Maciej Lewenstein, et. 11institutetext: ICFO - Institut de Ciencies
Fotoniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels (Barcelona), Spain
22institutetext: ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain
33institutetext: Physics Program, Guangdong Technion - Israel Institute of
Technology, Shantou, Guangdong 515063, China 44institutetext: Technion -
Israel Institute of Technology, Haifa, 32000, Israel 55institutetext:
Institute for Quantum Optics and Quantum Information of the Austrian Academy
of Sciences, A-6020 Innsbruck, Austria 66institutetext: Institute for
Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria
77institutetext: Center for Theoretical Physics, Polish Academy of Sciences,
Aleja Lotników 32/46, 02-668 Warsaw, Poland 88institutetext: Department of
Physics, University of Crete, P.O. Box 2208, GR-70013 Heraklion (Crete),
Greece 99institutetext: Department of Physics and Astronomy, Aarhus
University, DK-8000 Aarhus C, Denmark 1010institutetext: Max Born Institute
for Nonlinear Optics and Short Pulse Spectroscopy, Max Born Strasse 2a,
D-12489 Berlin, Germany 1111institutetext: Department of Physics, King’s
College London, Strand, London, UK 1212institutetext: Foundation for Research
and Technology-Hellas, Institute of Electronic Structure & Laser, GR- 70013
Heraklion (Crete), Greece 1313institutetext: ELI-ALPS, ELI-Hu Non-Profit Ltd.,
Dugonics tér 13, H-6720 Szeged, Hungary
1313email<EMAIL_ADDRESS>
# Attosecond Physics and Quantum Information Science

M. Lewenstein 1122 N. Baldelli 11 U. Bhattacharya 11 J. Biegert 1122 M.F.
Ciappina 3344 U. Elu 11 T. Grass 11 P.T. Grochowski 11556677 A. Johnson 11 Th.
Lamprou 881212 A.S. Maxwell 99 A. Ordóñez 11 E. Pisanty 10101111 J. Rivera-
Dean 11 P. Stammer 11 I. Tyulnev 22 P. Tzallas 12121313

###### Abstract

In this article, we will discuss a possibility of a symbiosis for attophysics
(AP) and quantum information (QI) and quantum technologies (QT). We will argue
that within few years AP will reach Technology Readiness Level (RTL) 4-5 in
QT, and will thus become a legitimate platform for QI and QT.

###### keywords:

quantum optics, laser-matter interaction, attosecond physics

##  1 Introduction

Contemporary Quantum Technologies face major difficulties in fault tolerant
quantum computing with error correction, and focus instead on various shades
of quantum simulation (Noisy Intermediate Scale Quantum, NISQ) devices [1],
analogue and digital Quantum Simulators [2], and quantum annealers [3]. There
is a clear need and quest for systems that, without necessarily simulating
quantum dynamics of some physical systems, can generate massive, controllable,
robust entangled and superposition states. This will, in particular, allow for
the use of decoherence in a controlled manner, enabling the use of these
states for quantum communications [4] (e.g. to achieve efficient transfer of
information in a safer and quicker way), quantum metrology [5], sensing and
diagnostics [6] (e.g. to precisely measure phase shifts of light fields, or to
diagnose quantum materials).

In this lecture, we propose an answer to these needs, by opening new avenues
for Quantum Information (QI) science via the symbiosis with Attophysics (AP)
and Quantum Optics (QO). To date, there are no existing platforms that can
bring processes at such short time-scales to Quantum Information systems. We
will illustrate how recent developments in AP aim at realizing a universal and
firmly established tool to offer completely unknown solutions and avenues for
QI. In particular, we will discuss how AP offers a set of stable and
reproducible methods to generate massively entangled states and massive
quantum superpositions [7, 8]. These methods apply, in the first place, to
fundamental Quantum Information science, but with the final goal of bringing
them to Quantum Technologies (QT). This will be accomplished by studying:

  * i)

The detection of topology, strongly correlated systems, chirality, etc. in AP;

  * ii)

The generation of entangled/quantum correlated states using conditioning
methods;

  * iii)

Strong-field physics and atto-second science driven by non-classical light;

  * iv)

The generation of entangled/quantum correlated states in Zerfall processes;

  * v)

Quantitative and measurable effects of decoherence in AP.

The lecture is organized around four projects that are being realized at ICFO:

  * •

European Research Council Advanced Grant ”NOvel Quantum SImulAtors (NOQIA)”,
run by M. Lewenstein.

  * •

European Research Council Advanced Grant ”Structural transformations and phase
transitions in real-time (TRANSFORMER)”, run by J. Biegert.

  * •

European Future and Emerging Technologies Project ”Optical Topological Logic”
(OPTOLOGIC), coordinated by J. Biegert (ICFO).

  * •

Not-yet-financed project Attophysics and Quantum information Science
(ATTOQUIS), coordinated by M. Lewenstein. This project summons experts in the
fields of QO, QI, and AP who will join forces to invent new mechanisms for
generating massive, controllable, robust, entangled light states, and
understand the process of decoherence in them, thus identifying new laws and
limits of QI. In a nutshell: Our results will set up the basis for a road map
toward a novel platform of AS for QT.

In this lecture we will focus on ATTOQUIS (to be read as ATTOKISS), the new
joint project of ICFO, FORTH, Technion, CEA, and IOTA toward a symbiosis of
attophysics and quantum information science. This is in particular aimed at
studying the generation of entangled/quantum correlated states using
conditioning methods. We will describe our joint efforts to generate
Schrödinger cat states (SCS) of photons conditioned on high harmonic
generation (HHG) [9] (see Fig. 1) and/or above threshold ionization [7, 10].
We will also talk about generation of topological order using laser pulses
with orbital angular momentum (OAM) [11]. We will also discuss HHG as a tool
to detect phase transitions and topological order in strongly correlated
systems [12, 13]. Finally, we will speculate about measurement of entanglement
of OAM of electrons in double ionization [14], and violation of Bell
inequalities by SCS. This paper is by no means a review – it is rather a
review of progress, based on a selection of relevant references, actually
mostly from external and collaborating groups. It follows the structure of the
invited lecture of M. Lewenstein at ATTOVIII Conference in Orlando [15].

Figure 1: High-photon number optical cat state created by conditioning on HHG.
(a) Theoretical prediction for the Wigner function according to [9, 7]. (b)
Experimental reconstruction of the Wigner function with mean photon number
$\langle n\rangle\approx 9.4\pm 0.1$, which represents an optical Schrödinger
cat state. For visualization reasons, the plots have been centered to zero.
The figure has been extracted from [7].

##  2 Detection of topology, strongly correlated systems, chirality, etc.

Recently, there is an explosion of interest in applying AP methods to diagnose
and detect topological order, strongly correlated systems, and chirality. The
primary role is played here by the process of high harmonic generation (HHG).

###  2.1 Detection of topological order with HHG

The pioneering work of R. Huber et consortes [16] studied lightwave-driven
Dirac currents in a topological surface band. It was noted that “Harnessing
the carrier wave of light as an alternating-current bias may enable
electronics at optical clock rates” [16, 17]. “Lightwave-driven currents have
been assumed to be essential for high-harmonic generation in solids” (cf. [18,
19]). They report angle-resolved photoemission spectroscopy with subcycle time
resolution that enables them to observe directly how the carrier wave of a
terahertz light pulse accelerates Dirac fermions in the band structure of the
topological surface state of Bi2Te3. This work opened a path toward all-
coherent lightwave-driven electronic devices.

Perhaps the first papers where HHG was considered for detection of topology
were the series of papers by D. Bauer with his collaborators on detection of
topological edge states in dimerized chains [20, 21]. In the first work, HHG
in the two topological phases of a finite, one-dimensional, periodic structure
is investigated using a self-consistent time-dependent density functional
theory approach. For harmonic photon energies smaller than the band gap, the
harmonic yield is found to differ by up to 14 orders of magnitude for the two
topological phases. Similar conclusions were obtained in the study of the Su-
Schrieffer-Heeger (SSH) chains that display topological edge states. The
authors calculated high-harmonic spectra of SSH chains that are coupled to an
external laser field of a frequency much smaller than the band gap. In recent
works, this method has been extended to the detection of Majorana fermions in
the Kitaev chain [22] and in quantum wires with proximity-induce p-wave
superconductivity [13] (see Fig. 2). Specifically, their harmonic emission
spectrum in strong fields is shown to exhibit spectral features due to
radiating edge modes, which characterize the spectrum and the density of
states in the topological phase, and which are absent in the trivial phase.
These features allow us to define an order parameter, obtained from emission
measurements, that unambiguously differentiates between the two phases. Local
probing provides insight into the localized and topologically protected nature
of the modes. The presented results establish that high harmonic spectroscopy
can be used as a novel all-optical tool for the detection of Majorana zero
modes.

In the seminal Nature Photonics paper, M. Ivanov et consortes discuss
topological strong-field physics on sub-laser-cycle timescale [23]. These
authors study response of the paradigmatic Haldane model [24] to ultrashort,
ultrastrong laser pulses. Here, they show that electrons tunnel differently in
trivial and topological insulators, for the same band structure, and identify
the key role of the Berry curvature in this process. These effects map onto
topologically dependent attosecond delays and helicities of emitted harmonics
that record the phase diagram of the system. As they say: “These findings
create new roadmaps in studies of topological systems, building on the
ubiquitous properties of the sub-laser-cycle strong-field response—a unique
mark of attosecond science”. Moreover, they show that strong fields can also
be used for the manipulation of topological properties of 2D materials,
relevant for valleytronic devices [25]. In a complementary approach, A. Chacón
et al. [26] studied circular dichroism in higher-order harmonic generation
from the Haldane model. These effect clearly heralds topological phases and
transitions in Chern insulators (see also A. Chacón’s contribution to [15] and
[27] for very recent findings).

Figure 2:  (a) The phase diagram of the proximity-induced p-wave
superconductor is obtained from an order parameter based on the harmonic
emission spectrum, exemplarily shown in (b) for two parameter choices, in the
topological and in the trival phase. At frequencies below the bandgap,
emission is suppressed in the trivial phase (red), whereas in the topological
phase emission is possible already at frequencies above half the bandgap due
to the edge state in the middle of the bandgap. The figures have been
reprinted from Ref. [13].

###  2.2 Detection of strongly correlated systems with HHG

This is another rapidly developing area. As pointed out in the recent review
[28]: “A hallmark effect of extreme nonlinear optics is the emission of high-
order harmonics of the laser from the bulk of materials. The discovery and
detailed study of this phenomenon over the course of the past decade has
offered a broad range of possibilities and seen the dawn of a new field of
extreme solid-state photonics. In particular, it opens the way to previously
inaccessible spectral ranges, as well as the development of novel solid-state
spectroscopy and microscopy techniques that enable detailed probing of the
electronic structure of solids.”

An example of pioneering studies of HHG in strongly correlated materials is
described in the paper by Silva et al. [29], where the authors show that HHG
can be used to time-resolve ultrafast many-body dynamics associated with an
optically driven phase transition to Mott insulator state, with accuracy far
exceeding one cycle of the driving light field. In Ref. [30], the authors
establish time-resolved high harmonic generation (tr-HHG) as a powerful
spectroscopy method for tracking photoinduced dynamics in strongly correlated
materials through a detailed investigation of the insulator-to-metal phase
transitions in vanadium dioxide. Further examples include the mentioned
studies of detection of Majorana fermions [13, 22].

Very recently, MBI group of Smirnova and Ivanov, using advanced techniques of
dynamical mean field theory to study the strong field response of the two-band
Hubbard model (see M. Ivanov’s contribution to [15, 31] proposed sub-cycle
multidimensional spectroscopy of strongly correlated materials). In this work
they introduce a new type of non-linear approach that allows to unravel the
sub-cycle dynamics of strongly correlated systems interacting with few-cycle
infrared pulses For the two-dimensional Hubbard model under the influence of
ultra-short, intense electric field transients, they demonstrate that our
approach can resolve pathways of charge and energy flow between localized and
delocalized many-body states on the sub-cycle timescale, including the
creation of a highly correlated state surviving after the end of the laser
pulse.

HHG may be also used for extraction of higher-order nonlinear electronic
response in solids [32]. Moreover, high harmonics may undergo spectacular
enhancement in solid state nano-structures, such as graphene heterostructures
[33]. Very recently, HHG was used to characterize quantum criticality in
strongly correlated systems [34]. By employing the exact diagonalization
method, we investigate the high-harmonic generation (HHG) of the correlated
systems under the strong laser irradiation. For the extended Hubbard model on
a periodic chain, HHG close to the quantum critical point (QCP) is more
significant compared to two neighboring gapped phases (i.e., charge-density-
wave and spin-density wave states), especially in low frequencies.

Another exciting prospect for HHG is to probe quantum phases in strongly
correlated high-temperature superconductors. In Ref. [12], we have
investigated the cuprate YBa2Cu3O7 (YBCO) in a broad temperature range from
80K to 300 K. The investigation probes different phases of the high-$T_{c}$
material (Fig. 3(b)). This work demonstrates that strong-field emission is
able to distinguish between various strongly correlated phases: The
superconducting phase is marked by a strong increase of emission at all odd
harmonic frequencies (3rd, 5th, and 7th), whereas the transition from the
pseudogap phase to the strange metallic phase is reflected by a intensity drop
only in the highest harmonics (7th) (Fig. 3(a)). These experimental findings
are reproduced by simulating the strong-field dynamics of a two-band model in
the BCS mean-field regime and in the presence of phenomenological scattering
terms (Fig. 3(c)).

Figure 3: (a) Harmonic spectra showing odd orders HH3, HH5 and HH7 for room
temperature (red) and at $T_{\text{c}}=90K$ (blue) for a mid-IR field strength
of 0.083 V/Å. Visible is already a blueshift of room-temperature harmonics
with increasing harmonic order and relative to the harmonics measured at
$T_{\text{c}}$. We observe a dramatic increase of HH7 amplitude upon cooling
into the SC phase. The inlay shows the theoretical prediction of the spectrum,
and the relative heights of the peaks matches well the behavior of the
experimental data. (b) Shown is the phase diagram for YBCO. Based on the value
of $T_{\text{c}}$ from the measurement, we determine a hole concentration of
$p\sim 0.15$. Based on this value, we mark the transition between the
different phases by the red arrow. In accord with measurement and theory,
$T^{\text{*}}$ marks the transition between strange metal and pseudogap phases
at 173 K. (c) Shown is the reflected mid-IR field together with harmonics HH3,
HH5 and HH7 as a function of temperature. These measurements are taken for a
mid-IR field strength of 0.083 V/Å. Results from the strong-field quasi-
Hubbard model are overlaid as solid lines. We observe a dramatic increase of
HH7 amplitude upon cooling into the SC phase at 88 K. All harmonic orders show
a clear turning point at the critical temperature $T_{\text{c}}$ and an
exponential increase in amplitude. More subtle, but still clearly discernible
is another critical point $T^{\text{*}}$ at 173 K, marking the transition from
strange metal to the pseudogap phase. The functional behaviour is reproduced
by the model. Figure adapted from [12].

All of there findings, discussed in this subsection, open a way to a regime of
imaging and manipulating strongly correlated materials at optical rates, far
beyond the multi-cycle approach employed in Floquet engineering of quantum
systems.

###  2.3 Detection of chirality with HHG

Detection of chirality is somewhat similar to the detection of topological
order with HHG, but mostly refers to molecular targets. Chirality is the norm
among biological molecules and the technological importance of distinguishing
molecules which differ only by their chirality cannot be overstated.
Absolutely seminal results along this direction in the context of ultrafast
imaging have been achieved by the group of O. Smirnova and collaborators [35]
(see [36] for a recent review). In particular, in Refs. [37, 38] they
introduce the concept of _locally chiral light_ – light which is chiral within
the electric-dipole approximation by means of its 3D chiral Lissajous figure –
and show how this type of chiral light can be synthesized and exploited to
dramatically enhance the contrast between HHG spectra in molecules with
opposite chiralities. They show that such contrast can be enhanced either at
the level of angle-integrated harmonic emission or at the level of direction
of harmonic emission, depending on the pattern of the light’s local chirality
over the interaction region. In Ref. [39], they address the role of the
transverse spin of the light (which emerges whenever light is confined to a
small region of space) in ultra-fast chiral imaging. They show that the
interaction of chiral molecules with a strong, tightly focused, few-cycle
pulse results in harmonics with a polarization that depends on the molecular
chirality.

In turn, we have recently studied the role of the orbital angular momentum
(OAM) of photoelectrons in ultrafast chiral imaging [40]. We found that chiral
molecules subject to strong-field ionization with a few-cycle, IR, linearly
polarized pulse serve as natural sources of _twisted_ photoelectrons, with the
twist depending on the molecular chirality. Besides exposing an alternative
road towards ultrafast chiral imaging, this effects suggests intriguing
perspectives regarding the role of the electron OAM in recollision-based
phenomena in chiral molecules.

##  3 Generation of topology, strongly correlated systems, chirality, etc.

While AP and HHG have proven to be great detection tools, ultrafast laser
pulse and attophysics methods may also be used for generation of topological
order, strongly correlated states, chirality, etc. These ideas are, of course,
very close to the attempts to generate room temperature high $T_{c}$
superconductivity [41], or Chern insulators [42], which belong more to the
domain of ultrafast, but Terahertz physics.

In fact, we have already some experience with laser induced phase transitions;
we have worked with the experimental group of S. Wall on Terahertz field
control of in-plane orbital order in La0.5Sr1.5MnO4 [43]. We have also started
to work on laser induce fluctuating bonds superconductivity [44] in the hole-
doped cuprates. But, for this section perhaps the most relevant is our recent
Phys. Rev. B Lett. [11], where we demonstrate the theoretical possibility of
generation of fermionic Chern insulator from twisted light with linear
polarization in graphene-like material.

Particularly interesting line of research connects generation of topology,
strongly correlated systems, chirality, etc. with the use of more complex
structured laser light, combining polarization and OAM effects. A perfect
example is the light, which knots fractional-order knots in the dynamics of
the electric field vector employing the polarization state of light and
superposition of the fundamental and doubled frequency [45]. Application of
strong laser pulsed of this form to atomic targets lead to “exotic”
conservation of torus-knot angular momentum in high-order harmonic generation
[46]. Combining two delayed circularly polarized pulses of frequency $\omega$
and $2\omega$, one can generate a light with a self-torque: extreme-
ultraviolet beams of HHG with time-varying orbital angular momentum [47].

As shown in Ref. [48], _locally chiral light_ [37, 38], another type of
structured light, can be used to efficiently (i.e. at the level of the
electric-dipole approximation) imprint 3D chirality on achiral matter, such as
atoms. Locally chiral light sculpts a chiral orbital out of the initially
achiral ground state. That is, it excites the electron into a superposition of
excited states such that the resulting orbital acquires a chiral shape.

##  4 Generation of massively quantum correlated states.

The generation of massively quantum correlated states is another spectacular
line of research of the ATTOQUIS platform. This direction it is driven by the
first and the very encouraging experimental and theoretical results
published/submitted on this subject within the last year [7, 8, 9, 10, 49, 51,
52]. In the following sub-sections we emphasize on the two main ideas
underlying this research direction. The ”QED and conditioning in strongly
laser driven interactions” (sub–section 4.1) and the ”QED of strong field
processes driven by non-classical light fields” (sub–section 4.2).

###  4.1 QED and conditioning in strongly laser driven interactions

The basic idea of this approach, which is schematically illustrated in Fig.
4(a), is the following (here we focus in conditioning on HHG):

  * •

We describe the intense laser–matter interaction fully quantum-
electrodynamically using coherent light states. In Fig. 4 (a) the coherent
state of the driving laser field is given by $|\alpha_{L}\rangle$.

  * •

During the interaction with the target, the driving pulse remains a coherent
state, but is modified due to the interactions with the matter. The resulted
driving field after the interaction is an amplitude shifted coherent state
$|\alpha_{L}+\chi_{L}\rangle$ and the generated harmonics are in coherent
state $|\chi_{q}\rangle$. These can be calculated using various methods. For
atomic and molecular targets one can use TDSE or SFA [53], for solids
sometimes the semiconductor Bloch equations [54], and for strongly correlated
systems more sophisticated methods.

  * •

After passing the target, the fundamental laser field in attenuated in a
coherent manner (resulting in $|\alpha+\chi\rangle$) in the range of
few–photons and conditioned to HHG by means of quantum spectrometer (QS)
method [9, 10, 55, 56]. The QS is a shot–to–shot IR vs HH photon
correlation–based method, which selects only the IR shots that are relevant to
the harmonic emission. It relies on photon statistics measurements and the
energy conservation, i.e., $q$ IR photons need to be absorbed from the IR
field for the generation of a photon of the $q_{th}$ harmonic. The output of
the QS contains only the IR shots conditioned on the HHG process, resulting in
the creation of non-classical light states. In case of intense laser–atom
interaction, this state is a coherent state superposition i.e. an optical
Schrödinger “cat” state of the form
$|\phi_{c}^{(IR)}\rangle=|\alpha+\chi\rangle-\xi|\alpha\rangle$, where $\xi$
is the overlap between $|\alpha+\chi\rangle$ and $|\alpha\rangle$.

  * •

The quantum character of the light field after the conditioning ($E_{in}$) can
be obtained by the measurement of its Wigner function. This can be archived by
means of a homodyne detection system and the quantum tomography method [57,
58]. The $E_{in}$ is spatiotemporally overlapped on a beam splitter (BS) with
the field of a local oscillator $E_{r}$ (with $E_{r}>>E_{in}$). The $E_{r}$ is
coming from the 2nd branch of the interferometer which introduces a
controllable phase shift $\varphi$ between the $E_{in}$ and $E_{r}$. The
fields after the BS are detected by a balanced differential photodetection
system consisting of two identical photodiodes (PD). This provides at each
value of $\varphi$ the photocurrent difference $i_{\varphi}$. The
characterization of the quantum state of light can be achieved by recording
for each shot the value of $i_{\varphi}$ as a function of $\varphi$. These
values are directly proportional to the measurement of the electric field
operator
$\hat{E}_{in}(\varphi)\propto\cos(\varphi)\hat{x}+\sin(\varphi)\hat{p}$ and
are used for the reconstruction of the Wigner function via Radon
transformation [59]. $\hat{x}=(\hat{a}+\hat{a}^{\dagger})/\sqrt{2}$ and
$\hat{p}=(\hat{a}-\hat{a}^{\dagger})/i\sqrt{2}$ are the non-commuting
quadrature field operators, and $\hat{a}$, $\hat{a}^{\dagger}$ are the photon
annihilation and creation operators, respectively.

  * •

The scheme has been used for the generation of high–photon number shifted
optical “cat” states (Fig. 1) and coherent state superposition with
controllable quantum features using intense–laser atom interaction [9, 7]. The
control of the quantum features is shown in Figs. 4(b) and (c) where an
optical “cat” state switches to a “kitten” for lower values of $\chi$. The
same scheme can be applied for conditioning on ATI [7, 10, 52], two-electron
ionization, etc. In our recent study [7], it has been theoretically shown that
conditioning on ATI process can also lead to the generation of optical cat
states. This, besides its fundamental interest associated with electron-photon
correlation during the ATI/HHG process, provides an additional “knob” of
controlling the quantum character of the optical cat states, a feature
extremely valuable for applications in quantum technology. Also, according to
our recent theoretical investigation performed for laser-atom interactions
[8], single and two-color driven laser-atom interactions can result in the
generation of “massively” entangled optical coherent states in the spectral
range from extreme ultraviolet (XUV) (shown in Fig. 4(a) as
$|\chi_{c}^{(q)}\rangle$) to the far infrared. Such states in combination with
passive linear optical elements (such as phase shifters, beam splitters, and
optical fibers) [50, 51] can be considered as unique resources for novel
applications in quantum technology.

Figure 4:  (a) Schematic illustration of the operation principle of the
conditioning approach. HHG is the area where the intense laser–matter
interaction takes place and the high harmonics are generated.
$|\alpha_{L}\rangle$ is the initial coherent state of the driving field.
$|\alpha_{L}+\chi_{L}\rangle$, $|\alpha+\chi\rangle$ are the states of the IR
field after the interaction and attenuation, respectively. $|\chi_{q}\rangle$
are the coherent states of the harmonic modes. $|\phi_{c}^{(IR)}\rangle$ and
$|\chi_{c}^{(q)}\rangle$ are the field states after conditioning on the HHG. A
scheme of the QT method used for the quantum state characterization.
$|\alpha_{r}\rangle$ is the state of the local oscillator reference field.
$\varphi$ is a controllable phase shift. BS is a beam splitter. PD are
identical IR photodiodes used from the balanced detector. $i_{\varphi}$ is the
$\varphi$ dependent output photocurrent difference used for the measurement of
the electric field operator and the reconstruction of the Wigner function. (b)
and (c) Reconstructed Wigner functions of an optical ”cat” and ”kitten” state,
repsectively. Panels (b) and (c) have been reproduced from Ref. [7].

###  4.2 QED of strong field processes driven by non-classical light fields

All the quantum optical studies of strong field driven processes have thus far
been approached by assuming a coherent state description for the driving
field. The corresponding field is classical, even if Hilbert space methods are
used for its description. However, in a recent work, presented at ATTO VIII by
Even Tzur et al. [60], this assumption was abandoned, and the process of HHG
driven by non-classical light fields has been presented for the first time.
They show, that the photon statistics of the driving field leaves its
signatures in the observed high harmonic spectrum. This will open the path for
investigating the interplay between non-classical properties of the driving
source, and how they are imprinted in the observables of the harmonic field
modes.

##  5 Studies of quantum correlated states/decoherence in Zerfall processes

###  5.1 Generation of entanglement in Zerfall processes

Finally, AP is a perfect playground to study generation of quantum
correlations and entanglement in Zerfall processes, i.e. processes of decay of
a “whole” in products. A prime example of this is non-sequential double
ionization (NSDI), where strong-field ionization followed by laser induced
recollision leads to double ionization of the target, see Fig. 5(a). Strong
correlation between the ionization products, in particular the two
photoelectrons, has long been known [61]. Previously, coherence and
interference between the two photoelectrons has been studied in great detail
[62, 63, 64, 65]. More recently, A. Maxwell et al. [14] studied generation of
entanglement in non-sequential double ionization, by using correlation in the
OAM of the outgoing electrons, for a mini-review on OAM in strong-field
ionization see [66, 67]. A key step, is to exploit the intermediate excited
state, and its inherent superposition over OAM, which is present for the
‘second’ electron for the low-intensity regime111This is known as the
recollision with subsequent ionization (RESI) mechanism.. Thus, simple
conservation laws dictate that the final OAM of the two electrons must be
anti-correlated, which at certain final momentum leads to a maximally
entangled qutrit, see Fig. 5(b), while remaining robust to incoherent effects,
such as focal averaging or decoherence with the ion.

Continuum products in a typical strong-field Zerfall processes will be
quantified through continuous variables, which can complicate analysis of
entanglement measures. However, the use of OAM makes the task of quantifying
and measuring entanglement much easier, the computations of the reduced
density matrix of ionized product in the OAM space becomes simple, and in the
case of NSDI, clearly exhibits entanglement. This also enables quantification
by the, so called, logarithmic negativity, which may be used for mixed states
to model incoherent effects.

Figure 5: Non-sequential double ionization (NSDI), a ‘Zerfall’ processes,
leading to entanglement between the orbital angular momentum (OAM) of the two
photoelectrons. In panel (a), an electron is ionized via a strong laser field
and recollides with the parent atom/ molecule. This leads to either: direct
ionization of a second electron in electron impact (EI) ionization, or
excitation with subsequent ionization (RESI) of a second electron. Panel (b)
focuses on RESI, the OAM superposition in the excited state is transferred to
the final state along with anti-correlation between the electrons.

###  5.2 Characterizing decoherence in Zerfall processes

Finally, studies of Zerfall processes offer a unique opportunity to understand
and characterize quantum and classical decoherence. This goes back to seminal
and pioneering theoretical works by R. Santra et al. [68, 69, 70], F. Martín
et al. [71], or M. Vacher et al. [72]. All of these works dealt essentially
with the single electron ionization process that might lead to entanglement of
the electron with the parent ion. Measuring reduced density matrix of the
electron provides thus information about decoherence and thus entanglement
Mark Vrakking et al. has developed both theoretical and the first experimental
ideas in this context [73, 74] (see also M. Vrakking’s contribution to [15]).
One of the main problems in this line of research is to make sure that the
source of decoherence is truly quantum. Indeed, by analyzing with great care
various classical and quantum models of decoherence in the process of single
electron ionization, Ch. Bourassin-Bouchet et al. demonstrated dominantly
classical sources of decoherence.

In contrast, in the recent work [75], A. L’Huillier et al. investigated
decoherence due to entanglement of the radial and angular degrees of freedom
of the photoelectron. They study two-photon ionization via the $2s2p$
autoionizing state in He using high spectral resolution photoelectron
interferometry. Combining experiment and theory, we show that the strong
dipole coupling of the $2s2p$ and $2p^{2}$ states results in the entanglement
of the angular and radial degrees of freedom. This translates, in angle
integrated measurements, into a dynamic loss of coherence during
autoionization.

##  6 Conclusions

This paper provides a progress review on a new emerging interdisciplinary area
of science, namely ATTOQUIS, in which ultrafast laser physics and attoscience
merge with quantum optics and quantum information science. We provide examples
of the ATTOQUIS research directions that are already at a step where
theoretical and proof-of-principle experimental investigations have been
developed. Based on these examples, we briefly discuss how the fully quantized
description of intense laser-matter interaction and high–harmonic generation,
in combination with conditioning methods, can be used for: i) the generation
of controllable massively entangled quantum state superpositions in all states
of matter, ii) the generation and detection of chirality and topological order
in strongly correlated systems, iii) the generation of high–harmonics using
intense quantum light, and iv) investigations quantum correlated states and
decoherence effects in atoms. In few words, the results demonstrated in the
aforementioned directions constitute the beginning of a very long ”story”.
They show the power of intense laser–matter interaction as a new resource for
novel investigations in attosecond science and quantum technology, a goal that
the ATTOQUIS platform strives for.

## Acknowledgments

M.L and group acknowledge support from ERC AdG NOQIA; Ministerio de Ciencia y
Innovation Agencia Estatal de Investigaciones (PGC2018-097027-B-
I00/10.13039/501100011033, CEX2019-000910-S/10.13039/501100011033, Plan
National FIDEUA PID2019-106901GB-I00, FPI, QUANTERA MAQS PCI2019-111828-2,
QUANTERA DYNAMITE PCI2022-132919, Proyectos de I+D+I “Retos Colaboración”
QUSPIN RTC2019-007196-7); European Union NextGenerationEU (PRTR C17.I1);
Fundació Cellex; Fundació Mir-Puig; Generalitat de Catalunya (European Social
Fund FEDER and CERCA program (AGAUR Grant No. 2017 SGR 134, QuantumCAT
U16-011424, co-funded by ERDF Operational Program of Catalonia 2014-2020);
Barcelona Supercomputing Center MareNostrum (FI-2022-1-0042); EU Horizon 2020
FET-OPEN OPTOlogic (Grant No 899794); National Science Centre, Poland
(Symfonia Grant No. 2016/20/W/ST4/00314); European Union’s Horizon 2020
research and innovation programme under the Marie-Skłodowska-Curie grant
agreement No 101029393 (STREDCH) and No 847648 (“La Caixa” Junior Leaders
fellowships ID100010434: LCF/BQ/PI19/11690013, LCF/BQ/PI20/11760031,
LCF/BQ/PR20/11770012, LCF/BQ/PR21/11840013); the Government of Spain
(FIS2020-TRANQI and Severo Ochoa CEX2019-000910-S). M. F. C. acknowledges
financial support from the Guangdong Province Science and Technology Major
Project (Future functional materials under extreme conditions -
2021B0301030005). J. B.and group acknowledge financial support from the
European Research Council for ERC Advanced Grant “TRANSFORMER” (788218) and
ERC Proof of Concept Grant “miniX” (840010), FET-OPEN “PETACom” (829153), FET-
OPEN “OPTOlogic” (899794), EIC-2021-PATHFINDEROPEN ”TwistedNano” (101046424),
Laserlab-Europe (654148), Marie Sklodowska-Curie ITN “smart-X” (860553), Plan
Nacional PID-PID2020-112664GB-I00-210901; National Science Centre, Poland
(Symfonia Grant No. 2016/20/W/ST4/00314) AGAUR for 2017 SGR 1639, “Severo
Ochoa” (SEV- 2015-0522), Fundació Cellex Barcelona, the CERCA Programme /
Generalitat de Catalunya, and the Alexander von Humboldt Foundation for the
Friedrich Wilhelm Bessel Prize. P. Tzallas group at FORTH acknowledges
LASERLABEUROPE V (H2020-EU.1.4.1.2 grant no. 871124), FORTH Synergy Grant
AgiIDA (grand no. 00133), the H2020 framework program for research and
innovation under the NEP-Europe-Pilot project (no. 101007417). ELI-ALPS is
supported by the European Union and co–financed by the European Regional
Development Fund (GINOP Grant No. 2.3.6-15-2015-00001). J.R-D. acknowledges
support from the Secretaria d’Universitats i Recerca del Departament d’Empresa
i Coneixement de la Generalitat de Catalunya, as well as the European Social
Fund (L’FSE inverteix en el teu futur)–FEDER. P. S. acknowledges funding from
the European Union’s Horizon 2020 research and innovation programme under the
Marie Sklodowska-Curie grant agreement No 847517. A. S. M. acknowledges
funding support from the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement SSFI No.$\backslash
887153$.

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# Real-Time Portrait Stylization on the Edge

Yanyu Li∗1 Xuan Shen∗1 Geng Yuan1 These authors contributed equally to this
work. Jiexiong Guan2
Wei Niu2 Hao Tang3 Bin Ren2 Yanzhi Wang1 1Northeastern University
2College of William & Mary
3CVL, ETH Zurich {li.yanyu, shen.xu, yuan.geng<EMAIL_ADDRESS>
{jguan<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work we demonstrate real-time portrait stylization, specifically,
translating self-portrait into cartoon or anime style on mobile devices. We
propose a latency-driven differentiable architecture search method,
maintaining realistic generative quality. With our framework, we obtain
$10\times$ computation reduction on the generative model and achieve real-time
video stylization on off-the-shelf smartphone using mobile GPUs.

##  1 Introduction

Thanks to hardware advancement, varieties of AI applications have been enabled
on portable smart devices, such as foreground segmentation, face recognition,
etc. In this work, we investigate portrait stylization which is a popular
feature in social media Apps, transferring self portraits into a desired
style, such as cartoon Andersson and Arvidsson (2020), anime Kim et al.
(2020); Li et al. (2021), etc.

Portrait stylization can be categorized as classic image-to-image translation,
which is often achieved by conditional Generative Adversarial Networks (GANs)
Isola et al. (2017); Zhu et al. (2017). In GAN training, a generator learns to
generate a fake instance and fool the discriminator, while the discriminator
takes true and generated images as input and learns to distinguish them.
Consequently, in the portrait stylization task, the generator is utilized to
map a portrait photo to the desired style domain during inference. Compared to
naive paired supervised training, GANs exhibits stunning generative quality,
e.g., sharp and realistic details, diverse features.

However, it still remains challenging to enable real-time face stylization to
process videos on a mobile device, and the reason comes two-fold. Firstly,
typically following an encoder-decoder design, image translation models suffer
from high computation complexity, especially on high-resolution images.
Secondly, GAN training is difficult and unstable, suffering from loss
divergence and mode collapse. As a result, existing compression techniques are
difficult to integrate into GAN training and preserve generative quality.

Figure 1: Demonstration of on-mobile real-time stylization for (a) real person
and (b) static image. The input video stream is captured by the mobile phone’s
camera in a real-time fashion. The full demo video is attached in the link .

In this work, we propose a compiler-aware differentiable architecture search
framework. We measure the latency of the building blocks with sufficient
configurations (channels, feature sizes), and train a neural network to
predict the latency. We show that a simple MLP speed model can make accurate
predictions. In order to search compact architectures, we integrate learnable
parameters in the GAN generator and regularize them by speed constraints to
reduce the model width and depth. With the speed model that maps architecture
parameters to latency, the speed penalty is differentiable so that we can
perform search with end-to-end training. Plus, different from prior work, we
do not select preserved/eliminated model weights by magnitude. Instead, we
apply straight through estimator on architecture parameters to sparsify them
into $\\{0,1\\}$. The benefit is two-fold. The remained weights are
represented by $1$s so that we can easily predict the latency of a certain
state. In addition, the gradients of pruned weights are completely zeroed out
so that their functionalities are preserved. As a result, pruned weights are
always ready to be reverted back to contribute to accuracy during exploration.
This is especially important in GAN search because of the unstable training
process.

Overall, our contributions include:

  * •

We develop a latency-driven differentiable architecture search for GANs. Our
sophisticated pipeline addresses the difficulty in GAN searching, achieves
unprecedented compression rate while preserving generative quality.

  * •

To the best of our knowledge, we are the first to achieve real-time portrait
stylization on mobile phones. Mobile demos are attached in the link111The link
for the demo video: https://youtu.be/2SFAVwaymvQ .

##  2 Background

#### Image-to-Image Style Transfer.

Isola et al. (2017); Liu et al. (2017); Huang et al. (2018); Lee et al.
(2018); Liu et al. (2019); Park et al. (2020); Kim et al. (2020); Li et al.
(2021); Chong and Forsyth (2021); Tang et al. (2019) aim to translate a image
from the source domain to match certain styles in the target domain, while
preserving semantics of the origin image. Early works train the generative
model with paired data Isola et al. (2017), however they cannot be applied to
more common unpaired datasets. Later work Zhu et al. (2017) proposed a cycle
consistency loss to train on unpaired data domains, inspiring lots of
successor research on image stylization Park et al. (2020); Kim et al. (2020);
Li et al. (2021); Chong and Forsyth (2021); Tang et al. (2021).

As for the domain of cartoon or anime, UGATIT Kim et al. (2020) released the
selfie2anime benchmark and develop an adaptive mixture of instance and layer
normalization. AniGAN Li et al. (2021) released a larger scale face2anime
dataset and studied style-guided anime translation. GNR Chong and Forsyth
(2021) refine the content and style to produce controllable and diverse
synthesis.

#### Compressing GANs.

Because of wide applications, the compression of GAN has drawn research
attention Fu et al. (2020); Wang et al. (2020); Li et al. (2020); Chen et al.
(2021); Jin et al. (2021). Recent work Li et al. (2020) proposed to
incorporate neural architecture search (NAS) and feature level knowledge
distillation (KD). Jin et al. (2021) integrated an Inception-like residual
block and performed self-distillation.

##  3 Compiler-Aware Architecture Search

Figure 2: Stylization results on face2anime dataset. Top: source image.
Middle: styles generated by baseline UGATIT model with 57.1 GMACs. Bottom: our
searched model under 5.56 GMACs.  Figure 3: Stylization results on
photo2cartoon dataset released (partly) by MiniVision. Top: source image. Note
that the source domain data (Asian women) comes from generated fake data by
StyleGAN2. Bottom: our searched model under 1.34 GMACs.

###  3.1 GAN Basics

We follow CycleGAN Zhu et al. (2017) and UGATIT Kim et al. (2020) paradigm to
develop our architecture search for stylization. According to CycleGAN, we
learn mapping functions between two unpaired domain
$X{=}\\{x_{i}\\}_{i=1}^{N}$ and $Y{=}\\{y_{j}\\}_{j=1}^{M}$. There are two
generators in inverse direction $G{:}X{\xrightarrow[]{}}Y$ and
$F{:}Y{\xrightarrow[]{}}X$, as well as two discriminators $D_{X}$ and $D_{Y}$.
Note that we refer $G$ as the generator mapping portraits to stylized images,
which is the only required model during inference.

#### Adversarial Loss.

We match the distribution of translated instances with the target domain as
follows.

$\small L_{gan}^{X}=\mathbb{E}_{y\sim Y}[D_{Y}^{2}(y)]+\mathbb{E}_{x\sim
X}[(1-D_{Y}(G(x)))^{2}].$ (1)

#### Cycle Consistency Loss.

To minimize reconstruction error,

$\small L_{cyc}=\mathbb{E}_{x\sim X}[|F(G(x))-x|_{1}]+\mathbb{E}_{y\sim
Y}[|G(F(y))-y|_{1}].$ (2)

Note that we also incorporate the identity loss $L_{id}$ and CAM loss
$L_{CAM}$ as proposed in UGATIT, for simplicity we skip detailed formulations
here and please refer to Kim et al. (2020).

Our overall GAN objective is:

$\small
L=\lambda_{1}L_{gan}^{X}+\lambda_{1}L_{gan}^{Y}+\lambda_{2}L_{cyc}+\lambda_{3}L_{id}+\lambda_{4}L_{CAM},$
(3)

where $\lambda_{1}{=}1,\lambda_{2}{=}10,\lambda_{3}{=}10,\lambda_{4}{=}1000$
are the hyperparameters to control each loss.

###  3.2 Layerwise Width Search

Width search is performed for each CONV layer. We choose the supernet from Zhu
et al. (2017); Kim et al. (2020), which is a commonly employed generator. In
order to create a learnable binary mask, we insert a depth-wise $1{\times}1$
CONV layer following each CONV layer to be pruned, as shown below,

$\bm{a}_{l}^{n}=\bm{v}_{l}^{n}\odot(\bm{w}_{l}^{n}\odot\bm{a}_{l-1}^{n}),$ (4)

where $\odot$ denotes the convolution operation.
$\bm{w}_{l}^{n}{\in}R^{o\times i\times k\times k}$ is the weight parameters in
the $l^{th}$ CONV layer of the $n^{th}$ block, with $o$ output channels, $i$
input channels, and kernels of size $k{\times}k$.
$\bm{a}_{l}^{n}{\in}R^{B\times o\times s\times s^{\prime}}$ represents the
output features of $l^{th}$ layer (with the trainable mask), with $o$ channels
and $s{\times}s^{\prime}$ feature size. $B$ denotes the batch size.
$\bm{v}_{l}^{n}{\in}R^{o\times 1\times 1\times 1}$ is the corresponding
weights of the depth-wise CONV layer (i.e., the mask layer).

Larger elements of $\bm{m}_{l}^{n}$ mean that the corresponding channels
should be preserved while smaller elements indicate that the corresponding
channels should be pruned. Formally, we use a threshold $t$ to convert
$\bm{m}_{l}^{n}$ into a binary mask as below,

$\bm{b}_{l}^{n}=\begin{cases}1,\bm{v}_{l}^{n}>t.\\\ 0,\bm{v}_{l}^{n}\leq
t.\end{cases}\text{(element-wise)},$ (5)

where $\bm{b}_{l}^{n}{\in}\\{0,1\\}^{o\times 1\times 1\times 1}$ is the
binarized $\bm{v}_{l}^{n}$. Typically, we initialize $\bm{v}_{l}^{n}$ with 1,
and the adjustable $t$ is set to 0.5. In order to make the mask differentiable
to enable backpropagation, we utilize Straight Through Estimator (STE) Bengio
et al. (2013); Chang et al. (2020) as shown below,

$\frac{\partial\mathcal{L}}{\partial\bm{b}_{l}^{n}}=\frac{\partial\mathcal{L}}{\partial\bm{v}_{l}^{n}}.$
(6)

Our trainable binary mask has the following advantages: (i) The mask can be
trained along with the network parameters via gradient descent, thus saving
search cost compared to NAS methods Zoph and Le (2017); Zhong et al. (2018).
(ii) Different from previous methods Han et al. (2015); Yu et al. (2017); He
et al. (2017); Guan et al. (2020), which determine the pruning according to
the parameter magnitudes, we decouple the parameter magnitudes of CONV or BN
layer from pruning, and utilize independent mask layers, thus the remained
parameters are not harmed. (iii) The discrete values can directly provide the
width information for each CONV layer, which is compatible with speed
prediction.

###  3.3 Length Search by Block

Note that although per-layer width search may also converge to zero width,
which eliminates the entire block, we find that there are usually a few
channels left in each block preventing us to remove the entire block. Plus,
the pruning indicator for each layer cannot represent the latency reduction of
entirely pruned blocks. Thus it is necessary to perform length search
separately from width search.

We construct two paths in each customized residual block, one is the masked
convolution block and the other is skip connection. In the aggregation layer
of the dual paths, we integrate binarized variables $\beta_{1}$ and
$\beta_{2}$, then the forward computation can be represented as follows,

$\small\bm{a}^{n}=\beta_{1}\cdot\bm{a}^{n-1}+\beta_{2}\cdot\bm{a}_{L}^{n}.$
(7)

The aggregation layer contains two trainable parameters $\alpha_{1}$ and
$\alpha_{2}$, shares similar STE recipe with width parameter. In the forward
pass, it selects the skip path or the masked convolution path based on the
relative relationship of $\alpha_{1}$ and $\alpha_{2}$,

$\displaystyle\bm{\beta}_{1}=0\ \text{and}\ \beta_{2}=1,\text{if}\
\alpha_{1}\leq\alpha_{2}.$ (8) $\displaystyle\bm{\beta}_{1}=1\ \text{and}\
\beta_{2}=0,\text{if}\ \alpha_{1}>\alpha_{2}.$ (9)

###  3.4 Speed Prediction with Speed Model

We take inference speed on mobile GPUs to constrain the optimization. A DNN-
based speed model is adopted to predict the inference speed of the block based
on its architecture configurations. Then the final predicted latency is
accumulated by the aggregated blocks so that we can compute the latency loss
$L_{latency}$ and integrate it into the searching pipeline.

$L_{search}=L+L_{latency}(\bm{v},\alpha).$ (10)

The trained speed model is accurate in predicting the speed of different layer
widths in the block (with 5% error at most).

Face2anime |  FID $\downarrow$ |  MACs ($\times 10^{9}$)
---|---|---
CycleGAN Zhu et al. (2017) | 50.09 | 56.8
UGATIT Kim et al. (2020) | 42.84 | 57.1
MUNIT Huang et al. (2018) | 43.75 | 77.3
FUNIT Liu et al. (2019) | 56.81 | -
DRIT Lee et al. (2018) | 70.59 | -
AniGAN Li et al. (2021) | 38.45 | -
Ours | 57.32 | 5.56
Table 1: Quantitative evaluation of our searched stylization model on
Face2anime dataset. Our model perform inference on video at 18 FPS on SAMSUNG
Galaxy S10 smartphone using mobile GPU.

##  4 Experiments and Demonstration

We conduct experiments on Face2anime dataset published by Li et al. (2021).
Face2anime consists of 17,796 images. We set input size to $256{\times}256$.
Further, we also demonstrate realistic quality on cartoon stylization released
by MiniVision, along with Asian women face training data generated by Karras
et al. (2020).

#### Experiment Setups.

As for the supernet, we search from Kim et al. (2020). Learning rate is set to
$2{\times}10^{-4}$ for both generator and discriminator, with Adam optimizer.
Leaning rate is fixed for the first 30k iterations and then linearly decayed
to zero in another 30k iterations.

#### Performance Evaluation.

We quantitatively compare our searched stylization model with representative
works in the quality of created images (FID) and computation costs (MACs). As
shown in Table 1, our generative model achieves 10$\times$ computation
reduction and preserves generative quality. Thanks to the significant
computation reduction, we achieve high quality real-time stylization on
mobile.

#### Results Visualization and On-mobile Demonstration.

Figure 2 shows the visualization of the created images on face2anime test
dataset and the comparison of our 5.56 GMACs model with the baseline UGATIT
model with 57.1 GMACs. Figure 3 shows the created images using photo2cartoon
test dataset. With simpler styles, our search method further reduce the model
size to 1.34 GMACs. We also demonstrate our method on the mobile device, as
shown in Figure 1. The full demo video is available in the link .

## Acknowledgments

This research is partially supported by National Science Foundation
CNS-1909172, CCF-1919117, and CCF-2047516 (CAREER).

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