# Quantum phenomena in attosecond science

Lidice Cruz-Rodriguez<EMAIL_ADDRESS>Department of Physics and
Astronomy, University College London, London WC1E 6BT, UK  Diptesh Dey
<EMAIL_ADDRESS>Department of Chemistry, Northwestern University,
Evanston, Illinois 60208, USA  Antonia Freibert<EMAIL_ADDRESS>hamburg.de Institute for Nanostructure and Solid-State Physics, Department of
Physics and Center for Free-Electron Laser Science, University of Hamburg,
22761 Hamburg, Germany  Philipp Stammer<EMAIL_ADDRESS>ICFO –
Institut de Ciencies Fotoniques, The Barcelona Institute of Science and
Technology, 08860 Castelldefels (Barcelona), Spain  Atominstitut, Technische
Universität Wien, 1020 Vienna, Austria

###### Abstract

Attosecond science has opened up new frontiers in our understanding of
processes happening on the intrinsic timescale of electrons. The ability to
manipulate and observe phenomena at the attosecond level has yielded
groundbreaking insights into processes such as electron dynamics and the
behavior of matter under extreme conditions. This interdisciplinary field
bridges various research areas such as quantum optics, quantum chemistry and
quantum information science facilitating a cohesive understanding. However,
despite many emerging successful applications, the discussion about intrinsic
quantum effects has mainly been ignored. In this Perspective, we explore the
latest advancements in quantum phenomena within attosecond science,
encompassing both experimental and theoretical progress. Specifically, in the
context of high-harmonic generation and above-threshold ionization, we focus
on discerning genuinely quantum observations and distinguishing them from
classical phenomena. Additionally, we illuminate the often overlooked yet
significant role of entanglement in attosecond processes, elucidating its
influence on experimental outcomes.

##  I Introduction

For over a century, investigating matter-field interactions has been a
fundamental aspect of physics and chemistry research. Matter exposed to high-
intensity laser sources, where the field strength is comparable to the binding
energy of the atom or molecule, undergo fascinating non-linear phenomena only
accessible in the strong-field regime. The interaction with strong-laser
fields leads to photoionization, producing photoelectrons via the above-
threshold ionization (ATI) or high-order ATI (HATI) mechanisms
agostini1979free ; becker2002above . Furthermore, the laser-driven electron
can return to the parent ion, emitting high-order harmonics (HHG)
ferray1988multiple  or producing a second photoelectron via the nonsequential
double ionization process (NSDI) l1983multiply . An overview of these
fundamental strong-field processes is illustrated in Fig. 1.

Figure 1: An illustration of the key strong-field processes: (a) high-harmonic
generation (HHG), (b) above-threshold ionization (ATI) and (c) nonsequential
double ionization (NSDI) along with the corresponding quantum states of light
and matter. The states of the driving field mode before and after interaction
are represented by $\ket{\alpha_{L}}$ and $\ket{\alpha_{L}+\chi_{L}}$,
respectively. The initial ground state and final continuum stateof the
electron are given by $\ket{g}$ and $\ket{\mathbf{v}}$. High-order harmonic
modes, due to HHG, are denoted by $\ket{\chi_{q}}$.

The observation of the HHG process led to the subsequent development of
extremely short-laser pulses of only a few hundred attoseconds ($1\,\rm
attosecond=10^{-18}\rm s$) PaulScience01 ; hentschel2001attosecond , creating
an entirely new field of research known as attosecond science
CorkumNaturePhys07 . Within this field, novel experimental techniques have
allowed us to look at the electron dynamics in their natural timescale with
impact in atomic, molecular, and solid-state physics calegari2016advances .

The development of novel theoretical approaches has accompanied the
advancement of experimental observations. A general challenge faced by
attosecond investigations is the presence of many degrees of freedom with
complex and, in general, unknown interactions. It is generally accepted that
the field-matter interaction is described by the time-dependent Schrödinger
equation (TDSE). However, in typical strong-field scenarios, the exact
solution of the TDSE is challenging and becomes unfeasible as the number of
degrees of freedom increases. Additionally, the physical interpretation of the
results from ab initio methods is non-trivial armstrong2021dialogue .
Henceforth, approximate methods have been constantly developed to describe
strong-field matter interaction, usually by treating the light field
classically stammer2023limitations . The most widespread method is the strong-
field approximation (SFA) amini2019symphony , which has been very successful
in modeling strong-field induced phenomena. However, as attosecond science
extends to more complex systems, the general quest for novel theoretical
approaches to account for electron-electron correlations or the coupling
between the electron and nuclear degrees of freedom is crucial. Also closely
related to the theoretical modeling of strong-field phenomena has been the
question of the quantum or classical origin of the experimental observations.
For example, in the process of NSDI, the role of electron-electron correlation
has been long known becker2012theories , but whether we are in the presence of
classical HoPRL05  or quantum hao2014quantum ; maxwell2016controlling ;
quan2017quantum  correlations has remained an open question.

Entanglement represents one of the most iconic departures from classical
physics. Most of the laser-induced phenomena involve the breaking of an
originally bound subsystem, leading to the creation of two or more subsystems
that can in general be entangled. However, over the years, the role of
entanglement in strong-field phenomena has mainly remained unexplored. Only
recently, new studies have focused on understanding the role of entanglement
in the photoionization process; both looking at the entanglement between the
photoelectron and the parent ion Vrakking2022 ; koll2022experimental ;
Nishi2019 ; Busto2022 ; ruberti2023bell ; Nabekawa2023 , the electron-electron
entanglement in the NSDI process christov2019phase ; maxwell2022entanglement
and the entanglement between electronic and nuclear degrees of
freedomMcKemmish2011 ; Vatasescu2013 ; Vatasescu2015 ; Izmaylov2017 ;
SanzVicario2017  and its impact on ultrafast photochemistry Blavier2022 .

Additionally, new approaches have been developed to achieve a full quantum
optical description of attosecond processes, including the matter and the
light field. Over the years, the theoretical models employed to describe
strong-laser field phenomena have relied on a quantum description of matter
while considering a classical laser field. Only recently, the bridge between
strong-field physics and quantum optics has been built by developing a fully
quantum description of the laser-matter interaction gorlach2020quantum ;
lewenstein2021generation ; stammer2023quantum . The HHG and ATI processes have
been studied within a complete quantum electrodynamics framework, showing the
appearance of non-classical and entangled states of light. Furthermore,
quantum optical approaches allow to consider non-classical states of light
driving the processes gorlach2023high ; stammer2023role ; even2023photon  and
challenge the widespread semi-classical models stammer2023limitations . These
recent developments have opened the possibility of connecting attosecond
physics with quantum information science and modern quantum technologies
bhattacharya2023strong ; lewenstein2022attosecond .

In this Perspective, we address how the notions of entanglement and non-
classical states of light emerge in processes on the attosecond time scale.
The paper will be structured as follows. In section II, we will address the
quantum phenomena of radiation in attoscience, focusing on the recently
developed full quantum optical descriptions, and consider the HHG and ATI
process under this new formulation. We will discuss the observables to witness
non-classicality, as well as the appearance of entanglement of the field
modes. In section III, we explore the recent analysis of the role of
entanglement in attosecond processes, with emphasis on ion-photoelectron
entanglement, nuclear-electronic entanglement, and electron-electron
entanglement. We conclude in section IV by providing an outlook where novel
quantum phenomena in attosecond science could emerge.

##  II Quantum phenomena of radiation in attoscience

In this section, we will focus on the two most prominent phenomena in
attosecond science, namely light scattering and photoionization by means of
the process of high-harmonic generation (HHG) and above-threshold ionization
(ATI), respectively. We particularly focus on the question which observations
are genuinely quantum and how to distinguish non-classical processes from its
classical counterparts. This is of particular current interest since the
theoretical approaches for the aforementioned processes have recently
developed from the semi-classical to the full quantum optical description
gorlach2020quantum ; lewenstein2021generation ; stammer2023quantum . The
additional degrees of freedom from the quantization of the electromagnetic
field allow to consider field observables not conceivable before, and using
Hilbert space constructs for the driving field as well as for the generated
harmonic radiation offer new possibilities to envision experiments for optical
quantum technologies o2009photonic ; gilchrist2004schrodinger .

###  II.1 High harmonic generation

It is known that classical radiation is generated from oscillating charge
currents scully1999quantum . In the semi-classical description of the HHG
process lewenstein1994theory  the electron is driven by an intense classical
electromagnetic field $E_{cl}(t)$, which is coupled to the electrons dipole
moment. The induced time-dependent dipole moment, is then analyzed to obtain
the spectrum of the scattered light in the HHG process. Here, the scattered
light is again obtained from a classical oscillating charge current,
emphasizing that the generated radiation is classical. However, within the
semi-classical description of attosecond processes the oberservables of the
light field is limited. In particular, field properties obtained from a
quantum optical perspective remain elusive and genuine non-classical features
of the field are hidden.

We will first briefly review the existing work on the quantum optical
description of radiation in attosecond processes, some of them going beyond
classical radiation properties, and then elaborate possible new avenues to
investigate the non-classical radiation properties in strongly driven systems.

####  II.1.1 Quantum optical formulation of HHG

To construct a full quantum optical formulation of the process of HHG, it is
inevitable to quantize the radiation field. A detailed quantum
electrodynamical quantization procedure and derivation of the effective
interaction Hamiltonian is given in stammer2023quantum . Constitutive in the
full quantum description is the electric field operator $E_{Q}(t)$ coupled to
the electron dipole moment, instead of the semi-classical coupling via the
classical oscillating electric field $E_{cl}(t)$. Furthermore, quantization of
the field requires to define the initial state boundary condition imposed by
the experiment. Assuming the HHG process is driven by classical coherent laser
light, described by a coherent state $\ket{\alpha_{L}}$, the classical
interaction can be recovered lewenstein2021generation ; stammer2023limitations
; stammer2023quantum2 . This classical interaction leads to product coherent
states in the harmonic field modes after the interaction

$\displaystyle\ket{\\{0_{q}\\}}\xrightarrow[\text{HHG}]{}\ket{\\{\chi_{q}\\}}=\bigotimes_{q}\ket{\chi_{q}},$
(1)

where $\ket{\\{0_{q}\\}}$ is the initial vacuum state of all harmonics. This
finding is in agreement with the semi-classical picture, and induces a shift
$\chi_{L}$ in the coherent state driving laser amplitude
lewenstein2021generation . The state after the interaction thus reads
$\ket{\alpha_{L}+\chi_{L}}$ and accounts for the depletion of the pump field.
In addition, the coherent state amplitudes of the harmonic field modes are
given by the Fourier transform of the dipole moment expectation value, i.e.
representing a classical oscillating charge current

$\displaystyle\chi_{q}\propto\int_{0}^{\infty}dt\bra{g}d(t)\ket{g}e^{i\omega_{q}t},$
(2)

which further manifests the validity of the semi-classical approach.

However, when solving the interaction of the radiation field with the dipole
moment of the electron, it is crucial to neglect the dipole moment
correlations sundaram1990high ; stammer2022theory  in order to obtain the
product coherent states in (1). This is because the exact operation for the
parametric process of HHG acting on the field state is given by

$\displaystyle
K_{\mathrm{HHG}}=\bra{g}\mathcal{T}\exp[-i\int_{0}^{\infty}dt\,d(t)E_{Q}(t)]\ket{g},$
(3)

which is in general hard to solve since the commutator of the dipole moment at
different times remains an operator in the electron Hilbert space
$[d(t_{1}),d(t_{2})]\in\mathcal{H}$, and therefore the different field modes
in the electric field operator mix stammer2022high ; stammer2022theory ;
stammer2023entanglement . Neglecting dipole moment correlations in (3), the
operator can be solved and leads to a multimode displacement operation in
phase-space

$\displaystyle K_{\mathrm{HHG}}$
$\displaystyle\simeq\mathcal{T}\exp[-i\int_{0}^{\infty}dt\,\bra{g}d(t)\ket{g}E_{Q}(t)]$
$\displaystyle=\prod_{q}D[\chi_{q}],$ (4)

where $D[\chi_{q}]=\exp[\chi_{q}b_{q}^{\dagger}-\chi_{q}^{*}b_{q}]$. Since the
approximate operation in (II.1.1) is linear in the field operators
$b_{q}^{(\dagger)}$ the modes do not couple and the field remains in product
coherent states as reported in lewenstein2021generation ; rivera2022strong .
Going beyond, and taking into account dipole moment correlations, can lead to
interesting effects such as field correlations pizzi2023light , mode squeezing
gorlach2020quantum ; stammer2023entanglement  or entanglement between the
modes stammer2023entanglement ; stammer2022theory . Further investigation
about the role of the aforementioned correlations could reveal novel radiation
properties of the generated harmonics.

####  II.1.2 Radiation properties: observables to witness non-classicality

With the quantum optical formulation of the HHG process we can consider field
observables beyond the HHG spectrum which is obtained from the oscillating
classical charge current of the electron. Recent investigations have shown
that there is a variety of interesting observations which further characterize
the field properties. These allow to obtain insights about the associated
quantum state of the field modes and to witness non-classical properties. We
refer to Table 1 for a collection of observables in HHG and a classification
of their genuine quantum origin and possible classical counterparts.

Beyond the spectrum, which has its classical counterpart in HHG, there are a
variety of different observables accessible within the quantum optical
description. One of the most prominent examples to witnesses non-classical
field states is the Wigner function $W(\beta)$, obtained from homodyne
measurements of the probe field together with a local oscillator
gerry2023introductory ; schleich2011quantum ; rivera2021new . Such homodyne
measurements allow to measure the field quadrature of the probe field and to
reconstruct it’s Wigner function (for experimental details related to HHG see
stammer2023quantum ; bhattacharya2023strong ). In the quantum optical context
the Wigner function is commonly used to classify the state into classical and
non-classical fields. Since classical probability distributions are strictly
positive, negativities in the Wigner function quasi-probability are witnesses
of non-classical light schleich2011quantum ; footnote1 . In the process of
HHG, there has been recent progress in studying the radiation properties of
the scattered light field using the Wigner function. For instance in
pizzi2023light  it was shown that initial matter correlations are imprinted in
the Wigner function, leading to non-Gaussian Wigner functions with
$W(\beta)\geq 0$, and allows for the engineering of the quantum states of the
harmonic field modes. Another direction to perform quantum state engineering
of light using the process of HHG has been developed in
lewenstein2021generation  and subsequent works rivera2022strong ;
stammer2022high ; stammer2022theory ; stammer2023quantum ;
lamprou2023nonlinear . Instead of using correlated matter systems, the state
engineering is approached with conditional measurements. Correlating the
harmonic radiation with the driving field allows to generate optical cat and
kitten states in the driving field mode lewenstein2021generation ;
rivera2022strong ; stammer2022high  or the harmonic field modes
stammer2022high ; stammer2022theory . All those methods allow to generate high
photon number non-classical field states ranging from the infrared to the
extreme ultraviolet regime. The intensities of the non-classical optical cat
states are even sufficient to drive non-linear processes lamprou2023nonlinear
and allow to perform quantum state engineering of the second harmonic field
obtained when driven by a non-classical field. Another prominent example of
non-classical field properties is the squeezing of a field quadrature.
Squeezing allows to manipulate the field fluctuations along a specific
quadrature, the most prominent being the amplitude or phase squeezing
walls1983squeezed . While the amplitude squeezing reduces the fluctuations in
the field intensity, and consequentially increases the phase fluctuations, the
phase squeezing increases the field intensity fluctuations, in contrast
leading to a better defined phase. For instance, classical coherent states
saturate the product of these two uncertainties, whereas the non-classicality
in squeezed states is manifested if the variance of one field quadrature is
below the level of the vacuum fluctuations walls1983squeezed . Squeezing of
the field in HHG has been reported recently due to dipole moment correlations
stammer2023entanglement  or when driven by squeezed light tzur2023generation .
An illustration of the aforementioned states in phase-space by means of their
Wigner function can be seen in Fig.2.

Moreover, quantum optics allows us to consider driving fields beyond the
classical perspective, and study the process of HHG driven by fields differing
from the conventional assumption of coherent laser light was initiated in
gorlach2023high . In there, it was shown that light fields with a higher
intensity fluctuation than classical coherent laser light, such as bright
squeezed vacuum or thermal states, lead to an extended cut-off in the HHG
spectrum. In contrast, intense photon number states have a well defined
intensity and therefore have the same cut-off as the conventional classical
coherent driving field. The well defined intensity of the photon number states
has the consequence of an arbitrary phase and leads to vanishing mean electric
field values at all times $E_{cl}(t)=0$. Consequentially, this poses the
question about the validity of the semi-classical perspective in such cases
stammer2023limitations , and the role of the optical phase of the driving
field in the process of HHG was studied in stammer2023role . This ultimately
raises the question of whether quantum optical coherence of the driving field,
in terms of non-vanishing off-diagonal density matrix elements, is necessary
to generate high-order harmonics. Indeed, it was recently shown in
stammer2023role  that quantum optical coherence is not needed to drive the HHG
process and that the generated harmonics are diagonal in their respective
photon number basis. This has the intriguing consequence that inferring the
coherence properties of the harmonics from the HHG spectrum alone is not
justified.

Figure 2: Examples of Wigner functions $W(\beta)$ for different states
considered in the quantum optical formulation of HHG. The top row shows
classical states: (a) coherent state $\ket{\alpha}$ and (b) phased mixed
coherent state
$\rho\propto\int_{0}^{2\pi}d\phi\outerproduct{\absolutevalue{\alpha}e^{i\phi}}{\absolutevalue{\alpha}e^{i\phi}}$.
In the bottom row are non-classical states (c) squeezed vacuum and (d) optical
cat state.

####  II.1.3 Entanglement of the field modes

Besides the Wigner function representation of a single field mode, there is a
crucial aspect of quantum theory when considering more than a single mode.
Considering all the field modes participating in the process of HHG we can
study entanglement within the total quantum state (for a brief introduction to
entanglement see section III). As shown in Eq. (1) the final state of all
field modes is a product state, which holds when neglecting dipole moment
correlations (see Eq. (II.1.1)). It was shown that this is related to the
assumption of negligible ground state depletion of the electron in the case of
moderate driving laser intensities stammer2022theory ; stammer2023entanglement
. However, as indicated in stammer2023quantum2  this approximation has the
consequence of neglecting high order terms, such that the field operators of
different modes do not mix. It was thus shown in stammer2023entanglement  that
going beyond the assumption of negligible continuum state population leads to
mode mixing, and thus, all field modes in the HHG process are, in general,
entangled. Using the process of HHG, therefore, allows the generation of
massive and bright entangled states using conditioning schemes stammer2022high
or resonant media yi2024generation .

Table 1: Overview on the recent findings for the full quantum optical
description of the HHG process and its classification into classical (x) and
quantum processes (✓), or observations which are yet not classified. In case a
classical analog exists, the respective counterpart is given.  Quantum
phenomena in HHG
---
|  Observable |  Classical counterpart | Quantum
HHG emission lewenstein2021generation ; rivera2022strong  |  HHG spectrum |  Dipole moment expectation value | x
Field mode entanglement stammer2022high ; stammer2022theory ; stammer2023entanglement  |  Correlation measurement (unknown) |  Unknown | ?111Although entanglement between the field modes is a feature of the full quantum optical description, the associated correlations in the observations can still have classical counterparts.
Mode squeezing stammer2023entanglement ; tzur2023generation  |  Homodyne detection |  x | ✓222Note that in tzur2023generation  only a fraction of the harmonic modes exhibit non-classical squeezing below the vacuum fluctuatons, and the residual harmonics are classical.
Quantum optical coherence stammer2023role  |  Spectrum & Photon statistics |  $\rho_{q}=\frac{1}{2\pi}e^{-\frac{\absolutevalue{\chi_{q}}^{2}}{2}}\int_{0}^{2\pi}d\phi\outerproduct{\absolutevalue{\chi_{q}}e^{i\phi}}{\absolutevalue{\chi_{q}}e^{i\phi}}$ | x333The phase mixed coherent state $\rho_{q}$ gives rise to the same HHG spectrum and photon statistics as the pure coherent state $\ket{\chi_{q}}$ of equal amplitude. The mixed state $\rho_{q}$ can equivalently be expressed in diagonal form in the photon number basis $\rho_{q}=e^{-\absolutevalue{\chi_{q}}^{2}}\sum_{n}\frac{\absolutevalue{\chi_{q}}^{2n}}{n!}\outerproduct{n}{n}$, and therefore does not exhibit quantum optical coherence stammer2023role .
Interferometric measurements |  Unknown | ?444Although the HHG spectrum and the photon statistics do not allow to infer on the quantum coherence properties of the harmonic radiation, it does not exclude possible interferometric measurements to reveal quantum coherence, i.e. the necessity of off-diagonal density matrix elements in the photon number basis to describe the phenomena.
Non-Gaussian Wigner function |  Optical cat states $W(\beta)\leq 0$ lewenstein2021generation  |  x | ✓
Wigner function with $W(\beta)\geq 0$ pizzi2023light  |  Mixed coherent state $\rho=\sum_{i}p_{i}\outerproduct{\alpha_{i}}{\alpha_{i}}$ | x555The fact that non-Gaussian Wigner functions refer to non-classical states only holds for pure quantum states hudson1974wigner , leading to negativities in the Wigner function.
HHG for $E_{cl}(t)=0$ stammer2023limitations ; stammer2023role  |  HHG spectrum |  Driven by phase mixed coherent state | x666The phase mixed coherent stat $\rho\propto\int_{0}^{2\pi}d\phi\outerproduct{\absolutevalue{\alpha}e^{i\phi}}{\absolutevalue{\alpha}e^{i\phi}}$ allows to generate high harmonics, which makes the assumption of non-vanishing electric fields unnecessary to describe the HHG process stammer2023role . Further, it was recently shown that HHG in the semi-classical description also occurs for vanishing classical electric fields weber2023quantum . However, the subtle difference here is that for the driving fields considered in stammer2023limitations ; stammer2023role , the electric field vanishes at all times.
Extended HHG cut-off gorlach2023high  |  HHG spectrum |  Driven by classical fluctuating fields | x777The extension of the HHG cut-off was shown to occur also for classical fields exhibiting large fluctuations in the field intensity, such as thermal states gorlach2023high .
Analog Simulation of HHG arguello2024analog  |  Absorption image for HHG spectrum |  HHG spectrum | x888Although quantum simulators buluta2009quantum  are a versatile platform to study quantum phenomena, the approached used in arguello2024analog  to simulate the HHG spectrum is based on classical interactions.

###  II.2 Above threshold ionization

For the process of above-threshold ionization (ATI), the semi-classical
description has thus far focused on the photoionization process alone.
However, due to the inherent limitations of such a classical description of
the light field, it is easily overlooked that the photoionization process can
be accompanied by the emission of radiation. In the following, we highlight
the insights obtained when analyzing the ATI process in light of a full
quantum optical perspective.

####  II.2.1 Quantum field coupled to ATI electrons

The photoionization mechanism in ATI can be well described by considering
classical driving fields, while the radiation properties are often neglected
in the process. However, recent advances in the full quantum optical
description of ATI stammer2023quantum  have revealed that the photoionization
process is accompanied by noteworthy consequences for the light field. For
instance, it was shown that the ATI process is accompanied by the emission of
radiation and that the driving laser field can experience variable changes
depending on the emission direction of the photoelectron stammer2023quantum .
Further, it was shown that quantizing the radiation field allows to study the
role of entanglement between the emitted electron and the simultaneously
emitted light field rivera2022light . This allows to study hybrid light-matter
entangled systems, which can potentially be used for quantum state engineering
of light. In milovsevic2023quantum  it was shown, that the light emission
during ATI can be controlled by varying the laser intensity, wavelength and by
considering different driving field polarizations.

####  II.2.2 ATI driven by non-classical light

As was shown for the process of HHG (see section II.1.2), driving fields
beyond the classical perspective leads to new observations. This is also the
case for ATI as shown in even2023photon , in which a bright squeezed state is
considered to induce the photoionization mechanism, with profound consequences
on the dynamical properties of the driven electron. This has measurable
consequences on the photoelectron momentum distribution of the ionized
electron fang2023strong . Furthermore, the observed extension of the HHG-
cutoff for driving fields with increased intensity fluctuations, the same
occurs in photoionization. It was shown in wang2023high  that the
photoelectron energy distribution experiences a similar extension of the
cutoff energy.

##  III Entanglement phenomena in attoscience

Entanglement Horodecki2009  describes an inherent quantum phenomena where the
components within a quantum system become interconnected in such a way that
the individual subsystem has no definite description. Consequently, the state
of a multipartite quantum system cannot be adequately represented by a single
product of wave functions $\ket{\psi_{i}}$ describing the individual parts,
i.e. $\ket{\Psi}\neq\bigotimes_{i}\ket{\psi_{i}}$. In particular, the
corresponding quantum correlations can reveal stronger correlations within the
composite system than classically allowed, and contradict the classical
assumption that any physical object possesses predetermined and individual
properties. Quantum entanglement is a fundamental component of quantum theory
and has been the subject of extensive research and fascination in the realm of
physics. In the following, we delve into the significance of quantum
entanglement in photoionization or photoionization-induced processes. These
processes form the foundation for a large variety of spectroscopic and imaging
techniques within the field of attosecond science.

Table 2: The table summarizes the different entanglement measures together
with the advantages and disadvantages as well as how they have been used in
the context of attosecond science.  Entanglement measure and witness in
attoscience
---
Entanglement Measure |  Relation to Entanglement |  Applications in Attoscience |  Pros and Cons
Purity: P = Tr(ρ_A^2)≤1, where $\rho_{A}=\Tr_{B}(\outerproduct{\psi_{AB}}{\psi_{AB}})$ is the reduced density matrix |  $1/d\leq P\leq 1$, with $d$ the dimension of the Hilbert space. For separable states $P=1$ and maximally entangled states $P=1/d$. |  Measure of the ion-photoelectron entanglement in photoinization process vrakking2021control ; Busto2022 ; Vrakking2022 ; Laurell2022  |  • entanglement measure for pure states • non-trivial extension to entangled mixed states
Entanglement entropy: S(ρ_A) = - Tr(ρ_A logρ_A ), where $\rho_{A}$ is the reduced density matrix |  $S(\rho_{A})=0$ for pure states $\rho_{A}$ and therefore separable $\outerproduct{\psi_{AB}}{\psi_{AB}}$ and $S(\rho_{A})$ maximal for maximally mixed $\rho_{A}$ and thus maximally entangled $\outerproduct{\psi_{AB}}{\psi_{AB}}$ |  Measure of the ion-photoelectron entanglement in photoinization process Ruberti2019 ; Nabekawa2023  |  • clear interpretation in terms of entropy and information theory • non-trivial extension to entangled mixed states
Logarithmic negativity: LN(ρ) =log∥ ρ^Γ∥_1 where $\rho^{\Gamma}$ is the partial transpose of $\rho$ and $\|A\|_{1}=\Tr[\sqrt{A^{\dagger}A}]$ denotes the trace norm |  Uses Positive Partial Transpose criterion: applies the partial transpose to the system and check if $\rho^{\Gamma}=(\mathds{1}_{A}\otimes T_{B})[\rho]\geq 0$ positive semi-definite (equivalent to positive eigenvalues). But not a sufficient condition for entanglement in arbitrary dimensions |  Quantifies entanglement between the OAM of the two photoelectrons in NSDI maxwell2022entanglement  |  • simple computation • directly applicable to mixed states • can not detect entanglement in arbitrary dimensions of the subsystems, i.e. state can be entanglement but still $LN(\rho)=0$
Entanglement witness $\mathcal{W}$ is an operator that is positive on all separable states |  Associated with observables whose expectation values are negative for entangled states, i.e. Tr(Wρ)¡0 Distinguish a subset of entangled states as non-separable. |  Quantifies entanglement between the OAM of the two photoelectrons in NSDI maxwell2022entanglement  |  • directly applicable to mixed states • versatile, can be tailored to specific system under consideration

###  III.1 Ion-photoelectron entanglement

Photoionization involves the disintegration of a complex many-electron quantum
system, like an atom or molecule, into an emitted photoelectron drifting away
and the remaining parent ion. The induced dynamics of the resulting ionic or
photoelectronic subsystem is often the focus in attosecond experiments. These
experiments often rely on interference effects, and the measurements typically
focus on probing the ultrafast dynamics either within the ion
SansoneNature2010 ; GoulielmakisNature10 ; CalegariScience14  or photoelectron
system PaulScience01 ; Kienberger2002 .
In most cases, theoretical investigations of photoionization-induced processes
treat the parent ion and photoelectron separately, overlooking their possible
quantum entanglement. For instance, in studies concerning ultrafast hole
dynamics and the associated charge migration in polyatomic molecules
CederbaumCPL99 ; Despre2015 ; Du2019 ; MatselyukhNatPhys2022 ;
FolorunsoJPCA2023 , a sudden coherent wave packet formation of the parent ion
is assumed yielding a well-defined initial state. However, the question how
such a coherent state is initially generated remains. Similarly, in studies
centered on the emitted photoelectron, the quantum correlations of the
combined system have largely been disregarded Guillemin2015 .
Nevertheless, quantum entanglement arising between the parent ion and
photoelectron during attosecond photoionization can significantly influence
the entire quantum system. Particularly, entanglement constrains coherence
properties, potentially hindering the observation of delay-dependent
observables sensitive to these coherences Carlstrom2018 ; RubertiPCCP2022 .
Although the precise role of ion-photoelectron entanglement in attosecond
science remains largely unexplored, recent experimental and theoretical
studies have increasingly focused on understanding the influence of quantum
entanglement on experimental observations.
For example, pioneering experimental research on the evolution of an electron
hole in ionized $\mathrm{Kr}$ atoms GoulielmakisNature10  already indicated
how the entanglement with the accompanying photoelectron might restrict
electronic coherence Smirnova2010 , further supported by theoretical work on
similar systems Pabst2011 ; Karamatskou2020 . Furthermore, entanglement acting
as a limiting factor for vibrational and electronic coherence of ionized
$\mathrm{H_{2}}$ molecules was demonstrated by several papers
vrakking2021control ; Vrakking2022 ; koll2022experimental ; Nishi2019 ;
Nabekawa2023 . The measure used to quantify the degree of entanglement has
been the purity of the reduced density matrix of one subsystem (see Table II).
In vrakking2021control , the purity of the reduced density matrices of the ion
and photoelectron was used to show how the degree of entanglement can be
controlled by changing the time delay between two ionizing extreme ultraviolet
(XUV) pulses. Fig. 3 (a) shows the purity as a function of the XUV-XUV time
delay for different pulse duration. This was followed by the experiment
koll2022experimental  showing how the vibrational coherence of the
$\mathrm{H_{2}}^{+}$ ions can be controlled by varying this time delay.
Additionally, it has been explored how using a chirped XUV pulse can also
affect the degree of entanglement Vrakking2022 . Fig 3(b) presents the purity
of the ion-photoelectron density matrix as a function of the chirped parameter
of the XUV pulse. Overall, these studies vrakking2021control ;
koll2022experimental ; Vrakking2022  extensively discuss the inherent relation
between quantum entanglement and coherence, and propose the potential
manipulation of these quantities by adjusting the spectral attributes of the
photoionizing pulse. The characteristics of the ionizing pulse, e.g. pulse
duration or mean photon energy will dictate the kinetic energy of the
photoelectron and thus its proximity to the parent ion that will result in
different interchannel couplings to the ionic states and therefore the
entanglement Pabst2011 ; RubertiPCCP2022 .

Entanglement naturally arises here since different combinations of
rovibrational states of the parent ion and angular momentum states of the
ejected electron are possible, such that the total wavefunction represents an
entangled state. To that end, a witness and quantification of this
entanglement is of current interest. Since the underlying process is
dynamical, its measurement will at first require the identification of
appropriate observables that are easy to detect and which allow to infer about
entanglement. A possible proof of quantum entanglement would be a (loophole
free) test of the violation of Bell inequalities Bell1964 , which in the case
of measuring ion-photoelectron entanglement, means detecting correlations in
the measurement of two non-commuting observables for the ion and the
photoelectron. Such a test has been recently proposed in the context of
attosecond science ruberti2023bell . The authors theoretically reported a
violation of Bell inequalities in the photoionization of argon atoms, with the
degree of freedom encoded in the spin angular momentum of the photoelectron.
The simulations, based on first principles, reported on correlations in the
measurements of the observables which violate the Bell inequality; while the
challenge lies in identifying the appropriate pairs of non-commuting
observables. On the experimental side, a considerable promise to quantify
entanglement due to spin-orbit interactions was demonstrated Laurell2022 .
Within this work a continuous-variable quantum state tomography of the
photoelectron was used – a protocol that is commonly used in quantum optics to
tomographically reconstruct the quantum state of a system, which they further
validated with density matrix calculations.

###  III.2 Nuclear-electronic entanglement

In the field of molecular dynamics, nuclear and electronic motion are
generally associated with different timescales justifying the conventional
practice of separating the slowly moving nuclei from the fast moving electrons
building the foundation of the Born-Oppenheimer (BO) approximation. This
approximation serves as fundamental framework, particularly in visualizing
potential pathways of photochemical and photophysical processes. Although
simulations involving excited electronic states often require to go beyond the
traditional BO picture by incorporating non-adiabatic coupling terms (if
electronic states cross each other), the initial step of partitioning the
molecular system into distinct, independent electronic and nuclear degrees of
freedom persists.

In investigations primarily centered on nuclear dynamics, this assumption
manifests in the concept of a potential energy surface (PES). These surfaces
represent solutions to the electronic problem alone and depend solely on the
nuclear position in a parametric manner. Subsequently, the nuclear problem is
solved, yielding the well-established depiction of nuclear wavepackets moving
on PESs (within and beyond the BO approximation). However, this sequential
resolution of the electronic and nuclear problems relies on the existence of
two disentangled sets of degrees of freedom.

In many theoretical inquiries into purely electronic phenomena in molecules,
such as hole migration, electron dynamics is often simulated at fixed nuclear
geometries CederbaumCPL99 ; RemaclePNAS06 ; Yong2022 . This practice again
overlooks the potential entanglement between the nuclear and electronic
degrees of freedom. Nevertheless, several studies have already indicated a
high correlation between electronic and nuclear dynamics, suggesting the
likelihood of electronic decoherence or even recoherence due to nuclear motion
VacherPRL17 ; Arnold2017 ; ArnoldPRL18 ; JiaJPCL19 ; DeyPRL22 . Specifically,
the dependence of quantum coherence on electronic-nuclear entanglement created
by non-adiabatic coupling has been demonstarted in Vatasescu2013 ;
Vatasescu2015 . While non-adiabatic couplings generally lead to both a
breakdown of the BO approximation and to entanglement, the actual degree of
electron-nuclear entanglement and the degree of validity of the BO
approximation are unrelated Izmaylov2017 . Moreover, recent work Blavier2022
delves into the time evolution of entanglement, utilizing a matrix
representation of the wave functions for the entangled systems. A singular
value decomposition of this matrix at each time will result in singular
values, whose time evolution will govern the time evolution of entanglement.
Their findings not only confirm the influence of non-adiabatic couplings on
molecular entanglement but also highlight the interaction with the exciting
pulse as a major contributing factor.

Thus, even under the assumption of complete unentangled parent ion and
photoelectron, with the ionic subsystem exhibiting full coherence, internal
entanglement may persist between the molecular electronic and nuclear degrees
of freedom RubertiPCCP2022 . This internal entanglement has implications for
electronic coherence and consequently impacts the outcomes of experiments in
the field of attoscience.

###  III.3 Electron-electron entanglement

Figure 3: The figure shows different entanglement measures and their
applications in attoscience. Panel (a) shows the purity of the reduced density
matrix of the ion-photoelectron system as a function of the time delay between
two XUV pulses for different pulse durations. The theoretical predictions show
how, by tuning the time delay, we can control the degree of entanglement from
low (high purity) to high (small purity), and thus have a high and low degree
of vibrational coherence, respectively. Panel (b), adapted from Vrakking2022 ,
shows the dependence of the purity of the ion-photoelectron density matrix as
a function of the chirp of the XUV pulse. The figure shows theoretical results
obtained from the solution of the time-dependent Schrödinger equation
indicating the transition from a pure to a mixed state and can be understood
as the consequence of the ion-photoelectron entanglement. Panels (c) and (d),
adapted from maxwell2022entanglement , present the logarithmic negativity
$E_{\mathcal{N}}$ and the average value of the entanglement witness
$\langle\mathcal{W}\rangle$, respectively as constructed in
maxwell2022entanglement . The figure shows an extensive search over different
targets and field intensities, the blue text corresponds to $\lambda=400\;\rm
nm$, and the red text to $\lambda=800\;\rm nm$. Panel (a) reprinted with
permission from vrakking2021control . Copyright 2021 by the American Physical
Society. Panel (b) reprinted from Vrakking2022  under the premises of the
Creative Commons Attribution 4.0 International License. Panels (c) and (d)
reprinted from maxwell2022entanglement  under the premises of the Creative
Commons Attribution 4.0 International License
https://creativecommons.org/licenses/by/4.0/

Electron-electron correlation has been observed from the early days of
attosecond science l1983multiply  in the context of laser-induced double and
multiple ionization. In particular, the process of non-sequential double
ionization (NSDI) has been extensively studied, revealing the role of
electron-electron correlation in the laser-induced dynamics WeberNat00 ;
becker2012theories  where the inter-electronic Coulomb interaction governs the
entanglement Akoury2007 .

The theoretical modeling of this phenomena has been controversial. Although
classical models have been quite successful in modeling NSDI HoPRL05 , it has
been debated whether we are in the presence of classical or quantum electron-
electron correlation. Several studies have revealed the onset of quantum
interference between the two photoelectrons hao2014quantum ;
maxwell2016controlling ; quan2017quantum . However, the study of electron-
electron entanglement has not been the center of attention. One important
reason is that typically measuring entanglement will require evaluating the
continuous variable density matrix or entanglement measures like the purity or
von Neumann entropy, which are very challenging to compute (see Table II for a
summary of the different measures of entanglement, the advantages and
disadvantages, and the applications in attoscience). A breakthrough in this
direction has been made in maxwell2022entanglement . They demonstrated the use
of orbital angular momentum (OAM) to quantify and measure entanglement in
NSDI. Here, entanglement arises due to the shared angular momentum during
recollision with subsequent ionization mechanism. The use of the OAM
significantly reduced the computational cost of evaluating the density matrix
and allowed a simple evaluation of the logarithmic negativity and entanglement
witnesses, exploring a wide range of targets and field intensities as shown in
Fig. 3 panels (c) and (d). In particular, the entanglement witness can be
decomposed into local measurements, which has a significant impact on
experimental entanglement detection. Thus, in NSDI, which provides a direct
manifestation of electron-electron correlation, the OAM entanglement of
photoelectrons can reveal its fundamental non-classical nature. A question
that remains open is how we can measure the OAM of the electrons. In the past
decade, new techniques have become available to measure the OAM for electron
vortex beams harvey2017stern ; noguchi2019efficient . As suggested in
maxwell2022entanglement , adding some of these techniques to the typical
reaction microscope, which is already suitable for studying the many-particle
entangled dynamics schmidt2021coltrims , could be one of the directions to
explore. Some challenges will arise using alkali metals, and further
theoretical studies should be performed to enhance entanglement in noble gas
targets.

##  IV Outlook

In this Perspective, we have presented an overview of quantum phenomena in
attosecond physics and particularly highlighted the emerging full quantum
descriptions and the increasing interest in entanglement in attosecond
processes. We can see that this rapidly growing field of quantum phenomena in
attosecond science provides new insights into the underlying dynamics and its
consequences to experimental observations. Moreover, this emerging field still
provides promising new perspectives for processes on the attosecond timescale
and can open many different research directions in this field. In the
following we present a list of potential future research directions to further
emphasize on the deep connection between quantum phenomena and attosecond
physics with the use for quantum information science lewenstein2022attosecond
.

  * •

Non-classical field properties: As illustrated in Sec. II the full quantum
optical description of attosecond processes has enabled to consider field
observables beyond the classical realm. Investigating novel field properties,
inaccessible before, is far from being uncovered and many different
perspectives can be considered. In particular, genuine quantum features such
as entanglement and squeezing of the optical field stammer2023entanglement ;
tzur2023generation  or driving the process by non-classical light
gorlach2023high ; stammer2023limitations ; stammer2023role ;
tzur2023generation  are only being considered very recently. Properties such
as the photon statistics or quantum noise as well as field correlation
functions are only a fraction of what can be explored within the Hilbert space
of the optical field modes inaccessible before.

  * •

Field-field entanglement: Interestingly, the derivation of the entangled field
state in stammer2023entanglement  does not rely on specific strong field
dynamics and is therefore generic to all parametric processes. We thus expect,
that this opens a new avenue for studying and engineering massive entangled
states of light when considering that Ref. pizzi2023light  indicated that
material correlations can be imprinted in the field properties. This might
allow to probe topological materials baldelli2022detecting ;
bera2023topological  or quantum phase transitions alcala2022high  using the
process of HHG.

  * •

Quantum state engineering of light: The quantum optical description of HHG
opened the avenue to use strong laser driven processes as a novel platform for
quantum state engineering of light stammer2023quantum ; bhattacharya2023strong
. Further understanding the role of quantum optical coherence stammer2023role
, the importance of the material correlations pizzi2023light ;
stammer2023entanglement  as well as bringing these approaches to different
targets such as molecular rivera2023quantum_molecule , plasma
lamprou2021plasma  or solid state systems rivera2024nonclassical ;
rivera2022quantum_wannierbloch ; gonoskov2022nonclassical  is of current
interest. Further controlling the properties of the generated non-classical
field states is of great importance and extending its range of application to
non-linear optics lamprou2023nonlinear  or quantum metrology
martos2023metrological  is of current interest.

  * •

Influence of entanglement on electronic coherence: In atomic or molecular
photoionization, ion-photoelectron entanglement limits the electronic
coherence in the ionic system GoulielmakisNature10 ; koll2022experimental .
The electronic coherence further depends on the spectral characteristics of
the ionizing pulse Pabst2011  and any internal entanglement between the
electronic and nuclear degrees of freedom in the ionic system Blavier2022 .
Understanding the role of entanglement is of particular interest in charge
migration studies to achieve charge-directed reactivity. Search for techniques
to quantify the entanglement is currently in progress, with a considerable
promise shown by the recent application of quantum state tomography in
attoscience Laurell2022 .

  * •

Entanglement in attosecond science: An alternative route to treat the
entanglement problem is the Bohmian framework pladevall2019applied . This
formulation offers a natural playground to distinguish between classical and
quantum phenomena, and entanglement has already been addressed in several
contexts. The applications of this theory to model entanglement in strong-
field-induced phenomena are yet to be explored. The initial steps have been
taken in christov2019phase  in the context of NSDI. Recently, an
implementation of Bohmian mechanics using the conditional wave function method
has been applied to model entanglement in multi-particle bosonic systems
elsayed2018entangled . The extension of such a model to multi-particle
fermionic systems represents a promising alternative to understanding
electron-electron entanglement in double and multi-ionization processes.

  * •

Conceive novel experiments: For all of the aforementioned cases, it is
inevitable to conceive novel experiments. On the one hand, to measure the
previously unexpected field properties as well as to witness the proposed
entanglement in attosecond processes koll2022experimental . This also includes
techniques for the reconstruction of quantum states of the field or the
electronic wavefunctions, which are currently limited to specific cases.

  * •

Bringing attosecond science towards optical quantum technologies: With the
flourishing connection between quantum information science and attosecond
physics, it seems likely that this symbiosis leads to the implementation of
optical quantum technologies lewenstein2022attosecond . This can emerge from
the recently developed quantum state engineering of light approaches or by
using entanglement or quantum coherence as a resource for technologies.

###### Acknowledgements.

The authors thank the organizers of the ‘Quantum Battles in Attoscience 2023’
conference for the opportunity and support during the battle preparation.
L.C.R. acknowledges financial support from the AQuA DIP project grant number
EP/J019143/1. A.F. acknowledges financial support from the Cluster of
Excellence ’CUI: Advanced Imaging of Matter’ of the Deutsche
Forschungsgemeinschaft (DFG) - EXC 2056 - project ID 390715994, from the
International Max Planck Graduate School for Ultrafast imaging $\&$ Structural
Dynamics (IMPRS-UFAST) and from the Christiane-Nüsslein-Vollhard-Foundation.
D.D. acknowledges funding from the National Science Foundation under Grant No.
CHE-2347622 and the European Union’s Horizon 2020 research and innovation
programme under the Marie Sklodowska-Curie Grant agreement No. 892554. P.S.
acknowledges funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No
847517. ICFO group acknowledges support from: ERC AdG NOQIA; MCIN/AEI
(PGC2018-0910.13039/501100011033, CEX2019-000910-S/10.13039/501100011033, Plan
National FIDEUA PID2019-106901GB-I00, Plan National STAMEENA
PID2022-139099NB-I00 project funded by MCIN/AEI/10.13039/501100011033 and by
the “European Union NextGenerationEU/PRTR” (PRTR-C17.I1), FPI); QUANTERA MAQS
PCI2019-111828-2); QUANTERA DYNAMITE PCI2022-132919 (QuantERA II Programme co-
funded by European Union’s Horizon 2020 program under Grant Agreement No
101017733), Ministry for Digital Transformation and of Civil Service of the
Spanish Government through the QUANTUM ENIA project call - Quantum Spain
project, and by the European Union through the Recovery, Transformation and
Resilience Plan - NextGenerationEU within the framework of the Digital Spain
2026 Agenda; Fundació Cellex; Fundació Mir-Puig; Generalitat de Catalunya
(European Social Fund FEDER and CERCA program, AGAUR Grant No. 2021 SGR 01452,
QuantumCAT U16-011424, co-funded by ERDF Operational Program of Catalonia
2014-2020); Barcelona Supercomputing Center MareNostrum (FI-2023-1-0013); EU
Quantum Flagship (PASQuanS2.1, 101113690, funded by the European Union. Views
and opinions expressed are however those of the author(s) only and do not
necessarily reflect those of the European Union or the European Commission.
Neither the European Union nor the granting authority can be held responsible
for them); EU Horizon 2020 FET-OPEN OPTOlogic (Grant No 899794); EU Horizon
Europe Program (Grant Agreement 101080086 — NeQST), results incorporated in
this standard have received funding from the European Innovation Council and
SMEs Executive Agency under the European Union’s Horizon Europe programme).,
ICFO Internal “QuantumGaudi” project; European Union’s Horizon 2020 program
under the Marie Sklodowska-Curie grant agreement No 847648; “La Caixa” Junior
Leaders fellowships, La Caixa” Foundation (ID 100010434): CF/BQ/PR23/11980043.
Views and opinions expressed are, however, those of the author(s) only and do
not necessarily reflect those of the European Union, European Commission,
European Climate, Infrastructure and Environment Executive Agency (CINEA), or
any other granting authority. Neither the European Union nor any granting
authority can be held responsible for them.

## Author contributions

The authors contributed equally to the writing of the manuscript.

## Competing interests

The authors declare no competing interests.

## References

  * [1] Pierre Agostini, F Fabre, Gérard Mainfray, Guillaume Petite, and N Ko Rahman.  Free-free transitions following six-photon ionization of xenon atoms.  Physical Review Letters, 42(17):1127, 1979.
  * [2] W Becker, Felix Grasbon, R Kopold, DB Milošević, GG Paulus, and Herbert Walther.  Above-threshold ionization: From classical features to quantum effects.  Advances in Atomic, molecular, and optical physics, 48:35–98, 2002\.
  * [3] M Ferray, Anne L’Huillier, XF Li, LA Lompre, G Mainfray, and C Manus.  Multiple-harmonic conversion of 1064 nm radiation in rare gases.  Journal of Physics B: Atomic, Molecular and Optical Physics, 21(3):L31, 1988.
  * [4] Anne l’Huillier, LA Lompre, G Mainfray, and C Manus.  Multiply charged ions induced by multiphoton absorption in rare gases at 0.53 $\mu$m.  Physical Review A, 27(5):2503, 1983.
  * [5] P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, Ph. Balcou, H. G. Muller, and P. Agostini.  Observation of a train of attosecond pulses from high harmonic generation.  Science, 292(5522):1689–1692, 2001.
  * [6] Michael Hentschel, Reinhard Kienberger, Ch Spielmann, Georg A Reider, Nenad Milosevic, Thomas Brabec, Paul Corkum, Ulrich Heinzmann, Markus Drescher, and Ferenc Krausz.  Attosecond metrology.  Nature, 414(6863):509–513, 2001.
  * [7] P. B. Corkum and Ferenc Krausz.  Attosecond science.  Nature Physics, 3(6):381–387, 2007.
  * [8] Francesca Calegari, Giuseppe Sansone, Salvatore Stagira, Caterina Vozzi, and Mauro Nisoli.  Advances in attosecond science.  Journal of Physics B: Atomic, Molecular and Optical Physics, 49(6):062001, 2016.
  * [9] Gregory SJ Armstrong, Margarita A Khokhlova, Marie Labeye, Andrew S Maxwell, Emilio Pisanty, and Marco Ruberti.  Dialogue on analytical and ab initio methods in attoscience.  The European Physical Journal D, 75(7):209, 2021.
  * [10] Philipp Stammer.  On the limitations of the semi-classical picture in high harmonic generation.  arXiv preprint arXiv:2308.15087, 2023.
  * [11] Kasra Amini, Jens Biegert, Francesca Calegari, Alexis Chacón, Marcelo F Ciappina, Alexandre Dauphin, Dmitry K Efimov, Carla Figueira de Morisson Faria, Krzysztof Giergiel, Piotr Gniewek, et al.  Symphony on strong field approximation.  Reports on Progress in Physics, 82(11):116001, 2019.
  * [12] Wilhelm Becker, XiaoJun Liu, Phay Jo Ho, and Joseph H Eberly.  Theories of photoelectron correlation in laser-driven multiple atomic ionization.  Reviews of Modern Physics, 84(3):1011, 2012.
  * [13] Phay J. Ho, R. Panfili, S. L. Haan, and J. H. Eberly.  Nonsequential double ionization as a completely classical photoelectric effect.  Phys. Rev. Lett., 94:093002, Mar 2005.
  * [14] XiaoLei Hao, Jing Chen, WeiDong Li, Bingbing Wang, Xiaodong Wang, and Wilhelm Becker.  Quantum effects in double ionization of argon below the threshold intensity.  Physical Review Letters, 112(7):073002, 2014.
  * [15] AS Maxwell and C Figueira de Morisson Faria.  Controlling below-threshold nonsequential double ionization via quantum interference.  Physical Review Letters, 116(14):143001, 2016.
  * [16] Wei Quan, XiaoLei Hao, YanLan Wang, YongJu Chen, ShaoGang Yu, SongPo Xu, ZhiLei Xiao, RenPing Sun, XuanYang Lai, ShiLin Hu, et al.  Quantum interference in laser-induced nonsequential double ionization.  Physical Review A, 96(3):032511, 2017.
  * [17] Marc J J Vrakking.  Ion-photoelectron entanglement in photoionization with chirped laser pulses.  Journal of Physics B: Atomic, Molecular and Optical Physics, 55(13):134001, jun 2022.
  * [18] Lisa-Marie Koll, Laura Maikowski, Lorenz Drescher, Tobias Witting, and Marc JJ Vrakking.  Experimental control of quantum-mechanical entanglement in an attosecond pump-probe experiment.  Physical Review Letters, 128(4):043201, 2022.
  * [19] Takanori Nishi, Erik Lötstedt, and Kaoru Yamanouchi.  Entanglement and coherence in photoionization of h 2 by an ultrashort xuv laser pulse.  Physical Review A, 100(1):013421, 2019.
  * [20] David Busto, Hugo Laurell, Daniel Finkelstein-Shapiro, Christiana Alexandridi, Marcus Isinger, Saikat Nandi, Richard J. Squibb, Margherita Turconi, Shiyang Zhong, Cord L. Arnold, Raimund Feifel, Mathieu Gisselbrecht, Pascal Salières, Tönu Pullerits, Fernando Martín, Luca Argenti, and Anne L’Huillier.  Probing electronic decoherence with high-resolution attosecond photoelectron interferometry.  The European Physical Journal D, 76(7), jul 2022.
  * [21] Marco Ruberti, Vitali Averbukh, and Florian Mintert.  Bell test of quantum entanglement in attosecond photoionization.  arXiv preprint arXiv:2312.05036, 2023.
  * [22] Yasuo Nabekawa and Katsumi Midorikawa.  Analysis of attosecond entanglement and coherence using feasible formulae.  Phys. Rev. Res., 5:033083, Aug 2023.
  * [23] Ivan P Christov.  Phase-locking mechanism in non-sequential double ionization.  Applied Physics B, 125(11):209, 2019.
  * [24] Andrew S Maxwell, Lars Bojer Madsen, and Maciej Lewenstein.  Entanglement of orbital angular momentum in non-sequential double ionization.  Nature Communications, 13(1):4706, 2022.
  * [25] Laura K. McKemmish, Ross H. McKenzie, Noel S. Hush, and Jeffrey R. Reimers.  Quantum entanglement between electronic and vibrational degrees of freedom in molecules.  The Journal of Chemical Physics, 135(24):244110, 12 2011.
  * [26] Mihaela Vatasescu.  Entanglement between electronic and vibrational degrees of freedom in a laser-driven molecular system.  Phys. Rev. A, 88:063415, Dec 2013.
  * [27] Mihaela Vatasescu.  Measures of electronic-vibrational entanglement and quantum coherence in a molecular system.  Phys. Rev. A, 92:042323, Oct 2015.
  * [28] Artur F. Izmaylov and Ignacio Franco.  Entanglement in the born–oppenheimer approximation.  Journal of Chemical Theory and Computation, 13(1):20–28, dec 2016\.
  * [29] José Luis Sanz-Vicario, Jhon Fredy Pérez-Torres, and Germán Moreno-Polo.  Electronic-nuclear entanglement in ${{\mathrm{H}}_{2}}^{+}$: Schmidt decomposition of non-born-oppenheimer wave functions expanded in nonorthogonal basis sets.  Phys. Rev. A, 96:022503, Aug 2017.
  * [30] Martin Blavier, R. D. Levine, and F. Remacle.  Time evolution of entanglement of electrons and nuclei and partial traces in ultrafast photochemistry.  Phys. Chem. Chem. Phys., 24:17516–17525, 2022.
  * [31] Alexey Gorlach, Ofer Neufeld, Nicholas Rivera, Oren Cohen, and Ido Kaminer.  The quantum-optical nature of high harmonic generation.  Nature communications, 11(1):4598, 2020.
  * [32] Maciej Lewenstein, Marcelo F Ciappina, Emilio Pisanty, Javier Rivera-Dean, Philipp Stammer, Th Lamprou, and Paraskevas Tzallas.  Generation of optical schrödinger cat states in intense laser–matter interactions.  Nature Physics, 17(10):1104–1108, 2021.
  * [33] Philipp Stammer, Javier Rivera-Dean, Andrew Maxwell, Theocharis Lamprou, Andrés Ordóñez, Marcelo F Ciappina, Paraskevas Tzallas, and Maciej Lewenstein.  Quantum electrodynamics of intense laser-matter interactions: a tool for quantum state engineering.  PRX Quantum, 4(1):010201, 2023.
  * [34] Alexey Gorlach, Matan Even Tzur, Michael Birk, Michael Krüger, Nicholas Rivera, Oren Cohen, and Ido Kaminer.  High-harmonic generation driven by quantum light.  Nature Physics, pages 1–8, 2023.
  * [35] Philipp Stammer.  On the role of the optical phase and coherence in high harmonic generation.  arXiv preprint arXiv:2309.05010, 2023.
  * [36] Matan Even Tzur, Michael Birk, Alexey Gorlach, Michael Krüger, Ido Kaminer, and Oren Cohen.  Photon-statistics force in ultrafast electron dynamics.  Nature Photonics, 17(6):501–509, 2023.
  * [37] Utso Bhattacharya, Theocharis Lamprou, Andrew Stephen Maxwell, Andres Ordonez, Emilio Pisanty, Javier Rivera-Dean, Philipp Stammer, Marcelo F Ciappina, Maciej Lewenstein, and Paraskevas Tzallas.  Strong–laser–field physics, non–classical light states and quantum information science.  Reports on Progress in Physics, 2023.
  * [38] M Lewenstein, N Baldelli, U Bhattacharya, J Biegert, MF Ciappina, U Elu, T Grass, PT Grochowski, A Johnson, Th Lamprou, et al.  Attosecond physics and quantum information science.  arXiv preprint arXiv:2208.14769, 2022.
  * [39] Jeremy L O’brien, Akira Furusawa, and Jelena Vučković.  Photonic quantum technologies.  Nature Photonics, 3(12):687–695, 2009.
  * [40] Alexei Gilchrist, Kae Nemoto, William J Munro, Timothy C Ralph, Scott Glancy, Samuel L Braunstein, and Gerard J Milburn.  Schrödinger cats and their power for quantum information processing.  Journal of Optics B: Quantum and Semiclassical Optics, 6(8):S828, 2004.
  * [41] Marlan O Scully and M Suhail Zubairy.  Quantum optics, 1999.
  * [42] Maciej Lewenstein, Ph Balcou, M Yu Ivanov, Anne L’huillier, and Paul B Corkum.  Theory of high-harmonic generation by low-frequency laser fields.  Physical Review A, 49(3):2117, 1994.
  * [43] Philipp Stammer and Maciej Lewenstein.  Quantum optics as applied quantum electrodynamics is back in town.  arXiv preprint arXiv:2306.07854, 2023.
  * [44] Bala Sundaram and Peter W Milonni.  High-order harmonic generation: simplified model and relevance of single-atom theories to experiment.  Physical Review A, 41(11):6571, 1990.
  * [45] Philipp Stammer.  Theory of entanglement and measurement in high-order harmonic generation.  Physical Review A, 106(5):L050402, 2022.
  * [46] Philipp Stammer, Javier Rivera-Dean, Theocharis Lamprou, Emilio Pisanty, Marcelo F Ciappina, Paraskevas Tzallas, and Maciej Lewenstein.  High photon number entangled states and coherent state superposition from the extreme ultraviolet to the far infrared.  Physical Review Letters, 128(12):123603, 2022.
  * [47] Philipp Stammer, Javier Rivera-Dean, Andrew S Maxwell, Theocharis Lamprou, Javier Argüello-Luengo, Paraskevas Tzallas, Marcelo F Ciappina, and Maciej Lewenstein.  Entanglement and squeezing of the optical field modes in high harmonic generation.  arXiv preprint arXiv:2310.15030, 2023.
  * [48] Javier Rivera-Dean, Th Lamprou, Emilio Pisanty, Philipp Stammer, Andrés F Ordóñez, AS Maxwell, MF Ciappina, M Lewenstein, and P Tzallas.  Strong laser fields and their power to generate controllable high-photon-number coherent-state superpositions.  Physical Review A, 105(3):033714, 2022.
  * [49] Andrea Pizzi, Alexey Gorlach, Nicholas Rivera, Andreas Nunnenkamp, and Ido Kaminer.  Light emission from strongly driven many-body systems.  Nature Physics, 19(4):551–561, 2023.
  * [50] Christopher C Gerry and Peter L Knight.  Introductory quantum optics.  Cambridge university press, 2023.
  * [51] Wolfgang P Schleich.  Quantum optics in phase space.  John Wiley & Sons, 2011.
  * [52] Javier Rivera-Dean, Philipp Stammer, Emilio Pisanty, Th Lamprou, Paraskevas Tzallas, Maciej Lewenstein, and Marcelo F Ciappina.  New schemes for creating large optical schrödinger cat states using strong laser fields.  Journal of Computational Electronics, 20:2111–2123, 2021.
  * [53] Note that this is not the only non-classical witness using phase-space representations [109], and also states with $W(\beta)\geq 0$ can be ne non-classical such as squeezed states [110].
  * [54] Theocharis Lamprou, Javier Rivera-Dean, Philipp Stammer, Maciej Lewenstein, and Paraskevas Tzallas.  Nonlinear optics using intense optical schr$\backslash$” odinger” cat” states.  arXiv preprint arXiv:2306.14480, 2023.
  * [55] Daniel F Walls.  Squeezed states of light.  nature, 306(5939):141–146, 1983.
  * [56] Matan Even Tzur, Michael Birk, Alexey Gorlach, Ido Kaminer, Michael Krueger, and Oren Cohen.  Generation of squeezed high-order harmonics.  arXiv preprint arXiv:2311.11257, 2023.
  * [57] Sili Yi, Ihar Babushkin, Olga Smirnova, and Misha Ivanov.  Generation of massively entangled bright states of light during harmonic generation in resonant media.  arXiv preprint arXiv:2401.02817, 2024.
  * [58] Robin L Hudson.  When is the wigner quasi-probability density non-negative?  Reports on Mathematical Physics, 6(2):249–252, 1974.
  * [59] Anne Weber, Margarita Khokhlova, and Emilio Pisanty.  Quantum tunnelling without a barrier.  arXiv preprint arXiv:2311.14826, 2023.
  * [60] Javier Argüello-Luengo, Javier Rivera-Dean, Philipp Stammer, Andrew S Maxwell, David M Weld, Marcelo F Ciappina, and Maciej Lewenstein.  Analog simulation of high-harmonic generation in atoms.  PRX Quantum, 5(1):010328, 2024.
  * [61] Iulia Buluta and Franco Nori.  Quantum simulators.  Science, 326(5949):108–111, 2009.
  * [62] Javier Rivera-Dean, Philipp Stammer, Andrew S Maxwell, Th Lamprou, Paraskevas Tzallas, Maciej Lewenstein, and Marcelo F Ciappina.  Light-matter entanglement after above-threshold ionization processes in atoms.  Physical Review A, 106(6):063705, 2022.
  * [63] DB Milošević.  Quantum theory of photon emission during strong-laser-field-induced ionization.  Physical Review A, 108(3):033110, 2023.
  * [64] Yiqi Fang, Feng-Xiao Sun, Qiongyi He, and Yunquan Liu.  Strong-field ionization of hydrogen atoms with quantum light.  Physical Review Letters, 130(25):253201, 2023.
  * [65] ShiJun Wang and XuanYang Lai.  High-order above-threshold ionization of an atom in intense quantum light.  Physical Review A, 108(6):063101, 2023.
  * [66] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki.  Quantum entanglement.  Rev. Mod. Phys., 81:865–942, Jun 2009.
  * [67] Marc JJ Vrakking.  Control of attosecond entanglement and coherence.  Physical Review Letters, 126(11):113203, 2021.
  * [68] H. Laurell, D. Finkelstein-Shapiro, C. Dittel, C. Guo, R. Demjaha, M. Ammitzböll, R. Weissenbilder, L. Neoričić, S. Luo, M. Gisselbrecht, C. L. Arnold, A. Buchleitner, T. Pullerits, A. L’Huillier, and D. Busto.  Continuous-variable quantum state tomography of photoelectrons.  Phys. Rev. Res., 4:033220, Sep 2022.
  * [69] M Ruberti.  Onset of ionic coherence and ultrafast charge dynamics in attosecond molecular ionisation.  Physical Chemistry Chemical Physics, 21(32):17584–17604, 2019.
  * [70] G. Sansone, F. Kelkensberg, J. F. Pérez-Torres, F. Morales, M. F. Kling, W. Siu, O. Ghafur, P. Johnsson, M. Swoboda, E. Benedetti, F. Ferrari, F. Lépine, J. L. Sanz-Vicario, S. Zherebtsov, I. Znakovskaya, A. L’Huillier, M. Yu. Ivanov, M. Nisoli, F. Martín, and M. J. J. Vrakking.  Electron localization following attosecond molecular photoionization.  Nature, 465(7299):763–766, 2010.
  * [71] Eleftherios Goulielmakis, Zhi-Heng Loh, Adrian Wirth, Robin Santra, Nina Rohringer, Vladislav S. Yakovlev, Sergey Zherebtsov, Thomas Pfeifer, Abdallah M. Azzeer, Matthias F. Kling, Stephen R. Leone, and Ferenc Krausz.  Real-time observation of valence electron motion.  Nature, 466(7307):739–743, 2010.
  * [72] F. Calegari, D. Ayuso, A. Trabattoni, L. Belshaw, S. De Camillis, S. Anumula, F. Frassetto, L. Poletto, A. Palacios, P. Decleva, J. B. Greenwood, F. Martín, and M. Nisoli.  Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses.  Science, 346(6207):336–339, 2014.
  * [73] R. Kienberger, M. Hentschel, M. Uiberacker, Ch. Spielmann, M. Kitzler, A. Scrinzi, M. Wieland, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz.  Steering attosecond electron wave packets with light.  Science, 297(5584):1144–1148, 2002.
  * [74] L.S. Cederbaum and J. Zobeley.  Ultrafast charge migration by electron correlation.  Chemical Physics Letters, 307(3):205–210, 1999.
  * [75] V. Despré, A. Marciniak, V. Loriot, M. C. E. Galbraith, A. Rouzée, M. J. J. Vrakking, F. Lépine, and A. I. Kuleff.  Attosecond hole migration in benzene molecules surviving nuclear motion.  The Journal of Physical Chemistry Letters, 6(3):426–431, 2015.  PMID: 26261959.
  * [76] Hongbo Du, Cody Covington, Stephen R. Leone, and Kálmán Varga.  Excited-state electronic coherence in vinyl bromide ions.  Phys. Rev. A, 100:053412, Nov 2019.
  * [77] Danylo T. Matselyukh, Victor Despré, Nikolay V. Golubev, Alexander I. Kuleff, and Hans Jakob Wörner.  Decoherence and revival in attosecond charge migration driven by non-adiabatic dynamics.  Nature Physics, 18(10):1206–1213, 2022.
  * [78] Aderonke S. Folorunso, François Mauger, Kyle A. Hamer, Denawakage D. Jayasinghe, Imam S. Wahyutama, Justin R. Ragains, Robert R. Jones, Louis F. DiMauro, Mette B. Gaarde, Kenneth J. Schafer, and Kenneth Lopata.  Attochemistry regulation of charge migration.  The Journal of Physical Chemistry A, 127(8):1894–1900, 2023.  PMID: 36791088.
  * [79] R. Guillemin, S. Sheinerman, R. Püttner, T. Marchenko, G. Goldsztejn, L. Journel, R. K. Kushawaha, D. Céolin, M. N. Piancastelli, and M. Simon.  Postcollision interaction effects in $kll$ auger spectra following argon $1s$ photoionization.  Phys. Rev. A, 92:012503, Jul 2015.
  * [80] Stefanos Carlström, Johan Mauritsson, Kenneth J Schafer, Anne L’Huillier, and Mathieu Gisselbrecht.  Quantum coherence in photo-ionisation with tailored xuv pulses.  Journal of Physics B: Atomic, Molecular and Optical Physics, 51(1):015201, nov 2017.
  * [81] Marco Ruberti, Serguei Patchkovskii, and Vitali Averbukh.  Quantum coherence in molecular photoionization.  Phys. Chem. Chem. Phys., 24:19673–19686, 2022.
  * [82] Olga Smirnova.  Attosecond prints of electrons.  Nature, 466(7307):701–702, aug 2010.
  * [83] Stefan Pabst, Loren Greenman, Phay J Ho, David A Mazziotti, and Robin Santra.  Decoherence in attosecond photoionization.  Physical Review Letters, 106(5):053003, 2011.
  * [84] Antonia Karamatskou, R Esteban Goetz, Christiane P Koch, and Robin Santra.  Suppression of hole decoherence in ultrafast photoionization.  Physical Review A, 101(4):043405, 2020.
  * [85] J. S. Bell.  On the einstein podolsky rosen paradox.  Physics Physique Fizika, 1:195–200, Nov 1964.
  * [86] F. Remacle and R. D. Levine.  An electronic time scale in chemistry.  Proceedings of the National Academy of Sciences, 103(18):6793–6798, 2006.
  * [87] Haiwang Yong, Shichao Sun, Bing Gu, and Shaul Mukamel.  Attosecond charge migration in molecules imaged by combined x-ray and electron diffraction.  Journal of the American Chemical Society, 144(45):20710–20716, 11 2022.
  * [88] Morgane Vacher, Michael J. Bearpark, Michael A. Robb, and João Pedro Malhado.  Electron dynamics upon ionization of polyatomic molecules: Coupling to quantum nuclear motion and decoherence.  Phys. Rev. Lett., 118:083001, Feb 2017.
  * [89] Caroline Arnold, Oriol Vendrell, and Robin Santra.  Electronic decoherence following photoionization: Full quantum-dynamical treatment of the influence of nuclear motion.  Phys. Rev. A, 95:033425, Mar 2017.
  * [90] Caroline Arnold, Oriol Vendrell, Ralph Welsch, and Robin Santra.  Control of nuclear dynamics through conical intersections and electronic coherences.  Phys. Rev. Lett., 120:123001, Mar 2018.
  * [91] Dongming Jia, Jörn Manz, and Yonggang Yang.  De- and recoherence of charge migration in ionized iodoacetylene.  The Journal of Physical Chemistry Letters, 10(15):4273–4277, 2019\.  PMID: 31287313.
  * [92] Diptesh Dey, Alexander I. Kuleff, and Graham A. Worth.  Quantum interference paves the way for long-lived electronic coherences.  Phys. Rev. Lett., 129:173203, Oct 2022.
  * [93] Th. Weber, H. Giessen, M. Weckenbrock, G. Urbasch, A. Staudte, L. Spielberger, O. Jagutzki, V. Mergel, M. Vollmer, and R. Dörner.  Correlated electron emission in multiphoton double ionization.  Nature, 405:658–661, 2000.
  * [94] Dominique Akoury, K Kreidi, T Jahnke, Th Weber, A Staudte, M Schoffler, N Neumann, J Titze, L Ph H Schmidt, A Czasch, et al.  The simplest double slit: interference and entanglement in double photoionization of h2.  Science, 318(5852):949–952, 2007.
  * [95] Tyler R Harvey, Vincenzo Grillo, and Benjamin J McMorran.  Stern-gerlach-like approach to electron orbital angular momentum measurement.  Physical Review A, 95(2):021801, 2017.
  * [96] Yuuki Noguchi, Shota Nakayama, Takafumi Ishida, Koh Saitoh, and Masaya Uchida.  Efficient measurement of the orbital-angular-momentum spectrum of an electron beam via a dammann vortex grating.  Physical Review Applied, 12(6):064062, 2019.
  * [97] Horst Schmidt-Böcking, Joachim Ullrich, Reinhard Dörner, and Charles Lewis Cocke.  The coltrims reaction microscope—the spyhole into the ultrafast entangled dynamics of atomic and molecular systems.  Annalen der Physik, 533(9):2100134, 2021.
  * [98] Niccolò Baldelli, Utso Bhattacharya, Daniel González-Cuadra, Maciej Lewenstein, and Tobias Graß.  Detecting majorana zero modes via strong field dynamics.  ACS omega, 7(50):47424–47430, 2022.
  * [99] Mohit Lal Bera, Jessica O de Almeida, Marlena Dziurawiec, Marcin Płodzień, Maciej M Maśka, Maciej Lewenstein, Tobias Grass, and Utso Bhattacharya.  Topological phase detection through high-harmonic spectroscopy in extended su-schrieffer-heeger chains.  Physical Review B, 108(21):214104, 2023.
  * [100] Jordi Alcalà, Utso Bhattacharya, Jens Biegert, Marcelo Ciappina, Ugaitz Elu, Tobias Graß, Piotr T Grochowski, Maciej Lewenstein, Anna Palau, Themistoklis PH Sidiropoulos, et al.  High-harmonic spectroscopy of quantum phase transitions in a high-tc superconductor.  Proceedings of the National Academy of Sciences, 119(40):e2207766119, 2022.
  * [101] Javier Rivera-Dean, Philipp Stammer, Andrew S Maxwell, Th Lamprou, Emilio Pisanty, Paraskevas Tzallas, Maciej Lewenstein, and Marcelo Fabián Ciappina.  Quantum optical analysis of high-order harmonic generation in $h_{2}^{+}$ molecular ions.  arXiv preprint arXiv:2307.12381, 2023.
  * [102] Theocharis Lamprou, Rodrigo Lopez-Martens, Stefan Haessler, Ioannis Liontos, Subhendu Kahaly, Javier Rivera-Dean, Philipp Stammer, Emilio Pisanty, Marcelo F Ciappina, Maciej Lewenstein, et al.  Quantum-optical spectrometry in relativistic laser–plasma interactions using the high-harmonic generation process: A proposal.  In Photonics, volume 8, page 192. MDPI, 2021.
  * [103] Javier Rivera-Dean, Philipp Stammer, Andrew S Maxwell, Th Lamprou, Andrés F Ordóñez, Emilio Pisanty, Paraskevas Tzallas, Maciej Lewenstein, and Marcelo Fabián Ciappina.  Nonclassical states of light after high-harmonic generation in semiconductors: A bloch-based perspective.  Physical Review B, 109(3):035203, 2024.
  * [104] Javier Rivera-Dean, Philipp Stammer, Andrew S Maxwell, Theocharis Lamprou, Andrés F Ordóñez, Emilio Pisanty, Paraskevas Tzallas, Maciej Lewenstein, and Marcelo F Ciappina.  Quantum optical analysis of high-harmonic generation in solids within a wannier-bloch picture.  arXiv preprint arXiv:2211.00033, 2022.
  * [105] Ivan Gonoskov, René Sondenheimer, Christian Hünecke, Daniil Kartashov, Ulf Peschel, and Stefanie Gräfe.  Nonclassical light generation and control from laser-driven semiconductor intraband excitations.  arXiv preprint arXiv:2211.06177, 2022.
  * [106] Tomás Fernández Martos, Maciej Lewenstein, Grzegorz Rajchel-Mieldzioć, and Philipp Stammer.  Metrological robustness of high photon number optical cat states.  arXiv preprint arXiv:2311.01371, 2023.
  * [107] Xavier Oriols Pladevall and Jordi Mompart.  Applied Bohmian mechanics: From nanoscale systems to cosmology.  CRC Press, 2019.
  * [108] Tarek A Elsayed, Klaus Mølmer, and Lars Bojer Madsen.  Entangled quantum dynamics of many-body systems using bohmian trajectories.  Scientific reports, 8(1):12704, 2018.
  * [109] Leonard Mandel.  Non-classical states of the electromagnetic field.  Physica scripta, 1986(T12):34, 1986.
  * [110] Rodney Loudon and Peter L Knight.  Squeezed light.  Journal of modern optics, 34(6-7):709–759, 1987.