∎

11institutetext: J. Rivera-Dean 22institutetext: ICFO - Institut de Ciencies
Fotoniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels (Barcelona), Spain 22email<EMAIL_ADDRESS>33institutetext: P. Stammer 44institutetext: Max Born Institute for Nonlinear
Optics and Short Pulse Spectroscopy, Max Born Strasse 2a, D-12489 Berlin,
Germany 55institutetext: ICFO - Institut de Ciencies Fotoniques, The Barcelona
Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
66institutetext: E. Pisanty 77institutetext: Max Born Institute for Nonlinear
Optics and Short Pulse Spectroscopy, Max Born Strasse 2a, D-12489 Berlin,
Germany 88institutetext: Th. Lamprou 99institutetext: Foundation for Research
and Technology-Hellas, Institute of Electronic Structure & Laser, GR-70013
Heraklion (Crete), Greece 1010institutetext: Department of Physics, University
of Crete, P.O. Box 2208, GR-71003 Heraklion (Crete), Greece 1111institutetext:
P. Tzallas 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 1414institutetext: M. Lewenstein 1515institutetext: ICFO -
Institut de Ciencies Fotoniques, The Barcelona Institute of Science and
Technology, 08860 Castelldefels (Barcelona), Spain 1616institutetext: ICREA,
Pg. Lluís Companys 23, 08010 Barcelona, Spain 1717institutetext: M. F.
Ciappina 1818institutetext: Physics Program, Guangdong Technion - Israel
Institute of Technology, Shantou, Guangdong 515063, China 1919institutetext:
Technion - Israel Institute of Technology, Haifa, 32000, Israel

# New schemes for creating large optical Schrödinger cat states using strong
laser fields

J. Rivera-Dean  P. Stammer  E. Pisanty  Th. Lamprou  P. Tzallas  M. Lewenstein
M. F. Ciappina

(Received: date / Accepted: date)

###### Abstract

Recently, using conditioning approaches on the high-harmonic generation
process induced by intense laser-atom interactions, we have developed a new
method for the generation of optical Schrödinger cat states Lewenstein2020 .
These quantum optical states have been proven to be very manageable as, by
modifying the conditions under which harmonics are generated, one can
interplay between _kitten_ and _genuine cat_ states. Here, we demonstrate that
this method can also be used for the development of new schemes towards the
creation of optical Schrödinger cat states, consisting of the superposition of
three distinct coherent states. Apart from the interest these kind of states
have on their own, we additionally propose a scheme for using them towards the
generation of large cat states involving the sum of two different coherent
states. The quantum properties of the obtained superpositions aim to
significantly increase the applicability of optical Schrödinger cat states for
quantum technology and quantum information processing.

###### Keywords:

Wigner functions Strong field physics Schrödinger cat states High-Harmonic
Generation

##  1 Introduction

In the last few years, the field of photonics has stood out as an important
stage for investigations in quantum technology OBrien2009 . Due to the unique
properties light has with respect to propagation and noise tolerance, quantum
optical states of light are considered as one of the most valuable tools for
quantum communication protocols Gisin2007 , quantum metrology Giovannetti2004
and quantum computation architectures NielsenChuangBook . Within this
direction, optical Schrödinger cat states, defined as the superposition of
distinct coherent states, have been proposed as a continuous-variable
candidate that could be employed in the mentioned studies Gilchrist2004 . For
this reason, obtaining cat states for which the overlap among the different
elements in the superposition is very small, the so-called _large_ cat states,
has become a central problem for its employment in quantum technologies.
Recently, we have demonstrated a new method based on intense laser-matter
interactions, that allows for the generation of tunable optical cat states
constituted by the superposition of two different coherent states (which we
shall refer to hereupon as _dead/alive_ cat states) Lewenstein2020 . Here, we
aim to take advantage of this approach in order to increase the number of
distinct terms in the considered superpositions. Furthermore, we will employ
the latter for generating enlarged versions of _dead/alive_ cat states.

At the core of our analysis lie the so-called strong-field physics processes.
Strong-field physics studies light-matter interactions in high-intensity
regimes, i.e. with field intensities of the order of $10^{14}$ to $10^{15}$
W/cm2, accessing phenomena previously unattainable Amini2019 ; Krausz2009 ;
Ciappina2017 . For instance, these processes have allowed for the generation
of attosecond pulses, both in the extreme ultraviolet Drescher2001  and in the
soft x-ray regimes Popmintchev2012 . Central to all this progress, from a
fundamental perspective, lies the so-called three-step model Corkum1993 ;
KulanderBook ; Lewenstein1994 , which describes the interaction of the systems
under consideration (typically atoms or molecules and, recently, solid
materials) with the high-intensity laser pulse. In this process, the electron
(1) tunnels out from the system, where it was initially bound, due to the
high-intensity laser field; (2) accelerates in the continuum driven by this
field; and (3) can recollide elastically or inelastically with the parent
system it originated from. Particularly interesting for the context of the
present paper are inelastic recollisions, which can lead to the High-Harmonic
Generation (HHG) process, where a high-energy photon is generated when the
electron recombines back to the ground state.

From a theoretical perspective, strong field processes have been extensively
studied using a semiclassical framework Lewenstein1994 , i.e. considering the
quantum aspects of the atomic, molecular or solid-state system while keeping a
classical behaviour for the laser field due to its high-photon number. Recent
approaches have attempted to study these phenomena, theoretically Sundaram1990
; Xu1993 ; Compagno1994 ; Gauthey1995 ; Becker1997 ; Dierstler2008 ;
Kominis2014 ; Bogatskaya2016spontaneous ; Bogatskaya2016polarization ;
Gonoskov2016 ; Bogatskaya2017 ; Bogatskaya2017spectroscopy ; Magunov2017 ;
RiveraDean2019 ; Gorlach2020 ; Yangaliev2020 ; Gombkoto2020  and
experimentally Tsatrafyllis2017 ; Tsatrafyllis2019 ; Bloch2019 , from a fully
quantum-mechanical perspective, that is, considering the quantum nature of
both the laser field and the atom. However, it was in Ref. Lewenstein2020
where we studied the depletion of the fundamental field and demonstrated,
theoretically and experimentally for the first time, the quantum nature of the
output light. In particular, we showed that, upon conditioning on the HHG
process, we can generate optical Schrödinger cat states.

This manuscript is structured as follows: in Sec. 2 we provide a brief
overview about optical Schrödinger cat states and how they can be produced.
Here, we focus our attention on the novelty of the method involving HHG
Lewenstein2020 , giving a brief summary about the underlying process. After
this, we review some basic concepts about the Wigner function, which
constitutes a fundamental tool for the experimental characterization of non-
classical states. In Sec. 3 we present two different methods for generating
superpositions of three distinct coherent states based on the initial quantum
HHG states, and introduce a procedure where the former are employed for
enhancing the latter. Finally, in Sec. 4, we present some concluding comments.
Our implementation is available in Ref. FigureMaker .

##  2 State of the art

###  2.1 Quantum-optical Schrödinger cat states

The concept of _cat states_ was initially coined by E. Schrödinger in his
famous _Gendankenexperiment_ Schrodinger1935  about quantum superpositions of
two classically distinguishable states, exemplified by the well-known cat
being in a dead and alive superposition. Within the context of quantum optics,
_cat states_ typically refers explicitly to states which are superpositions of
two distinct coherent states $\ket{\alpha_{1}}$ and $\ket{\alpha_{2}}$, i.e.

$\ket{\psi}=a_{1}\ket{\alpha_{1}}+a_{2}\ket{\alpha_{2}},$ (1)

where the $a_{i}$ are complex coefficients satisfying normalization
requirements. Besides their foundational interest, these states have proven to
be very useful for practical purposes in different fields Gilchrist2004 ;
Lvovsky2020 , such as quantum computation Ralph2003 , quantum information
Sanders1992 ; Jeong2003 ; Stobinska2007  and quantum metrology Munro2002 .

The actual generation of such quantum states is considered as one of the most
crucial tasks. Several methods have been proposed in the past towards this
direction, with the main purpose of generating states like the one shown in
Eq. (1), where the distance between both coherent states is sufficiently big
so that their overlap is negligible. In particular, we can find techniques
that condition the output of one of the ports of a beam splitter, which has
been fed with a squeezed and a vacuum state, to the extraction of an odd or an
even number of photons Dakna1997 ; Ourjoumtsev2006 . We also have methods that
employ cavities in which a trapped atom can be used to change the phase of an
input coherent state depending on the atomic quantum state Hacker2019 . The
latest achievement in this direction is the recently developed technique which
relies on conditioning measurements over high-harmonic generation processes
induced by intense laser-atom interactions Lewenstein2020 . This last method
constitutes the basis of the analysis presented in this manuscript.

The optical Schrödinger cat states given in Eq. (1) can be further generalized
so that they involve the superposition of more than two distinct coherent
states, that is,

$\ket{\psi}=\sum^{n\geq 2}_{k=1}a_{k}\ket{\alpha_{k}},$ (2)

where $\alpha_{i}\neq\alpha_{j}\forall i\neq j$. On the one hand, states like
the superposition presented above, where each $\alpha_{i}$ have the same
amplitude but differ by a constant phase, can actually be generated by using
Kerr nonlinearity and a beam splitter vanEnk2003 . On the other hand, those
for which the $\alpha_{i}$ differ in amplitude are more complicated to create,
but have proven to be useful theoretically in the context of Bell inequalities
violation Wenger2003 .

###  2.2 Schrödinger cat states via High Harmonic Generation

Strong-field physics studies light-matter interactions in the regime of high-
intensity electromagnetic fields, which allows us to witness highly non-linear
optical phenomena such as HHG. From the theoretical perspective, and
considering that a high number of photons interact with the medium under
consideration, one can justifiably treat the light classically and the matter
quantum mechanically Lewenstein1994 . However, by considering the quantum
nature of the field, we can uncover new properties of the strong-field physics
processes that are useful from a quantum optical perspective Kominis2014 . In
particular, it was shown in Lewenstein2020  that by considering the quantum
nature of the electromagnetic field in HHG, one is able to generate a new
class of Schrödinger cat states with quantum features that depend on the HHG
conditions. To show this, we consider as initial state of the whole system

$\ket{\Psi_{0}}=\ket{\text{g}}\otimes\ket{\alpha}\otimes\ket{\\{0\\}},$ (3)

where the first term represents the ground state of the atomic system, the
second one the coherent state of the input infrared electromagnetic field, and
the last one reflects that there are no excitations in all the harmonic modes,
i.e. they lie in a vacuum state. We model the dynamics of this system with the
Hamiltonian

$H=H_{\text{a}}+H_{f}+V_{\text{a-f}}$ (4)

where $H_{\text{a}}=\hat{P}^{2}/2+V(\hat{R})$ is the atomic/molecular
Hamiltonian (in atomic units), $H_{f}$ is the field-free Hamiltonian
containing all the harmonic modes up to the cutoff and
$V_{\text{a-f}}=\hat{E}(t)\cdot\hat{R}$ describes the atom-field interaction.
As we have shown previously Lewenstein2020 , after conditioning the final
Schrödinger equation to the HHG process, and by considering the strong-field
assumptions Lewenstein1994 , we get for the final quantum-optical state of the
system Lewenstein2020

$\ket{\psi}=\ket{\alpha+\delta\alpha}\bigotimes^{\text{cutoff}}_{q=2}\ket{\beta_{q}}.$
(5)

In this last expression, the input infrared field gets shifted by a
displacement $\delta\alpha$ that represents the amount of photons that have
been lost because of the electron-field interaction, while the harmonics
appear as non-zero coherent states. In order to obtain the Schrödinger cat
states, we perform a conditioning measurement that involves the generated
harmonics. A clearer picture of this operation emerges from the fact that the
excitation process is governed by the creation operator

$B^{\dagger}\propto
a^{\dagger}e^{i\omega_{L}t}+\sum_{q=2}^{\text{cutoff}}\sqrt{q}\
b^{\dagger}_{q}e^{iq\omega_{L}t},$ (6)

where $a^{\dagger}$ and $b_{q}^{\dagger}$ are respectively the creation
operators of the infrared mode and of its $q$th harmonic Lewenstein2020 . From
Eq. (6) we see that an excitation in the fundamental mode is accompanied by
excitations in the harmonic modes Gonoskov2016 ; Tsatrafyllis2017 . Thus,
experimental analysis of the generated harmonics can be understood as a
witness of the fact that a shift has been generated over the fundamental mode,
and projects the final coherent state onto everything that is not
$\ket{\alpha}$, i.e. the initial state. This conditioning is carried out
experimentally by means of the Quantum Tomography and Quantum Spectrometer
(QT/QS) approach Lewenstein2020 ; Tsatrafyllis2017 . We, thus, have after such
conditioning

$\displaystyle P\ket{\alpha+\delta\alpha}$
$\displaystyle=\big{(}\mathbbm{1}-\outerproduct{\alpha}{\alpha}\big{)}\ket{\alpha+\delta\alpha}$
(7) $\displaystyle=\ket{\alpha+\delta\alpha}-\xi\ket{\alpha},$

where

$\xi\equiv\innerproduct{\alpha}{\alpha+\delta\alpha}.$ (8)

According to our previous definition, this kind of superpositions corresponds
to an optical Schrödinger cat state. Notice that one of the greatest
advantages of these strong-field cat states is that they are controllable:
depending on the magnitude of $\delta\alpha$ one can move from a _kitten_
state to a _genuine cat_ state, with $\delta\alpha$ depending on the medium
used for HHG and on the specific intensity of the applied laser-field.
However, the appearance of $\xi$ in our equations imposes as well some
limitations over the size of the generated cats: if $\delta\alpha$ is very big
then $\xi\to 0$ and our cat turns into a shifted coherent state.

###  2.3 Characterizing quantum-optical states: the Wigner function

The Wigner function was first introduced by E. Wigner in 1932 Wigner1932
within the context of quantum corrections to thermodynamic equilibrium, but it
has found wide applicability in quantum optics SchleichBook ; ScullyBook . The
reason for this is that it provides a natural way of characterizing quantum
optical states within their quadrature representation $(x,p)$, both
theoretically and experimentally Smithey1993 . It was initially formulated as
a quantum analogue of the Liouville density, which gives the probability of
finding a particle at a given point of the phase space. However, due to the
uncertainty principle, there are some standard properties verified by the
Liouville density that are not satisfied by the Wigner distribution and, for
that reason, it is usually referred to as a _quasiprobability distribution_.

Starting from the conventional definition ($\hbar=1$)

$W(x,p)=\dfrac{1}{\pi}\int^{\infty}_{-\infty}\differential
y\matrixelement{x+y}{\rho}{x-y}e^{-i2py},$ (9)

where $\rho$ is a density matrix characterizing a certain quantum state, one
can prove the following properties for the Wigner function SchleichBook :

  1. (i)

it is normalized, i.e.

$\int^{\infty}_{-\infty}\\!\\!\\!\\!\differential
x\int^{\infty}_{-\infty}\\!\\!\\!\\!\differential p\ W(x,p)=1;$ (10)

  2. (ii)

its marginals recover the probability distributions of the corresponding state
along the different quadratures, i.e.

$\displaystyle\mathcal{P}(x)=\matrixelement{x}{\rho}{x}=\int^{\infty}_{-\infty}\\!\\!\\!\\!\differential
p\ W(x,p),$ (11)
$\displaystyle\mathcal{P}(p)=\matrixelement{p}{\rho}{p}=\int^{\infty}_{-\infty}\\!\\!\\!\\!\differential
x\ W(x,p);$

  3. (iii)

the overlap between two quantum states can be written in terms of their
respective Wigner function representations, i.e.

$\Tr(\rho_{1}\rho_{2})=\pi\int^{\infty}_{-\infty}\\!\\!\\!\\!\differential
x\int^{\infty}_{-\infty}\\!\\!\\!\\!\differential p\
W_{\rho_{1}}(x,p)W_{\rho_{2}}(x,p),$ (12)

where $W_{\rho_{1}}$ ($W_{\rho_{2}}$) is the Wigner function associated to the
quantum state $\rho_{1}$ ($\rho_{2}$). By setting this overlap to zero it
follows that the we can find situations where the Wigner function adopts
negative values, contrarily to what would happen with a well-defined
probability distribution;

  4. (iv)

it satisfies the following inequality

$-\dfrac{1}{\pi}\leq W(x,p)\leq\dfrac{1}{\pi}.$ (13)

It is well established that Gaussian states depict a positive Wigner function
that can be interpreted as proper probability densities Hudson1974  as
happens, for instance, with coherent or squeezed states (see Fig. 1 (a) and
(c)). In fact, for pure states, Gaussian states are the only ones which lead
to positive Wigner functions. This is not necessarily true for non-Gaussian
pure states such as Fock states (see Fig. 1 (c)), and may happen for non-
Gaussian mixed states. Thus, one of the main uses of the Wigner function
within quantum optics is to experimentally characterize non-classical states
of light, a task that is typically approached using homodyne detection
Smithey1993 . In this approach, one introduces in one of the two arms of an
interferometer a fully characterized coherent state $\ket{\alpha}$ and, on the
other arm, the state that wants to be studied. Then, by measuring both outputs
of the beam splitter and varying the phase of the reference coherent state
(known as the _local oscillator_), one can recover the probability
distribution along different points in the quadrature space of the unknown
state. This information can then be used to extract the Wigner function of the
state Smithey1993 .

Figure 1: Wigner functions of (a) a vacuum state, (b) a squeezed state and (c)
a Fock state with $n=1$. The first two cases correspond to Gaussian states
examples but whose Wigner function have clearly a different behaviour, while
the last one is a non-Gaussian state whose Wigner function depicts negative
values.

##  3 Results

In this section, employing the quantum High-Harmonic Generation (QHHG)
approach explained above, we present two schemes that generate superpositions
involving three coherent states of different amplitude. The first consists of
the use of an interferometer, where the QHHG process takes place in each arm,
while the second consists of a generalization of the conditioning measurements
that are performed with the QT/QS approach. Finally, we discuss a method for
using the generated coherent-state superpositions for obtaining enlarged
optical Schrödinger cat states _à la_ Eq. (1), i.e. _dead/alive_ cat states.

###  3.1 Quantum High-Harmonic Generation as an optical element

The method we present here considers the QHHG process as a modularized optical
element of an experimental setup. This naturally implies that this approach
could be extended to any other method that is able to generate Schrödinger cat
states from a coherent state source. In that case, the QHHG stage should be
substituted with the corresponding technique.

The configuration we consider here is shown in Fig. 2. It consists of two
50/50 beam splitters (BS), two quantum HHG elements (represented by QHHG), and
additional paths that add a delay to one of the beams with respect to the
other. From a theoretical point of view, we characterize each of the previous
elements as follows:

Figure 2: Optical setup consisting of two 50/50 beam splitters (BS1 and BS2),
two quantum High-Harmonic Generation systems and two mirrors that redirect one
of the beams. To BS1 arrives a coherent state of amplitude $\sqrt{2}\alpha$
which is used afterwards for high-harmonic generation in QHHG. After this
process, we get two quantum-optical Schrödinger cat states that are mixed
again in a second beam splitter (BS2).

  * •

Beam splitter. Assuming no losses, beam splitters can be described as a two-
mode unitary operator that transfer energy from the input modes to the
reflected and transmitted output modes. Thus, we can characterize this device
as

$B(\theta)=\exp[\theta(a_{1}a_{2}^{\dagger}-a_{1}^{\dagger}a_{2})],$ (14)

where $a_{1}$ ($a_{1}^{\dagger}$) and $a_{2}$ ($a_{2}^{\dagger}$) are the
annihilation (creation) operators of both input modes, and the mixing angle
$\theta$ determines the ratio between reflection and transmission. In
particular, if $\theta=\pi/4$, we get the so called 50/50 beam splitter. Thus,
given two input coherent states $\ket{\alpha_{1}}\otimes\ket{\alpha_{2}}$
entering into the system, the output is determined by

$\displaystyle B(\theta)\ket{\alpha_{1}}\ket{\alpha_{2}}$
$\displaystyle=\ket{\alpha_{1}\cos(\theta)+\alpha_{2}\sin(\theta)}$ (15)
$\displaystyle\hskip
11.38092pt\otimes\ket{-\alpha_{1}\sin(\theta)+\alpha_{2}\cos(\theta)}.$

  * •

Quantum High-Harmonic Generation. This element not only includes the medium
employed for HHG, but also the experimental setup used for performing the
conditioning. Therefore, according to our previous analysis, the effect of
this element over a given input state $\ket{\alpha}$ is obtained from Eq. (7)
as

$\text{QHHG}(\ket{\alpha})=\ket{\phi_{\text{cat}}},$ (16)

where $\ket{\phi_{\text{cat}}}$ is the normalized version of the state
described in Eq. (7).

  * •

Delay path. The objective of the delay path is to add an extra phase $\varphi$
to the corresponding field. In particular, if we start with a coherent state
$\ket{\alpha}$ we get after the delay

$\ket{\alpha}\xrightarrow[]{\text{Delay}}\ket{\alpha e^{i\varphi}}.$ (17)

With all this set, we consider the initial state of the system to be

$\ket{\psi_{0}}=\ket{\sqrt{2}\alpha}\ket{0},$ (18)

where we will assume in what follows that $\alpha$ and the obtained shift
$\delta\alpha$ are real quantities, positive and negative respectively. In
case of zero optical path difference between the two arms of the
interferometer, we obtain

$\displaystyle\ket{\psi_{\text{BS2}}}$
$\displaystyle=\dfrac{1}{\sqrt{N}}\bigg{[}\ket{0}\Big{(}\ket{\sqrt{2}(\alpha+\delta\alpha)}+\xi^{2}\ket{\sqrt{2}\alpha}\Big{)}$
(19) $\displaystyle\hskip
28.45274pt-\xi\Big{(}\ket{\tfrac{\delta\alpha}{\sqrt{2}}}+\ket{\tfrac{-\delta\alpha}{\sqrt{2}}}\Big{)}\ket{\sqrt{2}\alpha+\tfrac{\delta\alpha}{\sqrt{2}}}\bigg{]},$

where $N$ is a normalization constant.

Thus, we have a superposition of three distinct coherent states in both output
modes of the last beam splitter. As commented before, this approach can be
generalized to other setups that allow for the generation of optical
Schrödinger cat states. However, the main advantage of the method employed
here relies on the adaptability HHG provides over the generated cat state,
through the dependence of $\delta\alpha$ on the HHG medium that is used and on
the input laser intensity.

Figure 3: Wigner function for the three state superposition. Each column is
defined for the different measurements defined in Eq. (20), and are ordered
accordingly to such equation from left to right. Figures (a) to (c) correspond
to $\delta\tilde{\alpha}=-0.9$, while (d) to (e) correspond to
$\delta\tilde{\alpha}=-1.3$. The measurements are strongly defined by the
relative weights of the three state superposition, and in consequence this is
reflected in the final form of the obtained Wigner functions.

In order to recover an exact superposition of three coherent states in one of
the modes, we have to constrain the output of the other mode upon a suitable
measurement. Here, we will consider the situation in which such measurements
are done over the second mode. In particular, we will constrain it so that we
measure one of the following coherent states

$\Big{\\{}\ket{\sqrt{2}(\alpha+\delta\alpha)},\ket{\sqrt{2}\big{(}\alpha+\tfrac{\delta\alpha}{2}\big{)}},\ket{\sqrt{2}\alpha}\Big{\\}},$
(20)

that is, the possible coherent states that we get in the second output. In
principle, this conditioning could be done over any other possible coherent
state, but we restrict to these values as they are the ones we can potentially
get after the beam splitter. Because of this, and from a practical point of
view, we can perform this measurement by superposing the field obtained in the
second output with a coherent state that has one of the previous amplitudes,
but with a phase shift of half a period of the fundamental mode. Therefore,
for the case in which the amplitude of both fields are equal, we will obtain a
destructive interference which leads to a vacuum state. In consequence, we can
understand this approach as the measurement of a vacuum state after performing
a suitable displacement operation determined by one of the coherent states
given in Eq. (20).

The previous operation leads us to states in the first mode of the form

$\ket{\psi_{1}}=\dfrac{1}{\sqrt{N_{1}}}\Big{[}a\ket{0}+b\ket{\delta\tilde{\alpha}}+b\ket{-\delta\tilde{\alpha}}\Big{]},$
(21)

where we denote $\delta\tilde{\alpha}=\delta\alpha/\sqrt{2}$, and $a$ and $b$
are coefficients depending on $\delta\tilde{\alpha}$ and on the specific
measurement that we are performing. Depending on the latter, the final
superposition will take different forms.

In Fig. 3 we show the obtained Wigner functions for the different measurements
and for two values of $\delta\tilde{\alpha}$. In particular, figures (a) to
(c) correspond to $\delta\tilde{\alpha}=-0.9$ where the ordering, from left to
right, follows the one in Eq. (20), whereas figures (d) to (f) correspond to
$\delta\tilde{\alpha}=-1.3$. In all cases, with the exception of Fig. 3 (d)
where we recover a single coherent state, we can identify the same patterns
for the Wigner functions: all of them are symmetric, due to the structure of
the state in Eq. (21), and they all present negative regions that are
typically located in $x=\pm\delta\tilde{\alpha}$, unless for figures (c) and
(f) where we instead have identical local maxima.

The main difference between the different subplots in Fig. 3 is produced by
the difference in the values taken by the weights $a$ and $b$ that appear in
Eq. (21), depending on the measurement we are performing. Thus, as we increase
the value of $|\delta\tilde{\alpha}|$, the conditioned state in the first mode
tends to a coherent state for the first two measurements, in particular to a
vacuum state which is the term dominating the superposition. Within this
regime, the Wigner function corresponds to a Gaussian centered in $x=p=0$. For
the second measurement this is a process that takes place more slowly, as for
$\delta\tilde{\alpha}=-1.3$ we still get negative regions for the Wigner
function arising from the overlap between the different elements in the
superposition, contrarily to what happens with the first measurement (Figs. 3
(e) and (d) respectively). On the other hand, for the third measurement there
are no changes at all in the final superposition, which can be seen from the
fact that its Wigner function remains unperturbed upon changes in
$\delta\tilde{\alpha}$. This implies that weights between the vacuum state and
the rest of the superposition in Eq. (21) are almost equal. Moreover, one can
check that, in fact, is the $b$ coefficient in the previous equation the
dominating one, and in the limit of large values of $|\delta\tilde{\alpha}|$
the Wigner function coincides with the one coming from the symmetric
superposition between $\ket{\delta\tilde{\alpha}}$ and
$\ket{-\delta\tilde{\alpha}}$. However, in order to obtain such states one has
to perform the adequate measurement successfully, and this is highly
determined by which state of Eq. (20) are we using. In Fig. 4 we show the
success probability for each of the possible measurements as a function of
$\delta\tilde{\alpha}$. As the latter becomes bigger, only the first
measurement can be performed successfully, which leads to a vacuum state as
mentioned previously.

Figure 4: Probability of performing each of the measurements in Eq. (20)
successfully. From left to right in the previous equation, the first
measurement corresponds to the blue continuous line, the second to the orange
dashed line and the third one to the green dashed-dotted line. As the value of
$|\delta\tilde{\alpha}|$ increases, the first measurement dominates while for
small values it decays to zero as our three state superposition vanishes. The
horizontal grey lines highlight the probabilities associated to the values of
$|\delta\tilde{\alpha}|$ used in Fig. 3.

###  3.2 Generalization of the conditioning approach

One of the main problems regarding the previous approach is that, due to the
use of the second beam splitter (BS2), the distance between the terms in the
superpositions that we can reach are relatively small, as the initial
$\delta\alpha$ gets affected by a factor of $1/\sqrt{2}$. In top of this, as
we have shown in Fig. 4 for values of $|\delta\tilde{\alpha}|>1.3$, the
measurement performed with the first state in Eq. (20) are more likely and, in
consequence, the vacuum term of Eq. (21) dominates the superposition, leading
to a final Gaussian state. Thus, for this kind of superpositions to be useful
for practical purposes such as in Wenger2003 , it seems instrumental to
develop techniques that allow us to enlarge the distance between the different
coherent states in the superposition, while keeping a good ratio between their
probability amplitudes.

Figure 5: Experimental setup for the generalized approach. The state
$\ket{\alpha}$ undergoes two HHG processes in the media HHG1 and HHG2. Later
on, the harmonics $\ket{\text{HH}_{i}}$ generated in each of these
interactions are used for performing the conditioning measurements with the
Quantum Tomography and Quantum Spectrometer approach, which we refer to with
the QT/QS box. As shown in the text, with this set-up we can naturally
generate superpositions of three distinct coherent states.

The approach we consider here is based on the architecture shown in Fig. 5. In
this case, we have several HHG processes taking place but, unlike the set-up
of Fig. 2, all of them are mediated with the same mode. Therefore, each of the
harmonics that are generated will define projectors with respect to different
initial coherent states once the conditioning measurement is performed.
Depending on the number of HHG processes considered in this scheme, we can
naturally generate superpositions containing more than two coherent states
without involving any beam splitter, and thus without introducing a factor
$1/\sqrt{2}$ affecting their final intensity.

Let us illustrate this idea with the configuration shown in Fig. 5. In this
case we start with a coherent state $\ket{\alpha}$ which undergoes HHG in a
given medium, represented as HHG1. We will denote the generated shift as
$\delta\alpha_{1}$ and use $\ket{\text{HH}_{1}}$ as a shorthand notation for
the harmonic modes generated in this system. Hence, the process taking place
in this first medium is

$\ket{\alpha}\otimes\ket{\\{0\\}}\rightarrow\ket{\alpha+\delta\alpha_{1}}\otimes\ket{\text{HH}_{1}}.$
(22)

Afterwards, instead of performing the conditioning measurement (represented in
Fig. 5 with the QT/QS box), we use the fundamental shifted mode
$\ket{\alpha+\delta\alpha_{1}}$ in another medium HHG2. Denoting with
$\delta\alpha_{2}$ and $\ket{\text{HH}_{2}}$ the generated shift and the
harmonics respectively, we get in this case

$\displaystyle\ket{\alpha+\delta\alpha_{1}}\otimes\ket{\text{HH}_{1}}\otimes\ket{\\{0\\}}$
(23) $\displaystyle\hskip
42.67912pt\to\ket{\alpha+\delta\alpha_{1}+\delta\alpha_{2}}\otimes\ket{\text{HH}_{1}}\otimes\ket{\text{HH}_{2}}.$

Note that the harmonics $\ket{\text{HH}_{1}}$ and $\ket{\text{HH}_{2}}$ that
we have generated thus far belong to two different modes, as they are
spatially separated. Moreover, both harmonic modes are uncorrelated to each
other, but correlated with the fundamental mode via the creation operators
$B^{\dagger}_{1}$ and $B^{\dagger}_{2}$ given in Eq. (6), where the difference
between them is that they are defined with respect to two different creation
operators $b^{\dagger}_{q,i}$, $i=1,2$, affecting the $q$th harmonic in each
of the space-separated modes. This implies that if we now use the QT/QS
approach to perform a conditioning measurement with respect to
$\ket{\text{HH}_{2}}$, we should consider for the definition of the
corresponding projector $P_{2}$ the initial state that was used for its
generation. More explicitly, we have

$\displaystyle\ket{\phi_{\text{cat}}^{(2)}}$
$\displaystyle=P_{2}\ket{\alpha+\delta\alpha_{1}+\delta\alpha_{2}}$ (24)
$\displaystyle=(1-\outerproduct{\alpha+\delta\alpha_{1}}{\alpha+\delta\alpha_{1}})\ket{\alpha+\delta\alpha_{1}+\delta\alpha_{2}}$
$\displaystyle=\ket{\alpha+\delta\alpha_{1}+\delta\alpha_{2}}-\xi_{2}\ket{\alpha+\delta\alpha_{1}},$

where
$\xi_{2}=\innerproduct{\alpha+\delta\alpha_{1}}{\alpha+\delta\alpha_{1}+\delta\alpha_{2}}$.

Now, we use the harmonics generated in the first medium, i.e.
$\ket{\text{HH}_{1}}$, for performing a second conditioning. Similarly to what
we had before, in this case the conditioning projector $P_{1}$ refers to the
initial state used for the generation of such harmonics, that is,
$\ket{\alpha}$. We get after that operation

$\displaystyle\ket{\Phi}$
$\displaystyle=P_{1}\ket{\phi_{\text{cat}}^{(2)}}=(1-\outerproduct{\alpha}{\alpha})\ket{\phi_{\text{cat}}^{(2)}}$
(25)
$\displaystyle=\ket{\alpha+\delta\alpha_{1}+\delta\alpha_{2}}-\xi_{2}\ket{\alpha+\delta\alpha_{1}}-\xi_{1}\ket{\alpha},$

where
$\xi_{1}=\innerproduct{\alpha}{\alpha+\delta\alpha_{1}+\delta\alpha_{2}}-\xi_{2}\innerproduct{\alpha}{\alpha+\delta\alpha_{1}}$.
As we see, this method provides us with a very natural way of generating the
desired superposition of three coherent states.

However, one of the main drawbacks with respect to this configuration is that
the $\xi_{1}$ constant might be very small. To avoid this, we implement
between the $P_{1}$ and $P_{2}$ conditioning an amplification operation over
the cat state $\ket{\phi_{\text{cat}}^{(2)}}$, described by the photonic
displacement operator $D(\gamma)$ ScullyBook . Thus, if we set
$\delta\alpha_{1}=\delta\alpha_{2}\equiv\delta\alpha$ for the sake of
simplicity, and consider $\gamma=-3\delta\alpha$, we get after the
amplification process

$\ket{\tilde{\phi}_{\text{cat}}^{(2)}}=D(-3\delta\alpha)\ket{\phi_{\text{cat}}^{(2)}}=\ket{\alpha-\delta\alpha}-\xi_{2}\ket{\alpha-2\delta\alpha},$
(26)

and after the conditioning

$\ket{\Phi}=-\xi^{\prime}_{1}\ket{\alpha}+\ket{\alpha-\delta\alpha}-\xi_{2}\ket{\alpha-2\delta\alpha},$
(27)

where
$\xi^{\prime}_{1}=\xi_{2}(1-\innerproduct{\alpha}{\alpha-2\delta\alpha})$. We
can see that as $\delta\alpha$ increases, $\xi^{\prime}_{1}\to\xi_{2}$ and
therefore the obtained state is symmetric around $\ket{\alpha-\delta\alpha}$.
Nevertheless, one should be careful in the limit where $\delta\alpha$ is very
large because then $\xi_{1}^{\prime},\xi_{2}\to 0$ and we get only a coherent
state.

Figure 6: Wigner functions for the state obtained by the generalized
measurement approach shown in Fig. 5, for (a) $\delta\alpha=-0.9$, (b)
$\delta\alpha=-1.3$, (c) $\delta\alpha=-1.5$ and (d) $\delta\alpha=-2.0$. As
we can see, with this method we can achieve negativities of the Wigner
function for bigger values of $\delta\alpha$ compared to most successful
measurement of the previously described method. Furthermore, the symmetry of
the Wigner functions increases for bigger values of $\delta\alpha$. In these
plots we have set $\alpha=0$ so that the functions are centered in the origin.

The main distinction between Eq. (21) and Eq. (27) lies in the coefficients
that go along with the different coherent states in the sum, which now are
only determined by the value of $\delta\alpha$, and on the distance between
the three states in the superposition. While in the former the distance
between the two outermost coherent states is given by $\sqrt{2}\delta\alpha$,
in the latter is given by $2\delta\alpha$. The corresponding Wigner functions
obtained for this state are shown in Fig. 6 for values of
$\delta\alpha=-0.9,-1.3,-1.5$ and $-2.0$, respectively, from (a) to (d). As we
can see, the obtained Wigner functions bear some similarity with the ones
obtained in Figs. 3 (b) and (e), specially Figs. 6 (a) and (b). However, one
of the most important differences with respect to Eq. (21) is that the
obtained state is not completely symmetric, specially for small values of
$|\delta\alpha|$ for which we have $\xi_{1}^{\prime}\neq\xi_{2}$, as it can be
clearly seen in Fig. 6 (a) where the negative regions of the Wigner function
located in the positive part of the $x$ axis are slightly bigger than the ones
obtained in the negative side. As mentioned previously, for increasing values
of $|\delta\alpha|$ the obtained Wigner function becomes more symmetric since
$\xi_{1}^{\prime}\to\xi_{2}$. This is clearly seen in Figs. 6 (c) and (d),
where there is almost no difference between the negative regions of both
Wigner functions. It is remarkable as well that for values of
$\delta\alpha=-2.0$ we still get distinguishable negative regions for the
Wigner function, which motivates the use of this kind of three coherent state
superpositions to generate large _dead/alive_ cat states. However, and above
that value, the central coherent state becomes more important and the Wigner
function tends to a Gaussian, as seen in Fig. 6 (d).

###  3.3 Enlarged cat states

The two methods we have presented so far allow us to generate a superposition
of three coherent states. While in the first method the distance between the
two outermost states in the superposition is $\sqrt{2}\delta\alpha$, with the
second approach we are able to increase this distance by a factor of
$\sqrt{2}$. Given that the superposition of two well-separated coherent
states, i.e. large _dead/alive_ cat states, are essential for practical uses
Gilchrist2004 , it is natural to ask ourselves how can we use these three-
state superpositions to generate enlarged _dead/alive_ cat states.

Inspired by Laghout2013 ; Sychev2017 , a straightforward way of achieving this
objective is by means of a beam splitter and a conditioning measurement in one
of the output modes. Given that the first method for obtaining the three
coherent state superposition already contains a beam splitter, the final
enhancement will not be excessively big. Thus, in this section we will focus
mainly on the states obtained with the generalized measurement method as it
does not contain any extra beam splitter in its definition.

Starting from the state shown in Eq. (27) and postprocessing it with a 50/50
beam splitter whose other input is fed with $\ket{\alpha+\delta\alpha}$, we
get

$\displaystyle\ket{\Phi_{\text{BS}}}$
$\displaystyle=\dfrac{1}{\sqrt{N}}\Big{[}-\xi^{\prime}_{1}\ket{\tilde{\alpha}+\tfrac{\delta\alpha}{\sqrt{2}}}\ket{-\tfrac{\delta\alpha}{\sqrt{2}}}+\ket{\tilde{\alpha}}\ket{0}$
(28) $\displaystyle\hskip
36.98866pt-\xi_{2}\ket{\tilde{\alpha}-\tfrac{\delta\alpha}{\sqrt{2}}}\ket{\tfrac{\delta\alpha}{\sqrt{2}}}\Big{]},$

where $\tilde{\alpha}=\sqrt{2}(\alpha-\delta\alpha)$ and $N$ is a
normalization constant.

Figure 7: Probability of success when conditioning the results of the second
mode to obtain an even number of photons, versus
$\delta\tilde{\alpha}\equiv\delta\alpha\cos(\theta)$ (in absolute value). We
present the results for three different beam splitters characterized by mixing
angles $\theta=\pi/3$ (blue continuous line), $\theta=\pi/2$ (orange dashed
line) and $\theta=\pi/6$ (green dashed-dotted line). While the latter gives
smaller probabilities, the decay for large values of $|\delta\tilde{\alpha}|$
takes place more slowly which allow us to obtain bigger _dead/alive_ cat
states. The vertical continuous grey lines correspond to the values of
$|\delta\tilde{\alpha}|$ for which we get the different Wigner functions shown
in the the insets (i)-(iii) on the right. In these plots we have set
$\tilde{\alpha}=0$ so that the functions are centered in the origin.

If we now condition the second output to the measurement of an even number of
photons Lvovsky2020 ; Gerrits2010 , we obtain

$\displaystyle\ket{\Phi_{\text{even}}}$
$\displaystyle=\Big{(}\mathbbm{1}\otimes\sum_{n=1}^{\infty}\outerproduct{2n}{2n}\Big{)}\ket{\Phi_{\text{BS}}}$
(29) $\displaystyle=\ket{\phi_{\text{even}}(\tfrac{\pi}{4})}\otimes
e^{-\tfrac{|\delta\tilde{\alpha}|^{2}}{2}}\sum^{\infty}_{n=1}\dfrac{(\delta\tilde{\alpha})^{2n}}{\sqrt{2n!}}\ket{2n},$

where

$\ket{\phi_{\text{even}}(\tfrac{\pi}{4})}=\dfrac{1}{\sqrt{N}}\Big{[}\xi_{1}^{\prime}\ket{\tilde{\alpha}+\tfrac{\delta\alpha}{\sqrt{2}}}+\xi_{2}\ket{\tilde{\alpha}-\tfrac{\delta\alpha}{\sqrt{2}}}\Big{]}$
(30)

is the non-normalized optical Schrödinger cat state that we get for the first
mode.

The state in Eq. (29) is a separable state where the first mode has been
projected into a _dead/alive_ cat state and the distance between both coherent
states is $\sqrt{2}\delta\alpha$, i.e. a factor $\sqrt{2}$ bigger than the
distance between the two outermost coherent states in the inputted three state
superposition. Because the state in Eq. (29) is separable, the partial trace
with respect to the second mode naturally leads to a pure state.

One of the main advantages of this technique in comparison with the methods
presented in Laghout2013 ; Sychev2017  is that, with this approach, we can get
larger enhancements by suitably changing the beam splitter transmittance and
the coherent state entering one of its inputs, provided that the three
coherent state superposition is inputted in the other. This contrasts with the
previously cited methods since, in those cases, we are forced to use a 50/50
beam splitter, because the two inputs are given by identical _dead/alive_ cat
states. More specifically, for a beam splitter characterized by the mixing
angle $\theta$ and a coherent state of the form
$\ket{(\alpha+\delta\alpha)\tan(\theta)}$, one can show (see Appendix A) that
the enhanced and not normalized _even_ optical Schrödinger cat state
NoteEvenOdd  is given by

$\displaystyle\ket{\phi_{\text{even}}}$
$\displaystyle=\dfrac{1}{\sqrt{N}}\big{[}\xi^{\prime}_{1}\ket{\tilde{\alpha}(\theta)+\delta\alpha\cos\theta}$
(31) $\displaystyle\hskip
48.36958pt+\xi_{2}\ket{\tilde{\alpha}(\theta)-\delta\alpha\cos\theta}\big{]},$

where $\tilde{\alpha}\equiv(\alpha-\delta\alpha)/\cos\theta$. As we can see,
the distance between both coherent states is now given by $2\cos\theta$.

Figure 8: Probability of success when conditioning the results of the second
mode to obtain an odd number of photons, versus
$\delta\tilde{\alpha}\equiv\delta\alpha\cos(\theta)$ (in absolute value). We
present the results for three different beam splitters characterized by mixing
angles $\theta=\pi/3$ (blue continuous line), $\theta=\pi/2$ (orange dashed
line) and $\theta=\pi/6$ (green dashed-dotted line). The obtained results are
similar to the ones obtained with the even number of photons measurement, but
slightly enhanced. The vertical continuous grey lines correspond to the values
of $|\delta\tilde{\alpha}|$ for which we get the different Wigner functions
shown in the insets (i)-(iii) on the right. In these plots we have set
$\tilde{\alpha}=0$ so that the functions are centered in the origin.

One of the main concerns regarding these operations is the probability of
being successful when performing the conditioning measurement over one of the
beam splitter modes (see Appendix A for its analytic derivation). In Fig. 7 we
present this quantity for three beam splitters characterized by mixing angles
$\theta=\pi/3$ (blue continuous line), $\theta=\pi/2$ (orange dashed line) and
$\theta=\pi/6$ (green dashed-dotted line). The reflectivity/transmisivity
ratio in these three cases are 75/25, 50/50 and 25/75, respectively. As
$\delta\tilde{\alpha}\equiv\delta\alpha\cos\theta$ increases, the success
probability decays almost exponentially for the three angles. However,
depending on the specific values the decay behaves differently. Of particular
interest is the case of $\theta=\pi/6$, for which the distance between the
coherent states is the largest and given by $\sqrt{3}\delta\alpha$. In this
case, the maximum success probability is slightly smaller than 0.1, but the
decay for increasing values of $\delta\tilde{\alpha}$ takes place more slowly
than in the other cases. In fact, for $|\delta\tilde{\alpha}|\geq 1.5$, it
clearly surpasses the other three cases, with a success probability on the
order of $10^{-2}$ to $10^{-3}$ up to $|\delta\tilde{\alpha}|=2$.

On the other hand, instead of conditioning the second mode over the
measurement of an even number of photons, we can instead condition over an odd
number. In that situation, the final state is similar to the one shown in Eq.
(31) but with a relative phase of $\pi$ between both states in the
superposition, i.e. an _odd_ optical Schrödinger cat state. Furthermore, as
can be seen in Fig. 8, this measurement allow us to slightly enhance the final
probability of success.

In the insets located in the right part of Figs. 7 and 8 we present the
density plot of the measured Wigner functions in a displaced frame of
reference for different values of $|\delta\tilde{\alpha}|$, in particular
$|\delta\tilde{\alpha}|=1.0,1.5$ and $2.0$. They present the typical form of
even and odd _dead/alive_ cat states: as $|\delta\tilde{\alpha}|$ increases,
the two coherent states in the superposition get far away one from the other,
which leads to the appearance of two positive peaks in the Wigner function
located in $x=\pm\delta\tilde{\alpha}$, as can be seen clearly for the case
$|\delta\tilde{\alpha}|=2.0$ (inset (iii) in Figs. 7 and 8). In such case, the
maxima and minima appearing in between corresponds with their overlap, which
are interchanged for even and odd cat states: the negative parts in this
intermediate region appearing for the even cat state, turn into minima for the
odd cat state and viceversa. Furthermore, in this case we have performed our
calculations with the state in Eq. (27), which means that for small values of
$|\delta\tilde{\alpha}|$ the final Wigner function is not completely symmetric
as can be seen more clearly for the odd cat state at
$|\delta\tilde{\alpha}|=-1.0$ in Fig. 8 (i), for which the positive ring
around the negative minima is not symmetric.

##  4 Conclusions

In this work we have studied how can we use the _dead/alive_ cat states
generated with HHG in order to obtain superpositions containing three coherent
states, and we have characterized them by looking at their Wigner function.
While the first method is based on an interferometer approach, where in each
arm we perform HHG, the second one is based on sequential HHG processes over
the same mode. Here, we have analysed some particular configurations that
allow us to obtain a wide variety of states, as we could see from the analysis
of their Wigner functions which show different tendencies for increasing
values of the generated shift $\delta\alpha$ depending on the specific set-up.
However, there exists a large set of possibilities depending on where do we
implement the conditioning measurements, and also on the different optical
elements we can use in between.

Additionally, we demonstrated that we can use these three coherent state
superpositions for the generation of enlarged versions of even and odd
_dead/alive_ cat states, which could be very beneficial for practical purposes
in quantum computation and quantum information. However, the main drawback of
this method is that it relies on a heralding measurement over one of the
outputs of a beam splitter, whose success probability decays almost
exponentially as the desired _dead/alive_ cat state becomes larger. Regarding
this, possible ways of improving the present results might be oriented towards
the performance of more complex measurements over the second mode that can
enhance the final probability distribution. Therefore, this motivates further
investigation towards the generation of non-classical light states with HHG.

Finally, in this work we have highlighted the potential role of strong-field
physics towards quantum optics. Given the wide variety of strong-field
processes that one can generate in the laboratory apart from HHG, a natural
question that arises is how their quantum analysis could be advantageous for
practical purposes in fields like quantum computation and quantum information.
Here we have indirectly tackled this question by altering and improving the
HHG Schrödinger cat states. Nevertheless, other strong-field processes, like
Above-Threshold Ionization where the electron recollides elastically with its
parent ion Faria2020 , could lead to the desired superpositions more directly.
Thus, the above discussion stimulates theoretical and experimental research
towards the connection between strong-field physics and practical uses of
quantum information processing.

###### Acknowledgements.

ICFO group acknowledges support from ERC AdG NOQIA, Agencia Estatal de
Investigación (“Severo Ochoa” Center of Excellence CEX2019-000910-S, Plan
National FIDEUA PID2019-106901GB-I00/10.13039/501100011033, FPI), Fundació
Privada Cellex, Fundació Mir-Puig, and from Generalitat de Catalunya (AGAUR
Grant No. 2017 SGR 1341, CERCA program, QuantumCAT_U16-011424, co-funded by
ERDF Operational Program of Catalonia 2014-2020), MINECO-EU QUANTERA MAQS
(funded by State Research Agency (AEI)
PCI2019-111828-2/10.13039/501100011033), EU Horizon 2020 FET-OPEN OPTOLogic
(Grant No 899794), and the National Science Centre, Poland-Symfonia Grant No.
2016/20/W/ST4/00314, Marie Sklodowska-Curie grant STRETCH No 101029393. 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.T. group
acknowledges LASERLABEUROPE (H2020-EU.1.4.1.2 Grand ID 654148)), FORTH Synergy
Grant AgiIDA (Grand No.: 00133). 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).

## Declarations

Conflicts of interest The authors declare that they have no conflict of
interest

Code and data availability The code used for the generation of the presented
figures is made available in FigureMaker  under the Creative Commons
Attribution-ShareAlike 4.0 (CC BY-SA 4.0) license.

## References

  * (1) 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, _(accepted Nat. Phys.)_ (2021).
  * (2) J. O’Brien, A. Furusawa and J. Vučković, Photonic quantum technologies, _Nature Photon._ , 3 687-695 (2009). https://doi.org/10.1038/nphoton.2009.229
  * (3) N. Gisin and R. Thew, Quantum communication, _Nature Photon._ , 1 165-171 (2007). https://doi.org/10.1038/nphoton.2007.22
  * (4) V. Giovannetti, S. Lloyd and L. Maccone, Quantum-Enhanced Measurements: Beating the Standard Quantum Limit, _Science_ , 306 1330-1336 (2004). https://doi.org/10.1126/science.1104149
  * (5) M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, pp. 277-352. Cambridge University Press, Cambridge, United Kingdom (2010).
  * (6) 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, _J. Opt B: Quantum Semiclass. Opt._ , 6 S828 (2004). https://doi.org/10.1088/1464-4266/6/8/032
  * (7) K. Amini et al., Symphony on strong field approximation, _Rep. Prog. Phys._ , 82 116001 (2019). https://doi.org/10.1088/1361-6633/ab2bb1
  * (8) F. Krausz and M. Y. Ivanov, Attosecond physics, _Rev. Mod. Phys._ , 81 163-234 (2009). https://doi.org/10.1103/RevModPhys.81.163
  * (9) M. F. Ciappina et al., Attosecond physics at the nanoscale, _Rep. Prog. Phys._ , 80 054401 (2017). https://doi.org/10.1088/1361-6633/aa574e
  * (10) M. Drescher, M. Hentschel, R. Kienberger, G. Tempea, C. Spielmann, G. A. Reider, P. B. Corkum and F. Krausz, X-ray Pulses Approaching the Attosecond Frontier, _Science_ , 291 1923-7 (2001). https://doi.org/10.1126/science.1058561
  * (11) T. Popmintchev et al., Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers, _Science_ , 336 1287-1291 (2012). https://doi.org/10.1126/science.1218497
  * (12) P. B. Corkum, Plasma perspective on strong field multiple ionization, _Phys. Rev. Lett._ , 71 1994-1997 (1993). https://doi.org/10.1103/PhysRevLett.71.1994
  * (13) K. C. Kulander, K. J. Schafer and J. L. Krause, Dynamics of short-pulse excitation, ionization and harmonic conversion, pp. 95-110, Plenum, New York (1993).
  * (14) M. Lewenstein, Ph. 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 2117-2132 (1994). https://doi.org/10.1103/PhysRevA.49.2117
  * (15) B. Sundaram and P. W. Milonni, High-order harmonic generation: Simplified model and relevance of single-atom theories to experiment, _Phys. Rev. A_ , 41 6571-6573 (1990). https://doi.org/10.1103/PhysRevA.41.6571
  * (16) H. Xu, Non-perturbative theory of harmonic generation under a high-intensity laser field, _Z. Phys. D: At. Mol. Clusters_ , 28 27-36 (1993). https://doi.org/10.1007/BF01437452
  * (17) G. Compagno, K. Dietz and F. Persico, QED theory of harmonic emission by a strongly driven atom, _J. Phys. B: At. Mol. Opt. Phys._ , 27 4779-4815 (1994). https://doi.org/10.1088/0953-4075/27/19/031
  * (18) 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, _Phys. Rev. A_ , 52 525-540 (1995). https://doi.org/10.1103/PhysRevA.52.525
  * (19) W. Becker, A. Lohr, M. Kleber and M. Lewenstein, A unified theory of high-harmonic generation: Application to polarization properties of the harmonics, _Phys. Rev. A_ , 56 654-656 (1997). https://doi.org/10.1103/PhysRevA.56.645
  * (20) D. J. Diestler, Harmonic generation: Quantum-electrodynamical theory of the harmonic photon-number spectrum, _Phys. Rev. A_ , 78 033814 (2008). https://doi.org/10.1103/PhysRevA.78.033814
  * (21) 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 063827 (2014). https://doi.org/10.1103/PhysRevA.89.063827
  * (22) 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 32821 (2016). https://doi.org/10.1038/srep32821
  * (23) A. Bogatskaya, E. A. Volkova and A. M. Popov, Spontaneous transitions in atomic system in the presence of high-intensity laser field, _EPL_ , 116 14003 (2016). https://doi.org/10.1209/0295-5075/116/14003
  * (24) A. V. Bogatskaya, E. A. Volkova, V. Y. Kharin and A. M. Popov, Polarization response in extreme nonlinear optics: when can the semiclassical approach be used?, _Las. Phys. Lett._ , 13 045301 (2016). https://doi.org/10.1088/1612-2011/13/4/045301
  * (25) A. V. Bogatskaya, E. A. Volkova and A. M. Popov, Spontaneous emission of atoms in a strong laser field, _JETP_ , 125 587-596 (2017). https://doi.org/10.1134/S1063776117090114
  * (26) A. V. Bogatskaya, E. A. Volkova and A. M. Popov, Spectroscopy of the atomic system driven by high intensity laser field, arXiv:1701.05777 (2017).
  * (27) A. I. Magunov and V. V. Strelkov, S-matrix approach to the problem of high-harmonic generation in the field of intense laser wave, _Phy. Wave Phen._ , 25 24–29 (2017). https://doi.org/10.3103/S1541308X17010046
  * (28) J. Rivera-Dean, Quantum-optical analysis of high-order harmonic generation, _Master thesis_ (2019). http://hdl.handle.net/2117/168580
  * (29) A. Gorlach, O. Neufeld, N. Rivera, O. Cohen and I. Kaminer, The quantum-optical nature of high-harmonic generation, _Nat. Commun_ , 11 4598 (2020). https://doi.org/10.1038/s41467-020-18218-w
  * (30) D. N. Yangaliev, V. P. Krainov and O. I. Tolstikhin, Quantum theory of radiation by nonstationary systems with application to high-order harmonic generation, _Phys. Rev. A_ , 101 013410 (2020). https://doi.org/10.1103/PhysRevA.101.013410
  * (31) A. Gombkötő, S. Varró, P. Mati and, P. Földi, High-order harmonic generation as induced by a quantized field: Phase-space picture, _Phys. Rev. A_ , 101 013418 (2020). https://doi.org/10.1103/PhysRevA.101.013418
  * (32) N. Tsatrafillys, 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. Commun._ , 8 15170 (2017). https://doi.org/10.1038/ncomms15170
  * (33) N. Tsatrafillys, S. Kühn, M. Dumerque, P. Foldi, S. Kahaly, E. Cormier, I. A. Gonoskov, B. Kiss, K. Varju, S. Varro and P. Tzallas, Quantum Optical Signatures in a Strong Laser Pulse after Interaction with Semiconductors, _Phys. Rev. Lett._ , 122 193602 (2019). https://doi.org/10.1103/PhysRevLett.122.193602
  * (34) E. Bloch, S. Beaulieu, D. Descamps, S. Petit, F. Légaré, A. Magunov, T. Mairesse and V. Strelkov, Hyper-Raman lines emission concomitant with high-order harmonic generation, _New. J. Phys._ , 21 073006 (2019). https://doi.org/10.1088/1367-2630/ab28b9
  * (35) J. Rivera-Dean, P. Stammer, E. Pisanty, Th. Lamprou, P. Tzallas, M. Lewenstein and M. F. Ciappina, Figure-maker code for _New schemes for creating large optical Schrödinger cat states using strong laser fields_ , Zenodo:5031359 (2021). https://doi.org/10.5281/zenodo.5031359
  * (36) E. Schrödinger, Die gegenwärtige Situation in der Quantenmechanik, _Naturwissenschaften_ , 23 807-812 (1935). https://doi.org/10.1007/BF01491987
  * (37) A. I. Lvovsky, P. Grangier, A. Ourjoumtsev, V. Parigi, M. Sasaki and. R. Tualle-Brouri, Production and applications of non-Gaussian quantum states of light, arXiv:2006.16985 (2020).
  * (38) T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro and S. Glancy, Quantum computation with optical coherent states, _Phys. Rev. A_ , 68 042319 (2003). https://doi.org/10.1103/PhysRevA.68.042319
  * (39) B. C. Sanders, Entangled coherent states, _Phys. Rev. A_ , 45 6811 (1992). https://doi.org/10.1103/PhysRevA.45.6811
  * (40) H. Jeong, W. Son, M. S. Kim, D. Ahn and C̆. Brukner, Quantum nonlocality test for continuous-variable states with dichotomic variables, _Phys. Rev. A_ , 67 012106 (2003). https://doi.org/10.1103/PhysRevA.67.012106
  * (41) M. Stobińska, H. Jeong and T. C. Ralph, Violation of Bell’s inequality using classical measurements and nonlinear local operations, _Phys. Rev. A_ , 75 052105 (2007). https://doi.org/10.1103/PhysRevA.75.052105
  * (42) W. J. Munro, K. Nemoto, G. J. Milburn and S. L. Braunstein, Weak-force detection with superposed coherent states, _Phys. Rev. A_ , 66 023819 (2002). https://doi.org/10.1103/PhysRevA.66.023819
  * (43) M. Dakna, T. Anhut, T. Opatrný, L. Knöll and. D.-G. Welsch, Generating Schrödinger-cat-like states by means of conditional measurements on a beam splitter, _Phys. Rev. A_ , 55 3184 (1997). https://doi.org/10.1103/PhysRevA.55.3184
  * (44) A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat and P. Grangier, Generating Optical Schrödinger Kittens for Quantum Information Processing, _Science_ , 312 8386 (2006). https://doi.org/10.1126/science.1122858
  * (45) B. Hacker, S. Welte, S. Daiss, A. Shaukat, S. Ritter, L. Li and G. Rempe, Deterministic creation of entangled atom-light Schrödinger-cat states, _Nature Photon._ , 13 110-115 (2019). https://doi.org/10.1038/s41566-018-0339-5
  * (46) S. J. van Enk, Entanglement Capabilities in Infinite Dimensions: Multidimensional Entangled Coherent States, _Phys. Rev. Lett._ , 91 017902 (2003). https://doi.org/10.1103/PhysRevLett.91.017902
  * (47) 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 012105 (2003). https://doi.org/10.1103/PhysRevA.67.012105
  * (48) E. Wigner, On the Quantum Correction for Thermodynamic Equilibrium, _Phys. Rev._ , 40 749-759 (1932). https://doi.org/10.1103/PhysRev.40.749
  * (49) M. O. Scully and M. S. Zubairy, Quantum Optics, 72-94. Cambridge University Press, Cambridge, United Kingdom (2006).
  * (50) W. P. Schleich, Quantum Optics in Phase Space, 67-99. Wiley-VCH, Berlin, Germany (2001).
  * (51) D. T. Smithey, M. Beck, M. G. Raymer and A. Faridani, Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum, _Phys. Rev. Lett._ , 70 1244 (1993). https://doi.org/10.1103/PhysRevLett.70.1244
  * (52) R. L. Hudson, When is the Wigner quasi-probability density non-negative?, _Reports on Mathematical Physics_ , 6 249-252 (1974). https://doi.org/10.1016/0034-4877(74)90007-X
  * (53) A. Laghout, J. S. Neergaard-Nielsen, I. Rigas, C. Kragh, A. Tipsmark and U. L. Andersen, Amplification of realistic Schrödinger-cat-state-like states by homodyne heralding, _Phys. Rev. A_ , 87 043826 (2013). https://doi.org/10.1103/PhysRevA.87.043826
  * (54) D. V. Sychev, A. E. Ulanov, A. A. Pushkina, M. W. Richards, I. A. Fedorov and A. I. Lvovsky, Enlargement of optical Schrödinger’s cat states, _Nature Photon._ , 11 379-382 (2017). https://doi.org/10.1038/nphoton.2017.57
  * (55) T. Gerrits et al., Generation of optical coherent-state superpositions by number-resolved photon substraction from the squeezed vacuum, _Phys. Rev. A_ , 82 031802(R) (2010). https://doi.org/10.1103/PhysRevA.82.031802
  * (56) In the literature, even and odd cat states are defined as those that, once expanded in the Fock basis, can be written in terms of even or odd Fock states, respectively. In particular, they correspond to superpositions of the form $\ket{\alpha}\pm\ket{-\alpha}$, where the additive superposition leads to even cat states, and the negative superposition to odd cat states. Here, we use the _even_ and _odd_ cat states terminology in a more general basis for referring to even and odd superpositions of displaced number states with respect to some coherent state.
  * (57) C. Figueira de Morisson Faria and A. S. Maxwell, It is all about phases: ultrafast holographic photoelectron imaging, _Rep. Prog. Phys._ , 83 034401 (2020). https://doi.org/10.1088/1361-6633/ab5c91

##  Appendix A Probability of success in the generalization of the
conditioning approach

Here we present a more detailed analysis about the probability of success
plotted in Figs. 7 and 8. In the text we are interested in measuring even or
odd number of photons in one of the output modes of the beam splitter, as this
operation projects the other mode into a cat state, which differ in a relative
phase of $\pi$ depending on the specific measurement that is performed. Thus,
the set of positive operator-valued measurements characterizing the operations
that we can implement is

$\bigg{\\{}\mathbbm{1}\otimes\outerproduct{0}{0},\mathbbm{1}\otimes\sum_{n=1}^{\infty}\outerproduct{2n}{2n},\mathbbm{1}\otimes\sum_{n=0}^{\infty}\outerproduct{2n+1}{2n+1}\bigg{\\}},$
(32)

which project the second mode into the vacuum state, an even number of photons
and an odd number of photons respectively.

Each of the measurements described in Eq. (32) have a probability associated,
so we define the _probability of success_ $\mathcal{P}$ as the probability of
successfully performing one of such measurements. For instance, the
probability of successfully measuring an even number of photons is given by

$\mathcal{P}_{\text{even}}=\dfrac{1}{\mathcal{P}_{\text{total}}}\tr{P_{\text{even}}\rho},$
(33)

where $\rho$ is the density matrix characterizing the state of our system, and
$\mathcal{P}_{\text{total}}$ is given as the sum of all the possible
measurements that we can perform, i.e.

$\mathcal{P}_{\text{total}}=\mathcal{P}_{\text{zero}}+\mathcal{P}_{\text{even}}+\mathcal{P}_{\text{odd}}.$
(34)

With all this set, let us consider the architecture presented in the main text
used for the generation of enlarged cat states. Using a beam splitter
characterized by the mixing angle $\theta$ and a coherent state of the form
$\ket{(\alpha-\delta\alpha)\tan(\theta)}$, the final state after the beam
splitter is given by

$\displaystyle\ket{\Phi_{\text{BS}}}$
$\displaystyle=\dfrac{1}{\sqrt{N}}\Big{[}-\xi_{1}^{\prime}\ket{\tilde{\alpha}+\delta\alpha\cos\theta}\ket{-\delta\alpha\sin\theta}$
(35) $\displaystyle\hskip
31.2982pt+\ket{\tilde{\alpha}}\ket{0}-\xi_{2}\ket{\tilde{\alpha}-\delta\alpha\cos\theta}\ket{\delta\alpha\sin\theta}\Big{]},$

where $\tilde{\alpha}=(\alpha-\delta\alpha)/\cos\theta$. Thus, the measurement
of an even number of photons projects the state in Eq. (35) into

$\displaystyle\ket{\Phi_{\text{even}}}$
$\displaystyle=\dfrac{1}{\sqrt{N}}\Big{[}-\xi_{1}^{\prime}\ket{\tilde{\alpha}+\delta\alpha\cos\theta}-\xi_{2}\ket{\tilde{\alpha}-\delta\alpha\cos\theta}\Big{]}$
(36) $\displaystyle\hskip
14.22636pt\otimes\sum_{n=1}^{\infty}\dfrac{(\delta\alpha\sin\theta)^{2n}}{\sqrt{2n!}}e^{-\tfrac{|\delta\alpha\sin\theta|^{2}}{2}}\ket{2n}$
$\displaystyle\equiv\ket{\phi_{\text{even}}}\otimes\sum_{n=1}^{\infty}\dfrac{(\delta\alpha\sin\theta)^{2n}}{\sqrt{2n!}}e^{-\tfrac{|\delta\alpha\sin\theta|^{2}}{2}}\ket{2n};$

the measurement of an odd number of photons projects the state into

$\displaystyle\ket{\Phi_{\text{odd}}}$
$\displaystyle=\dfrac{1}{\sqrt{N}}\Big{[}\xi_{1}^{\prime}\ket{\tilde{\alpha}+\delta\alpha\cos\theta}-\xi_{2}\ket{\tilde{\alpha}-\delta\alpha\cos\theta}\Big{]}$
(37) $\displaystyle\hskip
14.22636pt\otimes\sum_{n=0}^{\infty}\dfrac{(\delta\alpha\sin\theta)^{2n+1}}{\sqrt{2n+1!}}e^{-\tfrac{|\delta\alpha\sin\theta|^{2}}{2}}\ket{2n+1}$
$\displaystyle\equiv\ket{\phi_{\text{odd}}}\otimes\sum_{n=0}^{\infty}\dfrac{(\delta\alpha\sin\theta)^{2n+1}}{\sqrt{2n+1!}}e^{-\tfrac{|\delta\alpha\sin\theta|^{2}}{2}}\ket{2n+1};$

and, finally, the measurement of zero number of photons leads to

$\displaystyle\ket{\Phi_{\text{zero}}}$
$\displaystyle=\dfrac{1}{\sqrt{N}}\Big{[}-\xi_{1}^{\prime}\innerproduct{0}{-\delta\alpha\sin\theta}\ket{\tilde{\alpha}+\delta\alpha\cos\theta}+\ket{\tilde{\alpha}}$
(38) $\displaystyle\hskip
31.2982pt-\xi_{2}\innerproduct{0}{\delta\alpha\sin\theta}\ket{\tilde{\alpha}-\delta\alpha\cos\theta}\Big{]}\otimes\ket{0}$
$\displaystyle\equiv\ket{\phi_{\text{zero}}}\otimes\ket{0}.$

Thus, the probability of success for each of the three measurements that we
can do is given by

$\mathcal{P}_{\text{even}}=\dfrac{\innerproduct{\phi_{\text{even}}}{\phi_{\text{even}}}}{\mathcal{P}_{\text{total}}}e^{-|\delta\alpha\sin\theta|^{2}}\big{[}-1+\cosh(\delta\alpha\sin\theta)\big{]},$
(39)
$\mathcal{P}_{\text{odd}}=\dfrac{\innerproduct{\phi_{\text{odd}}}{\phi_{\text{odd}}}}{\mathcal{P}_{\text{total}}}e^{-|\delta\alpha\sin\theta|^{2}}\sinh(\delta\alpha\sin\theta),$
(40)

and

$\mathcal{P}_{\text{zero}}=\dfrac{\innerproduct{\phi_{\text{zero}}}{\phi_{\text{zero}}}}{\mathcal{P}_{\text{total}}}.$
(41)

Note that the case of $\delta\alpha=0$ leads to a divergence in the obtained
equations. In that case the applied measurement does not make sense at all
because the case of $\delta\alpha=0$ corresponds to the situation in which we
do not have HHG processes taking place. Thus, no optical Schrödinger cat state
is being generated.