# Anisotropic molecular photoemission dynamics: Interpreting and accounting
for the nuclear motion

Antoine Desrier  Morgan Berkane  Camille Lévêque  Richard Taïeb  Jérémie
Caillat<EMAIL_ADDRESS>Sorbonne Université, CNRS,
Laboratoire de Chimie Physique-Matière et Rayonnement, LCPMR, F-75005 Paris,
France

###### Abstract

We investigate how vibration affects molecular photoemission dynamics, through
simulations on two-dimension asymmetric model molecules including the
electronic and nuclear motions in a fully correlated way. We show that a
slight anisotropy in the electron-ion momentum sharing is sufficient to
prevent one from unambigously characterizing the vibrationnaly averaged
photoemission dynamics in terms of stereo Wigner delays. We further show that
vibrational resolution can be retrieved in fixed-nuclei simulations, using
effective molecular conformations that are specific to each vibrational
channel. The optimal internuclear distances found empirically in 1-photon
processes can be identified a priori using simple physical arguments. They
also turn out to be efficient to simulate vibrationnally-resolved RABBIT
measurements and to account for interchannel coherences in broadband 1-photon
ionization.

##  I Introduction

From its early emergence as an applied branch of attosecond sciences to its
most recent and ongoing developments, attochemistry offers and envisions
unprecedented ways of exploring fundamental dynamics occurring in molecules on
their natural time scale [1, 2, 3, 4, 5, 6, 7]. In the broader perspective of
attosecond time-resolved spectroscopies, photoemission endorses a singular
role, either as a characterization tool [8, 9] or as the probed process itself
[10, 11, 12].

Revisiting a process as essential as molecular photoemission in the time
domain is made possible only through intertwined, long term interactions
between theory and experiments, that tackle and exploit the complexities and
richness specific to molecular systems both in the conception and in the
interpretation of simulations or measurements. It was pioneered by
investigating experimentally the intricate dynamics of resonant ionization in
N2 with vibrational resolution [13], using the ‘reconstruction of attosecond
beatings by interferences of two-photon transitions’ (RABBIT) scheme [8, 14].
The interpretation of the measured channel-selective RABBIT phases in terms of
transition delays was subsequently clarified by means of simulations on low-
dimensional model molecules [15, 16]. The experimental results were
furthermore reproduced and expanded recently in elaborate simulations aiming
at a quantitative agreement [17]. The anisotropy of ionization dynamics, which
is another essential feature of molecular photoemission, was considered first
in simulations of attosecond ‘streaking’ measurements [9, 18] on a model CO
molecule [19] and then explored experimentally, also on CO, using RABBIT [20].
Further experimental and theoretical investigations addressing anisotropy have
since then been reported on systems of increasing complexities [21, 22, 23,
24, 25, 26], see also [27] and references therein.

The imprints of nuclear motion on photemission dynamics in polyatomic
molecules is the subject of an increasing number of theoretical and
experimental investigations. Among the most recent studies, the authors of
Ref. [28] investigate theoretically how nuclear motion affects photoelectron
spectra in the context of RABBIT measurements. In this context, they establish
an expression of molecular RABBIT spectra in terms of a convolution between a
nuclear autocorrelation function and two-photon electronic transition matrix
elements, following an approach initially derived to simulate molecular
streaking [29]. In [30], the imprints of nuclear motion on attosecond
photoemission delays is investigated experimentally on the methane molecule
and its deutered counterpart, using RABBIT and an advanced theoretical
support. The authors find no significant isotopic effect, but a $\sim 20$ as
difference between dissociative and non-dissociative channels. A closely
related study on the same polyatomic molecules further highlighted how nuclear
motion impacts the coherence of the photoemission process [31], which is an
essential issue for most of the interferometric attosecond resolved pump-probe
schemes.

Nonetheless, investigations on smaller – diatomic – benchmark molecules [32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44] remain of great importance to
gradually and comprehensively explore the interplay of the molecular degrees
of freedom from a time-dependent perspective. On the theory side, simulations
on simple models with limited degrees of freedom [45] are essential to guide
and interpret experiments and elaborate simulations in terms of intuitive
pictures and practical notions.

Figure 1:  (Color online) Born-Oppenheimer energies of the model molecules
$\mathcal{A}$ [frame (a)], $\mathcal{B}$ [frame (b)] and $\mathcal{C}$ [frame
(c)] designed for the simulations, as a function of the internuclear distance
$R$. The energy scale refers here to the overall ground states of the neutral
molecules. For each molecule, the full grey and black curves correspond to the
neutral and ionic electronic ground states, respectively. In frames (b) and
(c), the dashed grey curve is an image of the neutral ground curve vertically
shifted towards the ionization threshold, displayed for comparison purpose.
The vertical dotted lines indicate the equilibrium distance of the neutral
$R_{\text{\tiny eq}}$. The vibrational levels $v$ presenting a significant
overlap with the ground state [$|F(v)|^{2}>10^{-2}$, see Eq. (21) and Fig. 7]
are indicated in the ionic curve of each molecule .

In the present article, we investigate numerically how vibration imprints the
attosecond dynamics of anisotropic molecular photoemission even when the
vibronic couplings are minimal – notably far from any vibronic resonance111It
comes as a companion paper to Ref. [27], which is focused on the definition
and RABBIT measurement of anisotropic molecular photoemission delays,
regardless of the nuclear motion.. Our study is based on low dimensional model
molecules allowing for extensive numerically exact simulations. The objectives
are two-fold: i) to study the interplay of molecular asymmetry and nuclear
motion on attosecond time-resolved photoemission, and ii) to identify a way to
account for the vibrational resolution in standard theoretical approaches with
fixed nuclei – keeping in mind that only the latter can be routinely applied
to the simulations of more realistic models.

The paper is organized as follows. Section II presents the model molecules and
the overall methodology followed in our study. In section III, we benchmark
the signatures of molecular asymmetry on 1-photon ionization dynamics with a
model displaying no effective nuclear motion during and upon photoemission. It
partly reproduces and expands the results presented and commented in [27]. In
section IV, we consider photoemission occurring with sensible nuclear motion.
We question the relevance of defining an orientation-resolved photoemission
delay for the photoelectron wave packet averaged over the open vibrationnal
channels. Then we consider vibrationnaly-resolved photoemission dynamics, and
investigate the possibility to reproduce it in a fixed-nuclei approach. In
Section IV.3 we address the capacities of the fixed nuclei approach to
reproduce coherent, vibrationnally-resolved, photoemission beyond the context
in which it was designed. The conclusions are presented in section V.

Unless stated otherwise, the equations are displayed in atomic units (a.u.)
all through the paper.

##  II Theoretical toolbox

###  II.1 Model molecules

Our simulations were performed on single-active electron diatomic model
molecules including correlated electronic and internuclear motions each in 1D.
They are similar to the ones used eg in [46, 47, 15].

####  II.1.1 Generic hamiltonian

The generic field-free hamiltonian of the model molecules reads

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\underbrace{-\frac{1}{2\mu}\frac{\partial^{2}}{\partial
R^{2}}+V_{\text{\tiny N-N}}(R)}_{H_{\text{\tiny
N}}}\underbrace{-\frac{1}{2}\frac{\partial^{2}}{\partial x^{2}}+V_{\text{\tiny
N-e}}(x,R)}_{H_{\text{\tiny e}}}$ (1)

where the two coordinates $x$ and $R$ are the electron position and the
internuclear distance, respectively, $\mu$ is the nuclei reduced mass,
$V_{\text{\tiny N-N}}(R)$ is the interatomic potential in absence of the
active electron, and $V_{\text{\tiny N-e}}(x,R)$ is the interaction potential
between the active electron and the molecular ion.

The interatomic potential $V_{\text{\tiny N-N}}$ for each molecule is defined
numerically on a discretized $R$-grid, while the electron-nuclei potential
$V_{\text{\tiny N-e}}$ is defined as an asymmetric two-center soft-Coulomb
potential,

$\displaystyle V_{\text{\tiny N-e}}(x,R)$ $\displaystyle=$
$\displaystyle\sum\limits_{j=1,2}-\frac{q_{j}}{\sqrt{(x-X_{j})^{2}+a(R)}}$ (2)

where

$\displaystyle X_{j}$ $\displaystyle=$
$\displaystyle(-1)^{j}\frac{\mu}{M_{j}}R$ (3)

are the positions of each nucleus ($j=1,2$), of mass $M_{j}$, with respect to
their center of mass. The $V_{\text{\tiny N-e}}$ potential is adjustable
through the effective charges $q_{j}>0$ (with the constraint $q_{1}+q_{2}=1$)
and the screening parameter $a(R)>0$ defined numerically over the same
$R$-grid as $V_{\text{\tiny N-N}}$. One should note that $V_{\text{\tiny
N-e}}$ accounts not only for the asymmetry related to the electronegativity
difference (through $q_{j}$) but also for the mass asymmetry (through
$X_{j}$).

In practice, we adjusted $V_{\text{\tiny N-N}}$ and $V_{\text{\tiny N-e}}$
empirically [45] for the energies of the model molecule to match some desired
energy curves within the Born-Oppenheimer (BO) approximation: $V_{\text{\tiny
N-N}}$ is directly the molecular ion energy curve, while the negative eigen-
energies of $H_{\text{\tiny e}}+V_{\text{\tiny N-N}}$ computed at each $R$
[see partition of $H_{0}$ in Eq. (1)] provide the electronic ground and
excited curves of the neutral. Within the BO framework, the ground state
energy $\varepsilon_{0}(R)$ of the electronic hamiltonian $H_{\text{\tiny e}}$
is the opposite of the vertical ionization potential for each $R$,

$\displaystyle\tilde{}E_{\text{{I}}}(R)$ $\displaystyle=$
$\displaystyle-\varepsilon_{0}(R),$ (4)

i.e., the energy gap between the ionic and the neutral ground state curves.

####  II.1.2 Model parameters

We considered three model molecules, hereafter referred to as $\mathcal{A}$,
$\mathcal{B}$ and $\mathcal{C}$ for simplicity. For each of them, the
effective charges were set to $q_{1}=0.33$ a.u. and $q_{2}=0.67$ a.u, and the
nuclei masses to $M_{1}=20.53$ u and $M_{2}=7.47$ u. The chosen masses are
such that $M_{1}+M_{2}=28$ u, i.e., the mass of CO – a prototypical
hereronuclear diatomics on which orientation- and time-resolved photoemission
simulations [19] and experiments [20] have already been performed. However,
our models are only loosely based on CO, as we, among other liberties,
increased the mass ratio $M_{2}/M_{1}$ in order to emphasize the consequences
of mass asymmetry on photoemission dynamics. The potentials were adjusted to
obtain the BO curves displayed in Fig. 1. The three molecular systems share
the same ground state energy curve, which mimics the one of CO [48], with an
equilibrium distance $R_{\text{\tiny eq}}=1.115$ Å.

In molecule $\mathcal{A}$, the ionic curve is a vertically shifted image of
the ground state. In molecule $\mathcal{B}$ (resp. $\mathcal{C}$), it is also
a replica of the ground state curve, translated towards lower (resp. larger)
values of $R$ with its minimum at $1.029$ Å (resp. $1.201$ Å). The energy
shifts of the ionic curves were adjusted to achieve
$\tilde{}E_{\text{{I}}}(R_{\text{\tiny eq}})\simeq 30$ eV for all molecules
($29.81$ eV, $29.18$ eV and $29.30$ eV for molecules $\mathcal{A}$,
$\mathcal{B}$ and $\mathcal{C}$, respectively). As an illustration, the
$x$-dependency of the electron-core potential for molecule $\mathcal{A}$ at
the equilibrium distance $R=R_{\text{\tiny eq}}$ can be found in Fig. 4(a).

Due to the relative positions of these energy curves, photoionization takes
place with no effective nuclear motion in molecule $\mathcal{A}$, while it
initiates a bond contraction in molecule $\mathcal{B}$ and an elongation in
molecule $\mathcal{C}$.

###  II.2 Methodology overview

We considered two different approaches for the simulations. The first one
consists in simulating the ‘complete’ vibronic dynamics of the model
molecules. To this end, we performed numerically converged, vibronically
correlated, simulations based on the resolution of the time-dependent
Schrödinger equation (TDSE). The second approach consists in adopting a
standard simplified treatment with fixed nuclei, based on the Born-Oppenheimer
(BO) approximation, where the photoemission dynamics are encoded in the
continuum solutions of the electronic time-independent Schrödinger equation
(TISE).

Our motivation is two-fold. On the one hand, the ‘complete’ vibronic
simulations highlight the impact of vibronic couplings on the dynamics of
molecular photoemission investigated with attosecond resolution. On the other
hand, by confronting these ‘exact’ dynamics to the ones obtained in the BO
framework, we assess the relevance of fixed-nuclei approaches to investigate
molecular photoemission process (see eg [49, 50, 19, 51, 23, 52]), when the
latter are performed with well-chosen molecular conformations. In all the
simulations, the $x$ origin was set a priori to coincide with the center of
mass of the molecule, consistently with the definition of the $V_{\text{\tiny
N-e}}$ potential [see Eqs. (2) and (3)].

The central quantities used in this work to characterize anisotropic
photoionization dynamics are orientation-resolved yields and stereo ionization
delays, which can be computed indifferently in both approaches.

####  II.2.1 ‘Complete’ simulations

The complete dynamics of the molecules were simulated by solving numerically
the vibronic TDSE

$\displaystyle i\frac{\partial\Phi(x,R,t)}{\partial t}$ $\displaystyle=$
$\displaystyle[H_{0}+W(t)]\Phi(x,R,t).$ (5)

Here $\Phi(x,R,t)$ is the propagated vibronic wave-function and $W(t)$
represents the dipole interaction of the molecule with the ionizing pulse,
implemented in the velocity gauge. The initial state is the ground vibronic
state $\Phi_{0}(x,R)$, obtained by imaginary time propagation, in all the
simulations. Numerically exact resolutions of the TDSE [Eq. (5)] were
performed using a split operator propagation based on a grid representation of
the $x$ coordinate, combined with an expansion over a set of eigenvectors of
$H_{\text{\tiny N}}$ for the $R$ coordinate, as detailed in [15, 45]. The
vector potentials of the ionizing pulses were assigned $\sin^{2}$ temporal
envelops with central photon energies $\omega_{\text{\small xuv}}$
corresponding to harmonics of a Ti:Sapphire laser ($\lambda_{0}=800$ nm,
$\omega_{0}=1.55$ eV) in the extreme ultraviolet (XUV). Case specific pulse
parameters are indicated further in the text. Simulations on molecule
$\mathcal{A}$ were performed by expanding the wave-function on a single
vibrationnal state ($v=0$ is the only open channel). For molecules
$\mathcal{B}$ and $\mathcal{C}$, the results shown in this paper were obtained
by including all vibrational channels up to $v=8$ and $17$, respectively,
which safely ensures convergence.

The analysis of these complete time-dependent simulations are mostly based on
the outgoing electron flux computed in the asymptotic $x$ region, where the
short-range components of $V_{\text{\tiny N-e}}$ (including the vibronic
couplings) vanish.

####  II.2.2 Simulations at fixed internuclear distances

In the BO framework, we treated 1-photon ionization by analyzing electronic
continuum wave-functions computed at fixed internuclear distances. Among the
solutions of the TISE

$\displaystyle H_{\text{\tiny e}}\psi_{\varepsilon}(x)$ $\displaystyle=$
$\displaystyle\varepsilon\psi_{\varepsilon}(x)$ (6)

in the degenerate continuum (at energies $\varepsilon>0$), we worked with the
wave-functions specifically selected by 1-photon transitions starting from the
ground state, hereafter referred to as SCWF for ‘selected continuum wave-
functions’. These real-valued continuum wave-functions are defined
unambiguously, and their analysis and interpretation are independant from
their definition and computation. The SCWF formalism was introduced in [53]
and later used in [37, 27]. Note that the electronic hamiltonian
$H_{\text{\tiny e}}$ [as defined in Eq. (1)], its eigenfunctions
$\psi_{\varepsilon}(x)$ and eigenvalues $\varepsilon$ depend parametrically on
the internuclear distance $R$ – although it does not appear explicitely in Eq.
(6) for the sake of readability.

Within this framework, the dynamics of photoemission are encoded in the
asymptotic features of the SCWF.

More computational details are provided along with the presentation of the
results.

##  III Photoemission from molecule $\mathcal{A}$: signatures of the potential
asymmetry

We present here the results obtained for molecule $\mathcal{A}$, which was
designed to benchmark our methodological approach. We first detail the
analysis of the time-dependent vibronic simulations, and then verify that the
latter are consistent with time-independent BO simulations at equilibrium
distance.

###  III.1 ‘Complete’ simulations

In these simulations, the XUV pulse intensities were set to $10^{12}$ W/cm2,
i.e., low enough to avoid any significant multi-photon process in 1-color
photoionization. The total pulse durations were set to $15.87$ fs, which
corresponds to 6 fundamental periods ($T_{0}=2\pi/\omega_{0}=2.67$ fs).

####  III.1.1 Photoelectron flux

Figure 2:  Outgoing electron flux $j(\theta_{\mbox{\tiny k}},r_{\text{\tiny
d}},t)$ computed at $r_{\text{\tiny d}}=800$ a.u. on the left
($\theta_{\mbox{\tiny k}}=180\degree$, red full curve) and right
($\theta_{\mbox{\tiny k}}=0\degree$, green dashed curve) sides of molecule
$\mathcal{A}$ ionized by a $\omega_{\text{\small xuv}}=35.67$ eV pulse, as a
function of propagation time $t$. The flux are normalized to 1 at the overall
maximum reached on the left side. The temporal envelop of the ionizing pulse
is represented by the blue filled curve.

In the complete time-dependent simulations, we characterized the ionization
dynamics through the outgoing photoelectron flux computed at a given detection
distance $r_{\text{\tiny d}}$ from the $x$-origin, on either side of the
molecule and averaged over $R$:

$\displaystyle j(\theta_{\mbox{\tiny k}},r_{\text{\tiny d}},t)$
$\displaystyle=$ $\displaystyle\cos\theta_{\mbox{\tiny k}}\times$
$\displaystyle\mathfrak{Im}\int\limits_{0}^{\infty}\,\Phi^{\star}(r_{\text{\tiny
d}}\cos\theta_{\mbox{\tiny k}},R,t)\Phi^{\prime}(r_{\text{\tiny
d}}\cos\theta_{\mbox{\tiny k}},R,t)\,{d}R$

where $\Phi^{\star}(x,R,t)$ is the complex conjugate of the vibronic wave-
function propagated according to Eq. (5), $\Phi^{\prime}(x,R,t)$ its
derivative with respect to $x$. Here and all through the paper,
$\theta_{\mbox{\tiny k}}$ represents the direction of photoemission,
restricted to two discrete values: $\theta_{\mbox{\tiny k}}=0\degree$
(emission towards the right, $x>0$) and $180\degree$ (towards the left,
$x<0$).

As an illustration, Figure 2 shows the flux computed at $r_{\text{\tiny
d}}=800$ a.u. (423 Å) on the left and right sides of molecule $\mathcal{A}$
ionized by a pulse of central frequency $\omega_{\text{\small
xuv}}=23\times\omega_{0}=35.67$ eV. The flux profiles follow the ionizing
pulse envelop (also shown), shifted by $\approx 30$ fs, which is consistent
with the time needed for a (nearly) free electron with the energy
$\omega_{\text{\small xuv}}-\tilde{}E_{\text{{I}}}(R_{\text{\tiny eq}})=5.86$
eV to cover the distance $r_{\text{\tiny d}}$. The flux maximum $\sim 1.5$
times larger on the left side than on the right side is a clear signature of
the photoemission anisotropy.

####  III.1.2 Orientation-resolved yields and delays

Figure 3:  Anisotropic photoemission from molecule $\mathcal{A}$. (a)
left/right ionization yields ratio computed in the fully correlated
simulations [Eq (9)] and in the BO framework [Eq (14)], resp. $\mathcal{R}$
(circles) and $\tilde{}\mathcal{R}$ (line); (b) stereo ionization delays
computed in the fully correlated simulations [Eq (11)] and in the BO framework
[Eq (16)], resp. $\Delta\tau$ (circles) and $\Delta\tau_{\mbox{\tiny\sc w}}$
(line). All data plotted as functions of the photoelectron energy
$\varepsilon$.

We thus computed the orientation-resolved yields,

$\displaystyle\mathcal{Y}(\theta_{\mbox{\tiny k}})$ $\displaystyle=$
$\displaystyle\int\,j(\theta_{\mbox{\tiny k}},r_{\text{\tiny d}},t)\,{d}t,$
(8)

in a series of simulations with field frequencies corresponding to harmonics
21 to 24, i.e., $\omega_{\text{\small xuv}}$ ranging from $32.57$ eV to
$37.23$ eV by $1.55$ eV steps. The corresponding yields ratios

$\displaystyle\mathcal{R}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{Y}(180\degree)}{\mathcal{Y}(0\degree)}$ (9)

are displayed in Fig. 3(a) (coloured circles) against the photoelectron energy
$\varepsilon$. It evolves smoothly slightly above 1.4, indicating that
photoemission is sensitive to the asymmetry of the ionic potential, with
little variations over the energy range covered in the simulations.

To characterize the asymmetry of photoemission in the time domain, we used the
average “time of flight” (TOF) [15, 53, 16] towards the detector in each
direction as

$\displaystyle\tau(\theta_{\mbox{\tiny k}})$ $\displaystyle=$
$\displaystyle\frac{\int\limits\,t\times j(\theta_{\mbox{\tiny
k}},r_{\text{\tiny d}},t)\,{d}t}{\int\limits\,j(\theta_{\mbox{\tiny
k}},r_{\text{\tiny d}},t)\,{d}t}.$ (10)

To further investigate the angular variations of the TOF computed as such, one
must take into consideration the arbitrary origin $x=0$ on either sides of
which the virtual detectors are located, and which does not coincide with the
average initial position of the electron. In the following, we will therefore
characterize these angular variations with the origin-corrected TOF
difference,

$\displaystyle\Delta\tau$ $\displaystyle=$
$\displaystyle\tau(180\degree)-\tau(0\degree)-2\frac{\langle
x\rangle_{0}}{\sqrt{2\varepsilon}},$ (11)

where $\langle x\rangle_{0}$ is the average initial electron position in the
ground state of the molecule $\Phi_{0}(x,R)$,

$\displaystyle\langle x\rangle_{0}=\langle\Phi_{0}|x|\Phi_{0}\rangle.$ (12)

The role of the last term on the r.h.s of Eq. (11) is discussed in details in
the companion paper [27] in the context of time-independent approaches. Its
justifications identically holds for time-dependent simulations. It
compensates a spurious shift appearing when computing ionization delays with
an arbitrary origin – while the photoelectron in average originates from
$\langle x\rangle_{0}$. We verified numerically that including this term
(after computing the TOFs) is equivalent to shifting the potential such that
$\langle x\rangle_{0}=0$ (prior to solving the TDSE). In the present
simulations, as well as in molecules $\mathcal{B}$ and $\mathcal{C}$, $\langle
x\rangle_{0}=-0.160$ Å.

The stereo ionization delays $\Delta\tau$ obtained for the considered set of
XUV frequencies are plotted in Fig. 3(b). Starting from $\sim-40$ as just
below $\varepsilon=2$ eV, its magnitude decays smoothly while the photon
energy increases, as could be expected in absence of significantly structured
continuum. Note that such attosecond delays cannot be resolved visually in
Fig. 2, since they are extremely small compared to the temporal spread of the
photoelectron wave-packet at detection ($\sim 15$ fs, approximately the XUV
pulse duration). They are nonetheless significant and we have ensured their
numerical convergence.

Among a series of standard numerical checks, we verified that the measured
stereo delays $\Delta\tau$ do not depend on the detection distance
$r_{\text{\tiny d}}$ (as long as it lies far enough from the ionic core),
which is a fundamental property of the short-range scattering delays.

###  III.2 Simulations at fixed internuclear distances

Here, we consider the molecule $\mathcal{A}$ at its equilibrium internuclear
distance, see Fig. 4(a). It corresponds to the 1D model molecule used for the
numerical experiments presented and analysed in [27].

####  III.2.1 Selected continuum wave-function

Figure 4:  Molecule $\mathcal{A}$ in the BO framework (equilibrium
internuclear distance). (a) Electron-nuclei potential $V_{\text{\tiny
N-e}}(x,R_{\text{\tiny eq}})$ as a function of the electron position $x$ (dark
yellow full curve). The ground state electronic wave-function is also shown
(light blue filled curve). (b) Electronic continuum wave-function (full curve)
selected at the energy $\varepsilon=5.86$ eV by a 1-photon transition from the
electronic ground state. Odd-parity reference wave-function (dashed curve)
used to define and compute the orientation-dependent phase-shifts. The
displayed continuum wave-functions are normalized such that their amplitudes
asymptotically converge to 1 on the left-hand side of the molecule. In this
Figure, the left/right discrimination and the parity refer to the arbitrary
$x=0$ origin (indicated by a vertical dotted line).

The BO approach relies on the analysis of the electronic continuum wave-
functions $\psi_{\varepsilon}(x)$ associated with the photoemission processes.
We thus computed the SCWF of molecule $\mathcal{A}$ by solving Eq. (6) over
the same energy range as in the full time-dependent simulations, with a fixed
internuclear distance set to $R_{\text{\tiny eq}}$.

The SCWF computed at the illustrative energy $\varepsilon=5.86$ eV
(corresponding to $\omega_{\text{\small xuv}}=23\times\omega_{0}=35.67$ eV, as
in Fig. 2) is shown in Fig. 4(b) (full curve). The pseudo-period of the
oscillations on both sides is consistent with a kinetic energy asymptotically
converging to $5.86$ eV. Here, the anisotropy of photoemission is clearly
visible in the asymmetry of the amplitudes of the SCWF on either sides of the
molecule.

####  III.2.2 Orientation-resolved yields and delays

For each pulse frequency considered in the fully-fledge approach, an
alternative evaluation of the orientation-dependent ionization yields is
provided, up to a global factor, as

$\displaystyle\tilde{}\mathcal{Y}(\theta_{\mbox{\tiny k}})$ $\displaystyle=$
$\displaystyle|A(R_{\text{\tiny eq}};\theta_{\mbox{\tiny k}})|^{2},$ (13)

where $A(R_{\text{\tiny eq}};,\theta_{\mbox{\tiny k}})$ is the asymptotic
amplitude of the SCWF computed with $R=R_{\text{\tiny eq}}$, at the average
photoelectron energy, on each side of the molecule. We evaluated these
amplitudes using Strömgren’s normalization procedure (see [54] and references
therein).

The corresponding yields ratio

$\displaystyle\tilde{}\mathcal{R}$ $\displaystyle=$
$\displaystyle\frac{\tilde{}\mathcal{Y}(180\degree)}{\tilde{}\mathcal{Y}(0\degree)}$
(14)

computed over the considered energy range is displayed as a full line in Fig.
3(a). The perfect agreement with the ratio $\mathcal{R}$ computed in the
vibronic time-dependent simulations [Eq. (9)] constitutes a first numerical
validation of our comparative approach.

From the time domain perspective, we analyzed the SCWF in terms of orientation
resolved Wigner delays [55], as detailed in [27]. In few words, the Wigner
delay is defined as the spectral derivative

$\displaystyle\tau_{\mbox{\tiny\sc w}}(\theta_{\mbox{\tiny k}})$
$\displaystyle=$ $\displaystyle\frac{\partial\eta(\theta_{\mbox{\tiny
k}})}{\partial\varepsilon}$ (15)

of the asymptotic phase shifts of the SCWF computed on each side of the model
molecule, with respect to an arbitrary intermediate reference (here, radial
Coulomb $s$-waves centered at $x=0$). As explained in [27], and consistently
with the ‘complete’ vibronic simulations, we characterized the angular
variations of the photoemission dynamics in the BO framework through the
origin-corrected stereo Wigner delay

$\displaystyle\Delta\tau_{\mbox{\tiny\sc w}}$ $\displaystyle=$
$\displaystyle\tau_{\mbox{\tiny\sc w}}(180\degree)-\tau_{\mbox{\tiny\sc
w}}(0\degree)-2\frac{\langle x\rangle_{0}}{\sqrt{2\varepsilon}}.$ (16)

Since the exact ground state of the molecule is accurately modelled by its BO
counterpart, the value of the initial average position $\langle x\rangle_{0}$
is the same here as in the ‘complete’ simulations. The stereo Wigner delays
computed in the considered energy range, already shown in Fig. 3 of [27], are
reported in Fig. 3(b) (full line). They perfectly agree with the stereo delays
computed in the ‘complete’ time-dependent simulations.

Apart from benchmarking our comparative approach, the results obtained with
molecule $\mathcal{A}$ put forwards the signatures of the left-right
electronegativity asymmetry of the molecule on the photoelectron dynamics. The
perfect agreement observed in Fig. 3 between the ‘complete’ and the fixed
nuclei approach comes as no surprise since (i) the equivalence between the
time-dependent and time-independant approaches has already been put forward by
Wigner when interpreting in the time-domain the group delay associated with a
scattering phase shift [55], and (ii) photoemission of molecule $\mathcal{A}$
occurs with no effective nuclear motion, i.e., the electron and nuclear
degrees of freedom, $x$ and $R$, are de facto uncoupled due to the relative
positions of the ground state and ionic curves [see Fig. 1(a)].

We now move on to the results obtained with the $\mathcal{B}$ and
$\mathcal{C}$ molecules, for which photoemission occurs along with sensible
nuclear dynamics.

##  IV Photoemission from molecules $\mathcal{B}$ and $\mathcal{C}$:
signatures of the vibronic couplings

The necessity to take the nuclear motion into consideration in near-threshold
photoemission from molecules $\mathcal{B}$ and $\mathcal{C}$ is a consequence
of the relative position shifts of their ground states and ionic BO curves
[see Fig. 1(b,c)]. We investigate how it affects the photoemission dynamics in
the present section by presenting and analyzing the orientation-resolved
photoemission yields and stereo delays in these two model molecules, and
addressing the relevance of fixed-nuclei simulations in this context.

###  IV.1 ‘Complete’ simulations

We used here the same pulse parameters as in the simulations with molecule
$\mathcal{A}$ (Sec. III.1).

####  IV.1.1 Signatures of electron-nuclei couplings

A striking signature of the electron-nuclei couplings in the photoemission
dynamics of $\mathcal{B}$ shows up in the delays inferred from the outgoing
flux. In contrast with the previous case, the stereo delays measured according
to Eqs. (10)–(11) strongly depend on the detection distance $r_{\text{\tiny
d}}$. This is illustrated in Table 1 with few representative cases. Not only
it evidences a rather dramatic dependency of $\Delta\tau$ with respect to
$r_{\text{\tiny d}}$, well above the numerical accuracy of the simulations,
but it turns out that $\Delta\tau$ monotonically diverges with increasing
$r_{\text{\tiny d}}$ (not shown). This tells us that, the delays computed as
such are irrelevant for an objective characterization of the ionization
dynamics.

Nevertheless their $r_{\text{\tiny d}}$-dependency bear a crucial information:
it suggests that the electron wave-packets travel on each side of the molecule
with slightly different average velocities. This is the signature of an
anisotropic momentum sharing between the nuclei and the active electron during
the concerted ionization/contraction of the molecular ion. It involves the
nuclear mass asymmetry since the lighter the nucleus, the more the
photoelectron is prone to share kinetic energy with it. The principle, which
applies to the average velocity, is sketched in Fig. 5. Assuming that only the
lighter nucleus (on the left) significantly moves, the photoelectron ends up
with a smaller velocity if it exits in the direction of motion of that nucleus
(towards the right) than in the other direction.

$\omega_{\text{\small xuv}}$ | $r_{\text{\tiny d}}$ | $\Delta\tau$
---|---|---
$32.6$ eV |  600 a.u. | 119 as
|  800 a.u. | 153 as
$34.1$ eV |  800 a.u. |  74 as
| 1000 a.u. |  89 as
Table 1:  Stereo delay $\Delta\tau$ computed according to Eq. (11) in molecule
$\mathcal{B}$ for an illustrative set of detection distances $r_{\text{\tiny
d}}$ and photon frequencies $\omega_{\text{\small xuv}}$.  Figure 5:  Sketch
of the asymmetric average momentum sharing between the photoelectron (empty
circle) and the nuclei (full circles) during the photoionization/early
contraction of molecule $\mathcal{B}$. For the sake of clarity, the velocity
asymmetries have been exagerated compared to the ones obtained in the actual
simulations. Figure 6:  Vibrationally resolved outgoing electron flux
$j_{v}(\theta_{\mbox{\tiny k}},r_{\text{\tiny d}},t)$ computed, in the fully
correlated simulations, at $r_{\text{\tiny d}}=800$ a.u. on the left
[$\theta_{\mbox{\tiny k}}=180\degree$, frame (a)] and right
[$\theta_{\mbox{\tiny k}}=0\degree$, frame (b)] sides of molecule
$\mathcal{B}$ ionized by $\omega_{\text{\small xuv}}=35.67$ eV pulse, as a
function of propagation time $t$, for the 5 first dominant channels. Each
curve is labelled according to its corresponding channel ($v=0-4$). The flux
are normalized to 1 at the overall maximum reached in the $v=2$ channel, on
the left side. The temporal envelop of the ionizing pulse is represented by
the blue filled curve.

Properly accounting for this momentum sharing hence requires giving up global
characterization and rather considering vibrationnally-resolved observables.
In the present context, they are derived from the channel-resolved wave-
packets

$\displaystyle\varphi_{v}(x,t)=\int\limits_{0}^{\infty}\,\chi_{v}(R)\Phi(x,R,t)\,{d}R$
(17)

in the vibronic TDSE simulations. In Eq. (17), $\chi_{v}(R)$ are the
vibrational eigenstates of the molecular ionic core hamiltonian
$H_{\text{\tiny N}}$ [see Eq. (1)]. The individual channel functions are
relevant as soon as they are uncoupled from each other, i.e., in the
asymptotic $x$ region where $V_{\text{\tiny N-e}}$ no longer depends
significantly on $R$ (typically beyond few 100 a.u.). Each channel is assigned
a specific ionization potential

$\displaystyle E_{\text{{I}}}(v)$ $\displaystyle=$
$\displaystyle\mathcal{E}^{(+)}_{v}-\mathcal{E}^{(0)}_{0},$ (18)

where $\mathcal{E}^{(+)}_{v}$ is the energy of the corresponding ionization
threshold (in the BO framework, that is the eigenvalue of $H_{\text{\tiny N}}$
associated with $\chi_{v}$) and $\mathcal{E}^{(0)}_{0}$ the vibronic ground
energy of the neutral molecule.

The key time-dependent observable becomes the vibrationally-resolved electron
flux

$\displaystyle j_{v}(\theta_{\mbox{\tiny k}},r_{\text{\tiny d}},t)$
$\displaystyle=$ $\displaystyle\cos\theta_{\mbox{\tiny k}}\times$
$\displaystyle\mathfrak{Im}\left\\{\varphi^{\star}_{v}(r_{\text{\tiny
d}}\cos\theta_{\mbox{\tiny k}},t)\varphi^{\prime}_{v}(r_{\text{\tiny
d}}\cos\theta_{\mbox{\tiny k}},t)\right\\}$

computed at a distance $r_{\text{\tiny d}}$ on each side of the molecule.
Vibrationnally-resolved anisotropic yields and delays can be computed out of
the flux $j_{v}(\theta_{\mbox{\tiny k}},r_{\text{\tiny d}},t)$ in a similar
fashion than in the integrated case, see Section III.

The flux computed at the distance $r_{\text{\tiny d}}=800$ a.u. on the left
(a) and right (b) sides of the $\mathcal{B}$ molecule submitted to a
$\omega_{\text{\small xuv}}=35.67$ eV pulse are shown in Fig. 6, for the first
few popupated channels. The flux maxima of the different channels are time-
shifted from each other, which is a manifestation of the energy conservation
law

$\displaystyle\varepsilon=\omega_{\text{\small xuv}}-E_{\text{{I}}}(v).$ (20)

The larger $E_{\text{{I}}}(v)$, the lower the photoelectron energy
$\varepsilon$ and the larger the time needed to reach the detector. As for
molecule $\mathcal{A}$, the asymmetry of molecule $\mathcal{B}$ results in
anisotropic ionization yields, visible when comparing the flux magnitudes on
the left and on the right. Similar results were obtained with molecule
$\mathcal{C}$ (not shown).

To get a better insight on the vibrational distributions upon ionization, we
display in Fig. 7, column i, the Franck-Condon (FC) factors for molecules
$\mathcal{B}$ (frame a) and $\mathcal{C}$ (frame b), defined as

$\displaystyle F(v)$ $\displaystyle=$
$\displaystyle\left|\int\limits_{0}^{\infty}\,\chi_{v}(R)\xi_{0}(R)\,{d}R\right|^{2},$
(21)

where $\xi_{0}(R)$ is the ground vibrational wave-function of the neutral
molecule treated in the BO framework. The FC factors peak at $v=1$ for both
model molecules, and extend significantly up to $v=4$ and $8$ for molecules
$\mathcal{B}$ and $\mathcal{C}$, respectively. They qualitatively reproduce
the actual $v$-resolved yields

$\displaystyle\mathcal{Y}(v;\theta_{\mbox{\tiny k}})$ $\displaystyle=$
$\displaystyle\int\,j_{v}(\theta_{\mbox{\tiny k}},r_{\text{\tiny
d}},t)\,{d}t,$ (22)

computed on the left (columns ii) and right (columns iii) sides of the
molecules ionized by the illustrative $\omega_{\text{\small xuv}}=35.67$ eV
pulse. For the sake of comparison, the displayed data are normalized such that
each set sums up to 1.

Figure 7:  Vibrational distribution of photoemission from molecules
$\mathcal{B}$ (a) and $\mathcal{C}$ (b) with a $\omega_{\text{\small
xuv}}=35.67$ eV pulse. (i): Franck-Condon factors $F(v)$ [Eq. (21)] ; (ii) and
(iii): normalized yields computed in the fully correlated simulations, resp.
$\mathcal{Y}(v;180\degree)$ and $\mathcal{Y}(v;0\degree)$ [Eq. (22)]; (iv) and
(v): normalized yields computed in the BO framework at the optimal
internuclear distances $R_{\text{\tiny opt}}(v)$, resp.
$\tilde{}\mathcal{Y}(v;180\degree)$ and $\tilde{}\mathcal{Y}(v;0\degree)$ [Eq.
(26)]. The displayed data are normalized such that each set sums up to 1, for
comparison purpose.  Figure 8:  Vibrationnally resolved anisotropic
photoemission from molecule $\mathcal{B}$ (left) and $\mathcal{C}$ (right).
All data are plotted as functions of the photoelectron energy $\varepsilon$.
Left/right yields ratio (top) computed in the fully correlated simulations and
in the BO framework, resp. $\mathcal{R}(v)$ (symbols) and
$\tilde{}\mathcal{R}(v)$ (lines); stereo delays (bottom) computed in the fully
correlated simulations and in the BO framework, resp. $\Delta\tau(v)$
(symbols) and $\Delta\tau_{\mbox{\tiny\sc w}}(v)$ (lines). In all frames, the
vibronic data are displayed for the first few significant vibrational
channels: $v=0$ (full circles), $1$ (empty circles), $2$ (full diamonds), $3$
(empty diamonds), $4$ (full triangles up), $5$ (empty triangles up), $6$ (full
triangles down), $7$ (empty triangles down) and $8$ (full squares). As
specified by the inset key, the displayed BO data were computed at a set of
$v$-dependent optimal distances [$R_{\text{\tiny opt}}(v)$] for molecule
$\mathcal{B}$, and of ‘vertical’ distances [$R_{\text{\tiny vrt}}(v)$] for
molecule $\mathcal{C}$ (see text and Tab. 2). The BO data computed at the
equilibrium distance ($R_{\text{\tiny eq}}$) are also shown for both
molecules.

A feature that cannot be included in the FC factors alone is obviously the
orientation dependency. In particular for molecule $\mathcal{B}$, the
vibrational distributions of the yields in Fig. 7(a) display slight yet
significant differences on the left and right emission sides. In spite of
being small, these discrepancies are sufficient to result in different average
photoelectron velocities, and to the above-mentioned $r_{\text{\tiny d}}$
dependance of the integrated left-right delay $\Delta\tau$. We emphasize that
the data in Fig. 7 are normalized to highlight the orientation-resolved
vibrational distributions, and therefore discards the vibrationally-resolved
magnitude of the ionization probability on each side of the molecules.

####  IV.1.2 Photoemission yields and delays

A finer insight on the $v$-resolved photoemission anisotropy is provided in
Fig. 8 which shows, for molecules $\mathcal{B}$ [frame (a)] and $\mathcal{C}$
[frame (b)], the yields ratio

$\displaystyle\mathcal{R}(v)$ $\displaystyle=$
$\displaystyle\frac{\mathcal{Y}(v;180\degree)}{\mathcal{Y}(v;0\degree)}$ (23)

obtained with several values of $\omega_{\text{\small xuv}}$ in the ‘complete’
simulations (symbols), as a function of the photoelectron energy
$\varepsilon$. Their values, comprised between $\sim 1.2$ and $\sim 1.7$,
follow for each $v$ the global trend observed with molecule $\mathcal{A}$, see
Fig. 3(a). However, they display a clear additional $v$-dependency for both
molecules $\mathcal{B}$ and $\mathcal{C}$.

We characterized the dynamics revealed in these simulations consistently with
the unresolved case, see Eq. (11). Here, the channel-resolved stereo delays
are defined as

$\displaystyle\Delta\tau(v)$ $\displaystyle=$
$\displaystyle\tau(v;180\degree)-\tau(v;0\degree)-2\frac{\langle
x\rangle_{0}}{\sqrt{2\varepsilon}}$ (24)

where the numerical TOF towards the virtual detector

$\displaystyle\tau(v;\theta_{\mbox{\tiny k}})$ $\displaystyle=$
$\displaystyle\frac{\int\limits\,t\times j_{v}(\theta_{\mbox{\tiny
k}},r_{\text{\tiny d}},t)\,{d}t}{\int\limits\,j_{v}(\theta_{\mbox{\tiny
k}},r_{\text{\tiny d}},t)\,{d}t},$ (25)

is now computed in each $v$-channel. In Eq. (24), one should keep in mind that
$\varepsilon$ implicitly depends on $v$ through Eq. (20), for a given
$\omega_{\text{\small xuv}}$. We have notably checked that the vibrationally
resolved stereo delays $\Delta\tau(v)$ do not depend on $r_{\text{\tiny d}}$,
in contrast with the $v$-integrated data commented before for molecule
$\mathcal{B}$ (Table 1). Indeed, for each vibrational channel, energy
conservation ensures a symmetric asymptotic electron velocity.

$v$
---
$E_{\text{{I}}}$ (eV)
$R_{\text{\tiny vrt}}$ (Å)
$R_{\text{\tiny opt}}$ (Å)

(a)
---
0 | 1 | 2 | 3 | 4
29.17 | 29.48 | 29.78 | 30.10 | 30.41
1.076 | 1.106 | 1.145 | 1.201 | 1.319
1.084 | 1.113 | 1.146 | 1.203 | 1.248

(b)
---
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
29.30 | 29.61 | 29.92 | 30.23 | 30.53 | 30.83 | 31.13 | 31.42 | 31.71
1.160 | 1.135 | 1.115 | 1.098 | 1.082 | 1.066 | 1.055 | 1.041 | 1.031
1.174 | 1.148 | 1.124 | 1.105 | 1.091 | 1.079 | 1.069 | 1.055 | 1.038

Table 2:  Channel-dependent ionization energy $E_{\text{{I}}}$ [Eq. (18)] and
representative internuclear distances $R_{\text{\tiny vrt}}$ and
$R_{\text{\tiny opt}}$ [see text and Eq. (27)] for molecules $\mathcal{B}$ (a)
and $\mathcal{C}$ (b) in the main vibrational ionization channels $v$.  Figure
9:  Vertical ionization potential $\tilde{}E_{\text{{I}}}(R)$ of molecules
$\mathcal{B}$ (a) and $\mathcal{C}$ (b) as a function of $R$ (blue full
curve), used to identify the internuclear distances $R_{\text{\tiny vrt}}(v)$
fulfilling Eq. (27) out of the $v$-dependent ionization potentials
$E_{\text{{I}}}(v)$.

The obtained $v$-dependent stereo delays are plotted as symbols in Fig. 8 for
molecules $\mathcal{B}$ [frame (c)] and $\mathcal{C}$ [frame (d)]. We first
note that they globally behave like the stereo delays reported for molecule
$\mathcal{A}$ in Fig. 3(b), with a comparable order of magnitude which decays
smoothly towards 0 when the photoelectron energy $\varepsilon$ increases.
Besides, a marked $v$-dependency is observed in molecule $\mathcal{B}$,
reminiscent of the behavior of the corresponding yields ratios [frame (a)]. In
molecule $\mathcal{C}$ however, the values of $\Delta\tau(v)$ appear to depend
mostly on $\varepsilon$ regardless of the vibrationnal channel, since all of
them follow a common spectral evolution. This is consistent with the
normalized vibrational distributions displayed in Fig. 7(b), which are almost
identical in the two directions, as pointed out before.

###  IV.2 Simulations at fixed internuclear distances

In this section, we investigate the possibilities to retrieve the orientation-
dependent data, including their $v$-dependencies, by analyzing electronic
continuum wave-functions obtained in simulations with fixed nuclei.

The yields ratio and delays computed for both molecules in the BO framework at
$R_{\text{\tiny eq}}$ according to Eq. (14) are displayed as a full black
lines in Fig. 8. They lie near the data obtained for the dominant channels in
the full TDSE simulations (between $v=1$ and $2$), and very close to the
equivalent results obtained with $\mathcal{A}$ [see Fig. 3(a)]. Nevertheless,
and quite obviously, simulations performed at a single fixed internuclar
distance cannot reproduce the non-trivial $v$-dependency observed in the full
simulations for the yields ratio and stereo delays of molecule $\mathcal{B}$,
and for the yields ratio of molecule $\mathcal{C}$.

Below, we use the simulations performed on molecule $\mathcal{B}$ to identify
and interprete $v$-dependent geometries that allow reproducing the results of
the vibronic simulations with the fixed nuclei approach. We then assess our
interpretation by applying it to fixed nuclei simulations on molecule
$\mathcal{C}$.

####  IV.2.1 Optimal conformations for molecule $\mathcal{B}$

By scanning through the support of the initial vibrational state $\xi_{0}(R)$
of molecule $\mathcal{B}$ ($R\sim 1.0-1.3$ Å), we identified empirically a set
of channel-specific optimal distances $R_{\text{\tiny opt}}(v)$ for which both
the yields ratio and the stereo delays in the BO approach reproduce the ones
of the complete vibronic simulations. The values of $R_{\text{\tiny opt}}(v)$
obtained are reported in Tab. 2(a). The orientation- and channel-resolved
yields, now computed in the BO framework at a given optimal conformation for
each channel,

$\displaystyle\tilde{}\mathcal{Y}(v;\theta_{\mbox{\tiny k}})$ $\displaystyle=$
$\displaystyle|A(R_{\text{\tiny opt}}(v);\theta_{\mbox{\tiny k}})|^{2}\times
F(v),$ (26)

are displayed for the left and right sides in columns iv and v of Fig. 7(a),
respectively. They reproduce very well the asymmetric $v$-dependency of the
actual yields $\mathcal{Y}(v;\theta_{\mbox{\tiny k}})$ (columns ii and iii).
The corresponding yields ratio $\tilde{}\mathcal{R}(v)$ and stereo delays
$\Delta\tau_{\mbox{\tiny\sc w}}(v)$, extracted from the SCWF computed at the
optimal conformation for each $v$ are displayed as full lines in frames (a)
and (c) of Fig. 8. They are in excellent agreement with the $v$-dependent data
obtained in the fully correlated simulations for both molecules, apart from
the $v=4$, as will be discussed below.

####  IV.2.2 Physical interpretation

The optimal molecular conformations can be interpreted by looking at the
$R$-dependent ionization potential $\tilde{}E_{\text{{I}}}(R)$ [Eq. (4)],
which for molecule $\mathcal{B}$ increases monotonically with $R$ within the
support of $\xi_{0}(R)$, see Fig. 9(a). On the same figure, we indicate the
internuclear distances $R_{\text{\tiny vrt}}(v)$ at which
$\tilde{}E_{\text{{I}}}$ matches the $v$-dependent ionization potential [Eq.
(18)],

$\displaystyle\tilde{}E_{\text{{I}}}[R_{\text{\tiny vrt}}(v)]$
$\displaystyle=$ $\displaystyle E_{\text{{I}}}(v).$ (27)

These “vertical” internuclear distances are reported in Table 2(a), for
comparison with the optimal distances found empirically. It turns out that
$R_{\text{\tiny opt}}$ matches $R_{\text{\tiny vrt}}$ within $\sim 1\%$ in all
the considered channels, except $v=4$ ($\sim 5\%$).

We repeated the same procedure for molecule $\mathcal{C}$. The optimal
internuclear distances, $R_{\text{\tiny opt}}(v)$, that we found empirically
are reported in Table 2(b), together with the ‘vertical’ ones, $R_{\text{\tiny
vrt}}(v)$, extracted from the ionization potential displayed in Fig. 9(b). It
is here also a monotonic function of $R$, with opposite variations than for
molecule $\mathcal{B}$ (as mentioned earlier, molecule $\mathcal{B}$ expands
upon ionization, while molecule $\mathcal{C}$ contracts upon ionization). The
BO data displayed as lines in Fig. 8(b) and (d) for molecule $\mathcal{C}$
were directly obtained with $R_{\text{\tiny vrt}}(v)$, for each channel. The
differences with the data obtained using $R_{\text{\tiny opt}}(v)$, not shown,
would be hardly discernible on that figure. The agreement between the
‘complete’ simulations and the fixed-nuclei approach is here excellent in all
considered channels – including the ionization ratios [frame (b)] which follow
spectral trends that significantly depend on $v$.

These results provide a simple physical criterion for selecting a priori a set
of optimal conformations which allow reproducing, in a fixed-nuclei framework,
the details of $v$-dependent molecular photoionization dynamics. It is
nevertheless predictable that these specific distances can be representative
of the $v$-dependent photoemission dynamics only when
$\tilde{}E_{\text{{I}}}(R)$ varies monotonically with $R$. In particular for
molecule $\mathcal{B}$ the $\tilde{}E_{\text{{I}}}(R)$ variations are less
pronounced as $R$ grows towards the $R_{\text{\tiny vrt}}$ of the highest
channels, within the studied range. Consistently, the data computed in the BO
framework for molecule $\mathcal{B}$ turned out to be less sensitive to small
$R$ variations beyond $R_{\text{\tiny opt}}(v=3)$ – to the extent that we
could not find an optimal distance for the $v=4$ channel with the same
accuracy as for other channels. This is particularly visible on the yields
ratio [Fig. 8(a)], where the BO data never reach the ‘exact’ ones obtained for
the highest ($v=4$) channel, the displayed BO data corresponding to
$R_{\text{\tiny opt}}(v=4)$ being the closest one could get. This however
concerns a minor channel [see Fig. 7(a)]. For molecule $\mathcal{C}$, the
narrower dispersions of the data can be directly related to the sharp
monotonic decay of $\tilde{}E_{\text{{I}}}$ when $R$ is increased, which
implies a narrow dispersion of $R_{\text{\tiny vrt}}(v)$.

###  IV.3 Illustrative applications of the optimal conformations

In this last section, we address the relevance of the optimal conformations
beyond the context in which they were identified. We investigate their
relevance first in simulations of interferometric RABBIT measurements, and
then to include electron-ion coherences during broadband 1-photon ionization

The RABBIT scheme, together with the attosecond streaking approach, provide
indirect ways to access photoemission dynamics222 See [27] and references
therein for discussions dedicated to the links between RABBIT measurements and
fundamental photoemission delays., the direct time domain approaches used in
our simulations having no experimental equivalent with attosecond resolution
(eg magnetic bottles have typical deadtimes of few 10 ns [56] and time-to-
digit convertor resolutions in the 100 ps range [57]).

These are interferometric techniques, where the time domain information is
extracted from the spectral variations of measured phases. Coherence is thus
an essential issue in the design and exploitation of these approaches, the
purpose of which is to reveal the fundamental dynamics of essentially quantum
processes.

####  IV.3.1 Intra-channel coherences: RABBIT interferometry

Figure 10:  Photoelectron spectra recorded in RABBIT simulations, on the left
(a) and right (b) sides of molecule $\mathcal{B}$. The spectral range is
focused on sideband 24, and each peak is associated with a specific ionization
channel $v=0-4$ (see peak labels). The different spectra were obtained with
different values of the pump-probe delay $\tau_{\text{\small xuv-ir}}$ (see
inset keys, where $T_{0}=2\times\pi/\omega_{0}$ is the laser period).  Figure
11:  Vibrationnally resolved anisotropic RABBIT simulations in molecule
$\mathcal{A}$ (left), $\mathcal{B}$ (center) and $\mathcal{C}$ (right). Data
derived from the analysis of sidebands 22 to 28 (right to left) plotted as
functions of the photoelectron energy $\varepsilon$. Left-right yields ratio
(top) computed in the fully correlated simulations and in the BO framework,
resp. $\mathcal{R}_{\text{\tiny rab}}(v)$ (symbols) and
$\tilde{\mathcal{R}}_{\text{\tiny rab}}(v)$ (crosses+guidelines); stereo
molecular delays (bottom) computed in the fully correlated simulations and in
the BO framework, resp. $\Delta\tau_{\mbox{\tiny mol}}(v)$ (symbols) and
$\Delta\tilde{\tau}_{\mbox{\tiny mol}}(v)$ (crosses+guideline). In each frame,
the data are displayed for the first few significant vibrational channels:
$v=0$ (full circles), $1$ (empty circles), $2$ (full diamonds), $3$ (empty
diamonds), $4$ (full triangles up), $5$ (empty triangles up), $6$ (full
triangles down), $7$ (empty triangles down) and $8$ (full squares).
Consistently with the results shown in Fig. 8, the displayed BO data were
obtained with the set of $v$-dependent optimal distances [$R_{\text{\tiny
opt}}(v)$] identified in 1-photon simulations for molecule $\mathcal{B}$, and
of ‘vertical’ distances [$R_{\text{\tiny vrt}}(v)$] for molecule $\mathcal{C}$
(see text and Tab. 2). They were also computed at the equilibrium distance
($R_{\text{\tiny eq}}$) for all molecules. The data displayed in frame (f) are
all multiplied by 4 for a better readability.

Following a standard 800-nm RABBIT scheme [8, 14], we simulated photoemission
from the three model molecules in presence of the fundamental IR field and a
comb of its odd harmonics $\mbox{HA}_{q}$ in the XUV domain (orders $q$ from
$21$ to $29$), with an adjustable pump-probe delay $\tau_{\text{\small xuv-
ir}}$. We computed vibrationally- and orientation-resolved photoelectron
spectra

$\displaystyle\sigma(v;\varepsilon,\theta_{\mbox{\tiny k}})$ $\displaystyle=$
$\displaystyle|a(v;\varepsilon,\theta_{\mbox{\tiny k}})|^{2}$ (28)

out of the final amplitudes corresponding to the ionized molecule,
$a(v;\varepsilon,\theta_{\mbox{\tiny k}})$. The latter were obtained in the
time-dependent simulations through a position-to-momentum Fourier transform of
the channel wave packets $\varphi_{v}(x,t)$ accumulated in the asymptotic
region at each time $t$ of the propagation (see Appendix C of [58]).

As expected, we obtained sideband (SB) peaks (orders $22$ to $28$) resulting
from 2-photon ${\text{{XUV}}}\pm{\text{{IR}}}$ transitions, the intensities of
which oscillate when $\tau_{\text{\small xuv-ir}}$ is tuned [59]. We used long
enough pulses (80 fs for the XUV and for the IR) to resolve the vibrational
channels in each sideband. This is illustrated in Fig. 10, which shows the
vibrationally-structured, orientation and delay-dependent $\mbox{SB}_{24}$
obtained with molecule $\mathcal{B}$. Similarly to the 1-photon case, the
relative peak intensities evidence a significantly larger photoemission
probability towards the left (a) than towards the right (b). Note however that
the orientation-dependency of the SB oscillations are too subtle to be
resolved visually in this figure. This is consistent with the ultrashort
timescale of the simulated non-resonant photoemission processes [27].

We thus performed a RABBIT analysis consisting in fitting the generic function
given in Eq. 8 of [27] to the $\tau_{\text{\small xuv-ir}}$ evolution of each
sideband peak associated with a given vibrational channel $v$, and a given
direction $\theta_{\mbox{\tiny k}}$. The fitting procedure gives access to the
orientation and channel-resolved phases of the peak oscillations
$\vartheta(v;\theta_{\mbox{\tiny k}})$, as well as to the $\tau_{\text{\small
xuv-ir}}$-averaged photoelectron yields $P(v;\theta_{\mbox{\tiny k}})$. Since
the XUV components in our simulations were all synchronized (i.e., they carry
no attochirp), $\vartheta(v;\theta_{\mbox{\tiny k}})$ directly corresponds to
the so-called “molecular phase”, see [27] and references therein.

We performed simulations both in the fully correlated approach, and in the BO
framework. The main results are shown in Fig. 11, for $\mbox{SB}_{22}$ to
$\mbox{SB}_{28}$ in molecules $\mathcal{A}$ (left), $\mathcal{B}$ (center) and
$\mathcal{C}$ (right). The disconnected symbols in the upper and lower frames
respectively correspond to the ratio of the $\tau_{\text{\small xuv-
ir}}$-averaged yields,

$\displaystyle\mathcal{R}_{\text{\tiny rab}}(v)$ $\displaystyle=$
$\displaystyle\frac{P(v;180\degree)}{P(v;0\degree)}$ (29)

and the “stereo molecular delay” [12, 19]

$\displaystyle\Delta\tau_{\mbox{\tiny mol}}(v)$ $\displaystyle=$
$\displaystyle\frac{\vartheta(v;180\degree)}{2\times\omega_{0}}-\frac{\vartheta(v;0\degree)}{2\times\omega_{0}}$
(30)

obtained in the fully correlated approach for the main $v$ channels. As in the
1-photon case, these quantities follow standard general trends which can be
summarized by looking at the single-channel results of molecule $\mathcal{A}$.
Its yields ratio $\mathcal{R}_{\text{\tiny rab}}(0)$ decays slowly, from $\sim
1.4$ down to $\sim 1.3$ over the covered energy range, which is reminiscent of
the 1-photon counterpart shown in Fig. 3(a). The stereo molecular delay
$\Delta\tau_{\mbox{\tiny mol}}(0)$ decays in magnitude, from $-10$ as at
$\mbox{SB}_{22}$ down to near $0$ as at $\mbox{SB}_{28}$. The relationship
between the latter and the 1-photon ionization delay is the subject of the
companion paper [27] and will not be further discussed here. Beyond the trends
commented above, the results for molecule $\mathcal{B}$ and $\mathcal{C}$
displays clear $v$-dependencies, both in the yields ratios and in the stereo
molecular delays, that are somehow more pronounced than in the 1-photon case.

The crosses with linear guidelines correspond to the equivalent quantities,
resp. $\tilde{\mathcal{R}}_{\text{\tiny rab}}(v)$ and
$\Delta\tilde{\tau}_{\mbox{\tiny mol}}(v)$, obtained in time-dependent
simulations at fixed internuclear distances. For molecule $\mathcal{A}$, we
used again the equilibrium distance $R_{\text{\tiny eq}}$, which
unsurprislingly perfectly reproduces the results of the ‘complete’
simulations. Note that these fixed nuclei data appear in Fig. 6 of [27]. For
molecule $\mathcal{B}$, we additionally show the BO results obtained with the
$v$-dependent optimal distances $R_{\text{\tiny opt}}$ identified in the
1-photon simulations (see Table 2), while for molecule $\mathcal{C}$ we used
the vertical internuclear distances $R_{\text{\tiny vrt}}(v)$. The data
obtained at $R_{\text{\tiny eq}}$ both with molecules $\mathcal{B}$ and
$\mathcal{C}$ are similar to the ones obtained with molecule $\mathcal{A}$.
Regarding the results provided at optimal conformations, one clearly sees that
the BO framework fails to reproduce the yields ratio for molecule
$\mathcal{B}$ in the minority $v=4$ channel [frame (b)], already discussed in
the 1-photon case. Apart from this, the data obtained in the BO framework,
including the stereo molecular delays in all channels for both molecules are
in excellent agreement with the ones obtained in the ‘complete’ simulations.
This illustrates the capacities of BO approaches with appropriate internuclear
distances to simulate vibrationnally-resolved RABBIT measurements with high
fidelity.

####  IV.3.2 Inter-channel coherences: broadband photoemission

Figure 12:  Photoelectron spectrum recorded on the left (a) and right (b)
sides of molecule $\mathcal{B}$, upon single-photon ionization with a 40 fs
light pulse of $37.23$ eV central frequency ($\mbox{HA}_{24}$ of a 800 nm
field). Thin gray lines: $v$-resolved spectra ($v=0-4$, see peak labels);
thick coloured lines: $v$-integrated spectra.  Figure 13:  Photoionization of
molecule $\mathcal{B}$ (top) and $\mathcal{C}$ (bottom) with a broadband XUV
pulse (same simulations as Fig. 12). Modulus of the final ion’s reduced
density matrix $\rho_{\text{\tiny ion}}(v,v^{\prime})$ [Eq.
(LABEL:eqn:rhoion)] in the complete simulations (left) and in simulations at
fixed conformations $R_{\text{\tiny vrt}}(v)$ (right). The values of the
matrix elements’ modulii are provided in the lower triangles (the powers of
ten are given in superscript). The matrices are displayed for the main open
channels. Each one is normalized to have a trace equal to 1, consistently with
the data displayed in Fig. 7. The value of the purity $p$ [Eq. (32)] is
indicated at the top of each matrix.

As a last case, we now highlight the capacities of fixed-nuclei approaches to
keep track of inter-channel coherences, when reconstructing the complete
vibronic wave-packet out of a set of simulations performed at the
$v$-dependent optimal conformations. To this end, we simulated the
photoionization of molecules $\mathcal{B}$ and $\mathcal{C}$ with a broad XUV
pulse of central frequency $\omega_{\text{\small xuv}}=37.23$ eV
($\mbox{HA}_{24}$ of a $800$ nm laser) and a duration of $8$ fs FWHM (3
fundamental IR cycles), in the 1-photon perturbative regime. The bandwidth of
this pulse overlaps few vibrationnal channels in the photoelectron spectra, as
shown in Fig. 12 for molecule $\mathcal{B}$. One therefore partially looses
the vibrationnal resolution when looking at the $v$-integrated spectra, in
contrast to the results obtained with narrower pulses, see for instance Fig.
10.

From the final channel-selective amplitudes in the continuum
$a(v;\varepsilon,\theta_{\mbox{\tiny k}})$, we computed the reduced density
matrix (RDM) of the ion in the final state as

$\displaystyle\rho_{\text{\tiny ion}}(v,v^{\prime})$ $\displaystyle=$
$\displaystyle\sum\limits_{\theta_{\mbox{\tiny
k}}=0\degree,180\degree}\int\limits_{\Delta}\,d\varepsilon\,\left[a(v;\varepsilon,\theta_{\mbox{\tiny
k}})\right]^{\star}a(v^{\prime};\varepsilon,\theta_{\mbox{\tiny k}})$

where $[\ ]^{\star}$ denotes the complex conjugate and the spectral
integration range $\Delta$ is restricted to the overall support of the
photoelectron spectrum. The modulus of the RDM computed in the complete
simulations for molecules $\mathcal{B}$ and $\mathcal{C}$ in the main
vibrationnal channels are respectively shown in frames (a) and (c) of Fig. 13.
For both molecules, we observe non vanishing off-diagonal elements which are
signatures of coherences between overlapping channels ($|v^{\prime}-v|\lesssim
3$).

In the fixed nuclei simulations, we reconstructed the final RDM of the ion out
of the amplitudes $\tilde{}a(R_{\text{\tiny
opt}}(v);\varepsilon,\theta_{\mbox{\tiny k}})$ obtained in time-dependent
simulations using the $v$-dependent vertical internuclear distances
$R_{\text{\tiny vrt}}(v)$ for both molecules. The results are shown in frames
(b) and (d) of Fig. 13 for molecules $\mathcal{B}$ and $\mathcal{C}$
respectively. The overall agreement with the complete simulations is excellent
both for the populations (diagonal elements) and the coherences. To further
evaluate the quality of these reconstructed vibrational RDM, we computed in
each simulation the purity of the final state

$\displaystyle p=\frac{\mbox{tr}(\rho_{\text{\tiny
ion}}^{2})}{\mbox{tr}(\rho_{\text{\tiny ion}})^{2}},$ (32)

where $\mbox{tr}()$ denotes the trace application. The purity values are
indicated in Fig. 13 at the top of each matrix333We verified that the
normalized matrices and the purities remain practically unchanged when looking
at the $\theta$-dependent RDM obtained as in Eq. (LABEL:eqn:rhoion) but
without angular integration (results not shown).. The purity in fixed-nuclei
results accurately match the ‘exact’ ones, within less than $1\%$, for both
molecules. It is worth noting that, when using the $R_{\text{\tiny opt}}(v)$
conformations rather than the $R_{\text{\tiny vrt}}(v)$ ones, the fixed-nuclei
simulations agree with the ‘exact’ ones with a $\sim 10\%$ accuracy only
(results not shown). On the one hand, this underlines the sensitivity of the
computed purity with respect to the conformations used in the fixed nuclei
approach. On the other hand, it emphasizes the remarkable efficiency of the
combined vertical conformations to reproduce the exact interchannel coherent
dynamics.

##  V Summary and conclusion

We studied numerically how nuclear motion affects the dynamics of orientation-
resolved photoemission in asymmetric diatomic model molecules presenting
minimal vibronic couplings. We considered near-threshold photoemission in
absence of any resonance, where the so-called stereo Wigner delays amount to
few tens of attoseconds or less.

We have shown that the intrinsic molecular asymmetry results in an anisotropic
electron-ion momentum sharing which, as slight as it may be, may prevent us
from assigning unambiguously a stereo Wigner delay to the channel-averaged
photoemission process. Indeed, a small asymmetry in the average photoelectron
energy leads to stereo delay values that diverge when the (virtual) detection
distance increases. This is circumvented by considering vibrationnally-
resolved photoemission. However, on the theory side, comprehensive time-
dependent vibronic simulations of molecules interacting with external fields
are restricted to smaller molecules such as H2 with a limited range of
physical and numerical parameters, or to simplified low dimensional model
molecules such as the ones used in the present work.

Therefore, we investigated ways of retrieving the vibrationnally-resolved
photoemission dynamics revealed by the ‘complete’ vibronic simulations, out of
more standard and broadly applicable time-independent approaches with fixed
nuclei. We found empirically that each vibrationnal channel could be assigned
an effective internuclear distance that reproduces the channel-resolved
orientation-dependent photoemission yields and delays with a good accuracy.
Furthermore, we identified a physical criterion, relying on the molecule’s
ionization energies, that allows selecting a priori the $v$-dependent
effective internuclear distances: It corresponds to the distance for which the
vertical ionization potential matches the exact channel-dependent ionization
potential. Identifying such unique effective molecular conformations is
expected to work efficiently as long as the vertical ionization potential
varies significantly when the molecular conformation is changed. Retrieving
photoemission dynamics with vibrational resolution out of fixed nuclei
simulation is also expected to work as long as photoemission takes place with
little vibronic correlation, typically in smooth continua with sufficiently
separated ionic vibrational levels.

Eventually, we assessed the relevence of the fixed-nuclei approach beyond the
context in which we identified the effective internuclear distances. We first
showed that it could be used to accurately simulate anisotropic
vibrationnally-resolved RABBIT interferometry, both in terms of phase and
amplitudes. Then, we investigated the fixed nuclei approach capacities to
account for interchannel coherences in broadband photoionization of our model
molecules, where the different channels significantly overlap. We showed that
the fixed-nuclei approach could satisfactorily reproduce both the ion’s
reduced density matrix and purity in the final state. This approach is thus of
particular interest to model attosecond resolved photoemission dynamics in
benchmark molecules without the need to resort to fully-fledged vibronic
approaches. It could be applied to simulate attosecond time-resolved
interferometry, highly non-linear processes such as strong field ionization
[60] or molecular high-order harmonic generation [46, 61, 62, 63, 64], and to
investigate ultrafast decoherence processes in molecule [35, 38, 44] of
crucial importance in attochemistry.

###### Acknowledgements.

This research received the financial support of the French National Research
Agency through Grants No. ANR-15-CE30-0001-CIMBAAD and ANR-20-CE30-0007-DECAP.

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# AdaBridge: Dynamic Data and Computation Reuse for Efficient Multi-task DNN
Co-evolution in Edge Systems

Lehao Wang1, Zhiwen Yu1,2, Sicong Liu1, Chenshu Wu3, Xiangrui Xu1, Bin Guo1

1Northwestern Polytechnical University, 2Harbin Engineering University, 3The
University of Hong Kong

###### Abstract

Running multi-task DNNs on mobiles is an emerging trend for various
applications like autonomous driving and mobile NLP. Mobile DNNs are often
compressed to fit the limited resources and thus suffer from degraded accuracy
and generalizability due to data drift. DNN evolution, _e.g._ , continuous
learning and domain adaptation, has been demonstrated effective in overcoming
these issues, mostly for single-task DNN, leaving multi-task DNN evolution an
important yet open challenge. To fill up this gap, we propose AdaBridge, which
exploits computational redundancies in multi-task DNNs as a unique opportunity
for dynamic data and computation reuse, thereby improving training efficacy
and resource efficiency among asynchronous multi-task co-evolution in edge
systems. Experimental evaluation shows that AdaBridge achieves 11% average
accuracy gain upon individual evolution baselines.

##  1 Introduction

Deploying multi-task DNNs on diverse mobile devices enhances the efficient
handling of various user interactions, resulting in a seamless and intelligent
experience. For instance, smartphones or tablets can engage with a virtual
assistant offering a range of services. However, due to limited memory and
computation resources, mobile DNNs are typically compressed, impacting their
generalization and making them susceptible to data drift [1], leading to a
drop in agnostic accuracy. To address this, several approaches for DNN
evolution have been explored, including continuous learning, online learning,
and source-dependent/free domain adaptation with cloud or edge assistance.
Edge-assisted continuous DNN evolution refines the mobile DNN with real-time
sensor data on the mobile client, under the guidance of a global model at the
edge server. This approach yields benefits such as increased accuracy,
additional resources, and reduced latency.

However, either individual or multi-task DNN evolution methods struggle to
balance evolution accuracy and efficiency in resource-constrained edge
systems. Applying them to multi-task cases poses the following challenges:

  * •

Each mobile client relying solely on local sensor data can impact evolution
generalization, especially when the data doesn’t satisfy the assumption of
independent and identically distributed (IID) or is imbalanced. Despite local
DNN, our key insight is that the sensor data from each mobile client has the
potential to enhance the evolution accuracy of DNNs on other clients. However,
uploading all client sensor data introduces transmission overhead and
complicates edge-evolution training. Moreover, resampling suitable sensor data
across multiple mobile models, especially in the absence of DNNs or data
distribution of other tasks, remains a significant challenge.

  * •

It is challenging to achieve efficient co-evolution of multiple asynchronous
tasks with limited edge memory and computing resources. We note computational
similarities and redundancies in retraining models from different clients,
serving as both burdens and opportunities for data and computation reuse. The
dynamic nature of asynchronous multi-task evolution requests makes these
redundancies and opportunities constantly changing. Also, balancing the
urgency and resource demands of multiple tasks complicates achieving average
high accuracy, low-cost, and flexible co-evolution of them.

These challenges motivate the design of AdaBridge. As far as we know, it is
the first work to adaptively bridge dynamic data and computation reuse across
algorithm and system co-designed levels. It can simultaneously improve
training efficacy and resource efficiency among asynchronous multi-task co-
evolution in edge systems.

##  2 AdaBridge Design

As shown in Figure 1, Core components of AdaBridge include a reuse-friendly
mobile sensor data resampling module and an asynchronous multi-task retraining
computation scheduling module. We present the detailed module design as
follows.

Figure 1: Illustration of AdaBridge’s system workflow.

Reuse-friendly mobile sensor data resampling. It is non-trivial for mobile
ends to dynamically resample locally sensed data useful for the evolution of
not only themselves but also other client DNNs. The reasons are two-fold: i)
prior research faces difficulties in rapidly evaluating the explicit data
contribution for accuracy gain of local DNNs with new, unlabeled sensor data
before actual training execution. This problem is more complex when predicting
implicit data contribution for accuracy gains for DNN evolution on other
clients that are locally inaccessible. Achieving accurate and real-time
profiling is challenging. ii) in the absence of distribution of sensor data
from other clients, determining the implicit complementarity between local
data and other clients’ data is intricate. Therefore, mobile clients first
conduct data shift type analysis based on runtime accuracy profiling for
multi-task DNNs, and then generate the $\mathrm{X}\to\mathrm{Y}$ mapping
function of other DNNs for data distribution comparison and resampling.

(a)

(b)

Figure 2: Comparison of accuracy fluctuation.

Asynchronous multi-task retraining computation scheduling. Upon receiving data
from mobile devices, the edge server faces challenges in improving data reuse
for multitask training, computation scheduling, and resource allocation. We
propose the following design: i) Plug-and-Play Asynchronous Multi-task
Computation Reusing: To achieve data reuse and accommodate flexible model
retraining fusion and separation, we introduce an adapter-based multi-task
learning framework. It asynchronously reuses learned features from different
subtasks and incorporates dynamic switching and incremental parameter updating
to optimize the limited representation capacity of compressed mobile DNNs. ii)
Memory I/O Cost-aware Underlying Computation Reuse: To further enhance I/O
efficiency in computation reuse, AdaBridge reorganizes data before training.
It reuses nearby features with high similarity, reducing the frequency of
reads and writes to high-level memory and decreasing computational loads.

##  3 Experiment

We have conducted a preliminary experiment in this section.

Experimental settings. We adopt two vehicles as mobile clients, _i.e._ ,
mobile unmanned cars (A), and unmanned aerial vehicles (UAV) (B), and deploy
compressed Faster-RCNN with ResNet50 on them, respectively. Two NVIDIA GeForce
RTX3080 GPUs with 10GB memory serve as the edge server. And they capture live
video while riding on open roads.

Deployment and results. We compare the evolution performance of independent
evolution with local data and AdaBridge’s co-evolution with data reuse.
Figure. 2 shows the results. AdaBridge performs better when facing data drift
with 11% improvement in lowest accuracy. This is because the co-evolved DNN
has better generalization ability. The co-evolved DNN outputted by AdaBridge
is stable in accuracy, contributing to the high flexibility and separability.

##  4 Conclusion

This paper presents AdaBridge, an end-to-end multi-task DNN co-evolution
system designed for resource-constrained edge systems. By establishing a
flexible adaptive reuse bridge between multiple tasks, the system efficiently
handles the continuous evolution requests of asynchronous mobile DNNs with
limited resources, aiming to overcome data drift.

## References

  * [1] Romil Bhardwaj, Zhengxu Xia, Ganesh Anantha- narayanan, Junchen Jiang, Yuanchao Shu, Nikolaos Karianakis, Kevin Hsieh, Paramvir Bahl, and Ion Stoica. Ekya: Continuous learning of video analytics models on edge compute servers. In Proceedings of NSDI, pages 119–135, 2022.