# Single-particle steering and nonlocality: The consecutive Stern-Gerlach
Experiments

E Benítez Rodríguez  E Piceno Martínez  L M Arévalo Aguilar
<EMAIL_ADDRESS>Facultad de Ciencias Físico Matemáticas, Benemérita
Universidad Autónoma de Puebla.

###### Abstract

Quantum nonlocality and quantum steering are fundamental correlations of
quantum systems which can not be created using classical resources only.
Nonlocality describes the ability to influence the possible results of
measurements carried out in distant systems, in quantum steering Alice
remotely steers Bob’s state. Research in nonlocality and steering possess a
fundamental interest for the development of quantum information and in many
applications requiring nonlocal resources like quantum key distribution. On
the other hand, the Stern-Gerlach experiment holds an important place in the
history, development and teaching of quantum mechanics and quantum
information. In particular, the thought experiment of consecutive Stern-
Gerlach Experiments is commonly used to exemplify the concept of non-
commutativity between quantum operators. However, to the best of our
knowledge, the consecutive Stern-Gerlach Experiments have not been treated in
a fully quantum manner yet, and it is a widely accepted idea that atoms
crossing consecutive Stern-Gerlach Experiments follow classical paths. Here we
demonstrate that two consecutive Stern-Gerlach Experiment generate nonlocality
and steering, these nonlocal effects strongly modify our usual understanding
of this experiment. Also, we discuss the implications of this result and its
relation with the entanglement. This suggests the use of quantum correlations,
of particles possessing mass, to generate nonlocal taks using this venerable
experiment.

Steering, Nonlocality, Stern-Gerlach Experiment, Consecutive Stern-Gerlach
Experiments

††preprint: APS/123-QED

##  I Introduction

Nonlocality, one of the fundamental features of quantum mechanics, refers to
the fact that for entangled states the result of a measurement on one
observable depends on the choice of measurement on the other observable –this
result manifests spacelike separation– [1, 2, 3, 4]; steering refers to the
ability to steer a state of one subsystem by measurements made in another
subsystem [5, 6, 7, 8, 9], it is a nonlocal property that differs from
nonlocality and nonseparability [5, 6]. Additionally, it was demonstrated that
nonlocality rests in the uncertainty principle and in the steering of physical
states[10, 11]. Besides, the nonlocality of the collapse of the wavefunction
of a single particle was experimentally proved for photons in reference [12],
for atoms see reference [13] for an experimental proof of nonclassical
collapses by using boxes games [14]. It is worthy of mention that steering was
experimentally proven in single photon experiment by Guerreiro et al. [15].
Additionally, it was shown that there is a strong relation between steering
and joint measurability [16, 17]. Nowadays it is understood that quantum
nonlocality is a fundamental resource for quantum information tasks which can
not be generated by using random data only [18]. Nevertheless, see the
interesting discussion over quantum nonlocality given by Khrennikov [19, 20,
21].

Moreover, although in some works it has been conjectured that single-particle
entanglement does not possess nonlocal correlations [22, 23], the fascinating
fact is that these nonlocal effects could be generated using the Stern-Gerlach
Experiment (SGE) by measuring the internal degree of freedom of the particle
that traverses it; i.e. by measuring $\hat{\sigma}_{z}$ or $\hat{\sigma}_{x}$,
the position of the particle could manifest itself at either of two different
places at (possible) spacelike distance [24]. These nonlocal properties of the
SGE were associated with the spreading of the wavefunction and named single-
particle steering [24]. Hence, depending on the kind of states, the spreading
of the wavefunction plays also a crucial role in the nonlocality of quantum
mechanics.

Furthermore, single-particle entanglement [23] has many applications in
quantum information like quantum key distribution [25, 26], bidirectional
quantum teleportation [27], swapping states [28] and entanglement
concentration [29]. Recently the single-particle entanglement between the spin
and orbital angular momentum (OAM) of photons was reported in metamaterials
[30] and the entanglement between polarization and OAM was studied in
reference [31]. Besides, as was mentioned earlier, steering was experimentally
proven using the single-photon entanglement [15], this effect was called
single-photon steering by Brunner 111Private communication .

On the other hand, the SGE stands as a pillar for the historical development
of quantum mechanics and nowadays constitutes an active field of research,
regarding both its experimental capabilities and theoretical studies [33, 34,
35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
54], and it also constitutes an important tool in the teaching of both quantum
mechanics and quantum information.

The quantum phenomena that is usually introduced in quantum mechanics courses
using the Stern-Gerlach experiment include the concept of spin and the non-
commutativity of quantum operators [55, 56, 57, 58]. For the latter is of most
importance the thought experiment of consecutive Stern-Gerlach Experiments
(CSGE), formally introduced by Feynman in his famous lectures [58],
regardeless of its first conception by Heisenberg in 1927 [59]. In quantum
information and computation theory the CSGE is used to exemplify the structure
of the qubit and the collapse of the wavefunction [60, 61].

Despite its importance, there has been a lack of quantum analyses of the CSGE,
at least that we are aware of. This is surprising given the amount of research
dedicated to describing and studying the usual SGE; in concrete, in recent
years there have been important advances describing the SGE in a completely
quantum manner [33, 34, 39, 40, 36, 37, 38, 47, 48, 49, 51]. In particular, it
was shown that the SGE behaves as an entangled device and not as a measurement
device as it was regarded for decades [33, 34]. Additionally, the violation of
Bell inequalities in the SGE was shown in reference [62], this violation of
the Bell inequalities was associated with quantum nonlocality, instead of
quantum contextuality, the difference is explained in reference [24].

In this article, we carry out the task of analyzing the CSGE using the tools
of quantum mechanics, and to that end we calculate the complete quantum
evolution of a spin-$1/2$ particle in such consecutive configuration by means
of the evolution operator method [63, 64]. This results also could serve to
define a basis of comparison for a possible experimental realization of this
experiment. The quantum treatment of the CSGE gives us the opportunity to
study the quantum characteristics of the system, such as the quantum
correlations and entanglement [62].

The paper is organized as follows. In Sec. II, starting from the time
dependent Schrödinger equation for the particle, we found its time evolution
applying a factorized evolution operator to the initial state of the particle,
similarly as done in [33, 34]. We calculate in this way the evolution in each
of the Stern-Gerlach apparatuses that are arranged consecutively, and arrive
at the result for the total evolution applying the sequential evolutions. In
Sec. III we analyze the steering produced by changing the measurement basis.
In Sec. IV we describe the quantum correlations of the CSGE and we test three
different Bell-type inequalities with which we found that this system is non-
local. In Sec. V we visualize the relation between the quantum correlations
and the entanglement in the CSGE by means of the works of Piceno et. al. [35]
and Roston et. al. [39]. Finally the main results obtained are recapitulated
in Sec.VI, that closes the paper with some concluding remarks.

##  II Dynamic evolution of the CSGE

The array of consecutive experiments that we take into account appears
schematized in FIG.1, this configuration has two Stern-Gerlach apparatuses –
each apparatus acts as an entangling device. The initial state entering to the
SGE in $x$ direction is a state previously prepared by other SGE; that is, the
initial state is given by the product of a spin up state (internal degree of
freedom) with a wavepacket (external degree of freedom), which we can write as
follows:

$\ket{\psi_{i}}=\frac{1}{(2\pi\sigma_{0}^{2})^{\frac{3}{4}}}\exp\left(-\frac{(x^{2}+y^{2}+z^{2})}{4\sigma_{0}^{2}}+ik_{y}y\right)\ket{\uparrow_{z}},$
(1)

with $\sigma_{0}$ the initial width of the wavepacket and $k_{y}$ the
component $y$ of the wave vector.

Figure 1: Scheme of the consecutive Stern-Gerlach experiments. The
inhomogeneity of the magnetic field in the first experiment is in direction
$x$, which is associated with a Hamiltonian, $\hat{H_{2}}$, and an evolution
operator, $\hat{U_{2}}$; the inhomogeneity of the magnetic field of the second
is in direction $z$, with its respective Hamiltonian, $\hat{H_{3}}$, and
evolution operator $\hat{U_{3}}$.

The quantum evolution of the Stern-Gerlach experiment has already been
described in [33], therefore, from this we know that if the initial state is a
product between a gaussian wavepacket and a spin DoF the resulting evolved
wavepacket widens with time, and gets translated in $z$ depending on the spin
component thanks to the interaction with the inhomogeneous magnetic field when
we select the spin component. With these considerations, the general packet of
Eq. (1) can be seen as that resulting from such preparation with a final width
$\sigma_{0}$ and taking the $z$ position of such state as our origin [33, 34].

To obtain the evolution as the state goes through the first experiment we have
the associated evolution operator from [33, 34],

$\begin{split}\hat{U}_{2}(t)&=\exp\left(-\frac{1}{6}\kappa\right)\exp\left[-\frac{it}{2m\hbar}\left(p_{y}^{2}+p_{z}^{2}\right)\right]\\\
&\times\exp\left[-\frac{it\mu_{c}}{\hbar}\left(B_{2}+b_{2}x\right)\sigma_{x}\right]\exp\left(\frac{it^{2}\mu_{c}b_{2}}{2m\hbar}p_{x}\sigma_{x}\right)\\\
&\times\exp\left(-\frac{it}{2m\hbar}p_{x}^{2}\right),\end{split}$ (2)

with $\kappa=(it^{2}\mu_{c}^{2}b_{2}^{2})/(m\hbar)$, $B_{2}$ the homogeneity
parameter of the magnetic field, $b_{2}$ the inhomogeneity parameter,
$\sigma_{x}$ the Pauli operator, $\mu_{c}=g{e}\hbar/(4m)$, $g$ the
gyromagnetic ratio, $m$ the mass of the particle and $e$ the unit charge.

Applying $\hat{U}_{2}(t)$ during certain time $t_{2}$ to $\ket{\psi_{i}}$ we
have obtained the middle state $\ket{\psi_{m}(t_{2})}$. See the Appendix A for
this calculation. Next, following our description, we apply the evolution
operator associated with the last SGE for a fixed time $t_{3}$, that is, the
operator in $z$ direction to the state $\ket{\psi_{m}(t_{2})}$. This operator
is given by

$\begin{split}\hat{U}_{3}(t)&=\exp\left(-\frac{1}{6}\kappa\right)\exp\left[-\frac{it}{2m\hbar}\left(p_{x}^{2}+p_{y}^{2}\right)\right]\\\
&\times\exp\left[-\frac{it\mu_{c}}{\hbar}\left(B_{3}+b_{3}z\right)\sigma_{z}\right]\exp\left(\frac{it^{2}\mu_{c}b}{2m\hbar}p_{z}\sigma_{z}\right)\\\
&\times\exp\left(-\frac{it}{2m\hbar}p_{z}^{2}\right)\end{split}$ (3)

Then we obtain the final state as

$\ket{\psi_{f}}=\ket{\psi_{+}}\ket{\uparrow_{z}}+\ket{\psi_{-}}\ket{\downarrow_{z}},$
(4)

in which we define

$\begin{split}\ket{\psi_{\pm}}&=M\exp\left(\mp
i\sqrt{2}k_{3}^{3}\tau_{3}z_{0}\right)\exp\left\\{-\frac{\left(Z\pm\left\\{\sqrt{2}k_{3}^{3}\tau_{3}^{2}+i2\sqrt{2}k_{3}^{3}\tau_{3}\left[1+i\left(\tau_{2}+\tau_{3}\right)\right]\right\\}\right)^{2}}{4\left[1+i\left(\tau_{2}+\tau_{3}\right)\right]}\right\\}\\\
&\times\left(\exp\left(-i\sqrt{2}k_{2}^{3}\tau_{2}x_{0}\right)\exp\left\\{-\frac{\left[X+\sqrt{2}k_{2}^{3}\tau_{2}^{2}+i2\sqrt{2}k_{2}^{3}\tau_{2}\left(1+i\tau_{2}\right)\right]^{2}}{4\left[1+i\left(\tau_{2}+\tau_{3}\right)\right]}\right\\}\right.\\\
&\quad\pm\left.\exp\left(i\sqrt{2}k_{2}^{3}\tau_{2}x_{0}\right)\exp\left\\{-\frac{\left[X-\sqrt{2}k_{2}^{3}\tau_{2}^{2}-i2\sqrt{2}k_{2}^{3}\tau_{2}\left(1+i\tau_{2}\right)\right]^{2}}{4\left[1+i\left(\tau_{2}+\tau_{3}\right)\right]}\right\\}\right),\end{split}$
(5)

with $M$ a normalization factor. Eq.(5) is dimensionless thanks to the
following definitions

$\displaystyle\tau_{2,3}$ $\displaystyle=\frac{\hbar
t_{2,3}}{2m\sigma_{0}^{2}},$ $\displaystyle k_{2,3}$
$\displaystyle=\sqrt{2}\sigma_{0}\left(\frac{m\mu_{c}b_{2,3}}{2\hbar^{2}}\right)^{1/3},$
$\displaystyle x_{0}$ $\displaystyle=\frac{B_{2}}{\sigma_{0}b_{2}},$
$\displaystyle z_{0}$ $\displaystyle=\frac{B_{3}}{\sigma_{0}b_{3}},$ (6)
$\displaystyle Z$ $\displaystyle=\frac{z}{\sigma_{0}},$ $\displaystyle X$
$\displaystyle=\frac{x}{\sigma_{0}}.$

The definitions in Eq. (II) make all the variables dimesionless and our
results completely comparable with those of other works that study the
entanglement in the SGE [39, 35]. We found that the final state is a
superposition state of the spin eigenstates, in the same way as the state
coming out from a single SGE [33, 34], and therefore does not represent a
state following a definite trajectory to the screen. This state presents
hybrid entanglement between the position DoFs and the spin DoF, congruently
with the known effect present in the SGE, the separation of spins. The
presence of the $Z$ variable in this entangled state attests the non-
commutativity between the different spin operators for the different spatial
orientations, this is in good accordance with the observations of the
semiclassical argument for the thought experiment. We also found an
interesting effect for the configuration proposed here in the presence of
entanglement also with the position DoF in the $X$ coordinate.

##  III Steering

As was mentioned earlier, steering is a manifestation of quantum correlations.
It was defined by Schrödinger (as a property of entangled systems) as the
possibility that by suitable measurements taken on one subsystem only, the
state of the other subsystem can be determined by the choice of the
measurement and without interacting with it. That is to say, it is possible to
steer the sate of a subsystem by choosing what kind of measurement to
implement on the other subsystem.

The actual understanding of steering comes from the operational definition
given by Wiseman et.al. [5], they define steering as a task, i.e. the task of
Alice is to convince Bob that she can prepare a bipartite entangled state. To
do it, Alice prepares a bipartite quantum state and sends one of them to Bob;
then, they measure their respective subsystem. If the correlations between
their measurement can be explained by a local hidden state model for Bob, then
Alice could have taken a pure state at random and sent it to Bob. But if the
measurements can not be explained by a local hidden state model, then Alice
steers Bob’s state. Wiseman et.al. [5] demonstrated that steering is stronger
than nonseparability and weaker than nonlocality. The existence of nonlocality
rules out local hidden variables and, in essence, steering rules out the
existence of local hidden state models [5]. Steering was experimentally proven
in single photon experiment by Guerreiro et al. [15]. In the case of pure
tripartite states, He and Reid have demonstrated that it is enough that each
party can be steered by one or both of the other two to certify steering [65].

In reference [24], it was demonstrated that in the case of a single SGE Alice
can steer Bob’s state depending on the measurement she chooses. In that paper
[24], a thought experiment where Alice is located in Tokyo and Bob in Paris
was posed. Hence, by choosing which one of the possible observables to
measure, Alice can steer Bob’s state. This confirms the nonlocality of the
entangled wavefunction of the SGE and reference [24] associates this
nonlocality with the spreading of the entangled wavefunction.

Figure 2: CSGE featuring the Einstein’s boxes, see reference [24]. The red box
is the oven, in violet a SGE in the $x$ direction is depicted, in blue the SGE
in $z$ direction is depicted, the red and green dots represent the fact that
there are not classical trajectories, see references [33, 34]. Alice could
communicate with Bob by using the classical channel in magenta. Moreover,
Alice is in full control of the SGE by using the classical channel in yellow,
and she possess the ability to turn it on and to choose between a single or
$N$ atoms. The red dots end in the $z$ axis in Tokyo with Alice and in Paris
with Bob. The green dots in the $x$ axis arrives at suitable places where
there are two friends of Alice.

In the case of the CSGE a similar situation –in fact more rich– can be
conceived. This is depicted in FIG (2), Alice in Tokyo is in full control of a
fully automatized CSGE located at a suitable place and can turn it on and off
and sent a single or many atoms one by one as she wishes.

Therefore, if Alice mades a check of $\ket{\downarrow_{z}}$ and she founds
$-\hbar/2$, then the entangled wavefunction given by Eq. (4), i.e.
$\ket{\psi_{f}}=\ket{\psi_{+}}\ket{\uparrow_{z}}+\ket{\psi_{-}}\ket{\downarrow_{z}},$
would collapse to $\ket{\psi_{-}}\ket{\downarrow_{z}}$ in Tokyo, notice that
$\ket{\psi_{-}}$ comprises a superposition state in $X$. On the other hand, if
she detected nothing, then the wave function would collapse to
$\ket{\psi_{+}}\ket{\uparrow_{z}}$ in Paris, see reference [24] for an
explanation of this effect in terms of the Einstein’s boxes, similarly,
$\ket{\psi_{+}}$ comprises a superposition state in $X$.

However, if Alice decides to measure in a different basis, for example
$\hat{\sigma_{x}}$, to ascertain the possibilities we must rewritte Eq. (4) in
the $\hat{\sigma_{x}}$ basis, getting

$\ket{\psi_{f}}=\left\\{\ket{\psi_{+}}+\ket{\psi_{-}}\right\\}\ket{\uparrow_{x}}+\left\\{\ket{\psi_{+}}-\ket{\psi_{-}}\right\\}\ket{\downarrow_{x}},$
(7)

then, the following situations can arise:

  1. i)

If she measures spin down in the basis of $\hat{\sigma}_{x}$ and obtains
$-\hbar/2$, then the wavefunction would collapse towards
$\left\\{\ket{\psi_{+}}-\ket{\psi_{-}}\right\\}\ket{\downarrow_{x}}$. Notice
that $\left\\{\ket{\psi_{+}}-\ket{\psi_{-}}\right\\}$ comprises a
superposition state in $Z$.

  2. ii)

If she checks spin up in the basis of $\hat{\sigma}_{x}$ and obtains
$\hbar/2$, then the wavefunction would collapse towards
$\left\\{\ket{\psi_{+}}+\ket{\psi_{-}}\right\\}\ket{\uparrow_{x}}$.

Therefore, we must conclude that Bob’s state is steered depending on the kind
of observable Alice decides to measure, confirming the nonlocality of the
wavefunction generated by the CSGE.

On the other hand, a similar procedure could be used by Bob in Paris to steer
Alice’s state. Or by one of the friends of Alice to steer states to Alice or
Bob. Therefore the He and Reid criterion cited above is fulfilled, i.e. the
CSGE possesses the nonlocal property of steering. As it can be seen in FIG.
(2) a richer situation that the one given in reference [24] arises, because in
the CSGE case there could be the participation of four people to address all
the possibilities. For example, if Alice’s Friend 2 in a different location
decides to check for $\hbar/2$ in the $\hat{\sigma_{x}}$ basis and obtains
nothing in their measurement, then the wavefunction would collapse towards the
state $\left\\{\ket{\psi_{+}}-\ket{\psi_{-}}\right\\}\ket{\downarrow_{x}}$.
This is equivalent to Alice measuring $\hat{\sigma_{x}}$ and obtaining
$-\hbar/2$ and is also equivalent to Friend 1 measuring $\hat{\sigma_{x}}$ and
obtaining $-\hbar/2$, both of them will collapse the wavefunction (by effects
of their measurements) towards
$\left\\{\ket{\psi_{+}}-\ket{\psi_{-}}\right\\}\ket{\downarrow_{x}}$, i.e. the
same function as that obtained by Friend 2 when testing $\hbar/2$ in the basis
$\hat{\sigma_{x}}$ and obtaining nothing.

##  IV Non-locality of the Consecutive Stern Gerlach experiments

The study of quantum correlations of quantum systems is strongly related with
non-clasical tasks [2, 3, 4, 39, 33, 34, 66, 62, 47, 35, 67, 68, 69, 70, 71,
18, 72, 73, 74] that open the way to important applications. We quantify the
quantum correlations of the CSGE with a correlation function for hybrid spin
systems that has already been tested for bipartite states [72, 62]

$\bm{C}(x,p_{x},z,p_{z},\theta)=\expectationvalue{\hat{W}(x,p_{x},z,p_{z})\hat{\sigma}(\theta)}{\psi_{f}},$
(8)

that is the generalized Banaszek-Wódkiewicz (BW) correlation function [73],
with

$\begin{split}\hat{W}&(x,p_{x},z,p_{z})=\int_{-\infty}^{\infty}dq_{x}dq_{z}\ket{x-\frac{1}{2}q_{x},z-\frac{1}{2}q_{z}}\\\
&\times\exp[-i(p_{z}q_{z}+p_{x}q_{x})/\hbar]\bra{x+\frac{1}{2}q_{x},z+\frac{1}{2}q_{z}},\end{split}$
(9)

the generalized Wigner operator, originally defined by Ben-Benjamin et. al.
[75], for coordinates $x$, $z$ with their respective moments $p_{x}$, $p_{z}$;
$\hat{\sigma}(\theta)$ is the usual Pauli operator for an arbitrary $\theta$
direction in the plane.

In this way we can calculate the correlation function for the final state of
the CSGE that is given by Eq. (4). The relation between the correlations and
entanglement in the CSGE will be discussed later in this paper.

###  IV.1 Results

Our quantum description of the experiments allows us to delve into the quantum
characteristics of the system, in this manner, we study the quantum
correlations that are present by means of the CHSH, Bell-Klyshko-Mermin and
Svetlichny inequalities.

####  IV.1.1 CHSH inequality

From Eq. (8) the correlation function of the CSGE has the following form:

$\begin{split}&\bm{C}(X,P_{x},Z,P_{z},\theta)=\exp[\omega^{\prime}_{z}(Z,P_{z})+\omega^{\prime}_{x}(X,P_{x})]\\\
&\times\Bigg{[}4\cos(\theta)\bigg{(}\exp[\omega_{z}]\Big{\\{}\exp[\omega_{x}]\cosh[d_{x}(X,P_{x})]\sinh[d_{z}(Z,P_{z})]\\\
&\qquad+\exp[-\omega_{x}]\cos[\delta_{x}(X,P_{x})]\cosh[d_{z}(Z,P_{z})]\Big{\\}}\bigg{)}\\\
&\ \ \
+4\sin(\theta)\bigg{(}\exp[-\omega_{z}]\Big{\\{}\exp[\omega_{x}]\sinh[d_{x}(X,P_{x})]\cos[\delta_{z}(Z,P_{z})]\\\
&\qquad+\exp[-\omega_{x}]\sin[\delta_{x}(X,P_{x})]\sin[\delta_{z}(Z,P_{z})]\Big{\\}}\bigg{)}\Bigg{]}\end{split}$
(10)

up to a normalization factor. The functions $\omega^{\prime}_{z}(Z,P_{z})$,
$\omega^{\prime}_{x}(X,P_{x})$, $\omega_{z}$, $\omega_{x}$, $d_{z}(Z,P_{z})$,
$d_{x}(X,P_{x})$, $\delta_{z}(Z,P_{z})$ and $\delta_{x}(X,P_{x})$ are real
valued functions. See the Appendix B for the complete definitions.

We have the extra dimensionless definitions given by

$\displaystyle P_{x}$ $\displaystyle=\frac{p_{x}\sigma_{0}}{\hbar},$
$\displaystyle P_{z}$ $\displaystyle=\frac{p_{z}\sigma_{0}}{\hbar}.$ (11)

The CHSH inequality [76, 68, 74, 72] that we use to test the existence of non-
locality between the $Z$-$\theta$ pair is as follows

$\begin{split}-2\leq&B_{CHSH}=\bm{C}(X,P_{x},Z,P_{z},\theta)+\bm{C}(X,P_{x},Z,P_{z},\theta^{\prime})\\\
&+\bm{C}(X,P_{x},Z^{\prime},P_{z},\theta)-\bm{C}(X,P_{x},Z^{\prime},P_{z},\theta^{\prime})\leq
2.\end{split}$ (12)

To study the non-locality between $z$ and $\theta$ we need to fix the other
variables, named $X$, $P_{x}$, $P_{z}$, $Z^{\prime}$ and $\theta^{\prime}$.

The plot of the correlation function for this case appears in FIG. 3 for the
values of the fixed parameters $X=0.1$, $P_{x}=0.129$, $P_{z}=0.049$,
$\tau_{2}=6.8$, $\tau_{3}=2.6$, $k_{2}=0.3$, $k_{3}=0.3$, $x_{0}=4$ and
$z_{0}=4$. The Bell function for this case, $B_{CHSH}$, is shown in FIG. 4 for
the primed quantities $Z^{\prime}=2.4$ and $\theta^{\prime}=\frac{\pi}{5}$. We
report the violation of the CHSH inequality for the $Z$,$\theta$ pair for a
minimum value of $B_{CHSH}$ of (negative) -2.62405.

Figure 3: Correlation function for the $Z$-$\theta$ pair taking the fixed
quantitites as $X=0.1$, $P_{x}=0.129$, $P_{z}=0.049$, $\tau_{2}=6.8$,
$\tau_{3}=2.6$, $k_{2}=0.3$, $k_{3}=0.3$, $x_{0}=4$ and $z_{0}=4$. Figure 4:
Plot of $B_{CHSH}$ for the primed quantities $Z^{\prime}=2.4$ and
$\theta^{\prime}=\frac{\pi}{5}$. We found a minimum of $-2.62405$ for this
function, in this way, there exists a violation for the inequiality (12) by an
amount of $0.62405$, i.e., around of $75\%$ of the maximal amount of
violation, $\sim 0.8284$, given by the Cirel’son’s bound[68, 74, 77].

####  IV.1.2 Bell-Klyshko-Mermin inequality

Our treatment for the study of correlations and the number of DOFs in the
system allows us to evaluate tripartite non-locality, the case of major
interest is the $X$-$Z$-$\theta$ triad. The Bell-Klyshko-Mermin inequality
that we use for this purpose is the following [68, 74]:

$\begin{split}-2&\leq
B_{BKM}=\bm{C}(X,P_{x},Z,P_{z},\theta^{\prime})+\bm{C}(X,P_{x},Z^{\prime},P_{z},\theta)\\\
&+\bm{C}(X^{\prime},P_{x},Z,P_{z},\theta)-\bm{C}(X^{\prime},P_{x},Z^{\prime},P_{z},\theta^{\prime})\leq
2.\end{split}$ (13)

In this case we need to fix the $P_{X}$, $P_{Z}$, $X^{\prime}$, $Z^{\prime}$
and $\theta^{\prime}$ variables. The verification of Eq. (13) can only be
handled numerically and it is not possible to visualize a plot of $B_{BKM}$
due to the number of DoFs in the Bell-Klyshko-Mermin inequality. Our study of
the inequality of Eq.(13) gives us as maximum value for $B_{BKM}$ the quantity
$2.43258$, demostrating the violation of the inequality (13) for the fixed
quantities $P_{x}=0.051$, $P_{z}=0.089$, $X^{\prime}=0.83$, $Z^{\prime}=3.3$
and $\theta^{\prime}=\frac{\pi}{2}$, $\tau_{2}=2.7$, $\tau_{3}=4.7$,
$k_{2}=0.3$, $k_{3}=0.3$, $x_{0}=4$ and $z_{0}=4$.

####  IV.1.3 Svetlichny Inequality

Our third test of non-locality is given by the Svetlichny inequality [78, 79,
71] for $X$, $Z$ and $\theta$. The Svetlichny inequality is given as

$\begin{split}-4&\leq
B_{S}=\bm{C}(X,P_{x},Z,P_{z},\theta)+\bm{C}(X,P_{x},Z,P_{z},\theta^{\prime})\\\
&\qquad+\bm{C}(X,P_{x},Z^{\prime},P_{z},\theta)-\bm{C}(X,P_{x},Z^{\prime},P_{z},\theta^{\prime})+\bm{C}(X^{\prime},P_{x},Z,P_{z},\theta)\\\
&\qquad-\bm{C}(X^{\prime},P_{x},Z,P_{z},\theta^{\prime})-\bm{C}(X^{\prime},P_{x},Z^{\prime},P_{z},\theta)-\bm{C}(X^{\prime},P_{x},Z^{\prime},P_{z},\theta^{\prime})\leq
4,\end{split}$ (14)

fixing $P_{x}$, $P_{z}$, $X^{\prime}$, $Z^{\prime}$ and $\theta^{\prime}$. Our
exhaustive study of the inequality of Eq. (14) permits us to report a possible
non-violation of the Svetlichny inequality for our system. We obtain the
maximum of $B_{S}\lessapprox 4$ for the fixed quantities $P_{x}=0.043$,
$P_{z}=0.066$, $X^{\prime}=0.52$, $Z^{\prime}=1.49$ and
$\theta^{\prime}=\frac{\pi}{5}$, $\tau_{2}=2.3$, $\tau_{3}=3.5$, $k_{2}=0.3$,
$k_{3}=0.3$, $x_{0}=4.5$ and $z_{0}=5.2$.

Nonetheless, we have tested our system with two weaker versions of the
Svetlichny inequality given by reference [71],

$\begin{split}B_{SV1}=&-\bm{C}(X,P_{x},Z,P_{z},\theta)+\bm{C}(X,P_{x},Z^{\prime},P_{z},\theta)\\\
&+\bm{C}(X^{\prime},P_{x},Z,P_{z},\theta)+\bm{C}(X^{\prime},P_{x},Z^{\prime},P_{z},\theta)+\bm{C}(X,P_{x},Z,P_{z},\theta^{\prime})\\\
&+\bm{C}(X,P_{x},Z^{\prime},P_{z},\theta^{\prime})+\bm{C}(X^{\prime},P_{x},Z,P_{z},\theta^{\prime})-\bm{C}(X^{\prime},P_{x},Z^{\prime},P_{z},\theta^{\prime})\leq
4,\end{split}$ (15)

and by inequality $185$ in reference [18],

$\begin{split}B_{SV2}=&-\bm{C}(X,P_{x},Z,P_{z},\theta)-\bm{C}(X,P_{x},Z^{\prime},P_{z},\theta)\\\
&+\bm{C}(X^{\prime},P_{x},Z,P_{z},\theta)-\bm{C}(X^{\prime},P_{x},Z^{\prime},P_{z},\theta)-\bm{C}(X,P_{x},Z,P_{z},\theta^{\prime})\\\
&+\bm{C}(X,P_{x},Z^{\prime},P_{z},\theta^{\prime})-\bm{C}(X^{\prime},P_{x},Z,P_{z},\theta^{\prime})-\bm{C}(X^{\prime},P_{x},Z^{\prime},P_{z},\theta^{\prime})\leq
4.\end{split}$ (16)

In both cases, for $B_{SV1}$ and $B_{SV2}$, we have not found any violation
for a wide variety of cases with different variables; after a wide search for
conditions to find a possible violation, this suggest a posible no-violation
of this quantities. This result is in concordance with the maximum found in
the Bell-Klyshko-Mermin inequality, Eq. (13), because the maximal violation is
not greater than $2\sqrt{2}$ for our case [74, 80]. Additionally, it is
convenient to recall that in certain cases stronger nonlocal effects are due
to weaker entangled states [81, 82, 83], in such cases the maximun entangled
states do not produce the maximun nonlocality [83].

##  V Relation between correlations and entanglement in the CSGE

The entanglement in the SGE has been widely studied in recent years, for
instance in [39, 47, 35]. One of the primal goals of these descriptions of
entanglement is the quantification of the same by means of entanglement
measures either in discrete, continous or hybrid sytems.

Our treatment of the quantum correlations is directly equiparable with the
study of the entanglement of the SGE by [39, 35] due to it maintaining the
same quantum description of the SGE of [35] and the same temporal dependence
in the correlation function, Eq. (10), and the entanglement of [35, 39].

For the case of the CHSH inequality, the maximum violation found in the
inequality of Eq. (12) is reached for $\tau_{2}=6.8$, $\tau_{3}=2.6$,
$k_{2}=0.3$, $k_{3}=0.3$. For values of the addimensional time $\tau_{2}=6.8$
with $k_{2}=0.3$, the maximum entanglement between $X$ and $\theta$ is almost
achieved after the first SGE, while for $\tau_{3}=2.6$ with $k_{3}=0.3$ in a
single SGE, the entanglement would be small; see FIG. 5. Therefore, in this
case, to get the maximum quantum nonlocality the atom does not need to expend
much time in the second SGE for the CSGE, the only thing that we can assure is
that in the presence of maximal quantum correlations exists entanglement.

We report a non-violation of the CHSH inequality of the CSGE for times,
$\tau_{2,3}$ where the entanglement is maximum, for example $\tau_{2,3}=10$
for $k_{2,3}=0.3$, a counterintuitive result at first glance but that has its
explanation in the dishtinguishablity that the SGE state presents when time
tends to infinity, a fact that is endorsed by the probability of the final
state of the SGE [33, 34], wich also could be calculated for the CSGE final
state of Eq. (LABEL:3mod).

We expect that for the case with more DoFs engaged, the entanglement in the
CSGE will have a similar behavior as the original SGE has, this is a
reasonable expectation. In this manner, we also report a non-violation of the
Bell-Klyshko-Mermin and the weaker Svetlichny inequalities when
$\tau_{2,3}\rightarrow\infty$, meaning that, in these cases, we anticipate the
lack of quantum correlations when the multipartite entanglement on those cases
will be maximum.

Figure 5: Entanglement entropy ($E$) of the SGE for various values of $k$
trough the time $\tau$. In the CSGE, we have $k_{2,3}$ and $\tau_{2,3}$, which
are equivalent to these non-labeled constants. In this way we can directly
compare the temporal behavior of the quantum correlations and entanglement in
the CSGE relying in the parameter $k$. Plot similar to the one given in [35].

##  VI Conclusions

In this work we have studied the non-locality and the steering in the CSGE. We
have demonstrated that the CSGE could be used to steer quantum states between
two different places. Our analysis suggest that the spreading of the entangled
wave function allows the particle to sense which observable is being measured
by Alice; hence, suggesting that the particle senses which observable is being
measured, for example $\hat{\sigma}_{x}$, $\hat{\sigma}_{z}$ or $\hat{x}$.

Additionally, we have found violation of the CHSH and Bell-Klyshko-Mermin
inequalities, however we have not found data suggesting that the evolved state
of the CSGE violate the strong Svetlichny inequality nor the two weaker forms
of the Svetlichny inequality. We then conclude the presence of non-locality in
the CSGE in the bipartite case but we can not assure the existence of
tripartite nonlocality nor real tripartite entanglement due to the lack of
violation of the Svetlichny inequalities [74, 80, 79]. This result does not
imply that a violation of the Svetlichny inequality can not exist for the CSGE
just that we can not provide a proof of violation of such inequality, further
research is necessary to find out whether of not it is violated by this state.

The striking result of the SGE with the spatial separation of the spins is
carried to the CSGE evolution, where our description of the evolved state
indicates the presence of entanglement between spin and position. The quantum
character of the SGE is then raised to attention when proving the correlations
between position and spin to be non-local, and therefore, non-classical in
nature. The quantum correlations of the hybrid tripartite state of the CSGE
were successfully characterized by a generalization of the BW correlation
function. The proposed function dichotomizes the spatial DoFs using the parity
operator and combines this description with the usual one for discrete
variables [72]. With these results, the CSGE presents an important case of
hybrid non-locality.

Furthermore, the violation of the CHSH inequality by the correlated
$Z$-$\theta$ pair, $Z$ being the direction of the last separation by the SGE
apparatus, was found to have a maximum value of $2.62$. This is a sizeable,
considerable violation, appreciably close to the maximum value for the
violation of the two-party CHSH inequality of $2\sqrt{2}$. Also, a very good
violation for this type of Bell operators [69, 70].

On the other hand, the maximal violation of the tripartite Bell-Klyshko-Mermin
inequality found for the CSGE state is 2.43 for the $X$-Z-$\theta$ triad. This
proves tripartite non-locality in the CSGE, but this inequality detects as
well the non-locality of the multiple two-party correlations. The violation
found is in this case far away from the possible maximum of 4 for the
tripartite inequality [74].

We can not establish the presence of genuine tripartite (three-way) non-
locality from the lack of violation of the Svetlichny inequalities, moreover,
real hybrid tripartite entanglement is also not assured in our system [71, 18,
80].

We can further correlate these results with the presence of entanglement in
the SGE. In the usual SGE, hybrid entanglement is created as soon as we have
interaction with the inhomogeneous field, with it increasing with the time of
evolution in the field until reaching the maximum possible entanglement [39,
35]. At this point, the correlations between position and spin are perfect. We
can reasonably expect these results to be carried over to the CSGE. However,
in the whole configuration space two-party non-locality has not been found for
parameters of the CSGE evolution where we expect maximal entanglement.
Tripartite non-locality in the form of a violation of the Bell-Klyshko-Mermin
inequality is also not found for the parameters that would give maximal
entanglement. This result is in concordance with previous studies where it is
shown that systems with maximum entanglement present slightly or null
nonlocality.

We can ensure that in the points where non-locality has been found, there
exists tripartite entanglement in the state of the CSGE, but this entanglement
is not perfect.

E. Benítez Rodríguez and E. Piceno Martínez thank CONACYT for PhD fellowship
support. We thank Vicerrectoría de Investigación y Estudios de Posgrado (BUAP)
for partial support. This work has not been supported by any scientific
agency.

##  Appendix A Dynamic evolution of the CSGE

In this part we will find the explicit evolution of the initial wavepacket as
it traverses or configuration of CSGE of FIG. 1.

The effective Hamiltonian related with the first experiment is [33, 34]

$\hat{H}_{2}=-\frac{\hbar^{2}}{2m}\nabla^{2}+\mu_{\text{c}}(\bm{\sigma}\cdot\bm{B}_{2}),$
(17)

with the effective inhomogeneous magnetic field
$\bm{B}_{2}=(B_{2}+b_{2}x)\hat{\imath}$, $\mu_{c}=ge\hbar/(4m)$, $g$ the
gyromagnetic ratio, $m$ the mass of the particle, $e$ the unit charge and
$\bm{\sigma}$ the Pauli matrices vector. In a similar way we have the
effective Hamiltonian associated with the second experiment

$\hat{H}_{3}=-\frac{\hbar^{2}}{2m}\nabla^{2}+\mu_{\text{c}}(\bm{\sigma}\cdot\bm{B}_{3})$
(18)

with the effective field $\bm{B}_{3}=(B_{3}+b_{3}z)\hat{k}$.

The initial state is given in Eq. (1) of the article as

$\ket{\psi_{i}}=\frac{1}{(2\pi\sigma_{0}^{2})^{\frac{3}{4}}}\exp\left(-\frac{(x^{2}+y^{2}+z^{2})}{4\sigma_{0}^{2}}+ik_{y}y\right)\ket{\uparrow_{z}}.$
(19)

To obtain the evolution through the first experiment we have the evolution
operator associated with the Hamiltonian $\hat{H_{2}}$ of Eq. (17), appearing
in Eq. (2) of the article

$\begin{split}\hat{U}_{2}(t)&=\exp\left(-\frac{1}{6}\kappa\right)\exp\left[-\frac{it}{2m\hbar}\left(p_{y}^{2}+p_{z}^{2}\right)\right]\\\
&\times\exp\left[-\frac{it\mu_{c}}{\hbar}\left(B_{2}+b_{2}x\right)\sigma_{x}\right]\exp\left(\frac{it^{2}\mu_{c}b_{2}}{2m\hbar}p_{x}\sigma_{x}\right)\\\
&\times\exp\left(-\frac{it}{2m\hbar}p_{x}^{2}\right),\end{split}$ (20)

with $\kappa=(it^{2}\mu_{c}^{2}b_{2}^{2})/(m\hbar)$, $B_{2}$ the homogeneity
parameter of the magnetic field, $b_{2}$ the inhomogeneity parameter,
$\sigma_{x}$ the Pauli operator. The evolution operator of this Eq. (20) is
factorized utilizing the Evolution Operator Factorization Method of [63, 64],
as done in [33, 34].

Applying $\hat{U}_{2}(t)$ for a certain time $t_{2}$ to $\ket{\psi_{i}}$ we
have the intermediate state $\ket{\psi_{m}}$,

$\begin{split}\hat{U}_{2}(t_{2})\ket{\psi_{i}}&=\ket{\psi_{m}(t_{2})}\\\
&=\frac{1}{\sqrt{2}}\exp\left(-\frac{1}{6}\kappa_{2}\right)\sigma_{0}^{\frac{3}{2}}\left(\sigma_{0}^{2}+\frac{i\hbar
t_{2}}{2m}\right)^{-\frac{3}{4}}\left\\{\left[(2\pi)^{\frac{1}{2}}\left(\sigma_{0}^{2}+\frac{\mathrm{i\hbar
t_{2}}}{2m}\right)^{\frac{1}{2}}\right]^{-\frac{1}{2}}\right\\}^{3}\\\
&\quad\times\exp(-k_{y}^{2}\sigma_{0}^{2})\exp\left\\{\frac{-[(y-2ik_{y}\sigma_{0}^{2})^{2}+z^{2}]}{4(\sigma_{0}^{2}+\frac{i\hbar
t_{2}}{2m})}\right\\}\\\
&\quad\times\left\\{\exp\left[-\frac{it_{2}\mu_{c}}{\hbar}(B_{2}+b_{2}x)\right]\exp\left[\frac{-(x+\frac{t^{2}_{2}\mu_{c}b_{2}}{2m})^{2}}{4(\sigma_{0}^{2}+\frac{i\hbar
t_{2}}{2m})}\right]\ket{\uparrow_{x}}\right.\\\
&\qquad+\left.\exp\left[\frac{it_{2}\mu_{c}}{\hbar}(B_{2}+b_{2}x)\right]\exp\left[\frac{-(x-\frac{t^{2}_{2}\mu_{c}b_{2}}{2m})^{2}}{4(\sigma_{0}^{2}+\frac{i\hbar
t_{2}}{2m})}\right]\ket{\downarrow_{x}}\right\\}\end{split}$ (21)

where we remember
$\ket{\uparrow_{z}}=\frac{1}{\sqrt{2}}(\ket{\uparrow_{x}}+\ket{\downarrow_{x}})$,
and $\kappa_{2}=\kappa(t_{2})$.

Now, following our description of the CSGE, we apply the evolution operator
associated with the last SGE for a fixed time $t_{3}$. That is, we apply the
evolution operator in direction $z$ direction, corresponding to the
Hamiltonian of Eq. (18), to the state of the Eq. (21). This operator is given
by

$\begin{split}\hat{U}_{3}(t)&=\exp\left(-\frac{1}{6}\kappa\right)\exp\left[-\frac{it}{2m\hbar}\left(p_{x}^{2}+p_{y}^{2}\right)\right]\\\
&\times\exp\left[-\frac{it\mu_{c}}{\hbar}\left(B_{3}+b_{3}z\right)\sigma_{z}\right]\exp\left(\frac{it^{2}\mu_{c}b}{2m\hbar}p_{z}\sigma_{z}\right)\\\
&\times\exp\left(-\frac{it}{2m\hbar}p_{z}^{2}\right)\end{split}$ (22)

Then we obtain the final state of the system, $\ket{\psi_{f}}$,

$\begin{split}\hat{U}_{3}(t_{3})&\ket{\psi_{m}(t_{2})}=\ket{\psi_{f}}\\\
&=A^{\prime}_{3}\times\exp\left\\{\frac{-(y-2ik_{y}\sigma_{0}^{2})^{2}}{4\left[\sigma_{0}^{2}+\frac{i\hbar(t_{2}+t_{3})}{2m}\right]}\right\\}\\\
&\times\left[\exp\left[-\frac{it_{3}\mu_{c}}{\hbar}(B_{3}+b_{3}z)\right]\exp\left\\{\frac{-\left(z+\frac{t^{2}_{3}\mu_{c}b_{3}}{2m}\right)^{2}}{4[\sigma_{0}^{2}+\frac{i\hbar(t_{2}+t_{3})}{2m}]}\right\\}\right.\\\
&\times\left(\exp\left\\{-\frac{it_{2}\mu_{c}}{\hbar}B_{2}\right\\}\exp\left\\{\frac{-\left[x+\frac{t^{2}_{2}\mu_{c}b_{2}}{2m}+2i\frac{t_{2}\mu_{c}b_{2}}{\hbar}(\sigma_{0}^{2}+\frac{i\hbar
t_{2}}{2m})\right]^{2}}{4[\sigma_{0}^{2}+\frac{i\hbar(t_{2}+t_{3})}{2m}]}\right\\}\right.\\\
&\quad+\left.\exp\left\\{\frac{it_{2}\mu_{c}}{\hbar}B_{2}\right\\}\exp\left\\{\frac{-\left[x-\frac{t^{2}_{2}\mu_{c}b_{2}}{2m}-2i\frac{t_{2}\mu_{c}b_{2}}{\hbar}(\sigma_{0}^{2}+\frac{i\hbar
t_{2}}{2m})\right]^{2}}{4[\sigma_{0}^{2}+\frac{i\hbar(t_{2}+t_{3})}{2m}]}\right\\}\right)\ket{\uparrow_{z}}\\\
&+\left.\exp\left[\frac{it_{3}\mu_{c}}{\hbar}(B_{3}+b_{3}z)\right]\exp\left\\{\frac{-\left(z-\frac{t^{2}_{3}\mu_{c}b_{3}}{2m}\right)^{2}}{4[\sigma_{0}^{2}+\frac{i\hbar(t_{2}+t_{3})}{2m}]}\right\\}\right.\\\
&\times\left(\exp\left\\{-\frac{it_{2}\mu_{c}}{\hbar}B_{2}\right\\}\exp\left\\{\frac{-\left[x+\frac{t^{2}_{2}\mu_{c}b_{2}}{2m}+2i\frac{t_{2}\mu_{c}b_{2}}{\hbar}(\sigma_{0}^{2}+\frac{i\hbar
t_{2}}{2m})\right]^{2}}{4[\sigma_{0}^{2}+\frac{i\hbar(t_{2}+t_{3})}{2m}]}\right\\}\right.\\\
&\quad-\left.\left.\exp\left\\{\frac{it_{2}\mu_{c}}{\hbar}B_{2}\right\\}\exp\left\\{\frac{-\left[x-\frac{t^{2}_{2}\mu_{c}b_{2}}{2m}-2i\frac{t_{2}\mu_{c}b_{2}}{\hbar}(\sigma_{0}^{2}+\frac{i\hbar
t_{2}}{2m})\right]^{2}}{4[\sigma_{0}^{2}+\frac{i\hbar(t_{2}+t_{3})}{2m}]}\right\\}\right)\ket{\downarrow_{z}}\right]\\\
&\\\ \end{split}$ (23)

with similarly $\kappa_{3}=\kappa(t_{3})$ and

$\begin{split}A^{\prime}_{3}&=\exp\left(\frac{-\kappa_{2}-\kappa_{3}}{6}\right)\frac{1}{2}\left[\frac{\sigma_{0}}{(2\pi)^{\frac{1}{2}}}\right]^{\frac{3}{2}}\\\
&\times\left(\sigma_{0}^{2}+\frac{i\hbar}{2m}(t_{2}+t_{3})\right)^{-\frac{3}{2}}\\\
&\times\exp\left(-k_{y}^{2}\sigma_{0}^{2}\right)\exp\left(\frac{it_{2}}{\hbar}\mu_{c}b_{2}\cdot\frac{t_{2}^{2}\mu_{c}b_{2}}{2m}\right)\\\
&\times\exp\left[-\left(\frac{t_{2}\mu_{c}b_{2}}{\hbar}\right)^{2}\left(\sigma_{0}^{2}+\frac{i\hbar
t_{2}}{2m}\right)\right]\end{split}$ (24)

To obtain Eq. (4) in the article, we rewrite the final state of Eq. (23) as
follows

$\ket{\psi_{f}}=\ket{\psi_{+}}\ket{\uparrow_{z}}+\ket{\psi_{-}}\ket{\downarrow_{z}},$
(25)

defining, as in Eq. (5) of the article,

$\begin{split}\ket{\psi_{\pm}}&=M\exp\left(\mp
i\sqrt{2}k_{3}^{2}\tau_{3}z_{0}\right)\exp\left\\{-\frac{\left(Z\pm\left\\{\sqrt{2}k_{3}^{3}\tau_{3}^{2}+i2\sqrt{2}k_{3}^{3}\tau_{3}\left[1+i\left(\tau_{2}+\tau_{3}\right)\right]\right\\}\right)^{2}}{4\left[1+i\left(\tau_{2}+\tau_{3}\right)\right]}\right\\}\\\
&\times\left(\exp\left(-i\sqrt{2}k_{2}^{3}\tau_{2}x_{0}\right)\exp\left\\{-\frac{\left[X+\sqrt{2}k_{2}^{3}\tau_{2}^{2}+i2\sqrt{2}k_{2}^{3}\tau_{2}\left(1+i\tau_{2}\right)\right]^{2}}{4\left[1+i\left(\tau_{2}+\tau_{3}\right)\right]}\right\\}\right.\\\
&\quad\pm\left.\exp\left(i\sqrt{2}k_{2}^{3}\tau_{2}x_{0}\right)\exp\left\\{-\frac{\left[X-\sqrt{2}k_{2}^{3}\tau_{2}^{2}-i2\sqrt{2}k_{2}^{3}\tau_{2}\left(1+i\tau_{2}\right)\right]^{2}}{4\left[1+i\left(\tau_{2}+\tau_{3}\right)\right]}\right\\}\right),\end{split}$
(26)

with $M$ the normalization factor given from Eqs. (21) and (24) by

$\begin{split}M&=\exp\left(\frac{-\kappa_{2}-\kappa_{3}}{6}\right)\frac{1}{2}\left[\frac{\sigma_{0}}{(2\pi)^{\frac{1}{2}}}\right]^{\frac{3}{2}}\left(\sigma_{0}^{2}+\frac{\mathrm{i}\hbar}{2m}(\tau_{2}+\tau_{3})\right)^{-\frac{3}{2}}\\\
&\times\exp\left(-k_{y}^{2}\sigma_{0}^{2}\right)\times\exp\left[-\left(\frac{t_{2}\mu_{c}b_{2}}{\hbar}\right)^{2}\sigma_{0}^{2}\right]\\\
&\times\exp\left[-\left(\frac{t_{3}\mu_{c}b_{3}}{\hbar}\right)^{2}\left(\sigma_{0}^{2}+\frac{i\hbar
t_{2}}{2m}\right)\right]\exp\left\\{\frac{-(y-2ik_{y}\sigma_{0}^{2})^{2}}{4\left[\sigma_{0}^{2}+\frac{i\hbar(t_{2}+t_{3})}{2m}\right]}\right\\},\end{split}$
(27)

and the dimensionless definitions of Eq. (II) in the article

$\displaystyle\tau_{2,3}$ $\displaystyle=\frac{\hbar
t_{2,3}}{2m\sigma_{0}^{2}},$ $\displaystyle k_{2,3}$
$\displaystyle=\sqrt{2}\sigma_{0}\left(\frac{m\mu_{c}b_{2,3}}{2\hbar^{2}}\right)^{1/3},$
$\displaystyle x_{0}$ $\displaystyle=\frac{B_{2}}{\sigma_{0}b_{2}},$
$\displaystyle z_{0}$ $\displaystyle=\frac{B_{3}}{\sigma_{0}b_{3}},$ (28)
$\displaystyle Z$ $\displaystyle=\frac{z}{\sigma_{0}},$ $\displaystyle X$
$\displaystyle=\frac{x}{\sigma_{0}}.$

##  Appendix B The correlation function for the CSGE

In the discussion surrounding Eq. (9) of the article, we found that the
correlation function for the CSGE can be written as

$\begin{split}&\bm{C}(X,P_{x},Z,P_{z},\theta)=\exp[\omega^{\prime}_{z}(Z,P_{z})+\omega^{\prime}_{x}(X,P_{x})]\\\
&\times\Bigg{[}4\cos(\theta)\bigg{(}\exp[\omega_{z}]\Big{\\{}\exp[\omega_{x}]\cosh[d_{x}(X,P_{x})]\sinh[d_{z}(Z,P_{z})]\\\
&\qquad+\exp[-\omega_{x}]\cos[\delta_{x}(X,P_{x})]\cosh[d_{z}(Z,P_{z})]\Big{\\}}\bigg{)}\\\
&\ \ \
+4\sin(\theta)\bigg{(}\exp[-\omega_{z}]\Big{\\{}\exp[\omega_{x}]\sinh[d_{x}(X,P_{x})]\cos[\delta_{z}(Z,P_{z})]\\\
&\qquad+\exp[-\omega_{x}]\sin[\delta_{x}(X,P_{x})]\sin[\delta_{z}(Z,P_{z})]\Big{\\}}\bigg{)}\Bigg{]}\end{split}$
(29)

up to a normalization factor.

We do so by defining the following functions that appear in Eq.(29):

$\begin{split}&\omega^{\prime}_{z}(Z,P_{z})=\frac{1}{4\left[1+\left(\tau_{2}+\tau_{3}\right)^{2}\right]}\left[-\left(\tau_{2}+\tau_{3}\right)^{2}\left(Z^{2}+2k_{3}^{6}\tau_{3}^{4}\right)\right]-2k_{3}^{6}\tau_{3}^{2}\left[1+\left(\tau_{2}+\tau_{3}\right)^{2}\right]\\\
&+2k_{3}^{6}\tau_{3}^{3}\left(\tau_{2}+\tau_{3}\right)-2\left[1+\left(\tau_{2}+\tau_{3}\right)^{2}\right]P_{z}^{2}-\frac{1}{4}Z^{2}+2P_{z}Z\left(\tau_{2}+\tau_{3}\right)\\\
&-\frac{1}{4\left[1+\left(\tau_{2}+\tau_{3}\right)^{2}\right]}\left(Z^{2}+2k_{3}^{6}\tau_{3}^{4}\right)-4k_{3}^{6}\tau_{3}^{2}\end{split}$
(30)

$\begin{split}&\omega^{\prime}_{x}(X,P_{x})=\frac{1}{4\left[1+\left(\tau_{2}+\tau_{3}\right)^{2}\right]}\Big{(}-\left(\tau_{2}+\tau_{3}\right)^{2}\left(X^{2}+2k_{2}^{6}\tau_{2}^{4}\right)\\\
&-8k_{2}^{6}\tau_{2}^{2}\left[4\left(1+\tau_{2}^{2}\right)^{2}-2\tau_{3}^{2}\left(1-\tau_{2}^{2}\right)+4\tau_{2}\tau_{3}\left(1+\tau_{2}^{2}\right)\right]\\\
&+8k_{2}^{6}\tau_{2}^{3}\left\\{4\left(\tau_{2}+\tau_{3}\right)-2\tau_{2}\left[1-\left(\tau_{2}+\tau_{3}\right)^{2}\right]\right\\}\Big{)}\\\
&-2\left[1+\left(\tau_{2}+\tau_{3}\right)^{2}\right]P_{x}^{2}-\frac{1}{4}X^{2}+2XP_{x}\left(\tau_{2}+\tau_{3}\right)\\\
&-\frac{1}{4\left[1+\left(\tau_{2}+\tau_{3}\right)^{2}\right]}\Big{\\{}X^{2}+2k_{2}^{6}\tau_{2}^{4}-8k_{2}^{6}\tau_{2}^{2}\left[2\left(1+\tau_{2}^{2}\right)+4\tau_{2}\tau_{3}\right]16k_{2}^{3}\tau_{2}^{3}\tau_{3}\Big{\\}}\end{split}$
(31)
$\begin{split}\omega_{z}&=-\frac{1}{2}k_{3}^{6}\tau_{3}^{4}-2k_{3}^{6}\tau_{3}^{2}\left[1+\left(\tau_{2}+\tau_{3}\right)^{2}\right]+2\left(\tau_{2}+\tau_{3}\right)k_{3}^{6}\tau_{3}^{3}\end{split}$
(32)

$\begin{split}\omega_{x}=-\frac{1}{2}k_{2}^{6}\tau_{2}^{4}-2k_{2}^{6}\tau_{2}^{2}\end{split}$
(33)

$\begin{split}d_{z}(Z,P_{z})&=\frac{1}{4\left[1+\left(\tau_{2}+\tau_{3}\right)^{2}\right]}\left[-2\sqrt{2}k_{3}^{3}\tau_{3}^{2}\left(\tau_{2}+\tau_{3}\right)^{2}Z\right]+\sqrt{2}k_{3}^{3}\tau_{3}\left(\tau_{2}+\tau_{3}\right)Z\\\
&-\frac{1}{\sqrt{2}}k_{3}^{3}\tau_{3}^{2}Z+\sqrt{2}k_{3}^{3}\tau_{3}\left(\tau_{2}+\tau_{3}\right)Z-\frac{1}{4\left[1+\left(\tau_{2}+\tau_{3}\right)^{2}\right]}\left[2\sqrt{2}k_{3}^{3}\tau_{3}^{2}Z\right]\\\
&+P_{z}\left\\{-4\sqrt{2}k_{3}^{3}\tau_{3}+\left(\tau_{2}+\tau_{3}\right)\left[2\sqrt{2}k_{3}^{3}\tau_{3}^{2}-4\sqrt{2}k_{3}^{3}\tau_{3}\left(\tau_{2}+\tau_{3}\right)\right]\right\\}\end{split}$
(34)

$\begin{split}d_{x}(X,P_{x})&=\frac{1}{4\left[1+\left(\tau_{2}+\tau_{3}\right)^{2}\right]}\\\
&\times\left(-2\sqrt{2}k_{2}^{3}\tau_{2}^{2}\left(\tau_{2}+\tau_{3}\right)^{2}X+2\sqrt{2}k_{2}^{3}\tau_{2}X\left\\{4\left(\tau_{2}+\tau_{3}\right)-2\tau_{2}\left[1-\left(\tau_{2}+\tau_{3}\right)^{2}\right]\right\\}\right)\\\
&-\frac{1}{\sqrt{2}}k_{2}^{3}\tau_{2}^{2}X+P_{x}\left[2\sqrt{2}k_{2}^{3}\tau_{2}^{2}\left(\tau_{2}+\tau_{3}\right)-4\sqrt{2}k_{2}^{3}\tau_{2}\left(1+\tau_{2}^{2}+\tau_{2}\tau_{3}\right)\right]\\\
&-\frac{1}{4\left[1+\left(\tau_{2}+\tau_{3}\right)^{2}\right]}\left(2\sqrt{2}k_{2}^{3}\tau_{2}^{2}X+8\sqrt{2}k_{2}^{3}\tau_{2}\tau_{3}X\right)\end{split}$
(35)

$\begin{split}\delta_{z}(Z,P_{z})&=-P_{z}2\sqrt{2}k_{3}^{3}\tau_{3}^{2}+2\sqrt{2}k_{3}^{3}\tau_{3}Z+2\sqrt{2}k_{3}^{3}\tau_{3}z_{0}\end{split}$
(36)
$\begin{split}&\delta_{x}(X,P_{x})=\frac{1}{4\left[1+\left(\tau_{2}+\tau_{3}\right)^{2}\right]}\left\\{2\sqrt{2}k_{2}^{3}\tau_{2}X\left[2-2\left(\tau_{2}+\tau_{3}\right)^{2}+4\tau_{2}\left(\tau_{2}+\tau_{3}\right)\right]\right\\}\\\
&+\sqrt{2}k_{2}^{3}\tau_{2}X-P_{x}\left(2\sqrt{2}k_{2}^{3}\tau_{2}^{2}+4\sqrt{2}k_{2}^{3}\tau_{2}\tau_{3}\right)\\\
&-\frac{1}{4\left[1+\left(\tau_{2}+\tau_{3}\right)^{2}\right]}\left\\{-4\sqrt{2}k_{2}^{3}\tau_{2}X\left[2+2\tau_{2}\left(\tau_{2}+\tau_{3}\right)\right]\right\\}+2\sqrt{2}k_{2}^{3}\tau_{2}x_{0}\end{split}$
(37)

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[remember picture,overlay]

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at (current page.south) Malaysian  User Group;

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at (current page.north)For graduate students and researchers;

[draw=Sienna,line width=1mm,rounded corners=.55cm,inner ysep=20pt, inner xsep=-5pt, text width=20cm, text=coverTextColor,font=, align=center] at (7,-0.7) 32pt0ptBeyond the Standard Model Cocktail ;

[text width=20cm, text=coverTextColor,font=,align=center] at (7,-4)  A modern and comprehensive review of the major open
puzzles in theoretical Particle Physics and Cosmology with a focus on Heavy Dark Matter

[anchor=north west, inner sep=0pt] at (-3.,-6.5) ;

[text width=18cm, text=coverTextColor,font=,] at (7.5,-20.5)  Yann Gouttenoire;
[text width=18cm, text=coverTextColor,font=,] at (7.5,-22.)  Géraldine Servant;
[text width=18cm, text=coverTextColor,font=,] at (7.5,-23.5)  Filippo Sala;

36pt46ptThe Wonderful Calmnessof Still Life Photos

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at (current page.south) Deutsches Elektronen-Synchrotron DESY and Hamburg University;

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[text width=18cm,fill=Sienna,text=white,font=,
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at (current page.north)PhD thesis written after a 3-year research study in DESY with the aim of receiving a doctoral degree from Hamburg University;
[draw=Sienna,line width=1mm,rounded corners=.55cm,inner ysep=15pt, inner xsep=-10pt, text width=20cm, text=coverTextColor,font=, align=center] at (7,-3.5) 32pt0ptBeyond the Standard Model Cocktail
Focus on Heavy Dark Matter;

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Supervisors’s Foreword

This PhD thesis manuscript  provides a remarkable and complete survey of the important questions at the interface between theoretical particle physics and cosmology.
It spans a very large range of physics in early universe cosmology. Out of the eight main chapters, four of them are reviews and the other four discuss new research results.  It goes  through all major open puzzles in theoretical particle physics beyond the Standard Model and presents the status of the art in the proposed solutions.
The clear logic in the organization of the material, the fact that both standard and more advanced computations are carried out in detail, and the extensive bibliography, make this volume a highly valuable reference for both researchers and students entering this field.

The first topic reflects many aspects in Dark Matter physics, from the production mechanism with non-trivial effects (Sommerfeld, bound state formation, dilution by entropy injection), to precise phenomenology and indirect detection signatures.
Over the past decades, Weakly-Interacting-Massive-Particles around the TeV scale have been among the most studied dark matter candidates. Motivated by the absence of experimental evidence for such particles, this thesis explores the possibility that dark matter is much heavier than what is conventionally assumed. The author shows that it is possible to have thermal dark matter with mass beyond the
unitarity upper limit of about 100 TeV, and motivates data analysis in this mass range by a plethora  of current telescopes, like ANTARES, and future ones, like CTA, LHAASO and KM3NeT.

The second topic is the analysis of supercooled confinement phase transitions during which the dark matter particle acquires its mass. It is massless outside the bubble and in the form of free quarks while it is composite and massive inside the bubble. This leads to very interesting dynamics that had never been addressed in this context before (string fragmentation,  cosmological deep inelastic scatterings) and that have implications for the final relic abundance of dark matter as a function of the amount of supercooling.  These effects  also affect the calculation of the bubble wall velocity, which is in turn relevant for the determination of the associated gravitational-wave signal and for the production of other early-universe relics.
Supercooled cosmological phase transitions have become a hot topic in the last few years. Yann Gouttenoire has contributed to bring new insights in this field.

The other line of investigation concerns gravitational waves from cosmic strings.
While the broad literature on this topic over the last two decades mainly focuses on the predictions within the standard cosmological paradigm, this work shows that the shape of the spectrum can be drastically different from usual predictions due to a variety of physical effects, such as non-standard equations of state due to intermediate matter eras or intermediate inflationary eras as motivated by either heavy unstable particles or supercooled cosmological phase transitions respectively.
It is shown how to use the potential measurement of a gravitational-wave spectrum produced by cosmic strings to extract information on very high energy physics and in particular derive bounds on the mass and lifetime of heavy unstable particles. These limits would extend well beyond what can presently be inferred from Big Bang Nucleosynthesis constraints.

Research in particle physics beyond the Standard Model has become highly interdisciplinary and much closer to cosmology.
In this thesis, Dr Yann Gouttenoire presents original work that illustrates the non-trivial connections between many topics, and provides a very comprehensive review of this thriving field. We are, in fact, already using this text as a precious reference for both our research work, and for the new students that we are introducing to particle cosmology.

Géraldine Servant and Filippo Sala.

This book provides a thorough survey of the important questions at the interface between theoretical particle physics and cosmology.
After discussing the theoretical and experimental physics revolution that led to the rise of the Standard Model in the past century, this volume reviews all major open puzzles, like the hierarchy problem, the small value of the cosmological constant, the matter-antimatter asymmetry, or the dark matter problem, and presents the state-of-the-art in the proposed solutions, with an extensive bibliography.
This manuscript emphasises the fields of thermal dark matter, cosmological first-order phase transitions and gravitational-wave signatures. Comprehensive and encyclopedic, this book could be a rich resource for both researchers and students entering the field.

Written after a 3-year doctoral program in DESY, in addition to the reviews composing two third of the material, one third presents the original PhD research work of the author.
Weakly Interacting Massive Particles (WIMPs) around the TeV scale have since long been among the best-motivated and most studied Dark Matter (DM) candidates. However, the absence of experimental evidence for such particles either in colliders, at telescopes or in underground laboratories, has stimulated model-building and  studies of new methods of detection beyond the WIMP paradigm. This thesis performs several  steps in this direction.

We are in particular interested in the case where DM is much heavier than a TeV. A well-known obstacle for such a realization is the unitarity bound on the annihilation cross-section which constrains the mass of thermal DM to be smaller than $\sim$100 TeV. However, the unitarity bound can be evaded in presence of entropy injection which dilutes the DM abundance. In this thesis, we investigate two possible sources of entropy injection.

First, we study the entropy injection following reheating after an early matter era, when a heavy spectator field, which we choose to be the DM mediator, dominating the energy density of the universe, decays into radiation. We study in detail the corresponding constraints from indirect detection, Cosmic Microwave Background (CMB) and 21-cm, and we show that experimentalists have interests to extend the thermal DM constraints to DM masses beyond the 10/100 TeV range.

Second, we study the entropy injection following reheating after an early stage of vacuum domination generated by a supercooled confining first-order phase transition. Considering the well-motivated scenario where DM is a composite state of a new  confining force, we found that a variety of new effects, e.g. string fragmentation and deep-inelastic-scattering in the early universe, play an important role for setting the final DM abundance. In both cases, we show that we can increase the DM mass up to the EeV scale, 4 orders of magnitude larger than the unitarity bound.

Such scenarios involve non standard cosmologies (either matter era or inflationary era inside the radiation era) and we show that these can be probed using the would-be imprints on the Gravitational-Waves (GW) spectrum from Cosmic Strings (CS) if observed with future GW detectors. In this thesis, we study in detail the computation of the GW spectrum from CS in the presence of non-standard cosmology and the associated constraints on DM models responsible for such a change of cosmology.

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This is with a sincere feeling of gratitude, that I am acknowledging Géraldine Servant. She first played an important role in my academic life when I was completing my master of theoretical physics in Paris and that the number of PhD subjects seemed infinite. Among the numerous researchers that I visited throughout that year, she was the one, whom enthusiasm for physics, as well as sympathy for the disoriented student I was, convinced me that I could confidently go into her supervision for a PhD, which I consider now as one of the best choices I have ever done in my life. I am indebted for the careful consideration she has manifested during the whole completion of the PhD,  for instance by planning a numerous amount of conferences and summer schools per year or by always being a precious help with administration.  I am indebted for the time she invested in crucial times, like during postdoc applications, and for dissecting every single word of this thesis manuscript with valuable comments and corrections added.
On the scientific side, I have been impressed by her ambition and determination throughout the completion of projects and by the resilience that she has manifested in presence of difficulties. Her formidable experience and her remarkable mastery in model-building were rewarding numerous times. She has been an extraordinary project designer, with my three main independent projects all being connected to each other in an elegant and coherent manner. I am very grateful to her for having successfully transmitted to me her enthusiasm for fantastic fields of research like cosmological first-order phase transitions and gravitational waves of primordial origin.

I am also indebted to Géraldine for having introduced me to Filippo Sala, who supervised my master thesis and who co-supervised my PhD thesis. The impact of Filippo on the young physicist I have become since I have met him is absolutely consequent. I have learnt so much from his critical thinking, constantly pushing me to be more skeptical about my claims and to think harder. Filippo is a careful thinker, a perfectionist, who gives a great attention to every single detail in his work. He has constantly been an inspiration, by showing me how to become more meticulous, more rigorous and more accurate. Besides that, I have always had a lot of fun thinking about physics with him, like when spending long days of confinement wondering about what happens when a quark approaches a wall boundary of confined phase. I also thank Filippo for having carefully read my thesis and for the numerous valuable comments added. Finally, I thank Filippo for having been helpful and diligent on the administrative side at multiple times.

I am deeply grateful to Géraldine, Filippo and Marco Cirelli for having invested so much time and energy - about 200 emails - into trying to build a cotutelle between Hamburg university and Sorbonne university. I would like to especially thank Marco for having moved heaven and earth in Sorbonne university and EDPIF doctoral school. I thank him for having addressed them a 11-page report to state that in 2020 it is impossible to have a PhD affiliated with two universities because the two doctoral certificate templates do not look the same, and because in France the doctoral certificate needs to be printed on a special parchment paper. I thank Brando Bellazzini and Benoit Estienne for having accepted to play the role of scientific tutor and mentor at EDPIF doctoral school.

I would like to thank Peera Simakachorn for the absolutely admirable dedication into our projects, which ended up with very thorough and detailed studies, with two of our common papers being 96 and 151 pages.
I thank Iason Baldes for many valuable physics discussions throughout our projects. I thank Kallia Petraki for teaching me Sommerfeld effects and bound state formation while hosting me in her office in LPTHE, and for many insightful physics discussions.

I thank Thomas Konstandin for having accepted to be a reviewer of the thesis, as well as Marco Cirelli, Dieter Horns and Günter Sigl for having accepted to be part of the jury. I also thank the anonymous third reviewer for having agreed to award the Summa Cum Laude. I thank Felix Giese and Marco Hufnagel for having translated the abstract in German.

To complete my PhD in the extremely rich and dynamic environment of DESY in Hamburg was a golden opportunity. With about 100 members in the theory group, we are exposed to a tremendous amount of seminars, conferences and interesting discussions. I am grateful to every colleagues and friends who have made the atmosphere so friendly and my time in Hamburg so unique.
The list would be long and I am sure that I don't need to cite their name for them to know what I think about them. I would simply thank my three partners in crime Adrien, Akshansh and Henrique for unforgettable moments, and Jorinde for her company during the difficult confinement phase. I also thank CrossFit WestGym and Erika for great times.

Je remercie mes amis Claude et Mathieu pour des moments incroyables durant notre master, merci en particulier à Claude pour les nuits passées à discuter de physique dans les escaliers du bâtiment G du campus de l'ENS Cachan.
Je remercie mon ami Sylvain pour avoir été un excellent binôme de travail pendant nos 2 ans à Orsay et pendant notre année d'agrégation. Une interaction scientifique de qualité et une discipline exceptionnelle nous ont permis de tenir le rythme effréné des 4 mois de préparation des épreuves orales, et je pense que l'on peut en être vraiment fier. Une pensée pour mes amis agrégatifs pour avoir rendu la préparation du concours si formidable. Merci à Arnaud Le Diffon pour des corrections de leçons de qualité.

Je remercie l'équipe enseignante du Magistère de Physique Fondamentale d'Orsay pour m'avoir autant subjugué. Ces deux années étaient incroyablement passionnantes, bien au-delà de mes attentes. La liste de personnes qui m'ont marquées durant ces deux ans est longue. Je voudrais simplement remercier Pélini, Jérémy, Bacchus et Leila pour tous nos moments de rire et de complicité.

Je remercie mes professeurs Bruno Dernoncourt et Jean-Paul Roux pour m'avoir captivé pour la physique ainsi que mon professeur de mathématiques Pierre-Jean Hormière pour avoir cru en mon potentiel et m'avoir recommandé le Magistère d'Orsay.
Je remercie ma bande de sup 1 crétins et l'internat Claude Fauriel pour avoir fait de nos deux ans de prépa une aventure humaine exceptionnelle, qui restera à jamais gravée dans nos mémoires. Merci Jonathan pour avoir transformé l'internat en école de spectacle.
Je remercie mon professeur de physique-chimie de première et terminale au lycée Albert Triboulet, Mr. Thibault,  pour ses qualités de pédagogie exceptionnelle.

Je remercie ma bande de riders pour des moments inoubliables dans les bois du Forez ou au cours des nombreux étés passés à Châtel. Je remercie Lois pour notre amitié sincère. Je remercie la bande des Rozierois, qui est comme une seconde famille. Je remercie Justine pour son soutien et son admiration sans faille alors que j'étais si focalisé sur les études. Je remercie mon frère, Houda, Édouard, Henrique, Sarah et Luis, mais aussi Eva, Élisa, Florian, Mathilde, Cyrielle, Clément, Simon et Antoine, pour leur compagnie durant le confinement à Rozier. Je remercie Bertand Phélut pour les cours de tennis. Je remercie Dounia pour sa grandeur d'âme.

Je dédie mon travail à ma famille. D'abord à ma maman, disparue en 2008, pour son amour inconditionnel et sa générosité inégalable. À mon papa pour son soutien inconditionnel, pour sa curiosité hors-du-commun, et  pour m'avoir transmis son intêret pour les grandes questions. Ensuite à mon frère pour notre complicité sans égal et notre admiration inconditionnelle l'un pour l'autre.
Je dédie également mon travail à Corinne, mes grand-parents, mes tantes, mes oncles, mes cousines et mes cousins pour leur gentillesse et leur soutien, et en particulier à mes grand-mères pour leur générosité infinie.

About the Author

During his Bachelor, Yann Gouttenoire had the opportunity to benefit from the excellent quality of courses at the Magistere of Fundamental Physics in Orsay, France.  The next year in 2015, he received a 2-year scholarship at École Normale Supérieure Paris-Saclay. The first year, he got awarded the “Agrégation” while in the second year he earned his MS in Theoretical Physics from École Normale Supérieure rue d'Ulm. In 2017, he joined DESY in Hamburg, one of the world's leading center for high-energy particle-physics research, where he studied the topics presented in this book. He defended his PhD on the 29th of September 2020 after which he got awarded the highest distinction from the University of Hamburg (“Summa Cum Laude”). In 2021, he earned an international Postdoctoral Fellowship in Tel Aviv University founded by the Azrieli Foundation.