Scattering description of Andreev molecules

J.-D. Pillet1,2, V. Benzoni1, J. Griesmar1, J.-L. Smirr1, Ç. Ö. Girit1*

1 $\Phi_{0}$, JEIP, USR 3573 CNRS, Collège de France, PSL Research University,
11, place Marcelin Berthelot, 75231 Paris Cedex 05, France

2 LSI, Ecole Polytechnique, CEA/DRF/IRAMIS, CNRS, Institut Polytechnique de
Paris, F-91128 Palaiseau, France

*<EMAIL_ADDRESS>
August 27, 2024

## Abstract

An Andreev molecule is a system of closely spaced superconducting weak links
accommodating overlapping Andreev Bound States. Recent theoretical proposals
have considered one-dimensional Andreev molecules with a single conduction
channel. Here we apply the scattering formalism and extend the analysis to
multiple conduction channels, a situation encountered in epitaxial
superconductor/semiconductor weak links. We obtain the multi-channel bound
state energy spectrum and quantify the contribution of the microscopic non-
local transport processes leading to the formation of Andreev molecules.

##  1 Introduction

The physical properties of Josephson junctions, both isolated and in
ensembles, are well understood and exploited in various fields such as
magnetometry and metrology [1]. Due to their quantum coherence and potential
for integration in large-scale circuits, Josephson junctions also serve as
superconducting qubits for quantum information and computation [2].

Recently we elucidated an unconventional coupling mechanism, historically
referred to as the “order-parameter” interaction [3] or quartets [4], between
two closely spaced Josephson junctions [5]. For two weak links separated on
the order of the superconducting coherence length $\xi_{0}$, this coupling
arises from the hybridization of quasiparticles to form a molecular, or multi-
weak link, Andreev Bound State (ABS). The results were obtained from an
analysis of the Bogolubiov-de Gennes (BdG) equations describing an
inhomogeneous superconductor in one dimension, with the “Andreev molecule”
comprised of two $\delta$-potential weak links separated by a finite
superconductor of length $l$. Although the BdG approach is sufficient to
develop an intuitive understanding of the phenomenon, it is unwieldy when
applied to complicated structures or to weak links with multiple conduction
channels.

In an isolated Josephson junction with multiple conduction channels, each
channel hosts independent ABS. The total supercurrent is given by the sum of
the contributions from each channel, a function of the overall superconducting
phase difference and the individual channel transmissions [6]. However, when
two multi-channel junctions are placed close to each other, each ABS at the
left junction can potentially couple to every ABS at the right junction to
form complex Andreev molecules. This situation is relevant since many quantum
conductors used in weak links have lateral dimensions comparable to or larger
than the Fermi wavelength and thus host multiple channels. For example, in
epitaxial superconductor/semiconductor nanowires [7] or 2D electron gases [8],
one can readily tune the number of channels with a local gate electrode [9,
10].

Here we apply the scattering matrix formalism to describe Andreev molecules
with multiple channels. First we formulate the problem for the case of two
weak links connected to three superconductors, introducing new terms
accounting for partial Andreev reflection at the finite central
superconductor. We identify the microscopic scattering processes, elastic
cotunneling and crossed Andreev reflection, which give rise to ABS
hybridization. After verifying that the results are consistent with the BdG
treatment of a single-channel molecule, we calculate the energy spectra of a
twenty-channel Andreev molecule. Finally we depict the extended quasiparticle
trajectories arising in an Andreev molecule and plot their probabilities as a
function of the size of the finite superconductor.

Figure 1: Scattering description of multi-channel Andreev molecules. (a) Plane
waves corresponding to electrons (black arrows) and holes (gray arrows)
scatter on superconductors (blue, red and magenta) and weak links. The left
(right) superconductor has phase $\varphi_{L,R}$ and the central
superconductor of length $l$ is grounded with phase zero. The ground
connection allows applying the phase differences $\varphi_{L,R}$ independently
by flowing different currents through each junction. The matrix $S_{S}$
describes Andreev scattering processes on the superconductors while $S_{L,R}$
describes normal scattering at the weak links. Only one channel is sketched.
(b) For $l\lesssim\xi_{0}$, an electron incident on the central superconductor
with energy $E$ less than the superconducting gap $\Delta$ can transmit across
(elastic co-tunneling, EC) with probability amplitude $t_{S}$ or be Andreev
reflected as a hole with amplitude $r_{S}$. (c) In the presence of scattering,
such that the left channel’s transmission probability is $\tau<1$, the
electron may also be backscattered (BS) or undergo crossed Andreev reflection
(CAR), which converts it into an outgoing hole. (d) The probabilities of
Andreev reflection, transmission, and their product, which factors into the
probability for CAR, is plotted as a function of $l/\xi_{0}$ for fixed energy
$E=0.1\Delta$. In a long superconductor, $l\gg\xi_{0}$, only Andreev
reflection occurs whereas for a short one, $l\ll\xi_{0}$, only elastic co-
tunneling occurs. At intermediate values $l\approx\xi_{0}$ and in the presence
of scattering there is a peak in the CAR probability which goes to zero
elsewhere.

##  2 Scattering Formalism

A convenient approach to treat conduction through mesoscopic systems is the
Landauer-Büttiker scattering formalism [11]. Matrices describe the scattering
of propagating electrons or holes on three different types of elements: weak
links, semi-infinite superconductors, and a superconductor of finite length,
Fig. 1(a). In this approach, electrons and holes ($e$ and $h$) are described
by an ensemble of waves propagating to the left or the right ($\leftarrow$ or
$\rightarrow$), which are connected to each other by normal scattering
processes at the left or right ($L$ or $R$) weak link or Andreev processes on
the three superconductors.

These waves can be labeled with two sets of eight coefficients

$\displaystyle\mathcal{A}=$
$\displaystyle\left(a_{Le}^{\rightarrow},a_{Le}^{\leftarrow},a_{Re}^{\rightarrow},a_{Re}^{\leftarrow},a_{Lh}^{\rightarrow},a_{Lh}^{\leftarrow},a_{Rh}^{\rightarrow},a_{Rh}^{\leftarrow}\right)^{T}$
(1) $\displaystyle\mathcal{B}=$
$\displaystyle\left(b_{Le}^{\leftarrow},b_{Le}^{\rightarrow},b_{Re}^{\leftarrow},b_{Re}^{\rightarrow},b_{Lh}^{\leftarrow},b_{Lh}^{\rightarrow},b_{Rh}^{\leftarrow},b_{Rh}^{\rightarrow}\right)^{T}$

where $\mathcal{A}$ describes waves propagating towards weak links with
amplitudes $a$ and $\mathcal{B}$ describes outgoing waves with amplitudes $b$.

The scattering equation for the weak links is given by
$\mathcal{B}=S_{N}\mathcal{A}$ with

$S_{N}=\left(\begin{array}[]{cccc}S_{L}&0&0&0\\\ 0&S_{R}&0&0\\\
0&0&S_{L}^{*}&0\\\ 0&0&0&S_{R}^{*}\end{array}\right).$ (2)

The individual normal scattering matrices at the left and right weak links are
$S_{L,R}$ for electrons and $S_{L,R}^{*}$ for holes. The specific form of
$S_{L,R}$ and $S_{L,R}^{*}$ will depend on the weak links. For example the
scattering matrix corresponding to a Dirac $\delta$-potential as used in the
BdG analysis of the Andreev molecule [5] is given by

$S=\left(\begin{array}[]{cc}\frac{-iu}{1+iu}&\frac{1}{1+iu}\\\
\frac{1}{1+iu}&\frac{-iu}{1+iu}\end{array}\right),$ (3)

where the constant $u$ is related to the strength of the $\delta$-potential,
$U_{0}$, and the Fermi velocity, $v_{F}$, by $u=U_{0}/\hbar v_{F}$. For
simplicity, in the following analysis for a multi-channel weak link we use
random symmetric unitary matrices for $S_{L}$ and $S_{R}$. These matrices can
in principle include additional scattering at the superconductor-weak link
interface. Other classes of scattering matrices corresponding to breaking
time-reversal symmetry or spin-rotation symmetry can be used to model the
effect of a magnetic field or spin-orbit interaction [12, 13, 14]. The
dimensions of $S_{N}$ is $8N\times 8N$ where $N$ is the number of channels.

It remains to determine scattering on the superconductors. In contrast to
scattering at the normal weak links, which need not preserve momentum,
scattering on the superconductors occur through Andreev processes which are
momentum-conserving when the Fermi energy is much larger than the
superconducting gap.

For the semi-infinite superconducting electrodes to the left and right, for
energies smaller than the superconducting gap ($\lvert E\rvert<\Delta$), the
only scattering process possible is Andreev reflection, in which an incident
electron is retroreflected as a hole and an incident hole is retroreflected as
an electron. This Andreev reflection amplitude is
$r_{A}=e^{-i(\alpha\pm\varphi_{L,R})}$, where $\varphi_{L,R}$ is the
superconducting phase of the left (right) superconductor and
$\alpha=\cos^{-1}\epsilon$ with $\epsilon=E/\Delta$[15]. Since the Andreev
reflection probability, $\lvert r_{A}\rvert^{2}$, is unity the semi-infinite
electrodes act as perfect phase-conjugating mirrors for electrons and holes
[16]. The phase shift acquired in reflection is the sum of $\alpha$, which is
energy dependent, and the superconducting phases $\varphi_{L,R}$.

As shown in Fig. 1(b), the situation is different for a superconductor of
finite length, in which an electron or hole can also propagate across and
emerge on the other side without being retroreflected. For example in Fig.
1(b) an electron incident on the central superconductor from the left with
amplitude $b_{Le}^{\rightarrow}$ and momentum $+k_{F}$ may either be
retroreflected as a left propagating hole of amplitude $a_{Lh}^{\leftarrow}$
or transmitted as a right propagating electron of amplitude
$a_{Re}^{\rightarrow}$, both particles having momentum $+k_{F}$.

When there is normal scattering in addition to a finite superconductor, such
as in Fig. 1(c) where the weak link has transmission probability $\tau<1$,
electrons and holes can also be backscattered (BS) and crossed-Andreev
reflected (CAR), which consists of tunneling through the superconductor and
conversion from electron to hole or vice-versa [17]. As depicted the CAR
process for an electron incident from the left corresponds to first an Andreev
reflection and then backscattering of the retroreflected hole, which then
traverses the finite superconductor and exits toward the right. This mechanism
can also be seen as the formation, in the central slab, of a Cooper pair
comprised of electrons from both left and right electrodes. The time-reversed
equivalent is known as Cooper-pair splitting. The CAR process, which does not
conserve momentum, requires backscattering in the normal weak links.

The probability amplitude associated with the process of partial Andreev
reflection, Fig. 1(d), can be found using the continuity of wavefunctions at
each interface. These wavefunctions are built from the electron and hole
eigenstates of an infinite superconductor ($\eta=e$ or $h$),

$\psi_{\eta\pm}^{\varphi}\left(x\right)=\left(u_{\eta}^{\varphi},v_{\eta}^{\varphi}\right)^{T}e^{\pm
ik_{\eta}x},$ (4)

where the coherence factors are given by

$\displaystyle u_{e,h}^{\varphi}$
$\displaystyle=\frac{e^{-i\varphi/2}}{\sqrt{2}}\left(1\pm\sqrt{1-\epsilon^{-2}}\right)^{1/2},$
$\displaystyle v_{e,h}^{\varphi}$
$\displaystyle=\mathrm{sgn}(\epsilon)\frac{e^{i\varphi/2}}{\sqrt{2}}\left(1\mp\sqrt{1-\epsilon^{-2}}\right)^{1/2},$

and $k_{e,h}$ are complex to account for bound states. If the superconducting
gap is much smaller than the Fermi energy $\Delta\ll E_{F}$, they can be
approximated as $k_{e,h}\approx k_{F}\pm i/\xi$ where $k_{F}$ is the Fermi
momentum in the normal state and the coherence length is a function of energy
$\xi^{-1}=\xi_{0}^{-1}\sqrt{1-\epsilon^{2}}\ll k_{F}$. Here $\xi_{0}=\hbar
v_{F}/\Delta$ is the bare superconducting coherence length, $v_{F}$ is the
Fermi velocity and $\epsilon=E/\Delta$ is the normalized energy.

If we focus on the subspace of waves with positive momentum the wavefunction
is given by

$\psi(x)=\begin{cases}b_{Le}^{\rightarrow}e^{ik_{F}\left(x+\frac{l}{2}\right)}(1,0)^{T}+a_{Lh}^{\leftarrow}e^{ik_{F}\left(x+\frac{l}{2}\right)}(0,1)^{T},&x<-l/2\\\
c_{e}^{+}e^{ik_{e}x}(u_{e}^{0},v_{e}^{0})^{T}+c_{h}^{+}e^{ik_{h}x}(u_{h}^{0},v_{h}^{0})^{T},&\lvert
x\rvert\leq l/2\\\
a_{Re}^{\rightarrow}e^{ik_{F}\left(x-\frac{l}{2}\right)}(1,0)^{T}+b_{Rh}^{\leftarrow}e^{ik_{F}\left(x-\frac{l}{2}\right)}(0,1)^{T},&x>l/2\end{cases}$

where the three regions are the finite superconducting slab ($\lvert
x\rvert\leq l/2$) and the normal conductors to the left ($x<-l/2$) and right
($x>l/2$) of the slab. The superconducting phase on the central superconductor
is fixed at zero and serves as the reference for the phase differences
$\varphi_{L,R}$ on the left and right superconductors. Each junction can be
shorted by a superconducting loop which allows tuning $\varphi_{L,R}$
independently with external magnetic fields. In addition this ground
connection allows an additional path for current flow such that the
supercurrents through the two weak links may be different.

In the normal regions ($x<-l/2$ or $x>l/2$) only electron or hole plane waves
are possible, with wavevectors $\pm k_{F}$ and coherence factors either
$(1,0)$ (electrons) or $(0,1)$ (holes). In the superconducting slab the
wavefunctions mix electrons and holes and may have an exponential, energy-
dependent envelope as a result of the complex wavevectors $k_{e,h}$.

Imposing boundary conditions at the slab edges $x=\pm l/2$ to preserve
continuity we have

$\displaystyle\left(\begin{array}[]{c}b_{Le}^{\rightarrow}\\\
a_{Lh}^{\leftarrow}\end{array}\right)$
$\displaystyle=e^{-\frac{ik_{F}l}{2}}\left(\begin{array}[]{cc}u_{e}^{0}e^{l/2\xi}&u_{h}^{0}e^{-l/2\xi}\\\
v_{e}^{0}e^{l/2\xi}&v_{h}^{0}e^{-l/2\xi}\end{array}\right)\left(\begin{array}[]{c}c_{e}^{+}\\\
c_{h}^{+}\end{array}\right),$
$\displaystyle\left(\begin{array}[]{c}a_{Re}^{\rightarrow}\\\
b_{Rh}^{\leftarrow}\end{array}\right)$
$\displaystyle=e^{+\frac{ik_{F}l}{2}}\left(\begin{array}[]{cc}u_{e}^{0}e^{-l/2\xi}&u_{h}^{0}e^{l/2\xi}\\\
v_{e}^{0}e^{-l/2\xi}&v_{h}^{0}e^{l/2\xi}\end{array}\right)\left(\begin{array}[]{c}c_{e}^{+}\\\
c_{h}^{+}\end{array}\right).$

By eliminating the coefficients $c_{e,h}^{+}$ we can relate incoming and
outgoing waves with a scattering matrix,

$\left(\begin{array}[]{c}a_{Lh}^{\leftarrow}\\\
a_{Re}^{\rightarrow}\end{array}\right)=\left(\begin{array}[]{cc}r_{S}&t_{S}^{-}\\\
t_{S}^{+}&r_{S}\end{array}\right)\left(\begin{array}[]{c}b_{Le}^{\rightarrow}\\\
b_{Rh}^{\leftarrow}\end{array}\right),$

where we define the Andreev transmission amplitude,

$t_{S}=\frac{e^{-l/\xi}\left(1-e^{-2i\alpha}\right)}{1-e^{-2l/\xi}e^{-2i\alpha}},$
(5)

with $t_{S}^{\pm}=t_{S}e^{\pm ik_{F}l}$, and the partial Andreev reflection
amplitude,

$r_{S}=\frac{e^{-i\alpha}\left(1-e^{-2l/\xi}\right)}{1-e^{-2l/\xi}e^{-2i\alpha}}.$
(6)

For the negative momentum wavefunction the substitution $k_{F}\rightarrow-
k_{F}$ yields the same scattering matrix with $t_{S}^{+}$ and $t_{S}^{-}$
swapped.

These amplitude satisfy $\lvert r_{S}\rvert^{2}+\lvert
t_{S}^{\pm}\rvert^{2}=1$ as expected from quasiparticle conservation. In a
realistic system with a three-dimensional central superconductor, the
wavefunctions $\psi$ (Eq. 4) will be spherical, the longitudinal part of the
wavevector can take any value between 0 and $k_{F}$, and the geometric factors
$e^{-l/\xi}$ describing the envelope of the probability amplitudes
$t_{S},r_{S}$ (Eqs. 5 and 6) will be different. In general the envelope will
decay faster and acquire additional dependence on the Fermi wavelength or the
mean free path [18, 19, 20]. This reduction can be understood from the
increase in scattering angle as the number of dimensions is increased.

The following analysis is limited to the one-dimensional case. For convenience
and visibility we set $k_{F}l$ to constant values in the scattering
coefficients while maintaining $k_{F}l\gg 1$. In principle each channel may
have a different phase factor resulting from interference but such offsets are
already included via the random unitary scattering matrices $S_{L,R}$ and do
not change the results qualitatively. In addition we have assumed that the
energy gap of the superconducting slab is the same as that of the
superconducting electrodes, effectively ignoring any inverse proximity effect
which is reasonable given that we consider typical semiconducting weak links.

In Fig. 1(d) we plot the Andreev reflection probability $\lvert
r_{S}\rvert^{2}$ and transmission probability $\lvert t_{S}\rvert^{2}$ for
fixed energy $\epsilon=0.1$ as a function of $l/\xi_{0}$. The likelihood of
elastic co-tunneling (EC), Fig. 1(b), in the absence of scattering at the weak
links ($\tau=1$) is quantified by $\lvert t_{S}\rvert^{2}$. As the
superconductor thickness goes to zero, $l/\xi_{0}\rightarrow 0$, Andreev
reflections are suppressed and all quasiparticles tunnel across,
$t_{S}\rightarrow 1$. Andreev processes are equally probable when
$l/\xi_{0}\approx 1$. As we extend the length of the central superconductor,
$l/\xi_{0}\rightarrow\infty$, one recovers the Andreev reflection amplitude of
a semi-infinite superconductor, $r_{S}\rightarrow r_{A}=e^{-i\alpha}$, and
transmission is quashed, $t_{S}\rightarrow 0$. The Andreev phase-conjugating
mirror is only perfect if it is much thicker than $\xi_{0}$, the
characteristic length scale for Andreev reflection.

Scattering at the weak links will also reduce elastic co-tunneling. If the
single-channel transmissions of the weak links are $\tau_{L,R}$, the first
order EC probability will be reduced to $\tau_{L}\tau_{R}|t_{S}|^{2}$. For
$\tau<1$, there will be higher order processes involving multiple reflections
at the barriers which will also transmit a particle across the superconductor.

Also plotted in Fig. 1(d) is the probability $\lvert r_{S}t_{S}\rvert^{2}$,
which is the Andreev scattering contribution to the first-order CAR process
depicted in Fig. 1(c). If the left interface has transmission probability
$\tau<1$, this CAR process requires normal barrier transmission ($\tau$), an
Andreev reflection ($\lvert r_{S}\rvert^{2}$), a normal reflection ($1-\tau$),
and an Andreev transmission ($\lvert t_{S}\rvert^{2}$). The Andreev
contribution, $\lvert t_{S}r_{S}\rvert^{2}$, is maximal at 0.25 for a
separation $l/\xi_{0}$ such that $\lvert t_{S}\rvert=\lvert r_{S}\rvert=0.5$
and the maximum of the normal part, $\tau(1-\tau)$, is also 0.25 for
$\tau=0.5$. Therefore the maximum likelihood of the first-order CAR process is
6.25%, with higher order processes contributing little as they scale as
$\tau^{n}(1-\tau)^{n}$. Ignoring higher order processes the likelihood of EC
in the presence of scattering at the left weak link, $\tau\lvert
t_{S}\rvert^{2}$, is approximately four times that of CAR for $\tau=0.5$ and
at a comparable separation $l/\xi_{0}\lesssim 1$ such that $|t_{S}|^{2}\approx
0.5$. The optimal separation $l/\xi_{0}$ to maximize CAR and EC depends on the
energy $\epsilon$ but the relative likelihood for CAR over EC remains
$(1-\tau)/2$. In a symmetric situation where both weak links have transmission
$\tau$, the first-order expressions above are reduced by a factor $\tau$.

In a similar fashion to the derivation of $S_{N}$, we use these results for
scattering from the three superconductors to define a matrix $S_{S}$ which
relates waves incident on the slab ($\mathcal{B}$) to the outgoing waves,
$\mathcal{A}=S_{S}\mathcal{B}$,

$S_{S}=\left(\begin{array}[]{cc}S_{ee}&S_{eh}e^{-i\Phi}\\\
S_{eh}e^{i\Phi}&S_{hh}\end{array}\right)\otimes\mathbb{I}_{N},$

with blocks $S_{eh}$ on the anti-diagonal for Andreev reflections,

$S_{eh}=\left(\begin{array}[]{cccc}r_{A}&0&0&0\\\ 0&r_{S}&0&0\\\
0&0&r_{S}&0\\\ 0&0&0&r_{A}\end{array}\right),$

and blocks $S_{ee}$ and $S_{hh}$ on the diagonal for tunneling through the
central superconducting slab,

$S_{ee}=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&0&t_{S}^{+}&0\\\
0&t_{S}^{+}&0&0\\\ 0&0&0&0\end{array}\right).$

$S_{hh}$ is obtained from $S_{ee}$ with the transformation
$t_{S}^{+}\rightarrow t_{S}^{-}$. The superconducting phases are contained in
the diagonal matrix
$\Phi=\mathrm{diag}\left(\varphi_{L},0,0,\varphi_{R}\right)$ and
$\mathbb{I}_{N}$ is the $N\times N$ identity matrix. The total size of
$S_{S}$, like $S_{N}$, is $8N\times 8N$, accounting for $N$ conduction
channels.

We combine the scattering equation for weak links,
$\mathcal{B}=S_{N}\mathcal{A}$, and for superconductors,
$\mathcal{A}=S_{S}\mathcal{B}$, in order to obtain the master equation,

$\mathcal{B}=S_{N}S_{S}\mathcal{B}.$ (7)

The scattering product $S_{N}S_{S}$ depends on energy $\epsilon$, the
scattering properties of the weak links ($S_{L,R}$), and the superconducting
phases $\varphi_{L,R}$. Eq. 7 is a unity eigenvalue problem in which solutions
of the characteristic equation,

$\det\left(\mathbb{I}_{8N}-S_{N}S_{S}\right)=0,$ (8)

gives the energy spectrum $\epsilon$, the scattering amplitudes $a$ and $b$,
and the corresponding wavefunctions of the Andreev molecule [21].

To verify correctness we numerically solved Eq. 8 for the spectra in the case
of a single channel Andreev molecule with symmetric $\delta$-function
barriers, i.e. $S_{L}$ and $S_{R}$ given by Eq. 3, and compared for agreement
with the Bogolubiov-de-Gennes solution for the same parameters [5].

##  3 Energy Spectra

In Fig. 2 we show the evolution in the energy spectra of a multi-channel
Andreev molecule as the size of the central superconductor is reduced. Spectra
are obtained by numerically solving the characteristic equation Eq. 8 for
fixed 20-channel random scattering matrices $S_{L,R}$ and fixed phase
$\varphi_{R}=3\pi/5$. Each channel of each weak link will have an effective
transmission $\tau$ which can be extracted from the scattering matrices
$S_{L,R}$. The spectra are plotted as a function of the left phase
$\varphi_{L}$ for four values of the separation $l/\xi_{0}$. Each conduction
channel of each junction hosts one pair of ABS and as a consequence there are
$4N=80$ lines, some of which are close to the gap edge and difficult to
distinguish.

For large separation, $l/\xi_{0}\gg 1$, there is no coupling between the two
weak links, and the spectral lines follow the standard ABS energy dispersion,

$E_{Ln,Rn}^{\pm}=\pm\Delta\sqrt{1-\tau_{Ln,Rn}\sin^{2}\left(\varphi_{L,R}/2\right)},$

where $\tau_{Ln,Rn}$ corresponds to the transmission of the $n$-th channel in
the left or right weak link. Since the right phase is fixed,
$\varphi_{R}=3\pi/5$, ABS corresponding to the right weak link (red) do not
disperse with $\varphi_{L}$, whereas those of the left junction (blue) dip
towards zero as $\varphi_{L}$ approaches $\pi$. There is no hybridization
between ABS at the right and left junctions and the spectral lines cross
without forming gaps.

As the junctions are brought closer, for $l/\xi_{0}=1,0.5,0.1$, multiple
avoided crossings materialize, signaling the formation of Andreev molecules.
Similarly to the one-channel case [5], the amplitude of the avoided crossings
increases as the separation is reduced and some discrete states are gradually
pushed out into the continuum.

At separation $l=0.1\xi_{0}$, where the Andreev molecule fuses into a single
weak link, only approximately half of the ABS remain in the gap and the states
have shifted in phase to the right by $\varphi_{R}=3\pi/5$.

Overall the spectra of Fig. 2 for the multi-channel case show qualitatively
the same behavior as for the Andreev molecule in the single channel case [5].
The most obvious global sign of hybridization remains the breaking of symmetry
about the phase $\varphi_{L}=\pi$. Since there are often phase offsets in
experiments it is difficult to verify that $\varphi_{L}=\pi$. One could
instead check for symmetry about the more easily identifiable point,
$\varphi_{L}=\varphi_{L}^{0}$, where the ABS are closest to zero in energy at
a fixed phase $\varphi_{R}$. The multi-channel spectra indicate that the
symmetry point $\varphi_{L}^{0}$ shifts from $\pi$ to $\pi+\varphi_{R}$ as the
separation $l/\xi_{0}$ goes from infinity to zero and that symmetry is broken
for $l\lesssim\xi_{0}$. Even though the spectra will become more dense as the
number of channels is increased, this symmetry breaking will be relevant
experimentally as long as $l\lesssim\xi_{0}$.

Figure 2: Energy spectra of a multi-channel Andreev molecules as a function of
separation $l/\xi_{0}$. Scattering for twenty-channel left and right weak
links is described by randomly generated symmetric unitary matrices $S_{L}$
and $S_{R}$ which are the same for each value of $l/\xi_{0}$. The
superconducting phase on the right weak link is fixed at $\varphi_{R}=3\pi/5$
and the left phase, $\varphi_{L}$, is varied. For $l\gg\xi_{0}$, the red lines
in the spectrum corresponds to Andreev Bound States (ABS) localized at the
right weak link and independent of $\varphi_{L}$, whereas the blue lines
correspond to ABS localized on the left weak link. The spectral lines are
distinct because the effective transmission of each channel, determined by
$S_{L,R}$, is random. As the separation $l/\xi_{0}$ is reduced, the red and
blue lines, now purple, form avoided crossings indicating the hybridization of
Andreev states and the formation of an Andreev molecule. For small separation
$l\ll\xi_{0}$ the spectrum transforms into that of a single twenty-channel
weak link shifted by $\varphi_{R}$. Note that $S_{L}\neq S_{R}$ and for
convenience the momentum is chosen such that $k_{F}l=0\pmod{2\pi}$, with
$k_{F}l\gg 1$.

##  4 Molecular Bound States

An eigenvector $\mathcal{B}_{0}$ which solves Eq. 7 corresponds to a closed
trajectory or bound state of the Andreev molecule, formed due to interference
between propagating and counterpropagating waves. There are three different
types of closed cycles, or orbits, with two non-trivial ones which can be
built from the EC and CAR processes shown in Fig. 1.

The trivial cycle is a conventional Andreev bound state at one of the weak
links and is represented in Fig. 3(a) where the central superconductor is
large, $l\gg\xi_{0}$. The closed orbit consists of two Andreev reflections at
the right weak link, with the left moving hole of amplitude
$b_{Rh}^{\leftarrow}$ being completely transformed into a right moving
electron of amplitude $b_{Re}^{\rightarrow}$ at the central superconductor
(purple). Since the Andreev transmission probability $t_{S}$ vanishes for
large $l/\xi_{0}$, Fig. 1(d), the incident hole cannot be transmitted through
the central superconductor. Likewise at the infinite left (blue) and right
(red) superconductors, only Andreev reflection is possible. A conventional ABS
does not connect particles on all three superconductors and therefore the
supercurrent associated with it only flows between two superconductors.

With a shorter central superconductor, Fig. 3(b), one has the first non-
trivial or “molecular” Andreev bound state: the loop passing through all three
superconductors. This orbit consists of two simultaneous EC processes, one
shown in Fig. 1(b), and the other its particle-conjugate dual in which a hole
propagates from right to left. Such a double elastic cotunneling (dEC) process
transports two electrons from the left to right superconductor. Since the
phases are fixed and all voltages are zero, this charge transfer corresponds
to a unidirectional supercurrent flowing across the device. dEC-type bound
states are probable when the normal scattering matrices have high channel
transmissions and the phases $\varphi_{L,R}$ have opposing signs and values
which result in an energy degeneracy in the limit
$l/\xi_{0}\rightarrow\infty$. In the case of a symmetric single-channel
Andreev molecule [5], dEC is maximal when the phases satisfy
$\varphi_{L}=-\varphi_{R}$.

Fig. 3(c) shows the dEC bound state probability as a function of $l/\xi_{0}$
determined by numerically solving the eigenvalue problem, Eq. 7, for the
lowest positive energy state of a symmetric, single-channel Andreev molecule
of transmission $\tau\approx 0.94$. In red we plot the probabilities $\lvert
b_{Re}^{\rightarrow}\rvert^{2}$ and $\lvert b_{Rh}^{\leftarrow}\rvert^{2}$
corresponding to the orbit shown in Fig. 3(a) or the right part of Fig. 3(b).
In blue we plot $\lvert b_{Le}^{\rightarrow}\rvert^{2}$ and $\lvert
b_{Lh}^{\leftarrow}\rvert^{2}$ which corresponds to the complementary orbit
passing through the left weak link in Fig. 3(b). The eigenvectors are
normalized so that the probabilities sum to 1 and the amplitudes $a$ are
related to the $b$’s by the scattering matrix $S_{N}$. To maximize dEC, the
phases are fixed at $\varphi_{R}=0.5\pi$ and $\varphi_{L}=-0.48\pi$. The
slight detuning of $\varphi_{L}$ from $-0.5\pi$ allows being sufficiently far
from degeneracy such that there is no mixing between left and right
eigenstates at $l/\xi_{0}=10$. In principle at exact degeneracy and
arbitrarily large $l/\xi_{0}$ a viable eigenstate can consist of equal weights
at the left and right weak links.

At large separation, $l/\xi_{0}\approx 10$, both probabilities at the right
weak link (red) are approximately $0.5$ whereas those at the red weak link
(blue) are almost zero, indicating that the eigenstate is a conventional ABS
as in Fig. 3(a).

As the separation is reduced, the weights at the left weak link (blue) start
to increase and those at the left weak link (red) decrease, indicating the
formation of a dEC state. The position of the step will depend on the detuning
of $\varphi_{L}$ from $-\varphi_{R}$. Near $l/\xi_{0}\approx 1$, the orbit is
approximately equally distributed between the left and right weak links. The
decomposition of dEC into two simultaneous EC processes leads to the
qualitatively similar form of the probabilities in blue with the EC
probability $\lvert t_{S}\rvert^{2}$ of Fig. 1(d).

Figure 3: Bound states of an Andreev molecule. (a) At large separation
$l\gg\xi_{0}$ the only eigenstate is a conventional Andreev Bound State (ABS),
shown here localized at the right weak link by Andreev reflections at the
central (purple) and right (red) superconductor. (b) At small separation
$l\lesssim\xi_{0}$ and for superconducting phases
$\varphi_{L}\approx-\varphi_{R}$ there is an additional trajectory, double
Elastic Co-tunneling (dEC), which extends across all three superconductors.
(c) The likelihood of dEC (blue lines) and ABS (red lines) trajectories are
plotted as a function of separation $l/\xi_{0}$ for $\varphi_{R}=0.5\pi$,
$\varphi_{L}=-0.48\pi$, $\tau\approx 0.94$ and $k_{F}l\gg
1,k_{F}l=0\pmod{2\pi}$. The dEC probability increases as the separation is
reduced. (d) A second “molecular” trajectory extending across all
superconductors is possible at small separation $l\lesssim\xi_{0}$ but for
superconducting phases $\varphi_{L}\approx\varphi_{R}$. This is called double
Crossed Andreev Reflection (dCAR) and differs from dEC by additional Andreev
reflections in the central superconductor. (e) The likelihood of dCAR and ABS
trajectories are plotted as a function of $l/\xi_{0}$. Parameters are the same
except for $\varphi_{L}=0.52\pi$ and $k_{F}l=\pi/2\pmod{2\pi}$. The dCAR
probability vanishes for large and small separation and is maximal at
$l\approx\xi_{0}$. Results are obtained by numerically solving the eigenvalue
equation, Eq. 7.

For even smaller separation both the red and blue probabilities decrease and
are compensated by an increase in the amplitudes $\lvert
b_{Le,Re}^{\leftarrow}\rvert^{2}$ and $\lvert
b_{Lh,Rh}^{\rightarrow}\rvert^{2}$ (not shown) of the counter-propagating
orbit given by reversing the directions of the arrows in Fig. 3(b). The
relative weight of these two trajectories will be determined by the value of
the phase difference $\varphi_{R}$. This can be understood by considering the
complementary time-reversed ABS trajectory to the one shown in Fig. 3(a). When
the phase $\varphi_{R}$ is zero or $\pi$, such that the supercurrent is zero,
these two trajectories have equal weights and compensate each other. At
extrema of the supercurrent one trajectory will dominate. This is why with our
choice of $\varphi_{R}=\pi/2$ the red probabilities in Fig. 3(c) approach 0.5
for large $l/\xi_{0}$, near a supercurrent maximum for the right weak link.
The situation is similar for a dEC orbit and when the separation approaches
zero, the total phase drop across the device is
$\varphi_{R}-\varphi_{L}\approx\pi$, so the dEC supercurrent vanishes and both
trajectories coexist. This is why all probabilities approach $1/8$ near
$l/\xi_{0}=0$ in Fig. 3(c), resulting in approximately equal clockwise and
counter-clockwise orbits. The additional splitting of the blue lines results
from normal scattering and is absent when $\tau=1$.

The second molecular bound state, dCAR, is shown in Fig. 3(d), and with
respect to the dEC orbit involves two additional quasiparticle conversions in
the central superconductor and a reversal of current direction at the left
weak link. During the conversion an incident electron of energy $E$ is
reflected as a hole of energy $-E$ which results in the crossing of
trajectories at the central superconductor and the twist relative to the dEC
diagram Fig. 3(b). dCAR describes supercurrent flowing from the central
superconductor to the outer ones and cannot occur for a floating central
island, or without a connection to ground.

The dCAR probability is plotted in Fig. 3(d) for the same $\varphi_{R}=\pi/2$
but with $\varphi_{L}=0.52\pi\approx\varphi_{R}$ in order to maximize the
effect while maintaining a detuning to avoid a trivial degeneracy. Note that
although the probabilities in red are identical to those for dEC, Fig. 3(c),
the probabilities in blue are $\lvert b_{Le}^{\leftarrow}\rvert^{2}$ and
$\lvert b_{Lh}^{\rightarrow}\rvert^{2}$ to take into account the reversal of
the trajectory on the left weak link. There is a non-physical numerical
instability at exactly $l/\xi_{0}=0$ so the x-axis extends from
$l/\xi_{0}=0.05$ to $10$. As expected at large separation $l/\xi_{0}=10$ the
eigenstate is an ABS localized at the right weak link.

As the separation is reduced the probability shifts to the left weak link,
much as with dEC. The increase in probability at the left weak link (blue
lines) occurs at smaller $l/\xi_{0}$ than for dEC, most likely a result of the
high value of transmission which leads to weak dCAR hybridization. After
reaching a maximum at $l/\xi_{0}\approx 1$ the blue lines take a sharp
downturn and approach zero as the separation is further reduced. The
probability for dCAR follows the Andreev reflection probability which vanishes
as $l/\xi_{0}\rightarrow 0$. As with dEC the probabilities describing
propagation through the right weak link, including the time-reversed ones not
shown, approach approximately the same value as $l/\xi_{0}\rightarrow 0$.
However since the probability of all trajectories at the left weak link must
vanish, the red lines approach a value of $1/4$ instead of $1/8$ as with dEC.
The additional splitting of the probabilities for $l/\xi_{0}\lesssim 1$ is
also due to imperfect transmission. Unsurprisingly, the overall shape of the
dCAR probabilities (blue lines) are similar to that of the CAR probability
plotted in Fig. 1(d).

In the general multi-channel, non-symmetric case and as a function of the
separation the eigenstates will be mixtures of conventional ABS and molecular
ABS. The phase configuration necessary for molecular orbits will coincide with
the position of level crossings in the large separation ABS energy spectrum
such as in Fig. 2 for $l=10\xi_{0}$.

##  5 Conclusion

Andreev molecules, or in general, arbitrary mesoscopic systems with
superconducting regions of size comparable to the coherence length can be
effectively modeled with the scattering approach incorporating the partial
Andreev reflection and transmission coefficients ($r_{S},t_{S}$). We validated
this formalism by checking for agreement with the Bogolubiov-de-Gennes results
for a single-channel Andreev molecule [5]. We then calculated the energy
spectrum of a multi-channel Andreev molecule, modeling the experimentally
relevant system of an epitaxial superconductor/semiconductor nanowire with
nanoscale weak links. The calculations show that Andreev Bound State
hybridization is robust and leads to observable consequences even in multi-
channel mesoscopic systems. In addition we have shown how to interpret the
formation of Andreev molecules in terms of the microscopic non-local
scattering processes of double elastic co-tunneling and double crossed Andreev
reflection. We quantified the probability for these processes and determined
the conditions to maximize them.

Although the formalism presented here has the advantage of simplicity, it has
several limitations. Our one-dimensional treatment ignores the lateral
extension of the central superconductor which, as mentioned above, results in
a larger overlap between ABS than expected in three dimensions. A smaller
overlap will lead to a reduction in the size of the avoided crossings in Fig.
2 as well as reducing the probabilities for dEC or dCAR states in Fig. 3.
However an analysis for a 3D finite superconductor has shown that the energy
gaps due to hybridization will remain measurably large, if not a significant
fraction of $\Delta$ [20]. We have also confined our treatment to short weak
links in which there is no additional accumulated phase. A sophisticated
treatment incorporating the quality of the nanowire-superconductor contact as
well as the lead resistance has attacked some of these shortcomings and
elucidated in detail the impact of the central lead on ABS hybridization [22].

The scattering formalism can easily be extended to more complicated structures
and take into account additional mechanisms such as spin-orbit interactions or
a magnetic field, relevant for Majorana bound states. It would also be
possible to model superconducting weak links with multi-junction nanowires
[23], where a quasiparticle incident on a short superconductor could be
Andreev transmitted in multiple directions. Yet another topology is Andreev
polymers, systems with chains or networks of short superconducting segments
connected by weak links, which would allow ABS hybridization across several
sites.

## Acknowledgements

We acknowledge insightful discussions with Yuli Nazarov and support from
Jeunes Equipes de l’Institut de Physique du Collège de France.

### Funding information

This research was supported by IDEX grant ANR-10-IDEX-0001-02 PSL and a Paris
“Programme Emergence(s)” Grant. This project has received funding from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement 636744).

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# Observing the launch of an Eddington wind in the luminous Seyfert galaxy
PG1211+143

Ken Pounds and Sergei Nayakshin
Department of Physics and Astronomy, University of Leicester, Leicester, LE1
7RH, UK

(Accepted ; Submitted )

###### Abstract

The luminous narrow line Seyfert galaxy PG1211+143 was the first non-BAL AGN
to reveal a powerful ionized wind, based on early observations with ESA’s XMM-
Newton X-ray Observatory. Subsequent observations, mainly with XMM-Newton and
the Japanese Suzaku Observatory, found such winds to be a common feature of
luminous AGN. Typical outflow velocities of v $\sim 0.1$c and flow momenta mv
$\sim\ L_{Edd}/c$ are consistent with winds being launched by continuum
driving from a disc when the local mass accretion rate is super-Eddington.
Here we report the launch of a new, ultra-fast outflow component in PG1211+143
near the end of a 5-week XMM-Newton observing campaign, and discuss its origin
in an ultra-fast inflow of similar velocity detected some 3 weeks earlier. We
find that the inflow lasted for at least 3 days and delivered some 10 Earth
mass of fresh material into the innermost region of the source. While this
mass by itself is insufficient to cause a complete inner disc restructuring,
we show that it is sufficient to disrupt the X-ray emitting corona of the
disc. We conclude that it is this coronal re-arrangement of the inner tens
gravitational radii in PG1211+143 that subsequently caused the launch of a new
wind.

###### keywords:

galaxies: active – galaxies: Seyfert: quasars: general – galaxies: individual:
PG1211+143 – X-ray: galaxies

††pagerange: Observing the launch of an Eddington wind in the luminous Seyfert
galaxy PG1211+143–LABEL:lastpage††pubyear: 2005

##  1 Introduction

X-ray spectra from an XMM-Newton observation of the narrow-line Seyfert galaxy
PG1211+143 in 2001 provided the first detection in a non-BAL AGN of strongly
blue-shifted absorption lines of highly ionized gas, corresponding to an
outflow velocity of 0.15$\pm$0.01c (Pounds et al. 2003, Pounds and Page 2006).
Further observations over several years with XMM-Newton, Chandra and Suzaku
showed the high velocity outflow to be persistent but of variable opacity (eg
Reeves et al. 2008). Evidence that the extended outflow in PG1211+143 was both
massive and energetic - with potential importance for galaxy feedback - came
from the detection of PCygni and other broad emission features obtained by
combining the 2001, 2004 and 2007 XMM-Newton EPIC spectra (Pounds and Reeves
2007, 2009).

Examination of archival data from XMM-Newton and Suzaku has since shown
similar ultra-fast, highly-ionized outflows (UFOs) to be relatively common in
nearby, luminous AGN (Tombesi 2010, 2011; Gofford 2013). The frequency of
these detections confirms a substantial covering factor and hence significant
mass and kinetic energy in such winds. Indeed, their integrated mechanical
energy may be substantially greater than required to disrupt the bulge gas in
the host galaxy, suggesting some winds are intermittent, or that much of the
energy in a persistent wind must be lost before reaching the star forming
region, perhaps by colliding with pre-ejecta, as seen in an XMM-Newton
observation of the low mass Seyfert NGC4051 (Pounds and Vaughan 2011, Pounds
and King 2013).

In order to further explore the velocity structure and evolution of the fast
wind in PG1211+143 an extended XMM-Newton observation was carried out during 7
spacecraft orbits over the period 2014 June 2 to 2014 July 9. Effective on-
target exposures for individual orbits ranged from $\sim$50 to $\sim$100 ks,
with a total duration of $\sim$650 ks. Full details of the XMM-Newton
observing log are given in Lobban et al.(2016), reporting the results of a
detailed timing analysis.

Swift (Gehrels et al. 2004) also observed PG 1211+143 with 43 snapshots, as
part of a Target of Opportunity (ToO) programme linked to the XMM-Newton
campaign. The Swift observations cover the period 2014 June 4 to 2014 August 4
and have typical durations of $\sim$ 1.5 ks. We have extracted x-ray light
curves from the X-ray Telescope (XRT; Burrows et al. 2005) and UV light curves
from the Ultra-Violet/Optical Telescope (UVOT; Roming et al. 2005), the latter
using the U, UVW1 and UVW2 filters. Full details of the Swift observations are
also presented in Lobban et al. (2016).

Figure 1 reproduces the orbital-mean x-ray fluxes from the XMM-Newton pn
camera (Strueder et al.2001), together with the first 17 snapshots from the
Swift XRT. Both data sets show a deep minimum flux near XMM-Newton orbit 2659
(day 16), when the ultra fast inflow was detected. The Swift light curve is
particularly interesting, with its softer x-ray bandwidth responding to both
column density and ionization changes in the line-of-sight flow, suggesting
the inflow seen in day 16 actually began some 3 days earlier followed by a
strong increase in x-ray emission, to a peak in orbit 2664 some 8 days later.
The high x-ray flux is then maintained for at least 4 further days, to orbit
2666, but has fallen substantially by the final observation (orbit 2670).

Figure 1: X-ray count rates from the pn camera for each of the 7 observations
in the 2014 XMM-Newton campaign, together with 17 overlapping observations
from the Swift satellite. XMM-Newton data points correspond to spacecraft
orbits 2652, 2659, 2661, 2663, 2664, 2666 and 2670. Both data sets show a deep
minimum flux near orbit 2659 (day16), followed by an increase to a peak near
orbit 2664 (day24). The earlier recovery of x-ray flux in the Swift data is
explained by the softer XRT energy band, compared with that of the XMM-Newton
pn camera, as the rising continuum increases the ionization state of the
inflowing matter.

Figure 2 shows x-ray spectra from the XMM-Newton pn camera over the same
interval, with orbits 2659(black), 2661(red), 2663(green) and 2664(blue),
plotted as a ratio to that of orbit 2652. The broad spectral band highlights
the strong soft x-ray absorption associated with the transient accretion event
in orbit 2659, which then falls over several days, with the continuum flux
continuing to increase to a new peak by orbit 2664.

Figure 2: Broad-band spectra from the XMM-Newton pn camera for orbits
2659(black), 2661(red), 2663(green) and 2664(blue) plotted as a ratio to that
of orbit 2652, illustrating strong soft x-ray absorption associated with the
transient line-of sight inflow first seen on day 16, which then falls over
several days as the continuum flux continues to increase to a new peak in
orbit 2664.

Published analysis of the 2014 XMM-Newton observation of PG1211+143 has
focussed on stacked X-ray spectra, where the high quality data have revealed a
complex velocity structure, with primary (high column density) outflow
components at v$\sim$0.067c, v$\sim$0.129c and v$\sim$0.187c (Pounds et
al.2016a, 2016b; hereafter P16a and P16B)). Given the limited spectral
resolution of the pn camera, detection of all 3 outflow velocities in the co-
aligned Reflection Grating Spectrometer (RGS: den Herder et al. 2001) was
important, while indicating the presence of co-moving higher density matter in
each flow component.

Notably, none of the outflow velocities in 2014 were consistent with the
powerful outflow of v$\sim$0.15c in the 2001 initial XMM-Newton observation,
while repeated observations of several AGN reported in the afore-mentioned
archival searches showed differing velocities weeks apart, implying that some
high velocity AGN winds may be relatively short-lived. An initial examination
of individual orbits during the 5-weeks XMM-Newton observation in 2014 found
the clearest spectral variability in the soft x-ray band, sensitive to both
column density and ionization state changes. A detailed inter-orbit analysis
of the RGS soft x-ray spectra has subsequently confirmed variability on
timescales of days, with the strongest outflow at v$\sim$0.06c clearly
resolved into distinct ionization (density) components (Reeves et al. 2018).

The present paper derives from an on-going orbit-by-orbit study of the harder
X-ray spectra from the EPIC pn (Strueder et al. 2001) and mos (Turner et al.
2001) cameras. One remarkable outcome already reported (Pounds et al. 2018)
was the detection of a transient ultra-fast inflow, with v$\sim$0.3c, during
the second XMM-Newton orbit in 2014. We now report the launch of a new high
velocity outflow component, with v$\sim$0.27c, a few weeks later, and note
that the similar velocities suggest the two events are physically linked.

We assume a redshift of $z=0.0809$ (Marziani et al. 1996), with a black hole
mass of $4\times 10^{7}$$\rm~{}M_{\odot}$ (Kaspi et al. 2000) indicating the
historical mean luminosity of PG1211+143 is close to Eddington. Spectral
modelling is based on the XSPEC package (Arnaud 1996) and includes absorption
due to the line-of-sight Galactic column NH $\sim 3\times 10^{20}$ (Murphy et
al. 1996). To allow comparison with previous analyses of the 2014 spectra we
again use photoionized absorption and emission grids of pre-computed spectra
based on the XSTAR code of Kallman et al. (1996).

##  2 Launch of a fourth high velocity outflow late in the 2014 campaign

As noted above, the unusually high statistical quality of the extended 2014
observation of PG1211+143 was important in resolving complex absorption
structure in the Fe K spectrum, identified in P16a with absorption line series
in FeXXV and XXVI corresponding to line-of-sight outflow velocities of
v$\sim$0.067c and v$\sim$0.129c, and evidence for a third velocity
v$\sim$0.187c subsequently confirmed in the soft X-ray spectrum reported in
(P16b. Table 1 lists the parameters of the 3 primary (high column) outflows
from the mean 2014 pn spectral fit.

Table 1: Parameters of the ultra-fast outflow obtained from a 2–10 keV spectral fit to the stacked 2014 pn data, with 3 photoionized absorbers defined by ionization parameter $\xi$ (erg cm s-1), column density $N_{\rm H}$ (cm-2) and outflow velocity ($v/c$). Absorber 4 relates to the new outflow reported in the present paper. Extracted luminosities for each photoionized component relate to the 2–10 keV observed energy range comp | log$\xi$ |  $N_{\rm H}$($10^{23}$) | $v/c$ | $L_{\rm abs}$ | $\Delta\chi^{2}$
---|---|---|---|---|---
abs 1 | 3.5$\pm$0.1 | 2.6$\pm$0.2 | 0.066$\pm$0.003 | 11$\times 10^{41}$ | 52/3
abs 2 | 3.9$\pm$0.6 | 1.5$\pm$0.2 | 0.129$\pm$0.006 | 6$\times 10^{41}$ | 18/3
abs 3 | 3.4$\pm$0.2 | 0.5$\pm$0.1 | 0.187$\pm$0.003 | 3$\times 10^{41}$ | 14/3
abs 4 | 2.9$\pm$0.2 | 1.0$\pm$0.3 | 0.27$\pm$0.01 | 4$\times 10^{41}$ | 17/3

Examination of pn data for individual orbits show the 3 outflow velocities to
remain as in Table 1, indicating all were in ’coasting’ phase.

However, in the final observation (orbit 2670) several additional absorption
lines appeared, notably at $\sim$8.1 kev and $\sim$8.6 keV and at 9-10 keV,
indicating absorption specific to orbit 2670. To quantify the new absorption
component, the spectral model from the stacked 2014 data (P16a) was applied to
the pn data from orbit 2670, with only the normalisation (amplitude) of X-ray
continuum and emission components free to change. Figure 3 (upper panel) shows
the additional absorption in the ratio of orbit 2670 data to the mean 2014
spectrum.

Adding a 4th photoionized absorber to the spectral model confirms a new high
column outflow at an observed blueshift of 0.185$\pm$0.005, with column
density NH 9.7$\pm$2.8$\times 10^{22}$cm-2 and ionisation parameter log$\xi$
$\sim$2.9. The corresponding outflow velocity in the AGN rest frame is
v$\sim$0.27$\pm$0.01c. Inclusion of the additional absorber (Figure 3, lower
panel) recovers an excellent spectral fit to the orbit 2670 data, with
$\Delta\chi^{2}$ of 17/3 and null probability of 4$\times 10^{-3}$.

Figure 3: (top) Absorption features specific to rev2670 are shown by fitting
the rev2070 pn data with the 2014 mean spectrum, allowing only the
normalisation of continuum and photoionized emission to vary. New absorption
lines are seen near 8.1 and 8.6 keV keV, with additional spectral structure
between 9 and 10 keV. (lower) Absorption profile of a 4th primary outflow
component with column density NH $\sim$ $10^{23}$, ionization parameter
log$\xi\sim$ 2.9 and observed blueshift $\sim$ 0.185

Comparison of the upper and lower panels of Figure 3 identifies the resonance
lines of He-like FeXXV and H-like FeXXVI blue-shifted to 8.1 and 8.6
keV,respectively, and the corresponding $\beta$ transitions both visible at
higher energies. The apparent broadening of the $\sim$8.1 keV line can be
explained by significant inner-shell absorption components in the low energy
wing of the Fe XXV resonance line, a feature often seen in similar AGN
spectra.

Independent support for the new high velocity outflow is provided by a similar
examination of the RGS data from orbit 2670, where a comparison with the mean
2014 spectrum (P16b) again reveals additional soft x-ray absorption specific
to orbit 2670. Modelling with a photionized absorber, as in P16b, finds a soft
x-ray absorber with column density NH $\sim$ $10^{22}$ and ionization
parameter log$\xi\sim$ 0.8, at an observed blueshift $\sim$ 0.183$\pm$0.003.

Figure 4 shows an array of soft x-ray absorption lines specific to orbit 2670,
identified with resonance transitions in OVII, OVIII, NVII and CVI, with a
common blueshift $\sim$ 0.185$\pm$0.006, corresponding to an outflow velocity
v$\sim$0.265$\pm$0.005c. The lower ionization parameter and column density
(compared with the pn data) again indicate higher density matter embedded in
the primary (high column) highly-ionized flow.

Figure 4: Additional soft x-ray absorption specific to the orbit 2670 RGS data
is identified with blue-shifted resonance lines of OVII, OVIII, NVII and CVI,
all consistent with an outflow velocity in the AGN rest frame of v$\sim$0.265

##  3 Discussion

Repeated XMM-Newton observations have demonstrated the highly variable nature
of the wind opacity in PG1211+143. The initial discovery of a strong outflow
at 0.15c in a 2001 XMM-Newton observation was followed by weaker detections in
2004 and 2007, and then the long 2014 campaign reporting a more complex
velocity structure. Assuming the highly ionized winds are driven by momentum
exchange with a super-Eddington radiation flux (King and Pounds 2003), an
explanation now supported in the quantitative agreement of data and theory
(King and Pounds 2015), the pattern of wind variability will also have direct
bearing on the nature of variable accretion in AGN.

Nixon et al. (2012) have shown how an accreting ’cloud’ approaching the disc
at an oblique angle to the black hole spin plane could cause the inner disk to
warp and break off, with subsequent collisions between rings of matter
precessing at different rates, leading to loss of angular momentum and direct
infall, potentially creating local disc regions of super-critical mass
accretion. The ultrafast infall detected during the 2014 XMM-Newton
observations of PG1211+143 was discussed in that context in Pounds et
al.(2018).

Observing the launch of a new primary (high column) ultrafast outflow
component, some 20 days later, is a striking illustration of continuum-driving
for a high velocity wind, as first described in the classic accretion disc
paper of Shakura and Sunyaev (1973), but - interestingly - not mentioned in
the original scientific case for XMM-Newton (Bleeker et al. 1982).

In that context, the near coincidence of a 0.3c infall velocity (with free-
fall location at $\sim$20 Rg) detected in rev 2659, and the wind launch seen
2-3 weeks later (with a similar escape velocity, and likely radial location)
strongly suggests a direct physical link. The increase in x-ray emission
between those two events (Fig.2) might then reflect the release of energy as
the added matter spreads out on the local viscous timescale, with an inward
accreting flow eventually reaching a radius where the mass rate is super-
Eddington, with excess matter being ejected as a new wind component.

To examine that idea quantitatively, we recall the transient inflow mass (in
line-of-sight) reported in P18 was $\sim$3.3$\times 10^{26}$g, accumulated
over 3000s, for a mean rate of $10^{23}$g $s^{-1}$. However, that measure is
only for matter in line of sight and sufficiently cool to be detected, and is
likely to be a significant underestimate. Furthermore, Fig. 1 suggests the
infall may have begun some 3 days earlier, increasing the observed mass dump
by a substantial factor. In principle the integrated mass in the new UFO could
have provided an alternative measure of the total mass dump, but unfortunately
no observations of PG1211+143 have been made since the 2014 XMM-Newton
campaign.

As noted above, Pounds et al.(2018) discussed mass deposition in the context
of the specific Nixon et al. (2012) scenario for mass deposition due to
colliding rings of a warped disc. An alternative scenario for gas deposition
in AGN, supported by simulations of clumpy in-homogeneous inflows is chaotic
or ballistic accretion, e.g., Hobbs et al. 2011, Faber & Dehnen (2018). In
this picture gas inflowing into the central parsec of AGN has a wide
distribution of angular momentum, in both orientation and magnitude. A small
fraction of the gas will have angular momentum small enough to fall into the
central region of AGN discs. Independently of how the gas is deposited on the
disc, the mass deposition rate due to a radial inflow is

$\dot{M}=\Delta\Omega R^{2}m_{p}n_{p}v_{r}\;,$ (1)

where $\Delta\Omega$ is the solid angle of the flow ($4\pi$ would correspond
to the full sky flow), $m_{p}$ and $n_{p}$ are the proton mass and particle
density in the flow, and $v_{r}$ is the radial velocity. Now $N_{\rm
H}=n_{p}\Delta R$, where $\Delta R<R$ is the radial thickness of the flow, and
we can eliminate $R$ through $v_{r}=v_{\rm ff}=(2GM/R)^{1/2}$, which yields
$R=R_{\rm S}(c/v_{r})^{2}$. We obtain

$\dot{M}=\Delta\Omega R_{\rm S}m_{p}cN_{H}\frac{R}{\Delta
R}\frac{c}{v_{r}}=0.003\;M_{\rm\odot}{\rm yr}^{-1}\;\Delta\Omega N_{23}\;,$
(2)

where $N_{23}=N_{H}/(10^{23})$. For comparison, the Eddington accretion rate
for the source is

${\dot{M}_{\rm Edd}}=\frac{{L_{\rm Edd}}}{0.1c^{2}}=0.8\;M_{\rm\odot}{\rm
yr}^{-1}$ (3)

We see immediately that it is unlikely that the ultra-fast inflow could have
significantly affected the bulk of the accretion flow, that is, its optically
thick UV-emitting part.

That prediction is supported by the UV fluxes from Swift over the same period
(Fig.5), showing no substantial flux variability for several days before and
after the ultra fast inflow on day 16. In contrast, the x-ray data from Swift
and XMM-Newton over the same period clearly show (Fig.1) enhanced soft x-ray
absorption between days 12 and 15, followed (Fig.2) by an increase to the
broad band x-ray emission from day 17.

Furthermore, making plausible assumptions about disc parameters in PG1211+143
the viscous timescale (on which mass accretion rate through the disc may vary)
is likely to be of order a few years at $R\sim 20R_{\rm g}$, too long for a
causal disc-mediated connection between the infall event and the launch of the
new wind reported here.

However, the inflow is certainly massive and powerful enough to have affected
the corona of the accretion disc in PG1211+143. Disc coronae must be Compton
optically thin to explain typical AGN spectra (e.g., Poutanen & Svensson
1996). The effect of absorption in a line-of-sight inflow on the Swift XRT
data (Fig. 2) suggests a substantial flow lasted for at least $\Delta
t=$3days, adding $\sim\dot{M}\Delta t/(\pi R^{2})/2\sim$ a few g/cm2, per side
of the disc, exceeding the expected amount of material in the corona.
Furthermore, the specific energy injected by the inflow into the coronal
region of the disc, $v_{r}^{2}/2=0.05c^{2}$ is one-two orders of magnitude
higher than that of the X-ray emitting coronal gas. We suggest the additional
material and extra energy introduced by the infalling gas is likely to have
caused a significant rearrangement in the corona, and the top layers of the
disc where most of the disc emission forms, and therefore propose the launch
of a new wind late in the 2014 campaign was a direct consequence of the
earlier fast infall of new matter.

Figure 5: UV observations from the Swift satellite showing no substantial flux
variability for several days before and after the ultra fast inflow observed
on day 16, in contrast to the x-ray spectra over the same period, which show
(Fig.1) enhanced soft x-ray absorption between days 12 and 15, followed
(Fig.2) by an increase to the broad band x-ray emission from day 17.

##  4 Conclusions

Our analysis indicates that the amount of material delivered into the
innermost tens of $R_{\rm g}$ by the inflow is likely to be small compared to
that in the optically thick disc in that region, a conclusion supported by the
lack of significant variability in the UV data from Swift (Fig.5), contrasting
with the increasing coronal x-ray emission from XMM-Newton orbits 2659 to 2664
(Fig.2). However, the additional mass and energy injected into the accretion
disc by the transient inflow are significant in comparison with the mass and
energy budgets of the coronal region of the inner disc. We therefore conclude
that the observed mass and energy injection early in the 2014 XMM-Newton
campaign most likely is the direct cause of the new wind launched some three
weeks later.

##  Acknowledgements

XMM-Newton is a space science mission developed and operated by the European
Space Agency. We acknowledge in particular the excellent work of ESA staff in
Madrid successfully planning and conducting the XMM-Newton observations. We
also thank the Swift PI for approving, and NASA mission planners for
scheduling, the additional observations used in this paper. XMM-Newton data
were reduced by our former colleague Andrew Lobban and Swift data were
provided by the UK Science Data Centre at the University of Leicester,
supported by the UK Science and Technology Facilities Council, with our
special thanks locally to Dr Kim Page.

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