metadata

11institutetext:  Centre for Theoretical Atomic, Molecular, and Optical
Physics, Queen’s University Belfast, Belfast BT7 1NN, United Kingdom 11email:
<EMAIL_ADDRESS>22institutetext: Max Born Institute for Nonlinear
Optics and Short Pulse Spectroscopy, Max-Born-Straße 2A, Berlin 12489, Germany
33institutetext: Department of Physics, Imperial College London, South
Kensington Campus, London SW7 2AZ, United Kingdom 44institutetext: PASTEUR,
Département de chimie, Ecole Normale Supérieure, PSL University, Sorbonne
Université, CNRS, 75005 Paris, France 55institutetext: Institut de Ciencies
Fotoniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels (Barcelona), Spain 66institutetext: Department of Physics and
Astronomy, University College London, Gower Street, London WC1E 6BT, United
Kingdom

# Dialogue on analytical and ab initio methods in attoscience

Gregory S. J. Armstrong  11 Margarita A. Khokhlova  2233 Marie Labeye 44
Andrew S. Maxwell  5566 Emilio Pisanty  2255 Marco Ruberti  33

###### Abstract

The perceived dichotomy between analytical and _ab initio_ approaches to
theory in attosecond science is often seen as a source of tension and
misconceptions. This Topical Review compiles the discussions held during a
round-table panel at the ‘Quantum Battles in Attoscience’ cecam virtual
workshop, to explore the sources of tension and attempt to dispel them. We
survey the main theoretical tools of attoscience—covering both analytical and
numerical methods—and we examine common misconceptions, including the
relationship between _ab initio_ approaches and the broader numerical methods,
as well as the role of numerical methods in ‘analytical’ techniques. We also
evaluate the relative advantages and disadvantages of analytical as well as
numerical and _ab initio_ methods, together with their role in scientific
discovery, told through the case studies of two representative attosecond
processes: non-sequential double ionisation and resonant high-harmonic
generation. We present the discussion in the form of a dialogue between two
hypothetical theoreticians, a numericist and an analytician, who introduce and
challenge the broader opinions expressed in the attoscience community.
Accepted Manuscript for Eur. Phys. J. D 75, 209 (2021), available as
arXiv:2101.09335 under CC BY.

###### pacs:

31.15.A-Ab initio calculations (electronic structure of atoms and molecules)
and 32.80.WrMultiphoton ionization and excitation and 42.50.HzStrong-field
excitation of optical transitions in quantum systems

## Introduction

Modern developments in laser technologies have kick-started the attosecond
revolution, which formed the field of attoscience, dealing with dynamics on
the attosecond (${10}^{-18}\text{\,}\mathrm{s}$) timescale brabec2000 ;
krausz2009 ; calegari2016 . Attosecond science was born with the study of
above-threshold ionisation (ATI) and high-order harmonic generation (HHG)
driven by strong laser pulses. As it has matured over the past three decades,
attoscience has given us access to phenomena which were previously thought to
be inaccessible—including the motion of valence electrons in atoms
Goulielmakis2010Aug , charge oscillations in molecules Calegari2014 , as well
as the direct observation of the electric-field oscillations of a laser pulse
Goulielmakis2004direct —and it has also spurred advances in ultrafast pulse
generation which have opened a completely new window into the dynamics of
matter.

The meteoric progress of attoscience has been fuelled, on the one hand, by
formidable experimental efforts, and, on the other hand, it has been supported
by a matching leap in our theoretical capabilities. These theoretical advances
have come in a wide variety, forming two opposing families of analytical and
numerical approaches. While these two families generally work together, the
dichotomy between analytical and numerical methods is sometimes perceived as a
source of tension within the attoscience community.

In this paper we present an exploration of this dichotomy, which collects the
arguments presented in the panel discussion ‘Quantum Battle 3 –- Numerical vs
Analytical Methods’ held during the online conference ‘Quantum Battles in
Attoscience’ battle . Our main purpose is to resolve the tension caused by
this dichotomy, by identifying the critical tension points, developing the
different viewpoints involved, and finding a common ground between them.

This process forms a natural dialogue between the analytical and numerical
perspectives. We delegate this dialogue to two hypothetical ‘combatants’— [1]

  * Analycia Hi, I’m Analycia Formuloff, and I am an attoscience theorist working with analytical approaches. Numerio Hello, my name is Numerio Codeman, and I’m a computational scientist working on _ab initio_ methods.

—who will voice the different views expressed during the panel discussion.

We follow the dialogue between Analycia and Numerio through three main
questions. First, in Section 1, we explore the scope and nature of analytical
and numerical methods, including the interchangeability of the terms
‘numerical’ and ‘ _ab initio_ ’. We then analyse, in Section 2, the relative
advantages and disadvantages of the two approaches, using non-sequential
double ionisation (NSDI) as a case study. Finally, in Section 3, we examine
their roles in scientific discovery, via the case study of resonant HHG. In
addition, in Section 4, we present some extra discussion points, as well as
our combatants’ responses to the questions raised by audience members, and a
summary of the responses to several polls taken during the live session.

##  1 ‘ _Ab initio_ ’ and analytical methods

A constructive discussion is always based on a good knowledge of the subject.
To this end, in this section we tackle the subtleties in the definitions of ‘
_ab initio_ ’, ‘numerical’ and ‘analytical’ methods: we detail their
differences, and we present a rough classification of the various theoretical
methods used in attosecond science. We first concentrate on Numerio’s
speciality, _ab initio_ methods, then we move to Analycia’s forte, analytical
theories. For each combatant, we first introduce their theoretical approach,
and then list the main methods in the corresponding toolset. After these
presentations, Numerio and Analycia discuss the friction points they have with
each other’s methods.

###  1.1 _Ab initio_ and numerical methods

In its dictionary sense, _ab initio_ is Latin for ‘from the beginning’. Thus,
a theoretical method can be defined to be _ab initio_ when it tackles the
description of a certain physical process starting from first principles,
i.e., using the most fundamental laws of nature that—according to our best
understanding—govern the physics of the phenomena that we aim to describe.

Within an _ab initio_ framework, the inputs of the theoretical calculation
should be limited to only well-known physical constants, with any interactions
kept as fundamental as possible. This means that no additional simplifications
or assumptions may be made on top of what we believe are the established laws
of nature. In other words, the specific aspects of the physical process of
interest need to be approached without using specially-tailored models.

_We now bring our combatants to the stage, to discuss the consequences of this
definition._

[1]

  * Analycia This is an extremely stringent definition, which will substantially limit the number of methods that can be classified in the _ab initio_ category. But, more importantly, it just delays the real question: what does ‘fundamental’ mean in this context?

Numerio The answer to this question is, in essence, a choice of ‘reference
frame’, within theory-space, which will frame our work. This choice is tightly
connected to the physical regime that we want to describe. We know that
attoscience, as part of atomic, molecular and optical physics, is ultimately
grounded on the Standard Model of elementary interactions in particle physics,
which gives—in principle—the ‘true’ fundamental laws. However, much of this
framework is largely irrelevant at the energies that concern us. Instead, we
are only interested in the quantum mechanics of electrons and atomic nuclei
interacting with each other and with light, and this gives us the freedom to
restrict ourselves to quantum electrodynamics (QED), or with its ‘friendlier’
face as light-matter interaction Cohen-Tannoudji1997 .

Analycia What does this mean, precisely? If QED is the right framework, that
means we must retain a fully relativistic approach as well as a full
quantisation of the electromagnetic field.

Numerio For most problems in attoscience, this would be overkill, as
relativistic effects are rarely relevant. Instead, it is generally acceptable
to work in the context of non-relativistic quantum mechanics, and to introduce
relativistic terms into the Hamiltonian at the required level of
approximation. These are the basic laws responsible for ‘a large part of
physics and the whole of chemistry’, as recognised by Dirac as early as 1929
dirac1929 .

Analycia I can see how this is appropriate, so long as spin-orbit and inner-
core effects are correctly accounted for. However, what about field
quantisation?

Numerio We normally deal with strong-field settings where laser pulses are in
coherent states comprising many trillions of photons, which means that a
classical description for electromagnetic radiation is suitable.

Analycia That typically works well, yes, but it is also important to keep in
mind that it can blind you to deep questions that lie outside of that
framework Lewenstein2020quantum . In any case, though: as ‘fundamental’, would
you be satisfied with a single-electron solution of the Schrödinger equation?

Numerio No, this would not be appropriate—cutting down to a single electron is
generally going too far. While this can be very convenient for numerical
reasons, restricting the dynamics to a single particle invariably requires
adjusting the interactions to account for the effect of the other electrons in
the system, via the introduction of a model potential. This approximation can
be validated using a number of techniques which can make it very solid, but it
always entails a semi-empirical step and, as such, it rules out the ‘ _ab
initio_ ’ label in its strict sense.

Analycia That leaves a many-body Hamiltonian of formidable complexity.

Numerio It does! Let me show you how it can be handled.

##### The ‘ _ab initio_ ’ toolset

The complexity of simulating the time-dependent Schrödinger equation (TDSE)
with the multi-electron Hamiltonian of atomic and molecular systems can be
tamed using a wide variety of approaches. Most of these are inherited from the
field of quantum chemistry, and they differentiate from each other via the
level of approximation they employ, and by the ranges of applicability where
they are accurate.

However, every approach in this space must face a challenging trade-off
between accuracy in capturing the relevant many-body effects, on the one hand,
and the computational cost that it requires, on the other. The key difficulty
here is the handling of electron-correlation effects, which are difficult to
manage at full rigour. Because of this, many methods adopt an ‘intermediate’
approach, which allows for lower computational expense, while at the same time
limiting the accuracy of the physical description.

Numerical methods thus form a hierarchy, schematised in Fig. 1, with rising
accuracy as more electron correlation effects are included:

  * •

Single-Active-Electron (SAE) approaches are the simplest numerical approaches
to the TDSE scrinzi2014 , though they are only _ab initio_ for atomic
hydrogen, and require model potentials to mimic larger systems. Nevertheless,
they can be used effectively to tackle problems where electron correlation
effects do not play a role, and their relative simplicity has allowed multiple
user-ready software packages that offer this functionality in strong-field
settings Bauer2006 ; Tulsky2020 ; Patchkovskii2016 ; Fritzsche2019 .

  * •

Density Functional Theory (DFT) allows an effective single-particle
description Ullrich2012 , widely considered as _ab initio_ , which still
includes electron correlation effects through the use of a suitable ‘exchange-
correlation functional’ Kohn1965 . Within attoscience, examples of approaches
which specifically target attosecond molecular ionisation dynamics include
real-time time-dependent (TD)-DFT DeGiovannini2013 ; Wopperer2017 ;
DeGiovannini2018  and time-dependent first-order perturbation theory static-
exchange DFT Calegari2014 ; Lara-Astasio2018 . More broadly, TD-DFT approaches
are robust enough that they appear in several user-ready software packages
Tancogne2020 ; Noda2019 ; Apra2020 ; Garcia2020Siesta  suitable for
attoscience.111For a detailed examination of the status (and shortcomings) of
DFT and TD-DFT, see the talks of N. Maitra, A. Schild and K. Lopata at Ref.
BIRS-CMO-Workshop

  * •

Non-equilibrium Green’s function theory also allows one to describe the many-
body problem from first principles by using effectively-single-particle
approaches Perfetto2018 ; Perfetto2018JPCL .

  * •

Quantum-chemistry approaches go beyond the SAE approximation and DFT to
include, directly, the effects of electron correlation szabo1996 . The
starting point for this is generally the Hartree-Fock (HF) mean-field
approach, though this is rarely sufficient on its own. Because of this,
quantum-chemistry methods climb the ladder all the way to the full
Configuration Interaction (CI) limit, a complete description of electron
correlation (which is generally so computationally-intensive that it is out of
reach in practice).

Most of the standard approaches of quantum chemistry were developed to
describe bound states of molecular systems szabo1996 ; Schirmer2018 , and they
have also proven to be highly successful for modelling band structures in
solid-state systems Evarestov2012 . Nevertheless, they often require
significant extensions to work well in attoscience, particularly regarding how
the ionisation continuum is handled. Recent examples of these extensions
include _ab initio_ methods based on the algebraic diagrammatic construction
(ADC) Ruberti2014JCP ; Simpson2016 ; Averbukh2018 ; Ruberti2018PCCP ;
ruberti2018  and its restricted-correlation-space extension (RCS-ADC)
Ruberti2019 ; Ruberti2019PCCP ; Ruberti2021 ; schwickert2020 , multi-reference
configuration interaction (MRCI) majety2015hacc ; Majety2015PRL , and multi-
configuration time-dependent Hartree Beck2000  and Hartree-Fock sukiasyan2009
methods, as well as restricted-active-space self-consistent-field (RAS-SCF)
Marante2017 ; Marante2017PRA ; Klinker2018  approaches.

  * •

Basis-set development is another crucial element of the numerical
implementation work for _ab initio_ methods in attoscience, since the physics
accessible to the method, as well as its computational cost, are often
determined by the basis set in use. Recent work on basis sets includes the
development of B-spline functions, both on their own Ruberti2014JCP ;
Simpson2016 ; Averbukh2018 ; Ruberti2018PCCP ; ruberti2018 ; Ruberti2019 ;
Ruberti2019PCCP ; Ruberti2021 ; schwickert2020 ; Toffoli2016 , and in hybrid
combinations with Gaussian-type orbitals (GTOs) Marante2017 ; Marante2017PRA ;
Klinker2018 , as well as finite-difference approaches clarke2018 ; brown2020 ;
Benda2020 , finite-element discrete-variable-representation functions
Miyagi2013 ; Miyagi2014 , grid-based methods sato2013 ; Greenman2010 , and
simple plane-waves NguyenDang2014 .

A more extensive (though still non-exhaustive) list of methods is shown in
Table 1. Here we focus on methods for the description of ultrafast electron
dynamics, happening on the attosecond time-scale. For numerical methods
tackling the (slower) nuclear motion in attoscience, we refer the reader to
Ref. Vrakking2018 .

Figure 1: Schematic representation of the hierarchy of methods to describe
electron correlation. headercellbackgroundFully ab initio methods
---
Time-Dependent B-spline Algebraic Diagrammatic Construction | Ruberti2014JCP ; Simpson2016 ; Averbukh2018 ; Ruberti2018PCCP ; ruberti2018  | TD B-spline ADC
Time-Dependent B-spline Restricted-Correlation-Space ADC | Ruberti2019 ; Ruberti2019PCCP ; Ruberti2021 ; schwickert2020  | TD B-spline RCS-ADC
Multi-Reference Configuration Interaction: TRECX | trecx ; majety2015hacc ; Majety2015PRL  | MRCI: TRECX
Restricted-Active-Space Self-Consistent-Field: XCHEM | Marante2017 ; Marante2017PRA ; Klinker2018  | RAS-SCF: XCHEM
Multi-Configuration Time-Dependent Hartree-Fock | Carette2013 ; Sawada2016  | MCTDHF
Time-Dependent Complete-Active-Space Self-Consistent-Field | sato2013  | TD CAS-SCF
Time-Dependent Restricted-Active-Space Self-Consistent-Field | Miyagi2013 ; Miyagi2014  | TD RAS-SCF
Time-Dependent Configuration Interaction Singles | Greenman2010 ; Toffoli2016 ; NguyenDang2014  | TD CIS
$R$-matrix with time dependence | clarke2018 ; brown2020 ; Benda2020  | RMT
Real-time Non-Equilibrium Green’s Function | Perfetto2018 ; Perfetto2018JPCL  | Real-time NEGF
headercellbackgroundDFT/hybrid/non-ab-initio methods
Real-time Time-Dependent Density Functional Theory | DeGiovannini2013 ; Wopperer2017 ; DeGiovannini2018  | Real-time TDDFT
Time-Dependent First-Order Perturbation Theory static-exchange DFT | Calegari2014 ; Lara-Astasio2018  |
Single-Active Electron Time-Dependent Schrödinger Equation | Bauer2006 ; Tulsky2020 ; Patchkovskii2016 ; Fritzsche2019  | SAE TDSE
Table 1:  Rough survey of numerical methods for attoscience and strong-field
physics.

###  1.2 Analytical methods

The use of analytical methods to describe strong-field phenomena has a long-
storied pedigree dating back to the 1960s keldysh1965 , before laser sources
could reach sufficient intensities to drive quantum systems beyond the lowest
order of perturbation theory. As a general rule, analytical methods are
approaches for which the governing equations can be solved directly, under
suitable approximations, and the solutions can be written down in ‘exact’ or
‘closed’ form.

_We return to our combatants on the stage, where Numerio is dissatisfied with
this definition._

[1]

  * Numerio That just seems like it is kicking the can down the road. What does ‘closed form’ mean?

Analycia As it turns out, when the term ‘closed form’ is placed under
examination, its precise meaning turns out to be rather elusive and ultimately
quite ambiguous berry2007 ; Borwein2013closed . That is to say: which ‘forms’
does the term ‘closed form’ actually include? Which ones does it exclude? Does
it stop at elementary functions, i.e., exponentials and logarithms? Or must it
cover special functions, like the Bessel functions? And if we do intend to
include special functions as part of the toolbox of analytical methods, which
special functions should be included? Do hypergeometric functions or Meijer
$G$ functions make the cut? What about newly-minted functions expressly
defined to encapsulate some hard numerical problem? (As one example of this,
take the recent proof of the integrability of the Rabi model of quantum optics
Braak2011 —should the functions defined in that work be considered special
functions?) More importantly: what does it really mean for a function to be a
‘special’ function?

Numerio But hasn’t this question been answered long ago?

Analycia Well, this is the kind of question where one could hope that we could
look to the mathematicians to provide an answer—say, by supplying an objective
classification of functions, from elementary through exponentials to the
‘higher’ transcendentals—as to where this class should stop. Unfortunately,
however, when such objective classifications are attempted, they run into a
bog of vague answers and incomplete taxonomies which leave out large classes
of useful functions. Ultimately, as Paul Turán put it andrews1999special ,
‘special’ functions are simply useful functions: they are a shared language
that we use to encapsulate and communicate concepts and patterns berry2007 ,
and their boundaries (and with it, the boundaries of analytical methods in
general) are subjective and a product of tradition and consensus.

Numerio This distinction seems like trivial semantics to me, to be honest.

Analycia At first glance, yes, but it is important to keep in mind that, as a
rule, special functions like the Bessel functions are defined as the solutions
of hard problems—canonical solutions of ordinary differential equations,
integrals that cannot be expressed in elementary terms, non-summable power
series—and when it comes to evaluating them in practice, they generally
require at least some degree of numerical calculation. In this regard, then,
what is to stop us from packaging up one of the numerical problems that face
us, be it a full TDSE simulation or one of its modular components, calling it
a special function, and declaring that methods that use it are ‘analytical’?

Numerio That sounds rather absurd.

Analycia Indeed it does, at face value, but it is not all that far from how
special functions are actually defined, i.e., as the solution of a tough
differential equation, or to dodge the fact that a given integral cannot be
evaluated in elementary terms by re-christening it as an integral
representation. More importantly, it encodes a serious question—what happens
when the ‘back end’ of analytical methods involves more numerical calculations
than the TDSE simulations they were intended to replace?

Numerio They should give the job to me, of course!

##### The analytical toolset

These issues aside, the analytical methods of strong-field physics, as
traditionally understood in the field, form a fairly well-defined set. This
set can be further subdivided into three main classes:

  * •

Fully quantum models, which retain the full coherence of the quantum-
mechanical framework. These frameworks date back to key conceptual leaps in
the early days of laser-matter interaction keldysh1965 ; Faisal1973 ;
Reiss1980 ; perelomov1966ionization ; ammosov1986tunnel , but they also
include applications of more standard perturbation-theory tools.

The central method in this category is known as the Strong-Field Approximation
(SFA) keldysh1965 ; Faisal1973 ; Reiss1980  (see Ref. Amini2019  for a recent
review), which builds on the solvability of the field-driven free-particle
problem, used to great effect for HHG Lewenstein1994Mar . The SFA is more
properly a family of related methods galstyan2016  with the key commonality of
taking the driving laser field as the dominant factor after the electron has
been released; in its fully-quantum version, it produces observables in the
form of highly-oscillatory time integrals.

  * •

Semiclassical models, which bridge the gap between the full quantum
description and the classical realm by incorporating recognizable trajectory
language but still keeping the quantum coherence of the different pathways
involved. The paradigmatic example is the quantum-orbit version of the SFA
Popruzhenko2014 , obtained by applying saddle-point approximations to the
SFA’s time integrals, which results in trajectory-centred models analogous to
Feynman path integrals salieres2001feynman  where the particles’ positions are
generally evaluated over complex-valued times Popruzhenko2014 ; ivanov2014
and are often complex-valued themselves torlina2012 ; Pisanty2016 ;
Pisanty2017 ; torlina2017 .

As a general structure, the relationship between semiclassical methods and the
full TDSE is the same as between ray optics and wave optics for light, a
correspondence that can be made rigorous as an eikonal limit Smirnova2008 .
This has the caveat that optical tunnelling requires evanescent waves in the
classically-forbidden region, with an optical counterpart in the use of
complex rays for evanescent light einziger1982 . The presence of these complex
values complicates the analysis, but it also presents its own opportunities
for insight Pisanty2020 .

The recent development of analytical methods has centred on correcting the SFA
to account in various ways for the interaction of the photoelectron with its
parent ion, from straightforward rescattering milosevic2007intensity  through
fuller Coulomb corrections Popruzhenko2008 ; torlina2012  and explicit path-
integral formulations Maxwell2017 , which now span a wide family of approaches
figueira2020 . On the other hand, it is important to keep in mind that there
are also multiple techniques, such as semiclassical propagators Zagoya2014 ,
which are independent of the SFA.

  * •

Fully classical models, which can retain a small core of quantum features
(most often, the tunnelling probability obtained from tunnelling theories
ammosov1986tunnel ) but which generally treat all the particle dynamics using
classical trajectories. This includes the paradigmatic Simple Man’s Model
Corkum1993 ; Kulander1993  for HHG, but it also covers much more elaborate
methods, often of a statistical kind, that look at classical trajectory
ensembles to understand the dynamics Panfili2001 , and in particular the
specific formulation as Classical Trajectory Monte Carlo dimitriou2004origin
‘shotgun’ approach to predicting photoelectron momentum and energy spectra.

These methods are summarised in Table 2.

headercellbackgroundFully quantum models
---
Lowest-order perturbation theory | semianalytic | Matrix elements must be taken from quantum chemistry
Strong field approximation Lewenstein1994Mar  | semianalytic | Intensive numerical integration
headercellbackgroundSemiclassical models
Quantum Orbit models Popruzhenko2014 ; figueira2020phases  | semianalytic | Saddle-point equations must be solved numerically
‘Improved’ SFA milosevic2007intensity  | semianalytic | Saddle-point equations must be solved numerically
Coulomb-Corrected SFA Popruzhenko2008  | semianalytic | Significant numerical integration
Path-integral approaches Maxwell2017  | not analytic | Numerical solutions for equations of motion
Semiclassical propagators Zagoya2014  | not analytic | Numerical solutions for equations of motion
headercellbackgroundFully classical models
Simple Man’s Model Corkum1993 ; Kulander1993  | semianalytic | Trajectories are only analytical for a few driving fields
Classical Ensemble Models Panfili2001  | not analytic | Equations of motion must be solved numerically
Classical Trajectory Monte Carlo dimitriou2004origin  | not analytic | Equations of motion must be solved numerically
Table 2:  Rough survey of analytical methods of strong-field physics and
attoscience.

###  1.3 Hybrid methods

In addition to the purely-numerical and purely-analytical approaches discussed
above, it is also possible to use hybrid approaches, which involve nontrivial
analytical manipulations coupled with numerical approaches that incur
significant computational expense.

This class of methods can include relatively simple variations on standard
themes, such as multi-channel SFA approaches that include transition
amplitudes and dipole transition matrix elements derived from quantum
chemistry Mairesse2010 , but it also includes long-standing pillars like
Molecular Dynamics and other rate-equation approaches—which use _ab initio_
potential-energy surfaces and cross-sections, but discard the quantum
coherence—that are now being applied within attoscience Shi2019 . Beyond these
simpler cases, there is also a wide variety of novel and creative methods,
such as the classical-ensemble back-propagation of TDSE results employed
recently to analyse tunnelling times Ni2016 , which hold significant promise
for the future of attoscience.

###  1.4 Friction points

_Now with a complete set of basic definitions, our combatants Analycia and
Numerio turn their discussion to more specific aspects of analytical and _ab
initio_ methods. _

####  1.4.1 Numerical $\neq$ _ab initio_

[1]

  * Analycia It seems to me that the classification of numerical and _ab initio_ methods, as presented in the ‘ _ab initio_ toolset’, is out of step with the definitions as originally stated. Are you using the terms ‘ _ab initio_ ’ and ‘numerical’ interchangeably?

Numerio You are right. It is important to emphasize the difference between
numerical methods and _ab initio_ methods. Both classes share and benefit from
the development and application of ‘computational thinking’, but strictly
speaking the latter category is a subset of the former. On the other hand, in
the literature, the two terms are often used interchangeably.

Analycia That may be, but then that is a problem with the literature. There
are many methods on the toolset that are very far from _ab initio_ as you
defined it.

The clearest examples of this are the methods based on the SAE approximation
Abusamha2010 ; Kukk2013 ; Saito2004 ; Gozem2015 ; Labeye2018 ;
vandenWildenberg2019 . This approach neglects, in an extremely crude way, the
two-body nature of the Coulomb electrostatic repulsion between the different
electrons, which is often called ‘electron correlation’. Should these methods
really be called ‘ _ab initio_ ’?

Numerio Most of these methods try—with varying degrees of success—to correct
for the neglect of electron correlations by introducing various
parameterisations of effective one-particle Hamiltonians. However, these
constructions are for the most part semi-empirical, and as such they introduce
significant physics beyond the fundamental laws, and definitely cannot be
called _ab initio_ methods.

Analycia It is good to see that laid out clearly. In a similar vein, what
about DFT and TD-DFT? I notice that many of the openly-available DFT packages
explicitly market themselves as being ‘ _ab initio_ ’ approaches Tancogne2020
; Noda2019 ; Apra2020 .

Numerio DFT is a rigorously _ab initio_ method, and it takes its validity from
strict theorems (originally for static systems Hohenberg1964 ; Kohn1965  and
subsequently extended to time-dependent ones runge1984 ; vanleeuwen1998 ;
vanleeuwen1999 ) that show that the complexity of the full multi-electron
wavefunction can be reduced to single-electron quantities. In brief, there
exists an ‘exchange-correlation functional’ that allows us to get multi-
electron rigour while calculating only single-electron densities.

Analycia That may be the case in the ideal world of mathematicians, but it
does not work in the real world. The formal DFT and TDDFT frameworks only work
if one knows what the exchange-correlation functional actually is, as well as
the functionals for any observables such as photoelectron spectra. In
practice, however, we can only guess at what those might be. I have a deep
respect for DFT and TDDFT: for large classes of systems it is our only viable
tool, and there is a large body of science which validates the functionals it
employs. Nevertheless, the methods for validating the ‘F’ in DFT are semi-
empirical, and do not have the full-sense rigour of _ab initio_.

Numerio Yes, those are fair points. However, it is worth noting that there
also exists a rigorous method to construct approximate parameterised
functionals. This is based on introducing parameters whose value can be fixed
by requiring them to satisfy the known exact properties of the functional.
These parameters are of universal nature in the sense that once they have been
determined, they are kept fixed for all systems to be calculated. Having said
this, in practice, when the DFT Hamiltonian ends up in the form of a semi-
empirical parameterisation Tong1997 ; Stener2005 ; Toffoli2013 , then this
takes it out of the _ab initio_ class.222It is worth noting that ‘ _ab initio_
’ can take different meanings in different fields. There are multiple
contexts, such as the study of condensed matter and other extended systems,
where wavefunction methods are not feasible, and DFT methods are the closest
approach to the _ab initio_ ideal as we have defined it here; most
descriptions of DFT as _ab initio_ appear in such contexts.

Analycia So, are there _any_ numerical methods which truly satisfy the _ab
initio_ definition?

Numerio Yes, there are. Most of the approaches based on quantum chemistry
possess potentially full _ab initio_ rigour. In practice, of course, for some
applications this full potential is not needed, and the degree of electron
correlation in the calculation can be restricted in order to reduce the
calculation time. However, even in those cases, there is still an _ab initio_
method underlying the computation.

That said, even within an _ab initio_ method, it is common to introduce semi-
empirical parametrisations of the Hamiltonian. This happens most often when we
cannot describe (or do not need to) every term in the Hamiltonian to an _ab
initio_ standard.

Analycia What kind of interactions would this approach apply to?

Numerio The introduction of pseudo-potentials can be used, for example, to
model the effect of core electrons in an atom or molecule. Another common case
is the effect of spin-orbit interactions in a semi-relativistic regime. This
can be seen as a non-_ab initio_ description of certain degrees of freedom or
interactions whose effect is not dominant within a given physical process or
regime. Sometimes this has a limited scope, but it can also extend out to what
we have described as ‘hybrid’ methods (such as Molecular Dynamics
simulations), which are not fully _ab initio_ but which nevertheless maintain
a very strong _ab initio_ identity.

Analycia This does not really paint a picture of a ‘single class’ of _ab
initio_ methods: instead, you have depicted a continuum of methods, which goes
smoothly from a full accounting of electron correlation down to restricted
numerical simulations which operate under substantial approximations.

Numerio I agree, and if you press me I should be able to organise these
methods on a spectrum, between approaches which are fully _ab initio_ and
techniques which are simply numerical approaches.

####  1.4.2 Analytical methods generally involve computation

_Having conceded that _ab initio_ methods span a rather large continuum,
Numerio strikes back at just how ‘analytical’ the analytical approaches really
are._

[1]

  * Numerio Since you are so keen to hold _ab initio_ methods to the ‘golden standard’ of the definition, it is only fair that we do the same for analytical methods. Many of the methods you have listed look rather heavy on the numerics to me, particularly on the fully-quantum side. To pick on something, perturbation theory is certainly purely analytical on its own, but those models often require accurate matrix elements for the transitions they describe, and those can only be obtained from quantum chemistry, often at great expense.

Analycia Yes, that is true—

Numerio interrupts Analycia

Numerio And is that not also the case even for the ‘stars’ of the show? The
SFA, in its time-integrated version, produces integrals which are highly
oscillatory, and this generally implies a significant computational cost.

Analycia I agree, the SFA and related methods often involve a large fraction
of numerical effort. Even for the quantum-orbit version, the key stages in the
calculation—the actual solution of the saddle-point equations—rely completely
on numerical methods. On the other hand, of course, this is typically at a
much lower computational cost than most TDSE simulations.

Numerio For most methods, that is quite clear. However, this lower
computational cost is much less clear for some of the more recent approaches
that implement Coulomb corrections on the SFA. The analytical complexity of
those methods can get very high—does that not come together with a higher
computational cost?

Analycia To be honest, the computational expense in some of the more complex
Coulomb-corrected approaches (in particular those that utilise ensembles of
quantum trajectories) to the SFA can, in fact, exceed that of some of the
simpler single-electron TDSE simulations.

Numerio I also notice that you have classified several classical trajectory
methods as ‘analytical’, including statistical ensemble and Monte Carlo
approaches that often involve substantial computation expense calculating
aggregates of millions of trajectories. More importantly, this goes beyond the
raw numbers—the Newtonian equations of motion are only solvable in closed form
for field-driven free particles. As soon as any sort of atomic or molecular
potential is included, one must turn to numerical integration.

Analycia Yes, that is also correct. The character of the method is different
to a direct simulation of the TDSE, but the numerical component cannot be
denied.

Numerio So, similarly to the continuum of ‘ _ab-initio_ -ness’ we agreed upon
earlier, what you are saying is that for analytical methods there is also a
continuous spectrum between fully analytical and exclusively numerical.

Analycia Yes, I suppose I am. We should then be able to place the theoretical
methods of attoscience on a two-dimensional spectrum depending on how much
they have an analytical and _ab initio_ character.

_Our two theorists sit down to chart the methods they have discussed so far,
and report their findings in Fig. 2._

Figure 2: Rough spectrum of the theoretical methods of attoscience, ranked by
their analytical (horizontal) and _ab initio_ (vertical) character.

####  1.4.3 Quantitative vs qualitative insights

[1]

  * Numerio It seems we have completely eliminated the dichotomy that we started with between analytical and ab initio methods.

Analycia So it seems, at least on the surface, but there is still a clear
difference between the two approaches. In this regard, I would like to make a
somewhat contentious claim: it is more important to distinguish methods
according to whether the insights we can obtain from them are of a more
quantitative nature or of a more qualitative one. It seems to me that it is
the spectrum between _those_ two extremes that carries more value.

Numerio Speaking of ‘qualitative methods’ is certainly unusual in the physical
sciences, and to me it feels like it carries some negative connotations.

Analycia Perhaps this is because that phrase has been mistakenly associated
too tightly with biological and social sciences, and physical scientists
sometimes want to distance themselves from that perception? If that is so,
then it is important to work to de-stigmatise that classification.

In any case, though, I am curious to know whether the attoscience community
agrees that this is a more important distinction.

The audience response to this poll is presented in Table 16, in Section 4
below.

####  1.4.4 Analytical $\neq$ approximate

[1]

  * Numerio You mentioned above that some of the ‘analytical’ methods of attoscience can involve substantial computational expense. What is the point of performing such computations, for approaches that can only ever be approximate.

Analycia This is a common misconception. ‘Analytical’ does not necessarily
mean ‘approximate’, and there are problems where analytical approaches can be
fully exact. Some of this list is limited to the canonical examples (the
particle in a box, the harmonic oscillator, the hydrogen atom, the free
particle driven by an electromagnetic field), but it is important to emphasise
that it also covers perturbation theory, which is exact in the regimes where
it holds. And, in that sense, it includes the exact solutions for single- and
few-photon ionisation and excitation processes, which are crucial to large
sections of attoscience, particularly when it comes to matter interacting with
XUV light.

Numerio With the caveat we discussed above, surely? The perturbation-theory
calculations are exact in their own right, but their domain of applicability
without numerical calculations is extremely limited.

####  1.4.5 _Ab initio_ $\neq$ exact

[1]

  * Analycia One striking aspect which is implicit in your description of _ab initio_ methods, and in how they are handled in the broader literature, is the implication that any _ab initio_ method is automatically exact.

Numerio No, that is inaccurate. The two descriptors are distinct and they
should not be considered as synonyms.

Analycia Perhaps the term is used to somehow overvalue the results of a
numerical simulation? It is easy to fall into the trap of thinking that a
result obtained in an _ab initio_ fashion is automatically quantitatively
accurate, but that is a misconception. The clearest examples of this
difference are simulations that work in reduced dimensionality, e.g. 1D or 2D,
but, more generally, plenty of _ab initio_ approaches make full use of
approximations when they are necessary.

Numerio That is true, and if a method’s approximations cannot be lifted then
it does not really fit the definition of _ab initio_. However, it is common to
use ‘lighter’, more flexible numerical methods—which use approximations to
reduce the cost—for more intensive investigations, while still benchmarking
them against an orthodox, fully _ab initio_ approach, and then we can be
confident in the accuracy of the more flexible methods.

Analycia But that is no different to how we benchmark analytical methods. What
is it about _ab initio_ approaches that singles them out as the ‘gold
standard’, then?

Numerio I would say that the key feature is the existence of a systematic way
to improve the accuracy which does not rely on any empirical fittings or
parametrisations, as the central part of the numerical convergence of the
method. When this is present, we can expect to get a description at the same
level of accuracy for the same physical observables, even if we change the
system in question, and we can also estimate the error we make in a systematic
way. Under these conditions, then, the _ab initio_ methods can achieve
fidelities that are so high that they can be considered to be fully
exact.333Having said this, it is important to notice a difference between
wavefunction-based and functional-based _ab intio_ methods. Although non-
semiempirical functionals can, in principle, be systematically improved, the
nature of this improvement will generally be highly system-specific and, in
contrast to wavefunction methods, a better description is not always
guaranteed for a particular system of interest.

####  1.4.6 The choice of basis set

[1]

  * Analycia You mentioned that a key part of ‘the _ab initio_ toolset’ is the development of suitable basis sets. This sounds odd to me: any two basis sets should be equivalent, so long as they are both complete—which, on the _ab initio_ side, corresponds to numerical convergence.

Numerio That is formally true, but it is not very useful in practice. The
basis set used to implement an _ab initio_ method—to formulate and solve the
Schrödinger equation—is a crucial factor in the numerical aspects, and it
determines the level of accuracy of the calculations as well as the
computational cost required to reach convergence to a stable solution that
captures the full physics of your problem.

More broadly, this is a source of approximation (and thus an entry point for
errors), as well as a powerful ally in the search for new physics. In short,
the choice of basis set largely determines the subspace of the solutions that
we can reasonably explore, and this in turn influences the physics that can be
investigated with the method.

Analycia You mentioned that many of the _ab initio_ approaches in attoscience
have their roots in quantum chemistry, and I understand that quantum chemists
have worked very hard at optimising basis sets for their work. Why can’t those
sets be used in attoscience?

Numerio The basis sets most commonly used in traditional quantum chemistry,
particularly Gaussian-type orbitals (GTOs), have indeed been highly optimised
by tailored fitting procedures over many years and, as a result, they have
enabled the flourishing of _ab initio_ quantum chemical methods szabo1996 .
There, the driving goal is to have accurate and fast numerical convergence for
the physical quantities that interest quantum chemists, such as ground-state
energies and electric polarizabilities.

Analycia Ah—and these goals do not align well with attoscience?

Numerio Exactly. These basis sets are generally poorly suited to describe free
electrons in the continuum. As such, traditional basis sets struggle when
describing molecular ionisation over a wide range of photoelectron kinetic
energies Ruberti2013 ; Ruberti2014 . By extension, this limits our ability to
describe general attosecond and strong-field physics.

Analycia So this is where the attoscience-specific development of basis sets
comes in, then.

Numerio Yes. For attoscience the key requirement is an accurate description of
wavefunctions with oscillatory behaviour far away from the parent molecular
region, and this drives the development when existing basis sets are
insufficient.

Analycia So what determines the choice of basis set in any given situation?

Numerio This depends on a number of factors—some down to numerical convenience
in the specific implementation, but also, often, determined by the physics
that the method seeks to describe. Within any particular _ab initio_
framework, the use of new basis sets allows us to explore different parts of
the Hilbert space of the system under investigation, and to look for new and
interesting solutions there.

Analycia This sounds reasonable enough, but it also speaks against the strict
definition of ‘ _ab initio_ ’ as you formulated it, which requires us not to
input any physics beyond the fundamental interactions. To the extent that the
basis-set choice determines the subspace where solutions will play out, that
represents an additional input about the physics which is built directly into
the code. This can then limit the reach of the method; one clear example of
this is the elimination of double-ionisation effects if a continuum with
doubly-ionised states is not included in the basis set. Given these
limitations, can we ever truly reach the _ab initio_ ideal?

Numerio When phrased in those terms, then I agree that it is an ideal, but
there is also no denying the practical reality that has been achieved in
describing the full complexity of quantum mechanics as regards attoscience.
And, I would argue, the methods we have available do offer systematic ways to
ensure convergence in a controlled fashion, and we can very well say that we
are approaching physics in an _ab initio_ way.

##  2 Advantages and disadvantages of analytical and numerical methods

In the previous section we developed, through our combatants Numerio and
Analycia, a framework that allows us to place the theoretical methods of
attoscience in a continuous spectrum: from analytical to numerical, and from
_ab initio_ to approximate, as well as from methods that offer qualitative
insights to ones whose output is most valuable in its quantitative aspects. In
this section we move on, to focus on the strengths and weaknesses of methods
across the theoretical spectrum established in Fig. 2. This analysis is
crucial, as it enables an impartial evaluation of different methods, which in
turn allows attoscientists to use the most suitable tools for the job at hand.
Understanding the advantages and disadvantages of different methods, as well
as their successes and shortcomings, allows us to highlight the most efficient
one—or the most effective combination—for the chosen application, and it is an
important guide in the development of hybrid methods.

###  2.1 Fundamental strengths and weaknesses

_Continuing the conversation Analycia and Numerio each make a case for their
respective methods, and attempt to scrutinise the shortcomings of each other’s
favoured methods. _ [1]

  * Numerio The main advantage that has struck me in recent years is the impressive progress in the application of numerical methods to problems of increasing complexity. A number of problems which were once well beyond our reach are now possible. This has been achieved both through the development and refinement of efficient computational methods, as well as the increasing availability of high-performance computing (HPC) platforms. Such methods can also act as benchmarks against which to test the validity of simpler, smaller-scale, or more approximate methods. Their other clear advantage is their generality, which enables their application to a variety of physical problems.

Analycia True, but despite these advantages, you must admit there can be a
heavy price to pay. As you mention, application of these methods can require
large-scale HPC resources, and such calculations can be extremely time-
consuming, even if optimised codes and efficient numerical methods are used.
It may not be possible to perform a large number of such calculations, which
then makes it infeasible to perform scans over laser parameters that are often
crucial to understand the physics. Additionally, an inherent difficulty in
many methods is the rapid increase in required numerical effort with the
number of degrees of freedom of the target system. This often restricts
methods, for instance, to the treatment of one active electron, or to
linearly-polarised laser fields. Releasing these restrictions, and others,
then incurs a significant computational cost.

Analytical methods, however, are not encumbered with many of the difficulties
encountered by numerical methods. Their inherent approximations afford them a
large speed advantage as well as a high degree of modularity. These qualities
allow them to provide an intuitive physical picture of the complex dynamics.
They may also avoid the unfavourable scaling properties with which numerical
methods can be saddled, allowing them to explore a more expansive parameter
space. This, coupled with the understanding they provide, can be used to
direct more resource-expensive numerical or experimental approaches.

Numerio Yes, I’m aware that analytical methods provide a number of advantages,
but the price tag is the required level of approximation that enables
analyticity. Approximation is a double-edged sword: ideally we would only
discard unnecessary details in order to highlight the important processes,
but, most commonly, we also end up discarding important details, and this may
imply that some physical processes are not accurately captured. Approximations
also often carry more restrictive regimes of validity, and this makes them
less general than _ab initio_ approaches. So, despite the advantage they may
have with regard to scaling properties, they can also be rather restricted in
some respects. This can often come in the form of rather unrealistic
assumptions, such as the assumption of monochromatic laser pulses.

_Unable to find common ground on their favoured methods, Numerio and Analycia
decide to look at the specific example of NSDI. _

###  2.2 In context: non-sequential double ionisation

_This case study on NSDI will explore the impact that the characteristics of
various methods can have in understanding a physical process. However, before
we rejoin our debating combatants, we will present a few of the key concepts
of NSDI._

Figure 3:  Singly- and doubly-charged helium ion yields as a function of laser
intensity, at a wavelength of $\lambda=$780\text{\,}\mathrm{n}\mathrm{m}$$,
showing the ‘knee’ structure associated with the transition between NSDI and
sequential double ionisation Walker1994 .333Reprinted with permission from
Ref. Walker1994 . © 1994 by the American Physical Society. Figure 4: Schematic
of the two main mechanisms in NSDI MaxwellThesis2019 . Panel (a) shows
electron impact ionisation (EI), where the recolliding electron mediates
direct double ionisation. Panel (b) shows the recollision excitation with
subsequent ionisation (RESI) mechanism, where the recolliding electron excites
the bound electron, which is subsequently released by the field.

NSDI has been studied using a wide variety of analytical and numerical
methods. These include both classical and quantum approaches, solving the
Newtonian equations of motions and the TDSE. This range of methods is a
testament to the difficulty in modelling this process, and thus makes it an
ideal case study.

##### What is NSDI?

Put simply, NSDI is a correlated double ionisation process, where the
recollision of a photoelectron with its parent ion leads to the ionisation of
a second electron. Historically, NSDI was discovered as an anomaly, where the
experimental ionisation rate did not agree with analytical computations for
sequential double ionisation for lower laser intensities, giving rise to the
famous ‘knee’ structure (see Fig. 3) LHuillier1983 ; Agostini1984 ;
Agostini1985 ; Fittinghoff1992 ; Kondo1993 ; Walker1994 . Originally, there
was contention over the precise mechanism, but over time the three-step model
Kuchiev1987 ; Corkum1993 ; Schafer1993  involving the laser-driven recollision
was accepted. The three steps of this model are (i) strong-field ionisation of
one electron, (ii) propagation of this electron in the continuum, and (iii)
laser-driven recollision and the release of two electrons. This classical
description is based on strong approximations, and it is generally considered
to be an analytical method (although, as we discussed in Section 1.4.2, it
generally relies on some numerical computations). In particular, the
exploitation of classical trajectories gives it the intuitive descriptive
power of an analytical method.

Within the three-step model, two main mechanisms have been identified for
NSDI. The first is electron impact (EI) ionisation, where the returning
electron has enough energy to release the second electron, leading to
simultaneous emission of both electrons, as depicted in Fig. 4(a). The
alternative mechanism is recollision with subsequent ionisation (RESI), which
occurs when the returning electron only has enough energy to excite the second
electron (but not remove it directly), and this second electron is
subsequently released by the strong field, leading to a delay between the
ionisation of the first and second electron, as shown in Fig. 4(b). The
separation of these mechanisms is best expressed by semi-analytic models based
on the SFA Becker1994 ; Becker1996 , where the mechanisms can be represented
as Feynman diagrams and linked to rescattering events Becker1999 .

##### The NSDI Toolset

Here we summarise some of the methods that are available to model NSDI. For
detailed reviews on these methods see Refs. becker2011 ; figueira2011 .

  * •

Three-step model: This simple and intuitive classical description neglects the
Coulomb potential and quantum effects Corkum1993 ; Schafer1993 ; Kuchiev1987 .
Nonetheless, this formulation has become the accepted mechanism of NSDI.

  * •

Classical Models: These can be split into those with some quantum ingredients
like a tunnelling rate Ye2008 ; Emmanouilidou2008 ; Emmanouilidou2009 ;
Brabec1996 ; Chen2000  and those that are fully classical, so that ionisation
only occurs by overcoming a potential barrier Panfili2001 ; Haan2008 ;
Haan2008PRL ; Ho2005 ; Haan2002 ; Panfili2002 ; Goreslavskii2001 ;
Popruzhenko2002 . The electron dynamics are approximated by classical
trajectories, which permits a clear and intuitive description. The
contributions of classes of trajectory can be analysed, which is crucial in
tracing the origin of certain physical processes. However, the model neglects
quantum phenomena such as interference maxwell2016 ; Hao2014 .

  * •

Semi-classical SFA: The Coulomb potential is neglected but the dynamics can be
understood via intuitive quantum orbits, and the different mechanisms can
easily be separated Becker1994 ; Becker1996 ; Becker1999 ; Becker2000 ;
Goreslavskii2001 ; Popruzhenko2002 ; Quan2009 ; Shaaran2010 ; Shaaran2012 ;
figueira2012 ; Maxwell2015 ; maxwell2016 . This also allows quantum effects
such as tunnelling and interference to be included, with interference effects
in NSDI being predicted Maxwell2015 ; maxwell2016 ; Hao2014  and measured
Liao2017  fairly recently.

  * •

Reduced-dimensionality TDSE simulations: Solution of the TDSE assuming that a
particular aspect of the motion can be restricted to the laser polarisation
axis. One-dimensional treatments restrict the entire electron motion to this
axis lein2000 , and two-dimensional treatments restrict the centre of mass
ruiz2006 , while treating electron correlation in full dimensionality. Similar
approximations are made in other methods, such as the multi-configurational
time-dependent Hartree method sukiasyan2009 , which treats NSDI with the
assumption of planar electron motion.

  * •

_Ab initio_ full dimensional TDSE simulation: Full quantum mechanical
treatment of a two-electron atom through direct solution of the time-dependent
close-coupling equations smyth1998 ; parker2001 ; pindzola1998 ; pindzola2007
; colgan2012 ; parker2006 . Such methods are computationally intensive,
although efficiency improvements have been made in recent years. To date,
these methods have not been extended to treat molecules or atoms other than
helium.

_We rejoin our two debating attoscientists, whose discussion has now moved on
to the specifics of different analytic and _ab initio_ methods in NSDI. The
discussion begins with a debate on the positive and negative aspects of a
direct _ab initio_ approach._

####  2.2.1 Full-dimensional numerical solution of the TDSE

[1]

  * Numerio As a numericist, I often feel that there is no substitute for solving the TDSE in its full dimensionality. In the context of NSDI, this is a daunting computational task, involving solution of many coupled radial equations—often thousands—on a two-dimensional grid. The first code development to do this began in the late 1990s smyth1998 ; parker2001 ; pindzola1998  and, by 2006, calculations could be carried out for double ionisation of helium at $390\text{\,}\mathrm{n}\mathrm{m}$ parker2006 . However, these calculations typically required enormous computational resources—an entire supercomputer, in fact—using all 16,000 cores available at that time on the UK’s national high-end computing platform (HECToR).

Following the literature over the next few years, I noted the development of a
number of similar approaches feist2008 ; feist2009 ; hu2010 ; nepstad2010 ;
hu2013 ; djiokap2012 ; djiokap2015 ; donsa2019 ; donsa2019prl . In NSDI
applications in particular, I was struck by the significant progress made in
reducing the scale of such calculations by the tsurff method scrinzi2012 ;
zielinski2016 ; trecx ; zhu2020 . This approach allowed calculations for
double ionisation of helium at $800\text{\,}\mathrm{n}\mathrm{m}$ to be
carried out using only 128 CPUs for around 10 days zielinski2016 . Fig. 4
shows a recent highlight of this work, the two-electron momentum distribution
for helium at $780\text{\,}\mathrm{n}\mathrm{m}$ zielinski2016 . The
calculation successfully displayed the expected minimum in the distribution
when both electrons attain equal momenta greater than $2U_{p}$. Watching these
developments unfold over the past 15 years, it has become clear to me that
even a daunting problem such as this is well within our grasp, and should be
attempted.

Figure 5: Two-electron momentum distribution for double ionisation of helium
at $780\text{\,}\mathrm{n}\mathrm{m}$, calculated using the tsurff method
zielinski2016 .444Reprinted with permission from Ref. zielinski2016 . © 2016
by the American Physical Society.

Analycia These are quite intensive calculations, so my first question would
be: is it always worth it? Calculations should not only be feasible—they
should also be justifiable. The large scale of each single calculation can be
a very limiting factor, since you may need further computations, perhaps to
perform intensity averaging, or to scan over a particular laser parameter.
Here you may encounter additional hurdles, since it is well known that the
computational cost can scale very unfavourably with certain laser parameters,
particularly wavelength. Even with the efficiency savings that you mention,
the method may struggle to perform calculations at longer wavelengths, or in
sufficient quantity to scan over experimental uncertainties.

Secondly, it’s true that significant progress has made these large-scale
calculations more tractable. However, this does not necessarily mean that the
results will be easy to analyse. Disentangling the complex web of physical
processes included in such calculations can be very difficult. This requires
tools and switches within the method, for example to evaluate the role of
certain interactions, and thereby aid your understanding. Even with such
analysis tools at hand, gaining strong physical insight may be an arduous
procedure, involving further large-scale calculations, and these may not even
be guaranteed to provide the insights you desire.

Numerio Absolutely, you have highlighted the main difficulties with _ab
initio_ methods that I have encountered. The scale of the calculations can
impose a limit on their scope, and their complexity can obscure
interpretation. On the other hand, simpler methods avoid these difficulties,
but they rely on approximations which need to be justified. For me, the ideal
tool would be a method with qualities representing the best of both worlds — a
method where many small-scale but accurate TDSE calculations could be carried
out to provide detailed interpretation. Although this is feasible in some
fields, in the context of NSDI currently it is not. However, equipped with an
arsenal of _ab initio_ methods, there is an opportunity to benchmark simpler
methods which fall short of a full _ab initio_ treatment. If their
approximations can be validated by such comparisons, then their interpretive
power will be valuable.

Analycia I think now we are beginning to agree.

_The debate above highlights that both calculation and interpretation are
important. Often, an _ab initio_ approach can provide a calculation, but
detailed interpretation may require analytical techniques. To discuss these
techniques further, the debate now moves to focus on the merits of analytical
methods used to study NSDI._

####  2.2.2 Analytical approaches

Figure 6: Comparison of experimental data (upper row) Kubel2014
55footnotemark: 5 with theoretical focal-averaged distributions (lower row)
selected from maxwell2016 .666Reprinted with permission from Ref. maxwell2016
. © 2016 by the American Physical Society. The left and right columns present
$16\text{\,}\mathrm{f}\mathrm{s}$ and $30\text{\,}\mathrm{f}\mathrm{s}$ laser
pulse lengths, respectively, with $\lambda=$800\text{\,}\mathrm{n}\mathrm{m}$$
($\omega=$0.057\text{\,}\mathrm{{a.u.}}$$) and
$I=${10}^{14}\text{\,}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$$
($U_{p}=$0.22\text{\,}\mathrm{{a.u.}}$$). Specific features associated with
quantum interference are marked by polygons in both upper and lower panels. It
was necessary to account for interference effects in the theoretical results
to get this agreement. ††footnotetext: Reproduced from Ref. Kubel2014  under a
CC BY license.

[1]

  * Analycia You see in my experience working on NSDI, descriptive power is often enabled by the high degree of modularity that analytical methods possess. This modularity may be harnessed to determine the physical origin of an effect by switching certain interactions on and off. Like intermediate-rigour numerical methods, the light computational demand means that large sets of individual calculations may be carried out where necessary.

A good example of the power of modularity in analytical models is the use of
interference in SFA models for NSDI to match experimental results maxwell2016
; Hao2014 . In Fig. 6 we see experimental results Kubel2014  for two pulse
lengths. The lower panels show the results of the SFA model maxwell2016  that
uses a superposition of different excited states in the RESI mechanism of
NSDI. Including interference leads to a good match, which provides strong
evidence for interference effects in NSDI. This was only possible because
interference effects could be switched on and off,777The consequences of this
difficulty were discussed in detail in Battle 2 of the Quantum Battles in
Attoscience conference battle2 , reported in a companion paper in this Special
Issue Amini2020 .  thereby allowing analysis of the different shapes and
structures within the distribution. Each of these shapes could then be
directly attributed to different excited states, which demonstrates the power
of the modularity of analytical methods in providing an intuitive
understanding of the physics.

Figure 7: Photoelectron spectra for NSDI in helium driven by a wavelength of
$800\text{\,}\mathrm{n}\mathrm{m}$ and intensity of
$4.5\text{\times}{10}^{14}\text{\,}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$
Staudte2007 , showing (a) the correlated momentum distribution, with a detail
(b) shown with superimposed results from a classical electron scattering
model, as well as (c) the electron energy spectrum of He2+ and
He+.888Reprinted with permission from Ref. Staudte2007 . © 2016 by the
American Physical Society.

Numerio The interpretive power is certainly valuable, and the availability of
switches such as these is often the key to a good physical understanding. My
main concern, however, is that the approximations may affect the accuracy of
the results. In particular, the SFA neglects the Coulomb potential, and it is
known that this influences the famous finger-like structure in NSDI seen in
Fig. 8, causing a suppression of two-electron ejection with equal momenta.
Furthermore, we would expect a host of other Coulomb effects just as there are
in single electron ionisation figueira2020phases . Thus, care must be taken
with the conclusions that you draw from such an analytical model. As I said
earlier, many numerical methods may not afford this degree of modularity, but
it would strengthen my confidence in the conclusions if an _ab initio_ method
also observed these effects. In this way, a numerical method could be guided
by analytical predictions to assess the accuracy of certain approximations.

Analycia This is a fair point, but the considerable speed advantage means that
you can often do additional checks and analysis to get around this problem.
The SFA model presented could be solved in five minutes on a desktop computer,
whereas, as you mentioned, _ab initio_ models will take days on hundreds of
cores. The fast SFA calculations can then account for additional factors such
as focal volume averaging, even though it increases the overall runtime by a
factor of ten or more. It can also perform scans through intensity and
frequency in a timely manner. Such scans can provide important insights, for
example in Fig. 9 where the contributions of various excited states is
monitored as a function of laser intensity and frequency. Their relative
contributions then explain the shapes appearing in various regions of the
momentum distribution. The extra analysis can increase the overall runtime by
factors of 100–1000, which is still perfectly manageable for the SFA, but
would be out of the question for most _ab initio_ methods.

Furthermore, there is always a place for analytical methods in performing
computationally inexpensive initial investigations, which then provide the
evidence needed to commit to using more expensive _ab initio_ or experimental
efforts. In recent work on interference in NSDI, motivated by predictions of
SFA models, experimental work was done to investigate interference effects
quan2017 .

Figure 8: Scan over intensity ($U_{p}$) and frequency ($\omega$), that
attributed preferential excitation to states with different orbital angular
momenta $l$ (s-, p- and d-states) in the RESI process for different pulse
lengths maxwell2016  to the shapes found in Kubel2014 . The contributions of
s-states are displayed in (a), (d) and (g), those of p-states in (b), (e) and
(h) and those of d-states in (c), (f) and (i).999Reprinted with permission
from Ref. maxwell2016 . © 2016 by the American Physical Society.

Numerio Yes, I agree in _some_ cases the extra analysis is beneficial.
However, _ab initio_ methods are still much more generalised than their
analytical counterparts. Take Fig. 4, where many different processes
contribute, including both the RESI and EI mechanisms, together with
sequential double ionisation. The presented SFA model includes only the RESI
mechanism.

Analycia There are two sides to this: it is nice to be able to clearly
separate EI and RESI in the SFA, but it is true that it introduces a lack of
flexibility.

With the goal of reaching some kind of agreement, I would posit that the
benefits of both types of models outweigh the negatives. In the case of
classical and semi-classical models, they have clearly led to huge leaps in
understanding for the mechanisms of NSDI. Furthermore, I would add that NSDI
in particular is a good candidate for hybrid models. Strongly correlated
dynamics and multi-electron effects are well-suited to an _ab initio_
approach, while the main ionisation dynamics are well-described by semi-
classical models.

That said, I would also like to know how the broader community feels about
this.

The audience response to this poll is presented as Poll 2 in Table 16, in
Section 4 below.

_Within the context of NSDI, our combatants have discussed the merits and
drawbacks of their respective approaches, and have begun to appreciate the
computational and interpretational qualities that analytical and _ab initio_
contribute. In the following section, we focus on how progress in scientific
discovery can be aided by both types of method._

##  3 Scientific discovery

The seed of a scientific discovery can be planted in the form of a bump or a
dip on a smooth curve of experimental data, as a whimsical term in the
denominator of some equation, or as a quirky splash in numerical results. In
other words, a scientific discovery can be triggered by experimental results
or theoretical ones, either analytical or numerical. As soon as something new
has been spotted, to become a real full-grown discovery it has to be examined
and explained by each of these aforementioned components, each branch of
research, and in the end there has to be an agreement among all of them.

In some cases, the initiating site is analytical and the others come next, as
in the case of optical tunnelling ionisation, predicted in 1965 keldysh1965
before its much later experimental observation in 1989 Corkum1989Mar
.101010Optical tunnelling ionisation also gives rise to the question of the
tunnelling time, which was discussed in depth during Battle 1 of the Quantum
Battles in Attoscience conference battle1 , reported in a companion paper in
this Special Issue Hofmann2021 . Sometimes, the role of a trigger is played by
numerical calculations, as for coherent multi-channel strong-field ionisation
Rohringer2009May , which was shortly followed by its experimental validation
Goulielmakis2010Aug . It can even be a little bit of both, as in the first
description of the RABBITT scheme in 1990 veniard_phase_1990 , or in single-
photon laser-enabled Auger decay (sp-LEAD), which was predicted in 2013
Cooper2013Aug , first observed in 2017 Iablonskyi2017Aug , and further
characterised in You2019 . There are also theoretical predictions—both
analytical, like molecular Auger interferometry Khokhlova2019Jun , and
numerical, like HHG in topological solids Bauer2018 ; Silva2019Dec ;
Chacon2020 —which have already sparked experimental efforts to confirm them,
but which are still waiting for their observations to come. On the other side,
we have discoveries which arise from experimental observations and are then
explained theoretically, such as NSDI, which was discussed in detail in
Section 2.

In this section we tell another story of scientific discovery in attoscience,
through the case study of resonant HHG, which also starts from recorded
experimental data.

###  3.1 Experimental kick-off

By the year 2000, HHG was already a full-grown discovery. It had been observed
mcpherson_studies_1987 ; ferray_mutliple_1988  and theoretically modelled
Kuchiev1987 ; Corkum1993 ; krause_hihg-order_1992 ; Lewenstein1994Mar  a
decade previously. After this breakthrough, many features of HHG were under
active investigation, both experimentally and theoretically. In particular,
resonances in the HHG spectrum had been extensively studied since the 1990s.
Some structures in the HHG spectra of atomic gases had been very early on
attributed to single-atom resonances lhuillier_coherence_1992 ; balcou_phase-
matching_1993 . Then more recent measurement toma_resonance-enhanced_1999  and
theoretical works figueira2002Jan ; Taieb2003Sep  explained these structures
with multiphoton resonances with bound excited states linked to enhancement of
specific electron trajectories that were recolliding multiple times with the
ionic core.

\begin{overpic}[width=130.08731pt]{figures/Ganeev2006figure.png}
\put(18.0,65.0){\bf(a)} \end{overpic}

\begin{overpic}[width=138.76157pt]{figures/Gilbertson2008figure.pdf}
\put(18.0,65.0){\bf(b)}
\end{overpic}\begin{overpic}[width=156.10345pt]{figures/Shiner2011figure.pdf}
\put(15.0,58.0){\bf(c)} \end{overpic}

Figure 9:  First observations of resonant HHG. (a) Figure taken from
Ganeev2006Jun : High-order harmonic spectra from (1) indium and (2) silver
plumes.1111footnotemark: 11 (b) Figure taken from Gilbertson2008Sep : Spectra
of the harmonic supercontinuum generated with double optical gating from the
different values of input pulse duration for a helium target
gas.121212Reprinted from Ref. Gilbertson2008Sep , with the permission of AIP
Publishing (c) Figure taken from Shiner2011Jun : Top, the raw HHG spectrum
from xenon at an intensity of
$1.9\text{\times}{10}^{14}\text{\,}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$.
Bottom, experimental HHG spectrum divided by the krypton wave packet (blue)
and the relativistic random-phase approximation (RRPA) calculation of the
xenon photoionisation cross-section from kutzner_extended_1989  (green). The
red and green symbols are PICS measurements from fahlman_xe_1984  and
becker_subshell_1989  respectively, each weighted using the anisotropy
parameter calculated in kutzner_extended_1989 .  ††footnotetext: Reprinted
with permission from Ref. Ganeev2006Jun  ©The Optical Society.

In this context, Ganeev et al. Ganeev2006Jun  first measured, in 2006, a
strong enhancement of a single harmonic, by two orders of magnitude, in the
HHG spectrum of plasma plumes. Their result is shown in Fig. 12(a). At that
time, they attributed this resonance to the multiple recolliding electron
trajectories that had previously been observed and modelled in atoms
toma_resonance-enhanced_1999 ; Taieb2003Sep , and they related these
trajectories to multiphoton resonances with excited states.

Then, in 2008, when studying the spectra of single attosecond pulses generated
in noble gases, Gilbertson et al. measured for the first time a strong
enhancement in the HHG spectrum of helium Gilbertson2008Sep , as shown on Fig.
12(b). Since they employed single attosecond pulses, they recorded continuous
spectra which allowed them to see the enhancement perfectly, as it would
otherwise fall between two harmonics if observed in an attosecond pulse train.
They did not give any explanation for this enhancement, as it was not their
main focus, but they observed that it appears at the energy of the $2s2p$
autoionising state (AIS) of helium.

Then in 2011, Shiner et al. measured a strong enhancement at
$100\text{\,}\mathrm{e}\mathrm{V}$ in the HHG spectrum of xenon gas
Shiner2011Jun ; Schmidt2012Mar , shown in Fig. 12(c). The experimental HHG
spectrum is displayed on the upper panel. From that spectrum the authors
extracted the photoionisation cross-section (PICS) by first dividing by the
spectrum of krypton (obtained at the same conditions), and then multiplying by
the photoionisation cross-section of krypton from Ref. huang_theoretical_1981
. The obtained experimental PICS is shown as a blue line in the lower panel.
The green curve is the photoionisation cross-section of xenon from Ref.
kutzner_extended_1989 . The very good agreement between the two curves,
combined with the qualitative agreement with a toy model including only the
$4d$ and $5p$ states of xenon, allowed the authors to relate the enhancement
at $100\text{\,}\mathrm{e}\mathrm{V}$ to the giant resonance of xenon.

Thus, by the late 2000s, there were observations of resonant enhancement
features in HHG from plasma plumes as well as few- and many-electron rare-gas
atoms—but no solid theoretical explanation.

_We now hand again the stage to our theoretical acquaintances, who have begun
discussing the ingredients required for a theoretical model for resonant HHG
and will guide us through the rest of the story._

###  3.2 Building the model

[1]

  * Numerio An explanation of the observed process demands the creation of a model, and this requires a thorough analysis of the experimental data revealing the same phenomenon and distinguishing its essential features. The essential feature of resonant HHG, which is common for all observations independently from the medium—gaseous or plasma—, is the enhancement of one or of a group of high harmonics. This does not sound like a lot to start with. However, this already gives a hint that the desired explanation has no connection to propagation effects. This fact restricts the model to an account of the single-particle response only.

Analycia There have been a number of attempts to create the model describing
resonant HHG. One group of theories is based on bound-bound transitions
Gaarde2001Jun ; figueira2002Jan ; Taieb2003Sep ; Ishikawa2003Jul , but it
cannot be applied for plateau harmonics due to the crucial role Ganeev2006Jun
played by the free-electron motion. Another group of theories mentions a
connection of the multi-electron excited states to the enhanced yield of
harmonics Milosevic2007Aug ; Frolov2009Jun ; Frolov2010Aug . In particular,
the enhancement of high harmonics generated in xenon Shiner2011Jun  was
associated Frolov2009Jun  with the region of the well-known ‘giant’ dipole
resonance in the photoionisation (photorecombination) cross-section of xenon
atoms.

Numerio This sounds closer to the ingredients that are likely required to
explain the phenomenon. Does this not get us closer to resolving the puzzle?

Analycia Indeed it does! After revealing a similar correspondence between
experimental HHG enhancements Gilbertson2008Sep ; Ganeev2006Jun  and
transitions with high oscillator strengths between the ground state and AIS of
the generating ions Chan1991Jul ; Duffy2001Mar , the model of resonant HHG was
forged in the form of the ‘four-step model’ Strelkov2010Mar .

Numerio It seems like there should be a vivid similarity with the common
three-step model for HHG, should there not?

Analycia Of course! The four-step model Strelkov2010Mar  extends the three-
step model Corkum1993 ; Schafer1993 ; Kuchiev1987  to include the resonant
harmonic emission along with the ‘classic’, nonresonant one. The first two
steps of the four-step model—(i) tunnelling ionisation, and (ii) free-electron
motion—repeat those of its forerunner. Then, if the energy of the electron
returning back to the parent ion is close to the one of the ground–AIS
transitions, the third step of the three-step model turns into two: (iii)
electron capture into the AIS, and (iv) relaxation from the AIS down to the
ground state, accompanied by the XUV emission.

Numerio Sure, that seems like a possible chain of events, but how can it lead
to a higher emission probability if it requires an extra step?

Analycia You are right that a substitution of one step by two of them should
intuitively cause a decrease in probability, but the combination of higher
probability for electron capture into the AIS (corresponding to the looser
localisation of the AIS), together with the high oscillator strength of the
transition between the AIS and the ground state, results in an increase of the
resonant harmonic yield by several orders of magnitude. Perhaps you could
argue this is similar to how NSDI may dominate over sequential double
ionisation despite having more steps, as discussed in Section 2.

_By this point a convincing model has been suggested; however, it is far from
an end of the story of ‘scientific discovery of resonant HHG’, and a series of
hurdles still has to be surmounted._

###  3.3 Challenging the model: numerical calculations

[1]

  * Numerio Alright, this model sounds physically reasonable enough, but we still need some actual proof that it’s describing the experiment properly. Small-scale numerical simulations were of great help in that matter. When building the four-step model, Strelkov also performed some TDSE simulations at the SAE level, and compared it with several experimental results for singly-ionised indium and tin, as shown in Fig. 13. The very good agreement shows that a single active electron is able to accurately model the process.

Analycia Sure, that is an important result, but in the paper, Strelkov also
made an analytical estimate of the enhancement using the oscillator strength
and lifetime of the resonant transition. The result of this estimate is shown
in Fig. 13 as blue squares for several singly-ionised atoms. The good
agreement both with experiment and with the TDSE calculations marks another
step in the confirmation of the four-step model.

Numerio Indeed, that was quite convincing already, but all these
considerations were time independent. When building a model in attosecond
science, it is often useful to have a dynamical point of view on the process
under study. Tudoroskaya and Lein investigated resonant HHG and the four-step
model using time-frequency analysis tudorovskaya_high-order_2011 . They solved
the SAE TDSE for 1D model potentials with a shape resonance that models an
AIS. They were able to reproduce an enhancement of more than two orders of
magnitude at the harmonic order corresponding to the shape resonance. Their
time-frequency analysis confirmed that the harmonic emission at resonance
starts when the electron returns to the ionic core. More interestingly, it
shows that the duration of the emission at resonance is much longer than the
emission duration at the other harmonic orders. More precisely, the emission
duration at resonance corresponds to the shape resonance lifetime, indicating
that the electron gets trapped in the resonance and emits from there, thus
validating the four-step model.

_After this convincing achievement, the model seems to be validated,
especially from Numerio’s point of view. But for Analycia the story is not
finished yet._

\begin{overpic}[scale={1.0},unit=1mm]{figures/strelkov-redone-2.pdf}
\put(15.0,38.0){\small\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Ganeev2006Jun}{\@@citephrase{(}}{\@@citephrase{)}}}}
\put(15.0,60.0){\small\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{ganeev_harmonic_2006}{\@@citephrase{(}}{\@@citephrase{)}}}}
\put(47.0,25.0){\small\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{ganeev_harmonic_2006}{\@@citephrase{(}}{\@@citephrase{)}}}}
\put(27.0,39.0){\small\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{suzuki_anomalous_2006}{\@@citephrase{(}}{\@@citephrase{)}}}}
\put(43.0,35.0){\small\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{suzuki_intense_2007}{\@@citephrase{(}}{\@@citephrase{)}}}}
\put(71.0,35.0){\small\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{ganeev_strong_2007}{\@@citephrase{(}}{\@@citephrase{)}}}}
\put(71.0,30.0){\small\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{ganeev_systematic_2007}{\@@citephrase{(}}{\@@citephrase{)}}}}
\put(85.0,18.0){\small\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{ganeev_systematic_2007}{\@@citephrase{(}}{\@@citephrase{)}}}}
\put(71.0,20.0){\small\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{ganeev_systematic_2007}{\@@citephrase{(}}{\@@citephrase{)}}}}
\put(43.0,20.0){\small\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{ganeev_systematic_2007}{\@@citephrase{(}}{\@@citephrase{)}}}}
\put(34.0,22.0){\small\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{ganeev_systematic_2007}{\@@citephrase{(}}{\@@citephrase{)}}}}
\put(27.0,46.5){\small\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{ganeev_systematic_2007}{\@@citephrase{(}}{\@@citephrase{)}}}}
\end{overpic} Figure 10:  Comparison of experimental measurements
Ganeev2006Jun ; ganeev_harmonic_2006 ; suzuki_anomalous_2006 ;
suzuki_intense_2007 ; ganeev_strong_2007 ; ganeev_systematic_2007  with
analytical theory and single-electron TDSE simulations Strelkov2010Mar  for
the enhancement factor in resonant HHG in plasma-plume ions, as reported in
Ref. Strelkov2010Mar .131313Adapted with permission from Ref. Strelkov2010Mar
. © 2010 by the American Physical Society.

###  3.4 Generalisation: analytical theory

[1]

  * Numerio Perfect, we now have the model of resonant HHG in our arsenal, which allows us to conduct a qualitative analysis and to make qualitative predictions. Moreover, we also possess quantitative answers based on SAE TDSE solutions for a number of generating particles in given laser fields. So I believe we have all we wanted then?

Analycia Not so fast! Even though there is a tool providing us with a
quantitative answer, it cannot be easily re-applied for a different generating
system or slightly different field parameters, in other words, there is a lack
of generality. This creates a strong demand for a computationally cheap and
more flexible tool.

Numerio Do you have some concrete solutions in mind?

Analycia This theoretical demand has been satisfied within the introduction of
the analytical theory of resonant HHG Strelkov2014May . The analytical theory
is built on two pillars: Lewenstein’s SFA-based theory Lewenstein1994Mar
(conventional for HHG), and Fano’s theory Fano1961Dec , which guides the
treatment of AISs originated from the configuration interaction.

Numerio I understand, each of these theories is indeed very successful in
describing the two physical processes at hand. But how do you combine them to
reproduce the experimental observations?

Analycia The resonant HHG theory delivers the answer—the spectrum of the
dipole moment of the system—as a product of the spectrum of the nonresonant
dipole moment and a Fano-like factor. The nonresonant dipole moment is the
same as in the well-known Lewenstein theory, which captures the field
configuration and the major characteristics of the ground state of the
generating particle. On the other hand, the Fano-like factor encodes the
resonance, and depends on the AIS’s features: its energy and its energy width,
as well as the dipole matrix element for the transition between the AIS and
the ground state.

As a result, the harmonic spectrum in the resonant case is identical to the
one in the nonresonant case far from the resonance, while in the vicinity of
the resonance it acquires a Lorentzian-like-shape profile due to the Fano-like
factor (see Fig. 14). This Fano-like profile around the resonance carries the
information about two major properties of the resonant harmonics—their
behaviour in amplitude and phase—which result in an enhancement and an
emission time delay of resonant harmonics, respectively.

Numerio Ok, I agree this analytical theory provides a much more general
picture of the process. But is it really that useful? I mean, what could we do
with resonant HHG?

Analycia The two features of resonant harmonics, amplitude and phase, provide
us with extra handles for improving the generation of attosecond pulses, an
intensity boost and an elongation of duration, and they also provide an
opportunity to study the structure of the AIS using the harmonic spectrum.

_With a robust framework in place, our scientists discuss the final obstacles
faced by the theory._

Figure 11:  Squared absolute value (red) and phase of the Fano-like factor
calculated analytically (solid) and numerically within SAE TDSE (with symbols)
for HHG in tin plasma plume Strelkov2014May .141414Reprinted with permission
from Ref. Strelkov2014May . © 2014 by the American Physical Society.

###  3.5 Closure: _ab initio_ calculations

[1]

  * Analycia Although the model and the analytical theory of resonant HHG coincide with the results of numerical TDSE calculations in the SAE approximation, this theory encountered significant resistance, both in conferences and in peer review, insofar as the model potential used in these calculations is artificial and does not reflect the fully multi-electron nature of AISs. What is your opinion regarding this issue?

Numerio I would say that, on the one hand, this is an instance of a broader
discussion regarding the role, advantages and disadvantages of the use of
model potentials in numerical calculations. On the other hand, however, this
limitation can be addressed using fully-_ab initio_ calculations, eliminating
this final uncertainty.

Recent first-principle calculations for resonant HHG by manganese atoms and
ions Wahyutama2019Jun  show the characteristic enhancement observed earlier in
the energy region around a group of AISs.

Analycia Finally! These results close the remaining questions in the
theoretical understanding and description of resonant HHG, and open a wide
front of study into the applications of this process, equipped with a full
toolset: analytical theory as well as numerical (SAE and _ab initio_)
calculations.

Numerio I agree, we are not always on great terms, but we really made a nice
team on this one!

The audience opinion on the necessity of combining different approaches is
presented as Poll 3 in Table 16 in Section 4 below.

_After this constructive exchange, the two agree to work more tightly together
from now on._

As we discuss below in Section 4.5 as a response to Audience Question 1, we
are not always necessarily after discoveries in our field, but also after
finding and solving interesting problems. Nonetheless, any scientific
production, or creative activity, before it can considered as scientific,
requires confrontation of different points of view. We argue here that this
confrontation is all the more efficient and constructive when it involves all
the different aspects of scientific work: experimental, analytical, numerical,
and _ab initio_. As we have seen at the start of this section, the initial
trigger can be pushed by any of them, but the actual scientific progress
generally happens afterwards, when they collaborate together.

##  4 Discussions

The dialogue between proponents of analytical and _ab initio_ approaches, as
we have followed it so far, opens a number of additional questions for deeper
examination. We now turn to these more specific points, as well as our
(combatants’) responses to the questions raised by audience members during the
talk.

During the online conference battle , in addition to the talk, several
questions were directed to the audience in the form of polls, both over the
Zoom platform as well as to a wider public over Twitter. We present in Table
16 a summary of the results of these polls.

_Our combatants return to the stage to resolve several still-itching questions
that remain from their conversation._

###  4.1 Is approximation a strength or a weakness?

_The degree of approximation made by a particular method is a typical source
of contention between numericists and analyticists. Here, Numerio and Analycia
discuss how they feel approximation should be characterised._

[1]

  * Analycia It has been said that approximation is a downside of analytical methods. However, I would like to argue—perhaps somewhat provocatively—that approximation is more of a strength. Approximation is what drives the interpretation—the qualitative picture—of a physical process as constructed by an analytical model. If you can remove all that is unnecessary, and still achieve reasonable agreement with _ab initio_ simulations or with experimental results, then this is when you actually start to gain some real understanding and interpretation of physical processes.

In other words, I do not think any method, analytical or numerical, is
scientifically useful by itself. Science stems from the comparison and
interplay of different methods, and particularly of different levels of
approximations.

Numerio I tend to agree that, as an ideal, this is where approximations can
really bring clarity to the table. However, in practice, most of the time when
we approximate we end up dropping some of the things that we would like to
retain. In that sense, approximation is both a blessing and a curse: it
simplifies the picture so we can better understand it, but we generally lose
out on some of the physics we want to describe. There is rarely a ‘happy
medium’ where approximation is purely a strength.

Having said that, though, I should also point out that these benefits and
disadvantages of approximation are equally applicable to numerical methods. If
an approximate calculation matches experiment or an _ab initio_ simulation in
full rigour, then we can be confident that we have captured the physics.

Analycia Wait! eagerly I think I see what you mean—there is no demand that
this approximate calculation needs to be analytical?

Numerio Yes. Moreover, approximation also mitigates some of the problems in
numerical methods regarding the complexity of interpretation. An overly-
complex method may yield information that is simply too fine-grained to be
analysed easily, but an approximate numerical method can strip away much of
that complexity by focusing on a suitable subspace of solutions, and if it
correctly matches the rigorous outcome then we can be confident that we
understand the physics.

_Numerio and Analycia agree to treat approximation as both a strength and a
weakness, and as a vital way to obtain new perspectives on physics, and move
on to the question of modularity in _ab initio_ methods. _

###  4.2 Modularity in ab initio methods

[1]

  * Numerio It is often thought that ab initio methods do not provide the level of modularity that analytical methods can. However, even if the solutions provided by ab initio methods are numerical, the Hamiltonian is typically comprised of a set of analytical terms. By switching these terms on and off, we can gain insights into their role in a particular aspect of the physics in question.

To give one example of this, in Fig. 12 I show an intensity scan of the degree
of coherence of the remaining ion in strong-field ionisation of CO2. To gain
physical insight, we can deactivate a number of interactions, and then compare
the result to the ‘true’ coherence. This comparison shows how the interplay of
different mechanisms contributes, in a non-trivial way, to the total
coherence.

Figure 12:  Modularity in ab initio calculations: the quantum coherence
between the $\sigma_{g}$ and $\sigma_{u}$ ionic states of CO2, as a function
of laser intensity, can be examined by switching different couplings on and
off, providing a valuable window into which effects are most essential.
Plotted from unpublished data obtained during the initial phase of the
research reported in ruberti2018 .

Analycia This is certainly a good demonstration of modularity within an ab
initio method! However, would you say this is a typical example? I imagine
that many _ab initio_ methods would struggle to match this level of
modularity. In this case, each individual calculation should come at a
reasonably small cost in terms of computational time and resource, so that
many calculations can be carried out. For some methods, the massive scale of
individual calculations means that this level of modularity cannot be
afforded.

More broadly, in numerical methods you cannot always do this type of switching
procedure, particularly when it comes to spatial or momentum interference
patterns battle2 ; Amini2020 . It is extremely rare to come across numerical
methods that are able to split between these, and provide clear assignments to
the different channels that are interfering. So sometimes, yes, you can switch
interactions on and off and assign things in a modular way, but _ab initio_
methods are often limited in the degree to which they can do this.

Numerio Yes, the degree of modularity often depends on the problem at hand.
Activating and deactivating Hamiltonian terms provides insight in certain
problems, but others will not be aided by this procedure. I suppose the ideal
situation would be to have a method which can solve a given problem using
reasonable computational resources, while keeping enough modularity to provide
the required physical understanding.

In that regard, one of the strongest tools is the use of approximations
specifically tailored to the situation—which is one clear instance of
approximation being a strength, as we have just agreed.

###  4.3 Are _both_ analytical and numerical methods required in scientific
discovery?

_Numerio and Analycia, agreeing that analytical and _ab initio_ methods are
not always used in equal measure, turn to discuss the impact that this has on
knowledge and discovery._

[1]

  * Numerio We have presented a case for analytical and numerical methods working best for scientific discovery when they are used in equal measure. However, is this always necessary? Take Fig. 13, where the analytical model matches experiment just as well as the TDSE model. My natural inclination in a case like this would be to carry out a large, multi-electron calculation for this problem—but since the analytical model has described the experiment so well, would it be worthwhile? In some cases, like this one, analytical methods can stand on their own, while in other cases it will not get you very far and you need to really crank the handle of big codes.

Analycia I agree—often one method will dominate, and it may be because it
works better or it may be due to historical reasons. However, while you can
prove wrong a model when its results do not match experiments, it does not
work the other way around: you can never prove that a physical model is right.
Therefore the agreement of different theoretical approaches is all the more
precious in that regards. In addition to what you said, analytical and _ab
initio_ methods are two different powerful tools, which lead to differences in
our understanding and interpretations. In different situations one method is
more useful than the other for advancing knowledge.

Numerio Yes, the methods we use will affect our understanding, but maybe we
should not be too hung up on this. We should mostly be driven by moving
between the discoveries of new knowledge. For instance, when we explore
different systems such as high-harmonic generation in liquids (e.g.
Flettner2003 ; Luu2018 ) or in solids (e.g. Ghimire2011 ; Vampa2017 ), we
start from the knowledge we had in the gas phase and push its limits and
extend this knowledge. Whether this knowledge originates from analytical, ab
initio, or experimental studies is ultimately not so important.

Analycia I can see what you are getting at, but in practice we cannot ignore
biases different methods imbue in our knowledge. Let us see what our community
thinks about this?

The audience responses to this question are presented as poll 3 in Table 16.

Numerio It seems it is a mixed bag, with most hedging their bets in how often
each method should be used.

Analycia We should take this result with a pinch of salt but perhaps we can
agree this means it is very contextual. Physical processes we study should be
attacked by exploring all of the approaches we have at hand.

Numerio Yes, but also we should prioritise these methods by their range of
applicability as well as the level of insight they elucidate.

_Our combatants decide that they will each include more methods in their
arsenal, as well as working together, to aid the process of scientific
discovery. However, Numerio still has one final bone to pick. _

headercellbackground 1 |  headercellbackgroundThe distinction of methods that provide quantitative vs qualitative insights is more useful in practice than the analytical and ab initio distinction? (Twitter link)
---|---
lg |  lg Agree |  lg 84% | Disagree | 16% |  lg |  lg |  |  |  lg Sample size |  lg 38
lg |  lg Agree |  lg 100% | Disagree | 0% |  lg |  lg |  |  |  lg Sample size |  lg 10
headercellbackground 2 |  headercellbackgroundWould you feel more confident in planning an experiment based on guidance from analytical theory, ab initio simulations or a hybrid model? (Twitter link)
lg |  lg Analytical |  lg 19% | ab initio | 38% |  lg hybrid |  lg 43% |  |  |  lg Sample size |  lg 42
lg |  lg Analytical |  lg 29% | ab initio | 57% |  lg hybrid |  lg 14% |  |  |  lg Sample size |  lg 7
headercellbackground 3 |  headercellbackgroundCombining analytical, numerical, ab initio and experimental methods in equal measure is the best route to discovery? (Twitter link)
lg |  lg Always |  lg 15% | Mostly | 41% |  lg Sometimes |  lg 44% | Never | 0% |  lg Sample size |  lg 46
lg |  lg Always |  lg 18% | Mostly | 36% |  lg Sometimes |  lg 36% | Never | 9% |  lg Sample size |  lg 11
headercellbackground 4 |  headercellbackgroundWhat do you agree with more:
(1) Computing (and quantum computing power) will advance so much we will not
need analytical methods anymore.
(2) We do not actually really need such computational power to understand
physics, as our understanding is more closely represented by analytical
models. (Twitter link)
lg |  lg I agree with (1) |  lg 11% | I agree with (2) | 34% |  lg I agree with neither |  lg 55% |  |  |  lg Sample size |  lg 43
lg |  lg I agree with (1) |  lg 50% | I agree with (2) | 50% |  lg I agree with neither |  lg 0% |  |  |  lg Sample size |  lg 2
Table 3:  Audience polls taken over the Zoom (upper rows) and Twitter (lower
rows) platform during the presentation.161616As part of the audience response
to this poll, J. Tennyson argued that this is a rhetorical question with the
wording placing the velvet on the answer ‘yes’, which should be considered
when interpreting the poll results.

###  4.4 The role of increasing computational power

_The consequences of increasing computational power are a common theme in the
development of modern physics mentioned time and time again. Numerio turns to
its role in attoscience. _

[1]

  * Numerio One aspect that was very apparent as we looked at the evolution of numerical and _ab initio_ methods in attoscience is that, even given the considerable challenges initially faced by the field, these methods have achieved many tasks that would have seemed completely impossible even a scant few years ago.

Going out on a limb, I would even claim that these improvements will continue
and accelerate, particularly once quantum computers become available, and that
these advancements will drastically reduce the need for analytical methods, or
even—despite their advantages, which we discussed earlier— eliminate it
altogether. And, I wonder, does our community agree with this?

The response to the poll is shown in poll 4, on Table 16.

Analycia I find it quite interesting that you should use a phrasing of the
form ‘computing power will make analytical theory obsolete’—because of how
_old_ that idea is. That concept dates back a full six decades berry2007 , to
when electronic computers were first being developed in the 1960s (to replace
_human_ computers). Within that context, it is understandable that people got
the impression that analytical theory—with its emphasis on special functions,
integral transforms and asymptotic methods—would be displaced by raw
computation.

However, over the past sixty years, time and time again the facts have
demonstrated the opposite: we now place a _higher_ value on special functions
and asymptotic methods than we did back then. Of course, it is possible that
at least some of the analytical methods of attoscience will be displaced by
raw simulations, at least of the single-electron TDSE, but whenever this
narrative starts to look appealing, it is important to take the long view and
keep this historical context in mind.

###  4.5 Audience Questions and Comments

Over the course of the panel discussion battle , questions and comments were
raised by the audience which helped challenge and develop the arguments being
fielded by the combatants. We present them here, voicing our answers through
Numerio and Analycia, and referencing answers already given above.

[1]

  * Reinhard Dörner Is our field really after discoveries? Is it not more about finding and solving interesting puzzles?

This question was motivated by the distinction made by Thomas Kuhn in his
famous book _The structure of scientific revolutions_ Kuhn2012 . Therein, he
argued that the times where the most progress is steadily made are times of
‘normal science’, where what scientists do is best described as solving
riddles with the tools of the paradigm they are working in Doerner2020 .

Numerio I agree that we are not looking for new fundamental laws. Here it is
instructive to connect back to the definition of ‘ _ab initio_ ’, particularly
to remind ourselves that we have fixed the fundamental, theoretical ‘reference
frame’. In attosecond physics, we are not yet looking for new fundamental
laws: we already have established fundamental laws, the ‘rules of the game’,
and we are looking for new solutions to the fundamental quantum-mechanical
equations of motion. The space of solutions is potentially infinite, as is the
amount of new physical phenomena yet to be described. In our case, we are
interested in understanding the physics of atoms and molecules, driven by
light-matter interactions, in new and unexplored regimes—and I would agree
that this can be described as finding and solving interesting puzzles.

Analycia I would disagree: the fact that something is not ‘fundamental’ does
not stop it from being a discovery. If nothing else, that viewpoint completely
disregards discoveries made in other sciences which are not ‘fundamental’. I
would say that it is still discovery if it is new knowledge.

Numerio Perhaps this is a matter of terminology: to me, speaking of
‘discovery’ entails finding new laws or entirely novel particles or
dimensions, which do not occur in attoscience. In our domain, the basic rules
are already set, and we are solving a puzzle which is as interesting as it is
difficult. There are many different ways of arranging the pieces of this
puzzle, with each one representing, in principle, a different physical
scenario that we can tackle with our theoretical methods, be they _ab initio_
, analytical, or hybrid. That said, we know only a limited set of such
scenarios and I agree that, when we find a new one, it can also be seen as a
discovery.

Analycia Yes, I see what you mean—but there is not always such a clear split
between rules and scenarios, i.e., between laws of physics and their
solutions. There is a level where we only have the fundamental laws, but there
are also higher levels of understanding and abstraction where the behaviour of
a set of solutions can become a ‘rule’, a law of physics, in itself. And, I
would argue, our role in attoscience is to discover these laws. However, I do
agree that our work mostly takes place within the fixed paradigm of a single
set of fundamental laws.

_The conclusion that Analycia and Numerio take away from this is that a
solution to a problem may still be interesting and useful, irrespective of
whether it is called a discovery._

[1]

  * Tom Meltzer Do you think the range of applicability of a model is not more important than whether it’s _ab initio_ , numerical, analytical, semiclassical and so on?

Numerio I agree that this is an important aspect of any method. That said, I
would also say that an even more important lens is whether the model gives us
insights of a qualitative or quantitative nature, as we argued in more detail
in Section 1.4.3.

Analycia I have a similar view on this. Here it is important to remark that
one of the big reasons why, I would argue, we should move away from the
‘analytical-versus-_ab initio_ ’ view is that, ultimately, it is impossible to
have a method which is truly _ab initio_. We discussed some of this in detail
in Section 1.4.6, regarding the impact of the choice of basis set has on a
method: to the extent that we must supply physics insights into that choice,
it takes the method away from the _ab initio_ ideal.

However, I would go beyond that, since there are many other ways in which the
base assumptions of how we phrase the problem—which often go unquestioned—can
affect the physics. The most obvious example in attoscience is macroscopic
effects coming from the propagation of light inside our sample, but there are
also other, more esoteric aspects—say, the appearance of collective quantum
effects such as superfluorescence Mercadier2019 ; Rohringer2020qbattles , or
effects coming from field quantisation Lewenstein2020quantum —which are ruled
out by the basic framing, and this takes us away from ever reaching the _ab
initio_ ideal.

Numerio Those are fair points, but there is also a danger of throwing out the
baby with the bathwater here, in discarding the valuable work done in pursuit
of that ideal. In that regard, I would argue that a better definition for ‘
_ab initio_ ’ could be ‘models with approximations that have well-defined
error bounds for an explicit parameter range, such that any neglected physics
will lie within these bounds’. We know what physics gets neglected, and we
should be able to quantify those well enough to know they are not relevant (as
well as the types of questions that become inaccessible); for any additional
sources of error, like the choice of basis set, the error must be
quantifiable.

Here is, I would say, where the range of applicability of the model is most
important, as it dictates whether those sources of error are quantifiable and
negligible—what one could call ‘allowed’ approximations—or into a regime with
unquantified approximations. This is then a major component that determines
where we can place our method on the qualitative-versus-quantitative spectrum.

_In summary, our combatants agree the range of applicability is a central
aspect to consider, which is essential for reshaping ideas on what is ‘ _ab
initio_ ’. However, they also assert that the characterisation of whether a
model provides qualitative or quantitative insights is the most important
feature to consider. _

[1]

  * (Anonymous) Can the single-configuration time-dependent Hartree-Fock method be used effectively to study multi-electron effects on atomic and molecular systems?

Numerio The method you mention (TDHF) can definitely be used to describe
multi-electron effects. It is equivalent, in its linear-response
reformulation, to the Random Phase Approximation with exchange (RPAX) method,
which has been widely used (mainly by the condensed-matter community) and
which can provide accurate molecular excitation energies and transition
moments. Using TDHF in its full time-dependent character to study non-
perturbative dynamics (beyond a perturbative approach) is certainly possible.

Analycia That sounds quite complicated for limited gain. Is it really worth
it?

Numerio This is a good method, but we also have available several multi-
configuration versions, including MCTDHF and TD CAS- and RAS-SCF, which are
generally more effective. This makes TDHF, in my opinion, only a
computationally-cheaper alternative which should be considered for large
systems that are not amenable to the full multi-configurational treatment.

[1]

  * Jens Biegert An experiment is like doing an _ab initio_ simulation in the sense that one can change the boundary conditions, but it does not necessarily allow you to disentangle what happens. However, analytic, semi-analytical and hybrid methods do allow insight.

Numerio It is true that _ab initio_ calculations are often characterised as
the theoretical analogue of experiments, and that analytical methods drill
down on the insightful details. However, in my experience, I feel that this is
a mischaracterisation, as I am aware of many instances where _ab initio_
methods were able to disentangle a variety of physical interactions, by virtue
of their modular properties.

Analycia Oh, really? I would be interested to hear more, as this is an area
where I always felt we analyticists held an advantage.

_Given the level of interest in this topic, the combatants broadened its scope
into the discussion given in Sec. 4.2._

## Author contributions

All the authors were involved in the preparation of the manuscript. All the
authors have read and approved the final manuscript.

## Author ORCID iDs

  1. Gregory S. J. Armstrong: 0000-0001-5949-2626

  2. Margarita A. Khokhlova: 0000-0002-5687-487X

  3. Andrew S. Maxwell: 0000-0002-6503-4661

  4. Emilio Pisanty: 0000-0003-0598-8524

  5. Marco Ruberti: 0000-0003-0424-3643

## Acknowledgements

We (the authors, and our combatants Analycia and Numerio) are deeply grateful
to the organisers of the Quantum Battles in Attoscience conference as well as
to the audience members for their active participation, and we are especially
grateful to the Battle’s ‘referee’, Stefanie Gräfe, for her insightful and
even-handed moderation during the panel discussion. We also thank S.D.
Bartlett, T. Rudolph and R.W. Spekkens Bartlett2006  as well as G. Galilei
Galilei1632  for inspiration for the format of this paper.

GSJA acknowledges funding from the UK Engineering and Physical Sciences
Research Council (EPSRC) under grant EP/T019530/1. MAK acknowledges funding
from the Alexander von Humboldt Foundation. ASM acknowledges grant
EP/P510270/1 funded by the UK EPSRC. ASM and EP acknowledge support from ERC
AdG NOQIA, Spanish Ministry of Economy and Competitiveness (“Severo Ochoa”
program for Centres of Excellence in R&D (CEX2019-000910-S), Plan National
FIDEUA PID2019-106901GB-I00/10.13039/501100011033, FPI), Fundació Privada
Cellex, Fundació Mir-Puig, and from Generalitat de Catalunya (AGAUR Grant No.
2017 SGR 1341, CERCA program, QuantumCAT _U16-011424, co-funded by the ERDF
Operational Program of Catalonia 2014-2020), MINECO-EU QUANTERA MAQS (funded
by State Research Agency (AEI) PCI2019-111828-2/10.13039/501100011033), EU
Horizon 2020 FET-OPEN OPTOLogic (Grant No 899794), Marie Sklodowska-Curie
grant STRETCH No. 101029393, and the National Science Centre, Poland-Symfonia
Grant No. 2016/20/W/ST4/00314. MR acknowledges funding from the EPSRC/DSTL
MURI grant EP/N018680/1.

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