# Quantum vacuum signatures in multi-color laser pulse collisions

Holger Gies<EMAIL_ADDRESS>Helmholtz-Institut Jena, Fröbelstieg 3,
07743 Jena, Germany  GSI Helmholtzzentrum für Schwerionenforschung,
Planckstraße 1, 64291 Darmstadt  Theoretisch-Physikalisches Institut, Abbe
Center of Photonics,
Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany
Felix Karbstein<EMAIL_ADDRESS>Helmholtz-Institut Jena,
Fröbelstieg 3, 07743 Jena, Germany  GSI Helmholtzzentrum für
Schwerionenforschung, Planckstraße 1, 64291 Darmstadt  Theoretisch-
Physikalisches Institut, Abbe Center of Photonics,
Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany
Leonhard Klar<EMAIL_ADDRESS>Helmholtz-Institut Jena, Fröbelstieg
3, 07743 Jena, Germany  GSI Helmholtzzentrum für Schwerionenforschung,
Planckstraße 1, 64291 Darmstadt  Theoretisch-Physikalisches Institut, Abbe
Center of Photonics,
Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany

###### Abstract

Quantum vacuum fluctuations give rise to effective non-linear interactions
between electromagnetic fields. A prominent signature of quantum vacuum
nonlinearities driven by macroscopic fields are signal photons differing in
characteristic properties such as frequency, propagation direction and
polarization from the driving fields. We devise a strategy for the efficient
tracing of the various vacuum-fluctuation-mediated interaction processes in
order to identify the most prospective signal photon channels. As an example,
we study the collision of up to four optical laser pulses and pay attention to
sum and difference frequency generation. We demonstrate how this information
can be used to enhance the signal photon yield in laser pulse collisions for a
given total laser energy.

##  I Introduction

Although the vacuum is the state characterized by the absence of real
particles it constitutes a portal to the particle degrees of freedom of the
underlying quantum field theory due to the omnipresence of quantum
fluctuations. In quantum electrodynamics (QED), which governs the interaction
of light and matter within the Standard Model of particle physics, these
fluctuations involve virtual electrons, positrons and photons. Electromagnetic
fields provide a promising means to probe these quantum vacuum fluctuations:
as electromagnetic fields couple to charges, vacuum fluctuations involving
charged particles mediate effective interactions between them Euler:1935zz ;
Heisenberg:1935qt ; Weisskopf:1936kd . The latter supplement Maxwell’s
classical theory of electrodynamics with nonlinear self-couplings of the
electromagnetic field. Using Heaviside-Lorentz units with $c=\hbar=1$, the
metric convention $g_{\mu\nu}=\mathrm{diag}(-,+,+,+)$. these quantum vacuum
nonlinearities are parametrically suppressed by inverse powers of
$m_{e}^{2}/e\simeq 1.3\times 10^{18}\,\frac{\rm V}{\rm m}\simeq 4.4\times
10^{9}\,{\rm T}$. Here $e$ is the elementary charge and $m_{e}$ the electron
mass, setting the reference scale the applied electromagnetic fields are
compared to. Due to the fact that the strongest macroscopic fields available
in the laboratory fulfill $E\simeq{\cal O}(10^{14})\frac{\rm V}{\rm m}$ and
$B\simeq{\cal O}(10^{6}){\rm T}$, the induced interactions are generically
very small and elusive in experiment.

All-optical probes provide a prominent route towards verifying QED vacuum
nonlinearities in a controlled laboratory experiment with macroscopic
electromagnetic fields. Various theoretical proposals studied in the
literature assume these electromagnetic fields to be delivered by high-
intensity lasers; see the Reviews Marklund:2006my ; DiPiazza:2011tq ;
Battesti:2012hf ; King:2015tba ; Hill:2017 ; Inada:2017lop ; Karbstein:2019oej
; Schoeffel:2020svx  and references therein. The basic idea is to look for
signal photons which are induced in the effective interaction of several
driving laser fields and differ in key properties, such as polarization,
frequency and propagation direction, from the photons constituting the latter.
However, separating the typically small signal from the large background in
general constitutes a major challenge Tommasini:2010fb ; King:2013am ;
Karbstein:2019dxo ; Karbstein:2020gzg . Aiming at a systematic enhancement of
photonic quantum vacuum signals, a detailed knowledge about the microscopic
origin of the prospective signal photon channels is indispensable. In this
article, we demonstrate how their microscopic origin can be efficiently
traced. Moreover, using a particular scenario envisioning the collision of
several high-intensity laser pulses as an example, we show how this
information can be used to enhance the signal photon yield for a given total
laser energy.

Our article is organized as follows: in Sec. II we briefly recall the vacuum
emission picture used in the present article to study all-optical signatures
of quantum vacuum nonlinearity in macroscopic electromagnetic fields as
provided by high-intensity lasers. Moreover, we introduce several definitions
relevant for the subsequent discussion. Thereafter, in Sec. III we apply the
formalism outlined in Sec. II to an exemplary experimental scenario based on
the collision of several high-intensity laser pulses. Following more general
considerations about a prospective collision geometry involving beams of
several colors, we analyze and outline the frequency and directional
characteristics of the attainable signals and assess the possibility of their
measurement against the background of the driving laser photons. In Sec. IV.1
we demonstrate in detail how such studies can be substantially enhanced and
simplified. These considerations highlight an aspect of the vacuum emission
approach which was not yet fully exploited in previous all-optical quantum
vacuum studies. Namely, we show how the microscopic origin of the signal can
be efficiently traced and analyzed. Moreover, we sketch how the information
extracted along these lines can be employed to enhance prospective signals.
Finally, we end with conclusions and a short outlook in Sec. V.

##  II Formalism

All-optical signatures of quantum vacuum nonlinearity can be efficiently
analyzed in terms of vacuum emission processes Galtsov:1971xm ;
Karbstein:2014fva : in the interaction region where the strong driving laser
fields overlap signal photons are generated. These signal photons are to be
detected far outside the interaction region and constitute the signature of
quantum vacuum nonlinearity in experiment. For state-of-the-art high-intensity
laser fields of optical and near-infrared frequencies $\omega\ll m_{e}$
reaching electric and magnetic peak field strengths
$\\{E,B\\}\ll\frac{e}{m_{e}^{2}}$, such a study can be based on the leading
contribution to the one-loop Heisenberg-Euler effective Lagrangian ${\cal
L}_{\rm HE}$ Euler:1935zz ; Heisenberg:1935qt ; Weisskopf:1936kd ;
Karplus:1950zz ; Schwinger:1951nm ; Ritus:1975cf ; Dittrich:1985yb ;
Dittrich:2000zu ; Dunne:2004nc ; Gies:2016yaa . The latter is a function of
the two scalar invariants of the electromagnetic field,

$\displaystyle\mathcal{F}$
$\displaystyle=\frac{1}{4}F^{\mu\nu}F_{\mu\nu}=\frac{1}{2}\left(\mathbf{B}^{2}-{\bf
E}^{2}\right)\,,$ $\displaystyle\mathcal{G}$
$\displaystyle=\frac{1}{4}\tilde{F}^{\mu\nu}F_{\mu\nu}=-\mathbf{B}\cdot{\bf
E}\,,$ (1)

with the metric convention $g_{\mu\nu}=\mathrm{diag}(-,+,+,+)$, and can be
decomposed as ${\cal L}_{\rm HE}={\cal L}_{\rm MW}+{\cal L}_{\rm int}$. Here,
${\cal L}_{\rm MW}=-{\cal F}$ is the classical Maxwell Lagrangian and

$\frac{\mathcal{L}_{\text{int}}}{m_{e}^{4}}=\frac{1}{360\pi^{2}}\Bigl{(}\frac{e}{m_{e}^{2}}\Bigr{)}^{4}\left(4\mathcal{F}^{2}+7\mathcal{G}^{2}\right)+\mathcal{O}\\!\left(\big{(}\tfrac{eF}{m_{e}^{2}}\bigr{)}^{6}\right)$
(2)

encodes the effective nonlinear interactions of the electromagnetic field
induced by QED vacuum fluctuations. As detailed in Refs. Karbstein:2014fva ;
Gies:2017ygp ; Karbstein:2019oej , the differential number of signal photons
${\rm d}^{3}N_{\left(p\right)}$ of polarization $p$ which have an energy ${\rm
k}=|\bf{k}|$ in the differential energy interval ${\rm dk}$ and are emitted
into the solid angle ${\rm d}\Omega$ around $\hat{{\bf k}}$ follows from the
zero-to-single signal photon transition amplitude ${\cal S}_{(p)}({\bf k})$ as

${\rm d}^{3}N_{(p)}=\frac{{\rm k^{2}}{\rm dk}\,{\rm
d}\Omega}{(2\pi)^{3}}\,|{\cal S}_{(p)}({\bf k})|^{2}\,.$ (3)

The signal photon amplitude can be determined from $\Gamma_{\rm
int}[\hat{a}(x)]=\int{\rm d}^{4}x\,{\cal L}_{\rm int}|_{F\to F+\hat{f}}$ upon
splitting the electromagnetic field as $F^{\mu\nu}\to
F^{\mu\nu}+\hat{f}^{\mu\nu}$ into a classical background field $F^{\mu\nu}$
and a operator-valued signal photon field $\hat{a}^{\mu}$, with field-strength
tensor
$\hat{f}^{\mu\nu}=\partial^{\mu}\hat{a}^{\nu}-\partial^{\nu}\hat{a}^{\mu}$. It
is given by

$\displaystyle S_{(p)}\left(\mathbf{k}\right)$
$\displaystyle=\bigl{\langle}\gamma_{\left(p\right)}\left(\mathbf{k}\right)\bigr{|}\Gamma_{\text{int}}[\hat{a}(x)]\bigl{|}0\bigr{\rangle}$
$\displaystyle=\mathrm{i}\frac{\epsilon^{*\mu}_{\left(p\right)}\bigl{(}\hat{\bf
k}\bigr{)}}{\sqrt{2k^{0}}}\int\mathrm{d}^{4}x\;\mathrm{e}^{\mathrm{i}k_{\alpha}x^{\alpha}}$
$\displaystyle\phantom{=}\times\Bigl{(}k^{\nu}F_{\nu\mu}\frac{\partial\mathcal{L}_{\text{int}}}{\partial\mathcal{F}}+k^{\nu}\tilde{F}_{\nu\mu}\frac{\partial\mathcal{L}_{\text{int}}}{\partial\mathcal{G}}\Bigr{)}\biggl{|}_{k^{0}={\rm
k}}\,,$ (4)

where $\epsilon^{\mu}_{\left(p\right)}\bigl{(}\hat{\bf k}\bigr{)}=(0,{\bf
e}_{(p)})$ is the polarization vector of the induced signal photon; $*$
denotes complex conjugation. We span the signal photon polarizations by two
transverse vectors ${\bf e}_{(p)}$ with $p\in\\{1,2\\}$, fulfilling $\hat{\bf
k}\times{\bf e}_{(p)}={\bf e}_{(p+1)}$ and ${\bf e}_{(3)}=-{\bf e}_{(1)}$. The
derivatives for ${\cal F}$ and ${\cal G}$ entering Eq. (4) follow
straightforwardly from Eq. (2),

$\left\\{\begin{array}[]{c}\\!\\!\frac{\partial\mathcal{L}_{\text{int}}}{\partial\mathcal{F}}\\!\\!\\\
\\!\\!\frac{\partial\mathcal{L}_{\text{int}}}{\partial\mathcal{G}}\\!\\!\end{array}\right\\}=\frac{1}{45}\frac{e^{2}}{4\pi}\Bigl{(}\frac{e}{m_{e}^{2}}\Bigr{)}^{2}\left\\{\begin{array}[]{c}\\!4\mathcal{F}\left(x\right)\\!\\!\\\
\\!7\mathcal{G}\left(x\right)\\!\\!\end{array}\right\\}+\mathcal{O}\left(\bigl{(}\tfrac{e\bar{F}}{m_{e}^{2}}\bigr{)}^{4}\right).$
(5)

Upon plugging these quantities into Eq. (4) and limiting ourselves to the
terms written out explicitly, we obtain

$\displaystyle S_{(p)}\left(\mathbf{k}\right)$
$\displaystyle=\frac{1}{\mathrm{i}}\frac{e^{2}}{4\pi}\frac{m_{e}^{2}}{45}\sqrt{\frac{\rm
k}{2}}\left(\frac{e}{m_{e}^{2}}\right)^{3}\int\mathrm{d}^{4}x\;\mathrm{e}^{\mathrm{ik}(\hat{\bf
k}-t)}$
$\displaystyle\phantom{=}\times\Bigl{(}4\left[\mathbf{e}_{\left(p\right)}\cdot{\bf
E}-\mathbf{e}_{\left(p+1\right)}\cdot\mathbf{B}\right]\mathcal{F}$
$\displaystyle\phantom{=\times\Bigl{(}}+7\left[\mathbf{e}_{\left(p\right)}\cdot\mathbf{B}+\mathbf{e}_{\left(p+1\right)}\cdot{\bf
E}\right]\mathcal{G}\Bigr{)}\,.$ (6)

In this article, we focus on the collision of $n+1$ linearly polarized
paraxial laser fields characterized by the electric and magnetic field vectors
${\bf E}_{i}={\cal E}_{i}(x)\,\hat{\bf E}_{i}$ and ${\bf B}_{i}={\cal
E}_{i}(x)\,\hat{\bf B}_{i}$, with $i\in\\{0,1,\ldots,n\\}$, fulfilling
$\hat{\bf E}_{i}\cdot\hat{\bf B}_{i}=0$. The associated unit wave vectors are
$\hat{\boldsymbol{\kappa}}_{i}=\hat{\bf E}_{i}\times\hat{\bf B}_{i}$. In this
case, the photon transition amplitude (6) can be expressed as
$S_{\left(p\right)}\left(\mathbf{k}\right)=\sum_{i,j,l}S_{\left(p\right);ijl}\left(\mathbf{k}\right)$,
with

$S_{\left(p\right);ijl}\left(\mathbf{k}\right)=\frac{1}{\mathrm{i}}\frac{e^{2}}{4\pi}\frac{m_{e}^{2}}{45}\sqrt{\frac{\rm
k}{2}}\,\Bigl{(}\frac{e}{m_{e}^{2}}\Bigr{)}^{3}\mathcal{I}_{ijl}\left(\mathbf{k}\right)g_{\left(p\right);ijl}\bigl{(}\hat{\mathbf{k}}\bigr{)}.$
(7)

Here, the entire dependence on the field profiles ${\cal E}_{i}(x)$ is encoded
in the quantity

$\mathcal{I}_{ijl}\left(\mathbf{k}\right)=\int\\!\mathrm{d}^{4}x\,\mathrm{e}^{\mathrm{ik}(\hat{\bf
k}\cdot\hat{\bf
x}-t)}\mathcal{E}_{i}(x)\mathcal{E}_{j}(x)\mathcal{E}_{l}(x)\,,$ (8)

and the dependence on the polarization assignments of the beams and the
collision geometry in

$\displaystyle g_{(p);ijl}(\hat{\bf k}\bigr{)}$
$\displaystyle=2\bigl{(}\mathbf{e}_{(p)}\\!\cdot\hat{\bf
E}_{l}-\mathbf{e}_{(p+1)}\\!\cdot\hat{\bf B}_{l}\bigr{)}\bigl{(}\hat{\bf
B}_{i}\cdot\hat{\bf B}_{j}-\hat{\bf E}_{i}\cdot\hat{\bf E}_{j}\bigr{)}$
$\displaystyle-\frac{7}{2}\big{(}\mathbf{e}_{(p)}\\!\cdot\hat{\bf
B}_{l}+\mathbf{e}_{(p+1)}\\!\cdot\hat{\bf E}_{l}\bigr{)}\bigl{(}\hat{\bf
B}_{i}\cdot\hat{\bf E}_{j}+\hat{\bf B}_{j}\cdot\hat{\bf E}_{i}\bigr{)}.$ (9)

In the following, we parameterize the emission directions of the signal
photons as $\hat{\bf
k}=(\cos\varphi\sin\vartheta,\sin\varphi\sin\vartheta,\cos\vartheta)$, such
that ${\rm d}^{2}\Omega={\rm d}\varphi\,{\rm d}\\!\cos\vartheta$. We choose
the vectors spanning the polarization basis of the signal photons as
$\mathbf{e}_{(1)}(\beta)=\hat{\bf
k}|_{\vartheta\to\vartheta+\frac{\pi}{2}}\cos\beta+\hat{\bf
k}|_{\vartheta=\frac{\pi}{2},\varphi\to\varphi+\frac{\pi}{2}}\sin\beta$ and
$\mathbf{e}_{(2)}(\beta)=\mathbf{e}_{(1)}(\beta+\pi/2)$, where $\beta$ is an a
priori arbitrary angle; its choice fixes a specific polarization basis.
Moreover, we introduce the number density $\rho_{(p)}(\varphi,\vartheta|{\rm
k}_{\rm min},{\rm k}_{\rm max})$ of signal photons of energies $\rm k$
constrained by ${\rm k}_{\rm min}\leq{\rm k}\leq{\rm k}_{\rm max}$. The latter
is obtained from Eq. (3) upon integration over energy as

$\rho_{(p)}(\varphi,\vartheta|{\rm k}_{\rm min},{\rm k}_{\rm
max})=\frac{1}{(2\pi)^{3}}\\!\int_{\mathrm{k}_{\rm min}}^{\mathrm{k}_{\rm
max}}\\!\\!\\!\mathrm{dk}\,{\rm
k}^{2}\bigl{|}S_{(p)}(\mathbf{k})\bigr{|}^{2}.$ (10)

In the present case, the modulus squared of the signal photon amplitude
entering Eq. (10) can be expressed as

$\bigl{|}S_{(p)}(\mathbf{k})\bigr{|}^{2}=\sum_{\ell,\ell^{\prime}}{\rm
Re}\bigl{\\{}S_{(p);\ell}(\mathbf{k})S^{*}_{(p);\ell^{\prime}}(\mathbf{k})\bigr{\\}}\,,$
(11)

where the sum runs over all sets $\ell=\\{i,j,l\\}$ and
$\ell^{\prime}=\\{i^{\prime},j^{\prime},l^{\prime}\\}$. From Eq. (10) the
number of signal photons of polarization $p$ and energies ${\rm k}_{\rm
min}\leq{\rm k}\leq{\rm k}_{\rm max}$ emitted into the solid angle ${\cal A}$
follows as

$N_{(p)}({\cal A}|{\rm k}_{\rm min},{\rm k}_{\rm max})=\int_{\cal A}{\rm
d}\Omega\,\rho_{(p)}(\varphi,\vartheta|{\rm k}_{\rm min},{\rm k}_{\rm
max})\,.$ (12)

Also note that the signal photon density and number accessible in a
polarization insensitive measurement follow upon summation over the two
transverse polarizations $p\in\\{1,2\\}$. They are given by
$\rho(\varphi,\vartheta|{\rm k}_{\rm min},{\rm k}_{\rm
max})=\sum_{p=1}^{2}\rho_{(p)}(\varphi,\vartheta|{\rm k}_{\rm min},{\rm
k}_{\rm max})$ and $N({\cal A}|{\rm k}_{\rm min},{\rm k}_{\rm
max})=\sum_{p=1}^{2}N_{(p)}({\cal A}|{\rm k}_{\rm min},{\rm k}_{\rm max})$,
respectively. Obviously, the total number of emitted signal photons is $N_{\rm
tot}=N(4\pi|0,\infty)$.

##  III Example Scenario

Let us apply the approach devised in Sec. II to a specific, experimentally
viable scenario involving the collision of several high-intensity laser
pulses. Our main focus is on the analysis and reconstruction of properties of
the microscopic scattering processes giving rise to the dominant signal photon
emission channels. Special attention is paid to signal photon contributions
which allow for a clear signal-to-background separation in experiment.
Prominent criteria facilitating such a separation are, e.g., a distinct
emission direction outside the forward cones of the driving beams, or a
frequency outside their spectra allowing for an unobstructed detection of the
signal.

###  III.1 Collision geometry and beam model

For definiteness, we use a collision geometry involving $n+1$ driving laser
pulses Klar:2020ych : beam $0$ collides with the apex of the regular pyramid
formed by the beam axes of $n$ additional laser beams; see Fig. 1 for an
illustration. Here, we focus on a scenario involving four driving laser
fields, i.e., $n=3$. These are envisioned to be generated by a single high-
intensity laser system of the ten petawatt class, such as available at ELI-NP
Lureau:2020 ; ELI-NP , by employing beam-splitting and frequency doubling
techniques. More specifically, we assume the initial laser system to deliver
pulses of energy $W=250\,{\rm J}$ and duration $\tau=25\,{\rm fs}$ at a
wavelength of $\lambda=800\,{\rm nm}$. While each of the four laser fields
generated in this way features exactly this pulse duration, we assume them to
have different frequencies: beams $0$ and $1$ are fundamental frequency beams
with $\omega_{0}=\frac{2\pi}{\lambda}\simeq 1.55\,\text{eV}$, beam $2$ is
frequency-doubled, and beam $3$ frequency-quadrupled.

Figure 1: Illustration of the scenario considered in the main text: three
high-intensity pulses of unit wave vectors $\hat{\boldsymbol{\kappa}}_{i}$,
with $i\in\\{1,2,3\\}$, are superposed to create a strongly peaked field
configuration in their overlap region; each pair of these beams encloses the
same angle $\theta$. An additional high-intensity laser pulse with
$\hat{\boldsymbol{\kappa}}_{0}$ collides with the apex of the pyramid formed
by beams $1$-$3$ under an angle of $\alpha$. The wave vector of the signal
photons is denoted by $\mathbf{k}$. It encodes the signal frequency in its
length (radius of the red sphere), and the emission direction in its
orientation parameterized by the angles $\vartheta$ and $\varphi$. The
possible polarizations of the signal are parameterized by the angle $\beta$
spanning a plane tangential to the sphere.

Moreover, we assume that each pair of the beams $1$-$3$ encloses the same
angle $\theta$, such that

$\hat{\boldsymbol{\kappa}}_{i}\cdot\hat{\boldsymbol{\kappa}}_{j}=(1-\cos\theta)\delta_{ij}+\cos\theta$
(13)

for $i,j\in\\{1,2,3\\}$ with Kronecker delta $\delta_{ij}$. Accordingly we
choose these unit wave vectors as

$\displaystyle\hat{\boldsymbol{\kappa}}_{i}=\left(\begin{array}[]{c}-a\cos\bigl{(}2\pi\frac{i-1}{3}\bigr{)}\vspace{1mm}\\\
-a\sin\bigl{(}2\pi\frac{i-1}{3}\bigr{)}\vspace{1mm}\\\
\sqrt{1-a^{2}}\end{array}\right),$ (17)

with $a=\sqrt{2(1-\cos{\theta})/3}$. On the other hand, the wave vector of the
beam $0$ colliding with the apex of the pyramid formed by beams $1$-$3$ is
$\hat{\boldsymbol{\kappa}}_{0}=-\hat{\mathbf{e}}_{z}$. The angle between this
and any other beam is $\alpha=\pi-\arctan(a/\sqrt{1-a^{2}})$. In the remainder
of this work, we set $\theta=\frac{\pi}{2}=90^{\circ}$, such that
$\alpha=\pi-\arctan\sqrt{2}\approx 125.26^{\circ}$.

We assume the driving laser fields to be well-described as paraxial Gaussian
beams. In order to facilitate a concise analytical analysis of the expected
signals, we resort to an infinite Rayleigh range approximation. This
approximation neglects the widening of the laser pulses as a function of the
longitudinal coordinate ${\bf x}\cdot\hat{\boldsymbol{\kappa}}_{i}$ measured
from the beam focus at ${\bf x}={\bf 0}$, where the temporal pulse envelope
reaches its maximum at $t=0$. This is a reasonable approximation of the
driving laser fields in the vicinity of their beam foci. We thus expect our
predictions for the attainable signal photon numbers to be quantitatively
accurate in the specific collision scenario considered here, where the
collision angles between each pair of laser beams
$\theta_{i,j}=\varangle(\hat{\boldsymbol{\kappa}}_{i},\hat{\boldsymbol{\kappa}}_{j})$
fulfill
$0^{\circ}\leq\varangle(\hat{\boldsymbol{\kappa}}_{i},\hat{\boldsymbol{\kappa}}_{j})\leq
130^{\circ}$ Gies:2017ygp . Note that, in principle, this analysis could also
be performed directly with paraxial Gaussian beams Gies:2017ygp  or even
generic laser fields fulfilling Maxwell’s equations in vacuo exactly
Blinne:2018nbd  even though this is technically or numerically more demanding.
Since all driving laser pulses are assumed to have the same pulse duration
$\tau$, the field profile of the $i$th laser beam can be approximated in the
interaction region as Gies:2017ygp

$\mathcal{E}_{i}\left(x\right)=\mathfrak{E}_{i}\,\mathrm{e}^{-(\frac{\bf
x\cdot\hat{\boldsymbol{\kappa}}_{i}-t}{\tau/2})^{2}}\,\mathrm{e}^{-\frac{{\bf
x}_{i}^{2}-({\bf
x}\cdot\hat{\boldsymbol{\kappa}_{i}})^{2}}{w_{i}^{2}}}\cos\bigl{(}\omega_{i}({\bf
x}\cdot\hat{\boldsymbol{\kappa}}_{i}-t)\bigr{)}\,,$ (18)

where $\mathfrak{E}_{i}$, $w_{i}$ and $\omega_{i}$ are the peak field
amplitude, beam waist and oscillation frequency, respectively. Throughout this
work, we consider only optimal laser pulse collisions, i.e., all laser beams
are focused on the same spot and reach their peak field values at the same
time.

Besides, we fix the linear polarizations of the laser fields, by choosing
$\hat{\mathbf{E}}_{0}=\mathbf{e}_{y}$ and
$\mathbf{e}_{y}\cdot\hat{\mathbf{E}}_{i}=0$ for $i\in\\{1,2,3\\}$. This choice
is motivated by the observation that the total number of signal photons
attainable in a polarization insensitive measurement is maximized for counter-
propagating beams with a relative polarization difference of $\pi/2$
Karbstein:2019bhp . We have explicitly checked that this is also the case for
our setup, where beam 0 can be considered as effectively counter-propagating
the combined field of beams 1-3. Together with the transversality condition
$\hat{\boldsymbol{\kappa}}_{i}\cdot\hat{\mathbf{E}}_{i}=0$ the above choice
determines the polarization vectors of all laser fields up to a sign. Ensuring
a positive sign for the $\rm x$ component, the polarization vectors of beams
$i\in\\{1,2,3\\}$ read

$\displaystyle\hat{\mathbf{E}}_{i}=\frac{1}{\sqrt{1-a^{2}\sin^{2}(2\pi\frac{i-1}{3})}}\left(\begin{array}[]{c}\sqrt{1-a^{2}}\\\
0\\\ a\cos(2\pi\frac{i-1}{3})\end{array}\right).$ (22)

The unit vectors for the associated magnetic fields are
$\hat{\mathbf{B}}_{i}=\hat{\boldsymbol{\kappa}}_{i}\times\hat{\mathbf{E}}_{i}$.

###  III.2 Beam splitting and losses

The peak field amplitude of a laser beam of pulse energy $W$ and duration
$\tau$ focused to a waist spot size of $w_{0}$ is Karbstein:2017jgh ;
Karbstein:2019oej

$\mathcal{E}_{\star}=\sqrt{8\sqrt{\frac{2}{\pi}}\frac{W}{\pi
w_{0}^{2}\tau}}\,.$ (23)

For a diffraction-limited Gaussian beam of wavelength $\lambda$, we have
$w_{0}\simeq\lambda$. Throughout this work, we assume all the individual laser
fields to be focused to the same waist spot size $w_{i}=w_{0}=\lambda$ and
measure their peak field amplitudes in units of the peak field ${\cal
E}_{\star}$ which could be achieved by focusing the initial laser pulse of
energy $W=250\,{\rm J}$ to its diffraction limit. In turn, we have
$\mathfrak{E}_{i}=A_{i}{\cal E}_{\star}$, where $A_{i}$ denote dimensionless
amplitudes, also accounting for potential losses. Analogously, we measure the
oscillation frequencies of the beams in units of $\omega_{0}$, such that
$\omega_{i}=\nu_{i}\omega_{0}$, with dimensionless amplitude $\nu_{i}$. In the
present case, here we have $\nu_{0}=\nu_{1}=1$, $\nu_{2}=2$ and $\nu_{3}=4$.

Each frequency doubling process comes with a loss: we conservatively estimate
the energy loss for the conversion process preserving the pulse duration as
$50\%$ Marcinkevicius:2004 . Correspondingly, the energies $W_{i}=A_{i}^{2}W$
of all beams do not add up to $W$ but to $W^{\rm eff}=\sum_{i=0}^{3}W_{i}<W$.
Only, for vanishing losses we would have $W^{\rm eff}=W$. Here, we assume the
beam splitting and higher harmonic generation to proceed in several steps.
First, the original laser pulse of energy $W$ is split into two parts: the
part with energy $W_{0}=(1-q_{0})W$ constitutes beam $0$, and the remainder of
energy $q_{0}W$ is to be subdivided further; the factor $0<q_{0}<1$ controls
the partitioning ratio. Second, the remaining energy $q_{0}W$ is again
partitioned into a fundamental frequency part of energy
$W_{1}=q_{0}(1-q_{1})W$ constituting beam $1$, and another one of energy
$q_{0}q_{1}W$ which undergoes frequency doubling; as above $0<q_{1}<1$.
Accounting for the loss of $50\%$ associated with the frequency doubling
process, the latter contribution results in a frequency-doubled pulse of
energy $\frac{1}{2}q_{0}q_{1}W$. In the last step, the procedure is repeated
for the frequency-doubled pulse with a partitioning factor of $0<q_{2}<1$.
This results in an energy of $W_{2}=\frac{1}{2}q_{0}q_{1}(1-q_{2})W$ for the
frequency-doubled beam $2$ and an energy of
$W_{3}=\frac{1}{4}q_{0}q_{1}q_{2}W$ for the frequency-quadrupled beam $3$.
Hence, in the present scenario we have $A_{0}=\sqrt{1-q_{0}}$,
$A_{1}=\sqrt{q_{0}(1-q_{1})}$, $A_{2}=\sqrt{q_{0}q_{1}(1-q_{2})/2}$ and
$A_{3}=\sqrt{q_{0}q_{1}q_{2}}/2$. Figure 2 illustrates this procedure for a
specific choice of the partition factors; see Fig. 3 below for the influence
of different choices of the partition factors on the signal photon numbers.

Figure 2: Schematic of the beam splitting and higher harmonic generation
processes invoked to create four beams $i\in\\{0,1,2,3\\}$ of pulse energy
$W_{i}$ from a single high-intensity laser system delivering pulses of energy
$W=250\,\text{J}$ and frequency $\omega_{0}=1.55\,\text{eV}$. The partitioning
proceeds in three stages, with the associated partition factors given by
$q_{0}$, $q_{1}$ and $q_{2}$. Beams $2$ and $3$ are frequency-doubled and
quadrupled, respectively. Each frequency doubling comes with a loss of $50\%$.
We depict the scenario with $q_{0}=5/6$. The values of $q_{1}$ and $q_{2}$ are
chosen such that $W_{1}=2W_{2}=4W_{3}$. Figure 3: Signal photon spectra
associated with the three different pulse-energy distributions of the driving
laser fields $(a)$-$(c)$ listed in Tab. 1. For each distribution we provide
the values of $N(4\pi|{\rm k_{min}},{\rm k_{max}})$ for the frequency regimes
$1.04\,\mathrm{eV}\lesssim{\rm k}\lesssim 2.06\,\mathrm{eV}$ (red),
$2.59\,\mathrm{eV}\lesssim{\rm k}\lesssim 3.61\,\mathrm{eV}$ (green) and
$5.69\,\mathrm{eV}\lesssim{\rm k}\lesssim 6.71\,\mathrm{eV}$ (blue). The
respective signal photon number is written inside the particular bar.

So far, we did not specify a particular choice of the dimensionless field
amplitudes $A_{i}$ of the driving laser beams. As detailed above, in an
experiment these amplitudes can be adjusted by choosing the partition factors
$q_{0}$, $q_{1}$ and $q_{2}$ accordingly. For definiteness, we choose
$q_{0}=\frac{5}{6}$ in the following. In this way, a substantial fraction of
the total laser energy the beam that collides with the apex of the pyramid
formed by the other beams.

We limit our discussion to three example distributions of the pulse energies
of beams $1$-$3$ forming the pyramid: either the pulse energy of each higher
frequency component is doubled (such that $W_{1}:W_{2}:W_{3}=1:2:4$), bisected
($W_{1}:W_{2}:W_{3}=4:2:1$), or quartered ($W_{1}:W_{2}:W_{3}=16:4:1$). The
explicit values of the required partition factors $q_{1}$ and $q_{2}$ are
listed in Tab. 1, together with the respective effective energy $W^{\rm eff}$
put into the interaction region by all four driving laser pulses.
Correspondingly, the associated energy loss is given by $W^{\rm loss}=W-W^{\rm
eff}$.

Table 1: Examples of different pulse-energy distributions for beams $1$-$3$. For each choice, we provide the values of the partition factors $q_{1}$ and $q_{2}$ required to ensure a given distribution for fixed $q_{0}=5/6$. $W^{\rm eff}$ is the effective energy put into the interaction region by all four beams. The initially available total laser energy is $W=250\,{\rm J}$, such that the associated energy loss is $W^{\rm loss}=W-W^{\rm eff}$. |  $W_{1}:W_{2}:W_{3}$ |  $q_{1}$ |  $q_{2}$ |  $W^{\rm eff}[{\rm J}]$ |  $W^{\rm loss}[{\rm J}]$
---|---|---|---|---|---
(a) |  $1:2:4$ |  $\;20/21\;$ |  $4/5$ |  111.11 |  138.89
(b) |  $4:2:1$ |  $2/3$ |  $1/2$ |  163.19 |  86.81
(c) |  $16:4:1$ |  $3/7$ |  $1/3$ |  197.92 |  52.08

Equation (12) allows to determine the number of signal photons $N(4\pi|{\rm
k_{min}},{\rm k_{max}})$ for each distribution. In the present scenario, we
find substantial signal photon contributions in the three distinct frequency
regimes $1.04\,\mathrm{eV}\lesssim{\rm k}\lesssim 2.06\,\mathrm{eV}$,
$2.59\,\mathrm{eV}\lesssim{\rm k}\lesssim 3.61\,\mathrm{eV}$ and
$5.69\,\mathrm{eV}\,\mathrm{eV}\lesssim{\rm k}\lesssim 6.71\,\mathrm{eV}$,
centered at the frequencies of the driving laser beams $\omega_{0}$,
$2\omega_{0}$ and $4\omega_{0}$, respectively. The width of each of these
regimes is $1.02\,{\rm eV}$ and has been chosen such as to cover the full
signal; cf. also the discussion in Sec. III.3 below. See Fig. 2 for the signal
photon numbers associated with these frequency regimes for the different
pulse-energy distributions considered in Tab. 1: in the first approach we
double the pulse energy put into each higher harmonic. This leads to a
relatively high number of signal photons with ${\rm k}\simeq 4\omega_{0}$ as
compared to the other regimes. Interestingly, even when bisecting the pulse
energy put into each higher harmonic, the number of signal photons with ${\rm
k}\simeq 4\omega_{0}$ slightly surpasses that for ${\rm k}\simeq 2\omega_{0}$.
However, most signal photons are induced at ${\rm k}\simeq\omega_{0}$. When
quartering the energy put into each higher harmonic most signal photons are
again found at ${\rm k}\simeq\omega_{0}$, but this time the amount of signal
photons with ${\rm k}\simeq 4\omega_{0}$ is smaller than that for ${\rm
k}\simeq 2\omega_{0}$. As is obvious from Tab. 1, higher pulse energies of the
frequency doubled and quartered beams imply larger losses.

Apart from these signals just, it is noteworthy we find a clear signal in the
frequency regime $7.24\,\mathrm{eV}\lesssim{\rm k}\lesssim 8.26\,{\rm eV}$
peaked at a frequency of $5\omega_{0}$. However, due to the substantially
smaller amount of signal photons associated with this frequency regime, we do
not display this signal in Fig. 3. For the pulse-energy distribution (a) we
count $1.07$, for (b) $2.81$, and for (c) $1.64$ signal photons per shot in
this frequency regime. On the other hand, the fact that this signal lies
outside the frequencies of the driving laser fields implies the possibility of
an essentially background-free detection.

The fact that the clearly discernible signal at $5\omega_{0}$ becomes maximum
for the pulse-energy distribution (b) with $q_{1}=2/3$ and $q_{2}=1/2$
motivates us to focus on this choice in the remainder of this article. For
completeness, we note that the relative amplitudes associated with this choice
are $A_{0}=1/\sqrt{6}$, $A_{1}=\sqrt{5/2}/3$, $A_{2}=\sqrt{5}/6$ and
$A_{3}=\sqrt{5/2}/6$.

###  III.3 Frequency and directional characteristics of the signal

First, we aim at resolving the frequency spectrum of the full signal in
detail. The full signal is obtained upon integration of Eq. (12) over all
emission directions, such that $\mathcal{A}=4\pi$, and summing over both
transverse polarizations $p$. To this end, we sample the signal photon number
$N(4\pi|{\rm k},{\rm k}+\Delta{\rm k})$ with a bin range of $\Delta{\rm
k}=0.02\,\text{eV}$. The results of this analysis are presented as histograms
in Fig. 4. The signal spectrum exhibits four pronounced maxima; the positions
of three maxima match the oscillation frequencies of the driving laser fields.
The additional maximum is centered around ${\rm k}\simeq 5\omega_{0}$. Adding
the contributions of all bins we obtain a total number of
$N_{\text{tot}}\simeq 5600$ signal photons.

Figure 4: Histogram of $N(4\pi|{\rm k},{\rm k}+\Delta{\rm k})$ in the
frequency regime $1\,{\rm eV}\lesssim{\rm k}\lesssim 8.3\,{\rm eV}$; the bins
range is $\Delta{\rm k}=0.02\,{\rm eV}$ and the signal photon number per bin
is given on the right axis. The left axis gives the differential number of
signal photons ${\rm d}N$ determined by performing Gaussian fits to the
histogram data (solid lines). The integrals of these curves reproduce the
signal photon numbers counted in the histograms reasonably well. In the
spectral regime highlighted here, the signal photons are predominantly induced
at frequencies ${\rm k}\simeq n\omega_{0}$ with $n\in\\{1,2,4\\}$ (dashed
vertical lines) matching those of the driving laser beams. In addition, we
encounter a signal peaked around $5\omega_{0}$.

In the present scenario, the positions of all the peaks can be understood in
terms of elastic Heinzl:2006xc ; DiPiazza:2006pr ; Tommasini:2010fb ;
Monden:2011 ; King:2012aw ; Dinu:2014tsa ; Karbstein:2015xra ;
Kadlecova:2019dxv ; Jeong:2020vpy  and manifestly inelastic Mckenna:1963 ;
Varfolomeev:1966 ; Rozanov:1996 ; Moulin:2002ya ; Lundstrom:2005za ;
DiPiazza:2005jc ; Lundin:2006wu ; Fedotov:2006ii ; Bernard:2010dx ;
Gies:2014jia ; Fillion-Gourdeau:2014uua ; Bohl:2015uba ; Gies:2017ezf ;
Aboushelbaya:2019ncg  sum or difference frequency generation processes
involving only the oscillation frequencies of the driving laser fields; cf. in
particular also Gies:2017ygp ; King:2018wtn . The reason is that the pulse
duration $\tau$ is much larger than the cycle durations
$1/(\nu_{i}\omega_{0})$. It is straightforward to verify that signal photon
emission in the formal limit of $\tau\rightarrow\infty$ is indeed restricted
to sharp delta peaks at the frequencies of the driving beams and combinations
thereof; in this limit the Fourier integral (8) over time results in Dirac
delta functions Gies:2017ygp . The finite width of the peaks in frequency
space is a consequence of the finiteness of $\tau$, and implies that the
frequency selection rules associated with the limit of $\tau\to\infty$ are
fulfilled only approximately.

While this is obvious for the quasi-elastic signal photon channels at
$\omega_{i}$, the signal with frequency ${\rm k}\simeq 5\omega_{0}$ outside
the frequency spectra of the driving beams highlighted in the inlay of Fig. 4
can be attributed to a sum and difference frequency generation process.

Besides the number of signal photons per bin, Fig. 4 shows the differential
number of signal photons ${\rm d}N/{\rm dk}$ extracted from these histograms:
upon dividing the signal photons numbers in a given bin by the bin range
$\Delta{\rm k}$, we assume their distribution in a given frequency range to be
well described by a Gaussian function. In all cases, the fitted peak values of
the Gaussians are close the frequencies $\nu_{i}\omega_{0}$. We find the peak
values at $(1.556\pm 4.8\times 10^{-5}){\rm eV}$, $(3.11\pm 2.8\times
10^{-5}){\rm eV}$, $(6.20\pm 4.9\times 10^{-6}){\rm eV}$, and $(7.74\pm
4.0\times 10^{-5}){\rm eV}$. The Gaussian standard deviations $\sigma_{G}$ are
$(84.47\pm 0.19)\times 10^{-3}\,\text{eV}$ for the $\omega_{0}$ signal,
$(87.01\pm 0.11)\times 10^{-3}\,\text{eV}$ for $2\omega_{0}$, $(86.97\pm
0.02)\times 10^{-3}\,\text{eV}$ for $4\omega_{0}$, and $(92.62\pm 0.18)\times
10^{-3}\,\text{eV}$ for $5\omega_{0}$. In the interval $\omega_{i}\pm
3\sigma_{G}$ $99\%$ of the Gaussian distributed signal is located. The values
for $\sigma_{G}$ extracted here are about an order of magnitude smaller than
the width of $1.02\,{\rm eV}$ employed to cover the full signal in the
previous section. Correspondingly, the above choice should indeed reliably
cover the full signal, while being still small enough to prevent an overlap of
the signals associated with other frequencies. Besides, the bin size
$\Delta{\rm k}$ should be sufficiently small to resolve potential deviations
from Gaussian distributions in the spectral domain.

The directional distribution of the signal photons of energies ${\rm k}_{\rm
min}\leq{\rm k}\leq{\rm k}_{\rm max}$ attainable in a polarization insensitive
measurement is encoded in the number density $\rho(\varphi,\vartheta|{\rm
k_{min}},{\rm k_{max}})$. Fig. 5 shows the Mollweide projection of the signal
photon density as a function of the azimuthal and polar angles $\varphi$ and
$\vartheta$, accounting for four different frequency regimes. This projection
maps the surface of a sphere onto a flat two dimensional chart conserving
areas. Obviously, it is not conformal.

Figure 5: Mollweide plot (longitude $\varphi$, latitude $\vartheta$) of the
signal photon density $\rho(\varphi,\vartheta|{\rm k}_{\rm min},{\rm k}_{\rm
max})$ highlighting four distinct frequency regimes ${\rm k}_{\rm min}\leq{\rm
k}\leq{\rm k}_{\rm max}$ in different colors, namely
$1.04\,\mathrm{eV}\leq{\rm k}\leq 2.06\,\mathrm{eV}$ (red),
$2.59\,\mathrm{eV}\leq{\rm k}\leq 3.61\,\mathrm{eV}$ (green),
$5.69\,\mathrm{eV}\leq{\rm k}\leq 6.71\,\mathrm{eV}$ (blue), and $7.24\,{\rm
eV}\leq{\rm k}\leq 8.26\,{\rm eV}$ (violet). All four color scales are linear
and are normalized to the maximum value in the respective frequency interval.

The four different frequency regimes are highlighted in different colors. We
adopt linear color scales which are normalized to the maximum value in a given
frequency regime. The brightest areas of a given color mark the dominant
emission directions of the signal photons. As to be expected, the signal
photons of frequencies close to $\omega_{0}$, $3\omega_{0}$ and $4\omega_{0}$,
respectively, are predominantly emitted in the forward cones of the driving
laser pulses featuring the same frequencies: apart from a
frequency-$\omega_{0}$ (red) peak at $\vartheta=180^{\circ}$, we observe three
distinct maxima at $\vartheta\approx 54.74^{\circ}$ which are separated by
$\approx 120^{\circ}$. These agree with the forward directions of the
additional – from left to right – $4\omega_{0}$ (blue), $\omega_{0}$ (red) and
$2\omega_{0}$ (green) beams. Additionally, we encounter a $5\omega_{0}$
(violet) signal at $\varphi\approx 23.26^{\circ}$ and $\vartheta\approx
50.85^{\circ}$.

Upon plugging these signal densities into Eq. (12) and integrating over the
full solid angle, we obtain the signal photon numbers in the respective
frequency regimes. This yields $N(4\pi|1.04\,{\rm eV},2.06\,{\rm eV})\simeq
3018.9$ signal photons in the regime around $\omega_{0}$, $N(4\pi|2.59\,{\rm
eV},3.61\,{\rm eV})\simeq 1240.8$ around $2\omega_{0}$, $N(4\pi|5.69\,{\rm
eV},6.71\,{\rm eV})\simeq 1337.6$ around $4\omega_{0}$ and $N(4\pi|7.24\,{\rm
eV},8.26\,{\rm eV})\simeq 2.81$ in the vicinity of $5\omega_{0}$; cf. also
Fig. 3.

We conclude this section by emphasizing that we have mainly focused on the
total numbers of signal photons induced in specific frequency intervals so far
and did not address the question of their measurability. This is particularly
unclear for the signals at $\omega_{0}$, $2\omega_{0}$ and $4\omega_{0}$ which
have been shown to be predominantly emitted into the forward directions of the
associated driving beams. In the next section we will address this question
and assess carefully which signal photon contributions could be isolated or
distinguished from the large background of the driving laser photons in
experiment.

###  III.4 Discernible signal photons

To assess if a specific signal can be discerned from the background of the
driving laser photons or not, we first have a look at the angular distribution
of the latter. Here we have modeled the driving lasers as Gaussian beams. The
far-field angular decay of a Gaussian beam made up of ${\cal N}_{i}$ photons
can be expressed as Karbstein:2019oej

${\rm d}^{2}\mathcal{N}_{i}(\varphi,\vartheta)={\rm d}^{2}\Omega\,\frac{{\cal
N}_{i}}{2\pi}(\omega_{i}w_{i})^{2}\,{\rm
e}^{-(\omega_{i}w_{i})^{2}\Theta_{i}^{2}(\varphi,\vartheta)/2}\,,$ (24)

where the angle $\Theta_{i}(\varphi,\vartheta)$ parameterizes the angular
decay of the laser photons measured from the forward beam axis
$\hat{\boldsymbol{\kappa}}_{i}$. As we consider Gaussian beams which feature a
rotational symmetry around the beam axis, a single angle parameter is
sufficient. In the present scenario, we have $\Theta_{0}=\vartheta-\pi$ and
$\Theta_{i}=-{\rm
arccos}\\{\cos\vartheta[\cos\alpha+\cos(\varphi-2\pi\frac{4-i}{3})\sin\alpha]\\}$
for beams $i\in\\{1,2,3\\}$. Moreover, recall that we have
$\mathcal{N}_{i}=WA_{i}^{2}/(\nu_{i}\omega_{0})$ and
$\omega_{i}w_{i}=2\pi\nu_{i}$; cf. Sec. III.2. In the following, we use the
notation ${\rm d}^{2}\mathcal{N}(\varphi,\vartheta)=\sum_{i=0}^{3}{\rm
d}^{2}\mathcal{N}_{i}(\varphi,\vartheta)$ for the differential number of
photons ${\cal N}=\sum_{i=1}^{3}{\cal N}_{i}$ constituting all laser beams.

Since we have assumed that all laser beams are focused to the same waist
$w_{i}=\lambda$, the far-field angular divergences of the beams scale as $\sim
1/\nu_{i}$. This implies that the beam with the largest value of $\nu_{i}$
features the smallest far-field divergence. At the same time, the effective
extent of the interaction region, determining the far-field divergences of the
quasi-elastically scattered signal photons should be similar for all
individual beams. The steeper decay of the laser photons constituting the
$4\omega_{0}$ background suggests that the signal photons arising from the
$4\omega_{0}$ beam should be more easily detectable than the analogous
contributions for the other beams.

To illustrate the major challenge of signal-to-background separation we
emphasize the huge background provided by the photons constituting the driving
laser pulses. Their total number per shot is as large as ${\cal N}\simeq
5.3\times 10^{20}$ to be contrasted with the number of ${\cal O}(10^{3})$
signal photons achievable in this setup; cf. Sec. III.2 above.

Figure 6 depicts on the directional characteristics of both the signal and
driving laser photons on a logarithmic scale. While the background of the
laser photons indeed dominates in most directions, the signal surpasses the
background in certain angular regions. This already hints at the principle
possibility of measuring the signal over the background in certain directions
– though it is unclear at this point whether there are enough discernible
signal photons yielding a sufficient statistics in a concrete experiment.

Figure 6: Directional characteristics (longitude $\varphi$, latitude
$\vartheta$) of the differential number of driving laser photons ${\rm
d}^{2}\mathcal{N}(\varphi,\vartheta)$ constituting the background and the
associated signal photon density $\rho(\varphi,\vartheta|0,\infty)$. Angular
regions where the signal dominates over the background are colored in green;
regions where the background dominates in red. In each color regime we adopt a
logarithmic color scale normalized to the maximum value.

The answer of this question requires a more detailed study. For this, we
proceed to a frequency-resolved analysis. More specifically, we search for
discernible signal photons in the four distinct frequency regimes around
$\omega_{0}$, $2\omega_{0}$, $4\omega_{0}$ and $5\omega_{0}$ introduced above.

####  $\omega_{0}$ regime

Isolating the $\omega_{0}$ signal from the background seems particularly
challenging: the signal photon density $\rho(\varphi,\vartheta|{\rm
k_{min}},{\rm k_{max}})$ features just two peaks in the frequency regime
delimited by ${\rm k_{min}=1.02\,eV}$ and ${\rm k_{max}=1.05\,eV}$, both of
which are coinciding with the forward directions of the two driving laser
beams of frequency $\omega_{0}$. At the same time, exactly these beams come
with the largest far-field divergences. Besides, particularly due to the large
energy put into beam $0$, the number of background photons is maximal in this
frequency regime. Though it might be an option to discern at least parts of
the signal from the background by advanced detection techniques, based on
analyses of the decay behavior or polarization details of both the background
and the signal photons, here we proceed to the other frequency regimes
suggesting more easily accessible signals; cf. below.

####  $2\omega_{0}$ regime

Next, we focus on the frequency regime centered around $2\omega_{0}\simeq
3.1\,{\rm eV}$ and constrained by ${\rm k_{min}}=2.59\,\text{eV}$ and ${\rm
k_{max}}=3.61\,\text{eV}$. As is clearly visible in Fig. 7, using the
numerical Newton method we identify a local maximum of the signal photon
density $\rho(\varphi,\vartheta|{\rm k_{min}},{\rm k_{max}})$ at
$(\varphi,\vartheta)\simeq(317.65^{\circ},101.77^{\circ})$. By comparison with
Fig. 5, this particular maximum is clearly separated from the forward beam
axis of the driving $2\omega_{0}$ beam constituting the main background in
this specific frequency regime. For an estimate of the quantitative number of
signal photons, we limit ourselves to the angular region
$\mathcal{A}^{(2\omega_{0})}=\\{(\varphi,\vartheta)|\varphi\in[314^{\circ},324^{\circ}],\vartheta\in[96^{\circ},106^{\circ}]\\}$
marked by the blue frame in Fig. 7. An integration over this angular region
results in
$N(\mathcal{A}^{(2\omega_{0})}|2.59\,\text{eV},3.61\,\text{eV})\simeq 62$
signal photons per shot. For completeness, we note that this value essentially
constitutes to the full number $N(\mathcal{A}^{(2\omega_{0})}|0,\infty)$ of
signal photons emitted into this angular regime.

Figure 7: Mollweide plot (longitude $\varphi$, latitude $\vartheta$) of the
signal photon density $\rho(\varphi,\vartheta|{\rm k}_{\rm min},{\rm k}_{\rm
max})$. We highlight the relevant angular domain (marked by a red frame) for
signal photon energies in the regime constrained by $2.59\,\mathrm{eV}\leq{\rm
k}\leq 3.61\,\mathrm{eV}$; the linear green color scale is normalized to its
maximum. The blue frame marks the angular region for which the number of
discernible signal photons quoted in the main text is determined.

Upon numerically integrating Eq. (24) for the $2\omega_{0}$ beam $i=2$ over
the same angular interval, we find
$\mathcal{N}(\mathcal{A}^{(2\omega_{0})})\simeq 0.01$ background photons per
shot. Of course, the other driving beams do not induce a background in this
frequency regime. This analysis implies that essentially all $\simeq 62$
photons with the considered directional characteristics are signal photons,
which can thus be clearly distinguished from the background.

####  $4\omega_{0}$ regime

Further, we turn to $4\omega_{0}$ frequency regime constrained by ${\rm
k_{min}}=5.69\,\text{eV}$ and ${\rm k_{max}}=6.71\,\text{eV}$. Also in this
regime we search for a local maximum of the signal photon density besides the
dominant one in the forward cone of driving $4\omega_{0}$ laser beam $i=3$. A
numerical analysis of the signal photon density $\rho(\varphi,\vartheta|{\rm
k_{min}},{\rm k_{max}})$ utilizing the Newton method allows us to identify a
local maximum with the desired properties at
$(\varphi,\vartheta)\simeq(49.43^{\circ},79.44^{\circ})$. See Fig. 8 for a
graphical illustration of the signal photon density in the relevant angular
regime.

Figure 8: Mollweide plot (longitude $\varphi$, latitude $\vartheta$) of the
signal photon density $\rho(\varphi,\vartheta|{\rm k}_{\rm min},{\rm k}_{\rm
max})$. We highlight the relevant angular domain (marked by a blue frame) for
signal photon energies in the regime constrained by $5.69\,\mathrm{eV}\leq{\rm
k}\leq 6.71\,\mathrm{eV}$; the linear blue color scale is normalized to its
maximum. The blue frame marks the angular region for which the number of
discernible signal photons quoted in the main text is determined.

An integration of the signal density over the area
$\mathcal{A}^{(4\omega_{0})}=\\{(\varphi,\vartheta)|\varphi\in[40^{\circ},56^{\circ}],\vartheta\in[76^{\circ},86^{\circ}]\\}$
highlighted in Fig. 8 results in $N(\mathcal{A}^{(4\omega_{0})})\approx 129$
signal photons per shot. We have explicitly checked that this is the full
number of signal photons emitted into this angular regime; there are no signal
photons of other frequencies. For comparison, in the same angular regime we
find $\mathcal{N}(\mathcal{A}^{(4\omega_{0})})\simeq 0.006$ driving laser
photons of frequency $4\omega_{0}$ per shot constituting the background.

####  $5\omega_{0}$ regime

As noted in Sec. III.3, apart from the signals just discussed, we also
identify a signal outside the frequencies of the driving laser beams in the
energy regime of $7.24\,{\rm eV}\leq{\rm k}\leq 8.26\,{\rm eV}$, featuring a
peak at about $5\omega_{0}\simeq 7.75\,{\rm eV}$. A numerical analysis of the
signal photon density $\rho(\varphi,\vartheta|{\rm k}_{\rm min},{\rm k}_{\rm
max})$ in this frequency regime unveils the existence of two pronounced maxima
signalizing two different main signal photon emission directions at (A):
$(\varphi,\vartheta)\simeq(23.26^{\circ},50.85^{\circ})$ and (B):
$(\varphi,\vartheta)\simeq(31.32^{\circ},31.01^{\circ})$, respectively.

In Fig. 9 we illustrate the signal photon density in the relevant angular
areas. The two maxima are located in the two angular areas marked by blue
frames. Maximum (A) is located in the upper blue frame delimiting the area
$\mathcal{A}^{(5\omega_{0},A)}=\\{(\varphi,\vartheta)|\varphi\in[14^{\circ},35^{\circ}],\vartheta\in[42^{\circ},60^{\circ}]\\}$,
and maximum (B) in the lower frame the area
$\mathcal{A}^{(5\omega_{0},B)}=\\{(\varphi,\vartheta)|\varphi\in[21^{\circ},42^{\circ}],\vartheta\in[25^{\circ},38^{\circ}]\\}$.

Figure 9: Mollweide plot (longitude $\varphi$, latitude $\vartheta$) of the
signal photon density $\rho(\varphi,\vartheta|{\rm k}_{\rm min},{\rm k}_{\rm
max})$. We highlight the relevant angular domain (marked by a red frame) for
signal photon energies in the regime constrained by $7.24\,\mathrm{eV}\leq{\rm
k}\leq 8.26\,\mathrm{eV}$; the linear violet color scale is normalized to its
maximum. The two blue frames mark the angular regions for which the number of
discernible signal photons quoted in the main text is determined.

Upon integration of the signal photon density over the two angular regimes
$\mathcal{A}^{(5\omega_{0},A)}$ and $\mathcal{A}^{(5\omega_{0},B)}$, we obtain
$N(\mathcal{A}^{(5\omega_{0},A)})\approx 2.3$ and
$N(\mathcal{A}^{(5\omega_{0},B)})\approx 0.5$ signal photons per shot. These
values essentially agree with the frequency-unresolved numbers of signal
photons emitted into the same angular regions, signaling the presence of
$5\omega_{0}$ photons only. We emphasize once again that there is no genuine
laser photon background in the $5\omega_{0}$ frequency regime since the
driving laser fields only contain frequencies in the vicinity of $\omega_{0}$,
$2\omega_{0}$ and $4\omega_{0}$.

We summarize the quantitative findings from the preceding sections in Tab. 2.
This table features the prospective signal photon numbers and numbers of
driving laser photons constituting the background for various frequencies and
emission directions.

Table 2: Prospective numbers of signal photons $N$ and driving laser photons $\mathcal{N}$ in different energy regimes and angular emission areas ${\cal A}$. See the main text for the definitions of the areas as well as further details. |  $2.59{\rm\,eV}\leq{\rm k}\leq 3.61{\rm\,eV}$ |  $5.69\,\text{eV}\leq{\rm k}\leq 6.71\,\text{eV}$ |  $7.24\,\text{eV}\leq{\rm k}\leq 8.26\,\text{eV}$
---|---|---|---
$N(4\pi)$ |  1240.80 |  1337.67 |  2.81
$\mathcal{N}(4\pi)$ |  $6.98\times 10^{19}$ |  $1.75\times 10^{19}$ |  0.00
$N(\mathcal{A}^{(2\omega_{0})})$ |  62.02 |  0.00 |  0.00
$\mathcal{N}(\mathcal{A}^{(2\omega_{0})})$ |  $10.13\times 10^{-3}$ |  0.00 |  0.00
$N(\mathcal{A}^{(4\omega_{0})})$ |  0.00 |  129.40 |  0.00
$\mathcal{N}(\mathcal{A}^{(4\omega_{0})})$ |  0.00 |  $5.91\times 10^{-3}$ |  0.00
$N(\mathcal{A}^{(5\omega_{0},A)})$ |  0.00 |  $8.19\times 10^{-5}$ |  2.31
$N(\mathcal{A}^{(5\omega_{0},B)})$ |  0.00 |  0.00 |  0.46

##  IV Channel analysis

In Sec. III we have identified several promising signals and demonstrated that
they are, in principle, discernible against the background of the photons of
the driving laser beams. To obtain these results we have relied on a rather
time consuming and brute force numerical evaluation of the signal photon
density $\rho_{(p)}(\varphi,\vartheta|{\rm k}_{\rm min},{\rm k}_{\rm max})$.
This quantity encodes information about all possible single photon emission
processes mediated by quantum vacuum fluctuations in the macroscopic field
driving the effect. In order to resolve different frequency regimes within
this approach, we have evaluated $\rho_{(p)}(\varphi,\vartheta|{\rm k}_{\rm
min},{\rm k}_{\rm max})$ for various values of ${\rm k}_{\rm min}$ and ${\rm
k}_{\rm max}$.

Subsequently, we demonstrate how these results can be obtained with
considerably less computational efforts, using the findings of the previous
section III as benchmarks for the new analysis carried out here. A channel
analysis for various three-pulse setups has been performed in Gies:2017ezf ;
King:2018wtn .

###  IV.1 Tracing the microscopic origin of the signal

Our starting point is the expression for the signal photon density in Eq.
(10), with the modulus squared of the zero-to-single signal transition
amplitude given by Eq. (11). Interchanging the integration over energy and the
summation over $\ell$ and $\ell^{\prime}$, the signal photon density can be
expressed as

$\rho_{(p)}(\varphi,\vartheta|{\rm k}_{\rm min},{\rm k}_{\rm
max})=\sum_{\ell,\ell^{\prime}}\rho_{(p);\ell,\ell^{\prime}}(\varphi,\vartheta|{\rm
k}_{\rm min},{\rm k}_{\rm max})\,,$ (25)

where the sum runs over all sets $\ell=\\{i,j,l\\}$,
$\ell^{\prime}=\\{i^{\prime},j^{\prime},l^{\prime}\\}$ and we have defined

$\displaystyle\rho_{(p);\ell,\ell^{\prime}}(\varphi,\vartheta|{\rm k}_{\rm
min},{\rm k}_{\rm max})$
$\displaystyle\quad\quad=\frac{1}{(2\pi)^{3}}\int_{\mathrm{k}_{\rm
min}}^{\mathrm{k}_{\rm max}}\mathrm{dk}\,{\rm k}^{2}\,{\rm
Re}\bigl{\\{}S_{(p);\ell}(\mathbf{k})S^{*}_{(p);\ell^{\prime}}(\mathbf{k})\bigr{\\}}.$
(26)

Accordingly, we introduce the signal photon number
$N_{(p);\ell,\ell^{\prime}}({\cal A}|{\rm k}_{\rm min},{\rm k}_{\rm max})$
following from Eq. (26) upon integration over the solid angle interval $\cal
A$. Obviously, $N_{(p)}=\sum_{\ell,\ell^{\prime}}N_{(p);\ell,\ell^{\prime}}$;
cf. Eq. (12).

While only the sum over all sets $\ell$ and $\ell^{\prime}$ constitutes the
physical density – off-diagonal terms with $\ell\neq\ell^{\prime}$ may even be
negative – this representation provides us with a convenient means to assess
the importance of individual contributions constituting the full density. As
demonstrated below, it allows to straightforwardly identify the single
interaction processes inducing a given vacuum emission channel. This
information can then be utilized to optimize the signal photon yield in this
channel, e.g., by changing the partitioning of the total available laser
energy into the different driving beams.

A closer look at the structure of $S_{(p);ijl}$ in Eqs. (7)-(9) unveils
certain symmetries which can be employed to reduce the number of contributions
to be evaluated explicitly. Most obviously, the Fourier integral (8) is
completely symmetric under permutations of the indices $i$, $j$, $l$. At the
same time, the function (9) which encodes the directional characteristics of
the interacting fields only exhibits a reduced symmetry: it is symmetric under
exchange of the first two indices, i.e., $g_{(p);ijl}(\hat{\bf
k}\bigr{)}=g_{(p);jil}(\hat{\bf k}\bigr{)}$, and vanishes if these two indices
agree, $g_{(p);iil}(\hat{\bf k}\bigr{)}=0$. Correspondingly, we have
$S_{(p);ijl}=S_{(p);jil}$ and $S_{(p);iil}=0$. This implies that the only non-
vanishing contributions with exactly two identical indices can be expressed in
terms of $S_{(p);iji}$, where $i\neq j$. On the other hand, given that all
indices are different, i.e., $i\neq j\neq l$, the only independent
contributions arise from $S_{(p);ijl}$.

Recall that when studying a collision scenario involving $n+1$ driving laser
fields each of the indices runs from $0$ to $n$. In this case the above
considerations result in $n(n+1)$ independent non-vanishing contributions to
$\sum_{\ell}S_{(p);\ell}$ due to terms for which all three indices disagree,
and another $n(n+1)$ ones from terms with two identical indices. Terms with
three indices identical vanish completely. Of course, the very same
considerations apply to $\sum_{\ell^{\prime}}S^{*}_{(p);\ell}$.

In the next step, we focus on an individual contribution with fixed
$\ell=\\{i,j,l\\}$, in order to identify the signal photon frequency
associated with this channel. To this end it is helpful to consider the formal
limit of $\tau\rightarrow\infty$ corresponding to the collision of
monochromatic laser beams with constant temporal envelope; cf. also
Gies:2017ygp . As noted in Sec. III.3, in this limit the temporal integration
in $\mathcal{I}_{ijl}$ results in a delta function ensuring the signal photon
energy $\rm k$ to be fully determined by the oscillation frequencies
$\omega_{i}=\nu_{i}\omega_{0}$ of the driving laser beams,

${\rm k}\to|\pm\omega_{i}\pm\omega_{j}\pm\omega_{l}|\,,$ (27)

where each sign can occur separately. A positive (negative) sign corresponds
to the emission (absorption) of a photon from the respective driving laser
beam. For instance, the contribution with $\ell=\\{0,3,0\\}$ triggers signal
photon frequencies of $2\omega_{0}$, $4\omega_{0}$ and $6\omega_{0}$. We
emphasize that, in order to obtain a sizable signal photon contribution of Eq.
(26) at a given frequency, both factors ${\cal S}_{(p);\ell}$ and ${\cal
S}^{*}_{(p);\ell^{\prime}}$ in Eq. (26) need to support this frequency. For
finite pulse durations $\tau$, the selection rules in Eq. (27) hold only
approximately. Nevertheless, this approximation is accurate as long as the
propagation directions of the driving beams are sufficiently different and
$\tau\omega_{0}\gg 1$, c.f. Sec. III.3. This is rather generic for scenarios
envisioning the collision of several quasi-monochromatic high-intensity laser
pulses. In this case the finite pulse envelopes blur the selection rules (27),
and the sharp delta peaks resulting from the temporal integration in
$\mathcal{I}_{ijl}$ for $\tau\to\infty$ are replaced by Gaussian peaks of a
finite spectral width scaling as $\sim 1/\tau$.

In the scenario considered here we have $\omega_{0}\tau\simeq 59\gg 1$, or
equivalently $1/\tau\simeq\omega_{0}/59\ll\omega_{0}$, which clearly hints at
the fact that the spectral width of these Gaussian peaks is much smaller than
the spectral separation of any two oscillation frequencies $\omega_{i}$ of the
driving laser fields. This is in line with the findings of Sec. III.3: here
Gaussian fits to the various peaks encountered in the differential number of
signal photons ${\rm d}N/{\rm dk}$ resulted in Gaussian standard deviations of
the same magnitude. For the largest extracted frequency width
($\sigma=0.0926\,{\rm eV}$), the associated full peak-width measured at $1\%$
of the peak-maximum is given by $\Delta\omega=4\sqrt{\log 100}\,\sigma\,\simeq
0.513\,\omega_{0}$.

Hence, instead of analyzing the signal photon spectrum by explicitly
segmenting the signal photon density in the spectral domain as done in Sec.
III.3, we can resolve its frequency spectrum by limiting the sum in Eq. (25)
to the relevant channels which induce non-vanishing contributions in the
vicinity of a given frequency.

If in addition the projections of the oscillation-period-averaged intensity
profiles on the beam axes of all other beams vary on scales much larger than
the wavelengths of the beams, also the wave vector of the signal photon $\bf
k$ should – to a good approximation – be determined by the wave vectors of the
photons comprising the beams in the plane-wave limit
$\\{\tau,w_{i}\\}\to\infty$ where focusing effects can be neglected. In this
case, we have

${\bf k}\simeq{\bf k}_{\rm
pw}=\pm\omega_{i}\hat{\boldsymbol{\kappa}}_{i}\pm\omega_{j}\hat{\boldsymbol{\kappa}}_{j}\pm\omega_{k}\hat{\boldsymbol{\kappa}}_{k}\,,$
(28)

where ${\bf k}_{\rm pw}$ denotes the corresponding wave vector in the plane-
wave limit. Under these conditions, only contributions with $i\neq j\neq l$,
i.e., manifestly inelastic signal photon contributions arising from the mixing
of three different driving waves, may result in clearly discernible signals.

To demonstrate this we first focus on the complementary case where two indices
agree. If, e.g., $i=l$ and $i\neq j$, Eq. (27) predicts signals either at
${\rm k}\simeq\omega_{j}$ or at ${\rm k}\simeq\omega_{j}\pm 2\omega_{i}$.
However, at the same time, Eq. (28) implies ${\bf
k}\simeq\pm\omega_{j}\hat{\boldsymbol{\kappa}_{j}}$ or ${\bf
k}\simeq\pm\omega_{j}\hat{\boldsymbol{\kappa}}_{j}\pm
2\omega_{i}\hat{\boldsymbol{\kappa}}_{i}$. From these findings it is obvious
that only the conditions ${\rm k}=|{\bf k}|=\omega_{j}$ are compatible with
each other for generic values of $\omega_{i}$ and $\omega_{j}$ as well as non-
collinear $\hat{\boldsymbol{\kappa}}_{i}$ and $\hat{\boldsymbol{\kappa}}_{j}$
as considered here. On the other hand, signal photons fulfilling ${\rm
k}=|{\bf k}|=\omega_{j}$ are expected to be predominantly emitted in the
forward direction $\hat{\boldsymbol{\kappa}}_{j}$ of the driving laser beam of
frequency $\omega_{j}$, rendering their experimental detection very
challenging. For completeness, note the principle possibility of a quantum
reflection signal in the opposite direction, which is completely negligible
for the present scenario where focusing effects are found to be subleading
Gies:2013yxa .

As an illustrative example we determine the signal photon number associated
with the channel $\ell=\ell^{\prime}=\\{0,3,0\\}$, yielding
$N_{030,030}(4\pi|5.69\,{\rm eV},6.71\,{\rm eV})\simeq 463.46$ photons per
shot in the frequency regime around $4\omega_{0}$. This signal is peaked at
$\hat{\boldsymbol{\kappa}}_{3}$. On the other hand, we find
$N_{030,030}(4\pi|0,\infty)\sim 463.46$, such that – as expected, and in line
with the arguments given above – obviously no signal photons at other
frequencies contribute to this channel.

In the remainder of this section our focus is on the manifestly inelastic
signal photon contributions associated with the 12 independent combinations to
the signal photon amplitude characterized by $i\neq j\neq l$. in particular,
we aim at verifying the signals arising in the angular regions
$\mathcal{A}^{(2\omega_{0})}$, $\mathcal{A}^{(4\omega_{0})}$ and
$\mathcal{A}^{(5\omega_{0},A/B)}$ introduced in Sec. III.4. This allows us to
explicitly restrict our analysis to channels giving rise to signal photons of
the desired energy and wave vectors pointing in the respective directions.

Tracking the $2\omega_{0}$ signal in $\mathcal{A}^{(2\omega_{0})}$, we analyze
all permutations of the indices $0$, $1$ and $2$. Microscopically, we expect
this signal to arise from a process involving the merging of two laser photons
from beams $0$ and $2$, respectively, and the absorption of a laser photon of
frequency $\omega_{1}$ from beam $1$. Resorting to the plane-wave
approximation, this results in a signal photon wave vector of modulus
$|\mathbf{k}_{\rm pw}|\approx 2.201\omega_{0}$ pointing at
$(\varphi,\vartheta)=(319.107^{\circ},101.07^{\circ})$. Obviously this value
is compatible with the condition $|{\rm k}-|{\bf k}_{\rm pw}||<\Delta\omega$
and ${\rm k}\simeq 2\omega_{0}$. It thus allows for a nonvanishing signal in
this parameter regime. A comparison with the dominant emission direction
determined numerically in Sec. III.4 unveils an excellent agreement; the
relative differences in the longitude and latitude are below $1\%$. The signal
associated with this channel is found to be peaked around ${\rm k}\simeq
2.015\omega_{0}$.

Restricting the sums over $\ell$ and $\ell^{\prime}$ in Eq. (25) to all
possible permutations of the indices $0$, $1$ and $2$ and integrating over the
full solid angle we obtain $N(4\pi|0,\infty)\simeq 78$ signal photons per
shot, while an explicit restriction to the angular region
$\mathcal{A}^{(2\omega_{0})}$ results in
$N(\mathcal{A}^{(2\omega_{0})}|0,\infty)\simeq 62$. To study the importance of
the individual contributions to this sum, in Tab. 3 (a) we explicitly list the
contributions of all nine termss constituting the signal photon number in this
parameter regime. This also allows to assess the relative importance of off-
diagonal terms with $\ell\neq\ell^{\prime}$. Exactly the same number is
obtained when the frequency regime is in addition restricted to $2.59\,{\rm
eV}\leq{\rm k}\leq 3.61{\rm eV}$. A comparison with the analogous number
extracted in Sec. III.4 establishes that all signal photons scattered into
this parameter regime are indeed emerging from the microscopic process
$\omega_{0}-\omega_{1}+\omega_{2}\to{\rm k}$.

Table 3: Comparison of various contributions $N_{\ell,\ell^{\prime}}({\cal
A}^{(n\omega_{0})}|0,\infty)$ to the signal photon number. Here, we highlight
several photon emission channels resulting in manifestly inelastically
scattered signal photons of frequency ${\rm k}\simeq n\omega_{0}$ with
$n\in\\{2,4,5\\}$.

(a): ${\rm k}\simeq 2\omega_{0}$ (b): ${\rm k}\simeq 4\omega_{0}$

$\ell$ $\ell^{\prime}$ |  $0\;1\;2$ |  $1\;2\;0$ |  $0\;2\;1$
---|---|---|---
$0\;1\;2$ |  1.557 |  0.327 |  6.516
$1\;2\;0$ |  0.327 |  2.143 |  5.122
$0\;2\;1$ |  6.516 |  5.122 |  34.394

$\ell$ $\ell^{\prime}$ |  $0\;1\;3$ |  $1\;3\;0$ |  $0\;3\;1$
---|---|---|---
$0\;1\;3$ |  0.437 |  0.754 |  4.917
$1\;3\;0$ |  0.754 |  12.814 |  18.851
$0\;3\;1$ |  4.917 |  18.851 |  67.105

(c): ${\rm k}\simeq 5\omega_{0}$ (d): ${\rm k}\simeq 5\omega_{0}$

$\ell$ $\ell^{\prime}$ |  $1\;2\;3$ |  $2\;3\;1$ |  $3\;1\;2$
---|---|---|---
$1\;2\;3$ |  0.052 |  -0.018 |  -0.206
$2\;3\;1$ |  -0.018 |  2.218 |  -0.242
$3\;1\;2$ |  -0.206 |  -0.242 |  0.973

$\ell$ $\ell^{\prime}$ |  $0\;2\;3$ |  $2\;3\;0$ |  $3\;0\;2$
---|---|---|---
$0\;2\;3$ |  0.228 |  -0.957 |  0.449
$2\;3\;0$ |  -0.957 |  5.464 |  -2.625
$3\;0\;2$ |  0.449 |  -2.625 |  3.034

The other signals can be analyzed along the same lines. For identifying the
microscopic process giving rise to the inelastic signal of frequency
$4\omega_{0}$, the channels associated with permutations of the indices $0$,
$1$ and $3$ need to be analyzed. The process responsible for this signal is
$\omega_{0}-\omega_{1}+\omega_{3}\to{\rm k}$. The corresponding signal photon
wave vector fulfills $|\mathbf{k}_{\rm pw}|\approx 3.813\omega_{0}$ and is
pointing at $(\varphi,\vartheta)\simeq(49.11^{\circ},78.93^{\circ})$. This
channel gives rise to $N(\mathcal{A}^{(4\omega_{0})}|0,\infty)\simeq 129$
signal photons per shot. For the individual contributions constituting this
number, see Tab. 3 (b). Finally, we turn to the two distinct signals with
frequencies around $5\omega_{0}$. These are triggered by the microscopic
processes $-\omega_{1}+\omega_{2}+\omega_{3}\to{\rm k}$ and
$-\omega_{0}+\omega_{2}+\omega_{3}\to{\rm k}$, respectively. We summarize the
detailed properties of these signals in Tab. 4; this table also includes the
parameters characterizing the $2\omega_{0}$ and $4\omega_{0}$ signals just
discussed. See Tabs. 3 (c) and 3 (d) for the individual contributions
constituting the signal photon numbers in these channels.

Table 4: Overview of the properties of the manifestly inelastic signal photon channels detailed in Sec. IV.1. For each signal photon frequency $\rm k\simeq n\omega_{0}$ with $n\in\\{2,4,5\\}$ we provide the longitude $\varphi$ and latitude $\vartheta$ characterizing the main emission direction as well as the number of signal photons per shot emitted into the solid angle $\mathcal{A}^{(n\omega_{0})}$. $\rm k$ | $2\omega_{0}$ | $4\omega_{0}$ |  $5\omega_{0}$, A |  $5\omega_{0}$, B
---|---|---|---|---
origin | $\ \omega_{0}-\omega_{1}+\omega_{2}\ $ | $\ \omega_{0}-\omega_{1}+\omega_{3}\ $ | $\ -\omega_{1}+\omega_{2}+\omega_{3}\ $ | $\ -\omega_{0}+\omega_{2}+\omega_{3}\ $
$\varphi$ | $319.11^{\circ}$ | $49.11^{\circ}$ | $23.41^{\circ}$ | $30.90^{\circ}$
$\vartheta$ | $101.07^{\circ}$ | $78.93^{\circ}$ | $50.95^{\circ}$ | $32.36^{\circ}$
$N(\mathcal{A}^{(n\omega_{0}}|0,\infty)$ | $62.02$ | $129.40$ | $2.31$ | $0.46$

###  IV.2 Implications of the channel analysis

In the preceding section, we have worked out a strategy allowing us to trace
all-optical signatures of quantum vacuum nonlinearity back to the underlying
four-wave mixing processes and thus infer information about their microscopic
origin. To enable a clear measurement of a photonic signature of quantum
vacuum nonlinearity it is desirable to maximize the signal at a given
frequency and emission direction such as to achieve the best possible signal-
to-background separation. A complete assessment of the question which signal
channel amounts to the most prospective one for an experimental verification,
of course, requires to account for many more details of a concrete
experimental set up, including, e.g., the sensitivity and efficiency of the
few photon detectors.

As the simultaneous measurement in several well-separated directions and at
several frequencies is, however, highly unlikely with state-of-the-art
technology, the typical challenge is to maximize the signal at a certain
frequency and emission direction. In this section, we sketch how the insights
obtained in Sec. IV.1 can be used to enhance a given signal photon channel.
Selecting a particularly promising signal, the channel analysis allows to
trace the microscopic origin of this signal and allows to modify the driving
laser fields such as to enhance the signal in this channel, e.g., by
redistributing the total available laser pulse energy into the individual
beams.

This is especially obvious for the manifestly inelastic signals analyzed in
detail in Sec. IV.1: while originating from the effective interaction of
different subsets of beams, each of these four signals (cf. Tab. 4) arises
from the mixing of precisely three different driving laser fields. Hence, in
order to increase the signal photon yield in any of these channels
individually, the driving laser beam which acts as a pure spectator can be
switch off and its energy instead be redistributed into the other beams
participating in the interaction.

Here, we illustrate this point using the example of a manifestly inelastic
${\rm k}\simeq 4\omega_{0}$ signal originating in the microscopic process
$\omega_{0}-\omega_{1}+\omega_{3}\to{\rm k}$, where $\omega_{1}=\omega_{0}$
and $\omega_{3}=4\omega_{0}$, respectively. Obviously, only beams $0$, $1$ and
$3$ are involved in this particular process. Let us now remove the spectator
beam $2$ and redistribute its energy into the other beams. Our choice for the
new beam energies is $\tilde{W}_{0}=\tilde{W}_{1}=\tilde{W}_{3}\approx
41.67\,{\rm J}$, maximizing the Fourier integral
$\tilde{\mathcal{I}}_{013}\propto\sqrt{\tilde{W}_{0}\tilde{W}_{1}\tilde{W}_{3}}$;
the other Fourier integrals do not support an inelastic channel or vanish for
$\tilde{W}_{2}=0$. The partition factors associated with this choice are
$\tilde{q}_{0}=5/6$, $\tilde{q}_{1}=4/5$ and $\tilde{q}_{2}=1$, resulting in
$\tilde{W}^{\rm eff}=\tilde{W}^{\rm loss}=1/2\,W$; cf. Sec. III.2. The new
result for the number of signal photons in the manifestly inelastic
$4\omega_{0}$ channel emitted into the angular area ${\cal A}^{(4\omega_{0})}$
can straightforwardly be obtained from the corresponding signal photon number
$N(\mathcal{A}^{(4\omega_{0})})\simeq 129$ determined in Sec. IV.1. It follows
upon rescaling this number with an overall factor of
$\tilde{W}_{0}\tilde{W}_{1}\tilde{W}_{3}/(W_{0}W_{1}W_{3})=36/25$, resulting
in
$\tilde{N}(\mathcal{A}^{(4\omega_{0})})=36/25N(\mathcal{A}^{(4\omega_{0})})\simeq
186$ signal photons per shot in this specific channel.

##  V Conclusions and Outlook

We have studied all-optical signatures of QED vacuum nonlinearity in the
collision of several high-intensity laser beams differing in frequency,
polarization and propagation direction. More specifically, we have focused on
an example scenario envisioning the collision of four laser pulses, all
originating from a single driving laser pulse, utilizing beam-splitting and
sum-difference frequency generation techniques. Such a scenario requires the
availability of a high-intensity laser system of the multi-petawatt class, and
thus will become possible at various state-of-the-art and upcoming high-
intensity laser facilities such as ELI-NP. While we base our considerations on
the availability of a single high-intensity laser of the ten petawatt class
with specific parameters, our results can straightforwardly be rescaled to
other laser parameters.

One of the goals of our study is to identify prospective signal channels
allowing for an efficient signal-to-background separation. To this end, we pay
special attention to the question of how to efficiently infer information
about the microscopic origin of prospective signatures of vacuum nonlinearity
by means of a channel analysis. This allows us to answer relevant questions,
such as which laser beams participate in the formation of a given signal, and
what is the specific interaction process inducing the latter. In addition, we
have explicitly demonstrated how this information can be used to enhance the
signal photon number in a given signal photon channel.

For completeness, note however the difficulty of an absolutely background-free
measurement in a real experimental set up. Any practical imperfection such as
a non-ideal vacuum in the vacuum chamber coming along with residual atoms and
molecules in the interaction region may give rise to higher-harmonic
backgrounds. Still, these backgrounds can, in principle, be monitored (e.g. by
rest-gas measurements, or geometric adjustments) and thus parametrically
controlled to a large degree. The full quantitative incorporation of such
effects is outside the scope of the present idealized analysis.

We are confident that the concepts outlined and applied in the present study
will prove very useful in future attempts at optimizing photonic signatures of
quantum vacuum nonlinearity for given experimental parameters and constraints.
Our formalism can also provide for an efficient basis to study recent
alternative suggestions Ahmadiniaz:2020kpl ; Sangal:2021qeg  for corresponding
discovery experiments beyond the optical regime.

###### Acknowledgements.

This work has been funded by the Deutsche Forschungsgemeinschaft (DFG) under
Grant Nos. 416607684 and 416611371 within the Research Unit FOR2783/1.

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