# Theory of entanglement and measurement in high harmonic generation

Philipp Stammer<EMAIL_ADDRESS>ICFO – Institut de Ciencies
Fotoniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels (Barcelona), Spain

###### Abstract

Quantum information science and intense laser matter interaction are two
apparently unrelated fields. However, the recent developments of the quantum
optical description of the intense laser driven process of high harmonic
generation allow to conceive new light engineering protocols. Here, we
introduce the notion of quantum information theory to intense laser driven
processes by providing the quantum mechanical description of measurement
protocols for high harmonic generation in atoms. We explicitly evaluate
conditioning experiments on individual optical field modes, and provide the
corresponding quantum operation for coherent states. The associated positive
operator-valued measures are obtained, and give rise to the quantum theory of
measurement for the generation of high dimensional entangled states, and
coherent state superposition with controllable non-classical features on the
attosecond timescale.

##  I Introduction

In the process of high harmonic generation (HHG), coherent radiation of higher
order harmonics of the driving laser frequency is generated [1, 2]. The
transfer of coherence, and energy, from the intense laser source to the
harmonic field modes (initially in the vacuum) is achieved by a highly
nonlinear interaction of the driving laser field with the HHG medium, in which
the electron is used as an intermediary between the optical modes. Until
recently this was mainly described in semi-classical terms, in which only the
electronic degrees of freedom are quantized [1], although there have been
early approaches to introduce a fully quantized description of the HHG process
[3, 4, 5, 6, 7, 8, 9]. However, recent advances in the quantum optical
analysis of HHG has established a new direction in the investigation of strong
field physics. This allows to study the quantum mechanical properties of the
harmonic radiation, or to take into account the backaction on the driving
field [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. In particular, it was
shown that conditioning procedures on processes induced by intense laser-
matter interaction can lead to the generation of high-photon number
controllable non-classical field states in a broad spectral range [16, 17, 18,
19].

However, a complete description of these measurement processes is missing thus
far. To make use of the full power of such protocols for quantum state
engineering in optical quantum technologies [21, 22, 23], the corresponding
theory, and the properties of the states of light are needed. One goal in this
direction is to construct the quantum theory of measurement for HHG, and to
obtain the generic structure of the entangled states of the electron and the
field modes in strong field processes. Both is presented in the present work
by introducing the natural language in which measurements are described. This
is done by providing the necessary measurement operations, and corresponding
positive operator-valued measure [24, 25]. The notion of measurement
operations is well known in quantum information science, and has lead to many
advances ranging from light-engineering protocols [26], describing open
quantum systems [27] or quantum control [28]. But, these methods play a new
role in the description of the interaction of an intense laser field with
matter. A quantum operation $\Phi_{n}(\cdot)$ is a completely positive map of
one quantum state to another $\rho\mapsto\rho_{n}=\Phi_{n}(\rho)/P(n)$, and
describe how the state $\rho$ changes during a process with outcome $n$, which
occurs with probability $P(n)=\Tr[\Phi_{n}(\rho)]$. The representation theorem
for quantum operations states that the operation is expressed as
$\Phi_{n}(\rho)=M_{n}\rho M_{n}^{\dagger}$, with linear operators $M_{n}$
satisfying $\sum_{n}M_{n}^{\dagger}M_{n}=\mathds{1}$. The operators $M_{n}$
correspond to generalized measurement operations, which for the special case
of projective (von Neumann) measurements are given by projection operators
$M_{n}=\outerproduct{n}{n}$. This concept of quantum operations naturally
leads to the notion of measurement theory via positive operator-valued measure
(POVM), in which each possible measurement outcome $n$ is associated to a
positive operator $E_{n}=M_{n}^{\dagger}M_{n}$. These operators are the
elements of the set $\\{E_{n}\\}$ defining the POVM, and allow to construct
all possible operations which can be performed on a quantum system. It will be
shown, that within this framework a conditioning measurement on the electron
allows to generate massive, and controllable, entangled and superposition
states of the optical field.

The main findings of the present work are: (i) We provide the detailed
description of the entangled state of the optical field modes in HHG with the
necessary quantum operation to separate it from idle processes. (ii) We
introduce a complete quantum theory of measurement for conditioning schemes in
HHG to perform quantum state engineering protocols. (iii) This ultimately
connects the domain of strong field physics with quantum information science,
which allows to generate novel states of light with previously inaccessible
photon numbers, frequencies, and timescales.

##  II Results

###  II.1 Entanglement in strong field physics

To describe the entangled state of the total system including the electron and
the optical field after strong field light matter interaction we consider an
initially uncorrelated state $\ket{\psi(t_{0})}=\ket{\phi_{i}}\ket{g}$. The
initial state of the optical field $\ket{\phi_{i}}=\otimes_{q}\ket{\phi_{q}}$
is assumed to be pure, and that the modes $q$ are uncorrelated, while the
electron is initially in the ground state $\ket{g}$. The formal solution of
the Schrödinger equation describing the interaction is given by

$\displaystyle\ket{\psi(t)}=U(t,t_{0})\ket{\phi_{i}}\ket{g}.$ (1)

In general, the field and the atom are correlated after the interaction, and
neither the total state
$\ket{\psi(t)}\neq\ket{\phi(t)}\otimes\ket{\epsilon(t)}$, where
$\ket{\epsilon(t)}$ describes the state of the electron, nor the optical field
modes $\ket{\phi(t)}\neq\otimes_{q}\ket{\phi_{q}(t)}$ are described as a
product state after the interaction. To make the correlation between the field
and the electron more apparent we introduce the identity on the electronic
subspace

$\displaystyle\mathds{1}=\outerproduct{g}{g}+\sum_{\mu}\outerproduct{\lambda_{\mu}}{\lambda_{\mu}}+\int
d\mathbf{v}\outerproduct{\mathbf{v}}{\mathbf{v}},$ (2)

where $\ket{\lambda_{\mu}}$ are the excited electronic bound states, and
$\ket{\mathbf{v}}$ are the continuum states with electron momentum
$\mathbf{v}$. The state (1) can now be written as

$\displaystyle\ket{\psi(t)}=$ $\displaystyle
K_{HHG}\ket{\phi_{i}}\ket{g}+\sum_{\mu}K_{EX}^{\mu}\ket{\phi_{i}}\ket{\lambda_{\mu}}$
(3) $\displaystyle+\int
d\mathbf{v}K_{ATI}(\mathbf{v})\ket{\phi_{i}}\ket{\mathbf{v}},$

where $K_{HHG}=\bra{g}U(t,t_{0})\ket{g}$,
$K_{EX}^{\mu}=\bra{\lambda_{\mu}}U(t,t_{0})\ket{g}$, and
$K_{ATI}(\mathbf{v})=\bra{\mathbf{v}}U(t,t_{0})\ket{g}$ are the operators
solely acting on the optical field state. The first term corresponds to the
electron conditioned on the ground state $\ket{g}$ leading to HHG [16], while
the third term corresponds to an electron in a continuum state
$\ket{\mathbf{v}}$ leading to the process of above-threshold ionization (ATI)
[29]. The second term, corresponding to the electron being in an excited bound
state $\ket{\lambda_{\mu}}$ is usually considered to be small compared to the
contribution of HHG and ATI. However, bound state resonances can lead to
important contributions in strong field ionization [30, 31, 32] or high
harmonic generation [33, 34], and are therefore included in the present
analysis. Hence, the interaction of the intense laser field with the atom
leads to the entangled state (3) in which the electron is either found in the
ground state, in an excited state or in a continuum state after the
interaction

$\displaystyle\ket{\psi}=\ket{\phi_{g}}\ket{g}+\sum_{\mu}\ket{\phi_{\mu}}\ket{\lambda_{\mu}}+\int
d\mathbf{v}\ket{\phi_{\mathbf{v}}}\ket{\mathbf{v}},$ (4)

where $\ket{\phi_{g}}=K_{HHG}\ket{\phi_{i}}$, and
$\ket{\phi_{\mathbf{v}}}=K_{ATI}(\mathbf{v})\ket{\phi_{i}}$ are the optical
field states corresponding to the electron conditioned on HHG or ATI,
respectively. The term corresponding to an electronic excitation is associated
to the field in the state $\ket{\phi_{\mu}}=K_{EX}^{\mu}\ket{\phi_{i}}$. This
is the generic entangled state between the optical field and the electron in
intense laser matter interaction. We only assumed an initial pure state of the
field in tensor product form, but no further specification was introduced. The
state is written in the form where experimental conditioning schemes on the
matter part (electronic ground state, excitation or ionization) become
apparent.

###  II.2 Entanglement in high harmonic generation

Since HHG is one of the most prominent processes in intense laser matter
interaction, we shall now describe the entanglement between the different
optical field modes, i.e. between the fundamental and the harmonic modes, in
more detail. To account for the process of HHG a conditioning measurement on
the electron has to be performed. Namely, we project onto the electronic
ground state $\Pi_{HHG}=\outerproduct{g}{g}$. This can, for instance, be done
by considering only the processes where no electron was detected, and assuming
that the excited state population is negligible. In all remaining events, the
electron is in the ground state, and the corresponding state is given by
$\Pi_{HHG}\ket{\psi}=\ket{\phi_{g}}\ket{g}.$ This conditioning effectively
eliminates all idle processes which do not correspond to HHG. Since the object
of interest is the optical field state $\ket{\phi_{g}}$, we trace over the
electronic degrees of freedom. To describe the entanglement between the
different optical field modes we shall now analyze the optical state in more
detail

$\displaystyle\ket{\phi_{g}}=K_{HHG}\ket{\phi_{i}}=\bra{g}U(t,t_{0})\ket{g}[\otimes_{q}\ket{\phi_{q}}].$
(5)

To model the HHG process we assume a single mode intense driving laser, which
we describe by a coherent state $\ket{\phi_{1}}=\ket{\alpha}$, while the
harmonic modes $q\in\\{2,...,N\\}$ are initially in the vacuum
$\ket{\\{0_{q}\\}}=\otimes_{q\geq 2}\ket{0_{q}}$. The field amplitude of the
driving laser is typically in the range of $\absolutevalue{\alpha}\sim
10^{7}$. The interaction of the field with the atomic medium conditioned on
the electronic ground state (leading to HHG) can then be described as an
effective multimode displacement operation (see Methods) [16, 18, 17]

$\displaystyle\ket{\phi_{g}}$
$\displaystyle=\prod_{q=1}^{N}D(\chi_{q})\ket{\phi_{i}}=\ket{\alpha+\chi_{1}}\bigotimes_{q=2}^{N}\ket{\chi_{q}},$
(6)

where
$\chi_{q}=-i\sqrt{q}\mathbf{\kappa}\expectationvalue{\mathbf{d}}(q\omega)$,
with the coupling constant $\mathbf{\kappa}$, which takes into account the
field polarization, and the Fourier component of the time dependent dipole
moment expectation value
$\expectationvalue{\mathbf{d}}(q\omega)=\int_{-\infty}^{\infty}dt\expectationvalue{\mathbf{d}}(t)e^{iq\omega
t}$. The shift of the fundamental mode $\chi_{1}$ accounts for the depletion
of the driving laser due to the generation of coherent radiation in the
harmonic modes with amplitudes $\chi_{q}$. The coherent properties of the
fundamental laser are transferred to the harmonic modes by virtue of the non-
linear interaction with the electron. The associated induced time-dependent
dipole moment $\expectationvalue{\mathbf{d}(t)}=\bra{g}\mathbf{d}(t)\ket{g}$
makes the respective shifts of the modes correlated. In fact, the actual mode
which is populated during the HHG process is a wavepacket mode taking into
account these correlations [16, 18], and can be described by the corresponding
number states $\ket{\tilde{n}}$. To account for the process of HHG, we express
the state in (6) in terms of the number states of the wavepacket mode

$\displaystyle\ket{\phi_{g}}=\sum_{\tilde{n}}\bra{\tilde{n}}\ket{\phi_{g}}\ket{\tilde{n}}.$
(7)

We can now consider two mutual exclusive events, which is either emitting
harmonic radiation or not. If harmonic radiation is generated the wavepacket
described by $\ket{\tilde{n}}$ is excited, and of course, if no harmonics are
generated the wavepacket is in its vacuum state $\ket{\tilde{0}}$. We can
therefore define the set of operators

$\displaystyle\mathcal{A}_{\tilde{n}}=\\{\Pi_{\tilde{0}},\,\Pi_{\tilde{n}\neq
0}\\},$ (8)

which either projects on the sub-space of all possible excitation of the
wavepacket mode

$\displaystyle\Pi_{\tilde{n}\neq 0}=\sum_{\tilde{n}\neq
0}\outerproduct{\tilde{n}}{\tilde{n}},$ (9)

or on its vacuum state

$\displaystyle\Pi_{\tilde{0}}=\outerproduct{\tilde{0}}{\tilde{0}}.$ (10)

This constitute the POVM elements for conditioning between the generation of
harmonic radiation, and no excitation of the corresponding wavepacket mode. We
shall now consider the more interesting case in which harmonic radiation is
generated, i.e. where an excitation of the wavepacket mode is present.
Projecting on $\Pi_{\tilde{n}\neq 0}$ we obtain

$\displaystyle\ket{\phi_{HH}}=\Pi_{\tilde{n}\neq
0}\ket{\phi_{g}}=\sum_{\begin{subarray}{c}\tilde{n}\neq
0\end{subarray}}\bra{\tilde{n}}\ket{\phi_{g}}\ket{\tilde{n}}.$ (11)

We use that $\sum_{\tilde{n}\neq
0}\outerproduct{\tilde{n}}{\tilde{n}}=\mathds{1}-\outerproduct{\tilde{0}}{\tilde{0}}$,
and that the vacuum state of the wavepacket mode is given by the state of the
field prior to the interaction with the atomic medium

$\displaystyle\ket{\tilde{0}}=\ket{\alpha}\bigotimes_{q=2}^{N}\ket{0_{q}},$
(12)

such that the state in (11) is given by

$\displaystyle\ket{\phi_{HH}}=\ket{\alpha+\chi_{1}}\bigotimes_{q=2}^{N}\ket{\chi_{q}}-\xi\ket{\alpha}\bigotimes_{q=2}^{N}\xi_{q}\ket{0_{q}},$
(13)

where $\xi=\bra{\alpha}\ket{\alpha+\chi_{1}}$, and
$\xi_{q}=\bra{0_{q}}\ket{\chi_{q}}$. This is the entangled state between all
modes of the optical field in the process of HHG, and is heralded by the
emission of harmonic radiation. The number of modes which are entangled is on
the order of the harmonic cutoff $\mathcal{O}(N)$, and presents a massively
entangled state of many field modes ($N>10$). Note that this entangled state
is naturally generated by the emission of harmonic radiation, and no
particular measurement has to be performed on the optical modes if ionization
and excitation events are eliminated. The generation of harmonic radiation by
itself serves as the origin of this entangled optical field state due to the
interaction with the atomic medium, and the corresponding correlations induced
between the different field modes. However, the crucial part is to
experimentally separate the process of HHG from all other idle processes such
as ionization or excitation, i.e. implementing the projection $\Pi_{HHG}$.
This can be done by measuring the electrons which might get ionized, and
neglecting all cases in which an electron was detected, i.e. only considering
the cases where no ionization takes place. Alternatively, as done in [16, 17],
the harmonic radiation can be measured together with a subsequent post-
processing procedure in which only those events are taken into account in
which the shift of the harmonic intensity is anti-correlated to the intensity
of the fundamental mode. This likewise allows to separate HHG from all idle
processes. The drawback of this scheme is that some fraction of the optical
field needs to be detected, and is thus inevitably interacting with additional
optical devices leading to decoherence and amplitude reduction. Nonetheless,
this scheme is very powerful to generate non-classical optical cat states with
high photon numbers [16, 18, 17, 19].

###  II.3 Quantum operations in HHG

In the following we introduce the quantum theory of measurement to describe
conditioning experiments in HHG. This can be seen as a new class of quantum
state engineering protocols of light. Suppose we perform a post-selection
measurement on the state $\rho_{HH}=\outerproduct{\phi_{HH}}{\phi_{HH}}$ from
HHG (13). In particular, we perform a measurement on $M$ field modes (denoted
by $q^{\prime}$), and post-select on the coherent states
$\ket{\\{\tilde{\chi}_{q^{\prime}}\\}}$. The state of the remaining $N-M$
field modes accordingly reads
$\rho^{\prime}=\bra{\\{\tilde{\chi}_{q^{\prime}}\\}}\rho_{HH}\ket{\\{\tilde{\chi}_{q^{\prime}}\\}}$.
Let us therefor define a quantum operation $\Phi_{\tilde{n}\neq
0}^{\tilde{\chi}}[\cdot]$, which corresponds to this post-selection
measurement on the modes $q^{\prime}$, and is acting on the initial state of
the remaining modes described by
$\rho_{0}=\Tr_{q^{\prime}}[\outerproduct{\phi_{i}}{\phi_{i}}]$. This quantum
operation for conditioning in the process of HHG reads

$\displaystyle\rho_{0}\mapsto\rho^{\prime}=\frac{\Phi_{\tilde{n}\neq
0}^{\tilde{\chi}}\left[\rho_{0}\right]}{P_{\tilde{n}\neq
0}^{\tilde{\chi}}}=\frac{M_{\tilde{n}\neq
0}^{\tilde{\chi}}\rho_{0}\left(M_{\tilde{n}\neq
0}^{\tilde{\chi}}\right)^{\dagger}}{\Tr\left[E_{\tilde{n}\neq
0}^{\tilde{\chi}}\rho_{0}\right]},$ (14)

where we have defined the effect

$\displaystyle E_{\tilde{n}\neq 0}^{\tilde{\chi}}=\left(M_{\tilde{n}\neq
0}^{\tilde{\chi}}\right)^{\dagger}M_{\tilde{n}\neq 0}^{\tilde{\chi}},$ (15)

with measurement operator

$\displaystyle M_{\tilde{n}\neq
0}^{\tilde{\chi}}=\bra{\\{\tilde{\chi}_{q^{\prime}}\\}}\Pi_{\tilde{n}\neq
0}\ket{\\{\chi_{q^{\prime}}\\}}\prod_{q\neq q^{\prime}}D(\chi_{q}).$ (16)

This measurement operator takes into account the shift of the field modes due
to the interaction with the HHG medium via $D(\chi_{q})$, and the conditioning
in terms of $\Pi_{\tilde{n}\neq 0}$ from the POVM in (8). The schematic
illustration of the experimental conditioning scheme can be seen in Fig. 1.
This allows to define the new POVM corresponding to the possible measurement
operation in terms of coherent states $\ket{\\{\tilde{\chi}_{q^{\prime}}\\}}$
performed on $M$ field modes in HHG

$\displaystyle\mathcal{X}_{HHG}=\left\\{\left.E_{\nu}^{\tilde{\chi}}=\left(M_{\nu}^{\tilde{\chi}}\right)^{\dagger}M_{\nu}^{\tilde{\chi}}\right|\nu\in\mathcal{A}_{\tilde{n}}\right\\},$
(17)

with POVM elements $E_{\nu}^{\tilde{\chi}}$, which satisfy the completeness
relation

$\displaystyle\frac{1}{\pi^{N-M}}\int\prod_{q^{\prime}}d^{2}\tilde{\chi}_{q^{\prime}}\sum_{\nu\in\mathcal{A}_{\tilde{n}}}\left(M_{\nu}^{\tilde{\chi}}\right)^{\dagger}M_{\nu}^{\tilde{\chi}}=\mathds{1},$
(18)

where we sum over $\nu\in\mathcal{A}_{\tilde{n}}$, the elements of the HHG
wavepacket POVM in (8). The measurement operation introduced in (14) is of
general kind, and permits to analyze various conditioning schemes in the
process of HHG. It allows to consider different initial states of the optical
field, e.g. HHG driven by multiple laser modes, non-classical or mixed states.
Note that for different driving laser fields the operation (6) on the modes
$K_{HHG}\simeq\prod_{q}D(\chi_{q})$, due to the interaction with the HHG
medium, is in general different. However, the quantum operation (14)
establishes the connection of the photonic platform of HHG towards quantum
information processing for light engineering protocols in terms of coherent
states. In the following we will show how this can be used to generate new
quantum states of light.

Figure 1: Schematic illustration of the conditioning experiment in HHG. The
initial product state $\ket{\alpha}\otimes\ket{\\{0_{q\geq 2}\\}}$ of the
optical field mode is shifted via $K_{HHG}\simeq\prod_{q}D(\chi_{q})$, and
correlated (via the wavepacket mode $\ket{\tilde{n}}$). Performing a
conditioning measurement on the harmonic modes $\ket{\\{\chi_{q}\\}}$ leads to
the quantum operation $\Phi_{\tilde{n}\neq 0}^{\chi}[\cdot]$ on the initial
coherent state of the driving laser.

###  II.4 High dimensional entangled states

With the general quantum operation introduced above we shall now provide the
explicit scheme to generate high dimensional and controllable entangled states
from HHG. For instance, the measurement in (14) can be performed on the
fundamental mode only ($q^{\prime}=1$), such that
$\rho_{\Omega}=\bra{\alpha+\chi_{1}}\rho_{HH}\ket{\alpha+\chi_{1}}$. The
corresponding quantum operation $\Phi_{\tilde{n}\neq
0}^{\alpha+\chi_{1}}[\cdot]$ on the harmonic modes read (up to normalization)

$\displaystyle\outerproduct{\\{0_{q}\\}}{\\{0_{q}\\}}\mapsto\rho_{\Omega}=\Phi_{\tilde{n}\neq
0}^{\alpha+\chi_{1}}\left[\outerproduct{\\{0_{q}\\}}{\\{0_{q}\\}}\right]$ (19)

with the corresponding measurement operator acting on the initial vacuum state
of the harmonic modes

$\displaystyle M_{\tilde{n}\neq 0}^{\alpha+\chi_{1}}$
$\displaystyle=\sum_{\tilde{n}\neq
0}\bra{\alpha+\chi_{1}}\ket{\tilde{n}}\bra{\tilde{n}}\ket{\alpha+\chi_{1}}\prod_{q=2}^{N}D(\chi_{q})$
$\displaystyle=\left[\mathds{1}-e^{-\absolutevalue{\chi_{1}}^{2}}\outerproduct{\\{0_{q}\\}}{\\{0_{q}\\}}\right]\prod_{q=2}^{N}D(\chi_{q}).$
(20)

The resulting state of the harmonic modes under this measurement operation can
be expressed as the pure state $\ket{\Psi_{\Omega}}=M_{\tilde{n}\neq
0}^{\alpha+\chi_{1}}\ket{\\{0_{q}\\}}$, and is given by

$\displaystyle\ket{\Psi_{\Omega}}=\bigotimes_{q=2}^{N}\ket{\chi_{q}}-e^{-\absolutevalue{\chi_{1}}^{2}}\bigotimes_{q=2}^{N}e^{-\frac{1}{2}\absolutevalue{\chi_{q}}^{2}}\ket{0_{q}}.$
(21)

This is a high dimensional entangled state of the harmonic field modes. It can
be particularly interesting when considering the regime of small harmonic
conversion efficiency, i.e. $\absolutevalue{\chi_{q}}<1$, such that the
coherent state of a single harmonic mode can be approximated as
$\ket{\chi_{q}}\simeq\ket{0_{q}}+\chi_{q}\ket{1_{q}}$. The state (21) can then
be written up to $\mathcal{O}(\absolutevalue{\chi_{q}}^{2})$, and reads

$\displaystyle\ket{\Psi_{\Omega}}=\sum_{q=2}^{N}\chi_{q}\ket{0}^{\otimes(q-2)}\ket{1_{q}}\ket{0}^{\otimes(N-q)}.$
(22)

Now, considering three harmonic modes $\\{q,r,s\\}$, the field has the form of
a $W$-state [35]

$\displaystyle\ket{\Psi_{\Omega_{3}}}=\chi_{q}\ket{1_{q}0_{r}0_{s}}+\chi_{r}\ket{0_{q}1_{r}0_{s}}+\chi_{s}\ket{0_{q}0_{r}1_{s}}.$
(23)

Since the optical field in the process of HHG can be tailored in an accurate
way it allows for very precise coherent control of the light engineering
protocols, and accordingly also on the multipartite entangled states.

###  II.5 Generation of optical cat states

To further explore the use of the quantum operation introduced above for
generating non-classical field states, we consider the case in which all
harmonic modes $q^{\prime}\geq 2$ are measured in (14). The resulting
operation on the initial state of the fundamental mode reads

$\displaystyle\outerproduct{\alpha}{\alpha}\mapsto\rho=\Phi_{\tilde{n}\neq
0}^{\chi}\left[\outerproduct{\alpha}{\alpha}\right],$ (24)

with the measurement operator

$\displaystyle M_{\tilde{n}\neq 0}^{\chi}$
$\displaystyle=\bra{\\{\chi_{q}\\}}\Pi_{\tilde{n}\neq
0}\ket{\\{\chi_{q}\\}}D(\chi_{1})$
$\displaystyle=\left[\mathds{1}-e^{-\Omega}\outerproduct{\alpha}{\alpha}\right]D(\chi_{1}),$
(25)

where $\Omega=\sum_{q=2}^{N}\absolutevalue{\chi_{q}}^{2}$. Applying this
operation on the initial state of the driving laser $\Phi_{\tilde{n}\neq
0}^{\chi}[\outerproduct{\alpha}{\alpha}]=\outerproduct{\Psi}{\Psi}$, we obtain
the pure state

$\displaystyle\ket{\Psi}=\ket{\alpha+\chi_{1}}-\bra{\alpha}\ket{\alpha+\chi_{1}}e^{-\Omega}\ket{\alpha},$
(26)

which can interpolate between an optical kitten or cat state with non-
classical signatures in the corresponding Wigner function [16, 17]. Note that
similar operation can be performed on other field modes in order to generate
optical cat states towards the XUV regime [18]. The influence of the harmonic
modes, which can be considered as an environment on which conditioning
measurements are performed, is captured in the decoherence factor $\Omega$.
The measurement operation in (II.5) consists of two contributions. First, the
displacement operation for the initial coherent state taking into account the
depletion of the fundamental mode due to HHG, and second, the operator
$P_{\tilde{n}}^{\chi}=\mathds{1}-e^{-\Omega}\outerproduct{\alpha}{\alpha}$,
which reflects the conditioning processes. This conditioning operation does
not constitute a projective measurement since $(P_{\tilde{n}}^{\chi})^{2}\neq
P_{\tilde{n}}^{\chi}$. But, we observe that for a small decoherence factor
$\Omega$ the operator $P_{\tilde{n}}^{\chi}$ is indeed given by a projective
measurement, such that
$(P_{\tilde{n}}^{\chi})^{2}=P_{\tilde{n}}^{\chi}=\mathds{1}-\outerproduct{\alpha}{\alpha}$.
This is the projective measurement introduced in [16], to interpret the
measured Wigner function after an experimental conditioning procedure on HHG.
Note that this projection is performed on the sub-space which is ”orthogonal”
to the initial state, i.e. the identity subtracted by the initial state of the
fundamental mode. In [18] it was shown that the environmental induced
decoherence factor scales as $\Omega\propto\mathcal{O}(1/N)$ with the harmonic
cutoff $N$. Since the harmonic cutoff usually extends towards values of
$N>10$, this scheme of generating optical cat states with large amplitudes is
robust against this source of decoherence.

However, we observe that the measurement operator (II.5) corresponds to the
set of operators $\\{M_{\nu}^{\tilde{\chi}}\\}$, where

$\displaystyle
M_{\nu}^{\tilde{\chi}}=\bra{\\{\tilde{\chi}_{q}\\}}\Pi_{\nu}\ket{\\{\chi_{q}\\}}D(\chi_{1}),$
(27)

which form the POVM defined by the set $\\{E_{\nu}^{\tilde{\chi}}\\}$ with
POVM elements

$\displaystyle E_{\nu}^{\tilde{\chi}}$
$\displaystyle=\left(M_{\nu}^{\tilde{\chi}}\right)^{\dagger}M_{\nu}^{\tilde{\chi}},$
(28)

satisfying the completeness relation

$\displaystyle\frac{1}{\pi^{N-1}}\int\prod_{q=2}^{N}d^{2}\tilde{\chi}_{q}\sum_{\nu\in\mathcal{A}_{\tilde{n}}}\left(M_{\nu}^{\tilde{\chi}}\right)^{\dagger}M_{\nu}^{\tilde{\chi}}=\mathds{1}.$
(29)

This shows that the conditioning experiments performed on HHG allow to
generate optical cat states over a broad spectral region.

##  III Discussion

We introduced the quantum theory of measurement to the process of HHG, which
establishes the connection between the two mainly unrelated fields of quantum
information science and intense laser matter interaction. Using the general
quantum operation in terms of coherent states allows to conceive new light
engineering protocols for the generation of high dimensional, and
controllable, optical entangled states. Using attosecond physics as a novel
platform for quantum information processing brings the advantage of the
intrinsic ultrafast time-scale, together with the high degree of coherent
control of the HHG mechanism by complex polarization states or spatial modes
of the driving field [36, 37, 38, 39]. This allows to tailor the continuous-
variable non-classical field states in the desired way for quantum information
processing protocols [40, 41, 42, 43, 44, 45], quantum computation [46] or
quantum key distribution [47]. Furthermore, the entangled coherent state can
be used for quantum metrology [48], or in correlation experiments towards the
use for fundamental tests of quantum theory by violating Bell-type
inequalities [49, 50, 51, 52], whereas the optical cat states allow for
spectroscopy with non-classical states of light [53, 54, 55]. Considering that
HHG from more complex materials such as solids or plasma [56, 57, 58, 59, 60],
strongly correlated [61], or topological systems [62, 63, 64, 65, 66] follows
very similar mechanism anticipates for further investigation of the non-
classical properties from the interaction with such materials. In particular,
entanglement in strong field driven processes, for instance, between the ion
and ionized electron [67], or in two-electron ionization [68] are of current
interest. The connection to quantum information science will ultimately help
to answer the question of the quantum mechanical properties in intense laser
matter interaction from atoms to complex materials, and how new quantum states
of light with the use for modern quantum technologies can be generated.

##  IV Methods

For the quantum optical description of HHG we consider the optical field modes
to be uncorrelated prior to the interaction with the atomic medium. The
initial optical state is given by
$\ket{\phi_{i}}=\ket{\alpha}\otimes\ket{\\{0_{q}\\}}$, where the intense
driving laser is assumed to be described by the coherent state $\ket{\alpha}$
and the harmonic modes are initially in the vacuum
$\ket{\\{0_{q}\\}}=\otimes_{q}\ket{0_{q}}$, where $q\in\\{2,...,N\\}$ with the
harmonic cutoff $N$. The electron is assumed to be in its ground state
$\ket{g}$, and the Hamiltonian describing the interaction with the optical
field in the length-gauge, and within the dipole approximation, in an
convenient frame is given by $H_{I}(t)=-\mathbf{d}(t)\cdot\mathbf{E}$. The
electric field operator
$\mathbf{E}(t)=-i\mathbf{\kappa}\sum_{q=1}^{N}\sqrt{q}\left(b_{q}^{\dagger}e^{iq\omega
t}-b_{q}e^{-iq\omega t}\right)$ is coupled to the dipole moment operator
$\mathbf{d}(t)=U^{\dagger}_{sc}(t,t_{0})\mathbf{d}U_{sc}(t,t_{0})$ in the
interaction picture with respect to the semi-classical Hamiltonian
$H_{sc}(t)=H_{a}-\mathbf{d}\cdot\mathbf{E}_{cl}(t)$, where $H_{a}$ is the
atomic Hamiltonian, and $\mathbf{E}_{cl}(t)$ is the classical part of the
optical field. The time evolution of the total system governed by the
Hamiltonian $H_{I}(t)$ is given by
$U(t,t_{0})=\mathcal{T}\exp{-i\int_{t_{0}}^{t}dt^{\prime}H_{I}(t^{\prime})}$.
To describe the evolution of the optical field conditioned on HHG it remains
to solve

$\displaystyle
K_{HHG}=\bra{g}\mathcal{T}\exp{i\int_{t_{0}}^{t}dt^{\prime}\mathbf{d}(t^{\prime})\cdot\mathbf{E}(t^{\prime})}\ket{g}.$
(30)

It was shown that this can be well approximated when neglecting the electric
correlations of the dipole moment [3, 16, 17], and hence, instead of the
dipole moment operator, the dipole moment expectation value is coupled to the
electric field

$\displaystyle
K_{HHG}\simeq\exp{i\int_{t_{0}}^{t}dt^{\prime}\expectationvalue{\mathbf{d}}(t^{\prime})\cdot\mathbf{E}(t^{\prime})},$
(31)

where
$\expectationvalue{\mathbf{d}}(t)=\bra{g}\mathbf{d}(t)\ket{g}=\bra{\epsilon(t)}\mathbf{d}\ket{\epsilon(t)}$
is the dipole moment expectation value. The electronic state
$\ket{\epsilon(t)}=U_{sc}(t,t_{0})\ket{g}$ is given by the conventional
solution of HHG in the semi-classical description [1]. The operator in (31)
can now be obtained exactly by noting that the electric field operator
$\mathbf{E}(t)$ is linear in the creation- and annihilation operators. In
fact, the solution of the operator in (31) is given by a multimode
displacement operator

$\displaystyle K_{HHG}=\prod_{q=1}^{N}e^{i\varphi_{q}}D(\chi_{q}),$ (32)

where the shift of the fundamental mode and the harmonic modes account for the
depletion of the driving laser amplitude and the generation of harmonic
radiation, respectively. The amplitude of the displacement in phase-space for
each mode $q$ is given by the respective Fourier component of the oscillating
time dependent dipole moment

$\displaystyle\chi_{q}=-i\sqrt{q}\mathbf{\kappa}\int_{-\infty}^{\infty}dt^{\prime}\expectationvalue{\mathbf{d}}(t^{\prime})e^{iq\omega
t^{\prime}},$ (33)

where we have considered $t_{0}\to-\infty$, and the asymptotic limit
$t\to\infty$. The phase shift of each mode is given by

$\displaystyle\varphi_{q}=q\mathbf{\kappa}^{2}\int_{-\infty}^{\infty}dt_{1}\int_{-\infty}^{t_{1}}dt_{2}\mathbf{d}(t_{1})\mathbf{d}(t_{2})\sin[q\omega(t_{1}-t_{2})].$
(34)

That the harmonic modes are described by coherent states is due to the fact
that the source for the coherent radiation of the harmonics is related to the
electron dipole moment expectation value which acts as a source of a classical
charge current.

###### Acknowledgements.

I thank Javier Rivera-Dean and Maciej Lewenstein for stimulating discussions
about entanglement in strong field driven processes, and Paris Tzallas for
helpful comments on the manuscript. This project has received 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; Agencia Estatal de Investigación (R$\&$D project
CEX2019-000910-S, funded by MCIN/ AEI/10.13039/501100011033, Plan National
FIDEUA PID2019-106901GB-I00, FPI, QUANTERA MAQS PCI2019-111828-2, Proyectos de
I+D+I “Retos Colaboración” QUSPIN RTC2019-007196-7); Fundació Cellex; Fundació
Mir-Puig; Generalitat de Catalunya through the CERCA program, AGAUR Grant No.
2017 SGR 134, QuantumCAT U16-011424, co-funded by ERDF Operational Program of
Catalonia 2014-2020; EU Horizon 2020 FET-OPEN OPTOLogic (Grant No 899794);
National Science Centre, Poland (Symfonia Grant No. 2016/20/W/ST4/00314);
Marie Skłodowska-Curie grant STREDCH No 101029393; “La Caixa” Junior Leaders
fellowships (ID100010434) and EU Horizon 2020 under Marie Skłodowska-Curie
grant agreement No 847648 (LCF/BQ/PI19/11690013, LCF/BQ/PI20/11760031,
LCF/BQ/PR20/11770012, LCF/BQ/PR21/11840013).

## References

  * [1] M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’huillier, and P. B. Corkum.  Theory of high-harmonic generation by low-frequency laser fields.  Phys. Rev. A, 49(3):2117, 1994.
  * [2] P. Salieres, A. L’huillier, P. Antoine, and M. Lewenstein.  Study of the spatial and temporal coherence of high order harmonics.  arXiv preprint quant-ph/9710060, 1997.
  * [3] B. Sundaram and P. W. Milonni.  High-order harmonic generation: simplified model and relevance of single-atom theories to experiment.  Phys. Rev. A, 41(11):6571, 1990.
  * [4] J.H. Eberly and M.V. Fedorov.  Spectrum of light scattered coherently or incoherently by a collection of atoms.  Physical Review A, 45(7):4706, 1992.
  * [5] H. Xu.  Non-perturbative theory of harmonic generation under a high-intensity laser field.  Zeitschrift für Physik D Atoms, Molecules and Clusters, 28(1):27–36, 1993.
  * [6] G. Compagno, K. Dietz, and F. Persico.  Qed theory of harmonic emission by a strongly driven atom.  Journal of Physics B: Atomic, Molecular and Optical Physics, 27(19):4779, 1994.
  * [7] F.I. Gauthey, C. H. Keitel, P. L. Knight, and A. Maquet.  Role of initial coherence in the generation of harmonics and sidebands from a strongly driven two-level atom.  Physical Review A, 52(1):525, 1995.
  * [8] W. Becker, A. Lohr, M. Kleber, and M. Lewenstein.  A unified theory of high-harmonic generation: Application to polarization properties of the harmonics.  Physical Review A, 56(1):645, 1997.
  * [9] D.J. Diestler.  Harmonic generation: Quantum-electrodynamical theory of the harmonic photon-number spectrum.  Physical Review A, 78(3):033814, 2008.
  * [10] I.K. Kominis, G. Kolliopoulos, D. Charalambidis, and P. Tzallas.  Quantum-optical nature of the recollision process in high-order-harmonic generation.  Phys. Rev. A, 89(6):063827, 2014.
  * [11] I.A. Gonoskov, N. Tsatrafyllis, I.K. Kominis, and P. Tzallas.  Quantum optical signatures in strong-field laser physics: Infrared photon counting in high-order-harmonic generation.  Sci. Rep., 6(1):1–9, 2016.
  * [12] N. Tsatrafyllis, I.K. Kominis, I.A. Gonoskov, and P. Tzallas.  High-order harmonics measured by the photon statistics of the infrared driving-field exiting the atomic medium.  Nat. Comm., 8(1):1–6, 2017.
  * [13] N. Tsatrafyllis, S. Kühn, M. Dumergue, P. Foldi, S. Kahaly, E. Cormier, I.A. Gonoskov, B. Kiss, K. Varju, S. Varro, et al.  Quantum optical signatures in a strong laser pulse after interaction with semiconductors.  Phys. Rev. Lett., 122(19):193602, 2019.
  * [14] Á. Gombkötő, S. Varró, P. Mati, and P. Földi.  High-order harmonic generation as induced by a quantized field: phase-space picture.  Physical Review A, 101(1):013418, 2020.
  * [15] A. Gorlach, O. Neufeld, N. Rivera, O. Cohen, and I. Kaminer.  The quantum-optical nature of high harmonic generation.  Nat. Comm., 11(1):1–11, 2020.
  * [16] M. Lewenstein, M.F. Ciappina, E. Pisanty, J. Rivera-Dean, P. Stammer, Th. Lamprou, and P. Tzallas.  Generation of optical schrödinger cat states in intense laser–matter interactions.  Nature Physics, 17(10):1104–1108, 2021.
  * [17] J. Rivera-Dean, Th. Lamprou, E. Pisanty, P. Stammer, A. F. Ordóñez, A. S. Maxwell, M. F. Ciappina, M. Lewenstein, and P. Tzallas.  Strong laser fields and their power to generate controllable high-photon-number coherent-state superpositions.  Phys. Rev. A, 105:033714, Mar 2022.
  * [18] P. Stammer, J. Rivera-Dean, Th. Lamprou, E. Pisanty, M. F. Ciappina, P. Tzallas, and M. Lewenstein.  High photon number entangled states and coherent state superposition from the extreme ultraviolet to the far infrared.  Phys. Rev. Lett., 128:123603, Mar 2022.
  * [19] J. Rivera-Dean, P. Stammer, E. Pisanty, Th. Lamprou, P. Tzallas, M. Lewenstein, and M. F. Ciappina.  New schemes for creating large optical schrödinger cat states using strong laser fields.  Journal of Computational Electronics, pages 1–13, 2021.
  * [20] Á. Gombkötő, P. Földi, and S. Varró.  Quantum-optical description of photon statistics and cross correlations in high-order harmonic generation.  Physical Review A, 104(3):033703, 2021.
  * [21] J. L. O’brien, A. Furusawa, and J. Vučković.  Photonic quantum technologies.  Nature Photonics, 3(12):687–695, 2009.
  * [22] I.A. Walmsley.  Quantum optics: Science and technology in a new light.  Science, 348(6234):525–530, 2015.
  * [23] A. Acín, I. Bloch, H. Buhrman, T. Calarco, C. Eichler, J. Eisert, D. Esteve, N. Gisin, S. J. Glaser, F. Jelezko, et al.  The quantum technologies roadmap: a european community view.  New Journal of Physics, 20(8):080201, 2018.
  * [24] K. Kraus, A. Böhm, J. D. Dollard, and W.H. Wootters.  States, effects, and operations: fundamental notions of quantum theory. lectures in mathematical physics at the university of texas at austin.  Lecture notes in physics, 190, 1983.
  * [25] M. A. Nielsen and I. Chuang.  Quantum computation and quantum information, 2002.
  * [26] C. C. Gerry and P. L. Knight.  Introductory quantum optics.  Cambridge university press, 2005.
  * [27] H.Peter. Breuer and F. Petruccione.  The theory of open quantum systems.  Oxford University Press on Demand, 2002.
  * [28] H. M. Wiseman and G. J. Milburn.  Quantum measurement and control.  Cambridge university press, 2009.
  * [29] W. Becker, F. Grasbon, R. Kopold, D.B. Milošević, G.G. Paulus, and H. Walther.  Above-threshold ionization: From classical features to quantum effects.  Advances in Atomic Molecular and Optical Physics, 48:35–98, 2002\.
  * [30] R.R. Freeman, P.H. Bucksbaum, H. Milchberg, S. Darack, D. Schumacher, and M.E. Geusic.  Above-threshold ionization with subpicosecond laser pulses.  Physical review letters, 59(10):1092, 1987.
  * [31] M.D. Perry, A. Szoke, and K.C. Kulander.  Resonantly enhanced above-threshold ionization of helium.  Physical review letters, 63(10):1058, 1989.
  * [32] P. Stammer, S. Patchkovskii, and F. Morales.  Evidence of ac-stark-shifted resonances in intense two-color circularly polarized laser fields.  Physical Review A, 101(3):033405, 2020.
  * [33] Á. Jiménez-Galán, N. Zhavoronkov, M. Schloz, F. Morales, and M. Ivanov.  Time-resolved high harmonic spectroscopy of dynamical symmetry breaking in bi-circular laser fields: the role of rydberg states.  Optics Express, 25(19):22880–22896, 2017.
  * [34] N. Mayer, S. Beaulieu, A. Jimenez-Galan, S. Patchkovskii, O. Kornilov, D. Descamps, S. Petit, O. Smirnova, Y. Mairesse, and M. Ivanov.  Role of spin-orbit coupling in high-order harmonic generation revealed by super-cycle rydberg trajectories.  arXiv preprint arXiv:2112.02981, 2021.
  * [35] W. Dür, G. Vidal, and J. I. Cirac.  Three qubits can be entangled in two inequivalent ways.  Physical Review A, 62(6):062314, 2000.
  * [36] A. Fleischer, O. Kfir, T. Diskin, P. Sidorenko, and O. Cohen.  Spin angular momentum and tunable polarization in high-harmonic generation.  Nat. Phot., 8(7):543–549, 2014.
  * [37] M. Ivanov and E. Pisanty.  High-harmonic generation: taking control of polarization.  2014\.
  * [38] Á. Jiménez-Galán, N. Zhavoronkov, D. Ayuso, F. Morales, S. Patchkovskii, M. Schloz, E. Pisanty, O. Smirnova, and M. Ivanov.  Control of attosecond light polarization in two-color bicircular fields.  Physical Review A, 97(2):023409, 2018.
  * [39] D. D. Hickstein, F. J. Dollar, P. Grychtol, J. L. Ellis, R. Knut, C. Hernández-García, D. Zusin, C. Gentry, J. M. Shaw, T. Fan, et al.  Non-collinear generation of angularly isolated circularly polarized high harmonics.  Nature Photonics, 9(11):743–750, 2015.
  * [40] S. Lloyd and S. L. Braunstein.  Quantum computation over continuous variables.  In Quantum information with continuous variables, pages 9–17. Springer, 1999.
  * [41] H. Jeong, M.S. Kim, and J. Lee.  Quantum-information processing for a coherent superposition state via a mixedentangled coherent channel.  Physical Review A, 64(5):052308, 2001.
  * [42] A. Gilchrist, K. Nemoto, W. J. Munro, T. C. Ralph, S. Glancy, S. L. Braunstein, and G. J. Milburn.  Schrödinger cats and their power for quantum information processing.  Journal of Optics B: Quantum and Semiclassical Optics, 6(8):S828, 2004.
  * [43] A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier.  Generating optical schrödinger kittens for quantum information processing.  Science, 312(5770):83–86, 2006.
  * [44] C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd.  Gaussian quantum information.  Reviews of Modern Physics, 84(2):621, 2012.
  * [45] B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf.  Deterministically encoding quantum information using 100-photon schrödinger cat states.  Science, 342(6158):607–610, 2013.
  * [46] T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy.  Quantum computation with optical coherent states.  Phys. Rev. A, 68(4):042319, 2003.
  * [47] P. Jouguet, S. Kunz-Jacques, A. Leverrier, P. Grangier, and E. Diamanti.  Experimental demonstration of long-distance continuous-variable quantum key distribution.  Nat. Phot., 7(5):378–381, 2013.
  * [48] J. Joo, W. J. Munro, and T. P. Spiller.  Quantum metrology with entangled coherent states.  Phys. Rev. Lett., 107(8):083601, 2011.
  * [49] A. Gilchrist, P. Deuar, and M.D. Reid.  Contradiction of quantum mechanics with local hidden variables for quadrature phase amplitude measurements.  Physical review letters, 80(15):3169, 1998.
  * [50] A. Gilchrist, P. Deuar, and M.D. Reid.  Contradiction of quantum mechanics with local hidden variables for quadrature phase measurements on pair-coherent states and squeezed macroscopic superpositions of coherent states.  Physical Review A, 60(6):4259, 1999.
  * [51] J. Wenger, M. Hafezi, F. Grosshans, R. Tualle-Brouri, and P. Grangier.  Maximal violation of bell inequalities using continuous-variable measurements.  Phys. Rev. A, 67(1):012105, 2003.
  * [52] R. García-Patrón, J. Fiurášek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier.  Proposal for a loophole-free bell test using homodyne detection.  Physical review letters, 93(13):130409, 2004.
  * [53] M. Kira, S. W. Koch, R. P. Smith, A. E. Hunter, and S. T. Cundiff.  Quantum spectroscopy with schrödinger-cat states.  Nature Physics, 7(10):799–804, 2011.
  * [54] K. E. Dorfman, F. Schlawin, and S. Mukamel.  Nonlinear optical signals and spectroscopy with quantum light.  Rev. Mod. Phys., 88(4):045008, 2016.
  * [55] S. Mukamel, M. Freyberger, W. Schleich, M. Bellini, A. Zavatta, G. Leuchs, C. Silberhorn, R. W. Boyd, L. L. Sánchez-Soto, A. Stefanov, et al.  Roadmap on quantum light spectroscopy.  Journal of Physics B: Atomic, Molecular and Optical Physics, 53(7):072002, 2020.
  * [56] G. Vampa, C.R. McDonald, G. Orlando, D.D. Klug, P.B. Corkum, and T. Brabec.  Theoretical analysis of high-harmonic generation in solids.  Physical review letters, 113(7):073901, 2014.
  * [57] G. Vampa, T.J. Hammond, N. Thiré, B.E. Schmidt, F. Légaré, C.R. McDonald, T. Brabec, and P.B. Corkum.  Linking high harmonics from gases and solids.  Nature, 522(7557):462–464, 2015.
  * [58] E. N. Osika, A. Chacón, L. Ortmann, N. Suárez, J. A. Pérez-Hernández, B. Szafran, M. F. Ciappina, F. Sols, A. S. Landsman, and M. Lewenstein.  Wannier-bloch approach to localization in high-harmonics generation in solids.  Physical Review X, 7(2):021017, 2017.
  * [59] S. Ghimire and D. A. Reis.  High-harmonic generation from solids.  Nature physics, 15(1):10–16, 2019.
  * [60] Th. Lamprou, R. Lopez-Martens, S. Haessler, I. Liontos, S. Kahaly, J. Rivera-Dean, P. Stammer, E. Pisanty, M. F. Ciappina, M. 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, 2021.
  * [61] REF Silva, Igor V Blinov, Alexey N Rubtsov, O Smirnova, and M Ivanov.  High-harmonic spectroscopy of ultrafast many-body dynamics in strongly correlated systems.  Nature Photonics, 12(5):266–270, 2018.
  * [62] D. Bauer and K. K. Hansen.  High-harmonic generation in solids with and without topological edge states.  Physical review letters, 120(17):177401, 2018.
  * [63] C. Jürß and D. Bauer.  High-harmonic generation in su-schrieffer-heeger chains.  Physical Review B, 99(19):195428, 2019.
  * [64] R.E.F. Silva, Á. Jiménez-Galán, B. Amorim, O. Smirnova, and M. Ivanov.  Topological strong-field physics on sub-laser-cycle timescale.  Nature Photonics, 13(12):849–854, 2019.
  * [65] A. Chacón, D. Kim, W. Zhu, S. P. Kelly, A. Dauphin, E. Pisanty, A. S. Maxwell, A. Picón, M. F. Ciappina, D. E. Kim, et al.  Circular dichroism in higher-order harmonic generation: Heralding topological phases and transitions in chern insulators.  Physical Review B, 102(13):134115, 2020.
  * [66] N. Baldelli, U. Bhattacharya, D. González-Cuadra, M. Lewenstein, and T. Graß.  Detecting majorana zero modes via strong field dynamics.  arXiv preprint arXiv:2202.03547, 2022.
  * [67] L.-M. Koll, L. Maikowski, L. Drescher, T. Witting, and M.J.J. Vrakking.  Experimental control of quantum-mechanical entanglement in an attosecond pump-probe experiment.  Physical Review Letters, 128(4):043201, 2022.
  * [68] A. S. Maxwell, L. B. Madsen, and M. Lewenstein.  Entanglement of orbital angular momentum in non-sequential double ionization.  arXiv preprint arXiv:2111.10148, 2021.