Phase gates acting on one photon
and the atom.

The control-Phase and the control-NOT operations between a photon and the atom
can be performed using the technique given in Ref. [31, 32, 33, 34]. For this
technique to work, the three-level atom should have the $\Lambda$-transition,
where the transition is allowed only between $\ket{\pm
1}\leftrightarrow\ket{0}$ and forbidden between
$\ket{1}\leftrightarrow\ket{-1}$ levels. Due to the conservation of angular
momentum, the right-circular polarized light interacts with
$\ket{1}\leftrightarrow\ket{0}$ transition and the left-circular with
$\ket{-1}\leftrightarrow\ket{0}$.

Figure 3: Protecting quantum information in an atomic system: The atomic state
space is spanned by the low-energy states $\ket{\pm 1}$ and the excited state
$\ket{0}$. The ancilla consists of two single-photons ($A$ and $B$) in
circularly polarization basis $\\{\ket{R},\ket{L}\\}$. The photons are
initiated in the states $(\ket{R}+\ket{L})/\sqrt{2}$. In order to implement
$U_{ABS}$ operation, we make two photons $A$ and $B$ interact with the atom
$S$ inside the cavity. Let the two photons be $\tau$ time apart where the
photon $B$ comes first. The interaction of the photon $B$ with the atom
results in the $C_{z}^{BS}$ operation. Then applying Hadamard operation on the
atom using STIRAP technique with the help of a classical laser pulses
($\Omega$) followed by the interaction of the photon $A$ yields $C_{x}^{AS}$
operation. Both operations together result in $U_{ABS}$. Afterward, the two
photons can be stored in an optical quantum memory [35, 36, 37] to be used
subsequently to implement $U^{\dagger}_{ABS}$ and $V_{ABS}$ operations.
Circulators [32] are used to direct the photons towards the cavity and optical
quantum memory. The electro-optic-modulator (EOM) [38, 39] is used to perform
Hadamard operation on the photons in order to convert $U_{ABS}$ into
$V_{ABS}$.

For an atom trapped inside a cavity in the strong coupling regime (Purcell
regime), in the steady state limit and at resonance, the relation between the
input $a_{\text{in}}$ and output $a_{\text{out}}$ light modes of the cavity
can be written as

$\displaystyle
a_{\text{out}}=\frac{-\kappa\gamma+4g^{2}}{\kappa\gamma+4g^{2}}a_{\text{in}},$
(15)

where $\kappa$ is the decay rate of the cavity, $g$ characterizes the coupling
strength between the cavity and the atom and $\gamma$ is the atomic decay rate
such that $\kappa\gg g\gg\gamma$. Therefore, if $g^{2}\gg\kappa\gamma$ then
$a_{\text{out}}\sim a_{\text{in}}$ whereas if $g^{2}\ll\kappa\gamma$ then
$a_{\text{out}}\sim-a_{\text{in}}$.

If we choose the right-circular polarization as the normal mode of the optical
cavity, then the transition $\ket{-1}\leftrightarrow\ket{0}$ is always
decoupled, i.e, $g=0$. Therefore, if the atom is prepared in the state
$\ket{-1}$ then it will reflect the photon with a $\pi$-phase, irrespective of
its polarization. However, if the atom is in the state $\ket{1}$ then the
photon will experience a $\sigma_{z}$ operation on its polarization states,
i.e., $\ket{L}\to-\ket{L}$ and $\ket{R}\to\ket{R}$. The general transformation
can be written as

$\ket{L}\otimes\ket{\pm 1}\to-\ket{L}\otimes\ket{\pm
1},\quad\ket{R}\otimes\ket{\pm 1}\to\pm\ket{R}\otimes\ket{\pm 1},$ (16)

which is exactly $-C_{z}$ operation. The $C_{z}$ operation can be converted
into the $C_{x}$ operation by applying Hadamard operation on the target qubit.
It can be done efficiently on the atomic system by the optimized-STIRAP
techniques where the excited state $\ket{0}$ is adiabaitcally eliminated [40,
41]. Hence, we can perform both the control-NOT and Control-Phase operations
on the photon plus atomic system.

Therefore, the unitary $U_{ABS}$ can be implemented by sequentially
interacting the two photons with the atom-cavity system. In the similar
fashion we can implement $V_{ABS}$ operator between the atom and the photons
by introducing $U_{ABS}$ along with the Hadamard operation on the polarization
states of the photons using HWP and the atomic states using STIRAP technique.

Discussions – Unlike the conventional noise-mitigating protocols that are, in
general, specific to the system, and the nature of the environment, the QCT
protocol can be applicable for an arbitrary channel on any system. The methods
to improve coherence time using dynamical-decoupling and feedback control are
particular to the nature of noise. While these protect local coherence to an
extent, they are not suitable for preserving the non-local coherence, such as
the quantum entanglement. In contrast, the QCT can protect both local and
global coherence for an indefinite time, irrespective of the nature of local
noises. There are techniques to protect quantum entanglement by creating it in
a decoherence-free-subspace. However, these are also specific to a small set
of entangled states and, unlike QCT, do not work for arbitrary states in
general. In communications, the quantum switch-based protocols to improve
channel capacity are, yet again, limited to a specific class of environmental
noise. But the QCT is capable of maximizing the capacity for arbitrary noisy
channels.

Another technique to mitigate the environmental effects is the SWAP protocol,
where one can transfer the noise effects from the system to the ancilla by
swapping their states. However, this requires the ancilla to be noise-free at
all times. On the contrary, the QCT protocol works even if the ancilla is
allowed to interact with a large class of noisy environments.

In summary, we have introduced a protocol that makes an arbitrary quantum
channel transparent for an arbitrary dimensional quantum system. Specifically,
we have given the implementation scheme on the optical systems in order to
perform perfect quantum communication using simple linear optical setups. We
have also presented a scheme to protect quantum information in an atomic
system inside a cavity.

The QCT protocol may open new avenues to protect quantum information and
correlations against environmental effects and enable perfect quantum
communications. For example, the protocol can be used to share entanglement
between distant parties; hence may replace quantum repeaters. Combining the
control over information, protection of quantum entanglement shared by distant
parties, and perfect communication makes it possible to build a perfect
quantum network. On the theoretical side, the protocol also identifies the
non-local superposition in evolutions as a quantum resource. Therefore, this
work is expected to initiate further theoretical investigations to
understanding the fundamental nature of these non-local resources and finding
possible applications in the context of quantum information theory.

Acknowledgments – R.G. thanks the Council of Scientific and Industrial
Research (CSIR), Government of India, for financial support through a research
fellowship (Award No. 09/947(0233)/2019-EMR-I). S.K.G. acknowledges the
financial support from Inter-disciplinary Cyber Physical Systems(ICPS) program
of the Department of Science and Technology, India, (Grant
No.:DST/ICPS/QuST/Theme-1/2019/12). M.N.B. gratefully acknowledges financial
supports from SERB-DST (CRG/2019/002199), Government of India. M.L.B.
thankfully acknowledges support from ERC AdG NOQIA, Agencia Estatal de
Investigación (“Severo Ochoa” Center of Excellence CEX2019-000910-S, Plan
National FIDEUA PID2019-106901GB-I00/10.13039/501100011033, FPI), Fundació
Privada Cellex, Fundació Mir-Puig, and from Generalitat de Catalunya (AGAUR
Grant No. 2017 SGR 1341, CERCA program, QuantumCAT_U16-011424, co-funded by
ERDF Operational Program of Catalonia 2014-2020), MINECO-EU QUANTERA MAQS
(funded by State Research Agency (AEI)
PCI2019-111828-2/10.13039/501100011033), EU Horizon 2020 FET-OPEN OPTOLogic
(Grant No 899794), and the National Science Centre, Poland-Symfonia Grant No.
2016/20/W/ST4/00314.

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